$( This is the Metamath database iset.mm. $)

$( Metamath is a formal language and associated computer program for
   archiving, verifying, and studying mathematical proofs, created by Norman
   Dwight Megill (1950--2021).  For more information, visit
   https://us.metamath.org and
   https://github.com/metamath/set.mm, and feel free to ask questions at
   https://groups.google.com/g/metamath. $)

$( New users may want to read https://us.metamath.org/ileuni/conventions.html
   to understand the label naming conventions used in iset.mm.  See also the
   Metamath program command "MM> HELP VERIFY MARKUP" for markup conventions. $)

$( To break this file into smaller modules, in the Metamath program type
   "MM> READ iset.mm" followed by "MM> WRITE SOURCE iset.mm / SPLIT".  To
   recombine, omit "/ SPLIT". $)

$( The database iset.mm was created by Mario Carneiro on 21-Jan-2015 from a
   fork of the database set.mm and has been continuously enriched since then
   (list of contributors below).  Many additions and updates included copying
   entire theorems and sections from set.mm in order to keep some consistency
   among these databases. $)


$( !
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Metamath source file for intuitionistic logic and set theory
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#

                           ~~ PUBLIC DOMAIN ~~
This work is waived of all rights, including copyright, according to the CC0
Public Domain Dedication.  https://creativecommons.org/publicdomain/zero/1.0/

Currently active maintainers: See the list in the CONTRIBUTING.md file of
https://github.com/metamath/set.mm.

Contributor list:

DA  David Abernethy
SA  Stefan Allan
TA  Thierry Arnoux
JA  Juha Arpiainen
JB  Jonathan Ben-Naim
GB  Gregory Bush
MC  Mario Carneiro
PC  Paul Chapman
DF  Drahflow
AD  Adrian Ducourtial
GD  Georgy Dunaev
SF  Scott Fenton
GG  Gino Giotto
JGH Jeff Hankins
AH  Anthony Hart
DH  David Harvey
CH  Chen-Pang He
JH  Jeff Hoffman
SJ  Szymon Jaroszewicz
BJ  Benoit Jubin
JK  Jim Kingdon
NRK Noah R Kingdon
WL  Wolf Lammen
GL  Gerard Lang
BL  Brendan Leahy
RL  Raph Levien
FL  Frederic Line
RFL Roy F. Longton
JM  Jeff Madsen
RM  Rodolfo Medina
NM  Norman Megill
JMG Jia Ming
MM  Mykola Mostovenko
MO  Mel L. O'Cat
SO  Stefan O'Rear
JO  Jason Orendorff
JP  Josh Purinton
SR  Steve Rodriguez
ATS Andrew Salmon
AS  Alan Sare
ES  Eric Schmidt
GS  Glauco Siliprandi
SS  Saveliy Skresanov
AWS Andrew W Swan
JU  Jarvin Udandy
AV  Alexander van der Vekens
DAW David A. Wheeler
JY  Jonathan Yan
FZ  Fan Zheng
JEH James E. Hanson

HTML code for accented names:
  BJ Benoît Jubin
  GL Gérard Lang
  FL Frédéric Liné

$)


$( See "MM> HELP VERIFY MARKUP" for help with modularization tags. $)
$( Begin $[ iset-header.mm $] $)
$( !
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Contents of this header
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

* Quick "How To"
* Bibliography
* Metamath syntax summary
* Other notes
* Acceptable shorter proofs


=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Quick "How To"
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

How to use this file under Windows 95/98/NT/2K/XP/Vista:

1. Download the program metamath.exe per the instructions on the
   Metamath home page (https://us.metamath.org) and put it in the same
   directory as this file.
2. In Windows Explorer, double-click on metamath.exe.
3. Type "read iset.mm" and press Enter.
4. Type "help" for a list of help topics, and "help demo" for some
   command examples.


=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Bibliography
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Bibliographical references are made by bracketing an identifer in a theorem's
comment, such as [RussellWhitehead].  These refer to HTML tags on the following
web pages:

  Logic and set theory - see https://us.metamath.org/mpegif/mmset.html#bib
  Hilbert space - see https://us.metamath.org/mpegif/mmhil.html#ref

A bracketed reference must be preceded by a theorem number, etc. and followed
by a page number.  See "MM> HELP WRITE BIBLIOGRAPHY" for details.


=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Metamath syntax summary
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

The HELP LANGUAGE command in the Metamath program will give you a quick
overview of Metamath.  The specification is found on pp. 92--95 of the
Metamath book.  The following syntax summary is provided for convenience
but may omit some details.

A Metamath database (set of one or more ASCII source files) is a sequence of
_tokens_, which are normally separated by spaces or line breaks.  The only
tokens that are built into the Metamath language are those (two-character
sequences) beginning with $, shown in the following. These tokens are called
_keywords_:

          $c ... $. - Constant declaration
          $v ... $. - Variable declaration
          $d ... $. - Disjoint (distinct) variable restriction
  <label> $f ... $. - "Floating" hypothesis (i.e. variable type declaration)
  <label> $e ... $. - "Essential" hypothesis (i.e. a logical assumption for a
                      theorem or axiom)
  <label> $a ... $. - Axiom or definition or syntax construction
  <label> $p ... $= ... $. - Theorem and its proof
          ${ ... $} - Block for defining the scope of the above statements
                      (except $a, $p which are forever active)
$)        $( ... $)
$(                  - Comments (may not be nested); see HELP LANGUAGE
                      for markup features
          $[ ... $] - Include a file

The above two-character sequences beginning with "$" are the only primitives
built into the Metamath language.  The only "logic" Metamath uses in its proof
verification algorithm is the substitution of expressions for variables while
checking for distinct variable violations.  Everything else, including the
axioms for logic, is defined in this database file.

All other tokens are user-defined, and their names are arbitrary.  There are
two kinds of user-defined tokens, called math symbols (or just symbols) and
labels.  A _symbol_ may contain any non-whitespace printable character except
"$".  A _label_ may contain only alphanumeric characters and the characters "."
(period), "-" (hyphen), and "_" (underscore).  Symbols and labels are
case-sensitive.  All labels (except in proofs) must be distinct.  A label may
not have the same name as a symbol (to simplify the coding of certain parsers
and translators).

Here is some more detail about the syntax:

  $c <symbollist> $.
      <symbollist> is a (whitespace-separated) list of distinct symbols that
      haven't been used before.
  $v <symbollist> $.
      <symbollist> is a list of distinct symbols that haven't been used yet
      in the current scope (see ${ ... $} below).
  $d <symbollist> $.
      <symbollist> is a (whitespace-separated) list of distinct symbols
      previously declared with $v in current scope.  It means that
      substitutions into these symbols may not have variables in common.
  <label> $f <symbollist> $.
      <symbollist> is a list of 2 symbols, the first of which must be
      previously declared with $c in the current scope.
  <label> $e <symbollist> $.
      <symbollist> is a list of 2 or more symbols, the first of which must be
      previously declared with $c in the current scope.
  <label> $a <symbollist> $.
      <symbollist> is a list of 2 or more symbols, the first of which must be
      previously declared with $c in the current scope.
  <label> $p <symbollist> $= <proof> $.
      <symbollist> is a list of 2 or more symbols, the first of which must be
      previously declared with $c in the current scope.  <proof> is either a
      whitespace-delimited sequence of previous labels (created by
      SAVE PROOF <label> /NORMAL) or a compressed proof (created by
      SAVE PROOF <label> /COMPRESSED).  After using SAVE PROOF, use
      WRITE SOURCE to save the database file to disk.
  ${ ... $}
      Block for scoping the above statements (except $a, $p which are forever
      active).  Currently, $c may not occur inside of a block.
$)
  $( <any text> $)
$(    Comment.  Note: <any text> may not contain adjacent "$" and ")"
      characters.
  $[ <filename> $]
      Insert contents of <filename> at this point.  If <filename> is current
      file or has been already been inserted, it will not be inserted again.

Inside of comments, it is recommended that labels be preceded with a tilde (~)
and math symbol tokens be enclosed in grave accents (` `).  This way the LaTeX
and HTML rendition of comments will be accurate, and (future) tools to globally
change labels and math symbols will also change them in comments.  Note that ``
inside of grave accents is interpreted as a single ` .  A special comment
containing $ t defines LaTeX and HTML symbols.  See HELP LANGUAGE and
HELP HTML for other markup features in comments.

The proofs in this file are in "compressed" format for storage efficiency.  The
Metamath program reads the compressed format directly.  This format is
described in an Appendix of the Metamath book.  It is not intended to be read
by humans.  For viewing proofs you should use the various SHOW PROOF commands
described in the Metamath book (or the online HELP).

The Metamath program does not normally affect any content of this file other
than proofs, i.e., the text between "$=" and "$." (and some rewrapping).  All
other content is user-created.  Proofs are created or modified with the PROVE
command.


=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Other notes
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

1. It is recommended that you be familiar with Chapters 2 and 4 of the
Metamath book to understand the Metamath language.  Chapters 2, 3 and 5 explain
how to use the Metamath program.  Chapter 3 gives an informal overview of what
this source file is all about.  Appendix A gives the standard mathematical
symbols corresponding to some of the ASCII tokens used in this file.

The ASCII tokens may seem cryptic at first, even if you are familiar with set
theory, but a review of the definition summary in Chapter 3 should quickly
enable you to see the correspondence to standard mathematical notation.  To
easily find the definition of a token, search for the first occurrences of the
token surrounded by spaces.  Some odd-looking ones include "-." for "not", and
"C_" for "is a subset of".  The Metamath program "MM> HELP TEX" command
explains how to obtain a LaTeX output to see the real mathematical symbols.
Let us know if you have better suggestions for naming ASCII tokens.

2. Logic and set theory provide a foundation for all of mathematics.  To learn
about them, you should study one or more of the references listed below.  The
textbooks provide a motivation for what we are doing, whereas Metamath lets you
see in detail all hidden and implicit steps.  Most standard theorems are
accompanied by citations.  Some closely followed texts include the following:

  Axioms of propositional calculus - [Margaris].
  Axioms of predicate calculus - [Megill] (System S3' in the article
      referenced).
  Theorems of propositional calculus - [WhiteheadRussell].
  Theorems of pure predicate calculus - [Margaris].
  Theorems of equality and substitution - [Monk2], [Tarski], [Megill].
  Axioms of set theory - [BellMachover].
  Development of set theory - [TakeutiZaring].  (The first part of [Quine]
      has a good explanation of the powerful device of "virtual" or
      class abstractions, which is essential to our development.)
  Construction of real and complex numbers - [Gleason]
  Theorems about real numbers - [Apostol]
  Intuitionistic logic and constructive mathematics - [Bauer] is an sampling
    of theorems and arguments which give a flavor of constructive mathematics.
    [Heyting] is a more comprehensive treatment of intuitionistic logic.

3. Convention:  All $a statements starting with "|-" have labels
starting with "ax-" (axioms) or "df-" (definitions).  "ax-" corresponds
to what is traditionally called an axiom.  "df-" introduces new symbols
or a new relationship among symbols that can be eliminated; they always
extend the definition of a wff or class.  Metamath blindly treats $a
statements as new given facts but does not try to justify them.  The
mmj2 program will justify the definitions as sound, except for four of them
(df-bi, df-clab, df-cleq, df-clel) that require a more complex metalogical
justification by hand.

4. Our method of definition, the axioms for predicate calculus, and the
development of substitution are somewhat different from those found in standard
texts.  The axioms were designed for direct derivation of standard results
without excessive use of metatheorems.  (See Theorem 9.7 of [Megill] for a
rigorous justification.)  Typically we are minimalist when introducing new
definitions; they are introduced only when a clear advantage becomes apparent
for reducing the number of symbols, shortening proofs, etc.  We generally avoid
the introduction of gratuitous definitions because each one requires associated
theorems and additional elimination steps in proofs.

5. Where possible, the notation attempts to conform to modern conventions, with
variations due to our choice of the axiom system or to make proofs shorter.
Listed below are some important conventions and how they correspond to textbook
language.  The notation is usually explained in more detail when first
introduced.

  Typically, Greek letters (ph = phi, ps = psi, ch = chi, etc.) are used for
      propositional (wff) variables; x,y,z,... for individual (i.e. set)
      variables; and A,B,C,... for class variables.
  "|-", meaning "It is provable that," is the first token of all assertions
      and hypotheses that aren't syntax constructions.  This is a standard
      convention in logic.  For us, it also prevents any ambiguity with
      statements that are syntax constructions, such as "wff -. ph".
  "$e |- ( ph -> A. x ph ) $." should be read "Assume variable x is
      (effectively) not free in wff phi."  Literally, this says "Assume it is
      provable that phi implies for all x phi."
  "|- ( -. A. x x = y -> ..." should be read "If x and y are distinct
      variables, then..."  This antecedent provides us with a technical
      device (called a "distinctor" in [Megill]) to avoid the need for the
      $d statement early in our development of predicate calculus, permitting
      unrestricted substitituions as conceptually simple as those in
      propositional calculus.  However, the $d eventually becomes a
      requirement, and after that this device is rarely used.
  "[ y / x ] ph" should be read "the wff that results when y is properly
      substituted for x in ph."
  "$d x y $." should be read "Assume x and y are distinct variables."
  "$d x ph $." should be read "Assume x does not occur in phi $."  Sometimes
      a theorem is proved with "$e |- ( ph -> A. x ph ) $." in place of
      "$d x ph $." when a more general result is desired; ~ ax-17 can be used
      to derive the $d version.  For an example of how to get from the $d
      version back to the $e version, see the proof of ~ euf from ~ df-eu .
  "$d x A $." should be read "Assume x is not a variable occurring in class A."
  "$d x A $.  $d x ps $.  $e |- ( x = A -> ( ph <-> ps ) ) $." is an idiom
      often used instead of explicit substitution, meaning "Assume psi results
      from the substitution of A for x in phi."
  "$e |- A e. _V $." should be read "Assume class A is a set (i.e. exists)."
      This is a convenient convention used by [Quine].
  "$d x y $.  $e |- y e. A -> A. x y e. A $." should be read "Assume x is
      (effectively) not a free variable in class A."
  "`' R" should be read "converse of (relation) R" and is the same as the more
      standard notation R^{-1}.
  "( f ` x )" should be read "the value of function f at x" and is the same as
      the more familiar f(x).
  The Deduction Theorem of standard logic is never used.  Instead, in set
      theory, we use other tricks to make a $e hypothesis become an antecedent.
      See the comment for Theorem ~ dedth below.


=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Acceptable shorter proofs
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Shorter proofs are welcome, and any shorter proof I accept will be acknowledged
in the theorem's description.  However, in some cases a proof may be "shorter"
or not depending on how it is formatted.  This section provides general
guidelines.

Usually I will automatically accept shorter proofs that (1) shorten the set.mm
file (with compressed proofs), (2) reduce the size of the HTML file generated
with SHOW STATEMENT xx / HTML, (3) use only existing, unmodified theorems in
the database (the order of theorems may be changed, though), and (4) use no
additional axioms.

Usually I will also automatically accept a _new_ theorem that is used to
shorten multiple proofs, if the total size of set.mm (including the comment of
the new theorem, not including the acknowledgment) decreases as a result.

In borderline cases, I typically place more importance on the number of
compressed proof steps and less on the length of the label section (since the
names are in principal arbitrary).  If two proofs have the same number of
compressed proof steps, I will typically give preference to the one with the
smaller number of different labels, or if these numbers are the same, the proof
with the fewest number of characters that the proofs happen to have by chance
when label lengths are included.

A few theorems have a longer proof than necessary in order to avoid the use of
certain axioms, for pedagogical purposes, and for other reasons.  Usually this
is clear from the theorem's description.  For example, ~ idALT shows a proof
directly from axioms.  Shorter proofs for such cases won't be accepted, of
course, unless the criteria described continues to be satisfied.

$)

$( End $[ iset-header.mm $] $)


$( Begin $[ iset-pred.mm $] $)

$( The following header is the first to appear in the Theorem List contents,
   because higher-level headers suppress all previous same-level or
   lower-level headers in the same comment area between $a and $p statements.
   See "MM> HELP WRITE THEOREM_LIST" for information about headers. $)


$(
###############################################################################
  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY
###############################################################################

  Logic can be defined as the "study of the principles of correct reasoning"
  (Merrilee H. Salmon's 1991 "Informal Reasoning and Informal Logic" in
  _Informal Reasoning and Education_ ) or as "a formal system using symbolic
  techniques and mathematical methods to establish truth-values" (the Oxford
  English Dictionary).

  This section formally defines the logic system we will use.  In particular,
  it defines symbols for declaring truthful statements, along with rules for
  deriving truthful statements from other truthful statements.  The system
  defined here is intuitionistic first-order logic with equality.

  We begin with a few housekeeping items in pre-logic, and then introduce
  propositional calculus (both its axioms and important theorems that can be
  derived from them).  Propositional calculus deals with general truths about
  well-formed formulas (wffs) regardless of how they are constructed.  This is
  followed by proofs that other axiomatizations of classical propositional
  calculus can be derived from the axioms we have chosen to use.

  We then define predicate calculus, which adds additional symbols and rules
  useful for discussing objects (beyond simply true or false).  In particular,
  it introduces the symbols ` = ` ("equals"), ` e. ` ("is a member of"), and `
  A. ` ("for all").  The first two are called "predicates".  A predicate
  specifies a true or false relationship between its two arguments.

$)


$(
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  Pre-logic
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  This section includes a few "housekeeping" mechanisms before we begin
  defining the basics of logic.

$)

  $( Declare the primitive constant symbols for propositional calculus. $)
  $c ( $.  $( Left parenthesis $)
  $c ) $.  $( Right parenthesis $)
  $c -> $.  $( Right arrow (read:  "implies") $)
  $c -. $.  $( Right handle (read:  "not") $)
  $c wff $.  $( Well-formed formula symbol (read:  "the following symbol
                sequence is a wff") $)
  $c |- $.  $( Turnstile (read:  "the following symbol sequence is provable" or
               'a proof exists for") $)

  $( Define the syntax and logical typecodes, and declare that our grammar is
     unambiguous (verifiable using the KLR parser, with compositing depth 5).
     (This $ j comment need not be read by verifiers, but is useful for parsers
     like mmj2.) $)
  $( $j
    syntax 'wff';
    syntax '|-' as 'wff';
    unambiguous 'klr 5';
  $)

  $( Declare typographical constant symbols that are not directly used in the
     formalism but are useful to explain it in comments. $)

  $c & $.  $( Ampersand (read: "and"). $)
  $c => $.  $( Double right arrow (read: "implies"). $)

  $( wff variable sequence:  ph ps ch th ta et ze si rh mu la ka $)
  $( Introduce some variable names we will use to represent well-formed
     formulas (wff's). $)
  $v ph $.  $( Greek phi $)
  $v ps $.  $( Greek psi $)
  $v ch $.  $( Greek chi $)
  $v th $.  $( Greek theta $)
  $v ta $.  $( Greek tau $)
  $v et $.  $( Greek eta $)
  $v ze $.  $( Greek zeta $)
  $v si $.  $( Greek sigma $)
  $v rh $.  $( Greek rho $)
  $v mu $.  $( Greek mu $)
  $v la $.  $( Greek lambda $)
  $v ka $.  $( Greek kappa $)

  $( Specify some variables that we will use to represent wff's.
     The fact that a variable represents a wff is relevant only to a theorem
     referring to that variable, so we may use $f hypotheses.  The symbol
     ` wff ` specifies that the variable that follows it represents a wff. $)
  $( Let variable ` ph ` be a wff. $)
  wph $f wff ph $.
  $( Let variable ` ps ` be a wff. $)
  wps $f wff ps $.
  $( Let variable ` ch ` be a wff. $)
  wch $f wff ch $.
  $( Let variable ` th ` be a wff. $)
  wth $f wff th $.
  $( Let variable ` ta ` be a wff. $)
  wta $f wff ta $.
  $( Let variable ` et ` be a wff. $)
  wet $f wff et $.
  $( Let variable ` ze ` be a wff. $)
  wze $f wff ze $.
  $( Let variable ` si ` be a wff. $)
  wsi $f wff si $.
  $( Let variable ` rh ` be a wff. $)
  wrh $f wff rh $.
  $( Let variable ` mu ` be a wff. $)
  wmu $f wff mu $.
  $( Let variable ` la ` be a wff. $)
  wla $f wff la $.
  $( Let variable ` ka ` be a wff. $)
  wka $f wff ka $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Inferences for assisting proof development
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The inference rules in this section will normally never appear in a completed
  proof.  They can be ignored if you are using this database to assist learning
  logic - please start with the statement ~ wn instead.

$)

  ${
    idi.1 $e |- ph $.
    $( (_Note_:  This inference rule and the next one, ~ a1ii , will normally
       never appear in a completed proof.  They can be ignored if you are using
       this database to assist learning logic; please start with the statement
       ~ wn instead.)

       This inference says "if ` ph ` is true then ` ph ` is true".  This
       inference requires no axioms for its proof, and is useful as a
       copy-paste mechanism during proof development in mmj2.  It is normally
       not referenced in the final version of a proof, since it is always
       redundant.  You can remove this using the metamath-exe (Metamath
       program) Proof Assistant using the "MM-PA> MINIMIZE__WITH *" command.
       This is the inference associated with ~ id , hence its name.
       (Contributed by Alan Sare, 31-Dec-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    idi $p |- ph $=
      (  ) B $.
  $}

  ${
    a1ii.1 $e |- ph $.
    a1ii.2 $e |- ps $.
    $( (_Note_:  This inference rule and the previous one, ~ idi , will
       normally never appear in a completed proof.)

       This is a technical inference to assist proof development.  It provides
       a temporary way to add an independent subproof to a proof under
       development, for later assignment to a normal proof step.

       The Metamath (Metamath-exe) program Proof Assistant requires proofs to
       be developed backwards from the conclusion with no gaps, and it has no
       mechanism that lets the user work on isolated subproofs.  This inference
       provides a workaround for this limitation.  It can be inserted at any
       point in a proof to allow an independent subproof to be developed on the
       side, for later use as part of the final proof.

       _Instructions_:  (1) Assign this inference to any unknown step in the
       proof.  Typically, the last unknown step is the most convenient, since
       "MM-PA> ASSIGN LAST" can be used.  This step will be replicated in
       hypothesis a1ii.1, from where the development of the main proof can
       continue.  (2) Develop the independent subproof backwards from
       hypothesis a1ii.2.  If desired, use a "MM-PA> LET STEP" command to
       pre-assign the conclusion of the independent subproof to a1ii.2.  (3)
       After the independent subproof is complete, use "MM-PA> IMPROVE ALL" to
       assign it automatically to an unknown step in the main proof that
       matches it.  (4) After the entire proof is complete, use "MM-PA>
       MINIMIZE__WITH *" to clean up (discard) all ~ a1ii references
       automatically.

       This inference was originally designed to assist importing partially
       completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can
       also be useful on its own.  Interestingly, no axioms are required for
       its proof.  It is the inference associated with ~ a1i .  (Contributed by
       NM, 7-Feb-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    a1ii $p |- ph $=
      (  ) C $.
  $}


$(
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  Propositional calculus
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  Propositional calculus deals with general truths about well-formed formulas
  (wffs) regardless of how they are constructed.  The simplest propositional
  truth is ` ( ph -> ph ) ` , which can be read "if something is true, then it
  is true" - rather trivial and obvious, but nonetheless it must be proved from
  the axioms (see Theorem ~ id ).

  Our system of propositional calculus consists of a few basic axioms and a
  unique rule of inference, modus ponens.

  The propositional calculus used here is the intuitionistic propositional
  calculus.

  All 194 axioms, definitions, and theorems for propositional calculus in
  _Principia Mathematica_ (specifically *1.2 through *5.75) are axioms or
  formally proven.  See the Bibliographic Cross-References at
  ~ https://us.metamath.org/ileuni/mmbiblio.html for a complete
  cross-reference from sources used to its formalization in the Intuitionistic
  Logic Explorer.

$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Recursively define primitive wffs for propositional calculus
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( If ` ph ` is a wff, so is ` -. ph ` or "not ` ph ` ".  Part of the
     recursive definition of a wff (well-formed formula).  Traditionally, Greek
     letters are used to represent wffs, and we follow this convention.  In
     propositional calculus, we define only wffs built up from other wffs,
     i.e., there is no starting or "atomic" wff.  Later, in predicate calculus,
     we will extend the basic wff definition by including atomic wffs ( ~ weq
     and ~ wel ). $)
  wn $a wff -. ph $.

  $( Register negation '-.' as a primitive expression (lacking a
     definition). $)
  $( $j primitive 'wn'; $)

  $( If ` ph ` and ` ps ` are wff's, so is ` ( ph -> ps ) ` or " ` ph ` implies
     ` ps ` ".  Part of the recursive definition of a wff.  The left-hand wff
     is called the antecedent, and the right-hand wff is called the consequent.
     In the case of ` ( ph -> ( ps -> ch ) ) ` , the middle ` ps ` may be
     informally called either an antecedent or part of the consequent depending
     on context. $)
  wi $a wff ( ph -> ps ) $.

  $( Register implication '->' as a primitive expression (lacking a
     definition). $)
  $( $j primitive 'wi'; $)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Propositional logic axioms for implication
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $( Minor premise for modus ponens. $)
    min $e |- ph $.
    $( Major premise for modus ponens. $)
    maj $e |- ( ph -> ps ) $.
    $( Rule of Modus Ponens.  The postulated inference rule of propositional
       calculus.  See, e.g., Rule 1 of [Hamilton] p. 73.  The rule says, "if
       ` ph ` is true, and ` ph ` implies ` ps ` , then ` ps ` must also be
       true".  This rule is sometimes called "detachment", since it detaches
       the minor premise from the major premise.  "Modus ponens" is short for
       "modus ponendo ponens", a Latin phrase that means "the mode that by
       affirming affirms" - remark in [Sanford] p. 39.  This rule is similar to
       the rule of modus tollens ~ mto .

       Note:  In some web page displays such as the Statement List, the
       symbols " ` & ` " and " ` => ` " informally indicate the relationship
       between the hypotheses and the assertion (conclusion), abbreviating the
       English words "and" and "implies".  They are not part of the formal
       language.  (Contributed by NM, 30-Sep-1992.) $)
    ax-mp $a |- ps $.
  $}

  $( Axiom _Simp_.  Axiom A1 of [Margaris] p. 49.  One of the axioms of
     propositional calculus.  This axiom is called _Simp_ or "the principle of
     simplification" in _Principia Mathematica_ (Theorem *2.02 of
     [WhiteheadRussell] p. 100) because "it enables us to pass from the joint
     assertion of ` ph ` and ` ps ` to the assertion of ` ph ` simply."

     The theorems of propositional calculus are also called _tautologies_.
     Although classical propositional logic tautologies can be proved using
     truth tables, there is no similarly simple system for intuitionistic
     propositional logic, so proving tautologies from axioms is the preferred
     approach.  (Contributed by NM, 5-Aug-1993.) $)
  ax-1 $a |- ( ph -> ( ps -> ph ) ) $.

  $( Axiom _Frege_.  Axiom A2 of [Margaris] p. 49.  This axiom "distributes" an
     antecedent over two consequents.  This axiom was part of Frege's original
     system and is known as _Frege_ in the literature.  It is also proved as
     Theorem *2.77 of [WhiteheadRussell] p. 108.  The other direction of this
     axiom also turns out to be true, as demonstrated by ~ pm5.41 .
     (Contributed by NM, 5-Aug-1993.) $)
  ax-2 $a |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical implication
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The results in this section are based on implication only, and only use
  ~ ax-1 , ~ ax-2 , and ~ ax-mp .  In an implication, the wff before the arrow
  is called the "antecedent" and the wff after the arrow is called the
  "consequent".

  We will use the following descriptive terms very loosely:  A "closed form" or
  "tautology" has no $e hypotheses.  An "inference" has one or more $e
  hypotheses.  A "deduction" is an inference in which the hypotheses and the
  conclusion share the same antecedent.

$)

  ${
    mp2b.1 $e |- ph $.
    mp2b.2 $e |- ( ph -> ps ) $.
    mp2b.3 $e |- ( ps -> ch ) $.
    $( A double modus ponens inference.  (Contributed by Mario Carneiro,
       24-Jan-2013.) $)
    mp2b $p |- ch $=
      ( ax-mp ) BCABDEGFG $.
  $}

  ${
    $( Premise for ~ a1i . $)
    a1i.1 $e |- ph $.
    $( Inference derived from Axiom ~ ax-1 .  See ~ a1d for an explanation of
       our informal use of the terms "inference" and "deduction".  See also the
       comment in ~ syld .  (Contributed by NM, 5-Aug-1993.) $)
    a1i $p |- ( ps -> ph ) $=
      ( wi ax-1 ax-mp ) ABADCABEF $.
  $}

  ${
    mp1i.a $e |- ph $.
    mp1i.b $e |- ( ph -> ps ) $.
    $( Drop and replace an antecedent.  (Contributed by Stefan O'Rear,
       29-Jan-2015.) $)
    mp1i $p |- ( ch -> ps ) $=
      ( ax-mp a1i ) BCABDEFG $.
  $}

  ${
    $( Premise for ~ a2i . $)
    a2i.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference derived from Axiom ~ ax-2 .  (Contributed by NM,
       5-Aug-1993.) $)
    a2i $p |- ( ( ph -> ps ) -> ( ph -> ch ) ) $=
      ( wi ax-2 ax-mp ) ABCEEABEACEEDABCFG $.
  $}

  ${
    imim2i.1 $e |- ( ph -> ps ) $.
    $( Inference adding common antecedents in an implication.  (Contributed by
       NM, 5-Aug-1993.) $)
    imim2i $p |- ( ( ch -> ph ) -> ( ch -> ps ) ) $=
      ( wi a1i a2i ) CABABECDFG $.
  $}

  ${
    mpd.1 $e |- ( ph -> ps ) $.
    mpd.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( A modus ponens deduction.  (Contributed by NM, 5-Aug-1993.) $)
    mpd $p |- ( ph -> ch ) $=
      ( wi a2i ax-mp ) ABFACFDABCEGH $.
  $}

  ${
    $( First of 2 premises for ~ syl . $)
    syl.1 $e |- ( ph -> ps ) $.
    $( Second of 2 premises for ~ syl . $)
    syl.2 $e |- ( ps -> ch ) $.
    $( An inference version of the transitive laws for implication ~ imim2 and
       ~ imim1 , which Russell and Whitehead call "the principle of the
       syllogism...because...the syllogism in Barbara is derived from them"
       (quote after Theorem *2.06 of [WhiteheadRussell] p. 101).  Some authors
       call this law a "hypothetical syllogism".  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by O'Cat, 20-Oct-2011.)  (Proof shortened
       by Wolf Lammen, 26-Jul-2012.) $)
    syl $p |- ( ph -> ch ) $=
      ( wi a1i mpd ) ABCDBCFAEGH $.
  $}

  ${
    mpi.1 $e |- ps $.
    mpi.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( A nested modus ponens inference.  (Contributed by NM, 5-Aug-1993.)
       (Proof shortened by Stefan Allan, 20-Mar-2006.) $)
    mpi $p |- ( ph -> ch ) $=
      ( a1i mpd ) ABCBADFEG $.
  $}

  ${
    mp2.1 $e |- ph $.
    mp2.2 $e |- ps $.
    mp2.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( A double modus ponens inference.  (Contributed by NM, 5-Apr-1994.)
       (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
    mp2 $p |- ch $=
      ( mpi ax-mp ) ACDABCEFGH $.
  $}

  ${
    3syl.1 $e |- ( ph -> ps ) $.
    3syl.2 $e |- ( ps -> ch ) $.
    3syl.3 $e |- ( ch -> th ) $.
    $( Inference chaining two syllogisms.  (Contributed by NM, 5-Aug-1993.) $)
    3syl $p |- ( ph -> th ) $=
      ( syl ) ACDABCEFHGH $.
  $}

  ${
    4syl.1 $e |- ( ph -> ps ) $.
    4syl.2 $e |- ( ps -> ch ) $.
    4syl.3 $e |- ( ch -> th ) $.
    4syl.4 $e |- ( th -> ta ) $.
    $( Inference chaining three syllogisms.  The use of this theorem is marked
       "discouraged" because it can cause the "minimize" command to have very
       long run times.  However, feel free to use "minimize 4syl /override" if
       you wish.  (Contributed by BJ, 14-Jul-2018.)
       (New usage is discouraged.) $)
    4syl $p |- ( ph -> ta ) $=
      ( 3syl syl ) ADEABCDFGHJIK $.
  $}

  $( Principle of identity.  Theorem *2.08 of [WhiteheadRussell] p. 101.  For
     another version of the proof directly from axioms, see ~ idALT .
     (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Stefan Allan,
     20-Mar-2006.) $)
  id $p |- ( ph -> ph ) $=
    ( wi ax-1 mpd ) AAABZAAACAECD $.

  $( Principle of identity.  Theorem *2.08 of [WhiteheadRussell] p. 101.  This
     version is proved directly from the axioms for demonstration purposes.
     This proof is a popular example in the literature and is identical, step
     for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a)
     of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of
     [Mendelson] p. 36.  It is also "Our first proof" in Hirst and Hirst's _A
     Primer for Logic and Proof_ p. 17 (PDF p. 23) at
     ~ http://www.mathsci.appstate.edu/~~jlh/primer/hirst.pdf .  For a shorter
     version of the proof that takes advantage of previously proved theorems,
     see ~ id .  (Contributed by NM, 5-Aug-1993.)
     (Proof modification is discouraged.)  Use ~ id instead.
     (New usage is discouraged.) $)
  idALT $p |- ( ph -> ph ) $=
    ( wi ax-1 ax-2 ax-mp ) AAABZBZFAACAFABBGFBAFCAFADEE $.

  $( Principle of identity with antecedent.  (Contributed by NM,
     26-Nov-1995.) $)
  idd $p |- ( ph -> ( ps -> ps ) ) $=
    ( wi id a1i ) BBCABDE $.

  ${
    a1d.1 $e |- ( ph -> ps ) $.
    $( Deduction introducing an embedded antecedent.  (The proof was revised by
       Stefan Allan, 20-Mar-2006.)

       _Naming convention_:  We often call a theorem a "deduction" and suffix
       its label with "d" whenever the hypotheses and conclusion are each
       prefixed with the same antecedent.  This allows us to use the theorem in
       places where (in traditional textbook formalizations) the standard
       Deduction Theorem would be used; here ` ph ` would be replaced with a
       conjunction ( ~ wa ) of the hypotheses of the would-be deduction.  By
       contrast, we tend to call the simpler version with no common antecedent
       an "inference" and suffix its label with "i"; compare Theorem ~ a1i .
       Finally, a "theorem" would be the form with no hypotheses; in this case
       the "theorem" form would be the original axiom ~ ax-1 .  We usually show
       the theorem form without a suffix on its label (e.g., ~ pm2.43 versus
       ~ pm2.43i versus ~ pm2.43d ).  (Contributed by NM, 5-Aug-1993.)
       (Revised by NM, 20-Mar-2006.) $)
    a1d $p |- ( ph -> ( ch -> ps ) ) $=
      ( wi ax-1 syl ) ABCBEDBCFG $.
  $}

  ${
    2a1d.1 $e |- ( ph -> ps ) $.
    $( Deduction introducing two antecedents.  Two applications of ~ a1d .
       Deduction associated with ~ 2a1 and ~ 2a1i .  (Contributed by BJ,
       10-Aug-2020.) $)
    2a1d $p |- ( ph -> ( ch -> ( th -> ps ) ) ) $=
      ( wi a1d ) ADBFCABDEGG $.
  $}

  ${
    a1i13.1 $e |- ( ps -> th ) $.
    $( Add two antecedents to a wff.  (Contributed by Jeff Hankins,
       4-Aug-2009.) $)
    a1i13 $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
      ( wi a1d a1i ) BCDFFABDCEGH $.
  $}

  $( A double form of ~ ax-1 .  Its associated inference is ~ 2a1i .  Its
     associated deduction is ~ 2a1d .  (Contributed by BJ, 10-Aug-2020.)
     (Proof shortened by Wolf Lammen, 1-Sep-2020.) $)
  2a1 $p |- ( ph -> ( ps -> ( ch -> ph ) ) ) $=
    ( id 2a1d ) AABCADE $.

  ${
    a2d.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( Deduction distributing an embedded antecedent.  (Contributed by NM,
       23-Jun-1994.) $)
    a2d $p |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $=
      ( wi ax-2 syl ) ABCDFFBCFBDFFEBCDGH $.
  $}

  ${
    2a1i.1 $e |- ch $.
    $( Add two antecedents to a wff.  (Contributed by Jeff Hankins,
       4-Aug-2009.)  (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
    2a1i $p |- ( ph -> ( ps -> ch ) ) $=
      ( a1i a1d ) ACBCADEF $.
  $}

  ${
    sylcom.1 $e |- ( ph -> ( ps -> ch ) ) $.
    sylcom.2 $e |- ( ps -> ( ch -> th ) ) $.
    $( Syllogism inference with commutation of antecedents.  (Contributed by
       NM, 29-Aug-2004.)  (Proof shortened by O'Cat, 2-Feb-2006.)  (Proof
       shortened by Stefan Allan, 23-Feb-2006.) $)
    sylcom $p |- ( ph -> ( ps -> th ) ) $=
      ( wi a2i syl ) ABCGBDGEBCDFHI $.
  $}

  ${
    syl5com.1 $e |- ( ph -> ps ) $.
    syl5com.2 $e |- ( ch -> ( ps -> th ) ) $.
    $( Syllogism inference with commuted antecedents.  (Contributed by NM,
       24-May-2005.) $)
    syl5com $p |- ( ph -> ( ch -> th ) ) $=
      ( a1d sylcom ) ACBDABCEGFH $.
  $}

  ${
    $( Premise for ~ com12 .  See ~ pm2.04 for the theorem form. $)
    com12.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference that swaps (commutes) antecedents in an implication.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       4-Aug-2012.) $)
    com12 $p |- ( ps -> ( ph -> ch ) ) $=
      ( id syl5com ) BBACBEDF $.
  $}

  ${
    syl11.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl11.2 $e |- ( th -> ph ) $.
    $( A syllogism inference.  Commuted form of an instance of ~ syl .
       (Contributed by BJ, 25-Oct-2021.) $)
    syl11 $p |- ( ps -> ( th -> ch ) ) $=
      ( wi syl com12 ) DBCDABCGFEHI $.
  $}

  ${
    syl5.1 $e |- ( ph -> ps ) $.
    syl5.2 $e |- ( ch -> ( ps -> th ) ) $.
    $( A syllogism rule of inference.  The second premise is used to replace
       the second antecedent of the first premise.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Wolf Lammen, 25-May-2013.) $)
    syl5 $p |- ( ch -> ( ph -> th ) ) $=
      ( syl5com com12 ) ACDABCDEFGH $.
  $}

  ${
    syl6.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl6.2 $e |- ( ch -> th ) $.
    $( A syllogism rule of inference.  The second premise is used to replace
       the consequent of the first premise.  (Contributed by NM, 5-Aug-1993.)
       (Proof shortened by Wolf Lammen, 30-Jul-2012.) $)
    syl6 $p |- ( ph -> ( ps -> th ) ) $=
      ( wi a1i sylcom ) ABCDECDGBFHI $.
  $}

  ${
    syl56.1 $e |- ( ph -> ps ) $.
    syl56.2 $e |- ( ch -> ( ps -> th ) ) $.
    syl56.3 $e |- ( th -> ta ) $.
    $( Combine ~ syl5 and ~ syl6 .  (Contributed by NM, 14-Nov-2013.) $)
    syl56 $p |- ( ch -> ( ph -> ta ) ) $=
      ( syl6 syl5 ) ABCEFCBDEGHIJ $.
  $}

  ${
    syl6com.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl6com.2 $e |- ( ch -> th ) $.
    $( Syllogism inference with commuted antecedents.  (Contributed by NM,
       25-May-2005.) $)
    syl6com $p |- ( ps -> ( ph -> th ) ) $=
      ( syl6 com12 ) ABDABCDEFGH $.
  $}

  ${
    mpcom.1 $e |- ( ps -> ph ) $.
    mpcom.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Modus ponens inference with commutation of antecedents.  (Contributed by
       NM, 17-Mar-1996.) $)
    mpcom $p |- ( ps -> ch ) $=
      ( com12 mpd ) BACDABCEFG $.
  $}

  ${
    syli.1 $e |- ( ps -> ( ph -> ch ) ) $.
    syli.2 $e |- ( ch -> ( ph -> th ) ) $.
    $( Syllogism inference with common nested antecedent.  (Contributed by NM,
       4-Nov-2004.) $)
    syli $p |- ( ps -> ( ph -> th ) ) $=
      ( com12 sylcom ) BACDECADFGH $.
  $}

  ${
    syl2im.1 $e |- ( ph -> ps ) $.
    syl2im.2 $e |- ( ch -> th ) $.
    syl2im.3 $e |- ( ps -> ( th -> ta ) ) $.
    $( Replace two antecedents.  Implication-only version of ~ syl2an .
       (Contributed by Wolf Lammen, 14-May-2013.) $)
    syl2im $p |- ( ph -> ( ch -> ta ) ) $=
      ( wi syl5 syl ) ABCEIFCDBEGHJK $.

    $( A commuted version of ~ syl2im .  Implication-only version of
       ~ syl2anr .  (Contributed by BJ, 20-Oct-2021.) $)
    syl2imc $p |- ( ch -> ( ph -> ta ) ) $=
      ( syl2im com12 ) ACEABCDEFGHIJ $.
  $}

  $( This theorem, called "Assertion," can be thought of as closed form of
     modus ponens ~ ax-mp .  Theorem *2.27 of [WhiteheadRussell] p. 104.
     (Contributed by NM, 5-Aug-1993.) $)
  pm2.27 $p |- ( ph -> ( ( ph -> ps ) -> ps ) ) $=
    ( wi id com12 ) ABCZABFDE $.

  ${
    mpdd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    mpdd.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( A nested modus ponens deduction.  (Contributed by NM, 12-Dec-2004.) $)
    mpdd $p |- ( ph -> ( ps -> th ) ) $=
      ( wi a2d mpd ) ABCGBDGEABCDFHI $.
  $}

  ${
    mpid.1 $e |- ( ph -> ch ) $.
    mpid.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( A nested modus ponens deduction.  (Contributed by NM, 14-Dec-2004.) $)
    mpid $p |- ( ph -> ( ps -> th ) ) $=
      ( a1d mpdd ) ABCDACBEGFH $.
  $}

  ${
    mpdi.1 $e |- ( ps -> ch ) $.
    mpdi.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( A nested modus ponens deduction.  (Contributed by NM, 16-Apr-2005.)
       (Proof shortened by O'Cat, 15-Jan-2008.) $)
    mpdi $p |- ( ph -> ( ps -> th ) ) $=
      ( wi a1i mpdd ) ABCDBCGAEHFI $.
  $}

  ${
    mpii.1 $e |- ch $.
    mpii.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( A doubly nested modus ponens inference.  (Contributed by NM,
       31-Dec-1993.)  (Proof shortened by Wolf Lammen, 31-Jul-2012.) $)
    mpii $p |- ( ph -> ( ps -> th ) ) $=
      ( a1i mpdi ) ABCDCBEGFH $.
  $}

  ${
    syld.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syld.2 $e |- ( ph -> ( ch -> th ) ) $.
    $( Syllogism deduction.

       Notice that ~ syld has the same form as ~ syl with ` ph ` added in front
       of each hypothesis and conclusion.  When all theorems referenced in a
       proof are converted in this way, we can replace ` ph ` with a hypothesis
       of the proof, allowing the hypothesis to be eliminated with ~ id and
       become an antecedent.  The Deduction Theorem for propositional calculus,
       e.g., Theorem 3 in [Margaris] p. 56, tells us that this procedure is
       always possible.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by
       O'Cat, 19-Feb-2008.)  (Proof shortened by Wolf Lammen, 3-Aug-2012.) $)
    syld $p |- ( ph -> ( ps -> th ) ) $=
      ( wi a1d mpdd ) ABCDEACDGBFHI $.
    $( Syllogism deduction.  Commuted form of ~ syld .  (Contributed by BJ,
       25-Oct-2021.) $)
    syldc $p |- ( ps -> ( ph -> th ) ) $=
      ( syld com12 ) ABDABCDEFGH $.
  $}

  ${
    mp2d.1 $e |- ( ph -> ps ) $.
    mp2d.2 $e |- ( ph -> ch ) $.
    mp2d.3 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( A double modus ponens deduction.  (Contributed by NM, 23-May-2013.)
       (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
    mp2d $p |- ( ph -> th ) $=
      ( mpid mpd ) ABDEABCDFGHI $.
  $}

  ${
    a1dd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction introducing a nested embedded antecedent.  (Contributed by NM,
       17-Dec-2004.)  (Proof shortened by O'Cat, 15-Jan-2008.) $)
    a1dd $p |- ( ph -> ( ps -> ( th -> ch ) ) ) $=
      ( wi ax-1 syl6 ) ABCDCFECDGH $.
  $}

  ${
    pm2.43i.1 $e |- ( ph -> ( ph -> ps ) ) $.
    $( Inference absorbing redundant antecedent.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by O'Cat, 28-Nov-2008.) $)
    pm2.43i $p |- ( ph -> ps ) $=
      ( id mpd ) AABADCE $.
  $}

  ${
    pm2.43d.1 $e |- ( ph -> ( ps -> ( ps -> ch ) ) ) $.
    $( Deduction absorbing redundant antecedent.  (Contributed by NM,
       18-Aug-1993.)  (Proof shortened by O'Cat, 28-Nov-2008.) $)
    pm2.43d $p |- ( ph -> ( ps -> ch ) ) $=
      ( id mpdi ) ABBCBEDF $.
  $}

  ${
    pm2.43a.1 $e |- ( ps -> ( ph -> ( ps -> ch ) ) ) $.
    $( Inference absorbing redundant antecedent.  (Contributed by NM,
       7-Nov-1995.)  (Proof shortened by O'Cat, 28-Nov-2008.) $)
    pm2.43a $p |- ( ps -> ( ph -> ch ) ) $=
      ( id mpid ) BABCBEDF $.
  $}

  ${
    pm2.43b.1 $e |- ( ps -> ( ph -> ( ps -> ch ) ) ) $.
    $( Inference absorbing redundant antecedent.  (Contributed by NM,
       31-Oct-1995.) $)
    pm2.43b $p |- ( ph -> ( ps -> ch ) ) $=
      ( pm2.43a com12 ) BACABCDEF $.
  $}

  $( Absorption of redundant antecedent.  Also called the "Contraction" or
     "Hilbert" axiom.  Theorem *2.43 of [WhiteheadRussell] p. 106.
     (Contributed by NM, 5-Aug-1993.)  (Proof shortened by O'Cat,
     15-Aug-2004.) $)
  pm2.43 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
    ( wi pm2.27 a2i ) AABCBABDE $.

  ${
    imim2d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction adding nested antecedents.  (Contributed by NM,
       5-Aug-1993.) $)
    imim2d $p |- ( ph -> ( ( th -> ps ) -> ( th -> ch ) ) ) $=
      ( wi a1d a2d ) ADBCABCFDEGH $.
  $}

  $( A closed form of syllogism (see ~ syl ).  Theorem *2.05 of
     [WhiteheadRussell] p. 100.  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 6-Sep-2012.) $)
  imim2 $p |- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) ) $=
    ( wi id imim2d ) ABDZABCGEF $.

  ${
    embantd.1 $e |- ( ph -> ps ) $.
    embantd.2 $e |- ( ph -> ( ch -> th ) ) $.
    $( Deduction embedding an antecedent.  (Contributed by Wolf Lammen,
       4-Oct-2013.) $)
    embantd $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
      ( wi imim2d mpid ) ABCGBDEACDBFHI $.
  $}

  ${
    3syld.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3syld.2 $e |- ( ph -> ( ch -> th ) ) $.
    3syld.3 $e |- ( ph -> ( th -> ta ) ) $.
    $( Triple syllogism deduction.  (Contributed by Jeff Hankins,
       4-Aug-2009.) $)
    3syld $p |- ( ph -> ( ps -> ta ) ) $=
      ( syld ) ABDEABCDFGIHI $.
  $}

  ${
    sylsyld.1 $e |- ( ph -> ps ) $.
    sylsyld.2 $e |- ( ph -> ( ch -> th ) ) $.
    sylsyld.3 $e |- ( ps -> ( th -> ta ) ) $.
    $( A double syllogism inference.  (Contributed by Alan Sare,
       20-Apr-2011.) $)
    sylsyld $p |- ( ph -> ( ch -> ta ) ) $=
      ( wi syl syld ) ACDEGABDEIFHJK $.
  $}

  ${
    imim12i.1 $e |- ( ph -> ps ) $.
    imim12i.2 $e |- ( ch -> th ) $.
    $( Inference joining two implications.  (Contributed by NM, 5-Aug-1993.)
       (Proof shortened by O'Cat, 29-Oct-2011.) $)
    imim12i $p |- ( ( ps -> ch ) -> ( ph -> th ) ) $=
      ( wi imim2i syl5 ) ABBCGDECDBFHI $.
  $}

  ${
    imim1i.1 $e |- ( ph -> ps ) $.
    $( Inference adding common consequents in an implication, thereby
       interchanging the original antecedent and consequent.  (Contributed by
       NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 4-Aug-2012.) $)
    imim1i $p |- ( ( ps -> ch ) -> ( ph -> ch ) ) $=
      ( id imim12i ) ABCCDCEF $.
  $}

  ${
    imim3i.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference adding three nested antecedents.  (Contributed by NM,
       19-Dec-2006.) $)
    imim3i $p |- ( ( th -> ph ) -> ( ( th -> ps ) -> ( th -> ch ) ) ) $=
      ( wi imim2i a2d ) DAFDBCABCFDEGH $.
  $}

  ${
    sylc.1 $e |- ( ph -> ps ) $.
    sylc.2 $e |- ( ph -> ch ) $.
    sylc.3 $e |- ( ps -> ( ch -> th ) ) $.
    $( A syllogism inference combined with contraction.  (Contributed by NM,
       4-May-1994.)  (Revised by NM, 13-Jul-2013.) $)
    sylc $p |- ( ph -> th ) $=
      ( syl2im pm2.43i ) ADABACDEFGHI $.
  $}

  ${
    syl3c.1 $e |- ( ph -> ps ) $.
    syl3c.2 $e |- ( ph -> ch ) $.
    syl3c.3 $e |- ( ph -> th ) $.
    syl3c.4 $e |- ( ps -> ( ch -> ( th -> ta ) ) ) $.
    $( A syllogism inference combined with contraction.  (Contributed by Alan
       Sare, 7-Jul-2011.) $)
    syl3c $p |- ( ph -> ta ) $=
      ( wi sylc mpd ) ADEHABCDEJFGIKL $.
  $}

  ${
    syl6mpi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl6mpi.2 $e |- th $.
    syl6mpi.3 $e |- ( ch -> ( th -> ta ) ) $.
    $( ~ syl6 combined with ~ mpi .  (Contributed by Alan Sare, 8-Jul-2011.)
       (Proof shortened by Wolf Lammen, 13-Sep-2012.) $)
    syl6mpi $p |- ( ph -> ( ps -> ta ) ) $=
      ( mpi syl6 ) ABCEFCDEGHIJ $.
  $}

  ${
    mpsyl.1 $e |- ph $.
    mpsyl.2 $e |- ( ps -> ch ) $.
    mpsyl.3 $e |- ( ph -> ( ch -> th ) ) $.
    $( Modus ponens combined with a syllogism inference.  (Contributed by Alan
       Sare, 20-Apr-2011.) $)
    mpsyl $p |- ( ps -> th ) $=
      ( a1i sylc ) BACDABEHFGI $.
  $}

  ${
    syl6c.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl6c.2 $e |- ( ph -> ( ps -> th ) ) $.
    syl6c.3 $e |- ( ch -> ( th -> ta ) ) $.
    $( Inference combining ~ syl6 with contraction.  (Contributed by Alan Sare,
       2-May-2011.) $)
    syl6c $p |- ( ph -> ( ps -> ta ) ) $=
      ( wi syl6 mpdd ) ABDEGABCDEIFHJK $.
  $}

  ${
    syldd.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    syldd.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.
    $( Nested syllogism deduction.  (Contributed by NM, 12-Dec-2004.)  (Proof
       shortened by Wolf Lammen, 11-May-2013.) $)
    syldd $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
      ( wi imim2 syl6c ) ABDEHCDHCEHGFDECIJ $.
  $}

  ${
    syl5d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl5d.2 $e |- ( ph -> ( th -> ( ch -> ta ) ) ) $.
    $( A nested syllogism deduction.  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Josh Purinton, 29-Dec-2000.)  (Proof shortened by O'Cat,
       2-Feb-2006.) $)
    syl5d $p |- ( ph -> ( th -> ( ps -> ta ) ) ) $=
      ( wi a1d syldd ) ADBCEABCHDFIGJ $.
  $}

  ${
    syl7.1 $e |- ( ph -> ps ) $.
    syl7.2 $e |- ( ch -> ( th -> ( ps -> ta ) ) ) $.
    $( A syllogism rule of inference.  The second premise is used to replace
       the third antecedent of the first premise.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Wolf Lammen, 3-Aug-2012.) $)
    syl7 $p |- ( ch -> ( th -> ( ph -> ta ) ) ) $=
      ( wi a1i syl5d ) CABDEABHCFIGJ $.
  $}

  ${
    syl6d.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    syl6d.2 $e |- ( ph -> ( th -> ta ) ) $.
    $( A nested syllogism deduction.  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Josh Purinton, 29-Dec-2000.)  (Proof shortened by O'Cat,
       2-Feb-2006.)  (Revised by NM, 3-Feb-2006.) $)
    syl6d $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
      ( wi a1d syldd ) ABCDEFADEHBGIJ $.
  $}

  ${
    syl8.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    syl8.2 $e |- ( th -> ta ) $.
    $( A syllogism rule of inference.  The second premise is used to replace
       the consequent of the first premise.  (Contributed by NM, 1-Aug-1994.)
       (Proof shortened by Wolf Lammen, 3-Aug-2012.) $)
    syl8 $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
      ( wi a1i syl6d ) ABCDEFDEHAGIJ $.
  $}

  ${
    syl9.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl9.2 $e |- ( th -> ( ch -> ta ) ) $.
    $( A nested syllogism inference with different antecedents.  (Contributed
       by NM, 5-Aug-1993.)  (Proof shortened by Josh Purinton, 29-Dec-2000.) $)
    syl9 $p |- ( ph -> ( th -> ( ps -> ta ) ) ) $=
      ( wi a1i syl5d ) ABCDEFDCEHHAGIJ $.
  $}

  ${
    syl9r.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl9r.2 $e |- ( th -> ( ch -> ta ) ) $.
    $( A nested syllogism inference with different antecedents.  (Contributed
       by NM, 5-Aug-1993.) $)
    syl9r $p |- ( th -> ( ph -> ( ps -> ta ) ) ) $=
      ( wi syl9 com12 ) ADBEHABCDEFGIJ $.
  $}

  ${
    imim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    imim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
    $( Deduction combining antecedents and consequents.  (Contributed by NM,
       7-Aug-1994.)  (Proof shortened by O'Cat, 30-Oct-2011.) $)
    imim12d $p |- ( ph -> ( ( ch -> th ) -> ( ps -> ta ) ) ) $=
      ( wi imim2d syl5d ) ABCCDHEFADECGIJ $.
  $}

  ${
    imim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction adding nested consequents.  (Contributed by NM, 3-Apr-1994.)
       (Proof shortened by Wolf Lammen, 12-Sep-2012.) $)
    imim1d $p |- ( ph -> ( ( ch -> th ) -> ( ps -> th ) ) ) $=
      ( idd imim12d ) ABCDDEADFG $.
  $}

  $( A closed form of syllogism (see ~ syl ).  Theorem *2.06 of
     [WhiteheadRussell] p. 100.  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 25-May-2013.) $)
  imim1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( wi id imim1d ) ABDZABCGEF $.

  $( Theorem *2.83 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.83 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ( ch -> th ) ) ->
                ( ph -> ( ps -> th ) ) ) ) $=
    ( wi imim1 imim3i ) BCECDEBDEABCDFG $.

  ${
    com3.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( Commutation of antecedents.  Swap 2nd and 3rd.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Wolf Lammen, 4-Aug-2012.) $)
    com23 $p |- ( ph -> ( ch -> ( ps -> th ) ) ) $=
      ( wi pm2.27 syl9 ) ABCDFCDECDGH $.

    $( Commutation of antecedents.  Rotate right.  (Contributed by NM,
       25-Apr-1994.) $)
    com3r $p |- ( ch -> ( ph -> ( ps -> th ) ) ) $=
      ( wi com23 com12 ) ACBDFABCDEGH $.

    $( Commutation of antecedents.  Swap 1st and 3rd.  (Contributed by NM,
       25-Apr-1994.)  (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
    com13 $p |- ( ch -> ( ps -> ( ph -> th ) ) ) $=
      ( com3r com23 ) CABDABCDEFG $.

    $( Commutation of antecedents.  Rotate left.  (Contributed by NM,
       25-Apr-1994.)  (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
    com3l $p |- ( ps -> ( ch -> ( ph -> th ) ) ) $=
      ( com3r ) CABDABCDEFF $.
  $}

  $( Swap antecedents.  Theorem *2.04 of [WhiteheadRussell] p. 100.
     (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
     12-Sep-2012.) $)
  pm2.04 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
    ( wi id com23 ) ABCDDZABCGEF $.

  ${
    com4.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
    $( Commutation of antecedents.  Swap 3rd and 4th.  (Contributed by NM,
       25-Apr-1994.) $)
    com34 $p |- ( ph -> ( ps -> ( th -> ( ch -> ta ) ) ) ) $=
      ( wi pm2.04 syl6 ) ABCDEGGDCEGGFCDEHI $.

    $( Commutation of antecedents.  Rotate left.  (Contributed by NM,
       25-Apr-1994.)  (Proof shortened by O'Cat, 15-Aug-2004.) $)
    com4l $p |- ( ps -> ( ch -> ( th -> ( ph -> ta ) ) ) ) $=
      ( wi com3l com34 ) BCADEABCDEGFHI $.

    $( Commutation of antecedents.  Rotate twice.  (Contributed by NM,
       25-Apr-1994.) $)
    com4t $p |- ( ch -> ( th -> ( ph -> ( ps -> ta ) ) ) ) $=
      ( com4l ) BCDAEABCDEFGG $.

    $( Commutation of antecedents.  Rotate right.  (Contributed by NM,
       25-Apr-1994.) $)
    com4r $p |- ( th -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) $=
      ( com4t com4l ) CDABEABCDEFGH $.

    $( Commutation of antecedents.  Swap 2nd and 4th.  (Contributed by NM,
       25-Apr-1994.)  (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
    com24 $p |- ( ph -> ( th -> ( ch -> ( ps -> ta ) ) ) ) $=
      ( wi com4t com13 ) CDABEGABCDEFHI $.

    $( Commutation of antecedents.  Swap 1st and 4th.  (Contributed by NM,
       25-Apr-1994.)  (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
    com14 $p |- ( th -> ( ps -> ( ch -> ( ph -> ta ) ) ) ) $=
      ( wi com4l com3r ) BCDAEGABCDEFHI $.
  $}

  ${
    com5.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $.
    $( Commutation of antecedents.  Swap 4th and 5th.  (Contributed by Jeff
       Hankins, 28-Jun-2009.) $)
    com45 $p |- ( ph -> ( ps -> ( ch -> ( ta -> ( th -> et ) ) ) ) ) $=
      ( wi pm2.04 syl8 ) ABCDEFHHEDFHHGDEFIJ $.

    $( Commutation of antecedents.  Swap 3rd and 5th.  (Contributed by Jeff
       Hankins, 28-Jun-2009.) $)
    com35 $p |- ( ph -> ( ps -> ( ta -> ( th -> ( ch -> et ) ) ) ) ) $=
      ( wi com34 com45 ) ABDECFHABDCEFABCDEFHGIJI $.

    $( Commutation of antecedents.  Swap 2nd and 5th.  (Contributed by Jeff
       Hankins, 28-Jun-2009.) $)
    com25 $p |- ( ph -> ( ta -> ( ch -> ( th -> ( ps -> et ) ) ) ) ) $=
      ( wi com24 com45 ) ADCEBFHADCBEFABCDEFHGIJI $.

    $( Commutation of antecedents.  Rotate left.  (Contributed by Jeff Hankins,
       28-Jun-2009.)  (Proof shortened by Wolf Lammen, 29-Jul-2012.) $)
    com5l $p |- ( ps -> ( ch -> ( th -> ( ta -> ( ph -> et ) ) ) ) ) $=
      ( wi com4l com45 ) BCDAEFABCDEFHGIJ $.

    $( Commutation of antecedents.  Swap 1st and 5th.  (Contributed by Jeff
       Hankins, 28-Jun-2009.)  (Proof shortened by Wolf Lammen,
       29-Jul-2012.) $)
    com15 $p |- ( ta -> ( ps -> ( ch -> ( th -> ( ph -> et ) ) ) ) ) $=
      ( wi com5l com4r ) BCDEAFHABCDEFGIJ $.

    $( Commutation of antecedents.  Rotate left twice.  (Contributed by Jeff
       Hankins, 28-Jun-2009.) $)
    com52l $p |- ( ch -> ( th -> ( ta -> ( ph -> ( ps -> et ) ) ) ) ) $=
      ( com5l ) BCDEAFABCDEFGHH $.

    $( Commutation of antecedents.  Rotate right twice.  (Contributed by Jeff
       Hankins, 28-Jun-2009.) $)
    com52r $p |- ( th -> ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) ) ) $=
      ( com52l com5l ) CDEABFABCDEFGHI $.

    $( Commutation of antecedents.  Rotate right.  (Contributed by Wolf Lammen,
       29-Jul-2012.) $)
    com5r $p |- ( ta -> ( ph -> ( ps -> ( ch -> ( th -> et ) ) ) ) ) $=
      ( com52l ) CDEABFABCDEFGHH $.
  $}

  $( Elimination of a nested antecedent.  (Contributed by Wolf Lammen,
     9-May-2013.) $)
  jarr $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $=
    ( wi ax-1 imim1i ) BABDCBAEF $.

  ${
    jarri.1 $e |- ( ( ph -> ps ) -> ch ) $.
    $( Inference associated with ~ jarr .  (Contributed by Wolf Lammen,
       20-Sep-2013.) $)
    jarri $p |- ( ps -> ch ) $=
      ( wi ax-1 syl ) BABECBAFDG $.
  $}

  ${
    pm2.86i.1 $e |- ( ( ph -> ps ) -> ( ph -> ch ) ) $.
    $( Inference based on ~ pm2.86 .  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Wolf Lammen, 3-Apr-2013.) $)
    pm2.86i $p |- ( ph -> ( ps -> ch ) ) $=
      ( wi ax-1 syl com12 ) BACBABEACEBAFDGH $.
  $}

  ${
    pm2.86d.1 $e |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $.
    $( Deduction based on ~ pm2.86 .  (Contributed by NM, 29-Jun-1995.)  (Proof
       shortened by Wolf Lammen, 3-Apr-2013.) $)
    pm2.86d $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
      ( wi ax-1 syl5 com23 ) ACBDCBCFABDFCBGEHI $.
  $}

  $( Converse of Axiom ~ ax-2 .  Theorem *2.86 of [WhiteheadRussell] p. 108.
     (Contributed by NM, 25-Apr-1994.)  (Proof shortened by Wolf Lammen,
     3-Apr-2013.) $)
  pm2.86 $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) ->
              ( ph -> ( ps -> ch ) ) ) $=
    ( wi id pm2.86d ) ABDACDDZABCGEF $.

  $( The Linearity Axiom of the infinite-valued sentential logic (L-infinity)
     of Lukasiewicz.  (Contributed by O'Cat, 12-Aug-2004.) $)
  loolin $p |- ( ( ( ph -> ps ) -> ( ps -> ph ) ) -> ( ps -> ph ) ) $=
    ( wi jarr pm2.43d ) ABCBACZCBAABFDE $.

  $( An alternate for the Linearity Axiom of the infinite-valued sentential
     logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, _Reports
     on Mathematical Logic_ 10, 129-137 (1978).  (Contributed by O'Cat,
     8-Aug-2004.) $)
  loowoz $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) ->
                 ( ( ps -> ph ) -> ( ps -> ch ) ) ) $=
    ( wi jarr a2d ) ABDACDZDBACABGEF $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical conjunction and logical equivalence
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare connectives for conjunction ('and') and biconditional. $)
  $c /\ $. $( Wedge (read:  'and') $)
  $c <-> $. $( Double arrow (read:  'if and only if' or
               'is logically equivalent to') $)

  $( Extend wff definition to include conjunction ('and'). $)
  wa $a wff ( ph /\ ps ) $.

  $( Extend our wff definition to include the biconditional connective. $)
  wb $a wff ( ph <-> ps ) $.

  $( Left 'and' elimination.  One of the axioms of propositional logic.  Use
     its alias ~ simpl instead for naming consistency with set.mm.
     (New usage is discouraged.)  (Contributed by Mario Carneiro,
     31-Jan-2015.) $)
  ax-ia1 $a |- ( ( ph /\ ps ) -> ph ) $.

  $( Right 'and' elimination.  One of the axioms of propositional logic.
     (Contributed by Mario Carneiro, 31-Jan-2015.)  Use its alias ~ simpr
     instead for naming consistency with set.mm.
     (New usage is discouraged.) $)
  ax-ia2 $a |- ( ( ph /\ ps ) -> ps ) $.

  $( 'And' introduction.  One of the axioms of propositional logic.
     (Contributed by Mario Carneiro, 31-Jan-2015.) $)
  ax-ia3 $a |- ( ph -> ( ps -> ( ph /\ ps ) ) ) $.

  $( Elimination of a conjunct.  Theorem *3.26 (Simp) of [WhiteheadRussell]
     p. 112.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 13-Nov-2012.) $)
  simpl $p |- ( ( ph /\ ps ) -> ph ) $=
    ( ax-ia1 ) ABC $.

  $( Elimination of a conjunct.  Theorem *3.27 (Simp) of [WhiteheadRussell]
     p. 112.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 13-Nov-2012.) $)
  simpr $p |- ( ( ph /\ ps ) -> ps ) $=
    ( ax-ia2 ) ABC $.

  ${
    simpli.1 $e |- ( ph /\ ps ) $.
    $( Inference eliminating a conjunct.  (Contributed by NM, 15-Jun-1994.) $)
    simpli $p |- ph $=
      ( wa simpl ax-mp ) ABDACABEF $.
  $}

  ${
    simpld.1 $e |- ( ph -> ( ps /\ ch ) ) $.
    $( Deduction eliminating a conjunct.  (Contributed by NM, 5-Aug-1993.) $)
    simpld $p |- ( ph -> ps ) $=
      ( wa simpl syl ) ABCEBDBCFG $.
  $}

  ${
    simpri.1 $e |- ( ph /\ ps ) $.
    $( Inference eliminating a conjunct.  (Contributed by NM, 15-Jun-1994.) $)
    simpri $p |- ps $=
      ( wa simpr ax-mp ) ABDBCABEF $.
  $}

  ${
    simprd.1 $e |- ( ph -> ( ps /\ ch ) ) $.
    $( Deduction eliminating a conjunct.  (Contributed by NM, 5-Aug-1993.)
       (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
    simprd $p |- ( ph -> ch ) $=
      ( wa simpr syl ) ABCECDBCFG $.
  $}

  ${
    exp.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Exportation inference.  (This theorem used to be labeled "exp" but was
       changed to "ex" so as not to conflict with the math token "exp", per the
       June 2006 Metamath spec change.)  (Contributed by NM, 5-Aug-1993.)
       (Proof shortened by Eric Schmidt, 22-Dec-2006.) $)
    ex $p |- ( ph -> ( ps -> ch ) ) $=
      ( wa ax-ia3 syl6 ) ABABECABFDG $.

    $( Exportation inference with commuted antecedents.  (Contributed by NM,
       25-May-2005.) $)
    expcom $p |- ( ps -> ( ph -> ch ) ) $=
      ( ex com12 ) ABCABCDEF $.
  $}

  $( This is our first definition, which introduces and defines the
     biconditional connective ` <-> ` , also called biimplication.  We define a
     wff of the form ` ( ph <-> ps ) ` as an abbreviation for
     ` ( ( ph -> ps ) /\ ( ps -> ph ) ) ` .

     Unlike most traditional developments, we have chosen not to have a
     separate symbol such as "Df." to mean "is defined as."  Instead, we will
     later use the biconditional connective for this purpose, as it allows us
     to use logic to manipulate definitions directly.  For an example of such a
     definition, see ~ df-3or .  This greatly simplifies many proofs since it
     eliminates the need for a separate mechanism for introducing and
     eliminating definitions.  Of course, we cannot use this mechanism to
     define the biconditional itself, since it hasn't been introduced yet.
     Instead, we use a more general form of definition, described as follows.

     In its most general form, a definition is simply an assertion that
     introduces a new symbol (or a new combination of existing symbols, as in
     ~ df-3an ) that is eliminable and does not strengthen the existing
     language.  The latter requirement means that the set of provable
     statements not containing the new symbol (or new combination) should
     remain exactly the same after the definition is introduced.  Our
     definition of the biconditional may look unusual compared to most
     definitions, but it strictly satisfies these requirements.

     The justification for our definition is that if we mechanically replace
     ` ( ph <-> ps ) ` (the definiendum i.e. the thing being defined) with
     ` ( ( ph -> ps ) /\ ( ps -> ph ) ) ` (the definiens i.e. the defining
     expression) in the definition, the definition becomes the previously
     proved theorem ~ biijust .  It is impossible to use ~ df-bi to prove any
     statement expressed in the original language that can't be proved from the
     original axioms, because if we simply replace each instance of ~ df-bi in
     the proof with the corresponding ~ biijust instance, we will end up with a
     proof from the original axioms.

     Note that from Metamath's point of view, a definition is just another
     axiom - i.e. an assertion we claim to be true - but from our high level
     point of view, we are are not strengthening the language.  To indicate
     this fact, we prefix definition labels with "df-" instead of "ax-".  (This
     prefixing is an informal convention that means nothing to the Metamath
     proof verifier; it is just for human readability.)

     ~ df-bi itself is a conjunction of two implications (to avoid using the
     biconditional in its own definition), but once we have the biconditional,
     we can prove ~ dfbi2 which uses the biconditional instead.

     Other definitions of the biconditional, such as ~ dfbi3dc , only hold for
     decidable propositions, not all propositions.  (Contributed by NM,
     5-Aug-1993.)  (Revised by Jim Kingdon, 24-Nov-2017.) $)
  df-bi $a |- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
        /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) ) $.

  $( Property of the biconditional connective.  (Contributed by NM,
     11-May-1999.)  (Revised by NM, 31-Jan-2015.) $)
  biimp $p |- ( ( ph <-> ps ) -> ( ph -> ps ) ) $=
    ( wb wi wa df-bi simpli simpld ) ABCZABDZBADZIJKEZDLIDABFGH $.

  $( Property of the biconditional connective.  (Contributed by NM,
     11-May-1999.) $)
  bi3 $p |- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) ) $=
    ( wi wb wa df-bi simpri ex ) ABCZBACZABDZKIJEZCLKCABFGH $.

  ${
    biimpi.1 $e |- ( ph <-> ps ) $.
    $( Infer an implication from a logical equivalence.  (Contributed by NM,
       5-Aug-1993.) $)
    biimpi $p |- ( ph -> ps ) $=
      ( wb wi biimp ax-mp ) ABDABECABFG $.
  $}

  ${
    sylbi.1 $e |- ( ph <-> ps ) $.
    sylbi.2 $e |- ( ps -> ch ) $.
    $( A mixed syllogism inference from a biconditional and an implication.
       Useful for substituting an antecedent with a definition.  (Contributed
       by NM, 3-Jan-1993.) $)
    sylbi $p |- ( ph -> ch ) $=
      ( biimpi syl ) ABCABDFEG $.
  $}

  ${
    sylib.1 $e |- ( ph -> ps ) $.
    sylib.2 $e |- ( ps <-> ch ) $.
    $( A mixed syllogism inference from an implication and a biconditional.
       (Contributed by NM, 3-Jan-1993.) $)
    sylib $p |- ( ph -> ch ) $=
      ( biimpi syl ) ABCDBCEFG $.
  $}

  ${
    sylbb.1 $e |- ( ph <-> ps ) $.
    sylbb.2 $e |- ( ps <-> ch ) $.
    $( A mixed syllogism inference from two biconditionals.  (Contributed by
       BJ, 30-Mar-2019.) $)
    sylbb $p |- ( ph -> ch ) $=
      ( biimpi sylbi ) ABCDBCEFG $.
  $}

  ${
    imp.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Importation inference.  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Eric Schmidt, 22-Dec-2006.) $)
    imp $p |- ( ( ph /\ ps ) -> ch ) $=
      ( wa simpl simpr sylc ) ABEABCABFABGDH $.

    $( Importation inference with commuted antecedents.  (Contributed by NM,
       25-May-2005.) $)
    impcom $p |- ( ( ps /\ ph ) -> ch ) $=
      ( com12 imp ) BACABCDEF $.
  $}

  ${
    impbii.1 $e |- ( ph -> ps ) $.
    impbii.2 $e |- ( ps -> ph ) $.
    $( Infer an equivalence from an implication and its converse.  (Contributed
       by NM, 5-Aug-1993.) $)
    impbii $p |- ( ph <-> ps ) $=
      ( wi wb bi3 mp2 ) ABEBAEABFCDABGH $.
  $}

  ${
    impbidd.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    impbidd.2 $e |- ( ph -> ( ps -> ( th -> ch ) ) ) $.
    $( Deduce an equivalence from two implications.  (Contributed by Rodolfo
       Medina, 12-Oct-2010.) $)
    impbidd $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
      ( wi wb bi3 syl6c ) ABCDGDCGCDHEFCDIJ $.
  $}

  ${
    impbid21d.1 $e |- ( ps -> ( ch -> th ) ) $.
    impbid21d.2 $e |- ( ph -> ( th -> ch ) ) $.
    $( Deduce an equivalence from two implications.  (Contributed by Wolf
       Lammen, 12-May-2013.) $)
    impbid21d $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
      ( wi a1i a1d impbidd ) ABCDBCDGGAEHADCGBFIJ $.
  $}

  ${
    impbid.1 $e |- ( ph -> ( ps -> ch ) ) $.
    impbid.2 $e |- ( ph -> ( ch -> ps ) ) $.
    $( Deduce an equivalence from two implications.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Wolf Lammen, 3-Nov-2012.) $)
    impbid $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wb impbid21d pm2.43i ) ABCFAABCDEGH $.
  $}

  $( Property of the biconditional connective.  (Contributed by NM,
     11-May-1999.)  (Proof shortened by Wolf Lammen, 11-Nov-2012.) $)
  biimpr $p |- ( ( ph <-> ps ) -> ( ps -> ph ) ) $=
    ( wb wi wa df-bi simpli simprd ) ABCZABDZBADZIJKEZDLIDABFGH $.

  $( Commutative law for equivalence.  (Contributed by Wolf Lammen,
     10-Nov-2012.) $)
  bicom1 $p |- ( ( ph <-> ps ) -> ( ps <-> ph ) ) $=
    ( wb biimpr biimp impbid ) ABCBAABDABEF $.

  ${
    bicomi.1 $e |- ( ph <-> ps ) $.
    $( Inference from commutative law for logical equivalence.  (Contributed by
       NM, 5-Aug-1993.)  (Revised by NM, 16-Sep-2013.) $)
    bicomi $p |- ( ps <-> ph ) $=
      ( wb bicom1 ax-mp ) ABDBADCABEF $.
  $}

  ${
    biimpri.1 $e |- ( ph <-> ps ) $.
    $( Infer a converse implication from a logical equivalence.  (Contributed
       by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 16-Sep-2013.) $)
    biimpri $p |- ( ps -> ph ) $=
      ( bicomi biimpi ) BAABCDE $.
  $}

  ${
    sylibr.1 $e |- ( ph -> ps ) $.
    sylibr.2 $e |- ( ch <-> ps ) $.
    $( A mixed syllogism inference from an implication and a biconditional.
       Useful for substituting a consequent with a definition.  (Contributed by
       NM, 5-Aug-1993.) $)
    sylibr $p |- ( ph -> ch ) $=
      ( biimpri syl ) ABCDCBEFG $.
  $}

  ${
    sylbir.1 $e |- ( ps <-> ph ) $.
    sylbir.2 $e |- ( ps -> ch ) $.
    $( A mixed syllogism inference from a biconditional and an implication.
       (Contributed by NM, 5-Aug-1993.) $)
    sylbir $p |- ( ph -> ch ) $=
      ( biimpri syl ) ABCBADFEG $.
  $}

  ${
    sylbbr.1 $e |- ( ph <-> ps ) $.
    sylbbr.2 $e |- ( ps <-> ch ) $.
    $( A mixed syllogism inference from two biconditionals.

       Note on the various syllogism-like statements in set.mm.  The
       hypothetical syllogism ~ syl infers an implication from two implications
       (and there are ~ 3syl and ~ 4syl for chaining more inferences).  There
       are four inferences inferring an implication from one implication and
       one biconditional: ~ sylbi , ~ sylib , ~ sylbir , ~ sylibr ; four
       inferences inferring an implication from two biconditionals: ~ sylbb ,
       ~ sylbbr , ~ sylbb1 , ~ sylbb2 ; four inferences inferring a
       biconditional from two biconditionals: ~ bitri , ~ bitr2i , ~ bitr3i ,
       ~ bitr4i (and more for chaining more biconditionals).  There are also
       closed forms and deduction versions of these, like, among many others,
       ~ syld , ~ syl5 , ~ syl6 , ~ mpbid , ~ bitrd , ~ bitrid , ~ bitrdi and
       variants.  (Contributed by BJ, 21-Apr-2019.) $)
    sylbbr $p |- ( ch -> ph ) $=
      ( biimpri sylibr ) CBABCEFDG $.
  $}

  ${
    sylbb1.1 $e |- ( ph <-> ps ) $.
    sylbb1.2 $e |- ( ph <-> ch ) $.
    $( A mixed syllogism inference from two biconditionals.  (Contributed by
       BJ, 21-Apr-2019.) $)
    sylbb1 $p |- ( ps -> ch ) $=
      ( biimpri sylib ) BACABDFEG $.
  $}

  ${
    sylbb2.1 $e |- ( ph <-> ps ) $.
    sylbb2.2 $e |- ( ch <-> ps ) $.
    $( A mixed syllogism inference from two biconditionals.  (Contributed by
       BJ, 21-Apr-2019.) $)
    sylbb2 $p |- ( ph -> ch ) $=
      ( biimpri sylbi ) ABCDCBEFG $.
  $}

  $( Join antecedents with conjunction.  Theorem *3.2 of [WhiteheadRussell]
     p. 111.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 12-Nov-2012.)  (Proof shortened by Jia Ming, 17-Nov-2020.) $)
  pm3.2 $p |- ( ph -> ( ps -> ( ph /\ ps ) ) ) $=
    ( ax-ia3 ) ABC $.

  $( Commutative law for equivalence.  Theorem *4.21 of [WhiteheadRussell]
     p. 117.  (Contributed by NM, 5-Aug-1993.)  (Revised by NM,
     11-Nov-2012.) $)
  bicom $p |- ( ( ph <-> ps ) <-> ( ps <-> ph ) ) $=
    ( wb bicom1 impbii ) ABCBACABDBADE $.

  ${
    bicomd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Commute two sides of a biconditional in a deduction.  (Contributed by
       NM, 5-Aug-1993.) $)
    bicomd $p |- ( ph -> ( ch <-> ps ) ) $=
      ( wb bicom sylib ) ABCECBEDBCFG $.
  $}

  ${
    impbid1.1 $e |- ( ph -> ( ps -> ch ) ) $.
    impbid1.2 $e |- ( ch -> ps ) $.
    $( Infer an equivalence from two implications.  (Contributed by NM,
       6-Mar-2007.) $)
    impbid1 $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wi a1i impbid ) ABCDCBFAEGH $.
  $}

  ${
    impbid2.1 $e |- ( ps -> ch ) $.
    impbid2.2 $e |- ( ph -> ( ch -> ps ) ) $.
    $( Infer an equivalence from two implications.  (Contributed by NM,
       6-Mar-2007.)  (Proof shortened by Wolf Lammen, 27-Sep-2013.) $)
    impbid2 $p |- ( ph -> ( ps <-> ch ) ) $=
      ( impbid1 bicomd ) ACBACBEDFG $.
  $}

  ${
    biimpd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduce an implication from a logical equivalence.  (Contributed by NM,
       5-Aug-1993.) $)
    biimpd $p |- ( ph -> ( ps -> ch ) ) $=
      ( wb wi biimp syl ) ABCEBCFDBCGH $.
  $}

  ${
    mpbi.min $e |- ph $.
    mpbi.maj $e |- ( ph <-> ps ) $.
    $( An inference from a biconditional, related to modus ponens.
       (Contributed by NM, 5-Aug-1993.) $)
    mpbi $p |- ps $=
      ( biimpi ax-mp ) ABCABDEF $.
  $}

  ${
    mpbir.min $e |- ps $.
    mpbir.maj $e |- ( ph <-> ps ) $.
    $( An inference from a biconditional, related to modus ponens.
       (Contributed by NM, 5-Aug-1993.) $)
    mpbir $p |- ph $=
      ( biimpri ax-mp ) BACABDEF $.
  $}

  ${
    mpbid.min $e |- ( ph -> ps ) $.
    mpbid.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( A deduction from a biconditional, related to modus ponens.  (Contributed
       by NM, 5-Aug-1993.) $)
    mpbid $p |- ( ph -> ch ) $=
      ( biimpd mpd ) ABCDABCEFG $.
  $}

  ${
    mpbii.min $e |- ps $.
    mpbii.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( An inference from a nested biconditional, related to modus ponens.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       25-Oct-2012.) $)
    mpbii $p |- ( ph -> ch ) $=
      ( a1i mpbid ) ABCBADFEG $.
  $}

  ${
    sylibd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    sylibd.2 $e |- ( ph -> ( ch <-> th ) ) $.
    $( A syllogism deduction.  (Contributed by NM, 3-Aug-1994.) $)
    sylibd $p |- ( ph -> ( ps -> th ) ) $=
      ( biimpd syld ) ABCDEACDFGH $.
  $}

  ${
    sylbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    sylbid.2 $e |- ( ph -> ( ch -> th ) ) $.
    $( A syllogism deduction.  (Contributed by NM, 3-Aug-1994.) $)
    sylbid $p |- ( ph -> ( ps -> th ) ) $=
      ( biimpd syld ) ABCDABCEGFH $.
  $}

  ${
    mpbidi.min $e |- ( th -> ( ph -> ps ) ) $.
    mpbidi.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( A deduction from a biconditional, related to modus ponens.  (Contributed
       by NM, 9-Aug-1994.) $)
    mpbidi $p |- ( th -> ( ph -> ch ) ) $=
      ( biimpd sylcom ) DABCEABCFGH $.
  $}

  ${
    biimtrid.1 $e |- ( ph <-> ps ) $.
    biimtrid.2 $e |- ( ch -> ( ps -> th ) ) $.
    $( A mixed syllogism inference from a nested implication and a
       biconditional.  Useful for substituting an embedded antecedent with a
       definition.  (Contributed by NM, 5-Aug-1993.) $)
    biimtrid $p |- ( ch -> ( ph -> th ) ) $=
      ( biimpi syl5 ) ABCDABEGFH $.
  $}

  ${
    biimtrrid.1 $e |- ( ps <-> ph ) $.
    biimtrrid.2 $e |- ( ch -> ( ps -> th ) ) $.
    $( A mixed syllogism inference from a nested implication and a
       biconditional.  (Contributed by NM, 5-Aug-1993.) $)
    biimtrrid $p |- ( ch -> ( ph -> th ) ) $=
      ( biimpri syl5 ) ABCDBAEGFH $.
  $}

  ${
    imbitrid.1 $e |- ( ph -> ps ) $.
    imbitrid.2 $e |- ( ch -> ( ps <-> th ) ) $.
    $( A mixed syllogism inference.  (Contributed by NM, 12-Jan-1993.) $)
    imbitrid $p |- ( ch -> ( ph -> th ) ) $=
      ( biimpd syl5 ) ABCDECBDFGH $.

    $( A mixed syllogism inference.  (Contributed by NM, 19-Jun-2007.) $)
    syl5ibcom $p |- ( ph -> ( ch -> th ) ) $=
      ( imbitrid com12 ) CADABCDEFGH $.
  $}

  ${
    imbitrrid.1 $e |- ( ph -> th ) $.
    imbitrrid.2 $e |- ( ch -> ( ps <-> th ) ) $.
    $( A mixed syllogism inference.  (Contributed by NM, 3-Apr-1994.)  (Revised
       by NM, 22-Sep-2013.) $)
    imbitrrid $p |- ( ch -> ( ph -> ps ) ) $=
      ( bicomd imbitrid ) ADCBECBDFGH $.

    $( A mixed syllogism inference.  (Contributed by NM, 20-Jun-2007.) $)
    syl5ibrcom $p |- ( ph -> ( ch -> ps ) ) $=
      ( imbitrrid com12 ) CABABCDEFGH $.
  $}

  ${
    biimprd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduce a converse implication from a logical equivalence.  (Contributed
       by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)
    biimprd $p |- ( ph -> ( ch -> ps ) ) $=
      ( id imbitrrid ) CBACCEDF $.
  $}

  ${
    biimpcd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduce a commuted implication from a logical equivalence.  (Contributed
       by NM, 3-May-1994.)  (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)
    biimpcd $p |- ( ps -> ( ph -> ch ) ) $=
      ( id syl5ibcom ) BBACBEDF $.

    $( Deduce a converse commuted implication from a logical equivalence.
       (Contributed by NM, 3-May-1994.)  (Proof shortened by Wolf Lammen,
       20-Dec-2013.) $)
    biimprcd $p |- ( ch -> ( ph -> ps ) ) $=
      ( id syl5ibrcom ) CBACCEDF $.
  $}

  ${
    imbitrdi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    imbitrdi.2 $e |- ( ch <-> th ) $.
    $( A mixed syllogism inference from a nested implication and a
       biconditional.  (Contributed by NM, 5-Aug-1993.) $)
    imbitrdi $p |- ( ph -> ( ps -> th ) ) $=
      ( biimpi syl6 ) ABCDECDFGH $.
  $}

  ${
    imbitrrdi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    imbitrrdi.2 $e |- ( th <-> ch ) $.
    $( A mixed syllogism inference from a nested implication and a
       biconditional.  Useful for substituting an embedded consequent with a
       definition.  (Contributed by NM, 5-Aug-1993.) $)
    imbitrrdi $p |- ( ph -> ( ps -> th ) ) $=
      ( biimpri syl6 ) ABCDEDCFGH $.
  $}

  ${
    biimtrdi.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    biimtrdi.2 $e |- ( ch -> th ) $.
    $( A mixed syllogism inference.  (Contributed by NM, 2-Jan-1994.) $)
    biimtrdi $p |- ( ph -> ( ps -> th ) ) $=
      ( biimpd syl6 ) ABCDABCEGFH $.
  $}

  ${
    biimtrrdi.1 $e |- ( ph -> ( ch <-> ps ) ) $.
    biimtrrdi.2 $e |- ( ch -> th ) $.
    $( A mixed syllogism inference.  (Contributed by NM, 18-May-1994.) $)
    biimtrrdi $p |- ( ph -> ( ps -> th ) ) $=
      ( biimprd syl6 ) ABCDACBEGFH $.
  $}

  ${
    syl7bi.1 $e |- ( ph <-> ps ) $.
    syl7bi.2 $e |- ( ch -> ( th -> ( ps -> ta ) ) ) $.
    $( A mixed syllogism inference from a doubly nested implication and a
       biconditional.  (Contributed by NM, 5-Aug-1993.) $)
    syl7bi $p |- ( ch -> ( th -> ( ph -> ta ) ) ) $=
      ( biimpi syl7 ) ABCDEABFHGI $.
  $}

  ${
    syl8ib.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    syl8ib.2 $e |- ( th <-> ta ) $.
    $( A syllogism rule of inference.  The second premise is used to replace
       the consequent of the first premise.  (Contributed by NM,
       1-Aug-1994.) $)
    syl8ib $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
      ( biimpi syl8 ) ABCDEFDEGHI $.
  $}

  ${
    mpbird.min $e |- ( ph -> ch ) $.
    mpbird.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( A deduction from a biconditional, related to modus ponens.  (Contributed
       by NM, 5-Aug-1993.) $)
    mpbird $p |- ( ph -> ps ) $=
      ( biimprd mpd ) ACBDABCEFG $.
  $}

  ${
    mpbiri.min $e |- ch $.
    mpbiri.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( An inference from a nested biconditional, related to modus ponens.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       25-Oct-2012.) $)
    mpbiri $p |- ( ph -> ps ) $=
      ( a1i mpbird ) ABCCADFEG $.
  $}

  ${
    sylibrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    sylibrd.2 $e |- ( ph -> ( th <-> ch ) ) $.
    $( A syllogism deduction.  (Contributed by NM, 3-Aug-1994.) $)
    sylibrd $p |- ( ph -> ( ps -> th ) ) $=
      ( biimprd syld ) ABCDEADCFGH $.
  $}

  ${
    sylbird.1 $e |- ( ph -> ( ch <-> ps ) ) $.
    sylbird.2 $e |- ( ph -> ( ch -> th ) ) $.
    $( A syllogism deduction.  (Contributed by NM, 3-Aug-1994.) $)
    sylbird $p |- ( ph -> ( ps -> th ) ) $=
      ( biimprd syld ) ABCDACBEGFH $.
  $}

  $( Principle of identity for logical equivalence.  Theorem *4.2 of
     [WhiteheadRussell] p. 117.  (Contributed by NM, 5-Aug-1993.) $)
  biid $p |- ( ph <-> ph ) $=
    ( id impbii ) AAABZDC $.

  $( Principle of identity with antecedent.  (Contributed by NM,
     25-Nov-1995.) $)
  biidd $p |- ( ph -> ( ps <-> ps ) ) $=
    ( wb biid a1i ) BBCABDE $.

  $( Two propositions are equivalent if they are both true.  Closed form of
     ~ 2th .  Equivalent to a ~ biimp -like version of the xor-connective.
     This theorem stays true, no matter how you permute its operands.  This is
     evident from its sharper version ` ( ph <-> ( ps <-> ( ph <-> ps ) ) ) ` .
     (Contributed by Wolf Lammen, 12-May-2013.) $)
  pm5.1im $p |- ( ph -> ( ps -> ( ph <-> ps ) ) ) $=
    ( ax-1 impbid21d ) ABABBACABCD $.

  ${
    2th.1 $e |- ph $.
    2th.2 $e |- ps $.
    $( Two truths are equivalent.  (Contributed by NM, 18-Aug-1993.) $)
    2th $p |- ( ph <-> ps ) $=
      ( a1i impbii ) ABBADEABCEF $.
  $}

  ${
    2thd.1 $e |- ( ph -> ps ) $.
    2thd.2 $e |- ( ph -> ch ) $.
    $( Two truths are equivalent (deduction form).  (Contributed by NM,
       3-Jun-2012.)  (Revised by NM, 29-Jan-2013.) $)
    2thd $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wb pm5.1im sylc ) ABCBCFDEBCGH $.
  $}

  ${
    ibi.1 $e |- ( ph -> ( ph <-> ps ) ) $.
    $( Inference that converts a biconditional implied by one of its arguments,
       into an implication.  (Contributed by NM, 17-Oct-2003.) $)
    ibi $p |- ( ph -> ps ) $=
      ( biimpd pm2.43i ) ABAABCDE $.
  $}

  ${
    ibir.1 $e |- ( ph -> ( ps <-> ph ) ) $.
    $( Inference that converts a biconditional implied by one of its arguments,
       into an implication.  (Contributed by NM, 22-Jul-2004.) $)
    ibir $p |- ( ph -> ps ) $=
      ( bicomd ibi ) ABABACDE $.
  $}

  ${
    ibd.1 $e |- ( ph -> ( ps -> ( ps <-> ch ) ) ) $.
    $( Deduction that converts a biconditional implied by one of its arguments,
       into an implication.  (Contributed by NM, 26-Jun-2004.) $)
    ibd $p |- ( ph -> ( ps -> ch ) ) $=
      ( wb biimp syli ) BABCECDBCFG $.
  $}

  $( Distribution of implication over biconditional.  Theorem *5.74 of
     [WhiteheadRussell] p. 126.  (Contributed by NM, 1-Aug-1994.)  (Proof
     shortened by Wolf Lammen, 11-Apr-2013.) $)
  pm5.74 $p |- ( ( ph -> ( ps <-> ch ) ) <->
               ( ( ph -> ps ) <-> ( ph -> ch ) ) ) $=
    ( wb wi biimp imim3i biimpr impbid pm2.86d impbidd impbii ) ABCDZEZABEZACEZ
    DZNOPMBCABCFGMCBABCHGIQABCQABCOPFJQACBOPHJKL $.

  ${
    pm5.74i.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Distribution of implication over biconditional (inference form).
       (Contributed by NM, 1-Aug-1994.) $)
    pm5.74i $p |- ( ( ph -> ps ) <-> ( ph -> ch ) ) $=
      ( wb wi pm5.74 mpbi ) ABCEFABFACFEDABCGH $.
  $}

  ${
    pm5.74ri.1 $e |- ( ( ph -> ps ) <-> ( ph -> ch ) ) $.
    $( Distribution of implication over biconditional (reverse inference form).
       (Contributed by NM, 1-Aug-1994.) $)
    pm5.74ri $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wb wi pm5.74 mpbir ) ABCEFABFACFEDABCGH $.
  $}

  ${
    pm5.74d.1 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
    $( Distribution of implication over biconditional (deduction form).
       (Contributed by NM, 21-Mar-1996.) $)
    pm5.74d $p |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $=
      ( wb wi pm5.74 sylib ) ABCDFGBCGBDGFEBCDHI $.
  $}

  ${
    pm5.74rd.1 $e |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $.
    $( Distribution of implication over biconditional (deduction form).
       (Contributed by NM, 19-Mar-1997.) $)
    pm5.74rd $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
      ( wi wb pm5.74 sylibr ) ABCFBDFGBCDGFEBCDHI $.
  $}

  ${
    bitri.1 $e |- ( ph <-> ps ) $.
    bitri.2 $e |- ( ps <-> ch ) $.
    $( An inference from transitive law for logical equivalence.  (Contributed
       by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 13-Oct-2012.) $)
    bitri $p |- ( ph <-> ch ) $=
      ( biimpi sylib biimpri sylibr impbii ) ACABCABDFEGCBABCEHDIJ $.
  $}

  ${
    bitr2i.1 $e |- ( ph <-> ps ) $.
    bitr2i.2 $e |- ( ps <-> ch ) $.
    $( An inference from transitive law for logical equivalence.  (Contributed
       by NM, 5-Aug-1993.) $)
    bitr2i $p |- ( ch <-> ph ) $=
      ( bitri bicomi ) ACABCDEFG $.
  $}

  ${
    bitr3i.1 $e |- ( ps <-> ph ) $.
    bitr3i.2 $e |- ( ps <-> ch ) $.
    $( An inference from transitive law for logical equivalence.  (Contributed
       by NM, 5-Aug-1993.) $)
    bitr3i $p |- ( ph <-> ch ) $=
      ( bicomi bitri ) ABCBADFEG $.
  $}

  ${
    bitr4i.1 $e |- ( ph <-> ps ) $.
    bitr4i.2 $e |- ( ch <-> ps ) $.
    $( An inference from transitive law for logical equivalence.  (Contributed
       by NM, 5-Aug-1993.) $)
    bitr4i $p |- ( ph <-> ch ) $=
      ( bicomi bitri ) ABCDCBEFG $.
  $}

  ${
    bitrd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $.
    $( Deduction form of ~ bitri .  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Wolf Lammen, 14-Apr-2013.) $)
    bitrd $p |- ( ph -> ( ps <-> th ) ) $=
      ( wi pm5.74i bitri pm5.74ri ) ABDABGACGADGABCEHACDFHIJ $.
  $}

  ${
    bitr2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitr2d.2 $e |- ( ph -> ( ch <-> th ) ) $.
    $( Deduction form of ~ bitr2i .  (Contributed by NM, 9-Jun-2004.) $)
    bitr2d $p |- ( ph -> ( th <-> ps ) ) $=
      ( bitrd bicomd ) ABDABCDEFGH $.
  $}

  ${
    bitr3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
    $( Deduction form of ~ bitr3i .  (Contributed by NM, 5-Aug-1993.) $)
    bitr3d $p |- ( ph -> ( ch <-> th ) ) $=
      ( bicomd bitrd ) ACBDABCEGFH $.
  $}

  ${
    bitr4d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitr4d.2 $e |- ( ph -> ( th <-> ch ) ) $.
    $( Deduction form of ~ bitr4i .  (Contributed by NM, 5-Aug-1993.) $)
    bitr4d $p |- ( ph -> ( ps <-> th ) ) $=
      ( bicomd bitrd ) ABCDEADCFGH $.
  $}

  ${
    bitrid.1 $e |- ( ph <-> ps ) $.
    bitrid.2 $e |- ( ch -> ( ps <-> th ) ) $.
    $( A syllogism inference from two biconditionals.  (Contributed by NM,
       12-Mar-1993.) $)
    bitrid $p |- ( ch -> ( ph <-> th ) ) $=
      ( wb a1i bitrd ) CABDABGCEHFI $.
  $}

  ${
    bitr2id.1 $e |- ( ph <-> ps ) $.
    bitr2id.2 $e |- ( ch -> ( ps <-> th ) ) $.
    $( A syllogism inference from two biconditionals.  (Contributed by NM,
       5-Aug-1993.) $)
    bitr2id $p |- ( ch -> ( th <-> ph ) ) $=
      ( bitrid bicomd ) CADABCDEFGH $.
  $}

  ${
    bitr3id.1 $e |- ( ps <-> ph ) $.
    bitr3id.2 $e |- ( ch -> ( ps <-> th ) ) $.
    $( A syllogism inference from two biconditionals.  (Contributed by NM,
       5-Aug-1993.) $)
    bitr3id $p |- ( ch -> ( ph <-> th ) ) $=
      ( bicomi bitrid ) ABCDBAEGFH $.
  $}

  ${
    bitr3di.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitr3di.2 $e |- ( ps <-> th ) $.
    $( A syllogism inference from two biconditionals.  (Contributed by NM,
       25-Nov-1994.) $)
    bitr3di $p |- ( ph -> ( ch <-> th ) ) $=
      ( bicomi bitr2id ) DBACBDFGEH $.
  $}

  ${
    bitrdi.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitrdi.2 $e |- ( ch <-> th ) $.
    $( A syllogism inference from two biconditionals.  (Contributed by NM,
       5-Aug-1993.) $)
    bitrdi $p |- ( ph -> ( ps <-> th ) ) $=
      ( wb a1i bitrd ) ABCDECDGAFHI $.
  $}

  ${
    bitr2di.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitr2di.2 $e |- ( ch <-> th ) $.
    $( A syllogism inference from two biconditionals.  (Contributed by NM,
       5-Aug-1993.) $)
    bitr2di $p |- ( ph -> ( th <-> ps ) ) $=
      ( bitrdi bicomd ) ABDABCDEFGH $.
  $}

  ${
    bitr4di.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitr4di.2 $e |- ( th <-> ch ) $.
    $( A syllogism inference from two biconditionals.  (Contributed by NM,
       5-Aug-1993.) $)
    bitr4di $p |- ( ph -> ( ps <-> th ) ) $=
      ( bicomi bitrdi ) ABCDEDCFGH $.
  $}

  ${
    bitr4id.2 $e |- ( ps <-> ch ) $.
    bitr4id.1 $e |- ( ph -> ( th <-> ch ) ) $.
    $( A syllogism inference from two biconditionals.  (Contributed by NM,
       25-Nov-1994.) $)
    bitr4id $p |- ( ph -> ( ps <-> th ) ) $=
      ( bicomi bitr2di ) ADCBFBCEGH $.
  $}

  ${
    3imtr3.1 $e |- ( ph -> ps ) $.
    3imtr3.2 $e |- ( ph <-> ch ) $.
    3imtr3.3 $e |- ( ps <-> th ) $.
    $( A mixed syllogism inference, useful for removing a definition from both
       sides of an implication.  (Contributed by NM, 10-Aug-1994.) $)
    3imtr3i $p |- ( ch -> th ) $=
      ( sylbir sylib ) CBDCABFEHGI $.
  $}

  ${
    3imtr4.1 $e |- ( ph -> ps ) $.
    3imtr4.2 $e |- ( ch <-> ph ) $.
    3imtr4.3 $e |- ( th <-> ps ) $.
    $( A mixed syllogism inference, useful for applying a definition to both
       sides of an implication.  (Contributed by NM, 5-Aug-1993.) $)
    3imtr4i $p |- ( ch -> th ) $=
      ( sylbi sylibr ) CBDCABFEHGI $.
  $}

  ${
    3imtr3d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3imtr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
    3imtr3d.3 $e |- ( ph -> ( ch <-> ta ) ) $.
    $( More general version of ~ 3imtr3i .  Useful for converting conditional
       definitions in a formula.  (Contributed by NM, 8-Apr-1996.) $)
    3imtr3d $p |- ( ph -> ( th -> ta ) ) $=
      ( sylibd sylbird ) ADBEGABCEFHIJ $.
  $}

  ${
    3imtr4d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3imtr4d.2 $e |- ( ph -> ( th <-> ps ) ) $.
    3imtr4d.3 $e |- ( ph -> ( ta <-> ch ) ) $.
    $( More general version of ~ 3imtr4i .  Useful for converting conditional
       definitions in a formula.  (Contributed by NM, 26-Oct-1995.) $)
    3imtr4d $p |- ( ph -> ( th -> ta ) ) $=
      ( sylibrd sylbid ) ADBEGABCEFHIJ $.
  $}

  ${
    3imtr3g.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3imtr3g.2 $e |- ( ps <-> th ) $.
    3imtr3g.3 $e |- ( ch <-> ta ) $.
    $( More general version of ~ 3imtr3i .  Useful for converting definitions
       in a formula.  (Contributed by NM, 20-May-1996.)  (Proof shortened by
       Wolf Lammen, 20-Dec-2013.) $)
    3imtr3g $p |- ( ph -> ( th -> ta ) ) $=
      ( biimtrrid imbitrdi ) ADCEDBACGFIHJ $.
  $}

  ${
    3imtr4g.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3imtr4g.2 $e |- ( th <-> ps ) $.
    3imtr4g.3 $e |- ( ta <-> ch ) $.
    $( More general version of ~ 3imtr4i .  Useful for converting definitions
       in a formula.  (Contributed by NM, 20-May-1996.)  (Proof shortened by
       Wolf Lammen, 20-Dec-2013.) $)
    3imtr4g $p |- ( ph -> ( th -> ta ) ) $=
      ( biimtrid imbitrrdi ) ADCEDBACGFIHJ $.
  $}

  ${
    3bitri.1 $e |- ( ph <-> ps ) $.
    3bitri.2 $e |- ( ps <-> ch ) $.
    3bitri.3 $e |- ( ch <-> th ) $.
    $( A chained inference from transitive law for logical equivalence.
       (Contributed by NM, 5-Aug-1993.) $)
    3bitri $p |- ( ph <-> th ) $=
      ( bitri ) ABDEBCDFGHH $.

    $( A chained inference from transitive law for logical equivalence.
       (Contributed by NM, 4-Aug-2006.) $)
    3bitrri $p |- ( th <-> ph ) $=
      ( bitr2i bitr3i ) DCAGABCEFHI $.
  $}

  ${
    3bitr2i.1 $e |- ( ph <-> ps ) $.
    3bitr2i.2 $e |- ( ch <-> ps ) $.
    3bitr2i.3 $e |- ( ch <-> th ) $.
    $( A chained inference from transitive law for logical equivalence.
       (Contributed by NM, 4-Aug-2006.) $)
    3bitr2i $p |- ( ph <-> th ) $=
      ( bitr4i bitri ) ACDABCEFHGI $.

    $( A chained inference from transitive law for logical equivalence.
       (Contributed by NM, 4-Aug-2006.) $)
    3bitr2ri $p |- ( th <-> ph ) $=
      ( bitr4i bitr2i ) ACDABCEFHGI $.
  $}

  ${
    3bitr3i.1 $e |- ( ph <-> ps ) $.
    3bitr3i.2 $e |- ( ph <-> ch ) $.
    3bitr3i.3 $e |- ( ps <-> th ) $.
    $( A chained inference from transitive law for logical equivalence.
       (Contributed by NM, 19-Aug-1993.) $)
    3bitr3i $p |- ( ch <-> th ) $=
      ( bitr3i bitri ) CBDCABFEHGI $.

    $( A chained inference from transitive law for logical equivalence.
       (Contributed by NM, 5-Aug-1993.) $)
    3bitr3ri $p |- ( th <-> ch ) $=
      ( bitr3i ) DBCGBACEFHH $.
  $}

  ${
    3bitr4i.1 $e |- ( ph <-> ps ) $.
    3bitr4i.2 $e |- ( ch <-> ph ) $.
    3bitr4i.3 $e |- ( th <-> ps ) $.
    $( A chained inference from transitive law for logical equivalence.  This
       inference is frequently used to apply a definition to both sides of a
       logical equivalence.  (Contributed by NM, 5-Aug-1993.) $)
    3bitr4i $p |- ( ch <-> th ) $=
      ( bitr4i bitri ) CADFABDEGHI $.

    $( A chained inference from transitive law for logical equivalence.
       (Contributed by NM, 2-Sep-1995.) $)
    3bitr4ri $p |- ( th <-> ch ) $=
      ( bitr4i bitr2i ) CADFABDEGHI $.
  $}

  ${
    3bitrd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    3bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $.
    3bitrd.3 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction from transitivity of biconditional.  (Contributed by NM,
       13-Aug-1999.) $)
    3bitrd $p |- ( ph -> ( ps <-> ta ) ) $=
      ( bitrd ) ABDEABCDFGIHI $.

    $( Deduction from transitivity of biconditional.  (Contributed by NM,
       4-Aug-2006.) $)
    3bitrrd $p |- ( ph -> ( ta <-> ps ) ) $=
      ( bitr2d bitr3d ) ADEBHABCDFGIJ $.
  $}

  ${
    3bitr2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    3bitr2d.2 $e |- ( ph -> ( th <-> ch ) ) $.
    3bitr2d.3 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction from transitivity of biconditional.  (Contributed by NM,
       4-Aug-2006.) $)
    3bitr2d $p |- ( ph -> ( ps <-> ta ) ) $=
      ( bitr4d bitrd ) ABDEABCDFGIHJ $.

    $( Deduction from transitivity of biconditional.  (Contributed by NM,
       4-Aug-2006.) $)
    3bitr2rd $p |- ( ph -> ( ta <-> ps ) ) $=
      ( bitr4d bitr2d ) ABDEABCDFGIHJ $.
  $}

  ${
    3bitr3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    3bitr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
    3bitr3d.3 $e |- ( ph -> ( ch <-> ta ) ) $.
    $( Deduction from transitivity of biconditional.  Useful for converting
       conditional definitions in a formula.  (Contributed by NM,
       24-Apr-1996.) $)
    3bitr3d $p |- ( ph -> ( th <-> ta ) ) $=
      ( bitr3d bitrd ) ADCEABDCGFIHJ $.

    $( Deduction from transitivity of biconditional.  (Contributed by NM,
       4-Aug-2006.) $)
    3bitr3rd $p |- ( ph -> ( ta <-> th ) ) $=
      ( bitr3d ) ACEDHABCDFGII $.
  $}

  ${
    3bitr4d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    3bitr4d.2 $e |- ( ph -> ( th <-> ps ) ) $.
    3bitr4d.3 $e |- ( ph -> ( ta <-> ch ) ) $.
    $( Deduction from transitivity of biconditional.  Useful for converting
       conditional definitions in a formula.  (Contributed by NM,
       18-Oct-1995.) $)
    3bitr4d $p |- ( ph -> ( th <-> ta ) ) $=
      ( bitr4d bitrd ) ADBEGABCEFHIJ $.

    $( Deduction from transitivity of biconditional.  (Contributed by NM,
       4-Aug-2006.) $)
    3bitr4rd $p |- ( ph -> ( ta <-> th ) ) $=
      ( bitr4d ) AEBDAECBHFIGI $.
  $}

  ${
    3bitr3g.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    3bitr3g.2 $e |- ( ps <-> th ) $.
    3bitr3g.3 $e |- ( ch <-> ta ) $.
    $( More general version of ~ 3bitr3i .  Useful for converting definitions
       in a formula.  (Contributed by NM, 4-Jun-1995.) $)
    3bitr3g $p |- ( ph -> ( th <-> ta ) ) $=
      ( bitr3id bitrdi ) ADCEDBACGFIHJ $.
  $}

  ${
    3bitr4g.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    3bitr4g.2 $e |- ( th <-> ps ) $.
    3bitr4g.3 $e |- ( ta <-> ch ) $.
    $( More general version of ~ 3bitr4i .  Useful for converting definitions
       in a formula.  (Contributed by NM, 5-Aug-1993.) $)
    3bitr4g $p |- ( ph -> ( th <-> ta ) ) $=
      ( bitrid bitr4di ) ADCEDBACGFIHJ $.
  $}

  ${
    bi3ant.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Construct a biconditional in antecedent position.  (Contributed by Wolf
       Lammen, 14-May-2013.) $)
    bi3ant $p |- ( ( ( th -> ta ) -> ph ) -> ( ( ( ta -> th ) -> ps ) ->
                                                ( ( th <-> ta ) -> ch ) ) ) $=
      ( wi wb biimp imim1i biimpr imim3i syl2im ) DEGZAGDEHZAGEDGZBGOBGOCGONADE
      IJOPBDEKJABCOFLM $.
  $}

  $( Express symmetries of theorems in terms of biconditionals.  (Contributed
     by Wolf Lammen, 14-May-2013.) $)
  bisym $p |- ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ( ps -> ph )
      -> ( th -> ch ) ) -> ( ( ph <-> ps ) -> ( ch <-> th ) ) ) ) $=
    ( wi wb bi3 bi3ant ) CDEDCECDFABCDGH $.

  $( The next three rules are useful for building up wff's around a
     definition, in order to make use of the definition. $)

  ${
    bi.a $e |- ( ph <-> ps ) $.
    $( Introduce an antecedent to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       6-Feb-2013.) $)
    imbi2i $p |- ( ( ch -> ph ) <-> ( ch -> ps ) ) $=
      ( wb a1i pm5.74i ) CABABECDFG $.
  $}

  ${
    bibi.a $e |- ( ph <-> ps ) $.
    $( Inference adding a biconditional to the left in an equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       7-May-2011.)  (Proof shortened by Wolf Lammen, 16-May-2013.) $)
    bibi2i $p |- ( ( ch <-> ph ) <-> ( ch <-> ps ) ) $=
      ( wb id bitrdi bitr4di impbii ) CAEZCBEZJCABJFDGKCBAKFDHI $.

    $( Inference adding a biconditional to the right in an equivalence.
       (Contributed by NM, 5-Aug-1993.) $)
    bibi1i $p |- ( ( ph <-> ch ) <-> ( ps <-> ch ) ) $=
      ( wb bicom bibi2i 3bitri ) ACECAECBEBCEACFABCDGCBFH $.

    ${
      bibi12.2 $e |- ( ch <-> th ) $.
      $( The equivalence of two equivalences.  (Contributed by NM,
         5-Aug-1993.) $)
      bibi12i $p |- ( ( ph <-> ch ) <-> ( ps <-> th ) ) $=
        ( wb bibi2i bibi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
    $}
  $}

  ${
    imbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction adding an antecedent to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.) $)
    imbi2d $p |- ( ph -> ( ( th -> ps ) <-> ( th -> ch ) ) ) $=
      ( wb a1d pm5.74d ) ADBCABCFDEGH $.

    $( Deduction adding a consequent to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       17-Sep-2013.) $)
    imbi1d $p |- ( ph -> ( ( ps -> th ) <-> ( ch -> th ) ) ) $=
      ( wi biimprd imim1d biimpd impbid ) ABDFCDFACBDABCEGHABCDABCEIHJ $.

    $( Deduction adding a biconditional to the left in an equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       19-May-2013.) $)
    bibi2d $p |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) ) $=
      ( wb wi pm5.74i bibi2i pm5.74 3bitr4i pm5.74ri ) ADBFZDCFZADGZABGZFOACGZF
      AMGANGPQOABCEHIADBJADCJKL $.

    $( Deduction adding a biconditional to the right in an equivalence.
       (Contributed by NM, 5-Aug-1993.) $)
    bibi1d $p |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> th ) ) ) $=
      ( wb bibi2d bicom 3bitr4g ) ADBFDCFBDFCDFABCDEGBDHCDHI $.
  $}

  ${
    imbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    imbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction joining two equivalences to form equivalence of implications.
       (Contributed by NM, 5-Aug-1993.) $)
    imbi12d $p |- ( ph -> ( ( ps -> th ) <-> ( ch -> ta ) ) ) $=
      ( wi imbi1d imbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
    $( Deduction joining two equivalences to form equivalence of
       biconditionals.  (Contributed by NM, 5-Aug-1993.) $)
    bibi12d $p |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> ta ) ) ) $=
      ( wb bibi1d bibi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  $( Theorem *4.84 of [WhiteheadRussell] p. 122.  (Contributed by NM,
     3-Jan-2005.) $)
  imbi1 $p |- ( ( ph <-> ps ) -> ( ( ph -> ch ) <-> ( ps -> ch ) ) ) $=
    ( wb id imbi1d ) ABDZABCGEF $.

  $( Theorem *4.85 of [WhiteheadRussell] p. 122.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 19-May-2013.) $)
  imbi2 $p |- ( ( ph <-> ps ) -> ( ( ch -> ph ) <-> ( ch -> ps ) ) ) $=
    ( wb id imbi2d ) ABDZABCGEF $.

  ${
    imbi1i.1 $e |- ( ph <-> ps ) $.
    $( Introduce a consequent to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       17-Sep-2013.) $)
    imbi1i $p |- ( ( ph -> ch ) <-> ( ps -> ch ) ) $=
      ( wb wi imbi1 ax-mp ) ABEACFBCFEDABCGH $.
  $}

  ${
    imbi12i.1 $e |- ( ph <-> ps ) $.
    imbi12i.2 $e |- ( ch <-> th ) $.
    $( Join two logical equivalences to form equivalence of implications.
       (Contributed by NM, 5-Aug-1993.) $)
    imbi12i $p |- ( ( ph -> ch ) <-> ( ps -> th ) ) $=
      ( wi imbi2i imbi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
  $}

  $( Theorem *4.86 of [WhiteheadRussell] p. 122.  (Contributed by NM,
     3-Jan-2005.) $)
  bibi1 $p |- ( ( ph <-> ps ) -> ( ( ph <-> ch ) <-> ( ps <-> ch ) ) ) $=
    ( wb id bibi1d ) ABDZABCGEF $.

  $( A wff is equivalent to itself with true antecedent.  (Contributed by NM,
     28-Jan-1996.) $)
  biimt $p |- ( ph -> ( ps <-> ( ph -> ps ) ) ) $=
    ( wi ax-1 pm2.27 impbid2 ) ABABCBADABEF $.

  $( Theorem *5.5 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.5 $p |- ( ph -> ( ( ph -> ps ) <-> ps ) ) $=
    ( wi biimt bicomd ) ABABCABDE $.

  ${
    a1bi.1 $e |- ph $.
    $( Inference introducing a theorem as an antecedent.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Wolf Lammen, 11-Nov-2012.) $)
    a1bi $p |- ( ps <-> ( ph -> ps ) ) $=
      ( wi wb biimt ax-mp ) ABABDECABFG $.
  $}

  $( Theorem *5.501 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.)  (Revised by NM, 24-Jan-2013.) $)
  pm5.501 $p |- ( ph -> ( ps <-> ( ph <-> ps ) ) ) $=
    ( wb pm5.1im biimp com12 impbid ) ABABCZABDHABABEFG $.

  $( Implication in terms of implication and biconditional.  (Contributed by
     NM, 31-Mar-1994.)  (Proof shortened by Wolf Lammen, 24-Jan-2013.) $)
  ibib $p |- ( ( ph -> ps ) <-> ( ph -> ( ph <-> ps ) ) ) $=
    ( wb pm5.501 pm5.74i ) ABABCABDE $.

  $( Implication in terms of implication and biconditional.  (Contributed by
     NM, 29-Apr-2005.)  (Proof shortened by Wolf Lammen, 21-Dec-2013.) $)
  ibibr $p |- ( ( ph -> ps ) <-> ( ph -> ( ps <-> ph ) ) ) $=
    ( wb pm5.501 bicom bitrdi pm5.74i ) ABBACZABABCHABDABEFG $.

  ${
    tbt.1 $e |- ph $.
    $( A wff is equivalent to its equivalence with truth.  (Contributed by NM,
       18-Aug-1993.)  (Proof shortened by Andrew Salmon, 13-May-2011.) $)
    tbt $p |- ( ps <-> ( ps <-> ph ) ) $=
      ( wb ibibr pm5.74ri ax-mp ) ABBADZDCABHABEFG $.
  $}

  $( Logical equivalence of commuted antecedents.  Part of Theorem *4.87 of
     [WhiteheadRussell] p. 122.  (Contributed by NM, 5-Aug-1993.) $)
  bi2.04 $p |- ( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) ) $=
    ( wi pm2.04 impbii ) ABCDDBACDDABCEBACEF $.

  $( Antecedent absorption implication.  Theorem *5.4 of [WhiteheadRussell]
     p. 125.  (Contributed by NM, 5-Aug-1993.) $)
  pm5.4 $p |- ( ( ph -> ( ph -> ps ) ) <-> ( ph -> ps ) ) $=
    ( wi pm2.43 ax-1 impbii ) AABCZCGABDGAEF $.

  $( Distributive law for implication.  Compare Theorem *5.41 of
     [WhiteheadRussell] p. 125.  (Contributed by NM, 5-Aug-1993.) $)
  imdi $p |- ( ( ph -> ( ps -> ch ) ) <->
               ( ( ph -> ps ) -> ( ph -> ch ) ) ) $=
    ( wi ax-2 pm2.86 impbii ) ABCDDABDACDDABCEABCFG $.

  $( Theorem *5.41 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 12-Oct-2012.) $)
  pm5.41 $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) <->
                ( ph -> ( ps -> ch ) ) ) $=
    ( wi imdi bicomi ) ABCDDABDACDDABCEF $.

  $( The antecedent of one side of a biconditional can be moved out of the
     biconditional to become the antecedent of the remaining biconditional.
     (Contributed by BJ, 1-Jan-2025.)  (Proof shortened by Wolf Lammen,
     5-Jan-2025.) $)
  imbibi $p |- ( ( ( ph -> ps ) <-> ch ) -> ( ph -> ( ps <-> ch ) ) ) $=
    ( wi wb pm5.4 imbi2 bitr3id pm5.74rd ) ABDZCEZABCJAJDKACDABFJCAGHI $.

  $( Simplify an implication between two implications when the antecedent of
     the first is a consequence of the antecedent of the second.  The reverse
     form is useful in producing the successor step in induction proofs.
     (Contributed by Paul Chapman, 22-Jun-2011.)  (Proof shortened by Wolf
     Lammen, 14-Sep-2013.) $)
  imim21b $p |- ( ( ps -> ph ) -> ( ( ( ph -> ch ) -> ( ps -> th ) ) <->
                                    ( ps -> ( ch -> th ) ) ) ) $=
    ( wi bi2.04 wb pm5.5 imbi1d imim2i pm5.74d bitrid ) ACEZBDEEBMDEZEBAEZBCDEZ
    EMBDFOBNPANPGBAMCDACHIJKL $.

  ${
    imp3.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( Importation deduction.  (Contributed by NM, 31-Mar-1994.) $)
    impd $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
      ( wa wi com3l imp com12 ) BCFADBCADGABCDEHIJ $.

    $( Importation deduction with commuted antecedents.  (Contributed by Peter
       Mazsa, 24-Sep-2022.)  (Proof shortened by Wolf Lammen, 22-Oct-2022.) $)
    impcomd $p |- ( ph -> ( ( ch /\ ps ) -> th ) ) $=
      ( com23 impd ) ACBDABCDEFG $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp31 $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
      ( wa wi imp ) ABFCDABCDGEHH $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp32 $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
      ( wa impd imp ) ABCFDABCDEGH $.
  $}

  ${
    exp3a.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( Exportation deduction.  (Contributed by NM, 20-Aug-1993.) $)
    expd $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
      ( wi wa com12 ex com3r ) BCADBCADFABCGDEHIJ $.

    $( A deduction version of exportation, followed by importation.
       (Contributed by NM, 6-Sep-2008.) $)
    expdimp $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
      ( wi expd imp ) ABCDFABCDEGH $.
  $}

  ${
    impancom.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
    $( Mixed importation/commutation inference.  (Contributed by NM,
       22-Jun-2013.) $)
    impancom $p |- ( ( ph /\ ch ) -> ( ps -> th ) ) $=
      ( wi ex com23 imp ) ACBDFABCDABCDFEGHI $.
  $}

  $( Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
  pm3.3 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) $=
    ( wa wi id expd ) ABDCEZABCHFG $.

  $( Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
  pm3.31 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph /\ ps ) -> ch ) ) $=
    ( wi id impd ) ABCDDZABCGEF $.

  $( Import-export theorem.  Part of Theorem *4.87 of [WhiteheadRussell]
     p. 122.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 24-Mar-2013.) $)
  impexp $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) $=
    ( wa wi pm3.3 pm3.31 impbii ) ABDCEABCEEABCFABCGH $.

  $( Join antecedents with conjunction.  Theorem *3.21 of [WhiteheadRussell]
     p. 111.  (Contributed by NM, 5-Aug-1993.) $)
  pm3.21 $p |- ( ph -> ( ps -> ( ps /\ ph ) ) ) $=
    ( wa pm3.2 com12 ) BABACBADE $.

  $( Theorem *3.22 of [WhiteheadRussell] p. 111.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 13-Nov-2012.) $)
  pm3.22 $p |- ( ( ph /\ ps ) -> ( ps /\ ph ) ) $=
    ( wa pm3.21 imp ) ABBACABDE $.

  $( Commutative law for conjunction.  Theorem *4.3 of [WhiteheadRussell]
     p. 118.  (Contributed by NM, 25-Jun-1998.)  (Proof shortened by Wolf
     Lammen, 4-Nov-2012.) $)
  ancom $p |- ( ( ph /\ ps ) <-> ( ps /\ ph ) ) $=
    ( wa pm3.22 impbii ) ABCBACABDBADE $.

  ${
    ancomd.1 $e |- ( ph -> ( ps /\ ch ) ) $.
    $( Commutation of conjuncts in consequent.  (Contributed by Jeff Hankins,
       14-Aug-2009.) $)
    ancomd $p |- ( ph -> ( ch /\ ps ) ) $=
      ( wa ancom sylib ) ABCECBEDBCFG $.
  $}

  ${
    ancoms.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Inference commuting conjunction in antecedent.  (Contributed by NM,
       21-Apr-1994.) $)
    ancoms $p |- ( ( ps /\ ph ) -> ch ) $=
      ( expcom imp ) BACABCDEF $.
  $}

  ${
    ancomsd.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( Deduction commuting conjunction in antecedent.  (Contributed by NM,
       12-Dec-2004.) $)
    ancomsd $p |- ( ph -> ( ( ch /\ ps ) -> th ) ) $=
      ( wa ancom biimtrid ) CBFBCFADCBGEH $.
  $}

  ${
    biancomi.1 $e |- ( ph <-> ( ch /\ ps ) ) $.
    $( Commuting conjunction in a biconditional.  (Contributed by Peter Mazsa,
       17-Jun-2018.) $)
    biancomi $p |- ( ph <-> ( ps /\ ch ) ) $=
      ( wa ancom bitr4i ) ACBEBCEDBCFG $.
  $}

  ${
    biancomd.1 $e |- ( ph -> ( ps <-> ( th /\ ch ) ) ) $.
    $( Commuting conjunction in a biconditional, deduction form.  (Contributed
       by Peter Mazsa, 3-Oct-2018.) $)
    biancomd $p |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $=
      ( wa ancom bitrdi ) ABDCFCDFEDCGH $.
  $}

  ${
    pm3.2i.1 $e |- ph $.
    pm3.2i.2 $e |- ps $.
    $( Infer conjunction of premises.  (Contributed by NM, 5-Aug-1993.) $)
    pm3.2i $p |- ( ph /\ ps ) $=
      ( wa pm3.2 mp2 ) ABABECDABFG $.
  $}

  $( Nested conjunction of antecedents.  (Contributed by NM, 5-Aug-1993.) $)
  pm3.43i $p |- ( ( ph -> ps ) ->
                ( ( ph -> ch ) -> ( ph -> ( ps /\ ch ) ) ) ) $=
    ( wa pm3.2 imim3i ) BCBCDABCEF $.

  ${
    simplbi.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Deduction eliminating a conjunct.  (Contributed by NM, 27-May-1998.) $)
    simplbi $p |- ( ph -> ps ) $=
      ( wa biimpi simpld ) ABCABCEDFG $.
  $}

  ${
    simprbi.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Deduction eliminating a conjunct.  (Contributed by NM, 27-May-1998.) $)
    simprbi $p |- ( ph -> ch ) $=
      ( wa biimpi simprd ) ABCABCEDFG $.
  $}

  ${
    adantr.1 $e |- ( ph -> ps ) $.
    $( Inference adding a conjunct to the right of an antecedent.  (Contributed
       by NM, 30-Aug-1993.) $)
    adantr $p |- ( ( ph /\ ch ) -> ps ) $=
      ( a1d imp ) ACBABCDEF $.
  $}

  ${
    adantl.1 $e |- ( ph -> ps ) $.
    $( Inference adding a conjunct to the left of an antecedent.  (Contributed
       by NM, 30-Aug-1993.)  (Proof shortened by Wolf Lammen, 23-Nov-2012.) $)
    adantl $p |- ( ( ch /\ ph ) -> ps ) $=
      ( adantr ancoms ) ACBABCDEF $.
  $}

  ${
    adantld.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction adding a conjunct to the left of an antecedent.  (Contributed
       by NM, 4-May-1994.)  (Proof shortened by Wolf Lammen, 20-Dec-2012.) $)
    adantld $p |- ( ph -> ( ( th /\ ps ) -> ch ) ) $=
      ( wa simpr syl5 ) DBFBACDBGEH $.
  $}

  ${
    adantrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction adding a conjunct to the right of an antecedent.  (Contributed
       by NM, 4-May-1994.) $)
    adantrd $p |- ( ph -> ( ( ps /\ th ) -> ch ) ) $=
      ( wa simpl syl5 ) BDFBACBDGEH $.
  $}

  ${
    impel.1 $e |- ( ph -> ( ps -> ch ) ) $.
    impel.2 $e |- ( th -> ps ) $.
    $( An inference for implication elimination.  (Contributed by Giovanni
       Mascellani, 23-May-2019.)  (Proof shortened by Wolf Lammen,
       2-Sep-2020.) $)
    impel $p |- ( ( ph /\ th ) -> ch ) $=
      ( syl5 imp ) ADCDBACFEGH $.
  $}

  ${
    mpan9.1 $e |- ( ph -> ps ) $.
    mpan9.2 $e |- ( ch -> ( ps -> th ) ) $.
    $( Modus ponens conjoining dissimilar antecedents.  (Contributed by NM,
       1-Feb-2008.)  (Proof shortened by Andrew Salmon, 7-May-2011.) $)
    mpan9 $p |- ( ( ph /\ ch ) -> th ) $=
      ( syl5 impcom ) CADABCDEFGH $.
  $}

  ${
    syldan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    syldan.2 $e |- ( ( ph /\ ch ) -> th ) $.
    $( A syllogism deduction with conjoined antecents.  (Contributed by NM,
       24-Feb-2005.)  (Proof shortened by Wolf Lammen, 6-Apr-2013.) $)
    syldan $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa expcom adantrd mpcom ) CABGDECADBACDFHIJ $.
  $}

  ${
    sylan.1 $e |- ( ph -> ps ) $.
    sylan.2 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A syllogism inference.  (Contributed by NM, 21-Apr-1994.)  (Proof
       shortened by Wolf Lammen, 22-Nov-2012.) $)
    sylan $p |- ( ( ph /\ ch ) -> th ) $=
      ( expcom mpan9 ) ABCDEBCDFGH $.
  $}

  ${
    sylanb.1 $e |- ( ph <-> ps ) $.
    sylanb.2 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A syllogism inference.  (Contributed by NM, 18-May-1994.) $)
    sylanb $p |- ( ( ph /\ ch ) -> th ) $=
      ( biimpi sylan ) ABCDABEGFH $.
  $}

  ${
    sylanbr.1 $e |- ( ps <-> ph ) $.
    sylanbr.2 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A syllogism inference.  (Contributed by NM, 18-May-1994.) $)
    sylanbr $p |- ( ( ph /\ ch ) -> th ) $=
      ( biimpri sylan ) ABCDBAEGFH $.
  $}

  ${
    sylan2.1 $e |- ( ph -> ch ) $.
    sylan2.2 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A syllogism inference.  (Contributed by NM, 21-Apr-1994.)  (Proof
       shortened by Wolf Lammen, 22-Nov-2012.) $)
    sylan2 $p |- ( ( ps /\ ph ) -> th ) $=
      ( adantl syldan ) BACDACBEGFH $.
  $}

  ${
    sylan2b.1 $e |- ( ph <-> ch ) $.
    sylan2b.2 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A syllogism inference.  (Contributed by NM, 21-Apr-1994.) $)
    sylan2b $p |- ( ( ps /\ ph ) -> th ) $=
      ( biimpi sylan2 ) ABCDACEGFH $.
  $}

  ${
    sylan2br.1 $e |- ( ch <-> ph ) $.
    sylan2br.2 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A syllogism inference.  (Contributed by NM, 21-Apr-1994.) $)
    sylan2br $p |- ( ( ps /\ ph ) -> th ) $=
      ( biimpri sylan2 ) ABCDCAEGFH $.
  $}

  ${
    syl2an.1 $e |- ( ph -> ps ) $.
    syl2an.2 $e |- ( ta -> ch ) $.
    syl2an.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A double syllogism inference.  (Contributed by NM, 31-Jan-1997.) $)
    syl2an $p |- ( ( ph /\ ta ) -> th ) $=
      ( sylan sylan2 ) EACDGABCDFHIJ $.

    $( A double syllogism inference.  (Contributed by NM, 17-Sep-2013.) $)
    syl2anr $p |- ( ( ta /\ ph ) -> th ) $=
      ( syl2an ancoms ) AEDABCDEFGHIJ $.
  $}

  ${
    syl2anb.1 $e |- ( ph <-> ps ) $.
    syl2anb.2 $e |- ( ta <-> ch ) $.
    syl2anb.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A double syllogism inference.  (Contributed by NM, 29-Jul-1999.) $)
    syl2anb $p |- ( ( ph /\ ta ) -> th ) $=
      ( sylanb sylan2b ) EACDGABCDFHIJ $.
  $}

  ${
    syl2anbr.1 $e |- ( ps <-> ph ) $.
    syl2anbr.2 $e |- ( ch <-> ta ) $.
    syl2anbr.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A double syllogism inference.  (Contributed by NM, 29-Jul-1999.) $)
    syl2anbr $p |- ( ( ph /\ ta ) -> th ) $=
      ( sylanbr sylan2br ) EACDGABCDFHIJ $.
  $}

  ${
    syland.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syland.2 $e |- ( ph -> ( ( ch /\ th ) -> ta ) ) $.
    $( A syllogism deduction.  (Contributed by NM, 15-Dec-2004.) $)
    syland $p |- ( ph -> ( ( ps /\ th ) -> ta ) ) $=
      ( wi expd syld impd ) ABDEABCDEHFACDEGIJK $.
  $}

  ${
    sylan2d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    sylan2d.2 $e |- ( ph -> ( ( th /\ ch ) -> ta ) ) $.
    $( A syllogism deduction.  (Contributed by NM, 15-Dec-2004.) $)
    sylan2d $p |- ( ph -> ( ( th /\ ps ) -> ta ) ) $=
      ( ancomsd syland ) ABDEABCDEFADCEGHIH $.
  $}

  ${
    syl2and.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl2and.2 $e |- ( ph -> ( th -> ta ) ) $.
    syl2and.3 $e |- ( ph -> ( ( ch /\ ta ) -> et ) ) $.
    $( A syllogism deduction.  (Contributed by NM, 15-Dec-2004.) $)
    syl2and $p |- ( ph -> ( ( ps /\ th ) -> et ) ) $=
      ( sylan2d syland ) ABCDFGADECFHIJK $.
  $}

  ${
    biimpa.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Inference from a logical equivalence.  (Contributed by NM,
       3-May-1994.) $)
    biimpa $p |- ( ( ph /\ ps ) -> ch ) $=
      ( biimpd imp ) ABCABCDEF $.

    $( Inference from a logical equivalence.  (Contributed by NM,
       3-May-1994.) $)
    biimpar $p |- ( ( ph /\ ch ) -> ps ) $=
      ( biimprd imp ) ACBABCDEF $.

    $( Inference from a logical equivalence.  (Contributed by NM,
       3-May-1994.) $)
    biimpac $p |- ( ( ps /\ ph ) -> ch ) $=
      ( biimpcd imp ) BACABCDEF $.

    $( Inference from a logical equivalence.  (Contributed by NM,
       3-May-1994.) $)
    biimparc $p |- ( ( ch /\ ph ) -> ps ) $=
      ( biimprcd imp ) CABABCDEF $.
  $}

  $( Introduction of antecedent as conjunct.  Theorem *4.73 of
     [WhiteheadRussell] p. 121.  (Contributed by NM, 30-Mar-1994.)  (Revised by
     NM, 24-Mar-2013.) $)
  iba $p |- ( ph -> ( ps <-> ( ps /\ ph ) ) ) $=
    ( wa pm3.21 simpl impbid1 ) ABBACABDBAEF $.

  $( Introduction of antecedent as conjunct.  (Contributed by NM, 5-Dec-1995.)
     (Revised by NM, 24-Mar-2013.) $)
  ibar $p |- ( ph -> ( ps <-> ( ph /\ ps ) ) ) $=
    ( wa pm3.2 simpr impbid1 ) ABABCABDABEF $.

  ${
    biantru.1 $e |- ph $.
    $( A wff is equivalent to its conjunction with truth.  (Contributed by NM,
       5-Aug-1993.) $)
    biantru $p |- ( ps <-> ( ps /\ ph ) ) $=
      ( wa wb iba ax-mp ) ABBADECABFG $.
  $}

  ${
    biantrur.1 $e |- ph $.
    $( A wff is equivalent to its conjunction with truth.  (Contributed by NM,
       3-Aug-1994.) $)
    biantrur $p |- ( ps <-> ( ph /\ ps ) ) $=
      ( wa wb ibar ax-mp ) ABABDECABFG $.
  $}

  ${
    biantrud.1 $e |- ( ph -> ps ) $.
    $( A wff is equivalent to its conjunction with truth.  (Contributed by NM,
       2-Aug-1994.)  (Proof shortened by Wolf Lammen, 23-Oct-2013.) $)
    biantrud $p |- ( ph -> ( ch <-> ( ch /\ ps ) ) ) $=
      ( wa wb iba syl ) ABCCBEFDBCGH $.

    $( A wff is equivalent to its conjunction with truth.  (Contributed by NM,
       1-May-1995.)  (Proof shortened by Andrew Salmon, 7-May-2011.) $)
    biantrurd $p |- ( ph -> ( ch <-> ( ps /\ ch ) ) ) $=
      ( wa wb ibar syl ) ABCBCEFDBCGH $.
  $}

  ${
    jca.1 $e |- ( ph -> ps ) $.
    jca.2 $e |- ( ph -> ch ) $.
    $( Deduce conjunction of the consequents of two implications ("join
       consequents with 'and'").  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Wolf Lammen, 25-Oct-2012.) $)
    jca $p |- ( ph -> ( ps /\ ch ) ) $=
      ( wa pm3.2 sylc ) ABCBCFDEBCGH $.
  $}

  ${
    jcad.1 $e |- ( ph -> ( ps -> ch ) ) $.
    jcad.2 $e |- ( ph -> ( ps -> th ) ) $.
    $( Deduction conjoining the consequents of two implications.  (Contributed
       by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
    jcad $p |- ( ph -> ( ps -> ( ch /\ th ) ) ) $=
      ( wa pm3.2 syl6c ) ABCDCDGEFCDHI $.
  $}

  ${
    jca2.1 $e |- ( ph -> ( ps -> ch ) ) $.
    jca2.2 $e |- ( ps -> th ) $.
    $( Inference conjoining the consequents of two implications.  (Contributed
       by Rodolfo Medina, 12-Oct-2010.) $)
    jca2 $p |- ( ph -> ( ps -> ( ch /\ th ) ) ) $=
      ( wi a1i jcad ) ABCDEBDGAFHI $.
  $}

  ${
    jca31.1 $e |- ( ph -> ps ) $.
    jca31.2 $e |- ( ph -> ch ) $.
    jca31.3 $e |- ( ph -> th ) $.
    $( Join three consequents.  (Contributed by Jeff Hankins, 1-Aug-2009.) $)
    jca31 $p |- ( ph -> ( ( ps /\ ch ) /\ th ) ) $=
      ( wa jca ) ABCHDABCEFIGI $.

    $( Join three consequents.  (Contributed by FL, 1-Aug-2009.) $)
    jca32 $p |- ( ph -> ( ps /\ ( ch /\ th ) ) ) $=
      ( wa jca ) ABCDHEACDFGII $.
  $}

  ${
    jcai.1 $e |- ( ph -> ps ) $.
    jcai.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction replacing implication with conjunction.  (Contributed by NM,
       5-Aug-1993.) $)
    jcai $p |- ( ph -> ( ps /\ ch ) ) $=
      ( mpd jca ) ABCDABCDEFG $.
  $}

  ${
    jctil.1 $e |- ( ph -> ps ) $.
    jctil.2 $e |- ch $.
    $( Inference conjoining a theorem to left of consequent in an implication.
       (Contributed by NM, 31-Dec-1993.) $)
    jctil $p |- ( ph -> ( ch /\ ps ) ) $=
      ( a1i jca ) ACBCAEFDG $.

    $( Inference conjoining a theorem to right of consequent in an implication.
       (Contributed by NM, 31-Dec-1993.) $)
    jctir $p |- ( ph -> ( ps /\ ch ) ) $=
      ( a1i jca ) ABCDCAEFG $.
  $}

  ${
    jctl.1 $e |- ps $.
    $( Inference conjoining a theorem to the left of a consequent.
       (Contributed by NM, 31-Dec-1993.)  (Proof shortened by Wolf Lammen,
       24-Oct-2012.) $)
    jctl $p |- ( ph -> ( ps /\ ph ) ) $=
      ( id jctil ) AABADCE $.

    $( Inference conjoining a theorem to the right of a consequent.
       (Contributed by NM, 18-Aug-1993.)  (Proof shortened by Wolf Lammen,
       24-Oct-2012.) $)
    jctr $p |- ( ph -> ( ph /\ ps ) ) $=
      ( id jctir ) AABADCE $.
  $}

  ${
    jctild.1 $e |- ( ph -> ( ps -> ch ) ) $.
    jctild.2 $e |- ( ph -> th ) $.
    $( Deduction conjoining a theorem to left of consequent in an implication.
       (Contributed by NM, 21-Apr-2005.) $)
    jctild $p |- ( ph -> ( ps -> ( th /\ ch ) ) ) $=
      ( a1d jcad ) ABDCADBFGEH $.
  $}

  ${
    jctird.1 $e |- ( ph -> ( ps -> ch ) ) $.
    jctird.2 $e |- ( ph -> th ) $.
    $( Deduction conjoining a theorem to right of consequent in an implication.
       (Contributed by NM, 21-Apr-2005.) $)
    jctird $p |- ( ph -> ( ps -> ( ch /\ th ) ) ) $=
      ( a1d jcad ) ABCDEADBFGH $.
  $}

  $( Conjoin antecedent to left of consequent.  (Contributed by NM,
     15-Aug-1994.) $)
  ancl $p |- ( ( ph -> ps ) -> ( ph -> ( ph /\ ps ) ) ) $=
    ( wa pm3.2 a2i ) ABABCABDE $.

  $( Conjoin antecedent to left of consequent.  Theorem *4.7 of
     [WhiteheadRussell] p. 120.  (Contributed by NM, 25-Jul-1999.)  (Proof
     shortened by Wolf Lammen, 24-Mar-2013.) $)
  anclb $p |- ( ( ph -> ps ) <-> ( ph -> ( ph /\ ps ) ) ) $=
    ( wa ibar pm5.74i ) ABABCABDE $.

  $( Theorem *5.42 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.42 $p |- ( ( ph -> ( ps -> ch ) ) <->
                ( ph -> ( ps -> ( ph /\ ch ) ) ) ) $=
    ( wi wa ibar imbi2d pm5.74i ) ABCDBACEZDACIBACFGH $.

  $( Conjoin antecedent to right of consequent.  (Contributed by NM,
     15-Aug-1994.) $)
  ancr $p |- ( ( ph -> ps ) -> ( ph -> ( ps /\ ph ) ) ) $=
    ( wa pm3.21 a2i ) ABBACABDE $.

  $( Conjoin antecedent to right of consequent.  (Contributed by NM,
     25-Jul-1999.)  (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
  ancrb $p |- ( ( ph -> ps ) <-> ( ph -> ( ps /\ ph ) ) ) $=
    ( wa iba pm5.74i ) ABBACABDE $.

  ${
    ancli.1 $e |- ( ph -> ps ) $.
    $( Deduction conjoining antecedent to left of consequent.  (Contributed by
       NM, 12-Aug-1993.) $)
    ancli $p |- ( ph -> ( ph /\ ps ) ) $=
      ( id jca ) AABADCE $.
  $}

  ${
    ancri.1 $e |- ( ph -> ps ) $.
    $( Deduction conjoining antecedent to right of consequent.  (Contributed by
       NM, 15-Aug-1994.) $)
    ancri $p |- ( ph -> ( ps /\ ph ) ) $=
      ( id jca ) ABACADE $.
  $}

  ${
    ancld.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction conjoining antecedent to left of consequent in nested
       implication.  (Contributed by NM, 15-Aug-1994.)  (Proof shortened by
       Wolf Lammen, 1-Nov-2012.) $)
    ancld $p |- ( ph -> ( ps -> ( ps /\ ch ) ) ) $=
      ( idd jcad ) ABBCABEDF $.
  $}

  ${
    ancrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction conjoining antecedent to right of consequent in nested
       implication.  (Contributed by NM, 15-Aug-1994.)  (Proof shortened by
       Wolf Lammen, 1-Nov-2012.) $)
    ancrd $p |- ( ph -> ( ps -> ( ch /\ ps ) ) ) $=
      ( idd jcad ) ABCBDABEF $.
  $}

  $( Conjoin antecedent to left of consequent in nested implication.
     (Contributed by NM, 10-Aug-1994.)  (Proof shortened by Wolf Lammen,
     14-Jul-2013.) $)
  anc2l $p |- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ps -> ( ph /\ ch ) ) ) ) $=
    ( wi wa pm5.42 biimpi ) ABCDDABACEDDABCFG $.

  $( Conjoin antecedent to right of consequent in nested implication.
     (Contributed by NM, 15-Aug-1994.) $)
  anc2r $p |- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ps -> ( ch /\ ph ) ) ) ) $=
    ( wi wa pm3.21 imim2d a2i ) ABCDBCAEZDACIBACFGH $.

  ${
    anc2li.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction conjoining antecedent to left of consequent in nested
       implication.  (Contributed by NM, 10-Aug-1994.)  (Proof shortened by
       Wolf Lammen, 7-Dec-2012.) $)
    anc2li $p |- ( ph -> ( ps -> ( ph /\ ch ) ) ) $=
      ( id jctild ) ABCADAEF $.
  $}

  ${
    anc2ri.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction conjoining antecedent to right of consequent in nested
       implication.  (Contributed by NM, 15-Aug-1994.)  (Proof shortened by
       Wolf Lammen, 7-Dec-2012.) $)
    anc2ri $p |- ( ph -> ( ps -> ( ch /\ ph ) ) ) $=
      ( id jctird ) ABCADAEF $.
  $}

  $( Theorem *3.41 of [WhiteheadRussell] p. 113.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.41 $p |- ( ( ph -> ch ) -> ( ( ph /\ ps ) -> ch ) ) $=
    ( wa simpl imim1i ) ABDACABEF $.

  $( Theorem *3.42 of [WhiteheadRussell] p. 113.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.42 $p |- ( ( ps -> ch ) -> ( ( ph /\ ps ) -> ch ) ) $=
    ( wa simpr imim1i ) ABDBCABEF $.

  $( Conjunction implies implication.  Theorem *3.4 of [WhiteheadRussell]
     p. 113.  (Contributed by NM, 31-Jul-1995.) $)
  pm3.4 $p |- ( ( ph /\ ps ) -> ( ph -> ps ) ) $=
    ( wa simpr a1d ) ABCBAABDE $.

  $( Conjunction with implication.  Compare Theorem *4.45 of [WhiteheadRussell]
     p. 119.  (Contributed by NM, 17-May-1998.) $)
  pm4.45im $p |- ( ph <-> ( ph /\ ( ps -> ph ) ) ) $=
    ( wi wa ax-1 ancli simpl impbii ) AABACZDAIABEFAIGH $.

  ${
    anim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    anim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
    $( Conjoin antecedents and consequents in a deduction.  (Contributed by NM,
       3-Apr-1994.)  (Proof shortened by Wolf Lammen, 18-Dec-2013.) $)
    anim12d $p |- ( ph -> ( ( ps /\ th ) -> ( ch /\ ta ) ) ) $=
      ( wa idd syl2and ) ABCDECEHZFGAKIJ $.
  $}

  ${
    anim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Add a conjunct to right of antecedent and consequent in a deduction.
       (Contributed by NM, 3-Apr-1994.) $)
    anim1d $p |- ( ph -> ( ( ps /\ th ) -> ( ch /\ th ) ) ) $=
      ( idd anim12d ) ABCDDEADFG $.

    $( Add a conjunct to left of antecedent and consequent in a deduction.
       (Contributed by NM, 5-Aug-1993.) $)
    anim2d $p |- ( ph -> ( ( th /\ ps ) -> ( th /\ ch ) ) ) $=
      ( idd anim12d ) ADDBCADFEG $.
  $}

  ${
    anim12i.1 $e |- ( ph -> ps ) $.
    anim12i.2 $e |- ( ch -> th ) $.
    $( Conjoin antecedents and consequents of two premises.  (Contributed by
       NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 14-Dec-2013.) $)
    anim12i $p |- ( ( ph /\ ch ) -> ( ps /\ th ) ) $=
      ( wa id syl2an ) ABDBDGZCEFJHI $.

    $( Variant of ~ anim12i with commutation.  (Contributed by Jonathan
       Ben-Naim, 3-Jun-2011.) $)
    anim12ci $p |- ( ( ph /\ ch ) -> ( th /\ ps ) ) $=
      ( wa anim12i ancoms ) CADBGCDABFEHI $.
  $}

  ${
    anim1i.1 $e |- ( ph -> ps ) $.
    $( Introduce conjunct to both sides of an implication.  (Contributed by NM,
       5-Aug-1993.) $)
    anim1i $p |- ( ( ph /\ ch ) -> ( ps /\ ch ) ) $=
      ( id anim12i ) ABCCDCEF $.

    $( Introduce conjunct to both sides of an implication.  (Contributed by
       Peter Mazsa, 24-Sep-2022.) $)
    anim1ci $p |- ( ( ph /\ ch ) -> ( ch /\ ps ) ) $=
      ( id anim12ci ) ABCCDCEF $.

    $( Introduce conjunct to both sides of an implication.  (Contributed by NM,
       5-Aug-1993.) $)
    anim2i $p |- ( ( ch /\ ph ) -> ( ch /\ ps ) ) $=
      ( id anim12i ) CCABCEDF $.
  $}

  ${
    anim12ii.1 $e |- ( ph -> ( ps -> ch ) ) $.
    anim12ii.2 $e |- ( th -> ( ps -> ta ) ) $.
    $( Conjoin antecedents and consequents in a deduction.  (Contributed by NM,
       11-Nov-2007.)  (Proof shortened by Wolf Lammen, 19-Jul-2013.) $)
    anim12ii $p |- ( ( ph /\ th ) -> ( ps -> ( ch /\ ta ) ) ) $=
      ( wa wi adantr adantl jcad ) ADHBCEABCIDFJDBEIAGKL $.
  $}

  $( Theorem *3.47 of [WhiteheadRussell] p. 113.  It was proved by Leibniz, and
     it evidently pleased him enough to call it 'praeclarum theorema' (splendid
     theorem).  (Contributed by NM, 12-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 7-Apr-2013.) $)
  anim12 $p |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) ->
                 ( ( ph /\ ch ) -> ( ps /\ th ) ) ) $=
    ( wi wa simpl simpr anim12d ) ABEZCDEZFABCDJKGJKHI $.

  $( Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.33 $p |- ( ( ( ph -> ps ) /\ ( ps -> ch ) ) -> ( ph -> ch ) ) $=
    ( wi imim1 imp ) ABDBCDACDABCEF $.

  $( Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.34 $p |- ( ( ( ps -> ch ) /\ ( ph -> ps ) ) -> ( ph -> ch ) ) $=
    ( wi imim2 imp ) BCDABDACDBCAEF $.

  $( Conjunctive detachment.  Theorem *3.35 of [WhiteheadRussell] p. 112.
     (Contributed by NM, 14-Dec-2002.) $)
  pm3.35 $p |- ( ( ph /\ ( ph -> ps ) ) -> ps ) $=
    ( wi pm2.27 imp ) AABCBABDE $.

  $( Theorem *5.31 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.31 $p |- ( ( ch /\ ( ph -> ps ) ) -> ( ph -> ( ps /\ ch ) ) ) $=
    ( wi wa pm3.21 imim2d imp ) CABDABCEZDCBIACBFGH $.

  ${
    imp4.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp4a $p |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) $=
      ( wi wa impexp imbitrrdi ) ABCDEGGCDHEGFCDEIJ $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp4b $p |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) ) $=
      ( wa wi imp4a imp ) ABCDGEHABCDEFIJ $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp4c $p |- ( ph -> ( ( ( ps /\ ch ) /\ th ) -> ta ) ) $=
      ( wa wi impd ) ABCGDEABCDEHFII $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp4d $p |- ( ph -> ( ( ps /\ ( ch /\ th ) ) -> ta ) ) $=
      ( wa imp4a impd ) ABCDGEABCDEFHI $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp41 $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $=
      ( wa wi imp imp31 ) ABGCDEABCDEHHFIJ $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp42 $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $=
      ( wa wi imp32 imp ) ABCGGDEABCDEHFIJ $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp43 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $=
      ( wa imp4b imp ) ABGCDGEABCDEFHI $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp44 $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ta ) $=
      ( wa imp4c imp ) ABCGDGEABCDEFHI $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.) $)
    imp45 $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ta ) $=
      ( wa imp4d imp ) ABCDGGEABCDEFHI $.
  $}

  ${
    imp5.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $.
    $( An importation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    imp5a $p |- ( ph -> ( ps -> ( ch -> ( ( th /\ ta ) -> et ) ) ) ) $=
      ( wi wa pm3.31 syl8 ) ABCDEFHHDEIFHGDEFJK $.

    $( An importation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    imp5d $p |- ( ( ( ph /\ ps ) /\ ch ) -> ( ( th /\ ta ) -> et ) ) $=
      ( wa wi imp31 impd ) ABHCHDEFABCDEFIIGJK $.

    $( An importation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    imp5g $p |- ( ( ph /\ ps ) -> ( ( ( ch /\ th ) /\ ta ) -> et ) ) $=
      ( wa wi imp imp4c ) ABHCDEFABCDEFIIIGJK $.

    $( An importation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    imp55 $p |- ( ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) /\ ta ) -> et ) $=
      ( wa wi imp4a imp42 ) ABCDHEFABCDEFIGJK $.

    $( An importation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    imp511 $p |- ( ( ph /\ ( ( ps /\ ( ch /\ th ) ) /\ ta ) ) -> et ) $=
      ( wa wi imp4a imp44 ) ABCDHEFABCDEFIGJK $.
  $}

  ${
    expimpd.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
    $( Exportation followed by a deduction version of importation.
       (Contributed by NM, 6-Sep-2008.) $)
    expimpd $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
      ( wi ex impd ) ABCDABCDFEGH $.
  $}

  ${
    exp31.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp31 $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
      ( wi wa ex ) ABCDFABGCDEHH $.
  $}

  ${
    exp32.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp32 $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
      ( wa ex expd ) ABCDABCFDEGH $.
  $}

  ${
    exp4a.1 $e |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp4a $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wa wi impexp imbitrdi ) ABCDGEHCDEHHFCDEIJ $.
  $}

  ${
    exp4b.1 $e |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.)  (Proof
       shortened by Wolf Lammen, 23-Nov-2012.) $)
    exp4b $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wa wi ex exp4a ) ABCDEABCDGEHFIJ $.
  $}

  ${
    exp4c.1 $e |- ( ph -> ( ( ( ps /\ ch ) /\ th ) -> ta ) ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp4c $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wi wa expd ) ABCDEGABCHDEFII $.
  $}

  ${
    exp4d.1 $e |- ( ph -> ( ( ps /\ ( ch /\ th ) ) -> ta ) ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp4d $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wa expd exp4a ) ABCDEABCDGEFHI $.
  $}

  ${
    exp41.1 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp41 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wi wa ex exp31 ) ABCDEGABHCHDEFIJ $.
  $}

  ${
    exp42.1 $e |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp42 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wi wa exp31 expd ) ABCDEGABCHDEFIJ $.
  $}

  ${
    exp43.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp43 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wa ex exp4b ) ABCDEABGCDGEFHI $.
  $}

  ${
    exp44.1 $e |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ta ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp44 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wi wa exp32 expd ) ABCDEGABCHDEFIJ $.
  $}

  ${
    exp45.1 $e |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ta ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.) $)
    exp45 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wa exp32 exp4a ) ABCDEABCDGEFHI $.
  $}

  ${
    expr.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( Export a wff from a right conjunct.  (Contributed by Jeff Hankins,
       30-Aug-2009.) $)
    expr $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
      ( wi exp32 imp ) ABCDFABCDEGH $.
  $}

  ${
    exp5c.1 $e |- ( ph -> ( ( ps /\ ch ) -> ( ( th /\ ta ) -> et ) ) ) $.
    $( An exportation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    exp5c $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
      ( wi wa exp4a expd ) ABCDEFHHABCIDEFGJK $.
  $}

  ${
    exp53.1 $e |- ( ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) /\ ta ) -> et ) $.
    $( An exportation inference.  (Contributed by Jeff Hankins,
       30-Aug-2009.) $)
    exp53 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
      ( wi wa ex exp43 ) ABCDEFHABICDIIEFGJK $.
  $}

  ${
    expl.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( Export a wff from a left conjunct.  (Contributed by Jeff Hankins,
       28-Aug-2009.) $)
    expl $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
      ( exp31 impd ) ABCDABCDEFG $.
  $}

  ${
    impr.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
    $( Import a wff into a right conjunct.  (Contributed by Jeff Hankins,
       30-Aug-2009.) $)
    impr $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
      ( wi ex imp32 ) ABCDABCDFEGH $.
  $}

  ${
    impl.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( Export a wff from a left conjunct.  (Contributed by Mario Carneiro,
       9-Jul-2014.) $)
    impl $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
      ( expd imp31 ) ABCDABCDEFG $.
  $}

  ${
    impac.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Importation with conjunction in consequent.  (Contributed by NM,
       9-Aug-1994.) $)
    impac $p |- ( ( ph /\ ps ) -> ( ch /\ ps ) ) $=
      ( wa ancrd imp ) ABCBEABCDFG $.
  $}

  ${
    exbiri.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
    $( Inference form of ~ exbir .  (Contributed by Alan Sare, 31-Dec-2011.)
       (Proof shortened by Wolf Lammen, 27-Jan-2013.) $)
    exbiri $p |- ( ph -> ( ps -> ( th -> ch ) ) ) $=
      ( wa biimpar exp31 ) ABDCABFCDEGH $.
  $}

  ${
    pm3.26bda.1 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
    $( Deduction eliminating a conjunct.  (Contributed by NM, 22-Oct-2007.) $)
    simprbda $p |- ( ( ph /\ ps ) -> ch ) $=
      ( wa biimpa simpld ) ABFCDABCDFEGH $.

    $( Deduction eliminating a conjunct.  (Contributed by NM, 22-Oct-2007.) $)
    simplbda $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa biimpa simprd ) ABFCDABCDFEGH $.
  $}

  ${
    pm3.26bi2.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Deduction eliminating a conjunct.  (Contributed by Alan Sare,
       31-Dec-2011.) $)
    simplbi2 $p |- ( ps -> ( ch -> ph ) ) $=
      ( wa biimpri ex ) BCAABCEDFG $.
  $}

  ${
    birani.1 $e |- ( ph <-> ps ) $.
    $( Inference adding a conjunct to the left-hand side of a biconditional.
       (Contributed by Matthew House, 22-May-2026.) $)
    birani $p |- ( ( ph /\ ch ) -> ps ) $=
      ( biimpi adantr ) ABCABDEF $.

    $( Inference adding a conjunct to the left-hand side of a biconditional.
       (Contributed by Matthew House, 22-May-2026.) $)
    bilani $p |- ( ( ch /\ ph ) -> ps ) $=
      ( biimpi adantl ) ABCABDEF $.

    $( Inference adding a conjunct to the right-hand side of a biconditional.
       (Contributed by Matthew House, 22-May-2026.) $)
    biranri $p |- ( ( ps /\ ch ) -> ph ) $=
      ( biimpri adantr ) BACABDEF $.

    $( Inference adding a conjunct to the right-hand side of a biconditional.
       (Contributed by Matthew House, 22-May-2026.) $)
    bilanri $p |- ( ( ch /\ ps ) -> ph ) $=
      ( biimpri adantl ) BACABDEF $.
  $}

  ${
    simpl2im.1 $e |- ( ph -> ( ps /\ ch ) ) $.
    simpl2im.2 $e |- ( ch -> th ) $.
    $( Implication from an eliminated conjunct implied by the antecedent.
       (Contributed by BJ/AV, 5-Apr-2021.) $)
    simpl2im $p |- ( ph -> th ) $=
      ( wa simpr 3syl ) ABCGCDEBCHFI $.
  $}

  ${
    simplbiim.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    simplbiim.2 $e |- ( ch -> th ) $.
    $( Implication from an eliminated conjunct equivalent to the antecedent.
       (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
    simplbiim $p |- ( ph -> th ) $=
      ( wa adantl sylbi ) ABCGDECDBFHI $.
  $}

  $( A theorem similar to the standard definition of the biconditional.
     Definition of [Margaris] p. 49.  (Contributed by NM, 5-Aug-1993.)
     (Revised by NM, 31-Jan-2015.) $)
  dfbi2 $p |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) $=
    ( wb wi wa df-bi simpli simpri impbii ) ABCZABDBADEZJKDZKJDZABFZGLMNHI $.

  $( Implication in terms of biconditional and conjunction.  Theorem *4.71 of
     [WhiteheadRussell] p. 120.  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 2-Dec-2012.) $)
  pm4.71 $p |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) ) $=
    ( wa wi wb simpl biantru anclb dfbi2 3bitr4i ) AABCZDZLKADZCABDAKEMLABFGABH
    AKIJ $.

  $( Implication in terms of biconditional and conjunction.  Theorem *4.71 of
     [WhiteheadRussell] p. 120 (with conjunct reversed).  (Contributed by NM,
     25-Jul-1999.) $)
  pm4.71r $p |- ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) ) $=
    ( wi wa wb pm4.71 ancom bibi2i bitri ) ABCAABDZEABADZEABFJKAABGHI $.

  ${
    pm4.71i.1 $e |- ( ph -> ps ) $.
    $( Inference converting an implication to a biconditional with conjunction.
       Inference from Theorem *4.71 of [WhiteheadRussell] p. 120.  (Contributed
       by NM, 4-Jan-2004.) $)
    pm4.71i $p |- ( ph <-> ( ph /\ ps ) ) $=
      ( wi wa wb pm4.71 mpbi ) ABDAABEFCABGH $.
  $}

  ${
    pm4.71ri.1 $e |- ( ph -> ps ) $.
    $( Inference converting an implication to a biconditional with conjunction.
       Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct
       reversed).  (Contributed by NM, 1-Dec-2003.) $)
    pm4.71ri $p |- ( ph <-> ( ps /\ ph ) ) $=
      ( wi wa wb pm4.71r mpbi ) ABDABAEFCABGH $.
  $}

  ${
    pm4.71rd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction converting an implication to a biconditional with conjunction.
       Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120.  (Contributed
       by Mario Carneiro, 25-Dec-2016.) $)
    pm4.71d $p |- ( ph -> ( ps <-> ( ps /\ ch ) ) ) $=
      ( wi wa wb pm4.71 sylib ) ABCEBBCFGDBCHI $.

    $( Deduction converting an implication to a biconditional with conjunction.
       Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120.  (Contributed
       by NM, 10-Feb-2005.) $)
    pm4.71rd $p |- ( ph -> ( ps <-> ( ch /\ ps ) ) ) $=
      ( wi wa wb pm4.71r sylib ) ABCEBCBFGDBCHI $.
  $}

  $( Theorem *4.24 of [WhiteheadRussell] p. 117.  (Contributed by NM,
     3-Jan-2005.)  (Revised by NM, 14-Mar-2014.) $)
  pm4.24 $p |- ( ph <-> ( ph /\ ph ) ) $=
    ( id pm4.71i ) AAABC $.

  $( Idempotent law for conjunction.  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 14-Mar-2014.) $)
  anidm $p |- ( ( ph /\ ph ) <-> ph ) $=
    ( wa pm4.24 bicomi ) AAABACD $.

  ${
    anidms.1 $e |- ( ( ph /\ ph ) -> ps ) $.
    $( Inference from idempotent law for conjunction.  (Contributed by NM,
       15-Jun-1994.) $)
    anidms $p |- ( ph -> ps ) $=
      ( ex pm2.43i ) ABAABCDE $.
  $}

  $( Conjunction idempotence with antecedent.  (Contributed by Roy F. Longton,
     8-Aug-2005.) $)
  anidmdbi $p |- ( ( ph -> ( ps /\ ps ) ) <-> ( ph -> ps ) ) $=
    ( wa anidm imbi2i ) BBCBABDE $.

  ${
    anasss.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( Associative law for conjunction applied to antecedent (eliminates
       syllogism).  (Contributed by NM, 15-Nov-2002.) $)
    anasss $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
      ( exp31 imp32 ) ABCDABCDEFG $.
  $}

  ${
    anassrs.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( Associative law for conjunction applied to antecedent (eliminates
       syllogism).  (Contributed by NM, 15-Nov-2002.) $)
    anassrs $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
      ( exp32 imp31 ) ABCDABCDEFG $.
  $}

  $( Associative law for conjunction.  Theorem *4.32 of [WhiteheadRussell]
     p. 118.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 24-Nov-2012.) $)
  anass $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) ) $=
    ( wa id anassrs anasss impbii ) ABDCDZABCDDZABCJJEFABCIIEGH $.

  ${
    sylanl1.1 $e |- ( ph -> ps ) $.
    sylanl1.2 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 10-Mar-2005.) $)
    sylanl1 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
      ( wa anim1i sylan ) ACHBCHDEABCFIGJ $.
  $}

  ${
    sylanl2.1 $e |- ( ph -> ch ) $.
    sylanl2.2 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 1-Jan-2005.) $)
    sylanl2 $p |- ( ( ( ps /\ ph ) /\ th ) -> ta ) $=
      ( wa anim2i sylan ) BAHBCHDEACBFIGJ $.
  $}

  ${
    sylanr1.1 $e |- ( ph -> ch ) $.
    sylanr1.2 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 9-Apr-2005.) $)
    sylanr1 $p |- ( ( ps /\ ( ph /\ th ) ) -> ta ) $=
      ( wa anim1i sylan2 ) ADHBCDHEACDFIGJ $.
  $}

  ${
    sylanr2.1 $e |- ( ph -> th ) $.
    sylanr2.2 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 9-Apr-2005.) $)
    sylanr2 $p |- ( ( ps /\ ( ch /\ ph ) ) -> ta ) $=
      ( wa anim2i sylan2 ) CAHBCDHEADCFIGJ $.
  $}

  ${
    sylani.1 $e |- ( ph -> ch ) $.
    sylani.2 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
    $( A syllogism inference.  (Contributed by NM, 2-May-1996.) $)
    sylani $p |- ( ps -> ( ( ph /\ th ) -> ta ) ) $=
      ( wi a1i syland ) BACDEACHBFIGJ $.
  $}

  ${
    sylan2i.1 $e |- ( ph -> th ) $.
    sylan2i.2 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
    $( A syllogism inference.  (Contributed by NM, 1-Aug-1994.) $)
    sylan2i $p |- ( ps -> ( ( ch /\ ph ) -> ta ) ) $=
      ( wi a1i sylan2d ) BADCEADHBFIGJ $.
  $}

  ${
    syl2ani.1 $e |- ( ph -> ch ) $.
    syl2ani.2 $e |- ( et -> th ) $.
    syl2ani.3 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
    $( A syllogism inference.  (Contributed by NM, 3-Aug-1999.) $)
    syl2ani $p |- ( ps -> ( ( ph /\ et ) -> ta ) ) $=
      ( sylan2i sylani ) ABCFEGFBCDEHIJK $.
  $}

  ${
    sylan9.1 $e |- ( ph -> ( ps -> ch ) ) $.
    sylan9.2 $e |- ( th -> ( ch -> ta ) ) $.
    $( Nested syllogism inference conjoining dissimilar antecedents.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       7-May-2011.) $)
    sylan9 $p |- ( ( ph /\ th ) -> ( ps -> ta ) ) $=
      ( wi syl9 imp ) ADBEHABCDEFGIJ $.
  $}

  ${
    sylan9r.1 $e |- ( ph -> ( ps -> ch ) ) $.
    sylan9r.2 $e |- ( th -> ( ch -> ta ) ) $.
    $( Nested syllogism inference conjoining dissimilar antecedents.
       (Contributed by NM, 5-Aug-1993.) $)
    sylan9r $p |- ( ( th /\ ph ) -> ( ps -> ta ) ) $=
      ( wi syl9r imp ) DABEHABCDEFGIJ $.
  $}

  ${
    syl2anc.1 $e |- ( ph -> ps ) $.
    syl2anc.2 $e |- ( ph -> ch ) $.
    syl2anc.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( Syllogism inference combined with contraction.  (Contributed by NM,
       16-Mar-2012.) $)
    syl2anc $p |- ( ph -> th ) $=
      ( ex sylc ) ABCDEFBCDGHI $.
  $}

  ${
    syl2anc2.1 $e |- ( ph -> ps ) $.
    syl2anc2.2 $e |- ( ps -> ch ) $.
    syl2anc2.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( Double syllogism inference combined with contraction.  (Contributed by
       BTernaryTau, 29-Sep-2023.) $)
    syl2anc2 $p |- ( ph -> th ) $=
      ( syl syl2anc ) ABCDEABCEFHGI $.
  $}

  ${
    sylancl.1 $e |- ( ph -> ps ) $.
    sylancl.2 $e |- ch $.
    sylancl.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( Syllogism inference combined with modus ponens.  (Contributed by Jeff
       Madsen, 2-Sep-2009.) $)
    sylancl $p |- ( ph -> th ) $=
      ( a1i syl2anc ) ABCDECAFHGI $.
  $}

  ${
    sylancr.1 $e |- ps $.
    sylancr.2 $e |- ( ph -> ch ) $.
    sylancr.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( Syllogism inference combined with modus ponens.  (Contributed by Jeff
       Madsen, 2-Sep-2009.) $)
    sylancr $p |- ( ph -> th ) $=
      ( a1i syl2anc ) ABCDBAEHFGI $.
  $}

  ${
    sylanblc.1 $e |- ( ph -> ps ) $.
    sylanblc.2 $e |- ch $.
    sylanblc.3 $e |- ( ( ps /\ ch ) <-> th ) $.
    $( Syllogism inference combined with a biconditional.  (Contributed by BJ,
       25-Apr-2019.) $)
    sylanblc $p |- ( ph -> th ) $=
      ( wa biimpi sylancl ) ABCDEFBCHDGIJ $.
  $}

  ${
    sylanblrc.1 $e |- ( ph -> ps ) $.
    sylanblrc.2 $e |- ch $.
    sylanblrc.3 $e |- ( th <-> ( ps /\ ch ) ) $.
    $( Syllogism inference combined with a biconditional.  (Contributed by BJ,
       25-Apr-2019.) $)
    sylanblrc $p |- ( ph -> th ) $=
      ( wa biimpri sylancl ) ABCDEFDBCHGIJ $.
  $}

  ${
    sylanbrc.1 $e |- ( ph -> ps ) $.
    sylanbrc.2 $e |- ( ph -> ch ) $.
    sylanbrc.3 $e |- ( th <-> ( ps /\ ch ) ) $.
    $( Syllogism inference.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    sylanbrc $p |- ( ph -> th ) $=
      ( wa jca sylibr ) ABCHDABCEFIGJ $.
  $}

  ${
    sylancb.1 $e |- ( ph <-> ps ) $.
    sylancb.2 $e |- ( ph <-> ch ) $.
    sylancb.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A syllogism inference combined with contraction.  (Contributed by NM,
       3-Sep-2004.) $)
    sylancb $p |- ( ph -> th ) $=
      ( syl2anb anidms ) ADABCDAEFGHI $.
  $}

  ${
    sylancbr.1 $e |- ( ps <-> ph ) $.
    sylancbr.2 $e |- ( ch <-> ph ) $.
    sylancbr.3 $e |- ( ( ps /\ ch ) -> th ) $.
    $( A syllogism inference combined with contraction.  (Contributed by NM,
       3-Sep-2004.) $)
    sylancbr $p |- ( ph -> th ) $=
      ( syl2anbr anidms ) ADABCDAEFGHI $.
  $}

  ${
    sylancom.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    sylancom.2 $e |- ( ( ch /\ ps ) -> th ) $.
    $( Syllogism inference with commutation of antecents.  (Contributed by NM,
       2-Jul-2008.) $)
    sylancom $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa simpr syl2anc ) ABGCBDEABHFI $.
  $}

  ${
    mpdan.1 $e |- ( ph -> ps ) $.
    mpdan.2 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 23-May-1999.)
       (Proof shortened by Wolf Lammen, 22-Nov-2012.) $)
    mpdan $p |- ( ph -> ch ) $=
      ( id syl2anc ) AABCAFDEG $.
  $}

  ${
    mpancom.1 $e |- ( ps -> ph ) $.
    mpancom.2 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( An inference based on modus ponens with commutation of antecedents.
       (Contributed by NM, 28-Oct-2003.)  (Proof shortened by Wolf Lammen,
       7-Apr-2013.) $)
    mpancom $p |- ( ps -> ch ) $=
      ( id syl2anc ) BABCDBFEG $.
  $}

  ${
    mpidan.1 $e |- ( ph -> ch ) $.
    mpidan.2 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( A deduction which "stacks" a hypothesis.  (Contributed by Stanislas
       Polu, 9-Mar-2020.)  (Proof shortened by Wolf Lammen, 28-Mar-2021.) $)
    mpidan $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa adantr mpdan ) ABGCDACBEHFI $.
  $}

  ${
    mpan.1 $e |- ph $.
    mpan.2 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 30-Aug-1993.)
       (Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
    mpan $p |- ( ps -> ch ) $=
      ( a1i mpancom ) ABCABDFEG $.
  $}

  ${
    mpan2.1 $e |- ps $.
    mpan2.2 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 16-Sep-1993.)
       (Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
    mpan2 $p |- ( ph -> ch ) $=
      ( a1i mpdan ) ABCBADFEG $.
  $}

  ${
    mp2an.1 $e |- ph $.
    mp2an.2 $e |- ps $.
    mp2an.3 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       13-Apr-1995.) $)
    mp2an $p |- ch $=
      ( mpan ax-mp ) BCEABCDFGH $.
  $}

  ${
    mp4an.1 $e |- ph $.
    mp4an.2 $e |- ps $.
    mp4an.3 $e |- ch $.
    mp4an.4 $e |- th $.
    mp4an.5 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
    $( An inference based on modus ponens.  (Contributed by Jeff Madsen,
       15-Jun-2011.) $)
    mp4an $p |- ta $=
      ( wa pm3.2i mp2an ) ABKCDKEABFGLCDHILJM $.
  $}

  ${
    mpan2d.1 $e |- ( ph -> ch ) $.
    mpan2d.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( A deduction based on modus ponens.  (Contributed by NM, 12-Dec-2004.) $)
    mpan2d $p |- ( ph -> ( ps -> th ) ) $=
      ( expd mpid ) ABCDEABCDFGH $.
  $}

  ${
    mpand.1 $e |- ( ph -> ps ) $.
    mpand.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( A deduction based on modus ponens.  (Contributed by NM, 12-Dec-2004.)
       (Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
    mpand $p |- ( ph -> ( ch -> th ) ) $=
      ( ancomsd mpan2d ) ACBDEABCDFGH $.
  $}

  ${
    mpani.1 $e |- ps $.
    mpani.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 10-Apr-1994.)
       (Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
    mpani $p |- ( ph -> ( ch -> th ) ) $=
      ( a1i mpand ) ABCDBAEGFH $.
  $}

  ${
    mpan2i.1 $e |- ch $.
    mpan2i.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 10-Apr-1994.)
       (Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
    mpan2i $p |- ( ph -> ( ps -> th ) ) $=
      ( a1i mpan2d ) ABCDCAEGFH $.
  $}

  ${
    mp2ani.1 $e |- ps $.
    mp2ani.2 $e |- ch $.
    mp2ani.3 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       12-Dec-2004.) $)
    mp2ani $p |- ( ph -> th ) $=
      ( mpani mpi ) ACDFABCDEGHI $.
  $}

  ${
    mp2and.1 $e |- ( ph -> ps ) $.
    mp2and.2 $e |- ( ph -> ch ) $.
    mp2and.3 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( A deduction based on modus ponens.  (Contributed by NM, 12-Dec-2004.) $)
    mp2and $p |- ( ph -> th ) $=
      ( mpand mpd ) ACDFABCDEGHI $.
  $}

  ${
    mpanl1.1 $e |- ph $.
    mpanl1.2 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 16-Aug-1994.)
       (Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
    mpanl1 $p |- ( ( ps /\ ch ) -> th ) $=
      ( wa jctl sylan ) BABGCDBAEHFI $.
  $}

  ${
    mpanl2.1 $e |- ps $.
    mpanl2.2 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 16-Aug-1994.)
       (Proof shortened by Andrew Salmon, 7-May-2011.) $)
    mpanl2 $p |- ( ( ph /\ ch ) -> th ) $=
      ( wa jctr sylan ) AABGCDABEHFI $.
  $}

  ${
    mpanl12.1 $e |- ph $.
    mpanl12.2 $e |- ps $.
    mpanl12.3 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       13-Jul-2005.) $)
    mpanl12 $p |- ( ch -> th ) $=
      ( mpanl1 mpan ) BCDFABCDEGHI $.
  $}

  ${
    mpanr1.1 $e |- ps $.
    mpanr1.2 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 3-May-1994.)
       (Proof shortened by Andrew Salmon, 7-May-2011.) $)
    mpanr1 $p |- ( ( ph /\ ch ) -> th ) $=
      ( anassrs mpanl2 ) ABCDEABCDFGH $.
  $}

  ${
    mpanr2.1 $e |- ch $.
    mpanr2.2 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 3-May-1994.)
       (Proof shortened by Andrew Salmon, 7-May-2011.)  (Proof shortened by
       Wolf Lammen, 7-Apr-2013.) $)
    mpanr2 $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa jctr sylan2 ) BABCGDBCEHFI $.
  $}

  ${
    mpanr12.1 $e |- ps $.
    mpanr12.2 $e |- ch $.
    mpanr12.3 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       24-Jul-2009.) $)
    mpanr12 $p |- ( ph -> th ) $=
      ( mpanr1 mpan2 ) ACDFABCDEGHI $.
  $}

  ${
    mpanlr1.1 $e |- ps $.
    mpanlr1.2 $e |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 30-Dec-2004.)
       (Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
    mpanlr1 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
      ( wa jctl sylanl2 ) CABCHDECBFIGJ $.
  $}

  ${
    mpbirand.1 $e |- ( ph -> ch ) $.
    mpbirand.2 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
    $( Detach truth from conjunction in biconditional.  (Contributed by Glauco
       Siliprandi, 3-Mar-2021.) $)
    mpbirand $p |- ( ph -> ( ps <-> th ) ) $=
      ( wa biantrurd bitr4d ) ABCDGDFACDEHI $.
  $}

  ${
    mpbiran2d.1 $e |- ( ph -> th ) $.
    mpbiran2d.2 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
    $( Detach truth from conjunction in biconditional.  Deduction form.
       (Contributed by Peter Mazsa, 24-Sep-2022.) $)
    mpbiran2d $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wa biantrud bitr4d ) ABCDGCFADCEHI $.
  $}

  ${
    pm5.74da.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
    $( Distribution of implication over biconditional (deduction form).
       (Contributed by NM, 4-May-2007.) $)
    pm5.74da $p |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $=
      ( wb ex pm5.74d ) ABCDABCDFEGH $.
  $}

  $( Distribution of implication with conjunction.  (Contributed by NM,
     31-May-1999.)  (Proof shortened by Wolf Lammen, 6-Dec-2012.) $)
  imdistan $p |- ( ( ph -> ( ps -> ch ) ) <->
                ( ( ph /\ ps ) -> ( ph /\ ch ) ) ) $=
    ( wi wa pm5.42 impexp bitr4i ) ABCDDABACEZDDABEIDABCFABIGH $.

  ${
    imdistani.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Distribution of implication with conjunction.  (Contributed by NM,
       1-Aug-1994.) $)
    imdistani $p |- ( ( ph /\ ps ) -> ( ph /\ ch ) ) $=
      ( wa anc2li imp ) ABACEABCDFG $.
  $}

  ${
    imdistanri.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Distribution of implication with conjunction.  (Contributed by NM,
       8-Jan-2002.) $)
    imdistanri $p |- ( ( ps /\ ph ) -> ( ch /\ ph ) ) $=
      ( com12 impac ) BACABCDEF $.
  $}

  ${
    imdistand.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( Distribution of implication with conjunction (deduction form).
       (Contributed by NM, 27-Aug-2004.) $)
    imdistand $p |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) ) $=
      ( wi wa imdistan sylib ) ABCDFFBCGBDGFEBCDHI $.
  $}

  ${
    imdistanda.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
    $( Distribution of implication with conjunction (deduction version with
       conjoined antecedent).  (Contributed by Jeff Madsen, 19-Jun-2011.) $)
    imdistanda $p |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) ) $=
      ( wi ex imdistand ) ABCDABCDFEGH $.
  $}

  ${
    syldanl.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    syldanl.2 $e |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $.
    $( A syllogism deduction with conjoined antecedents.  (Contributed by Jeff
       Madsen, 20-Jun-2011.) $)
    syldanl $p |- ( ( ( ph /\ ps ) /\ th ) -> ta ) $=
      ( wa ex imdistani sylan ) ABHACHDEABCABCFIJGK $.
  $}

  ${
    pm5.32d.1 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
    $( Distribution of implication over biconditional (deduction form).
       (Contributed by NM, 29-Oct-1996.)  (Revised by NM, 31-Jan-2015.) $)
    pm5.32d $p |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) ) $=
      ( wa wb wi biimp syl6 imdistand biimpr impbid ) ABCFBDFABCDABCDGZCDHECDIJ
      KABDCABNDCHECDLJKM $.

    $( Distribution of implication over biconditional (deduction form).
       (Contributed by NM, 25-Dec-2004.) $)
    pm5.32rd $p |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) ) $=
      ( wa pm5.32d ancom 3bitr4g ) ABCFBDFCBFDBFABCDEGCBHDBHI $.
  $}

  ${
    pm5.32da.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
    $( Distribution of implication over biconditional (deduction form).
       (Contributed by NM, 9-Dec-2006.) $)
    pm5.32da $p |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) ) $=
      ( wb ex pm5.32d ) ABCDABCDFEGH $.
  $}

  $( Distribution of implication over biconditional.  Theorem *5.32 of
     [WhiteheadRussell] p. 125.  (Contributed by NM, 1-Aug-1994.)  (Revised by
     NM, 31-Jan-2015.) $)
  pm5.32 $p |- ( ( ph -> ( ps <-> ch ) ) <->
               ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) $=
    ( wb wi wa id pm5.32d ibar bibi12d biimprcd impbii ) ABCDZEZABFZACFZDZNABCN
    GHAMQABOCPABIACIJKL $.

  ${
    pm5.32i.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Distribution of implication over biconditional (inference form).
       (Contributed by NM, 1-Aug-1994.) $)
    pm5.32i $p |- ( ( ph /\ ps ) <-> ( ph /\ ch ) ) $=
      ( wb wi wa pm5.32 mpbi ) ABCEFABGACGEDABCHI $.

    $( Distribution of implication over biconditional (inference form).
       (Contributed by NM, 12-Mar-1995.) $)
    pm5.32ri $p |- ( ( ps /\ ph ) <-> ( ch /\ ph ) ) $=
      ( wa pm5.32i ancom 3bitr4i ) ABEACEBAECAEABCDFBAGCAGH $.
  $}

  ${
    biadan2.1 $e |- ( ph -> ps ) $.
    biadan2.2 $e |- ( ps -> ( ph <-> ch ) ) $.
    $( Add a conjunction to an equivalence.  (Contributed by Jeff Madsen,
       20-Jun-2011.) $)
    biadan2 $p |- ( ph <-> ( ps /\ ch ) ) $=
      ( wa pm4.71ri pm5.32i bitri ) ABAFBCFABDGBACEHI $.
  $}

  ${
    bi.aa $e |- ( ph <-> ps ) $.
    $( Introduce a left conjunct to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       16-Nov-2013.) $)
    anbi2i $p |- ( ( ch /\ ph ) <-> ( ch /\ ps ) ) $=
      ( wb a1i pm5.32i ) CABABECDFG $.

    $( Introduce a right conjunct to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       16-Nov-2013.) $)
    anbi1i $p |- ( ( ph /\ ch ) <-> ( ps /\ ch ) ) $=
      ( wb a1i pm5.32ri ) CABABECDFG $.

    $( Variant of ~ anbi2i with commutation.  (Contributed by Jonathan
       Ben-Naim, 3-Jun-2011.)  (Proof shortened by Andrew Salmon,
       14-Jun-2011.) $)
    anbi2ci $p |- ( ( ph /\ ch ) <-> ( ch /\ ps ) ) $=
      ( wa anbi1i ancom bitri ) ACEBCECBEABCDFBCGH $.
  $}

  ${
    anbi12.1 $e |- ( ph <-> ps ) $.
    anbi12.2 $e |- ( ch <-> th ) $.
    $( Conjoin both sides of two equivalences.  (Contributed by NM,
       5-Aug-1993.) $)
    anbi12i $p |- ( ( ph /\ ch ) <-> ( ps /\ th ) ) $=
      ( wa anbi1i anbi2i bitri ) ACGBCGBDGABCEHCDBFIJ $.

    $( Variant of ~ anbi12i with commutation.  (Contributed by Jonathan
       Ben-Naim, 3-Jun-2011.) $)
    anbi12ci $p |- ( ( ph /\ ch ) <-> ( th /\ ps ) ) $=
      ( wa anbi12i ancom bitri ) ACGBDGDBGABCDEFHBDIJ $.
  $}

  ${
    sylan9bb.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    sylan9bb.2 $e |- ( th -> ( ch <-> ta ) ) $.
    $( Nested syllogism inference conjoining dissimilar antecedents.
       (Contributed by NM, 4-Mar-1995.) $)
    sylan9bb $p |- ( ( ph /\ th ) -> ( ps <-> ta ) ) $=
      ( wa wb adantr adantl bitrd ) ADHBCEABCIDFJDCEIAGKL $.
  $}

  ${
    sylan9bbr.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    sylan9bbr.2 $e |- ( th -> ( ch <-> ta ) ) $.
    $( Nested syllogism inference conjoining dissimilar antecedents.
       (Contributed by NM, 4-Mar-1995.) $)
    sylan9bbr $p |- ( ( th /\ ph ) -> ( ps <-> ta ) ) $=
      ( wb sylan9bb ancoms ) ADBEHABCDEFGIJ $.
  $}

  ${
    anbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction adding a left conjunct to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       16-Nov-2013.) $)
    anbi2d $p |- ( ph -> ( ( th /\ ps ) <-> ( th /\ ch ) ) ) $=
      ( wb a1d pm5.32d ) ADBCABCFDEGH $.

    $( Deduction adding a right conjunct to both sides of a logical
       equivalence.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
       Lammen, 16-Nov-2013.) $)
    anbi1d $p |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ th ) ) ) $=
      ( wb a1d pm5.32rd ) ADBCABCFDEGH $.
  $}

  $( Introduce a right conjunct to both sides of a logical equivalence.
     Theorem *4.36 of [WhiteheadRussell] p. 118.  (Contributed by NM,
     3-Jan-2005.) $)
  anbi1 $p |- ( ( ph <-> ps ) -> ( ( ph /\ ch ) <-> ( ps /\ ch ) ) ) $=
    ( wb id anbi1d ) ABDZABCGEF $.

  $( Introduce a left conjunct to both sides of a logical equivalence.
     (Contributed by NM, 16-Nov-2013.) $)
  anbi2 $p |- ( ( ph <-> ps ) -> ( ( ch /\ ph ) <-> ( ch /\ ps ) ) ) $=
    ( wb id anbi2d ) ABDZABCGEF $.

  ${
    anbi1cd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Introduce a proposition as left conjunct on the left-hand side and right
       conjunct on the right-hand side of an equivalence.  Deduction form.
       (Contributed by Peter Mazsa, 22-May-2021.) $)
    anbi1cd $p |- ( ph -> ( ( th /\ ps ) <-> ( ch /\ th ) ) ) $=
      ( wa anbi2d biancomd ) ADBFCDABCDEGH $.
  $}

  ${
    bianass.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( An inference to merge two lists of conjuncts.  (Contributed by Giovanni
       Mascellani, 23-May-2019.) $)
    bianass $p |- ( ( et /\ ph ) <-> ( ( et /\ ps ) /\ ch ) ) $=
      ( wa anbi2i anass bitr4i ) DAFDBCFZFDBFCFAJDEGDBCHI $.

    $( An inference to merge two lists of conjuncts.  (Contributed by Peter
       Mazsa, 24-Sep-2022.) $)
    bianassc $p |- ( ( et /\ ph ) <-> ( ( ps /\ et ) /\ ch ) ) $=
      ( wa bianass ancom anbi1i bitri ) DAFDBFZCFBDFZCFABCDEGKLCDBHIJ $.
  $}

  $( Swap two conjuncts.  (Contributed by Peter Mazsa, 18-Sep-2022.) $)
  an21 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
    ( wa biid bianassc bicomi ) BACDZDABDCDHACBHEFG $.

  $( Theorem *4.22 of [WhiteheadRussell] p. 117.  (Contributed by NM,
     3-Jan-2005.) $)
  bitr $p |- ( ( ( ph <-> ps ) /\ ( ps <-> ch ) ) -> ( ph <-> ch ) ) $=
    ( wb bibi1 biimpar ) ABDACDBCDABCEF $.

  ${
    anbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    anbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction joining two equivalences to form equivalence of conjunctions.
       (Contributed by NM, 5-Aug-1993.) $)
    anbi12d $p |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ ta ) ) ) $=
      ( wa anbi1d anbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  $( Modus ponens mixed with several conjunctions.  (Contributed by Jim
     Kingdon, 7-Jan-2018.) $)
  mpan10 $p |- ( ( ( ( ph -> ps ) /\ ch ) /\ ph ) -> ( ps /\ ch ) ) $=
    ( wi wa ancom anbi2i anass 3bitr4i id imp anim1i sylbi ) ABDZCEAEZNAEZCEZBC
    ENCAEZENACEZEOQRSNCAFGNCAHNACHIPBCNABNJKLM $.

  $( Theorem *5.3 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Andrew Salmon, 7-May-2011.) $)
  pm5.3 $p |- ( ( ( ph /\ ps ) -> ch ) <->
               ( ( ph /\ ps ) -> ( ph /\ ch ) ) ) $=
    ( wa wi impexp imdistan bitri ) ABDZCEABCEEIACDEABCFABCGH $.

  ${
    adant2.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       4-May-1994.)  (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
    adantll $p |- ( ( ( th /\ ph ) /\ ps ) -> ch ) $=
      ( wa simpr sylan ) DAFABCDAGEH $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       4-May-1994.)  (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
    adantlr $p |- ( ( ( ph /\ th ) /\ ps ) -> ch ) $=
      ( wa simpl sylan ) ADFABCADGEH $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       4-May-1994.)  (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
    adantrl $p |- ( ( ph /\ ( th /\ ps ) ) -> ch ) $=
      ( wa simpr sylan2 ) DBFABCDBGEH $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       4-May-1994.)  (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
    adantrr $p |- ( ( ph /\ ( ps /\ th ) ) -> ch ) $=
      ( wa simpl sylan2 ) BDFABCBDGEH $.
  $}

  ${
    adantl2.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       26-Dec-2004.)  (Proof shortened by Wolf Lammen, 2-Dec-2012.) $)
    adantlll $p |- ( ( ( ( ta /\ ph ) /\ ps ) /\ ch ) -> th ) $=
      ( wa simpr sylanl1 ) EAGABCDEAHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       26-Dec-2004.)  (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
    adantllr $p |- ( ( ( ( ph /\ ta ) /\ ps ) /\ ch ) -> th ) $=
      ( wa simpl sylanl1 ) AEGABCDAEHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       26-Dec-2004.)  (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
    adantlrl $p |- ( ( ( ph /\ ( ta /\ ps ) ) /\ ch ) -> th ) $=
      ( wa simpr sylanl2 ) EBGABCDEBHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       26-Dec-2004.)  (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
    adantlrr $p |- ( ( ( ph /\ ( ps /\ ta ) ) /\ ch ) -> th ) $=
      ( wa simpl sylanl2 ) BEGABCDBEHFI $.
  $}

  ${
    adantr2.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       26-Dec-2004.)  (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
    adantrll $p |- ( ( ph /\ ( ( ta /\ ps ) /\ ch ) ) -> th ) $=
      ( wa simpr sylanr1 ) EBGABCDEBHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       26-Dec-2004.)  (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
    adantrlr $p |- ( ( ph /\ ( ( ps /\ ta ) /\ ch ) ) -> th ) $=
      ( wa simpl sylanr1 ) BEGABCDBEHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       26-Dec-2004.)  (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
    adantrrl $p |- ( ( ph /\ ( ps /\ ( ta /\ ch ) ) ) -> th ) $=
      ( wa simpr sylanr2 ) ECGABCDECHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       26-Dec-2004.)  (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
    adantrrr $p |- ( ( ph /\ ( ps /\ ( ch /\ ta ) ) ) -> th ) $=
      ( wa simpl sylanr2 ) CEGABCDCEHFI $.
  $}

  ${
    ad2ant.1 $e |- ( ph -> ps ) $.
    $( Deduction adding two conjuncts to antecedent.  (Contributed by NM,
       19-Oct-1999.)  (Proof shortened by Wolf Lammen, 20-Nov-2012.) $)
    ad2antrr $p |- ( ( ( ph /\ ch ) /\ th ) -> ps ) $=
      ( adantr adantlr ) ADBCABDEFG $.

    $( Deduction adding two conjuncts to antecedent.  (Contributed by NM,
       19-Oct-1999.)  (Proof shortened by Wolf Lammen, 20-Nov-2012.) $)
    ad2antlr $p |- ( ( ( ch /\ ph ) /\ th ) -> ps ) $=
      ( adantr adantll ) ADBCABDEFG $.

    $( Deduction adding two conjuncts to antecedent.  (Contributed by NM,
       19-Oct-1999.) $)
    ad2antrl $p |- ( ( ch /\ ( ph /\ th ) ) -> ps ) $=
      ( wa adantr adantl ) ADFBCABDEGH $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by NM,
       19-Oct-1999.) $)
    ad2antll $p |- ( ( ch /\ ( th /\ ph ) ) -> ps ) $=
      ( wa adantl ) DAFBCABDEGG $.

    $( Deduction adding three conjuncts to antecedent.  (Contributed by NM,
       28-Jul-2012.) $)
    ad3antrrr $p |- ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) -> ps ) $=
      ( wa adantr ad2antrr ) ACGBDEABCFHI $.

    $( Deduction adding three conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.) $)
    ad3antlr $p |- ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) -> ps ) $=
      ( wa ad2antlr adantr ) CAGDGBEABCDFHI $.

    $( Deduction adding 4 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.) $)
    ad4antr $p |- ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) -> ps ) $=
      ( wa ad3antrrr adantr ) ACHDHEHBFABCDEGIJ $.

    $( Deduction adding 4 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.) $)
    ad4antlr $p |- ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) -> ps ) $=
      ( wa ad3antlr adantr ) CAHDHEHBFABCDEGIJ $.

    $( Deduction adding 5 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.) $)
    ad5antr $p |- ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) -> ps ) $=
      ( wa ad4antr adantr ) ACIDIEIFIBGABCDEFHJK $.

    $( Deduction adding 5 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.) $)
    ad5antlr $p |- ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) -> ps ) $=
      ( wa ad4antlr adantr ) CAIDIEIFIBGABCDEFHJK $.

    $( Deduction adding 6 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.) $)
    ad6antr $p |- ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) -> ps ) $=
      ( wa ad5antr adantr ) ACJDJEJFJGJBHABCDEFGIKL $.

    $( Deduction adding 6 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.) $)
    ad6antlr $p |- ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) -> ps ) $=
      ( wa ad5antlr adantr ) CAJDJEJFJGJBHABCDEFGIKL $.

    $( Deduction adding 7 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.) $)
    ad7antr $p |- ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) -> ps ) $=
      ( wa ad6antr adantr ) ACKDKEKFKGKHKBIABCDEFGHJLM $.

    $( Deduction adding 7 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.) $)
    ad7antlr $p |- ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) -> ps ) $=
      ( wa ad6antlr adantr ) CAKDKEKFKGKHKBIABCDEFGHJLM $.

    $( Deduction adding 8 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.) $)
    ad8antr $p |- ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
      ( wa ad7antr adantr ) ACLDLELFLGLHLILBJABCDEFGHIKMN $.

    $( Deduction adding 8 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.) $)
    ad8antlr $p |- ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
      ( wa ad7antlr adantr ) CALDLELFLGLHLILBJABCDEFGHIKMN $.

    $( Deduction adding 9 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.) $)
    ad9antr $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
      ( wa ad8antr adantr ) ACMDMEMFMGMHMIMJMBKABCDEFGHIJLNO $.

    $( Deduction adding 9 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.) $)
    ad9antlr $p |- ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
      ( wa ad8antlr adantr ) CAMDMEMFMGMHMIMJMBKABCDEFGHIJLNO $.

    $( Deduction adding 10 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.) $)
    ad10antr $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
      ( wa ad9antr adantr ) ACNDNENFNGNHNINJNKNBLABCDEFGHIJKMOP $.

    $( Deduction adding 10 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.) $)
    ad10antlr $p |- ( ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
      ( wa ad9antlr adantr ) CANDNENFNGNHNINJNKNBLABCDEFGHIJKMOP $.
  $}

  ${
    ad2ant2.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Deduction adding two conjuncts to antecedent.  (Contributed by NM,
       8-Jan-2006.) $)
    ad2ant2l $p |- ( ( ( th /\ ph ) /\ ( ta /\ ps ) ) -> ch ) $=
      ( wa adantrl adantll ) AEBGCDABCEFHI $.

    $( Deduction adding two conjuncts to antecedent.  (Contributed by NM,
       8-Jan-2006.) $)
    ad2ant2r $p |- ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) -> ch ) $=
      ( wa adantrr adantlr ) ABEGCDABCEFHI $.

    $( Deduction adding two conjuncts to antecedent.  (Contributed by NM,
       23-Nov-2007.) $)
    ad2ant2lr $p |- ( ( ( th /\ ph ) /\ ( ps /\ ta ) ) -> ch ) $=
      ( wa adantrr adantll ) ABEGCDABCEFHI $.

    $( Deduction adding two conjuncts to antecedent.  (Contributed by NM,
       24-Nov-2007.) $)
    ad2ant2rl $p |- ( ( ( ph /\ th ) /\ ( ta /\ ps ) ) -> ch ) $=
      ( wa adantrl adantlr ) AEBGCDABCEFHI $.
  $}

  ${
    adantl3r.1 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $.
    $( Deduction adding 1 conjunct to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.) $)
    adantl3r $p |- ( ( ( ( ( ph /\ et ) /\ ps ) /\ ch ) /\ th ) -> ta ) $=
      ( wa id adantlr sylanl1 ) AFHBHABHZCDEABLFLIJGK $.
  $}

  ${
    ad4ant2.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad4ant13 $p |- ( ( ( ( ph /\ th ) /\ ps ) /\ ta ) -> ch ) $=
      ( wa adantr adantllr ) ABECDABGCEFHI $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad4ant14 $p |- ( ( ( ( ph /\ th ) /\ ta ) /\ ps ) -> ch ) $=
      ( wa adantlr ) ADGBCEABCDFHH $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad4ant23 $p |- ( ( ( ( th /\ ph ) /\ ps ) /\ ta ) -> ch ) $=
      ( wa adantr adantlll ) ABECDABGCEFHI $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad4ant24 $p |- ( ( ( ( th /\ ph ) /\ ta ) /\ ps ) -> ch ) $=
      ( adantlr adantlll ) AEBCDABCEFGH $.
  $}

  ${
    adantl4r.1 $e |- ( ( ( ( ( ph /\ si ) /\ rh ) /\ mu ) /\ la ) -> ka ) $.
    $( Deduction adding 1 conjunct to antecedent.  (Contributed by Thierry
       Arnoux, 11-Feb-2018.) $)
    adantl4r $p |- ( ( ( ( ( ( ph /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la )
      -> ka ) $=
      ( wa wi ex adantl3r imp ) ABICIDIEIFGACDEFGJBACIDIEIFGHKLM $.
  $}

  ${
    ad5ant2.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.) $)
    ad5ant12 $p |- ( ( ( ( ( ph /\ ps ) /\ th ) /\ ta ) /\ et ) -> ch ) $=
      ( wa ad3antrrr ) ABHCDEFGI $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant13 $p |- ( ( ( ( ( ph /\ th ) /\ ps ) /\ ta ) /\ et ) -> ch ) $=
      ( wa adantlr ad2antrr ) ADHBHCEFABCDGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant14 $p |- ( ( ( ( ( ph /\ th ) /\ ta ) /\ ps ) /\ et ) -> ch ) $=
      ( wa adantlr ad4ant13 ) ADHBCEFABCDGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant15 $p |- ( ( ( ( ( ph /\ th ) /\ ta ) /\ et ) /\ ps ) -> ch ) $=
      ( wa adantlr ad4ant14 ) ADHBCEFABCDGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant23 $p |- ( ( ( ( ( th /\ ph ) /\ ps ) /\ ta ) /\ et ) -> ch ) $=
      ( wa adantll ad2antrr ) DAHBHCEFABCDGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant24 $p |- ( ( ( ( ( th /\ ph ) /\ ta ) /\ ps ) /\ et ) -> ch ) $=
      ( wa adantll ad4ant13 ) DAHBCEFABCDGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant25 $p |- ( ( ( ( ( th /\ ph ) /\ ta ) /\ et ) /\ ps ) -> ch ) $=
      ( wa adantll ad4ant14 ) DAHBCEFABCDGIJ $.
  $}

  ${
    adantl5r.1 $e |- ( ( ( ( ( ( ph /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la )
      -> ka ) $.
    $( Deduction adding 1 conjunct to antecedent.  (Contributed by Thierry
       Arnoux, 11-Feb-2018.) $)
    adantl5r $p |- ( ( ( ( ( ( ( ph /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu )
      /\ la ) -> ka ) $=
      ( wa wi ex adantl4r imp ) ABJCJDJEJFJGHABCDEFGHKACJDJEJFJGHILMN $.
  $}

  ${
    adantl6r.1 $e |- ( ( ( ( ( ( ( ph /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu )
      /\ la ) -> ka ) $.
    $( Deduction adding 1 conjunct to antecedent.  (Contributed by Thierry
       Arnoux, 11-Feb-2018.) $)
    adantl6r $p |- ( ( ( ( ( ( ( ( ph /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh )
      /\ mu ) /\ la ) -> ka ) $=
      ( wa wi ex adantl5r imp ) ABKCKDKEKFKGKHIABCDEFGHILACKDKEKFKGKHIJMNO $.
  $}

  $( Simplification of a conjunction.  (Contributed by NM, 18-Mar-2007.) $)
  simpll $p |- ( ( ( ph /\ ps ) /\ ch ) -> ph ) $=
    ( id ad2antrr ) AABCADE $.

  ${
    simplld.1 $e |- ( ph -> ( ( ps /\ ch ) /\ th ) ) $.
    $( Deduction form of ~ simpll , eliminating a double conjunct.
       (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
    simplld $p |- ( ph -> ps ) $=
      ( wa simpld ) ABCABCFDEGG $.
  $}

  $( Simplification of a conjunction.  (Contributed by NM, 20-Mar-2007.) $)
  simplr $p |- ( ( ( ph /\ ps ) /\ ch ) -> ps ) $=
    ( id ad2antlr ) BBACBDE $.

  ${
    simplrd.1 $e |- ( ph -> ( ( ps /\ ch ) /\ th ) ) $.
    $( Deduction eliminating a double conjunct.  (Contributed by Glauco
       Siliprandi, 11-Dec-2019.) $)
    simplrd $p |- ( ph -> ch ) $=
      ( wa simpld simprd ) ABCABCFDEGH $.
  $}

  $( Simplification of a conjunction.  (Contributed by NM, 21-Mar-2007.) $)
  simprl $p |- ( ( ph /\ ( ps /\ ch ) ) -> ps ) $=
    ( id ad2antrl ) BBACBDE $.

  ${
    simprld.1 $e |- ( ph -> ( ps /\ ( ch /\ th ) ) ) $.
    $( Deduction eliminating a double conjunct.  (Contributed by Glauco
       Siliprandi, 11-Dec-2019.) $)
    simprld $p |- ( ph -> ch ) $=
      ( wa simprd simpld ) ACDABCDFEGH $.
  $}

  $( Simplification of a conjunction.  (Contributed by NM, 21-Mar-2007.) $)
  simprr $p |- ( ( ph /\ ( ps /\ ch ) ) -> ch ) $=
    ( id ad2antll ) CCABCDE $.

  ${
    simprrd.1 $e |- ( ph -> ( ps /\ ( ch /\ th ) ) ) $.
    $( Deduction form of ~ simprr , eliminating a double conjunct.
       (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
    simprrd $p |- ( ph -> th ) $=
      ( wa simprd ) ACDABCDFEGG $.
  $}

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.) $)
  simplll $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ph ) $=
    ( wa simpl ad2antrr ) ABEACDABFG $.

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.) $)
  simpllr $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ps ) $=
    ( wa simpr ad2antrr ) ABEBCDABFG $.

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.) $)
  simplrl $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ps ) $=
    ( wa simpl ad2antlr ) BCEBADBCFG $.

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.) $)
  simplrr $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ch ) $=
    ( wa simpr ad2antlr ) BCECADBCFG $.

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.) $)
  simprll $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ps ) $=
    ( wa simpl ad2antrl ) BCEBADBCFG $.

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.) $)
  simprlr $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ch ) $=
    ( wa simpr ad2antrl ) BCECADBCFG $.

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.) $)
  simprrl $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ch ) $=
    ( wa simpl ad2antll ) CDECABCDFG $.

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.) $)
  simprrr $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> th ) $=
    ( wa simpr ad2antll ) CDEDABCDFG $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-4l $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta ) -> ph ) $=
    ( wa simplll adantr ) ABFCFDFAEABCDGH $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-4r $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta ) -> ps ) $=
    ( wa simpllr adantr ) ABFCFDFBEABCDGH $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-5l $p |- ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) -> ph ) $=
    ( wa simp-4l adantr ) ABGCGDGEGAFABCDEHI $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-5r $p |- ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) -> ps ) $=
    ( wa simp-4r adantr ) ABGCGDGEGBFABCDEHI $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-6l $p |- ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) -> ph ) $=
    ( wa simp-5l adantr ) ABHCHDHEHFHAGABCDEFIJ $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-6r $p |- ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) -> ps ) $=
    ( wa simp-5r adantr ) ABHCHDHEHFHBGABCDEFIJ $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-7l $p |- ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) -> ph ) $=
    ( wa simp-6l adantr ) ABICIDIEIFIGIAHABCDEFGJK $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-7r $p |- ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) -> ps ) $=
    ( wa simp-6r adantr ) ABICIDIEIFIGIBHABCDEFGJK $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-8l $p |- ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) -> ph ) $=
    ( wa simp-7l adantr ) ABJCJDJEJFJGJHJAIABCDEFGHKL $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-8r $p |- ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) -> ps ) $=
    ( wa simp-7r adantr ) ABJCJDJEJFJGJHJBIABCDEFGHKL $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-9l $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ph ) $=
    ( wa simp-8l adantr ) ABKCKDKEKFKGKHKIKAJABCDEFGHILM $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-9r $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
    ( wa simp-8r adantr ) ABKCKDKEKFKGKHKIKBJABCDEFGHILM $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-10l $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ph ) $=
    ( wa simp-9l adantr ) ABLCLDLELFLGLHLILJLAKABCDEFGHIJMN $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-10r $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
    ( wa simp-9r adantr ) ABLCLDLELFLGLHLILJLBKABCDEFGHIJMN $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-11l $p |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ph ) $=
    ( wa simp-10l adantr ) ABMCMDMEMFMGMHMIMJMKMALABCDEFGHIJKNO $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.) $)
  simp-11r $p |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
    ( wa simp-10r adantr ) ABMCMDMEMFMGMHMIMJMKMBLABCDEFGHIJKNO $.

  $( Theorem *4.87 of [WhiteheadRussell] p. 122.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Eric Schmidt, 26-Oct-2006.) $)
  pm4.87 $p |- ( ( ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) /\
                ( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) ) ) /\
                ( ( ps -> ( ph -> ch ) ) <-> ( ( ps /\ ph ) -> ch ) ) ) $=
    ( wa wi wb impexp bi2.04 pm3.2i bicomi ) ABDCEABCEEZFZKBACEEZFZDMBADCEZFLNA
    BCGABCHIOMBACGJI $.

  ${
    a2and.1 $e |- ( ph -> ( ( ps /\ rh ) -> ( ta -> th ) ) ) $.
    a2and.2 $e |- ( ph -> ( ( ps /\ rh ) -> ch ) ) $.
    $( Deduction distributing a conjunction as embedded antecedent.
       (Contributed by AV, 25-Oct-2019.)  (Proof shortened by Wolf Lammen,
       19-Jan-2020.) $)
    a2and $p |- ( ph -> ( ( ( ps /\ ch ) -> ta )
                            -> ( ( ps /\ rh ) -> th ) ) ) $=
      ( wa wi expd imdistand imp embantd ex com23 ) ABFIZBCIZEJZDAQSDJAQIREDAQR
      ABFCABFCHKLMAQEDJGMNOP $.
  $}

  ${
    animpimp2impd.1 $e |- ( ( ps /\ ph ) -> ( ch -> ( th -> et ) ) ) $.
    animpimp2impd.2 $e |- ( ( ps /\ ( ph /\ th ) ) -> ( et -> ta ) ) $.
    $( Deduction deriving nested implications from conjunctions.  (Contributed
       by AV, 21-Aug-2022.) $)
    animpimp2impd $p |- ( ph -> ( ( ps -> ch ) -> ( ps -> ( th -> ta ) ) ) ) $=
      ( wi wa expr a2d syld expcom ) ABCDEIZBACOIBAJZCDFIOGPDFEBADFEIHKLMNL $.
  $}

  $( Introduce one conjunct as an antecedent to the other.  "abai" stands for
     "and, biconditional, and, implication".  (Contributed by NM, 12-Aug-1993.)
     (Proof shortened by Wolf Lammen, 7-Dec-2012.) $)
  abai $p |- ( ( ph /\ ps ) <-> ( ph /\ ( ph -> ps ) ) ) $=
    ( wi biimt pm5.32i ) ABABCABDE $.

  $( Swap two conjuncts.  Note that the first digit (1) in the label refers to
     the outer conjunct position, and the next digit (2) to the inner conjunct
     position.  (Contributed by NM, 12-Mar-1995.) $)
  an12 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
    ( wa ancom anbi1i anass 3bitr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH $.

  $( A rearrangement of conjuncts.  (Contributed by NM, 12-Mar-1995.)  (Proof
     shortened by Wolf Lammen, 25-Dec-2012.) $)
  an32 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ph /\ ch ) /\ ps ) ) $=
    ( wa anass an12 ancom 3bitri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.

  $( A rearrangement of conjuncts.  (Contributed by NM, 24-Jun-2012.)  (Proof
     shortened by Wolf Lammen, 31-Dec-2012.) $)
  an13 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ch /\ ( ps /\ ph ) ) ) $=
    ( wa an12 anass ancom 3bitr2i ) ABCDDBACDDBADZCDCIDABCEBACFICGH $.

  $( A rearrangement of conjuncts.  (Contributed by NM, 24-Jun-2012.)  (Proof
     shortened by Wolf Lammen, 31-Dec-2012.) $)
  an31 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ch /\ ps ) /\ ph ) ) $=
    ( wa an13 anass 3bitr4i ) ABCDDCBADDABDCDCBDADABCEABCFCBAFG $.

  ${
    an12s.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( Swap two conjuncts in antecedent.  The label suffix "s" means that
       ~ an12 is combined with ~ syl (or a variant).  (Contributed by NM,
       13-Mar-1996.) $)
    an12s $p |- ( ( ps /\ ( ph /\ ch ) ) -> th ) $=
      ( wa an12 sylbi ) BACFFABCFFDBACGEH $.

    $( Inference commuting a nested conjunction in antecedent.  (Contributed by
       NM, 24-May-2006.)  (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
    ancom2s $p |- ( ( ph /\ ( ch /\ ps ) ) -> th ) $=
      ( wa pm3.22 sylan2 ) CBFABCFDCBGEH $.

    $( Swap two conjuncts in antecedent.  (Contributed by NM, 31-May-2006.) $)
    an13s $p |- ( ( ch /\ ( ps /\ ph ) ) -> th ) $=
      ( exp32 com13 imp32 ) CBADABCDABCDEFGH $.
  $}

  ${
    an32s.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( Swap two conjuncts in antecedent.  (Contributed by NM, 13-Mar-1996.) $)
    an32s $p |- ( ( ( ph /\ ch ) /\ ps ) -> th ) $=
      ( wa an32 sylbi ) ACFBFABFCFDACBGEH $.

    $( Inference commuting a nested conjunction in antecedent.  (Contributed by
       NM, 24-May-2006.)  (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
    ancom1s $p |- ( ( ( ps /\ ph ) /\ ch ) -> th ) $=
      ( wa pm3.22 sylan ) BAFABFCDBAGEH $.

    $( Swap two conjuncts in antecedent.  (Contributed by NM, 31-May-2006.) $)
    an31s $p |- ( ( ( ch /\ ps ) /\ ph ) -> th ) $=
      ( exp31 com13 imp31 ) CBADABCDABCDEFGH $.
  $}

  ${
    anass1rs.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( Commutative-associative law for conjunction in an antecedent.
       (Contributed by Jeff Madsen, 19-Jun-2011.) $)
    anass1rs $p |- ( ( ( ph /\ ch ) /\ ps ) -> th ) $=
      ( anassrs an32s ) ABCDABCDEFG $.
  $}

  $( Absorption into embedded conjunct.  (Contributed by NM, 4-Sep-1995.)
     (Proof shortened by Wolf Lammen, 16-Nov-2013.) $)
  anabs1 $p |- ( ( ( ph /\ ps ) /\ ph ) <-> ( ph /\ ps ) ) $=
    ( wa simpl pm4.71i bicomi ) ABCZGACGAABDEF $.

  $( Absorption into embedded conjunct.  (Contributed by NM, 20-Jul-1996.)
     (Proof shortened by Wolf Lammen, 9-Dec-2012.) $)
  anabs5 $p |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) ) $=
    ( wa ibar bicomd pm5.32i ) AABCZBABGABDEF $.

  $( Absorption into embedded conjunct.  (Contributed by NM, 20-Jul-1996.)
     (Proof shortened by Wolf Lammen, 17-Nov-2013.) $)
  anabs7 $p |- ( ( ps /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) ) $=
    ( wa simpr pm4.71ri bicomi ) ABCZBGCGBABDEF $.

  ${
    anabsan.1 $e |- ( ( ( ph /\ ph ) /\ ps ) -> ch ) $.
    $( Absorption of antecedent with conjunction.  (Contributed by NM,
       24-Mar-1996.)  (Revised by NM, 18-Nov-2013.) $)
    anabsan $p |- ( ( ph /\ ps ) -> ch ) $=
      ( wa pm4.24 sylanb ) AAAEBCAFDG $.
  $}

  ${
    anabss1.1 $e |- ( ( ( ph /\ ps ) /\ ph ) -> ch ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       20-Jul-1996.)  (Proof shortened by Wolf Lammen, 31-Dec-2012.) $)
    anabss1 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( an32s anabsan ) ABCABACDEF $.
  $}

  ${
    anabss4.1 $e |- ( ( ( ps /\ ph ) /\ ps ) -> ch ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       20-Jul-1996.) $)
    anabss4 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( anabss1 ancoms ) BACBACDEF $.
  $}

  ${
    anabss5.1 $e |- ( ( ph /\ ( ph /\ ps ) ) -> ch ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       10-May-1994.)  (Proof shortened by Wolf Lammen, 1-Jan-2013.) $)
    anabss5 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( anassrs anabsan ) ABCAABCDEF $.
  $}

  ${
    anabsi5.1 $e |- ( ph -> ( ( ph /\ ps ) -> ch ) ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       11-Jun-1995.)  (Proof shortened by Wolf Lammen, 18-Nov-2013.) $)
    anabsi5 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( wa imp anabss5 ) ABCAABECDFG $.
  $}

  ${
    anabsi6.1 $e |- ( ph -> ( ( ps /\ ph ) -> ch ) ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       14-Aug-2000.) $)
    anabsi6 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( ancomsd anabsi5 ) ABCABACDEF $.
  $}

  ${
    anabsi7.1 $e |- ( ps -> ( ( ph /\ ps ) -> ch ) ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       20-Jul-1996.)  (Proof shortened by Wolf Lammen, 18-Nov-2013.) $)
    anabsi7 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( anabsi6 ancoms ) BACBACDEF $.
  $}

  ${
    anabsi8.1 $e |- ( ps -> ( ( ps /\ ph ) -> ch ) ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       26-Sep-1999.) $)
    anabsi8 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( anabsi5 ancoms ) BACBACDEF $.
  $}

  ${
    anabss7.1 $e |- ( ( ps /\ ( ph /\ ps ) ) -> ch ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       20-Jul-1996.)  (Proof shortened by Wolf Lammen, 19-Nov-2013.) $)
    anabss7 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( anassrs anabss4 ) ABCBABCDEF $.
  $}

  ${
    anabsan2.1 $e |- ( ( ph /\ ( ps /\ ps ) ) -> ch ) $.
    $( Absorption of antecedent with conjunction.  (Contributed by NM,
       10-May-2004.)  (Revised by NM, 1-Jan-2013.) $)
    anabsan2 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( an12s anabss7 ) ABCABBCDEF $.
  $}

  ${
    anabss3.1 $e |- ( ( ( ph /\ ps ) /\ ps ) -> ch ) $.
    $( Absorption of antecedent into conjunction.  (Contributed by NM,
       20-Jul-1996.)  (Proof shortened by Wolf Lammen, 1-Jan-2013.) $)
    anabss3 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( anasss anabsan2 ) ABCABBCDEF $.
  $}

  $( Rearrangement of 4 conjuncts.  (Contributed by NM, 10-Jul-1994.) $)
  an4 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
              ( ( ph /\ ch ) /\ ( ps /\ th ) ) ) $=
    ( wa an12 anbi2i anass 3bitr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHAC
    LHI $.

  $( Rearrangement of 4 conjuncts.  (Contributed by NM, 7-Feb-1996.) $)
  an42 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
                 ( ( ph /\ ch ) /\ ( th /\ ps ) ) ) $=
    ( wa an4 ancom anbi2i bitri ) ABECDEEACEZBDEZEJDBEZEABCDFKLJBDGHI $.

  $( Rearrangement of 4 conjuncts.  (Contributed by Rodolfo Medina,
     24-Sep-2010.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
  an43 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
               ( ( ph /\ th ) /\ ( ps /\ ch ) ) ) $=
    ( wa an42 bicomi ) ADEBCEEABECDEEADBCFG $.

  $( A rearrangement of conjuncts.  (Contributed by Rodolfo Medina,
     25-Sep-2010.) $)
  an3 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ( ph /\ th ) ) $=
    ( wa an43 simplbi ) ABECDEEADEBCEABCDFG $.

  ${
    an4s.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
    $( Inference rearranging 4 conjuncts in antecedent.  (Contributed by NM,
       10-Aug-1995.) $)
    an4s $p |- ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) -> ta ) $=
      ( wa an4 sylbi ) ACGBDGGABGCDGGEACBDHFI $.
  $}

  ${
    an41r3s.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
    $( Inference rearranging 4 conjuncts in antecedent.  (Contributed by NM,
       10-Aug-1995.) $)
    an42s $p |- ( ( ( ph /\ ch ) /\ ( th /\ ps ) ) -> ta ) $=
      ( wa an4s ancom2s ) ACGBDEABCDEFHI $.
  $}

  $( Distribution of conjunction over conjunction.  (Contributed by NM,
     14-Aug-1995.) $)
  anandi $p |- ( ( ph /\ ( ps /\ ch ) ) <->
               ( ( ph /\ ps ) /\ ( ph /\ ch ) ) ) $=
    ( wa anidm anbi1i an4 bitr3i ) ABCDZDAADZIDABDACDDJAIAEFAABCGH $.

  $( Distribution of conjunction over conjunction.  (Contributed by NM,
     24-Aug-1995.) $)
  anandir $p |- ( ( ( ph /\ ps ) /\ ch ) <->
               ( ( ph /\ ch ) /\ ( ps /\ ch ) ) ) $=
    ( wa anidm anbi2i an4 bitr3i ) ABDZCDICCDZDACDBCDDJCICEFABCCGH $.

  ${
    anandis.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) -> ta ) $.
    $( Inference that undistributes conjunction in the antecedent.
       (Contributed by NM, 7-Jun-2004.) $)
    anandis $p |- ( ( ph /\ ( ps /\ ch ) ) -> ta ) $=
      ( wa an4s anabsan ) ABCFDABACDEGH $.
  $}

  ${
    anandirs.1 $e |- ( ( ( ph /\ ch ) /\ ( ps /\ ch ) ) -> ta ) $.
    $( Inference that undistributes conjunction in the antecedent.
       (Contributed by NM, 7-Jun-2004.) $)
    anandirs $p |- ( ( ( ph /\ ps ) /\ ch ) -> ta ) $=
      ( wa an4s anabsan2 ) ABFCDACBCDEGH $.
  $}

  ${
    syl2an2.1 $e |- ( ph -> ps ) $.
    syl2an2.2 $e |- ( ( ch /\ ph ) -> th ) $.
    syl2an2.3 $e |- ( ( ps /\ th ) -> ta ) $.
    $( ~ syl2an with antecedents in standard conjunction form.  (Contributed by
       Alan Sare, 27-Aug-2016.) $)
    syl2an2 $p |- ( ( ch /\ ph ) -> ta ) $=
      ( wa syl2an anabss7 ) CAEABDECAIFGHJK $.
  $}

  ${
    syl2an2r.1 $e |- ( ph -> ps ) $.
    syl2an2r.2 $e |- ( ( ph /\ ch ) -> th ) $.
    syl2an2r.3 $e |- ( ( ps /\ th ) -> ta ) $.
    $( ~ syl2anr with antecedents in standard conjunction form.  (Contributed
       by Alan Sare, 27-Aug-2016.) $)
    syl2an2r $p |- ( ( ph /\ ch ) -> ta ) $=
      ( wa syl2an anabss5 ) ACEABDEACIFGHJK $.
  $}

  ${
    impbida.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    impbida.2 $e |- ( ( ph /\ ch ) -> ps ) $.
    $( Deduce an equivalence from two implications.  (Contributed by NM,
       17-Feb-2007.) $)
    impbida $p |- ( ph -> ( ps <-> ch ) ) $=
      ( ex impbid ) ABCABCDFACBEFG $.
  $}

  $( Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.45 $p |- ( ( ph -> ps ) -> ( ( ph /\ ch ) -> ( ps /\ ch ) ) ) $=
    ( wi id anim1d ) ABDZABCGEF $.

  ${
    im2an9.1 $e |- ( ph -> ( ps -> ch ) ) $.
    im2an9.2 $e |- ( th -> ( ta -> et ) ) $.
    $( Deduction joining nested implications to form implication of
       conjunctions.  (Contributed by NM, 29-Feb-1996.) $)
    im2anan9 $p |- ( ( ph /\ th ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) ) $=
      ( wa wi adantr adantl anim12d ) ADIBCEFABCJDGKDEFJAHLM $.

    $( Deduction joining nested implications to form implication of
       conjunctions.  (Contributed by NM, 29-Feb-1996.) $)
    im2anan9r $p |- ( ( th /\ ph ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) ) $=
      ( wa wi im2anan9 ancoms ) ADBEICFIJABCDEFGHKL $.
  $}

  ${
    anim12dan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    anim12dan.2 $e |- ( ( ph /\ th ) -> ta ) $.
    $( Conjoin antecedents and consequents in a deduction.  (Contributed by
       Mario Carneiro, 12-May-2014.) $)
    anim12dan $p |- ( ( ph /\ ( ps /\ th ) ) -> ( ch /\ ta ) ) $=
      ( wa ex anim12d imp ) ABDHCEHABCDEABCFIADEGIJK $.
  $}

  $( Two propositions are equivalent if they are both true.  Theorem *5.1 of
     [WhiteheadRussell] p. 123.  (Contributed by NM, 21-May-1994.) $)
  pm5.1 $p |- ( ( ph /\ ps ) -> ( ph <-> ps ) ) $=
    ( wb pm5.501 biimpa ) ABABCABDE $.

  $( Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113.  (Contributed by NM,
     3-Jan-2005.)  (Revised by NM, 27-Nov-2013.) $)
  pm3.43 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) ->
                ( ph -> ( ps /\ ch ) ) ) $=
    ( wi wa pm3.43i imp ) ABDACDABCEDABCFG $.

  $( Distributive law for implication over conjunction.  Compare Theorem *4.76
     of [WhiteheadRussell] p. 121.  (Contributed by NM, 3-Apr-1994.)  (Proof
     shortened by Wolf Lammen, 27-Nov-2013.) $)
  jcab $p |- ( ( ph -> ( ps /\ ch ) ) <->
                ( ( ph -> ps ) /\ ( ph -> ch ) ) ) $=
    ( wa wi simpl imim2i simpr jca pm3.43 impbii ) ABCDZEZABEZACEZDMNOLBABCFGLC
    ABCHGIABCJK $.

  $( Theorem *4.76 of [WhiteheadRussell] p. 121.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.76 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) <->
                ( ph -> ( ps /\ ch ) ) ) $=
    ( wa wi jcab bicomi ) ABCDEABEACEDABCFG $.

  $( Theorem *4.38 of [WhiteheadRussell] p. 118.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.38 $p |- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) ->
                ( ( ph /\ ps ) <-> ( ch /\ th ) ) ) $=
    ( wb wa simpl simpr anbi12d ) ACEZBDEZFACBDJKGJKHI $.

  ${
    bi2an9.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bi2an9.2 $e |- ( th -> ( ta <-> et ) ) $.
    $( Deduction joining two equivalences to form equivalence of conjunctions.
       (Contributed by NM, 31-Jul-1995.) $)
    bi2anan9 $p |- ( ( ph /\ th ) -> ( ( ps /\ ta ) <-> ( ch /\ et ) ) ) $=
      ( wa anbi1d anbi2d sylan9bb ) ABEICEIDCFIABCEGJDEFCHKL $.

    $( Deduction joining two equivalences to form equivalence of conjunctions.
       (Contributed by NM, 19-Feb-1996.) $)
    bi2anan9r $p |- ( ( th /\ ph ) -> ( ( ps /\ ta ) <-> ( ch /\ et ) ) ) $=
      ( wa wb bi2anan9 ancoms ) ADBEICFIJABCDEFGHKL $.

    $( Deduction joining two biconditionals with different antecedents.
       (Contributed by NM, 12-May-2004.) $)
    bi2bian9 $p |- ( ( ph /\ th ) -> ( ( ps <-> ta ) <-> ( ch <-> et ) ) ) $=
      ( wa wb adantr adantl bibi12d ) ADIBCEFABCJDGKDEFJAHLM $.
  $}

  $( Theorem *5.33 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.33 $p |- ( ( ph /\ ( ps -> ch ) ) <->
                ( ph /\ ( ( ph /\ ps ) -> ch ) ) ) $=
    ( wi wa ibar imbi1d pm5.32i ) ABCDABEZCDABICABFGH $.

  $( Theorem *5.36 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.36 $p |- ( ( ph /\ ( ph <-> ps ) ) <-> ( ps /\ ( ph <-> ps ) ) ) $=
    ( wb id pm5.32ri ) ABCZABFDE $.

  ${
    bianabs.1 $e |- ( ph -> ( ps <-> ( ph /\ ch ) ) ) $.
    $( Absorb a hypothesis into the second member of a biconditional.
       (Contributed by FL, 15-Feb-2007.) $)
    bianabs $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wa ibar bitr4d ) ABACECDACFG $.
  $}

  ${
    biadani.1 $e |- ( ph -> ps ) $.
    $( An implication implies to the equivalence of some implied equivalence
       and some other equivalence involving a conjunction.  (Contributed by BJ,
       4-Mar-2023.) $)
    biadani $p |- ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) ) $=
      ( wb wi wa pm5.32 pm4.71ri bibi1i bitr4i ) BACEFBAGZBCGZEAMEBACHALMABDIJK
      $.

    biadanii.2 $e |- ( ps -> ( ph <-> ch ) ) $.
    $( Inference associated with ~ biadani .  Add a conjunction to an
       equivalence.  (Contributed by Jeff Madsen, 20-Jun-2011.)  (Proof
       shortened by BJ, 4-Mar-2023.) $)
    biadanii $p |- ( ph <-> ( ps /\ ch ) ) $=
      ( wb wi wa biadani mpbi ) BACFGABCHFEABCDIJ $.
  $}

  ${
    biadanid.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    biadanid.2 $e |- ( ( ph /\ ch ) -> ( ps <-> th ) ) $.
    $( Deduction associated with ~ biadani .  Add a conjunction to an
       equivalence.  (Contributed by Thierry Arnoux, 16-Jun-2024.) $)
    biadanid $p |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $=
      ( wa biimpa an32s mpdan jca biimpar anasss impbida ) ABCDGABGZCDEOCDEACBD
      ACGZBDFHIJKACDBPBDFLMN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical negation (intuitionistic)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( 'Not' introduction.  One of the axioms of propositional logic.
     (Contributed by Mario Carneiro, 31-Jan-2015.)  Use its alias ~ pm2.01
     instead.  (New usage is discouraged.) $)
  ax-in1 $a |- ( ( ph -> -. ph ) -> -. ph ) $.

  $( 'Not' elimination.  One of the axioms of propositional logic.
     (Contributed by Mario Carneiro, 31-Jan-2015.) $)
  ax-in2 $a |- ( -. ph -> ( ph -> ps ) ) $.

  $( Reductio ad absurdum.  Theorem *2.01 of [WhiteheadRussell] p. 100.  This
     is valid intuitionistically (in the terminology of Section 1.2 of [Bauer]
     p. 482 it is a proof of negation not a proof by contradiction); compare
     with ~ pm2.18dc which only holds for some propositions.  Also called weak
     Clavius law.  (Contributed by Mario Carneiro, 12-May-2015.) $)
  pm2.01 $p |- ( ( ph -> -. ph ) -> -. ph ) $=
    ( ax-in1 ) AB $.

  $( From a wff and its negation, anything is true.  Theorem *2.21 of
     [WhiteheadRussell] p. 104.  Also called the Duns Scotus law.  (Contributed
     by Mario Carneiro, 12-May-2015.) $)
  pm2.21 $p |- ( -. ph -> ( ph -> ps ) ) $=
    ( ax-in2 ) ABC $.

  ${
    pm2.01d.1 $e |- ( ph -> ( ps -> -. ps ) ) $.
    $( Deduction based on reductio ad absurdum.  (Contributed by NM,
       18-Aug-1993.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
    pm2.01d $p |- ( ph -> -. ps ) $=
      ( wn wi pm2.01 syl ) ABBDZEHCBFG $.
  $}

  ${
    pm2.21d.1 $e |- ( ph -> -. ps ) $.
    $( A contradiction implies anything.  Deduction from ~ pm2.21 .
       (Contributed by NM, 10-Feb-1996.) $)
    pm2.21d $p |- ( ph -> ( ps -> ch ) ) $=
      ( wn wi pm2.21 syl ) ABEBCFDBCGH $.
  $}

  ${
    pm2.21dd.1 $e |- ( ph -> ps ) $.
    pm2.21dd.2 $e |- ( ph -> -. ps ) $.
    $( A contradiction implies anything.  Deduction from ~ pm2.21 .
       (Contributed by Mario Carneiro, 9-Feb-2017.) $)
    pm2.21dd $p |- ( ph -> ch ) $=
      ( pm2.21d mpd ) ABCDABCEFG $.
  $}

  $( Theorem *2.24 of [WhiteheadRussell] p. 104.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.24 $p |- ( ph -> ( -. ph -> ps ) ) $=
    ( wn pm2.21 com12 ) ACABABDE $.

  ${
    pm2.24d.1 $e |- ( ph -> ps ) $.
    $( Deduction version of ~ pm2.24 .  (Contributed by NM, 30-Jan-2006.)
       (Revised by Mario Carneiro, 31-Jan-2015.) $)
    pm2.24d $p |- ( ph -> ( -. ps -> ch ) ) $=
      ( wn wi pm2.24 syl ) ABBECFDBCGH $.
  $}

  ${
    pm2.24i.1 $e |- ph $.
    $( Inference version of ~ pm2.24 .  (Contributed by NM, 20-Aug-2001.)
       (Revised by Mario Carneiro, 31-Jan-2015.) $)
    pm2.24i $p |- ( -. ph -> ps ) $=
      ( wn pm2.21 mpi ) ADABCABEF $.
  $}

  ${
    con2d.1 $e |- ( ph -> ( ps -> -. ch ) ) $.
    $( A contraposition deduction.  (Contributed by NM, 19-Aug-1993.)  (Revised
       by NM, 12-Feb-2013.) $)
    con2d $p |- ( ph -> ( ch -> -. ps ) ) $=
      ( wn wi ax-in2 syl6 com23 pm2.01 ) ACBBEZFKABCKABCECKFDCKGHIBJH $.
  $}

  ${
    mt2d.1 $e |- ( ph -> ch ) $.
    mt2d.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
    $( Modus tollens deduction.  (Contributed by NM, 4-Jul-1994.) $)
    mt2d $p |- ( ph -> -. ps ) $=
      ( wn con2d mpd ) ACBFDABCEGH $.
  $}

  ${
    nsyl3.1 $e |- ( ph -> -. ps ) $.
    nsyl3.2 $e |- ( ch -> ps ) $.
    $( A negated syllogism inference.  (Contributed by NM, 1-Dec-1995.)
       (Revised by NM, 13-Jun-2013.) $)
    nsyl3 $p |- ( ch -> -. ph ) $=
      ( wn wi a1i mt2d ) CABEABFGCDHI $.
  $}

  ${
    con2i.a $e |- ( ph -> -. ps ) $.
    $( A contraposition inference.  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by O'Cat, 28-Nov-2008.)  (Proof shortened by Wolf Lammen,
       13-Jun-2013.) $)
    con2i $p |- ( ps -> -. ph ) $=
      ( id nsyl3 ) ABBCBDE $.
  $}

  ${
    nsyl.1 $e |- ( ph -> -. ps ) $.
    nsyl.2 $e |- ( ch -> ps ) $.
    $( A negated syllogism inference.  (Contributed by NM, 31-Dec-1993.)
       (Proof shortened by Wolf Lammen, 2-Mar-2013.) $)
    nsyl $p |- ( ph -> -. ch ) $=
      ( nsyl3 con2i ) CAABCDEFG $.
  $}

  $( Double negation introduction.  Theorem *2.12 of [WhiteheadRussell] p. 101.
     The converse need not hold.  It holds exactly for stable propositions (by
     definition, see ~ df-stab ) and in particular for decidable propositions
     (see ~ notnotrdc ).  See also ~ notnotnot .  (Contributed by NM,
     28-Dec-1992.)  (Proof shortened by Wolf Lammen, 2-Mar-2013.) $)
  notnot $p |- ( ph -> -. -. ph ) $=
    ( wn id con2i ) ABZAECD $.

  ${
    notnotd.1 $e |- ( ph -> ps ) $.
    $( Deduction associated with ~ notnot and ~ notnoti .  (Contributed by
       Jarvin Udandy, 2-Sep-2016.)  Avoid biconditional.  (Revised by Wolf
       Lammen, 27-Mar-2021.) $)
    notnotd $p |- ( ph -> -. -. ps ) $=
      ( wn notnot syl ) ABBDDCBEF $.
  $}

  ${
    con3d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( A contraposition deduction.  (Contributed by NM, 5-Aug-1993.)  (Revised
       by NM, 31-Jan-2015.) $)
    con3d $p |- ( ph -> ( -. ch -> -. ps ) ) $=
      ( wn notnot syl6 con2d ) ABCEZABCIEDCFGH $.
  $}

  ${
    con3i.a $e |- ( ph -> ps ) $.
    $( A contraposition inference.  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Wolf Lammen, 20-Jun-2013.) $)
    con3i $p |- ( -. ps -> -. ph ) $=
      ( wn id nsyl ) BDZBAGECF $.
  $}

  ${
    con3rr3.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Rotate through consequent right.  (Contributed by Wolf Lammen,
       3-Nov-2013.) $)
    con3rr3 $p |- ( -. ch -> ( ph -> -. ps ) ) $=
      ( wn con3d com12 ) ACEBEABCDFG $.
  $}

  $( Triple negation is equivalent to negation.  (Contributed by Jim Kingdon,
     28-Jul-2018.) $)
  notnotnot $p |- ( -. -. -. ph <-> -. ph ) $=
    ( wn notnot con3i impbii ) ABZBZBFAGACDFCE $.

  ${
    con3dimp.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Variant of ~ con3d with importation.  (Contributed by Jonathan Ben-Naim,
       3-Jun-2011.) $)
    con3dimp $p |- ( ( ph /\ -. ch ) -> -. ps ) $=
      ( wn con3d imp ) ACEBEABCDFG $.
  $}

  ${
    pm2.01da.1 $e |- ( ( ph /\ ps ) -> -. ps ) $.
    $( Deduction based on reductio ad absurdum.  (Contributed by Mario
       Carneiro, 9-Feb-2017.) $)
    pm2.01da $p |- ( ph -> -. ps ) $=
      ( wn ex pm2.01d ) ABABBDCEF $.
  $}

  $( In classical logic, this is just a restatement of ~ pm3.2 .  In
     intuitionistic logic, it still holds, but is weaker than pm3.2.
     (Contributed by Mario Carneiro, 12-May-2015.) $)
  pm3.2im $p |- ( ph -> ( ps -> -. ( ph -> -. ps ) ) ) $=
    ( wn wi pm2.27 con2d ) AABCZDBAGEF $.

  ${
    expi.1 $e |- ( -. ( ph -> -. ps ) -> ch ) $.
    $( An exportation inference.  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by O'Cat, 28-Nov-2008.) $)
    expi $p |- ( ph -> ( ps -> ch ) ) $=
      ( wn wi pm3.2im syl6 ) ABABEFECABGDH $.
  $}

  ${
    pm2.65i.1 $e |- ( ph -> ps ) $.
    pm2.65i.2 $e |- ( ph -> -. ps ) $.
    $( Inference for proof by contradiction.  (Contributed by NM, 18-May-1994.)
       (Proof shortened by Wolf Lammen, 11-Sep-2013.) $)
    pm2.65i $p |- -. ph $=
      ( wn wi nsyl3 pm2.01 ax-mp ) AAEZFJABADCGAHI $.
  $}

  ${
    mt2.1 $e |- ps $.
    mt2.2 $e |- ( ph -> -. ps ) $.
    $( A rule similar to modus tollens.  (Contributed by NM, 19-Aug-1993.)
       (Proof shortened by Wolf Lammen, 10-Sep-2013.) $)
    mt2 $p |- -. ph $=
      ( a1i pm2.65i ) ABBACEDF $.
  $}

  $( Theorem used to justify definition of intuitionistic biconditional
     ~ df-bi .  (Contributed by NM, 24-Nov-2017.) $)
  biijust $p  |- ( ( ( ( ph -> ps ) /\ ( ps -> ph ) )
        -> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
    /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) )
           -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) ) $=
    ( wi wa id pm3.2i ) ABCBACDZGCZHGEZIF $.

  $( Contraposition.  Theorem *2.16 of [WhiteheadRussell] p. 103.  (Contributed
     by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 13-Feb-2013.) $)
  con3 $p |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) ) $=
    ( wi id con3d ) ABCZABFDE $.

  $( Contraposition.  Theorem *2.03 of [WhiteheadRussell] p. 100.  (Contributed
     by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
  con2 $p |- ( ( ph -> -. ps ) -> ( ps -> -. ph ) ) $=
    ( wn wi id con2d ) ABCDZABGEF $.

  ${
    mt2i.1 $e |- ch $.
    mt2i.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
    $( Modus tollens inference.  (Contributed by NM, 26-Mar-1995.)  (Proof
       shortened by Wolf Lammen, 15-Sep-2012.) $)
    mt2i $p |- ( ph -> -. ps ) $=
      ( a1i mt2d ) ABCCADFEG $.
  $}

  ${
    negbi.1 $e |- ph $.
    $( Infer double negation.  (Contributed by NM, 27-Feb-2008.) $)
    notnoti $p |- -. -. ph $=
      ( wn notnot ax-mp ) AACCBADE $.
  $}

  ${
    pm2.21i.1 $e |- -. ph $.
    $( A contradiction implies anything.  Inference from ~ pm2.21 .
       (Contributed by NM, 16-Sep-1993.)  (Revised by Mario Carneiro,
       31-Jan-2015.) $)
    pm2.21i $p |- ( ph -> ps ) $=
      ( wn wi pm2.21 ax-mp ) ADABECABFG $.
  $}

  ${
    pm2.24ii.1 $e |- ph $.
    pm2.24ii.2 $e |- -. ph $.
    $( A contradiction implies anything.  Inference from ~ pm2.24 .
       (Contributed by NM, 27-Feb-2008.) $)
    pm2.24ii $p |- ps $=
      ( pm2.21i ax-mp ) ABCABDEF $.
  $}

  ${
    nsyld.1 $e |- ( ph -> ( ps -> -. ch ) ) $.
    nsyld.2 $e |- ( ph -> ( ta -> ch ) ) $.
    $( A negated syllogism deduction.  (Contributed by NM, 9-Apr-2005.) $)
    nsyld $p |- ( ph -> ( ps -> -. ta ) ) $=
      ( wn con3d syld ) ABCGDGEADCFHI $.
  $}

  ${
    nsyli.1 $e |- ( ph -> ( ps -> ch ) ) $.
    nsyli.2 $e |- ( th -> -. ch ) $.
    $( A negated syllogism inference.  (Contributed by NM, 3-May-1994.) $)
    nsyli $p |- ( ph -> ( th -> -. ps ) ) $=
      ( wn con3d syl5 ) DCGABGFABCEHI $.
  $}

  ${
    nsyl4.1 $e |- ( ph -> ps ) $.
    nsyl4.2 $e |- ( -. ph -> ch ) $.
    $( A negated syllogism inference.  (Contributed by Wolf Lammen,
       20-May-2024.) $)
    nsyl5 $p |- ( -. ps -> ch ) $=
      ( wn con3i syl ) BFAFCABDGEH $.
  $}

  ${
    jc.1 $e |- ( ph -> ps ) $.
    jc.2 $e |- ( ph -> ch ) $.
    $( Inference joining the consequents of two premises.  (Contributed by NM,
       5-Aug-1993.) $)
    jc $p |- ( ph -> -. ( ps -> -. ch ) ) $=
      ( wn wi pm3.2im sylc ) ABCBCFGFDEBCHI $.
  $}

  $( Theorem joining the consequents of two premises.  Theorem 8 of [Margaris]
     p. 60.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Josh
     Purinton, 29-Dec-2000.) $)
  jcn $p |- ( ph -> ( -. ps -> -. ( ph -> ps ) ) ) $=
    ( wi pm2.27 con3d ) AABCBABDE $.

  ${
    jcnd.1 $e |- ( ph -> ps ) $.
    jcnd.2 $e |- ( ph -> -. ch ) $.
    $( Deduction joining the consequents of two premises.  (Contributed by
       Glauco Siliprandi, 11-Dec-2019.)  (Proof shortened by Wolf Lammen,
       10-Apr-2024.) $)
    jcnd $p |- ( ph -> -. ( ps -> ch ) ) $=
      ( wn wi jcn sylc ) ABCFBCGFDEBCHI $.
  $}

  $( Contrapositive of ~ ax-1 .  (Contributed by BJ, 28-Oct-2023.) $)
  conax1 $p |- ( -. ( ph -> ps ) -> -. ps ) $=
    ( wi ax-1 con3i ) BABCBADE $.

  $( Weakening of ~ conax1 .  General instance of ~ pm2.51 and of ~ pm2.52 .
     (Contributed by BJ, 28-Oct-2023.) $)
  conax1k $p |- ( -. ( ph -> ps ) -> ( ch -> -. ps ) ) $=
    ( wi wn conax1 a1d ) ABDEBECABFG $.

  $( Theorem *2.51 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.51 $p |- ( -. ( ph -> ps ) -> ( ph -> -. ps ) ) $=
    ( conax1k ) ABAC $.

  $( Theorem *2.52 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
  pm2.52 $p |- ( -. ( ph -> ps ) -> ( -. ph -> -. ps ) ) $=
    ( wn conax1k ) ABACD $.

  $( Exportation theorem ~ pm3.3 (closed form of ~ ex ) expressed with
     primitive connectives.  (Contributed by NM, 5-Aug-1993.) $)
  expt $p |- ( ( -. ( ph -> -. ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) $=
    ( wn wi pm3.2im imim1d com12 ) AABDEDZCEBCEABICABFGH $.

  $( Elimination of a nested antecedent.  (Contributed by Wolf Lammen,
     10-May-2013.) $)
  jarl $p |- ( ( ( ph -> ps ) -> ch ) -> ( -. ph -> ch ) ) $=
    ( wn wi pm2.21 imim1i ) ADABECABFG $.

  $( Theorem *2.65 of [WhiteheadRussell] p. 107.  Proof by contradiction.
     Proofs, such as this one, which assume a proposition, here ` ph ` , derive
     a contradiction, and therefore conclude ` -. ph ` , are valid
     intuitionistically (and can be called "proof of negation", for example by
     Section 1.2 of [Bauer] p. 482).  By contrast, proofs which assume
     ` -. ph ` , derive a contradiction, and conclude ` ph ` , such as
     ~ condandc , are only valid for decidable propositions.  (Contributed by
     NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 8-Mar-2013.) $)
  pm2.65 $p |- ( ( ph -> ps ) -> ( ( ph -> -. ps ) -> -. ph ) ) $=
    ( wi wn pm2.27 con2d a2i ) ABCAABDZCZABIDAIBAHEFGF $.

  ${
    pm2.65d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    pm2.65d.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
    $( Deduction for proof by contradiction.  (Contributed by NM, 26-Jun-1994.)
       (Proof shortened by Wolf Lammen, 26-May-2013.) $)
    pm2.65d $p |- ( ph -> -. ps ) $=
      ( nsyld pm2.01d ) ABABCBEDFG $.
  $}

  ${
    pm2.65da.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    pm2.65da.2 $e |- ( ( ph /\ ps ) -> -. ch ) $.
    $( Deduction for proof by contradiction.  (Contributed by NM,
       12-Jun-2014.) $)
    pm2.65da $p |- ( ph -> -. ps ) $=
      ( ex wn pm2.65d ) ABCABCDFABCGEFH $.
  $}

  ${
    mto.1 $e |- -. ps $.
    mto.2 $e |- ( ph -> ps ) $.
    $( The rule of modus tollens.  (Contributed by NM, 19-Aug-1993.)  (Proof
       shortened by Wolf Lammen, 11-Sep-2013.) $)
    mto $p |- -. ph $=
      ( wn a1i pm2.65i ) ABDBEACFG $.
  $}

  ${
    mtod.1 $e |- ( ph -> -. ch ) $.
    mtod.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Modus tollens deduction.  (Contributed by NM, 3-Apr-1994.)  (Proof
       shortened by Wolf Lammen, 11-Sep-2013.) $)
    mtod $p |- ( ph -> -. ps ) $=
      ( wn a1d pm2.65d ) ABCEACFBDGH $.
  $}

  ${
    mtoi.1 $e |- -. ch $.
    mtoi.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Modus tollens inference.  (Contributed by NM, 5-Jul-1994.)  (Proof
       shortened by Wolf Lammen, 15-Sep-2012.) $)
    mtoi $p |- ( ph -> -. ps ) $=
      ( wn a1i mtod ) ABCCFADGEH $.
  $}

  ${
    mtand.1 $e |- ( ph -> -. ch ) $.
    mtand.2 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( A modus tollens deduction.  (Contributed by Jeff Hankins,
       19-Aug-2009.) $)
    mtand $p |- ( ph -> -. ps ) $=
      ( ex mtod ) ABCDABCEFG $.
  $}

  $( Equivalence property for negation.  Closed form.  (Contributed by BJ,
     27-Jan-2020.) $)
  notbi $p |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) ) $=
    ( wb wn biimpr con3d biimp impbid ) ABCZADBDIBAABEFIABABGFH $.

  ${
    notbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalence property for negation.  Deduction form.  (Contributed by NM,
       21-May-1994.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
    notbid $p |- ( ph -> ( -. ps <-> -. ch ) ) $=
      ( wb wn notbi syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    notbii.1 $e |- ( ph <-> ps ) $.
    $( Equivalence property for negation.  Inference form.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
    notbii $p |- ( -. ph <-> -. ps ) $=
      ( wb wn notbi ax-mp ) ABDAEBEDCABFG $.
  $}

  $( Contraposition.  Bidirectional version of ~ con2 .  (Contributed by NM,
     5-Aug-1993.) $)
  con2b $p |- ( ( ph -> -. ps ) <-> ( ps -> -. ph ) ) $=
    ( wn wi con2 impbii ) ABCDBACDABEBAEF $.

  $( A conjunction with a negated conjunction.  (Contributed by AV,
     8-Mar-2022.)  (Proof shortened by Wolf Lammen, 1-Apr-2022.) $)
  annotanannot $p |- ( ( ph /\ -. ( ph /\ ps ) ) <-> ( ph /\ -. ps ) ) $=
    ( wa wn ibar bicomd notbid pm5.32i ) AABCZDBDAIBABIABEFGH $.

  ${
    mtbi.1 $e |- -. ph $.
    mtbi.2 $e |- ( ph <-> ps ) $.
    $( An inference from a biconditional, related to modus tollens.
       (Contributed by NM, 15-Nov-1994.)  (Proof shortened by Wolf Lammen,
       25-Oct-2012.) $)
    mtbi $p |- -. ps $=
      ( biimpri mto ) BACABDEF $.
  $}

  ${
    mtbir.1 $e |- -. ps $.
    mtbir.2 $e |- ( ph <-> ps ) $.
    $( An inference from a biconditional, related to modus tollens.
       (Contributed by NM, 15-Nov-1994.)  (Proof shortened by Wolf Lammen,
       14-Oct-2012.) $)
    mtbir $p |- -. ph $=
      ( bicomi mtbi ) BACABDEF $.
  $}

  ${
    mtbid.min $e |- ( ph -> -. ps ) $.
    mtbid.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( A deduction from a biconditional, similar to modus tollens.
       (Contributed by NM, 26-Nov-1995.) $)
    mtbid $p |- ( ph -> -. ch ) $=
      ( biimprd mtod ) ACBDABCEFG $.
  $}

  ${
    mtbird.min $e |- ( ph -> -. ch ) $.
    mtbird.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( A deduction from a biconditional, similar to modus tollens.
       (Contributed by NM, 10-May-1994.) $)
    mtbird $p |- ( ph -> -. ps ) $=
      ( biimpd mtod ) ABCDABCEFG $.
  $}

  ${
    mtbii.min $e |- -. ps $.
    mtbii.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( An inference from a biconditional, similar to modus tollens.
       (Contributed by NM, 27-Nov-1995.) $)
    mtbii $p |- ( ph -> -. ch ) $=
      ( biimprd mtoi ) ACBDABCEFG $.
  $}

  ${
    mtbiri.min $e |- -. ch $.
    mtbiri.maj $e |- ( ph -> ( ps <-> ch ) ) $.
    $( An inference from a biconditional, similar to modus tollens.
       (Contributed by NM, 24-Aug-1995.) $)
    mtbiri $p |- ( ph -> -. ps ) $=
      ( biimpd mtoi ) ABCDABCEFG $.
  $}

  ${
    sylnib.1 $e |- ( ph -> -. ps ) $.
    sylnib.2 $e |- ( ps <-> ch ) $.
    $( A mixed syllogism inference from an implication and a biconditional.
       (Contributed by Wolf Lammen, 16-Dec-2013.) $)
    sylnib $p |- ( ph -> -. ch ) $=
      ( wb a1i mtbid ) ABCDBCFAEGH $.
  $}

  ${
    sylnibr.1 $e |- ( ph -> -. ps ) $.
    sylnibr.2 $e |- ( ch <-> ps ) $.
    $( A mixed syllogism inference from an implication and a biconditional.
       Useful for substituting an consequent with a definition.  (Contributed
       by Wolf Lammen, 16-Dec-2013.) $)
    sylnibr $p |- ( ph -> -. ch ) $=
      ( bicomi sylnib ) ABCDCBEFG $.
  $}

  ${
    sylnbi.1 $e |- ( ph <-> ps ) $.
    sylnbi.2 $e |- ( -. ps -> ch ) $.
    $( A mixed syllogism inference from a biconditional and an implication.
       Useful for substituting an antecedent with a definition.  (Contributed
       by Wolf Lammen, 16-Dec-2013.) $)
    sylnbi $p |- ( -. ph -> ch ) $=
      ( wn notbii sylbi ) AFBFCABDGEH $.
  $}

  ${
    sylnbir.1 $e |- ( ps <-> ph ) $.
    sylnbir.2 $e |- ( -. ps -> ch ) $.
    $( A mixed syllogism inference from a biconditional and an implication.
       (Contributed by Wolf Lammen, 16-Dec-2013.) $)
    sylnbir $p |- ( -. ph -> ch ) $=
      ( bicomi sylnbi ) ABCBADFEG $.
  $}

  ${
    xchnxbi.1 $e |- ( -. ph <-> ps ) $.
    xchnxbi.2 $e |- ( ph <-> ch ) $.
    $( Replacement of a subexpression by an equivalent one.  (Contributed by
       Wolf Lammen, 27-Sep-2014.) $)
    xchnxbi $p |- ( -. ch <-> ps ) $=
      ( wn notbii bitr3i ) CFAFBACEGDH $.
  $}

  ${
    xchnxbir.1 $e |- ( -. ph <-> ps ) $.
    xchnxbir.2 $e |- ( ch <-> ph ) $.
    $( Replacement of a subexpression by an equivalent one.  (Contributed by
       Wolf Lammen, 27-Sep-2014.) $)
    xchnxbir $p |- ( -. ch <-> ps ) $=
      ( bicomi xchnxbi ) ABCDCAEFG $.
  $}

  ${
    xchbinx.1 $e |- ( ph <-> -. ps ) $.
    xchbinx.2 $e |- ( ps <-> ch ) $.
    $( Replacement of a subexpression by an equivalent one.  (Contributed by
       Wolf Lammen, 27-Sep-2014.) $)
    xchbinx $p |- ( ph <-> -. ch ) $=
      ( wn notbii bitri ) ABFCFDBCEGH $.
  $}

  ${
    xchbinxr.1 $e |- ( ph <-> -. ps ) $.
    xchbinxr.2 $e |- ( ch <-> ps ) $.
    $( Replacement of a subexpression by an equivalent one.  (Contributed by
       Wolf Lammen, 27-Sep-2014.) $)
    xchbinxr $p |- ( ph <-> -. ch ) $=
      ( bicomi xchbinx ) ABCDCBEFG $.
  $}

  ${
    mt2bi.1 $e |- ph $.
    $( A false consequent falsifies an antecedent.  (Contributed by NM,
       19-Aug-1993.)  (Proof shortened by Wolf Lammen, 12-Nov-2012.) $)
    mt2bi $p |- ( -. ps <-> ( ps -> -. ph ) ) $=
      ( wn wi a1bi con2b bitri ) BDZAIEBADEAICFABGH $.
  $}

  $( Modus-tollens-like theorem.  (Contributed by NM, 7-Apr-2001.)  (Revised by
     Mario Carneiro, 31-Jan-2015.) $)
  mtt $p |- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) ) $=
    ( wn wi pm2.21 con3 com12 impbid2 ) ACZBCZBADZBAEKIJBAFGH $.

  $( Express conjunction in terms of implication.  One direction of Theorem
     *4.61 of [WhiteheadRussell] p. 120.  The converse holds for decidable
     propositions, as can be seen at ~ annimdc .  (Contributed by Jim Kingdon,
     24-Dec-2017.) $)
  annimim $p |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) ) $=
    ( wn wi pm2.27 con3 syl imp ) ABCZABDZCZAJBDIKDABEJBFGH $.

  $( One direction of Theorem *4.65 of [WhiteheadRussell] p. 120.  The converse
     holds in classical logic.  (Contributed by Jim Kingdon, 28-Jul-2018.) $)
  pm4.65r $p |- ( ( -. ph /\ -. ps ) -> -. ( -. ph -> ps ) ) $=
    ( wn annimim ) ACBD $.

  $( Express implication in terms of conjunction.  The converse only holds
     given a decidability condition; see ~ imandc .  (Contributed by Jim
     Kingdon, 24-Dec-2017.) $)
  imanim $p |- ( ( ph -> ps ) -> -. ( ph /\ -. ps ) ) $=
    ( wn wa wi annimim con2i ) ABCDABEABFG $.

  $( Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.37 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ( ph /\ -. ch ) -> -. ps ) ) $=
    ( wa wi wn pm3.3 con3 syl6 impd ) ABDCEZACFZBFZKABCELMEABCGBCHIJ $.

  $( Express implication in terms of conjunction.  (Contributed by NM,
     9-Apr-1994.)  (Revised by Mario Carneiro, 1-Feb-2015.) $)
  imnan $p |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) ) $=
    ( wn wi wa pm3.2im imp con2i pm3.2 con3rr3 impbii ) ABCDZABEZCMLABLCABFGHAB
    MABIJK $.

  ${
    imnani.1 $e |- -. ( ph /\ ps ) $.
    $( Express implication in terms of conjunction.  (Contributed by Mario
       Carneiro, 28-Sep-2015.) $)
    imnani $p |- ( ph -> -. ps ) $=
      ( wn wi wa imnan mpbir ) ABDEABFDCABGH $.
  $}

  $( Theorem to move a conjunct in and out of a negation.  (Contributed by NM,
     9-Nov-2003.) $)
  nan $p |- ( ( ph -> -. ( ps /\ ch ) ) <-> ( ( ph /\ ps ) -> -. ch ) ) $=
    ( wa wn wi impexp imnan imbi2i bitr2i ) ABDCEZFABKFZFABCDEZFABKGLMABCHIJ $.

  ${
    mpnanrd.1 $e |- ( ph -> ps ) $.
    mpnanrd.2 $e |- ( ph -> -. ( ps /\ ch ) ) $.
    $( Eliminate the right side of a negated conjunction in an implication.
       (Contributed by ML, 17-Oct-2020.) $)
    mpnanrd $p |- ( ph -> -. ch ) $=
      ( wn wa wi imnan sylibr mpd ) ABCFZDABCGFBLHEBCIJK $.
  $}

  $( Law of noncontradiction.  Theorem *3.24 of [WhiteheadRussell] p. 111 (who
     call it the "law of contradiction").  (Contributed by NM, 16-Sep-1993.)
     (Revised by Mario Carneiro, 2-Feb-2015.) $)
  pm3.24 $p |- -. ( ph /\ -. ph ) $=
    ( wn wi wa notnot imnan mpbi ) AABZBCAHDBAEAHFG $.

  $( Theorem *4.15 of [WhiteheadRussell] p. 117.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 18-Nov-2012.) $)
  pm4.15 $p |- ( ( ( ph /\ ps ) -> -. ch ) <-> ( ( ps /\ ch ) -> -. ph ) ) $=
    ( wa wn wi con2b nan bitr2i ) BCDZAEFAJEFABDCEFJAGABCHI $.

  $( Two propositions are equivalent if they are both false.  Theorem *5.21 of
     [WhiteheadRussell] p. 124.  (Contributed by NM, 21-May-1994.)  (Revised by
     Mario Carneiro, 31-Jan-2015.) $)
  pm5.21 $p |- ( ( -. ph /\ -. ps ) -> ( ph <-> ps ) ) $=
    ( wn wa simpl pm2.21d simpr impbid ) ACZBCZDZABKABIJEFKBAIJGFH $.

  $( Two propositions are equivalent if they are both false.  Closed form of
     ~ 2false .  Equivalent to a ~ biimpr -like version of the xor-connective.
     (Contributed by Wolf Lammen, 13-May-2013.)  (Revised by Mario Carneiro,
     31-Jan-2015.) $)
  pm5.21im $p |- ( -. ph -> ( -. ps -> ( ph <-> ps ) ) ) $=
    ( wn wb pm5.21 ex ) ACBCABDABEF $.

  $( The negation of a wff is equivalent to the wff's equivalence to falsehood.
     (Contributed by Juha Arpiainen, 19-Jan-2006.)  (Revised by Mario Carneiro,
     31-Jan-2015.) $)
  nbn2 $p |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) ) $=
    ( wn wb pm5.21im wi biimpr mtt imbitrrid impbid ) ACZBCZABDZABEMLKBAFABGABH
    IJ $.

  $( Transfer negation via an equivalence.  (Contributed by NM, 3-Oct-2007.)
     (Proof shortened by Wolf Lammen, 28-Jan-2013.) $)
  bibif $p |- ( -. ps -> ( ( ph <-> ps ) <-> -. ph ) ) $=
    ( wn wb nbn2 bicom bitr2di ) BCACBADABDBAEBAFG $.

  ${
    nbn.1 $e |- -. ph $.
    $( The negation of a wff is equivalent to the wff's equivalence to
       falsehood.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
       Lammen, 3-Oct-2013.) $)
    nbn $p |- ( -. ps <-> ( ps <-> ph ) ) $=
      ( wb wn bibif ax-mp bicomi ) BADZBEZAEIJDCBAFGH $.
  $}

  ${
    nbn3.1 $e |- ph $.
    $( Transfer falsehood via equivalence.  (Contributed by NM,
       11-Sep-2006.) $)
    nbn3 $p |- ( -. ps <-> ( ps <-> -. ph ) ) $=
      ( wn notnoti nbn ) ADBACEF $.
  $}

  ${
    2false.1 $e |- -. ph $.
    2false.2 $e |- -. ps $.
    $( Two falsehoods are equivalent.  (Contributed by NM, 4-Apr-2005.)
       (Revised by Mario Carneiro, 31-Jan-2015.) $)
    2false $p |- ( ph <-> ps ) $=
      ( pm2.21i impbii ) ABABCEBADEF $.
  $}

  ${
    2falsed.1 $e |- ( ph -> -. ps ) $.
    2falsed.2 $e |- ( ph -> -. ch ) $.
    $( Two falsehoods are equivalent (deduction form).  (Contributed by NM,
       11-Oct-2013.) $)
    2falsed $p |- ( ph -> ( ps <-> ch ) ) $=
      ( pm2.21d impbid ) ABCABCDFACBEFG $.
  $}

  ${
    pm5.21ni.1 $e |- ( ph -> ps ) $.
    pm5.21ni.2 $e |- ( ch -> ps ) $.
    $( Two propositions implying a false one are equivalent.  (Contributed by
       NM, 16-Feb-1996.)  (Proof shortened by Wolf Lammen, 19-May-2013.) $)
    pm5.21ni $p |- ( -. ps -> ( ph <-> ch ) ) $=
      ( wn con3i 2falsed ) BFACABDGCBEGH $.

    pm5.21nii.3 $e |- ( ps -> ( ph <-> ch ) ) $.
    $( Eliminate an antecedent implied by each side of a biconditional.
       (Contributed by NM, 21-May-1999.)  (Revised by Mario Carneiro,
       31-Jan-2015.) $)
    pm5.21nii $p |- ( ph <-> ch ) $=
      ( wb syl ibi ibir impbii ) ACACABACGZDFHICACBLEFHJK $.
  $}

  ${
    pm5.21ndd.1 $e |- ( ph -> ( ch -> ps ) ) $.
    pm5.21ndd.2 $e |- ( ph -> ( th -> ps ) ) $.
    pm5.21ndd.3 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
    $( Eliminate an antecedent implied by each side of a biconditional,
       deduction version.  (Contributed by Paul Chapman, 21-Nov-2012.)
       (Revised by Mario Carneiro, 31-Jan-2015.) $)
    pm5.21ndd $p |- ( ph -> ( ch <-> th ) ) $=
      ( wb syld ibd bicom1 syl6 impbid ) ACDACDACBCDHZEGIJADCADNDCHADBNFGICDKLJ
      M $.
  $}

  $( Theorem *5.19 of [WhiteheadRussell] p. 124.  (Contributed by NM,
     3-Jan-2005.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
  pm5.19 $p |- -. ( ph <-> -. ph ) $=
    ( wn wb biimp pm2.01d id mpbird pm2.65i ) AABZCZAJAIJAAIDEZJFGKH $.

  $( Theorem *4.8 of [WhiteheadRussell] p. 122.  This one holds for all
     propositions, but compare with ~ pm4.81dc which requires a decidability
     condition.  (Contributed by NM, 3-Jan-2005.) $)
  pm4.8 $p |- ( ( ph -> -. ph ) <-> -. ph ) $=
    ( wn wi pm2.01 ax-1 impbii ) AABZCGADGAEF $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical disjunction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare the connective for disjunction ('or'). $)
  $c \/ $. $( Vee (read:  'or') $)

  $( Extend wff definition to include disjunction ('or'). $)
  wo $a wff ( ph \/ ps ) $.

  $( Definition of 'or'.  One of the axioms of propositional logic.
     (Contributed by Mario Carneiro, 31-Jan-2015.)  Use its alias ~ jaob
     instead.  (New usage is discouraged.) $)
  ax-io $a |-
             ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) ) $.

  $( Disjunction of antecedents.  Compare Theorem *4.77 of [WhiteheadRussell]
     p. 121.  Alias of ~ ax-io .  (Contributed by NM, 30-May-1994.)  (Revised
     by Mario Carneiro, 31-Jan-2015.) $)
  jaob $p |- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) ) $=
    ( ax-io ) ABCD $.

  $( Introduction of a disjunct.  Axiom *1.3 of [WhiteheadRussell] p. 96.
     (Contributed by NM, 30-Aug-1993.)  (Revised by NM, 31-Jan-2015.) $)
  olc $p |- ( ph -> ( ps \/ ph ) ) $=
    ( wo wi wa id jaob mpbi simpri ) BBACZDZAJDZJJDKLEJFBJAGHI $.

  $( Introduction of a disjunct.  Theorem *2.2 of [WhiteheadRussell] p. 104.
     (Contributed by NM, 30-Aug-1993.)  (Revised by NM, 31-Jan-2015.) $)
  orc $p |- ( ph -> ( ph \/ ps ) ) $=
    ( wo wi wa id jaob mpbi simpli ) AABCZDZBJDZJJDKLEJFAJBGHI $.

  $( Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107.
     (Contributed by NM, 3-Jan-2005.)  (Revised by NM, 9-Dec-2012.) $)
  pm2.67-2 $p |- ( ( ( ph \/ ch ) -> ps ) -> ( ph -> ps ) ) $=
    ( wo orc imim1i ) AACDBACEF $.

  $( Absorption of disjunction into equivalence.  (Contributed by NM,
     6-Aug-1995.)  (Proof shortened by Wolf Lammen, 3-Nov-2013.) $)
  oibabs $p |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) <-> ( ph <-> ps ) ) $=
    ( wo wb wi pm2.67-2 ibd olc imim1i ibibr sylibr impbid ax-1 impbii ) ABCZAB
    DZEZPQABQABAPBFGQBPEBAEBOPBAHIBAJKLPOMN $.

  $( Theorem *3.44 of [WhiteheadRussell] p. 113.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
  pm3.44 $p |- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) ->
                ( ( ps \/ ch ) -> ph ) ) $=
    ( wo wi wa jaob biimpri ) BCDAEBAECAEFBACGH $.

  ${
    jaoi.1 $e |- ( ph -> ps ) $.
    jaoi.2 $e |- ( ch -> ps ) $.
    $( Inference disjoining the antecedents of two implications.  (Contributed
       by NM, 5-Apr-1994.)  (Revised by NM, 31-Jan-2015.) $)
    jaoi $p |- ( ( ph \/ ch ) -> ps ) $=
      ( wi wo pm3.44 mp2an ) ABFCBFACGBFDEBACHI $.
  $}

  ${
    jaod.1 $e |- ( ph -> ( ps -> ch ) ) $.
    jaod.2 $e |- ( ph -> ( th -> ch ) ) $.
    $( Deduction disjoining the antecedents of two implications.  (Contributed
       by NM, 18-Aug-1994.)  (Revised by NM, 4-Apr-2013.) $)
    jaod $p |- ( ph -> ( ( ps \/ th ) -> ch ) ) $=
      ( wo wi com12 jaoi ) BDGACBACHDABCEIADCFIJI $.

    jaod.3 $e |- ( ph -> ( ps \/ th ) ) $.
    $( Eliminate a disjunction in a deduction.  (Contributed by Mario Carneiro,
       29-May-2016.) $)
    mpjaod $p |- ( ph -> ch ) $=
      ( wo jaod mpd ) ABDHCGABCDEFIJ $.
  $}

  ${
    jaao.1 $e |- ( ph -> ( ps -> ch ) ) $.
    jaao.2 $e |- ( th -> ( ta -> ch ) ) $.
    $( Inference conjoining and disjoining the antecedents of two implications.
       (Contributed by NM, 30-Sep-1999.) $)
    jaao $p |- ( ( ph /\ th ) -> ( ( ps \/ ta ) -> ch ) ) $=
      ( wa wi adantr adantl jaod ) ADHBCEABCIDFJDECIAGKL $.

    $( Inference disjoining and conjoining the antecedents of two implications.
       (Contributed by Stefan Allan, 1-Nov-2008.) $)
    jaoa $p |- ( ( ph \/ th ) -> ( ( ps /\ ta ) -> ch ) ) $=
      ( wa wi adantrd adantld jaoi ) ABEHCIDABCEFJDECBGKL $.
  $}

  $( Implication in terms of disjunction.  One direction of theorem *4.6 of
     [WhiteheadRussell] p. 120.  The converse holds for decidable propositions,
     as seen at ~ imordc .  (Contributed by Jim Kingdon, 21-Jul-2018.) $)
  imorr $p |- ( ( -. ph \/ ps ) -> ( ph -> ps ) ) $=
    ( wn wi ax-in2 ax-1 jaoi ) ACABDBABEBAFG $.

  $( Theorem *2.53 of [WhiteheadRussell] p. 107.  This holds
     intuitionistically, although its converse does not (see ~ pm2.54dc ).
     (Contributed by NM, 3-Jan-2005.)  (Revised by NM, 31-Jan-2015.) $)
  pm2.53 $p |- ( ( ph \/ ps ) -> ( -. ph -> ps ) ) $=
    ( wn wi pm2.24 ax-1 jaoi ) AACZBDBABEBHFG $.

  ${
    ori.1 $e |- ( ph \/ ps ) $.
    $( Infer implication from disjunction.  (Contributed by NM, 11-Jun-1994.)
       (Revised by Mario Carneiro, 31-Jan-2015.) $)
    ori $p |- ( -. ph -> ps ) $=
      ( wo wn wi pm2.53 ax-mp ) ABDAEBFCABGH $.
  $}

  ${
    ord.1 $e |- ( ph -> ( ps \/ ch ) ) $.
    $( Deduce implication from disjunction.  (Contributed by NM, 18-May-1994.)
       (Revised by Mario Carneiro, 31-Jan-2015.) $)
    ord $p |- ( ph -> ( -. ps -> ch ) ) $=
      ( wo wn wi pm2.53 syl ) ABCEBFCGDBCHI $.
  $}

  $( Elimination of disjunction by denial of a disjunct.  Theorem *2.55 of
     [WhiteheadRussell] p. 107.  (Contributed by NM, 12-Aug-1994.)  (Proof
     shortened by Wolf Lammen, 21-Jul-2012.) $)
  orel1 $p |- ( -. ph -> ( ( ph \/ ps ) -> ps ) ) $=
    ( wo wn pm2.53 com12 ) ABCADBABEF $.

  $( Elimination of disjunction by denial of a disjunct.  Theorem *2.56 of
     [WhiteheadRussell] p. 107.  (Contributed by NM, 12-Aug-1994.)  (Proof
     shortened by Wolf Lammen, 5-Apr-2013.) $)
  orel2 $p |- ( -. ph -> ( ( ps \/ ph ) -> ps ) ) $=
    ( wn idd pm2.21 jaod ) ACZBBAGBDABEF $.

  $( Axiom *1.4 of [WhiteheadRussell] p. 96.  (Contributed by NM, 3-Jan-2005.)
     (Revised by NM, 15-Nov-2012.) $)
  pm1.4 $p |- ( ( ph \/ ps ) -> ( ps \/ ph ) ) $=
    ( wo olc orc jaoi ) ABACBABDBAEF $.

  $( Commutative law for disjunction.  Theorem *4.31 of [WhiteheadRussell]
     p. 118.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 15-Nov-2012.) $)
  orcom $p |- ( ( ph \/ ps ) <-> ( ps \/ ph ) ) $=
    ( wo pm1.4 impbii ) ABCBACABDBADE $.

  ${
    orcomd.1 $e |- ( ph -> ( ps \/ ch ) ) $.
    $( Commutation of disjuncts in consequent.  (Contributed by NM,
       2-Dec-2010.) $)
    orcomd $p |- ( ph -> ( ch \/ ps ) ) $=
      ( wo orcom sylib ) ABCECBEDBCFG $.
  $}

  ${
    orcoms.1 $e |- ( ( ph \/ ps ) -> ch ) $.
    $( Commutation of disjuncts in antecedent.  (Contributed by NM,
       2-Dec-2012.) $)
    orcoms $p |- ( ( ps \/ ph ) -> ch ) $=
      ( wo pm1.4 syl ) BAEABECBAFDG $.
  $}

  ${
    orci.1 $e |- ph $.
    $( Deduction introducing a disjunct.  (Contributed by NM, 19-Jan-2008.)
       (Revised by Mario Carneiro, 31-Jan-2015.) $)
    orci $p |- ( ph \/ ps ) $=
      ( wo orc ax-mp ) AABDCABEF $.

    $( Deduction introducing a disjunct.  (Contributed by NM, 19-Jan-2008.)
       (Revised by Mario Carneiro, 31-Jan-2015.) $)
    olci $p |- ( ps \/ ph ) $=
      ( wo olc ax-mp ) ABADCABEF $.
  $}

  ${
    orcd.1 $e |- ( ph -> ps ) $.
    $( Deduction introducing a disjunct.  (Contributed by NM, 20-Sep-2007.) $)
    orcd $p |- ( ph -> ( ps \/ ch ) ) $=
      ( wo orc syl ) ABBCEDBCFG $.

    $( Deduction introducing a disjunct.  (Contributed by NM, 11-Apr-2008.)
       (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
    olcd $p |- ( ph -> ( ch \/ ps ) ) $=
      ( orcd orcomd ) ABCABCDEF $.
  $}

  ${
    orcs.1 $e |- ( ( ph \/ ps ) -> ch ) $.
    $( Deduction eliminating disjunct. _Notational convention_:  We sometimes
       suffix with "s" the label of an inference that manipulates an
       antecedent, leaving the consequent unchanged.  The "s" means that the
       inference eliminates the need for a syllogism ( ~ syl ) -type inference
       in a proof.  (Contributed by NM, 21-Jun-1994.) $)
    orcs $p |- ( ph -> ch ) $=
      ( wo orc syl ) AABECABFDG $.
  $}

  ${
    olcs.1 $e |- ( ( ph \/ ps ) -> ch ) $.
    $( Deduction eliminating disjunct.  (Contributed by NM, 21-Jun-1994.)
       (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
    olcs $p |- ( ps -> ch ) $=
      ( orcoms orcs ) BACABCDEF $.
  $}

  $( Theorem *2.07 of [WhiteheadRussell] p. 101.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.07 $p |- ( ph -> ( ph \/ ph ) ) $=
    ( olc ) AAB $.

  $( Theorem *2.45 of [WhiteheadRussell] p. 106.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.45 $p |- ( -. ( ph \/ ps ) -> -. ph ) $=
    ( wo orc con3i ) AABCABDE $.

  $( Theorem *2.46 of [WhiteheadRussell] p. 106.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.46 $p |- ( -. ( ph \/ ps ) -> -. ps ) $=
    ( wo olc con3i ) BABCBADE $.

  $( Theorem *2.47 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.47 $p |- ( -. ( ph \/ ps ) -> ( -. ph \/ ps ) ) $=
    ( wo wn pm2.45 orcd ) ABCDADBABEF $.

  $( Theorem *2.48 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.48 $p |- ( -. ( ph \/ ps ) -> ( ph \/ -. ps ) ) $=
    ( wo wn pm2.46 olcd ) ABCDBDAABEF $.

  $( Theorem *2.49 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.49 $p |- ( -. ( ph \/ ps ) -> ( -. ph \/ -. ps ) ) $=
    ( wo wn pm2.46 olcd ) ABCDBDADABEF $.

  $( Theorem *2.67 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.)  (Revised by NM, 9-Dec-2012.) $)
  pm2.67 $p |- ( ( ( ph \/ ps ) -> ps ) -> ( ph -> ps ) ) $=
    ( pm2.67-2 ) ABBC $.

  $( A wff is equivalent to its disjunction with falsehood.  Theorem *4.74 of
     [WhiteheadRussell] p. 121.  (Contributed by NM, 23-Mar-1995.)  (Proof
     shortened by Wolf Lammen, 18-Nov-2012.) $)
  biorf $p |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) ) $=
    ( wn wo olc orel1 impbid2 ) ACBABDBAEABFG $.

  $( A wff is equivalent to its negated disjunction with falsehood.
     (Contributed by NM, 9-Jul-2012.) $)
  biortn $p |- ( ph -> ( ps <-> ( -. ph \/ ps ) ) ) $=
    ( wn wo wb notnot biorf syl ) AACZCBIBDEAFIBGH $.

  ${
    biorfi.1 $e |- -. ph $.
    $( A wff is equivalent to its disjunction with falsehood.  (Contributed by
       NM, 23-Mar-1995.) $)
    biorfi $p |- ( ps <-> ( ps \/ ph ) ) $=
      ( wn wo wb orc orel2 impbid2 ax-mp ) ADZBBAEZFCKBLBAGABHIJ $.
  $}

  $( Theorem *2.621 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.)  (Revised by NM, 13-Dec-2013.) $)
  pm2.621 $p |- ( ( ph -> ps ) -> ( ( ph \/ ps ) -> ps ) ) $=
    ( wi id idd jaod ) ABCZABBGDGBEF $.

  $( Theorem *2.62 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 13-Dec-2013.) $)
  pm2.62 $p |- ( ( ph \/ ps ) -> ( ( ph -> ps ) -> ps ) ) $=
    ( wi wo pm2.621 com12 ) ABCABDBABEF $.

  ${
    imorri.1 $e |- ( -. ph \/ ps ) $.
    $( Infer implication from disjunction.  (Contributed by Jonathan Ben-Naim,
       3-Jun-2011.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
    imorri $p |- ( ph -> ps ) $=
      ( wn wo wi pm2.21 ax-1 jaoi ax-mp ) ADZBEABFZCKLBABGBAHIJ $.
  $}

  $( One direction of theorem *4.52 of [WhiteheadRussell] p. 120.  The converse
     also holds in classical logic.  (Contributed by Jim Kingdon,
     27-Jul-2018.) $)
  pm4.52im $p |- ( ( ph /\ -. ps ) -> -. ( -. ph \/ ps ) ) $=
    ( wn wa wi wo annimim imorr nsyl ) ABCDABEACBFABGABHI $.

  $( One direction of theorem *4.53 of [WhiteheadRussell] p. 120.  The converse
     also holds in classical logic.  (Contributed by Jim Kingdon,
     27-Jul-2018.) $)
  pm4.53r $p |- ( ( -. ph \/ ps ) -> -. ( ph /\ -. ps ) ) $=
    ( wn wa wo pm4.52im con2i ) ABCDACBEABFG $.

  $( Negated disjunction in terms of conjunction.  This version of DeMorgan's
     law is a biconditional for all propositions (not just decidable ones),
     unlike ~ oranim , ~ anordc , or ~ ianordc .  Compare Theorem *4.56 of
     [WhiteheadRussell] p. 120.  (Contributed by NM, 5-Aug-1993.)  (Revised by
     Mario Carneiro, 31-Jan-2015.) $)
  ioran $p |- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) ) $=
    ( wo wn wa pm2.45 pm2.46 jca simpl con2i simpr jaoi impbii ) ABCZDZADZBDZEZ
    OPQABFABGHNRARDBRAPQIJRBPQKJLJM $.

  $( Theorem *3.14 of [WhiteheadRussell] p. 111.  One direction of De Morgan's
     law).  The biconditional holds for decidable propositions as seen at
     ~ ianordc .  The converse holds for decidable propositions, as seen at
     ~ pm3.13dc .  (Contributed by NM, 3-Jan-2005.)  (Revised by Mario
     Carneiro, 31-Jan-2015.) $)
  pm3.14 $p |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) ) $=
    ( wn wa simpl con3i simpr jaoi ) ACABDZCBCIAABEFIBABGFH $.

  $( Theorem *3.1 of [WhiteheadRussell] p. 111.  The converse holds for
     decidable propositions, as seen at ~ anordc .  (Contributed by NM,
     3-Jan-2005.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
  pm3.1 $p |- ( ( ph /\ ps ) -> -. ( -. ph \/ -. ps ) ) $=
    ( wn wo wa pm3.14 con2i ) ACBCDABEABFG $.

  $( Disjunction of antecedents.  Compare Theorem *3.44 of [WhiteheadRussell]
     p. 113.  (Contributed by NM, 5-Apr-1994.)  (Proof shortened by Wolf
     Lammen, 4-Apr-2013.) $)
  jao $p |- ( ( ph -> ps ) -> ( ( ch -> ps ) -> ( ( ph \/ ch ) -> ps ) ) ) $=
    ( wi wo pm3.44 ex ) ABDCBDACEBDBACFG $.

  $( Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96.  (Contributed by NM,
     3-Jan-2005.)  (Revised by NM, 10-Mar-2013.) $)
  pm1.2 $p |- ( ( ph \/ ph ) -> ph ) $=
    ( id jaoi ) AAAABZDC $.

  $( Idempotent law for disjunction.  Theorem *4.25 of [WhiteheadRussell]
     p. 117.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew
     Salmon, 16-Apr-2011.)  (Proof shortened by Wolf Lammen, 10-Mar-2013.) $)
  oridm $p |- ( ( ph \/ ph ) <-> ph ) $=
    ( wo pm1.2 pm2.07 impbii ) AABAACADE $.

  $( Theorem *4.25 of [WhiteheadRussell] p. 117.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.25 $p |- ( ph <-> ( ph \/ ph ) ) $=
    ( wo oridm bicomi ) AABAACD $.

  ${
    orim12i.1 $e |- ( ph -> ps ) $.
    orim12i.2 $e |- ( ch -> th ) $.
    $( Disjoin antecedents and consequents of two premises.  (Contributed by
       NM, 6-Jun-1994.)  (Proof shortened by Wolf Lammen, 25-Jul-2012.) $)
    orim12i $p |- ( ( ph \/ ch ) -> ( ps \/ th ) ) $=
      ( wo orcd olcd jaoi ) ABDGCABDEHCDBFIJ $.
  $}

  ${
    orim1i.1 $e |- ( ph -> ps ) $.
    $( Introduce disjunct to both sides of an implication.  (Contributed by NM,
       6-Jun-1994.) $)
    orim1i $p |- ( ( ph \/ ch ) -> ( ps \/ ch ) ) $=
      ( id orim12i ) ABCCDCEF $.

    $( Introduce disjunct to both sides of an implication.  (Contributed by NM,
       6-Jun-1994.) $)
    orim2i $p |- ( ( ch \/ ph ) -> ( ch \/ ps ) ) $=
      ( id orim12i ) CCABCEDF $.
  $}

  ${
    orbi2i.1 $e |- ( ph <-> ps ) $.
    $( Inference adding a left disjunct to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       12-Dec-2012.) $)
    orbi2i $p |- ( ( ch \/ ph ) <-> ( ch \/ ps ) ) $=
      ( wo biimpi orim2i biimpri impbii ) CAECBEABCABDFGBACABDHGI $.

    $( Inference adding a right disjunct to both sides of a logical
       equivalence.  (Contributed by NM, 5-Aug-1993.) $)
    orbi1i $p |- ( ( ph \/ ch ) <-> ( ps \/ ch ) ) $=
      ( wo orcom orbi2i 3bitri ) ACECAECBEBCEACFABCDGCBFH $.
  $}

  ${
    orbi12i.1 $e |- ( ph <-> ps ) $.
    orbi12i.2 $e |- ( ch <-> th ) $.
    $( Infer the disjunction of two equivalences.  (Contributed by NM,
       5-Aug-1993.) $)
    orbi12i $p |- ( ( ph \/ ch ) <-> ( ps \/ th ) ) $=
      ( wo orbi2i orbi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
  $}

  $( Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96.  (Contributed by NM,
     3-Jan-2005.) $)
  pm1.5 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ps \/ ( ph \/ ch ) ) ) $=
    ( wo orc olcd olc orim2i jaoi ) ABACDZDBCDAJBACEFCJBCAGHI $.

  $( Swap two disjuncts.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by
     Wolf Lammen, 14-Nov-2012.) $)
  or12 $p |- ( ( ph \/ ( ps \/ ch ) ) <-> ( ps \/ ( ph \/ ch ) ) ) $=
    ( wo pm1.5 impbii ) ABCDDBACDDABCEBACEF $.

  $( Associative law for disjunction.  Theorem *4.33 of [WhiteheadRussell]
     p. 118.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew
     Salmon, 26-Jun-2011.) $)
  orass $p |- ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) ) $=
    ( wo orcom or12 orbi2i 3bitri ) ABDZCDCIDACBDZDABCDZDICECABFJKACBEGH $.

  $( Theorem *2.31 of [WhiteheadRussell] p. 104.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.31 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ( ph \/ ps ) \/ ch ) ) $=
    ( wo orass biimpri ) ABDCDABCDDABCEF $.

  $( Theorem *2.32 of [WhiteheadRussell] p. 105.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.32 $p |- ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ( ps \/ ch ) ) ) $=
    ( wo orass biimpi ) ABDCDABCDDABCEF $.

  $( A rearrangement of disjuncts.  (Contributed by NM, 18-Oct-1995.)  (Proof
     shortened by Andrew Salmon, 26-Jun-2011.) $)
  or32 $p |- ( ( ( ph \/ ps ) \/ ch ) <-> ( ( ph \/ ch ) \/ ps ) ) $=
    ( wo orass or12 orcom 3bitri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.

  $( Rearrangement of 4 disjuncts.  (Contributed by NM, 12-Aug-1994.) $)
  or4 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <->
                ( ( ph \/ ch ) \/ ( ps \/ th ) ) ) $=
    ( wo or12 orbi2i orass 3bitr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHAC
    LHI $.

  $( Rearrangement of 4 disjuncts.  (Contributed by NM, 10-Jan-2005.) $)
  or42 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <->
                 ( ( ph \/ ch ) \/ ( th \/ ps ) ) ) $=
    ( wo or4 orcom orbi2i bitri ) ABECDEEACEZBDEZEJDBEZEABCDFKLJBDGHI $.

  $( Distribution of disjunction over disjunction.  (Contributed by NM,
     25-Feb-1995.) $)
  orordi $p |- ( ( ph \/ ( ps \/ ch ) ) <->
               ( ( ph \/ ps ) \/ ( ph \/ ch ) ) ) $=
    ( wo oridm orbi1i or4 bitr3i ) ABCDZDAADZIDABDACDDJAIAEFAABCGH $.

  $( Distribution of disjunction over disjunction.  (Contributed by NM,
     25-Feb-1995.) $)
  orordir $p |- ( ( ( ph \/ ps ) \/ ch ) <->
               ( ( ph \/ ch ) \/ ( ps \/ ch ) ) ) $=
    ( wo oridm orbi2i or4 bitr3i ) ABDZCDICCDZDACDBCDDJCICEFABCCGH $.

  $( Theorem *2.3 of [WhiteheadRussell] p. 104.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.3 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ph \/ ( ch \/ ps ) ) ) $=
    ( wo pm1.4 orim2i ) BCDCBDABCEF $.

  $( Theorem *2.41 of [WhiteheadRussell] p. 106.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.41 $p |- ( ( ps \/ ( ph \/ ps ) ) -> ( ph \/ ps ) ) $=
    ( wo olc id jaoi ) BABCZGBADGEF $.

  $( Theorem *2.42 of [WhiteheadRussell] p. 106.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.42 $p |- ( ( -. ph \/ ( ph -> ps ) ) -> ( ph -> ps ) ) $=
    ( wn wi pm2.21 id jaoi ) ACABDZHABEHFG $.

  $( Theorem *2.4 of [WhiteheadRussell] p. 106.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.4 $p |- ( ( ph \/ ( ph \/ ps ) ) -> ( ph \/ ps ) ) $=
    ( wo orc id jaoi ) AABCZGABDGEF $.

  $( Theorem *4.44 of [WhiteheadRussell] p. 119.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.44 $p |- ( ph <-> ( ph \/ ( ph /\ ps ) ) ) $=
    ( wa wo orc id simpl jaoi impbii ) AAABCZDAJEAAJAFABGHI $.

  $( Theorem *4.56 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.56 $p |- ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) ) $=
    ( wo wn wa ioran bicomi ) ABCDADBDEABFG $.

  $( Disjunction in terms of conjunction (DeMorgan's law).  One direction of
     Theorem *4.57 of [WhiteheadRussell] p. 120.  The converse does not hold
     intuitionistically but does hold in classical logic.  (Contributed by Jim
     Kingdon, 25-Jul-2018.) $)
  oranim $p |- ( ( ph \/ ps ) -> -. ( -. ph /\ -. ps ) ) $=
    ( wn wa wo pm4.56 biimpi con2i ) ACBCDZABEZIJCABFGH $.

  $( Implication distributes over disjunction.  One direction of Theorem *4.78
     of [WhiteheadRussell] p. 121.  The converse holds in classical logic.
     (Contributed by Jim Kingdon, 15-Jan-2018.) $)
  pm4.78i $p |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) ->
       ( ph -> ( ps \/ ch ) ) ) $=
    ( wi wo orc imim2i olc jaoi ) ABDABCEZDACDBJABCFGCJACBHGI $.

  ${
    mtord.1 $e |- ( ph -> -. ch ) $.
    mtord.2 $e |- ( ph -> -. th ) $.
    mtord.3 $e |- ( ph -> ( ps -> ( ch \/ th ) ) ) $.
    $( A modus tollens deduction involving disjunction.  (Contributed by Jeff
       Hankins, 15-Jul-2009.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
    mtord $p |- ( ph -> -. ps ) $=
      ( wo wn pm2.21d jaod syld pm2.01d ) ABABCDHBIZGACNDACNEJADNFJKLM $.
  $}

  $( Theorem *4.45 of [WhiteheadRussell] p. 119.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.45 $p |- ( ph <-> ( ph /\ ( ph \/ ps ) ) ) $=
    ( wo orc pm4.71i ) AABCABDE $.

  $( Theorem *3.48 of [WhiteheadRussell] p. 114.  (Contributed by NM,
     28-Jan-1997.)  (Revised by NM, 1-Dec-2012.) $)
  pm3.48 $p |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) ->
               ( ( ph \/ ch ) -> ( ps \/ th ) ) ) $=
    ( wi wo orc imim2i olc jaao ) ABEABDFZCDECBKABDGHDKCDBIHJ $.

  ${
    orim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    orim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
    $( Disjoin antecedents and consequents in a deduction.  (Contributed by NM,
       10-May-1994.) $)
    orim12d $p |- ( ph -> ( ( ps \/ th ) -> ( ch \/ ta ) ) ) $=
      ( wi wo pm3.48 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
  $}

  ${
    orim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Disjoin antecedents and consequents in a deduction.  (Contributed by NM,
       23-Apr-1995.) $)
    orim1d $p |- ( ph -> ( ( ps \/ th ) -> ( ch \/ th ) ) ) $=
      ( idd orim12d ) ABCDDEADFG $.

    $( Disjoin antecedents and consequents in a deduction.  (Contributed by NM,
       23-Apr-1995.) $)
    orim2d $p |- ( ph -> ( ( th \/ ps ) -> ( th \/ ch ) ) ) $=
      ( idd orim12d ) ADDBCADFEG $.
  $}

  $( Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97.  (Contributed by NM,
     3-Jan-2005.) $)
  orim2 $p |- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
    ( wi id orim2d ) BCDZBCAGEF $.

  ${
    orbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction adding a left disjunct to both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       31-Jan-2015.) $)
    orbi2d $p |- ( ph -> ( ( th \/ ps ) <-> ( th \/ ch ) ) ) $=
      ( wo biimpd orim2d biimprd impbid ) ADBFDCFABCDABCEGHACBDABCEIHJ $.

    $( Deduction adding a right disjunct to both sides of a logical
       equivalence.  (Contributed by NM, 5-Aug-1993.) $)
    orbi1d $p |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ th ) ) ) $=
      ( wo orbi2d orcom 3bitr4g ) ADBFDCFBDFCDFABCDEGBDHCDHI $.
  $}

  $( Theorem *4.37 of [WhiteheadRussell] p. 118.  (Contributed by NM,
     3-Jan-2005.) $)
  orbi1 $p |- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) ) $=
    ( wb id orbi1d ) ABDZABCGEF $.

  ${
    orbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    orbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction joining two equivalences to form equivalence of disjunctions.
       (Contributed by NM, 5-Aug-1993.) $)
    orbi12d $p |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) ) $=
      ( wo orbi1d orbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  $( Theorem *5.61 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 30-Jun-2013.) $)
  pm5.61 $p |- ( ( ( ph \/ ps ) /\ -. ps ) <-> ( ph /\ -. ps ) ) $=
    ( wn wo biorf orcom bitr2di pm5.32ri ) BCZABDZAIABADJBAEBAFGH $.

  ${
    jaoian.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    jaoian.2 $e |- ( ( th /\ ps ) -> ch ) $.
    $( Inference disjoining the antecedents of two implications.  (Contributed
       by NM, 23-Oct-2005.) $)
    jaoian $p |- ( ( ( ph \/ th ) /\ ps ) -> ch ) $=
      ( wo wi ex jaoi imp ) ADGBCABCHDABCEIDBCFIJK $.
  $}

  ${
    jao1i.1 $e |- ( ps -> ( ch -> ph ) ) $.
    $( Add a disjunct in the antecedent of an implication.  (Contributed by
       Rodolfo Medina, 24-Sep-2010.) $)
    jao1i $p |- ( ( ph \/ ps ) -> ( ch -> ph ) ) $=
      ( wi ax-1 jaoi ) ACAEBACFDG $.
  $}

  ${
    jaodan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    jaodan.2 $e |- ( ( ph /\ th ) -> ch ) $.
    $( Deduction disjoining the antecedents of two implications.  (Contributed
       by NM, 14-Oct-2005.) $)
    jaodan $p |- ( ( ph /\ ( ps \/ th ) ) -> ch ) $=
      ( wo ex jaod imp ) ABDGCABCDABCEHADCFHIJ $.

    jaodan.3 $e |- ( ph -> ( ps \/ th ) ) $.
    $( Eliminate a disjunction in a deduction.  A translation of natural
       deduction rule ` \/ ` E ( ` \/ ` elimination).  (Contributed by Mario
       Carneiro, 29-May-2016.) $)
    mpjaodan $p |- ( ph -> ch ) $=
      ( wo jaodan mpdan ) ABDHCGABCDEFIJ $.
  $}

  $( Theorem *4.77 of [WhiteheadRussell] p. 121.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.77 $p |- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) <->
                ( ( ps \/ ch ) -> ph ) ) $=
    ( wo wi wa jaob bicomi ) BCDAEBAECAEFBACGH $.

  $( Theorem *2.63 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.63 $p |- ( ( ph \/ ps ) -> ( ( -. ph \/ ps ) -> ps ) ) $=
    ( wo wn pm2.53 idd jaod ) ABCZADBBABEHBFG $.

  $( Theorem *2.64 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.64 $p |- ( ( ph \/ ps ) -> ( ( ph \/ -. ps ) -> ph ) ) $=
    ( wn wo wi ax-1 orel2 jaoi com12 ) ABCZDABDZAAKAEJAKFBAGHI $.

  $( Theorem *5.53 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.53 $p |- ( ( ( ( ph \/ ps ) \/ ch ) -> th ) <->
                ( ( ( ph -> th ) /\ ( ps -> th ) ) /\ ( ch -> th ) ) ) $=
    ( wo wi wa jaob anbi1i bitri ) ABEZCEDFKDFZCDFZGADFBDFGZMGKDCHLNMADBHIJ $.

  $( Theorem *2.38 of [WhiteheadRussell] p. 105.  (Contributed by NM,
     6-Mar-2008.) $)
  pm2.38 $p |- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ch \/ ph ) ) ) $=
    ( wi id orim1d ) BCDZBCAGEF $.

  $( Theorem *2.36 of [WhiteheadRussell] p. 105.  (Contributed by NM,
     6-Mar-2008.) $)
  pm2.36 $p |- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ch \/ ph ) ) ) $=
    ( wo wi pm1.4 pm2.38 syl5 ) ABDBADBCECADABFABCGH $.

  $( Theorem *2.37 of [WhiteheadRussell] p. 105.  (Contributed by NM,
     6-Mar-2008.) $)
  pm2.37 $p |- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ph \/ ch ) ) ) $=
    ( wi wo pm2.38 pm1.4 syl6 ) BCDBAECAEACEABCFCAGH $.

  $( Theorem *2.73 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.73 $p |- ( ( ph -> ps ) -> ( ( ( ph \/ ps ) \/ ch ) ->
                ( ps \/ ch ) ) ) $=
    ( wi wo pm2.621 orim1d ) ABDABEBCABFG $.

  $( Theorem *2.74 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Mario Carneiro, 31-Jan-2015.) $)
  pm2.74 $p |- ( ( ps -> ph ) -> ( ( ( ph \/ ps ) \/ ch ) ->
                ( ph \/ ch ) ) ) $=
    ( wi wo idd id jaod orim1d ) BADZABEACJAABJAFJGHI $.

  $( Theorem *2.76 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
  pm2.76 $p |- ( ( ph \/ ( ps -> ch ) ) ->
                ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
    ( wo wi orc a1d orim2 jaoi ) AABDZACDZEBCEAKJACFGABCHI $.

  $( Theorem *2.75 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 4-Jan-2013.) $)
  pm2.75 $p |- ( ( ph \/ ps ) ->
                ( ( ph \/ ( ps -> ch ) ) -> ( ph \/ ch ) ) ) $=
    ( wi wo pm2.76 com12 ) ABCDEABEACEABCFG $.

  $( Theorem *2.8 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Mario Carneiro, 31-Jan-2015.) $)
  pm2.8 $p |- ( ( ph \/ ps ) -> ( ( -. ps \/ ch ) -> ( ph \/ ch ) ) ) $=
    ( wo wn pm1.4 ord orim1d ) ABDZBEACIBAABFGH $.

  $( Theorem *2.81 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.81 $p |- ( ( ps -> ( ch -> th ) ) -> ( ( ph \/ ps ) ->
                ( ( ph \/ ch ) -> ( ph \/ th ) ) ) ) $=
    ( wi wo orim2 pm2.76 syl6 ) BCDEZEABFAJFACFADFEABJGACDHI $.

  $( Theorem *2.82 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.82 $p |- ( ( ( ph \/ ps ) \/ ch ) -> ( ( ( ph \/ -. ch ) \/ th ) ->
                ( ( ph \/ ps ) \/ th ) ) ) $=
    ( wo wn wi ax-1 pm2.24 orim2d jaoi orim1d ) ABEZCEACFZEZMDMOMGCMOHCNBACBIJK
    L $.

  ${
    pm3.2ni.1 $e |- -. ph $.
    pm3.2ni.2 $e |- -. ps $.
    $( Infer negated disjunction of negated premises.  (Contributed by NM,
       4-Apr-1995.) $)
    pm3.2ni $p |- -. ( ph \/ ps ) $=
      ( wo id pm2.21i jaoi mto ) ABEACAABAFBADGHI $.
  $}

  $( Absorption of redundant internal disjunct.  Compare Theorem *4.45 of
     [WhiteheadRussell] p. 119.  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 28-Feb-2014.) $)
  orabs $p |- ( ph <-> ( ( ph \/ ps ) /\ ph ) ) $=
    ( wo orc pm4.71ri ) AABCABDE $.

  $( Absorb a disjunct into a conjunct.  (Contributed by Roy F. Longton,
     23-Jun-2005.)  (Proof shortened by Wolf Lammen, 10-Nov-2013.) $)
  oranabs $p |- ( ( ( ph \/ -. ps ) /\ ps ) <-> ( ph /\ ps ) ) $=
    ( wn wo biortn orcom bitr2di pm5.32ri ) BABCZDZABAIADJBAEIAFGH $.

  $( Distributive law for disjunction.  Theorem *4.41 of [WhiteheadRussell]
     p. 119.  (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
     31-Jan-2015.) $)
  ordi $p |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) ) $=
    ( wa wo simpl orim2i simpr jca orc adantl adantr olc jaoian jaodan impbii )
    ABCDZEZABEZACEZDRSTQBABCFGQCABCHGISARCARSAQJZKACRBARCUALQAMNOP $.

  $( Distributive law for disjunction.  (Contributed by NM, 12-Aug-1994.) $)
  ordir $p |- ( ( ( ph /\ ps ) \/ ch ) <->
              ( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) $=
    ( wa wo ordi orcom anbi12i 3bitr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMKNL
    ACGBCGHI $.

  $( Distributive law for conjunction.  Theorem *4.4 of [WhiteheadRussell]
     p. 118.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 5-Jan-2013.) $)
  andi $p |- ( ( ph /\ ( ps \/ ch ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) ) $=
    ( wo wa orc olc jaodan anim2i jaoi impbii ) ABCDZEZABEZACEZDZABPCNOFONGHNMO
    BLABCFICLACBGIJK $.

  $( Distributive law for conjunction.  (Contributed by NM, 12-Aug-1994.) $)
  andir $p |- ( ( ( ph \/ ps ) /\ ch ) <->
              ( ( ph /\ ch ) \/ ( ps /\ ch ) ) ) $=
    ( wo wa andi ancom orbi12i 3bitr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMKNL
    ACGBCGHI $.

  $( Double distributive law for disjunction.  (Contributed by NM,
     12-Aug-1994.) $)
  orddi $p |- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <->
              ( ( ( ph \/ ch ) /\ ( ph \/ th ) ) /\
              ( ( ps \/ ch ) /\ ( ps \/ th ) ) ) ) $=
    ( wa wo ordir ordi anbi12i bitri ) ABECDEZFAKFZBKFZEACFADFEZBCFBDFEZEABKGLN
    MOACDHBCDHIJ $.

  $( Double distributive law for conjunction.  (Contributed by NM,
     12-Aug-1994.) $)
  anddi $p |- ( ( ( ph \/ ps ) /\ ( ch \/ th ) ) <->
              ( ( ( ph /\ ch ) \/ ( ph /\ th ) ) \/
              ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) ) $=
    ( wo wa andir andi orbi12i bitri ) ABECDEZFAKFZBKFZEACFADFEZBCFBDFEZEABKGLN
    MOACDHBCDHIJ $.

  $( Prove formula-building rules for the biconditional connective. $)

  $( Theorem *4.39 of [WhiteheadRussell] p. 118.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.39 $p |- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) ->
                ( ( ph \/ ps ) <-> ( ch \/ th ) ) ) $=
    ( wb wa simpl simpr orbi12d ) ACEZBDEZFACBDJKGJKHI $.

  $( Conjunction implies disjunction with one common formula (1/4).
     (Contributed by BJ, 4-Oct-2019.) $)
  animorl $p |- ( ( ph /\ ps ) -> ( ph \/ ch ) ) $=
    ( wa simpl orcd ) ABDACABEF $.

  $( Conjunction implies disjunction with one common formula (2/4).
     (Contributed by BJ, 4-Oct-2019.) $)
  animorr $p |- ( ( ph /\ ps ) -> ( ch \/ ps ) ) $=
    ( wa simpr olcd ) ABDBCABEF $.

  $( Conjunction implies disjunction with one common formula (3/4).
     (Contributed by BJ, 4-Oct-2019.) $)
  animorlr $p |- ( ( ph /\ ps ) -> ( ch \/ ph ) ) $=
    ( wa simpl olcd ) ABDACABEF $.

  $( Conjunction implies disjunction with one common formula (4/4).
     (Contributed by BJ, 4-Oct-2019.) $)
  animorrl $p |- ( ( ph /\ ps ) -> ( ps \/ ch ) ) $=
    ( wa simpr orcd ) ABDBCABEF $.

  $( Implication in terms of biconditional and disjunction.  Theorem *4.72 of
     [WhiteheadRussell] p. 121.  (Contributed by NM, 30-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 30-Jan-2013.) $)
  pm4.72 $p |- ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) ) $=
    ( wi wo wb olc pm2.621 impbid2 orc biimpr syl5 impbii ) ABCZBABDZEZMBNBAFAB
    GHANOBABIBNJKL $.

  $( Theorem *5.16 of [WhiteheadRussell] p. 124.  (Contributed by NM,
     3-Jan-2005.)  (Revised by Mario Carneiro, 31-Jan-2015.) $)
  pm5.16 $p |- -. ( ( ph <-> ps ) /\ ( ph <-> -. ps ) ) $=
    ( wb wn wa pm5.19 simpl simpr bitr3d mto ) ABCZABDZCZEZBLCBFNABLKMGKMHIJ $.

  $( A disjunction with a true formula is equivalent to that true formula.
     (Contributed by NM, 23-May-1999.) $)
  biort $p |- ( ph -> ( ph <-> ( ph \/ ps ) ) ) $=
    ( wo id orc 2thd ) AAABCADABEF $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Stable propositions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare connective for stability. $)
  $c STAB $.

  $( Extend wff definition to include stability. $)
  wstab $a wff STAB ph $.

  $( Propositions where a double-negative can be removed are called stable.
     See Chapter 2 [Moschovakis] p. 2.

     Our notation for stability is a connective ` STAB ` which we place before
     the formula in question.  For example, ` STAB x = y ` corresponds
     to " ` x = y ` is stable".

     (Contributed by David A. Wheeler, 13-Aug-2018.) $)
  df-stab $a |- ( STAB ph <-> ( -. -. ph -> ph ) ) $.

  ${
    stbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( The equivalent of a stable proposition is stable.  (Contributed by Jim
       Kingdon, 12-Aug-2022.) $)
    stbid $p |- ( ph -> ( STAB ps <-> STAB ch ) ) $=
      ( wn wi wstab notbid imbi12d df-stab 3bitr4g ) ABEZEZBFCEZEZCFBGCGAMOBCAL
      NABCDHHDIBJCJK $.
  $}

  $( Every negated formula is stable.  (Contributed by David A. Wheeler,
     13-Aug-2018.) $)
  stabnot $p |- STAB -. ph $=
    ( wn wstab wi notnotnot biimpi df-stab mpbir ) ABZCIBBZIDJIAEFIGH $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Decidable propositions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare connective for decidability. $)
  $c DECID $.

  $( Extend wff definition to include decidability. $)
  wdc $a wff DECID ph $.

  $( Propositions which are known to be true or false are called decidable.
     The (classical) Law of the Excluded Middle corresponds to the principle
     that all propositions are decidable, but even given intuitionistic logic,
     particular kinds of propositions may be decidable (for example, the
     proposition that two natural numbers are equal will be decidable under
     most sets of axioms).

     Our notation for decidability is a connective ` DECID ` which we place
     before the formula in question.  For example, ` DECID x = y ` corresponds
     to " ` x = y ` is decidable".

     We could transform intuitionistic logic to classical logic by adding
     unconditional forms of ~ condc , ~ exmiddc , ~ peircedc , or ~ notnotrdc ,
     any of which would correspond to the assertion that all propositions are
     decidable.

     (Contributed by Jim Kingdon, 11-Mar-2018.) $)
  df-dc $a |- ( DECID ph <-> ( ph \/ -. ph ) ) $.

  $( Law of excluded middle, for a decidable proposition.  The law of the
     excluded middle is also called the principle of _tertium non datur_.
     Theorem *2.11 of [WhiteheadRussell] p. 101.  It says that something is
     either true or not true; there are no in-between values of truth.  The key
     way in which intuitionistic logic differs from classical logic is that
     intuitionistic logic says that excluded middle only holds for some
     propositions, and classical logic says that it holds for all propositions.
     (Contributed by Jim Kingdon, 12-May-2018.) $)
  exmiddc $p |- ( DECID ph -> ( ph \/ -. ph ) ) $=
    ( wdc wn wo df-dc biimpi ) ABAACDAEF $.

  $( Commuted law of the excluded middle for a decidable proposition.  Based on
     theorem *2.1 of [WhiteheadRussell] p. 101.  (Contributed by Jim Kingdon,
     25-Mar-2018.) $)
  pm2.1dc $p |- ( DECID ph -> ( -. ph \/ ph ) ) $=
    ( wdc wn wo df-dc orcom bitri biimpi ) ABZACZADZIAJDKAEAJFGH $.

  ${
    dcbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalence property for decidability.  Deduction form.  (Contributed by
       Jim Kingdon, 7-Sep-2019.) $)
    dcbid $p |- ( ph -> ( DECID ps <-> DECID ch ) ) $=
      ( wn wo wdc notbid orbi12d df-dc 3bitr4g ) ABBEZFCCEZFBGCGABCLMDABCDHIBJC
      JK $.
  $}

  $( Equivalence property for decidability.  Closed form.  (Contributed by BJ,
     27-Jan-2020.) $)
  dcbiit $p |- ( ( ph <-> ps ) -> ( DECID ph <-> DECID ps ) ) $=
    ( wb id dcbid ) ABCZABFDE $.

  ${
    dcbii.1 $e |- ( ph <-> ps ) $.
    $( Equivalence property for decidability.  Inference form.  (Contributed by
       Jim Kingdon, 28-Mar-2018.) $)
    dcbii $p |- ( DECID ph <-> DECID ps ) $=
      ( wb wdc dcbiit ax-mp ) ABDAEBEDCABFG $.
  $}

  $( An implication between two decidable propositions is decidable.
     (Contributed by Jim Kingdon, 28-Mar-2018.) $)
  dcim $p |- ( DECID ph -> ( DECID ps -> DECID ( ph -> ps ) ) ) $=
    ( wn wo wi df-dc wa anbi2i andi bitri pm3.4 annimim orim12i sylbi sylibr ex
    wdc ax-in2 a1d orc syl6 jaoi ) AQAACZDBQZABEZQZEZAFAUGUCAUDUFAUDGZUEUECZDZU
    FUHABGZABCZGZDZUJUHABULDZGUNUDUOABFHABULIJUKUEUMUIABKABLMNUEFZOPUCUDUEUFUCU
    EUDABRSUEUJUFUEUITUPOUAUBN $.

  $( The negation of a decidable proposition is decidable.  The converse need
     not hold, but does hold for negated propositions, see ~ dcnn .
     (Contributed by Jim Kingdon, 25-Mar-2018.) $)
  dcn $p |- ( DECID ph -> DECID -. ph ) $=
    ( wn wo wdc notnot orim2i orcoms df-dc 3imtr4i ) AABZCJJBZCZADJDJALAKJAEFGA
    HJHI $.

  $( Double negation elimination for a decidable proposition.  The converse,
     ~ notnot , holds for all propositions, not just decidable ones.  This is
     Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability
     condition added.  (Contributed by Jim Kingdon, 11-Mar-2018.) $)
  notnotrdc $p |- ( DECID ph -> ( -. -. ph -> ph ) ) $=
    ( wdc wn wo wi df-dc orcom bitri pm2.53 sylbi ) ABZACZADZLCAEKALDMAFALGHLAI
    J $.

  $( Decidability implies stability.  The converse need not hold.  (Contributed
     by David A. Wheeler, 13-Aug-2018.) $)
  dcstab $p |- ( DECID ph -> STAB ph ) $=
    ( wdc wn wi wstab notnotrdc df-stab sylibr ) ABACCADAEAFAGH $.

  $( A formula is decidable if and only if its negation is decidable and it is
     stable (that is, it is testable and stable).  (Contributed by David A.
     Wheeler, 13-Aug-2018.)  (Proof shortened by BJ, 28-Oct-2023.) $)
  stdcndc $p |- ( ( STAB ph /\ DECID -. ph ) <-> DECID ph ) $=
    ( wstab wn wdc wa wo df-stab df-dc pm2.36 imp syl2anb sylibr dcstab dcn jca
    wi impbii ) ABZACZDZEZADZUAASFZUBRSCZAPZSUDFZUCTAGSHUEUFUCSUDAIJKAHLUBRTAMA
    NOQ $.

  $( Obsolete version of ~ stdcndc as of 28-Oct-2023.  (Contributed by David A.
     Wheeler, 13-Aug-2018.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  stdcndcOLD $p |- ( ( STAB ph /\ DECID -. ph ) <-> DECID ph ) $=
    ( wstab wn wdc wa wo exmiddc adantl df-stab biimpi orim2d adantr mpd orcomd
    wi df-dc sylibr dcstab dcn jca impbii ) ABZACZDZEZADZUEAUCFUFUEUCAUEUCUCCZF
    ZUCAFZUDUHUBUCGHUBUHUIOUDUBUGAUCUBUGAOAIJKLMNAPQUFUBUDARASTUA $.

  $( A formula is stable if and only if the decidability of its negation
     implies its decidability.  Note that the right-hand side of this
     biconditional is the converse of ~ dcn .  (Contributed by BJ,
     18-Nov-2023.) $)
  stdcn $p |- ( STAB ph <-> ( DECID -. ph -> DECID ph ) ) $=
    ( wstab wn wdc wi wa stdcndc biimpi ex wo imim1i orel2 sylcom df-dc imbi12i
    olc df-stab 3imtr4i impbii ) ABZACZDZADZEZTUBUCTUBFUCAGHIUAUACZJZAUAJZEZUEA
    EUDTUHUEUGAUEUFUGUEUAPKUAALMUBUFUCUGUANANOAQRS $.

  $( Decidability of the negation of a proposition is equivalent to
     decidability of its double negation.  See also ~ dcn .  The relation
     between ~ dcn and ~ dcnn is analogous to that between ~ notnot and
     ~ notnotnot (and directly stems from it).  Using the notion of "testable
     proposition" (proposition whose negation is decidable), ~ dcnn means that
     a proposition is testable if and only if its negation is testable, and
     ~ dcn means that decidability implies testability.  (Contributed by David
     A. Wheeler, 6-Dec-2018.)  (Proof shortened by BJ, 25-Nov-2023.) $)
  dcnn $p |- ( DECID -. ph <-> DECID -. -. ph ) $=
    ( wn wdc dcn wstab wi stabnot stdcn mpbi impbii ) ABZCZKBCZKDKEMLFAGKHIJ $.

  $( Obsolete proof of ~ dcnnOLD as of 25-Nov-2023.  (Contributed by David A.
     Wheeler, 6-Dec-2018.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  dcnnOLD $p |- ( DECID -. ph <-> DECID -. -. ph ) $=
    ( wn wo wdc notnotnot orbi2i orcom bitri df-dc 3bitr4ri ) ABZBZLBZCZKLCZLDK
    DNLKCOMKLAEFLKGHLIKIJ $.

  $( Double negation of decidability of a formula.  See also comment of ~ nndc
     to avoid a pitfall that could come from the label "nnexmid".  This theorem
     can also be proved from ~ bj-nnor as in ~ bj-nndcALT .  (Contributed by
     BJ, 9-Oct-2019.) $)
  nnexmid $p |- -. -. ( ph \/ -. ph ) $=
    ( wn wo wa pm3.24 ioran mtbir ) AABZCBHHBDHEAHFG $.

  $( Double negation of decidability of a formula.  Intuitionistic logic
     refutes the negation of decidability (but does not prove decidability) of
     any formula.

     This should not trick the reader into thinking that ` -. -. EXMID ` is
     provable in intuitionistic logic.  Indeed, if we could quantify over
     formula metavariables, then generalizing ~ nnexmid over ` ph ` would
     give " ` |- A. ph -. -. DECID ph ` ", but ` EXMID `
     is " ` A. ph DECID ph ` ", so proving ` -. -. EXMID ` would amount to
     proving " ` -. -. A. ph DECID ph ` ", which is not implied by the above
     theorem.  Indeed, the converse of ~ nnal does not hold.  Since our system
     does not allow quantification over formula metavariables, we can reproduce
     this argument by representing formulas as subsets of ` ~P 1o ` , like we
     do in our definition of ` EXMID ` ( ~ df-exmid ): then, we can prove
     ` A. x e. ~P 1o -. -. DECID x = 1o ` but we cannot prove
     ` -. -. A. x e. ~P 1o DECID x = 1o ` because the converse of ~ nnral does
     not hold.

     Actually, ` -. -. EXMID ` is not provable in intuitionistic logic since
     intuitionistic logic has models satisfying ` -. EXMID ` and
     noncontradiction holds ( ~ pm3.24 ).  (Contributed by BJ, 9-Oct-2019.)
     Add explanation on non-provability of ` -. -. EXMID ` .  (Revised by BJ,
     11-Aug-2024.) $)
  nndc $p |- -. -. DECID ph $=
    ( wdc wn wo nnexmid df-dc notbii mtbir ) ABZCAACDZCAEIJAFGH $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Theorems of decidable propositions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Many theorems of logic hold in intuitionistic logic just as they do in
  classical (non-intuitionistic) logic, for all propositions.  Other theorems
  only hold for decidable propositions, such as the law of the excluded middle
  ( ~ df-dc ), double negation elimination ( ~ notnotrdc ), or contraposition
  ( ~ condc ).  Our goal is to prove all well-known or important classical
  theorems, but with suitable decidability conditions so that the proofs follow
  from intuitionistic axioms.  This section is focused on such proofs, given
  decidability conditions.

  Many theorems of this section actually hold for stable propositions (see
  ~ df-stab ).  Decidable propositions are stable ( ~ dcstab ), but the
  converse need not hold.

$)

  $( Contraposition when the antecedent is a negated stable proposition.  See
     comment of ~ condc .  (Contributed by BJ, 18-Nov-2023.)  (Proof shortened
     by BJ, 11-Nov-2024.) $)
  const $p |- ( STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) ) $=
    ( wn wi wstab con2 df-stab biimpi syl9r ) ACZBCDBJCZAEZAJBFLKADAGHI $.

  $( Contraposition of a decidable proposition.

     This theorem swaps or "transposes" the order of the consequents when
     negation is removed.  An informal example is that the statement "if there
     are no clouds in the sky, it is not raining" implies the statement "if it
     is raining, there are clouds in the sky".  This theorem (without the
     decidability condition, of course) is called _Transp_ or "the principle of
     transposition" in _Principia Mathematica_ (Theorem *2.17 of
     [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49.  We will
     also use the term "contraposition" for this principle, although the reader
     is advised that in the field of philosophical logic, "contraposition" has
     a different technical meaning.

     (Contributed by Jim Kingdon, 13-Mar-2018.)  (Proof shortened by BJ,
     18-Nov-2023.) $)
  condc $p |- ( DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) ) $=
    ( wdc wstab wn wi dcstab const syl ) ACADAEBEFBAFFAGABHI $.

  $( Obsolete proof of ~ condc as of 18-Nov-2023.  (Contributed by Jim Kingdon,
     13-Mar-2018.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  condcOLD $p |- ( DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) ) $=
    ( wdc wn wo wi df-dc ax-1 a1d pm2.27 ax-in2 syl6 jaoi sylbi ) ACAADZEOBDZFZ
    BAFZFZAGASOARQABHIOQPROPJBAKLMN $.

  $( Proof by contradiction for a decidable proposition.  Based on Theorem
     *2.18 of [WhiteheadRussell] p. 103 (also called Clavius law).
     Intuitionistically it requires a decidability assumption, but compare with
     ~ pm2.01 which does not.  (Contributed by Jim Kingdon, 24-Mar-2018.) $)
  pm2.18dc $p |- ( DECID ph -> ( ( -. ph -> ph ) -> ph ) ) $=
    ( wdc wn wi pm2.21 a2i condc syl5 pm2.43d ) ABZACZADZALKLCZDJLADKAMAMEFALGH
    I $.

  $( Contraposition for a decidable proposition.  Based on theorem *2.15 of
     [WhiteheadRussell] p. 102.  (Contributed by Jim Kingdon, 29-Mar-2018.) $)
  con1dc $p |- ( DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) ) $=
    ( wn wi wdc notnot imim2i condc syl5 ) ACZBDJBCZCZDAEKADBLJBFGAKHI $.

  ${
    con4biddc.1 $e |- ( ph -> ( DECID ps -> ( DECID ch ->
        ( -. ps <-> -. ch ) ) ) ) $.
    $( A contraposition deduction.  (Contributed by Jim Kingdon,
       18-May-2018.) $)
    con4biddc $p |- ( ph -> ( DECID ps -> ( DECID ch -> ( ps <-> ch ) ) ) ) $=
      ( wdc wb wa wi wn biimpr syl8 condc syl6 imp31 biimp imim2d sylcom impbid
      a2i exp31 ) ABEZCEZBCFAUAGUBGBCAUAUBBCHZAUAUBCIZBIZHZHUBUCHAUAUBUEUDFZUFD
      UEUDJKUBUFUCCBLSMNAUAUBCBHZAUAUBUEUDHZHUBUHHAUAUBUGUIDUEUDOKUAUIUHUBBCLPQ
      NRT $.
  $}

  ${
    impidc.1 $e |- ( DECID ch -> ( ph -> ( ps -> ch ) ) ) $.
    $( An importation inference for a decidable consequent.  (Contributed by
       Jim Kingdon, 30-Apr-2018.) $)
    impidc $p |- ( DECID ch -> ( -. ( ph -> -. ps ) -> ch ) ) $=
      ( wdc wn wi wa imp con3d ex com23 con1dc mpd ) CEZCFZABFZGZGRFCGOAPQOAPQG
      OAHBCOABCGDIJKLCRMN $.
  $}

  $( Simplification given a decidable proposition.  Similar to Theorem *3.27
     (Simp) of [WhiteheadRussell] p. 112.  (Contributed by Jim Kingdon,
     30-Apr-2018.) $)
  simprimdc $p |- ( DECID ps -> ( -. ( ph -> -. ps ) -> ps ) ) $=
    ( wi wdc idd a1i impidc ) ABBABBCCBDABEFG $.

  $( Simplification for a decidable proposition.  Similar to Theorem *3.26
     (Simp) of [WhiteheadRussell] p. 112.  (Contributed by Jim Kingdon,
     29-Mar-2018.) $)
  simplimdc $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ph ) ) $=
    ( wdc wn wi pm2.21 con1dc mpi ) ACADABEZEIDAEABFAIGH $.

  ${
    pm2.61ddc.1 $e |- ( ph -> ( ps -> ch ) ) $.
    pm2.61ddc.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
    $( Deduction eliminating a decidable antecedent.  (Contributed by Jim
       Kingdon, 4-May-2018.) $)
    pm2.61ddc $p |- ( DECID ps -> ( ph -> ch ) ) $=
      ( wdc wn wo wi df-dc com12 jaoi sylbi ) BFBBGZHACIZBJBONABCDKANCEKLM $.
  $}

  $( Case elimination for a decidable proposition.  Based on theorem *2.6 of
     [WhiteheadRussell] p. 107.  (Contributed by Jim Kingdon, 25-Mar-2018.) $)
  pm2.6dc $p |- ( DECID ph ->
      ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) ) $=
    ( wdc wn wi wo wa pm2.1dc pm3.44 syl5com expd ) ACZADZBEZABEZBLMAFNOGBAHBMA
    IJK $.

  ${
    jadc.1 $e |- ( DECID ph -> ( -. ph -> ch ) ) $.
    jadc.2 $e |- ( ps -> ch ) $.
    $( Inference forming an implication from the antecedents of two premises,
       where a decidable antecedent is negated.  (Contributed by Jim Kingdon,
       25-Mar-2018.) $)
    jadc $p |- ( DECID ph -> ( ( ph -> ps ) -> ch ) ) $=
      ( wi wdc imim2i wn pm2.6dc mpd syl5 ) ABFACFZAGZCBCAEHNAICFMCFDACJKL $.
  $}

  ${
    jaddc.1 $e |- ( ph -> ( DECID ps -> ( -. ps -> th ) ) ) $.
    jaddc.2 $e |- ( ph -> ( ch -> th ) ) $.
    $( Deduction forming an implication from the antecedents of two premises,
       where a decidable antecedent is negated.  (Contributed by Jim Kingdon,
       26-Mar-2018.) $)
    jaddc $p |- ( ph -> ( DECID ps -> ( ( ps -> ch ) -> th ) ) ) $=
      ( wi wdc imim2d wn pm2.6dc sylcom syl5d ) ABCGBDGZBHZDACDBFIAOBJDGNDGEBDK
      LM $.
  $}

  $( Case elimination for a decidable proposition.  Theorem *2.61 of
     [WhiteheadRussell] p. 107 under a decidability condition.  (Contributed by
     Jim Kingdon, 29-Mar-2018.) $)
  pm2.61dc $p |- ( DECID ph ->
                   ( ( ph -> ps ) -> ( ( -. ph -> ps ) -> ps ) ) ) $=
    ( wdc wn wi pm2.6dc com23 ) ACADBEABEBABFG $.

  $( Negating an implication for a decidable antecedent.  General instance of
     Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition.
     (Contributed by Jim Kingdon, 29-Mar-2018.) $)
  pm2.5gdc $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ch ) ) ) $=
    ( wdc wi wn wa simplimdc imp pm2.24d ex ) ADZABEFZAFCELMGACLMAABHIJK $.

  $( Negating an implication for a decidable antecedent.  Theorem *2.5 of
     [WhiteheadRussell] p. 107 under a decidability condition.  (Contributed by
     Jim Kingdon, 29-Mar-2018.) $)
  pm2.5dc $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) ) $=
    ( pm2.5gdc ) ABBC $.

  $( A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a
     decidability condition.  (Contributed by BJ, 28-Oct-2023.) $)
  pm2.521gdc $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ( ch -> ph ) ) ) $=
    ( wdc wi wn pm2.5gdc condc syld ) ADABEFAFCFZECAEABJGACHI $.

  $( Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional
     decidability condition.  Note that by replacing in proof ~ pm2.52 with
     ~ conax1k , we obtain a proof of the more general instance where the last
     occurrence of ` ph ` is replaced with any ` ch ` .  (Contributed by Jim
     Kingdon, 5-May-2018.) $)
  pm2.521dc $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ( ps -> ph ) ) ) $=
    ( pm2.521gdc ) ABBC $.

  $( Alternate proof of ~ pm2.521dc .  (Contributed by Jim Kingdon,
     5-May-2018.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  pm2.521dcALT $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ( ps -> ph ) ) ) $=
    ( wi wn wdc pm2.52 condc syl5 ) ABCDADBDCAEBACABFABGH $.

  $( Contraposition.  Theorem *4.1 of [WhiteheadRussell] p. 116, but for a
     decidable proposition.  (Contributed by Jim Kingdon, 24-Apr-2018.) $)
  con34bdc $p |- ( DECID ps -> ( ( ph -> ps ) <-> ( -. ps -> -. ph ) ) ) $=
    ( wdc wi wn con3 condc impbid2 ) BCABDBEAEDABFBAGH $.

  $( Double negation equivalence for a decidable proposition.  Like Theorem
     *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent.
     The forward direction, ~ notnot , holds for all propositions, not just
     decidable ones.  (Contributed by Jim Kingdon, 13-Mar-2018.) $)
  notnotbdc $p |- ( DECID ph -> ( ph <-> -. -. ph ) ) $=
    ( wdc wn notnot notnotrdc impbid2 ) ABAACCADAEF $.

  $( Contraposition.  (Contributed by Jim Kingdon, 4-Apr-2018.) $)
  con1biimdc $p |- ( DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps <-> ph ) ) ) $=
    ( wdc wn wb wi biimp con1dc syl5 biimpr con2d a1i impbidd ) ACZADZBEZBDZAPO
    BFNQAFOBGABHIPAQFFNPBAOBJKLM $.

  $( Contraposition.  (Contributed by Jim Kingdon, 17-Apr-2018.) $)
  con1bidc $p |- ( DECID ph -> ( DECID ps ->
      ( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) ) ) $=
    ( wdc wn wb wa wi con1biimdc adantr adantl impbid ex ) ACZBCZADBEZBDAEZEMNF
    OPMOPGNABHINPOGMBAHJKL $.

  $( Contraposition.  (Contributed by Jim Kingdon, 17-Apr-2018.) $)
  con2bidc $p |- ( DECID ph -> ( DECID ps ->
      ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) ) $=
    ( wdc wn wb wa con1bidc imp bicom 3bitr3g bicomd ex ) ACZBCZABDZEZBADZEZEMN
    FZRPSQBEZOAEZRPMNTUAEABGHQBIOAIJKL $.

  ${
    con1biddc.1 $e |- ( ph -> ( DECID ps -> ( -. ps <-> ch ) ) ) $.
    $( A contraposition deduction.  (Contributed by Jim Kingdon,
       4-Apr-2018.) $)
    con1biddc $p |- ( ph -> ( DECID ps -> ( -. ch <-> ps ) ) ) $=
      ( wdc wn wb con1biimdc sylcom ) ABEBFCGCFBGDBCHI $.
  $}

  ${
    con1biidc.1 $e |- ( DECID ph -> ( -. ph <-> ps ) ) $.
    $( A contraposition inference.  (Contributed by Jim Kingdon,
       15-Mar-2018.) $)
    con1biidc $p |- ( DECID ph -> ( -. ps <-> ph ) ) $=
      ( wdc wn notnotbdc notbid bitrd bicomd ) ADZABEZJAAEZEKAFJLBCGHI $.
  $}

  $( Contraposition.  Bidirectional version of ~ con1dc .  (Contributed by NM,
     5-Aug-1993.) $)
  con1bdc $p |- ( DECID ph -> ( DECID ps ->
                  ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) ) $=
    ( wdc wn wi wb wa con1dc adantr adantl impbid ex ) ACZBCZADBEZBDAEZFMNGOPMO
    PENABHINPOEMBAHJKL $.

  ${
    con2biidc.1 $e |- ( DECID ps -> ( ph <-> -. ps ) ) $.
    $( A contraposition inference.  (Contributed by Jim Kingdon,
       15-Mar-2018.) $)
    con2biidc $p |- ( DECID ps -> ( ps <-> -. ph ) ) $=
      ( wdc wn bicomd con1biidc ) BDZAEBBAHABECFGF $.
  $}

  ${
    con2biddc.1 $e |- ( ph -> ( DECID ch -> ( ps <-> -. ch ) ) ) $.
    $( A contraposition deduction.  (Contributed by Jim Kingdon,
       11-Apr-2018.) $)
    con2biddc $p |- ( ph -> ( DECID ch -> ( ch <-> -. ps ) ) ) $=
      ( wdc wn wb bicom imbitrdi con1biddc ) ACEZBFZCGCLGACBAKBCFZGMBGDBMHIJLCH
      I $.
  $}

  ${
    condandc.1 $e |- ( ( ph /\ -. ps ) -> ch ) $.
    condandc.2 $e |- ( ( ph /\ -. ps ) -> -. ch ) $.
    $( Proof by contradiction.  This only holds for decidable propositions, as
       it is part of the family of theorems which assume ` -. ps ` , derive a
       contradiction, and therefore conclude ` ps ` .  By contrast, assuming
       ` ps ` , deriving a contradiction, and therefore concluding ` -. ps ` ,
       as in ~ pm2.65 , is valid for all propositions.  (Contributed by Jim
       Kingdon, 13-May-2018.) $)
    condandc $p |- ( DECID ps -> ( ph -> ps ) ) $=
      ( wn wdc pm2.65da notnotrdc syl5 ) ABFZFBGBAKCDEHBIJ $.
  $}

  ${
    bijadc.1 $e |- ( ph -> ( ps -> ch ) ) $.
    bijadc.2 $e |- ( -. ph -> ( -. ps -> ch ) ) $.
    $( Combine antecedents into a single biconditional.  This inference is
       reminiscent of ~ jadc .  (Contributed by Jim Kingdon, 4-May-2018.) $)
    bijadc $p |- ( DECID ps -> ( ( ph <-> ps ) -> ch ) ) $=
      ( wb biimpr syli wn biimp con3d pm2.61ddc ) ABFZBCBMACABGDHBIMAICMABABJKE
      HL $.
  $}

  $( Relationship between an equivalence and an equivalence with some negation,
     for decidable propositions.  Based on theorem *5.18 of [WhiteheadRussell]
     p. 124.  Given decidability, we can consider ` -. ( ph <-> -. ps ) ` to
     represent "negated exclusive-or".  (Contributed by Jim Kingdon,
     4-Apr-2018.) $)
  pm5.18dc $p |- ( DECID ph -> ( DECID ps ->
        ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) ) $=
    ( wdc wn wo wb wi df-dc wa pm5.501 a1d con1biddc imp adantr bitr2d dcn nbn2
    ex syl5 jaoi sylbi ) ACAADZEBCZABFZABDZFZDZFZGZAHAUIUBAUCUHAUCIUGBUDAUCUGBF
    ABUFAUEUFFUCAUEJKLMABUDFUCABJNORUBUCUHUBUCIUGUEUDUBUCUGUEFZUCUECZUBUJBPUBUE
    UFUBUEDUFFUKAUEQKLSMUBUEUDFUCABQNORTUA $.

  $( Definition of 'and' in terms of negation and implication, for decidable
     propositions.  The forward direction holds for all propositions, and can
     (basically) be found at ~ pm3.2im .  (Contributed by Jim Kingdon,
     30-Apr-2018.) $)
  dfandc $p |- ( DECID ph -> ( DECID ps ->
      ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) ) $=
    ( wdc wa wn wi wb pm3.2im imp simplimdc adantr simprimdc adantl jca impbid2
    ex ) ACZBCZABDZABEZFEZGQRDZSUAABUAABHIUBUASUBUADABUBUAAQUAAFRATJKIUBUABRUAB
    FQABLMINPOP $.

  $( A decidable proposition or its triple negation is true.  Theorem *2.13 of
     [WhiteheadRussell] p. 101 with decidability condition added.  (Contributed
     by Jim Kingdon, 13-May-2018.) $)
  pm2.13dc $p |- ( DECID ph -> ( ph \/ -. -. -. ph ) ) $=
    ( wdc wn wo df-dc notnotrdc con3d orim2d biimtrid pm2.43i ) ABZAACZCZCZDZKA
    LDKOAEKLNAKMAAFGHIJ $.

  $( Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions.
     (Contributed by Jim Kingdon, 1-May-2018.) $)
  pm4.63dc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( ph -> -. ps ) <-> ( ph /\ ps ) ) ) ) $=
    ( wdc wn wi wa wb dfandc imp bicomd ex ) ACZBCZABDEDZABFZGLMFONLMONGABHIJK
    $.

  $( Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions.
     (Contributed by Jim Kingdon, 1-May-2018.) $)
  pm4.67dc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( -. ph -> -. ps ) <-> ( -. ph /\ ps ) ) ) ) $=
    ( wdc wn wi wa wb dcn pm4.63dc syl ) ACADZCBCKBDEDKBFGEAHKBIJ $.

  $( Express implication in terms of conjunction.  Theorem 3.4(27) of [Stoll]
     p. 176.  (Contributed by NM, 12-Mar-1993.)  (Proof shortened by Wolf
     Lammen, 30-Oct-2012.) $)
  imanst $p |- ( STAB ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) ) $=
    ( wstab wi wn wa notnot df-stab biimpi impbid2 imbi2d imnan bitrdi ) BCZABD
    ABEZEZDAOFENBPANBPBGNPBDBHIJKAOLM $.

  $( Express implication in terms of conjunction.  Theorem 3.4(27) of [Stoll]
     p. 176, with an added decidability condition.  The forward direction,
     ~ imanim , holds for all propositions, not just decidable ones.
     (Contributed by Jim Kingdon, 25-Apr-2018.) $)
  imandc $p |- ( DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) ) $=
    ( wdc wstab wi wn wa wb dcstab imanst syl ) BCBDABEABFGFHBIABJK $.

  $( Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability
     condition.  (Contributed by Jim Kingdon, 24-Apr-2018.) $)
  pm4.14dc $p |- ( DECID ch ->
      ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) ) ) $=
    ( wdc wi wn wa con34bdc imbi2d impexp 3bitr4g ) CDZABCEZEACFZBFZEZEABGCEANG
    OELMPABCHIABCJANOJK $.

  $( Deriving disjunction from implication for a decidable proposition.  Based
     on theorem *2.54 of [WhiteheadRussell] p. 107.  The converse, ~ pm2.53 ,
     holds whether the proposition is decidable or not.  (Contributed by Jim
     Kingdon, 26-Mar-2018.) $)
  pm2.54dc $p |- ( DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) ) $=
    ( wdc wn wi wo dcn notnotrdc orc syl6 a1d olc a1i jaddc mpd ) ACZADZCZQBEAB
    FZEAGPQBSPQDZSERPTASAHABIJKBSEPBALMNO $.

  $( Definition of disjunction in terms of negation and implication for a
     decidable proposition.  Based on definition of [Margaris] p. 49.  One
     direction, ~ pm2.53 , holds for all propositions, not just decidable ones.
     (Contributed by Jim Kingdon, 26-Mar-2018.) $)
  dfordc $p |- ( DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) ) $=
    ( wdc wo wn wi pm2.53 pm2.54dc impbid2 ) ACABDAEBFABGABHI $.

  $( Elimination of disjunction based on a disjunction, for a decidable
     proposition.  Based on theorem *2.25 of [WhiteheadRussell] p. 104.
     (Contributed by NM, 3-Jan-2005.) $)
  pm2.25dc $p |- ( DECID ph -> ( ph \/ ( ( ph \/ ps ) -> ps ) ) ) $=
    ( wdc wn wo wi df-dc orel1 orim2i sylbi ) ACAADZEAABEBFZEAGKLAABHIJ $.

  $( Concluding disjunction from implication for a decidable proposition.
     Based on theorem *2.68 of [WhiteheadRussell] p. 108.  Converse of ~ pm2.62
     and one half of ~ dfor2dc .  (Contributed by Jim Kingdon, 27-Mar-2018.) $)
  pm2.68dc $p |- ( DECID ph -> ( ( ( ph -> ps ) -> ps ) -> ( ph \/ ps ) ) ) $=
    ( wi wn wdc wo jarl pm2.54dc syl5 ) ABCBCADBCAEABFABBGABHI $.

  $( Disjunction expressed in terms of implication only, for a decidable
     proposition.  Based on theorem *5.25 of [WhiteheadRussell] p. 124.
     (Contributed by Jim Kingdon, 27-Mar-2018.) $)
  dfor2dc $p |- ( DECID ph -> ( ( ph \/ ps ) <-> ( ( ph -> ps ) -> ps ) ) ) $=
    ( wdc wo wi pm2.62 pm2.68dc impbid2 ) ACABDABEBEABFABGH $.

  $( Simplify an implication between implications, for a decidable proposition.
     (Contributed by Jim Kingdon, 18-Mar-2018.) $)
  imimorbdc $p |- ( DECID ps -> ( ( ( ps -> ch ) -> ( ph -> ch ) ) <->
                  ( ph -> ( ps \/ ch ) ) ) ) $=
    ( wdc wi wo bi2.04 dfor2dc imbi2d bitr4id ) BDZBCEZACEEALCEZEABCFZELACGKNMA
    BCHIJ $.

  $( Implication in terms of disjunction for a decidable proposition.  Based on
     theorem *4.6 of [WhiteheadRussell] p. 120.  The reverse direction,
     ~ imorr , holds for all propositions.  (Contributed by Jim Kingdon,
     20-Apr-2018.) $)
  imordc $p |- ( DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) ) $=
    ( wdc wi wn wo notnotbdc imbi1d wb dcn dfordc syl bitr4d ) ACZABDAEZEZBDZOB
    FZNAPBAGHNOCRQIAJOBKLM $.

  $( Implication in terms of disjunction.  Like Theorem *4.62 of
     [WhiteheadRussell] p. 120, but for a decidable antecedent.  (Contributed
     by Jim Kingdon, 21-Apr-2018.) $)
  pm4.62dc $p |- ( DECID ph -> ( ( ph -> -. ps ) <-> ( -. ph \/ -. ps ) ) ) $=
    ( wn imordc ) ABCD $.

  $( Negated conjunction in terms of disjunction (DeMorgan's law).  Theorem
     *4.51 of [WhiteheadRussell] p. 120, but where one proposition is
     decidable.  The reverse direction, ~ pm3.14 , holds for all propositions,
     but the equivalence only holds where one proposition is decidable.
     (Contributed by Jim Kingdon, 21-Apr-2018.) $)
  ianordc $p |- ( DECID ph -> ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) ) $=
    ( wa wn wi wdc wo imnan pm4.62dc bitr3id ) ABCDABDZEAFADKGABHABIJ $.

  $( Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability
     condition.  The reverse direction, ~ pm2.53 , holds for all propositions.
     (Contributed by Jim Kingdon, 2-May-2018.) $)
  pm4.64dc $p |- ( DECID ph -> ( ( -. ph -> ps ) <-> ( ph \/ ps ) ) ) $=
    ( wdc wo wn wi dfordc bicomd ) ACABDAEBFABGH $.

  $( Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability
     condition.  (Contributed by Jim Kingdon, 2-May-2018.) $)
  pm4.66dc $p |- ( DECID ph -> ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) ) ) $=
    ( wn pm4.64dc ) ABCD $.

  $( Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions.
     One form of DeMorgan's law.  (Contributed by Jim Kingdon, 2-May-2018.) $)
  pm4.54dc $p |- ( DECID ph -> ( DECID ps ->
      ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) ) $=
    ( wdc wn wa wo wb wi dcn dfandc syl imp pm4.66dc adantr notbid bitrd ex ) A
    CZBCZADZBEZABDZFZDZGRSEZUATUBHZDZUDRSUAUGGZRTCSUHHAITBJKLUEUFUCRUFUCGSABMNO
    PQ $.

  $( Equivalence between a disjunction of two implications, and a conjunction
     and an implication.  Based on theorem *4.79 of [WhiteheadRussell] p. 121
     but with additional decidability antecedents.  (Contributed by Jim
     Kingdon, 28-Mar-2018.) $)
  pm4.79dc $p |- ( DECID ph -> ( DECID ps ->
                 ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <->
                   ( ( ps /\ ch ) -> ph ) ) ) ) $=
    ( wdc wi wo wa wb id jaoa simplimdc pm3.3 syl9 dcim pm2.54dc syl6 syl5d imp
    wn impbid2 expcom ) BDZADZBAEZCAEZFZBCGAEZHUBUCGUFUGUDBAUECUDIUEIJUBUCUGUFE
    UBUGUDSZUEEZUCUFUBUHBUGUEBAKBCALMUBUCUDDUIUFEBANUDUEOPQRTUA $.

  $( Two ways of stating exclusive-or which are equivalent for a decidable
     proposition.  Based on theorem *5.17 of [WhiteheadRussell] p. 124.
     (Contributed by Jim Kingdon, 16-Apr-2018.) $)
  pm5.17dc $p |- ( DECID ps ->
      ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) ) $=
    ( wn wb wdc wo wa bicom dfbi2 orcom dfordc bitr2id imnan a1i anbi12d bitrid
    wi ) ABCZDRADZBEZABFZABGCZGZARHSRAQZARQZGTUCRAITUDUAUEUBUABAFTUDABJBAKLUEUB
    DTABMNOPL $.

  $( Reverse distribution of disjunction over implication, given decidability.
     Based on theorem *2.85 of [WhiteheadRussell] p. 108.  (Contributed by Jim
     Kingdon, 1-Apr-2018.) $)
  pm2.85dc $p |- ( DECID ph -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) ->
                ( ph \/ ( ps -> ch ) ) ) ) $=
    ( wdc wn wo wi df-dc orc a1d olc imim1i orel1 syl9r syl6 jaoi sylbi ) ADAAE
    ZFABFZACFZGZABCGZFZGZAHAUDRAUCUAAUBIJRUAUBUCUABTRCBSTBAKLACMNUBAKOPQ $.

  $( Disjunction distributes over implication.  The forward direction,
     ~ pm2.76 , is valid intuitionistically.  The reverse direction holds if
     ` ph ` is decidable, as can be seen at ~ pm2.85dc .  (Contributed by Jim
     Kingdon, 1-Apr-2018.) $)
  orimdidc $p |- ( DECID ph -> ( ( ph \/ ( ps -> ch ) ) <->
                ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) ) $=
    ( wdc wi wo pm2.76 pm2.85dc impbid2 ) ADABCEFABFACFEABCGABCHI $.

  $( Decidable proposition version of theorem *2.26 of [WhiteheadRussell]
     p. 104.  (Contributed by Jim Kingdon, 20-Apr-2018.) $)
  pm2.26dc $p |- ( DECID ph -> ( -. ph \/ ( ( ph -> ps ) -> ps ) ) ) $=
    ( wdc wi wn wo pm2.27 imordc mpbii ) ACAABDBDZDAEJFABGAJHI $.

  $( Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions.
     This one needs a decidability condition, but compare with ~ pm4.8 which
     holds for all propositions.  (Contributed by Jim Kingdon, 4-Jul-2018.) $)
  pm4.81dc $p |- ( DECID ph -> ( ( -. ph -> ph ) <-> ph ) ) $=
    ( wdc wn wi pm2.18dc pm2.24 impbid1 ) ABACADAAEAAFG $.

  $( A decidable proposition or its negation implies a second proposition.
     Based on theorem *5.11 of [WhiteheadRussell] p. 123.  (Contributed by Jim
     Kingdon, 29-Mar-2018.) $)
  pm5.11dc $p |- ( DECID ph -> ( DECID ps ->
                   ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) ) $=
    ( wdc wi wn wo dcim pm2.5dc pm2.54dc syl5com syld ) ACZBCABDZCZMAEBDZFZABGL
    MEODNPABHMOIJK $.

  $( Excluded middle with antecedents for a decidable consequent.  Based on
     theorem *5.12 of [WhiteheadRussell] p. 123.  (Contributed by Jim Kingdon,
     30-Mar-2018.) $)
  pm5.12dc $p |- ( DECID ps -> ( ( ph -> ps ) \/ ( ph -> -. ps ) ) ) $=
    ( wdc wn wo wi df-dc ax-1 orim12i sylbi ) BCBBDZEABFZAKFZEBGBLKMBAHKAHIJ $.

  $( A decidable proposition is implied by or implies other propositions.
     Based on theorem *5.14 of [WhiteheadRussell] p. 123.  (Contributed by Jim
     Kingdon, 30-Mar-2018.) $)
  pm5.14dc $p |- ( DECID ps -> ( ( ph -> ps ) \/ ( ps -> ch ) ) ) $=
    ( wdc wn wo wi df-dc ax-1 ax-in2 orim12i sylbi ) BDBBEZFABGZBCGZFBHBNMOBAIB
    CJKL $.

  $( An implication holds in at least one direction, where one proposition is
     decidable.  Based on theorem *5.13 of [WhiteheadRussell] p. 123.
     (Contributed by Jim Kingdon, 30-Mar-2018.) $)
  pm5.13dc $p |- ( DECID ps -> ( ( ph -> ps ) \/ ( ps -> ph ) ) ) $=
    ( pm5.14dc ) ABAC $.

  $( A disjunction is equivalent to one of its disjuncts, given a decidable
     disjunct.  Based on theorem *5.55 of [WhiteheadRussell] p. 125.
     (Contributed by Jim Kingdon, 30-Mar-2018.) $)
  pm5.55dc $p |- ( DECID ph ->
        ( ( ( ph \/ ps ) <-> ph ) \/ ( ( ph \/ ps ) <-> ps ) ) ) $=
    ( wdc wn wo wb df-dc biort bicomd biorf orim12i sylbi ) ACAADZEABEZAFZNBFZE
    AGAOMPAANABHIMBNABJIKL $.

  $( Peirce's theorem for a decidable proposition.  This odd-looking theorem
     can be seen as an alternative to ~ exmiddc , ~ condc , or ~ notnotrdc in
     the sense of expressing the "difference" between an intuitionistic system
     of propositional calculus and a classical system.  In intuitionistic
     logic, it only holds for decidable propositions.  (Contributed by Jim
     Kingdon, 3-Jul-2018.) $)
  peircedc $p |- ( DECID ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) ) $=
    ( wdc wn wo wi df-dc ax-1 pm2.21 imim1i com12 jaoi sylbi ) ACAADZEABFZAFZAF
    ZAGAQNAPHPNANOAABIJKLM $.

  $( The Inversion Axiom of the infinite-valued sentential logic (L-infinity)
     of Lukasiewicz, but where one of the propositions is decidable.  Using
     ~ dfor2dc , we can see that this expresses "disjunction is commutative".
     Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability
     condition).  (Contributed by NM, 12-Aug-2004.) $)
  looinvdc $p |- ( DECID ph ->
      ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) ) $=
    ( wi wdc imim1 peircedc syl9r ) ABCZBCBACHACADAHBAEABFG $.


$(
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  Miscellaneous theorems of propositional calculus
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$)

  ${
    pm5.21nd.1 $e |- ( ( ph /\ ps ) -> th ) $.
    pm5.21nd.2 $e |- ( ( ph /\ ch ) -> th ) $.
    pm5.21nd.3 $e |- ( th -> ( ps <-> ch ) ) $.
    $( Eliminate an antecedent implied by each side of a biconditional.
       (Contributed by NM, 20-Nov-2005.)  (Proof shortened by Wolf Lammen,
       4-Nov-2013.) $)
    pm5.21nd $p |- ( ph -> ( ps <-> ch ) ) $=
      ( ex wb wi a1i pm5.21ndd ) ADBCABDEHACDFHDBCIJAGKL $.
  $}

  $( Theorem *5.35 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.35 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) ->
                ( ph -> ( ps <-> ch ) ) ) $=
    ( wi wa pm5.1 pm5.74rd ) ABDZACDZEABCHIFG $.

  $( A conjunction is equivalent to one of its conjuncts, given a decidable
     conjunct.  Based on theorem *5.54 of [WhiteheadRussell] p. 125.
     (Contributed by Jim Kingdon, 30-Mar-2018.) $)
  pm5.54dc $p |- ( DECID ph ->
                   ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) ) ) $=
    ( wdc wa wb wn df-dc simpr ax-ia3 impbid2 simpl ax-in2 orim12i sylbi orcomd
    wo ) ACZABDZBEZRAEZQAAFZPSTPAGASUATARBABHABIJUARAABKARLJMNO $.

  ${
    baib.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Move conjunction outside of biconditional.  (Contributed by NM,
       13-May-1999.) $)
    baib $p |- ( ps -> ( ph <-> ch ) ) $=
      ( wa ibar bitr4id ) BABCECDBCFG $.

    $( Move conjunction outside of biconditional.  (Contributed by NM,
       11-Jul-1994.) $)
    baibr $p |- ( ps -> ( ch <-> ph ) ) $=
      ( baib bicomd ) BACABCDEF $.

    $( Move conjunction outside of biconditional.  (Contributed by Mario
       Carneiro, 11-Sep-2015.) $)
    rbaib $p |- ( ch -> ( ph <-> ps ) ) $=
      ( wa ancom bitri baib ) ACBABCECBEDBCFGH $.

    $( Move conjunction outside of biconditional.  (Contributed by Mario
       Carneiro, 11-Sep-2015.) $)
    rbaibr $p |- ( ch -> ( ps <-> ph ) ) $=
      ( wa ancom bitri baibr ) ACBABCECBEDBCFGH $.
  $}

  ${
    baibd.1 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
    $( Move conjunction outside of biconditional.  (Contributed by Mario
       Carneiro, 11-Sep-2015.) $)
    baibd $p |- ( ( ph /\ ch ) -> ( ps <-> th ) ) $=
      ( wa ibar bicomd sylan9bb ) ABCDFZCDECDJCDGHI $.

    $( Move conjunction outside of biconditional.  (Contributed by Mario
       Carneiro, 11-Sep-2015.) $)
    rbaibd $p |- ( ( ph /\ th ) -> ( ps <-> ch ) ) $=
      ( wa iba bicomd sylan9bb ) ABCDFZDCEDCJDCGHI $.
  $}

  $( Theorem *5.44 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.44 $p |- ( ( ph -> ps ) -> ( ( ph -> ch ) <->
                ( ph -> ( ps /\ ch ) ) ) ) $=
    ( wa wi jcab baibr ) ABCDEABEACEABCFG $.

  $( Conjunction in antecedent versus disjunction in consequent, for a
     decidable proposition.  Theorem *5.6 of [WhiteheadRussell] p. 125, with
     decidability condition added.  The reverse implication holds for all
     propositions (see ~ pm5.6r ).  (Contributed by Jim Kingdon,
     2-Apr-2018.) $)
  pm5.6dc $p |- ( DECID ps ->
        ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( ps \/ ch ) ) ) ) $=
    ( wdc wn wa wi wo impexp dfordc imbi2d bitr4id ) BDZABEZFCGANCGZGABCHZGANCI
    MPOABCJKL $.

  $( Conjunction in antecedent versus disjunction in consequent.  One direction
     of Theorem *5.6 of [WhiteheadRussell] p. 125.  If ` ps ` is decidable, the
     converse also holds (see ~ pm5.6dc ).  (Contributed by Jim Kingdon,
     4-Aug-2018.) $)
  pm5.6r $p |- ( ( ph -> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) -> ch ) ) $=
    ( wo wi wn pm2.53 imim2i impd ) ABCDZEABFZCJKCEABCGHI $.

  ${
    orcanai.1 $e |- ( ph -> ( ps \/ ch ) ) $.
    $( Change disjunction in consequent to conjunction in antecedent.
       (Contributed by NM, 8-Jun-1994.) $)
    orcanai $p |- ( ( ph /\ -. ps ) -> ch ) $=
      ( wn ord imp ) ABECABCDFG $.
  $}

  ${
    intnan.1 $e |- -. ph $.
    $( Introduction of conjunct inside of a contradiction.  (Contributed by NM,
       16-Sep-1993.) $)
    intnan $p |- -. ( ps /\ ph ) $=
      ( wa simpr mto ) BADACBAEF $.

    $( Introduction of conjunct inside of a contradiction.  (Contributed by NM,
       3-Apr-1995.) $)
    intnanr $p |- -. ( ph /\ ps ) $=
      ( wa simpl mto ) ABDACABEF $.
  $}

  ${
    intnand.1 $e |- ( ph -> -. ps ) $.
    $( Introduction of conjunct inside of a contradiction.  (Contributed by NM,
       10-Jul-2005.) $)
    intnand $p |- ( ph -> -. ( ch /\ ps ) ) $=
      ( wa simpr nsyl ) ABCBEDCBFG $.

    $( Introduction of conjunct inside of a contradiction.  (Contributed by NM,
       10-Jul-2005.) $)
    intnanrd $p |- ( ph -> -. ( ps /\ ch ) ) $=
      ( wa simpl nsyl ) ABBCEDBCFG $.
  $}

  ${
    dcand.1 $e |- ( ph -> DECID ps ) $.
    dcand.2 $e |- ( ph -> DECID ch ) $.
    $( A conjunction of two decidable propositions is decidable.  (Contributed
       by Jim Kingdon, 12-Apr-2018.)  (Revised by BJ, 14-Nov-2024.) $)
    dcand $p |- ( ph -> DECID ( ps /\ ch ) ) $=
      ( wa wn wo wdc df-dc id intnanrd orim2i sylbi syl intnand sylanbrc sylibr
      ordir ) ABCFZTGZHZTIABUAHZCUAHZUBABIZUCDUEBBGZHUCBJUFUABUFBCUFKLMNOACIZUD
      EUGCCGZHUDCJUHUACUHCBUHKPMNOBCUASQTJR $.
  $}

  $( A conjunction of two decidable propositions is decidable.  (Contributed by
     Jim Kingdon, 12-Apr-2018.) $)
  dcan $p |- ( ( DECID ph /\ DECID ps ) -> DECID ( ph /\ ps ) ) $=
    ( wdc wa simpl simpr dcand ) ACZBCZDABHIEHIFG $.

  $( A conjunction of two decidable propositions is decidable, expressed in a
     curried form as compared to ~ dcan .  This is deprecated; it's trivial to
     recreate with ~ ex , but it's here in case someone is using this older
     form.  (Contributed by Jim Kingdon, 12-Apr-2018.)
     (New usage is discouraged.) $)
  dcan2 $p |- ( DECID ph -> ( DECID ps -> DECID ( ph /\ ps ) ) ) $=
    ( wdc wa dcan ex ) ACBCABDCABEF $.

  $( A disjunction of two decidable propositions is decidable.  (Contributed by
     Jim Kingdon, 21-Apr-2018.) $)
  dcor $p |- ( DECID ph -> ( DECID ps -> DECID ( ph \/ ps ) ) ) $=
    ( wdc wn wo wi df-dc orc orcd sylibr a1d wa olc adantl ioran biimpri jaodan
    olcd sylan2b ex jaoi sylbi ) ACAADZEBCZABEZCZFZAGAUGUCAUFUDAUEUEDZEZUFAUEUH
    ABHIUEGZJKUCUDUFUDUCBBDZEUFBGUCBUFUKUCBLZUIUFULUEUHBUEUCBAMNIUJJUCUKLZUIUFU
    MUHUEUHUMABOPRUJJQSTUAUB $.

  $( An equivalence of two decidable propositions is decidable.  (Contributed
     by Jim Kingdon, 12-Apr-2018.) $)
  dcbi $p |- ( DECID ph -> ( DECID ps -> DECID ( ph <-> ps ) ) ) $=
    ( wdc wi wa wb dcim com12 dcan ex syl6c dfbi2 dcbii imbitrrdi ) ACZBCZABDZB
    ADZEZCZABFZCOPQCZRCZTABGPOUCBAGHUBUCTQRIJKUASABLMN $.

  $( Express conjunction in terms of implication.  The forward direction,
     ~ annimim , is valid for all propositions, but as an equivalence, it
     requires a decidability condition.  (Contributed by Jim Kingdon,
     25-Apr-2018.) $)
  annimdc $p |- ( DECID ph -> ( DECID ps ->
      ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) ) $=
    ( wdc wn wa wi wb imandc adantl dcim imp dcan sylan2 con2bidc sylc mpbid ex
    dcn ) ACZBCZABDZEZABFZDGZSTEZUCUBDGZUDTUFSABHIUEUCCZUBCZUFUDGSTUGABJKTSUACU
    HBRAUALMUCUBNOPQ $.

  $( Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions.
     (Contributed by Jim Kingdon, 2-May-2018.) $)
  pm4.55dc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) ) ) $=
    ( wdc wn wa wo wb pm4.54dc imp dcor impel dcan sylan con2bidc sylc 3bitr2rd
    dcn mpbird ex ) ACZBCZADZBEZDZABDZFZGTUAEZUFUDUFUDUGUFUDGZUCUFDGZTUAUIABHIU
    GUFCZUCCZUHUIGTUECUJUAAUEJBQKTUBCUAUKAQUBBLMUFUCNORZULULPS $.

  $( Disjunction in terms of conjunction (De Morgan's law), for decidable
     propositions.  Compare Theorem *4.57 of [WhiteheadRussell] p. 120.
     (Contributed by Jim Kingdon, 13-Dec-2021.) $)
  orandc $p |- ( ( DECID ph /\ DECID ps ) ->
      ( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) ) $=
    ( wdc wa wn wo wb pm4.56 dcn dcan syl2an dcor imp con2bidc sylc mpbii ) ACZ
    BCZDZAEZBEZDZABFZEGZUCUBEGZABHSUBCZUCCZUDUEGQTCUACUFRAIBITUAJKQRUGABLMUBUCN
    OP $.

  ${
    mpbiran.1 $e |- ps $.
    mpbiran.2 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Detach truth from conjunction in biconditional.  (Contributed by NM,
       27-Feb-1996.)  (Revised by NM, 9-Jan-2015.) $)
    mpbiran $p |- ( ph <-> ch ) $=
      ( wa biantrur bitr4i ) ABCFCEBCDGH $.
  $}

  ${
    mpbiran2.1 $e |- ch $.
    mpbiran2.2 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Detach truth from conjunction in biconditional.  (Contributed by NM,
       22-Feb-1996.)  (Revised by NM, 9-Jan-2015.) $)
    mpbiran2 $p |- ( ph <-> ps ) $=
      ( wa biantru bitr4i ) ABCFBECBDGH $.
  $}

  ${
    mpbir2an.1 $e |- ps $.
    mpbir2an.2 $e |- ch $.
    mpbiran2an.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Detach a conjunction of truths in a biconditional.  (Contributed by NM,
       10-May-2005.)  (Revised by NM, 9-Jan-2015.) $)
    mpbir2an $p |- ph $=
      ( mpbiran mpbir ) ACEABCDFGH $.
  $}

  ${
    mpbi2and.1 $e |- ( ph -> ps ) $.
    mpbi2and.2 $e |- ( ph -> ch ) $.
    mpbi2and.3 $e |- ( ph -> ( ( ps /\ ch ) <-> th ) ) $.
    $( Detach a conjunction of truths in a biconditional.  (Contributed by NM,
       6-Nov-2011.)  (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
    mpbi2and $p |- ( ph -> th ) $=
      ( wa jca mpbid ) ABCHDABCEFIGJ $.
  $}

  ${
    mpbir2and.1 $e |- ( ph -> ch ) $.
    mpbir2and.2 $e |- ( ph -> th ) $.
    mpbir2and.3 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
    $( Detach a conjunction of truths in a biconditional.  (Contributed by NM,
       6-Nov-2011.)  (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
    mpbir2and $p |- ( ph -> ps ) $=
      ( wa jca mpbird ) ABCDHACDEFIGJ $.
  $}

  $( Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition.
     (Contributed by Jim Kingdon, 12-May-2018.) $)
  pm5.62dc $p |- ( DECID ps ->
      ( ( ( ph /\ ps ) \/ -. ps ) <-> ( ph \/ -. ps ) ) ) $=
    ( wdc wn wo wa wb df-dc ordir simplbi simplbi2 com12 impbid2 sylbi ) BCBBDZ
    EZABFOEZAOEZGBHPQRQRPABOIZJRPQQRPSKLMN $.

  $( Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition.
     (Contributed by Jim Kingdon, 12-May-2018.) $)
  pm5.63dc $p |- ( DECID ph ->
      ( ( ph \/ ps ) <-> ( ph \/ ( -. ph /\ ps ) ) ) ) $=
    ( wdc wo wn wa wi df-dc ordi simplbi2 sylbi simprbi impbid1 ) ACZABDZAAEZBF
    DZNAPDZOQGAHQROAPBIZJKQROSLM $.

  ${
    bianfi.1 $e |- -. ph $.
    $( A wff conjoined with falsehood is false.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Wolf Lammen, 26-Nov-2012.) $)
    bianfi $p |- ( ph <-> ( ps /\ ph ) ) $=
      ( wa intnan 2false ) ABADCABCEF $.
  $}

  ${
    bianfd.1 $e |- ( ph -> -. ps ) $.
    $( A wff conjoined with falsehood is false.  (Contributed by NM,
       27-Mar-1995.)  (Proof shortened by Wolf Lammen, 5-Nov-2013.) $)
    bianfd $p |- ( ph -> ( ps <-> ( ps /\ ch ) ) ) $=
      ( wa intnanrd 2falsed ) ABBCEDABCDFG $.
  $}

  $( Theorem *4.43 of [WhiteheadRussell] p. 119.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 26-Nov-2012.) $)
  pm4.43 $p |- ( ph <-> ( ( ph \/ ps ) /\ ( ph \/ -. ps ) ) ) $=
    ( wn wa wo pm3.24 biorfi ordi bitri ) AABBCZDZEABEAJEDKABFGABJHI $.

  $( Theorem *4.82 of [WhiteheadRussell] p. 122.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.82 $p |- ( ( ( ph -> ps ) /\ ( ph -> -. ps ) ) <-> -. ph ) $=
    ( wi wn wa pm2.65 imp pm2.21 jca impbii ) ABCZABDZCZEADZKMNABFGNKMABHALHIJ
    $.

  $( Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions.
     As with other case elimination theorems, like ~ pm2.61dc , it only holds
     for decidable propositions.  (Contributed by Jim Kingdon, 12-May-2018.) $)
  pm4.83dc $p |- ( DECID ph ->
      ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) <-> ps ) ) $=
    ( wdc wi wn wa wo df-dc pm3.44 com12 sylbi ax-1 jca impbid1 ) ACZABDZAEZBDZ
    FZBOAQGZSBDAHSTBBAQIJKBPRBALBQLMN $.

  $( A transitive law of equivalence.  Compare Theorem *4.22 of
     [WhiteheadRussell] p. 117.  (Contributed by NM, 18-Aug-1993.) $)
  biantr $p |- ( ( ( ph <-> ps ) /\ ( ch <-> ps ) ) -> ( ph <-> ch ) ) $=
    ( wb id bibi2d biimparc ) CBDZACDABDHCBAHEFG $.

  $( Disjunction distributes over the biconditional, for a decidable
     proposition.  Based on an axiom of system DS in Vladimir Lifschitz, "On
     calculational proofs" (1998),
     ~ http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384 .
     (Contributed by Jim Kingdon, 2-Apr-2018.) $)
  orbididc $p |- ( DECID ph -> ( ( ph \/ ( ps <-> ch ) ) <->
                ( ( ph \/ ps ) <-> ( ph \/ ch ) ) ) ) $=
    ( wdc wi wo wa wb orimdidc anbi12d dfbi2 orbi2i ordi bitri 3bitr4g ) ADZABC
    EZFZACBEZFZGZABFZACFZEZUCUBEZGABCHZFZUBUCHPRUDTUEABCIACBIJUGAQSGZFUAUFUHABC
    KLAQSMNUBUCKO $.

  $( Disjunction distributes over the biconditional, for a decidable
     proposition.  Based on theorem *5.7 of [WhiteheadRussell] p. 125.  This
     theorem is similar to ~ orbididc .  (Contributed by Jim Kingdon,
     2-Apr-2018.) $)
  pm5.7dc $p |- ( DECID ch -> ( ( ( ph \/ ch ) <-> ( ps \/ ch ) ) <->
               ( ch \/ ( ph <-> ps ) ) ) ) $=
    ( wdc wb wo orbididc orcom bibi12i bitr2di ) CDCABEFCAFZCBFZEACFZBCFZECABGK
    MLNCAHCBHIJ $.

  $( Dijkstra-Scholten's Golden Rule for calculational proofs.  (Contributed by
     NM, 10-Jan-2005.) $)
  bigolden $p |- ( ( ( ph /\ ps ) <-> ph ) <-> ( ps <-> ( ph \/ ps ) ) ) $=
    ( wi wa wb wo pm4.71 pm4.72 bicom 3bitr3ri ) ABCAABDZEBABFEKAEABGABHAKIJ $.

  $( Conjunction in terms of disjunction (DeMorgan's law).  Theorem *4.5 of
     [WhiteheadRussell] p. 120, but where the propositions are decidable.  The
     forward direction, ~ pm3.1 , holds for all propositions, but the
     equivalence only holds given decidability.  (Contributed by Jim Kingdon,
     21-Apr-2018.) $)
  anordc $p |- ( DECID ph -> ( DECID ps ->
      ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) ) $=
    ( wdc wa wn wo wb dcan ex ianordc 3bitr2rd a1d con2biddc syld ) ACZBCZABDZC
    ZQAEBEFZEGOPRABHIOSQOSQEZGROTSTSABJZUAUAKLMN $.

  $( Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable
     propositions.  The converse, ~ pm3.1 , holds for all propositions, not
     just decidable ones.  (Contributed by Jim Kingdon, 22-Apr-2018.) $)
  pm3.11dc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) ) ) $=
    ( wdc wn wo wa wi wb anordc imp biimprd ex ) ACZBCZADBDEDZABFZGMNFPOMNPOHAB
    IJKL $.

  $( Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable
     propositions.  (Contributed by Jim Kingdon, 22-Apr-2018.) $)
  pm3.12dc $p |- ( DECID ph -> ( DECID ps ->
      ( ( -. ph \/ -. ps ) \/ ( ph /\ ps ) ) ) ) $=
    ( wdc wn wo wa wi pm3.11dc imp wb dcn dcor syl2im dfordc syl6 mpbird ex ) A
    CZBCZADZBDZEZABFZEZRSFUDUBDUCGZRSUEABHIRSUDUEJZRSUBCZUFRTCSUACUGAKBKTUALMUB
    UCNOIPQ $.

  $( Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable
     propositions.  The converse, ~ pm3.14 , holds for all propositions.
     (Contributed by Jim Kingdon, 22-Apr-2018.) $)
  pm3.13dc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) ) ) $=
    ( wdc wn wo wa wi dcn dcor syl2im pm3.11dc con1dc syl6c ) ACZBCZADZBDZEZCZR
    DABFZGTDRGNPCOQCSAHBHPQIJABKRTLM $.

  $( DN_1 for decidable propositions.  Without the decidability conditions,
     DN_1 can serve as a single axiom for Boolean algebra.  See
     ~ http://www-unix.mcs.anl.gov/~~mccune/papers/basax/v12.pdf .
     (Contributed by Jim Kingdon, 22-Apr-2018.) $)
  dn1dc $p |- ( ( DECID ph /\ ( DECID ps /\ ( DECID ch /\ DECID th ) ) ) ->
      ( -. ( -. ( -. ( ph \/ ps ) \/ ch ) \/
            -. ( ph \/ -. ( -. ch \/ -. ( ch \/ th ) ) ) ) <-> ch ) ) $=
    ( wo wn wa wdc wi pm2.45 imnan mpbi biorfi wb dcor imp anordc sylc dcn syl
    orcom ordir 3bitri pm4.45 simprrl ad2antll bitrid orbi2d anbi2d syl6 syldan
    adantrr bitrd bitr2id ) CABEZFZCEZACEZGZAHZBHZCHZDHZGZGZGZUQFACFZCDEZFZEZFZ
    EZFEFZCCUPAGZEVNCEUSVNCUPAFIVNFABJUPAKLMCVNUAUPACUBUCVFUSUQVLGZVMVFURVLUQVF
    CVKACCVHGZVFVKCDUDVFVBVHHZVPVKNUTVAVBVCUEZVDVQUTVAVBVCVQCDOPUFZCVHQRUGUHUIV
    FUQHZVLHZVOVMNVFUPHZVBVTUTVAWBVDUTVAWBUTVAUOHWBABOUOSUJPULVRUPCORUTVEVKHZWA
    VFVJHZWCVFVGHZVIHZWDVFVBWEVRCSTVFVQWFVSVHSTVGVIORVJSTUTWCWAAVKOPUKUQVLQRUMU
    N $.

  $( Decidable proposition version of theorem *5.71 of [WhiteheadRussell]
     p. 125.  (Contributed by Roy F. Longton, 23-Jun-2005.)  (Modified for
     decidability by Jim Kingdon, 19-Apr-2018.) $)
  pm5.71dc $p |- ( DECID ps ->
      ( ( ps -> -. ch ) -> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph /\ ch ) ) ) ) $=
    ( wn wo wa wb wi wdc orel2 orc impbid1 anbi1d a1i pm2.21 pm5.32rd jadc ) BC
    DZABEZCFACFGZBDZTHBIUASACUASABAJABKLMNRCSACSAGOPQ $.

  $( Theorem *5.75 of [WhiteheadRussell] p. 126.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Andrew Salmon, 7-May-2011.)  (Proof
     shortened by Wolf Lammen, 23-Dec-2012.) $)
  pm5.75 $p |- ( ( ( ch -> -. ps ) /\ ( ph <-> ( ps \/ ch ) ) ) ->
                ( ( ph /\ -. ps ) <-> ch ) ) $=
    ( wo wb wn wa wi anbi1 anbi1i pm5.61 bitrdi pm4.71 biimpi bicomd sylan9bbr
    orcom bitri ) ABCDZEZABFZGZCUAGZCUAHZCTUBSUAGZUCASUAIUECBDZUAGUCSUFUABCQJCB
    KRLUDCUCUDCUCECUAMNOP $.

  $( Removal of conjunct from one side of an equivalence.  (Contributed by NM,
     5-Aug-1993.) $)
  bimsc1 $p |- ( ( ( ph -> ps ) /\ ( ch <-> ( ps /\ ph ) ) )
               -> ( ch <-> ph ) ) $=
    ( wi wa wb simpr ancr impbid2 bibi2d biimpa ) ABDZCBAEZFCAFLMACLMABAGABHIJK
    $.

  ${
    ccase.1 $e |- ( ( ph /\ ps ) -> ta ) $.
    ccase.2 $e |- ( ( ch /\ ps ) -> ta ) $.
    ccase.3 $e |- ( ( ph /\ th ) -> ta ) $.
    ccase.4 $e |- ( ( ch /\ th ) -> ta ) $.
    $( Inference for combining cases.  (Contributed by NM, 29-Jul-1999.)
       (Proof shortened by Wolf Lammen, 6-Jan-2013.) $)
    ccase $p |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta ) $=
      ( wo jaoian jaodan ) ACJBEDABECFGKADECHIKL $.
  $}

  ${
    ccased.1 $e |- ( ph -> ( ( ps /\ ch ) -> et ) ) $.
    ccased.2 $e |- ( ph -> ( ( th /\ ch ) -> et ) ) $.
    ccased.3 $e |- ( ph -> ( ( ps /\ ta ) -> et ) ) $.
    ccased.4 $e |- ( ph -> ( ( th /\ ta ) -> et ) ) $.
    $( Deduction for combining cases.  (Contributed by NM, 9-May-2004.) $)
    ccased $p |- ( ph -> ( ( ( ps \/ th ) /\ ( ch \/ ta ) ) -> et ) ) $=
      ( wo wa wi com12 ccase ) BDKCEKLAFBCDEAFMABCLFGNADCLFHNABELFINADELFJNON
      $.
  $}

  ${
    ccase2.1 $e |- ( ( ph /\ ps ) -> ta ) $.
    ccase2.2 $e |- ( ch -> ta ) $.
    ccase2.3 $e |- ( th -> ta ) $.
    $( Inference for combining cases.  (Contributed by NM, 29-Jul-1999.) $)
    ccase2 $p |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta ) $=
      ( adantr adantl ccase ) ABCDEFCEBGIDEAHJDECHJK $.
  $}

  ${
    niabn.1 $e |- ph $.
    $( Miscellaneous inference relating falsehoods.  (Contributed by NM,
       31-Mar-1994.) $)
    niabn $p |- ( -. ps -> ( ( ch /\ ps ) <-> -. ph ) ) $=
      ( wa wn simpr pm2.24i pm5.21ni ) CBEBAFCBGABDHI $.
  $}

  $( Alternate version of ~ dedlema .  (Contributed by NM, 2-Apr-1994.)  (Proof
     shortened by Andrew Salmon, 7-May-2011.)  (Proof shortened by Wolf Lammen,
     4-Dec-2012.) $)
  dedlem0a $p |- ( ph -> ( ps <-> ( ( ch -> ph ) -> ( ps /\ ph ) ) ) ) $=
    ( wa wi iba wb ax-1 biimt syl bitrd ) ABBADZCAEZLEZABFAMLNGACHMLIJK $.

  $( Lemma for ~ iftrue .  (Contributed by NM, 26-Jun-2002.)  (Proof shortened
     by Andrew Salmon, 7-May-2011.) $)
  dedlema $p |- ( ph -> ( ps <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) ) $=
    ( wa wn wo orc expcom wi simpl a1i pm2.24 adantld jaod impbid ) ABBADZCAEZD
    ZFZBASPRGHAPBRPBIABAJKAQBCABLMNO $.

  $( Lemma for ~ iffalse .  (Contributed by NM, 15-May-1999.)  (Proof shortened
     by Andrew Salmon, 7-May-2011.) $)
  dedlemb $p |- ( -. ph -> ( ch <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) ) $=
    ( wn wa wo olc expcom pm2.21 adantld wi simpl a1i jaod impbid ) ADZCBAEZCPE
    ZFZCPSRQGHPQCRPACBACIJRCKPCPLMNO $.

  $( One direction of Theorem *4.42 of [WhiteheadRussell] p. 119.  (Contributed
     by Jim Kingdon, 4-Aug-2018.) $)
  pm4.42r $p |- ( ( ( ph /\ ps ) \/ ( ph /\ -. ps ) ) -> ph ) $=
    ( wa wn simpl jaoi ) ABCAABDZCABEAGEF $.

  ${
    ninba.1 $e |- ph $.
    $( Miscellaneous inference relating falsehoods.  (Contributed by NM,
       31-Mar-1994.) $)
    ninba $p |- ( -. ps -> ( -. ph <-> ( ch /\ ps ) ) ) $=
      ( wn wa niabn bicomd ) BECBFAEABCDGH $.
  $}

  ${
    prlem1.1 $e |- ( ph -> ( et <-> ch ) ) $.
    prlem1.2 $e |- ( ps -> -. th ) $.
    $( A specialized lemma for set theory (to derive the Axiom of Pairing).
       (Contributed by NM, 18-Oct-1995.)  (Proof shortened by Andrew Salmon,
       13-May-2011.)  (Proof shortened by Wolf Lammen, 5-Jan-2013.) $)
    prlem1 $p |- ( ph -> ( ps ->
                  ( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) ) ) $=
      ( wa wo wi biimprd adantld pm2.21d adantrd jaao ex ) ABBCIZDEIZJFKARFBSAC
      FBAFCGLMBDFEBDFHNOPQ $.
  $}

  $( A specialized lemma for set theory (to derive the Axiom of Pairing).
     (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
     13-May-2011.)  (Proof shortened by Wolf Lammen, 9-Dec-2012.) $)
  prlem2 $p |- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <->
              ( ( ph \/ ch ) /\ ( ( ph /\ ps ) \/ ( ch /\ th ) ) ) ) $=
    ( wa wo simpl orim12i pm4.71ri ) ABEZCDEZFACFJAKCABGCDGHI $.

  ${
    oplem1.1 $e |- ( ph -> ( ps \/ ch ) ) $.
    oplem1.2 $e |- ( ph -> ( th \/ ta ) ) $.
    oplem1.3 $e |- ( ps <-> th ) $.
    oplem1.4 $e |- ( ch -> ( th <-> ta ) ) $.
    $( A specialized lemma for set theory (ordered pair theorem).  (Contributed
       by NM, 18-Oct-1995.)  (Proof shortened by Wolf Lammen, 8-Dec-2012.)
       (Proof shortened by Mario Carneiro, 2-Feb-2015.) $)
    oplem1 $p |- ( ph -> ps ) $=
      ( wo idd wi ax-1 biimprcd jaoi syl imbitrrdi jaod mpd ) ABCJBFABBCABKACDB
      ADEJCDLZGDTEDCMCDEINOPHQRS $.
  $}

  $( Lemma used in construction of real numbers.  (Contributed by NM,
     4-Sep-1995.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
  rnlem $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
  ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) /\ ( ( ph /\ th ) /\ ( ps /\ ch ) ) ) ) $=
    ( wa an4 biimpi an42 biimpri jca adantl impbii ) ABECDEEZACEBDEEZADEBCEEZEM
    NOMNABCDFGOMADBCHZIJOMNOMPGKL $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The conditional operator for propositions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This subsection introduces the conditional operator for propositions, denoted
  by ` if- ( ph , ps , ch ) ` (see ~ df-ifp ).  It is the analogue for
  propositions of the conditional operator for classes, denoted by
  ` if ( ph , A , B ) ` (see ~ df-if ).

$)

  $( Comma.  Also used later for adders, pairs, tuples, etc. $)
  $c , $.

  $( Symbol for the conditional operator for propositions. $)
  $c if- $.

  $( Extend wff notation to include the conditional operator for
     propositions. $)
  wif $a wff if- ( ph , ps , ch ) $.

  $( Definition of the conditional operator for propositions.  The expression
     ` if- ( ph , ps , ch ) ` is read "if ` ph ` then ` ps ` else ` ch ` ".
     See ~ dfifp2dc , ~ dfifp3dc , ~ dfifp4dc and ~ dfifp5dc for alternate
     definitions.

     This definition (in the form of ~ dfifp2dc ) appears in Section II.24 of
     [Church] p. 129 (Definition D12 page 132), where it is called "conditioned
     disjunction".  Church's ` [ ps , ph , ch ] ` corresponds to our
     ` if- ( ph , ps , ch ) ` (note the permutation of the first two
     variables).

     This form was chosen as the definition rather than ~ dfifp2dc for
     compatibility with intuitionistic logic development: with this form, it is
     clear that ` if- ( ph , ps , ch ) ` implies decidability of ` ph `
     ( ~ ifpdc ), which is most often what is wanted.

     (Contributed by BJ, 22-Jun-2019.) $)
  df-ifp $a |-
            ( if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) ) $.

  $( The conditional operator for propositions implies decidability.
     (Contributed by Jim Kingdon, 25-Jan-2026.) $)
  ifpdc $p |- ( if- ( ph , ps , ch ) -> DECID ph ) $=
    ( wa wn wo wif wdc simpl orim12i df-ifp df-dc 3imtr4i ) ABDZAEZCDZFAOFABCGA
    HNAPOABIOCIJABCKALM $.

  $( Forward direction of ~ dfifp2dc .  This direction does not require
     decidability.  (Contributed by Jim Kingdon, 25-Jan-2026.) $)
  ifp2 $p |- ( if- ( ph , ps , ch ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) $=
    ( wif wa wn wo wi df-ifp pm3.4 pm2.24 adantr jca ax-in2 jaoi sylbi ) ABCDAB
    EZAFZCEZGABHZRCHZEZABCIQUBSQTUAABJAUABACKLMSTUARTCABNLRCJMOP $.

  $( Alternate definition of the conditional operator for decidable
     propositions.  The value of ` if- ( ph , ps , ch ) ` is "if ` ph ` then
     ` ps ` , and if not ` ph ` then ` ch ` ".  This is the definition used in
     Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of
     ~ df-ifp ).  (Contributed by BJ, 22-Jun-2019.) $)
  dfifp2dc $p |- ( DECID ph ->
      ( if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) ) $=
    ( wdc wif wi wn wa ifp2 wo exmiddc simpl simprl jcai orcd simprr olcd sylan
    jaoian df-ifp sylibr ex impbid2 ) ADZABCEZABFZAGZCFZHZABCIUDUIUEUDUIHABHZUG
    CHZJZUEUDAUGJUIULAKAUIULUGAUIHZUJUKUMABAUILAUFUHMNOUGUIHZUKUJUNUGCUGUILUGUF
    UHPNQSRABCTUAUBUC $.

  $( Alternate definition of the conditional operator for propositions.
     (Contributed by BJ, 30-Sep-2019.) $)
  dfifp3dc $p |- ( DECID ph ->
      ( if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) ) ) $=
    ( wdc wif wi wn wa wo dfifp2dc pm4.64dc anbi2d bitrd ) ADZABCEABFZAGCFZHOAC
    IZHABCJNPQOACKLM $.

  $( Alternate definition of the conditional operator for propositions.
     (Contributed by BJ, 30-Sep-2019.) $)
  dfifp4dc $p |- ( DECID ph ->
      ( if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) ) ) $=
    ( wdc wif wi wo wa wn dfifp3dc imordc anbi1d bitrd ) ADZABCEABFZACGZHAIBGZP
    HABCJNOQPABKLM $.

  $( Alternate definition of the conditional operator for propositions.
     (Contributed by BJ, 2-Oct-2019.) $)
  dfifp5dc $p |- ( DECID ph ->
      ( if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( -. ph -> ch ) ) ) ) $=
    ( wdc wif wi wn wa wo dfifp2dc imordc anbi1d bitrd ) ADZABCEABFZAGZCFZHPBIZ
    QHABCJNORQABKLM $.

  $( Define the biconditional as conditional logic operator.  (Contributed by
     RP, 20-Apr-2020.)  (Proof shortened by Wolf Lammen, 30-Apr-2024.) $)
  ifpdfbidc $p |- ( DECID ph ->
      ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) ) ) $=
    ( wdc wi wa wn wb wif con34bdc anbi2d dfbi2 a1i dfifp2dc 3bitr4d ) ACZABDZB
    ADZEZPAFBFZDZEABGZABSHOQTPBAIJUARGOABKLABSMN $.

  $( The conditional operator is implied by the conjunction of its possible
     outputs.  Dual statement of ~ ifpor .  (Contributed by BJ,
     30-Sep-2019.) $)
  anifpdc $p |- ( DECID ph -> ( ( ps /\ ch ) -> if- ( ph , ps , ch ) ) ) $=
    ( wa wif wdc wn wo olc anim12i dfifp4dc imbitrrid ) BCDABCEAFAGZBHZACHZDBNC
    OBMICAIJABCKL $.

  $( The conditional operator implies the disjunction of its possible outputs.
     Dual statement of ~ anifpdc .  (Contributed by BJ, 1-Oct-2019.) $)
  ifpor $p |- ( if- ( ph , ps , ch ) -> ( ps \/ ch ) ) $=
    ( wif wa wn wo df-ifp simpr orim12i sylbi ) ABCDABEZAFZCEZGBCGABCHLBNCABIMC
    IJK $.

  $( Conditional operator for the negation of a proposition.  (Contributed by
     BJ, 30-Sep-2019.)  (Proof shortened by Wolf Lammen, 5-May-2024.) $)
  ifpnst $p |- ( STAB ph ->
      ( if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) ) ) $=
    ( wstab wif wn wdc ifpdc adantl wa biimpi sylan2 wi wo dfifp5dc biancomd wb
    stdcndc dcn dfifp3dc syl bitr4d pm5.21nd ) ADZABCEZAFZCBEZAGZUEUHUDABCHIUGU
    DUFGZUHUFCBHUDUIJUHARKLUHUEUFCMZUFBNZJZUGUHUEUJUKABCOPUHUIUGULQASUFCBTUAUBU
    C $.

  $( Value of the conditional operator for propositions when its first argument
     is true.  Analogue for propositions of ~ iftrue .  This is essentially
     ~ dedlema .  (Contributed by BJ, 20-Sep-2019.)  (Proof shortened by Wolf
     Lammen, 10-Jul-2020.) $)
  ifptru $p |- ( ph -> ( if- ( ph , ps , ch ) <-> ps ) ) $=
    ( wif wa wn wo df-ifp ancom orbi12i bitri dedlema bitr4id ) AABCDZBAEZCAFZE
    ZGZBNABEZPCEZGRABCHSOTQABIPCIJKABCLM $.

  $( Value of the conditional operator for propositions when its first argument
     is false.  Analogue for propositions of ~ iffalse .  This is essentially
     ~ dedlemb .  (Contributed by BJ, 20-Sep-2019.)  (Proof shortened by Wolf
     Lammen, 25-Jun-2020.) $)
  ifpfal $p |- ( -. ph -> ( if- ( ph , ps , ch ) <-> ch ) ) $=
    ( wn wif wa wo df-ifp ancom orbi12i bitri dedlemb bitr4id ) ADZABCEZBAFZCNF
    ZGZCOABFZNCFZGRABCHSPTQABINCIJKABCLM $.

  $( Value of the conditional operator for propositions when the same
     proposition is returned in either case.  Analogue for propositions of
     ~ ifiddc .  (Contributed by BJ, 20-Sep-2019.) $)
  ifpiddc $p |- ( DECID ph -> ( if- ( ph , ps , ps ) <-> ps ) ) $=
    ( wdc wn wo wif wb exmiddc ifptru ifpfal jaoi syl ) ACAADZEABBFBGZAHANMABBI
    ABBJKL $.

  ${
    ifpbi123d.1 $e |- ( ph -> ( ps <-> ta ) ) $.
    ifpbi123d.2 $e |- ( ph -> ( ch <-> et ) ) $.
    ifpbi123d.3 $e |- ( ph -> ( th <-> ze ) ) $.
    $( Equivalence deduction for conditional operator for propositions.
       (Contributed by AV, 30-Dec-2020.)  (Proof shortened by Wolf Lammen,
       17-Apr-2024.) $)
    ifpbi123d $p |- ( ph -> ( if- ( ps , ch , th )
                              <-> if- ( ta , et , ze ) ) ) $=
      ( wa wn wo wif anbi12d notbid orbi12d df-ifp 3bitr4g ) ABCKZBLZDKZMEFKZEL
      ZGKZMBCDNEFGNATUCUBUEABECFHIOAUAUDDGABEHPJOQBCDREFGRS $.
  $}

  ${
    ifpbi23d.1 $e |- ( ph -> ( ch <-> et ) ) $.
    ifpbi23d.2 $e |- ( ph -> ( th <-> ze ) ) $.
    $( Equivalence deduction for conditional operator for propositions.
       Convenience theorem for a frequent case.  (Contributed by Wolf Lammen,
       28-Apr-2024.) $)
    ifpbi23d $p |- ( ph -> ( if- ( ps , ch , th )
                              <-> if- ( ps , et , ze ) ) ) $=
      ( biidd ifpbi123d ) ABCDBEFABIGHJ $.
  $}

  ${
    1fpid3.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( The value of the conditional operator for propositions is its third
       argument if the first and second argument imply the third argument.
       (Contributed by AV, 4-Apr-2021.) $)
    1fpid3 $p |- ( if- ( ph , ps , ch ) -> ch ) $=
      ( wif wa wn wo df-ifp simpr jaoi sylbi ) ABCEABFZAGZCFZHCABCIMCODNCJKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Abbreviated conjunction and disjunction of three wff's
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Extend wff definition to include 3-way disjunction ('or'). $)
  w3o $a wff ( ph \/ ps \/ ch ) $.
  $( Extend wff definition to include 3-way conjunction ('and'). $)
  w3a $a wff ( ph /\ ps /\ ch ) $.

  $( These abbreviations help eliminate parentheses to aid readability. $)

  $( Define disjunction ('or') of 3 wff's.  Definition *2.33 of
     [WhiteheadRussell] p. 105.  This abbreviation reduces the number of
     parentheses and emphasizes that the order of bracketing is not important
     by virtue of the associative law ~ orass .  (Contributed by NM,
     8-Apr-1994.) $)
  df-3or $a |- ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) ) $.

  $( Define conjunction ('and') of 3 wff.s.  Definition *4.34 of
     [WhiteheadRussell] p. 118.  This abbreviation reduces the number of
     parentheses and emphasizes that the order of bracketing is not important
     by virtue of the associative law ~ anass .  (Contributed by NM,
     8-Apr-1994.) $)
  df-3an $a |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) $.

  $( Associative law for triple disjunction.  (Contributed by NM,
     8-Apr-1994.) $)
  3orass $p |- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) ) $=
    ( w3o wo df-3or orass bitri ) ABCDABECEABCEEABCFABCGH $.

  $( Associative law for triple conjunction.  (Contributed by NM,
     8-Apr-1994.) $)
  3anass $p |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) ) $=
    ( w3a wa df-3an anass bitri ) ABCDABECEABCEEABCFABCGH $.

  $( Rotation law for triple conjunction.  (Contributed by NM, 8-Apr-1994.) $)
  3anrot $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ch /\ ph ) ) $=
    ( wa w3a ancom 3anass df-3an 3bitr4i ) ABCDZDJADABCEBCAEAJFABCGBCAHI $.

  $( Rotation law for triple disjunction.  (Contributed by NM, 4-Apr-1995.) $)
  3orrot $p |- ( ( ph \/ ps \/ ch ) <-> ( ps \/ ch \/ ph ) ) $=
    ( wo w3o orcom 3orass df-3or 3bitr4i ) ABCDZDJADABCEBCAEAJFABCGBCAHI $.

  $( Commutation law for triple conjunction.  (Contributed by NM,
     21-Apr-1994.) $)
  3ancoma $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ph /\ ch ) ) $=
    ( wa w3a ancom anbi1i df-3an 3bitr4i ) ABDZCDBADZCDABCEBACEJKCABFGABCHBACHI
    $.

  $( Commutation law for triple conjunction.  (Contributed by NM,
     21-Apr-1994.) $)
  3ancomb $p |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ch /\ ps ) ) $=
    ( w3a 3ancoma 3anrot bitri ) ABCDBACDACBDABCEBACFG $.

  $( Commutation law for triple disjunction.  (Contributed by Scott Fenton,
     20-Apr-2011.) $)
  3orcomb $p |- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ch \/ ps ) ) $=
    ( wo w3o orcom orbi2i 3orass 3bitr4i ) ABCDZDACBDZDABCEACBEJKABCFGABCHACBHI
    $.

  $( Reversal law for triple conjunction.  (Contributed by NM, 21-Apr-1994.) $)
  3anrev $p |- ( ( ph /\ ps /\ ch ) <-> ( ch /\ ps /\ ph ) ) $=
    ( w3a 3ancoma 3anrot bitr4i ) ABCDBACDCBADABCECBAFG $.

  $( Convert triple conjunction to conjunction, then commute.  (Contributed by
     Jonathan Ben-Naim, 3-Jun-2011.) $)
  3anan32 $p |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ch ) /\ ps ) ) $=
    ( w3a wa df-3an an32 bitri ) ABCDABECEACEBEABCFABCGH $.

  $( Convert triple conjunction to conjunction, then commute.  (Contributed by
     Jonathan Ben-Naim, 3-Jun-2011.)  (Proof shortened by Andrew Salmon,
     14-Jun-2011.) $)
  3anan12 $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
    ( w3a wa 3ancoma 3anass bitri ) ABCDBACDBACEEABCFBACGH $.

  $( Distribution of triple conjunction over conjunction.  (Contributed by
     David A. Wheeler, 4-Nov-2018.) $)
  anandi3 $p |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )
    $=
    ( w3a wa 3anass anandi bitri ) ABCDABCEEABEACEEABCFABCGH $.

  $( Distribution of triple conjunction over conjunction.  (Contributed by
     David A. Wheeler, 4-Nov-2018.) $)
  anandi3r $p |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ( ch /\ ps ) ) )
    $=
    ( w3a wa 3anan32 anandir bitri ) ABCDACEBEABECBEEABCFACBGH $.

  $( Negated triple disjunction as triple conjunction.  (Contributed by Scott
     Fenton, 19-Apr-2011.) $)
  3ioran $p |- ( -. ( ph \/ ps \/ ch ) <-> ( -. ph /\ -. ps /\ -. ch ) ) $=
    ( wo wn wa w3o w3a ioran anbi1i df-3or xchnxbir df-3an 3bitr4i ) ABDZEZCEZF
    ZAEZBEZFZQFABCGZESTQHPUAQABIJOCDRUBOCIABCKLSTQMN $.

  $( Simplification of triple conjunction.  (Contributed by NM,
     21-Apr-1994.) $)
  3simpa $p |- ( ( ph /\ ps /\ ch ) -> ( ph /\ ps ) ) $=
    ( w3a wa df-3an simplbi ) ABCDABECABCFG $.

  $( Simplification of triple conjunction.  (Contributed by NM,
     21-Apr-1994.) $)
  3simpb $p |- ( ( ph /\ ps /\ ch ) -> ( ph /\ ch ) ) $=
    ( w3a wa 3ancomb 3simpa sylbi ) ABCDACBDACEABCFACBGH $.

  $( Simplification of triple conjunction.  (Contributed by NM, 21-Apr-1994.)
     (Proof shortened by Andrew Salmon, 13-May-2011.) $)
  3simpc $p |- ( ( ph /\ ps /\ ch ) -> ( ps /\ ch ) ) $=
    ( w3a wa 3anrot 3simpa sylbi ) ABCDBCADBCEABCFBCAGH $.

  $( Simplification of triple conjunction.  (Contributed by NM,
     21-Apr-1994.) $)
  simp1 $p |- ( ( ph /\ ps /\ ch ) -> ph ) $=
    ( w3a 3simpa simpld ) ABCDABABCEF $.

  $( Simplification of triple conjunction.  (Contributed by NM,
     21-Apr-1994.) $)
  simp2 $p |- ( ( ph /\ ps /\ ch ) -> ps ) $=
    ( w3a 3simpa simprd ) ABCDABABCEF $.

  $( Simplification of triple conjunction.  (Contributed by NM,
     21-Apr-1994.) $)
  simp3 $p |- ( ( ph /\ ps /\ ch ) -> ch ) $=
    ( w3a 3simpc simprd ) ABCDBCABCEF $.

  $( Simplification rule.  (Contributed by Jeff Hankins, 17-Nov-2009.) $)
  simpl1 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ph ) $=
    ( w3a simp1 adantr ) ABCEADABCFG $.

  $( Simplification rule.  (Contributed by Jeff Hankins, 17-Nov-2009.) $)
  simpl2 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ps ) $=
    ( w3a simp2 adantr ) ABCEBDABCFG $.

  $( Simplification rule.  (Contributed by Jeff Hankins, 17-Nov-2009.) $)
  simpl3 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ch ) $=
    ( w3a simp3 adantr ) ABCECDABCFG $.

  $( Simplification rule.  (Contributed by Jeff Hankins, 17-Nov-2009.) $)
  simpr1 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ps ) $=
    ( w3a simp1 adantl ) BCDEBABCDFG $.

  $( Simplification rule.  (Contributed by Jeff Hankins, 17-Nov-2009.) $)
  simpr2 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ch ) $=
    ( w3a simp2 adantl ) BCDECABCDFG $.

  $( Simplification rule.  (Contributed by Jeff Hankins, 17-Nov-2009.) $)
  simpr3 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> th ) $=
    ( w3a simp3 adantl ) BCDEDABCDFG $.

  ${
    3simp1i.1 $e |- ( ph /\ ps /\ ch ) $.
    $( Infer a conjunct from a triple conjunction.  (Contributed by NM,
       19-Apr-2005.) $)
    simp1i $p |- ph $=
      ( w3a simp1 ax-mp ) ABCEADABCFG $.

    $( Infer a conjunct from a triple conjunction.  (Contributed by NM,
       19-Apr-2005.) $)
    simp2i $p |- ps $=
      ( w3a simp2 ax-mp ) ABCEBDABCFG $.

    $( Infer a conjunct from a triple conjunction.  (Contributed by NM,
       19-Apr-2005.) $)
    simp3i $p |- ch $=
      ( w3a simp3 ax-mp ) ABCECDABCFG $.
  $}

  ${
    3simp1d.1 $e |- ( ph -> ( ps /\ ch /\ th ) ) $.
    $( Deduce a conjunct from a triple conjunction.  (Contributed by NM,
       4-Sep-2005.) $)
    simp1d $p |- ( ph -> ps ) $=
      ( w3a simp1 syl ) ABCDFBEBCDGH $.

    $( Deduce a conjunct from a triple conjunction.  (Contributed by NM,
       4-Sep-2005.) $)
    simp2d $p |- ( ph -> ch ) $=
      ( w3a simp2 syl ) ABCDFCEBCDGH $.

    $( Deduce a conjunct from a triple conjunction.  (Contributed by NM,
       4-Sep-2005.) $)
    simp3d $p |- ( ph -> th ) $=
      ( w3a simp3 syl ) ABCDFDEBCDGH $.
  $}

  ${
    3simp1bi.1 $e |- ( ph <-> ( ps /\ ch /\ th ) ) $.
    $( Deduce a conjunct from a triple conjunction.  (Contributed by Jonathan
       Ben-Naim, 3-Jun-2011.) $)
    simp1bi $p |- ( ph -> ps ) $=
      ( w3a biimpi simp1d ) ABCDABCDFEGH $.

    $( Deduce a conjunct from a triple conjunction.  (Contributed by Jonathan
       Ben-Naim, 3-Jun-2011.) $)
    simp2bi $p |- ( ph -> ch ) $=
      ( w3a biimpi simp2d ) ABCDABCDFEGH $.

    $( Deduce a conjunct from a triple conjunction.  (Contributed by Jonathan
       Ben-Naim, 3-Jun-2011.) $)
    simp3bi $p |- ( ph -> th ) $=
      ( w3a biimpi simp3d ) ABCDABCDFEGH $.
  $}

  ${
    3adant.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       16-Jul-1995.) $)
    3adant1 $p |- ( ( th /\ ph /\ ps ) -> ch ) $=
      ( w3a wa 3simpc syl ) DABFABGCDABHEI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       16-Jul-1995.) $)
    3adant2 $p |- ( ( ph /\ th /\ ps ) -> ch ) $=
      ( w3a wa 3simpb syl ) ADBFABGCADBHEI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       16-Jul-1995.) $)
    3adant3 $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
      ( w3a wa 3simpa syl ) ABDFABGCABDHEI $.
  $}

  ${
    3ad2ant.1 $e |- ( ph -> ch ) $.
    $( Deduction adding conjuncts to an antecedent.  (Contributed by NM,
       21-Apr-2005.) $)
    3ad2ant1 $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
      ( adantr 3adant2 ) ADCBACDEFG $.

    $( Deduction adding conjuncts to an antecedent.  (Contributed by NM,
       21-Apr-2005.) $)
    3ad2ant2 $p |- ( ( ps /\ ph /\ th ) -> ch ) $=
      ( adantr 3adant1 ) ADCBACDEFG $.

    $( Deduction adding conjuncts to an antecedent.  (Contributed by NM,
       21-Apr-2005.) $)
    3ad2ant3 $p |- ( ( ps /\ th /\ ph ) -> ch ) $=
      ( adantl 3adant1 ) DACBACDEFG $.
  $}

  $( Simplification of triple conjunction.  (Contributed by NM, 9-Nov-2011.) $)
  simp1l $p |- ( ( ( ph /\ ps ) /\ ch /\ th ) -> ph ) $=
    ( wa simpl 3ad2ant1 ) ABECADABFG $.

  $( Simplification of triple conjunction.  (Contributed by NM, 9-Nov-2011.) $)
  simp1r $p |- ( ( ( ph /\ ps ) /\ ch /\ th ) -> ps ) $=
    ( wa simpr 3ad2ant1 ) ABECBDABFG $.

  $( Simplification of triple conjunction.  (Contributed by NM, 9-Nov-2011.) $)
  simp2l $p |- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ps ) $=
    ( wa simpl 3ad2ant2 ) BCEABDBCFG $.

  $( Simplification of triple conjunction.  (Contributed by NM, 9-Nov-2011.) $)
  simp2r $p |- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ch ) $=
    ( wa simpr 3ad2ant2 ) BCEACDBCFG $.

  $( Simplification of triple conjunction.  (Contributed by NM, 9-Nov-2011.) $)
  simp3l $p |- ( ( ph /\ ps /\ ( ch /\ th ) ) -> ch ) $=
    ( wa simpl 3ad2ant3 ) CDEACBCDFG $.

  $( Simplification of triple conjunction.  (Contributed by NM, 9-Nov-2011.) $)
  simp3r $p |- ( ( ph /\ ps /\ ( ch /\ th ) ) -> th ) $=
    ( wa simpr 3ad2ant3 ) CDEADBCDFG $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp11 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ph ) $=
    ( w3a simp1 3ad2ant1 ) ABCFDAEABCGH $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp12 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ps ) $=
    ( w3a simp2 3ad2ant1 ) ABCFDBEABCGH $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp13 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ch ) $=
    ( w3a simp3 3ad2ant1 ) ABCFDCEABCGH $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp21 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> ps ) $=
    ( w3a simp1 3ad2ant2 ) BCDFABEBCDGH $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp22 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> ch ) $=
    ( w3a simp2 3ad2ant2 ) BCDFACEBCDGH $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp23 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> th ) $=
    ( w3a simp3 3ad2ant2 ) BCDFADEBCDGH $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp31 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> ch ) $=
    ( w3a simp1 3ad2ant3 ) CDEFACBCDEGH $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp32 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> th ) $=
    ( w3a simp2 3ad2ant3 ) CDEFADBCDEGH $.

  $( Simplification of doubly triple conjunction.  (Contributed by NM,
     17-Nov-2011.) $)
  simp33 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> ta ) $=
    ( w3a simp3 3ad2ant3 ) CDEFAEBCDEGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpll1 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ph ) $=
    ( w3a wa simpl1 adantr ) ABCFDGAEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpll2 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ps ) $=
    ( w3a wa simpl2 adantr ) ABCFDGBEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpll3 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ch ) $=
    ( w3a wa simpl3 adantr ) ABCFDGCEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simplr1 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ph ) $=
    ( w3a wa simpr1 adantr ) DABCFGAEDABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simplr2 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ps ) $=
    ( w3a wa simpr2 adantr ) DABCFGBEDABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simplr3 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ch ) $=
    ( w3a wa simpr3 adantr ) DABCFGCEDABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simprl1 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph ) $=
    ( w3a wa simpl1 adantl ) ABCFDGAEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simprl2 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps ) $=
    ( w3a wa simpl2 adantl ) ABCFDGBEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simprl3 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch ) $=
    ( w3a wa simpl3 adantl ) ABCFDGCEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simprr1 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
    ( w3a wa simpr1 adantl ) DABCFGAEDABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simprr2 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
    ( w3a wa simpr2 adantl ) DABCFGBEDABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simprr3 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
    ( w3a wa simpr3 adantl ) DABCFGCEDABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl1l $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ph ) $=
    ( wa w3a simp1l adantr ) ABFCDGAEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl1r $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ps ) $=
    ( wa w3a simp1r adantr ) ABFCDGBEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl2l $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ph ) $=
    ( wa w3a simp2l adantr ) CABFDGAECABDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl2r $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ps ) $=
    ( wa w3a simp2r adantr ) CABFDGBECABDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl3l $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ph ) $=
    ( wa w3a simp3l adantr ) CDABFGAECDABHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl3r $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ps ) $=
    ( wa w3a simp3r adantr ) CDABFGBECDABHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr1l $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph ) $=
    ( wa w3a simp1l adantl ) ABFCDGAEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr1r $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps ) $=
    ( wa w3a simp1r adantl ) ABFCDGBEABCDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr2l $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph ) $=
    ( wa w3a simp2l adantl ) CABFDGAECABDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr2r $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps ) $=
    ( wa w3a simp2r adantl ) CABFDGBECABDHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr3l $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph ) $=
    ( wa w3a simp3l adantl ) CDABFGAECDABHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr3r $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps ) $=
    ( wa w3a simp3r adantl ) CDABFGBECDABHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1ll $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th /\ ta ) -> ph ) $=
    ( wa simpll 3ad2ant1 ) ABFCFDAEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1lr $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th /\ ta ) -> ps ) $=
    ( wa simplr 3ad2ant1 ) ABFCFDBEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1rl $p |- ( ( ( ch /\ ( ph /\ ps ) ) /\ th /\ ta ) -> ph ) $=
    ( wa simprl 3ad2ant1 ) CABFFDAECABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1rr $p |- ( ( ( ch /\ ( ph /\ ps ) ) /\ th /\ ta ) -> ps ) $=
    ( wa simprr 3ad2ant1 ) CABFFDBECABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2ll $p |- ( ( th /\ ( ( ph /\ ps ) /\ ch ) /\ ta ) -> ph ) $=
    ( wa simpll 3ad2ant2 ) ABFCFDAEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2lr $p |- ( ( th /\ ( ( ph /\ ps ) /\ ch ) /\ ta ) -> ps ) $=
    ( wa simplr 3ad2ant2 ) ABFCFDBEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2rl $p |- ( ( th /\ ( ch /\ ( ph /\ ps ) ) /\ ta ) -> ph ) $=
    ( wa simprl 3ad2ant2 ) CABFFDAECABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2rr $p |- ( ( th /\ ( ch /\ ( ph /\ ps ) ) /\ ta ) -> ps ) $=
    ( wa simprr 3ad2ant2 ) CABFFDBECABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3ll $p |- ( ( th /\ ta /\ ( ( ph /\ ps ) /\ ch ) ) -> ph ) $=
    ( wa simpll 3ad2ant3 ) ABFCFDAEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3lr $p |- ( ( th /\ ta /\ ( ( ph /\ ps ) /\ ch ) ) -> ps ) $=
    ( wa simplr 3ad2ant3 ) ABFCFDBEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3rl $p |- ( ( th /\ ta /\ ( ch /\ ( ph /\ ps ) ) ) -> ph ) $=
    ( wa simprl 3ad2ant3 ) CABFFDAECABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3rr $p |- ( ( th /\ ta /\ ( ch /\ ( ph /\ ps ) ) ) -> ps ) $=
    ( wa simprr 3ad2ant3 ) CABFFDBECABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl11 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ph ) $=
    ( w3a simp11 adantr ) ABCGDEGAFABCDEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl12 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ps ) $=
    ( w3a simp12 adantr ) ABCGDEGBFABCDEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl13 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ch ) $=
    ( w3a simp13 adantr ) ABCGDEGCFABCDEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl21 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ph ) $=
    ( w3a simp21 adantr ) DABCGEGAFDABCEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl22 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ps ) $=
    ( w3a simp22 adantr ) DABCGEGBFDABCEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl23 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ch ) $=
    ( w3a simp23 adantr ) DABCGEGCFDABCEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl31 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ph ) $=
    ( w3a simp31 adantr ) DEABCGGAFDEABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl32 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ps ) $=
    ( w3a simp32 adantr ) DEABCGGBFDEABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpl33 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ch ) $=
    ( w3a simp33 adantr ) DEABCGGCFDEABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr11 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ph ) $=
    ( w3a simp11 adantl ) ABCGDEGAFABCDEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr12 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ps ) $=
    ( w3a simp12 adantl ) ABCGDEGBFABCDEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr13 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ch ) $=
    ( w3a simp13 adantl ) ABCGDEGCFABCDEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr21 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ph ) $=
    ( w3a simp21 adantl ) DABCGEGAFDABCEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr22 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ps ) $=
    ( w3a simp22 adantl ) DABCGEGBFDABCEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr23 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ch ) $=
    ( w3a simp23 adantl ) DABCGEGCFDABCEHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr31 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
    ( w3a simp31 adantl ) DEABCGGAFDEABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr32 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
    ( w3a simp32 adantl ) DEABCGGBFDEABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simpr33 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
    ( w3a simp33 adantl ) DEABCGGCFDEABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1l1 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ph ) $=
    ( w3a wa simpl1 3ad2ant1 ) ABCGDHEAFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1l2 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ps ) $=
    ( w3a wa simpl2 3ad2ant1 ) ABCGDHEBFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1l3 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ch ) $=
    ( w3a wa simpl3 3ad2ant1 ) ABCGDHECFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1r1 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ph ) $=
    ( w3a wa simpr1 3ad2ant1 ) DABCGHEAFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1r2 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ps ) $=
    ( w3a wa simpr2 3ad2ant1 ) DABCGHEBFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp1r3 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ch ) $=
    ( w3a wa simpr3 3ad2ant1 ) DABCGHECFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2l1 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ph ) $=
    ( w3a wa simpl1 3ad2ant2 ) ABCGDHEAFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2l2 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ps ) $=
    ( w3a wa simpl2 3ad2ant2 ) ABCGDHEBFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2l3 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ch ) $=
    ( w3a wa simpl3 3ad2ant2 ) ABCGDHECFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2r1 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ph ) $=
    ( w3a wa simpr1 3ad2ant2 ) DABCGHEAFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2r2 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ps ) $=
    ( w3a wa simpr2 3ad2ant2 ) DABCGHEBFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp2r3 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ch ) $=
    ( w3a wa simpr3 3ad2ant2 ) DABCGHECFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3l1 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph ) $=
    ( w3a wa simpl1 3ad2ant3 ) ABCGDHEAFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3l2 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps ) $=
    ( w3a wa simpl2 3ad2ant3 ) ABCGDHEBFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3l3 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch ) $=
    ( w3a wa simpl3 3ad2ant3 ) ABCGDHECFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3r1 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
    ( w3a wa simpr1 3ad2ant3 ) DABCGHEAFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3r2 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
    ( w3a wa simpr2 3ad2ant3 ) DABCGHEBFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp3r3 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
    ( w3a wa simpr3 3ad2ant3 ) DABCGHECFDABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp11l $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta /\ et ) -> ph ) $=
    ( wa w3a simp1l 3ad2ant1 ) ABGCDHEAFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp11r $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta /\ et ) -> ps ) $=
    ( wa w3a simp1r 3ad2ant1 ) ABGCDHEBFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp12l $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta /\ et ) -> ph ) $=
    ( wa w3a simp2l 3ad2ant1 ) CABGDHEAFCABDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp12r $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta /\ et ) -> ps ) $=
    ( wa w3a simp2r 3ad2ant1 ) CABGDHEBFCABDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp13l $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta /\ et ) -> ph ) $=
    ( wa w3a simp3l 3ad2ant1 ) CDABGHEAFCDABIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp13r $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta /\ et ) -> ps ) $=
    ( wa w3a simp3r 3ad2ant1 ) CDABGHEBFCDABIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp21l $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) /\ et ) -> ph ) $=
    ( wa w3a simp1l 3ad2ant2 ) ABGCDHEAFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp21r $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) /\ et ) -> ps ) $=
    ( wa w3a simp1r 3ad2ant2 ) ABGCDHEBFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp22l $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) /\ et ) -> ph ) $=
    ( wa w3a simp2l 3ad2ant2 ) CABGDHEAFCABDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp22r $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) /\ et ) -> ps ) $=
    ( wa w3a simp2r 3ad2ant2 ) CABGDHEBFCABDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp23l $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) /\ et ) -> ph ) $=
    ( wa w3a simp3l 3ad2ant2 ) CDABGHEAFCDABIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp23r $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) /\ et ) -> ps ) $=
    ( wa w3a simp3r 3ad2ant2 ) CDABGHEBFCDABIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp31l $p |- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph ) $=
    ( wa w3a simp1l 3ad2ant3 ) ABGCDHEAFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp31r $p |- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps ) $=
    ( wa w3a simp1r 3ad2ant3 ) ABGCDHEBFABCDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp32l $p |- ( ( ta /\ et /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph ) $=
    ( wa w3a simp2l 3ad2ant3 ) CABGDHEAFCABDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp32r $p |- ( ( ta /\ et /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps ) $=
    ( wa w3a simp2r 3ad2ant3 ) CABGDHEBFCABDIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp33l $p |- ( ( ta /\ et /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph ) $=
    ( wa w3a simp3l 3ad2ant3 ) CDABGHEAFCDABIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp33r $p |- ( ( ta /\ et /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps ) $=
    ( wa w3a simp3r 3ad2ant3 ) CDABGHEBFCDABIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp111 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ph ) $=
    ( w3a simp11 3ad2ant1 ) ABCHDEHFAGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp112 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ps ) $=
    ( w3a simp12 3ad2ant1 ) ABCHDEHFBGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp113 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ch ) $=
    ( w3a simp13 3ad2ant1 ) ABCHDEHFCGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp121 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ph ) $=
    ( w3a simp21 3ad2ant1 ) DABCHEHFAGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp122 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ps ) $=
    ( w3a simp22 3ad2ant1 ) DABCHEHFBGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp123 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ch ) $=
    ( w3a simp23 3ad2ant1 ) DABCHEHFCGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp131 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ph ) $=
    ( w3a simp31 3ad2ant1 ) DEABCHHFAGDEABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp132 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ps ) $=
    ( w3a simp32 3ad2ant1 ) DEABCHHFBGDEABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp133 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ch ) $=
    ( w3a simp33 3ad2ant1 ) DEABCHHFCGDEABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp211 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ph ) $=
    ( w3a simp11 3ad2ant2 ) ABCHDEHFAGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp212 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ps ) $=
    ( w3a simp12 3ad2ant2 ) ABCHDEHFBGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp213 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ch ) $=
    ( w3a simp13 3ad2ant2 ) ABCHDEHFCGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp221 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ph ) $=
    ( w3a simp21 3ad2ant2 ) DABCHEHFAGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp222 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ps ) $=
    ( w3a simp22 3ad2ant2 ) DABCHEHFBGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp223 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ch ) $=
    ( w3a simp23 3ad2ant2 ) DABCHEHFCGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp231 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ph ) $=
    ( w3a simp31 3ad2ant2 ) DEABCHHFAGDEABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp232 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ps ) $=
    ( w3a simp32 3ad2ant2 ) DEABCHHFBGDEABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp233 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ch ) $=
    ( w3a simp33 3ad2ant2 ) DEABCHHFCGDEABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp311 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ph ) $=
    ( w3a simp11 3ad2ant3 ) ABCHDEHFAGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp312 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ps ) $=
    ( w3a simp12 3ad2ant3 ) ABCHDEHFBGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp313 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ch ) $=
    ( w3a simp13 3ad2ant3 ) ABCHDEHFCGABCDEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp321 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ph ) $=
    ( w3a simp21 3ad2ant3 ) DABCHEHFAGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp322 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ps ) $=
    ( w3a simp22 3ad2ant3 ) DABCHEHFBGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp323 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ch ) $=
    ( w3a simp23 3ad2ant3 ) DABCHEHFCGDABCEIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp331 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
    ( w3a simp31 3ad2ant3 ) DEABCHHFAGDEABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp332 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
    ( w3a simp32 3ad2ant3 ) DEABCHHFBGDEABCIJ $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.) $)
  simp333 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
    ( w3a simp33 3ad2ant3 ) DEABCHHFCGDEABCIJ $.

  ${
    3adantl.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       24-Feb-2005.) $)
    3adantl1 $p |- ( ( ( ta /\ ph /\ ps ) /\ ch ) -> th ) $=
      ( w3a wa 3simpc sylan ) EABGABHCDEABIFJ $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       24-Feb-2005.) $)
    3adantl2 $p |- ( ( ( ph /\ ta /\ ps ) /\ ch ) -> th ) $=
      ( w3a wa 3simpb sylan ) AEBGABHCDAEBIFJ $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       24-Feb-2005.) $)
    3adantl3 $p |- ( ( ( ph /\ ps /\ ta ) /\ ch ) -> th ) $=
      ( w3a wa 3simpa sylan ) ABEGABHCDABEIFJ $.
  $}

  ${
    3adantr.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       27-Apr-2005.) $)
    3adantr1 $p |- ( ( ph /\ ( ta /\ ps /\ ch ) ) -> th ) $=
      ( w3a wa 3simpc sylan2 ) EBCGABCHDEBCIFJ $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       27-Apr-2005.) $)
    3adantr2 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
      ( w3a wa 3simpb sylan2 ) BECGABCHDBECIFJ $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       27-Apr-2005.) $)
    3adantr3 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
      ( w3a wa 3simpa sylan2 ) BCEGABCHDBCEIFJ $.
  $}

  ${
    3ad2antl.1 $e |- ( ( ph /\ ch ) -> th ) $.
    $( Deduction adding conjuncts to antecedent.  (Contributed by NM,
       4-Aug-2007.) $)
    3ad2antl1 $p |- ( ( ( ph /\ ps /\ ta ) /\ ch ) -> th ) $=
      ( adantlr 3adantl2 ) AECDBACDEFGH $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by NM,
       4-Aug-2007.) $)
    3ad2antl2 $p |- ( ( ( ps /\ ph /\ ta ) /\ ch ) -> th ) $=
      ( adantlr 3adantl1 ) AECDBACDEFGH $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by NM,
       4-Aug-2007.) $)
    3ad2antl3 $p |- ( ( ( ps /\ ta /\ ph ) /\ ch ) -> th ) $=
      ( adantll 3adantl1 ) EACDBACDEFGH $.

    $( Deduction adding a conjuncts to antecedent.  (Contributed by NM,
       25-Dec-2007.) $)
    3ad2antr1 $p |- ( ( ph /\ ( ch /\ ps /\ ta ) ) -> th ) $=
      ( adantrr 3adantr3 ) ACBDEACDBFGH $.

    $( Deduction adding a conjuncts to antecedent.  (Contributed by NM,
       27-Dec-2007.) $)
    3ad2antr2 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
      ( adantrl 3adantr3 ) ABCDEACDBFGH $.

    $( Deduction adding a conjuncts to antecedent.  (Contributed by NM,
       30-Dec-2007.) $)
    3ad2antr3 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
      ( adantrl 3adantr1 ) AECDBACDEFGH $.
  $}

  ${
    3anibar.1 $e |- ( ( ph /\ ps /\ ch ) -> ( th <-> ( ch /\ ta ) ) ) $.
    $( Remove a hypothesis from the second member of a biconditional.
       (Contributed by FL, 22-Jul-2008.) $)
    3anibar $p |- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) $=
      ( w3a wa simp3 biantrurd bitr4d ) ABCGZDCEHEFLCEABCIJK $.
  $}

  $( Introduction in triple disjunction.  (Contributed by NM, 4-Apr-1995.) $)
  3mix1 $p |- ( ph -> ( ph \/ ps \/ ch ) ) $=
    ( wo w3o orc 3orass sylibr ) AABCDZDABCEAIFABCGH $.

  $( Introduction in triple disjunction.  (Contributed by NM, 4-Apr-1995.) $)
  3mix2 $p |- ( ph -> ( ps \/ ph \/ ch ) ) $=
    ( w3o 3mix1 3orrot sylibr ) AACBDBACDACBEBACFG $.

  $( Introduction in triple disjunction.  (Contributed by NM, 4-Apr-1995.) $)
  3mix3 $p |- ( ph -> ( ps \/ ch \/ ph ) ) $=
    ( w3o 3mix1 3orrot sylib ) AABCDBCADABCEABCFG $.

  ${
    3mixi.1 $e |- ph $.
    $( Introduction in triple disjunction.  (Contributed by Mario Carneiro,
       6-Oct-2014.) $)
    3mix1i $p |- ( ph \/ ps \/ ch ) $=
      ( w3o 3mix1 ax-mp ) AABCEDABCFG $.

    $( Introduction in triple disjunction.  (Contributed by Mario Carneiro,
       6-Oct-2014.) $)
    3mix2i $p |- ( ps \/ ph \/ ch ) $=
      ( w3o 3mix2 ax-mp ) ABACEDABCFG $.

    $( Introduction in triple disjunction.  (Contributed by Mario Carneiro,
       6-Oct-2014.) $)
    3mix3i $p |- ( ps \/ ch \/ ph ) $=
      ( w3o 3mix3 ax-mp ) ABCAEDABCFG $.
  $}

  ${
    3mixd.1 $e |- ( ph -> ps ) $.
    $( Deduction introducing triple disjunction.  (Contributed by Scott Fenton,
       8-Jun-2011.) $)
    3mix1d $p |- ( ph -> ( ps \/ ch \/ th ) ) $=
      ( w3o 3mix1 syl ) ABBCDFEBCDGH $.

    $( Deduction introducing triple disjunction.  (Contributed by Scott Fenton,
       8-Jun-2011.) $)
    3mix2d $p |- ( ph -> ( ch \/ ps \/ th ) ) $=
      ( w3o 3mix2 syl ) ABCBDFEBCDGH $.

    $( Deduction introducing triple disjunction.  (Contributed by Scott Fenton,
       8-Jun-2011.) $)
    3mix3d $p |- ( ph -> ( ch \/ th \/ ps ) ) $=
      ( w3o 3mix3 syl ) ABCDBFEBCDGH $.
  $}

  ${
    3pm3.2i.1 $e |- ph $.
    3pm3.2i.2 $e |- ps $.
    3pm3.2i.3 $e |- ch $.
    $( Infer conjunction of premises.  (Contributed by NM, 10-Feb-1995.) $)
    3pm3.2i $p |- ( ph /\ ps /\ ch ) $=
      ( w3a wa pm3.2i df-3an mpbir2an ) ABCGABHCABDEIFABCJK $.
  $}

  $( ~ pm3.2 for a triple conjunction.  (Contributed by Alan Sare,
     24-Oct-2011.) $)
  pm3.2an3 $p |- ( ph -> ( ps -> ( ch -> ( ph /\ ps /\ ch ) ) ) ) $=
    ( wa w3a wi pm3.2 ex df-3an bicomi syl8ib ) ABCABDZCDZABCEZABCMFLCGHNMABCIJ
    K $.

  ${
    3jca.1 $e |- ( ph -> ps ) $.
    3jca.2 $e |- ( ph -> ch ) $.
    3jca.3 $e |- ( ph -> th ) $.
    $( Join consequents with conjunction.  (Contributed by NM, 9-Apr-1994.) $)
    3jca $p |- ( ph -> ( ps /\ ch /\ th ) ) $=
      ( wa w3a jca31 df-3an sylibr ) ABCHDHBCDIABCDEFGJBCDKL $.
  $}

  ${
    3jcad.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3jcad.2 $e |- ( ph -> ( ps -> th ) ) $.
    3jcad.3 $e |- ( ph -> ( ps -> ta ) ) $.
    $( Deduction conjoining the consequents of three implications.
       (Contributed by NM, 25-Sep-2005.) $)
    3jcad $p |- ( ph -> ( ps -> ( ch /\ th /\ ta ) ) ) $=
      ( w3a wa imp 3jca ex ) ABCDEIABJCDEABCFKABDGKABEHKLM $.
  $}

  ${
    mpbir3an.1 $e |- ps $.
    mpbir3an.2 $e |- ch $.
    mpbir3an.3 $e |- th $.
    mpbir3an.4 $e |- ( ph <-> ( ps /\ ch /\ th ) ) $.
    $( Detach a conjunction of truths in a biconditional.  (Contributed by NM,
       16-Sep-2011.)  (Revised by NM, 9-Jan-2015.) $)
    mpbir3an $p |- ph $=
      ( w3a 3pm3.2i mpbir ) ABCDIBCDEFGJHK $.
  $}

  ${
    mpbir3and.1 $e |- ( ph -> ch ) $.
    mpbir3and.2 $e |- ( ph -> th ) $.
    mpbir3and.3 $e |- ( ph -> ta ) $.
    mpbir3and.4 $e |- ( ph -> ( ps <-> ( ch /\ th /\ ta ) ) ) $.
    $( Detach a conjunction of truths in a biconditional.  (Contributed by
       Mario Carneiro, 11-May-2014.) $)
    mpbir3and $p |- ( ph -> ps ) $=
      ( w3a 3jca mpbird ) ABCDEJACDEFGHKIL $.
  $}

  ${
    syl3anbrc.1 $e |- ( ph -> ps ) $.
    syl3anbrc.2 $e |- ( ph -> ch ) $.
    syl3anbrc.3 $e |- ( ph -> th ) $.
    syl3anbrc.4 $e |- ( ta <-> ( ps /\ ch /\ th ) ) $.
    $( Syllogism inference.  (Contributed by Mario Carneiro, 11-May-2014.) $)
    syl3anbrc $p |- ( ph -> ta ) $=
      ( w3a 3jca sylibr ) ABCDJEABCDFGHKIL $.
  $}

  ${
    syl21anbrc.1 $e |- ( ph -> ps ) $.
    syl21anbrc.2 $e |- ( ph -> ch ) $.
    syl21anbrc.3 $e |- ( ph -> th ) $.
    syl21anbrc.4 $e |- ( ta <-> ( ( ps /\ ch ) /\ th ) ) $.
    $( Syllogism inference.  (Contributed by Peter Mazsa, 18-Sep-2022.) $)
    syl21anbrc $p |- ( ph -> ta ) $=
      ( wa jca31 sylibr ) ABCJDJEABCDFGHKIL $.
  $}

  ${
    3imp3i2an.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    3imp3i2an.2 $e |- ( ( ph /\ ch ) -> ta ) $.
    3imp3i2an.3 $e |- ( ( th /\ ta ) -> et ) $.
    $( An elimination deduction.  (Contributed by Alan Sare, 17-Oct-2017.)
       (Proof shortened by Wolf Lammen, 13-Apr-2022.) $)
    3imp3i2an $p |- ( ( ph /\ ps /\ ch ) -> et ) $=
      ( w3a 3adant2 syl2anc ) ABCJDEFGACEBHKIL $.
  $}

  ${
    3anim123i.1 $e |- ( ph -> ps ) $.
    3anim123i.2 $e |- ( ch -> th ) $.
    3anim123i.3 $e |- ( ta -> et ) $.
    $( Join antecedents and consequents with conjunction.  (Contributed by NM,
       8-Apr-1994.) $)
    3anim123i $p |- ( ( ph /\ ch /\ ta ) -> ( ps /\ th /\ et ) ) $=
      ( w3a 3ad2ant1 3ad2ant2 3ad2ant3 3jca ) ACEJBDFACBEGKCADEHLEAFCIMN $.
  $}

  ${
    3animi.1 $e |- ( ph -> ps ) $.
    $( Add two conjuncts to antecedent and consequent.  (Contributed by Jeff
       Hankins, 16-Aug-2009.) $)
    3anim1i $p |- ( ( ph /\ ch /\ th ) -> ( ps /\ ch /\ th ) ) $=
      ( id 3anim123i ) ABCCDDECFDFG $.

    $( Add two conjuncts to antecedent and consequent.  (Contributed by AV,
       21-Nov-2019.) $)
    3anim2i $p |- ( ( ch /\ ph /\ th ) -> ( ch /\ ps /\ th ) ) $=
      ( id 3anim123i ) CCABDDCFEDFG $.

    $( Add two conjuncts to antecedent and consequent.  (Contributed by Jeff
       Hankins, 19-Aug-2009.) $)
    3anim3i $p |- ( ( ch /\ th /\ ph ) -> ( ch /\ th /\ ps ) ) $=
      ( id 3anim123i ) CCDDABCFDFEG $.
  $}

  ${
    bi3.1 $e |- ( ph <-> ps ) $.
    bi3.2 $e |- ( ch <-> th ) $.
    bi3.3 $e |- ( ta <-> et ) $.
    $( Join 3 biconditionals with conjunction.  (Contributed by NM,
       21-Apr-1994.) $)
    3anbi123i $p |- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) ) $=
      ( wa w3a anbi12i df-3an 3bitr4i ) ACJZEJBDJZFJACEKBDFKOPEFABCDGHLILACEMBD
      FMN $.

    $( Join 3 biconditionals with disjunction.  (Contributed by NM,
       17-May-1994.) $)
    3orbi123i $p |- ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) $=
      ( wo w3o orbi12i df-3or 3bitr4i ) ACJZEJBDJZFJACEKBDFKOPEFABCDGHLILACEMBD
      FMN $.
  $}

  ${
    3anbi1i.1 $e |- ( ph <-> ps ) $.
    $( Inference adding two conjuncts to each side of a biconditional.
       (Contributed by NM, 8-Sep-2006.) $)
    3anbi1i $p |- ( ( ph /\ ch /\ th ) <-> ( ps /\ ch /\ th ) ) $=
      ( biid 3anbi123i ) ABCCDDECFDFG $.

    $( Inference adding two conjuncts to each side of a biconditional.
       (Contributed by NM, 8-Sep-2006.) $)
    3anbi2i $p |- ( ( ch /\ ph /\ th ) <-> ( ch /\ ps /\ th ) ) $=
      ( biid 3anbi123i ) CCABDDCFEDFG $.

    $( Inference adding two conjuncts to each side of a biconditional.
       (Contributed by NM, 8-Sep-2006.) $)
    3anbi3i $p |- ( ( ch /\ th /\ ph ) <-> ( ch /\ th /\ ps ) ) $=
      ( biid 3anbi123i ) CCDDABCFDFEG $.
  $}

  ${
    3imp.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( Importation inference.  (Contributed by NM, 8-Apr-1994.) $)
    3imp $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( w3a wa df-3an imp31 sylbi ) ABCFABGCGDABCHABCDEIJ $.
  $}

  ${
    3impa.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( Importation from double to triple conjunction.  (Contributed by NM,
       20-Aug-1995.) $)
    3impa $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( exp31 3imp ) ABCDABCDEFG $.
  $}

  ${
    ex3.1 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $.
    $( Apply ~ ex to a hypothesis with a 3-right-nested conjunction antecedent,
       with the antecedent of the assertion being a triple conjunction rather
       than a 2-right-nested conjunction.  (Contributed by Alan Sare,
       22-Apr-2018.) $)
    ex3 $p |- ( ( ph /\ ps /\ ch ) -> ( th -> ta ) ) $=
      ( wi wa ex 3impa ) ABCDEGABHCHDEFIJ $.
  $}

  ${
    3imp31.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( The importation inference ~ 3imp with commutation of the first and third
       conjuncts of the assertion relative to the hypothesis.  (Contributed by
       Alan Sare, 11-Sep-2016.) $)
    3imp31 $p |- ( ( ch /\ ps /\ ph ) -> th ) $=
      ( com13 3imp ) CBADABCDEFG $.

    $( Importation inference.  (Contributed by Alan Sare, 17-Oct-2017.) $)
    3imp231 $p |- ( ( ps /\ ch /\ ph ) -> th ) $=
      ( com3l 3imp ) BCADABCDEFG $.

    $( The importation inference ~ 3imp with commutation of the first and
       second conjuncts of the assertion relative to the hypothesis.
       (Contributed by Alan Sare, 11-Sep-2016.)  (Revised to shorten ~ 3com12
       by Wolf Lammen, 23-Jun-2022.) $)
    3imp21 $p |- ( ( ps /\ ph /\ ch ) -> th ) $=
      ( com13 3imp231 ) CBADABCDEFG $.
  $}

  ${
    3impb.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( Importation from double to triple conjunction.  (Contributed by NM,
       20-Aug-1995.) $)
    3impb $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( exp32 3imp ) ABCDABCDEFG $.
  $}

  ${
    3impia.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
    $( Importation to triple conjunction.  (Contributed by NM, 13-Jun-2006.) $)
    3impia $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( wi ex 3imp ) ABCDABCDFEGH $.
  $}

  ${
    3impib.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( Importation to triple conjunction.  (Contributed by NM, 13-Jun-2006.) $)
    3impib $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( expd 3imp ) ABCDABCDEFG $.
  $}

  ${
    3exp.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( Exportation inference.  (Contributed by NM, 30-May-1994.) $)
    3exp $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
      ( w3a pm3.2an3 syl8 ) ABCABCFDABCGEH $.

    $( Exportation from triple to double conjunction.  (Contributed by NM,
       20-Aug-1995.) $)
    3expa $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
      ( 3exp imp31 ) ABCDABCDEFG $.

    $( Exportation from triple to double conjunction.  (Contributed by NM,
       20-Aug-1995.) $)
    3expb $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
      ( 3exp imp32 ) ABCDABCDEFG $.

    $( Exportation from triple conjunction.  (Contributed by NM,
       19-May-2007.) $)
    3expia $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
      ( wi 3exp imp ) ABCDFABCDEGH $.

    $( Exportation from triple conjunction.  (Contributed by NM,
       19-May-2007.) $)
    3expib $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
      ( 3exp impd ) ABCDABCDEFG $.

    $( Commutation in antecedent.  Swap 1st and 3rd.  (Contributed by NM,
       28-Jan-1996.)  (Proof shortened by Andrew Salmon, 13-May-2011.) $)
    3com12 $p |- ( ( ps /\ ph /\ ch ) -> th ) $=
      ( w3a 3ancoma sylbi ) BACFABCFDBACGEH $.

    $( Commutation in antecedent.  Swap 1st and 3rd.  (Contributed by NM,
       28-Jan-1996.) $)
    3com13 $p |- ( ( ch /\ ps /\ ph ) -> th ) $=
      ( w3a 3anrev sylbi ) CBAFABCFDCBAGEH $.

    $( Commutation in antecedent.  Swap 2nd and 3rd.  (Contributed by NM,
       28-Jan-1996.) $)
    3com23 $p |- ( ( ph /\ ch /\ ps ) -> th ) $=
      ( 3exp com23 3imp ) ACBDABCDABCDEFGH $.

    $( Commutation in antecedent.  Rotate left.  (Contributed by NM,
       28-Jan-1996.) $)
    3coml $p |- ( ( ps /\ ch /\ ph ) -> th ) $=
      ( 3com23 3com13 ) ACBDABCDEFG $.

    $( Commutation in antecedent.  Rotate right.  (Contributed by NM,
       28-Jan-1996.) $)
    3comr $p |- ( ( ch /\ ph /\ ps ) -> th ) $=
      ( 3coml ) BCADABCDEFF $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       16-Feb-2008.) $)
    3adant3r1 $p |- ( ( ph /\ ( ta /\ ps /\ ch ) ) -> th ) $=
      ( 3expb 3adantr1 ) ABCDEABCDFGH $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       17-Feb-2008.) $)
    3adant3r2 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
      ( 3expb 3adantr2 ) ABCDEABCDFGH $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       18-Feb-2008.) $)
    3adant3r3 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
      ( 3expb 3adantr3 ) ABCDEABCDFGH $.
  $}

  ${
    ad4ant3.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad4ant123 $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ta ) -> th ) $=
      ( wa 3expa adantr ) ABGCGDEABCDFHI $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad4ant124 $p |- ( ( ( ( ph /\ ps ) /\ ta ) /\ ch ) -> th ) $=
      ( wa 3expa adantlr ) ABGCDEABCDFHI $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad4ant134 $p |- ( ( ( ( ph /\ ta ) /\ ps ) /\ ch ) -> th ) $=
      ( 3expa adantllr ) ABCDEABCDFGH $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad4ant234 $p |- ( ( ( ( ta /\ ph ) /\ ps ) /\ ch ) -> th ) $=
      ( 3expa adantlll ) ABCDEABCDFGH $.
  $}

  ${
    3an1rs.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
    $( Swap conjuncts.  (Contributed by NM, 16-Dec-2007.) $)
    3an1rs $p |- ( ( ( ph /\ ps /\ th ) /\ ch ) -> ta ) $=
      ( w3a wi ex 3exp com34 3imp imp ) ABDGCEABDCEHABCDEABCDEHABCGDEFIJKLM $.
  $}

  ${
    3imp1.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
    $( Importation to left triple conjunction.  (Contributed by NM,
       24-Feb-2005.) $)
    3imp1 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $=
      ( w3a wi 3imp imp ) ABCGDEABCDEHFIJ $.

    $( Importation deduction for triple conjunction.  (Contributed by NM,
       26-Oct-2006.) $)
    3impd $p |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $=
      ( w3a wi com4l 3imp com12 ) BCDGAEBCDAEHABCDEFIJK $.

    $( Importation to right triple conjunction.  (Contributed by NM,
       26-Oct-2006.) $)
    3imp2 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $=
      ( w3a 3impd imp ) ABCDGEABCDEFHI $.
  $}

  ${
    3exp1.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
    $( Exportation from left triple conjunction.  (Contributed by NM,
       24-Feb-2005.) $)
    3exp1 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wi w3a ex 3exp ) ABCDEGABCHDEFIJ $.
  $}

  ${
    3expd.1 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
    $( Exportation deduction for triple conjunction.  (Contributed by NM,
       26-Oct-2006.) $)
    3expd $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wi w3a com12 3exp com4r ) BCDAEBCDAEGABCDHEFIJK $.
  $}

  ${
    3exp2.1 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
    $( Exportation from right triple conjunction.  (Contributed by NM,
       26-Oct-2006.) $)
    3exp2 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( w3a ex 3expd ) ABCDEABCDGEFHI $.
  $}

  ${
    exp5o.1 $e |- ( ( ph /\ ps /\ ch ) -> ( ( th /\ ta ) -> et ) ) $.
    $( A triple exportation inference.  (Contributed by Jeff Hankins,
       8-Jul-2009.) $)
    exp5o $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
      ( wi w3a expd 3exp ) ABCDEFHHABCIDEFGJK $.
  $}

  ${
    exp516.1 $e |- ( ( ( ph /\ ( ps /\ ch /\ th ) ) /\ ta ) -> et ) $.
    $( A triple exportation inference.  (Contributed by Jeff Hankins,
       8-Jul-2009.) $)
    exp516 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
      ( wi w3a exp31 3expd ) ABCDEFHABCDIEFGJK $.
  $}

  ${
    exp520.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta ) ) -> et ) $.
    $( A triple exportation inference.  (Contributed by Jeff Hankins,
       8-Jul-2009.) $)
    exp520 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
      ( w3a wa ex exp5o ) ABCDEFABCHDEIFGJK $.
  $}

  ${
    3anassrs.1 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
    $( Associative law for conjunction applied to antecedent (eliminates
       syllogism).  (Contributed by Mario Carneiro, 4-Jan-2017.) $)
    3anassrs $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $=
      ( 3exp2 imp41 ) ABCDEABCDEFGH $.
  $}

  ${
    3adant1l.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.) $)
    3adant1l $p |- ( ( ( ta /\ ph ) /\ ps /\ ch ) -> th ) $=
      ( wa 3expb adantll 3impb ) EAGBCDABCGDEABCDFHIJ $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.) $)
    3adant1r $p |- ( ( ( ph /\ ta ) /\ ps /\ ch ) -> th ) $=
      ( wa 3expb adantlr 3impb ) AEGBCDABCGDEABCDFHIJ $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.) $)
    3adant2l $p |- ( ( ph /\ ( ta /\ ps ) /\ ch ) -> th ) $=
      ( wa 3com12 3adant1l ) EBGACDBACDEABCDFHIH $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.) $)
    3adant2r $p |- ( ( ph /\ ( ps /\ ta ) /\ ch ) -> th ) $=
      ( wa 3com12 3adant1r ) BEGACDBACDEABCDFHIH $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.) $)
    3adant3l $p |- ( ( ph /\ ps /\ ( ta /\ ch ) ) -> th ) $=
      ( wa 3com13 3adant1l ) ECGBADCBADEABCDFHIH $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.) $)
    3adant3r $p |- ( ( ph /\ ps /\ ( ch /\ ta ) ) -> th ) $=
      ( wa 3com13 3adant1r ) CEGBADCBADEABCDFHIH $.
  $}

  ${
    ad5ant.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant245 $p |- ( ( ( ( ( ta /\ ph ) /\ et ) /\ ps ) /\ ch ) -> th ) $=
      ( wa 3adant1l ad4ant134 ) EAHBCDFABCDEGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant234 $p |- ( ( ( ( ( ta /\ ph ) /\ ps ) /\ ch ) /\ et ) -> th ) $=
      ( wa ad4ant234 adantr ) EAHBHCHDFABCDEGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 14-Apr-2022.) $)
    ad5ant235 $p |- ( ( ( ( ( ta /\ ph ) /\ ps ) /\ et ) /\ ch ) -> th ) $=
      ( wa ad4ant234 adantlr ) EAHBHCDFABCDEGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
    ad5ant123 $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ ta ) /\ et ) -> th ) $=
      ( wa 3expa ad2antrr ) ABHCHDEFABCDGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
    ad5ant124 $p |- ( ( ( ( ( ph /\ ps ) /\ ta ) /\ ch ) /\ et ) -> th ) $=
      ( wa ad4ant124 adantr ) ABHEHCHDFABCDEGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
    ad5ant125 $p |- ( ( ( ( ( ph /\ ps ) /\ ta ) /\ et ) /\ ch ) -> th ) $=
      ( wa wi 3expia 2a1d imp41 ) ABHZEFCDMCDIEFABCDGJKL $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
    ad5ant134 $p |- ( ( ( ( ( ph /\ ta ) /\ ps ) /\ ch ) /\ et ) -> th ) $=
      ( wa ad4ant134 adantr ) AEHBHCHDFABCDEGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
    ad5ant135 $p |- ( ( ( ( ( ph /\ ta ) /\ ps ) /\ et ) /\ ch ) -> th ) $=
      ( wa ad4ant134 adantlr ) AEHBHCDFABCDEGIJ $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
    ad5ant145 $p |- ( ( ( ( ( ph /\ ta ) /\ et ) /\ ps ) /\ ch ) -> th ) $=
      ( wa ad4ant134 adantllr ) AEHBCDFABCDEGIJ $.
  $}

  ${
    sylXanc.1 $e |- ( ph -> ps ) $.
    sylXanc.2 $e |- ( ph -> ch ) $.
    sylXanc.3 $e |- ( ph -> th ) $.
    ${
      syl12anc.4 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
      $( Syllogism combined with contraction.  (Contributed by Jeff Hankins,
         1-Aug-2009.) $)
      syl12anc $p |- ( ph -> ta ) $=
        ( wa jca32 syl ) ABCDJJEABCDFGHKIL $.
    $}

    ${
      syl21anc.4 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
      $( Syllogism combined with contraction.  (Contributed by Jeff Hankins,
         1-Aug-2009.) $)
      syl21anc $p |- ( ph -> ta ) $=
        ( wa jca31 syl ) ABCJDJEABCDFGHKIL $.
    $}

    ${
      syl111anc.4 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl3anc $p |- ( ph -> ta ) $=
        ( w3a 3jca syl ) ABCDJEABCDFGHKIL $.
    $}

    sylXanc.4 $e |- ( ph -> ta ) $.
    ${
      syl22anc.5 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) ) -> et ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl22anc $p |- ( ph -> et ) $=
        ( wa jca syl12anc ) ABCLDEFABCGHMIJKN $.
    $}

    ${
      syl13anc.5 $e |- ( ( ps /\ ( ch /\ th /\ ta ) ) -> et ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl13anc $p |- ( ph -> et ) $=
        ( w3a 3jca syl2anc ) ABCDELFGACDEHIJMKN $.
    $}

    ${
      syl31anc.5 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl31anc $p |- ( ph -> et ) $=
        ( w3a 3jca syl2anc ) ABCDLEFABCDGHIMJKN $.
    $}

    ${
      syl112anc.5 $e |- ( ( ps /\ ch /\ ( th /\ ta ) ) -> et ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl112anc $p |- ( ph -> et ) $=
        ( wa jca syl3anc ) ABCDELFGHADEIJMKN $.
    $}

    ${
      syl121anc.5 $e |- ( ( ps /\ ( ch /\ th ) /\ ta ) -> et ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl121anc $p |- ( ph -> et ) $=
        ( wa jca syl3anc ) ABCDLEFGACDHIMJKN $.
    $}

    ${
      syl211anc.5 $e |- ( ( ( ps /\ ch ) /\ th /\ ta ) -> et ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl211anc $p |- ( ph -> et ) $=
        ( wa jca syl3anc ) ABCLDEFABCGHMIJKN $.
    $}

    sylXanc.5 $e |- ( ph -> et ) $.
    ${
      syl23anc.6 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) ) -> ze ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl23anc $p |- ( ph -> ze ) $=
        ( wa jca syl13anc ) ABCNDEFGABCHIOJKLMP $.
    $}

    ${
      syl32anc.6 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) ) -> ze ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl32anc $p |- ( ph -> ze ) $=
        ( wa jca syl31anc ) ABCDEFNGHIJAEFKLOMP $.
    $}

    ${
      syl122anc.6 $e |- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et ) ) -> ze ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl122anc $p |- ( ph -> ze ) $=
        ( wa jca syl121anc ) ABCDEFNGHIJAEFKLOMP $.
    $}

    ${
      syl212anc.6 $e |- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et ) ) -> ze ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl212anc $p |- ( ph -> ze ) $=
        ( wa jca syl211anc ) ABCDEFNGHIJAEFKLOMP $.
    $}

    ${
      syl221anc.6 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ et ) -> ze ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl221anc $p |- ( ph -> ze ) $=
        ( wa jca syl211anc ) ABCDENFGHIADEJKOLMP $.
    $}

    ${
      syl113anc.6 $e |- ( ( ps /\ ch /\ ( th /\ ta /\ et ) ) -> ze ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl113anc $p |- ( ph -> ze ) $=
        ( w3a 3jca syl3anc ) ABCDEFNGHIADEFJKLOMP $.
    $}

    ${
      syl131anc.6 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ et ) -> ze ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl131anc $p |- ( ph -> ze ) $=
        ( w3a 3jca syl3anc ) ABCDENFGHACDEIJKOLMP $.
    $}

    ${
      syl311anc.6 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ et ) -> ze ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl311anc $p |- ( ph -> ze ) $=
        ( w3a 3jca syl3anc ) ABCDNEFGABCDHIJOKLMP $.
    $}

    sylXanc.6 $e |- ( ph -> ze ) $.
    ${
      syl33anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) )
           -> si ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl33anc $p |- ( ph -> si ) $=
        ( w3a 3jca syl13anc ) ABCDPEFGHABCDIJKQLMNOR $.
    $}

    ${
      syl222anc.7 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ ( et /\ ze ) )
           -> si ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl222anc $p |- ( ph -> si ) $=
        ( wa jca syl221anc ) ABCDEFGPHIJKLAFGMNQOR $.
    $}

    ${
      syl123anc.7 $e |- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et /\ ze ) )
           -> si ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl123anc $p |- ( ph -> si ) $=
        ( wa jca syl113anc ) ABCDPEFGHIACDJKQLMNOR $.
    $}

    ${
      syl132anc.7 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ ( et /\ ze ) )
           -> si ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Jul-2012.) $)
      syl132anc $p |- ( ph -> si ) $=
        ( wa jca syl131anc ) ABCDEFGPHIJKLAFGMNQOR $.
    $}

    ${
      syl213anc.7 $e |- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) )
           -> si ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl213anc $p |- ( ph -> si ) $=
        ( wa jca syl113anc ) ABCPDEFGHABCIJQKLMNOR $.
    $}

    ${
      syl231anc.7 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ze )
           -> si ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl231anc $p |- ( ph -> si ) $=
        ( wa jca syl131anc ) ABCPDEFGHABCIJQKLMNOR $.
    $}

    ${
      syl312anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze ) )
           -> si ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Jul-2012.) $)
      syl312anc $p |- ( ph -> si ) $=
        ( wa jca syl311anc ) ABCDEFGPHIJKLAFGMNQOR $.
    $}

    ${
      syl321anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ze )
           -> si ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Jul-2012.) $)
      syl321anc $p |- ( ph -> si ) $=
        ( wa jca syl311anc ) ABCDEFPGHIJKAEFLMQNOR $.
    $}

    sylXanc.7 $e |- ( ph -> si ) $.
    ${
      syl133anc.8 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ ( et /\ ze /\ si ) )
           -> rh ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl133anc $p |- ( ph -> rh ) $=
        ( w3a 3jca syl131anc ) ABCDEFGHRIJKLMAFGHNOPSQT $.
    $}

    ${
      syl313anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze /\ si ) )
           -> rh ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl313anc $p |- ( ph -> rh ) $=
        ( w3a 3jca syl311anc ) ABCDEFGHRIJKLMAFGHNOPSQT $.
    $}

    ${
      syl331anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ si )
           -> rh ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl331anc $p |- ( ph -> rh ) $=
        ( w3a 3jca syl311anc ) ABCDEFGRHIJKLAEFGMNOSPQT $.
    $}

    ${
      syl223anc.8 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ ( et /\ ze /\ si )
          ) -> rh ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl223anc $p |- ( ph -> rh ) $=
        ( wa jca syl213anc ) ABCDERFGHIJKADELMSNOPQT $.
    $}

    ${
      syl232anc.8 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si )
          ) -> rh ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl232anc $p |- ( ph -> rh ) $=
        ( wa jca syl231anc ) ABCDEFGHRIJKLMNAGHOPSQT $.
    $}

    ${
      syl322anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ( ze /\ si )
          ) -> rh ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl322anc $p |- ( ph -> rh ) $=
        ( wa jca syl321anc ) ABCDEFGHRIJKLMNAGHOPSQT $.
    $}

    sylXanc.8 $e |- ( ph -> rh ) $.
    ${
      syl233anc.9 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si /\
          rh ) ) -> mu ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl233anc $p |- ( ph -> mu ) $=
        ( wa jca syl133anc ) ABCTDEFGHIJABCKLUAMNOPQRSUB $.
    $}

    ${
      syl323anc.9 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ( ze /\ si /\
          rh ) ) -> mu ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl323anc $p |- ( ph -> mu ) $=
        ( wa jca syl313anc ) ABCDEFTGHIJKLMAEFNOUAPQRSUB $.
    $}

    ${
      syl332anc.9 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ ( si /\
          rh ) ) -> mu ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl332anc $p |- ( ph -> mu ) $=
        ( wa jca syl331anc ) ABCDEFGHITJKLMNOPAHIQRUASUB $.
    $}

    sylXanc.9 $e |- ( ph -> mu ) $.
    ${
      syl333anc.10 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze )
          /\ ( si /\ rh /\ mu ) ) -> la ) $.
      $( A syllogism inference combined with contraction.  (Contributed by NM,
         10-Mar-2012.) $)
      syl333anc $p |- ( ph -> la ) $=
        ( w3a 3jca syl331anc ) ABCDEFGHIJUBKLMNOPQAHIJRSTUCUAUD $.
    $}
  $}

  ${
    syl3an1.1 $e |- ( ph -> ps ) $.
    syl3an1.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an1 $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
      ( w3a 3anim1i syl ) ACDHBCDHEABCDFIGJ $.
  $}

  ${
    syl3an2.1 $e |- ( ph -> ch ) $.
    syl3an2.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an2 $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
      ( wi 3exp syl5 3imp ) BADEACBDEHFBCDEGIJK $.
  $}

  ${
    syl3an3.1 $e |- ( ph -> th ) $.
    syl3an3.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an3 $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
      ( 3exp syl7 3imp ) BCAEADBCEFBCDEGHIJ $.
  $}

  ${
    syl3an1b.1 $e |- ( ph <-> ps ) $.
    syl3an1b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an1b $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
      ( biimpi syl3an1 ) ABCDEABFHGI $.
  $}

  ${
    syl3an2b.1 $e |- ( ph <-> ch ) $.
    syl3an2b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an2b $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
      ( biimpi syl3an2 ) ABCDEACFHGI $.
  $}

  ${
    syl3an3b.1 $e |- ( ph <-> th ) $.
    syl3an3b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an3b $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
      ( biimpi syl3an3 ) ABCDEADFHGI $.
  $}

  ${
    syl3an1br.1 $e |- ( ps <-> ph ) $.
    syl3an1br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an1br $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
      ( biimpri syl3an1 ) ABCDEBAFHGI $.
  $}

  ${
    syl3an2br.1 $e |- ( ch <-> ph ) $.
    syl3an2br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an2br $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
      ( biimpri syl3an2 ) ABCDECAFHGI $.
  $}

  ${
    syl3an3br.1 $e |- ( th <-> ph ) $.
    syl3an3br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.) $)
    syl3an3br $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
      ( biimpri syl3an3 ) ABCDEDAFHGI $.
  $}

  ${
    syl3an.1 $e |- ( ph -> ps ) $.
    syl3an.2 $e |- ( ch -> th ) $.
    syl3an.3 $e |- ( ta -> et ) $.
    syl3an.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
    $( A triple syllogism inference.  (Contributed by NM, 13-May-2004.) $)
    syl3an $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
      ( w3a 3anim123i syl ) ACELBDFLGABCDEFHIJMKN $.
  $}

  ${
    syl3anb.1 $e |- ( ph <-> ps ) $.
    syl3anb.2 $e |- ( ch <-> th ) $.
    syl3anb.3 $e |- ( ta <-> et ) $.
    syl3anb.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
    $( A triple syllogism inference.  (Contributed by NM, 15-Oct-2005.) $)
    syl3anb $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
      ( w3a 3anbi123i sylbi ) ACELBDFLGABCDEFHIJMKN $.
  $}

  ${
    syl3anbr.1 $e |- ( ps <-> ph ) $.
    syl3anbr.2 $e |- ( th <-> ch ) $.
    syl3anbr.3 $e |- ( et <-> ta ) $.
    syl3anbr.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
    $( A triple syllogism inference.  (Contributed by NM, 29-Dec-2011.) $)
    syl3anbr $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
      ( bicomi syl3anb ) ABCDEFGBAHLDCILFEJLKM $.
  $}

  ${
    syld3an3.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    syld3an3.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 20-May-2007.) $)
    syld3an3 $p |- ( ( ph /\ ps /\ ch ) -> ta ) $=
      ( w3a simp1 simp2 syl3anc ) ABCHABDEABCIABCJFGK $.
  $}

  ${
    syld3an1.1 $e |- ( ( ch /\ ps /\ th ) -> ph ) $.
    syld3an1.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 7-Jul-2008.) $)
    syld3an1 $p |- ( ( ch /\ ps /\ th ) -> ta ) $=
      ( 3com13 syld3an3 ) DBCEDBCAECBDAFHABDEGHIH $.
  $}

  ${
    syld3an2.1 $e |- ( ( ph /\ ch /\ th ) -> ps ) $.
    syld3an2.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 20-May-2007.) $)
    syld3an2 $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
      ( 3com23 syld3an3 ) ADCEADCBEACDBFHABDEGHIH $.
  $}

  ${
    syl3anl1.1 $e |- ( ph -> ps ) $.
    syl3anl1.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
    $( A syllogism inference.  (Contributed by NM, 24-Feb-2005.) $)
    syl3anl1 $p |- ( ( ( ph /\ ch /\ th ) /\ ta ) -> et ) $=
      ( w3a 3anim1i sylan ) ACDIBCDIEFABCDGJHK $.
  $}

  ${
    syl3anl2.1 $e |- ( ph -> ch ) $.
    syl3anl2.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
    $( A syllogism inference.  (Contributed by NM, 24-Feb-2005.) $)
    syl3anl2 $p |- ( ( ( ps /\ ph /\ th ) /\ ta ) -> et ) $=
      ( w3a wi ex syl3an2 imp ) BADIEFABCDEFJGBCDIEFHKLM $.
  $}

  ${
    syl3anl3.1 $e |- ( ph -> th ) $.
    syl3anl3.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
    $( A syllogism inference.  (Contributed by NM, 24-Feb-2005.) $)
    syl3anl3 $p |- ( ( ( ps /\ ch /\ ph ) /\ ta ) -> et ) $=
      ( w3a 3anim3i sylan ) BCAIBCDIEFADBCGJHK $.
  $}

  ${
    syl3anl.1 $e |- ( ph -> ps ) $.
    syl3anl.2 $e |- ( ch -> th ) $.
    syl3anl.3 $e |- ( ta -> et ) $.
    syl3anl.4 $e |- ( ( ( ps /\ th /\ et ) /\ ze ) -> si ) $.
    $( A triple syllogism inference.  (Contributed by NM, 24-Dec-2006.) $)
    syl3anl $p |- ( ( ( ph /\ ch /\ ta ) /\ ze ) -> si ) $=
      ( w3a 3anim123i sylan ) ACEMBDFMGHABCDEFIJKNLO $.
  $}

  ${
    syl3anr1.1 $e |- ( ph -> ps ) $.
    syl3anr1.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
    $( A syllogism inference.  (Contributed by NM, 31-Jul-2007.) $)
    syl3anr1 $p |- ( ( ch /\ ( ph /\ th /\ ta ) ) -> et ) $=
      ( w3a 3anim1i sylan2 ) ADEICBDEIFABDEGJHK $.
  $}

  ${
    syl3anr2.1 $e |- ( ph -> th ) $.
    syl3anr2.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
    $( A syllogism inference.  (Contributed by NM, 1-Aug-2007.) $)
    syl3anr2 $p |- ( ( ch /\ ( ps /\ ph /\ ta ) ) -> et ) $=
      ( w3a ancoms syl3anl2 ) BAEICFABDECFGCBDEIFHJKJ $.
  $}

  ${
    syl3anr3.1 $e |- ( ph -> ta ) $.
    syl3anr3.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
    $( A syllogism inference.  (Contributed by NM, 23-Aug-2007.) $)
    syl3anr3 $p |- ( ( ch /\ ( ps /\ th /\ ph ) ) -> et ) $=
      ( w3a 3anim3i sylan2 ) BDAICBDEIFAEBDGJHK $.
  $}

  ${
    syldbl2.1 $e |- ( ( ph /\ ps ) -> ( ps -> th ) ) $.
    $( Stacked hypotheseis implies goal.  (Contributed by Stanislas Polu,
       9-Mar-2020.) $)
    syldbl2 $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa com12 anabsi7 ) ABCABEBCDFG $.
  $}

  ${
    3impdi.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) -> th ) $.
    $( Importation inference (undistribute conjunction).  (Contributed by NM,
       14-Aug-1995.) $)
    3impdi $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( anandis 3impb ) ABCDABCDEFG $.
  $}

  ${
    3impdir.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ ps ) ) -> th ) $.
    $( Importation inference (undistribute conjunction).  (Contributed by NM,
       20-Aug-1995.) $)
    3impdir $p |- ( ( ph /\ ch /\ ps ) -> th ) $=
      ( anandirs 3impa ) ACBDACBDEFG $.
  $}

  ${
    3anidm12.1 $e |- ( ( ph /\ ph /\ ps ) -> ch ) $.
    $( Inference from idempotent law for conjunction.  (Contributed by NM,
       7-Mar-2008.) $)
    3anidm12 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( 3expib anabsi5 ) ABCAABCDEF $.
  $}

  ${
    3anidm13.1 $e |- ( ( ph /\ ps /\ ph ) -> ch ) $.
    $( Inference from idempotent law for conjunction.  (Contributed by NM,
       7-Mar-2008.) $)
    3anidm13 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( 3com23 3anidm12 ) ABCABACDEF $.
  $}

  ${
    3anidm23.1 $e |- ( ( ph /\ ps /\ ps ) -> ch ) $.
    $( Inference from idempotent law for conjunction.  (Contributed by NM,
       1-Feb-2007.) $)
    3anidm23 $p |- ( ( ph /\ ps ) -> ch ) $=
      ( 3expa anabss3 ) ABCABBCDEF $.
  $}

  ${
    syl2an3an.1 $e |- ( ph -> ps ) $.
    syl2an3an.2 $e |- ( ph -> ch ) $.
    syl2an3an.3 $e |- ( th -> ta ) $.
    syl2an3an.4 $e |- ( ( ps /\ ch /\ ta ) -> et ) $.
    $( ~ syl3an with antecedents in standard conjunction form.  (Contributed by
       Alan Sare, 31-Aug-2016.) $)
    syl2an3an $p |- ( ( ph /\ th ) -> et ) $=
      ( syl3an 3anidm12 ) ADFABACDEFGHIJKL $.
  $}

  ${
    syl2an23an.1 $e |- ( ph -> ps ) $.
    syl2an23an.2 $e |- ( ph -> ch ) $.
    syl2an23an.3 $e |- ( ( th /\ ph ) -> ta ) $.
    syl2an23an.4 $e |- ( ( ps /\ ch /\ ta ) -> et ) $.
    $( Deduction related to ~ syl3an with antecedents in standard conjunction
       form.  (Contributed by Alan Sare, 31-Aug-2016.) $)
    syl2an23an $p |- ( ( th /\ ph ) -> et ) $=
      ( wa wi 3exp sylc syl5 anabsi7 ) DAFDAKEAFIABCEFLGHBCEFJMNOP $.
  $}

  ${
    3ori.1 $e |- ( ph \/ ps \/ ch ) $.
    $( Infer implication from triple disjunction.  (Contributed by NM,
       26-Sep-2006.) $)
    3ori $p |- ( ( -. ph /\ -. ps ) -> ch ) $=
      ( wn wa wo ioran w3o df-3or mpbi ori sylbir ) AEBEFABGZECABHNCABCINCGDABC
      JKLM $.
  $}

  $( Disjunction of 3 antecedents.  (Contributed by NM, 8-Apr-1994.) $)
  3jao $p |- ( ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) ->
              ( ( ph \/ ch \/ th ) -> ps ) ) $=
    ( w3o wo wi w3a df-3or jao syl6 3imp biimtrid ) ACDEACFZDFZABGZCBGZDBGZHBAC
    DIPQROBGZPQNBGRSGABCJNBDJKLM $.

  $( Disjunction of 3 antecedents.  (Contributed by NM, 13-Sep-2011.) $)
  3jaob $p |- ( ( ( ph \/ ch \/ th ) -> ps ) <->
              ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) ) $=
    ( w3o wi w3a 3mix1 imim1i 3mix2 3mix3 3jca 3jao impbii ) ACDEZBFZABFZCBFZDB
    FZGPQRSAOBACDHICOBCADJIDOBDACKILABCDMN $.

  ${
    3jaoi.1 $e |- ( ph -> ps ) $.
    3jaoi.2 $e |- ( ch -> ps ) $.
    3jaoi.3 $e |- ( th -> ps ) $.
    $( Disjunction of 3 antecedents (inference).  (Contributed by NM,
       12-Sep-1995.) $)
    3jaoi $p |- ( ( ph \/ ch \/ th ) -> ps ) $=
      ( wi w3a w3o 3pm3.2i 3jao ax-mp ) ABHZCBHZDBHZIACDJBHNOPEFGKABCDLM $.
  $}

  ${
    3jaod.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3jaod.2 $e |- ( ph -> ( th -> ch ) ) $.
    3jaod.3 $e |- ( ph -> ( ta -> ch ) ) $.
    $( Disjunction of 3 antecedents (deduction).  (Contributed by NM,
       14-Oct-2005.) $)
    3jaod $p |- ( ph -> ( ( ps \/ th \/ ta ) -> ch ) ) $=
      ( wi w3o 3jao syl3anc ) ABCIDCIECIBDEJCIFGHBCDEKL $.
  $}

  ${
    3jaoian.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    3jaoian.2 $e |- ( ( th /\ ps ) -> ch ) $.
    3jaoian.3 $e |- ( ( ta /\ ps ) -> ch ) $.
    $( Disjunction of 3 antecedents (inference).  (Contributed by NM,
       14-Oct-2005.) $)
    3jaoian $p |- ( ( ( ph \/ th \/ ta ) /\ ps ) -> ch ) $=
      ( w3o wi ex 3jaoi imp ) ADEIBCABCJDEABCFKDBCGKEBCHKLM $.
  $}

  ${
    3jaodan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    3jaodan.2 $e |- ( ( ph /\ th ) -> ch ) $.
    3jaodan.3 $e |- ( ( ph /\ ta ) -> ch ) $.
    $( Disjunction of 3 antecedents (deduction).  (Contributed by NM,
       14-Oct-2005.) $)
    3jaodan $p |- ( ( ph /\ ( ps \/ th \/ ta ) ) -> ch ) $=
      ( w3o ex 3jaod imp ) ABDEICABCDEABCFJADCGJAECHJKL $.
  $}

  ${
    mpjao3dan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    mpjao3dan.2 $e |- ( ( ph /\ th ) -> ch ) $.
    mpjao3dan.3 $e |- ( ( ph /\ ta ) -> ch ) $.
    mpjao3dan.4 $e |- ( ph -> ( ps \/ th \/ ta ) ) $.
    $( Eliminate a 3-way disjunction in a deduction.  (Contributed by Thierry
       Arnoux, 13-Apr-2018.) $)
    mpjao3dan $p |- ( ph -> ch ) $=
      ( wo jaodan w3o df-3or sylib mpjaodan ) ABDJZCEABCDFGKHABDELPEJIBDEMNO $.
  $}

  ${
    3jaao.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3jaao.2 $e |- ( th -> ( ta -> ch ) ) $.
    3jaao.3 $e |- ( et -> ( ze -> ch ) ) $.
    $( Inference conjoining and disjoining the antecedents of three
       implications.  (Contributed by Jeff Hankins, 15-Aug-2009.)  (Proof
       shortened by Andrew Salmon, 13-May-2011.) $)
    3jaao $p |- ( ( ph /\ th /\ et ) -> ( ( ps \/ ta \/ ze ) -> ch ) ) $=
      ( w3a wi 3ad2ant1 3ad2ant2 3ad2ant3 3jaod ) ADFKBCEGADBCLFHMDAECLFINFAGCL
      DJOP $.
  $}

  $( Triple disjunction implies negated triple conjunction.  (Contributed by
     Jim Kingdon, 23-Dec-2018.) $)
  3ianorr $p |- ( ( -. ph \/ -. ps \/ -. ch ) -> -. ( ph /\ ps /\ ch ) ) $=
    ( wn w3a simp1 con3i simp2 simp3 3jaoi ) ADABCEZDBDCDKAABCFGKBABCHGKCABCIGJ
    $.

  ${
    syl3an9b.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    syl3an9b.2 $e |- ( th -> ( ch <-> ta ) ) $.
    syl3an9b.3 $e |- ( et -> ( ta <-> ze ) ) $.
    $( Nested syllogism inference conjoining 3 dissimilar antecedents.
       (Contributed by NM, 1-May-1995.) $)
    syl3an9b $p |- ( ( ph /\ th /\ et ) -> ( ps <-> ze ) ) $=
      ( wb wa sylan9bb 3impa ) ADFBGKADLBEFGABCDEHIMJMN $.
  $}

  ${
    bi3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bi3d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    bi3d.3 $e |- ( ph -> ( et <-> ze ) ) $.
    $( Deduction joining 3 equivalences to form equivalence of disjunctions.
       (Contributed by NM, 20-Apr-1994.) $)
    3orbi123d $p |- ( ph -> ( ( ps \/ th \/ et ) <-> ( ch \/ ta \/ ze ) ) ) $=
      ( wo w3o orbi12d df-3or 3bitr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
      NCEGNO $.

    $( Deduction joining 3 equivalences to form equivalence of conjunctions.
       (Contributed by NM, 22-Apr-1994.) $)
    3anbi123d $p |- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ ze ) ) ) $=
      ( wa w3a anbi12d df-3an 3bitr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
      NCEGNO $.
  $}

  ${
    3anbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    3anbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction conjoining and adding a conjunct to equivalences.
       (Contributed by NM, 8-Sep-2006.) $)
    3anbi12d $p |- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ et ) ) ) $=
      ( biidd 3anbi123d ) ABCDEFFGHAFIJ $.

    $( Deduction conjoining and adding a conjunct to equivalences.
       (Contributed by NM, 8-Sep-2006.) $)
    3anbi13d $p |- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ ta ) ) ) $=
      ( biidd 3anbi123d ) ABCFFDEGAFIHJ $.

    $( Deduction conjoining and adding a conjunct to equivalences.
       (Contributed by NM, 8-Sep-2006.) $)
    3anbi23d $p |- ( ph -> ( ( et /\ ps /\ th ) <-> ( et /\ ch /\ ta ) ) ) $=
      ( biidd 3anbi123d ) AFFBCDEAFIGHJ $.
  $}

  ${
    3anbi1d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction adding conjuncts to an equivalence.  (Contributed by NM,
       8-Sep-2006.) $)
    3anbi1d $p |- ( ph -> ( ( ps /\ th /\ ta ) <-> ( ch /\ th /\ ta ) ) ) $=
      ( biidd 3anbi12d ) ABCDDEFADGH $.

    $( Deduction adding conjuncts to an equivalence.  (Contributed by NM,
       8-Sep-2006.) $)
    3anbi2d $p |- ( ph -> ( ( th /\ ps /\ ta ) <-> ( th /\ ch /\ ta ) ) ) $=
      ( biidd 3anbi12d ) ADDBCEADGFH $.

    $( Deduction adding conjuncts to an equivalence.  (Contributed by NM,
       8-Sep-2006.) $)
    3anbi3d $p |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) ) $=
      ( biidd 3anbi13d ) ADDBCEADGFH $.
  $}

  ${
    3anim123d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    3anim123d.2 $e |- ( ph -> ( th -> ta ) ) $.
    3anim123d.3 $e |- ( ph -> ( et -> ze ) ) $.
    $( Deduction joining 3 implications to form implication of conjunctions.
       (Contributed by NM, 24-Feb-2005.) $)
    3anim123d $p |- ( ph -> ( ( ps /\ th /\ et ) -> ( ch /\ ta /\ ze ) ) ) $=
      ( wa w3a anim12d df-3an 3imtr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
      NCEGNO $.

    $( Deduction joining 3 implications to form implication of disjunctions.
       (Contributed by NM, 4-Apr-1997.) $)
    3orim123d $p |- ( ph -> ( ( ps \/ th \/ et ) -> ( ch \/ ta \/ ze ) ) ) $=
      ( wo w3o orim12d df-3or 3imtr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
      NCEGNO $.
  $}

  $( Rearrangement of 6 conjuncts.  (Contributed by NM, 13-Mar-1995.) $)
  an6 $p |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <->
              ( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) ) $=
    ( w3a wa df-3an anbi12i an4 anbi1i 3bitri bitr4i ) ABCGZDEFGZHZADHZBEHZHZCF
    HZHZRSUAGQABHZCHZDEHZFHZHUCUEHZUAHUBOUDPUFABCIDEFIJUCCUEFKUGTUAABDEKLMRSUAI
    N $.

  $( Analog of ~ an4 for triple conjunction.  (Contributed by Scott Fenton,
     16-Mar-2011.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
  3an6 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) /\ ( ta /\ et ) ) <->
                ( ( ph /\ ch /\ ta ) /\ ( ps /\ th /\ et ) ) ) $=
    ( w3a wa an6 bicomi ) ACEGBDFGHABHCDHEFHGACEBDFIJ $.

  $( Analog of ~ or4 for triple conjunction.  (Contributed by Scott Fenton,
     16-Mar-2011.) $)
  3or6 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) \/ ( ta \/ et ) ) <->
                ( ( ph \/ ch \/ ta ) \/ ( ps \/ th \/ et ) ) ) $=
    ( wo w3o or4 orbi1i bitr2i df-3or orbi12i 3bitr4i ) ABGZCDGZGZEFGZGZACGZEGZ
    BDGZFGZGZOPRHACEHZBDFHZGUDTUBGZRGSTEUBFIUGQRACBDIJKOPRLUEUAUFUCACELBDFLMN
    $.

  ${
    mp3an1.1 $e |- ph $.
    mp3an1.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       21-Nov-1994.) $)
    mp3an1 $p |- ( ( ps /\ ch ) -> th ) $=
      ( wa 3expb mpan ) ABCGDEABCDFHI $.
  $}

  ${
    mp3an2.1 $e |- ps $.
    mp3an2.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       21-Nov-1994.) $)
    mp3an2 $p |- ( ( ph /\ ch ) -> th ) $=
      ( 3expa mpanl2 ) ABCDEABCDFGH $.
  $}

  ${
    mp3an3.1 $e |- ch $.
    mp3an3.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       21-Nov-1994.) $)
    mp3an3 $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa 3expia mpi ) ABGCDEABCDFHI $.
  $}

  ${
    mp3an12.1 $e |- ph $.
    mp3an12.2 $e |- ps $.
    mp3an12.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       13-Jul-2005.) $)
    mp3an12 $p |- ( ch -> th ) $=
      ( mp3an1 mpan ) BCDFABCDEGHI $.
  $}

  ${
    mp3an13.1 $e |- ph $.
    mp3an13.2 $e |- ch $.
    mp3an13.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       14-Jul-2005.) $)
    mp3an13 $p |- ( ps -> th ) $=
      ( mp3an3 mpan ) ABDEABCDFGHI $.
  $}

  ${
    mp3an23.1 $e |- ps $.
    mp3an23.2 $e |- ch $.
    mp3an23.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       14-Jul-2005.) $)
    mp3an23 $p |- ( ph -> th ) $=
      ( mp3an3 mpan2 ) ABDEABCDFGHI $.
  $}

  ${
    mp3an1i.1 $e |- ps $.
    mp3an1i.2 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 5-Jul-2005.) $)
    mp3an1i $p |- ( ph -> ( ( ch /\ th ) -> ta ) ) $=
      ( wa wi w3a com12 mp3an1 ) CDHAEBCDAEIFABCDJEGKLK $.
  $}

  ${
    mp3anl1.1 $e |- ph $.
    mp3anl1.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       24-Feb-2005.) $)
    mp3anl1 $p |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $=
      ( wa wi w3a ex mp3an1 imp ) BCHDEABCDEIFABCJDEGKLM $.
  $}

  ${
    mp3anl2.1 $e |- ps $.
    mp3anl2.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       24-Feb-2005.) $)
    mp3anl2 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
      ( wa wi w3a ex mp3an2 imp ) ACHDEABCDEIFABCJDEGKLM $.
  $}

  ${
    mp3anl3.1 $e |- ch $.
    mp3anl3.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       24-Feb-2005.) $)
    mp3anl3 $p |- ( ( ( ph /\ ps ) /\ th ) -> ta ) $=
      ( wa wi w3a ex mp3an3 imp ) ABHDEABCDEIFABCJDEGKLM $.
  $}

  ${
    mp3anr1.1 $e |- ps $.
    mp3anr1.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 4-Nov-2006.) $)
    mp3anr1 $p |- ( ( ph /\ ( ch /\ th ) ) -> ta ) $=
      ( wa w3a ancoms mp3anl1 ) CDHAEBCDAEFABCDIEGJKJ $.
  $}

  ${
    mp3anr2.1 $e |- ch $.
    mp3anr2.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       24-Nov-2006.) $)
    mp3anr2 $p |- ( ( ph /\ ( ps /\ th ) ) -> ta ) $=
      ( wa w3a ancoms mp3anl2 ) BDHAEBCDAEFABCDIEGJKJ $.
  $}

  ${
    mp3anr3.1 $e |- th $.
    mp3anr3.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       19-Oct-2007.) $)
    mp3anr3 $p |- ( ( ph /\ ( ps /\ ch ) ) -> ta ) $=
      ( wa w3a ancoms mp3anl3 ) BCHAEBCDAEFABCDIEGJKJ $.
  $}

  ${
    mp3an.1 $e |- ph $.
    mp3an.2 $e |- ps $.
    mp3an.3 $e |- ch $.
    mp3an.4 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM,
       14-May-1999.) $)
    mp3an $p |- th $=
      ( mp3an1 mp2an ) BCDFGABCDEHIJ $.
  $}

  ${
    mpd3an3.2 $e |- ( ( ph /\ ps ) -> ch ) $.
    mpd3an3.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 8-Nov-2007.) $)
    mpd3an3 $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa 3expa mpdan ) ABGCDEABCDFHI $.
  $}

  ${
    mpd3an23.1 $e |- ( ph -> ps ) $.
    mpd3an23.2 $e |- ( ph -> ch ) $.
    mpd3an23.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( An inference based on modus ponens.  (Contributed by NM, 4-Dec-2006.) $)
    mpd3an23 $p |- ( ph -> th ) $=
      ( id syl3anc ) AABCDAHEFGI $.
  $}

  ${
    mp3and.1 $e |- ( ph -> ps ) $.
    mp3and.2 $e |- ( ph -> ch ) $.
    mp3and.3 $e |- ( ph -> th ) $.
    mp3and.4 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
    $( A deduction based on modus ponens.  (Contributed by Mario Carneiro,
       24-Dec-2016.) $)
    mp3and $p |- ( ph -> ta ) $=
      ( w3a 3jca mpd ) ABCDJEABCDFGHKIL $.
  $}

  ${
    mp3an12i.1 $e |- ph $.
    mp3an12i.2 $e |- ps $.
    mp3an12i.3 $e |- ( ch -> th ) $.
    mp3an12i.4 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
    $( ~ mp3an with antecedents in standard conjunction form and with one
       hypothesis an implication.  (Contributed by Alan Sare, 28-Aug-2016.) $)
    mp3an12i $p |- ( ch -> ta ) $=
      ( mp3an12 syl ) CDEHABDEFGIJK $.
  $}

  ${
    mp3an2i.1 $e |- ph $.
    mp3an2i.2 $e |- ( ps -> ch ) $.
    mp3an2i.3 $e |- ( ps -> th ) $.
    mp3an2i.4 $e |- ( ( ph /\ ch /\ th ) -> ta ) $.
    $( ~ mp3an with antecedents in standard conjunction form and with two
       hypotheses which are implications.  (Contributed by Alan Sare,
       28-Aug-2016.) $)
    mp3an2i $p |- ( ps -> ta ) $=
      ( mp3an1 syl2anc ) BCDEGHACDEFIJK $.
  $}

  ${
    mp3an3an.1 $e |- ph $.
    mp3an3an.2 $e |- ( ps -> ch ) $.
    mp3an3an.3 $e |- ( th -> ta ) $.
    mp3an3an.4 $e |- ( ( ph /\ ch /\ ta ) -> et ) $.
    $( ~ mp3an with antecedents in standard conjunction form and with two
       hypotheses which are implications.  (Contributed by Alan Sare,
       28-Aug-2016.) $)
    mp3an3an $p |- ( ( ps /\ th ) -> et ) $=
      ( mp3an1 syl2an ) BCEFDHIACEFGJKL $.
  $}

  ${
    mp3an2ani.1 $e |- ph $.
    mp3an2ani.2 $e |- ( ps -> ch ) $.
    mp3an2ani.3 $e |- ( ( ps /\ th ) -> ta ) $.
    mp3an2ani.4 $e |- ( ( ph /\ ch /\ ta ) -> et ) $.
    $( An elimination deduction.  (Contributed by Alan Sare, 17-Oct-2017.) $)
    mp3an2ani $p |- ( ( ps /\ th ) -> et ) $=
      ( wa mp3an3an anabss5 ) BDFABCBDKEFGHIJLM $.
  $}

  ${
    biimp3a.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
    $( Infer implication from a logical equivalence.  Similar to ~ biimpa .
       (Contributed by NM, 4-Sep-2005.) $)
    biimp3a $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( wa biimpa 3impa ) ABCDABFCDEGH $.

    $( Infer implication from a logical equivalence.  Similar to ~ biimpar .
       (Contributed by NM, 2-Jan-2009.) $)
    biimp3ar $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
      ( exbiri 3imp ) ABDCABCDEFG $.
  $}

  ${
    3anandis.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) )
                      -> ta ) $.
    $( Inference that undistributes a triple conjunction in the antecedent.
       (Contributed by NM, 18-Apr-2007.) $)
    3anandis $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $=
      ( w3a wa simpl simpr1 simpr2 simpr3 syl222anc ) ABCDGZHABACADEANIZABCDJOA
      BCDKOABCDLFM $.
  $}

  ${
    3anandirs.1 $e |- ( ( ( ph /\ th ) /\ ( ps /\ th ) /\ ( ch /\ th ) )
                      -> ta ) $.
    $( Inference that undistributes a triple conjunction in the antecedent.
       (Contributed by NM, 25-Jul-2006.)  (Revised by NM, 18-Apr-2007.) $)
    3anandirs $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $=
      ( w3a wa simpl1 simpr simpl2 simpl3 syl222anc ) ABCGZDHADBDCDEABCDINDJZAB
      CDKOABCDLOFM $.
  $}

  ${
    ecased.1 $e |- ( ph -> -. ch ) $.
    ecased.2 $e |- ( ph -> ( ps \/ ch ) ) $.
    $( Deduction form of disjunctive syllogism.  (Contributed by Jim Kingdon,
       9-Dec-2017.) $)
    ecased $p |- ( ph -> ps ) $=
      ( wn wo wa jca orel2 imp syl ) ACFZBCGZHBAMNDEIMNBCBJKL $.
  $}

  ${
    ecase23d.1 $e |- ( ph -> -. ch ) $.
    ecase23d.2 $e |- ( ph -> -. th ) $.
    ecase23d.3 $e |- ( ph -> ( ps \/ ch \/ th ) ) $.
    $( Variation of ~ ecased with three disjuncts instead of two.  (Contributed
       by NM, 22-Apr-1994.)  (Revised by Jim Kingdon, 9-Dec-2017.) $)
    ecase23d $p |- ( ph -> ps ) $=
      ( wo w3o df-3or sylib ecased ) ABCEABCHZDFABCDIMDHGBCDJKLL $.
  $}

  ${
    ecase2d.1 $e |- ( ph -> ps ) $.
    ecase2d.2 $e |- ( ph -> -. ( ps /\ ch ) ) $.
    ecase2d.3 $e |- ( ph -> -. ( ps /\ th ) ) $.
    ecase2d.4 $e |- ( ph -> ( ta \/ ( ch \/ th ) ) ) $.
    $( Deduction for elimination by cases.  (Contributed by NM, 21-Apr-1994.)
       (Proof shortened by Wolf Lammen, 19-Sep-2024.) $)
    ecase2d $p |- ( ph -> ta ) $=
      ( wo wn mpnanrd ioran sylanbrc ecased ) AECDJZACKDKPKABCFGLABDFHLCDMNIO
      $.
  $}

  ${
    3biorfd.1 $e |- ( ph -> -. th ) $.
    $( A disjunction is equivalent to a threefold disjunction with single
       falsehood, analogous to ~ biorf .  (Contributed by Alexander van der
       Vekens, 8-Sep-2017.) $)
    3bior1fd $p |- ( ph -> ( ( ch \/ ps ) <-> ( th \/ ch \/ ps ) ) ) $=
      ( wo w3o wn wb biorf syl 3orass bitr4di ) ACBFZDNFZDCBGADHNOIEDNJKDCBLM
      $.

    $( A disjunction is equivalent to a threefold disjunction with single
       falsehood of a conjunction.  (Contributed by Alexander van der Vekens,
       8-Sep-2017.) $)
    3bior1fand $p |- ( ph -> ( ( ch \/ ps )
                       <-> ( ( th /\ ta ) \/ ch \/ ps ) ) ) $=
      ( wa intnanrd 3bior1fd ) ABCDEGADEFHI $.

    3biorfd.2 $e |- ( ph -> -. ch ) $.
    $( A wff is equivalent to its threefold disjunction with double falsehood,
       analogous to ~ biorf .  (Contributed by Alexander van der Vekens,
       8-Sep-2017.) $)
    3bior2fd $p |- ( ph -> ( ps <-> ( th \/ ch \/ ps ) ) ) $=
      ( wo w3o wn wb biorf syl 3bior1fd bitrd ) ABCBGZDCBHACIBOJFCBKLABCDEMN $.
  $}

  ${
    3biantd.1 $e |- ( ph -> th ) $.
    $( A conjunction is equivalent to a threefold conjunction with single
       truth, analogous to ~ biantrud .  (Contributed by Alexander van der
       Vekens, 26-Sep-2017.) $)
    3biant1d $p |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ch /\ ps ) ) ) $=
      ( wa w3a biantrurd 3anass bitr4di ) ACBFZDKFDCBGADKEHDCBIJ $.
  $}

  ${
    intn3and.1 $e |- ( ph -> -. ps ) $.
    $( Introduction of a triple conjunct inside a contradiction.  (Contributed
       by FL, 27-Dec-2007.)  (Proof shortened by Andrew Salmon,
       26-Jun-2011.) $)
    intn3an1d $p |- ( ph -> -. ( ps /\ ch /\ th ) ) $=
      ( w3a simp1 nsyl ) ABBCDFEBCDGH $.

    $( Introduction of a triple conjunct inside a contradiction.  (Contributed
       by FL, 27-Dec-2007.)  (Proof shortened by Andrew Salmon,
       26-Jun-2011.) $)
    intn3an2d $p |- ( ph -> -. ( ch /\ ps /\ th ) ) $=
      ( w3a simp2 nsyl ) ABCBDFECBDGH $.

    $( Introduction of a triple conjunct inside a contradiction.  (Contributed
       by FL, 27-Dec-2007.)  (Proof shortened by Andrew Salmon,
       26-Jun-2011.) $)
    intn3an3d $p |- ( ph -> -. ( ch /\ th /\ ps ) ) $=
      ( w3a simp3 nsyl ) ABCDBFECDBGH $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  True and false constants
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Universal quantifier for use by df-tru
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-

  Even though it is not ordinarily part of propositional calculus, the
  universal quantifier ` A. ` is introduced here so that the soundness of
  Definition ~ df-tru can be checked by the same algorithm that is used for
  predicate calculus.  Its first real use is in Axiom ~ ax-5 in the
  predicate calculus section below.  For those who want propositional calculus
  to be self-contained, i.e., to use wff variables only, the alternate
  Definition ~ dftru2 may be adopted and this subsection moved down to the
  start of the subsection with ~ wex below.  However, the use of ~ dftru2 as a
  definition requires a more elaborate definition checking algorithm that we
  prefer to avoid.

$)

  $( Declare new symbols needed for predicate calculus. $)
  $c A. $.  $( "inverted A" universal quantifier (read:  "for all") $)
  $c setvar $.  $( Individual variable type (read:  "the following is an
                   individual (set) variable") $)

  $( Add 'setvar' as a typecode for bound variables. $)
  $( $j syntax 'setvar'; bound 'setvar'; $)

  ${
    $v x $.
    $( Let ` x ` be an individual variable (temporary declaration). $)
    vx.wal $f setvar x $.
    $( Extend wff definition to include the universal quantifier ("for all").
       ` A. x ph ` is read " ` ph ` (phi) is true for all ` x ` ".  Typically,
       in its final application ` ph ` would be replaced with a wff containing
       a (free) occurrence of the variable ` x ` , for example ` x = y ` .  In
       a universe with a finite number of objects, "for all" is equivalent to a
       big conjunction (AND) with one wff for each possible case of ` x ` .
       When the universe is infinite (as with set theory), such a
       propositional-calculus equivalent is not possible because an infinitely
       long formula has no meaning, but conceptually the idea is the same. $)
    wal $a wff A. x ph $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Equality predicate for use by df-tru
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-

  Even though it is not ordinarily part of propositional calculus, the equality
  predicate ` = ` is introduced here so that the soundness of definition
  ~ df-tru can be checked by the same algorithm as is used for predicate
  calculus.  Its first real use is in Axiom ~ ax-8 in the predicate calculus
  section below.  For those who want propositional calculus to be
  self-contained, i.e., to use wff variables only, the alternate definition
  ~ dftru2 may be adopted and this subsection moved down to just above ~ weq
  below.  However, the use of ~ dftru2 as a definition requires a more
  elaborate definition checking algorithm that we prefer to avoid.

$)

  $c class $.

  $( Add 'class' as a typecode. $)
  $( $j syntax 'class'; $)

  ${
    $v x $.
    $( Let ` x ` be an individual variable (temporary declaration). $)
    vx.cv $f setvar x $.
    $( This syntax construction states that a variable ` x ` , which has been
       declared to be a setvar variable by $f statement vx, is also a class
       expression.  This can be justified informally as follows.  We know that
       the class builder ` { y | y e. x } ` is a class by ~ cab .  Since (when
       ` y ` is distinct from ` x ` ) we have ` x = { y | y e. x } ` by
       ~ cvjust , we can argue that the syntax " ` class x ` " can be viewed as
       an abbreviation for " ` class { y | y e. x } ` ".  See the discussion
       under the definition of class in [Jech] p. 4 showing that "Every set can
       be considered to be a class."

       While it is tempting and perhaps occasionally useful to view ~ cv as a
       "type conversion" from a setvar variable to a class variable, keep in
       mind that ~ cv is intrinsically no different from any other
       class-building syntax such as ~ cab , ~ cun , or ~ c0 .

       For a general discussion of the theory of classes and the role of ~ cv ,
       see ~ https://us.metamath.org/mpeuni/mmset.html#class .

       (The description above applies to set theory, not predicate calculus.
       The purpose of introducing ` class x ` here, and not in set theory where
       it belongs, is to allow us to express i.e.  "prove" the ~ weq of
       predicate calculus from the ~ wceq of set theory, so that we don't
       overload the ` = ` connective with two syntax definitions.  This is done
       to prevent ambiguity that would complicate some Metamath parsers.) $)
    cv $a class x $.
  $}

  $( Declare the equality predicate symbol. $)
  $c = $.  $( Equal sign (read:  'is equal to') $)

  ${
    $v A $.
    $v B $.
    $( Temporary declarations of ` A ` and ` B ` . $)
    cA.wceq $f class A $.
    cB.wceq $f class B $.
    $( Extend wff definition to include class equality.

       For a general discussion of the theory of classes, see
       ~ https://us.metamath.org/mpeuni/mmset.html#class .

       (The purpose of introducing ` wff A = B ` here, and not in set theory
       where it belongs, is to allow us to express i.e.  "prove" the ~ weq of
       predicate calculus in terms of the ~ wceq of set theory, so that we
       don't "overload" the ` = ` connective with two syntax definitions.  This
       is done to prevent ambiguity that would complicate some Metamath
       parsers.  For example, some parsers - although not the Metamath program
       - stumble on the fact that the ` = ` in ` x = y ` could be the ` = ` of
       either ~ weq or ~ wceq , although mathematically it makes no difference.
       The class variables ` A ` and ` B ` are introduced temporarily for the
       purpose of this definition but otherwise not used in predicate calculus.
       See ~ df-cleq for more information on the set theory usage of
       ~ wceq .) $)
    wceq $a wff A = B $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Define the true and false constants
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c T. $.

  $( ` T. ` is a wff. $)
  wtru $a wff T. $.

  ${
    $v x $.
    $v y $.
    $( Temporary declarations of ` x ` and ` y ` for local use by ~ df-tru .
       These will be redeclared globally in the predicate calculus section. $)
    vx.tru $f setvar x $.
    vy.tru $f setvar y $.
    $( Soundness justification theorem for ~ df-tru .  (Contributed by Mario
       Carneiro, 17-Nov-2013.)  (Revised by NM, 11-Jul-2019.) $)
    trujust $p |- ( ( A. x x = x -> A. x x = x )
              <-> ( A. y y = y -> A. y y = y ) ) $=
      ( cv wceq wal wi id 2th ) ACZIDAEZJFBCZKDBEZLFJGLGH $.

    $( Definition of the truth value "true", or "verum", denoted by ` T. ` .
       This is a tautology, as proved by ~ tru .  In this definition, an
       instance of ~ id is used as the definiens, although any tautology, such
       as an axiom, can be used in its place.  This particular ~ id instance
       was chosen so this definition can be checked by the same algorithm that
       is used for predicate calculus.  This definition should be referenced
       directly only by ~ tru , and other proofs should depend on ~ tru
       (directly or indirectly) instead of this definition, since there are
       many alternate ways to define ` T. ` .  (Contributed by Anthony Hart,
       13-Oct-2010.)  (Revised by NM, 11-Jul-2019.)
       (New usage is discouraged.) $)
    df-tru $a |- ( T. <-> ( A. x x = x -> A. x x = x ) ) $.

    $( The truth value ` T. ` is provable.  (Contributed by Anthony Hart,
       13-Oct-2010.) $)
    tru $p |- T. $=
      ( vx.tru wtru cv wceq wal wi id df-tru mpbir ) BACZJDAEZKFKGAHI $.
  $}

  $c F. $.

  $( ` F. ` is a wff. $)
  wfal $a wff F. $.

  $( Definition of the truth value "false", or "falsum", denoted by ` F. ` .
     See also ~ df-tru .  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  df-fal $a |- ( F. <-> -. T. ) $.

  $( The truth value ` F. ` is refutable.  (Contributed by Anthony Hart,
     22-Oct-2010.)  (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) $)
  fal $p |- -. F. $=
    ( wfal wtru wn tru notnoti df-fal mtbir ) ABCBDEFG $.

  $( An alternate definition of "true".  (Contributed by Anthony Hart,
     13-Oct-2010.)  (Revised by BJ, 12-Jul-2019.)
     (New usage is discouraged.) $)
  dftru2 $p |- ( T. <-> ( ph -> ph ) ) $=
    ( wtru wi tru id 2th ) BAACDAEF $.

  ${
    mptru.1 $e |- ( T. -> ph ) $.
    $( Eliminate ` T. ` as an antecedent.  A proposition implied by ` T. ` is
       true.  (Contributed by Mario Carneiro, 13-Mar-2014.) $)
    mptru $p |- ph $=
      ( wtru tru ax-mp ) CADBE $.
  $}

  $( A proposition is equivalent to itself being equivalent to ` T. ` .
     (Contributed by Anthony Hart, 14-Aug-2011.) $)
  tbtru $p |- ( ph <-> ( ph <-> T. ) ) $=
    ( wtru tru tbt ) BACD $.

  $( The negation of a proposition is equivalent to itself being equivalent to
     ` F. ` .  (Contributed by Anthony Hart, 14-Aug-2011.) $)
  nbfal $p |- ( -. ph <-> ( ph <-> F. ) ) $=
    ( wfal fal nbn ) BACD $.

  ${
    bitru.1 $e |- ph $.
    $( A theorem is equivalent to truth.  (Contributed by Mario Carneiro,
       9-May-2015.) $)
    bitru $p |- ( ph <-> T. ) $=
      ( wtru tru 2th ) ACBDE $.
  $}

  ${
    bifal.1 $e |- -. ph $.
    $( A contradiction is equivalent to falsehood.  (Contributed by Mario
       Carneiro, 9-May-2015.) $)
    bifal $p |- ( ph <-> F. ) $=
      ( wfal fal 2false ) ACBDE $.
  $}

  $( The truth value ` F. ` implies anything.  Also called the principle of
     explosion, or "ex falso quodlibet".  (Contributed by FL, 20-Mar-2011.)
     (Proof shortened by Anthony Hart, 1-Aug-2011.) $)
  falim $p |- ( F. -> ph ) $=
    ( wfal fal pm2.21i ) BACD $.

  $( The truth value ` F. ` implies anything.  (Contributed by Mario Carneiro,
     9-Feb-2017.) $)
  falimd $p |- ( ( ph /\ F. ) -> ps ) $=
    ( wfal falim adantl ) CBABDE $.

  $( Anything implies ` T. ` .  (Contributed by FL, 20-Mar-2011.)  (Proof
     shortened by Anthony Hart, 1-Aug-2011.) $)
  trud $p |- ( ph -> T. ) $=
    ( wtru tru a1i ) BACD $.

  $( True can be removed from a conjunction.  (Contributed by FL, 20-Mar-2011.)
     (Proof shortened by Wolf Lammen, 21-Jul-2019.) $)
  truan $p |- ( ( T. /\ ph ) <-> ph ) $=
    ( wtru wa tru biantrur bicomi ) ABACBADEF $.

  $( Given falsum, we can define the negation of a wff ` ph ` as the statement
     that a contradiction follows from assuming ` ph ` .  (Contributed by Mario
     Carneiro, 9-Feb-2017.)  (Proof shortened by Wolf Lammen, 21-Jul-2019.) $)
  dfnot $p |- ( -. ph <-> ( ph -> F. ) ) $=
    ( wfal wn wi wb fal mtt ax-mp ) BCACABDEFBAGH $.

  ${
    inegd.1 $e |- ( ( ph /\ ps ) -> F. ) $.
    $( Negation introduction rule from natural deduction.  (Contributed by
       Mario Carneiro, 9-Feb-2017.) $)
    inegd $p |- ( ph -> -. ps ) $=
      ( wfal wi wn ex dfnot sylibr ) ABDEBFABDCGBHI $.
  $}

  ${
    pm2.21fal.1 $e |- ( ph -> ps ) $.
    pm2.21fal.2 $e |- ( ph -> -. ps ) $.
    $( If a wff and its negation are provable, then falsum is provable.
       (Contributed by Mario Carneiro, 9-Feb-2017.) $)
    pm2.21fal $p |- ( ph -> F. ) $=
      ( wfal pm2.21dd ) ABECDF $.
  $}

  $( Negation inferred from embedded conjunct.  (Contributed by NM,
     20-Aug-1993.)  (Proof rewritten by Jim Kingdon, 4-May-2018.) $)
  pclem6 $p |- ( ( ph <-> ( ps /\ -. ph ) ) -> -. ps ) $=
    ( wn wa wb wfal biimp pm3.4 com12 ax-ia3 biimpr syl9 impbidd pm5.19 pm2.21i
    wi syl9r syl6com dfnot sylibr ) ABACZDZEZBFPBCBUCAUAEZFBUCAUAUCAUBBUAAUBGUB
    BUABUAHIQBUAUBUCABUAJAUBKLMUDFANORBST $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical 'xor'
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare connective for exclusive disjunction ('xor'). $)
  $c \/_ $. $( Underlined 'vee' (read:  'xor') $)

  $( Extend wff definition to include exclusive disjunction ('xor'). $)
  wxo $a wff ( ph \/_ ps ) $.

  $( Define exclusive disjunction (logical 'xor').  Return true if either the
     left or right, but not both, are true.  Contrast with ` /\ ` ( ~ wa ),
     ` \/ ` ( ~ wo ), and ` -> ` ( ~ wi ) .  (Contributed by FL, 22-Nov-2010.)
     (Modified by Jim Kingdon, 1-Mar-2018.) $)
  df-xor $a |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) $.

  $( One way of defining exclusive or.  Equivalent to ~ df-xor .  (Contributed
     by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) $)
  xoranor $p |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) ) $=
    ( wxo wo wn wa df-xor ax-ia3 con3d olc syl6 pm3.21 orc jaoi imdistani sylbi
    wi pm3.14 anim2i sylibr impbii ) ABCZABDZAEZBEZDZFZUBUCABFZEZFZUGABGZUCUIUF
    AUIUFQBAUIUEUFABUHABHIUEUDJKBUIUDUFBAUHBALIUDUEMKNOPUGUJUBUFUIUCABRSUKTUA
    $.

  $( This tautology shows that xor is really exclusive.  (Contributed by FL,
     22-Nov-2010.)  (Proof rewritten by Jim Kingdon, 5-May-2018.) $)
  excxor $p |- ( ( ph \/_ ps ) <->
       ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) $=
    ( wxo wn wa wo xoranor andi orcom pm3.24 biorfi andir pm5.61 orbi12i 3bitri
    3bitr4ri ancom orbi2i ) ABCZBADZEZABDZEZFZUCUAFUCTBEZFSABFZTUBFEUFTEZUFUBEZ
    FUDABGUFTUBHUGUAUHUCUAATEZFUIUAFUAUGUAUIIUIUAAJKABTLPABMNOUAUCIUAUEUCBTQRO
    $.

  $( XOR implies OR. (Contributed by BJ, 19-Apr-2019.) $)
  xoror $p |- ( ( ph \/_ ps ) -> ( ph \/ ps ) ) $=
    ( wxo wo wn xoranor simplbi ) ABCABDAEBEDABFG $.

  ${
    xorbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction joining an equivalence and a left operand to form equivalence
       of exclusive-or.  (Contributed by Jim Kingdon, 7-Oct-2018.) $)
    xorbi2d $p |- ( ph -> ( ( th \/_ ps ) <-> ( th \/_ ch ) ) ) $=
      ( wo wa wn wxo orbi2d anbi2d notbid anbi12d df-xor 3bitr4g ) ADBFZDBGZHZG
      DCFZDCGZHZGDBIDCIAPSRUAABCDEJAQTABCDEKLMDBNDCNO $.

    $( Deduction joining an equivalence and a right operand to form equivalence
       of exclusive-or.  (Contributed by Jim Kingdon, 7-Oct-2018.) $)
    xorbi1d $p |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ th ) ) ) $=
      ( wo wa wn wxo orbi1d anbi1d notbid anbi12d df-xor 3bitr4g ) ABDFZBDGZHZG
      CDFZCDGZHZGBDICDIAPSRUAABCDEJAQTABCDEKLMBDNCDNO $.
  $}

  ${
    xorbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    xorbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction joining two equivalences to form equivalence of exclusive-or.
       (Contributed by Jim Kingdon, 7-Oct-2018.) $)
    xorbi12d $p |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) ) $=
      ( wxo xorbi1d xorbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  ${
    xorbi12.1 $e |- ( ph <-> ps ) $.
    xorbi12.2 $e |- ( ch <-> th ) $.
    $( Equality property for XOR. (Contributed by Mario Carneiro,
       4-Sep-2016.) $)
    xorbi12i $p |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) ) $=
      ( wxo wb wtru a1i xorbi12d mptru ) ACGBDGHIABCDABHIEJCDHIFJKL $.
  $}

  $( A consequence of exclusive or.  In classical logic the converse also
     holds.  (Contributed by Jim Kingdon, 8-Mar-2018.) $)
  xorbin $p  |- ( ( ph \/_ ps ) -> ( ph <-> -. ps ) ) $=
    ( wn wo wa wi df-xor imnan biimpri adantl sylbi pm2.53 orcoms adantr impbid
    wxo ) ABPZABCZQABDZABECZEZARFZABGZTUBSUBTABHIJKQUARAFZUCSUDTBAUDBALMNKO $.

  $( One direction of ~ pm5.18dc , which holds for all propositions, not just
     decidable propositions.  (Contributed by Jim Kingdon, 10-Mar-2018.) $)
  pm5.18im $p |- ( ( ph <-> ps ) -> -. ( ph <-> -. ps ) ) $=
    ( wb wn pm5.19 bibi1 notbid mpbiri ) ABCZABDZCZDBJCZDBEIKLABJFGH $.

  $( A consequence of exclusive or.  For decidable propositions this is an
     equivalence, as seen at ~ xornbidc .  (Contributed by Jim Kingdon,
     10-Mar-2018.) $)
  xornbi $p  |- ( ( ph \/_ ps ) -> -. ( ph <-> ps ) ) $=
    ( wxo wn wb xorbin pm5.18im con2i syl ) ABCABDEZABEZDABFKJABGHI $.

  $( Two ways to express "exclusive or" between decidable propositions.
     (Contributed by Jim Kingdon, 12-Apr-2018.) $)
  xor3dc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) ) $=
    ( wdc wb wn wa dcn dcbi syl5 imp pm5.18dc a1d con2biddc mpd bicomd ex ) ACZ
    BCZABDZEZABEZDZDQRFZUBTUCUBCZUBTDQRUDRUACQUDBGAUAHIJUCSUBUCSUBEDZUDQRUEABKJ
    LMNOP $.

  $( ` \/_ ` is commutative.  (Contributed by David A. Wheeler, 6-Oct-2018.) $)
  xorcom $p |- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) ) $=
    ( wo wa wn wxo orcom ancom notbii anbi12i df-xor 3bitr4i ) ABCZABDZEZDBACZB
    ADZEZDABFBAFMPORABGNQABHIJABKBAKL $.

  $( A decidable proposition is equivalent to a decidable proposition or its
     negation.  Based on theorem *5.15 of [WhiteheadRussell] p. 124.
     (Contributed by Jim Kingdon, 18-Apr-2018.) $)
  pm5.15dc $p |- ( DECID ph -> ( DECID ps ->
      ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) ) ) $=
    ( wdc wb wn wo wa wi xor3dc imp biimpd dcbi dfordc syl mpbird ex ) ACZBCZAB
    DZABEDZFZQRGZUASEZTHZUBUCTQRUCTDABIJKUBSCZUAUDDQRUEABLJSTMNOP $.

  $( Two ways to express "exclusive or" between decidable propositions.
     (Contributed by Jim Kingdon, 17-Apr-2018.) $)
  xor2dc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) ) ) $=
    ( wdc wb wn wo wa xor3dc imp pm5.17dc adantl bitr4d ex ) ACZBCZABDEZABFABGE
    GZDNOGPABEDZQNOPRDABHIOQRDNABJKLM $.

  $( Exclusive or is equivalent to negated biconditional for decidable
     propositions.  (Contributed by Jim Kingdon, 27-Apr-2018.) $)
  xornbidc $p  |- ( DECID ph -> ( DECID ps ->
      ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) ) ) $=
    ( wdc wxo wb wn wa wo df-xor xor2dc imp bitr4id ex ) ACZBCZABDZABEFZENOGPAB
    HABGFGZQABINOQREABJKLM $.

  $( Two ways to express "exclusive or" between decidable propositions.
     Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable
     propositions.  (Contributed by Jim Kingdon, 5-May-2018.) $)
  xordc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) ) $=
    ( wdc wb wn wa wo wxo xornbidc imp excxor ancom orbi2i bitri bitr3di ex ) A
    CZBCZABDEZABEFZBAEZFZGZDQRFABHZSUCQRUDSDABIJUDTUABFZGUCABKUEUBTUABLMNOP $.

  $( Exclusive or implies the left proposition is decidable.  (Contributed by
     Jim Kingdon, 12-Mar-2018.) $)
  xordc1 $p |- ( ( ph \/_ ps ) -> DECID ph ) $=
    ( wo wa wn wxo wdc andir simpl imnan ancom xchbinxr pm3.35 sylan2br orim12i
    wi sylbi df-xor df-dc 3imtr4i ) ABCABDZEZDZAAEZCZABFAGUCAUBDZBUBDZCUEABUBHU
    FAUGUDAUBIUBBBUDPZUDUHBADUABAJABKLBUDMNOQABRAST $.

  $( Move negation outside of biconditional, for decidable propositions.
     Compare Theorem *5.18 of [WhiteheadRussell] p. 124.  (Contributed by Jim
     Kingdon, 18-Apr-2018.) $)
  nbbndc $p |- ( DECID ph -> ( DECID ps ->
      ( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) ) ) $=
    ( wdc wn wb wa xor3dc imp con2bidc bitrd bicom bitr2di ex ) ACZBCZADZBEZABE
    DZENOFZRBPEZQSRABDEZTNORUAEABGHNOUATEABIHJBPKLM $.

  $( Associative law for the biconditional, for decidable propositions.

     The classical version (without the decidability conditions) is an axiom of
     system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure
     and Applied Logic, 113:207-224, 2002,
     ~ http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805 , and,
     interestingly, was not included in _Principia Mathematica_ but was
     apparently first noted by Jan Lukasiewicz circa 1923.  (Contributed by Jim
     Kingdon, 4-May-2018.) $)
  biassdc $p |- ( DECID ph -> ( DECID ps -> ( DECID ch ->
      ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) ) ) ) $=
    ( wdc wb wn wo wa wi df-dc pm5.501 bibi1d bitr3d a1d nbbndc imp adantl nbn2
    adantr ex jaoi sylbi expd ) ADZBDZCDZABEZCEZABCEZEZEZUDAAFZGUEUFHZUKIZAJAUN
    ULAUKUMAUIUHUJABUGCABKLAUIKMNULUMUKULUMHZUIFZUHUJUOBFZCEZUPUHUMURUPEZULUEUF
    USBCOPQULURUHEUMULUQUGCABRLSMULUPUJEUMAUIRSMTUAUBUC $.

  $( Lukasiewicz's shortest axiom for equivalential calculus (but modified to
     require decidable propositions).  Storrs McCall, ed., _Polish Logic
     1920-1939_ (Oxford, 1967), p. 96.  (Contributed by Jim Kingdon,
     5-May-2018.) $)
  bilukdc $p |- ( ( ( DECID ph /\ DECID ps ) /\ DECID ch ) ->
      ( ( ph <-> ps ) <-> ( ( ch <-> ps ) <-> ( ph <-> ch ) ) ) ) $=
    ( wdc wa wb bicom bibi1i biassdc imp31 bitrid ancom1s dcbi imp adantr simpr
    syl9 syl3c mpbid simplr adantlr bitr4d ) ADZBDZEZCDZEZABFZCBACFZFZFZCBFUIFZ
    UGUHCFZUJFZUHUKFZUDUCUFUNUMBAFZCFZUDUCEUFEUJUHUPCABGHUDUCUFUQUJFBACIJKLUGUH
    DZUFUJDZUNUOFUEURUFUCUDURABMNOUEUFPZUCUDUFUSUCUFUIDZUDUSACMZBUIMQJUHCUJIRSU
    GUFUDVAULUKFUTUCUDUFTUCUFVAUDUCUFVAVBNUACBUIIRUB $.

  $( An alternate definition of the biconditional for decidable propositions.
     Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability
     conditions.  (Contributed by Jim Kingdon, 5-May-2018.) $)
  dfbi3dc $p |- ( DECID ph -> ( DECID ps ->
      ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) ) ) $=
    ( wdc wb wa wn dcn xordc imp sylan2 pm5.18dc notnotbdc anbi2d ancom orbi12d
    wo a1i adantl 3bitr4d ex ) ACZBCZABDZABEZAFZBFZEZPZDUAUBEAUFDFZAUFFZEZUFUEE
    ZPZUCUHUBUAUFCZUIUMDZBGUAUNUOAUFHIJUAUBUCUIDABKIUBUHUMDUAUBUDUKUGULUBBUJABL
    MUGULDUBUEUFNQORST $.

  $( Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable
     propositions.  (Contributed by Jim Kingdon, 5-May-2018.) $)
  pm5.24dc $p |- ( DECID ph -> ( DECID ps ->
      ( -. ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <->
                ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) ) $=
    ( wdc wa wn wo wb dfbi3dc imp notbid xordc bitr3d ex ) ACZBCZABDAEZBEZDFZEZ
    AQDBPDFZGNODZABGZEZSTUAUBRNOUBRGABHIJNOUCTGABKILM $.

  $( Conjunction distributes over exclusive-or, for decidable propositions.
     This is one way to interpret the distributive law of multiplication over
     addition in modulo 2 arithmetic.  (Contributed by Jim Kingdon,
     14-Jul-2018.) $)
  xordidc $p |- ( DECID ph -> ( DECID ps -> ( DECID ch ->
    ( ( ph /\ ( ps \/_ ch ) ) <->
      ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) ) ) ) $=
    ( wdc wxo wa wb wn dcbi imp wi annimdc pm5.32 notbii bitrdi sylan2 xornbidc
    adantl anbi2d dcan adantrr adantrl sylc 3bitr4d exp32 ) ADZBDZCDZABCEZFZABF
    ZACFZEZGUFUGUHFZFZABCGZHZFZUKULGZHZUJUMUNUFUPDZURUTGUGUHVABCIJUFVAFURAUPKZH
    ZUTUFVAURVCGAUPLJVBUSABCMNOPUOUIUQAUNUIUQGZUFUGUHVDBCQJRSUOUKDZULDZUMUTGUFU
    GVEUHABTUAUFUHVFUGACTUBUKULQUCUDUE $.

  $( Conjunction distributes over exclusive-or.  (Contributed by Mario Carneiro
     and Jim Kingdon, 7-Oct-2018.) $)
  anxordi $p |- ( ( ph /\ ( ps \/_ ch ) ) <->
      ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) $=
    ( wxo wa simpl wo wn df-xor simplbi jaoi syl ibar xorbi12d bitr3d pm5.21nii
    ) ABCDZEZAABEZACEZDZAQFUASTGZAUAUBSTEHSTIJSATABFACFKLAQRUAAQMABSCTABMACMNOP
    $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Truth tables: Operations on true and false constants
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  For classical logic, truth tables can be used to define propositional
  logic operations, by showing the results of those operations for all
  possible combinations of true ( ` T. ` ) and false ( ` F. ` ).

  Although the intuitionistic logic connectives are not as simply defined,
  ` T. ` and ` F. ` do play similar roles as in classical logic and most
  theorems from classical logic continue to hold.

  Here we show that our definitions and axioms produce equivalent results for
  ` T. ` and ` F. ` as we would get from truth tables for
  ` /\ ` (conjunction aka logical 'and') ~ wa ,
  ` \/ ` (disjunction aka logical inclusive 'or') ~ wo ,
  ` -> ` (implies) ~ wi ,
  ` -. ` (not) ~ wn ,
  ` <-> ` (logical equivalence) ~ df-bi , and
  ` \/_ ` (exclusive or) ~ df-xor .
$)

  $( A ` /\ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  truantru $p |- ( ( T. /\ T. ) <-> T. ) $=
    ( wtru anidm ) AB $.

  $( A ` /\ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  truanfal $p |- ( ( T. /\ F. ) <-> F. ) $=
    ( wfal truan ) AB $.

  $( A ` /\ ` identity.  (Contributed by David A. Wheeler, 23-Feb-2018.) $)
  falantru $p |- ( ( F. /\ T. ) <-> F. ) $=
    ( wfal wtru wa simpl falim impbii ) ABCZAABDGEF $.

  $( A ` /\ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  falanfal $p |- ( ( F. /\ F. ) <-> F. ) $=
    ( wfal anidm ) AB $.

  $( A ` \/ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  truortru $p |- ( ( T. \/ T. ) <-> T. ) $=
    ( wtru oridm ) AB $.

  $( A ` \/ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  truorfal $p |- ( ( T. \/ F. ) <-> T. ) $=
    ( wtru wfal wo tru orci bitru ) ABCABDEF $.

  $( A ` \/ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  falortru $p |- ( ( F. \/ T. ) <-> T. ) $=
    ( wfal wtru wo tru olci bitru ) ABCBADEF $.

  $( A ` \/ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  falorfal $p |- ( ( F. \/ F. ) <-> F. ) $=
    ( wfal oridm ) AB $.

  $( A ` -> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  truimtru $p |- ( ( T. -> T. ) <-> T. ) $=
    ( wtru wi id bitru ) AABACD $.

  $( A ` -> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  truimfal $p |- ( ( T. -> F. ) <-> F. ) $=
    ( wfal wtru wi tru a1bi bicomi ) ABACBADEF $.

  $( A ` -> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  falimtru $p |- ( ( F. -> T. ) <-> T. ) $=
    ( wfal wtru wi falim bitru ) ABCBDE $.

  $( A ` -> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  falimfal $p |- ( ( F. -> F. ) <-> T. ) $=
    ( wfal wi id bitru ) AABACD $.

  $( A ` -. ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  nottru $p |- ( -. T. <-> F. ) $=
    ( wfal wtru wn df-fal bicomi ) ABCDE $.

  $( A ` -. ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  notfal $p |- ( -. F. <-> T. ) $=
    ( wfal wn fal bitru ) ABCD $.

  $( A ` <-> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  trubitru $p |- ( ( T. <-> T. ) <-> T. ) $=
    ( wtru wb biid bitru ) AABACD $.

  $( A ` <-> ` identity.  (Contributed by David A. Wheeler, 23-Feb-2018.) $)
  trubifal $p |- ( ( T. <-> F. ) <-> F. ) $=
    ( wtru wfal wb wi wa dfbi2 truimfal falimtru anbi12i falantru 3bitri ) ABCA
    BDZBADZEBAEBABFLBMAGHIJK $.

  $( A ` <-> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  falbitru $p |- ( ( F. <-> T. ) <-> F. ) $=
    ( wfal wtru wb bicom trubifal bitri ) ABCBACAABDEF $.

  $( A ` <-> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  falbifal $p |- ( ( F. <-> F. ) <-> T. ) $=
    ( wfal wb biid bitru ) AABACD $.

  $( A ` \/_ ` identity.  (Contributed by David A. Wheeler, 2-Mar-2018.) $)
  truxortru $p |- ( ( T. \/_ T. ) <-> F. ) $=
    ( wtru wxo wo wa wn df-xor oridm nottru anidm xchnxbir anbi12i truan 3bitri
    wfal ) AABAACZAADZEZDANDNAAFOAQNAGANPHAIJKNLM $.

  $( A ` \/_ ` identity.  (Contributed by David A. Wheeler, 2-Mar-2018.) $)
  truxorfal $p |- ( ( T. \/_ F. ) <-> T. ) $=
    ( wtru wfal wxo wo wa wn df-xor truorfal notfal truan xchnxbir anidm 3bitri
    anbi12i ) ABCABDZABEZFZEAAEAABGOAQAHBAPIBJKNALM $.

  $( A ` \/_ ` identity.  (Contributed by David A. Wheeler, 2-Mar-2018.) $)
  falxortru $p |- ( ( F. \/_ T. ) <-> T. ) $=
    ( wfal wtru wo wa wn df-xor falortru notfal falantru xchnxbir anbi12i anidm
    wxo 3bitri ) ABMABCZABDZEZDBBDBABFOBQBGABPHIJKBLN $.

  $( A ` \/_ ` identity.  (Contributed by David A. Wheeler, 2-Mar-2018.) $)
  falxorfal $p |- ( ( F. \/_ F. ) <-> F. ) $=
    ( wfal wxo wo wa wn wtru df-xor oridm notfal anidm xchnxbir falantru 3bitri
    anbi12i ) AABAACZAADZEZDAFDAAAGOAQFAHAFPIAJKNLM $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Stoic logic indemonstrables (Chrysippus of Soli)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The Greek Stoics developed a system of logic.  The Stoic Chrysippus, in
  particular, was often considered one of the greatest logicians of antiquity.
  Stoic logic is different from Aristotle's system, since it focuses on
  propositional logic, though later thinkers did combine the systems of the
  Stoics with Aristotle.  Jan Lukasiewicz reports, "For anybody familiar with
  mathematical logic it is self-evident that the Stoic dialectic is the ancient
  form of modern propositional logic"
  ( _On the history of the logic of proposition_ by Jan Lukasiewicz (1934),
  translated in: _Selected Works_ - Edited by Ludwik Borkowski -
  Amsterdam, North-Holland, 1970 pp. 197-217,
  referenced in "History of Logic"
  ~ https://www.historyoflogic.com/logic-stoics.htm ).
  For more about Aristotle's system, see barbara and related theorems.

  A key part of the Stoic logic system is a set of five "indemonstrables"
  assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is
  difficult to assign specific ideas to specific thinkers.  The indemonstrables
  are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and
  [Hitchcock] p. 5.  These indemonstrables are modus ponendo ponens (modus
  ponens) ~ ax-mp , modus tollendo tollens (modus tollens) ~ mto , modus
  ponendo tollens I ~ mptnan , modus ponendo tollens II ~ mptxor , and modus
  tollendo ponens (exclusive-or version) ~ mtpxor .  The first is an axiom, the
  second is already proved; in this section we prove the other three.  Since we
  assume or prove all of indemonstrables, the system of logic we use here is as
  at least as strong as the set of Stoic indemonstrables.  Note that modus
  tollendo ponens ~ mtpxor originally used exclusive-or, but over time the name
  modus tollendo ponens has increasingly referred to an inclusive-or variation,
  which is proved in ~ mtpor .  This set of indemonstrables is not the entire
  system of Stoic logic.

$)

  ${
    $( Minor premise for modus ponendo tollens 1. $)
    mptnan.min $e |- ph $.
    $( Major premise for modus ponendo tollens 1. $)
    mptnan.maj $e |- -. ( ph /\ ps ) $.
    $( Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic.
       See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and
       rule A3 in [Hitchcock] p. 5.  Sanford describes this rule second (after
       ~ mptxor ) as a "safer, and these days much more common" version of
       modus ponendo tollens because it avoids confusion between inclusive-or
       and exclusive-or.  (Contributed by David A. Wheeler, 3-Jul-2016.) $)
    mptnan $p |- -. ps $=
      ( wn imnani ax-mp ) ABECABDFG $.
  $}

  ${
    $( Minor premise for modus ponendo tollens 2. $)
    mptxor.min $e |- ph $.
    $( Major premise for modus ponendo tollens 2. $)
    mptxor.maj $e |- ( ph \/_ ps ) $.
    $( Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic.
       Note that this uses exclusive-or ` \/_ ` .  See rule 2 on
       [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in
       [Hitchcock] p. 5 .  (Contributed by David A. Wheeler, 2-Mar-2018.) $)
    mptxor $p |- -. ps $=
      ( wo wa wn wxo df-xor mpbi simpri mptnan ) ABCABEZABFGZABHMNFDABIJKL $.
  $}

  ${
    $( Minor premise for modus tollendo ponens (inclusive-or version). $)
    mtpor.min $e |- -. ph $.
    $( Major premise for modus tollendo ponens (inclusive-or version). $)
    mtpor.max $e |- ( ph \/ ps ) $.
    $( Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism.
       This is similar to ~ mtpxor , one of the five original "indemonstrables"
       in Stoic logic.  However, in Stoic logic this rule used exclusive-or,
       while the name modus tollendo ponens often refers to a variant of the
       rule that uses inclusive-or instead.  The rule says, "if ` ph ` is not
       true, and ` ph ` or ` ps ` (or both) are true, then ` ps ` must be
       true".  An alternate phrasing is, "Once you eliminate the impossible,
       whatever remains, no matter how improbable, must be the truth". --
       Sherlock Holmes (Sir Arthur Conan Doyle, 1890:  The Sign of the Four,
       ch. 6).  (Contributed by David A. Wheeler, 3-Jul-2016.)  (Proof
       shortened by Wolf Lammen, 11-Nov-2017.) $)
    mtpor $p |- ps $=
      ( wn ori ax-mp ) AEBCABDFG $.
  $}

  ${
    $( Minor premise for modus tollendo ponens (original exclusive-or version).
    $)
    mtpxor.min $e |- -. ph $.
    $( Major premise for modus tollendo ponens (original exclusive-or version).
    $)
    mtpxor.maj $e |- ( ph \/_ ps ) $.
    $( Modus tollendo ponens (original exclusive-or version), aka disjunctive
       syllogism, similar to ~ mtpor , one of the five "indemonstrables" in
       Stoic logic.  The rule says, "if ` ph ` is not true, and either ` ph `
       or ` ps ` (exclusively) are true, then ` ps ` must be true".  Today the
       name "modus tollendo ponens" often refers to a variant, the inclusive-or
       version as defined in ~ mtpor .  See rule 3 on [Lopez-Astorga] p. 12
       (note that the "or" is the same as ~ mptxor , that is, it is
       exclusive-or ~ df-xor ), rule 3 of [Sanford] p. 39 (where it is not as
       clearly stated which kind of "or" is used but it appears to be in the
       same sense as ~ mptxor ), and rule A5 in [Hitchcock] p. 5 (exclusive-or
       is expressly used).  (Contributed by David A. Wheeler, 4-Jul-2016.)
       (Proof shortened by Wolf Lammen, 11-Nov-2017.)  (Proof shortened by BJ,
       19-Apr-2019.) $)
    mtpxor $p |- ps $=
      ( wxo wo xoror ax-mp mtpor ) ABCABEABFDABGHI $.
  $}

  ${
    $( Premise for Stoic logic thema 1. $)
    stoic1.1 $e |- ( ( ph /\ ps ) -> th ) $.
    $( Stoic logic Thema 1 (part a).

       The first thema of the four Stoic logic themata, in its basic form, was:

       "When from two (assertibles) a third follows, then from either of them
       together with the contradictory of the conclusion the contradictory of
       the other follows."  (Apuleius Int. 209.9-14), see [Bobzien] p. 117 and
       ~ https://plato.stanford.edu/entries/logic-ancient/

       We will represent thema 1 as two very similar rules ~ stoic1a and
       ~ stoic1b to represent each side.  (Contributed by David A. Wheeler,
       16-Feb-2019.)  (Proof shortened by Wolf Lammen, 21-May-2020.) $)
    stoic1a $p |- ( ( ph /\ -. th ) -> -. ps ) $=
      ( ex con3dimp ) ABCABCDEF $.

    $( Stoic logic Thema 1 (part b).  The other part of thema 1 of Stoic logic;
       see ~ stoic1a .  (Contributed by David A. Wheeler, 16-Feb-2019.) $)
    stoic1b $p |- ( ( ps /\ -. th ) -> -. ph ) $=
      ( ancoms stoic1a ) BACABCDEF $.
  $}

  ${
    $( Premise 1 for Stoic logic thema 2 version a. $)
    stoic2a.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Premise 2 for Stoic logic thema 2 version a. $)
    stoic2a.2 $e |- ( ( ph /\ ch ) -> th ) $.
    $( Stoic logic Thema 2 version a.

       Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic
       logic thema 2 as follows:  "When from two assertibles a third follows,
       and from the third and one (or both) of the two another follows, then
       this other follows from the first two."

       Bobzien uses constructs such as ` ph ` , ` ps |- ch ` ; in Metamath we
       will represent that construct as ` ph /\ ps -> ch ` .

       This version a is without the phrase "or both"; see ~ stoic2b for the
       version with the phrase "or both".  We already have this rule as
       ~ syldan , so here we show the equivalence and discourage its use.
       (New usage is discouraged.)  (Contributed by David A. Wheeler,
       17-Feb-2019.) $)
    stoic2a $p |- ( ( ph /\ ps ) -> th ) $=
      ( syldan ) ABCDEFG $.
  $}

  ${
    $( Premise 1 for Stoic logic thema 2 version b. $)
    stoic2b.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Premise 2 for Stoic logic thema 2 version b. $)
    stoic2b.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( Stoic logic Thema 2 version b.  See ~ stoic2a .

       Version b is with the phrase "or both".  We already have this rule as
       ~ mpd3an3 , so here we prove the equivalence and discourage its use.
       (New usage is discouraged.)  (Contributed by David A. Wheeler,
       17-Feb-2019.) $)
    stoic2b $p |- ( ( ph /\ ps ) -> th ) $=
      ( mpd3an3 ) ABCDEFG $.
  $}

  ${
    $( Premise 1 for Stoic logic thema 3. $)
    stoic3.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Premise 2 for Stoic logic thema 3. $)
    stoic3.2 $e |- ( ( ch /\ th ) -> ta ) $.
    $( Stoic logic Thema 3.

       Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic thema 3.

       "When from two (assemblies) a third follows, and from the one that
       follows (i.e., the third) together with another, external external
       assumption, another follows, then other follows from the first two and
       the externally co-assumed one.  (Simp.  Cael. 237.2-4)" (Contributed by
       David A. Wheeler, 17-Feb-2019.) $)
    stoic3 $p |- ( ( ph /\ ps /\ th ) -> ta ) $=
      ( wa sylan 3impa ) ABDEABHCDEFGIJ $.
  $}

  ${
    $( Premise 1 for Stoic logic thema 4a. $)
    stoic4a.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Premise 2 for Stoic logic thema 4a. $)
    stoic4a.2 $e |- ( ( ch /\ ph /\ th ) -> ta ) $.
    $( Stoic logic Thema 4 version a.

       Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic
       logic thema 4:  "When from two assertibles a third follows, and from the
       third and one (or both) of the two and one (or more) external
       assertible(s) another follows, then this other follows from the first
       two and the external(s)."

       We use ` th ` to represent the "external" assertibles.  This is version
       a, which is without the phrase "or both"; see ~ stoic4b for the version
       with the phrase "or both".  (Contributed by David A. Wheeler,
       17-Feb-2019.) $)
    stoic4a $p |- ( ( ph /\ ps /\ th ) -> ta ) $=
      ( w3a 3adant3 simp1 simp3 syl3anc ) ABDHCADEABCDFIABDJABDKGL $.
  $}

  ${
    $( Premise 1 for Stoic logic thema 4b. $)
    stoic4b.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Premise 2 for Stoic logic thema 4b. $)
    stoic4b.2 $e |- ( ( ( ch /\ ph /\ ps ) /\ th ) -> ta ) $.
    $( Stoic logic Thema 4 version b.

       This is version b, which is with the phrase "or both".  See ~ stoic4a
       for more information.  (Contributed by David A. Wheeler,
       17-Feb-2019.) $)
    stoic4b $p |- ( ( ph /\ ps /\ th ) -> ta ) $=
      ( w3a 3adant3 simp1 simp2 simp3 syl31anc ) ABDHCABDEABCDFIABDJABDKABDLGM
      $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical implication (continued)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    syl6an.1 $e |- ( ph -> ps ) $.
    syl6an.2 $e |- ( ph -> ( ch -> th ) ) $.
    syl6an.3 $e |- ( ( ps /\ th ) -> ta ) $.
    $( A syllogism deduction combined with conjoining antecedents.
       (Contributed by Alan Sare, 28-Oct-2011.) $)
    syl6an $p |- ( ph -> ( ch -> ta ) ) $=
      ( wa jctild syl6 ) ACBDIEACDBGFJHK $.
  $}

  ${
    syl10.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl10.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.
    syl10.3 $e |- ( ch -> ( ta -> et ) ) $.
    $( A nested syllogism inference.  (Contributed by Alan Sare,
       17-Jul-2011.) $)
    syl10 $p |- ( ph -> ( ps -> ( th -> et ) ) ) $=
      ( wi syl6 syldd ) ABDEFHABCEFJGIKL $.
  $}

  ${
    a1ddd.1 $e |- ( ph -> ( ps -> ( ch -> ta ) ) ) $.
    $( Triple deduction introducing an antecedent to a wff.  Deduction
       associated with ~ a1dd .  Double deduction associated with ~ a1d .
       Triple deduction associated with ~ ax-1 and ~ a1i .  (Contributed by
       Jeff Hankins, 4-Aug-2009.) $)
    a1ddd $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wi ax-1 syl8 ) ABCEDEGFEDHI $.
  $}

  $( Exportation implication also converting head from biconditional to
     conditional.  (Contributed by Alan Sare, 31-Dec-2011.) $)
  exbir $p |- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) ->
              ( ph -> ( ps -> ( th -> ch ) ) ) ) $=
    ( wa wb wi biimpr imim2i expd ) ABEZCDFZGABDCGZLMKCDHIJ $.

  $( ~ impexp with a 3-conjunct antecedent.  (Contributed by Alan Sare,
     31-Dec-2011.) $)
  3impexp $p |- ( ( ( ph /\ ps /\ ch ) -> th ) <->
                ( ph -> ( ps -> ( ch -> th ) ) ) ) $=
    ( w3a wi id 3expd 3impd impbii ) ABCEDFZABCDFFFZKABCDKGHLABCDLGIJ $.

  $( ~ 3impexp with biconditional consequent of antecedent that is commuted in
     consequent.  (Contributed by Alan Sare, 31-Dec-2011.) $)
  3impexpbicom $p |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <->
                     ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) $=
    ( w3a wb wi bicom imbi2 biimpcd mpi 3expd 3impexp biimpri imbitrrdi impbii
    ) ABCFZDEGZHZABCEDGZHHHZTABCUATSUAGZRUAHZDEIZUCTUDSUARJKLMUBRUASUDUBABCUANO
    UEPQ $.

  ${
    3impexpbicomi.1 $e |- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) $.
    $( Deduction form of ~ 3impexpbicom .  (Contributed by Alan Sare,
       31-Dec-2011.) $)
    3impexpbicomi $p |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) $=
      ( wb w3a bicomd 3exp ) ABCEDGABCHDEFIJ $.
  $}

  $( Closed form of ~ ancoms .  (Contributed by Alan Sare, 31-Dec-2011.) $)
  ancomsimp $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) ) $=
    ( wa ancom imbi1i ) ABDBADCABEF $.

  ${
    expcomd.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( Deduction form of ~ expcom .  (Contributed by Alan Sare,
       22-Jul-2012.) $)
    expcomd $p |- ( ph -> ( ch -> ( ps -> th ) ) ) $=
      ( expd com23 ) ABCDABCDEFG $.
  $}

  ${
    expdcom.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( Commuted form of ~ expd .  (Contributed by Alan Sare, 18-Mar-2012.) $)
    expdcom $p |- ( ps -> ( ch -> ( ph -> th ) ) ) $=
      ( expd com3l ) ABCDABCDEFG $.
  $}

  $( Implication form of ~ simplbi2com .  (Contributed by Alan Sare,
     22-Jul-2012.) $)
  simplbi2comg $p |- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) ) $=
    ( wa wb biimpr expcomd ) ABCDZEBCAAHFG $.

  ${
    simplbi2com.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( A deduction eliminating a conjunct, similar to ~ simplbi2 .
       (Contributed by Alan Sare, 22-Jul-2012.)  (Proof shortened by Wolf
       Lammen, 10-Nov-2012.) $)
    simplbi2com $p |- ( ch -> ( ps -> ph ) ) $=
      ( simplbi2 com12 ) BCAABCDEF $.
  $}

  ${
    syl6ci.1 $e |- ( ph -> ( ps -> ch ) ) $.
    syl6ci.2 $e |- ( ph -> th ) $.
    syl6ci.3 $e |- ( ch -> ( th -> ta ) ) $.
    $( A syllogism inference combined with contraction.  (Contributed by Alan
       Sare, 18-Mar-2012.) $)
    syl6ci $p |- ( ph -> ( ps -> ta ) ) $=
      ( a1d syl6c ) ABCDEFADBGIHJ $.
  $}

  ${
    mpisyl.1 $e |- ( ph -> ps ) $.
    mpisyl.2 $e |- ch $.
    mpisyl.3 $e |- ( ps -> ( ch -> th ) ) $.
    $( A syllogism combined with a modus ponens inference.  (Contributed by
       Alan Sare, 25-Jul-2011.) $)
    mpisyl $p |- ( ph -> th ) $=
      ( mpi syl ) ABDEBCDFGHI $.
  $}

  ${
    dcfromnotnotr.1 $e |- ( ph <-> ( ps \/ -. ps ) ) $.
    dcfromnotnotr.2 $e |- ( -. -. ph -> ph ) $.
    $( The decidability of a proposition ` ps ` follows from a suitable
       instance of double negation elimination (DNE).  Therefore, if we were to
       introduce DNE as a general principle (without the decidability condition
       in ~ notnotrdc ), then we could prove that every proposition is
       decidable, giving us the classical system of propositional calculus
       (since DNE itself is classically valid).  (Contributed by Adrian
       Ducourtial, 6-Oct-2025.) $)
    dcfromnotnotr $p |- DECID ps
        $=
      ( wdc wn wo nnexmid notbii 3imtr3i ax-mp df-dc mpbir ) BEBBFGZNFZFZNBHAFZ
      FAPNDQOANCIICJKBLM $.
  $}

  ${
    dcfromcon.1 $e |- ( ph <-> ( ch \/ -. ch ) ) $.
    dcfromcon.2 $e |- ( ps <-> T. ) $.
    dcfromcon.3 $e |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $.
    $( The decidability of a proposition ` ch ` follows from a suitable
       instance of the principle of contraposition.  Therefore, if we were to
       introduce contraposition as a general principle (without the
       decidability condition in ~ condc ), then we could prove that every
       proposition is decidable, giving us the classical system of
       propositional calculus (since the principle of contraposition is itself
       classically valid).  (Contributed by Adrian Ducourtial, 6-Oct-2025.) $)
    dcfromcon $p |- DECID ch
          $=
      ( wdc wn wo wtru nnexmid pm2.21i notbii imbi12i 3imtr3i ax-mp mptru df-dc
      wi mpbir ) CGCCHIZUAUAHZJHZSZJUASZUBUCCKLAHZBHZSBASUDUEFUFUBUGUCAUADMBJEM
      NBJAUAEDNOPQCRT $.
  $}

  ${
    dcfrompeirce.1 $e |- ( ph <-> ( ch \/ -. ch ) ) $.
    dcfrompeirce.2 $e |- ( ps <-> F. ) $.
    dcfrompeirce.3 $e |- ( ( ( ph -> ps ) -> ph ) -> ph ) $.
    $( The decidability of a proposition ` ch ` follows from a suitable
       instance of Peirce's law.  Therefore, if we were to introduce Peirce's
       law as a general principle (without the decidability condition in
       ~ peircedc ), then we could prove that every proposition is decidable,
       giving us the classical system of propositional calculus (since Perice's
       law is itself classically valid).  (Contributed by Adrian Ducourtial,
       6-Oct-2025.) $)
    dcfrompeirce $p |- DECID ch
          $=
      ( wdc wn wo wfal wi pm2.67-2 dfnot olcd imbi12i 3imtr3i ax-mp df-dc mpbir
      sylibr ) CGCCHZIZUBJKZUBKZUBUCUACUCCJKUACJUALCMTNABKZAKAUDUBFUEUCAUBAUBBJ
      DEODODPQCRS $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Predicate calculus mostly without distinct variables
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Universal quantifier (continued)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The universal quantifier was introduced above in ~ wal for use by ~ df-tru .
  See the comments in that section.  In this section, we continue with the
  first "real" use of it.
$)

  $( Declare some names for individual variables. $)
  $v x $.
  $v y $.
  $v z $.
  $v w $.
  $v v $.
  $v u $.
  $v t $.
  $( Let ` x ` be an individual variable. $)
  vx $f setvar x $.
  $( Let ` y ` be an individual variable. $)
  vy $f setvar y $.
  $( Let ` z ` be an individual variable. $)
  vz $f setvar z $.
  $( Let ` w ` be an individual variable. $)
  vw $f setvar w $.
  $( Let ` v ` be an individual variable. $)
  vv $f setvar v $.
  $( Let ` u ` be an individual variable. $)
  vu $f setvar u $.
  $( Let ` t ` be an individual variable. $)
  vt $f setvar t $.

  $( Axiom of Quantified Implication.  Axiom C4 of [Monk2] p. 105.
     (Contributed by NM, 5-Aug-1993.) $)
  ax-5 $a |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $.

  $( Axiom of Quantifier Commutation.  This axiom says universal quantifiers
     can be swapped.  One of the predicate logic axioms which do not involve
     equality.  Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint).
     Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2]
     p. 113.  (Contributed by NM, 5-Aug-1993.) $)
  ax-7 $a |- ( A. x A. y ph -> A. y A. x ph ) $.

  ${
    ax-g.1 $e |- ph $.
    $( Rule of Generalization.  The postulated inference rule of predicate
       calculus.  See, e.g., Rule 2 of [Hamilton] p. 74.  This rule says that
       if something is unconditionally true, then it is true for all values of
       a variable.  For example, if we have proved ` x = x ` , we can conclude
       ` A. x x = x ` or even ` A. y x = x ` .  Theorem ~ spi shows we can go
       the other way also: in other words we can add or remove universal
       quantifiers from the beginning of any theorem as required.  (Contributed
       by NM, 5-Aug-1993.) $)
    ax-gen $a |- A. x ph $.
  $}

  ${
    gen2.1 $e |- ph $.
    $( Generalization applied twice.  (Contributed by NM, 30-Apr-1998.) $)
    gen2 $p |- A. x A. y ph $=
      ( wal ax-gen ) ACEBACDFF $.
  $}

  ${
    mpg.1 $e |- ( A. x ph -> ps ) $.
    mpg.2 $e |- ph $.
    $( Modus ponens combined with generalization.  (Contributed by NM,
       24-May-1994.) $)
    mpg $p |- ps $=
      ( wal ax-gen ax-mp ) ACFBACEGDH $.
  $}

  ${
    mpgbi.1 $e |- ( A. x ph <-> ps ) $.
    mpgbi.2 $e |- ph $.
    $( Modus ponens on biconditional combined with generalization.
       (Contributed by NM, 24-May-1994.)  (Proof shortened by Stefan Allan,
       28-Oct-2008.) $)
    mpgbi $p |- ps $=
      ( wal ax-gen mpbi ) ACFBACEGDH $.
  $}

  ${
    mpgbir.1 $e |- ( ph <-> A. x ps ) $.
    mpgbir.2 $e |- ps $.
    $( Modus ponens on biconditional combined with generalization.
       (Contributed by NM, 24-May-1994.)  (Proof shortened by Stefan Allan,
       28-Oct-2008.) $)
    mpgbir $p |- ph $=
      ( wal ax-gen mpbir ) ABCFBCEGDH $.
  $}

  ${
    a7s.1 $e |- ( A. x A. y ph -> ps ) $.
    $( Swap quantifiers in an antecedent.  (Contributed by NM, 5-Aug-1993.) $)
    a7s $p |- ( A. y A. x ph -> ps ) $=
      ( wal ax-7 syl ) ACFDFADFCFBADCGEH $.
  $}

  ${
    alimi.1 $e |- ( ph -> ps ) $.
    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 5-Aug-1993.) $)
    alimi $p |- ( A. x ph -> A. x ps ) $=
      ( wi wal ax-5 mpg ) ABEACFBCFECABCGDH $.

    $( Inference doubly quantifying both antecedent and consequent.
       (Contributed by NM, 3-Feb-2005.) $)
    2alimi $p |- ( A. x A. y ph -> A. x A. y ps ) $=
      ( wal alimi ) ADFBDFCABDEGG $.
  $}

  $( Theorem 19.20 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.)
     (Proof shortened by O'Cat, 30-Mar-2008.) $)
  alim $p |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $=
    ( ax-5 ) ABCD $.

  ${
    al2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference quantifying antecedent, nested antecedent, and consequent.
       (Contributed by NM, 5-Aug-1993.) $)
    al2imi $p |- ( A. x ph -> ( A. x ps -> A. x ch ) ) $=
      ( wal wi alimi alim syl ) ADFBCGZDFBDFCDFGAKDEHBCDIJ $.
  $}

  ${
    alanimi.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Variant of ~ al2imi with conjunctive antecedent.  (Contributed by Andrew
       Salmon, 8-Jun-2011.) $)
    alanimi $p |- ( ( A. x ph /\ A. x ps ) -> A. x ch ) $=
      ( wal ex al2imi imp ) ADFBDFCDFABCDABCEGHI $.
  $}

  $c F/ $.  $( The not-free symbol. $)

  $( Extend wff definition to include the not-free predicate. $)
  wnf $a wff F/ x ph $.

  $( Define the not-free predicate for wffs.  This is read " ` x ` is not free
     in ` ph ` ".  Not-free means that the value of ` x ` cannot affect the
     value of ` ph ` , e.g., any occurrence of ` x ` in ` ph ` is effectively
     bound by a "for all" or something that expands to one (such as "there
     exists").  In particular, substitution for a variable not free in a wff
     does not affect its value ( ~ sbf ).  An example of where this is used is
     ~ stdpc5 .  See ~ nf2 for an alternate definition which does not involve
     nested quantifiers on the same variable.

     Nonfreeness is a commonly used condition, so it is useful to have a
     notation for it.  Surprisingly, there is no common formal notation for it,
     so here we devise one.  Our definition lets us work with the notion of
     nonfreeness within the logic itself rather than as a metalogical side
     condition.

     To be precise, our definition really means "effectively not free", because
     it is slightly less restrictive than the usual textbook definition for
     "not free" (which considers syntactic freedom).  For example, ` x ` is
     effectively not free in the expression ` x = x ` (even though ` x ` is
     syntactically free in it, so would be considered "free" in the usual
     textbook definition) because the value of ` x ` in the formula ` x = x `
     does not affect the truth of that formula (and thus substitutions will not
     change the result), see ~ nfequid .  (Contributed by Mario Carneiro,
     11-Aug-2016.) $)
  df-nf $a |- ( F/ x ph <-> A. x ( ph -> A. x ph ) ) $.

  ${
    nfi.1 $e |- ( ph -> A. x ph ) $.
    $( Deduce that ` x ` is not free in ` ph ` from the definition.
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nfi $p |- F/ x ph $=
      ( wnf wal wi df-nf mpgbir ) ABDAABEFBABGCH $.
  $}

  ${
    hbth.1 $e |- ph $.
    $( No variable is (effectively) free in a theorem.

       This and later "hypothesis-building" lemmas, with labels starting
       "hb...", allow us to construct proofs of formulas of the form
       ` |- ( ph -> A. x ph ) ` from smaller formulas of this form.  These are
       useful for constructing hypotheses that state " ` x ` is (effectively)
       not free in ` ph ` ".  (Contributed by NM, 5-Aug-1993.) $)
    hbth $p |- ( ph -> A. x ph ) $=
      ( wal ax-gen a1i ) ABDAABCEF $.

    $( No variable is (effectively) free in a theorem.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    nfth $p |- F/ x ph $=
      ( hbth nfi ) ABABCDE $.
  $}

  ${
    nfnth.1 $e |- -. ph $.
    $( No variable is (effectively) free in a non-theorem.  (Contributed by
       Mario Carneiro, 6-Dec-2016.) $)
    nfnth $p |- F/ x ph $=
      ( wal pm2.21i nfi ) ABAABDCEF $.
  $}

  $( The true constant has no free variables.  (This can also be proven in one
     step with ~ nfv , but this proof does not use ~ ax-17 .)  (Contributed by
     Mario Carneiro, 6-Oct-2016.) $)
  nftru $p |- F/ x T. $=
    ( wtru tru nfth ) BACD $.

  ${
    alimdh.1 $e |- ( ph -> A. x ph ) $.
    alimdh.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.20 of [Margaris] p. 90.  (Contributed by NM,
       4-Jan-2002.) $)
    alimdh $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
      ( wal wi al2imi syl ) AADGBDGCDGHEABCDFIJ $.
  $}

  $( Theorem 19.15 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
  albi $p |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) ) $=
    ( wb wal biimp al2imi biimpr impbid ) ABDZCEACEBCEJABCABFGJBACABHGI $.

  ${
    alrimih.1 $e |- ( ph -> A. x ph ) $.
    alrimih.2 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.)  (New usage is discouraged.) $)
    alrimih $p |- ( ph -> A. x ps ) $=
      ( wal alimi syl ) AACFBCFDABCEGH $.
  $}

  ${
    albii.1 $e |- ( ph <-> ps ) $.
    $( Inference adding universal quantifier to both sides of an equivalence.
       (Contributed by NM, 7-Aug-1994.) $)
    albii $p |- ( A. x ph <-> A. x ps ) $=
      ( wb wal albi mpg ) ABEACFBCFECABCGDH $.

    $( Inference adding 2 universal quantifiers to both sides of an
       equivalence.  (Contributed by NM, 9-Mar-1997.) $)
    2albii $p |- ( A. x A. y ph <-> A. x A. y ps ) $=
      ( wal albii ) ADFBDFCABDEGG $.
  $}

  ${
    hbxfrbi.1 $e |- ( ph <-> ps ) $.
    hbxfrbi.2 $e |- ( ps -> A. x ps ) $.
    $( A utility lemma to transfer a bound-variable hypothesis builder into a
       definition.  (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
    hbxfrbi $p |- ( ph -> A. x ph ) $=
      ( wal albii 3imtr4i ) BBCFAACFEDABCDGH $.
  $}

  ${
    nfbii.1 $e |- ( ph <-> ps ) $.
    $( Equality theorem for not-free.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfbii $p |- ( F/ x ph <-> F/ x ps ) $=
      ( wal wi wnf albii imbi12i df-nf 3bitr4i ) AACEZFZCEBBCEZFZCEACGBCGMOCABL
      NDABCDHIHACJBCJK $.

    ${
      nfxfr.2 $e |- F/ x ps $.
      $( A utility lemma to transfer a bound-variable hypothesis builder into a
         definition.  (Contributed by Mario Carneiro, 11-Aug-2016.) $)
      nfxfr $p |- F/ x ph $=
        ( wnf nfbii mpbir ) ACFBCFEABCDGH $.
    $}

    ${
      nfxfrd.2 $e |- ( ch -> F/ x ps ) $.
      $( A utility lemma to transfer a bound-variable hypothesis builder into a
         definition.  (Contributed by Mario Carneiro, 24-Sep-2016.) $)
      nfxfrd $p |- ( ch -> F/ x ph ) $=
        ( wnf nfbii sylibr ) CBDGADGFABDEHI $.
    $}
  $}

  ${
    alcoms.1 $e |- ( A. x A. y ph -> ps ) $.
    $( Swap quantifiers in an antecedent.  (Contributed by NM, 11-May-1993.) $)
    alcoms $p |- ( A. y A. x ph -> ps ) $=
      ( wal ax-7 syl ) ACFDFADFCFBADCGEH $.
  $}

  ${
    hbal.1 $e |- ( ph -> A. x ph ) $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
       (Contributed by NM, 5-Aug-1993.) $)
    hbal $p |- ( A. y ph -> A. x A. y ph ) $=
      ( wal alimi ax-7 syl ) ACEZABEZCEIBEAJCDFACBGH $.
  $}

  $( Theorem 19.5 of [Margaris] p. 89.  (Contributed by NM, 5-Aug-1993.) $)
  alcom $p |- ( A. x A. y ph <-> A. y A. x ph ) $=
    ( wal ax-7 impbii ) ACDBDABDCDABCEACBEF $.

  ${
    alrimdh.1 $e |- ( ph -> A. x ph ) $.
    alrimdh.2 $e |- ( ps -> A. x ps ) $.
    alrimdh.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.21 of [Margaris] p. 90.  (Contributed by NM,
       10-Feb-1997.)  (Proof shortened by Andrew Salmon, 13-May-2011.) $)
    alrimdh $p |- ( ph -> ( ps -> A. x ch ) ) $=
      ( wal alimdh syl5 ) BBDHACDHFABCDEGIJ $.
  $}

  ${
    albidh.1 $e |- ( ph -> A. x ph ) $.
    albidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for universal quantifier (deduction form).
       (Contributed by NM, 5-Aug-1993.) $)
    albidh $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
      ( wb wal alrimih albi syl ) ABCGZDHBDHCDHGALDEFIBCDJK $.
  $}

  $( Theorem 19.26 of [Margaris] p. 90.  Also Theorem *10.22 of
     [WhiteheadRussell] p. 119.  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 4-Jul-2014.) $)
  19.26 $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) ) $=
    ( wa wal simpl alimi simpr jca id alanimi impbii ) ABDZCEZACEZBCEZDNOPMACAB
    FGMBCABHGIABMCMJKL $.

  $( Theorem 19.26 of [Margaris] p. 90 with two quantifiers.  (Contributed by
     NM, 3-Feb-2005.) $)
  19.26-2 $p |- ( A. x A. y ( ph /\ ps ) <->
                ( A. x A. y ph /\ A. x A. y ps ) ) $=
    ( wa wal 19.26 albii bitri ) ABEDFZCFADFZBDFZEZCFKCFLCFEJMCABDGHKLCGI $.

  $( Theorem 19.26 of [Margaris] p. 90 with triple conjunction.  (Contributed
     by NM, 13-Sep-2011.) $)
  19.26-3an $p |- ( A. x ( ph /\ ps /\ ch )
                   <-> ( A. x ph /\ A. x ps /\ A. x ch ) ) $=
    ( wa wal w3a 19.26 anbi1i bitri df-3an albii 3bitr4i ) ABEZCEZDFZADFZBDFZEZ
    CDFZEZABCGZDFQRTGPNDFZTEUANCDHUCSTABDHIJUBODABCKLQRTKM $.

  $( Theorem 19.33 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
  19.33 $p |- ( ( A. x ph \/ A. x ps ) -> A. x ( ph \/ ps ) ) $=
    ( wal wo orc alimi olc jaoi ) ACDABEZCDBCDAJCABFGBJCBAHGI $.

  $( Theorem *11.21 in [WhiteheadRussell] p. 160.  (Contributed by Andrew
     Salmon, 24-May-2011.) $)
  alrot3 $p |- ( A. x A. y A. z ph <-> A. y A. z A. x ph ) $=
    ( wal alcom albii bitri ) ADEZCEBEIBEZCEABEDEZCEIBCFJKCABDFGH $.

  $( Rotate 4 universal quantifiers twice.  (Contributed by NM, 2-Feb-2005.)
     (Proof shortened by Wolf Lammen, 28-Jun-2014.) $)
  alrot4 $p |- ( A. x A. y A. z A. w ph <-> A. z A. w A. x A. y ph ) $=
    ( wal alrot3 albii alcom 3bitri ) AEFDFCFZBFACFZEFZDFZBFMBFZDFLBFEFZDFKNBAC
    DEGHMBDIOPDLBEIHJ $.

  $( Split a biconditional and distribute quantifier.  (Contributed by NM,
     18-Aug-1993.) $)
  albiim $p |- ( A. x ( ph <-> ps ) <->
             ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) ) $=
    ( wb wal wi wa dfbi2 albii 19.26 bitri ) ABDZCEABFZBAFZGZCEMCENCEGLOCABHIMN
    CJK $.

  $( Split a biconditional and distribute 2 quantifiers.  (Contributed by NM,
     3-Feb-2005.) $)
  2albiim $p |- ( A. x A. y ( ph <-> ps ) <->
             ( A. x A. y ( ph -> ps ) /\ A. x A. y ( ps -> ph ) ) ) $=
    ( wb wal wi wa albiim albii 19.26 bitri ) ABEDFZCFABGDFZBAGDFZHZCFNCFOCFHMP
    CABDIJNOCKL $.

  ${
    hband.1 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    hband.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    $( Deduction form of bound-variable hypothesis builder ~ hban .
       (Contributed by NM, 2-Jan-2002.) $)
    hband $p |- ( ph -> ( ( ps /\ ch ) -> A. x ( ps /\ ch ) ) ) $=
      ( wa wal anim12d 19.26 imbitrrdi ) ABCGZBDHZCDHZGLDHABMCNEFIBCDJK $.
  $}

  ${
    hb3and.1 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    hb3and.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    hb3and.3 $e |- ( ph -> ( th -> A. x th ) ) $.
    $( Deduction form of bound-variable hypothesis builder ~ hb3an .
       (Contributed by NM, 17-Feb-2013.) $)
    hb3and $p |- ( ph -> ( ( ps /\ ch /\ th ) -> A. x ( ps /\ ch /\ th ) ) ) $=
      ( w3a wal 3anim123d 19.26-3an imbitrrdi ) ABCDIZBEJZCEJZDEJZINEJABOCPDQFG
      HKBCDELM $.
  $}

  ${
    hbald.1 $e |- ( ph -> A. y ph ) $.
    hbald.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Deduction form of bound-variable hypothesis builder ~ hbal .
       (Contributed by NM, 2-Jan-2002.) $)
    hbald $p |- ( ph -> ( A. y ps -> A. x A. y ps ) ) $=
      ( wal alimdh ax-7 syl6 ) ABDGZBCGZDGKCGABLDEFHBDCIJ $.
  $}

  $( Declare the existential quantifier symbol. $)
  $c E. $.   $( Backwards E (read:  "there exists") $)

  $( Extend wff definition to include the existential quantifier ("there
     exists"). $)
  wex $a wff E. x ph $.

  $( ` x ` is bound in ` E. x ph ` .  One of the axioms of predicate logic.
     (Contributed by Mario Carneiro, 31-Jan-2015.) $)
  ax-ie1 $a |- ( E. x ph -> A. x E. x ph ) $.

  $( Define existential quantification. ` E. x ph ` means "there exists at
     least one set ` x ` such that ` ph ` is true".  One of the axioms of
     predicate logic.  (Contributed by Mario Carneiro, 31-Jan-2015.) $)
  ax-ie2 $a |- ( A. x ( ps -> A. x ps ) ->
               ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $.

  $( ` x ` is not free in ` E. x ph ` .  (Contributed by NM, 5-Aug-1993.) $)
  hbe1 $p |- ( E. x ph -> A. x E. x ph ) $=
    ( ax-ie1 ) ABC $.

  $( ` x ` is not free in ` E. x ph ` .  (Contributed by Mario Carneiro,
     11-Aug-2016.) $)
  nfe1 $p |- F/ x E. x ph $=
    ( wex hbe1 nfi ) ABCBABDE $.

  $( Closed form of Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
     7-Nov-2005.)  (Revised by Mario Carneiro, 1-Feb-2015.) $)
  19.23ht $p |- ( A. x ( ps -> A. x ps ) ->
               ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
    ( ax-ie2 ) ABCD $.

  ${
    19.23h.1 $e |- ( ps -> A. x ps ) $.
    $( Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.)
       (Revised by Mario Carneiro, 1-Feb-2015.) $)
    19.23h $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
      ( wal wi wex wb ax-gen 19.23ht ax-mp ) BBCEFZCEABFCEACGBFHLCDIABCJK $.
  $}

  $( Theorem 19.7 of [Margaris] p. 89.  To read this intuitionistically, think
     of it as "if ` ph ` can be refuted for all ` x ` , then it is not possible
     to find an ` x ` for which ` ph ` holds" (and likewise for the converse).
     Comparing this with ~ dfexdc illustrates that statements which look
     similar (to someone used to classical logic) can be different
     intuitionistically due to different placement of negations.  (Contributed
     by NM, 5-Aug-1993.)  (Revised by NM, 1-Feb-2015.)  (Revised by Mario
     Carneiro, 12-May-2015.) $)
  alnex $p |- ( A. x -. ph <-> -. E. x ph ) $=
    ( wfal wi wal wex wn fal pm2.21i 19.23h dfnot albii 3bitr4i ) ACDZBEABFZCDA
    GZBEOGACBCCBEHIJPNBAKLOKM $.

  ${
    nex.1 $e |- -. ph $.
    $( Generalization rule for negated wff.  (Contributed by NM,
       18-May-1994.) $)
    nex $p |- -. E. x ph $=
      ( wn wex alnex mpgbi ) ADABEDBABFCG $.
  $}

  $( Defining ` E. x ph ` given decidability.  It is common in classical logic
     to define ` E. x ph ` as ` -. A. x -. ph ` but in intuitionistic logic
     without a decidability condition, that is only an implication not an
     equivalence, as seen at ~ exalim .  (Contributed by Jim Kingdon,
     15-Mar-2018.) $)
  dfexdc $p |- ( DECID E. x ph
                 -> ( E. x ph <-> -. A. x -. ph ) ) $=
    ( wn wal wex wb wdc alnex a1i con2biidc ) ACBDZABEZKLCFLGABHIJ $.

  $( One direction of a classical definition of existential quantification.
     One direction of Definition of [Margaris] p. 49.  For a decidable
     proposition, this is an equivalence, as seen as ~ dfexdc .  (Contributed
     by Jim Kingdon, 29-Jul-2018.) $)
  exalim $p |- ( E. x ph -> -. A. x -. ph ) $=
    ( wn wal wex alnex biimpi con2i ) ACBDZABEZIJCABFGH $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Equality predicate (continued)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The equality predicate was introduced above in ~ wceq for use by ~ df-tru .
  See the comments in that section.  In this section, we continue with the
  first "real" use of it.

$)

  $( Extend wff definition to include atomic formulas using the equality
     predicate.

     (Instead of introducing ~ weq as an axiomatic statement, as was done in an
     older version of this database, we introduce it by "proving" a special
     case of set theory's more general ~ wceq .  This lets us avoid overloading
     the ` = ` connective, thus preventing ambiguity that would complicate
     certain Metamath parsers.  However, logically ~ weq is considered to be a
     primitive syntax, even though here it is artificially "derived" from
     ~ wceq .  Note:  To see the proof steps of this syntax proof, type "show
     proof weq /all" in the Metamath program.)  (Contributed by NM,
     24-Jan-2006.) $)
  weq $p wff x = y $=
    ( cv wceq ) ACBCD $.

  $( Axiom of Equality.  One of the equality and substitution axioms of
     predicate calculus with equality.  This is similar to, but not quite, a
     transitive law for equality (proved later as ~ equtr ).  Axiom scheme C8'
     in [Megill] p. 448 (p. 16 of the preprint).  Also appears as Axiom C7 of
     [Monk2] p. 105.

     Axioms ~ ax-8 through ~ ax-16 are the axioms having to do with equality,
     substitution, and logical properties of our binary predicate ` e. ` (which
     later in set theory will mean "is a member of").  Note that all axioms
     except ~ ax-16 and ~ ax-17 are still valid even when ` x ` , ` y ` , and
     ` z ` are replaced with the same variable because they do not have any
     distinct variable (Metamath's $d) restrictions.  Distinct variable
     restrictions are required for ~ ax-16 and ~ ax-17 only.  (Contributed by
     NM, 5-Aug-1993.) $)
  ax-8 $a |- ( x = y -> ( x = z -> y = z ) ) $.

  $( Axiom of Quantifier Substitution.  One of the equality and substitution
     axioms of predicate calculus with equality.  Appears as Lemma L12 in
     [Megill] p. 445 (p. 12 of the preprint).

     The original version of this axiom was ~ ax-10o ("o" for "old") and was
     replaced with this shorter ~ ax-10 in May 2008.  The old axiom is proved
     from this one as Theorem ~ ax10o .  Conversely, this axiom is proved from
     ~ ax-10o as Theorem ~ ax10 .  (Contributed by NM, 5-Aug-1993.) $)
  ax-10 $a |- ( A. x x = y -> A. y y = x ) $.

  $( Axiom of Variable Substitution.  One of the 5 equality axioms of predicate
     calculus.  The final consequent ` A. x ( x = y -> ph ) ` is a way of
     expressing " ` y ` substituted for ` x ` in wff ` ph ` " (cf. ~ sb6 ).  It
     is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105,
     from which it can be proved by cases.

     Variants of this axiom which are equivalent in classical logic but which
     have not been shown to be equivalent for intuitionistic logic are
     ~ ax11v , ~ ax11v2 and ~ ax-11o .  (Contributed by NM, 5-Aug-1993.) $)
  ax-11 $a |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $.

  $( Axiom of Quantifier Introduction.  One of the equality and substitution
     axioms of predicate calculus with equality.  Informally, it says that
     whenever ` z ` is distinct from ` x ` and ` y ` , and ` x = y ` is true,
     then ` x = y ` quantified with ` z ` is also true.  In other words, ` z `
     is irrelevant to the truth of ` x = y ` .  Axiom scheme C9' in [Megill]
     p. 448 (p. 16 of the preprint).  It apparently does not otherwise appear
     in the literature but is easily proved from textbook predicate calculus by
     cases.

     This axiom has been modified from the original ~ ax12 for compatibility
     with intuitionistic logic.  (Contributed by Mario Carneiro, 31-Jan-2015.)
     Use its alias ~ ax12or instead, for labeling consistency.
     (New usage is discouraged.) $)
  ax-i12 $a |- ( A. z z = x \/ ( A. z z = y \/
                 A. z ( x = y -> A. z x = y ) ) ) $.

  $( Alias for ~ ax-i12 , to be used in place of it for labeling consistency.
     (Contributed by NM, 3-Feb-2015.) $)
  ax12or $p |- ( A. z z = x \/ ( A. z z = y \/
                 A. z ( x = y -> A. z x = y ) ) ) $=
    ( ax-i12 ) ABCD $.

  $( Axiom of bundling.  The general idea of this axiom is that two variables
     are either distinct or non-distinct.  That idea could be expressed as
     ` A. z z = x \/ -. A. z z = x ` .  However, we instead choose an axiom
     which has many of the same consequences, but which is different with
     respect to a universe which contains only one object. ` A. z z = x ` holds
     if ` z ` and ` x ` are the same variable, likewise for ` z ` and ` y ` ,
     and ` A. x A. z ( x = y -> A. z x = y ) ` holds if ` z ` is distinct from
     the others (and the universe has at least two objects).

     As with other statements of the form "x is decidable (either true or
     false)", this does not entail the full Law of the Excluded Middle (which
     is the proposition that all statements are decidable), but instead merely
     the assertion that particular kinds of statements are decidable (or in
     this case, an assertion similar to decidability).

     This axiom implies ~ ax-i12 as can be seen at ~ axi12 .  Whether ~ ax-bndl
     can be proved from the remaining axioms including ~ ax-i12 is not known.

     The reason we call this "bundling" is that a statement without a distinct
     variable constraint "bundles" together two statements, one in which the
     two variables are the same and one in which they are different.
     (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.) $)
  ax-bndl $a |- ( A. z z = x \/
                       ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) ) $.

  $( Axiom of Specialization.  A quantified wff implies the wff without a
     quantifier (i.e. an instance, or special case, of the generalized wff).
     In other words if something is true for all ` x ` , it is true for any
     specific ` x ` (that would typically occur as a free variable in the wff
     substituted for ` ph ` ).  (A free variable is one that does not occur in
     the scope of a quantifier: ` x ` and ` y ` are both free in ` x = y ` ,
     but only ` x ` is free in ` A. y x = y ` .)  Axiom scheme C5' in [Megill]
     p. 448 (p. 16 of the preprint).  Also appears as Axiom B5 of [Tarski]
     p. 67 (under his system S2, defined in the last paragraph on p. 77).

     Note that the converse of this axiom does not hold in general, but a
     weaker inference form of the converse holds and is expressed as rule
     ~ ax-gen .  Conditional forms of the converse are given by ~ ax12 ,
     ~ ax-16 , and ~ ax-17 .

     Unlike the more general textbook Axiom of Specialization, we cannot choose
     a variable different from ` x ` for the special case.  For use, that
     requires the assistance of equality axioms, and we deal with it later
     after we introduce the definition of proper substitution - see ~ stdpc4 .

     (Contributed by NM, 5-Aug-1993.) $)
  ax-4 $a |- ( A. x ph -> ph ) $.

  $( Specialization.  Another name for ~ ax-4 .  (Contributed by NM,
     21-May-2008.) $)
  sp $p |- ( A. x ph -> ph ) $=
    ( ax-4 ) ABC $.

  $( Rederive the original version of the axiom from ~ ax-i12 .  (Contributed
     by Mario Carneiro, 3-Feb-2015.) $)
  ax12 $p |- ( -. A. z z = x -> ( -. A. z z = y ->
              ( x = y -> A. z x = y ) ) ) $=
    ( cv wceq wal wn wi wo ax12or ori ord ax-4 syl6 ) CDZADZECFZGZOBDZECFZGPSEZ
    UACFHZCFZUBRTUCQTUCIABCJKLUBCMN $.

  $( Bound-variable hypothesis builder for ` x = x ` .  This theorem tells us
     that any variable, including ` x ` , is effectively not free in
     ` x = x ` , even though ` x ` is technically free according to the
     traditional definition of free variable.

     The proof uses only ~ ax-8 and ~ ax-i12 on top of (the FOL analogue of)
     modal logic KT. This shows that this can be proved without ~ ax-i9 , even
     though Theorem ~ equid cannot.  A shorter proof using ~ ax-i9 is
     obtainable from ~ equid and ~ hbth .  (Contributed by NM, 13-Jan-2011.)
     (Proof shortened by Wolf Lammen, 23-Mar-2014.) $)
  hbequid $p |- ( x = x -> A. y x = x ) $=
    ( cv wceq wal wi wo ax12or ax-8 pm2.43i alimi a1d ax-4 jaoi ax-mp ) BCACZDZ
    BEZRPPDZSBEZFZBEZGZGUAAABHRUAUCRTSQSBQSBAAIJKLZRUAUBUDUABMNNO $.

  $( Proof that ~ ax-i12 follows from ~ ax-bndl .  So that we can track which
     theorems rely on ~ ax-bndl , proofs should reference ~ ax12or rather than
     this theorem.  (Contributed by Jim Kingdon, 17-Aug-2018.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  axi12 $p |- ( A. z z = x \/ ( A. z z = y \/
                 A. z ( x = y -> A. z x = y ) ) ) $=
    ( cv wceq wal wi wo ax-bndl sp orim2i ax-mp ) CDZADZECFZMBDZECFZNPEZRCFGCFZ
    AFZHZHOQSHZHABCIUAUBOTSQSAJKKL $.

  $( Commutation law for identical variable specifiers.  The antecedent and
     consequent are true when ` x ` and ` y ` are substituted with the same
     variable.  Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).
     (Contributed by NM, 5-Aug-1993.) $)
  alequcom $p |- ( A. x x = y -> A. y y = x ) $=
    ( ax-10 ) ABC $.

  ${
    alequcoms.1 $e |- ( A. x x = y -> ph ) $.
    $( A commutation rule for identical variable specifiers.  (Contributed by
       NM, 5-Aug-1993.) $)
    alequcoms $p |- ( A. y y = x -> ph ) $=
      ( weq wal alequcom syl ) CBECFBCEBFACBGDH $.
  $}

  ${
    nalequcoms.1 $e |- ( -. A. x x = y -> ph ) $.
    $( A commutation rule for distinct variable specifiers.  (Contributed by
       NM, 2-Jan-2002.)  (Revised by Mario Carneiro, 2-Feb-2015.) $)
    nalequcoms $p |- ( -. A. y y = x -> ph ) $=
      ( cv wceq wal wn alequcom con3i syl ) CEZBEZFCGZHMLFBGZHAONBCIJDK $.
  $}

  $( Consequence of the definition of not-free.  (Contributed by Mario
     Carneiro, 26-Sep-2016.) $)
  nfr $p |- ( F/ x ph -> ( ph -> A. x ph ) ) $=
    ( wnf wal wi df-nf sp sylbi ) ABCAABDEZBDIABFIBGH $.

  ${
    nfri.1 $e |- F/ x ph $.
    $( Consequence of the definition of not-free.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    nfri $p |- ( ph -> A. x ph ) $=
      ( wnf wal wi nfr ax-mp ) ABDAABEFCABGH $.
  $}

  ${
    nfrd.1 $e |- ( ph -> F/ x ps ) $.
    $( Consequence of the definition of not-free in a context.  (Contributed by
       Mario Carneiro, 11-Aug-2016.) $)
    nfrd $p |- ( ph -> ( ps -> A. x ps ) ) $=
      ( wnf wal wi nfr syl ) ABCEBBCFGDBCHI $.
  $}

  ${
    alimd.1 $e |- F/ x ph $.
    alimd.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.20 of [Margaris] p. 90.  (Contributed by Mario
       Carneiro, 24-Sep-2016.) $)
    alimd $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
      ( nfri alimdh ) ABCDADEGFH $.
  $}

  ${
    alrimi.1 $e |- F/ x ph $.
    alrimi.2 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Contributed by Mario
       Carneiro, 24-Sep-2016.) $)
    alrimi $p |- ( ph -> A. x ps ) $=
      ( nfri alrimih ) ABCACDFEG $.
  $}

  ${
    nfd.1 $e |- F/ x ph $.
    nfd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Deduce that ` x ` is not free in ` ps ` in a context.  (Contributed by
       Mario Carneiro, 24-Sep-2016.) $)
    nfd $p |- ( ph -> F/ x ps ) $=
      ( wal wi wnf nfri alrimih df-nf sylibr ) ABBCFGZCFBCHAMCACDIEJBCKL $.
  $}

  ${
    nfdh.1 $e |- ( ph -> A. x ph ) $.
    nfdh.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Deduce that ` x ` is not free in ` ps ` in a context.  (Contributed by
       Mario Carneiro, 24-Sep-2016.) $)
    nfdh $p |- ( ph -> F/ x ps ) $=
      ( nfi nfd ) ABCACDFEG $.
  $}

  ${
    nfrimi.1 $e |- F/ x ph $.
    nfrimi.2 $e |- F/ x ( ph -> ps ) $.
    $( Moving an antecedent outside ` F/ ` .  (Contributed by Jim Kingdon,
       23-Mar-2018.) $)
    nfrimi $p |- ( ph -> F/ x ps ) $=
      ( wal wi nfri ax-5 syl2im pm2.86i nfd ) ABCDABBCFZABGZNCFAACFMNCEHACDHABC
      IJKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom ax-17 - first use of the $d distinct variable statement
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x ph $.
    $( Axiom to quantify a variable over a formula in which it does not occur.
       Axiom C5 in [Megill] p. 444 (p. 11 of the preprint).  Also appears as
       Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of
       [Monk2] p. 113.

       (Contributed by NM, 5-Aug-1993.) $)
    ax-17 $a |- ( ph -> A. x ph ) $.
  $}

  ${
    $d x ps $.
    $( ~ ax-17 with antecedent.  (Contributed by NM, 1-Mar-2013.) $)
    a17d $p |- ( ph -> ( ps -> A. x ps ) ) $=
      ( wal wi ax-17 a1i ) BBCDEABCFG $.
  $}

  ${
    $d x ph $.
    $( If ` x ` is not present in ` ph ` , then ` x ` is not free in ` ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nfv $p |- F/ x ph $=
      ( ax-17 nfi ) ABABCD $.
  $}

  ${
    $d x ps $.
    $( ~ nfv with antecedent.  Useful in proofs of deduction versions of
       bound-variable hypothesis builders such as ~ nfimd .  (Contributed by
       Mario Carneiro, 6-Oct-2016.) $)
    nfvd $p |- ( ph -> F/ x ps ) $=
      ( wnf nfv a1i ) BCDABCEF $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce Axiom of Existence
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Existence.  One of the equality and substitution axioms of
     predicate calculus with equality.  One thing this axiom tells us is that
     at least one thing exists (although ~ ax-4 and possibly others also tell
     us that, i.e. they are not valid in the empty domain of a "free logic").
     In this form (not requiring that ` x ` and ` y ` be distinct) it was used
     in an axiom system of Tarski (see Axiom B7' in footnote 1 of
     [KalishMontague] p. 81.)  Another name for this theorem is ~ a9e , which
     has additional remarks.  (Contributed by Mario Carneiro, 31-Jan-2015.) $)
  ax-i9 $a |- E. x x = y $.

  $( Derive ~ ax-9 from ~ ax-i9 , the modified version for intuitionistic
     logic.  Although ~ ax-9 does hold intuistionistically, in intuitionistic
     logic it is weaker than ~ ax-i9 .  (Contributed by NM, 3-Feb-2015.) $)
  ax-9 $p |- -. A. x -. x = y $=
    ( cv wceq wn wal wex ax-i9 notnoti alnex mtbir ) ACBCDZEAFLAGZEMABHILAJK $.

  $( ~ equid with some quantification and negation without using ~ ax-4 or
     ~ ax-17 .  (Contributed by NM, 13-Jan-2011.)  (Proof shortened by Wolf
     Lammen, 27-Feb-2014.) $)
  equidqe $p |- -. A. y -. x = x $=
    ( weq wn wal ax-9 ax-8 pm2.43i con3i alimi mto ) AACZDZBEBACZDZBEBAFMOBNLNL
    BAAGHIJK $.

  $( A special case of ~ ax-4 without using ~ ax-4 or ~ ax-17 .  (Contributed
     by NM, 13-Jan-2011.) $)
  ax4sp1 $p |- ( A. y -. x = x -> -. x = x ) $=
    ( weq wn wal equidqe pm2.21i ) AACDZBEHABFG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Additional intuitionistic axioms
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( ` x ` is not free in ` A. x ph ` .  One of the axioms of predicate logic.
     (Contributed by Mario Carneiro, 31-Jan-2015.) $)
  ax-ial $a |- ( A. x ph -> A. x A. x ph ) $.

  $( Axiom of quantifier collection.  (Contributed by Mario Carneiro,
     31-Jan-2015.) $)
  ax-i5r $a |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) ) $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Predicate calculus including ax-4, without distinct variables
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    spi.1 $e |- A. x ph $.
    $( Inference reversing generalization (specialization).  (Contributed by
       NM, 5-Aug-1993.) $)
    spi $p |- ph $=
      ( wal ax-4 ax-mp ) ABDACABEF $.
  $}

  ${
    sps.1 $e |- ( ph -> ps ) $.
    $( Generalization of antecedent.  (Contributed by NM, 5-Aug-1993.) $)
    sps $p |- ( A. x ph -> ps ) $=
      ( wal sp syl ) ACEABACFDG $.
  $}

  ${
    spsd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction generalizing antecedent.  (Contributed by NM, 17-Aug-1994.) $)
    spsd $p |- ( ph -> ( A. x ps -> ch ) ) $=
      ( wal sp syl5 ) BDFBACBDGEH $.
  $}

  ${
    nfbidf.1 $e |- F/ x ph $.
    nfbidf.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( An equality theorem for effectively not free.  (Contributed by Mario
       Carneiro, 4-Oct-2016.) $)
    nfbidf $p |- ( ph -> ( F/ x ps <-> F/ x ch ) ) $=
      ( wal wi wnf nfri albidh imbi12d df-nf 3bitr4g ) ABBDGZHZDGCCDGZHZDGBDICD
      IAPRDADEJZABCOQFABCDSFKLKBDMCDMN $.
  $}

  $( ` x ` is not free in ` A. x ph ` .  Example in Appendix in [Megill] p. 450
     (p. 19 of the preprint).  Also Lemma 22 of [Monk2] p. 114.  (Contributed
     by NM, 5-Aug-1993.) $)
  hba1 $p |- ( A. x ph -> A. x A. x ph ) $=
    ( ax-ial ) ABC $.

  $( ` x ` is not free in ` A. x ph ` .  (Contributed by Mario Carneiro,
     11-Aug-2016.) $)
  nfa1 $p |- F/ x A. x ph $=
    ( wal hba1 nfi ) ABCBABDE $.

  ${
    axc4i.1 $e |- ( A. x ph -> ps ) $.
    $( Inference version of ~ 19.21 .  (Contributed by NM, 3-Jan-1993.) $)
    axc4i $p |- ( A. x ph -> A. x ps ) $=
      ( wal nfa1 alrimi ) ACEBCACFDG $.
  $}

  ${
    a5i.1 $e |- ( A. x ph -> ps ) $.
    $( Inference generalizing a consequent.  (Contributed by NM,
       5-Aug-1993.) $)
    a5i $p |- ( A. x ph -> A. x ps ) $=
      ( wal wi hba1 ax-5 syl5 mpg ) ACEZBFZKBCEZFCKKCELCEMACGKBCHIDJ $.
  $}

  $( ` x ` is not free in ` F/ x ph ` .  (Contributed by Mario Carneiro,
     11-Aug-2016.) $)
  nfnf1 $p |- F/ x F/ x ph $=
    ( wnf wal wi df-nf nfa1 nfxfr ) ABCAABDEZBDBABFIBGH $.

  ${
    hb.1 $e |- ( ph -> A. x ph ) $.
    hb.2 $e |- ( ps -> A. x ps ) $.
    $( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
       ` ( ph -> ps ) ` .  (Contributed by NM, 5-Aug-1993.)  (Proof shortened
       by O'Cat, 3-Mar-2008.)  (Revised by Mario Carneiro, 2-Feb-2015.) $)
    hbim $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
      ( wi wal ax-4 imim12i ax-i5r imim1i alimi 3syl ) ABFZACGZBCGZFOBFZCGNCGOA
      BPACHEIABCJQNCAOBDKLM $.

    $( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
       ` ( ph \/ ps ) ` .  (Contributed by NM, 5-Aug-1993.)  (Revised by NM,
       2-Feb-2015.) $)
    hbor $p |- ( ( ph \/ ps ) -> A. x ( ph \/ ps ) ) $=
      ( wo wal orc alimi syl olc jaoi ) AABFZCGZBAACGNDAMCABHIJBBCGNEBMCBAKIJL
      $.

    $( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
       ` ( ph /\ ps ) ` .  (Contributed by NM, 5-Aug-1993.)  (Proof shortened
       by Mario Carneiro, 2-Feb-2015.) $)
    hban $p |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) ) $=
      ( wa wal anim12i 19.26 sylibr ) ABFZACGZBCGZFKCGALBMDEHABCIJ $.

    $( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
       ` ( ph <-> ps ) ` .  (Contributed by NM, 5-Aug-1993.) $)
    hbbi $p |- ( ( ph <-> ps ) -> A. x ( ph <-> ps ) ) $=
      ( wb wi wa dfbi2 hbim hban hbxfrbi ) ABFABGZBAGZHCABIMNCABCDEJBACEDJKL $.

    hb.3 $e |- ( ch -> A. x ch ) $.
    $( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
       ` ( ph \/ ps \/ ch ) ` .  (Contributed by NM, 14-Sep-2003.) $)
    hb3or $p |- ( ( ph \/ ps \/ ch ) -> A. x ( ph \/ ps \/ ch ) ) $=
      ( w3o wo df-3or hbor hbxfrbi ) ABCHABIZCIDABCJMCDABDEFKGKL $.

    $( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
       ` ( ph /\ ps /\ ch ) ` .  (Contributed by NM, 14-Sep-2003.) $)
    hb3an $p |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) ) $=
      ( w3a wa df-3an hban hbxfrbi ) ABCHABIZCIDABCJMCDABDEFKGKL $.
  $}

  $( Lemma 24 of [Monk2] p. 114.  (Contributed by NM, 29-May-2008.) $)
  hba2 $p |- ( A. y A. x ph -> A. x A. y A. x ph ) $=
    ( wal hba1 hbal ) ABDBCABEF $.

  $( Lemma 23 of [Monk2] p. 114.  (Contributed by NM, 29-May-2008.) $)
  hbia1 $p |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> A. x ps ) ) $=
    ( wal hba1 hbim ) ACDBCDCACEBCEF $.

  ${
    19.3h.1 $e |- ( ph -> A. x ph ) $.
    $( A wff may be quantified with a variable not free in it.  Theorem 19.3 of
       [Margaris] p. 89.  (Contributed by NM, 5-Aug-1993.)  (Revised by NM,
       21-May-2007.) $)
    19.3h $p |- ( A. x ph <-> ph ) $=
      ( wal ax-4 impbii ) ABDAABECF $.
  $}

  ${
    19.3.1 $e |- F/ x ph $.
    $( A wff may be quantified with a variable not free in it.  Theorem 19.3 of
       [Margaris] p. 89.  (Contributed by NM, 5-Aug-1993.)  (Revised by Mario
       Carneiro, 24-Sep-2016.) $)
    19.3 $p |- ( A. x ph <-> ph ) $=
      ( wal sp nfri impbii ) ABDAABEABCFG $.
  $}

  ${
    19.16.1 $e |- F/ x ph $.
    $( Theorem 19.16 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
    19.16 $p |- ( A. x ( ph <-> ps ) -> ( ph <-> A. x ps ) ) $=
      ( wal wb 19.3 albi bitr3id ) AACEABFCEBCEACDGABCHI $.
  $}

  ${
    19.17.1 $e |- F/ x ps $.
    $( Theorem 19.17 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
    19.17 $p |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> ps ) ) $=
      ( wb wal albi 19.3 bitrdi ) ABECFACFBCFBABCGBCDHI $.
  $}

  ${
    19.21h.1 $e |- ( ph -> A. x ph ) $.
    $( Theorem 19.21 of [Margaris] p. 90.  The hypothesis can be thought of
       as " ` x ` is not free in ` ph ` ".  New proofs should use ~ 19.21
       instead.  (Contributed by NM, 5-Aug-1993.)
       (New usage is discouraged.) $)
    19.21h $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
      ( wi wal alim syl5 hba1 hbim ax-4 imim2i alrimih impbii ) ABEZCFZABCFZEZA
      ACFPQDABCGHROCAQCDBCIJQBABCKLMN $.
  $}

  ${
    19.21bi.1 $e |- ( ph -> A. x ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.) $)
    19.21bi $p |- ( ph -> ps ) $=
      ( wal ax-4 syl ) ABCEBDBCFG $.
  $}

  ${
    19.21bbi.1 $e |- ( ph -> A. x A. y ps ) $.
    $( Inference removing double quantifier.  (Contributed by NM,
       20-Apr-1994.) $)
    19.21bbi $p |- ( ph -> ps ) $=
      ( wal 19.21bi ) ABDABDFCEGG $.
  $}

  ${
    19.27h.1 $e |- ( ps -> A. x ps ) $.
    $( Theorem 19.27 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
    19.27h $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
      ( wa wal 19.26 19.3h anbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
  $}

  ${
    19.27.1 $e |- F/ x ps $.
    $( Theorem 19.27 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
    19.27 $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
      ( wa wal 19.26 19.3 anbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
  $}

  ${
    19.28h.1 $e |- ( ph -> A. x ph ) $.
    $( Theorem 19.28 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
    19.28h $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
      ( wa wal 19.26 19.3h anbi1i bitri ) ABECFACFZBCFZEALEABCGKALACDHIJ $.
  $}

  ${
    19.28.1 $e |- F/ x ph $.
    $( Theorem 19.28 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
    19.28 $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
      ( wa wal 19.26 19.3 anbi1i bitri ) ABECFACFZBCFZEALEABCGKALACDHIJ $.
  $}

  ${
    nfan1.1 $e |- F/ x ph $.
    nfan1.2 $e |- ( ph -> F/ x ps ) $.
    $( A closed form of ~ nfan .  (Contributed by Mario Carneiro,
       3-Oct-2016.) $)
    nfan1 $p |- F/ x ( ph /\ ps ) $=
      ( wa wal nfrd imdistani nfri 19.28h sylibr nfi ) ABFZCNABCGZFNCGABOABCEHI
      ABCACDJKLM $.
  $}

  ${
    nfan.1 $e |- F/ x ph $.
    nfan.2 $e |- F/ x ps $.
    $( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
       ` ( ph /\ ps ) ` .  (Contributed by Mario Carneiro, 11-Aug-2016.)
       (Proof shortened by Wolf Lammen, 13-Jan-2018.) $)
    nfan $p |- F/ x ( ph /\ ps ) $=
      ( wnf a1i nfan1 ) ABCDBCFAEGH $.

    nfan.3 $e |- F/ x ch $.
    $( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
       ` ( ph /\ ps /\ ch ) ` .  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nf3an $p |- F/ x ( ph /\ ps /\ ch ) $=
      ( w3a wa df-3an nfan nfxfr ) ABCHABIZCIDABCJMCDABDEFKGKL $.
  $}

  ${
    nford.1 $e |- ( ph -> F/ x ps ) $.
    nford.2 $e |- ( ph -> F/ x ch ) $.
    $( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
       in ` ( ps \/ ch ) ` .  (Contributed by Jim Kingdon, 29-Oct-2019.) $)
    nford $p |- ( ph -> F/ x ( ps \/ ch ) ) $=
      ( wo wal wi wnf wa df-nf anbi12i biimpi syl2anc 19.26 sylibr alimi imim2i
      orc olc jaao syl ) ABCGZUDDHZIZDHZUDDJABBDHZIZCCDHZIZKZDHZUGAUIDHZUKDHZKZ
      UMABDJZCDJZUPEFUQURKUPUQUNURUOBDLCDLMNOUIUKDPQULUFDUIBUEUKCUHUEBBUDDBCTRS
      UJUECCUDDCBUARSUBRUCUDDLQ $.
  $}

  ${
    nfand.1 $e |- ( ph -> F/ x ps ) $.
    nfand.2 $e |- ( ph -> F/ x ch ) $.
    $( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
       in ` ( ps /\ ch ) ` .  (Contributed by Mario Carneiro, 7-Oct-2016.) $)
    nfand $p |- ( ph -> F/ x ( ps /\ ch ) ) $=
      ( wa wal wi wnf jca df-nf anbi12i 19.26 bitr4i anim12 imbitrrdi alimi syl
      sylbi sylibr ) ABCGZUBDHZIZDHZUBDJABDJZCDJZGZUEAUFUGEFKUHBBDHZIZCCDHZIZGZ
      DHZUEUHUJDHZULDHZGUNUFUOUGUPBDLCDLMUJULDNOUMUDDUMUBUIUKGUCBUICUKPBCDNQRTS
      UBDLUA $.

    nfand.3 $e |- ( ph -> F/ x th ) $.
    $( Deduction form of bound-variable hypothesis builder ~ nf3an .
       (Contributed by NM, 17-Feb-2013.)  (Revised by Mario Carneiro,
       16-Oct-2016.) $)
    nf3and $p |- ( ph -> F/ x ( ps /\ ch /\ th ) ) $=
      ( w3a wa df-3an nfand nfxfrd ) BCDIBCJZDJAEBCDKANDEABCEFGLHLM $.
  $}

  ${
    hbim1.1 $e |- ( ph -> A. x ph ) $.
    hbim1.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( A closed form of ~ hbim .  (Contributed by NM, 5-Aug-1993.) $)
    hbim1 $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
      ( wi wal a2i 19.21h sylibr ) ABFZABCGZFKCGABLEHABCDIJ $.
  $}

  ${
    nfim1.1 $e |- F/ x ph $.
    nfim1.2 $e |- ( ph -> F/ x ps ) $.
    $( A closed form of ~ nfim .  (Contributed by NM, 5-Aug-1993.)  (Revised by
       Mario Carneiro, 24-Sep-2016.)  (Proof shortened by Wolf Lammen,
       2-Jan-2018.) $)
    nfim1 $p |- F/ x ( ph -> ps ) $=
      ( wi nfri nfrd hbim1 nfi ) ABFCABCACDGABCEHIJ $.
  $}

  ${
    nfim.1 $e |- F/ x ph $.
    nfim.2 $e |- F/ x ps $.
    $( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
       ` ( ph -> ps ) ` .  (Contributed by Mario Carneiro, 11-Aug-2016.)
       (Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
    nfim $p |- F/ x ( ph -> ps ) $=
      ( wnf a1i nfim1 ) ABCDBCFAEGH $.
  $}

  ${
    hbimd.1 $e |- ( ph -> A. x ph ) $.
    hbimd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    hbimd.3 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    $( Deduction form of bound-variable hypothesis builder ~ hbim .
       (Contributed by NM, 1-Jan-2002.)  (Revised by NM, 2-Feb-2015.) $)
    hbimd $p |- ( ph -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) ) $=
      ( wi wal imim2d ax-4 imim1i ax-i5r syl syl6 imim1d alimdh syld ) ABCHZBDI
      ZCHZDIZSDIASBCDIZHZUBACUCBGJUDTUCHUBTBUCBDKLBCDMNOAUASDEABTCFPQR $.
  $}

  ${
    nfor.1 $e |- F/ x ph $.
    nfor.2 $e |- F/ x ps $.
    $( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
       ` ( ph \/ ps ) ` .  (Contributed by Jim Kingdon, 11-Mar-2018.) $)
    nfor $p |- F/ x ( ph \/ ps ) $=
      ( wo nfri hbor nfi ) ABFCABCACDGBCEGHI $.
  $}

  ${
    hbbid.1 $e |- ( ph -> A. x ph ) $.
    hbbid.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    hbbid.3 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    $( Deduction form of bound-variable hypothesis builder ~ hbbi .
       (Contributed by NM, 1-Jan-2002.) $)
    hbbid $p |- ( ph -> ( ( ps <-> ch ) -> A. x ( ps <-> ch ) ) ) $=
      ( wi wa wal wb hbimd anim12d dfbi2 albiim 3imtr4g ) ABCHZCBHZIQDJZRDJZIBC
      KZUADJAQSRTABCDEFGLACBDEGFLMBCNBCDOP $.
  $}

  ${
    nfal.1 $e |- F/ x ph $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.)  Remove dependency on
       ~ ax-4 .  (Revised by GG, 25-Aug-2024.) $)
    nfal $p |- F/ x A. y ph $=
      ( wnf wal wi df-nf biimpi alimi ax-7 ax-5 syl6 3syl sylibr mpg ) ABEZACFZ
      BEZCQCFZRRBFZGZBFZSTAABFZGZBFZCFUECFZBFUCQUFCQUFABHIJUECBKUGUBBUGRUDCFUAA
      UDCLACBKMJNRBHODP $.
    $( $j usage 'nfal' avoids 'ax-4'; $)

    $( If ` x ` is not free in ` ph ` , it is not free in ` F/ y ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.)  (Proof shortened by Wolf
       Lammen, 30-Dec-2017.) $)
    nfnf $p |- F/ x F/ y ph $=
      ( wnf wal wi df-nf nfal nfim nfxfr ) ACEAACFZGZCFBACHMBCALBDABCDIJIK $.
  $}

  $( Closed form of ~ nfal .  (Contributed by Jim Kingdon, 11-May-2018.) $)
  nfalt $p |- ( A. y F/ x ph -> F/ x A. y ph ) $=
    ( wal wi wnf alim alcom imbitrdi alimi df-nf albii bitri 3imtr4i ) AABDZEZC
    DZBDZACDZSBDZEZBDABFZCDZSBFQUABQSOCDTAOCGACBHIJUCPBDZCDRUBUDCABKLPCBHMSBKN
    $.

  $( Lemma 24 of [Monk2] p. 114.  (Contributed by Mario Carneiro,
     24-Sep-2016.) $)
  nfa2 $p |- F/ x A. y A. x ph $=
    ( wal nfa1 nfal ) ABDBCABEF $.

  $( Lemma 23 of [Monk2] p. 114.  (Contributed by Mario Carneiro,
     24-Sep-2016.) $)
  nfia1 $p |- F/ x ( A. x ph -> A. x ps ) $=
    ( wal nfa1 nfim ) ACDBCDCACEBCEF $.

  $( Closed form of Theorem 19.21 of [Margaris] p. 90.  (Contributed by NM,
     27-May-1997.)  (New usage is discouraged.) $)
  19.21ht $p |- ( A. x ( ph -> A. x ph ) ->
               ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) ) $=
    ( wal wi alim imim2d com12 sps hba1 ax-4 a1i hbimd imim2i alimi syl6 impbid
    ) AACDZEZCDZABEZCDZABCDZEZSUBUDECUBSUDUBRUCAABCFGHITUDUDCDUBTAUCCSCJSCKUCUC
    CDETBCJLMUDUACUCBABCKNOPQ $.

  $( Closed form of Theorem 19.21 of [Margaris] p. 90.  (Contributed by NM,
     27-May-1997.) $)
  19.21t $p |- ( F/ x ph ->
               ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) ) $=
    ( wnf wal wi wb df-nf 19.21ht sylbi ) ACDAACEFCEABFCEABCEFGACHABCIJ $.

  ${
    19.21.1 $e |- F/ x ph $.
    $( Theorem 19.21 of [Margaris] p. 90.  The hypothesis can be thought of
       as " ` x ` is not free in ` ph ` ".  (Contributed by NM, 5-Aug-1993.)
       (Revised by Mario Carneiro, 24-Sep-2016.) $)
    19.21 $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
      ( wnf wi wal wb 19.21t ax-mp ) ACEABFCGABCGFHDABCIJ $.
  $}

  ${
    stdpc5.1 $e |- F/ x ph $.
    $( An axiom scheme of standard predicate calculus that emulates Axiom 5 of
       [Mendelson] p. 69.  The hypothesis ` F/ x ph ` can be thought of as
       emulating " ` x ` is not free in ` ph ` ".  With this definition, the
       meaning of "not free" is less restrictive than the usual textbook
       definition; for example ` x ` would not (for us) be free in ` x = x ` by
       ~ nfequid .  This theorem scheme can be proved as a metatheorem of
       Mendelson's axiom system, even though it is slightly stronger than his
       Axiom 5.  (Contributed by NM, 22-Sep-1993.)  (Revised by Mario Carneiro,
       12-Oct-2016.)  (Proof shortened by Wolf Lammen, 1-Jan-2018.) $)
    stdpc5 $p |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) $=
      ( wi wal 19.21 biimpi ) ABECFABCFEABCDGH $.
  $}

  ${
    nfimd.1 $e |- ( ph -> F/ x ps ) $.
    nfimd.2 $e |- ( ph -> F/ x ch ) $.
    $( If in a context ` x ` is not free in ` ps ` and ` ch ` , then it is not
       free in ` ( ps -> ch ) ` .  (Contributed by Mario Carneiro,
       24-Sep-2016.)  (Proof shortened by Wolf Lammen, 30-Dec-2017.) $)
    nfimd $p |- ( ph -> F/ x ( ps -> ch ) ) $=
      ( wnf wi wal nfnf1 nfri nfr imim2d 19.21t biimprd syl9r alrimdh imbitrrdi
      df-nf sylc ) ABDGZCDGZBCHZDGZEFUAUBUCUCDIZHZDIUDUAUBUFDUADBDJKUBDCDJKUBUC
      BCDIZHZUAUEUBCUGBCDLMUAUEUHBCDNOPQUCDSRT $.
  $}

  ${
    aaanh.1 $e |- ( ph -> A. y ph ) $.
    aaanh.2 $e |- ( ps -> A. x ps ) $.
    $( Rearrange universal quantifiers.  (Contributed by NM, 12-Aug-1993.) $)
    aaanh $p |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) ) $=
      ( wa wal 19.28h albii hbal 19.27h bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJ
      AOCBCDFKLM $.
  $}

  ${
    aaan.1 $e |- F/ y ph $.
    aaan.2 $e |- F/ x ps $.
    $( Rearrange universal quantifiers.  (Contributed by NM, 12-Aug-1993.) $)
    aaan $p |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) ) $=
      ( wa wal 19.28 albii nfal 19.27 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
      CBCDFKLM $.
  $}

  ${
    nfbid.1 $e |- ( ph -> F/ x ps ) $.
    nfbid.2 $e |- ( ph -> F/ x ch ) $.
    $( If in a context ` x ` is not free in ` ps ` and ` ch ` , then it is not
       free in ` ( ps <-> ch ) ` .  (Contributed by Mario Carneiro,
       24-Sep-2016.)  (Proof shortened by Wolf Lammen, 29-Dec-2017.) $)
    nfbid $p |- ( ph -> F/ x ( ps <-> ch ) ) $=
      ( wb wi wa dfbi2 nfimd nfand nfxfrd ) BCGBCHZCBHZIADBCJANODABCDEFKACBDFEK
      LM $.
  $}

  ${
    nfbi.1 $e |- F/ x ph $.
    nfbi.2 $e |- F/ x ps $.
    $( If ` x ` is not free in ` ph ` and ` ps ` , then it is not free in
       ` ( ph <-> ps ) ` .  (Contributed by Mario Carneiro, 11-Aug-2016.)
       (Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
    nfbi $p |- F/ x ( ph <-> ps ) $=
      ( wb wnf wtru a1i nfbid mptru ) ABFCGHABCACGHDIBCGHEIJK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The existential quantifier
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( If a wff is true, then it is true for at least one instance.  Special case
     of Theorem 19.8 of [Margaris] p. 89.  (Contributed by NM, 5-Aug-1993.) $)
  19.8a $p |- ( ph -> E. x ph ) $=
    ( wex wi wal id hbe1 19.23h mpbir spi ) AABCZDZBLBEKKDKFAKBABGHIJ $.

  ${
    19.8ad.1 $e |- ( ph -> ps ) $.
    $( If a wff is true, it is true for at least one instance.  Deduction form
       of ~ 19.8a .  (Contributed by DAW, 13-Feb-2017.) $)
    19.8ad $p |- ( ph -> E. x ps ) $=
      ( wex 19.8a syl ) ABBCEDBCFG $.
  $}

  ${
    19.23bi.1 $e |- ( E. x ph -> ps ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.) $)
    19.23bi $p |- ( ph -> ps ) $=
      ( wex 19.8a syl ) AACEBACFDG $.
  $}

  ${
    exlimih.1 $e |- ( ps -> A. x ps ) $.
    exlimih.2 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Andrew Salmon, 13-May-2011.) $)
    exlimih $p |- ( E. x ph -> ps ) $=
      ( wi wex 19.23h mpgbi ) ABFACGBFCABCDHEI $.
  $}

  ${
    exlimi.1 $e |- F/ x ps $.
    exlimi.2 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90.  (Contributed by Mario
       Carneiro, 24-Sep-2016.) $)
    exlimi $p |- ( E. x ph -> ps ) $=
      ( nfri exlimih ) ABCBCDFEG $.
  $}

  ${
    exlimd2.1 $e |- ( ph -> A. x ph ) $.
    exlimd2.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    exlimd2.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.23 of [Margaris] p. 90.  Similar to ~ exlimdh
       but with one slightly different hypothesis.  (Contributed by Jim
       Kingdon, 30-Dec-2017.) $)
    exlimd2 $p |- ( ph -> ( E. x ps -> ch ) ) $=
      ( wal wi wex alrimih 19.23ht biimpd sylc ) ACCDHIZDHZBCIZDHZBDJCIZAODEFKA
      QDEGKPRSBCDLMN $.
  $}

  ${
    exlimdh.1 $e |- ( ph -> A. x ph ) $.
    exlimdh.2 $e |- ( ch -> A. x ch ) $.
    exlimdh.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
       28-Jan-1997.) $)
    exlimdh $p |- ( ph -> ( E. x ps -> ch ) ) $=
      ( wi wal wex alrimih 19.23h sylib ) ABCHZDIBDJCHANDEGKBCDFLM $.
  $}

  ${
    exlimd.1 $e |- F/ x ph $.
    exlimd.2 $e |- F/ x ch $.
    exlimd.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.9 of [Margaris] p. 89.  (Contributed by Mario
       Carneiro, 24-Sep-2016.)  (Proof rewritten by Jim Kingdon,
       18-Jun-2018.) $)
    exlimd $p |- ( ph -> ( E. x ps -> ch ) ) $=
      ( nfri exlimdh ) ABCDADEHCDFHGI $.
  $}

  ${
    $d x ps $.
    exlimiv.1 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90.

       This inference, along with our many variants is used to implement a
       metatheorem called "Rule C" that is given in many logic textbooks.  See,
       for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or
       Rule C in Hirst and Hirst's _A Primer for Logic and Proof_ p. 59 (PDF
       p. 65) at ~ http://www.mathsci.appstate.edu/~~jlh/primer/hirst.pdf .

       In informal proofs, the statement "Let C be an element such that..."
       almost always means an implicit application of Rule C.

       In essence, Rule C states that if we can prove that some element ` x `
       exists satisfying a wff, i.e. ` E. x ph ( x ) ` where ` ph ( x ) ` has
       ` x ` free, then we can use ` ph ( ` C ` ) ` as a hypothesis for the
       proof where C is a new (ficticious) constant not appearing previously in
       the proof, nor in any axioms used, nor in the theorem to be proved.  The
       purpose of Rule C is to get rid of the existential quantifier.

       We cannot do this in Metamath directly.  Instead, we use the original
       ` ph ` (containing ` x ` ) as an antecedent for the main part of the
       proof.  We eventually arrive at ` ( ph -> ps ) ` where ` ps ` is the
       theorem to be proved and does not contain ` x ` .  Then we apply
       ~ exlimiv to arrive at ` ( E. x ph -> ps ) ` .  Finally, we separately
       prove ` E. x ph ` and detach it with modus ponens ~ ax-mp to arrive at
       the final theorem ` ps ` .  (Contributed by NM, 5-Aug-1993.)  (Revised
       by NM, 25-Jul-2012.) $)
    exlimiv $p |- ( E. x ph -> ps ) $=
      ( ax-17 exlimih ) ABCBCEDF $.
  $}

  $( Theorem 19.22 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.)
     (Proof shortened by Wolf Lammen, 4-Jul-2014.) $)
  exim $p |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) ) $=
    ( wi wal wex hba1 hbe1 19.8a imim2i sps exlimdh ) ABDZCEABCFZCMCGBCHMANDCBN
    ABCIJKL $.

  ${
    eximi.1 $e |- ( ph -> ps ) $.
    $( Inference adding existential quantifier to antecedent and consequent.
       (Contributed by NM, 5-Aug-1993.) $)
    eximi $p |- ( E. x ph -> E. x ps ) $=
      ( wi wex exim mpg ) ABEACFBCFECABCGDH $.

    $( Inference adding 2 existential quantifiers to antecedent and consequent.
       (Contributed by NM, 3-Feb-2005.) $)
    2eximi $p |- ( E. x E. y ph -> E. x E. y ps ) $=
      ( wex eximi ) ADFBDFCABDEGG $.
  $}

  ${
    eximii.1 $e |- E. x ph $.
    eximii.2 $e |- ( ph -> ps ) $.
    $( Inference associated with ~ eximi .  (Contributed by BJ, 3-Feb-2018.) $)
    eximii $p |- E. x ps $=
      ( wex eximi ax-mp ) ACFBCFDABCEGH $.
  $}

  $( A transformation of quantifiers and logical connectives.  (Contributed by
     NM, 19-Aug-1993.) $)
  alinexa $p |- ( A. x ( ph -> -. ps ) <-> -. E. x ( ph /\ ps ) ) $=
    ( wn wi wal wa wex imnan albii alnex bitri ) ABDEZCFABGZDZCFNCHDMOCABIJNCKL
    $.

  $( Theorem 19.18 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
  exbi $p |- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) ) $=
    ( wb wal wex wi biimp alimi exim syl biimpr impbid ) ABDZCEZACFZBCFZOABGZCE
    PQGNRCABHIABCJKOBAGZCEQPGNSCABLIBACJKM $.

  ${
    exbii.1 $e |- ( ph <-> ps ) $.
    $( Inference adding existential quantifier to both sides of an equivalence.
       (Contributed by NM, 24-May-1994.) $)
    exbii $p |- ( E. x ph <-> E. x ps ) $=
      ( wb wex exbi mpg ) ABEACFBCFECABCGDH $.

    $( Inference adding 2 existential quantifiers to both sides of an
       equivalence.  (Contributed by NM, 16-Mar-1995.) $)
    2exbii $p |- ( E. x E. y ph <-> E. x E. y ps ) $=
      ( wex exbii ) ADFBDFCABDEGG $.

    $( Inference adding 3 existential quantifiers to both sides of an
       equivalence.  (Contributed by NM, 2-May-1995.) $)
    3exbii $p |- ( E. x E. y E. z ph <-> E. x E. y E. z ps ) $=
      ( wex exbii 2exbii ) AEGBEGCDABEFHI $.
  $}

  $( Commutation of conjunction inside an existential quantifier.  (Contributed
     by NM, 18-Aug-1993.) $)
  exancom $p |- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) ) $=
    ( wa ancom exbii ) ABDBADCABEF $.

  ${
    alrimdd.1 $e |- F/ x ph $.
    alrimdd.2 $e |- ( ph -> F/ x ps ) $.
    alrimdd.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.21 of [Margaris] p. 90.  (Contributed by Mario
       Carneiro, 24-Sep-2016.) $)
    alrimdd $p |- ( ph -> ( ps -> A. x ch ) ) $=
      ( wal nfrd alimd syld ) ABBDHCDHABDFIABCDEGJK $.
  $}

  ${
    alrimd.1 $e |- F/ x ph $.
    alrimd.2 $e |- F/ x ps $.
    alrimd.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.21 of [Margaris] p. 90.  (Contributed by Mario
       Carneiro, 24-Sep-2016.) $)
    alrimd $p |- ( ph -> ( ps -> A. x ch ) ) $=
      ( wnf a1i alrimdd ) ABCDEBDHAFIGJ $.
  $}

  ${
    eximdh.1 $e |- ( ph -> A. x ph ) $.
    eximdh.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Contributed by NM,
       20-May-1996.) $)
    eximdh $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
      ( wi wal wex alrimih exim syl ) ABCGZDHBDICDIGAMDEFJBCDKL $.
  $}

  ${
    eximd.1 $e |- F/ x ph $.
    eximd.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Contributed by Mario
       Carneiro, 24-Sep-2016.) $)
    eximd $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
      ( nfri eximdh ) ABCDADEGFH $.
  $}

  ${
    nexd.1 $e |- ( ph -> A. x ph ) $.
    nexd.2 $e |- ( ph -> -. ps ) $.
    $( Deduction for generalization rule for negated wff.  (Contributed by NM,
       2-Jan-2002.) $)
    nexd $p |- ( ph -> -. E. x ps ) $=
      ( wn wal wex alrimih alnex sylib ) ABFZCGBCHFALCDEIBCJK $.
  $}

  ${
    exbidh.1 $e |- ( ph -> A. x ph ) $.
    exbidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for existential quantifier (deduction form).
       (Contributed by NM, 5-Aug-1993.) $)
    exbidh $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
      ( wb wal wex alrimih exbi syl ) ABCGZDHBDICDIGAMDEFJBCDKL $.
  $}

  ${
    albid.1 $e |- F/ x ph $.
    albid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for universal quantifier (deduction form).
       (Contributed by Mario Carneiro, 24-Sep-2016.) $)
    albid $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
      ( nfri albidh ) ABCDADEGFH $.
  $}

  ${
    exbid.1 $e |- F/ x ph $.
    exbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for existential quantifier (deduction form).
       (Contributed by Mario Carneiro, 24-Sep-2016.) $)
    exbid $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
      ( nfri exbidh ) ABCDADEGFH $.
  $}

  $( Simplification of an existentially quantified conjunction.  (Contributed
     by Rodolfo Medina, 25-Sep-2010.)  (Proof shortened by Andrew Salmon,
     29-Jun-2011.) $)
  exsimpl $p |- ( E. x ( ph /\ ps ) -> E. x ph ) $=
    ( wa simpl eximi ) ABDACABEF $.

  $( Simplification of an existentially quantified conjunction.  (Contributed
     by Rodolfo Medina, 25-Sep-2010.)  (Proof shortened by Andrew Salmon,
     29-Jun-2011.) $)
  exsimpr $p |- ( E. x ( ph /\ ps ) -> E. x ps ) $=
    ( wa simpr eximi ) ABDBCABEF $.

  $( Theorem 19.6 of [Margaris] p. 89, given a decidability condition.  The
     forward direction holds for all propositions, as seen at ~ alexim .
     (Contributed by Jim Kingdon, 2-Jun-2018.) $)
  alexdc $p |- ( A. x DECID ph -> ( A. x ph <-> -. E. x -. ph ) ) $=
    ( wdc wal wn wex nfa1 wb notnotbdc sps albid alnex bitrdi ) ACZBDZABDAEZEZB
    DPBFEOAQBNBGNAQHBAIJKPBLM $.

  $( Theorem 19.29 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.)
     (Proof shortened by Andrew Salmon, 13-May-2011.) $)
  19.29 $p |- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) $=
    ( wal wex wa wi pm3.2 alimi exim syl imp ) ACDZBCEZABFZCEZMBOGZCDNPGAQCABHI
    BOCJKL $.

  $( Variation of Theorem 19.29 of [Margaris] p. 90.  (Contributed by NM,
     18-Aug-1993.) $)
  19.29r $p |- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ ps ) ) $=
    ( wal wex wa 19.29 ancom exancom 3imtr4i ) BCDZACEZFBAFCELKFABFCEBACGLKHABC
    IJ $.

  $( Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification.
     (Contributed by NM, 3-Feb-2005.) $)
  19.29r2 $p |- ( ( E. x E. y ph /\ A. x A. y ps ) ->
             E. x E. y ( ph /\ ps ) ) $=
    ( wex wal wa 19.29r eximi syl ) ADEZCEBDFZCFGKLGZCEABGDEZCEKLCHMNCABDHIJ $.

  $( Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification.
     (Contributed by NM, 11-Feb-2005.) $)
  19.29x $p |- ( ( E. x A. y ph /\ A. x E. y ps ) ->
             E. x E. y ( ph /\ ps ) ) $=
    ( wal wex wa 19.29r 19.29 eximi syl ) ADEZCFBDFZCEGLMGZCFABGDFZCFLMCHNOCABD
    IJK $.

  $( Forward direction of Theorem 19.35 of [Margaris] p. 90.  The converse
     holds for classical logic but not (for all propositions) in intuitionistic
     logic.  (Contributed by Mario Carneiro, 2-Feb-2015.) $)
  19.35-1 $p |- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) ) $=
    ( wal wi wex wa 19.29 pm3.35 eximi syl expcom ) ACDZABEZCFZBCFZMOGANGZCFPAN
    CHQBCABIJKL $.

  ${
    19.35i.1 $e |- E. x ( ph -> ps ) $.
    $( Inference from Theorem 19.35 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.)  (Revised by NM, 2-Feb-2015.) $)
    19.35i $p |- ( A. x ph -> E. x ps ) $=
      ( wi wex wal 19.35-1 ax-mp ) ABECFACGBCFEDABCHI $.
  $}

  $( Theorem 19.25 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.)
     (Revised by NM, 2-Feb-2015.) $)
  19.25 $p |- ( A. y E. x ( ph -> ps ) ->
              ( E. y A. x ph -> E. y E. x ps ) ) $=
    ( wi wex wal 19.35-1 alimi exim syl ) ABECFZDGACGZBCFZEZDGMDFNDFELODABCHIMN
    DJK $.

  $( Theorem 19.30 of [Margaris] p. 90, with an additional decidability
     condition.  (Contributed by Jim Kingdon, 21-Jul-2018.) $)
  19.30dc $p |- ( DECID E. x ps ->
      ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) ) $=
    ( wex wdc wn wo wal df-dc olc a1d alnex orel2 al2imi sylbir syl6 jaoi sylbi
    wi orc ) BCDZEUAUAFZGABGZCHZACHZUAGZSZUAIUAUGUBUAUFUDUAUEJKUBUDUEUFUBBFZCHU
    DUESBCLUHUCACBAMNOUEUATPQR $.

  $( Theorem 19.43 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.)
     (Proof shortened by Mario Carneiro, 2-Feb-2015.) $)
  19.43 $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) ) $=
    ( wo wex hbe1 hbor 19.8a orim12i exlimih orc eximi olc jaoi impbii ) ABDZCE
    ZACEZBCEZDZPTCRSCACFBCFGARBSACHBCHIJRQSAPCABKLBPCBAMLNO $.

  $( The antecedent provides a condition implying the converse of ~ 19.33 .
     Compare Theorem 19.33 of [Margaris] p. 90.  This variation of ~ 19.33bdc
     is intuitionistically valid without a decidability condition.
     (Contributed by Mario Carneiro, 2-Feb-2015.) $)
  19.33b2 $p |- ( ( -. E. x ph \/ -. E. x ps ) ->
               ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) $=
    ( wex wn wo wal orcom alnex orbi12i bitr4i wi pm2.53 orcoms al2imi biimtrid
    orim12d com12 19.33 impbid1 ) ACDEZBCDEZFZABFZCGZACGZBCGZFZUEUCUHUCBEZCGZAE
    ZCGZFZUEUHUCUBUAFUMUAUBHUJUBULUABCIACIJKUEUJUFULUGUDUIACBAUIALBAMNOUDUKBCAB
    MOQPRABCST $.

  $( Converse of ~ 19.33 given ` -. ( E. x ph /\ E. x ps ) ` and a decidability
     condition.  Compare Theorem 19.33 of [Margaris] p. 90.  For a version
     which does not require a decidability condition, see ~ 19.33b2
     (Contributed by Jim Kingdon, 23-Apr-2018.) $)
  19.33bdc $p |- ( DECID E. x ph -> ( -. ( E. x ph /\ E. x ps ) ->
               ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) ) $=
    ( wex wdc wa wn wo wal wb ianordc 19.33b2 biimtrdi ) ACDZENBCDZFGNGOGHABHCI
    ACIBCIHJNOKABCLM $.

  $( Theorem 19.40 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
  19.40 $p |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) ) $=
    ( wa wex exsimpl simpr eximi jca ) ABDZCEACEBCEABCFJBCABGHI $.

  $( Theorem *11.42 in [WhiteheadRussell] p. 163.  Theorem 19.40 of [Margaris]
     p. 90 with 2 quantifiers.  (Contributed by Andrew Salmon, 24-May-2011.) $)
  19.40-2 $p |- ( E. x E. y ( ph /\ ps ) ->
        ( E. x E. y ph /\ E. x E. y ps ) ) $=
    ( wa wex 19.40 eximi syl ) ABEDFZCFADFZBDFZEZCFKCFLCFEJMCABDGHKLCGI $.

  $( Add/remove a conjunct in the scope of an existential quantifier.
     (Contributed by Raph Levien, 3-Jul-2006.) $)
  exintrbi $p |- ( A. x ( ph -> ps ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) ) $=
    ( wi wal wa wb wex pm4.71 albii exbi sylbi ) ABDZCEAABFZGZCEACHNCHGMOCABIJA
    NCKL $.

  $( Introduce a conjunct in the scope of an existential quantifier.
     (Contributed by NM, 11-Aug-1993.) $)
  exintr $p |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) ) $=
    ( wi wal wex wa exintrbi biimpd ) ABDCEACFABGCFABCHI $.

  $( Theorem *10.3 in [WhiteheadRussell] p. 150.  (Contributed by Andrew
     Salmon, 8-Jun-2011.) $)
  alsyl $p |- ( ( A. x ( ph -> ps ) /\ A. x ( ps -> ch ) ) ->
        A. x ( ph -> ch ) ) $=
    ( wi pm3.33 alanimi ) ABEBCEACEDABCFG $.

  ${
    hbex.1 $e |- ( ph -> A. x ph ) $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
       (Contributed by NM, 5-Aug-1993.)  (Revised by NM, 2-Feb-2015.) $)
    hbex $p |- ( E. y ph -> A. x E. y ph ) $=
      ( wex wal hbe1 hbal 19.8a alrimih exlimih ) AACEZBFCLCBACGHALBDACIJK $.
  $}

  ${
    nfex.1 $e |- F/ x ph $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.)  (Proof shortened by Wolf
       Lammen, 30-Dec-2017.) $)
    nfex $p |- F/ x E. y ph $=
      ( wex nfri hbex nfi ) ACEBABCABDFGH $.
  $}

  $( Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables.
     (Contributed by O'Cat, 31-Mar-2008.) $)
  19.2 $p |- ( A. x ph -> E. y ph ) $=
    ( wex 19.8a sps ) AACDBACEF $.

  ${
    i19.24.1 $e |- ( ( A. x ph -> E. x ps ) -> E. x ( ph -> ps ) ) $.
    $( Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis.  The
       hypothesis is the converse of ~ 19.35-1 , and is a theorem of classical
       logic, but in intuitionistic logic it will only be provable for some
       propositions.  (Contributed by Jim Kingdon, 22-Jul-2018.) $)
    i19.24 $p |- ( ( A. x ph -> A. x ps ) -> E. x ( ph -> ps ) ) $=
      ( wal wi wex 19.2 imim2i syl ) ACEZBCEZFKBCGZFABFCGLMKBCCHIDJ $.

    $( Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis.  The
       hypothesis is the converse of ~ 19.35-1 , and is a theorem of classical
       logic, but in intuitionistic logic it will only be provable for some
       propositions.  (Contributed by Jim Kingdon, 22-Jul-2018.) $)
    i19.39 $p |- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) ) $=
      ( wex wi wal 19.2 imim1i syl ) ACEZBCEZFACGZLFABFCEMKLACCHIDJ $.
  $}

  $( A closed version of one direction of ~ 19.9 .  (Contributed by NM,
     5-Aug-1993.) $)
  19.9ht $p |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) ) $=
    ( wal wi wex id ax-gen 19.23ht mpbii ) AABCDBCAADZBCABEADJBAFGAABHI $.

  $( A closed version of ~ 19.9 .  (Contributed by NM, 5-Aug-1993.)  (Revised
     by Mario Carneiro, 24-Sep-2016.)  (Proof shortended by Wolf Lammen,
     30-Dec-2017.) $)
  19.9t $p |- ( F/ x ph -> ( E. x ph <-> ph ) ) $=
    ( wnf wex wal wi df-nf 19.9ht sylbi 19.8a impbid1 ) ABCZABDZALAABEFBEMAFABG
    ABHIABJK $.

  ${
    19.9h.1 $e |- ( ph -> A. x ph ) $.
    $( A wff may be existentially quantified with a variable not free in it.
       Theorem 19.9 of [Margaris] p. 89.  (Contributed by FL, 24-Mar-2007.) $)
    19.9h $p |- ( E. x ph <-> ph ) $=
      ( wex wal wi 19.9ht mpg 19.8a impbii ) ABDZAAABEFKAFBABGCHABIJ $.
  $}

  ${
    19.9.1 $e |- F/ x ph $.
    $( A wff may be existentially quantified with a variable not free in it.
       Theorem 19.9 of [Margaris] p. 89.  (Contributed by FL, 24-Mar-2007.)
       (Revised by Mario Carneiro, 24-Sep-2016.)  (Proof shortened by Wolf
       Lammen, 30-Dec-2017.) $)
    19.9 $p |- ( E. x ph <-> ph ) $=
      ( nfri 19.9h ) ABABCDE $.
  $}

  $( One direction of Theorem 19.6 of [Margaris] p. 89.  The converse holds
     given a decidability condition, as seen at ~ alexdc .  (Contributed by Jim
     Kingdon, 2-Jul-2018.) $)
  alexim $p |- ( A. x ph -> -. E. x -. ph ) $=
    ( wal wn wex wfal wi pm2.24 alimi exim syl nfv 19.9 imbitrdi dfnot sylibr )
    ABCZADZBEZFGSDQSFBEZFQRFGZBCSTGAUABAFHIRFBJKFBFBLMNSOP $.

  $( One direction of Theorem 19.14 of [Margaris] p. 90.  The converse holds in
     classical but not in intuitionistic logic.  (Contributed by Jim Kingdon,
     15-Jul-2018.) $)
  exnalim $p |- ( E. x -. ph -> -. A. x ph ) $=
    ( wal wn wex alexim con2i ) ABCADBEABFG $.

  $( A transformation of quantifiers and logical connectives.  The converse
     holds in classical but not in intuitionistic logic.  (Contributed by Jim
     Kingdon, 15-Jul-2018.) $)
  exanaliim $p |- ( E. x ( ph /\ -. ps ) -> -. A. x ( ph -> ps ) ) $=
    ( wn wa wex wi wal annimim eximi exnalim syl ) ABDEZCFABGZDZCFNCHDMOCABIJNC
    KL $.

  $( A relationship between two quantifiers and negation.  (Contributed by Jim
     Kingdon, 27-Aug-2018.) $)
  alexnim $p |- ( A. x E. y -. ph -> -. E. x A. y ph ) $=
    ( wn wex wal exnalim alimi alnex sylib ) ADCEZBFACFZDZBFLBEDKMBACGHLBIJ $.

  $( The double negation of a universal quantification implies the universal
     quantification of the double negation.  The converse holds in classical
     but not in intuitionistic logic.  (Contributed by BJ, 24-Nov-2023.) $)
  nnal $p |- ( -. -. A. x ph -> A. x -. -. ph ) $=
    ( wal wn wex exnalim con3i alnex sylibr ) ABCDZDADZBEZDKDBCLJABFGKBHI $.

  ${
    hbn.1 $e |- ( ph -> A. x ph ) $.
    $( If ` x ` is not free in ` ph ` , then it is not free in ` -. ph ` .
       This theorem does not depend on ~ ax-ial , contrary to ~ hbn1 and
       ~ hbnt .  (Contributed by NM, 5-Aug-1993.)  Remove dependency on
       ~ ax-ial .  (Revised by GD, 27-Jan-2018.) $)
    hbn $p |- ( -. ph -> A. x -. ph ) $=
      ( wn wex wal id exlimih con3i alnex sylibr ) ADZABEZDLBFMAAABCAGHIABJK $.
    $( $j usage 'hbn' avoids 'ax-ial'; $)
  $}

  $( Quantified Negation.  Axiom C5-2 of [Monk2] p. 113.  The setvar ` x ` is
     not free in ` -. A. x ph ` .  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 18-Aug-2014.)  (Proof shortened by GD,
     27-Jan-2018.) $)
  hbn1 $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
    ( wal ax-ial hbn ) ABCBABDE $.

  $( Closed form of bound-variable hypothesis builder ~ hbn .  (Contributed by
     NM, 5-Aug-1993.)  (Revised by NM, 2-Feb-2015.) $)
  hbnt $p |- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) ) $=
    ( wn wal wi ax-4 con3i hbn1 syl con3 al2imi syl5 ) ACZABDZCZBDZANEZBDMBDMOP
    NAABFGABHIQOMBANJKL $.

  ${
    hbnOLD.1 $e |- ( ph -> A. x ph ) $.
    $( Obsolete proof of ~ hbn as of 2-May-2026.  (Contributed by NM,
       5-Aug-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    hbnOLD $p |- ( -. ph -> A. x -. ph ) $=
      ( wal wi wn hbnt mpg ) AABDEAFZIBDEBABGCH $.
  $}

  ${
    hbnd.1 $e |- ( ph -> A. x ph ) $.
    hbnd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Deduction form of bound-variable hypothesis builder ~ hbn .
       (Contributed by NM, 3-Jan-2002.) $)
    hbnd $p |- ( ph -> ( -. ps -> A. x -. ps ) ) $=
      ( wal wi wn alrimih hbnt syl ) ABBCFGZCFBHZMCFGALCDEIBCJK $.
  $}

  $( If ` x ` is not free in ` ph ` , then it is not free in ` -. ph ` .
     (Contributed by Mario Carneiro, 24-Sep-2016.)  (Proof shortened by Wolf
     Lammen, 28-Dec-2017.)  (Revised by BJ, 24-Jul-2019.) $)
  nfnt $p |- ( F/ x ph -> F/ x -. ph ) $=
    ( wnf wn nfnf1 wal wi df-nf hbnt sylbi nfd ) ABCZADZBABELAABFGBFMMBFGABHABI
    JK $.

  ${
    nfnd.1 $e |- ( ph -> F/ x ps ) $.
    $( Deduction associated with ~ nfnt .  (Contributed by Mario Carneiro,
       24-Sep-2016.) $)
    nfnd $p |- ( ph -> F/ x -. ps ) $=
      ( wnf wn nfnt syl ) ABCEBFCEDBCGH $.
  $}

  ${
    nfn.1 $e |- F/ x ph $.
    $( Inference associated with ~ nfnt .  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfn $p |- F/ x -. ph $=
      ( wnf wn nfnt ax-mp ) ABDAEBDCABFG $.
  $}

  ${
    nfdc.1 $e |- F/ x ph $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` DECID ph ` .
       (Contributed by Jim Kingdon, 11-Mar-2018.) $)
    nfdc $p |- F/ x DECID ph $=
      ( wdc wn wo df-dc nfn nfor nfxfr ) ADAAEZFBAGAKBCABCHIJ $.
  $}

  $( The analog in our predicate calculus of axiom 5 of modal logic S5.
     (Contributed by NM, 5-Oct-2005.) $)
  modal-5 $p |- ( -. A. x -. ph -> A. x -. A. x -. ph ) $=
    ( wn hbn1 ) ACBD $.

  ${
    19.9d.1 $e |- ( ps -> F/ x ph ) $.
    $( A deduction version of one direction of ~ 19.9 .  (Contributed by NM,
       5-Aug-1993.)  (Revised by Mario Carneiro, 24-Sep-2016.) $)
    19.9d $p |- ( ps -> ( E. x ph -> ph ) ) $=
      ( wex wnf wb 19.9t syl biimpd ) BACEZABACFKAGDACHIJ $.
  $}

  ${
    19.9hd.1 $e |- ( ps -> A. x ps ) $.
    19.9hd.2 $e |- ( ps -> ( ph -> A. x ph ) ) $.
    $( A deduction version of one direction of ~ 19.9 .  This is an older
       variation of this theorem; new proofs should use ~ 19.9d .  (Contributed
       by NM, 5-Aug-1993.)  (New usage is discouraged.) $)
    19.9hd $p |- ( ps -> ( E. x ph -> ph ) ) $=
      ( wal wi wex alimi 19.9ht 3syl ) BBCFAACFGZCFACHAGDBLCEIACJK $.
  $}

  $( One direction of Theorem 19.11 of [Margaris] p. 89.  (Contributed by NM,
     5-Aug-1993.) $)
  excomim $p |- ( E. x E. y ph -> E. y E. x ph ) $=
    ( wex 19.8a 2eximi hbe1 hbex 19.9h sylib ) ACDBDABDZCDZBDLAKBCABEFLBKBCABGH
    IJ $.

  $( Theorem 19.11 of [Margaris] p. 89.  (Contributed by NM, 5-Aug-1993.) $)
  excom $p |- ( E. x E. y ph <-> E. y E. x ph ) $=
    ( wex excomim impbii ) ACDBDABDCDABCEACBEF $.

  $( Theorem 19.12 of [Margaris] p. 89.  Assuming the converse is a mistake
     sometimes made by beginners!  (Contributed by NM, 5-Aug-1993.) $)
  19.12 $p |- ( E. x A. y ph -> A. y E. x ph ) $=
    ( wal wex hba1 hbex ax-4 eximi alrimih ) ACDZBEABECKCBACFGKABACHIJ $.

  ${
    19.19.1 $e |- F/ x ph $.
    $( Theorem 19.19 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
    19.19 $p |- ( A. x ( ph <-> ps ) -> ( ph <-> E. x ps ) ) $=
      ( wex wb wal 19.9 exbi bitr3id ) AACEABFCGBCEACDHABCIJ $.
  $}

  ${
    19.21-2.1 $e |- F/ x ph $.
    19.21-2.2 $e |- F/ y ph $.
    $( Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers.  (Contributed
       by NM, 4-Feb-2005.) $)
    19.21-2 $p |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) ) $=
      ( wi wal 19.21 albii bitri ) ABGDHZCHABDHZGZCHAMCHGLNCABDFIJAMCEIK $.
  $}

  $( An alternate definition of ~ df-nf , which does not involve nested
     quantifiers on the same variable.  (Contributed by Mario Carneiro,
     24-Sep-2016.) $)
  nf2 $p |- ( F/ x ph <-> ( E. x ph -> A. x ph ) ) $=
    ( wnf wal wi wex df-nf nfa1 nfri 19.23h bitri ) ABCAABDZEBDABFLEABGALBLBABH
    IJK $.

  $( An alternate definition of ~ df-nf .  (Contributed by Mario Carneiro,
     24-Sep-2016.) $)
  nf3 $p |- ( F/ x ph <-> A. x ( E. x ph -> ph ) ) $=
    ( wnf wex wal wi nf2 nfe1 nfri 19.21h bitr4i ) ABCABDZABEFLAFBEABGLABLBABHI
    JK $.

  $( Variable ` x ` is effectively not free in ` ph ` iff ` ph ` is always true
     or always false, given a decidability condition.  The reverse direction,
     ~ nf4r , holds for all propositions.  (Contributed by Jim Kingdon,
     21-Jul-2018.) $)
  nf4dc $p |- ( DECID E. x ph -> ( F/ x ph <-> ( A. x ph \/ A. x -. ph ) ) ) $=
    ( wex wdc wnf wn wal wo nf2 imordc bitrid orcom alnex orbi2i bitr4i bitrdi
    wi ) ABCZDZABEZRFZABGZHZUBAFBGZHZTRUBQSUCABIRUBJKUCUBUAHUEUAUBLUDUAUBABMNOP
    $.

  $( If ` ph ` is always true or always false, then variable ` x ` is
     effectively not free in ` ph ` .  The converse holds given a decidability
     condition, as seen at ~ nf4dc .  (Contributed by Jim Kingdon,
     21-Jul-2018.) $)
  nf4r $p |- ( ( A. x ph \/ A. x -. ph ) -> F/ x ph ) $=
    ( wal wn wo wex wnf orcom alnex orbi2i bitr4i wi imorr nf2 sylibr sylbir )
    ABCZADBCZEZABFZDZQEZABGZUBQUAESUAQHRUAQABIJKUBTQLUCTQMABNOP $.

  ${
    19.36i.1 $e |- F/ x ps $.
    19.36i.2 $e |- E. x ( ph -> ps ) $.
    $( Inference from Theorem 19.36 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.)  (Revised by NM, 2-Feb-2015.) $)
    19.36i $p |- ( A. x ph -> ps ) $=
      ( wal wex 19.35i id exlimi syl ) ACFBCGBABCEHBBCDBIJK $.
  $}

  ${
    19.36-1.1 $e |- F/ x ps $.
    $( Closed form of ~ 19.36i .  One direction of Theorem 19.36 of [Margaris]
       p. 90.  The converse holds in classical logic, but does not hold (for
       all propositions) in intuitionistic logic.  (Contributed by Jim Kingdon,
       20-Jun-2018.) $)
    19.36-1 $p |- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) ) $=
      ( wi wex wal 19.35-1 19.9 imbitrdi ) ABECFACGBCFBABCHBCDIJ $.
  $}

  ${
    19.37-1.1 $e |- F/ x ph $.
    $( One direction of Theorem 19.37 of [Margaris] p. 90.  The converse holds
       in classical logic but not, in general, here.  (Contributed by Jim
       Kingdon, 21-Jun-2018.) $)
    19.37-1 $p |- ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) ) $=
      ( wal wi wex 19.3 19.35-1 biimtrrid ) AACEABFCGBCGACDHABCIJ $.
  $}

  ${
    $d x ph $.
    19.37aiv.1 $e |- E. x ( ph -> ps ) $.
    $( Inference from Theorem 19.37 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.) $)
    19.37aiv $p |- ( ph -> E. x ps ) $=
      ( wi wex nfv 19.37-1 ax-mp ) ABECFABCFEDABCACGHI $.
  $}

  $( Theorem 19.38 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
  19.38 $p |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) ) $=
    ( wex wal wi hbe1 hba1 hbim 19.8a ax-4 imim12i alrimih ) ACDZBCEZFABFCNOCAC
    GBCHIANOBACJBCKLM $.

  $( Closed form of Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
     7-Nov-2005.)  (Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
  19.23t $p |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
    ( wnf wi wal wex exim 19.9t biimpd syl9r nfr imim2d 19.38 syl6 impbid ) BCD
    ZABECFZACGZBEZRSBCGZQBABCHQUABBCIJKQTSBCFZERQBUBSBCLMABCNOP $.

  ${
    19.23.1 $e |- F/ x ps $.
    $( Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.)
       (Revised by Mario Carneiro, 24-Sep-2016.) $)
    19.23 $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
      ( wnf wi wal wex wb 19.23t ax-mp ) BCEABFCGACHBFIDABCJK $.
  $}

  ${
    19.32dc.1 $e |- F/ x ph $.
    $( Theorem 19.32 of [Margaris] p. 90, where ` ph ` is decidable.
       (Contributed by Jim Kingdon, 4-Jun-2018.) $)
    19.32dc $p |- ( DECID ph ->
        ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) ) $=
      ( wdc wn wi wal wo wb nfn 19.21 a1i nfdc dfordc albid 3bitr4d ) AEZAFZBGZ
      CHZSBCHZGZABIZCHAUBIUAUCJRSBCACDKLMRUDTCACDNABOPAUBOQ $.
  $}

  ${
    19.32r.1 $e |- F/ x ph $.
    $( One direction of Theorem 19.32 of [Margaris] p. 90.  The converse holds
       if ` ph ` is decidable, as seen at ~ 19.32dc .  (Contributed by Jim
       Kingdon, 28-Jul-2018.) $)
    19.32r $p |- ( ( ph \/ A. x ps ) -> A. x ( ph \/ ps ) ) $=
      ( wo wal orc alrimi olc alimi jaoi ) AABEZCFBCFALCDABGHBLCBAIJK $.
  $}

  ${
    19.31r.1 $e |- F/ x ps $.
    $( One direction of Theorem 19.31 of [Margaris] p. 90.  The converse holds
       in classical logic, but not intuitionistic logic.  (Contributed by Jim
       Kingdon, 28-Jul-2018.) $)
    19.31r $p |- ( ( A. x ph \/ ps ) -> A. x ( ph \/ ps ) ) $=
      ( wal wo 19.32r orcom albii 3imtr4i ) BACEZFBAFZCEKBFABFZCEBACDGKBHMLCABH
      IJ $.
  $}

  ${
    19.44.1 $e |- F/ x ps $.
    $( Theorem 19.44 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
    19.44 $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ ps ) ) $=
      ( wo wex 19.43 19.9 orbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
  $}

  ${
    19.45.1 $e |- F/ x ph $.
    $( Theorem 19.45 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
    19.45 $p |- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) ) $=
      ( wo wex 19.43 19.9 orbi1i bitri ) ABECFACFZBCFZEALEABCGKALACDHIJ $.
  $}

  $( Theorem 19.34 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.) $)
  19.34 $p |- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) ) $=
    ( wal wex wo 19.2 orim1i 19.43 sylibr ) ACDZBCEZFACEZLFABFCEKMLACCGHABCIJ
    $.

  ${
    19.41h.1 $e |- ( ps -> A. x ps ) $.
    $( Theorem 19.41 of [Margaris] p. 90.  New proofs should use ~ 19.41
       instead.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew
       Salmon, 25-May-2011.)  (New usage is discouraged.) $)
    19.41h $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
      ( wa wex 19.40 id exlimih anim2i syl pm3.21 eximdh impcom impbii ) ABEZCF
      ZACFZBEZQRBCFZESABCGTBRBBCDBHIJKBRQBAPCDBALMNO $.
  $}

  ${
    19.41.1 $e |- F/ x ps $.
    $( Theorem 19.41 of [Margaris] p. 90.  (Contributed by NM, 5-Aug-1993.)
       (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof shortened by
       Wolf Lammen, 12-Jan-2018.) $)
    19.41 $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
      ( wa wex 19.40 19.9 anbi2i sylib pm3.21 eximd impcom impbii ) ABEZCFZACFZ
      BEZPQBCFZERABCGSBQBCDHIJBQPBAOCDBAKLMN $.
  $}

  ${
    19.42h.1 $e |- ( ph -> A. x ph ) $.
    $( Theorem 19.42 of [Margaris] p. 90.  New proofs should use ~ 19.42
       instead.  (Contributed by NM, 18-Aug-1993.)
       (New usage is discouraged.) $)
    19.42h $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
      ( wa wex 19.41h exancom ancom 3bitr4i ) BAECFBCFZAEABECFAKEBACDGABCHAKIJ
      $.
  $}

  ${
    19.42.1 $e |- F/ x ph $.
    $( Theorem 19.42 of [Margaris] p. 90.  (Contributed by NM, 18-Aug-1993.) $)
    19.42 $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
      ( wa wex 19.41 exancom ancom 3bitr4i ) BAECFBCFZAEABECFAKEBACDGABCHAKIJ
      $.
  $}

  $( Swap 1st and 3rd existential quantifiers.  (Contributed by NM,
     9-Mar-1995.) $)
  excom13 $p |- ( E. x E. y E. z ph <-> E. z E. y E. x ph ) $=
    ( wex excom exbii 3bitri ) ADEZCEBEIBEZCEABEZDEZCEKCEDEIBCFJLCABDFGKCDFH $.

  $( Rotate existential quantifiers.  (Contributed by NM, 17-Mar-1995.) $)
  exrot3 $p |- ( E. x E. y E. z ph <-> E. y E. z E. x ph ) $=
    ( wex excom13 excom bitri ) ADECEBEABEZCEDEIDECEABCDFIDCGH $.

  $( Rotate existential quantifiers twice.  (Contributed by NM, 9-Mar-1995.) $)
  exrot4 $p |- ( E. x E. y E. z E. w ph <-> E. z E. w E. x E. y ph ) $=
    ( wex excom13 exbii bitri ) AEFDFCFZBFACFZDFEFZBFKBFEFDFJLBACDEGHKBEDGI $.

  ${
    nexr.1 $e |- -. E. x ph $.
    $( Inference from ~ 19.8a .  (Contributed by Jeff Hankins, 26-Jul-2009.) $)
    nexr $p |- -. ph $=
      ( wex 19.8a mto ) AABDCABEF $.
  $}

  ${
    exan.1 $e |- ( E. x ph /\ ps ) $.
    $( Place a conjunct in the scope of an existential quantifier.
       (Contributed by NM, 18-Aug-1993.)  (Proof shortened by Andrew Salmon,
       25-May-2011.) $)
    exan $p |- E. x ( ph /\ ps ) $=
      ( wex wal wa hbe1 19.28h mpgbi 19.29r ax-mp ) ACEZBCFGZABGCEMBGNCMBCACHID
      JABCKL $.
  $}

  ${
    hbexd.1 $e |- ( ph -> A. y ph ) $.
    hbexd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Deduction form of bound-variable hypothesis builder ~ hbex .
       (Contributed by NM, 2-Jan-2002.) $)
    hbexd $p |- ( ph -> ( E. y ps -> A. x E. y ps ) ) $=
      ( wex wal eximdh 19.12 syl6 ) ABDGZBCHZDGLCHABMDEFIBDCJK $.
  $}

  ${
    eeor.1 $e |- F/ y ph $.
    eeor.2 $e |- F/ x ps $.
    $( Rearrange existential quantifiers.  (Contributed by NM, 8-Aug-1994.) $)
    eeor $p |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) ) $=
      ( wo wex 19.45 exbii nfex 19.44 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
      CBCDFKLM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Equality theorems without distinct variables
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( At least one individual exists.  This is not a theorem of free logic,
     which is sound in empty domains.  For such a logic, we would add this
     theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10).  In the
     system consisting of ~ ax-5 through ~ ax-14 and ~ ax-17 , all axioms other
     than ~ ax-9 are believed to be theorems of free logic, although the system
     without ~ ax-9 is probably not complete in free logic.  (Contributed by
     NM, 5-Aug-1993.)  (Revised by NM, 3-Feb-2015.) $)
  a9e $p |- E. x x = y $=
    ( ax-i9 ) ABC $.

  ${
    $d x y $.
    $( At least one individual exists.  Weaker version of ~ a9e .  (Contributed
       by NM, 3-Aug-2017.) $)
    a9ev $p |- E. x x = y $=
      ( ax-i9 ) ABC $.
  $}

  $( An implication related to substitution.  (Contributed by NM, 5-Aug-1993.)
     (Revised by NM, 3-Feb-2015.) $)
  ax9o $p |- ( A. x ( x = y -> A. x ph ) -> ph ) $=
    ( cv wceq wex wal wi a9e wa 19.29r hba1 pm3.35 exlimih ax-4 syl mpan ) BDCD
    EZBFZRABGZHZBGZABCISUBJRUAJZBFZARUABKUDTAUCTBABLRTMNABOPPQ $.

  ${
    $d x y $.
    spimfv.nf $e |- F/ x ps $.
    spimfv.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Specialization, using implicit substitution.  Version of ~ spim with a
       disjoint variable condition.  See ~ spimv for another variant.
       (Contributed by NM, 10-Jan-1993.)  (Revised by BJ, 31-May-2019.) $)
    spimfv $p |- ( A. x ph -> ps ) $=
      ( weq wi a9ev eximii 19.36i ) ABCECDGABHCCDIFJK $.
  $}

  ${
    $d x y $.
    chvarfv.nf $e |- F/ x ps $.
    chvarfv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    chvarfv.2 $e |- ph $.
    $( Implicit substitution of ` y ` for ` x ` into a theorem.  Version of
       ~ chvar with a disjoint variable condition.  (Contributed by Raph
       Levien, 9-Jul-2003.)  (Revised by BJ, 31-May-2019.) $)
    chvarfv $p |- ps $=
      ( weq biimpd spimfv mpg ) ABCABCDECDHABFIJGK $.
  $}

  ${
    $d x y $.
    $( Identity law for equality (reflexivity).  Lemma 6 of [Tarski] p. 68.
       This is often an axiom of equality in textbook systems, but we don't
       need it as an axiom since it can be proved from our other axioms.

       This proof is similar to Tarski's and makes use of a dummy variable
       ` y ` .  It also works in intuitionistic logic, unlike some other
       possible ways of proving this theorem.  (Contributed by NM,
       1-Apr-2005.) $)
    equid $p |- x = x $=
      ( vy weq wex a9e ax-17 ax-8 pm2.43i exlimih ax-mp ) BACZBDAACZBAEKLBLBFKL
      BAAGHIJ $.
  $}

  $( Bound-variable hypothesis builder for ` x = x ` .  This theorem tells us
     that any variable, including ` x ` , is effectively not free in
     ` x = x ` , even though ` x ` is technically free according to the
     traditional definition of free variable.  (Contributed by NM,
     13-Jan-2011.)  (Revised by NM, 21-Aug-2017.) $)
  nfequid $p |- F/ y x = x $=
    ( weq equid nfth ) AACBADE $.

  $( One of the two equality axioms of standard predicate calculus, called
     reflexivity of equality.  (The other one is ~ stdpc7 .)  Axiom 6 of
     [Mendelson] p. 95.  Mendelson doesn't say why he prepended the redundant
     quantifier, but it was probably to be compatible with free logic (which is
     valid in the empty domain).  (Contributed by NM, 16-Feb-2005.) $)
  stdpc6 $p |- A. x x = x $=
    ( weq equid ax-gen ) AABAACD $.

  $( Commutative law for equality.  Lemma 7 of [Tarski] p. 69.  (Contributed by
     NM, 5-Aug-1993.) $)
  equcomi $p |- ( x = y -> y = x ) $=
    ( weq equid ax-8 mpi ) ABCAACBACADABAEF $.

  ${
    $d x y $.
    $( A commuted form of ~ a9ev .  The naming reflects how axioms were
       numbered in the Metamath Proof Explorer as of 2020 (a numbering which we
       eventually plan to adopt here too, but until this happens everywhere
       only some theorems will have it).  (Contributed by BJ, 7-Dec-2020.) $)
    ax6evr $p |- E. x y = x $=
      ( weq a9ev equcomi eximii ) ABCBACAABDABEF $.
  $}

  $( Commutative law for equality.  (Contributed by NM, 20-Aug-1993.) $)
  equcom $p |- ( x = y <-> y = x ) $=
    ( weq equcomi impbii ) ABCBACABDBADE $.

  ${
    equcomd.1 $e |- ( ph -> x = y ) $.
    $( Deduction form of ~ equcom , symmetry of equality.  For the versions for
       classes, see ~ eqcom and ~ eqcomd .  (Contributed by BJ, 6-Oct-2019.) $)
    equcomd $p |- ( ph -> y = x ) $=
      ( weq equcom sylib ) ABCECBEDBCFG $.
  $}

  ${
    equcoms.1 $e |- ( x = y -> ph ) $.
    $( An inference commuting equality in antecedent.  Used to eliminate the
       need for a syllogism.  (Contributed by NM, 5-Aug-1993.) $)
    equcoms $p |- ( y = x -> ph ) $=
      ( weq equcomi syl ) CBEBCEACBFDG $.
  $}

  $( A transitive law for equality.  (Contributed by NM, 23-Aug-1993.) $)
  equtr $p |- ( x = y -> ( y = z -> x = z ) ) $=
    ( weq wi ax-8 equcoms ) BCDACDEBABACFG $.

  $( A transitive law for equality.  Lemma L17 in [Megill] p. 446 (p. 14 of the
     preprint).  (Contributed by NM, 23-Aug-1993.) $)
  equtrr $p |- ( x = y -> ( z = x -> z = y ) ) $=
    ( weq equtr com12 ) CADABDCBDCABEF $.

  $( A transitive law for equality.  (Contributed by NM, 12-Aug-1993.)  (Proof
     shortened by Andrew Salmon, 25-May-2011.) $)
  equtr2 $p |- ( ( x = z /\ y = z ) -> x = y ) $=
    ( weq wi equtrr equcoms impcom ) BCDACDZABDZIJECBCBAFGH $.

  $( An equivalence law for equality.  (Contributed by NM, 5-Aug-1993.) $)
  equequ1 $p |- ( x = y -> ( x = z <-> y = z ) ) $=
    ( weq ax-8 equtr impbid ) ABDACDBCDABCEABCFG $.

  $( An equivalence law for equality.  (Contributed by NM, 5-Aug-1993.) $)
  equequ2 $p |- ( x = y -> ( z = x <-> z = y ) ) $=
    ( weq equtrr wi equcoms impbid ) ABDCADZCBDZABCEJIFBABACEGH $.

  ${
    ax11i.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    ax11i.2 $e |- ( ps -> A. x ps ) $.
    $( Inference that has ~ ax-11 (without ` A. y ` ) as its conclusion and
       does not require ~ ax-10 , ~ ax-11 , or ~ ax12 for its proof.  The
       hypotheses may be eliminable without one or more of these axioms in
       special cases.  Proof similar to Lemma 16 of [Tarski] p. 70.
       (Contributed by NM, 20-May-2008.) $)
    ax11i $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
      ( weq wi wal biimprcd alrimih biimtrdi ) CDGZABMAHZCIEBNCFMABEJKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axioms ax-10 and ax-11
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Show that ~ ax-10o can be derived from ~ ax-10 .  An open problem is
     whether this theorem can be derived from ~ ax-10 and the others when
     ~ ax-11 is replaced with ~ ax-11o .  See Theorem ~ ax10 for the
     rederivation of ~ ax-10 from ~ ax10o .

     Normally, ~ ax10o should be used rather than ~ ax-10o , except by theorems
     specifically studying the latter's properties.  (Contributed by NM,
     16-May-2008.) $)
  ax10o $p |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $=
    ( weq wal wi ax-10 ax-11 equcoms sps pm2.27 al2imi sylsyld ) BCDZBECBDZCEAB
    EZOAFZCEZACEBCGNPRFZBSCBACBHIJOQACOAKLM $.

  $( Axiom ~ ax-10o ("o" for "old") was the original version of ~ ax-10 ,
     before it was discovered (in May 2008) that the shorter ~ ax-10 could
     replace it.  It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of
     the preprint).

     This axiom is redundant, as shown by Theorem ~ ax10o .

     Normally, ~ ax10o should be used rather than ~ ax-10o , except by theorems
     specifically studying the latter's properties.  (Contributed by NM,
     5-Aug-1993.)  (New usage is discouraged.) $)
  ax-10o $a |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $.

  $( Rederivation of ~ ax-10 from original version ~ ax-10o .  See Theorem
     ~ ax10o for the derivation of ~ ax-10o from ~ ax-10 .

     This theorem should not be referenced in any proof.  Instead, use ~ ax-10
     above so that uses of ~ ax-10 can be more easily identified.  (Contributed
     by NM, 16-May-2008.)  (New usage is discouraged.) $)
  ax10 $p |- ( A. x x = y -> A. y y = x ) $=
    ( weq wal ax-10o pm2.43i equcomi alimi syl ) ABCZADZJBDZBACZBDKLJABEFJMBABG
    HI $.

  $( All variables are effectively bound in an identical variable specifier.
     (Contributed by NM, 5-Aug-1993.)  (Revised by NM, 3-Feb-2015.) $)
  hbae $p |- ( A. x x = y -> A. z A. x x = y ) $=
    ( cv wceq wal wi ax12or ax10o alequcoms pm2.43i syl5 ax-4 imim1i jaoi ax-mp
    wo sps a5i ax-7 syl ) ADZBDZEZAFZUDCFZAFUECFUDUFACDZUBECFZUGUCECFZUDUFGZCFZ
    QZQUEUFGZABCHUHUMULUMACUDACIJUIUMUKUMBCUEUDBFZUCUGEBFUFUEUNUDABIKUDBCILJUJU
    MCUEUDUFUDAMNROOPSUDACTUA $.

  $( All variables are effectively bound in an identical variable specifier.
     (Contributed by Mario Carneiro, 11-Aug-2016.) $)
  nfae $p |- F/ z A. x x = y $=
    ( weq wal hbae nfi ) ABDAECABCFG $.

  ${
    hbalequs.1 $e |- ( A. z A. x x = y -> ph ) $.
    $( Rule that applies ~ hbae to antecedent.  (Contributed by NM,
       5-Aug-1993.) $)
    hbaes $p |- ( A. x x = y -> ph ) $=
      ( weq wal hbae syl ) BCFBGZJDGABCDHEI $.
  $}

  $( All variables are effectively bound in a distinct variable specifier.
     Lemma L19 in [Megill] p. 446 (p. 14 of the preprint).  (Contributed by NM,
     5-Aug-1993.) $)
  hbnae $p |- ( -. A. x x = y -> A. z -. A. x x = y ) $=
    ( weq wal hbae hbn ) ABDAECABCFG $.

  $( All variables are effectively bound in a distinct variable specifier.
     (Contributed by Mario Carneiro, 11-Aug-2016.) $)
  nfnae $p |- F/ z -. A. x x = y $=
    ( weq wal nfae nfn ) ABDAECABCFG $.

  ${
    hbnalequs.1 $e |- ( A. z -. A. x x = y -> ph ) $.
    $( Rule that applies ~ hbnae to antecedent.  (Contributed by NM,
       5-Aug-1993.) $)
    hbnaes $p |- ( -. A. x x = y -> ph ) $=
      ( weq wal wn hbnae syl ) BCFBGHZKDGABCDIEJ $.
  $}

  ${
    naecoms.1 $e |- ( -. A. x x = y -> ph ) $.
    $( A commutation rule for distinct variable specifiers.  (Contributed by
       NM, 2-Jan-2002.) $)
    naecoms $p |- ( -. A. y y = x -> ph ) $=
      ( cv wceq wal wn ax-10 con3i syl ) CEZBEZFCGZHMLFBGZHAONBCIJDK $.
  $}

  $( Lemma used in proofs of substitution properties.  (Contributed by NM,
     5-Aug-1993.)  (Proof shortened by Mario Carneiro, 20-May-2014.) $)
  equs4 $p |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) ) $=
    ( cv wceq wi wal wa wex a9e 19.29 mpan2 ancl imp eximi syl ) BDCDEZAFZBGZRQ
    HZBIZQAHZBISQBIUABCJRQBKLTUBBRQUBQAMNOP $.

  ${
    equsalh.1 $e |- ( ps -> A. x ps ) $.
    equsalh.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( A useful equivalence related to substitution.  New proofs should use
       ~ equsal instead.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by
       Andrew Salmon, 12-Aug-2011.)  (New usage is discouraged.) $)
    equsalh $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
      ( weq wi wal 19.3h bitr4di pm5.74i albii a1d alrimih ax9o impbii bitr4i )
      CDGZAHZCISBCIZHZCIZBTUBCSAUASABUAFBCEJKLMBUCBUBCEBUASENOBCDPQR $.
  $}

  ${
    equsal.1 $e |- F/ x ps $.
    equsal.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( A useful equivalence related to substitution.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Andrew Salmon, 12-Aug-2011.)  (Revised
       by Mario Carneiro, 3-Oct-2016.)  (Proof shortened by Wolf Lammen,
       5-Feb-2018.) $)
    equsal $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
      ( weq wi wal wex 19.23 pm5.74i albii a9e a1bi 3bitr4i ) CDGZBHZCIQCJZBHQA
      HZCIBQBCEKTRCQABFLMSBCDNOP $.
  $}

  ${
    equsex.1 $e |- ( ps -> A. x ps ) $.
    equsex.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( A useful equivalence related to substitution.  (Contributed by NM,
       5-Aug-1993.)  (Revised by NM, 3-Feb-2015.) $)
    equsex $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( cv wceq wa wex biimpa exlimih a9e idd biimprcd jcad eximdh mpi impbii )
      CGDGHZAIZCJZBUABCETABFKLBTCJUBCDMBTUACEBTTABTNTABFOPQRS $.
  $}

  ${
    $d x y $.  $d x ps $.
    equsalvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ equsex with two disjoint variable conditions.  (Contributed
       by BJ, 31-May-2019.)  (Proof shortened by Wolf Lammen, 23-Oct-2023.) $)
    equsexvw $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( ax-17 equsex ) ABCDBCFEG $.
  $}

  ${
    equsexd.1 $e |- ( ph -> A. x ph ) $.
    equsexd.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    equsexd.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Deduction form of ~ equsex .  (Contributed by Jim Kingdon,
       29-Dec-2017.) $)
    equsexd $p |- ( ph -> ( E. x ( x = y /\ ps ) <-> ch ) ) $=
      ( cv wceq wa wex wb wi biimp imim2i 3syl wal a1i imp pm3.31 exlimd2 19.26
      a9e jca anim12 syl imbitrrdi anabsi5 idd biimpr pm2.04 jcad eximdh mpi ex
      impbid ) ADIEIJZBKZDLZCAUSCDFGAURBCMZNZURBCNZNUSCNHVAVCURBCOPURBCUAQUBACU
      TACKZURDLUTDEUDVDURUSDACVDDRZAVDADRZCDRZKZVEAAVFNZCVGNZKVDVHNAVIVJVIAFSGU
      EAVFCVGUFUGACDUCUHUIVDURURBACURURNZCVKNACURUJSTACURBNZAVBURCBNZNCVLNHVAVM
      URBCUKPURCBULQTUMUNUOUPUQ $.
  $}

  ${
    dral1.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
       (Contributed by NM, 24-Nov-1994.) $)
    dral1 $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $=
      ( weq wal hbae biimpd alimdh ax10o syld biimprd wi alequcoms impbid ) CDF
      CGZACGZBDGZQRBCGSQABCCDCHQABEIJBCDKLQSADGZRQBADCDDHQABEMJTRNDCADCKOLP $.
  $}

  ${
    dral2.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
       (Contributed by NM, 27-Feb-2005.) $)
    dral2 $p |- ( A. x x = y -> ( A. z ph <-> A. z ps ) ) $=
      ( weq wal hbae albidh ) CDGCHABECDEIFJ $.
  $}

  ${
    drex2.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
       (Contributed by NM, 27-Feb-2005.) $)
    drex2 $p |- ( A. x x = y -> ( E. z ph <-> E. z ps ) ) $=
      ( weq wal hbae exbidh ) CDGCHABECDEIFJ $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       (Contributed by Mario Carneiro, 4-Oct-2016.) $)
    drnf1 $p |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) $=
      ( weq wal wi wnf dral1 imbi12d df-nf 3bitr4g ) CDFCGZAACGZHZCGBBDGZHZDGAC
      IBDIPRCDNABOQEABCDEJKJACLBDLM $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       (Contributed by Mario Carneiro, 4-Oct-2016.) $)
    drnf2 $p |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) ) $=
      ( weq wal wi wnf dral2 imbi12d df-nf 3bitr4g ) CDGCHZAAEHZIZEHBBEHZIZEHAE
      JBEJQSCDEOABPRFABCDEFKLKAEMBEMN $.
  $}

  $( Closed theorem form of ~ spim .  (Contributed by NM, 15-Jan-2008.)
     (New usage is discouraged.) $)
  spimth $p |- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) ->
              ( A. x ph -> ps ) ) $=
    ( wal wi weq wa imim2 imim2d imp com23 al2imi ax9o syl6 ) BBCEZFZCDGZABFZFZ
    HZCEACERPFZCEBUAAUBCUARAPQTRAPFZFQSUCRBPAIJKLMBCDNO $.

  $( Closed theorem form of ~ spim .  (Contributed by NM, 15-Jan-2008.)
     (Revised by Mario Carneiro, 17-Oct-2016.)  (Proof shortened by Wolf
     Lammen, 24-Feb-2018.) $)
  spimt $p |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) ->
              ( A. x ph -> ps ) ) $=
    ( cv wceq wi wal wex wnf a9e exim mpi 19.35-1 syl 19.9t biimpd sylan9r ) CE
    DEFZABGZGCHZACHZBCIZBCJZBUATCIZUBUCGUASCIUECDKSTCLMABCNOUDUCBBCPQR $.

  ${
    spimh.1 $e |- ( ps -> A. x ps ) $.
    spimh.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Specialization, using implicit substitition.  Compare Lemma 14 of
       [Tarski] p. 70.  The ~ spim series of theorems requires that only one
       direction of the substitution hypothesis hold.  (Contributed by NM,
       5-Aug-1993.)  (Revised by NM, 8-May-2008.)
       (New usage is discouraged.) $)
    spimh $p |- ( A. x ph -> ps ) $=
      ( wal weq wi syl6com alimi ax9o syl ) ACGCDHZBCGZIZCGBAPCNABOFEJKBCDLM $.
  $}

  ${
    spim.1 $e |- F/ x ps $.
    spim.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Specialization, using implicit substitution.  Compare Lemma 14 of
       [Tarski] p. 70.  The ~ spim series of theorems requires that only one
       direction of the substitution hypothesis hold.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Mario Carneiro, 3-Oct-2016.)  (Proof rewritten
       by Jim Kingdon, 10-Jun-2018.) $)
    spim $p |- ( A. x ph -> ps ) $=
      ( nfri spimh ) ABCDBCEGFH $.
  $}

  ${
    spimeh.1 $e |- ( ph -> A. x ph ) $.
    spimeh.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Existential introduction, using implicit substitition.  Compare Lemma 14
       of [Tarski] p. 70.  (Contributed by NM, 7-Aug-1994.)  (Revised by NM,
       3-Feb-2015.)  (New usage is discouraged.) $)
    spimeh $p |- ( ph -> E. x ps ) $=
      ( cv wceq wex a9e com12 eximdh mpi ) ACGDGHZCIBCICDJANBCENABFKLM $.
  $}

  ${
    spimed.1 $e |- ( ch -> F/ x ph ) $.
    spimed.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Deduction version of ~ spime .  (Contributed by NM, 5-Aug-1993.)
       (Revised by Mario Carneiro, 3-Oct-2016.)  (Proof shortened by Wolf
       Lammen, 19-Feb-2018.) $)
    spimed $p |- ( ch -> ( ph -> E. x ps ) ) $=
      ( wal wex nfrd weq wi a9e eximii 19.35i syl6 ) CAADHBDICADFJABDDEKABLDDEM
      GNOP $.
  $}

  ${
    spime.1 $e |- F/ x ph $.
    spime.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Existential introduction, using implicit substitution.  Compare Lemma 14
       of [Tarski] p. 70.  (Contributed by NM, 7-Aug-1994.)  (Revised by Mario
       Carneiro, 3-Oct-2016.)  (Proof shortened by Wolf Lammen, 6-Mar-2018.) $)
    spime $p |- ( ph -> E. x ps ) $=
      ( wex wi wtru wnf a1i spimed mptru ) ABCGHABICDACJIEKFLM $.
  $}

  ${
    cbv3.1 $e |- F/ y ph $.
    cbv3.2 $e |- F/ x ps $.
    cbv3.3 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because proofs are encouraged to use the
       weaker ~ cbv3v if possible.  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Wolf Lammen, 12-May-2018.)  (New usage is discouraged.) $)
    cbv3 $p |- ( A. x ph -> A. y ps ) $=
      ( wal nfal spim alrimi ) ACHBDADCEIABCDFGJK $.
  $}

  ${
    cbv3h.1 $e |- ( ph -> A. y ph ) $.
    cbv3h.2 $e |- ( ps -> A. x ps ) $.
    cbv3h.3 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       25-May-2011.)  (Proof shortened by Wolf Lammen, 12-May-2018.) $)
    cbv3h $p |- ( A. x ph -> A. y ps ) $=
      ( nfi cbv3 ) ABCDADEHBCFHGI $.
  $}

  ${
    $d x y $.
    cbv3v.nf1 $e |- F/ y ph $.
    cbv3v.nf2 $e |- F/ x ps $.
    cbv3v.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbv3 with a disjoint variable condition.  (Contributed by
       NM, 5-Aug-1993.)  (Revised by BJ, 31-May-2019.) $)
    cbv3v $p |- ( A. x ph -> A. y ps ) $=
      ( cbv3 ) ABCDEFGH $.
  $}

  ${
    cbv1.1 $e |- F/ x ph $.
    cbv1.2 $e |- F/ y ph $.
    cbv1.3 $e |- ( ph -> F/ y ps ) $.
    cbv1.4 $e |- ( ph -> F/ x ch ) $.
    cbv1.5 $e |- ( ph -> ( x = y -> ( ps -> ch ) ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Revised to format hypotheses to common style.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Mario Carneiro, 3-Oct-2016.)  (Revised by Wolf
       Lammen, 13-May-2018.) $)
    cbv1 $p |- ( ph -> ( A. x ps -> A. y ch ) ) $=
      ( wal wi nfim1 weq com12 a2d cbv3 19.21 3imtr3i pm2.86i ) ABDKZCEKZABLZDK
      ACLZEKAUALAUBLUCUDDEABEGHMACDFIMDENZABCAUEBCLJOPQABDFRACEGRST $.
  $}

  ${
    cbv1h.1 $e |- ( ph -> ( ps -> A. y ps ) ) $.
    cbv1h.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    cbv1h.3 $e |- ( ph -> ( x = y -> ( ps -> ch ) ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen,
       13-May-2018.) $)
    cbv1h $p |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) ) $=
      ( wal nfa1 nfa2 wi sp sps syl nfd weq cbv1 ) AEIZDIZBCDESDJZAEDKZTBEUBTAB
      BEILSADAEMNZFOPTCDUATACCDILUCGOPTADEQBCLLUCHOR $.
  $}

  ${
    $d x y $.
    cbv1v.1 $e |- F/ x ph $.
    cbv1v.2 $e |- F/ y ph $.
    cbv1v.3 $e |- ( ph -> F/ y ps ) $.
    cbv1v.4 $e |- ( ph -> F/ x ch ) $.
    cbv1v.5 $e |- ( ph -> ( x = y -> ( ps -> ch ) ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 5-Aug-1993.)  (Revised by BJ, 16-Jun-2019.) $)
    cbv1v $p |- ( ph -> ( A. x ps -> A. y ch ) ) $=
      ( wal wi nfim1 weq com12 a2d cbv3v 19.21 3imtr3i pm2.86i ) ABDKZCEKZABLZD
      KACLZEKAUALAUBLUCUDDEABEGHMACDFIMDENZABCAUEBCLJOPQABDFRACEGRST $.
  $}

  ${
    cbv2h.1 $e |- ( ph -> ( ps -> A. y ps ) ) $.
    cbv2h.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    cbv2h.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 5-Aug-1993.) $)
    cbv2h $p |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) ) $=
      ( wal weq wb wi biimp syl6 cbv1h equcomi biimpr syl56 a7s impbid ) AEIDIB
      DIZCEIZABCDEFGADEJZBCKZBCLHBCMNOAUBUALEDACBEDGFEDJUCAUDCBLEDPHBCQROST $.
  $}

  ${
    cbv2.1 $e |- F/ x ph $.
    cbv2.2 $e |- F/ y ph $.
    cbv2.3 $e |- ( ph -> F/ y ps ) $.
    cbv2.4 $e |- ( ph -> F/ x ch ) $.
    cbv2.5 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Revised to align format of hypotheses to common style.  (Contributed by
       NM, 5-Aug-1993.)  (Revised by Mario Carneiro, 3-Oct-2016.)  (Revised by
       Wolf Lammen, 13-May-2018.) $)
    cbv2 $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
      ( wal wb nfri nfal syl nfrd cbv2h ) AAEKZDKZBDKCEKLARSAEGMRDADEFNMOABCDEA
      BEHPACDIPJQO $.
  $}

  ${
    $d x y $.
    cbv2w.1 $e |- F/ x ph $.
    cbv2w.2 $e |- F/ y ph $.
    cbv2w.3 $e |- ( ph -> F/ y ps ) $.
    cbv2w.4 $e |- ( ph -> F/ x ch ) $.
    cbv2w.5 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbv2 with a disjoint variable condition.  (Contributed by
       NM, 5-Aug-1993.)  (Revised by GG, 10-Jan-2024.) $)
    cbv2w $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
      ( wal weq wb wi biimp syl6 cbv1v equcomi biimpr syl56 impbid ) ABDKCEKABC
      DEFGHIADELZBCMZBCNJBCOPQACBEDGFIHEDLUBAUCCBNEDRJBCSTQUA $.
  $}

  ${
    $d x y $.
    cbvalv1.nf1 $e |- F/ y ph $.
    cbvalv1.nf2 $e |- F/ x ps $.
    cbvalv1.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbval with a disjoint variable condition.  See ~ cbvalvw
       for a version with two disjoint variable conditions, and ~ cbvalv for
       another variant.  (Contributed by NM, 13-May-1993.)  (Revised by BJ,
       31-May-2019.) $)
    cbvalv1 $p |- ( A. x ph <-> A. y ps ) $=
      ( wal weq biimpd cbv3v wi biimprd equcoms impbii ) ACHBDHABCDEFCDIZABGJKB
      ADCFEBALCDPABGMNKO $.
    $( $j usage 'cbvalv1' avoids 'ax-io' 'ax-10' 'ax-i12' 'ax-bndl'; $)

    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvex with a disjoint variable condition.  See ~ cbvexvw
       for a version with two disjoint variable conditions, and ~ cbvexv for
       another variant.  (Contributed by NM, 21-Jun-1993.)  (Revised by BJ,
       31-May-2019.) $)
    cbvexv1 $p |- ( E. x ph <-> E. y ps ) $=
      ( wex nfex weq wa nfri bicomd equcoms equsex exsimpr sylbir exlimi impbii
      wb ) ACHZBDHZAUBCBCDFIADCJZBKDHUBBADCADELBATCDCDJZABGMNOUCBDPQRBUADADCEIB
      UDAKCHUAABCDBCFLGOUDACPQRS $.
    $( $j usage 'cbvexv1' avoids 'ax-io' 'ax-10' 'ax-i12' 'ax-bndl'; $)
  $}

  ${
    cbvalh.1 $e |- ( ph -> A. y ph ) $.
    cbvalh.2 $e |- ( ps -> A. x ps ) $.
    cbvalh.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitition.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       25-May-2011.) $)
    cbvalh $p |- ( A. x ph <-> A. y ps ) $=
      ( wal weq biimpd cbv3h wb equcoms biimprd impbii ) ACHBDHABCDEFCDIABGJKBA
      DCFEDCIABABLCDGMNKO $.
  $}

  ${
    cbval.1 $e |- F/ y ph $.
    cbval.2 $e |- F/ x ps $.
    cbval.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       3-Oct-2016.) $)
    cbval $p |- ( A. x ph <-> A. y ps ) $=
      ( nfri cbvalh ) ABCDADEHBCFHGI $.
  $}

  ${
    cbvexh.1 $e |- ( ph -> A. y ph ) $.
    cbvexh.2 $e |- ( ps -> A. x ps ) $.
    cbvexh.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitition.
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       3-Feb-2015.) $)
    cbvexh $p |- ( E. x ph <-> E. y ps ) $=
      ( wex hbex cv wceq wa wb bicomd equcoms equsex simpr eximi sylbir exlimih
      impbii ) ACHZBDHZAUCCBCDFIADJZCJZKZBLZDHUCBADCEBAMCDUEUDKZABGNOPUGBDUFBQR
      STBUBDADCEIBUHALZCHUBABCDFGPUIACUHAQRSTUA $.
  $}

  ${
    cbvex.1 $e |- F/ y ph $.
    cbvex.2 $e |- F/ x ps $.
    cbvex.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 5-Aug-1993.) $)
    cbvex $p |- ( E. x ph <-> E. y ps ) $=
      ( nfri cbvexh ) ABCDADEHBCFHGI $.
  $}

  ${
    chvar.1 $e |- F/ x ps $.
    chvar.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    chvar.3 $e |- ph $.
    $( Implicit substitution of ` y ` for ` x ` into a theorem.  (Contributed
       by Raph Levien, 9-Jul-2003.)  (Revised by Mario Carneiro,
       3-Oct-2016.) $)
    chvar $p |- ps $=
      ( weq biimpd spim mpg ) ABCABCDECDHABFIJGK $.
  $}

  $( A variable introduction law for equality.  Lemma 15 of [Monk2] p. 109,
     however we do not require ` z ` to be distinct from ` x ` and ` y `
     (making the proof longer).  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Andrew Salmon, 25-May-2011.) $)
  equvini $p |- ( x = y -> E. z ( x = z /\ z = y ) ) $=
    ( cv wceq wal wi wo wex ax12or equcomi alimi a9e jctir a1d 19.29 syl6 ax-mp
    wa jaoi eximi 2a1i anc2ri 19.29r ax-8 anc2li equcoms com12 exim syl mpi sps
    imim2i ) CDZADZEZCFZUNBDZEZCFZUOUREZVACFZGZCFZHZHVAUOUNEZUSSZCIZGZABCJUQVIV
    EUQVAVFCFZUSCIZSZVHUQVLVAUQVJVKUPVFCCAKZLCBMNOVFUSCPQUTVIVDUTVAVFCIZUTSVHUT
    VAVNUTVAVNUPCIZVNCAMZUPVFCVMUARUBUCVFUSCUDQVCVICVBVHVAVBVOVHVPVBUPVGGZCFVOV
    HGVAVQCUPVAVGVAVGGACVFVAUSACBUEUFUGUHLUPVGCUIUJUKUMULTTR $.

  $( A variable elimination law for equality with no distinct variable
     requirements.  (Compare ~ equvini .)  (Contributed by NM, 1-Mar-2013.)
     (Revised by NM, 3-Feb-2015.) $)
  equveli $p |- ( A. z ( z = x <-> z = y ) -> x = y ) $=
    ( cv wceq wb wal wi wa albiim ax12or equequ1 imbi12d equid biimtrdi adantrd
    wo sps ax-4 jaoi dral2 a1bi biimpri dral1 mpi equcomi adantld hbequid hbimd
    syl hba1 a1i a5i equtr ax-8 imim12d ax-gen 19.26 spimth sylbir sylancl mpii
    ax-mp sylbi ) CDZADZEZVEBDZEZFCGVGVIHZCGZVIVGHZCGZIZVFVHEZVGVICJVGCGZVICGZV
    OVOCGHZCGZQZQVNVOHZABCKVPWAVTVPVKVOVMVPVKVFVFEZVOHZCGZVOVJWCCACVGVJWCFCVGVG
    WBVIVOCAALCABLMRUAWCVOCVOWCWBVOANZUBUCROPVQWAVSVQVMVOVKVQVMVHVHEZVHVFEZHZBG
    ZVOVLWHCBVIVLWHFCVIVIWFVGWGCBBLCBALMRUDWIWGVOWIWFWGBNWHBSUEBAUFUJOUGVSVKVOV
    MVSVKWBVOWEVSWCWDHZCGZVGVJWCHHZCGZVKWCHZVRWJCVSWBVOCVRCUKWBWBCGHVSACUHULVRC
    SUIUMWLCVGWBVGVIVOCAAUNCABUOUPUQWKWMIWJWLICGWNWJWLCURVJWCCAUSUTVAVBPTTVCVD
    $.

  ${
    nfald.1 $e |- F/ y ph $.
    nfald.2 $e |- ( ph -> F/ x ps ) $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
       (Contributed by Mario Carneiro, 24-Sep-2016.)  (Proof shortened by Wolf
       Lammen, 6-Jan-2018.) $)
    nfald $p |- ( ph -> F/ x A. y ps ) $=
      ( wnf wal nfri alrimih nfnf1 nfal hba1 sp nfrd hbald nfd syl ) ABCGZDHZBD
      HZCGASDADEIFJTUACSCDBCKLTBCDSDMTBCSDNOPQR $.

    $( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
       (Contributed by Mario Carneiro, 24-Sep-2016.)  (Proof rewritten by Jim
       Kingdon, 7-Feb-2018.) $)
    nfexd $p |- ( ph -> F/ x E. y ps ) $=
      ( wex wal wnf nfri df-nf sylib alrimih alcom exim alimi syl 19.12 imim2i
      wi sylibr ) ABDGZUBCHZTZCHZUBCIAUBBCHZDGZTZCHZUEABUFTZDHZCHZUIAUJCHZDHULA
      UMDADEJABCIUMFBCKLMUJDCNLUKUHCBUFDOPQUHUDCUGUCUBBDCRSPQUBCKUA $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Substitution (without distinct variables)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c [ $. $( Left bracket $)
  $c / $. $( Division. $)
  $c ] $.  $( Right bracket $)

  $( Extend wff definition to include proper substitution (read "the wff that
     results when ` y ` is properly substituted for ` x ` in wff ` ph ` ").
     (Contributed by NM, 24-Jan-2006.) $)
  wsb $a wff [ y / x ] ph $.

  $( Define proper substitution.  Remark 9.1 in [Megill] p. 447 (p. 15 of the
     preprint).  For our notation, we use ` [ y / x ] ph ` to mean "the wff
     that results when ` y ` is properly substituted for ` x ` in the wff
     ` ph ` ".  We can also use ` [ y / x ] ph ` in place of the "free for"
     side condition used in traditional predicate calculus; see, for example,
     ~ stdpc4 .

     Our notation was introduced in Haskell B. Curry's _Foundations of
     Mathematical Logic_ (1977), p. 316 and is frequently used in textbooks of
     lambda calculus and combinatory logic.  This notation improves the common
     but ambiguous notation, " ` ph ( y ) ` is the wff that results when ` y `
     is properly substituted for ` x ` in ` ph ( x ) ` ".  For example, if the
     original ` ph ( x ) ` is ` x = y ` , then ` ph ( y ) ` is ` y = y ` , from
     which we obtain that ` ph ( x ) ` is ` x = x ` .  So what exactly does
     ` ph ( x ) ` mean?  Curry's notation solves this problem.

     In most books, proper substitution has a somewhat complicated recursive
     definition with multiple cases based on the occurrences of free and bound
     variables in the wff.  Instead, we use a single formula that is exactly
     equivalent and gives us a direct definition.  We later prove that our
     definition has the properties we expect of proper substitution (see
     Theorems ~ sbequ , ~ sbcom2 and ~ sbid2v ).

     Note that our definition is valid even when ` x ` and ` y ` are replaced
     with the same variable, as ~ sbid shows.  We achieve this by having ` x `
     free in the first conjunct and bound in the second.  We can also achieve
     this by using a dummy variable, as the alternate definition ~ dfsb7 shows
     (which some logicians may prefer because it doesn't mix free and bound
     variables).  Another alternate definition which uses a dummy variable is
     ~ dfsb7a .

     When ` x ` and ` y ` are distinct, we can express proper substitution with
     the simpler expressions of ~ sb5 and ~ sb6 .

     In classical logic, another possible definition is
     ` ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ` but we do not have an
     intuitionistic proof that this is equivalent.

     There are no restrictions on any of the variables, including what
     variables may occur in wff ` ph ` .  (Contributed by NM, 5-Aug-1993.) $)
  df-sb $a |- ( [ y / x ] ph <->
              ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) $.

  ${
    sbimi.1 $e |- ( ph -> ps ) $.
    $( Infer substitution into antecedent and consequent of an implication.
       (Contributed by NM, 25-Jun-1998.) $)
    sbimi $p |- ( [ y / x ] ph -> [ y / x ] ps ) $=
      ( weq wi wa wex wsb imim2i anim2i eximi anim12i df-sb 3imtr4i ) CDFZAGZQA
      HZCIZHQBGZQBHZCIZHACDJBCDJRUATUCABQEKSUBCABQELMNACDOBCDOP $.
  $}

  ${
    sbbii.1 $e |- ( ph <-> ps ) $.
    $( Infer substitution into both sides of a logical equivalence.
       (Contributed by NM, 5-Aug-1993.) $)
    sbbii $p |- ( [ y / x ] ph <-> [ y / x ] ps ) $=
      ( wsb biimpi sbimi biimpri impbii ) ACDFBCDFABCDABEGHBACDABEIHJ $.
  $}

  $( One direction of a simplified definition of substitution.  (Contributed by
     NM, 5-Aug-1993.) $)
  sb1 $p |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) $=
    ( wsb weq wi wa wex df-sb simprbi ) ABCDBCEZAFKAGBHABCIJ $.

  $( One direction of a simplified definition of substitution.  (Contributed by
     NM, 5-Aug-1993.) $)
  sb2 $p |- ( A. x ( x = y -> ph ) -> [ y / x ] ph ) $=
    ( weq wi wal wa wex wsb ax-4 equs4 df-sb sylanbrc ) BCDZAEZBFONAGBHABCIOBJA
    BCKABCLM $.

  $( An equality theorem for substitution.  (Contributed by NM, 5-Aug-1993.) $)
  sbequ1 $p |- ( x = y -> ( ph -> [ y / x ] ph ) ) $=
    ( weq wsb wa wi wex pm3.4 19.8a df-sb sylanbrc ex ) BCDZAABCEZNAFZNAGPBHONA
    IPBJABCKLM $.

  $( An equality theorem for substitution.  (Contributed by NM, 5-Aug-1993.) $)
  sbequ2 $p |- ( x = y -> ( [ y / x ] ph -> ph ) ) $=
    ( wsb weq wi wa wex df-sb simpl com12 biimtrid ) ABCDBCEZAFZMAGBHZGZMAABCIP
    MANOJKL $.

  $( One of the two equality axioms of standard predicate calculus, called
     substitutivity of equality.  (The other one is ~ stdpc6 .)  Translated to
     traditional notation, it can be read:  " ` x = y -> ( ph ( x ` ,
     ` x ) -> ph ( x ` , ` y ) ) ` , provided that ` y ` is free for ` x ` in
     ` ph ( x ` , ` y ) ` ".  Axiom 7 of [Mendelson] p. 95.  (Contributed by
     NM, 15-Feb-2005.) $)
  stdpc7 $p |- ( x = y -> ( [ x / y ] ph -> ph ) ) $=
    ( wsb wi sbequ2 equcoms ) ACBDAECBACBFG $.

  $( An equality theorem for substitution.  (Contributed by NM, 5-Aug-1993.) $)
  sbequ12 $p |- ( x = y -> ( ph <-> [ y / x ] ph ) ) $=
    ( weq wsb sbequ1 sbequ2 impbid ) BCDAABCEABCFABCGH $.

  $( An equality theorem for substitution.  (Contributed by NM, 6-Oct-2004.)
     (Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
  sbequ12r $p |- ( x = y -> ( [ x / y ] ph <-> ph ) ) $=
    ( wsb wb weq sbequ12 bicomd equcoms ) ACBDZAECBCBFAJACBGHI $.

  $( An equality theorem for substitution.  (Contributed by NM, 5-Aug-1993.) $)
  sbequ12a $p |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) ) $=
    ( weq wsb sbequ12 wb equcoms bitr3d ) BCDAABCEACBEZABCFAJGCBACBFHI $.

  $( An identity theorem for substitution.  Remark 9.1 in [Megill] p. 447 (p.
     15 of the preprint).  (Contributed by NM, 5-Aug-1993.) $)
  sbid $p |- ( [ x / x ] ph <-> ph ) $=
    ( wsb weq wb equid sbequ12 ax-mp bicomi ) AABBCZBBDAJEBFABBGHI $.

  $( The specialization axiom of standard predicate calculus.  It states that
     if a statement ` ph ` holds for all ` x ` , then it also holds for the
     specific case of ` y ` (properly) substituted for ` x ` .  Translated to
     traditional notation, it can be read:  " ` A. x ph ( x ) -> ph ( y ) ` ,
     provided that ` y ` is free for ` x ` in ` ph ( x ) ` ".  Axiom 4 of
     [Mendelson] p. 69.  (Contributed by NM, 5-Aug-1993.) $)
  stdpc4 $p |- ( A. x ph -> [ y / x ] ph ) $=
    ( wal weq wi wsb ax-1 alimi sb2 syl ) ABDBCEZAFZBDABCGAMBALHIABCJK $.

  ${
    sbh.1 $e |- ( ph -> A. x ph ) $.
    $( Substitution for a variable not free in a wff does not affect it.
       (Contributed by NM, 5-Aug-1993.)  (Revised by NM, 17-Oct-2004.) $)
    sbh $p |- ( [ y / x ] ph <-> ph ) $=
      ( wsb weq wex wa sb1 19.41h sylib simprd wal stdpc4 syl impbii ) ABCEZAQB
      CFZBGZAQRAHBGSAHABCIRABDJKLAABMQDABCNOP $.
  $}

  ${
    sbf.1 $e |- F/ x ph $.
    $( Substitution for a variable not free in a wff does not affect it.
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       4-Oct-2016.) $)
    sbf $p |- ( [ y / x ] ph <-> ph ) $=
      ( nfri sbh ) ABCABDEF $.
  $}

  $( Substitution has no effect on a bound variable.  (Contributed by NM,
     1-Jul-2005.) $)
  sbf2 $p |- ( [ y / x ] A. x ph <-> A. x ph ) $=
    ( wal nfa1 sbf ) ABDBCABEF $.

  ${
    sb6x.1 $e |- ( ph -> A. x ph ) $.
    $( Equivalence involving substitution for a variable not free.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       12-Aug-2011.) $)
    sb6x $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
      ( wsb weq wi wal sbh biidd equsalh bitr4i ) ABCEABCFZAGBHABCDIAABCDMAJKL
      $.
  $}

  ${
    nfs1f.1 $e |- F/ x ph $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nfs1f $p |- F/ x [ y / x ] ph $=
      ( wsb nfri sbh nfxfr ) ABCEABABCABDFGDH $.
  $}

  ${
    hbs1f.1 $e |- ( ph -> A. x ph ) $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` .
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       25-May-2011.) $)
    hbs1f $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
      ( wsb sbh hbxfrbi ) ABCEABABCDFDG $.
  $}

  $( Substitution does not change an identical variable specifier.
     (Contributed by NM, 5-Aug-1993.)  (Revised by NM, 21-Dec-2004.) $)
  sbequ5 $p |- ( [ w / z ] A. x x = y <-> A. x x = y ) $=
    ( weq wal nfae sbf ) ABEAFCDABCGH $.

  $( Substitution does not change a distinctor.  (Contributed by NM,
     5-Aug-1993.)  (Revised by NM, 14-May-2005.) $)
  sbequ6 $p |- ( [ w / z ] -. A. x x = y <-> -. A. x x = y ) $=
    ( weq wal wn nfnae sbf ) ABEAFGCDABCHI $.

  ${
    sbt.1 $e |- ph $.
    $( A substitution into a theorem remains true.  (See ~ chvar and ~ chvarv
       for versions using implicit substitition.)  (Contributed by NM,
       21-Jan-2004.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    sbt $p |- [ y / x ] ph $=
      ( wsb nfth sbf mpbir ) ABCEADABCABDFGH $.
  $}

  $( Substitution applied to an atomic wff.  (Contributed by NM,
     5-Aug-1993.) $)
  equsb1 $p |- [ y / x ] x = y $=
    ( weq wi wsb sb2 id mpg ) ABCZIDIABEAIABFIGH $.

  $( Substitution applied to an atomic wff.  (Contributed by NM,
     5-Aug-1993.) $)
  equsb2 $p |- [ y / x ] y = x $=
    ( weq wi wsb sb2 equcomi mpg ) ABCBACZDIABEAIABFABGH $.

  ${
    sbiedh.1 $e |- ( ph -> A. x ph ) $.
    sbiedh.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
    sbiedh.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Conversion of implicit substitution to explicit substitution (deduction
       version of ~ sbieh ).  New proofs should use ~ sbied instead.
       (Contributed by NM, 30-Jun-1994.)  (Proof shortened by Andrew Salmon,
       25-May-2011.)  (New usage is discouraged.) $)
    sbiedh $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
      ( wsb wex weq wa sb1 wb wi biimp syl6 impd syld wal eximdh syl5 com23 sb2
      19.9hd biimpr alimdh impbid ) ABDEIZCAUICDJZCUIDEKZBLZDJAUJBDEMAULCDFAUKB
      CAUKBCNZBCOHBCPQRUAUBCADFGUESACCDTZUIGAUNUKBOZDTUIACUODFAUKCBAUKUMCBOHBCU
      FQUCUGBDEUDQSUH $.
  $}

  ${
    sbied.1 $e |- F/ x ph $.
    sbied.2 $e |- ( ph -> F/ x ch ) $.
    sbied.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Conversion of implicit substitution to explicit substitution (deduction
       version of ~ sbie ).  (Contributed by NM, 30-Jun-1994.)  (Revised by
       Mario Carneiro, 4-Oct-2016.) $)
    sbied $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
      ( nfri nfrd sbiedh ) ABCDEADFIACDGJHK $.
  $}

  ${
    $d x ph $.  $d x ch $.
    sbiedv.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
    $( Conversion of implicit substitution to explicit substitution (deduction
       version of ~ sbie ).  (Contributed by NM, 7-Jan-2017.) $)
    sbiedv $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
      ( nfv nfvd weq wb ex sbied ) ABCDEADGACDHADEIBCJFKL $.
  $}

  ${
    sbieh.1 $e |- ( ps -> A. x ps ) $.
    sbieh.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit substitution.  New
       proofs should use ~ sbie instead.  (Contributed by NM, 30-Jun-1994.)
       (New usage is discouraged.) $)
    sbieh $p |- ( [ y / x ] ph <-> ps ) $=
      ( wi wsb wb id hbth wal a1i weq sbiedh ax-mp ) AAGZACDHBIAJZQABCDQCRKBBCL
      GQEMCDNABIGQFMOP $.
  $}

  ${
    sbie.1 $e |- F/ x ps $.
    sbie.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit substitution.
       (Contributed by NM, 30-Jun-1994.)  (Revised by Mario Carneiro,
       4-Oct-2016.)  (Revised by Wolf Lammen, 30-Apr-2018.) $)
    sbie $p |- ( [ y / x ] ph <-> ps ) $=
      ( nfri sbieh ) ABCDBCEGFH $.
  $}

  ${
    $d x y $.
    sbiev.1 $e |- F/ x ps $.
    sbiev.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit substitution.  Version
       of ~ sbie with a disjoint variable condition.  (Contributed by Wolf
       Lammen, 18-Jan-2023.) $)
    sbiev $p |- ( [ y / x ] ph <-> ps ) $=
      ( sbie ) ABCDEFG $.
  $}

  ${
    $d x y $.
    equsalv.nf $e |- F/ x ps $.
    equsalv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( An equivalence related to implicit substitution.  Version of ~ equsal
       with a disjoint variable condition.  (Contributed by NM, 2-Jun-1993.)
       (Revised by BJ, 31-May-2019.) $)
    equsalv $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
      ( weq wi wal wex 19.23 pm5.74i albii a9ev a1bi 3bitr4i ) CDGZBHZCIQCJZBHQ
      AHZCIBQBCEKTRCQABFLMSBCDNOP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Theorems using axiom ax-11
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( A property related to substitution that unlike ~ equs5 doesn't require a
     distinctor antecedent.  (Contributed by NM, 2-Feb-2007.) $)
  equs5a $p |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) ) $=
    ( weq wal wa wi hba1 ax-11 imp exlimih ) BCDZACEZFLAGZBEZBNBHLMOABCIJK $.

  $( A property related to substitution that unlike ~ equs5 doesn't require a
     distinctor antecedent.  (Contributed by NM, 2-Feb-2007.)  (Revised by NM,
     3-Feb-2015.) $)
  equs5e $p |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) ) $=
    ( cv wceq wa wex wal wi 19.8a hbe1 syl anim2i eximi equs5a ) BDCDEZAFZBGPAC
    GZCHZFZBGPRIBHQTBASPARSACJACKLMNRBCOL $.

  $( Analogue to ~ ax-11 but for existential quantification.  (Contributed by
     Mario Carneiro and Jim Kingdon, 31-Dec-2017.)  (Proved by Mario Carneiro,
     9-Feb-2018.) $)
  ax11e $p |- ( x = y -> ( E. x ( x = y /\ ph ) -> E. y ph ) ) $=
    ( cv wceq wa wex wi equs5e 19.21bi com12 ) BDCDEZAFBGZLACGZMLNHBABCIJK $.

  $( Quantifier Substitution for existential quantifiers.  Analogue to ~ ax10o
     but for ` E. ` rather than ` A. ` .  (Contributed by Jim Kingdon,
     21-Dec-2017.) $)
  ax10oe $p |- ( A. x x = y -> ( E. x ps -> E. y ps ) ) $=
    ( cv wceq wal wex wa wi ax-ia3 alimi exim syl ax11e sps syld ) BDCDEZBFZABG
    ZQAHZBGZACGZRATIZBFSUAIQUCBQAJKATBLMQUAUBIBABCNOP $.

  ${
    drex1.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
       (Contributed by NM, 27-Feb-2005.)  (Revised by NM, 3-Feb-2015.) $)
    drex1 $p |- ( A. x x = y -> ( E. x ph <-> E. y ps ) ) $=
      ( cv wceq wal wex wa hbae ax-4 biantrurd bitr2d exbidh wi sylbird equcomi
      ax11e sps bitr3d alequcoms impbid ) CFZDFZGZCHZACIZBDIZUGUHUFBJZCIZUIUGUJ
      ACCDCKUGABUJEUGUFBUFCLMNOUFUKUIPCBCDSTQUGUIUEUDGZAJZDIZUHUGUMBDCDDKUGAUMB
      UGULAUFULCCDRTMEUAOUNUHPZDCULUODADCSTUBQUC $.
  $}

  $( Formula-building lemma for use with the Distinctor Reduction Theorem.
     Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).  (Contributed
     by NM, 5-Aug-1993.) $)
  drsb1 $p |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) ) $=
    ( weq wal wi wa wex wsb wb equequ1 sps imbi1d anbi1d drex1 anbi12d 3bitr4g
    df-sb ) BCEZBFZBDEZAGZUBAHZBIZHCDEZAGZUFAHZCIZHABDJACDJUAUCUGUEUIUAUBUFATUB
    UFKBBCDLMZNUDUHBCUAUBUFAUJOPQABDSACDSR $.

  ${
    exdistrfor.1 $e |- ( A. x x = y \/ A. x F/ y ph ) $.
    $( Distribution of existential quantifiers, with a bound-variable
       hypothesis saying that ` y ` is not free in ` ph ` , but ` x ` can be
       free in ` ph ` (and there is no distinct variable condition on ` x ` and
       ` y ` ).  (Contributed by Jim Kingdon, 25-Feb-2018.) $)
    exdistrfor $p |- ( E. x E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) ) $=
      ( weq wal wnf wo wa wex biidd drex1 drex2 hbe1 19.9h 19.8a anim2i eximi
      wi sylbi biimtrrdi ax-ial 19.40 19.9t biimpd anim1d syl5 sps eximdh ax-mp
      jaoi ) CDFCGZADHZCGZIABJZDKZCKZABDKZJZCKZTZEUMVBUOUMURUPCKZCKZVAVCUQCDCUP
      UPCDUMUPLMNVDVCVAVCCUPCOPUPUTCBUSABDQRSUAUBUOUQUTCUNCUCUNUQUTTCUQADKZUSJU
      NUTABDUDUNVEAUSUNVEAADUEUFUGUHUIUJULUK $.
  $}

  $( A version of ~ sb4 that doesn't require a distinctor antecedent.
     (Contributed by NM, 2-Feb-2007.) $)
  sb4a $p |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) ) $=
    ( wal wsb weq wa wex wi sb1 equs5a syl ) ACDZBCEBCFZMGBHNAIBDMBCJABCKL $.

  ${
    equs45f.1 $e |- ( ph -> A. y ph ) $.
    $( Two ways of expressing substitution when ` y ` is not free in ` ph ` .
       (Contributed by NM, 25-Apr-2008.) $)
    equs45f $p |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) $=
      ( weq wa wex wi wal anim2i eximi equs5a syl equs4 impbii ) BCEZAFZBGZPAHB
      IZRPACIZFZBGSQUABATPDJKABCLMABCNO $.

    $( Equivalence for substitution when ` y ` is not free in ` ph ` .
       (Contributed by NM, 5-Aug-1993.)  (Revised by NM, 30-Apr-2008.) $)
    sb6f $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
      ( wsb weq wi wal sbimi sb4a syl sb2 impbii ) ABCEZBCFAGBHZNACHZBCEOAPBCDI
      ABCJKABCLM $.

    $( Equivalence for substitution when ` y ` is not free in ` ph ` .
       (Contributed by NM, 5-Aug-1993.)  (Revised by NM, 18-May-2008.) $)
    sb5f $p |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) $=
      ( wsb weq wi wal wa wex sb6f equs45f bitr4i ) ABCEBCFZAGBHNAIBJABCDKABCDL
      M $.
  $}

  $( One direction of a simplified definition of substitution that unlike ~ sb4
     doesn't require a distinctor antecedent.  (Contributed by NM,
     2-Feb-2007.) $)
  sb4e $p |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) ) $=
    ( wsb weq wa wex wi wal sb1 equs5e syl ) ABCDBCEZAFBGMACGHBIABCJABCKL $.

  $( Special case of a bound-variable hypothesis builder for substitution.
     (Contributed by NM, 2-Feb-2007.) $)
  hbsb2a $p |- ( [ y / x ] A. y ph -> A. x [ y / x ] ph ) $=
    ( wal wsb weq wi sb4a sb2 a5i syl ) ACDBCEBCFAGZBDABCEZBDABCHLMBABCIJK $.

  $( Special case of a bound-variable hypothesis builder for substitution.
     (Contributed by NM, 2-Feb-2007.) $)
  hbsb2e $p |- ( [ y / x ] ph -> A. x [ y / x ] E. y ph ) $=
    ( wsb weq wex wi wal sb4e sb2 a5i syl ) ABCDBCEACFZGZBHMBCDZBHABCINOBMBCJKL
    $.

  ${
    hbsb3.1 $e |- ( ph -> A. y ph ) $.
    $( If ` y ` is not free in ` ph ` , ` x ` is not free in ` [ y / x ] ph ` .
       (Contributed by NM, 5-Aug-1993.) $)
    hbsb3 $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
      ( wsb wal sbimi hbsb2a syl ) ABCEZACFZBCEJBFAKBCDGABCHI $.
  $}

  ${
    nfs1.1 $e |- F/ y ph $.
    $( If ` y ` is not free in ` ph ` , ` x ` is not free in ` [ y / x ] ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nfs1 $p |- F/ x [ y / x ] ph $=
      ( wsb nfri hbsb3 nfi ) ABCEBABCACDFGH $.
  $}

  ${
    sbcof2.1 $e |- ( ph -> A. x ph ) $.
    $( Version of ~ sbco where ` x ` is not free in ` ph ` .  (Contributed by
       Jim Kingdon, 28-Dec-2017.) $)
    sbcof2 $p |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) $=
      ( wsb weq wi wal hbsb3 sb6f imbi2i albii bitri alimi wex 3syl wa jca sb5f
      eximi ax-11 equcomi imim1i imim2i pm2.43d syl6 a2i sylbi ax-i9 mpi ax-ial
      exim 19.9h sylib sb2 simpl 19.8a anim1i ax11e syl5 imdistani anbi2i exbii
      sb1 sylibr impbii ) ACBEZBCEZABCEZVHBCFZVJAGZBHZGZBHZVLVIVHVJCBFZAGZCHZGZ
      BHZVNVHVJVGGZBHVSVGBCACBDIZJVTVRBVGVQVJACBDJKLMVRVMBVJVQVLVJVQVJVPGZBHVLV
      PBCUAWBVKBWBVJAVPVKVJVJVOABCUBZUCUDUENUFUGNUHVNVLBOZVLVNVJBOWDBCUIVJVLBUL
      UJVLBVKBUKUMUNABCUOPVIVJVOAQZCOZQZBOZVHVIVJAQZBOZVJWJQZBOWHABCVDWIWKBWIVJ
      WJVJAUPZWIBUQRTWKWGBVJWJWFWJVJWEQZBOVJWFWIWMBWIVJWEWLVJVOAWCURRTWEBCUSUTV
      ATPVHVJVGQZBOWHVGBCWASWNWGBVGWFVJACBDSVBVCMVEVF $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Predicate calculus with distinct variables
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the axiom of distinct variables ax-16
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x ps $.
    spimv.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( A version of ~ spim with a distinct variable requirement instead of a
       bound-variable hypothesis.  (Contributed by NM, 5-Aug-1993.) $)
    spimv $p |- ( A. x ph -> ps ) $=
      ( nfv spim ) ABCDBCFEG $.
  $}

  ${
    $v f $.
    $( Define a temporary individual variable. $)
    aev.vf $f setvar f $.
    $d f u v $.  $d f u x y $.  $d u w $.
    $( A "distinctor elimination" lemma with no restrictions on variables in
       the consequent, proved without using ~ ax-16 .  (Contributed by NM,
       8-Nov-2006.)  (Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
    aev $p |- ( A. x x = y -> A. z w = v ) $=
      ( aev.vf vu weq wal hbae ax-8 alrimih equcomi syl6 alequcoms a5i alequcom
      spimv 3syl ) ABHZAIZDEHZCABCJUAFBHZFIZGEHZGIZUBUAUCFABFJTUCAFAFBKRLUDFGHZ
      FIZEGHZEIUFUCUGFUGBFBFHZUGBGBGHUJGFHUGBGFKGFMNROPUHUIEFGEJUGUIFEFEGKRLEGQ
      SUEUBGDGDEKRSL $.
  $}

  ${
    $d x y $.  $d z ph $.
    $( Theorem showing that ~ ax-16 is redundant if ~ ax-17 is included in the
       axiom system.  The important part of the proof is provided by ~ aev .

       See ~ ax16ALT for an alternate proof that does not require ~ ax-10 or
       ~ ax12 .

       This theorem should not be referenced in any proof.  Instead, use
       ~ ax-16 below so that theorems needing ~ ax-16 can be more easily
       identified.  (Contributed by NM, 8-Nov-2006.) $)
    ax16 $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
      ( vz weq wal wi aev wsb ax-17 sbequ12 biimpcd alimdh hbsb3 stdpc7 syl6com
      cbv3h syl ) BCEBFBDEZDFZAABFZGBCDBDHATABDIZDFUAASUBDADJZSAUBABDKLMUBADBAB
      DUCNUCADBOQPR $.
  $}

  ${
    $d x y $.
    $( Axiom of Distinct Variables.  The only axiom of predicate calculus
       requiring that variables be distinct (if we consider ~ ax-17 to be a
       metatheorem and not an axiom).  Axiom scheme C16' in [Megill] p. 448 (p.
       16 of the preprint).  It apparently does not otherwise appear in the
       literature but is easily proved from textbook predicate calculus by
       cases.  It is a somewhat bizarre axiom since the antecedent is always
       false in set theory, but nonetheless it is technically necessary as you
       can see from its uses.

       This axiom is redundant if we include ~ ax-17 ; see Theorem ~ ax16 .

       This axiom is obsolete and should no longer be used.  It is proved above
       as Theorem ~ ax16 .  (Contributed by NM, 5-Aug-1993.)
       (New usage is discouraged.) $)
    ax-16 $a |- ( A. x x = y -> ( ph -> A. x ph ) ) $.
  $}

  ${
    $d z x $.
    $( Quantifier introduction when one pair of variables is distinct.
       (Contributed by NM, 2-Jan-2002.) $)
    dveeq2 $p |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $=
      ( weq wal wn wi wo ax12or orcom orbi2i mpbi orass mpbir orel2 mpi ax16 sp
      jaoi syl ) ABDAEZFZACDAEZCBDZUDAEGZAEZHZUEUBUGUAHZUGUHUCUFUAHZHZUCUAUFHZH
      UJCBAIUKUIUCUAUFJKLUCUFUAMNUAUGOPUCUEUFUDACQUEARST $.
  $}

  ${
    $d z x $.
    $( Quantifier introduction when one pair of variables is distinct.  Like
       ~ dveeq2 but connecting ` A. x x = y ` by a disjunction rather than
       negation and implication makes the theorem stronger in intuitionistic
       logic.  (Contributed by Jim Kingdon, 1-Feb-2018.) $)
    dveeq2or $p |- ( A. x x = y \/ F/ x z = y ) $=
      ( weq wal wi wnf ax12or orass mpbir pm1.4 orim1i ax-mp mpbi ax16 a5i jaoi
      wo id orim2i df-nf biimpri ) ABDAEZCBDZUDAEFZAEZRZUCUDAGZRUCACDZAEZUFRZRZ
      UGUCUJRZUFRZULUJUCRZUFRZUNUPUJUGRCBAHUJUCUFIJUOUMUFUJUCKLMUCUJUFINUKUFUCU
      JUFUFUIUEAUDACOPUFSQTMUFUHUCUHUFUDAUAUBTM $.
  $}

  ${
    $d x z $.  $d y z $.
    dvelimfALT2.1 $e |- ( ph -> A. x ph ) $.
    dvelimfALT2.2 $e |- ( ps -> A. z ps ) $.
    dvelimfALT2.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
    dvelimfALT2.4 $e |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $.
    $( Proof of ~ dvelimf using ~ dveeq2 (shown as the last hypothesis) instead
       of ~ ax12 .  This shows that ~ ax12 could be replaced by ~ dveeq2 (the
       last hypothesis).  (Contributed by Andrew Salmon, 21-Jul-2011.) $)
    dvelimfALT2 $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( cv wceq wal wn wi ax-17 hbn1 a1i hbimd hbald equsalh albii 3imtr3g ) CJ
      DJZKZCLMZEJUCKZANZELZUHCLBBCLUEUGCEUEEOUEUFACUDCPIAACLNUEFQRSABEDGHTZUHBC
      UIUAUB $.
  $}

  ${
    $d z x $.
    $( A lemma for proving conditionless ZFC axioms.  (Contributed by NM,
       8-Jan-2002.) $)
    nd5 $p |- ( -. A. y y = x -> ( z = y -> A. x z = y ) ) $=
      ( cv wceq wal wi dveeq2 nalequcoms ) CDBDEZJAFGABABCHI $.
  $}

  ${
    $d x ch $.  $d x ph $.
    exlimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
       27-Apr-1994.) $)
    exlimdv $p |- ( ph -> ( E. x ps -> ch ) ) $=
      ( ax-17 exlimdh ) ABCDADFCDFEG $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    ax11v2.1 $e |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) $.
    $( Recovery of ~ ax11o from ~ ax11v without using ~ ax-11 .  The hypothesis
       is even weaker than ~ ax11v , with ` z ` both distinct from ` x ` _and_
       not occurring in ` ph ` .  Thus the hypothesis provides an alternate
       axiom that can be used in place of ~ ax11o .  (Contributed by NM,
       2-Feb-2007.) $)
    ax11v2 $p |- ( -. A. x x = y ->
                 ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
      ( weq wal wn wex wi a9e wa wb equequ2 adantl dveeq2 imp hba1 imbi1d sps
      albidh syl imbi2d imbi12d mpbii ex exlimdv mpi ) BCFZBGHZDCFZDIUIAUIAJZBG
      ZJZJZDCKUJUKUODUJUKUOUJUKLZBDFZAUQAJZBGZJZJUOEUPUQUIUTUNUKUQUIMUJDCBNZOUP
      USUMAUPUKBGZUSUMMUJUKVBBCDPQVBURULBUKBRUKURULMBUKUQUIAVASTUAUBUCUDUEUFUGU
      H $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    ax11a2.1 $e |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) ) $.
    $( Derive ~ ax-11o from a hypothesis in the form of ~ ax-11 .  The
       hypothesis is even weaker than ~ ax-11 , with ` z ` both distinct from
       ` x ` and not occurring in ` ph ` .  Thus the hypothesis provides an
       alternate axiom that can be used in place of ~ ax11o .  (Contributed by
       NM, 2-Feb-2007.) $)
    ax11a2 $p |- ( -. A. x x = y ->
                 ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
      ( wal weq wi ax-17 syl5 ax11v2 ) ABCDAADFBDGZLAHBFADIEJK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the obsolete axiom of variable substitution ax-11o
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x z $.  $d y z $.  $d z ph $.
    $( Derivation of set.mm's original ~ ax-11o from the shorter ~ ax-11 that
       has replaced it.

       An open problem is whether this theorem can be proved without relying on
       ~ ax-16 or ~ ax-17 .

       Normally, ~ ax11o should be used rather than ~ ax-11o , except by
       theorems specifically studying the latter's properties.  (Contributed by
       NM, 3-Feb-2007.) $)
    ax11o $p |- ( -. A. x x = y ->
               ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
      ( vz ax-11 ax11a2 ) ABCDABDEF $.
  $}

  $( Axiom ~ ax-11o ("o" for "old") was the original version of ~ ax-11 ,
     before it was discovered (in Jan. 2007) that the shorter ~ ax-11 could
     replace it.  It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of
     the preprint).  It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of
     [Monk2] p. 105, from which it can be proved by cases.  To understand this
     theorem more easily, think of " ` -. A. x x = y -> ` ..." as informally
     meaning "if ` x ` and ` y ` are distinct variables, then..."  The
     antecedent becomes false if the same variable is substituted for ` x ` and
     ` y ` , ensuring the theorem is sound whenever this is the case.  In some
     later theorems, we call an antecedent of the form ` -. A. x x = y ` a
     "distinctor."

     This axiom is redundant, as shown by Theorem ~ ax11o .

     This axiom is obsolete and should no longer be used.  It is proved above
     as Theorem ~ ax11o .  (Contributed by NM, 5-Aug-1993.)
     (New usage is discouraged.) $)
  ax-11o $a |- ( -. A. x x = y ->
             ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  More theorems related to ax-11 and substitution
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x ph $.
    albidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for universal quantifier (deduction form).
       (Contributed by NM, 5-Aug-1993.) $)
    albidv $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
      ( ax-17 albidh ) ABCDADFEG $.

    $( Formula-building rule for existential quantifier (deduction form).
       (Contributed by NM, 5-Aug-1993.) $)
    exbidv $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
      ( ax-17 exbidh ) ABCDADFEG $.
  $}

  $( A bidirectional version of ~ ax-11o .  (Contributed by NM,
     30-Jun-2006.) $)
  ax11b $p |- ( ( -. A. x x = y /\ x = y ) ->
              ( ph <-> A. x ( x = y -> ph ) ) ) $=
    ( weq wal wn wa wi ax11o imp ax-4 com12 adantl impbid ) BCDZBEFZOGAOAHZBEZP
    OARHABCIJORAHPROAQBKLMN $.

  ${
    $d x y $.  $d x z $.  $d y z $.  $d ph z $.
    $( This is a version of ~ ax-11o when the variables are distinct.  Axiom
       (C8) of [Monk2] p. 105.  (Contributed by NM, 5-Aug-1993.)  (Revised by
       Jim Kingdon, 15-Dec-2017.) $)
    ax11v $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
      ( vz cv wceq wex wi wal a9e ax-17 ax-11 syl5 equequ2 imbi1d albidv imbi2d
      imbi12d mpbii exlimiv ax-mp ) DEZCEZFZDGBEZUCFZAUFAHZBIZHZHZDCJUDUJDUDUEU
      BFZAUKAHZBIZHZHUJAADIUKUMADKABDLMUDUKUFUNUIDCBNZUDUMUHAUDULUGBUDUKUFAUOOP
      QRSTUA $.
  $}

  ${
    $d x y $.  $d x z $.  $d y z $.  $d ph z $.
    $( Analogue to ~ ax11v for existential quantification.  (Contributed by Jim
       Kingdon, 9-Jan-2018.) $)
    ax11ev $p |- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) ) $=
      ( vz cv wceq wex wa wi a9e ax11e ax-17 19.9h equequ2 anbi1d exbidv imbi1d
      imbitrdi imbi12d mpbii exlimiv ax-mp ) DEZCEZFZDGBEZUDFZUGAHZBGZAIZIZDCJU
      EUKDUEUFUCFZULAHZBGZAIZIUKULUNADGAABDKADADLMRUEULUGUOUJDCBNZUEUNUIAUEUMUH
      BUEULUGAUPOPQSTUAUB $.
  $}

  $( Lemma used in proofs of substitution properties.  (Contributed by NM,
     5-Aug-1993.) $)
  equs5 $p |- ( -. A. x x = y ->
             ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) ) $=
    ( weq wal wn wa wi hbnae hba1 ax11o impd exlimdh ) BCDZBEFZNAGNAHZBEZBBCBIP
    BJONAQABCKLM $.

  ${
    $d x z $.  $d y z $.  $d ph z $.

    $( Lemma used in proofs of substitution properties.  Like ~ equs5 but, in
       intuitionistic logic, replacing negation and implication with
       disjunction makes this a stronger result.  (Contributed by Jim Kingdon,
       2-Feb-2018.) $)
    equs5or $p |- ( A. x x = y \/
             ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) ) $=
      ( vz weq wex wal wa wi wo a9e wnf dveeq2or nfnf1 nfri ax11v equequ2 ax-mp
      wb hba1 adantl nfr imp imbi1d sps albidh syl imbi2d imbi12d mpbii alrimih
      ex imp4a 19.21t mpbid 19.23h imbitrdi orim2i pm2.76 olcs exlimiv ) DCEZDF
      BCEZBGZVCAHZBFVCAIZBGZIZJZDCKVBVIDVDVBVIVDVBVHIZJZVDVBJVIIVDVBBLZJVKBCDMV
      LVJVDVLVBVEVGIZBGZVHVLVBVMIZBGVBVNIVLVOBVLBVBBNOVLVBVCAVGVLVBVCAVGIZIZVLV
      BHZBDEZAVSAIZBGZIZIVQABDPVRVSVCWBVPVBVSVCSVLDCBQZUAVRWAVGAVRVBBGZWAVGSVLV
      BWDVBBUBUCWDVTVFBVBBTVBVTVFSBVBVSVCAWCUDUEUFUGUHUIUJULUMUKVBVMBUNUOVEVGBV
      FBTUPUQURRVDVBVHUSRUTVAR $.
  $}

  $( One direction of a simplified definition of substitution when variables
     are distinct.  (Contributed by NM, 5-Aug-1993.) $)
  sb3 $p |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) ) $=
    ( weq wal wn wa wex wi wsb equs5 sb2 syl6 ) BCDZBEFNAGBHNAIBEABCJABCKABCLM
    $.

  $( One direction of a simplified definition of substitution when variables
     are distinct.  (Contributed by NM, 5-Aug-1993.) $)
  sb4 $p |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) ) $=
    ( wsb weq wa wex wal wn wi sb1 equs5 syl5 ) ABCDBCEZAFBGNBHINAJBHABCKABCLM
    $.

  $( One direction of a simplified definition of substitution when variables
     are distinct.  Similar to ~ sb4 but stronger in intuitionistic logic.
     (Contributed by Jim Kingdon, 2-Feb-2018.) $)
  sb4or $p |- ( A. x x = y \/
      A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) ) $=
    ( weq wal wa wex wi wo wsb equs5or nfe1 nfa1 nfim sb1 imim1i alrimih orim2i
    nfri ax-mp ) BCDZBEZUAAFZBGZUAAHZBEZHZIUBABCJZUFHZBEZIABCKUGUJUBUGUIBUGBUDU
    FBUCBLUEBMNSUHUDUFABCOPQRT $.

  $( Simplified definition of substitution when variables are distinct.
     (Contributed by NM, 27-May-1997.) $)
  sb4b $p |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) $=
    ( weq wal wn wsb wi sb4 sb2 impbid1 ) BCDZBEFABCGLAHBEABCIABCJK $.

  $( Simplified definition of substitution when variables are distinct,
     expressed via disjunction.  (Contributed by Jim Kingdon, 18-Mar-2018.) $)
  sb4bor $p |- ( A. x x = y \/
      A. x ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) $=
    ( weq wal wsb wi wo wb sb4or sb2 wa df-bi simpri mpan2 alimi orim2i ax-mp )
    BCDZBEZABCFZSAGBEZGZBEZHTUAUBIZBEZHABCJUDUFTUCUEBUCUBUAGZUEABCKUEUCUGLZGUHU
    EGUAUBMNOPQR $.

  $( Bound-variable hypothesis builder for substitution.  (Contributed by NM,
     5-Aug-1993.) $)
  hbsb2 $p |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) ) $=
    ( weq wal wn wsb wi sb4 sb2 a5i syl6 ) BCDZBEFABCGZMAHZBENBEABCIONBABCJKL
    $.

  $( Bound-variable hypothesis builder for substitution.  Similar to ~ hbsb2
     but in intuitionistic logic a disjunction is stronger than an implication.
     (Contributed by Jim Kingdon, 2-Feb-2018.) $)
  nfsb2or $p |- ( A. x x = y \/ F/ x [ y / x ] ph ) $=
    ( weq wal wsb wi wnf sb4or sb2 a5i imim2i alimi df-nf sylibr orim2i ax-mp
    wo ) BCDZBEZABCFZSAGZBEZGZBEZRTUABHZRABCIUEUFTUEUAUABEZGZBEUFUDUHBUCUGUAUBU
    ABABCJKLMUABNOPQ $.

  ${
    sbequilem.1 $e |- ( ph \/ ( ps -> ( ch -> th ) ) ) $.
    sbequilem.2 $e |- ( ta \/ ( ps -> ( th -> et ) ) ) $.
    $( Propositional logic lemma used in the ~ sbequi proof.  (Contributed by
       Jim Kingdon, 1-Feb-2018.) $)
    sbequilem $p |- ( ph \/ ( ta \/ ( ps -> ( ch -> et ) ) ) ) $=
      ( wo wi wa pm3.2i andi mpbi andir orbi12i orim2i ax-mp orim1i orass simpr
      pm3.43 pm3.33 syl6 sylbir simpl mpbir orcom orbi1i ) AEIZBCFJZJZIZAEULIIE
      AIZULIZUMUOEAULIZIZEABDFJZJZKZULIZIZUQAEKBCDJZJZEKIZVAIZVBVEUTVDUSKZIZIZV
      FAVDIZEKZVJUSKZIZVIVJEUSIZKVMVJVNGHLVJEUSMNVKVEVLVHAVDEOZAVDUSOPNVHVAVEVG
      ULUTVGBVCURKUKBVCURUBCDFUCUDQQRVEEVAVEVKEVOVJEUAUESRVAUPEUTAULAUSUFSQREAU
      LTUGUNUJULEAUHUINAEULTN $.
  $}

  $( An equality theorem for substitution.  (Contributed by NM, 5-Aug-1993.)
     (Proof modified by Jim Kingdon, 1-Feb-2018.) $)
  sbequi $p |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) $=
    ( weq wal wsb wi wo wex nfsb2or wa stdpc7 sbequ1 sylan9 orim2i ax-mp biimpd
    wnf sps equvini eximi 19.35-1 3syl syl9 19.9t sbequilem sbequ2 adantr drsb1
    nfr ax-1 alequcoms sylan9r syld ex orim1i pm1.2 syl jaoi ) DBEZDFZDCEZDFZBC
    EZADBGZADCGZHZHZIZIVIVBVEVFVGDJZVDVGVBVFDSZIVBVEVFVKHHZIADBKVLVMVBVLVFVFDFZ
    VEVKVFDUKVEBDEZVCLZDJVHDJVNVKHBCDUAVPVHDVOVFAVCVGABDMADCNZOUBVFVGDUCUDUEPQV
    DVKVGHZIZVDVEVRHZIVDVGDSZIVSADCKWAVRVDWAVKVGVGDUFRPQVRVTVDVRVEULPQUGVBVIVJV
    BVEVHVBVELVFAVGVBVFAHZVEVAWBDADBUHTUIVEAABCGZVBVGABCNWCVGHBDVOBFWCVGABDCUJR
    UMUNUOUPVJVIVIIVIVDVIVIVDVEVHVDVELVFAVGVDVFACBGZVEAVDVFWDADCBUJRABCMOVDAVGH
    ZVEVCWEDVQTUIUOUPUQVIURUSUTQ $.

  $( An equality theorem for substitution.  Used in proof of Theorem 9.7 in
     [Megill] p. 449 (p. 16 of the preprint).  (Contributed by NM,
     5-Aug-1993.) $)
  sbequ $p |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) $=
    ( weq wsb sbequi wi equcoms impbid ) BCEADBFZADCFZABCDGLKHCBACBDGIJ $.

  $( Formula-building lemma for use with the Distinctor Reduction Theorem.
     Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).  (Contributed
     by NM, 27-Feb-2005.) $)
  drsb2 $p |- ( A. x x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) $=
    ( weq wsb wb sbequ sps ) BCEADBFADCFGBABCDHI $.

  $( A specialization theorem, mostly the same as Theorem 19.8 of [Margaris]
     p. 89.  (Contributed by NM, 5-Aug-1993.)  (Proof rewritten by Jim Kingdon,
     29-Dec-2017.) $)
  spsbe $p |- ( [ y / x ] ph -> E. x ph ) $=
    ( wsb weq wa wex sb1 simpr eximi syl ) ABCDBCEZAFZBGABGABCHMABLAIJK $.

  $( Specialization of implication.  (Contributed by NM, 5-Aug-1993.)  (Proof
     rewritten by Jim Kingdon, 21-Jan-2018.) $)
  spsbim $p |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
    ( wi wal weq wa wex wsb imim2 sps id anim2d alimi syl anim12d df-sb 3imtr4g
    exim ) ABEZCFZCDGZAEZUCAHZCIZHUCBEZUCBHZCIZHACDJBCDJUBUDUGUFUIUAUDUGECABUCK
    LUBUEUHEZCFUFUIEUAUJCUAABUCUAMNOUEUHCTPQACDRBCDRS $.

  $( Specialization of biconditional.  (Contributed by NM, 5-Aug-1993.)  (Proof
     rewritten by Jim Kingdon, 21-Jan-2018.) $)
  spsbbi $p |- ( A. x ( ph <-> ps ) -> ( [ y / x ] ph <-> [ y / x ] ps ) ) $=
    ( wi wal wa wsb wb spsbim anim12i albiim dfbi2 3imtr4i ) ABECFZBAECFZGACDHZ
    BCDHZEZRQEZGABICFQRIOSPTABCDJBACDJKABCLQRMN $.

  ${
    sbbidh.1 $e |- ( ph -> A. x ph ) $.
    sbbidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction substituting both sides of a biconditional.  New proofs should
       use ~ sbbid instead.  (Contributed by NM, 5-Aug-1993.)
       (New usage is discouraged.) $)
    sbbidh $p |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) ) $=
      ( wb wal wsb alrimih spsbbi syl ) ABCHZDIBDEJCDEJHANDFGKBCDELM $.
  $}

  ${
    sbbid.1 $e |- F/ x ph $.
    sbbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction substituting both sides of a biconditional.  (Contributed by
       NM, 30-Jun-1993.) $)
    sbbid $p |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) ) $=
      ( wb wal wsb alrimi spsbbi syl ) ABCHZDIBDEJCDEJHANDFGKBCDELM $.
  $}

  $( Elimination of equality from antecedent after substitution.  (Contributed
     by NM, 5-Aug-1993.)  (Proof revised by Jim Kingdon, 20-Jan-2018.) $)
  sbequ8 $p |- ( [ y / x ] ph <-> [ y / x ] ( x = y -> ph ) ) $=
    ( weq wi wa wex wsb pm5.4 simpl pm3.35 jca pm3.4 impbii exbii anbi12i df-sb
    3bitr4ri ) BCDZSAEZEZSTFZBGZFTSAFZBGZFTBCHABCHUATUCUESAIUBUDBUBUDUBSASTJSAK
    LUDSTSAJSAMLNOPTBCQABCQR $.

  $( Substitution has no effect on a nonfree variable.  (Contributed by NM,
     30-May-2009.)  (Revised by Mario Carneiro, 12-Oct-2016.)  (Proof shortened
     by Wolf Lammen, 3-May-2018.) $)
  sbft $p |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) ) $=
    ( wnf wsb wex spsbe 19.9t imbitrid wal nfr stdpc4 syl6 impbid ) ABDZABCEZAP
    ABFOAABCGABHIOAABJPABKABCLMN $.

  ${
    sbid2h.1 $e |- ( ph -> A. x ph ) $.
    $( An identity law for substitution.  (Contributed by NM, 5-Aug-1993.) $)
    sbid2h $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
      ( wsb sbcof2 sbh bitri ) ACBEBCEABCEAABCDFABCDGH $.
  $}

  ${
    sbid2.1 $e |- F/ x ph $.
    $( An identity law for substitution.  (Contributed by NM, 5-Aug-1993.)
       (Revised by Mario Carneiro, 6-Oct-2016.) $)
    sbid2 $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
      ( nfri sbid2h ) ABCABDEF $.
  $}

  $( An idempotent law for substitution.  (Contributed by NM, 30-Jun-1994.)
     (Proof rewritten by Jim Kingdon, 21-Jan-2018.) $)
  sbidm $p |- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph ) $=
    ( wsb weq wi wa df-sb simplbi sbimi sbequ8 sylibr ax-1 pm4.24 ax-ie1 19.41h
    wex sb1 bitr4i exbii anim2i anim1i eximi sylbi anass anbi2i sylanbrc impbii
    sylib syl ) ABCDZBCDZUKULBCEZAFZBCDUKUKUNBCUKUNUMAGZBQZABCHZIJABCKLUKUMUKFU
    MUKGZBQZULUKUMMUKUPUSABCRUPUMUNUPGZGZBQZUSUPUMUNGZUPGZBQZVBUPUOUPGZBQZVEUPU
    PUPGVGUPNUOUPBUOBOPSVFVDBUOVCUPAUNUMAUMMUAUBUCUDVDVABUMUNUPUETUIURVABUKUTUM
    UQUFTLUJUKBCHUGUH $.

  ${
    sb5rf.1 $e |- ( ph -> A. y ph ) $.
    $( Reversed substitution.  (Contributed by NM, 3-Feb-2005.)  (Proof
       shortened by Andrew Salmon, 25-May-2011.) $)
    sb5rf $p |- ( ph <-> E. y ( y = x /\ [ y / x ] ph ) ) $=
      ( weq wsb wa wex sbid2h sb1 sylbir stdpc7 imp exlimih impbii ) ACBEZABCFZ
      GZCHZAQCBFSACBDIQCBJKRACDPQAACBLMNO $.

    $( Reversed substitution.  (Contributed by NM, 5-Aug-1993.)  (Proof
       shortened by Andrew Salmon, 25-May-2011.) $)
    sb6rf $p |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) ) $=
      ( weq wsb wi wal sbequ1 equcoms com12 alrimih sb2 sbid2h sylib impbii ) A
      CBEZABCFZGZCHZASCDQARARGBCABCIJKLTRCBFARCBMACBDNOP $.
  $}

  ${
    sb8h.1 $e |- ( ph -> A. y ph ) $.
    $( Substitution of variable in universal quantifier.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Jim Kingdon, 15-Jan-2018.) $)
    sb8h $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
      ( wsb hbsb3 sbequ12 cbvalh ) AABCEBCDABCDFABCGH $.
  $}

  ${
    sb8eh.1 $e |- ( ph -> A. y ph ) $.
    $( Substitution of variable in existential quantifier.  (Contributed by NM,
       12-Aug-1993.)  (Proof rewritten by Jim Kingdon, 15-Jan-2018.) $)
    sb8eh $p |- ( E. x ph <-> E. y [ y / x ] ph ) $=
      ( wsb hbsb3 sbequ12 cbvexh ) AABCEBCDABCDFABCGH $.
  $}

  ${
    sb8e.1 $e |- F/ y ph $.
    $( Substitution of variable in universal quantifier.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof shortened
       by Jim Kingdon, 15-Jan-2018.) $)
    sb8 $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
      ( wsb nfs1 sbequ12 cbval ) AABCEBCDABCDFABCGH $.

    $( Substitution of variable in existential quantifier.  (Contributed by NM,
       12-Aug-1993.)  (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof
       shortened by Jim Kingdon, 15-Jan-2018.) $)
    sb8e $p |- ( E. x ph <-> E. y [ y / x ] ph ) $=
      ( wsb nfs1 sbequ12 cbvex ) AABCEBCDABCDFABCGH $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Predicate calculus with distinct variables (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.  $d z ph $.
    ax16i.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
    ax16i.2 $e |- ( ps -> A. x ps ) $.
    $( Inference with ~ ax-16 as its conclusion, that does not require
       ~ ax-10 , ~ ax-11 , or ~ ax12 for its proof.  The hypotheses may be
       eliminable without one or more of these axioms in special cases.
       (Contributed by NM, 20-May-2008.) $)
    ax16i $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
      ( weq wal wi ax-17 ax-8 cbv3h spimv equid mpi syl syl5com alimdh mpcom
      alimi biimpcd biimprd syl6com ) CDHZCIZCEHZEIZAACIZJUFEDHZEIZUHUEUJCEUEEK
      ZUJCKCEDLMUKECHZEIZUHUEUKUNUJUEECECDLNUEUJUMEULUEDCHZUJUMUECCHUOCOCDCLPUJ
      DEHZUOUMJUJEEHZUPEOZEDELPDECLQRSTUMUGEUMUQUGURECELPZUAQQAUHBEIUIAUGBEAEKZ
      UGABFUBSBAECGUTUMUGBAJUSUGABFUCQMUDQ $.
  $}

  ${
    $d x y z $.  $d z ph $.
    $( Version of ~ ax16 that does not require ~ ax-10 or ~ ax12 for its proof.
       (Contributed by NM, 17-May-2008.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ax16ALT $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
      ( vz wsb sbequ12 ax-17 hbsb3 ax16i ) AABDEBCDABDFABDADGHI $.
  $}

  ${
    $d x ps $.
    spv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Specialization, using implicit substitition.  (Contributed by NM,
       30-Aug-1993.) $)
    spv $p |- ( A. x ph -> ps ) $=
      ( weq biimpd spimv ) ABCDCDFABEGH $.
  $}

  ${
    $d x ph $.
    spimev.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Distinct-variable version of ~ spime .  (Contributed by NM,
       5-Aug-1993.) $)
    spimev $p |- ( ph -> E. x ps ) $=
      ( nfv spime ) ABCDACFEG $.
  $}

  ${
    $d x ps $.
    speiv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    speiv.2 $e |- ps $.
    $( Inference from existential specialization, using implicit substitition.
       (Contributed by NM, 19-Aug-1993.) $)
    speiv $p |- E. x ph $=
      ( wex weq biimprd spimev ax-mp ) BACGFBACDCDHABEIJK $.
  $}

  ${
    $d x z $.  $d y z $.
    $( A variable introduction law for equality.  Lemma 15 of [Monk2] p. 109.
       (Contributed by NM, 5-Aug-1993.) $)
    equvin $p |- ( x = y <-> E. z ( x = z /\ z = y ) ) $=
      ( weq wa wex equvini ax-17 equtr imp exlimih impbii ) ABDZACDZCBDZEZCFABC
      GPMCMCHNOMACBIJKL $.
  $}

  ${
    $d x y $.
    $( A generalization of Axiom ~ ax-16 .  (Contributed by NM, 5-Aug-1993.)
       (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    a16g $p |- ( A. x x = y -> ( ph -> A. z ph ) ) $=
      ( weq wal aev ax16 biidd dral1 biimprd sylsyld ) BCEBFDBEDFZAABFZADFZBCDD
      BGABCHMONAADBMAIJKL $.

    $( A generalization of Axiom ~ ax-16 .  (Contributed by NM, 5-Aug-1993.) $)
    a16gb $p |- ( A. x x = y -> ( ph <-> A. z ph ) ) $=
      ( weq wal a16g ax-4 impbid1 ) BCEBFAADFABCDGADHI $.

    $( If there is only one element in the universe, then everything satisfies
       ` F/ ` .  (Contributed by Mario Carneiro, 7-Oct-2016.) $)
    a16nf $p |- ( A. x x = y -> F/ z ph ) $=
      ( weq wal nfae a16g nfd ) BCEBFADBCDGABCDHI $.
  $}

  ${
    $d x ph $.  $d y ph $.
    2albidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for 2 existential quantifiers (deduction form).
       (Contributed by NM, 4-Mar-1997.) $)
    2albidv $p |- ( ph -> ( A. x A. y ps <-> A. x A. y ch ) ) $=
      ( wal albidv ) ABEGCEGDABCEFHH $.

    $( Formula-building rule for 2 existential quantifiers (deduction form).
       (Contributed by NM, 1-May-1995.) $)
    2exbidv $p |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) ) $=
      ( wex exbidv ) ABEGCEGDABCEFHH $.
  $}

  ${
    $d x ph $.  $d y ph $.  $d z ph $.
    3exbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for 3 existential quantifiers (deduction form).
       (Contributed by NM, 1-May-1995.) $)
    3exbidv $p |- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) ) $=
      ( wex exbidv 2exbidv ) ABFHCFHDEABCFGIJ $.
  $}

  ${
    $d x ph $.  $d y ph $.  $d z ph $.  $d w ph $.
    4exbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for 4 existential quantifiers (deduction form).
       (Contributed by NM, 3-Aug-1995.) $)
    4exbidv $p |- ( ph ->
                     ( E. x E. y E. z E. w ps <-> E. x E. y E. z E. w ch ) ) $=
      ( wex 2exbidv ) ABGIFICGIFIDEABCFGHJJ $.
  $}

  ${
    $d x ph $.
    $( Special case of Theorem 19.9 of [Margaris] p. 89.  (Contributed by NM,
       28-May-1995.)  (Revised by NM, 21-May-2007.) $)
    19.9v $p |- ( E. x ph <-> ph ) $=
      ( ax-17 19.9h ) ABABCD $.
  $}

  ${
    exlimdd.1 $e |- F/ x ph $.
    exlimdd.2 $e |- F/ x ch $.
    exlimdd.3 $e |- ( ph -> E. x ps ) $.
    exlimdd.4 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Existential elimination rule of natural deduction.  (Contributed by
       Mario Carneiro, 9-Feb-2017.) $)
    exlimdd $p |- ( ph -> ch ) $=
      ( wex ex exlimd mpd ) ABDICGABCDEFABCHJKL $.
  $}

  ${
    $d x ph $.
    $( Special case of Theorem 19.21 of [Margaris] p. 90. _Notational
       convention_:  We sometimes suffix with "v" the label of a theorem
       eliminating a hypothesis such as ` ( ph -> A. x ph ) ` in ~ 19.21 via
       the use of distinct variable conditions combined with ~ ax-17 .
       Conversely, we sometimes suffix with "f" the label of a theorem
       introducing such a hypothesis to eliminate the need for the distinct
       variable condition; e.g., ~ euf derived from ~ df-eu .  The "f" stands
       for "not free in" which is less restrictive than "does not occur in".
       (Contributed by NM, 5-Aug-1993.) $)
    19.21v $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
      ( ax-17 19.21h ) ABCACDE $.
  $}

  ${
    $d x ph $.
    alrimiv.1 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.) $)
    alrimiv $p |- ( ph -> A. x ps ) $=
      ( ax-17 alrimih ) ABCACEDF $.
  $}

  ${
    $d x ph $.  $d y ph $.
    alrimivv.1 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Contributed by NM,
       31-Jul-1995.) $)
    alrimivv $p |- ( ph -> A. x A. y ps ) $=
      ( wal alrimiv ) ABDFCABDEGG $.
  $}

  ${
    $d x ph $.  $d x ps $.
    alrimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.21 of [Margaris] p. 90.  (Contributed by NM,
       10-Feb-1997.) $)
    alrimdv $p |- ( ph -> ( ps -> A. x ch ) ) $=
      ( ax-17 alrimdh ) ABCDADFBDFEG $.
  $}

  ${
    $d x ph $.
    nfdv.1 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Apply the definition of not-free in a context.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    nfdv $p |- ( ph -> F/ x ps ) $=
      ( wal wi wnf alrimiv df-nf sylibr ) ABBCEFZCEBCGAKCDHBCIJ $.
  $}

  ${
    $d x ph $.  $d y ph $.
    $( Quantification of two variables over a formula in which they do not
       occur.  (Contributed by Alan Sare, 12-Apr-2011.) $)
    2ax17 $p |- ( ph -> A. x A. y ph ) $=
      ( id alrimivv ) AABCADE $.
  $}

  ${
    $d x ph $.
    alimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.20 of [Margaris] p. 90.  (Contributed by NM,
       3-Apr-1994.) $)
    alimdv $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
      ( ax-17 alimdh ) ABCDADFEG $.

    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Contributed by NM,
       27-Apr-1994.) $)
    eximdv $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
      ( ax-17 eximdh ) ABCDADFEG $.
  $}

  ${
    $d x ph $.  $d y ph $.
    2alimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Contributed by NM,
       27-Apr-2004.) $)
    2alimdv $p |- ( ph -> ( A. x A. y ps -> A. x A. y ch ) ) $=
      ( wal alimdv ) ABEGCEGDABCEFHH $.

    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Contributed by NM,
       3-Aug-1995.) $)
    2eximdv $p |- ( ph -> ( E. x E. y ps -> E. x E. y ch ) ) $=
      ( wex eximdv ) ABEGCEGDABCEFHH $.
  $}

  ${
    $d x ps $.
    $( Special case of Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
       28-Jun-1998.) $)
    19.23v $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
      ( ax-17 19.23h ) ABCBCDE $.
  $}

  ${
    $d x ps $.  $d y ps $.
    $( Theorem 19.23 of [Margaris] p. 90 extended to two variables.
       (Contributed by NM, 10-Aug-2004.) $)
    19.23vv $p |- ( A. x A. y ( ph -> ps ) <-> ( E. x E. y ph -> ps ) ) $=
      ( wi wal wex 19.23v albii bitri ) ABEDFZCFADGZBEZCFLCGBEKMCABDHILBCHJ $.
  $}

  ${
    $d x y $.  $d x ph $.
    $( If a formula does not contain a variable ` x ` , then it is equivalent
       to the corresponding prototype of substitution with a fresh variable
       (see ~ sb6 ).  (Contributed by BJ, 23-Jul-2023.) $)
    equsv $p |- ( A. x ( x = y -> ph ) <-> ph ) $=
      ( weq wi wal wex 19.23v a9ev a1bi bitr4i ) BCDZAEBFLBGZAEALABHMABCIJK $.
  $}

  ${
    $d x ph $.
    sbbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction substituting both sides of a biconditional, with ` ph ` and
       ` x ` disjoint.  See also ~ sbbid .  (Contributed by Wolf Lammen,
       6-May-2023.)  (Proof shortened by Steven Nguyen, 6-Jul-2023.) $)
    sbbidv $p |- ( ph -> ( [ t / x ] ps <-> [ t / x ] ch ) ) $=
      ( wb wal wsb alrimiv spsbbi syl ) ABCGZDHBDEICDEIGAMDFJBCDEKL $.
  $}

  ${
    $d x y $.
    $( Two equivalent ways of expressing the proper substitution of ` y ` for
       ` x ` in ` ph ` , when ` x ` and ` y ` are distinct.  Theorem 6.2 of
       [Quine] p. 40.  The proof does not involve ~ df-sb .  (Contributed by
       NM, 14-Apr-2008.) $)
    sb56 $p |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) $=
      ( weq wi wal hba1 ax11v ax-4 com12 impbid equsex ) ABCDZAEZBFZBCNBGMAOABC
      HOMANBIJKL $.

    $( Equivalence for substitution.  Compare Theorem 6.2 of [Quine] p. 40.
       Also proved as Lemmas 16 and 17 of [Tarski] p. 70.  (Contributed by NM,
       18-Aug-1993.)  (Revised by NM, 14-Apr-2008.) $)
    sb6 $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
      ( weq wi wa wex wal wsb sb56 anbi2i df-sb ax-4 pm4.71ri 3bitr4i ) BCDZAEZ
      PAFBGZFQQBHZFABCISRSQABCJKABCLSQQBMNO $.

    $( Equivalence for substitution.  Similar to Theorem 6.1 of [Quine] p. 40.
       (Contributed by NM, 18-Aug-1993.)  (Revised by NM, 14-Apr-2008.) $)
    sb5 $p |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) $=
      ( wsb weq wi wal wa wex sb6 sb56 bitr4i ) ABCDBCEZAFBGMAHBIABCJABCKL $.

    $( Version of ~ sbn where ` x ` and ` y ` are distinct.  (Contributed by
       Jim Kingdon, 18-Dec-2017.) $)
    sbnv $p |- ( [ y / x ] -. ph <-> -. [ y / x ] ph ) $=
      ( wn wsb weq wa wex wi wal sb6 alinexa bitri sb5 xchbinxr ) ADZBCEZBCFZAG
      BHZABCEQRPIBJSDPBCKRABLMABCNO $.

    $( Version of ~ sban where ` x ` and ` y ` are distinct.  (Contributed by
       Jim Kingdon, 24-Dec-2017.) $)
    sbanv $p |- ( [ y / x ] ( ph /\ ps ) <->
                  ( [ y / x ] ph /\ [ y / x ] ps ) ) $=
      ( wa wsb weq wi wal sb6 anbi12i 19.26 pm4.76 albii 3bitr2i bitr4i ) ABEZC
      DFCDGZQHZCIZACDFZBCDFZEZQCDJUCRAHZCIZRBHZCIZEUDUFEZCITUAUEUBUGACDJBCDJKUD
      UFCLUHSCRABMNOP $.

    $( Version of ~ sbor where ` x ` and ` y ` are distinct.  (Contributed by
       Jim Kingdon, 3-Feb-2018.) $)
    sborv $p |- ( [ y / x ] ( ph \/ ps ) <->
                  ( [ y / x ] ph \/ [ y / x ] ps ) ) $=
      ( wo wsb weq wa wex sb5 andi exbii 19.43 3bitri orbi12i bitr4i ) ABEZCDFZ
      CDGZAHZCIZSBHZCIZEZACDFZBCDFZERSQHZCITUBEZCIUDQCDJUGUHCSABKLTUBCMNUEUAUFU
      CACDJBCDJOP $.

    $( Forward direction of ~ sbimv .  (Contributed by Jim Kingdon,
       25-Dec-2017.) $)
    sbi1v $p |- ( [ y / x ] ( ph -> ps )
                      -> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
      ( wsb weq wi wal sb6 ax-2 al2imi sb2 syl6 sylbi biimtrid ) ACDECDFZAGZCHZ
      ABGZCDEZBCDEZACDITPSGZCHZRUAGSCDIUCRPBGZCHUAUBQUDCPABJKBCDLMNO $.

    $( Reverse direction of ~ sbimv .  (Contributed by Jim Kingdon,
       18-Jan-2018.) $)
    sbi2v $p |- ( ( [ y / x ] ph -> [ y / x ] ps )
                      -> [ y / x ] ( ph -> ps ) ) $=
      ( weq wa wex wi wal wsb 19.38 pm3.3 pm2.04 syli alimi syl sb5 sb6 imbi12i
      3imtr4i ) CDEZAFZCGZUABHZCIZHZUAABHZHZCIZACDJZBCDJZHUGCDJUFUBUDHZCIUIUBUD
      CKULUHCUAULAUDHUGUAAUDLAUABMNOPUJUCUKUEACDQBCDRSUGCDRT $.

    $( Intuitionistic proof of ~ sbim where ` x ` and ` y ` are distinct.
       (Contributed by Jim Kingdon, 18-Jan-2018.) $)
    sbimv $p |- ( [ y / x ] ( ph -> ps )
                  <-> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
      ( wi wsb sbi1v sbi2v impbii ) ABECDFACDFBCDFEABCDGABCDHI $.
  $}

  ${
    $d x ph $.
    $( Substitution for a variable not occurring in a proposition.  See ~ sbf
       for a version without disjoint variable condition on ` x , ph ` .  If
       one adds a disjoint variable condition on ` x , t ` , then ~ sbv can be
       proved directly by chaining ~ equsv with ~ sb6 .  (Contributed by BJ,
       22-Dec-2020.) $)
    sbv $p |- ( [ t / x ] ph <-> ph ) $=
      ( ax-17 sbh ) ABCABDE $.
  $}

  ${
    $d x y $.
    sblimv.1 $e |- ( ps -> A. x ps ) $.
    $( Version of ~ sblim where ` x ` and ` y ` are distinct.  (Contributed by
       Jim Kingdon, 19-Jan-2018.) $)
    sblimv $p |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) ) $=
      ( wi wsb sbimv sbh imbi2i bitri ) ABFCDGACDGZBCDGZFLBFABCDHMBLBCDEIJK $.
  $}

  ${
    $d ph y $.  $d ps x $.
    $( Theorem *11.53 in [WhiteheadRussell] p. 164.  (Contributed by Andrew
       Salmon, 24-May-2011.) $)
    pm11.53 $p |- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) ) $=
      ( wi wal wex 19.21v albii ax-17 hbal 19.23h bitri ) ABEDFZCFABDFZEZCFACGO
      ENPCABDHIAOCBCDBCJKLM $.
  $}

  ${
    $d x ps $.  $d y ps $.
    exlimivv.1 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
       1-Aug-1995.) $)
    exlimivv $p |- ( E. x E. y ph -> ps ) $=
      ( wex exlimiv ) ADFBCABDEGG $.
  $}

  ${
    $d x ch $.  $d x ph $.  $d y ch $.  $d y ph $.
    exlimdvv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.23 of [Margaris] p. 90.  (Contributed by NM,
       31-Jul-1995.) $)
    exlimdvv $p |- ( ph -> ( E. x E. y ps -> ch ) ) $=
      ( wex exlimdv ) ABEGCDABCEFHH $.
  $}

  ${
    $d x ch $.  $d x ph $.
    exlimddv.1 $e |- ( ph -> E. x ps ) $.
    exlimddv.2 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Existential elimination rule of natural deduction.  (Contributed by
       Mario Carneiro, 15-Jun-2016.) $)
    exlimddv $p |- ( ph -> ch ) $=
      ( wex ex exlimdv mpd ) ABDGCEABCDABCFHIJ $.
  $}

  ${
    $d x ps $.
    $( Theorem 19.27 of [Margaris] p. 90.  (Contributed by NM, 3-Jun-2004.) $)
    19.27v $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
      ( ax-17 19.27h ) ABCBCDE $.
  $}

  ${
    $d x ph $.
    $( Theorem 19.28 of [Margaris] p. 90.  (Contributed by NM, 25-Mar-2004.) $)
    19.28v $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
      ( ax-17 19.28h ) ABCACDE $.
  $}

  ${
    $d x ps $.
    19.36aiv.1 $e |- E. x ( ph -> ps ) $.
    $( Inference from Theorem 19.36 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.) $)
    19.36aiv $p |- ( A. x ph -> ps ) $=
      ( nfv 19.36i ) ABCBCEDF $.
  $}

  ${
    $d x ps $.
    $( Special case of Theorem 19.41 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.) $)
    19.41v $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
      ( ax-17 19.41h ) ABCBCDE $.
  $}

  ${
    $d x ps $.  $d y ps $.
    $( Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers.  (Contributed by
       NM, 30-Apr-1995.) $)
    19.41vv $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x E. y ph /\ ps ) ) $=
      ( wa wex 19.41v exbii bitri ) ABEDFZCFADFZBEZCFKCFBEJLCABDGHKBCGI $.
  $}

  ${
    $d x ps $.  $d y ps $.  $d z ps $.
    $( Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers.  (Contributed by
       NM, 30-Apr-1995.) $)
    19.41vvv $p |- ( E. x E. y E. z ( ph /\ ps ) <->
                     ( E. x E. y E. z ph /\ ps ) ) $=
      ( wa wex 19.41vv exbii 19.41v bitri ) ABFEGDGZCGAEGDGZBFZCGMCGBFLNCABDEHI
      MBCJK $.
  $}

  ${
    $d w ps $.  $d x ps $.  $d y ps $.  $d z ps $.
    $( Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers.  (Contributed by
       FL, 14-Jul-2007.) $)
    19.41vvvv $p |- ( E. w E. x E. y E. z ( ph /\ ps ) <->
                     ( E. w E. x E. y E. z ph /\ ps ) ) $=
      ( wa wex 19.41vvv exbii 19.41v bitri ) ABGEHDHCHZFHAEHDHCHZBGZFHNFHBGMOFA
      BCDEIJNBFKL $.
  $}

  ${
    $d x ph $.
    $( Special case of Theorem 19.42 of [Margaris] p. 90.  (Contributed by NM,
       5-Aug-1993.) $)
    19.42v $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
      ( ax-17 19.42h ) ABCACDE $.
  $}

  ${
    $d x y $.  $d x ps $.
    spvv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ spv with a disjoint variable condition.  (Contributed by
       BJ, 31-May-2019.) $)
    spvv $p |- ( A. x ph -> ps ) $=
      ( spv ) ABCDEF $.
  $}

  ${
    $d x y $.  $d x ps $.
    chvarvv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    chvarvv.2 $e |- ph $.
    $( Version of ~ chvarv with a disjoint variable condition.  (Contributed by
       BJ, 31-May-2019.) $)
    chvarvv $p |- ps $=
      ( spvv mpg ) ABCABCDEGFH $.
  $}

  ${
    $d y ph $.
    $( Distribution of existential quantifiers.  (Contributed by NM,
       9-Mar-1995.) $)
    exdistr $p |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) ) $=
      ( wa wex 19.42v exbii ) ABEDFABDFECABDGH $.
  $}

  ${
    $d y ph $.  $d x ps $.  $d x y $.
    $( Distribute a pair of existential quantifiers (over disjoint variables)
       over a conjunction.  Combination of ~ 19.41v and ~ 19.42v .  For a
       version with fewer disjoint variable conditions but requiring more
       axioms, see ~ eeanv .  (Contributed by BJ, 30-Sep-2022.) $)
    exdistrv $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) ) $=
      ( wa wex exdistr 19.41v bitri ) ABEDFCFABDFZECFACFJEABCDGAJCHI $.
  $}

  ${
    $d x ph $.  $d y ph $.
    $( Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers.  (Contributed by
       NM, 16-Mar-1995.) $)
    19.42vv $p |- ( E. x E. y ( ph /\ ps ) <-> ( ph /\ E. x E. y ps ) ) $=
      ( wa wex exdistr 19.42v bitri ) ABEDFCFABDFZECFAJCFEABCDGAJCHI $.
  $}

  ${
    $d x ph $.  $d y ph $.  $d z ph $.
    $( Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers.  (Contributed by
       NM, 21-Sep-2011.) $)
    19.42vvv $p |- ( E. x E. y E. z ( ph /\ ps )
                       <-> ( ph /\ E. x E. y E. z ps ) ) $=
      ( wa wex 19.42vv exbii 19.42v bitri ) ABFEGDGZCGABEGDGZFZCGAMCGFLNCABDEHI
      AMCJK $.
  $}

  ${
    $d w ph $.  $d x ph $.  $d y ph $.  $d z ph $.
    $( Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers.  (Contributed by
       Jim Kingdon, 23-Nov-2019.) $)
    19.42vvvv $p |- ( E. w E. x E. y E. z ( ph /\ ps ) <->
                     ( ph /\ E. w E. x E. y E. z ps ) ) $=
      ( wa wex 19.42vv 2exbii bitri ) ABGEHDHZCHFHABEHDHZGZCHFHAMCHFHGLNFCABDEI
      JAMFCIK $.
  $}

  ${
    $d y ph $.  $d z ph $.
    $( Distribution of existential quantifiers.  (Contributed by NM,
       17-Mar-1995.) $)
    exdistr2 $p |- ( E. x E. y E. z ( ph /\ ps ) <->
                   E. x ( ph /\ E. y E. z ps ) ) $=
      ( wa wex 19.42vv exbii ) ABFEGDGABEGDGFCABDEHI $.
  $}

  ${
    $d y ph $.  $d z ph $.  $d z ps $.
    $( Distribution of existential quantifiers.  (Contributed by NM,
       9-Mar-1995.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3exdistr $p |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <->
                E. x ( ph /\ E. y ( ps /\ E. z ch ) ) ) $=
      ( w3a wex wa 3anass 2exbii 19.42vv exdistr anbi2i 3bitri exbii ) ABCGZFHE
      HZABCFHIEHZIZDRABCIZIZFHEHAUAFHEHZITQUBEFABCJKAUAEFLUCSABCEFMNOP $.
  $}

  ${
    $d y ph $.  $d z ph $.  $d w ph $.  $d z ps $.  $d w ps $.  $d w ch $.
    $( Distribution of existential quantifiers.  (Contributed by NM,
       9-Mar-1995.) $)
    4exdistr $p |- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
                E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) ) $=
      ( wa wex anass exbii 19.42v anbi2i 3bitri bitri ) ABICDIZIZHJZGJZFJZABCDH
      JIZGJIZFJIZEUAAUCIZFJUDTUEFTABUBIZIZGJAUFGJZIUESUGGSABQIZIZHJZUGRUJHABQKL
      UKAUIHJZIABQHJZIZIUGAUIHMULUNABQHMNUNUFAUMUBBCDHMNNOPLAUFGMUHUCABUBGMNOLA
      UCFMPL $.
  $}

  ${
    $d y ph $.  $d x ps $.
    cbvalv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitition.
       (Contributed by NM, 5-Aug-1993.) $)
    cbvalv $p |- ( A. x ph <-> A. y ps ) $=
      ( ax-17 cbvalh ) ABCDADFBCFEG $.

    $( Rule used to change bound variables, using implicit substitition.
       (Contributed by NM, 5-Aug-1993.) $)
    cbvexv $p |- ( E. x ph <-> E. y ps ) $=
      ( ax-17 cbvexh ) ABCDADFBCFEG $.
  $}

  ${
    $d x y $.  $d x ps $.  $d y ph $.
    cbvalvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable.  See ~ cbvalv for a version with fewer disjoint
       variable conditions.  (Contributed by NM, 9-Apr-2017.)  Avoid ~ ax-7 .
       (Revised by GG, 25-Aug-2024.) $)
    cbvalvw $p |- ( A. x ph <-> A. y ps ) $=
      ( wal spv alrimiv weq wb equcoms biimprd spimv impbii ) ACFZBDFZOBDABCDEG
      HPACBADCDCIABABJCDEKLMHN $.
    $( $j usage 'cbvalvw' avoids 'ax-7'; $)

    $( Change bound variable.  See ~ cbvexv for a version with fewer disjoint
       variable conditions.  (Contributed by NM, 19-Apr-2017.)  Avoid ~ ax-7 .
       (Revised by GG, 25-Aug-2024.) $)
    cbvexvw $p |- ( E. x ph <-> E. y ps ) $=
      ( wex wi weq biimpd equcoms spimev exlimiv biimprd impbii ) ACFZBDFZAPCAB
      DCABGCDCDHZABEIJKLBODBACDQABEMKLN $.
    $( $j usage 'cbvexvw' avoids 'ax-7'; $)
  $}

  ${
    $d y x $.  $d y z $.  $d w x $.  $d w z $.
    cbval2.1 $e |- F/ z ph $.
    cbval2.2 $e |- F/ w ph $.
    cbval2.3 $e |- F/ x ps $.
    cbval2.4 $e |- F/ y ps $.
    cbval2.5 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 22-Dec-2003.)  (Revised by Mario Carneiro,
       6-Oct-2016.)  (Proof shortened by Wolf Lammen, 22-Apr-2018.) $)
    cbval2 $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
      ( wal nfal weq wi nfv nfim wb cbval 19.21v pm5.74d 3bitr3i pm5.74ri
      expcom ) ADLZBFLZCEAEDGMBCFIMCENZUEUFUGAOZDLUGBOZFLUGUEOUGUFOUHUIDFUGAFUG
      FPHQUGBDUGDPJQDFNZUGABUGUJABRKUDUASUGADTUGBFTUBUCS $.

    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 14-Sep-2003.)  (Revised by Mario Carneiro,
       6-Oct-2016.) $)
    cbvex2 $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
      ( wex nfex weq wb wa nfv nfan cbvex 19.42v wi expcom pm5.32d pm5.32 mpbir
      3bitr3i ) ADLZBFLZCEAEDGMBCFIMCENZUGUHOUAUIUGPZUIUHPZOUIAPZDLUIBPZFLUJUKU
      LUMDFUIAFUIFQHRUIBDUIDQJRDFNZUIABUIUNABOKUBUCSUIADTUIBFTUFUIUGUHUDUES $.
  $}

  ${
    $d z w ph $.  $d x y ps $.  $d x w $.  $d z y $.
    cbval2v.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 4-Feb-2005.) $)
    cbval2v $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
      ( nfv cbval2 ) ABCDEFAEHAFHBCHBDHGI $.

    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 26-Jul-1995.) $)
    cbvex2v $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
      ( nfv cbvex2 ) ABCDEFAEHAFHBCHBDHGI $.
  $}

  ${
    $d x ph $.  $d x ch $.
    cbvald.1 $e |- F/ y ph $.
    cbvald.2 $e |- ( ph -> F/ y ps ) $.
    cbvald.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Deduction used to change bound variables, using implicit substitution,
       particularly useful in conjunction with ~ dvelim .  (Contributed by NM,
       2-Jan-2002.)  (Revised by Mario Carneiro, 6-Oct-2016.)  (Revised by Wolf
       Lammen, 13-May-2018.) $)
    cbvald $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
      ( nfv wnf a1i cbv2 ) ABCDEADIFGCDJACDIKHL $.
  $}

  ${
    $d x ph $.  $d x ch $.
    cbvexdh.1 $e |- ( ph -> A. y ph ) $.
    cbvexdh.2 $e |- ( ph -> ( ps -> A. y ps ) ) $.
    cbvexdh.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Deduction used to change bound variables, using implicit substitition,
       particularly useful in conjunction with ~ dvelim .  (Contributed by NM,
       2-Jan-2002.)  (Proof rewritten by Jim Kingdon, 30-Dec-2017.) $)
    cbvexdh $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
      ( wex ax-17 cv wceq wa wb wi equsexd simpr eximi biimtrrdi wal bicom1 syl
      hbex equcomi imim12i exlimdh eximdh 19.12 syl6 a1i exlimd2 impbid ) ABDIZ
      CEIZABUNDADJZCDECDJZUCABEKZDKZLZCMZEIUNACBEDFGAURUQLZBCNZOUSCBNZOHUSVAVBV
      CEDUDBCUAUEUBPUTCEUSCQRSUFACUMEFAUMBETZDIUMETABVDDUOGUGBDEUHUIACVABMZDIUM
      ABCDEUOCCDTOAUPUJHPVEBDVABQRSUKUL $.
  $}

  ${
    $d x ph $.  $d x ch $.
    cbvexd.1 $e |- F/ y ph $.
    cbvexd.2 $e |- ( ph -> F/ y ps ) $.
    cbvexd.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Deduction used to change bound variables, using implicit substitution,
       particularly useful in conjunction with ~ dvelim .  (Contributed by NM,
       2-Jan-2002.)  (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof rewritten
       by Jim Kingdon, 10-Jun-2018.) $)
    cbvexd $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
      ( nfri nfrd cbvexdh ) ABCDEAEFIABEGJHK $.
  $}

  ${
    $d ps y $.  $d ch x $.  $d ph x $.  $d ph y $.
    cbvaldva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
    $( Rule used to change the bound variable in a universal quantifier with
       implicit substitution.  Deduction form.  (Contributed by David Moews,
       1-May-2017.) $)
    cbvaldva $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
      ( nfv nfvd weq wb ex cbvald ) ABCDEAEGABEHADEIBCJFKL $.

    $( Rule used to change the bound variable in an existential quantifier with
       implicit substitution.  Deduction form.  (Contributed by David Moews,
       1-May-2017.) $)
    cbvexdva $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
      ( nfv nfvd weq wb ex cbvexd ) ABCDEAEGABEHADEIBCJFKL $.
  $}

  ${
    $d ps y $.  $d ch x $.  $d ph x y $.
    cbvaldvaw.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
    $( Rule used to change the bound variable in a universal quantifier with
       implicit substitution.  Deduction form.  Version of ~ cbvaldva with a
       disjoint variable condition.  (Contributed by David Moews, 1-May-2017.)
       (Revised by GG, 10-Jan-2024.)  (Revised by Wolf Lammen, 10-Feb-2024.) $)
    cbvaldvaw $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
      ( wal wi weq wb ancoms pm5.74da cbvalvw 19.21v 3bitr3i pm5.74ri ) ABDGZCE
      GZABHZDGACHZEGAQHARHSTDEDEIZABCAUABCJFKLMABDNACENOP $.

    $( Rule used to change the bound variable in an existential quantifier with
       implicit substitution.  Deduction form.  Version of ~ cbvexdva with a
       disjoint variable condition.  (Contributed by David Moews, 1-May-2017.)
       (Revised by GG, 10-Jan-2024.)  (Revised by Wolf Lammen, 10-Feb-2024.) $)
    cbvexdvaw $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
      ( cbvexdva ) ABCDEFG $.
  $}

  ${
    $d w z ph $.  $d x y ps $.  $d w x y z $.
    cbval2vw.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 4-Feb-2005.)  (Revised by GG, 10-Jan-2024.) $)
    cbval2vw $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
      ( wal weq cbvaldvaw cbvalvw ) ADHBFHCECEIABDFGJK $.

    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 26-Jul-1995.)  (Revised by GG, 10-Jan-2024.) $)
    cbvex2vw $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
      ( wex weq cbvexdvaw cbvexvw ) ADHBFHCECEIABDFGJK $.
  $}

  ${
    $v f $.
    $v g $.
    $( Define temporary individual variables. $)
    cbvex4v.vf $f setvar f $.
    cbvex4v.vg $f setvar g $.
    $d w z ch $.  $d u v ph $.  $d x y ps $.  $d f g ps $.  $d f w $.
    $d g z $.  $d u v w z $.  $d u w x z $.  $d v w y z $.  $d w x y z $.
    cbvex4v.1 $e |- ( ( x = v /\ y = u ) -> ( ph <-> ps ) ) $.
    cbvex4v.2 $e |- ( ( z = f /\ w = g ) -> ( ps <-> ch ) ) $.
    $( Rule used to change bound variables, using implicit substitition.
       (Contributed by NM, 26-Jul-1995.) $)
    cbvex4v $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $=
      ( wex weq wa 2exbidv cbvex2v 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZINH
      NUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $.
  $}

  ${
    eean.1 $e |- F/ y ph $.
    eean.2 $e |- F/ x ps $.
    $( Rearrange existential quantifiers.  (Contributed by NM, 27-Oct-2010.)
       (Revised by Mario Carneiro, 6-Oct-2016.) $)
    eean $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) ) $=
      ( wa wex 19.42 exbii nfex 19.41 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
      CBCDFKLM $.
  $}

  ${
    $d y ph $.  $d x ps $.
    $( Rearrange existential quantifiers.  (Contributed by NM, 26-Jul-1995.) $)
    eeanv $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) ) $=
      ( nfv eean ) ABCDADEBCEF $.
  $}

  ${
    $d y ph $.  $d z ph $.  $d x z ps $.  $d x y ch $.
    $( Rearrange existential quantifiers.  (Contributed by NM, 26-Jul-1995.)
       (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    eeeanv $p |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <->
                 ( E. x ph /\ E. y ps /\ E. z ch ) ) $=
      ( w3a wex wa df-3an 3exbii eeanv exbii anbi1i 19.41v 3bitr4i 3bitri ) ABC
      GZFHEHDHABIZCIZFHEHZDHSEHZCFHZIZDHZADHZBEHZUCGZRTDEFABCJKUAUDDSCEFLMUBDHZ
      UCIUFUGIZUCIUEUHUIUJUCABDELNUBUCDOUFUGUCJPQ $.
  $}

  ${
    $d z ph $.  $d w ph $.  $d x ps $.  $d y ps $.  $d y z $.  $d w x $.
    $( Rearrange existential quantifiers.  (Contributed by NM, 31-Jul-1995.) $)
    ee4anv $p |- ( E. x E. y E. z E. w ( ph /\ ps ) <->
                  ( E. x E. y ph /\ E. z E. w ps ) ) $=
      ( wa wex excom exbii eeanv 2exbii 3bitri ) ABGFHZEHDHZCHNDHZEHZCHADHZBFHZ
      GZEHCHRCHSEHGOQCNDEIJPTCEABDFKLRSCEKM $.
  $}

  ${
    $v s $.
    $( Define a temporary individual variable. $)
    ee8anv.vs $f setvar s $.

    $d v ph $.  $d u ph $.  $d t ph $.  $d s ph $.  $d x ps $.  $d y ps $.
    $d z ps $.  $d w ps $.  $d s x $.  $d s y $.  $d s z $.  $d t w $.
    $d t x $.  $d t y $.  $d u w $.  $d u x $.  $d u z $.  $d v w $.  $d v y $.
    $d v z $.
    $( Rearrange existential quantifiers.  (Contributed by Jim Kingdon,
       23-Nov-2019.) $)
    ee8anv $p |- ( E. x E. y E. z E. w E. v E. u E. t E. s ( ph /\ ps ) <->
                  ( E. x E. y E. z E. w ph /\ E. v E. u E. t E. s ps ) ) $=
      ( wa wex exrot4 2exbii ee4anv 3bitri ) ABKJLILZHLGLFLELZDLCLQFLELZHLGLZDL
      CLAFLELZBJLILZKZHLGLZDLCLUADLCLUBHLGLKRTCDQEFGHMNTUDCDSUCGHABEFIJONNUAUBC
      DGHOP $.
  $}

  ${
    $d x ph $.
    nexdv.1 $e |- ( ph -> -. ps ) $.
    $( Deduction for generalization rule for negated wff.  (Contributed by NM,
       5-Aug-1993.) $)
    nexdv $p |- ( ph -> -. E. x ps ) $=
      ( ax-17 nexd ) ABCACEDF $.
  $}

  ${
    $d x ps $.
    chv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    chv.2 $e |- ph $.
    $( Implicit substitution of ` y ` for ` x ` into a theorem.  (Contributed
       by NM, 20-Apr-1994.) $)
    chvarv $p |- ps $=
      ( spv mpg ) ABCABCDEGFH $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  More substitution theorems
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

$( The theorems in this section make use of the $d statement. $)

  ${
    $d x y $.
    $( ` x ` is not free in ` [ y / x ] ph ` when ` x ` and ` y ` are distinct.
       (Contributed by NM, 5-Aug-1993.)  (Proof by Jim Kingdon, 16-Dec-2017.)
       (New usage is discouraged.) $)
    hbs1 $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
      ( wsb weq wi wal sb6 ax-ial hbxfrbi ) ABCDBCEAFZBGBABCHKBIJ $.

    $( ` x ` is not free in ` [ y / x ] ph ` when ` x ` and ` y ` are distinct.
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nfs1v $p |- F/ x [ y / x ] ph $=
      ( wsb hbs1 nfi ) ABCDBABCEF $.
  $}

  ${
    $d y ph $.
    $( Two ways of expressing " ` x ` is (effectively) not free in ` ph ` ."
       (Contributed by NM, 29-May-2009.) $)
    sbhb $p |- ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) $=
      ( wal wi wsb ax-17 sb8h imbi2i 19.21v bitr4i ) AABDZEAABCFZCDZEAMECDLNAAB
      CACGHIAMCJK $.
  $}

  ${
    $d x z $.  $d y z $.
    hbsbv.1 $e |- ( ph -> A. z ph ) $.
    $( This is a version of ~ hbsb with an extra distinct variable constraint,
       on ` z ` and ` x ` .  (Contributed by Jim Kingdon, 25-Dec-2017.) $)
    hbsbv $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $=
      ( wsb weq wi wa wex df-sb ax-17 hbim hban hbex hbxfrbi ) ABCFBCGZAHZQAIZB
      JZIDABCKRTDQADQDLZEMSDBQADUAENONP $.
  $}

  ${
    $d x y $.  $d y z $.
    nfsbxy.1 $e |- F/ z ph $.
    $( Similar to ~ hbsb but with an extra distinct variable constraint, on
       ` x ` and ` y ` .  (Contributed by Jim Kingdon, 19-Mar-2018.) $)
    nfsbxy $p |- F/ z [ y / x ] ph $=
      ( weq wal wi wo wsb wnf ax-bndl nfs1v drsb1 drnf2 mpbii a16nf df-nf albii
      jaoi wa wex sb5 nfa1 sp a1i nfand nfexd nfxfrd sylbir ax-mp ) DBFDGZDCFDG
      ZBCFZUNDGHDGZBGZIZIABCJZDKZBCDLULUSUQULADCJZDKUSADCMUTURDBDADBCNOPUMUSUPU
      RDCDQUPUNDKZBGZUSVAUOBUNDRSURUNAUAZBUBVBDABCUCVBVCDBVABUDVBUNADVABUEADKVB
      EUFUGUHUIUJTTUK $.
  $}

  ${
    $d x y $.  $d y z $.
    $( Closed form of ~ nfsbxy .  (Contributed by Jim Kingdon, 9-May-2018.) $)
    nfsbxyt $p |- ( A. x F/ z ph -> F/ z [ y / x ] ph ) $=
      ( weq wal wi wo wnf wsb ax-bndl nfs1v drsb1 drnf2 mpbii a1d wa nfa1 jaoi
      sp a16nf df-nf albii wex sb5 nfan adantr adantl nfand nfexd nfxfrd sylbir
      ex ax-mp ) DBEDFZDCEDFZBCEZUQDFGDFZBFZHZHADIZBFZABCJZDIZGZBCDKUOVEUTUOVDV
      BUOADCJZDIVDADCLVFVCDBDADBCMNOPUPVEUSUPVDVBVCDCDUAPUSUQDIZBFZVEVGURBUQDUB
      UCVHVBVDVCUQAQZBUDVHVBQZDABCUEVJVIDBVHVBBVGBRVABRUFVJUQADVHVGVBVGBTUGVBVA
      VHVABTUHUIUJUKUMULSSUN $.
  $}

  ${
    $d x z $.  $d y z $.
    sbco2vlem.1 $e |- ( ph -> A. z ph ) $.
    $( This is a version of ~ sbco2 where ` z ` is distinct from ` x ` and from
       ` y ` .  It is a lemma on the way to proving ~ sbco2v which only
       requires that ` z ` and ` x ` be distinct.  (Contributed by Jim Kingdon,
       25-Dec-2017.)  Remove one disjoint variable condition.  (Revised by Jim
       Kingdon, 3-Feb-2018.) $)
    sbco2vlem $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( wsb hbsbv sbequ sbieh ) ABDFABCFDCABCDEGADCBHI $.
  $}

  ${
    $d x z w $.  $d y w $.  $d z w $.  $d ph w $.
    sbco2vh.1 $e |- ( ph -> A. z ph ) $.
    $( This is a version of ~ sbco2 where ` z ` is distinct from ` x ` .
       (Contributed by Jim Kingdon, 12-Feb-2018.) $)
    sbco2vh $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( vw wsb sbco2vlem sbbii ax-17 3bitr3i ) ABDGZDFGZFCGABFGZFCGLDCGABCGMNFC
      ABFDEHILDCFLFJHABCFAFJHK $.
  $}

  ${
    $d w y z $.  $d w ph $.  $d w x $.
    nfsb.1 $e |- F/ z ph $.
    $( If ` z ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` when
       ` y ` and ` z ` are distinct.  (Contributed by Mario Carneiro,
       11-Aug-2016.)  (Proof rewritten by Jim Kingdon, 19-Mar-2018.) $)
    nfsb $p |- F/ z [ y / x ] ph $=
      ( vw wsb wnf nfsbxy ax-17 sbco2vh nfbii mpbi ) ABFGZFCGZDHABCGZDHNFCDABFD
      EIIOPDABCFAFJKLM $.
  $}

  ${
    $d x z $.  $d y z $.
    nfsbv.nf $e |- F/ z ph $.
    $( If ` z ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` when
       ` z ` is distinct from ` x ` and ` y ` .  Version of ~ nfsb requiring
       more disjoint variables.  (Contributed by Wolf Lammen, 7-Feb-2023.)
       Remove disjoint variable condition on ` x , y ` .  (Revised by Steven
       Nguyen, 13-Aug-2023.)  Reduce axiom usage.  (Revised by GG,
       25-Aug-2024.) $)
    nfsbv $p |- F/ z [ y / x ] ph $=
      ( wsb weq wi wa wex df-sb nfv nfim nfan nfex nfxfr ) ABCFBCGZAHZQAIZBJZID
      ABCKRTDQADQDLZEMSDBQADUAENONP $.
    $( $j usage 'nfsbv' avoids 'ax-io' 'ax-8' 'ax-10' 'ax-11' 'ax-i12'
       'ax-bndl' 'ax-i9'; $)
  $}

  ${
    $d x z $.  $d y z $.
    sbco2v.1 $e |- F/ z ph $.
    $( Version of ~ sbco2 with disjoint variable conditions.  (Contributed by
       Wolf Lammen, 29-Apr-2023.) $)
    sbco2v $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( wsb nfsbv sbequ sbiev ) ABDFABCFDCABCDEGADCBHI $.
  $}

  ${
    $d y z $.
    hbsb.1 $e |- ( ph -> A. z ph ) $.
    $( If ` z ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` when
       ` y ` and ` z ` are distinct.  (Contributed by NM, 12-Aug-1993.)  (Proof
       rewritten by Jim Kingdon, 22-Mar-2018.) $)
    hbsb $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $=
      ( wsb nfi nfsb nfri ) ABCFDABCDADEGHI $.
  $}

  ${
    $d x z $.  $d x y $.
    $( Lemma for ~ equsb3 .  (Contributed by NM, 4-Dec-2005.)  (Proof shortened
       by Andrew Salmon, 14-Jun-2011.) $)
    equsb3lem $p |- ( [ y / x ] x = z <-> y = z ) $=
      ( cv wceq ax-17 equequ1 sbieh ) ADCDZEBDIEZABJAFABCGH $.
  $}

  ${
    $d w x z $.  $d w y $.
    $( Substitution applied to an atomic wff.  (Contributed by Raph Levien and
       FL, 4-Dec-2005.) $)
    equsb3 $p |- ( [ y / x ] x = z <-> y = z ) $=
      ( vw weq wsb equsb3lem sbbii ax-17 sbco2vh 3bitr3i ) ACEZADFZDBFDCEZDBFLA
      BFBCEMNDBADCGHLABDLDIJDBCGK $.
  $}

  ${
    $d z ph $.  $d z ps $.  $d z x $.  $d z y $.

    $( Negation inside and outside of substitution are equivalent.
       (Contributed by NM, 5-Aug-1993.)  (Proof rewritten by Jim Kingdon,
       3-Feb-2018.) $)
    sbn $p |- ( [ y / x ] -. ph <-> -. [ y / x ] ph ) $=
      ( vz wn wsb sbnv sbbii bitri ax-17 hbn sbco2vh notbii 3bitr3i ) AEZBDFZDC
      FZABDFZDCFZEZOBCFABCFZEQREZDCFTPUBDCABDGHRDCGIOBCDADADJZKLSUAABCDUCLMN $.

    $( Implication inside and outside of substitution are equivalent.
       (Contributed by NM, 5-Aug-1993.)  (Proof rewritten by Jim Kingdon,
       3-Feb-2018.) $)
    sbim $p |- ( [ y / x ] ( ph -> ps ) <->
        ( [ y / x ] ph -> [ y / x ] ps ) ) $=
      ( vz wi wsb sbimv sbbii bitri ax-17 sbco2vh imbi12i 3bitr3i ) ABFZCEGZEDG
      ZACEGZEDGZBCEGZEDGZFZOCDGACDGZBCDGZFQRTFZEDGUBPUEEDABCEHIRTEDHJOCDEOEKLSU
      CUAUDACDEAEKLBCDEBEKLMN $.

    $( Logical OR inside and outside of substitution are equivalent.
       (Contributed by NM, 29-Sep-2002.)  (Proof rewritten by Jim Kingdon,
       3-Feb-2018.) $)
    sbor $p |- ( [ y / x ] ( ph \/ ps ) <->
        ( [ y / x ] ph \/ [ y / x ] ps ) ) $=
      ( vz wo wsb sborv sbbii bitri ax-17 sbco2vh orbi12i 3bitr3i ) ABFZCEGZEDG
      ZACEGZEDGZBCEGZEDGZFZOCDGACDGZBCDGZFQRTFZEDGUBPUEEDABCEHIRTEDHJOCDEOEKLSU
      CUAUDACDEAEKLBCDEBEKLMN $.

    $( Conjunction inside and outside of a substitution are equivalent.
       (Contributed by NM, 5-Aug-1993.)  (Proof rewritten by Jim Kingdon,
       3-Feb-2018.) $)
    sban $p |- ( [ y / x ] ( ph /\ ps ) <->
        ( [ y / x ] ph /\ [ y / x ] ps ) ) $=
      ( vz wa wsb sbanv sbbii bitri ax-17 sbco2vh anbi12i 3bitr3i ) ABFZCEGZEDG
      ZACEGZEDGZBCEGZEDGZFZOCDGACDGZBCDGZFQRTFZEDGUBPUEEDABCEHIRTEDHJOCDEOEKLSU
      CUAUDACDEAEKLBCDEBEKLMN $.
  $}

  ${
    sbrim.1 $e |- ( ph -> A. x ph ) $.
    $( Substitution with a variable not free in antecedent affects only the
       consequent.  (Contributed by NM, 5-Aug-1993.) $)
    sbrim $p |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) $=
      ( wi wsb sbim sbh imbi1i bitri ) ABFCDGACDGZBCDGZFAMFABCDHLAMACDEIJK $.
  $}

  ${
    sblim.1 $e |- F/ x ps $.
    $( Substitution with a variable not free in consequent affects only the
       antecedent.  (Contributed by NM, 14-Nov-2013.)  (Revised by Mario
       Carneiro, 4-Oct-2016.) $)
    sblim $p |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) ) $=
      ( wi wsb sbim sbf imbi2i bitri ) ABFCDGACDGZBCDGZFLBFABCDHMBLBCDEIJK $.
  $}

  $( Conjunction inside and outside of a substitution are equivalent.
     (Contributed by NM, 14-Dec-2006.) $)
  sb3an $p |- ( [ y / x ] ( ph /\ ps /\ ch ) <->
              ( [ y / x ] ph /\ [ y / x ] ps /\ [ y / x ] ch ) ) $=
    ( w3a wsb wa df-3an sbbii sban anbi1i bitr4i 3bitri ) ABCFZDEGABHZCHZDEGPDE
    GZCDEGZHZADEGZBDEGZSFZOQDEABCIJPCDEKTUAUBHZSHUCRUDSABDEKLUAUBSIMN $.

  $( Equivalence inside and outside of a substitution are equivalent.
     (Contributed by NM, 5-Aug-1993.) $)
  sbbi $p |- ( [ y / x ] ( ph <-> ps )
     <-> ( [ y / x ] ph <-> [ y / x ] ps ) ) $=
    ( wb wsb wi wa dfbi2 sbbii sbim anbi12i sban 3bitr4i bitri ) ABEZCDFABGZBAG
    ZHZCDFZACDFZBCDFZEZPSCDABIJQCDFZRCDFZHUAUBGZUBUAGZHTUCUDUFUEUGABCDKBACDKLQR
    CDMUAUBINO $.

  ${
    sblbis.1 $e |- ( [ y / x ] ph <-> ps ) $.
    $( Introduce left biconditional inside of a substitution.  (Contributed by
       NM, 19-Aug-1993.) $)
    sblbis $p |- ( [ y / x ] ( ch <-> ph ) <-> ( [ y / x ] ch <-> ps ) ) $=
      ( wb wsb sbbi bibi2i bitri ) CAGDEHCDEHZADEHZGLBGCADEIMBLFJK $.
  $}

  ${
    sbrbis.1 $e |- ( [ y / x ] ph <-> ps ) $.
    $( Introduce right biconditional inside of a substitution.  (Contributed by
       NM, 18-Aug-1993.) $)
    sbrbis $p |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> [ y / x ] ch ) ) $=
      ( wb wsb sbbi bibi1i bitri ) ACGDEHADEHZCDEHZGBMGACDEILBMFJK $.
  $}

  ${
    sbrbif.1 $e |- ( ch -> A. x ch ) $.
    sbrbif.2 $e |- ( [ y / x ] ph <-> ps ) $.
    $( Introduce right biconditional inside of a substitution.  (Contributed by
       NM, 18-Aug-1993.) $)
    sbrbif $p |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> ch ) ) $=
      ( wb wsb sbrbis sbh bibi2i bitri ) ACHDEIBCDEIZHBCHABCDEGJNCBCDEFKLM $.
  $}

  ${
    $d y z $.
    sbco2yz.1 $e |- F/ z ph $.
    $( This is a version of ~ sbco2 where ` z ` is distinct from ` y ` .  It is
       a lemma on the way to proving ~ sbco2 which has no distinct variable
       constraints.  (Contributed by Jim Kingdon, 19-Mar-2018.) $)
    sbco2yz $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( wsb nfsb nfri sbequ sbieh ) ABDFABCFZDCKDABCDEGHADCBIJ $.
  $}

  ${
    $d w z $.  $d w x $.  $d w y $.  $d ph w $.
    sbco2h.1 $e |- ( ph -> A. z ph ) $.
    $( A composition law for substitution.  (Contributed by NM, 30-Jun-1994.)
       (Proof rewritten by Jim Kingdon, 19-Mar-2018.) $)
    sbco2h $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( vw wsb nfi sbco2yz sbbii nfv 3bitr3i ) ABDGZDFGZFCGABFGZFCGMDCGABCGNOFC
      ABFDADEHIJMDCFMFKIABCFAFKIL $.
  $}

  ${
    sbco2.1 $e |- F/ z ph $.
    $( A composition law for substitution.  (Contributed by NM, 30-Jun-1994.)
       (Revised by Mario Carneiro, 6-Oct-2016.) $)
    sbco2 $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( nfri sbco2h ) ABCDADEFG $.
  $}

  ${
    sbco2d.1 $e |- ( ph -> A. x ph ) $.
    sbco2d.2 $e |- ( ph -> A. z ph ) $.
    sbco2d.3 $e |- ( ph -> ( ps -> A. z ps ) ) $.
    $( A composition law for substitution.  (Contributed by NM, 5-Aug-1993.) $)
    sbco2d $p |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) ) $=
      ( wsb wi hbim1 sbco2h sbrim sbbii bitri 3bitr3i pm5.74ri ) ABCEIZEDIZBCDI
      ZABJZCEIZEDIZUACDIASJZATJUACDEABEGHKLUCARJZEDIUDUBUEEDABCEFMNAREDGMOABCDF
      MPQ $.
  $}

  ${
    $d x z $.
    sbco2vd.1 $e |- ( ph -> A. x ph ) $.
    sbco2vd.2 $e |- ( ph -> A. z ph ) $.
    sbco2vd.3 $e |- ( ph -> ( ps -> A. z ps ) ) $.
    $( Version of ~ sbco2d with a distinct variable constraint between ` x `
       and ` z ` .  (Contributed by Jim Kingdon, 19-Feb-2018.) $)
    sbco2vd $p |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) ) $=
      ( wsb wi hbim1 sbco2vh sbrim sbbii bitri 3bitr3i pm5.74ri ) ABCEIZEDIZBCD
      IZABJZCEIZEDIZUACDIASJZATJUACDEABEGHKLUCARJZEDIUDUBUEEDABCEFMNAREDGMOABCD
      FMPQ $.
  $}

  $( A composition law for substitution.  (Contributed by NM, 5-Aug-1993.) $)
  sbco $p |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) $=
    ( wsb wb weq equsb2 sbequ12 bicomd sbimi ax-mp sbbi mpbi ) ACBDZAEZBCDZNBCD
    ABCDECBFZBCDPBCGQOBCQANACBHIJKNABCLM $.

  ${
    $d x y $.
    $( Version of ~ sbco3 with a distinct variable constraint between ` x ` and
       ` y ` .  (Contributed by Jim Kingdon, 19-Feb-2018.) $)
    sbco3v $p |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) $=
      ( wsb nfs1v nfri sbco2vh sbco sbbii bitr3i ) ABCEZCDELCBEZBDEACBEZBDELCDB
      LBABCFGHMNBDACBIJK $.
  $}

  $( Relationship between composition and commutativity for substitution.
     (Contributed by Jim Kingdon, 28-Feb-2018.) $)
  sbcocom $p |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) $=
    ( wsb wb weq equsb1 sbequ sbimi ax-mp sbbi mpbi ) ABCEZABDEZFZCDEZNCDEOCDEF
    CDGZCDEQCDHRPCDACDBIJKNOCDLM $.

  ${
    $d x z $.
    $( Version of ~ sbcom with a distinct variable constraint between ` x ` and
       ` z ` .  (Contributed by Jim Kingdon, 28-Feb-2018.) $)
    sbcomv $p |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph ) $=
      ( wsb sbco3v sbcocom 3bitr3i ) ABDEDCEADBEBCEABCEDCEADCEBCEABDCFABDCGADBC
      GH $.
  $}

  ${
    $d x y $.  $d y z $.
    $( Version of ~ sbcom with distinct variable constraints between ` x ` and
       ` y ` , and ` y ` and ` z ` .  (Contributed by Jim Kingdon,
       21-Mar-2018.) $)
    sbcomxyyz $p |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph ) $=
      ( weq wal wi wo wsb ax-ial drsb1 sbbidh bitr3d sbequ12 sps wnf nfs1v nfrd
      wb a1i ax-bndl hbae df-nf albii nfsb nfr wa nfnf1 nfa1 nfan nfri sylan9bb
      ex adantl sbiedh syld bicomd sylbir jaoi ax-mp ) DBEZDFZDCEZDFZBCEZVEDFZG
      DFZBFZHZHABCIZDCIZADCIZBCIZSZBCDUAVBVNVIVBVLDCIVKVMVBVLVJDCVADJADBCKLVLDB
      CKMVDVNVHVDVJVKVMVCVJVKSDVJDCNZOVDAVLBCDCBUBVCAVLSDADCNOLMVHVEDPZBFZVNVPV
      GBVEDUCUDVQVMVKVQVLVKBCVPBJVQVKBVKBPVQVJDCBABCQUETRVPVEVLVKSZGBVPVEVFVRVE
      DUFVPVFVRVPVFUGZAVKDCVSDVPVFDVEDUHVEDUIUJUKVSVKDVKDPVSVJDCQTRVFVCAVKSZGZV
      PVEWADVEVCVTVEAVJVCVKABCNVOULUMOUNUOUMUPOUOUQURUSUSUT $.
  $}

  ${
    $d x z $.  $d y z $.
    $( Version of ~ sbco3 with distinct variable constraints between ` x ` and
       ` z ` , and ` y ` and ` z ` .  Lemma for proving ~ sbco3 .  (Contributed
       by Jim Kingdon, 22-Mar-2018.) $)
    sbco3xzyz $p |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) $=
      ( wsb sbcomxyyz sbcocom 3bitr4i ) ABDECDEACDEBDEABCECDEACBEBDEABDCFABCDGA
      CBDGH $.
  $}

  ${
    $d w x $.  $d w y $.  $d w ph $.
    $( A composition law for substitution.  (Contributed by NM, 5-Aug-1993.)
       (Proof rewritten by Jim Kingdon, 22-Mar-2018.) $)
    sbco3 $p |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) $=
      ( vw wsb sbco3xzyz sbbii ax-17 sbco2h 3bitr3i ) ABCFZCEFZEDFACBFZBEFZEDFL
      CDFNBDFMOEDABCEGHLCDELEIJNBDENEIJK $.
  $}

  $( A commutativity law for substitution.  (Contributed by NM, 27-May-1997.)
     (Proof rewritten by Jim Kingdon, 22-Mar-2018.) $)
  sbcom $p |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph ) $=
    ( wsb sbco3 sbcocom 3bitr3i ) ABDEDCEADBEBCEABCEDCEADCEBCEABDCFABDCGADBCGH
    $.

  ${
    $d w y z $.  $d w ph $.  $d w x $.
    $( Closed form of ~ nfsb .  (Contributed by Jim Kingdon, 9-May-2018.) $)
    nfsbt $p |- ( A. x F/ z ph -> F/ z [ y / x ] ph ) $=
      ( vw wnf wal wsb ax-17 nfsbxyt alimi syl nfv sbco2 nfbii sylib ) ADFBGZQE
      GZABCHZDFZQEIRABEHZECHZDFZTRUADFZEGUCQUDEABEDJKUAECDJLUBSDABCEAEMNOPL $.
  $}

  ${
    $d y z $.
    nfsbd.1 $e |- F/ x ph $.
    nfsbd.2 $e |- ( ph -> F/ z ps ) $.
    $( Deduction version of ~ nfsb .  (Contributed by NM, 15-Feb-2013.) $)
    nfsbd $p |- ( ph -> F/ z [ y / x ] ps ) $=
      ( wal wnf wsb nfri alimi nfsbt 3syl ) AACHBEIZCHBCDJEIACFKAOCGLBCDEMN $.
  $}

  ${
    $d x y $.
    $( Like ~ sb9 but with a distinct variable constraint between ` x ` and
       ` y ` .  (Contributed by Jim Kingdon, 28-Feb-2018.) $)
    sb9v $p |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) $=
      ( wsb hbs1 weq wb sbequ12 equcoms bitr3d cbvalh ) ACBDZABCDZBCACBEABCEBCF
      ALMALGCBACBHIABCHJK $.
  $}

  ${
    $d w x $.  $d w y $.  $d w ph $.
    $( Commutation of quantification and substitution variables.  (Contributed
       by NM, 5-Aug-1993.)  (Proof rewritten by Jim Kingdon, 23-Mar-2018.) $)
    sb9 $p |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) $=
      ( vw wsb wal sb9v sbcom albii 3bitri ax-17 sbco2h 3bitr3ri ) ABDEZDCEZCFZ
      ACDEZDBEZBFZABCEZCFACBEZBFPNCDEZDFQBDEZDFSNCDGUBUCDABDCHIQDBGJOTCABCDADKZ
      LIRUABACBDUDLIM $.
  $}

  $( Commutation of quantification and substitution variables.  (Contributed by
     NM, 5-Aug-1993.)  (Proof rewritten by Jim Kingdon, 23-Mar-2018.) $)
  sb9i $p |- ( A. x [ x / y ] ph -> A. y [ y / x ] ph ) $=
    ( wsb wal sb9 biimpi ) ACBDBEABCDCEABCFG $.

  ${
    $d x y z $.  $d y z ph $.
    $( Two ways of expressing " ` x ` is (effectively) not free in ` ph ` ."
       (Contributed by G&eacute;rard Lang, 14-Nov-2013.)  (Revised by Mario
       Carneiro, 6-Oct-2016.) $)
    sbnf2 $p |- ( F/ x ph
       <-> A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) ) $=
      ( wsb wb wal wi wnf 2albiim sbhb albii alcom 3bitri nfv nfs1v sblim bitri
      wa sb8 df-nf anbi12i anidm 3bitr2ri ) ABCEZABDEZFDGCGUEUFHZDGCGZUFUEHZDGZ
      CGZSABIZULSULUEUFCDJULUHULUKULAUFHZBGZDGZUGCGZDGUHULAABGHZBGZUMDGZBGUOABU
      AZUQUSBABDKLUMBDMNUNUPDUNUMBCEZCGUPUMBCUMCOTVAUGCAUFBCABDPQLRLUGDCMNULAUE
      HZBGZCGZUKULURVBCGZBGVDUTUQVEBABCKLVBBCMNVCUJCVCVBBDEZDGUJVBBDVBDOTVFUIDA
      UEBDABCPQLRLRUBULUCUD $.
  $}

  ${
    $d y z $.
    hbsbd.1 $e |- ( ph -> A. x ph ) $.
    hbsbd.2 $e |- ( ph -> A. z ph ) $.
    hbsbd.3 $e |- ( ph -> ( ps -> A. z ps ) ) $.
    $( Deduction version of ~ hbsb .  (Contributed by NM, 15-Feb-2013.)  (Proof
       rewritten by Jim Kingdon, 23-Mar-2018.) $)
    hbsbd $p |- ( ph -> ( [ y / x ] ps -> A. z [ y / x ] ps ) ) $=
      ( wsb nfi wi wnf nfdh nfim1 nfsb sbrim nfbii mpbi nfrimi nfrd ) ABCDIZEAU
      AEAEGJZABKZCDIZELAUAKZELUCCDEABEUBABEGHMNOUDUEEABCDFPQRST $.
  $}

  ${
    $d x y z $.  $d w y $.
    $( Equivalence for double substitution.  (Contributed by NM,
       3-Feb-2005.) $)
    2sb5 $p |- ( [ z / x ] [ w / y ] ph <->
               E. x E. y ( ( x = z /\ y = w ) /\ ph ) ) $=
      ( wsb weq wa wex sb5 19.42v anass exbii anbi2i 3bitr4ri bitri ) ACEFZBDFB
      DGZQHZBIRCEGZHAHZCIZBIQBDJSUBBRTAHZHZCIRUCCIZHUBSRUCCKUAUDCRTALMQUERACEJN
      OMP $.

    $( Equivalence for double substitution.  (Contributed by NM,
       3-Feb-2005.) $)
    2sb6 $p |- ( [ z / x ] [ w / y ] ph <->
               A. x A. y ( ( x = z /\ y = w ) -> ph ) ) $=
      ( wsb weq wi wal wa sb6 19.21v impexp albii imbi2i 3bitr4ri bitri ) ACEFZ
      BDFBDGZRHZBISCEGZJAHZCIZBIRBDKTUCBSUAAHZHZCISUDCIZHUCTSUDCLUBUECSUAAMNRUF
      SACEKOPNQ $.
  $}

  ${
    $d w x z $.  $d x y z $.
    $( Lemma for proving ~ sbcom2 .  It is the same as ~ sbcom2 but with
       additional distinct variable constraints on ` x ` and ` y ` , and on
       ` w ` and ` z ` .  (Contributed by Jim Kingdon, 19-Feb-2018.) $)
    sbcom2v $p |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) $=
      ( weq wal wsb alcom bi2.04 albii 19.21v bitri 3bitr3i sb6 imbi2i 3bitr4i
      wi ) DEFZBCFZARZBGZRZDGZTSARZDGZRZBGZABCHZDEHZADEHZBCHZTUERZBGZDGUMDGZBGU
      DUHUMDBIUNUCDUNSUARZBGUCUMUPBTSAJKSUABLMKUOUGBTUEDLKNUJSUIRZDGUDUIDEOUQUC
      DUIUBSABCOPKMULTUKRZBGUHUKBCOURUGBUKUFTADEOPKMQ $.
  $}

  ${
    $d v w x z $.  $d v y z $.  $d v ph $.
    $( Lemma for proving ~ sbcom2 .  It is the same as ~ sbcom2v but removes
       the distinct variable constraint on ` x ` and ` y ` .  (Contributed by
       Jim Kingdon, 19-Feb-2018.) $)
    sbcom2v2 $p |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) $=
      ( vv wsb sbcom2v sbbii bitri ax-17 sbco2vh 3bitr3i ) ABFGZFCGZDEGZADEGZBF
      GZFCGZABCGZDEGQBCGPNDEGZFCGSNFCDEHUARFCABFDEHIJOTDEABCFAFKLIQBCFQFKLM $.
  $}

  ${
    $d x z $.  $d v x w $.  $d v y z $.  $d v ph $.
    $( Commutativity law for substitution.  Used in proof of Theorem 9.7 of
       [Megill] p. 449 (p. 16 of the preprint).  (Contributed by NM,
       27-May-1997.)  (Proof modified to be intuitionistic by Jim Kingdon,
       19-Feb-2018.) $)
    sbcom2 $p |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) $=
      ( vv wsb sbcom2v2 sbbii bitri ax-17 sbco2vh 3bitr3i ) ABCGZDFGZFEGZADFGZF
      EGZBCGZNDEGADEGZBCGPQBCGZFEGSOUAFEABCDFHIQBCFEHJNDEFNFKLRTBCADEFAFKLIM $.
  $}

  ${
    $d x y $.
    $( Equivalence for substitution.  (Contributed by NM, 5-Aug-1993.) $)
    sb6a $p |- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) ) $=
      ( wsb weq wi wal sb6 wb sbequ12 equcoms pm5.74i albii bitri ) ABCDBCEZAFZ
      BGOACBDZFZBGABCHPRBOAQAQICBACBJKLMN $.
  $}

  ${
    $d x y $.  $d x w $.  $d y z $.  $d z w $.
    2sb5rf.1 $e |- ( ph -> A. z ph ) $.
    2sb5rf.2 $e |- ( ph -> A. w ph ) $.
    $( Reversed double substitution.  (Contributed by NM, 3-Feb-2005.) $)
    2sb5rf $p |- ( ph <->
                E. z E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) ) $=
      ( weq wsb wex sb5rf 19.42v sbcom2 anbi2i anass bitri exbii hbsbv 3bitr4ri
      wa ) ADBHZABDIZTZDJUAECHZTZACEIBDIZTZEJZDJABDFKUCUHDUAUDUBCEIZTZTZEJUAUJE
      JZTUHUCUAUJELUGUKEUGUEUITUKUFUIUEACEBDMNUAUDUIOPQUBULUAUBCEABDEGRKNSQP $.

    $( Reversed double substitution.  (Contributed by NM, 3-Feb-2005.) $)
    2sb6rf $p |- ( ph <->
                A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) ) $=
      ( weq wsb wi wal wa sb6rf 19.21v sbcom2 imbi2i impexp bitri albii hbsbv
      3bitr4ri ) ADBHZABDIZJZDKUBECHZLZACEIBDIZJZEKZDKABDFMUDUIDUBUEUCCEIZJZJZE
      KUBUKEKZJUIUDUBUKENUHULEUHUFUJJULUGUJUFACEBDOPUBUEUJQRSUCUMUBUCCEABDEGTMP
      UASR $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    $( An alternate definition of proper substitution ~ df-sb .  By introducing
       a dummy variable ` z ` in the definiens, we are able to eliminate any
       distinct variable restrictions among the variables ` x ` , ` y ` , and
       ` ph ` of the definiendum.  No distinct variable conflicts arise because
       ` z ` effectively insulates ` x ` from ` y ` .  To achieve this, we use
       a chain of two substitutions in the form of ~ sb5 , first ` z ` for
       ` x ` then ` y ` for ` z ` .  Compare Definition 2.1'' of [Quine] p. 17.
       Theorem ~ sb7f provides a version where ` ph ` and ` z ` don't have to
       be distinct.  (Contributed by NM, 28-Jan-2004.) $)
    dfsb7 $p |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
      ( wsb weq wa wex sb5 sbbii ax-17 sbco2vh 3bitr3i ) ABDEZDCEBDFAGBHZDCEABC
      EDCFOGDHNODCABDIJABCDADKLODCIM $.
  $}

  ${
    $d x z $.  $d y z $.
    sb7f.1 $e |- ( ph -> A. z ph ) $.
    $( This version of ~ dfsb7 does not require that ` ph ` and ` z ` be
       disjoint.  This permits it to be used as a definition for substitution
       in a formalization that omits the logically redundant axiom ~ ax-17 ,
       i.e., that does not have the concept of a variable not occurring in a
       formula.  (Contributed by NM, 26-Jul-2006.)  (Proof shortened by Andrew
       Salmon, 25-May-2011.) $)
    sb7f $p |- ( [ y / x ] ph <->
               E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
      ( wsb weq wa wex sb5 sbbii sbco2vh 3bitr3i ) ABDFZDCFBDGAHBIZDCFABCFDCGOH
      DINODCABDJKABCDELODCJM $.
  $}

  ${
    $d x z $.  $d y z $.
    sb7af.1 $e |- F/ z ph $.
    $( An alternate definition of proper substitution ~ df-sb .  Similar to
       ~ dfsb7a but does not require that ` ph ` and ` z ` be distinct.
       Similar to ~ sb7f in that it involves a dummy variable ` z ` , but
       expressed in terms of ` A. ` rather than ` E. ` .  (Contributed by Jim
       Kingdon, 5-Feb-2018.) $)
    sb7af $p |- ( [ y / x ] ph
                      <-> A. z
                             ( z = y
                             -> A. x ( x = z -> ph ) ) ) $=
      ( wsb weq wi wal sb6 sbbii sbco2 3bitr3i ) ABDFZDCFBDGAHBIZDCFABCFDCGOHDI
      NODCABDJKABCDELODCJM $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    $( An alternate definition of proper substitution ~ df-sb .  Similar to
       ~ dfsb7 in that it involves a dummy variable ` z ` , but expressed in
       terms of ` A. ` rather than ` E. ` .  For a version which only requires
       ` F/ z ph ` rather than ` z ` and ` ph ` being distinct, see ~ sb7af .
       (Contributed by Jim Kingdon, 5-Feb-2018.) $)
    dfsb7a $p |- ( [ y / x ] ph
                      <-> A. z
                             ( z = y
                             -> A. x ( x = z -> ph ) ) ) $=
      ( nfv sb7af ) ABCDADEF $.
  $}

  ${
    $d x y $.
    sb10f.1 $e |- ( ph -> A. x ph ) $.
    $( Hao Wang's identity axiom P6 in Irving Copi, _Symbolic Logic_ (5th ed.,
       1979), p. 328.  In traditional predicate calculus, this is a sole axiom
       for identity from which the usual ones can be derived.  (Contributed by
       NM, 9-May-2005.) $)
    sb10f $p |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) ) $=
      ( weq wsb wa wex hbsb sbequ equsex bicomi ) BCFADBGZHBIADCGZNOBCADCBEJABC
      DKLM $.
  $}

  ${
    $d x ph $.
    $( An identity law for substitution.  Used in proof of Theorem 9.7 of
       [Megill] p. 449 (p. 16 of the preprint).  (Contributed by NM,
       5-Aug-1993.) $)
    sbid2v $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
      ( ax-17 sbid2h ) ABCABDE $.
  $}

  ${
    $d x y $.  $d x ph $.
    $( Elimination of substitution.  (Contributed by NM, 5-Aug-1993.) $)
    sbelx $p |- ( ph <-> E. x ( x = y /\ [ x / y ] ph ) ) $=
      ( ax-17 sb5rf ) ACBABDE $.
  $}

  ${
    $( Note:  A more general case could also be proved with
       "$d x z $.  $d y w $.  $d x ph $.  $d y ph $.", but with more
       difficulty. $)
    $d x y z $.  $d w y $.  $d x y ph $.
    $( Elimination of double substitution.  (Contributed by NM, 5-Aug-1993.) $)
    sbel2x $p |- ( ph <-> E. x E. y ( ( x = z /\ y = w ) /\
                     [ y / w ] [ x / z ] ph ) ) $=
      ( weq wsb wa wex sbelx anbi2i exbii exdistr 3bitr4i anass 2exbii bitr4i )
      ABDFZCEFZADBGZECGZHZHZCIBIZRSHUAHZCIBIRTHZBIRUBCIZHZBIAUDUFUHBTUGRTCEJKLA
      BDJRUBBCMNUEUCBCRSUAOPQ $.
  $}

  ${
    $d x y z $.
    $( Move universal quantifier in and out of substitution.  Identical to
       ~ sbal except that it has an additional distinct variable constraint on
       ` y ` and ` z ` .  (Contributed by Jim Kingdon, 29-Dec-2017.) $)
    sbalyz $p |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) $=
      ( wal wsb nfa1 nfsbxy ax-4 sbimi alrimi weq wi sb6 albii alcom nfv stdpc5
      bitri alimi sb2 syl sylbi impbii ) ABEZCDFZACDFZBEZUFUGBUECDBABGHUEACDABI
      JKUHCDLZAMZBEZCEZUFUHUJCEZBEULUGUMBACDNOUJBCPSULUIUEMZCEUFUKUNCUIABUIBQRT
      UECDUAUBUCUD $.
  $}

  ${
    $d x y w $.  $d x z w $.  $d w ph $.
    $( Move universal quantifier in and out of substitution.  (Contributed by
       NM, 5-Aug-1993.)  (Proof rewritten by Jim Kingdon, 12-Feb-2018.) $)
    sbal $p |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) $=
      ( vw wal wsb sbalyz sbbii bitri ax-17 sbco2vh albii 3bitr3i ) ABFZCEGZEDG
      ZACEGZEDGZBFZOCDGACDGZBFQRBFZEDGTPUBEDABCEHIRBEDHJOCDEOEKLSUABACDEAEKLMN
      $.
  $}

  ${
    $d x y $.  $d y z $.
    $( Lemma for proving ~ sbal1 .  Same as ~ sbal1 but with an additional
       disjoint variable condition on ` y , z ` .  (Contributed by Jim Kingdon,
       23-Feb-2018.) $)
    sbal1yz $p |- ( -. A. x x = z ->
             ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
      ( weq wal wn wsb wi wb wo dveeq2or equcom nfbii 19.21t sylbi imbi1i albii
      wnf sb6 orim2i ax-mp ori albidv alcom bitri bitr4i bitr2i 3bitr3g bicomd
      ) BDEBFZGZACDHZBFZABFZCDHZULDCEZAIZBFZCFZUQUOIZCFZUNUPULUSVACUKUSVAJZUKCD
      EZBSZKUKVCKBDCLVEVCUKVEUQBSVCVDUQBCDMZNUQABOPUAUBUCUDUTURCFZBFUNURCBUEUMV
      GBUMVDAIZCFVGACDTVHURCVDUQAVFQRUFRUGUPVDUOIZCFVBUOCDTVIVACVDUQUOVFQRUHUIU
      J $.
  $}

  ${
    $d x y $.  $d w x $.  $d w y $.  $d w z $.  $d ph w $.

    $( A theorem used in elimination of disjoint variable conditions on
       ` x , y ` by replacing it with a distinctor ` -. A. x x = z ` .
       (Contributed by NM, 5-Aug-1993.)  (Proof rewitten by Jim Kingdon,
       24-Feb-2018.) $)
    sbal1 $p |- ( -. A. x x = z ->
             ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
      ( vw weq wal wn wsb sbal sbbii sbal1yz bitrid ax-17 sbco2vh albii 3bitr3g
      ) BDFBGHZABGZCEIZEDIZACEIZEDIZBGZSCDIACDIZBGUAUBBGZEDIRUDTUFEDABCEJKUBBED
      LMSCDESENOUCUEBACDEAENOPQ $.
  $}

  ${
    $d x y z $.
    $( Move existential quantifier in and out of substitution.  Identical to
       ~ sbex except that it has an additional disjoint variable condition on
       ` y , z ` .  (Contributed by Jim Kingdon, 29-Dec-2017.) $)
    sbexyz $p |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph ) $=
      ( wex wsb weq wa sb5 exdistr excom 3bitr2i exbii bitr4i ) ABEZCDFZCDGZAHZ
      CEZBEZACDFZBEPQOHCERBECETOCDIQACBJRCBKLUASBACDIMN $.
  $}

  ${
    $d x y w $.  $d x z w $.  $d w ph $.
    $( Move existential quantifier in and out of substitution.  (Contributed by
       NM, 27-Sep-2003.)  (Proof rewritten by Jim Kingdon, 12-Feb-2018.) $)
    sbex $p |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph ) $=
      ( vw wex wsb sbexyz sbbii bitri ax-17 sbco2vh exbii 3bitr3i ) ABFZCEGZEDG
      ZACEGZEDGZBFZOCDGACDGZBFQRBFZEDGTPUBEDABCEHIRBEDHJOCDEOEKLSUABACDEAEKLMN
      $.
  $}

  ${
    $d x z $.  $d y z $.
    sbalv.1 $e |- ( [ y / x ] ph <-> ps ) $.
    $( Quantify with new variable inside substitution.  (Contributed by NM,
       18-Aug-1993.) $)
    sbalv $p |- ( [ y / x ] A. z ph <-> A. z ps ) $=
      ( wal wsb sbal albii bitri ) AEGCDHACDHZEGBEGAECDILBEFJK $.
  $}

  ${
    $d v w ph $.  $d v w x $.  $d v w y $.
    $( Lemma for ~ sbco4 .  It replaces the temporary variable ` v ` with
       another temporary variable ` w ` .  (Contributed by Jim Kingdon,
       26-Sep-2018.) $)
    sbco4lem $p |- ( [ x / v ] [ y / x ] [ v / y ] ph <->
        [ x / w ] [ y / x ] [ w / y ] ph ) $=
      ( wsb sbcom2 sbbii nfv sbco2 bitri sbid2 3bitr3i ) ACDFZDEFZBCFZEDFZDBFZO
      EDFZBCFZDBFACEFZBCFZEBFZNBCFZDBFQTDBOBCEDGHRUBEDFZDBFUCQUEDBPUBEDOUABCACE
      DADIJHHHUBEBDUBDIJKTUDDBSNBCNEDNEILHHM $.
  $}

  ${
    $d t u v ph $.  $d t u v x $.  $d t u v y $.  $d w ph $.  $d w x $.
    $d w y $.  $d t w $.
    $( Two ways of exchanging two variables.  Both sides of the biconditional
       exchange ` x ` and ` y ` , either via two temporary variables ` u ` and
       ` v ` , or a single temporary ` w ` .  (Contributed by Jim Kingdon,
       25-Sep-2018.) $)
    sbco4 $p |- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <->
        [ x / w ] [ y / x ] [ w / y ] ph ) $=
      ( vt wsb sbcom2 nfv sbco2 sbbii bitr3i sbco4lem 3bitri ) ACEHZBFHZEBHFCHZ
      PBCHZEBHZACGHBCHGBHACDHBCHDBHRQFCHZEBHTQFCEBIUASEBPBCFPFJKLMABCGENABCDGNO
      $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( An equivalent expression for existence.  (Contributed by NM,
       2-Feb-2005.) $)
    exsb $p |- ( E. x ph <-> E. y A. x ( x = y -> ph ) ) $=
      ( wex wsb weq wi wal ax-17 sb8eh sb6 exbii bitri ) ABDABCEZCDBCFAGBHZCDAB
      CACIJNOCABCKLM $.
  $}

  ${
    $d x y z $.  $d y w $.  $d z w ph $.
    $( An equivalent expression for double existence.  (Contributed by NM,
       2-Feb-2005.) $)
    2exsb $p |- ( E. x E. y ph <->
                  E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) ) $=
      ( wex weq wi wal exsb exbii excom bitri impexp albii 19.21v bitr2i 3bitri
      wa ) ACFZBFZCEGZAHZCIZBFZEFZBDGZUBSAHZCIZBIZDFZEFUJEFDFUAUDEFZBFUFTULBACE
      JKUDBELMUEUKEUEUGUDHZBIZDFUKUDBDJUNUJDUMUIBUIUGUCHZCIUMUHUOCUGUBANOUGUCCP
      QOKMKUJEDLR $.
  $}

  ${
    $d z ps $.  $d x z $.  $d y z $.
    dvelimALT.1 $e |- ( ph -> A. x ph ) $.
    dvelimALT.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( Version of ~ dvelim that doesn't use ~ ax-10 .  Because it has different
       distinct variable constraints than ~ dvelim and is used in important
       proofs, it would be better if it had a name which does not end in ALT
       (ideally more close to set.mm naming).  (Contributed by NM,
       17-May-2008.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    dvelimALT $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( weq wal wn wi wnf nfv wo ax12or orcom orbi2i mpbi a1i nfimd orass mpbir
      nfa1 ax16ALT nfd nfi df-nf id sylbir orim1i ax-mp ori nfald ax-17 equsalh
      jaoi nfbii sylib nfrd ) CDHCIZJZBCVAEDHZAKZEIZCLBCLVAVCCEVAEMUTVCCLZVEUTN
      ZUTVENCEHZCIZVBVBCIKCIZNZUTNZVFVKVHVIUTNZNZVHUTVINZNVMEDCOVNVLVHUTVIPQRVH
      VIUTUAUBVJVEUTVHVEVIVHVBACVHVBCVGCUCVBCEUDUEACLZVHACFUFZSTVIVBCLZVEVBCUGV
      QVBACVQUHVOVQVPSTUIUPUJUKVEUTPRULUMVDBCABEDBEUNGUOUQURUS $.
  $}

  ${
    $d x z $.
    dvelimfv.1 $e |- ( ph -> A. x ph ) $.
    dvelimfv.2 $e |- ( ps -> A. z ps ) $.
    dvelimfv.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( Like ~ dvelimf but with a distinct variable constraint on ` x ` and
       ` z ` .  (Contributed by Jim Kingdon, 6-Mar-2018.) $)
    dvelimfv $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( weq wal wn wi wnf nfnae wo ax12or orcom mpbi a1i nfimd orass mpbir nfae
      orbi2i ax16ALT nfd nfi df-nf sylbir jaoi orim1i ax-mp nfald equsalh nfbii
      id ori sylib nfrd ) CDICJZKZBCVAEDIZALZEJZCMBCMVAVCCECDENUTVCCMZVEUTOZUTV
      EOCEICJZVBVBCJLCJZOZUTOZVFVJVGVHUTOZOZVGUTVHOZOVLEDCPVMVKVGUTVHQUDRVGVHUT
      UAUBVIVEUTVGVEVHVGVBACVGVBCCECUCVBCEUEUFACMZVGACFUGZSTVHVBCMZVEVBCUHVPVBA
      CVPUPVNVPVOSTUIUJUKULVEUTQRUQUMVDBCABEDGHUNUOURUS $.
  $}

  ${
    $d w x $.  $d w y $.  $d w z $.  $d ph w $.
    hbsb4.1 $e |- ( ph -> A. z ph ) $.
    $( A variable not free remains so after substitution with a distinct
       variable.  (Contributed by NM, 5-Aug-1993.)  (Proof rewritten by Jim
       Kingdon, 23-Mar-2018.) $)
    hbsb4 $p |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) $=
      ( vw wsb hbsb sbequ dvelimALT ) ABFGABCGDCFABFDEHAFCBIJ $.
  $}

  $( A variable not free remains so after substitution with a distinct variable
     (closed form of ~ hbsb4 ).  (Contributed by NM, 7-Apr-2004.)  (Proof
     shortened by Andrew Salmon, 25-May-2011.) $)
  hbsb4t $p |- ( A. x A. z ( ph -> A. z ph ) ->
               ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) ) $=
    ( weq wal wn wsb wi hba1 hbsb4 spsbim sps ax-4 sbimi alimi a1i imim12d syl5
    a7s ) DCEDFGADFZBCHZUBDFZIZAUAIZDFBFABCHZUFDFZIZUABCDADJKUEUDUHIDBUEBFZDFZU
    FUBUCUGUIUFUBIDAUABCLMUCUGIUJUBUFDUAABCADNOPQRTS $.

  $( A variable not free remains so after substitution with a distinct variable
     (closed form of ~ hbsb4 ).  (Contributed by NM, 7-Apr-2004.)  (Revised by
     Mario Carneiro, 4-Oct-2016.)  (Proof rewritten by Jim Kingdon,
     9-May-2018.) $)
  nfsb4t $p |- ( A. x F/ z ph ->
                 ( -. A. z z = y -> F/ z [ y / x ] ph ) ) $=
    ( wnf wal weq wn wsb wa nfnf1 nfal nfnae nfan df-nf albii hbsb4t sylbi imp
    wi nfd ex ) ADEZBFZDCGDFHZABCIZDEUDUEJUFDUDUEDUCDBADKLDCDMNUDUEUFUFDFTZUDAA
    DFTDFZBFUEUGTUCUHBADOPABCDQRSUAUB $.

  ${
    dvelimf.1 $e |- ( ph -> A. x ph ) $.
    dvelimf.2 $e |- ( ps -> A. z ps ) $.
    dvelimf.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( Version of ~ dvelim without any variable restrictions.  (Contributed by
       NM, 1-Oct-2002.) $)
    dvelimf $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( weq wal wn wsb hbsb4 sbieh albii 3imtr3g ) CDICJKAEDLZQCJBBCJAEDCFMABED
      GHNZQBCROP $.
  $}

  ${
    dvelimdf.1 $e |- F/ x ph $.
    dvelimdf.2 $e |- F/ z ph $.
    dvelimdf.3 $e |- ( ph -> F/ x ps ) $.
    dvelimdf.4 $e |- ( ph -> F/ z ch ) $.
    dvelimdf.5 $e |- ( ph -> ( z = y -> ( ps <-> ch ) ) ) $.
    $( Deduction form of ~ dvelimf .  This version may be useful if we want to
       avoid ~ ax-17 and use ~ ax-16 instead.  (Contributed by NM, 7-Apr-2004.)
       (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof shortened by Wolf
       Lammen, 11-May-2018.) $)
    dvelimdf $p |- ( ph -> ( -. A. x x = y -> F/ x ch ) ) $=
      ( weq wal wn wsb wnf wi alrimi nfsb4t syl sbied nfbidf sylibd ) ADELDMNZB
      FEOZDPZCDPABDPZFMUDUFQAUGFHIRBFEDSTAUECDGABCFEHJKUAUBUC $.
  $}

  ${
    $d z ps $.
    dvelim.1 $e |- ( ph -> A. x ph ) $.
    dvelim.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( This theorem can be used to eliminate a distinct variable restriction on
       ` x ` and ` z ` and replace it with the "distinctor" ` -. A. x x = y `
       as an antecedent. ` ph ` normally has ` z ` free and can be read
       ` ph ( z ) ` , and ` ps ` substitutes ` y ` for ` z ` and can be read
       ` ph ( y ) ` .  We don't require that ` x ` and ` y ` be distinct: if
       they aren't, the distinctor will become false (in multiple-element
       domains of discourse) and "protect" the consequent.

       To obtain a closed-theorem form of this inference, prefix the hypotheses
       with ` A. x A. z ` , conjoin them, and apply ~ dvelimdf .

       Other variants of this theorem are ~ dvelimf (with no distinct variable
       restrictions) and ~ dvelimALT (that avoids ~ ax-10 ).  (Contributed by
       NM, 23-Nov-1994.) $)
    dvelim $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( ax-17 dvelimf ) ABCDEFBEHGI $.
  $}

  ${
    $d z ps $.  $d x z $.
    dvelimor.1 $e |- F/ x ph $.
    dvelimor.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( Disjunctive distinct variable constraint elimination.  A user of this
       theorem starts with a formula ` ph ` (containing ` z ` ) and a distinct
       variable constraint between ` x ` and ` z ` .  The theorem makes it
       possible to replace the distinct variable constraint with the disjunct
       ` A. x x = y ` ( ` ps ` is just a version of ` ph ` with ` y `
       substituted for ` z ` ).  (Contributed by Jim Kingdon, 11-May-2018.) $)
    dvelimor $p |- ( A. x x = y \/ F/ x ps ) $=
      ( weq wal wi wnf wo ax-bndl orcom orbi2i mpbi orass mpbir nfae ax-mp jaoi
      a16nf alrimi df-nf id nfimd sylbir alimi orim1i nfalt ax-17 equsalh nfbii
      a1i sylib orim2i ) CDHCIZEDHZAJZCKZEIZLZUQBCKZLVAUQLZVBCEHCIZURURCIJCIZEI
      ZLZUQLZVDVIVEVGUQLZLZVEUQVGLZLVKEDCMVLVJVEUQVGNOPVEVGUQQRVHVAUQVEVAVGVEUT
      ECEESUSCECUBUCVFUTEVFURCKZUTURCUDVMURACVMUEACKVMFUNUFUGUHUAUITVAUQNPVAVCU
      QVAUSEIZCKVCUSCEUJVNBCABEDBEUKGULUMUOUPT $.
  $}

  ${
    $d z x $.
    $( Quantifier introduction when one pair of variables is distinct.
       (Contributed by NM, 2-Jan-2002.)  (Proof rewritten by Jim Kingdon,
       19-Feb-2018.) $)
    dveeq1 $p |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $=
      ( weq wal wn dveeq2 equcom albii 3imtr3g ) ABDAEFCBDZKAEBCDZLAEABCGCBHZKL
      AMIJ $.
  $}

  ${
    $d z y $.  $d z x $.
    $( Move quantifier in and out of substitution.  (Contributed by NM,
       2-Jan-2002.) $)
    sbal2 $p |- ( -. A. x x = y ->
             ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
      ( weq wal wn wi wsb hbnae wb dveeq1 alimi hbnaes 19.21ht syl albidh alcom
      bitr3di sb6 albii 3bitr4g ) BCEBFGZCDEZABFZHZCFZUDAHZCFZBFZUECDIACDIZBFUC
      UHBFZCFUGUJUCULUFCBCCJUCUDUDBFHZBFZULUFKUNBCBUCUMBBCDLMNUDABOPQUHCBRSUECD
      TUKUIBACDTUAUB $.
  $}

  ${
    $d w x $.  $d w y $.  $d w z $.  $d ph w $.
    nfsb4or.1 $e |- F/ z ph $.
    $( A variable not free remains so after substitution with a distinct
       variable.  (Contributed by Jim Kingdon, 11-May-2018.) $)
    nfsb4or $p |- ( A. z z = y \/ F/ z [ y / x ] ph ) $=
      ( vw wsb nfsb sbequ dvelimor ) ABFGABCGDCFABFDEHAFCBIJ $.
  $}

  ${
    nfd2.1 $e |- ( ph -> ( E. x ps -> A. x ps ) ) $.
    $( Deduce that ` x ` is not free in ` ps ` in a context.  (Contributed by
       Wolf Lammen, 16-Sep-2021.) $)
    nfd2 $p |- ( ph -> F/ x ps ) $=
      ( wex wal wi wnf nf2 sylibr ) ABCEBCFGBCHDBCIJ $.
  $}

  $( Dual statement of ~ hbe1 .  (Contributed by Wolf Lammen, 15-Sep-2021.) $)
  hbe1a $p |- ( E. x A. x ph -> A. x ph ) $=
    ( wal wex wi wnf nfa1 nf3 mpbi spi ) ABCZBDKEZBKBFLBCABGKBHIJ $.

  $( One direction of nf5 .  (Contributed by Wolf Lammen, 16-Sep-2021.) $)
  nf5-1 $p |- ( A. x ( ph -> A. x ph ) -> F/ x ph ) $=
    ( wal wi wex exim hbe1a syl6 nfd2 ) AABCZDBCZABKABEJBEJAJBFABGHI $.

  ${
    nf5d.1 $e |- F/ x ph $.
    nf5d.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Deduce that ` x ` is not free in ` ps ` in a context.  (Contributed by
       Mario Carneiro, 24-Sep-2016.) $)
    nf5d $p |- ( ph -> F/ x ps ) $=
      ( wal wi wnf alrimi nf5-1 syl ) ABBCFGZCFBCHALCDEIBCJK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Existential uniqueness
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new symbols needed for uniqueness notation. $)
  $c E! $.  $( Backwards E exclamation point. $)
  $c E* $.  $( Backwards E superscript *. $)

  $( Extend wff definition to include existential uniqueness ("there exists a
     unique ` x ` such that ` ph ` "). $)
  weu $a wff E! x ph $.

  $( Extend wff definition to include uniqueness ("there exists at most one
     ` x ` such that ` ph ` "). $)
  wmo $a wff E* x ph $.

  ${
    $d w x y $.  $d x z $.  $d y ph $.  $d w z ph $.
    $( A soundness justification theorem for ~ df-eu , showing that the
       definition is equivalent to itself with its dummy variable renamed.
       Note that ` y ` and ` z ` needn't be distinct variables.  (Contributed
       by NM, 11-Mar-2010.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
    eujust $p |- ( E. y A. x ( ph <-> x = y )
        <-> E. z A. x ( ph <-> x = z ) ) $=
      ( vw cv wceq wb wal wex equequ2 bibi2d albidv cbvexv bitri ) ABFZCFZGZHZB
      IZCJAPEFZGZHZBIZEJAPDFZGZHZBIZDJTUDCEQUAGZSUCBUIRUBACEBKLMNUDUHEDUAUEGZUC
      UGBUJUBUFAEDBKLMNO $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( Define existential uniqueness, i.e., "there exists exactly one ` x `
       such that ` ph ` ".  Definition 10.1 of [BellMachover] p. 97; also
       Definition *14.02 of [WhiteheadRussell] p. 175.  Other possible
       definitions are given by ~ eu1 , ~ eu2 , ~ eu3 , and ~ eu5 (which in
       some cases we show with a hypothesis ` ph -> A. y ph ` in place of a
       distinct variable condition on ` y ` and ` ph ` ).  Double uniqueness is
       tricky: ` E! x E! y ph ` does not mean "exactly one ` x ` and one
       ` y ` " (see ~ 2eu4 ).  (Contributed by NM, 5-Aug-1993.) $)
    df-eu $a |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) $.
  $}

  $( Define "there exists at most one ` x ` such that ` ph ` ".  Here we define
     it in terms of existential uniqueness.  Notation of [BellMachover] p. 460,
     whose definition we show as ~ mo3 .  For another possible definition see
     ~ mo4 .  (Contributed by NM, 5-Aug-1993.) $)
  df-mo $a |- ( E* x ph <-> ( E. x ph -> E! x ph ) ) $.

  ${
    $d x y z $.  $d ph z $.
    euf.1 $e |- ( ph -> A. y ph ) $.
    $( A version of the existential uniqueness definition with a hypothesis
       instead of a distinct variable condition.  (Contributed by NM,
       12-Aug-1993.) $)
    euf $p |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) $=
      ( vz weu weq wb wal wex df-eu ax-17 hbbi hbal equequ2 bibi2d albidv bitri
      cbvexh ) ABFABEGZHZBIZEJABCGZHZBIZCJABEKUBUEECUACBATCDTCLMNUEELECGZUAUDBU
      FTUCAECBOPQSR $.
  $}

  ${
    $d x y $.  $d y ph $.  $d y ps $.  $d y ch $.
    eubidh.1 $e |- ( ph -> A. x ph ) $.
    eubidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for unique existential quantifier (deduction
       form).  (Contributed by NM, 9-Jul-1994.) $)
    eubidh $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
      ( vy weq wb wal wex weu bibi1d albidh exbidv df-eu 3bitr4g ) ABDGHZIZDJZG
      KCRIZDJZGKBDLCDLATUBGASUADEABCRFMNOBDGPCDGPQ $.
  $}

  ${
    $d x y $.  $d y ph $.  $d y ps $.  $d y ch $.
    eubid.1 $e |- F/ x ph $.
    eubid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for unique existential quantifier (deduction
       form).  (Contributed by NM, 9-Jul-1994.) $)
    eubid $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
      ( vy weq wb wal wex weu bibi1d albid exbidv df-eu 3bitr4g ) ABDGHZIZDJZGK
      CRIZDJZGKBDLCDLATUBGASUADEABCRFMNOBDGPCDGPQ $.
  $}

  ${
    $d x ph $.
    eubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for unique existential quantifier (deduction
       form).  (Contributed by NM, 9-Jul-1994.) $)
    eubidv $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
      ( nfv eubid ) ABCDADFEG $.
  $}

  ${
    eubii.1 $e |- ( ph <-> ps ) $.
    $( Introduce unique existential quantifier to both sides of an equivalence.
       (Contributed by NM, 9-Jul-1994.)  (Revised by Mario Carneiro,
       6-Oct-2016.) $)
    eubii $p |- ( E! x ph <-> E! x ps ) $=
      ( weu wb wtru a1i eubidv mptru ) ACEBCEFGABCABFGDHIJ $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( Bound-variable hypothesis builder for uniqueness.  (Contributed by NM,
       9-Jul-1994.) $)
    hbeu1 $p |- ( E! x ph -> A. x E! x ph ) $=
      ( vy weu weq wb wal wex df-eu hba1 hbex hbxfrbi ) ABDABCEFZBGZCHBABCINBCM
      BJKL $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( Bound-variable hypothesis builder for uniqueness.  (Contributed by NM,
       9-Jul-1994.)  (Revised by Mario Carneiro, 7-Oct-2016.) $)
    nfeu1 $p |- F/ x E! x ph $=
      ( vy weu weq wb wal wex df-eu nfa1 nfex nfxfr ) ABDABCEFZBGZCHBABCINBCMBJ
      KL $.
  $}

  $( Bound-variable hypothesis builder for "at most one".  (Contributed by NM,
     8-Mar-1995.)  (Revised by Mario Carneiro, 7-Oct-2016.) $)
  nfmo1 $p |- F/ x E* x ph $=
    ( wmo wex weu wi df-mo nfe1 nfeu1 nfim nfxfr ) ABCABDZABEZFBABGLMBABHABIJK
    $.

  ${
    $d w y z $.  $d ph z w $.  $d w x z $.
    sb8eu.1 $e |- F/ y ph $.
    $( Variable substitution in unique existential quantifier.  (Contributed by
       NM, 7-Aug-1994.)  (Revised by Mario Carneiro, 7-Oct-2016.) $)
    sb8eu $p |- ( E! x ph <-> E! y [ y / x ] ph ) $=
      ( vz vw weq wb wal wex wsb weu nfv sb8 sbbi nfsb equsb3 nfxfr nfbi df-eu
      sbequ cbval sblbis albii 3bitri exbii 3bitr4i ) ABEGZHZBIZEJABCKZCEGZHZCI
      ZEJABLUKCLUJUNEUJUIBFKZFIUIBCKZCIUNUIBFUIFMNUOUPFCUOABFKZUHBFKZHCAUHBFOUQ
      URCABFCDPURFEGZCBFEQUSCMRSRUPFMUIFCBUAUBUPUMCUHULABCBCEQUCUDUEUFABETUKCET
      UG $.

    $( Variable substitution for "at most one".  (Contributed by Alexander van
       der Vekens, 17-Jun-2017.) $)
    sb8mo $p |- ( E* x ph <-> E* y [ y / x ] ph ) $=
      ( wex weu wi wsb wmo sb8e sb8eu imbi12i df-mo 3bitr4i ) ABEZABFZGABCHZCEZ
      QCFZGABIQCIORPSABCDJABCDKLABMQCMN $.
  $}

  ${
    $d x y z $.  $d z ph $.  $d z ps $.
    nfeudv.1 $e |- F/ y ph $.
    nfeudv.2 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfeu .  Similar to ~ nfeud but has the additional
       constraint that ` x ` and ` y ` must be distinct.  (Contributed by Jim
       Kingdon, 25-May-2018.) $)
    nfeudv $p |- ( ph -> F/ x E! y ps ) $=
      ( vz cv wceq wb wal wex wnf weu nfv a1i nfbid nfald nfexd df-eu sylibr
      nfbii ) ABDHGHIZJZDKZGLZCMBDNZCMAUECGAGOAUDCDEABUCCFUCCMAUCCOPQRSUGUFCBDG
      TUBUA $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.  $d z ps $.
    nfeud.1 $e |- F/ y ph $.
    nfeud.2 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfeu .  (Contributed by NM, 15-Feb-2013.)
       (Revised by Mario Carneiro, 7-Oct-2016.)  (Proof rewritten by Jim
       Kingdon, 25-May-2018.) $)
    nfeud $p |- ( ph -> F/ x E! y ps ) $=
      ( vz weu wsb nfv sb8eu nfsbd nfeudv nfxfrd ) BDHBDGIZGHACBDGBGJKAOCGAGJAB
      DGCEFLMN $.

    $( Bound-variable hypothesis builder for "at most one".  (Contributed by
       Mario Carneiro, 14-Nov-2016.) $)
    nfmod $p |- ( ph -> F/ x E* y ps ) $=
      ( wmo wex weu wi df-mo nfexd nfeud nfimd nfxfrd ) BDGBDHZBDIZJACBDKAPQCAB
      CDEFLABCDEFMNO $.
  $}

  ${
    $d x y z $.  $d z ph $.
    nfeuv.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for existential uniqueness.  This is
       similar to ~ nfeu but has the additional condition that ` x ` and ` y `
       must be distinct.  (Contributed by Jim Kingdon, 23-May-2018.) $)
    nfeuv $p |- F/ x E! y ph $=
      ( vz weu wnf weq wb wal wex nfv nfbi nfal nfex df-eu nfbii mpbir ) ACFZBG
      ACEHZIZCJZEKZBGUBBEUABCATBDTBLMNOSUCBACEPQR $.
  $}

  ${
    $d y z $.  $d x z $.  $d z ph $.
    nfeu.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for existential uniqueness.  Note that
       ` x ` and ` y ` needn't be distinct.  (Contributed by NM, 8-Mar-1995.)
       (Revised by Mario Carneiro, 7-Oct-2016.)  (Proof rewritten by Jim
       Kingdon, 23-May-2018.) $)
    nfeu $p |- F/ x E! y ph $=
      ( vz weu wsb nfv sb8eu nfsb nfeuv nfxfr ) ACFACEGZEFBACEAEHIMBEACEBDJKL
      $.

    $( Bound-variable hypothesis builder for "at most one".  (Contributed by
       NM, 9-Mar-1995.) $)
    nfmo $p |- F/ x E* y ph $=
      ( wmo wnf wtru nftru a1i nfmod mptru ) ACEBFGABCCHABFGDIJK $.
  $}

  ${
    hbeu.1 $e |- ( ph -> A. x ph ) $.
    $( Bound-variable hypothesis builder for uniqueness.  Note that ` x ` and
       ` y ` needn't be distinct.  (Contributed by NM, 8-Mar-1995.)  (Proof
       rewritten by Jim Kingdon, 24-May-2018.) $)
    hbeu $p |- ( E! y ph -> A. x E! y ph ) $=
      ( weu nfi nfeu nfri ) ACEBABCABDFGH $.
  $}

  ${
    hbeud.1 $e |- ( ph -> A. x ph ) $.
    hbeud.2 $e |- ( ph -> A. y ph ) $.
    hbeud.3 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Deduction version of ~ hbeu .  (Contributed by NM, 15-Feb-2013.)  (Proof
       rewritten by Jim Kingdon, 25-May-2018.) $)
    hbeud $p |- ( ph -> ( E! y ps -> A. x E! y ps ) ) $=
      ( weu nfi nfd nfeud nfrd ) ABDHCABCDADFIABCACEIGJKL $.
  $}

  ${
    $d w y z $.  $d ph z w $.  $d w x z $.
    sb8euh.1 $e |- ( ph -> A. y ph ) $.
    $( Variable substitution in unique existential quantifier.  (Contributed by
       NM, 7-Aug-1994.)  (Revised by Andrew Salmon, 9-Jul-2011.) $)
    sb8euh $p |- ( E! x ph <-> E! y [ y / x ] ph ) $=
      ( vz vw weq wb wal wex wsb ax-17 sb8h sbbi hbsb equsb3 hbxfrbi hbbi df-eu
      weu sbequ cbvalh sblbis albii 3bitri exbii 3bitr4i ) ABEGZHZBIZEJABCKZCEG
      ZHZCIZEJABTUKCTUJUNEUJUIBFKZFIUIBCKZCIUNUIBFUIFLMUOUPFCUOABFKZUHBFKZHCAUH
      BFNUQURCABFCDOURFEGZCBFEPUSCLQRQUPFLUIFCBUAUBUPUMCUHULABCBCEPUCUDUEUFABES
      UKCESUG $.
  $}

  ${
    cbveu.1 $e |- F/ y ph $.
    cbveu.2 $e |- F/ x ps $.
    cbveu.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 25-Nov-1994.)  (Revised by Mario Carneiro,
       7-Oct-2016.) $)
    cbveu $p |- ( E! x ph <-> E! y ps ) $=
      ( weu wsb sb8eu sbie eubii bitri ) ACHACDIZDHBDHACDEJNBDABCDFGKLM $.
  $}

  ${
    $d x y $.
    eu1.1 $e |- ( ph -> A. y ph ) $.
    $( An alternate way to express uniqueness used by some authors.  Exercise
       2(b) of [Margaris] p. 110.  (Contributed by NM, 20-Aug-1993.) $)
    eu1 $p |- ( E! x ph <->
                E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) ) $=
      ( wsb weu weq wb wal wex wi wa hbs1 euf sb8euh equcom albii sb6rf 3bitr4i
      imbi2i anbi12i ancom albiim exbii ) ABCEZCFUECBGZHCIZBJABFAUEBCGZKZCIZLZB
      JUECBABCMNABCDOUKUGBUJALUEUFKZCIZUFUEKCIZLUKUGUJUMAUNUIULCUHUFUEBCPTQABCD
      RUAAUJUBUEUFCUCSUDS $.
  $}

  ${
    euor.1 $e |- ( ph -> A. x ph ) $.
    $( Introduce a disjunct into a unique existential quantifier.  (Contributed
       by NM, 21-Oct-2005.) $)
    euor $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $=
      ( wn weu wo hbn biorf eubidh biimpa ) AEZBCFABGZCFLBMCACDHABIJK $.
  $}

  ${
    $d x ph $.
    $( Introduce a disjunct into a unique existential quantifier.  (Contributed
       by NM, 23-Mar-1995.) $)
    euorv $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $=
      ( ax-17 euor ) ABCACDE $.
  $}

  ${
    $d x y $.
    mon.1 $e |- F/ y ph $.
    $( There is at most one of something which does not exist.  (Contributed by
       Jim Kingdon, 2-Jul-2018.) $)
    mo2n $p |- ( -. E. x ph -> E. y A. x ( ph -> x = y ) ) $=
      ( wex wsb weq wi wal sb8e wn alnex nfs1v sbequ1 equcoms con3d cbv3 pm2.21
      nfn alimi 19.8a 3syl sylbir sylnbi ) ABEABCFZCEZABCGZHZBIZCEZABCDJUFKUEKZ
      CIZUJUECLULAKZBIUIUJUKUMCBUEBABCMSACDSCBGAUEAUEHBCABCNOPQUMUHBAUGRTUICUAU
      BUCUD $.
  $}

  $( There is at most one of something which does not exist.  (Contributed by
     Jim Kingdon, 5-Jul-2018.) $)
  mon $p |- ( -. E. x ph -> E* x ph ) $=
    ( wex wn weu wi wmo ax-in2 df-mo sylibr ) ABCZDKABEZFABGKLHABIJ $.

  ${
    $d x y $.  $d ph y $.
    $( Existential uniqueness implies existence.  (Contributed by NM,
       15-Sep-1993.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
    euex $p |- ( E! x ph -> E. x ph ) $=
      ( vy weu wsb weq wi wal wa wex ax-17 eu1 exsimpl sylbi ) ABDAABCEBCFGCHZI
      BJABJABCACKLAOBMN $.
  $}

  ${
    $d x y $.
    eumo0.1 $e |- ( ph -> A. y ph ) $.
    $( Existential uniqueness implies "at most one".  (Contributed by NM,
       8-Jul-1994.) $)
    eumo0 $p |- ( E! x ph -> E. y A. x ( ph -> x = y ) ) $=
      ( weu weq wb wal wex wi euf biimp alimi eximi sylbi ) ABEABCFZGZBHZCIAPJZ
      BHZCIABCDKRTCQSBAPLMNO $.
  $}

  $( Existential uniqueness implies "at most one".  (Contributed by NM,
     23-Mar-1995.)  (Proof rewritten by Jim Kingdon, 27-May-2018.) $)
  eumo $p |- ( E! x ph -> E* x ph ) $=
    ( weu wex wi wmo ax-1 df-mo sylibr ) ABCZABDZJEABFJKGABHI $.

  ${
    eumoi.1 $e |- E! x ph $.
    $( "At most one" inferred from existential uniqueness.  (Contributed by NM,
       5-Apr-1995.) $)
    eumoi $p |- E* x ph $=
      ( weu wmo eumo ax-mp ) ABDABECABFG $.
  $}

  ${
    mobidh.1 $e |- ( ph -> A. x ph ) $.
    mobidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for "at most one" quantifier (deduction form).
       (Contributed by NM, 8-Mar-1995.) $)
    mobidh $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
      ( wex weu wi wmo exbidh eubidh imbi12d df-mo 3bitr4g ) ABDGZBDHZICDGZCDHZ
      IBDJCDJAPRQSABCDEFKABCDEFLMBDNCDNO $.
  $}

  ${
    mobid.1 $e |- F/ x ph $.
    mobid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for "at most one" quantifier (deduction form).
       (Contributed by NM, 8-Mar-1995.) $)
    mobid $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
      ( wex weu wi wmo exbid eubid imbi12d df-mo 3bitr4g ) ABDGZBDHZICDGZCDHZIB
      DJCDJAPRQSABCDEFKABCDEFLMBDNCDNO $.
  $}

  ${
    $d x ph $.
    mobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for "at most one" quantifier (deduction form).
       (Contributed by Mario Carneiro, 7-Oct-2016.) $)
    mobidv $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
      ( nfv mobid ) ABCDADFEG $.
  $}

  ${
    mobii.1 $e |- ( ps <-> ch ) $.
    $( Formula-building rule for "at most one" quantifier (inference form).
       (Contributed by NM, 9-Mar-1995.)  (Revised by Mario Carneiro,
       17-Oct-2016.) $)
    mobii $p |- ( E* x ps <-> E* x ch ) $=
      ( wmo wb wtru a1i mobidv mptru ) ACEBCEFGABCABFGDHIJ $.
  $}

  $( Bound-variable hypothesis builder for "at most one".  (Contributed by NM,
     8-Mar-1995.) $)
  hbmo1 $p |- ( E* x ph -> A. x E* x ph ) $=
    ( wmo wex weu wi df-mo hbe1 hbeu1 hbim hbxfrbi ) ABCABDZABEZFBABGLMBABHABIJ
    K $.

  ${
    hbmo.1 $e |- ( ph -> A. x ph ) $.
    $( Bound-variable hypothesis builder for "at most one".  (Contributed by
       NM, 9-Mar-1995.) $)
    hbmo $p |- ( E* y ph -> A. x E* y ph ) $=
      ( wmo wex weu wi df-mo hbex hbeu hbim hbxfrbi ) ACEACFZACGZHBACINOBABCDJA
      BCDKLM $.
  $}

  ${
    cbvmo.1 $e |- F/ y ph $.
    cbvmo.2 $e |- F/ x ps $.
    cbvmo.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 9-Mar-1995.)  (Revised by Andrew Salmon,
       8-Jun-2011.) $)
    cbvmo $p |- ( E* x ph <-> E* y ps ) $=
      ( wex weu wi wmo cbvex cbveu imbi12i df-mo 3bitr4i ) ACHZACIZJBDHZBDIZJAC
      KBDKQSRTABCDEFGLABCDEFGMNACOBDOP $.
  $}

  ${
    $d x y z $.  $d ph z $.  $d ps z $.
    cbvmow.1 $e |- F/ y ph $.
    cbvmow.2 $e |- F/ x ps $.
    cbvmow.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvmo with a disjoint variable condition.  (Contributed by
       NM, 9-Mar-1995.)  (Revised by GG, 23-May-2024.) $)
    cbvmow $p |- ( E* x ph <-> E* y ps ) $=
      ( cbvmo ) ABCDEFGH $.
  $}

  ${
    $d x y z $.  $d ph z $.
    mo23.1 $e |- F/ y ph $.
    $( An implication between two definitions of "there exists at most one."
       (Contributed by Jim Kingdon, 25-Jun-2018.) $)
    mo23 $p |- ( E. y A. x ( ph -> x = y ) ->
               A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
      ( vz weq wi wal wex wsb wa nfv nfim nfal equequ2 imbi2d albidv cbvex nfri
      nfs1v sbequ2 ax-8 imim12d cbv3 ancli sylibr anim12 equtr2 syl6 2alimi syl
      aaanh exlimiv sylbir ) ABCFZGZBHZCIABEFZGZBHZEIAABCJZKZUOGZCHBHZUTUQECUSC
      BAURCDURCLMZNUQELECFZUSUPBVFURUOAECBOPQRUTVDEUTUSVACEFZGZKZCHBHZVDUTUTVHC
      HZKVJUTVKUSVHBCVEVAVGBABCTVGBLMZUOVAAURVGABCUABCEUBUCUDUEUSVHBCUSCVESVHBV
      LSULUFVIVCBCVIVBURVGKUOAURVAVGUGBCEUHUIUJUKUMUN $.
  $}

  ${
    $d x y $.
    mor.1 $e |- F/ y ph $.
    $( Converse of ~ mo23 with an additional ` E. x ph ` condition.
       (Contributed by Jim Kingdon, 25-Jun-2018.) $)
    mor $p |- ( E. x ph ->
        ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ->
          E. y A. x ( ph -> x = y ) ) ) $=
      ( wex wsb wa weq wi sb8e impexp bi2.04 bitri 2albii nfs1v nfri eximi alim
      wal alimi a7s exim syl syl5com biimtrid sylbi ) ABEABCFZCEZAUGGBCHZIZCSBS
      ZAUIIZBSZCEZIABCDJUKUGULIZCSBSZUHUNUJUOBCUJAUGUIIIUOAUGUIKAUGUILMNUHUGBSZ
      CEZUPUNUGUQCUGBABCOPQUPUQUMIZCSZURUNIUOUTCBUOBSUSCUGULBRTUAUQUMCUBUCUDUEU
      F $.
  $}

  ${
    $d x y $.
    modc.1 $e |- F/ y ph $.
    $( Equivalent definitions of "there exists at most one," given decidable
       existence.  (Contributed by Jim Kingdon, 1-Jul-2018.) $)
    modc $p |- ( DECID E. x ph ->
        ( E. y A. x ( ph -> x = y ) <->
          A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
      ( wex wdc weq wi wal wsb wa mo23 wn wo exmiddc mor mo2n a1d jaoi syl
      impbid2 ) ABEZFZABCGZHBICEZAABCJKUDHCIBIZABCDLUCUBUBMZNUFUEHZUBOUBUHUGABC
      DPUGUEUFABCDQRSTUA $.
  $}

  ${
    $d x y $.
    eu2.1 $e |- F/ y ph $.
    $( An alternate way of defining existential uniqueness.  Definition 6.10 of
       [TakeutiZaring] p. 26.  (Contributed by NM, 8-Jul-1994.) $)
    eu2 $p |- ( E! x ph <->
        ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
      ( weu wex wsb wa weq wal euex nfri eumo0 mo23 syl jca 19.29r impexp albii
      wi 19.21 bitri anbi2i abai bitr4i exbii sylib eu1 sylibr impbii ) ABEZABF
      ZAABCGZHBCIZTZCJZBJZHZUKULUQABKUKAUNTBJCFUQABCACDLZMABCDNOPURAUMUNTZCJZHZ
      BFZUKURAUPHZBFVCAUPBQVDVBBVDAAVATZHVBUPVEAUPAUTTZCJVEUOVFCAUMUNRSAUTCDUAU
      BUCAVAUDUEUFUGABCUSUHUIUJ $.
  $}

  ${
    $d x y $.
    eu3h.1 $e |- ( ph -> A. y ph ) $.
    $( An alternate way to express existential uniqueness.  (Contributed by NM,
       8-Jul-1994.)  (New usage is discouraged.) $)
    eu3h $p |- ( E! x ph <->
                ( E. x ph /\ E. y A. x ( ph -> x = y ) ) ) $=
      ( weu wex weq wi wal euex eumo0 jca wsb nfi mo23 anim2i eu2 sylibr impbii
      wa ) ABEZABFZABCGZHBICFZTZUAUBUDABJABCDKLUEUBAABCMTUCHCIBIZTUAUDUFUBABCAC
      DNZOPABCUGQRS $.
  $}

  ${
    $d x y $.
    eu3.1 $e |- F/ y ph $.
    $( An alternate way to express existential uniqueness.  (Contributed by NM,
       8-Jul-1994.) $)
    eu3 $p |- ( E! x ph <->
                ( E. x ph /\ E. y A. x ( ph -> x = y ) ) ) $=
      ( nfri eu3h ) ABCACDEF $.
  $}

  $( Uniqueness in terms of "at most one".  (Contributed by NM, 23-Mar-1995.)
     (Proof rewritten by Jim Kingdon, 27-May-2018.) $)
  eu5 $p |- ( E! x ph <-> ( E. x ph /\ E* x ph ) ) $=
    ( weu wex wmo wa euex eumo jca wi df-mo biimpi imp ancoms impbii ) ABCZABDZ
    ABEZFPQRABGABHIRQPRQPRQPJABKLMNO $.

  $( Existence implies "at most one" is equivalent to uniqueness.  (Contributed
     by NM, 5-Apr-2004.) $)
  exmoeu2 $p |- ( E. x ph -> ( E* x ph <-> E! x ph ) ) $=
    ( weu wex wmo eu5 baibr ) ABCABDABEABFG $.

  $( Absorption of existence condition by "at most one".  (Contributed by NM,
     4-Nov-2002.) $)
  moabs $p |- ( E* x ph <-> ( E. x ph -> E* x ph ) ) $=
    ( wex weu wi wmo pm5.4 df-mo imbi2i 3bitr4ri ) ABCZKABDZEZEMKABFZENKLGNMKAB
    HZIOJ $.

  $( If existence is decidable, something exists or at most one exists.
     (Contributed by Jim Kingdon, 30-Jun-2018.) $)
  exmodc $p |- ( DECID E. x ph -> ( E. x ph \/ E* x ph ) ) $=
    ( wex wdc wn wo wmo df-dc weu wi pm2.21 df-mo sylibr orim2i sylbi ) ABCZDPP
    EZFPABGZFPHQRPQPABIZJRPSKABLMNO $.

  $( There is at most one of something which does not exist.  Unlike ~ exmodc
     there is no decidability condition.  (Contributed by Jim Kingdon,
     22-Sep-2018.) $)
  exmonim $p |- ( -. E. x ph -> E* x ph ) $=
    ( wex wn weu wi wmo pm2.21 df-mo sylibr ) ABCZDKABEZFABGKLHABIJ $.

  ${
    $d x y $.
    mo2r.1 $e |- F/ y ph $.
    $( A condition which implies "at most one".  (Contributed by Jim Kingdon,
       2-Jul-2018.) $)
    mo2r $p |- ( E. y A. x ( ph -> x = y ) -> E* x ph ) $=
      ( weq wi wal wex weu wmo nfri eu3h simplbi2com df-mo sylibr ) ABCEFBGCHZA
      BHZABIZFABJRQPABCACDKLMABNO $.
  $}

  ${
    $d x y $.
    mo3h.1 $e |- ( ph -> A. y ph ) $.
    $( Alternate definition of "at most one".  Definition of [BellMachover]
       p. 460, except that definition has the side condition that ` y ` not
       occur in ` ph ` in place of our hypothesis.  (Contributed by NM,
       8-Mar-1995.)  (New usage is discouraged.) $)
    mo3h $p |- ( E* x ph <->
               A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
      ( wmo wsb wa weq wi wal wex weu nfi eu2 imbi2i df-mo anclb 3bitr4i sylibr
      19.38 19.21 albii anabs5 pm3.31 biimtrrid 2alimi sylbi simplbi2com impbii
      syl ) ABEZAABCFZGZBCHZIZCJZBJZUKABKZUQIZUQURABLZIZURURUQGZIUKUSUTVBURABCA
      CDMZNZOABPZURUQQRUSAUOIZCJZBJZUQUSAUPIZBJVHAUPBTVGVIBAUOCVCUAUBSVFUOBCUMA
      UMGVFUNAULUCAUMUNUDUEUFUJUGUQVAUKUTURUQVDUHVESUI $.
  $}

  ${
    $d x y $.
    mo3.1 $e |- F/ y ph $.
    $( Alternate definition of "at most one".  Definition of [BellMachover]
       p. 460, except that definition has the side condition that ` y ` not
       occur in ` ph ` in place of our hypothesis.  (Contributed by NM,
       8-Mar-1995.) $)
    mo3 $p |- ( E* x ph <->
               A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
      ( nfri mo3h ) ABCACDEF $.
  $}

  ${
    $d x y $.
    mo2dc.1 $e |- F/ y ph $.
    $( Alternate definition of "at most one" where existence is decidable.
       (Contributed by Jim Kingdon, 2-Jul-2018.) $)
    mo2dc $p |- ( DECID E. x ph ->
        ( E* x ph <-> E. y A. x ( ph -> x = y ) ) ) $=
      ( wex wdc wmo wsb wa weq wi wal nfri mo3h modc bitr4id ) ABEFABGAABCHIBCJ
      ZKCLBLAQKBLCEABCACDMNABCDOP $.
  $}

  ${
    euan.1 $e |- ( ph -> A. x ph ) $.
    $( Introduction of a conjunct into unique existential quantifier.
       (Contributed by NM, 19-Feb-2005.)  (Proof shortened by Andrew Salmon,
       9-Jul-2011.) $)
    euan $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $=
      ( wa weu wex wmo simpl exlimih adantr simpr eximi hbe1 a1d impbid2 mobidh
      ancrd biimpa eu5 jca32 anbi2i 3imtr4i ibar eubidh impbii ) ABEZCFZABCFZEZ
      UGCGZUGCHZEZABCGZBCHZEZEUHUJUMAUNUOUKAULUGACDABIJZKUKUNULUGBCABLZMKUKULUO
      UKUGBCUGCNUKUGBURUKBAUKABUQORPQSUAUGCTUIUPABCTUBUCAUIUHABUGCDABUDUESUF $.
  $}

  ${
    $d x ph $.
    $( Introduction of a conjunct into unique existential quantifier.
       (Contributed by NM, 23-Mar-1995.) $)
    euanv $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $=
      ( ax-17 euan ) ABCACDE $.
  $}

  $( Introduce or eliminate a disjunct in a unique existential quantifier.
     (Contributed by NM, 21-Oct-2005.)  (Proof shortened by Andrew Salmon,
     9-Jul-2011.) $)
  euor2 $p |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) ) $=
    ( wex wn wo hbe1 hbn wb 19.8a con3i orel1 olc impbid1 syl eubidh ) ACDZEZAB
    FZBCQCACGHRAEZSBIAQACJKTSBABLBAMNOP $.

  ${
    $d w x z $.  $d w y z $.  $d w ph $.
    $( Substitution into "at most one".  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    sbmo $p |- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph ) $=
      ( vw wsb wa cv wceq wal wmo nfv sblim sban imbi1i sbcom2 anbi2i sbalv mo3
      wi 3bitri sbbii 3bitr4i ) AADEFZGZDHEHIZTZEJZDJZBCFABCFZUJDEFZGZUFTZEJZDJ
      ADKZBCFUJDKUHUNBCDUGUMBCEUGBCFUEBCFZUFTUJUDBCFZGZUFTUMUEUFBCUFBLMUPURUFAU
      DBCNOURULUFUQUKUJADEBCPQOUARRUOUIBCADEAELSUBUJDEUJELSUC $.
  $}

  ${
    $d x y $.  $d y ph $.
    mo4f.1 $e |- F/ x ps $.
    mo4f.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( "At most one" expressed using implicit substitution.  (Contributed by
       NM, 10-Apr-2004.) $)
    mo4f $p |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) ) $=
      ( wmo wsb wa weq wi wal ax-17 mo3h sbie anbi2i imbi1i 2albii bitri ) ACGA
      ACDHZIZCDJZKZDLCLABIZUBKZDLCLACDADMNUCUECDUAUDUBTBAABCDEFOPQRS $.
  $}

  ${
    $d x y $.  $d y ph $.  $d x ps $.
    mo4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( "At most one" expressed using implicit substitution.  (Contributed by
       NM, 26-Jul-1995.) $)
    mo4 $p |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) ) $=
      ( nfv mo4f ) ABCDBCFEG $.
  $}

  ${
    $d x y $.  $d y ph $.  $d x ps $.
    eu4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Uniqueness using implicit substitution.  (Contributed by NM,
       26-Jul-1995.) $)
    eu4 $p |- ( E! x ph <-> ( E. x ph /\
             A. x A. y ( ( ph /\ ps ) -> x = y ) ) ) $=
      ( weu wex wmo wa weq wi wal eu5 mo4 anbi2i bitri ) ACFACGZACHZIQABICDJKDL
      CLZIACMRSQABCDENOP $.
  $}

  $( Existence in terms of "at most one" and uniqueness.  (Contributed by Jim
     Kingdon, 3-Jul-2018.) $)
  exmoeudc $p |- ( DECID E. x ph -> ( E. x ph <-> ( E* x ph -> E! x ph ) ) ) $=
    ( wex wdc wmo weu wi df-mo com12 biimpri euex imim12i peircedc syl5 impbid2
    biimpi ) ABCZDZQABEZABFZGZSQTSQTGZABHZPIUAUBQGRQUBSTQSUBUCJABKLQTMNO $.

  ${
    $d x y $.  $d y ph $.  $d y ps $.
    $( "At most one" is preserved through implication (notice wff reversal).
       (Contributed by NM, 22-Apr-1995.) $)
    moim $p |- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) ) $=
      ( vy wi wal wsb wa weq nfa1 ax-4 spsbim anim12d imim1d alimdv alimd ax-17
      wmo mo3h 3imtr4g ) ABEZCFZBBCDGZHZCDIZEZDFZCFAACDGZHZUEEZDFZCFBCRACRUBUGU
      KCUACJUBUFUJDUBUIUDUEUBABUHUCUACKABCDLMNOPBCDBDQSACDADQST $.
  $}

  ${
    moimi.1 $e |- ( ph -> ps ) $.
    $( "At most one" is preserved through implication (notice wff reversal).
       (Contributed by NM, 15-Feb-2006.) $)
    moimi $p |- ( E* x ps -> E* x ph ) $=
      ( wi wmo moim mpg ) ABEBCFACFECABCGDH $.
  $}

  ${
    $d x y $.  $d x y ph $.  $d y ps $.
    $( Move antecedent outside of "at most one".  (Contributed by NM,
       28-Jul-1995.) $)
    moimv $p |- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) ) $=
      ( vy wi wmo wsb weq wal ax-1 a1i sbimi nfv sbf sbim 3imtr3i anim12d ax-17
      wa mo3h imim1d 2alimdv 3imtr4g com12 ) AABEZCFZBCFZAUEUECDGZSZCDHZEZDICIB
      BCDGZSZUJEZDICIUFUGAUKUNCDAUMUIUJABUEULUHBUEEZABAJKZACDGUOCDGAULUHEAUOCDU
      PLACDACMNBUECDOPQUAUBUECDUEDRTBCDBDRTUCUD $.
  $}

  $( Uniqueness implies "at most one" through implication.  (Contributed by NM,
     22-Apr-1995.) $)
  euimmo $p |- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) ) $=
    ( weu wmo wi wal eumo moim syl5 ) BCDBCEABFCGACEBCHABCIJ $.

  $( Add existential unique existential quantifiers to an implication.  Note
     the reversed implication in the antecedent.  (Contributed by NM,
     19-Oct-2005.)  (Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
  euim $p |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) ) $=
    ( wex wi wal wa weu wmo ax-1 euimmo anim12ii eu5 imbitrrdi ) ACDZABECFZGBCH
    ZOACIZGACHOQOPROQJABCKLACMN $.

  $( "At most one" is still the case when a conjunct is added.  (Contributed by
     NM, 22-Apr-1995.) $)
  moan $p |- ( E* x ph -> E* x ( ps /\ ph ) ) $=
    ( wa simpr moimi ) BADACBAEF $.

  ${
    moani.1 $e |- E* x ph $.
    $( "At most one" is still true when a conjunct is added.  (Contributed by
       NM, 9-Mar-1995.) $)
    moani $p |- E* x ( ps /\ ph ) $=
      ( wmo wa moan ax-mp ) ACEBAFCEDABCGH $.
  $}

  $( "At most one" is still the case when a disjunct is removed.  (Contributed
     by NM, 5-Apr-2004.) $)
  moor $p |- ( E* x ( ph \/ ps ) -> E* x ph ) $=
    ( wo orc moimi ) AABDCABEF $.

  $( "At most one" imports disjunction to conjunction.  (Contributed by NM,
     5-Apr-2004.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
  mooran1 $p |- ( ( E* x ph \/ E* x ps ) -> E* x ( ph /\ ps ) ) $=
    ( wmo wa simpl moimi moan jaoi ) ACDABEZCDBCDJACABFGBACHI $.

  $( "At most one" exports disjunction to conjunction.  (Contributed by NM,
     5-Apr-2004.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
  mooran2 $p |- ( E* x ( ph \/ ps ) -> ( E* x ph /\ E* x ps ) ) $=
    ( wo wmo moor olc moimi jca ) ABDZCEACEBCEABCFBJCBAGHI $.

  ${
    $d x y $.  $d y ph $.  $d y ps $.
    moanim.1 $e |- F/ x ph $.
    $( Introduction of a conjunct into at-most-one quantifier.  (Contributed by
       NM, 3-Dec-2001.) $)
    moanim $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
      ( vy wsb wa weq wi wal wmo anandi imbi1i impexp sbf anbi1i ax-17 3bitr4ri
      sban mo3h bitr2i anbi2i 3bitr3i 2albii 19.21 19.21v albii imbi2i ) ABBCEF
      ZGZCEHZIZIZEJZCJZABGZUPCEFZGZUKIZEJCJABCKZIZUPCKUMUSCEAUJGZUKIUPAUIGZGZUK
      IUMUSVBVDUKABUILMAUJUKNVDURUKVCUQUPUQACEFZUIGVCABCESVEAUIACEDOPUAUBMUCUDA
      ULEJZIZCJAVFCJZIUOVAAVFCDUEUNVGCAULEUFUGUTVHABCEBEQTUHRUPCEUPEQTR $.
  $}

  ${
    $d x ph $.
    $( Introduction of a conjunct into at-most-one quantifier.  (Contributed by
       NM, 23-Mar-1995.) $)
    moanimv $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
      ( nfv moanim ) ABCACDE $.
  $}

  $( Nested at-most-one and unique existential quantifiers.  (Contributed by
     NM, 25-Jan-2006.) $)
  moaneu $p |- E* x ( ph /\ E! x ph ) $=
    ( weu wa wmo wi eumo nfeu1 moanim mpbir ancom mobii ) AABCZDZBEMADZBEZPMABE
    FABGMABABHIJNOBAMKLJ $.

  $( Nested at-most-one quantifiers.  (Contributed by NM, 25-Jan-2006.) $)
  moanmo $p |- E* x ( ph /\ E* x ph ) $=
    ( wmo wa wi id nfmo1 moanim mpbir ancom mobii ) AABCZDZBCLADZBCZOLLELFLABAB
    GHIMNBALJKI $.

  ${
    $d x y $.  $d y ph $.  $d y ps $.
    $( "At most one" picks a variable value, eliminating an existential
       quantifier.  (Contributed by NM, 27-Jan-1997.) $)
    mopick $p |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) ) $=
      ( vy wa wex wmo wi wsb hbs1 hban weq sbequ12 anbi12d cbvexh wal mo3h ax-4
      ax-17 sylbi sps sbequ2 imim2i expd com4t imp syl5 exlimiv impcom ) ABEZCF
      ZACGZABHZUKACDIZBCDIZEZDFULUMHZUJUPCDUJDSUNUOCACDJBCDJKCDLZAUNBUOACDMBCDM
      NOUPUQDULAUNEZURHZUPUMULUTDPZCPUTACDADSQVAUTCUTDRUATUNUOUTUMHUTAUNUOBUTAU
      NUOBHZURVBUSBCDUBUCUDUEUFUGUHTUI $.
  $}

  $( Existential uniqueness "picks" a variable value for which another wff is
     true.  If there is only one thing ` x ` such that ` ph ` is true, and
     there is also an ` x ` (actually the same one) such that ` ph ` and ` ps `
     are both true, then ` ph ` implies ` ps ` regardless of ` x ` .  This
     theorem can be useful for eliminating existential quantifiers in a
     hypothesis.  Compare Theorem *14.26 in [WhiteheadRussell] p. 192.
     (Contributed by NM, 10-Jul-1994.) $)
  eupick $p |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) ) $=
    ( weu wmo wa wex wi eumo mopick sylan ) ACDACEABFCGABHACIABCJK $.

  $( Version of ~ eupick with closed formulas.  (Contributed by NM,
     6-Sep-2008.) $)
  eupicka $p |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) ) $=
    ( weu wa wex wi hbeu1 hbe1 hban eupick alrimih ) ACDZABEZCFZEABGCMOCACHNCIJ
    ABCKL $.

  $( Existential uniqueness "pick" showing wff equivalence.  (Contributed by
     NM, 25-Nov-1994.) $)
  eupickb $p |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) ->
               ( ph <-> ps ) ) $=
    ( weu wa wex w3a wi eupick 3adant2 3simpc pm3.22 eximi anim2i 3syl impbid )
    ACDZBCDZABEZCFZGZABQTABHRABCIJUARTERBAEZCFZEBAHQRTKTUCRSUBCABLMNBACIOP $.

  $( Theorem *14.26 in [WhiteheadRussell] p. 192.  (Contributed by Andrew
     Salmon, 11-Jul-2011.) $)
  eupickbi $p |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) ) $=
    ( weu wa wex wi wal eupicka ex hba1 wb ancl simpl impbid1 sps euex biimtrdi
    eubidh com12 impbid ) ACDZABEZCFZABGZCHZUBUDUFABCIJUFUBUDUFUBUCCDUDUFAUCCUE
    CKUEAUCLCUEAUCABMABNOPSUCCQRTUA $.

  $( "At most one" can show the existence of a common value.  In this case we
     can infer existence of conjunction from a conjunction of existence, and it
     is one way to achieve the converse of ~ 19.40 .  (Contributed by NM,
     5-Apr-2004.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
  mopick2 $p |- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) ->
                E. x ( ph /\ ps /\ ch ) ) $=
    ( wmo wex w3a hbmo1 hbe1 mopick ancld anim1d df-3an imbitrrdi eximdh 3impia
    wa hban ) ADEZABQZDFZACQZDFABCGZDFSUAQZUBUCDSUADADHTDIRUDUBTCQUCUDATCUDABAB
    DJKLABCMNOP $.

  ${
    moexexdc.1 $e |- F/ y ph $.
    $( "At most one" double quantification.  (Contributed by Jim Kingdon,
       5-Jul-2018.) $)
    moexexdc $p |- ( DECID E. x ph ->
        ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) ) $=
      ( wex wdc wn wo wmo wal wa wi df-dc hbmo1 hba1 hbmo hbim exlimih a1d hbe1
      nfri mopick com3r alrimdh moim spsd syl6 hbex exsimpl con3i mon jaoi impd
      ex syl sylbi ) ACFZGURURHZIZACJZBDJZCKZLABLZCFZDJZMURNUTVAVCVFURVAVCVFMZM
      ZUSAVHCVAVGCACOVCVFCVBCPVECDVDCUAQRRAVAVEBMZDKZVGAVAVIDADEUBZADCVKQVAVEAB
      VAVEABMABCUCUOUDUEVJVBVFCVEBDUFUGUHSUSVGVAUSVFVCUSVEDFZHVFVLURVEURDADCVKU
      IABCUJSUKVEDULUPTTUMUNUQ $.
  $}

  ${
    euexex.1 $e |- F/ y ph $.
    $( Existential uniqueness and "at most one" double quantification.
       (Contributed by Jim Kingdon, 28-Dec-2018.) $)
    euexex $p |-
        ( ( E! x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) $=
      ( weu wmo wal wa wex wi eu5 nfmo1 nfa1 nfe1 nfmo nfim mopick ex imp com3r
      alrimd moim spsd syl6 exlimi sylbi ) ACFZBDGZCHZABIZCJZDGZUHACJZACGZIUJUM
      KZACLUNUOUPAUOUPKCUOUPCACMUJUMCUICNULCDUKCOPQQAUOULBKZDHZUPAUOUQDEADCEPUO
      ULABUOULABKABCRSUAUBURUIUMCULBDUCUDUEUFTUGT $.
  $}

  $( Double quantification with "at most one".  (Contributed by NM,
     3-Dec-2001.) $)
  2moex $p |- ( E* x E. y ph -> A. y E* x ph ) $=
    ( wex wmo hbe1 hbmo 19.8a moimi alrimih ) ACDZBEABECKCBACFGAKBACHIJ $.

  $( Double quantification with existential uniqueness.  (Contributed by NM,
     3-Dec-2001.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
  2euex $p |- ( E! x E. y ph -> E. y E! x ph ) $=
    ( wex weu wmo wa excom hbe1 hbmo wi 19.8a moimi df-mo sylib eximdh biimtrid
    eu5 impcom sylbi ) ACDZBEUABDZUABFZGABEZCDZUABRUCUBUEUBABDZCDUCUEABCHUCUFUD
    CUACBACIJUCABFUFUDKAUABACLMABNOPQST $.

  $( Double quantification with existential uniqueness and "at most one."
     (Contributed by NM, 3-Dec-2001.) $)
  2eumo $p |- ( E! x E* y ph -> E* x E! y ph ) $=
    ( weu wmo wi euimmo eumo mpg ) ACDZACEZFKBDJBEFBJKBGACHI $.

  $( Double existential uniqueness.  (Contributed by NM, 3-Dec-2001.) $)
  2eu2ex $p |- ( E! x E! y ph -> E. x E. y ph ) $=
    ( weu wex euex eximi syl ) ACDZBDIBEACEZBEIBFIJBACFGH $.

  $( A condition allowing swap of "at most one" and existential quantifiers.
     (Contributed by Jim Kingdon, 6-Jul-2018.) $)
  2moswapdc $p |- ( DECID E. x E. y ph -> ( A. x E* y ph ->
      ( E* x E. y ph -> E* y E. x ph ) ) ) $=
    ( wex wdc wmo wa wi nfe1 moexexdc expcomd 19.8a pm4.71ri exbii mobii imbi2i
    wal imbitrrdi ) ACDZBDEZACFBQZSBFZSAGZBDZCFZHUBABDZCFZHTUBUAUESABCACIJKUGUE
    UBUFUDCAUCBASACLMNOPR $.

  $( A condition allowing swap of uniqueness and existential quantifiers.
     (Contributed by Jim Kingdon, 7-Jul-2018.) $)
  2euswapdc $p |- ( DECID E. x E. y ph -> ( A. x E* y ph ->
      ( E! x E. y ph -> E! y E. x ph ) ) ) $=
    ( wex wdc wmo wal weu wi wa excomim a1i 2moswapdc imp anim12d eu5 3imtr4g
    ex ) ACDZBDZEZACFBGZSBHZABDZCHZIUAUBJZTSBFZJUDCDZUDCFZJUCUEUFTUHUGUITUHIUFA
    BCKLUAUBUGUIIABCMNOSBPUDCPQR $.

  $( Double existential uniqueness implies double unique existential
     quantification.  (Contributed by NM, 3-Dec-2001.) $)
  2exeu $p |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph ) $=
    ( wex wmo weu excom hbe1 hbmo 19.41h 19.8a moimi anim2i eximi sylbir sylanb
    wa simpl eu5 anbi12i adantl anim12i ancoms exbii mobii bitri 3imtr4i ) ACDZ
    BDZUHBEZQZABDZCDZULCEZQZQUHACEZQZBDZUQBEZQZUHBFZULCFZQACFZBFZUOUKUTUOURUKUS
    UMUIUNURACBGUIUNQUHUNQZBDURUHUNBULBCABHIJVEUQBUNUPUHAULCABKLMNOPUJUSUIUQUHB
    UHUPRLUAUBUCVAUKVBUOUHBSULCSTVDVCBDZVCBEZQUTVCBSVFURVGUSVCUQBACSZUDVCUQBVHU
    ETUFUG $.

  ${
    $d x y z w $.  $d z w ph $.
    $( This theorem provides us with a definition of double existential
       uniqueness ("exactly one ` x ` and exactly one ` y ` ").  Naively one
       might think (incorrectly) that it could be defined by ` E! x E! y ph ` .
       See ~ 2exeu for a one-way implication.  (Contributed by NM,
       3-Dec-2001.) $)
    2eu4 $p |- ( ( E! x E. y ph /\ E! y E. x ph ) <->
      ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) ) $=
      ( wex weu wa weq wi wal ax-17 eu3h anbi12i anbi2i bitri 19.3h 19.26 albii
      hba1 excom anidm jcab 3bitr4ri alcom bitr4i 19.23v 2albii hbe1 hbim aaanh
      an4 3bitri 2exbii eeanv bitr2i ) ACFZBGZABFZCGZHUQBFZUQBDIZJZBKZDFZHZUSCF
      ZUSCEIZJZCKZEFZHZHVAVGHZVEVKHZHVAAVBVHHJZCKZBKZEFDFZHURVFUTVLUQBDUQDLMUSC
      EUSELMNVAVEVGVKULVMVAVNVRVMVAVAHVAVGVAVAACBUAOVAUBPVRVDVJHZEFDFVNVQVSDEVQ
      AVBJZCKZAVHJZBKZHZCKZBKZVCVIHZCKBKVSVQWAWBCKZBKZHZBKZWFWABKZWIBKZHWLWIHZW
      KVQWMWIWLWIBWHBTQOWAWIBRVQWAWHHZBKWNVPWOBVPVTWBHZCKWOVOWPCAVBVHUCSVTWBCRP
      SWAWHBRPUDWEWJBWEWACKZWCCKZHWJWAWCCRWQWAWRWIWACVTCTQWBCBUENPSUFWDWGBCWAVC
      WCVIAVBCUGAVHBUGNUHVCVIBCUQVBCACUIVBCLUJUSVHBABUIVHBLUJUKUMUNVDVJDEUOUPNU
      M $.
  $}

  $( Two equivalent expressions for double existential uniqueness.
     (Contributed by NM, 19-Feb-2005.) $)
  2eu7 $p |- ( ( E! x E. y ph /\ E! y E. x ph ) <->
             E! x E! y ( E. x ph /\ E. y ph ) ) $=
    ( wex weu wa hbe1 hbeu euan ancom eubii 3bitri 3bitr4ri ) ABDZCEZACDZFZBEOP
    BEZFNPFZCEZBEROFOPBNBCABGHITQBTPNFZCEPOFQSUACNPJKPNCACGIPOJLKROJM $.

  ${
    $d x y z $.
    $( Equality has existential uniqueness.  (Contributed by Stefan Allan,
       4-Dec-2008.) $)
    euequ1 $p |- E! x x = y $=
      ( vz weq weu wex wa wi wal a9e equtr2 gen2 equequ1 eu4 mpbir2an ) ABDZAEP
      AFPCBDZGACDHZCIAIABJRACACBKLPQACACBMNO $.
  $}

  ${
    $d x y $.
    $( Two ways to express "only one thing exists".  The left-hand side
       requires only one variable to express this.  Both sides are false in set
       theory.  (Contributed by NM, 5-Apr-2004.) $)
    exists1 $p |- ( E! x x = x <-> A. x x = y ) $=
      ( weq weu wb wal wex df-eu equid tbt bicom bitri albii exbii hbae 3bitr2i
      19.9h ) AACZADRABCZEZAFZBGSAFZBGUBRABHUBUABSTASSRETRSAIJSRKLMNUBBABBOQP
      $.
  $}

  ${
    $d x y $.
    $( A condition implying that at least two things exist.  (Contributed by
       NM, 10-Apr-2004.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
    exists2 $p |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x ) $=
      ( vy wex wn cv wceq weu wal hbeu1 hba1 wi exists1 ax16 sylbi com12 alexim
      exlimdh syl6 con2d imp ) ABDZAEBDZBFZUDGZBHZEUBUFUCUBUFABIZUCEUFUBUGUFAUG
      BUEBJABKUFUDCFGBIAUGLBCMABCNORPABQSTUA $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Aristotelian logic: Assertic syllogisms
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Model the Aristotelian assertic syllogisms using modern notation.
  This section shows that the Aristotelian assertic syllogisms can be proven
  with our axioms of logic, and also provides generally useful theorems.

  In antiquity Aristotelian logic and Stoic logic
  (see ~ mptnan ) were the leading logical systems.
  Aristotelian logic became the leading system in medieval Europe;
  this section models this system (including later refinements to it).
  Aristotle defined syllogisms very generally
  ("a discourse in which certain (specific) things having been supposed,
  something different from the things supposed results of necessity
  because these things are so")
  Aristotle, _Prior Analytics_ 24b18-20.
  However, in _Prior Analytics_ he limits himself to
  categorical syllogisms that consist of three categorical propositions
  with specific structures.  The syllogisms are the valid subset of
  the possible combinations of these structures.
  The medieval schools used vowels to identify the types of terms
  (a=all, e=none, i=some, and o=some are not), and named the different
  syllogisms with Latin words that had the vowels in the intended order.

  "There is a surprising amount of scholarly debate
  about how best to formalize Aristotle's syllogisms..." according to
  _Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic_,
  Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28.
  For example, Lukasiewicz believes it is important to note that
  "Aristotle does not introduce singular terms or premisses into his system".
  Lukasiewicz also believes that Aristotelian syllogisms are
  predicates (having a true/false value), not inference rules:
  "The characteristic sign of an inference is the word 'therefore'...
  no syllogism is formulated by Aristotle primarily as an inference,
  but they are all implications."
  Jan Lukasiewicz, _Aristotle's Syllogistic from the Standpoint of
  Modern Formal Logic_, Second edition, Oxford, 1957, page 1-2.
  Lukasiewicz devised a specialized prefix notation for representing
  Aristotelian syllogisms instead of using standard predicate logic notation.

  We instead translate each Aristotelian syllogism into an inference rule,
  and each rule is defined using standard predicate logic notation and
  predicates.  The predicates are represented by wff variables
  that may depend on the quantified variable ` x ` .
  Our translation is essentially identical to the one
  use in Rini page 18, Table 2 "Non-Modal Syllogisms in
  Lower Predicate Calculus (LPC)", which uses
  standard predicate logic with predicates.  Rini states,
  "the crucial point is that we capture the meaning Aristotle intends,
  and the method by which we represent that meaning is less important."
  There are two differences: we make the existence criteria explicit, and
  we use ` ph ` , ` ps ` , and ` ch ` in the order they appear
  (a common Metamath convention).
  Patzig also uses standard predicate logic notation and predicates
  (though he interprets them as conditional propositions, not as
  inference rules); see
  Gunther Patzig, _Aristotle's Theory of the Syllogism_ second edition, 1963,
  English translation by Jonathan Barnes, 1968, page 38.
  Terms such as "all" and "some" are translated into predicate logic
  using the aproach devised by Frege and Russell.
  "Frege (and Russell) devised an ingenious procedure for regimenting
  binary quantifiers like "every" and "some" in terms of unary quantifiers
  like "everything" and "something": they formalized sentences of the form
  "Some A is B" and "Every A is B" as
  exists x (Ax and Bx) and all x (Ax implies Bx), respectively."
  "Quantifiers and Quantification", _Stanford Encyclopedia of Philosophy_,
  ~ https://plato.stanford.edu/entries/quantification/ .
  See _Principia Mathematica_ page 22 and *10 for more information
  (especially *10.3 and *10.26).

  Expressions of the form "no ` ph ` is ` ps ` " are consistently translated as
  ` A. x ( ph -> -. ps ) ` .  These can also be expressed as
  ` -. E. x ( ph /\ ps ) ` , per ~ alinexa .
  We translate "all ` ph ` is ` ps ` " to ` A. x ( ph -> ps ) ` ,
  "some ` ph ` is ` ps ` " to ` E. x ( ph /\ ps ) ` , and
  "some ` ph ` is not ` ps ` " to ` E. x ( ph /\ -. ps ) ` .
  It is traditional to use the singular verb "is", not the plural
  verb "are", in the generic expressions.
  By convention the major premise is listed first.

  In traditional Aristotelian syllogisms the predicates
  have a restricted form ("x is a ..."); those predicates
  could be modeled in modern notation by constructs such as
  ` x = A ` , ` x e. A ` , or ` x C_ A ` .
  Here we use wff variables instead of specialized restricted forms.
  This generalization makes the syllogisms more useful
  in more circumstances.  In addition, these expressions make
  it clearer that the syllogisms of Aristolean logic are the
  forerunners of predicate calculus.  If we used restricted forms
  like ` x e. A ` instead, we would not only unnecessarily limit
  their use, but we would also need to use set and class axioms,
  making their relationship to predicate calculus less clear.

  There are some widespread misconceptions about the existential
  assumptions made by Aristotle (aka "existential import").
  Aristotle was not trying to develop something exactly corresponding
  to modern logic.  Aristotle devised "a companion-logic for science.
  He relegates fictions like fairy godmothers and mermaids and unicorns to
  the realms of poetry and literature. In his mind, they exist outside the
  ambit of science. This is why he leaves no room for such nonexistent
  entities in his logic.  This is a thoughtful choice, not an inadvertent
  omission. Technically, Aristotelian science is a search for definitions,
  where a definition is "a phrase signifying a thing's essence."
  (Topics, I.5.102a37, Pickard-Cambridge.)...
  Because nonexistent entities cannot be anything, they do not, in
  Aristotle's mind, possess an essence...  This is why he leaves
  no place for fictional entities like goat-stags (or unicorns)."
  Source: Louis F. Groarke, "Aristotle: Logic",
  section 7. (Existential Assumptions),
  _Internet Encyclopedia of Philosophy_ (A Peer-Reviewed Academic Resource),
  ~ https://iep.utm.edu/aristotle-log/ .
  Thus, some syllogisms have "extra" existence
  hypotheses that do not directly appear in Aristotle's original materials
  (since they were always assumed); they are added where they are needed.
  This affects ~ barbari , ~ celaront , ~ cesaro , ~ camestros , ~ felapton ,
  ~ darapti , ~ calemos , ~ fesapo , and ~ bamalip .

  These are only the _assertic_ syllogisms.
  Aristotle also defined modal syllogisms that deal with modal
  qualifiers such as "necessarily" and "possibly".
  Historically Aristotelian modal syllogisms were not as widely used.
  For more about modal syllogisms in a modern context, see Rini as well as
  _Aristotle's Modal Syllogistic_ by Marko Malink, Harvard
  University Press, November 2013.  We do not treat them further here.

  Aristotelian logic is essentially the forerunner of predicate calculus
  (as well as set theory since it discusses membership in groups),
  while Stoic logic is essentially the forerunner of propositional calculus.
$)

  $( Figure 1.  Aristotelian syllogisms are grouped by "figures",
     which doesn't matter for our purposes but is a reasonable way
     to order them. $)

  ${
    $( Major premise for the Aristotelian syllogism "Barbara", e.g.,
       "All men are mortal". By convention, the major premise is first. $)
    barbara.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Barbara, e.g., "Socrates is a man". $)
    barbara.min $e |- A. x ( ch -> ph ) $.
    $( "Barbara", one of the fundamental syllogisms of Aristotelian logic.  All
       ` ph ` is ` ps ` , and all ` ch ` is ` ph ` , therefore all ` ch ` is
       ` ps ` .  (In Aristotelian notation, AAA-1:  MaP and SaM therefore SaP.)
       For example, given "All men are mortal" and "Socrates is a man", we can
       prove "Socrates is mortal".  If H is the set of men, M is the set of
       mortal beings, and S is Socrates, these word phrases can be represented
       as ` A. x ( x e. H -> x e. M ) ` (all men are mortal) and
       ` A. x ( x = S -> x e. H ) ` (Socrates is a man) therefore
       ` A. x ( x = S -> x e. M ) ` (Socrates is mortal).  Russell and
       Whitehead note that the "syllogism in Barbara is derived..." from
       ~ syl .  (quote after Theorem *2.06 of [WhiteheadRussell] p. 101).  Most
       of the proof is in ~ alsyl .  There are a legion of sources for Barbara,
       including ~ https://www.friesian.com/aristotl.htm ,
       ~ https://plato.stanford.edu/entries/aristotle-logic/ , and
       ~ https://en.wikipedia.org/wiki/Syllogism .  (Contributed by David A.
       Wheeler, 24-Aug-2016.) $)
    barbara $p |- A. x ( ch -> ps ) $=
      ( wi wal alsyl mp2an ) CAGDHABGDHCBGDHFECABDIJ $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Celarent", e.g.,
       "No reptiles have fur". $)
    celarent.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Celarent, e.g., "All snakes are reptiles". $)
    celarent.min $e |- A. x ( ch -> ph ) $.
    $( "Celarent", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` , and all ` ch ` is ` ph ` , therefore no ` ch ` is ` ps ` .  (In
       Aristotelian notation, EAE-1:  MeP and SaM therefore SeP.) For example,
       given the "No reptiles have fur" and "All snakes are reptiles",
       therefore "No snakes have fur".  Example from
       ~ https://en.wikipedia.org/wiki/Syllogism .  (Contributed by David A.
       Wheeler, 24-Aug-2016.)  (Revised by David A. Wheeler, 2-Sep-2016.) $)
    celarent $p |- A. x ( ch -> -. ps ) $=
      ( wn barbara ) ABGCDEFH $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Darii", e.g.,
       "All rabbits have fur". $)
    darii.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Darii, e.g., "Some pets are rabbits." $)
    darii.min $e |- E. x ( ch /\ ph ) $.
    $( "Darii", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , and some ` ch ` is ` ph ` , therefore some ` ch ` is ` ps ` .
       (In Aristotelian notation, AII-1:  MaP and SiM therefore SiP.) For
       example, given "All rabbits have fur" and "Some pets are rabbits",
       therefore "Some pets have fur".  Example from
       ~ https://en.wikipedia.org/wiki/Syllogism .  (Contributed by David A.
       Wheeler, 24-Aug-2016.) $)
    darii $p |- E. x ( ch /\ ps ) $=
      ( wa wi spi anim2i eximii ) CAGCBGDFABCABHDEIJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Ferio" ("Ferioque"),
       e.g., "No homework is fun". $)
    ferio.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Ferio, e.g., "Some reading is homework." $)
    ferio.min $e |- E. x ( ch /\ ph ) $.
    $( "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic.  No
       ` ph ` is ` ps ` , and some ` ch ` is ` ph ` , therefore some ` ch ` is
       not ` ps ` .  (In Aristotelian notation, EIO-1:  MeP and SiM therefore
       SoP.) For example, given "No homework is fun" and "Some reading is
       homework", therefore "Some reading is not fun".  This is essentially a
       logical axiom in Aristotelian logic.  Example from
       ~ https://en.wikipedia.org/wiki/Syllogism .  (Contributed by David A.
       Wheeler, 24-Aug-2016.)  (Revised by David A. Wheeler, 2-Sep-2016.) $)
    ferio $p |- E. x ( ch /\ -. ps ) $=
      ( wn darii ) ABGCDEFH $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Barbari", e.g.,
       e.g., "All men are mortal". $)
    barbari.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Barbari, e.g., "All Greeks are men." $)
    barbari.min $e |- A. x ( ch -> ph ) $.
    $( Existence premise for Barbari, e.g., "Greeks exist." $)
    barbari.e $e |- E. x ch $.
    $( "Barbari", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , all ` ch ` is ` ph ` , and some ` ch ` exist, therefore some
       ` ch ` is ` ps ` .  (In Aristotelian notation, AAI-1:  MaP and SaM
       therefore SiP.) For example, given "All men are mortal", "All Greeks are
       men", and "Greeks exist", therefore "Some Greeks are mortal".  Note the
       existence hypothesis (to prove the "some" in the conclusion).  Example
       from ~ https://en.wikipedia.org/wiki/Syllogism .  (Contributed by David
       A. Wheeler, 27-Aug-2016.)  (Revised by David A. Wheeler,
       30-Aug-2016.) $)
    barbari $p |- E. x ( ch /\ ps ) $=
      ( wa wi barbara spi ancli eximii ) CCBHDGCBCBIDABCDEFJKLM $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Celaront", e.g.,
       e.g., "No reptiles have fur". $)
    celaront.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Celaront, e.g., "All Snakes are reptiles." $)
    celaront.min $e |- A. x ( ch -> ph ) $.
    $( Existence premise for Celaront, e.g., "Snakes exist." $)
    celaront.e $e |- E. x ch $.
    $( "Celaront", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` , all ` ch ` is ` ph ` , and some ` ch ` exist, therefore some
       ` ch ` is not ` ps ` .  (In Aristotelian notation, EAO-1:  MeP and SaM
       therefore SoP.) For example, given "No reptiles have fur", "All snakes
       are reptiles.", and "Snakes exist.", prove "Some snakes have no fur".
       Note the existence hypothesis.  Example from
       ~ https://en.wikipedia.org/wiki/Syllogism .  (Contributed by David A.
       Wheeler, 27-Aug-2016.)  (Revised by David A. Wheeler, 2-Sep-2016.) $)
    celaront $p |- E. x ( ch /\ -. ps ) $=
      ( wn barbari ) ABHCDEFGI $.
  $}

  $( Figure 2 $)

  ${
    $( Major premise for the Aristotelian syllogism "Cesare" $)
    cesare.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Cesare $)
    cesare.min $e |- A. x ( ch -> ps ) $.
    $( "Cesare", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` , and all ` ch ` is ` ps ` , therefore no ` ch ` is ` ph ` .  (In
       Aristotelian notation, EAE-2:  PeM and SaM therefore SeP.) Related to
       ~ celarent .  (Contributed by David A. Wheeler, 27-Aug-2016.)  (Revised
       by David A. Wheeler, 13-Nov-2016.) $)
    cesare $p |- A. x ( ch -> -. ph ) $=
      ( wn wi spi nsyl3 ax-gen ) CAGHDABCABGHDEICBHDFIJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Camestres" $)
    camestres.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Camestres $)
    camestres.min $e |- A. x ( ch -> -. ps ) $.
    $( "Camestres", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , and no ` ch ` is ` ps ` , therefore no ` ch ` is ` ph ` .  In
       Aristotelian notation, AEE-2:  PaM and SeM therefore SeP. (Contributed
       by David A. Wheeler, 28-Aug-2016.)  (Revised by David A. Wheeler,
       2-Sep-2016.) $)
    camestres $p |- A. x ( ch -> -. ph ) $=
      ( wn wi spi nsyl ax-gen ) CAGHDCBACBGHDFIABHDEIJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Festino" $)
    festino.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Festino $)
    festino.min $e |- E. x ( ch /\ ps ) $.
    $( "Festino", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` , and some ` ch ` is ` ps ` , therefore some ` ch ` is not
       ` ph ` .  (In Aristotelian notation, EIO-2:  PeM and SiM therefore SoP.)
       (Contributed by David A. Wheeler, 25-Nov-2016.) $)
    festino $p |- E. x ( ch /\ -. ph ) $=
      ( wa wn wi spi con2i anim2i eximii ) CBGCAHZGDFBNCABABHIDEJKLM $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Baroco" $)
    baroco.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Baroco $)
    baroco.min $e |- E. x ( ch /\ -. ps ) $.
    $( "Baroco", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , and some ` ch ` is not ` ps ` , therefore some ` ch ` is not
       ` ph ` .  (In Aristotelian notation, AOO-2:  PaM and SoM therefore SoP.)
       For example, "All informative things are useful", "Some websites are not
       useful", therefore "Some websites are not informative".  (Contributed by
       David A. Wheeler, 28-Aug-2016.) $)
    baroco $p |- E. x ( ch /\ -. ph ) $=
      ( wn wa wi spi con3i anim2i eximii ) CBGZHCAGZHDFNOCABABIDEJKLM $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Cesaro" $)
    cesaro.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Cesaro $)
    cesaro.min $e |- A. x ( ch -> ps ) $.
    $( Existence premise for Cesaro $)
    cesaro.e $e |- E. x ch $.
    $( "Cesaro", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` , all ` ch ` is ` ps ` , and ` ch ` exist, therefore some ` ch `
       is not ` ph ` .  In Aristotelian notation, EAO-2:  PeM and SaM therefore
       SoP. (Contributed by David A. Wheeler, 28-Aug-2016.)  (Revised by David
       A. Wheeler, 2-Sep-2016.) $)
    cesaro $p |- E. x ( ch /\ -. ph ) $=
      ( wn wa wi spi nsyl3 ancli eximii ) CCAHZIDGCOABCABHJDEKCBJDFKLMN $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Camestros" $)
    camestros.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for  $)
    camestros.min $e |- A. x ( ch -> -. ps ) $.
    $( Existence premise for Camestros $)
    camestros.e $e |- E. x ch $.
    $( "Camestros", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , no ` ch ` is ` ps ` , and ` ch ` exist, therefore some ` ch `
       is not ` ph ` .  (In Aristotelian notation, AEO-2:  PaM and SeM
       therefore SoP.) For example, "All horses have hooves", "No humans have
       hooves", and humans exist, therefore "Some humans are not horses".
       (Contributed by David A. Wheeler, 28-Aug-2016.)  (Revised by David A.
       Wheeler, 2-Sep-2016.) $)
    camestros $p |- E. x ( ch /\ -. ph ) $=
      ( wn wa wi spi nsyl ancli eximii ) CCAHZIDGCOCBACBHJDFKABJDEKLMN $.
  $}

  $( Figure 3 $)

  ${
    $( Major premise for the Aristotelian syllogism "Datisi" $)
    datisi.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for  $)
    datisi.min $e |- E. x ( ph /\ ch ) $.
    $( "Datisi", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , and some ` ph ` is ` ch ` , therefore some ` ch ` is ` ps ` .
       (In Aristotelian notation, AII-3:  MaP and MiS therefore SiP.)
       (Contributed by David A. Wheeler, 28-Aug-2016.) $)
    datisi $p |- E. x ( ch /\ ps ) $=
      ( wa simpr wi spi adantr jca eximii ) ACGZCBGDFNCBACHABCABIDEJKLM $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Disamis" $)
    disamis.maj $e |- E. x ( ph /\ ps ) $.
    $( Minor premise for  $)
    disamis.min $e |- A. x ( ph -> ch ) $.
    $( "Disamis", one of the syllogisms of Aristotelian logic.  Some ` ph ` is
       ` ps ` , and all ` ph ` is ` ch ` , therefore some ` ch ` is ` ps ` .
       (In Aristotelian notation, IAI-3:  MiP and MaS therefore SiP.)
       (Contributed by David A. Wheeler, 28-Aug-2016.) $)
    disamis $p |- E. x ( ch /\ ps ) $=
      ( wa wi spi anim1i eximii ) ABGCBGDEACBACHDFIJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Ferison" $)
    ferison.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for  $)
    ferison.min $e |- E. x ( ph /\ ch ) $.
    $( "Ferison", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` , and some ` ph ` is ` ch ` , therefore some ` ch ` is not
       ` ps ` .  (In Aristotelian notation, EIO-3:  MeP and MiS therefore SoP.)
       (Contributed by David A. Wheeler, 28-Aug-2016.)  (Revised by David A.
       Wheeler, 2-Sep-2016.) $)
    ferison $p |- E. x ( ch /\ -. ps ) $=
      ( wn datisi ) ABGCDEFH $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Bocardo" $)
    bocardo.maj $e |- E. x ( ph /\ -. ps ) $.
    $( Minor premise for  $)
    bocardo.min $e |- A. x ( ph -> ch ) $.
    $( "Bocardo", one of the syllogisms of Aristotelian logic.  Some ` ph ` is
       not ` ps ` , and all ` ph ` is ` ch ` , therefore some ` ch ` is not
       ` ps ` .  (In Aristotelian notation, OAO-3:  MoP and MaS therefore SoP.)
       For example, "Some cats have no tails", "All cats are mammals",
       therefore "Some mammals have no tails".  A reorder of ~ disamis ; prefer
       using that instead.  (Contributed by David A. Wheeler, 28-Aug-2016.)
       (New usage is discouraged.) $)
    bocardo $p |- E. x ( ch /\ -. ps ) $=
      ( wn disamis ) ABGCDEFH $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Felapton" $)
    felapton.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for  $)
    felapton.min $e |- A. x ( ph -> ch ) $.
    $( Existence premise for Felapton $)
    felapton.e $e |- E. x ph $.
    $( "Felapton", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` , all ` ph ` is ` ch ` , and some ` ph ` exist, therefore some
       ` ch ` is not ` ps ` .  (In Aristotelian notation, EAO-3:  MeP and MaS
       therefore SoP.) For example, "No flowers are animals" and "All flowers
       are plants", therefore "Some plants are not animals".  (Contributed by
       David A. Wheeler, 28-Aug-2016.)  (Revised by David A. Wheeler,
       2-Sep-2016.) $)
    felapton $p |- E. x ( ch /\ -. ps ) $=
      ( wn wa wi spi jca eximii ) ACBHZIDGACNACJDFKANJDEKLM $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Darapti" $)
    darapti.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for  $)
    darapti.min $e |- A. x ( ph -> ch ) $.
    $( Existence premise for Darapti $)
    darapti.e $e |- E. x ph $.
    $( "Darapti", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , all ` ph ` is ` ch ` , and some ` ph ` exist, therefore some
       ` ch ` is ` ps ` .  (In Aristotelian notation, AAI-3:  MaP and MaS
       therefore SiP.) For example, "All squares are rectangles" and "All
       squares are rhombuses", therefore "Some rhombuses are rectangles".
       (Contributed by David A. Wheeler, 28-Aug-2016.) $)
    darapti $p |- E. x ( ch /\ ps ) $=
      ( wa wi spi jca eximii ) ACBHDGACBACIDFJABIDEJKL $.
  $}

  $( Figure 4 $)

  ${
    $( Major premise for the Aristotelian syllogism "Calemes" $)
    calemes.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for  $)
    calemes.min $e |- A. x ( ps -> -. ch ) $.
    $( "Calemes", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , and no ` ps ` is ` ch ` , therefore no ` ch ` is ` ph ` .  In
       Aristotelian notation, AEE-4:  PaM and MeS therefore SeP. (Contributed
       by David A. Wheeler, 28-Aug-2016.)  (Revised by David A. Wheeler,
       2-Sep-2016.) $)
    calemes $p |- A. x ( ch -> -. ph ) $=
      ( wn wi spi con2i nsyl ax-gen ) CAGHDCBABCBCGHDFIJABHDEIKL $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Dimatis" $)
    dimatis.maj $e |- E. x ( ph /\ ps ) $.
    $( Minor premise for  $)
    dimatis.min $e |- A. x ( ps -> ch ) $.
    $( "Dimatis", one of the syllogisms of Aristotelian logic.  Some ` ph ` is
       ` ps ` , and all ` ps ` is ` ch ` , therefore some ` ch ` is ` ph ` .
       (In Aristotelian notation, IAI-4:  PiM and MaS therefore SiP.) For
       example, "Some pets are rabbits.", "All rabbits have fur", therefore
       "Some fur bearing animals are pets".  Like ~ darii with positions
       interchanged.  (Contributed by David A. Wheeler, 28-Aug-2016.) $)
    dimatis $p |- E. x ( ch /\ ph ) $=
      ( wa wi spi adantl simpl jca eximii ) ABGZCAGDENCABCABCHDFIJABKLM $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Fresison" $)
    fresison.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for  $)
    fresison.min $e |- E. x ( ps /\ ch ) $.
    $( "Fresison", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` (PeM), and some ` ps ` is ` ch ` (MiS), therefore some ` ch ` is
       not ` ph ` (SoP).  In Aristotelian notation, EIO-4:  PeM and MiS
       therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.)  (Revised
       by David A. Wheeler, 2-Sep-2016.) $)
    fresison $p |- E. x ( ch /\ -. ph ) $=
      ( wa wn simpr wi spi con2i adantr jca eximii ) BCGZCAHZGDFPCQBCIBQCABABHJ
      DEKLMNO $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Calemos" $)
    calemos.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for  $)
    calemos.min $e |- A. x ( ps -> -. ch ) $.
    $( Existence premise for Calemos $)
    calemos.e $e |- E. x ch $.
    $( "Calemos", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` (PaM), no ` ps ` is ` ch ` (MeS), and ` ch ` exist, therefore
       some ` ch ` is not ` ph ` (SoP).  In Aristotelian notation, AEO-4:  PaM
       and MeS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.)
       (Revised by David A. Wheeler, 2-Sep-2016.) $)
    calemos $p |- E. x ( ch /\ -. ph ) $=
      ( wn wa wi spi con2i nsyl ancli eximii ) CCAHZIDGCPCBABCBCHJDFKLABJDEKMNO
      $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Fesapo" $)
    fesapo.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for  $)
    fesapo.min $e |- A. x ( ps -> ch ) $.
    $( Existence premise for Fesapo $)
    fesapo.e $e |- E. x ps $.
    $( "Fesapo", one of the syllogisms of Aristotelian logic.  No ` ph ` is
       ` ps ` , all ` ps ` is ` ch ` , and ` ps ` exist, therefore some ` ch `
       is not ` ph ` .  In Aristotelian notation, EAO-4:  PeM and MaS therefore
       SoP. (Contributed by David A. Wheeler, 28-Aug-2016.)  (Revised by David
       A. Wheeler, 2-Sep-2016.) $)
    fesapo $p |- E. x ( ch /\ -. ph ) $=
      ( wn wa wi spi con2i jca eximii ) BCAHZIDGBCOBCJDFKABABHJDEKLMN $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Bamalip" $)
    bamalip.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for  $)
    bamalip.min $e |- A. x ( ps -> ch ) $.
    $( Existence premise for Bamalip $)
    bamalip.e $e |- E. x ph $.
    $( "Bamalip", one of the syllogisms of Aristotelian logic.  All ` ph ` is
       ` ps ` , all ` ps ` is ` ch ` , and ` ph ` exist, therefore some ` ch `
       is ` ph ` .  (In Aristotelian notation, AAI-4:  PaM and MaS therefore
       SiP.) Like ~ barbari .  (Contributed by David A. Wheeler,
       28-Aug-2016.) $)
    bamalip $p |- E. x ( ch /\ ph ) $=
      ( wa wi spi syl ancri eximii ) ACAHDGACABCABIDEJBCIDFJKLM $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  First-order logic with one non-logical binary predicate
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#

  This section adds one non-logical binary predicate to the first-order logic
  developed until here.  We call it "the membership predicate" since it will
  be used in the next part as the membership predicate of set theory, but in
  this section, it has no other property than being "a binary predicate".
  "Non-logical" means that it does not belong to the logic.  In our logic (and
  in most treatments), the only logical predicate is the equality predicate
  (see ~ weq ).

$)

  $( Declare the membership predicate symbol. $)
  $c e. $.  $( Stylized lowercase Greek letter epsilon. $)

  ${
    $v A $.
    $v B $.
    wcel.cA $f class A $.
    wcel.cB $f class B $.
    $( Extend wff definition to include the membership connective between
       classes.

       For a general discussion of the theory of classes, see
       ~ mmset.html#class .

       The purpose of introducing ` wff A e. B ` here is to allow us to prove
       the ~ wel of predicate calculus in terms of the ~ wceq of set theory, so
       that we do not overload the ` e. ` connective with two syntax
       definitions.  This is done to prevent ambiguity that would complicate
       some Metamath parsers.  The class variables ` A ` and ` B ` are
       introduced temporarily for the purpose of this definition but otherwise
       not used in predicate calculus.  See ~ df-clab for more information on
       the set theory usage of ~ wcel . $)
    wcel $a wff A e. B $.
  $}

  $( Extend wff definition to include atomic formulas with the membership
     predicate.  This is read either " ` x ` is an element of ` y ` ",
     or " ` x ` is a member of ` y ` ", or " ` x ` belongs to ` y ` ",
     or " ` y ` contains ` x ` ".  Note:  The phrase " ` y ` includes
     ` x ` " means " ` x ` is a subset of ` y ` "; to use it also for
     ` x e. y ` , as some authors occasionally do, is poor form and causes
     confusion, according to George Boolos (1992 lecture at MIT).

     This syntactical construction introduces a binary non-logical predicate
     symbol ` e. ` into our predicate calculus.  We will eventually use it for
     the membership predicate of set theory, but that is irrelevant at this
     point: the predicate calculus axioms for ` e. ` apply to any arbitrary
     binary predicate symbol.  "Non-logical" means that the predicate is
     presumed to have additional properties beyond the realm of predicate
     calculus, although these additional properties are not specified by
     predicate calculus itself but rather by the axioms of a theory (in our
     case set theory) added to predicate calculus.  "Binary" means that the
     predicate has two arguments.

     Instead of introducing ~ wel as an axiomatic statement, as was done in an
     older version of this database, we introduce it by "proving" a special
     case of set theory's more general ~ wcel .  This lets us avoid overloading
     the ` e. ` connective, thus preventing ambiguity that would complicate
     certain Metamath parsers.  However, logically ~ wel is considered to be a
     primitive syntax, even though here it is artificially "derived" from
     ~ wcel .  Note:  To see the proof steps of this syntax proof, type "MM>
     SHOW PROOF wel / ALL" in the Metamath program.  (Contributed by NM,
     24-Jan-2006.) $)
  wel $p wff x e. y $=
    ( cv wcel ) ACBCD $.

  $( Axiom of left equality for the binary predicate ` e. ` .  One of the
     equality and substitution axioms for a non-logical predicate in our
     predicate calculus with equality.  It substitutes equal variables into the
     left-hand side of the ` e. ` binary predicate.  Axiom scheme C12' in
     [Megill] p. 448 (p. 16 of the preprint).  It is a special case of Axiom B8
     (p. 75) of system S2 of [Tarski] p. 77.  "Non-logical" means that the
     predicate is not a primitive of predicate calculus proper but instead is
     an extension to it.  "Binary" means that the predicate has two arguments.
     In a system of predicate calculus with equality, like ours, equality is
     not usually considered to be a non-logical predicate.  In systems of
     predicate calculus without equality, it typically would be.  (Contributed
     by NM, 5-Aug-1993.) $)
  ax-13 $a |- ( x = y -> ( x e. z -> y e. z ) ) $.

  $( Axiom of right equality for the binary predicate ` e. ` .  One of the
     equality and substitution axioms for a non-logical predicate in our
     predicate calculus with equality.  It substitutes equal variables into the
     right-hand side of the ` e. ` binary predicate.  Axiom scheme C13' in
     [Megill] p. 448 (p. 16 of the preprint).  It is a special case of Axiom B8
     (p. 75) of system S2 of [Tarski] p. 77.  (Contributed by NM,
     5-Aug-1993.) $)
  ax-14 $a |- ( x = y -> ( z e. x -> z e. y ) ) $.

  $( An identity law for the non-logical predicate.  (Contributed by NM,
     5-Aug-1993.) $)
  elequ1 $p |- ( x = y -> ( x e. z <-> y e. z ) ) $=
    ( weq wel ax-13 wi equcoms impbid ) ABDACEZBCEZABCFKJGBABACFHI $.

  $( An identity law for the non-logical predicate.  (Contributed by NM,
     5-Aug-1993.) $)
  elequ2 $p |- ( x = y -> ( z e. x <-> z e. y ) ) $=
    ( weq wel ax-14 wi equcoms impbid ) ABDCAEZCBEZABCFKJGBABACFHI $.

  ${
    $d x z $.  $d y z $.
    $( When the class variables of set theory are replaced with setvar
       variables, this theorem of predicate calculus is the result.  This
       theorem provides part of the justification for the consistency of that
       definition, which "overloads" the setvar variables in ~ wel with the
       class variables in ~ wcel .  (Contributed by NM, 28-Jan-2004.) $)
    cleljust $p |- ( x e. y <-> E. z ( z = x /\ z e. y ) ) $=
      ( weq wel wa wex ax-17 elequ1 equsex bicomi ) CADCBEZFCGABEZLMCAMCHCABIJK
      $.
  $}

  ${
    $d w x z $.  $d w y $.
    $( Substitution for the first argument of the non-logical predicate in an
       atomic formula.  See ~ elsb2 for substitution for the second argument.
       (Contributed by NM, 7-Nov-2006.)  (Proof shortened by Andrew Salmon,
       14-Jun-2011.) $)
    elsb1 $p |- ( [ y / x ] x e. z <-> y e. z ) $=
      ( vw wel wsb ax-17 elequ1 sbieh sbbii sbco2h bitr3i wb equsb1 sbimi ax-mp
      weq sbbi mpbi sbh 3bitri ) ACEZABFZDCEZDBFZBCEZDBFZUFUCUDDAFZABFUEUHUBABU
      DUBDAUBDGDACHIJUDDBAUDAGKLUDUFMZDBFZUEUGMDBQZDBFUJDBNUKUIDBDBCHOPUDUFDBRS
      UFDBUFDGTUA $.
  $}

  ${
    $d w x z $.  $d w y $.
    $( Substitution for the second argument of the non-logical predicate in an
       atomic formula.  See ~ elsb1 for substitution for the first argument.
       (Contributed by Rodolfo Medina, 3-Apr-2010.)  (Proof shortened by Andrew
       Salmon, 14-Jun-2011.) $)
    elsb2 $p |- ( [ y / x ] z e. x <-> z e. y ) $=
      ( vw wel wsb ax-17 elequ2 sbieh sbbii sbco2h bitr3i wb equsb1 sbimi ax-mp
      weq sbbi mpbi sbh 3bitri ) CAEZABFZCDEZDBFZCBEZDBFZUFUCUDDAFZABFUEUHUBABU
      DUBDAUBDGDACHIJUDDBAUDAGKLUDUFMZDBFZUEUGMDBQZDBFUJDBNUKUIDBDBCHOPUDUFDBRS
      UFDBUFDGTUA $.
  $}

  ${
    $d w z x $.  $d w y $.
    $( Quantifier introduction when one pair of variables is disjoint.
       (Contributed by NM, 2-Jan-2002.) $)
    dveel1 $p |- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) ) $=
      ( vw wel ax-17 elequ1 dvelimf ) DCEZBCEZABDIAFJDFDBCGH $.
  $}

  ${
    $d w z x $.  $d w y $.
    $( Quantifier introduction when one pair of variables is disjoint.
       (Contributed by NM, 2-Jan-2002.) $)
    dveel2 $p |- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) ) $=
      ( vw wel ax-17 elequ2 dvelimf ) CDEZCBEZABDIAFJDFDBCGH $.
  $}

$( End $[ iset-pred.mm $] $)


$(
###############################################################################
  SET THEORY
###############################################################################

  Set theory uses the formalism of propositional and predicate calculus to
  assert properties of arbitrary mathematical objects called "sets".  A set can
  be an element of another set, and this relationship is indicated by the
  ` e. `
  symbol.  Starting with the simplest mathematical object, called the empty
  set, set theory builds up more and more complex structures whose existence
  follows from the axioms, eventually resulting in extremely complicated sets
  that we identify with the real numbers and other familiar mathematical
  objects.

  Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel
  (IZF) system, mostly following the IZF axioms as laid out in [Crosilla].
  Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not
  as easy to formalize in Metamath because the statement of some of its axioms
  uses the notion of "bounded formula".  Since Metamath has, purposefully, a
  very weak metalogic, that notion must be developed in the logic itself.  This
  is similar to our treatment of substitution ( ~ df-sb ) and our definition of
  the nonfreeness predicate ( ~ df-nf ), whereas substitution and bound and
  free variables are ordinarily defined in the metalogic.  The development of
  CZF has begun in BJ's mathbox, see ~ wbd .

$)


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  IZF Set Theory - start with the Axiom of Extensionality
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Extensionality
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.
    $( Axiom of Extensionality.  It states that two sets are identical if they
       contain the same elements.  Axiom 1 of [Crosilla] p.  "Axioms of CZF and
       IZF" (with unnecessary quantifiers removed).

       Set theory can also be formulated with a _single_ primitive predicate
       ` e. ` on top of traditional predicate calculus _without_ equality.  In
       that case the Axiom of Extensionality becomes
       ` ( A. w ( w e. x <-> w e. y ) -> ( x e. z -> y e. z ) ) ` , and
       equality ` x = y ` is _defined_ as ` A. w ( w e. x <-> w e. y ) ` .  All
       of the usual axioms of equality then become theorems of set theory.
       See, for example, Axiom 1 of [TakeutiZaring] p. 8.

       To use the above "equality-free" version of Extensionality with
       Metamath's logical axioms, we would rewrite ~ ax-8 through ~ ax-16 with
       equality expanded according to the above definition.  Some of those
       axioms could be proved from set theory and would be redundant.  Not all
       of them are redundant, since our axioms of predicate calculus make
       essential use of equality for the proper substitution that is a
       primitive notion in traditional predicate calculus.  A study of such an
       axiomatization would be an interesting project for someone exploring the
       foundations of logic.

       It is important to understand that strictly speaking, all of our set
       theory axioms are really schemes that represent an infinite number of
       actual axioms.  This is inherent in the design of Metamath
       ("metavariable math"), which manipulates only metavariables.  For
       example, the metavariable ` x ` in ~ ax-ext can represent any actual
       variable _v1_, _v2_, _v3_,... .  Distinct variable restrictions ($d)
       prevent us from substituting say _v1_ for both ` x ` and ` z ` .  This
       is in contrast to typical textbook presentations that present actual
       axioms (except for axioms which involve wff metavariables).  In
       practice, though, the theorems and proofs are essentially the same.  The
       $d restrictions make each of the infinite axioms generated by the
       ~ ax-ext scheme exactly logically equivalent to each other and in
       particular to the actual axiom of the textbook version.  (Contributed by
       NM, 5-Aug-1993.) $)
    ax-ext $a |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $.
  $}

  ${
    $d z x w $.  $d z y w $.
    $( A generalization of the Axiom of Extensionality in which ` x ` and ` y `
       need not be distinct.  (Contributed by NM, 15-Sep-1993.)  (Proof
       shortened by Andrew Salmon, 12-Aug-2011.) $)
    axext3 $p |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $=
      ( vw wel wb wal weq wi elequ2 bibi1d albidv equequ1 imbi12d ax-ext chvarv
      ) CDEZCBEZFZCGZDBHZICAEZRFZCGZABHZIDADAHZTUDUAUEUFSUCCUFQUBRDACJKLDABMNDB
      COP $.

    $( A bidirectional version of Extensionality.  Although this theorem
       "looks" like it is just a definition of equality, it requires the Axiom
       of Extensionality for its proof under our axiomatization.  See the
       comments for ~ ax-ext .  (Contributed by NM, 14-Nov-2008.) $)
    axext4 $p |- ( x = y <-> A. z ( z e. x <-> z e. y ) ) $=
      ( weq wel wb wal elequ2 alrimiv axext3 impbii ) ABDZCAECBEFZCGLMCABCHIABC
      JK $.
  $}

  ${
    $d x y z $.  $d ph z $.
    bm1.1.1 $e |- F/ x ph $.
    $( Any set defined by a property is the only set defined by that property.
       Theorem 1.1 of [BellMachover] p. 462.  (Contributed by NM,
       30-Jun-1994.) $)
    bm1.1 $p |- ( E. x A. y ( y e. x <-> ph ) ->
                  E! x A. y ( y e. x <-> ph ) ) $=
      ( vz wel wb wal wex wsb wa weq wi weu nfv nfbi nfal elequ2 bibi1d albidv
      sbie 19.26 biantr alimi ax-ext syl sylbir sylan2b gen2 jctr eu2 sylibr )
      CBFZAGZCHZBIZUPUOUOBEJZKBELZMZEHBHZKUOBNUPUTUSBEUQUOCEFZAGZCHZURUOVCBEVBB
      CVAABVABODPQURUNVBCURUMVAABECRSTUAUOVCKUNVBKZCHZURUNVBCUBVEUMVAGZCHURVDVF
      CUMAVAUCUDBECUEUFUGUHUIUJUOBEUOEOUKUL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Class abstractions (a.k.a. class builders)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constants use in class definition. $)
  $c { $. $( Left brace $)
  $c | $.  $( Vertical bar $)
  $c } $. $( Right brace $)
  $( --- Start of old code before overloading prevention patch. $)
  $(
  @c class @. @( Class variable type @)
  $)
  $( --- End of old code before overloading prevention patch. $)

  $( Declare symbols as variables $)
  $v ./\ $.
  $v .\/ $.
  $v .<_ $.
  $v .< $.
  $v .+ $.
  $v .- $.
  $v .X. $.
  $v ./ $.
  $v .^ $.
  $v .0. $.
  $v .1. $.
  $v .|| $.
  $v .# $.
  $v .~ $.
  $v ._|_ $.
  $v .+^ $.
  $v .+b $.
  $v .(+) $.
  $v .* $.
  $v .x. $.
  $v .xb $.
  $v ., $.
  $v .(x) $.
  $v .o. $.
  $v .0b $.

  $( Declare variable symbols that will be used to represent classes.  Note
     that later on ` R ` , ` S ` , ` F ` and ` G ` denote relations and
     functions, but these letters serve as mnemonics only and in fact behave
     no differently from the variables ` A ` through ` D ` . $)
  $v A $.
  $v B $.
  $v C $.
  $v D $.
  $v P $.
  $v Q $.
  $v R $.
  $v S $.
  $v T $.
  $v U $.

  $( Introduce the class builder or class abstraction notation ("the class of
     sets ` x ` such that ` ph ` is true").  Our class variables ` A ` ,
     ` B ` , etc. range over class builders (sometimes implicitly).  Note that
     a setvar variable can be expressed as a class builder per Theorem
     ~ cvjust , justifying the assignment of setvar variables to class
     variables via the use of ~ cv . $)
  cab $a class { x | ph } $.

  $( --- Start of old code before overloading prevention patch. $)
  $(
  @( A setvar variable is a class expression.  The syntax " ` class x ` " can
     be
     viewed as an abbreviation for " ` class { y | y e. x } ` " (a special case
     of ~ cab ), where ` y ` is distinct from ` x ` .  See the discussion under
     the definition of class in [Jech] p. 4.  Note that ` { y | y e. x } = x `
     by ~ cvjust . @)
  cv @a class x @.
  $)
  $( --- End of old code before overloading prevention patch. $)
  $( $j primitive 'cv' 'wceq' 'wcel' 'cab'; $)

  $( Let ` A ` be a class variable. $)
  cA $f class A $.
  $( Let ` B ` be a class variable. $)
  cB $f class B $.
  $( Let ` C ` be a class variable. $)
  cC $f class C $.

  $( Define a connective symbol for use as a class variable. $)
  c.pa $f class .|| $.
  c.ap $f class .# $.

  $( Let ` D ` be a class variable. $)
  cD $f class D $.

  $( Define a connective symbol for use as a class variable. $)
  c.dv $f class ./ $.

  $( Let ` P ` be a class variable. $)
  cP $f class P $.

  $( Define a connective symbol for use as a class variable. $)
  c.pl $f class .+ $.

  $( Define a connective symbol for use as a class variable. $)
  c.pd $f class .+^ $.

  $( Define a connective symbol for use as a class variable. $)
  c.pb $f class .+b $.

  $( Define a connective symbol for use as a class variable. $)
  c.po $f class .(+) $.

  $( Let ` Q ` be a class variable. $)
  cQ $f class Q $.

  $( Define a connective symbol for use as a class variable. $)
  c.sm $f class .~ $.

  $( Let ` R ` be a class variable. $)
  cR $f class R $.
  $( Let ` S ` be a class variable. $)
  cS $f class S $.

  $( Define a connective symbol for use as a class variable. $)
  c.lt $f class .< $.

  $( Define a connective symbol for use as a class variable. $)
  c.xb $f class .xb $.

  $( Let ` T ` be a class variable. $)
  cT $f class T $.

  $( Define a connective symbol for use as a class variable. $)
  c.x $f class .x. $.

  $( Define a connective symbol for use as a class variable. $)
  c.xp $f class .X. $.

  $( Define a connective symbol for use as a class variable. $)
  c.xo $f class .(x) $.

  $( Let ` U ` be a class variable. $)
  cU $f class U $.

  $( Define a connective symbol for use as a class variable. $)
  c.1 $f class .1. $.

  $v e $.
  $v f $.
  $v g $.
  $v h $.
  $v i $.
  $v j $.
  $v k $.
  $v m $.
  $v n $.
  $v o $.
  $v E $.
  $v F $.
  $v G $.
  $v H $.
  $v I $.
  $v J $.
  $v K $.
  $v L $.
  $v M $.
  $v N $.
  $v V $.
  $v W $.
  $v X $.
  $v Y $.
  $v Z $.
  $v O $.
  $v s $.
  $v r $.
  $v q $.
  $v p $.
  $v a $.
  $v b $.
  $v c $.
  $v d $.
  $v l $.

  $( Let ` e ` be an individual variable. $)
  ve $f setvar e $.
  $( Let ` f ` be an individual variable. $)
  vf $f setvar f $.
  $( Let ` g ` be an individual variable. $)
  vg $f setvar g $.
  $( Let ` h ` be an individual variable. $)
  vh $f setvar h $.
  $( Let ` i ` be an individual variable. $)
  vi $f setvar i $.
  $( Let ` j ` be an individual variable. $)
  vj $f setvar j $.
  $( Let ` k ` be an individual variable. $)
  vk $f setvar k $.
  $( Let ` m ` be an individual variable. $)
  vm $f setvar m $.
  $( Let ` n ` be an individual variable. $)
  vn $f setvar n $.
  $( Let ` o ` be an individual variable. $)
  vo $f setvar o $.
  $( Let ` E ` be a class variable. $)
  cE $f class E $.

  $( Define a connective symbol for use as a class variable. $)
  c.ex $f class .^ $.

  $( Let ` F ` be a class variable. $)
  cF $f class F $.
  $( Let ` G ` be a class variable. $)
  cG $f class G $.
  $( Let ` H ` be a class variable. $)
  cH $f class H $.

  $( Define a connective symbol for use as a class variable. $)
  c.xi $f class ., $.

  $( Let ` I ` be a class variable. $)
  cI $f class I $.

  $( Define a connective symbol for use as a class variable. $)
  c.as $f class .* $.

  $( Let ` J ` be a class variable. $)
  cJ $f class J $.

  $( Define a connective symbol for use as a class variable. $)
  c.or $f class .\/ $.

  $( Let ` K ` be a class variable. $)
  cK $f class K $.
  $( Let ` L ` be a class variable. $)
  cL $f class L $.

  $( Define a connective symbol for use as a class variable. $)
  c.le $f class .<_ $.

  $( Let ` M ` be a class variable. $)
  cM $f class M $.

  $( Define a connective symbol for use as a class variable. $)
  c.an $f class ./\ $.

  $( Define a connective symbol for use as a class variable. $)
  c.mi $f class .- $.

  $( Let ` N ` be a class variable. $)
  cN $f class N $.

  $( Define a connective symbol for use as a class variable. $)
  c.pe $f class ._|_ $.

  $( Let ` O ` be a class variable. $)
  cO $f class O $.
  $( Let ` V ` be a class variable. $)
  cV $f class V $.
  $( Let ` W ` be a class variable. $)
  cW $f class W $.
  $( Let ` X ` be a class variable. $)
  cX $f class X $.
  $( Let ` Y ` be a class variable. $)
  cY $f class Y $.

  $( Define a connective symbol for use as a class variable. $)
  c.0 $f class .0. $.
  $( Define a connective symbol for use as a class variable. $)
  c.0b $f class .0b $.
  $( Define a connective symbol for use as a class variable. $)
  c.op $f class .o. $.

  $( Let ` Z ` be a class variable. $)
  cZ $f class Z $.
  $( Let ` s ` be an individual variable. $)
  vs $f setvar s $.
  $( Let ` r ` be an individual variable. $)
  vr $f setvar r $.
  $( Let ` q ` be an individual variable. $)
  vq $f setvar q $.
  $( Let ` p ` be an individual variable. $)
  vp $f setvar p $.
  $( Let ` a ` be an individual variable. $)
  va $f setvar a $.
  $( Let ` b ` be an individual variable. $)
  vb $f setvar b $.
  $( Let ` c ` be an individual variable. $)
  vc $f setvar c $.
  $( Let ` d ` be an individual variable. $)
  vd $f setvar d $.
  $( Let ` l ` be an individual variable. $)
  vl $f setvar l $.

  $( --- Start of old code before overloading prevention patch. $)
  $(
  @( Extend wff definition to include class equality. @)
  wceq @a wff A = B @.
  $)
  $( --- End of old code before overloading prevention patch. $)

  $( --- Start of old code before overloading prevention patch. $)
  $(
  @( Extend wff definition to include the membership connective between
     classes. @)
  wcel @a wff A e. B @.
  $)
  $( --- End of old code before overloading prevention patch. $)

  $( Define class abstraction notation (so-called by Quine), also called a
     "class builder" in the literature. ` x ` and ` y ` need not be distinct.
     Definition 2.1 of [Quine] p. 16.  Typically, ` ph ` will have ` y ` as a
     free variable, and " ` { y | ph } ` " is read "the class of all sets ` y `
     such that ` ph ( y ) ` is true".  We do not define ` { y | ph } ` in
     isolation but only as part of an expression that extends or "overloads"
     the ` e. ` relationship.

     This is our first use of the ` e. ` symbol to connect classes instead of
     sets.  The syntax definition ~ wcel , which extends or "overloads" the
     ~ wel definition connecting setvar variables, requires that both sides of
     ` e. ` be a class.  In ~ df-cleq and ~ df-clel , we introduce a new kind
     of variable (class variable) that can substituted with expressions such as
     ` { y | ph } ` .  In the present definition, the ` x ` on the left-hand
     side is a setvar variable.  Syntax definition ~ cv allows us to substitute
     a setvar variable ` x ` for a class variable: all sets are classes by
     ~ cvjust (but not necessarily vice-versa).  For a full description of how
     classes are introduced and how to recover the primitive language, see the
     discussion in Quine (and under ~ abeq2 for a quick overview).

     Because class variables can be substituted with compound expressions and
     setvar variables cannot, it is often useful to convert a theorem
     containing a free setvar variable to a more general version with a class
     variable.

     This is called the "axiom of class comprehension" by [Levy] p. 338, who
     treats the theory of classes as an extralogical extension to our logic and
     set theory axioms.  He calls the construction ` { y | ph } ` a "class
     term".

     For a general discussion of the theory of classes, see
     ~ https://us.metamath.org/mpeuni/mmset.html#class .  (Contributed by NM,
     5-Aug-1993.) $)
  df-clab $a |- ( x e. { y | ph } <-> [ x / y ] ph ) $.

  $( Simplification of class abstraction notation when the free and bound
     variables are identical.  (Contributed by NM, 5-Aug-1993.) $)
  abid $p |- ( x e. { x | ph } <-> ph ) $=
    ( cv cab wcel wsb df-clab sbid bitri ) BCABDEABBFAABBGABHI $.

  ${
    $d x y $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by NM, 5-Aug-1993.) $)
    hbab1 $p |- ( y e. { x | ph } -> A. x y e. { x | ph } ) $=
      ( cv cab wcel wsb df-clab hbs1 hbxfrbi ) CDABEFABCGBACBHABCIJ $.

    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by Mario Carneiro, 11-Aug-2016.) $)
    nfsab1 $p |- F/ x y e. { x | ph } $=
      ( cv cab wcel hbab1 nfi ) CDABEFBABCGH $.
  $}

  ${
    $d x z $.
    hbab.1 $e |- ( ph -> A. x ph ) $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by NM, 1-Mar-1995.) $)
    hbab $p |- ( z e. { y | ph } -> A. x z e. { y | ph } ) $=
      ( cv cab wcel wsb df-clab hbsb hbxfrbi ) DFACGHACDIBADCJACDBEKL $.
  $}

  ${
    $d x z $.
    nfsab.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by Mario Carneiro, 11-Aug-2016.) $)
    nfsab $p |- F/ x z e. { y | ph } $=
      ( cv cab wcel nfri hbab nfi ) DFACGHBABCDABEIJK $.
  $}

  ${
    $d x A $.  $d x B $.  $d x y z $.
    df-cleq.1 $e |- ( A. x ( x e. y <-> x e. z ) -> y = z ) $.
    $( Define the equality connective between classes.  Definition 2.7 of
       [Quine] p. 18.  Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4
       provides its justification and methods for eliminating it.  Note that
       its elimination will not necessarily result in a single wff in the
       original language but possibly a "scheme" of wffs.

       This is an example of a somewhat "risky" definition, meaning that it has
       a more complex than usual soundness justification (outside of Metamath),
       because it "overloads" or reuses the existing equality symbol rather
       than introducing a new symbol.  This allows us to make statements that
       may not hold for the original symbol.  For example, it permits us to
       deduce ` y = z <-> A. x ( x e. y <-> x e. z ) ` , which is not a theorem
       of logic but rather presupposes the Axiom of Extensionality (see Theorem
       ~ axext4 ).  We therefore include this axiom as a hypothesis, so that
       the use of Extensionality is properly indicated.

       We could avoid this complication by introducing a new symbol, say =_2,
       in place of ` = ` .  This would also have the advantage of making
       elimination of the definition straightforward, so that we could
       eliminate Extensionality as a hypothesis.  We would then also have the
       advantage of being able to identify in various proofs exactly where
       Extensionality truly comes into play rather than just being an artifact
       of a definition.  One of our theorems would then be ` x ` =_2
       ` y <-> x = y ` by invoking Extensionality.

       However, to conform to literature usage, we retain this overloaded
       definition.  This also makes some proofs shorter and probably easier to
       read, without the constant switching between two kinds of equality.

       See also comments under ~ df-clab , ~ df-clel , and ~ abeq2 .

       In the form of ~ dfcleq , this is called the "axiom of extensionality"
       by [Levy] p. 338, who treats the theory of classes as an extralogical
       extension to our logic and set theory axioms.

       For a general discussion of the theory of classes, see
       ~ https://us.metamath.org/mpeuni/mmset.html#class .  (Contributed by NM,
       15-Sep-1993.) $)
    df-cleq $a |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $.
  $}

  ${
    $d x A $.  $d x B $.  $d x y z $.
    $( The same as ~ df-cleq with the hypothesis removed using the Axiom of
       Extensionality ~ ax-ext .  (Contributed by NM, 15-Sep-1993.) $)
    dfcleq $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
      ( vy vz ax-ext df-cleq ) ADEBCDEAFG $.
  $}

  ${
    $d x y z $.
    $( Every set is a class.  Proposition 4.9 of [TakeutiZaring] p. 13.  This
       theorem shows that a setvar variable can be expressed as a class
       abstraction.  This provides a motivation for the class syntax
       construction ~ cv , which allows us to substitute a setvar variable for
       a class variable.  See also ~ cab and ~ df-clab .  Note that this is not
       a rigorous justification, because ~ cv is used as part of the proof of
       this theorem, but a careful argument can be made outside of the
       formalism of Metamath, for example as is done in Chapter 4 of Takeuti
       and Zaring.  See also the discussion under the definition of class in
       [Jech] p. 4 showing that "Every set can be considered to be a class."
       (Contributed by NM, 7-Nov-2006.) $)
    cvjust $p |- x = { y | y e. x } $=
      ( vz cv wcel cab wceq wb dfcleq wsb df-clab elsb1 bitr2i mpgbir ) ADZBDOE
      ZBFZGCDZOEZRQEZHCCOQITPBCJSPCBKBCALMN $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Define the membership connective between classes.  Theorem 6.3 of
       [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we
       adopt as a definition.  See these references for its metalogical
       justification.  Note that like ~ df-cleq it extends or "overloads" the
       use of the existing membership symbol, but unlike ~ df-cleq it does not
       strengthen the set of valid wffs of logic when the class variables are
       replaced with setvar variables (see ~ cleljust ), so we don't include
       any set theory axiom as a hypothesis.  See also comments about the
       syntax under ~ df-clab .

       This is called the "axiom of membership" by [Levy] p. 338, who treats
       the theory of classes as an extralogical extension to our logic and set
       theory axioms.

       For a general discussion of the theory of classes, see
       ~ https://us.metamath.org/mpeuni/mmset.html#class .  (Contributed by NM,
       5-Aug-1993.) $)
    df-clel $a |- ( A e. B <-> E. x ( x = A /\ x e. B ) ) $.
  $}

  ${
    $d x A $.  $d x B $.
    eqriv.1 $e |- ( x e. A <-> x e. B ) $.
    $( Infer equality of classes from equivalence of membership.  (Contributed
       by NM, 5-Aug-1993.) $)
    eqriv $p |- A = B $=
      ( wceq cv wcel wb dfcleq mpgbir ) BCEAFZBGKCGHAABCIDJ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    eqrdv.1 $e |- ( ph -> ( x e. A <-> x e. B ) ) $.
    $( Deduce equality of classes from equivalence of membership.  (Contributed
       by NM, 17-Mar-1996.) $)
    eqrdv $p |- ( ph -> A = B ) $=
      ( cv wcel wb wal wceq alrimiv dfcleq sylibr ) ABFZCGNDGHZBICDJAOBEKBCDLM
      $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    eqrdav.1 $e |- ( ( ph /\ x e. A ) -> x e. C ) $.
    eqrdav.2 $e |- ( ( ph /\ x e. B ) -> x e. C ) $.
    eqrdav.3 $e |- ( ( ph /\ x e. C ) -> ( x e. A <-> x e. B ) ) $.
    $( Deduce equality of classes from an equivalence of membership that
       depends on the membership variable.  (Contributed by NM, 7-Nov-2008.) $)
    eqrdav $p |- ( ph -> A = B ) $=
      ( cv wcel wa biimpd impancom mpd wi exbiri com23 imp impbida eqrdv ) ABCD
      ABIZCJZUADJZAUBKUAEJZUCFAUDUBUCAUDKUBUCHLMNAUCKUDUBGAUCUDUBOAUDUCUBAUDUBU
      CHPQRNST $.
  $}

  ${
    $d x A $.
    $( Law of identity (reflexivity of class equality).  Theorem 6.4 of [Quine]
       p. 41.

       This law is thought to have originated with Aristotle (_Metaphysics_,
       Zeta, 17, 1041 a, 10-20).  (Thanks to Stefan Allan and BJ for this
       information.)  (Contributed by NM, 5-Aug-1993.)  (Revised by BJ,
       14-Oct-2017.) $)
    eqid $p |- A = A $=
      ( vx cv wcel biid eqriv ) BAABCADEF $.
  $}

  $( Class identity law with antecedent.  (Contributed by NM, 21-Aug-2008.) $)
  eqidd $p |- ( ph -> A = A ) $=
    ( wceq eqid a1i ) BBCABDE $.

  ${
    $d x A $.  $d x B $.
    $( Commutative law for class equality.  Theorem 6.5 of [Quine] p. 41.
       (Contributed by NM, 5-Aug-1993.) $)
    eqcom $p |- ( A = B <-> B = A ) $=
      ( vx cv wcel wb wal wceq bicom albii dfcleq 3bitr4i ) CDZAEZMBEZFZCGONFZC
      GABHBAHPQCNOIJCABKCBAKL $.
  $}

  ${
    eqcoms.1 $e |- ( A = B -> ph ) $.
    $( Inference applying commutative law for class equality to an antecedent.
       (Contributed by NM, 5-Aug-1993.) $)
    eqcoms $p |- ( B = A -> ph ) $=
      ( wceq eqcom sylbi ) CBEBCEACBFDG $.
  $}

  ${
    eqcomi.1 $e |- A = B $.
    $( Inference from commutative law for class equality.  (Contributed by NM,
       5-Aug-1993.) $)
    eqcomi $p |- B = A $=
      ( wceq eqcom mpbi ) ABDBADCABEF $.
  $}

  ${
    neqcomd.1 $e |- ( ph -> -. A = B ) $.
    $( Commute an inequality.  (Contributed by Rohan Ridenour, 3-Aug-2023.) $)
    neqcomd $p |- ( ph -> -. B = A ) $=
      ( wceq eqcom sylnib ) ABCECBEDBCFG $.
  $}

  ${
    eqcomd.1 $e |- ( ph -> A = B ) $.
    $( Deduction from commutative law for class equality.  (Contributed by NM,
       15-Aug-1994.) $)
    eqcomd $p |- ( ph -> B = A ) $=
      ( wceq eqcom sylib ) ABCECBEDBCFG $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Equality implies equivalence of equalities.  (Contributed by NM,
       5-Aug-1993.) $)
    eqeq1 $p |- ( A = B -> ( A = C <-> B = C ) ) $=
      ( vx wceq cv wcel wb wal dfcleq biimpi 19.21bi bibi1d albidv 3bitr4g ) AB
      EZDFZAGZQCGZHZDIQBGZSHZDIACEBCEPTUBDPRUASPRUAHZDPUCDIDABJKLMNDACJDBCJO $.
  $}

  ${
    eqeq1i.1 $e |- A = B $.
    $( Inference from equality to equivalence of equalities.  (Contributed by
       NM, 5-Aug-1993.) $)
    eqeq1i $p |- ( A = C <-> B = C ) $=
      ( wceq wb eqeq1 ax-mp ) ABEACEBCEFDABCGH $.
  $}

  ${
    eqeq1d.1 $e |- ( ph -> A = B ) $.
    $( Deduction from equality to equivalence of equalities.  (Contributed by
       NM, 27-Dec-1993.) $)
    eqeq1d $p |- ( ph -> ( A = C <-> B = C ) ) $=
      ( wceq wb eqeq1 syl ) ABCFBDFCDFGEBCDHI $.
  $}

  $( Equality implies equivalence of equalities.  (Contributed by NM,
     5-Aug-1993.) $)
  eqeq2 $p |- ( A = B -> ( C = A <-> C = B ) ) $=
    ( wceq eqeq1 eqcom 3bitr4g ) ABDACDBCDCADCBDABCECAFCBFG $.

  ${
    eqeq2i.1 $e |- A = B $.
    $( Inference from equality to equivalence of equalities.  (Contributed by
       NM, 5-Aug-1993.) $)
    eqeq2i $p |- ( C = A <-> C = B ) $=
      ( wceq wb eqeq2 ax-mp ) ABECAECBEFDABCGH $.
  $}

  ${
    eqeq2d.1 $e |- ( ph -> A = B ) $.
    $( Deduction from equality to equivalence of equalities.  (Contributed by
       NM, 27-Dec-1993.) $)
    eqeq2d $p |- ( ph -> ( C = A <-> C = B ) ) $=
      ( wceq wb eqeq2 syl ) ABCFDBFDCFGEBCDHI $.
  $}

  $( Equality relationship among 4 classes.  (Contributed by NM,
     3-Aug-1994.) $)
  eqeq12 $p |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) ) $=
    ( wceq eqeq1 eqeq2 sylan9bb ) ABEACEBCECDEBDEABCFCDBGH $.

  ${
    eqeq12i.1 $e |- A = B $.
    eqeq12i.2 $e |- C = D $.
    $( A useful inference for substituting definitions into an equality.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       25-May-2011.) $)
    eqeq12i $p |- ( A = C <-> B = D ) $=
      ( wceq wb eqeq12 mp2an ) ABGCDGACGBDGHEFABCDIJ $.

    $( Theorem eqeq12i is the congruence law for equality. $)
    $( $j congruence 'eqeq12i'; $)
  $}

  ${
    eqeq12d.1 $e |- ( ph -> A = B ) $.
    eqeq12d.2 $e |- ( ph -> C = D ) $.
    $( A useful inference for substituting definitions into an equality.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       25-May-2011.) $)
    eqeq12d $p |- ( ph -> ( A = C <-> B = D ) ) $=
      ( wceq wb eqeq12 syl2anc ) ABCHDEHBDHCEHIFGBCDEJK $.
  $}

  ${
    eqeqan12d.1 $e |- ( ph -> A = B ) $.
    eqeqan12d.2 $e |- ( ps -> C = D ) $.
    $( A useful inference for substituting definitions into an equality.
       (Contributed by NM, 9-Aug-1994.)  (Proof shortened by Andrew Salmon,
       25-May-2011.) $)
    eqeqan12d $p |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) ) $=
      ( wceq wb eqeq12 syl2an ) ACDIEFICEIDFIJBGHCDEFKL $.
  $}

  ${
    eqeqan12rd.1 $e |- ( ph -> A = B ) $.
    eqeqan12rd.2 $e |- ( ps -> C = D ) $.
    $( A useful inference for substituting definitions into an equality.
       (Contributed by NM, 9-Aug-1994.) $)
    eqeqan12rd $p |- ( ( ps /\ ph ) -> ( A = C <-> B = D ) ) $=
      ( wceq wb eqeqan12d ancoms ) ABCEIDFIJABCDEFGHKL $.
  $}

  $( Transitive law for class equality.  Proposition 4.7(3) of [TakeutiZaring]
     p. 13.  (Contributed by NM, 25-Jan-2004.) $)
  eqtr $p |- ( ( A = B /\ B = C ) -> A = C ) $=
    ( wceq eqeq1 biimpar ) ABDACDBCDABCEF $.

  $( A transitive law for class equality.  (Contributed by NM, 20-May-2005.)
     (Proof shortened by Andrew Salmon, 25-May-2011.) $)
  eqtr2 $p |- ( ( A = B /\ A = C ) -> B = C ) $=
    ( wceq eqcom eqtr sylanb ) ABDBADACDBCDABEBACFG $.

  $( A transitive law for class equality.  (Contributed by NM, 20-May-2005.) $)
  eqtr3 $p |- ( ( A = C /\ B = C ) -> A = B ) $=
    ( wceq eqcom eqtr sylan2b ) BCDACDCBDABDBCEACBFG $.

  ${
    eqtri.1 $e |- A = B $.
    eqtri.2 $e |- B = C $.
    $( An equality transitivity inference.  (Contributed by NM, 5-Aug-1993.) $)
    eqtri $p |- A = C $=
      ( wceq eqeq2i mpbi ) ABFACFDBCAEGH $.
  $}

  ${
    eqtr2i.1 $e |- A = B $.
    eqtr2i.2 $e |- B = C $.
    $( An equality transitivity inference.  (Contributed by NM,
       21-Feb-1995.) $)
    eqtr2i $p |- C = A $=
      ( eqtri eqcomi ) ACABCDEFG $.
  $}

  ${
    eqtr3i.1 $e |- A = B $.
    eqtr3i.2 $e |- A = C $.
    $( An equality transitivity inference.  (Contributed by NM, 6-May-1994.) $)
    eqtr3i $p |- B = C $=
      ( eqcomi eqtri ) BACABDFEG $.
  $}

  ${
    eqtr4i.1 $e |- A = B $.
    eqtr4i.2 $e |- C = B $.
    $( An equality transitivity inference.  (Contributed by NM, 5-Aug-1993.) $)
    eqtr4i $p |- A = C $=
      ( eqcomi eqtri ) ABCDCBEFG $.
  $}

  $( Register '=' as an equality for its type (class). $)
  $( $j equality 'wceq' from 'eqid' 'eqcomi' 'eqtri'; $)

  ${
    3eqtri.1 $e |- A = B $.
    3eqtri.2 $e |- B = C $.
    3eqtri.3 $e |- C = D $.
    $( An inference from three chained equalities.  (Contributed by NM,
       29-Aug-1993.) $)
    3eqtri $p |- A = D $=
      ( eqtri ) ABDEBCDFGHH $.

    $( An inference from three chained equalities.  (Contributed by NM,
       3-Aug-2006.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3eqtrri $p |- D = A $=
      ( eqtri eqtr2i ) ACDABCEFHGI $.
  $}

  ${
    3eqtr2i.1 $e |- A = B $.
    3eqtr2i.2 $e |- C = B $.
    3eqtr2i.3 $e |- C = D $.
    $( An inference from three chained equalities.  (Contributed by NM,
       3-Aug-2006.) $)
    3eqtr2i $p |- A = D $=
      ( eqtr4i eqtri ) ACDABCEFHGI $.

    $( An inference from three chained equalities.  (Contributed by NM,
       3-Aug-2006.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3eqtr2ri $p |- D = A $=
      ( eqtr4i eqtr2i ) ACDABCEFHGI $.
  $}

  ${
    3eqtr3i.1 $e |- A = B $.
    3eqtr3i.2 $e |- A = C $.
    3eqtr3i.3 $e |- B = D $.
    $( An inference from three chained equalities.  (Contributed by NM,
       6-May-1994.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3eqtr3i $p |- C = D $=
      ( eqtr3i ) BCDABCEFHGH $.

    $( An inference from three chained equalities.  (Contributed by NM,
       15-Aug-2004.) $)
    3eqtr3ri $p |- D = C $=
      ( eqtr3i ) BDCGABCEFHH $.
  $}

  ${
    3eqtr4i.1 $e |- A = B $.
    3eqtr4i.2 $e |- C = A $.
    3eqtr4i.3 $e |- D = B $.
    $( An inference from three chained equalities.  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3eqtr4i $p |- C = D $=
      ( eqtr4i ) CADFDBAGEHH $.

    $( An inference from three chained equalities.  (Contributed by NM,
       2-Sep-1995.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3eqtr4ri $p |- D = C $=
      ( eqtr4i ) DACDBAGEHFH $.
  $}

  ${
    eqtrd.1 $e |- ( ph -> A = B ) $.
    eqtrd.2 $e |- ( ph -> B = C ) $.
    $( An equality transitivity deduction.  (Contributed by NM, 5-Aug-1993.) $)
    eqtrd $p |- ( ph -> A = C ) $=
      ( wceq eqeq2d mpbid ) ABCGBDGEACDBFHI $.
  $}

  ${
    eqtr2d.1 $e |- ( ph -> A = B ) $.
    eqtr2d.2 $e |- ( ph -> B = C ) $.
    $( An equality transitivity deduction.  (Contributed by NM,
       18-Oct-1999.) $)
    eqtr2d $p |- ( ph -> C = A ) $=
      ( eqtrd eqcomd ) ABDABCDEFGH $.
  $}

  ${
    eqtr3d.1 $e |- ( ph -> A = B ) $.
    eqtr3d.2 $e |- ( ph -> A = C ) $.
    $( An equality transitivity equality deduction.  (Contributed by NM,
       18-Jul-1995.) $)
    eqtr3d $p |- ( ph -> B = C ) $=
      ( eqcomd eqtrd ) ACBDABCEGFH $.
  $}

  ${
    eqtr4d.1 $e |- ( ph -> A = B ) $.
    eqtr4d.2 $e |- ( ph -> C = B ) $.
    $( An equality transitivity equality deduction.  (Contributed by NM,
       18-Jul-1995.) $)
    eqtr4d $p |- ( ph -> A = C ) $=
      ( eqcomd eqtrd ) ABCDEADCFGH $.
  $}

  ${
    3eqtrd.1 $e |- ( ph -> A = B ) $.
    3eqtrd.2 $e |- ( ph -> B = C ) $.
    3eqtrd.3 $e |- ( ph -> C = D ) $.
    $( A deduction from three chained equalities.  (Contributed by NM,
       29-Oct-1995.) $)
    3eqtrd $p |- ( ph -> A = D ) $=
      ( eqtrd ) ABCEFACDEGHII $.

    $( A deduction from three chained equalities.  (Contributed by NM,
       4-Aug-2006.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3eqtrrd $p |- ( ph -> D = A ) $=
      ( eqtrd eqtr2d ) ABDEABCDFGIHJ $.
  $}

  ${
    3eqtr2d.1 $e |- ( ph -> A = B ) $.
    3eqtr2d.2 $e |- ( ph -> C = B ) $.
    3eqtr2d.3 $e |- ( ph -> C = D ) $.
    $( A deduction from three chained equalities.  (Contributed by NM,
       4-Aug-2006.) $)
    3eqtr2d $p |- ( ph -> A = D ) $=
      ( eqtr4d eqtrd ) ABDEABCDFGIHJ $.

    $( A deduction from three chained equalities.  (Contributed by NM,
       4-Aug-2006.) $)
    3eqtr2rd $p |- ( ph -> D = A ) $=
      ( eqtr4d eqtr2d ) ABDEABCDFGIHJ $.
  $}

  ${
    3eqtr3d.1 $e |- ( ph -> A = B ) $.
    3eqtr3d.2 $e |- ( ph -> A = C ) $.
    3eqtr3d.3 $e |- ( ph -> B = D ) $.
    $( A deduction from three chained equalities.  (Contributed by NM,
       4-Aug-1995.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3eqtr3d $p |- ( ph -> C = D ) $=
      ( eqtr3d ) ACDEABCDFGIHI $.

    $( A deduction from three chained equalities.  (Contributed by NM,
       14-Jan-2006.) $)
    3eqtr3rd $p |- ( ph -> D = C ) $=
      ( eqtr3d ) ACEDHABCDFGII $.
  $}

  ${
    3eqtr4d.1 $e |- ( ph -> A = B ) $.
    3eqtr4d.2 $e |- ( ph -> C = A ) $.
    3eqtr4d.3 $e |- ( ph -> D = B ) $.
    $( A deduction from three chained equalities.  (Contributed by NM,
       4-Aug-1995.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    3eqtr4d $p |- ( ph -> C = D ) $=
      ( eqtr4d ) ADBEGAECBHFII $.

    $( A deduction from three chained equalities.  (Contributed by NM,
       21-Sep-1995.) $)
    3eqtr4rd $p |- ( ph -> D = C ) $=
      ( eqtr4d ) AEBDAECBHFIGI $.
  $}

  ${
    eqtrid.1 $e |- A = B $.
    eqtrid.2 $e |- ( ph -> B = C ) $.
    $( An equality transitivity deduction.  (Contributed by NM,
       21-Jun-1993.) $)
    eqtrid $p |- ( ph -> A = C ) $=
      ( wceq a1i eqtrd ) ABCDBCGAEHFI $.
  $}

  ${
    eqtr2id.1 $e |- A = B $.
    eqtr2id.2 $e |- ( ph -> B = C ) $.
    $( An equality transitivity deduction.  (Contributed by NM,
       29-Mar-1998.) $)
    eqtr2id $p |- ( ph -> C = A ) $=
      ( eqtrid eqcomd ) ABDABCDEFGH $.
  $}

  ${
    eqtr3id.1 $e |- B = A $.
    eqtr3id.2 $e |- ( ph -> B = C ) $.
    $( An equality transitivity deduction.  (Contributed by NM, 5-Aug-1993.) $)
    eqtr3id $p |- ( ph -> A = C ) $=
      ( eqcomi eqtrid ) ABCDCBEGFH $.
  $}

  ${
    eqtr3di.1 $e |- ( ph -> A = B ) $.
    eqtr3di.2 $e |- A = C $.
    $( An equality transitivity deduction.  (Contributed by NM,
       29-Mar-1998.) $)
    eqtr3di $p |- ( ph -> B = C ) $=
      ( eqcomi eqtr2id ) ADBCBDFGEH $.
  $}

  ${
    eqtrdi.1 $e |- ( ph -> A = B ) $.
    eqtrdi.2 $e |- B = C $.
    $( An equality transitivity deduction.  (Contributed by NM, 5-Aug-1993.) $)
    eqtrdi $p |- ( ph -> A = C ) $=
      ( wceq a1i eqtrd ) ABCDECDGAFHI $.
  $}

  ${
    eqtr2di.1 $e |- ( ph -> A = B ) $.
    eqtr2di.2 $e |- B = C $.
    $( An equality transitivity deduction.  (Contributed by NM,
       29-Mar-1998.) $)
    eqtr2di $p |- ( ph -> C = A ) $=
      ( eqtrdi eqcomd ) ABDABCDEFGH $.
  $}

  ${
    eqtr4di.1 $e |- ( ph -> A = B ) $.
    eqtr4di.2 $e |- C = B $.
    $( An equality transitivity deduction.  (Contributed by NM, 5-Aug-1993.) $)
    eqtr4di $p |- ( ph -> A = C ) $=
      ( eqcomi eqtrdi ) ABCDEDCFGH $.
  $}

  ${
    eqtr4id.2 $e |- A = B $.
    eqtr4id.1 $e |- ( ph -> C = B ) $.
    $( An equality transitivity deduction.  (Contributed by NM,
       29-Mar-1998.) $)
    eqtr4id $p |- ( ph -> A = C ) $=
      ( eqcomi eqtr2di ) ADCBFBCEGH $.
  $}

  ${
    sylan9eq.1 $e |- ( ph -> A = B ) $.
    sylan9eq.2 $e |- ( ps -> B = C ) $.
    $( An equality transitivity deduction.  (Contributed by NM, 8-May-1994.)
       (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    sylan9eq $p |- ( ( ph /\ ps ) -> A = C ) $=
      ( wceq eqtr syl2an ) ACDHDEHCEHBFGCDEIJ $.
  $}

  ${
    sylan9req.1 $e |- ( ph -> B = A ) $.
    sylan9req.2 $e |- ( ps -> B = C ) $.
    $( An equality transitivity deduction.  (Contributed by NM,
       23-Jun-2007.) $)
    sylan9req $p |- ( ( ph /\ ps ) -> A = C ) $=
      ( eqcomd sylan9eq ) ABCDEADCFHGI $.
  $}

  ${
    sylan9eqr.1 $e |- ( ph -> A = B ) $.
    sylan9eqr.2 $e |- ( ps -> B = C ) $.
    $( An equality transitivity deduction.  (Contributed by NM, 8-May-1994.) $)
    sylan9eqr $p |- ( ( ps /\ ph ) -> A = C ) $=
      ( wceq sylan9eq ancoms ) ABCEHABCDEFGIJ $.
  $}

  ${
    3eqtr3g.1 $e |- ( ph -> A = B ) $.
    3eqtr3g.2 $e |- A = C $.
    3eqtr3g.3 $e |- B = D $.
    $( A chained equality inference, useful for converting from definitions.
       (Contributed by NM, 15-Nov-1994.) $)
    3eqtr3g $p |- ( ph -> C = D ) $=
      ( eqtr3id eqtrdi ) ADCEADBCGFIHJ $.
  $}

  ${
    3eqtr3a.1 $e |- A = B $.
    3eqtr3a.2 $e |- ( ph -> A = C ) $.
    3eqtr3a.3 $e |- ( ph -> B = D ) $.
    $( A chained equality inference, useful for converting from definitions.
       (Contributed by Mario Carneiro, 6-Nov-2015.) $)
    3eqtr3a $p |- ( ph -> C = D ) $=
      ( eqtrid eqtr3d ) ABDEGABCEFHIJ $.
  $}

  ${
    3eqtr4g.1 $e |- ( ph -> A = B ) $.
    3eqtr4g.2 $e |- C = A $.
    3eqtr4g.3 $e |- D = B $.
    $( A chained equality inference, useful for converting to definitions.
       (Contributed by NM, 5-Aug-1993.) $)
    3eqtr4g $p |- ( ph -> C = D ) $=
      ( eqtrid eqtr4di ) ADCEADBCGFIHJ $.
  $}

  ${
    3eqtr4a.1 $e |- A = B $.
    3eqtr4a.2 $e |- ( ph -> C = A ) $.
    3eqtr4a.3 $e |- ( ph -> D = B ) $.
    $( A chained equality inference, useful for converting to definitions.
       (Contributed by NM, 2-Feb-2007.)  (Proof shortened by Andrew Salmon,
       25-May-2011.) $)
    3eqtr4a $p |- ( ph -> C = D ) $=
      ( eqtrdi eqtr4d ) ADCEADBCGFIHJ $.
  $}

  ${
    eq2tr.1 $e |- ( A = C -> D = F ) $.
    eq2tr.2 $e |- ( B = D -> C = G ) $.
    $( A compound transitive inference for class equality.  (Contributed by NM,
       22-Jan-2004.) $)
    eq2tri $p |- ( ( A = C /\ B = F ) <-> ( B = D /\ A = G ) ) $=
      ( wceq wa ancom eqeq2d pm5.32i 3bitr3i ) ACIZBDIZJPOJOBEIZJPAFIZJOPKOPQOD
      EBGLMPORPCFAHLMN $.
  $}

  ${
    $d z x $.  $d z y $.  $d z A $.
    $( Weaker version of ~ eleq1 (but more general than ~ elequ1 ) not
       depending on ~ ax-ext nor ~ df-cleq .  (Contributed by BJ,
       24-Jun-2019.) $)
    eleq1w $p |- ( x = y -> ( x e. A <-> y e. A ) ) $=
      ( vz weq cv wcel wa wex equequ2 anbi1d exbidv df-clel 3bitr4g ) ABEZDAEZD
      FCGZHZDIDBEZQHZDIAFZCGBFZCGORTDOPSQABDJKLDUACMDUBCMN $.
    $( $j usage 'eleq1w' avoids 'ax-io' 'ax-10' 'ax-i12' 'ax-bndl'
      'ax-ext' 'df-cleq'; $)

    $( Weaker version of ~ eleq2 (but more general than ~ elequ2 ) not
       depending on ~ ax-ext nor ~ df-cleq .  (Contributed by BJ,
       29-Sep-2019.) $)
    eleq2w $p |- ( x = y -> ( A e. x <-> A e. y ) ) $=
      ( vz cv wceq wcel wa wex elequ2 anbi2d exbidv df-clel 3bitr4g ) AEZBEZFZD
      EZCFZROGZHZDISRPGZHZDICOGCPGQUAUCDQTUBSABDJKLDCOMDCPMN $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Equality implies equivalence of membership.  (Contributed by NM,
       5-Aug-1993.) $)
    eleq1 $p |- ( A = B -> ( A e. C <-> B e. C ) ) $=
      ( vx wceq cv wcel wa wex eqeq2 anbi1d exbidv df-clel 3bitr4g ) ABEZDFZAEZ
      PCGZHZDIPBEZRHZDIACGBCGOSUADOQTRABPJKLDACMDBCMN $.

    $( Equality implies equivalence of membership.  (Contributed by NM,
       5-Aug-1993.) $)
    eleq2 $p |- ( A = B -> ( C e. A <-> C e. B ) ) $=
      ( vx wceq cv wcel wa wex wal dfcleq biimpi 19.21bi anbi2d df-clel 3bitr4g
      wb exbidv ) ABEZDFZCEZTAGZHZDIUATBGZHZDICAGCBGSUCUEDSUBUDUASUBUDQZDSUFDJD
      ABKLMNRDCAODCBOP $.
  $}

  $( Equality implies equivalence of membership.  (Contributed by NM,
     31-May-1999.) $)
  eleq12 $p |- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) ) $=
    ( wceq wcel eleq1 eleq2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.

  ${
    eleq1i.1 $e |- A = B $.
    $( Inference from equality to equivalence of membership.  (Contributed by
       NM, 5-Aug-1993.) $)
    eleq1i $p |- ( A e. C <-> B e. C ) $=
      ( wceq wcel wb eleq1 ax-mp ) ABEACFBCFGDABCHI $.

    $( Inference from equality to equivalence of membership.  (Contributed by
       NM, 5-Aug-1993.) $)
    eleq2i $p |- ( C e. A <-> C e. B ) $=
      ( wceq wcel wb eleq2 ax-mp ) ABECAFCBFGDABCHI $.

    ${
      eleq12i.2 $e |- C = D $.
      $( Inference from equality to equivalence of membership.  (Contributed by
         NM, 31-May-1994.) $)
      eleq12i $p |- ( A e. C <-> B e. D ) $=
        ( wcel eleq2i eleq1i bitri ) ACGADGBDGCDAFHABDEIJ $.

      $( Theorem eleq12i is the congruence law for elementhood. $)
      $( $j congruence 'eleq12i'; $)
    $}
  $}

  ${
    eleq1d.1 $e |- ( ph -> A = B ) $.
    $( Deduction from equality to equivalence of membership.  (Contributed by
       NM, 5-Aug-1993.) $)
    eleq1d $p |- ( ph -> ( A e. C <-> B e. C ) ) $=
      ( wceq wcel wb eleq1 syl ) ABCFBDGCDGHEBCDIJ $.

    $( Deduction from equality to equivalence of membership.  (Contributed by
       NM, 27-Dec-1993.) $)
    eleq2d $p |- ( ph -> ( C e. A <-> C e. B ) ) $=
      ( wceq wcel wb eleq2 syl ) ABCFDBGDCGHEBCDIJ $.

    ${
      eleq12d.2 $e |- ( ph -> C = D ) $.
      $( Deduction from equality to equivalence of membership.  (Contributed by
         NM, 31-May-1994.) $)
      eleq12d $p |- ( ph -> ( A e. C <-> B e. D ) ) $=
        ( wcel eleq2d eleq1d bitrd ) ABDHBEHCEHADEBGIABCEFJK $.
    $}
  $}

  $( A transitive-type law relating membership and equality.  (Contributed by
     NM, 9-Apr-1994.) $)
  eleq1a $p |- ( A e. B -> ( C = A -> C e. B ) ) $=
    ( wceq wcel eleq1 biimprcd ) CADCBEABECABFG $.

  ${
    eqeltr.1 $e |- A = B $.
    eqeltr.2 $e |- B e. C $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       NM, 5-Aug-1993.) $)
    eqeltri $p |- A e. C $=
      ( wcel eleq1i mpbir ) ACFBCFEABCDGH $.
  $}

  ${
    eqeltrr.1 $e |- A = B $.
    eqeltrr.2 $e |- A e. C $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       NM, 5-Aug-1993.) $)
    eqeltrri $p |- B e. C $=
      ( eqcomi eqeltri ) BACABDFEG $.
  $}

  ${
    eleqtr.1 $e |- A e. B $.
    eleqtr.2 $e |- B = C $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       NM, 5-Aug-1993.) $)
    eleqtri $p |- A e. C $=
      ( wcel eleq2i mpbi ) ABFACFDBCAEGH $.
  $}

  ${
    eleqtrr.1 $e |- A e. B $.
    eleqtrr.2 $e |- C = B $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       NM, 5-Aug-1993.) $)
    eleqtrri $p |- A e. C $=
      ( eqcomi eleqtri ) ABCDCBEFG $.
  $}

  ${
    eqeltrd.1 $e |- ( ph -> A = B ) $.
    eqeltrd.2 $e |- ( ph -> B e. C ) $.
    $( Substitution of equal classes into membership relation, deduction form.
       (Contributed by Raph Levien, 10-Dec-2002.) $)
    eqeltrd $p |- ( ph -> A e. C ) $=
      ( wcel eleq1d mpbird ) ABDGCDGFABCDEHI $.
  $}

  ${
    eqeltrrd.1 $e |- ( ph -> A = B ) $.
    eqeltrrd.2 $e |- ( ph -> A e. C ) $.
    $( Deduction that substitutes equal classes into membership.  (Contributed
       by NM, 14-Dec-2004.) $)
    eqeltrrd $p |- ( ph -> B e. C ) $=
      ( eqcomd eqeltrd ) ACBDABCEGFH $.
  $}

  ${
    eleqtrd.1 $e |- ( ph -> A e. B ) $.
    eleqtrd.2 $e |- ( ph -> B = C ) $.
    $( Deduction that substitutes equal classes into membership.  (Contributed
       by NM, 14-Dec-2004.) $)
    eleqtrd $p |- ( ph -> A e. C ) $=
      ( wcel eleq2d mpbid ) ABCGBDGEACDBFHI $.
  $}

  ${
    eleqtrrd.1 $e |- ( ph -> A e. B ) $.
    eleqtrrd.2 $e |- ( ph -> C = B ) $.
    $( Deduction that substitutes equal classes into membership.  (Contributed
       by NM, 14-Dec-2004.) $)
    eleqtrrd $p |- ( ph -> A e. C ) $=
      ( eqcomd eleqtrd ) ABCDEADCFGH $.
  $}

  ${
    3eltr3.1 $e |- A e. B $.
    3eltr3.2 $e |- A = C $.
    3eltr3.3 $e |- B = D $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       Mario Carneiro, 6-Jan-2017.) $)
    3eltr3i $p |- C e. D $=
      ( eleqtri eqeltrri ) ACDFABDEGHI $.
  $}

  ${
    3eltr4.1 $e |- A e. B $.
    3eltr4.2 $e |- C = A $.
    3eltr4.3 $e |- D = B $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       Mario Carneiro, 6-Jan-2017.) $)
    3eltr4i $p |- C e. D $=
      ( eleqtrri eqeltri ) CADFABDEGHI $.
  $}

  ${
    3eltr3d.1 $e |- ( ph -> A e. B ) $.
    3eltr3d.2 $e |- ( ph -> A = C ) $.
    3eltr3d.3 $e |- ( ph -> B = D ) $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       Mario Carneiro, 6-Jan-2017.) $)
    3eltr3d $p |- ( ph -> C e. D ) $=
      ( eleqtrd eqeltrrd ) ABDEGABCEFHIJ $.
  $}

  ${
    3eltr4d.1 $e |- ( ph -> A e. B ) $.
    3eltr4d.2 $e |- ( ph -> C = A ) $.
    3eltr4d.3 $e |- ( ph -> D = B ) $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       Mario Carneiro, 6-Jan-2017.) $)
    3eltr4d $p |- ( ph -> C e. D ) $=
      ( eleqtrrd eqeltrd ) ADBEGABCEFHIJ $.
  $}

  ${
    3eltr3g.1 $e |- ( ph -> A e. B ) $.
    3eltr3g.2 $e |- A = C $.
    3eltr3g.3 $e |- B = D $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       Mario Carneiro, 6-Jan-2017.) $)
    3eltr3g $p |- ( ph -> C e. D ) $=
      ( wcel eleq12i sylib ) ABCIDEIFBDCEGHJK $.
  $}

  ${
    3eltr4g.1 $e |- ( ph -> A e. B ) $.
    3eltr4g.2 $e |- C = A $.
    3eltr4g.3 $e |- D = B $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       Mario Carneiro, 6-Jan-2017.) $)
    3eltr4g $p |- ( ph -> C e. D ) $=
      ( wcel eleq12i sylibr ) ABCIDEIFDBECGHJK $.
  $}

  ${
    eqeltrid.1 $e |- A = B $.
    eqeltrid.2 $e |- ( ph -> B e. C ) $.
    $( B membership and equality inference.  (Contributed by NM,
       4-Jan-2006.) $)
    eqeltrid $p |- ( ph -> A e. C ) $=
      ( wceq a1i eqeltrd ) ABCDBCGAEHFI $.
  $}

  ${
    eqeltrrid.1 $e |- B = A $.
    eqeltrrid.2 $e |- ( ph -> B e. C ) $.
    $( B membership and equality inference.  (Contributed by NM,
       4-Jan-2006.) $)
    eqeltrrid $p |- ( ph -> A e. C ) $=
      ( eqcomi eqeltrid ) ABCDCBEGFH $.
  $}

  ${
    eleqtrid.1 $e |- A e. B $.
    eleqtrid.2 $e |- ( ph -> B = C ) $.
    $( B membership and equality inference.  (Contributed by NM,
       4-Jan-2006.) $)
    eleqtrid $p |- ( ph -> A e. C ) $=
      ( wcel a1i eleqtrd ) ABCDBCGAEHFI $.
  $}

  ${
    eleqtrrid.1 $e |- A e. B $.
    eleqtrrid.2 $e |- ( ph -> C = B ) $.
    $( B membership and equality inference.  (Contributed by NM,
       4-Jan-2006.) $)
    eleqtrrid $p |- ( ph -> A e. C ) $=
      ( eqcomd eleqtrid ) ABCDEADCFGH $.
  $}

  ${
    eqeltrdi.1 $e |- ( ph -> A = B ) $.
    eqeltrdi.2 $e |- B e. C $.
    $( A membership and equality inference.  (Contributed by NM,
       4-Jan-2006.) $)
    eqeltrdi $p |- ( ph -> A e. C ) $=
      ( wcel a1i eqeltrd ) ABCDECDGAFHI $.
  $}

  ${
    eqeltrrdi.1 $e |- ( ph -> B = A ) $.
    eqeltrrdi.2 $e |- B e. C $.
    $( A membership and equality inference.  (Contributed by NM,
       4-Jan-2006.) $)
    eqeltrrdi $p |- ( ph -> A e. C ) $=
      ( eqcomd eqeltrdi ) ABCDACBEGFH $.
  $}

  ${
    eleqtrdi.1 $e |- ( ph -> A e. B ) $.
    eleqtrdi.2 $e |- B = C $.
    $( A membership and equality inference.  (Contributed by NM,
       4-Jan-2006.) $)
    eleqtrdi $p |- ( ph -> A e. C ) $=
      ( wceq a1i eleqtrd ) ABCDECDGAFHI $.
  $}

  ${
    eleqtrrdi.1 $e |- ( ph -> A e. B ) $.
    eleqtrrdi.2 $e |- C = B $.
    $( A membership and equality inference.  (Contributed by NM,
       24-Apr-2005.) $)
    eleqtrrdi $p |- ( ph -> A e. C ) $=
      ( eqcomi eleqtrdi ) ABCDEDCFGH $.
  $}

  ${
    eleq2s.1 $e |- ( A e. B -> ph ) $.
    eleq2s.2 $e |- C = B $.
    $( Substitution of equal classes into a membership antecedent.
       (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
    eleq2s $p |- ( A e. C -> ph ) $=
      ( wcel eleq2i sylbi ) BDGBCGADCBFHEI $.
  $}

  ${
    eqneltrd.1 $e |- ( ph -> A = B ) $.
    eqneltrd.2 $e |- ( ph -> -. B e. C ) $.
    $( If a class is not an element of another class, an equal class is also
       not an element.  Deduction form.  (Contributed by David Moews,
       1-May-2017.) $)
    eqneltrd $p |- ( ph -> -. A e. C ) $=
      ( wcel eleq1d mtbird ) ABDGCDGFABCDEHI $.
  $}

  ${
    eqneltrrd.1 $e |- ( ph -> A = B ) $.
    eqneltrrd.2 $e |- ( ph -> -. A e. C ) $.
    $( If a class is not an element of another class, an equal class is also
       not an element.  Deduction form.  (Contributed by David Moews,
       1-May-2017.) $)
    eqneltrrd $p |- ( ph -> -. B e. C ) $=
      ( wcel eleq1d mtbid ) ABDGCDGFABCDEHI $.
  $}

  ${
    neleqtrd.1 $e |- ( ph -> -. C e. A ) $.
    neleqtrd.2 $e |- ( ph -> A = B ) $.
    $( If a class is not an element of another class, it is also not an element
       of an equal class.  Deduction form.  (Contributed by David Moews,
       1-May-2017.) $)
    neleqtrd $p |- ( ph -> -. C e. B ) $=
      ( wcel eleq2d mtbid ) ADBGDCGEABCDFHI $.
  $}

  ${
    neleqtrrd.1 $e |- ( ph -> -. C e. B ) $.
    neleqtrrd.2 $e |- ( ph -> A = B ) $.
    $( If a class is not an element of another class, it is also not an element
       of an equal class.  Deduction form.  (Contributed by David Moews,
       1-May-2017.) $)
    neleqtrrd $p |- ( ph -> -. C e. A ) $=
      ( wcel eleq2d mtbird ) ADBGDCGEABCDFHI $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    cleqh.1 $e |- ( y e. A -> A. x y e. A ) $.
    cleqh.2 $e |- ( y e. B -> A. x y e. B ) $.
    $( Establish equality between classes, using bound-variable hypotheses
       instead of distinct variable conditions.  See also ~ cleqf .
       (Contributed by NM, 5-Aug-1993.) $)
    cleqh $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
      ( wceq cv wcel wb wal dfcleq ax-17 wi wa dfbi2 hbim weq eleq1 cbv3h hban
      hbxfrbi bibi12d biimpd equcoms biimprd impbii bitr4i ) CDGBHZCIZUIDIZJZBK
      ZAHZCIZUNDIZJZAKZBCDLURUMUQULABUQBMZULUJUKNZUKUJNZOAUJUKPUTVAAUJUKAEFQUKU
      JAFEQUAUBZABRZUQULVCUOUJUPUKUNUICSUNUIDSUCZUDTULUQBAVBUSBARUQULUQULJABVDU
      EUFTUGUH $.
  $}

  $( A way of showing two classes are not equal.  (Contributed by NM,
     1-Apr-1997.) $)
  nelneq $p |- ( ( A e. C /\ -. B e. C ) -> -. A = B ) $=
    ( wcel wceq eleq1 biimpcd con3dimp ) ACDZABEZBCDZJIKABCFGH $.

  $( A way of showing two classes are not equal.  (Contributed by NM,
     12-Jan-2002.) $)
  nelneq2 $p |- ( ( A e. B /\ -. A e. C ) -> -. B = C ) $=
    ( wcel wceq eleq2 biimpcd con3dimp ) ABDZBCEZACDZJIKBCAFGH $.

  ${
    $d x y $.  $d x A $.
    $( Lemma for ~ eqsb1 .  (Contributed by Rodolfo Medina, 28-Apr-2010.)
       (Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
    eqsb1lem $p |- ( [ y / x ] x = A <-> y = A ) $=
      ( cv wceq nfv eqeq1 sbie ) ADZCEBDZCEZABKAFIJCGH $.
  $}

  ${
    $d w x A $.  $d w y $.
    $( Substitution for the left-hand side in an equality.  Class version of
       ~ equsb3 .  (Contributed by Rodolfo Medina, 28-Apr-2010.) $)
    eqsb1 $p |- ( [ y / x ] x = A <-> y = A ) $=
      ( vw cv wceq wsb eqsb1lem sbbii nfv sbco2 3bitr3i ) AECFZADGZDBGDECFZDBGM
      ABGBECFNODBADCHIMABDMDJKDBCHL $.
  $}

  ${
    $d w x A $.  $d w y $.
    $( Substitution for the first argument of the membership predicate in an
       atomic formula (class version of ~ elsb1 ).  (Contributed by Rodolfo
       Medina, 28-Apr-2010.)  (Proof shortened by Andrew Salmon,
       14-Jun-2011.) $)
    clelsb1 $p |- ( [ y / x ] x e. A <-> y e. A ) $=
      ( vw cv wcel wsb nfv sbco2 eleq1 sbie sbbii 3bitr3i ) DEZCFZDAGZABGODBGAE
      ZCFZABGBEZCFZODBAOAHIPRABORDARDHNQCJKLOTDBTDHNSCJKM $.
  $}

  ${
    $d w x A $.  $d w y $.
    $( Substitution for the second argument of the membership predicate in an
       atomic formula (class version of ~ elsb2 ).  (Contributed by Jim
       Kingdon, 22-Nov-2018.) $)
    clelsb2 $p |- ( [ y / x ] A e. x <-> A e. y ) $=
      ( vw cv wcel wsb nfv sbco2 eleq2 sbie sbbii 3bitr3i ) CDEZFZDAGZABGODBGCA
      EZFZABGCBEZFZODBAOAHIPRABORDARDHNQCJKLOTDBTDHNSCJKM $.
  $}

  ${
    hbxfr.1 $e |- A = B $.
    hbxfr.2 $e |- ( y e. B -> A. x y e. B ) $.
    $( A utility lemma to transfer a bound-variable hypothesis builder into a
       definition.  See ~ hbxfrbi for equivalence version.  (Contributed by NM,
       21-Aug-2007.) $)
    hbxfreq $p |- ( y e. A -> A. x y e. A ) $=
      ( cv wcel eleq2i hbxfrbi ) BGZCHKDHACDKEIFJ $.
  $}

  ${
    $d y A $.  $d x z $.
    hblem.1 $e |- ( y e. A -> A. x y e. A ) $.
    $( Change the free variable of a hypothesis builder.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Andrew Salmon, 11-Jul-2011.) $)
    hblem $p |- ( z e. A -> A. x z e. A ) $=
      ( cv wcel wsb wal hbsb clelsb1 albii 3imtr3i ) BFDGZBCHZOAICFDGZPAINBCAEJ
      BCDKZOPAQLM $.
  $}

  ${
    $d x A y $.  $d ph y $.
    $( Equality of a class variable and a class abstraction (also called a
       class builder).  Theorem 5.1 of [Quine] p. 34.  This theorem shows the
       relationship between expressions with class abstractions and expressions
       with class variables.  Note that ~ abbi and its relatives are among
       those useful for converting theorems with class variables to equivalent
       theorems with wff variables, by first substituting a class abstraction
       for each class variable.

       Class variables can always be eliminated from a theorem to result in an
       equivalent theorem with wff variables, and vice-versa.  The idea is
       roughly as follows.  To convert a theorem with a wff variable ` ph `
       (that has a free variable ` x ` ) to a theorem with a class variable
       ` A ` , we substitute ` x e. A ` for ` ph ` throughout and simplify,
       where ` A ` is a new class variable not already in the wff.  Conversely,
       to convert a theorem with a class variable ` A ` to one with ` ph ` , we
       substitute ` { x | ph } ` for ` A ` throughout and simplify, where ` x `
       and ` ph ` are new set and wff variables not already in the wff.  For
       more information on class variables, see Quine pp. 15-21 and/or Takeuti
       and Zaring pp. 10-13.  (Contributed by NM, 5-Aug-1993.) $)
    abeq2 $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
      ( vy cab wceq cv wcel wb wal ax-17 hbab1 cleqh abid bibi2i albii bitri )
      CABEZFBGZCHZSRHZIZBJTAIZBJBDCRDGCHBKABDLMUBUCBUAATABNOPQ $.
  $}

  ${
    $d x A $.
    $( Equality of a class variable and a class abstraction.  (Contributed by
       NM, 20-Aug-1993.) $)
    abeq1 $p |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) ) $=
      ( cab wceq cv wcel wb wal abeq2 eqcom bicom albii 3bitr4i ) CABDZEBFCGZAH
      ZBIOCEAPHZBIABCJOCKRQBAPLMN $.
  $}

  ${
    abeqi.1 $e |- A = { x | ph } $.
    $( Equality of a class variable and a class abstraction (inference form).
       (Contributed by NM, 3-Apr-1996.) $)
    abeq2i $p |- ( x e. A <-> ph ) $=
      ( cv wcel cab eleq2i abid bitri ) BEZCFKABGZFACLKDHABIJ $.
  $}

  ${
    abeqri.1 $e |- { x | ph } = A $.
    $( Equality of a class variable and a class abstraction (inference form).
       (Contributed by NM, 31-Jul-1994.) $)
    abeq1i $p |- ( ph <-> x e. A ) $=
      ( cv cab wcel abid eleq2i bitr3i ) ABEZABFZGKCGABHLCKDIJ $.
  $}

  ${
    abeqd.1 $e |- ( ph -> A = { x | ps } ) $.
    $( Equality of a class variable and a class abstraction (deduction).
       (Contributed by NM, 16-Nov-1995.) $)
    abeq2d $p |- ( ph -> ( x e. A <-> ps ) ) $=
      ( cv wcel cab eleq2d abid bitrdi ) ACFZDGLBCHZGBADMLEIBCJK $.
  $}

  ${
    $d ph y $.  $d ps y $.  $d x y $.
    $( Equivalent wff's correspond to equal class abstractions.  (Contributed
       by NM, 25-Nov-2013.)  (Revised by Mario Carneiro, 11-Aug-2016.) $)
    abbibcom $p |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } ) $=
      ( vy cab wceq wcel wal dfcleq nfsab1 nfbi nfv wsb df-clab sbequ12r bitrid
      cv wb bibi12d cbval bitr2i ) ACEZBCEZFDQZUBGZUDUCGZRZDHABRZCHDUBUCIUGUHDC
      UEUFCACDJBCDJKUHDLUDCQFZUEAUFBUEACDMUIAADCNADCOPUFBCDMUIBBDCNBDCOPSTUA $.
  $}

  ${
    $d x A $.
    abbiri.1 $e |- ( x e. A <-> ph ) $.
    $( Equality of a class variable and a class abstraction (inference form).
       (Contributed by NM, 5-Aug-1993.) $)
    abbi2i $p |- A = { x | ph } $=
      ( cab wceq cv wcel wb abeq2 mpgbir ) CABEFBGCHAIBABCJDK $.
  $}

  ${
    abbii.1 $e |- ( ph <-> ps ) $.
    $( Equivalent wff's yield equal class abstractions (inference form).
       (Contributed by NM, 5-Aug-1993.) $)
    abbii $p |- { x | ph } = { x | ps } $=
      ( wb cab wceq abbibcom mpgbi ) ABEACFBCFGCABCHDI $.

    $( Theorem abbii is the congruence law for class abstraction. $)
    $( $j congruence 'abbii'; $)
  $}

  ${
    abbid.1 $e |- F/ x ph $.
    abbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal class abstractions (deduction form).
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       7-Oct-2016.) $)
    abbid $p |- ( ph -> { x | ps } = { x | ch } ) $=
      ( wb wal cab wceq alrimi abbibcom sylib ) ABCGZDHBDICDIJANDEFKBCDLM $.
  $}

  ${
    $d ph y $.  $d ps y $.  $d x y $.
    $( Equal class abstractions require equivalent formulas, and conversely.
       (Contributed by NM, 25-Nov-2013.)  (Revised by Mario Carneiro,
       11-Aug-2016.)  Remove dependency on ~ ax-8 and ~ df-clel (by avoiding
       use of ~ cleqh ).  (Revised by BJ, 23-Jun-2019.)  Definitial form.
       (Revised by Wolf Lammen, 23-Feb-2025.) $)
    abbib $p |- ( { x | ph } = { x | ps } <-> A. x ( ph <-> ps ) ) $=
      ( vy cab wceq cv wcel wal dfcleq nfsab1 nfbi nfv weq wsb df-clab sbequ12r
      wb bitrid bibi12d cbvalv1 bitri ) ACEZBCEZFDGZUCHZUEUDHZRZDIABRZCIDUCUDJU
      HUIDCUFUGCACDKBCDKLUIDMDCNZUFAUGBUFACDOUJAADCPADCQSUGBCDOUJBBDCPBDCQSTUAU
      B $.
    $( $j usage 'abbib' avoids 'ax-13' 'df-clel' ; $)
  $}

  ${
    $d ph y $.  $d ps y $.  $d x y $.
    $( Equivalent formulas yield equal class abstractions (closed form).  This
       is the backward implication of ~ abbib , proved from fewer axioms, and
       hence is independently named.  (Contributed by BJ and WL and SN,
       20-Aug-2023.) $)
    abbi $p |- ( A. x ( ph <-> ps ) -> { x | ph } = { x | ps } ) $=
      ( vy wb wal cab wsb cv wcel spsbbi df-clab 3bitr4g eqrdv ) ABECFZDACGZBCG
      ZOACDHBCDHDIZPJRQJABCDKADCLBDCLMN $.
    $( $j usage 'abbi' avoids 'ax-i12' 'ax-bndl' 'ax-13' 'df-clel' ; $)
  $}

  ${
    $d x ph $.
    abbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal class abstractions (deduction form).
       (Contributed by NM, 10-Aug-1993.) $)
    abbidv $p |- ( ph -> { x | ps } = { x | ch } ) $=
      ( nfv abbid ) ABCDADFEG $.
  $}

  ${
    $d x A $.  $d ph x $.
    abbirdv.1 $e |- ( ph -> ( x e. A <-> ps ) ) $.
    $( Deduction from a wff to a class abstraction.  (Contributed by NM,
       9-Jul-1994.) $)
    abbi2dv $p |- ( ph -> A = { x | ps } ) $=
      ( cv wcel wb wal cab wceq alrimiv abeq2 sylibr ) ACFDGBHZCIDBCJKAOCELBCDM
      N $.
  $}

  ${
    $d x A $.  $d ph x $.
    abbildv.1 $e |- ( ph -> ( ps <-> x e. A ) ) $.
    $( Deduction from a wff to a class abstraction.  (Contributed by NM,
       9-Jul-1994.) $)
    abbi1dv $p |- ( ph -> { x | ps } = A ) $=
      ( cv wcel wb wal cab wceq alrimiv abeq1 sylibr ) ABCFDGHZCIBCJDKAOCELBCDM
      N $.
  $}

  ${
    $d x A $.
    $( A simplification of class abstraction.  Theorem 5.2 of [Quine] p. 35.
       (Contributed by NM, 26-Dec-1993.) $)
    abid2 $p |- { x | x e. A } = A $=
      ( cv wcel cab biid abbi2i eqcomi ) BACBDZAEIABIFGH $.
  $}

  ${
    $d x z $.  $d y z $.  $d ph z $.
    sb8ab.1 $e |- F/ y ph $.
    $( Substitution of variable in class abstraction.  (Contributed by Jim
       Kingdon, 27-Sep-2018.) $)
    sb8ab $p |- { x | ph } = { y | [ y / x ] ph } $=
      ( vz cab wsb cv wcel sbco2 df-clab 3bitr4ri eqriv ) EABFZABCGZCFZOCEGABEG
      EHZPIQNIABECDJOECKAEBKLM $.
  $}

  ${
    $d x y z $.  $d ph z $.  $d ps z $.
    cbvabw.1 $e |- F/ y ph $.
    cbvabw.2 $e |- F/ x ps $.
    cbvabw.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ cbvab with a disjoint variable condition.  (Contributed by
       GG, 10-Jan-2024.)  Reduce axiom usage.  (Revised by GG, 25-Aug-2024.) $)
    cbvabw $p |- { x | ph } = { y | ps } $=
      ( vz cab wsb cv wcel nfsbv weq wi wal equequ2 sb6 sbiev df-clab 3bitr4g
      imbi1d albidv sbbii bitr3i 3bitr4i eqriv ) HACIZBDIZACHJZBDHJZHKZUHLULUIL
      UJACDJZDHJUKUMUJDHACHDEMDHNZCDNZAOZCPCHNZAOZCPUMUJUNUPURCUNUOUQADHCQUBUCA
      CDRACHRUASUMBDHABCDFGSUDUEAHCTBHDTUFUG $.
    $( $j usage 'cbvabw' avoids 'ax-io' 'ax-10' 'ax-i12' 'ax-bndl'; $)
  $}

  ${
    $d x z $.  $d y z $.  $d ph z $.  $d ps z $.
    cbvab.1 $e |- F/ y ph $.
    cbvab.2 $e |- F/ x ps $.
    cbvab.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by Andrew Salmon, 11-Jul-2011.) $)
    cbvab $p |- { x | ph } = { y | ps } $=
      ( vz cab wsb cv wcel nfsb weq wb equcoms bicomd sbie sbequ df-clab eqriv
      bitr3id 3bitr4i ) HACIZBDIZACHJBDHJZHKZUDLUGUELAUFCHBDHCFMABDCJCHNUFBADCE
      DCNABABOCDGPQRBCHDSUBRAHCTBHDTUCUA $.
  $}

  ${
    $d y ph $.  $d x ps $.
    cbvabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 26-May-1999.) $)
    cbvabv $p |- { x | ph } = { y | ps } $=
      ( nfv cbvab ) ABCDADFBCFEG $.
  $}

  ${
    $d x A y $.  $d ph y $.
    $( Membership of a class variable in a class abstraction.  (Contributed by
       NM, 23-Dec-1993.) $)
    clelab $p |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) ) $=
      ( vy cv wceq cab wcel wex wsb df-clab anbi2i exbii df-clel nfv nfs1v nfan
      wa eqeq1 sbequ12 anbi12d cbvex 3bitr4i ) DEZCFZUDABGZHZRZDIUEABDJZRZDICUF
      HBEZCFZARZBIUHUJDUGUIUEADBKLMDCUFNUMUJBDUMDOUEUIBUEBOABDPQUKUDFULUEAUIUKU
      DCSABDTUAUBUC $.
  $}

  ${
    $d y A $.  $d y ph $.  $d x y $.
    $( Membership of a class abstraction in another class.  (Contributed by NM,
       17-Jan-2006.) $)
    clabel $p |- ( { x | ph } e. A <->
                 E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) ) $=
      ( cab wcel cv wceq wa wex wb wal df-clel abeq2 anbi2ci exbii bitri ) ABEZ
      DFCGZRHZSDFZIZCJUABGSFAKBLZIZCJCRDMUBUDCTUCUAABSNOPQ $.
  $}

  ${
    $d z A $.  $d z x $.  $d z y $.
    $( The right-hand side of the second equality is a way of representing
       proper substitution of ` y ` for ` x ` into a class variable.
       (Contributed by NM, 14-Sep-2003.) $)
    sbab $p |- ( x = y -> A = { z | [ y / x ] z e. A } ) $=
      ( weq cv wcel wsb sbequ12 abbi2dv ) ABECFDGZABHCDKABIJ $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Elementary properties of class abstractions
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d x y A $.  $d ph x y $.  $d ps y $.
    eqabdv.1 $e |- ( ph -> ( x e. A <-> ps ) ) $.
    $( Deduction from a wff to a class abstraction.  (Contributed by NM,
       9-Jul-1994.)  (Revised by Wolf Lammen, 6-May-2023.) $)
    eqabdv $p |- ( ph -> A = { x | ps } ) $=
      ( vy cab cv wcel wsb sbbidv clelsb1 bicomi df-clab 3bitr4g eqrdv ) AFDBCG
      ZACHDIZCFJZBCFJFHZDIZTQIARBCFEKSUACFDLMBFCNOP $.
  $}

  ${
    $d x A $.  $d ph x $.
    eqabcdv.1 $e |- ( ph -> ( ps <-> x e. A ) ) $.
    $( Deduction from a wff to a class abstraction.  (Contributed by NM,
       9-Jul-1994.)  (Proof shortened by Wolf Lammen, 16-Nov-2019.) $)
    eqabcdv $p |- ( ph -> { x | ps } = A ) $=
      ( cab cv wcel bicomd eqabdv eqcomd ) ADBCFABCDABCGDHEIJK $.
  $}

  ${
    $d x A $.
    eqabi.1 $e |- ( x e. A <-> ph ) $.
    $( Equality of a class variable and a class abstraction (inference form).
       (Contributed by NM, 26-May-1993.)  (Revised by Wolf Lammen,
       6-May-2023.) $)
    eqabi $p |- A = { x | ph } $=
      ( cab wceq wtru cv wcel wb a1i eqabdv mptru ) CABEFGABCBHCIAJGDKLM $.
  $}

  ${
    $d x A $.
    $( Every class is equal to a class abstraction (the class of sets belonging
       to it).  Theorem 5.2 of [Quine] p. 35.  This is a generalization to
       classes of ~ cvjust .  The proof does not rely on ~ cvjust , so ~ cvjust
       could be proved as a special instance of it.  Note however that ~ abid1
       necessarily relies on ~ df-clel , whereas ~ cvjust does not.

       This theorem requires ~ ax-ext , ~ df-clab , ~ df-cleq , ~ df-clel , but
       to prove that any specific class term not containing class variables is
       a setvar or is equal to a class abstraction does not require these
       $a-statements.  This last fact is a metatheorem, consequence of the fact
       that the only $a-statements with typecode ` class ` are ~ cv , ~ cab ,
       and statements corresponding to defined class constructors.

       Note on the simultaneous presence in set.mm of this ~ abid1 and its
       commuted form ~ abid2 :  It is rare that two forms so closely related
       both appear in set.mm.  Indeed, such equalities are generally used in
       later proofs as parts of transitive inferences, and with the many
       variants of ~ eqtri (search for *eqtr*), it would be rare that either
       one would shorten a proof compared to the other.  There is typically a
       choice between what we call a "definitional form", where the shorter
       expression is on the LHS (left-hand side), and a "computational form",
       where the shorter expression is on the RHS (right-hand side).  An
       example is ~ df-2 versus ~ 1p1e2 .  We do not need ~ 1p1e2 , but because
       it occurs "naturally" in computations, it can be useful to have it
       directly, together with a uniform set of 1-digit operations like
       ~ 1p2e3 , etc.  In most cases, we do not need both a definitional and a
       computational forms.  A definitional form would favor consistency with
       genuine definitions, while a computational form is often more natural.
       The situation is similar with biconditionals in propositional calculus:
       see for instance ~ pm4.24 and ~ anidm , while other biconditionals
       generally appear in a single form (either definitional, but more often
       computational).  In the present case, the equality is important enough
       that both ~ abid1 and ~ abid2 are in set.mm.

       (Contributed by NM, 26-Dec-1993.)  (Revised by BJ, 10-Nov-2020.) $)
    abid1 $p |- A = { x | x e. A } $=
      ( cv wcel biid eqabi ) ACBDZABGEF $.
  $}

  ${
    $d x A y $.  $d ph y $.
    $( One direction of ~ eqabb .  (Contributed by Wolf Lammen,
       13-Feb-2025.) $)
    eqab $p |- ( A. x ( x e. A <-> ph ) -> A = { x | ph } ) $=
      ( cv wcel wb wal cab abid1 abbi eqtrid ) BDCEZAFBGCLBHABHBCILABJK $.

    $( Equality of a class variable and a class abstraction (also called a
       class builder).  Theorem 5.1 of [Quine] p. 34.  This theorem shows the
       relationship between expressions with class abstractions and expressions
       with class variables.  Note that ~ abbib and its relatives are among
       those useful for converting theorems with class variables to equivalent
       theorems with wff variables, by first substituting a class abstraction
       for each class variable.

       Class variables can always be eliminated from a theorem to result in an
       equivalent theorem with wff variables, and vice-versa.  The idea is
       roughly as follows.  To convert a theorem with a wff variable ` ph `
       (that has a free variable ` x ` ) to a theorem with a class variable
       ` A ` , we substitute ` x e. A ` for ` ph ` throughout and simplify,
       where ` A ` is a new class variable not already in the wff.  An example
       is the conversion of ~ sepgi to ~ inex1 (look at the instance of ~ sepgi
       that occurs in the proof of ~ inex1 ).  Conversely, to convert a theorem
       with a class variable ` A ` to one with ` ph ` , we substitute
       ` { x | ph } ` for ` A ` throughout and simplify, where ` x ` and ` ph `
       are new setvar and wff variables not already in the wff.  For more
       information on class variables, see Quine pp. 15-21 and/or Takeuti and
       Zaring pp. 10-13.

       (Contributed by NM, 26-May-1993.)  (Proof shortened by Wolf Lammen,
       12-Feb-2025.) $)
    eqabb $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
      ( cab wceq cv wcel wb wal abid1 eqeq1i abbib bitri ) CABDZEBFCGZBDZNEOAHB
      ICPNBCJKOABLM $.
  $}

  ${
    $d x y $.  $d y A $.  $d ph y $.  $d ps x $.
    eqabbw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ eqabb using implicit substitution.  (Contributed by GG and
       AV, 18-Sep-2024.) $)
    eqabbw $p |- ( A = { x | ph } <-> A. y ( y e. A <-> ps ) ) $=
      ( cab wceq cv wcel wb wal cbvabv eqeq2i eqabb bitri ) EACGZHEBDGZHDIEJBKD
      LQREABCDFMNBDEOP $.

    $( Version of ~ eqabcb using implicit substitution, which requires fewer
       axioms.  (Contributed by TM, 24-Jan-2026.) $)
    eqabcbw $p |- ( { x | ph } = A <-> A. y ( ps <-> y e. A ) ) $=
      ( cab wceq cv wcel wb wal eqabbw eqcom bicom albii 3bitr4i ) EACGZHDIEJZB
      KZDLREHBSKZDLABCDEFMRENUATDBSOPQ $.
    $( $j usage 'eqabcbw' avoids 'df-clel'; $)
  $}

  ${
    $d x A $.
    $( Equality of a class variable and a class abstraction.  Commuted form of
       ~ eqabb .  (Contributed by NM, 20-Aug-1993.) $)
    eqabcb $p |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) ) $=
      ( cab wceq cv wcel wb wal eqabb eqcom bicom albii 3bitr4i ) CABDZEBFCGZAH
      ZBIOCEAPHZBIABCJOCKRQBAPLMN $.
  $}

  ${
    eqabrd.1 $e |- ( ph -> A = { x | ps } ) $.
    $( Equality of a class variable and a class abstraction (deduction form of
       ~ eqabb ).  (Contributed by NM, 16-Nov-1995.) $)
    eqabrd $p |- ( ph -> ( x e. A <-> ps ) ) $=
      ( cv wcel cab eleq2d abid bitrdi ) ACFZDGLBCHZGBADMLEIBCJK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Class form not-free predicate
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c F/_ $.  $( Underlined not-free symbol. $)

  $( Extend wff definition to include the not-free predicate for classes. $)
  wnfc $a wff F/_ x A $.

  ${
    $d x y z $.  $d y z A $.
    $( Justification theorem for ~ df-nfc .  (Contributed by Mario Carneiro,
       13-Oct-2016.) $)
    nfcjust $p |- ( A. y F/ x y e. A <-> A. z F/ x z e. A ) $=
      ( cv wcel wnf weq nfv eleq1 nfbidf cbvalv ) BEZDFZAGCEZDFZAGBCBCHZNPAQAIM
      ODJKL $.
  $}

  ${
    $d x y $.  $d y A $.
    $( Define the not-free predicate for classes.  This is read " ` x ` is not
       free in ` A ` ".  Not-free means that the value of ` x ` cannot affect
       the value of ` A ` , e.g., any occurrence of ` x ` in ` A ` is
       effectively bound by a quantifier or something that expands to one (such
       as "there exists at most one").  It is defined in terms of the not-free
       predicate ~ df-nf for wffs; see that definition for more information.
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    df-nfc $a |- ( F/_ x A <-> A. y F/ x y e. A ) $.

    ${
      nfci.1 $e |- F/ x y e. A $.
      $( Deduce that a class ` A ` does not have ` x ` free in it.
         (Contributed by Mario Carneiro, 11-Aug-2016.) $)
      nfci $p |- F/_ x A $=
        ( wnfc cv wcel wnf df-nfc mpgbir ) ACEBFCGAHBABCIDJ $.
    $}

    ${
      nfcii.1 $e |- ( y e. A -> A. x y e. A ) $.
      $( Deduce that a class ` A ` does not have ` x ` free in it.
         (Contributed by Mario Carneiro, 11-Aug-2016.) $)
      nfcii $p |- F/_ x A $=
        ( cv wcel nfi nfci ) ABCBECFADGH $.
    $}

    $( Consequence of the not-free predicate.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfcr $p |- ( F/_ x A -> F/ x y e. A ) $=
      ( wnfc cv wcel wnf wal df-nfc sp sylbi ) ACDBECFAGZBHLABCILBJK $.
  $}

  ${
    $d x y z $.  $d z A $.
    nfcri.1 $e |- F/_ x A $.
    $( Consequence of the not-free predicate.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfcrii $p |- ( y e. A -> A. x y e. A ) $=
      ( vz cv wcel wnfc wnf nfcr ax-mp nfri hblem ) AEBCEFCGZAACHNAIDAECJKLM $.

    $( Consequence of the not-free predicate.  (Note that unlike ~ nfcr , this
       does not require ` y ` and ` A ` to be disjoint.)  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    nfcri $p |- F/ x y e. A $=
      ( cv wcel nfcrii nfi ) BECFAABCDGH $.
  $}

  ${
    $d x y $.  $d y A $.
    nfcd.1 $e |- F/ y ph $.
    nfcd.2 $e |- ( ph -> F/ x y e. A ) $.
    $( Deduce that a class ` A ` does not have ` x ` free in it.  (Contributed
       by Mario Carneiro, 11-Aug-2016.) $)
    nfcd $p |- ( ph -> F/_ x A ) $=
      ( cv wcel wnf wal wnfc alrimi df-nfc sylibr ) ACGDHBIZCJBDKAOCEFLBCDMN $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    nfceqi.1 $e |- A = B $.
    $( Equality theorem for class not-free.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfceqi $p |- ( F/_ x A <-> F/_ x B ) $=
      ( vy cv wcel wnf wal wnfc eleq2i nfbii albii df-nfc 3bitr4i ) EFZBGZAHZEI
      PCGZAHZEIABJACJRTEQSABCPDKLMAEBNAECNO $.

    ${
      nfcxfr.2 $e |- F/_ x B $.
      $( A utility lemma to transfer a bound-variable hypothesis builder into a
         definition.  (Contributed by Mario Carneiro, 11-Aug-2016.) $)
      nfcxfr $p |- F/_ x A $=
        ( wnfc nfceqi mpbir ) ABFACFEABCDGH $.
    $}

    ${
      nfcxfrd.2 $e |- ( ph -> F/_ x B ) $.
      $( A utility lemma to transfer a bound-variable hypothesis builder into a
         definition.  (Contributed by Mario Carneiro, 11-Aug-2016.) $)
      nfcxfrd $p |- ( ph -> F/_ x A ) $=
        ( wnfc nfceqi sylibr ) ABDGBCGFBCDEHI $.
    $}
  $}

  ${
    $d x y $.  $d A y $.  $d B y $.  $d ph y $.
    nfceqdf.1 $e |- F/ x ph $.
    nfceqdf.2 $e |- ( ph -> A = B ) $.
    $( An equality theorem for effectively not free.  (Contributed by Mario
       Carneiro, 14-Oct-2016.) $)
    nfceqdf $p |- ( ph -> ( F/_ x A <-> F/_ x B ) ) $=
      ( vy cv wcel wnf wal wnfc eleq2d nfbidf albidv df-nfc 3bitr4g ) AGHZCIZBJ
      ZGKRDIZBJZGKBCLBDLATUBGASUABEACDRFMNOBGCPBGDPQ $.
  $}

  ${
    $d x y A $.
    $( If ` x ` is disjoint from ` A ` , then ` x ` is not free in ` A ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nfcv $p |- F/_ x A $=
      ( vy cv wcel nfv nfci ) ACBCDBEAFG $.

    $( If ` x ` is disjoint from ` A ` , then ` x ` is not free in ` A ` .
       (Contributed by Mario Carneiro, 7-Oct-2016.) $)
    nfcvd $p |- ( ph -> F/_ x A ) $=
      ( wnfc nfcv a1i ) BCDABCEF $.
  $}

  ${
    $d x y $.  $d y A $.  $d y ph $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by Mario Carneiro, 11-Aug-2016.) $)
    nfab1 $p |- F/_ x { x | ph } $=
      ( vy cab nfsab1 nfci ) BCABDABCEF $.

    $( ` x ` is bound in ` F/_ x A ` .  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfnfc1 $p |- F/ x F/_ x A $=
      ( vy wnfc cv wcel wnf wal df-nfc nfnf1 nfal nfxfr ) ABDCEBFZAGZCHAACBINAC
      MAJKL $.
  $}

  ${
    $d w x $.  $d w A $.  $d w y $.
    clelsb1f.1 $e |- F/_ x A $.
    $( Substitution for the first argument of the membership predicate in an
       atomic formula (class version of ~ elsb1 ).  (Contributed by Rodolfo
       Medina, 28-Apr-2010.)  (Proof shortened by Andrew Salmon, 14-Jun-2011.)
       (Revised by Thierry Arnoux, 13-Mar-2017.) $)
    clelsb1f $p |- ( [ y / x ] x e. A <-> y e. A ) $=
      ( vw cv wcel wsb nfcri sbco2 nfv eleq1w sbie sbbii 3bitr3i ) EFCGZEAHZABH
      PEBHAFCGZABHBFCGZPEBAAECDIJQRABPREAREKEACLMNPSEBSEKEBCLMO $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    nfab.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by Mario Carneiro, 11-Aug-2016.) $)
    nfab $p |- F/_ x { y | ph } $=
      ( vz cab nfsab nfci ) BEACFABCEDGH $.
  $}

  $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
     by Mario Carneiro, 14-Oct-2016.) $)
  nfaba1 $p |- F/_ x { y | A. x ph } $=
    ( wal nfa1 nfab ) ABDBCABEF $.

  ${
    $d x z $.  $d y z $.  $d z A $.  $d z B $.
    nfnfc.1 $e |- F/_ x A $.
    $( Hypothesis builder for ` F/_ y A ` .  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfnfc $p |- F/ x F/_ y A $=
      ( vz wnfc cv wcel wnf wal df-nfc nfcri nfnf nfal nfxfr ) BCFEGCHZBIZEJABE
      CKQAEPABAECDLMNO $.

    nfeq.2 $e |- F/_ x B $.
    $( Hypothesis builder for equality.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfeq $p |- F/ x A = B $=
      ( vz wceq cv wcel wb wal dfcleq nfcri nfbi nfal nfxfr ) BCGFHZBIZQCIZJZFK
      AFBCLTAFRSAAFBDMAFCEMNOP $.

    $( Hypothesis builder for elementhood.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfel $p |- F/ x A e. B $=
      ( vz wcel cv wceq wa wex df-clel nfcv nfeq nfcri nfan nfex nfxfr ) BCGFHZ
      BIZSCGZJZFKAFBCLUBAFTUAAASBASMDNAFCEOPQR $.
  $}

  ${
    $d x B $.
    nfeq1.1 $e |- F/_ x A $.
    $( Hypothesis builder for equality, special case.  (Contributed by Mario
       Carneiro, 10-Oct-2016.) $)
    nfeq1 $p |- F/ x A = B $=
      ( nfcv nfeq ) ABCDACEF $.

    $( Hypothesis builder for elementhood, special case.  (Contributed by Mario
       Carneiro, 10-Oct-2016.) $)
    nfel1 $p |- F/ x A e. B $=
      ( nfcv nfel ) ABCDACEF $.
  $}

  ${
    $d x A $.
    nfeq2.1 $e |- F/_ x B $.
    $( Hypothesis builder for equality, special case.  (Contributed by Mario
       Carneiro, 10-Oct-2016.) $)
    nfeq2 $p |- F/ x A = B $=
      ( nfcv nfeq ) ABCABEDF $.

    $( Hypothesis builder for elementhood, special case.  (Contributed by Mario
       Carneiro, 10-Oct-2016.) $)
    nfel2 $p |- F/ x A e. B $=
      ( nfcv nfel ) ABCABEDF $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    nfeqd.1 $e |- ( ph -> F/_ x A ) $.
    $( Consequence of the not-free predicate.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfcrd $p |- ( ph -> F/ x y e. A ) $=
      ( wnfc cv wcel wnf nfcr syl ) ABDFCGDHBIEBCDJK $.

    $d y ph $.
    nfeqd.2 $e |- ( ph -> F/_ x B ) $.
    $( Hypothesis builder for equality.  (Contributed by Mario Carneiro,
       7-Oct-2016.) $)
    nfeqd $p |- ( ph -> F/ x A = B ) $=
      ( vy wceq cv wcel wb wal dfcleq nfv nfcrd nfbid nfald nfxfrd ) CDHGIZCJZS
      DJZKZGLABGCDMAUBBGAGNATUABABGCEOABGDFOPQR $.

    $( Hypothesis builder for elementhood.  (Contributed by Mario Carneiro,
       7-Oct-2016.) $)
    nfeld $p |- ( ph -> F/ x A e. B ) $=
      ( vy wcel cv wceq wa wex df-clel nfv nfcvd nfeqd nfcrd nfand nfexd nfxfrd
      ) CDHGIZCJZUADHZKZGLABGCDMAUDBGAGNAUBUCBABUACABUAOEPABGDFQRST $.
  $}

  ${
    $d w x $.  $d w y $.  $d w z $.  $d w A $.  $d w B $.
    drnfc1.1 $e |- ( A. x x = y -> A = B ) $.
    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       (Contributed by Mario Carneiro, 8-Oct-2016.) $)
    drnfc1 $p |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) ) $=
      ( vw weq wal cv wcel wnf wnfc eleq2d drnf1 dral2 df-nfc 3bitr4g ) ABGAHZF
      IZCJZAKZFHSDJZBKZFHACLBDLUAUCABFTUBABRCDSEMNOAFCPBFDPQ $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       (Contributed by Mario Carneiro, 8-Oct-2016.) $)
    drnfc2 $p |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) ) $=
      ( vw weq wal cv wcel wnf wnfc eleq2d drnf2 dral2 df-nfc 3bitr4g ) ABHAIZG
      JZDKZCLZGITEKZCLZGICDMCEMUBUDABGUAUCABCSDETFNOPCGDQCGEQR $.
  $}

  ${
    $d x y z $.  $d z ph $.  $d z ps $.
    nfabdw.1 $e |- F/ y ph $.
    nfabdw.2 $e |- ( ph -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for a class abstraction.  Version of
       ~ nfabd with a disjoint variable condition.  (Contributed by Mario
       Carneiro, 8-Oct-2016.)  (Revised by GG, 10-Jan-2024.) $)
    nfabdw $p |- ( ph -> F/_ x { y | ps } ) $=
      ( vz cab nfv cv wcel wsb df-clab wnf wal alrimi wi weq nfim1 nfal nfa1 wb
      sb6 a1i biimpri axc4i biimtrdi nf5d sbequ12 imbi2d equsalv bicomi sp nfim
      nfnf1 nfxfr pm5.5 nfbidf mpbii syl nfxfrd nfcd ) ACGBDHZAGIGJVCKBDGLZACBG
      DMABCNZDOZVDCNZAVEDEFPVFVFVDQZCNVGVHDGRZVFBQZQZDOZCVLVHVJVHDGVFVDDVEDUAZV
      FVDDVMVFVDVIBQZDOZVDDOVDVOUBVFBDGUCZUDVNVDDVDVOVPUEUFUGUHSVIBVDVFBDGUIUJU
      KULVKCDVIVJCVICIVFBCVECDBCUOTZVEDUMSUNTUPVFVHVDCVQVFVDUQURUSUTVAVB $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.  $d z ps $.
    nfabd.1 $e |- F/ y ph $.
    nfabd.2 $e |- ( ph -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by Mario Carneiro, 8-Oct-2016.) $)
    nfabd $p |- ( ph -> F/_ x { y | ps } ) $=
      ( vz cab nfv cv wcel wsb df-clab nfsbd nfxfrd nfcd ) ACGBDHZAGIGJQKBDGLAC
      BGDMABDGCEFNOP $.
  $}

  ${
    $d w x $.  $d w y $.  $d w z $.  $d w A $.  $d w B $.  $d w ph $.
    dvelimdc.1 $e |- F/ x ph $.
    dvelimdc.2 $e |- F/ z ph $.
    dvelimdc.3 $e |- ( ph -> F/_ x A ) $.
    dvelimdc.4 $e |- ( ph -> F/_ z B ) $.
    dvelimdc.5 $e |- ( ph -> ( z = y -> A = B ) ) $.
    $( Deduction form of ~ dvelimc .  (Contributed by Mario Carneiro,
       8-Oct-2016.) $)
    dvelimdc $p |- ( ph -> ( -. A. x x = y -> F/_ x B ) ) $=
      ( vw weq wal wn wnfc wa nfv wcel nfcrd cv wnf wceq wb eleq2 syl6 dvelimdf
      imp nfcd ex ) ABCMBNOZBFPAUKQZBLFULLRAUKLUAZFSZBUBAUMESZUNBCDGHABLEITADLF
      JTADCMEFUCUOUNUDKEFUMUEUFUGUHUIUJ $.
  $}

  ${
    dvelimc.1 $e |- F/_ x A $.
    dvelimc.2 $e |- F/_ z B $.
    dvelimc.3 $e |- ( z = y -> A = B ) $.
    $( Version of ~ dvelim for classes.  (Contributed by Mario Carneiro,
       8-Oct-2016.) $)
    dvelimc $p |- ( -. A. x x = y -> F/_ x B ) $=
      ( weq wal wn wnfc wi wtru nftru a1i wceq dvelimdc mptru ) ABIAJKAELMNABCD
      EAOCOADLNFPCELNGPCBIDEQMNHPRS $.
  $}

  ${
    $d x z $.  $d y z $.
    $( If ` x ` and ` y ` are distinct, then ` x ` is not free in ` y ` .
       (Contributed by Mario Carneiro, 8-Oct-2016.) $)
    nfcvf $p |- ( -. A. x x = y -> F/_ x y ) $=
      ( vz cv nfcv weq id dvelimc ) ABCCDZBDZAIECJECBFGH $.

    $( If ` x ` and ` y ` are distinct, then ` y ` is not free in ` x ` .
       (Contributed by Mario Carneiro, 5-Dec-2016.) $)
    nfcvf2 $p |- ( -. A. x x = y -> F/_ y x ) $=
      ( cv wnfc nfcvf naecoms ) BACDBABAEF $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    cleqf.1 $e |- F/_ x A $.
    cleqf.2 $e |- F/_ x B $.
    $( Establish equality between classes, using bound-variable hypotheses
       instead of distinct variable conditions.  See also ~ cleqh .
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       7-Oct-2016.) $)
    cleqf $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
      ( vy wceq cv wcel wb wal dfcleq nfv nfcri nfbi eleq1 bibi12d cbval bitr4i
      ) BCGFHZBIZTCIZJZFKAHZBIZUDCIZJZAKFBCLUGUCAFUGFMUAUBAAFBDNAFCENOUDTGUEUAU
      FUBUDTBPUDTCPQRS $.
  $}

  ${
    abid2f.1 $e |- F/_ x A $.
    $( A simplification of class abstraction.  Theorem 5.2 of [Quine] p. 35.
       (Contributed by NM, 5-Sep-2011.)  (Revised by Mario Carneiro,
       7-Oct-2016.) $)
    abid2f $p |- { x | x e. A } = A $=
      ( cv wcel cab wceq wb wal nfab1 cleqf abid bibi2i albii bitri biid mpgbir
      eqcomi ) BADZBEZAFZBUAGZTTHZAUBTSUAEZHZAIUCAIABUACTAJKUEUCAUDTTTALMNOTPQR
      $.
  $}

  ${
    $d v A $.  $d x z v $.  $d y z v $.  $d v ph $.
    sbabel.1 $e |- F/_ x A $.
    $( Theorem to move a substitution in and out of a class abstraction.
       (Contributed by NM, 27-Sep-2003.)  (Revised by Mario Carneiro,
       7-Oct-2016.) $)
    sbabel $p |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A ) $=
      ( vv cv cab wceq wcel wa wex wsb wb wal sbf abeq2 sbbii 3bitr4i sbex sban
      nfv sbrbis sbalv nfcri anbi12i bitri exbii df-clel ) GHZADIZJZUKEKZLZGMZB
      CNZUKABCNZDIZJZUNLZGMZULEKZBCNUSEKUQUOBCNZGMVBUOGBCUAVDVAGVDUMBCNZUNBCNZL
      VAUMUNBCUBVEUTVFUNDHUKKZAOZDPZBCNVGUROZDPVEUTVHVJBCDVGVGABCVGBCVGBUCQUDUE
      UMVIBCADUKRSURDUKRTUNBCBGEFUFQUGUHUIUHVCUPBCGULEUJSGUSEUJT $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Negated equality and membership
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Negated equality
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c =/= $. $( Not equal to (equal sign with slash through it). $)

  $( Extend wff notation to include inequality. $)
  wne $a wff A =/= B $.

  $( Define inequality.  (Contributed by NM, 5-Aug-1993.) $)
  df-ne $a |- ( A =/= B <-> -. A = B ) $.

  ${
    neii.1 $e |- A =/= B $.
    $( Inference associated with ~ df-ne .  (Contributed by BJ, 7-Jul-2018.) $)
    neii $p |- -. A = B $=
      ( wne wceq wn df-ne mpbi ) ABDABEFCABGH $.
  $}

  ${
    neir.1 $e |- -. A = B $.
    $( Inference associated with ~ df-ne .  (Contributed by BJ, 7-Jul-2018.) $)
    neir $p |- A =/= B $=
      ( wne wceq wn df-ne mpbir ) ABDABEFCABGH $.
  $}

  $( Negation of inequality.  (Contributed by Jim Kingdon, 23-Dec-2018.) $)
  nner $p |- ( A = B -> -. A =/= B ) $=
    ( wne wceq wn df-ne biimpi con2i ) ABCZABDZIJEABFGH $.

  $( Negation of inequality where equality is decidable.  (Contributed by Jim
     Kingdon, 15-May-2018.) $)
  nnedc $p |- ( DECID A = B -> ( -. A =/= B <-> A = B ) ) $=
    ( wceq wdc wne wn wb df-ne a1i con2biidc bicomd ) ABCZDZLABEZFNLNLFGMABHIJK
    $.

  ${
    dcned.eq $e |- ( ph -> DECID A = B ) $.
    $( Decidable equality implies decidable negated equality.  (Contributed by
       Jim Kingdon, 3-May-2020.) $)
    dcned $p |- ( ph -> DECID A =/= B ) $=
      ( wceq wn wdc wne dcn syl df-ne dcbii sylibr ) ABCEZFZGZBCHZGANGPDNIJQOBC
      KLM $.
  $}

  ${
    neqned.1 $e |- ( ph -> -. A = B ) $.
    $( If it is not the case that two classes are equal, they are unequal.
       Converse of ~ neneqd .  One-way deduction form of ~ df-ne .
       (Contributed by David Moews, 28-Feb-2017.)  Allow a shortening of
       ~ necon3bi .  (Revised by Wolf Lammen, 22-Nov-2019.) $)
    neqned $p |- ( ph -> A =/= B ) $=
      ( wceq wn wne df-ne sylibr ) ABCEFBCGDBCHI $.
  $}

  $( From non-equality to inequality.  (Contributed by Glauco Siliprandi,
     11-Dec-2019.) $)
  neqne $p |- ( -. A = B -> A =/= B ) $=
    ( wceq wn id neqned ) ABCDZABGEF $.

  $( No class is unequal to itself.  (Contributed by Stefan O'Rear,
     1-Jan-2015.)  (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
  neirr $p |- -. A =/= A $=
    ( wne wn wceq eqid notnoti df-ne notbii mpbir ) AABZCAADZCZCKAEFJLAAGHI $.

  $( A contradiction concerning equality implies anything.  (Contributed by
     Alexander van der Vekens, 25-Jan-2018.) $)
  eqneqall $p |- ( A = B -> ( A =/= B -> ph ) ) $=
    ( wne wceq wn df-ne pm2.24 biimtrid ) BCDBCEZFJABCGJAHI $.

  $( Decidable equality expressed in terms of ` =/= ` .  Basically the same as
     ~ df-dc .  (Contributed by Jim Kingdon, 14-Mar-2020.) $)
  dcne $p |- ( DECID A = B <-> ( A = B \/ A =/= B ) ) $=
    ( wceq wdc wn wo wne df-dc df-ne orbi2i bitr4i ) ABCZDLLEZFLABGZFLHNMLABIJK
    $.

  $( Law of noncontradiction with equality and inequality.  (Contributed by NM,
     3-Feb-2012.) $)
  nonconne $p |- -. ( A = B /\ A =/= B ) $=
    ( wceq wne wa wn pm3.24 df-ne anbi2i mtbir ) ABCZABDZEKKFZEKGLMKABHIJ $.

  $( Equality theorem for inequality.  (Contributed by NM, 19-Nov-1994.) $)
  neeq1 $p |- ( A = B -> ( A =/= C <-> B =/= C ) ) $=
    ( wceq wn wne eqeq1 notbid df-ne 3bitr4g ) ABDZACDZEBCDZEACFBCFKLMABCGHACIB
    CIJ $.

  $( Equality theorem for inequality.  (Contributed by NM, 19-Nov-1994.) $)
  neeq2 $p |- ( A = B -> ( C =/= A <-> C =/= B ) ) $=
    ( wceq wn wne eqeq2 notbid df-ne 3bitr4g ) ABDZCADZECBDZECAFCBFKLMABCGHCAIC
    BIJ $.

  ${
    neeq1i.1 $e |- A = B $.
    $( Inference for inequality.  (Contributed by NM, 29-Apr-2005.) $)
    neeq1i $p |- ( A =/= C <-> B =/= C ) $=
      ( wceq wne wb neeq1 ax-mp ) ABEACFBCFGDABCHI $.

    $( Inference for inequality.  (Contributed by NM, 29-Apr-2005.) $)
    neeq2i $p |- ( C =/= A <-> C =/= B ) $=
      ( wceq wne wb neeq2 ax-mp ) ABECAFCBFGDABCHI $.

    neeq12i.2 $e |- C = D $.
    $( Inference for inequality.  (Contributed by NM, 24-Jul-2012.) $)
    neeq12i $p |- ( A =/= C <-> B =/= D ) $=
      ( wne neeq2i neeq1i bitri ) ACGADGBDGCDAFHABDEIJ $.
  $}

  ${
    neeq1d.1 $e |- ( ph -> A = B ) $.
    $( Deduction for inequality.  (Contributed by NM, 25-Oct-1999.) $)
    neeq1d $p |- ( ph -> ( A =/= C <-> B =/= C ) ) $=
      ( wceq wne wb neeq1 syl ) ABCFBDGCDGHEBCDIJ $.

    $( Deduction for inequality.  (Contributed by NM, 25-Oct-1999.) $)
    neeq2d $p |- ( ph -> ( C =/= A <-> C =/= B ) ) $=
      ( wceq wne wb neeq2 syl ) ABCFDBGDCGHEBCDIJ $.

    neeq12d.2 $e |- ( ph -> C = D ) $.
    $( Deduction for inequality.  (Contributed by NM, 24-Jul-2012.) $)
    neeq12d $p |- ( ph -> ( A =/= C <-> B =/= D ) ) $=
      ( wne neeq1d neeq2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  ${
    neneqd.1 $e |- ( ph -> A =/= B ) $.
    $( Deduction eliminating inequality definition.  (Contributed by Jonathan
       Ben-Naim, 3-Jun-2011.) $)
    neneqd $p |- ( ph -> -. A = B ) $=
      ( wne wceq wn df-ne sylib ) ABCEBCFGDBCHI $.
  $}

  $( From inequality to non-equality.  (Contributed by Glauco Siliprandi,
     11-Dec-2019.) $)
  neneq $p |- ( A =/= B -> -. A = B ) $=
    ( wne id neneqd ) ABCZABFDE $.

  ${
    eqnetr.1 $e |- A = B $.
    eqnetr.2 $e |- B =/= C $.
    $( Substitution of equal classes into an inequality.  (Contributed by NM,
       4-Jul-2012.) $)
    eqnetri $p |- A =/= C $=
      ( wne neeq1i mpbir ) ACFBCFEABCDGH $.
  $}

  ${
    eqnetrd.1 $e |- ( ph -> A = B ) $.
    eqnetrd.2 $e |- ( ph -> B =/= C ) $.
    $( Substitution of equal classes into an inequality.  (Contributed by NM,
       4-Jul-2012.) $)
    eqnetrd $p |- ( ph -> A =/= C ) $=
      ( wne neeq1d mpbird ) ABDGCDGFABCDEHI $.
  $}

  ${
    eqnetrr.1 $e |- A = B $.
    eqnetrr.2 $e |- A =/= C $.
    $( Substitution of equal classes into an inequality.  (Contributed by NM,
       4-Jul-2012.) $)
    eqnetrri $p |- B =/= C $=
      ( eqcomi eqnetri ) BACABDFEG $.
  $}

  ${
    eqnetrrd.1 $e |- ( ph -> A = B ) $.
    eqnetrrd.2 $e |- ( ph -> A =/= C ) $.
    $( Substitution of equal classes into an inequality.  (Contributed by NM,
       4-Jul-2012.) $)
    eqnetrrd $p |- ( ph -> B =/= C ) $=
      ( eqcomd eqnetrd ) ACBDABCEGFH $.
  $}

  ${
    neeqtr.1 $e |- A =/= B $.
    neeqtr.2 $e |- B = C $.
    $( Substitution of equal classes into an inequality.  (Contributed by NM,
       4-Jul-2012.) $)
    neeqtri $p |- A =/= C $=
      ( wne neeq2i mpbi ) ABFACFDBCAEGH $.
  $}

  ${
    neeqtrd.1 $e |- ( ph -> A =/= B ) $.
    neeqtrd.2 $e |- ( ph -> B = C ) $.
    $( Substitution of equal classes into an inequality.  (Contributed by NM,
       4-Jul-2012.) $)
    neeqtrd $p |- ( ph -> A =/= C ) $=
      ( wne neeq2d mpbid ) ABCGBDGEACDBFHI $.
  $}

  ${
    neeqtrr.1 $e |- A =/= B $.
    neeqtrr.2 $e |- C = B $.
    $( Substitution of equal classes into an inequality.  (Contributed by NM,
       4-Jul-2012.) $)
    neeqtrri $p |- A =/= C $=
      ( eqcomi neeqtri ) ABCDCBEFG $.
  $}

  ${
    neeqtrrd.1 $e |- ( ph -> A =/= B ) $.
    neeqtrrd.2 $e |- ( ph -> C = B ) $.
    $( Substitution of equal classes into an inequality.  (Contributed by NM,
       4-Jul-2012.) $)
    neeqtrrd $p |- ( ph -> A =/= C ) $=
      ( eqcomd neeqtrd ) ABCDEADCFGH $.
  $}

  ${
    eqnetrrid.1 $e |- B = A $.
    eqnetrrid.2 $e |- ( ph -> B =/= C ) $.
    $( B chained equality inference for inequality.  (Contributed by NM,
       6-Jun-2012.) $)
    eqnetrrid $p |- ( ph -> A =/= C ) $=
      ( wne neeq1i sylib ) ACDGBDGFCBDEHI $.
  $}

  ${
    3netr3d.1 $e |- ( ph -> A =/= B ) $.
    3netr3d.2 $e |- ( ph -> A = C ) $.
    3netr3d.3 $e |- ( ph -> B = D ) $.
    $( Substitution of equality into both sides of an inequality.  (Contributed
       by NM, 24-Jul-2012.) $)
    3netr3d $p |- ( ph -> C =/= D ) $=
      ( wne neeq12d mpbid ) ABCIDEIFABDCEGHJK $.
  $}

  ${
    3netr4d.1 $e |- ( ph -> A =/= B ) $.
    3netr4d.2 $e |- ( ph -> C = A ) $.
    3netr4d.3 $e |- ( ph -> D = B ) $.
    $( Substitution of equality into both sides of an inequality.  (Contributed
       by NM, 24-Jul-2012.) $)
    3netr4d $p |- ( ph -> C =/= D ) $=
      ( wne neeq12d mpbird ) ADEIBCIFADBECGHJK $.
  $}

  ${
    3netr3g.1 $e |- ( ph -> A =/= B ) $.
    3netr3g.2 $e |- A = C $.
    3netr3g.3 $e |- B = D $.
    $( Substitution of equality into both sides of an inequality.  (Contributed
       by NM, 24-Jul-2012.) $)
    3netr3g $p |- ( ph -> C =/= D ) $=
      ( wne neeq12i sylib ) ABCIDEIFBDCEGHJK $.
  $}

  ${
    3netr4g.1 $e |- ( ph -> A =/= B ) $.
    3netr4g.2 $e |- C = A $.
    3netr4g.3 $e |- D = B $.
    $( Substitution of equality into both sides of an inequality.  (Contributed
       by NM, 14-Jun-2012.) $)
    3netr4g $p |- ( ph -> C =/= D ) $=
      ( wne neeq12i sylibr ) ABCIDEIFDBECGHJK $.
  $}

  ${
    necon3abii.1 $e |- ( A = B <-> ph ) $.
    $( Deduction from equality to inequality.  (Contributed by NM,
       9-Nov-2007.) $)
    necon3abii $p |- ( A =/= B <-> -. ph ) $=
      ( wne wceq df-ne xchbinx ) BCEBCFABCGDH $.
  $}

  ${
    necon3bbii.1 $e |- ( ph <-> A = B ) $.
    $( Deduction from equality to inequality.  (Contributed by NM,
       13-Apr-2007.) $)
    necon3bbii $p |- ( -. ph <-> A =/= B ) $=
      ( wne wn wceq bicomi necon3abii ) BCEAFABCABCGDHIH $.
  $}

  ${
    necon3bii.1 $e |- ( A = B <-> C = D ) $.
    $( Inference from equality to inequality.  (Contributed by NM,
       23-Feb-2005.) $)
    necon3bii $p |- ( A =/= B <-> C =/= D ) $=
      ( wne wceq wn necon3abii df-ne bitr4i ) ABFCDGZHCDFLABEICDJK $.
  $}

  ${
    necon3abid.1 $e |- ( ph -> ( A = B <-> ps ) ) $.
    $( Deduction from equality to inequality.  (Contributed by NM,
       21-Mar-2007.) $)
    necon3abid $p |- ( ph -> ( A =/= B <-> -. ps ) ) $=
      ( wne wceq wn df-ne notbid bitrid ) CDFCDGZHABHCDIALBEJK $.
  $}

  ${
    necon3bbid.1 $e |- ( ph -> ( ps <-> A = B ) ) $.
    $( Deduction from equality to inequality.  (Contributed by NM,
       2-Jun-2007.) $)
    necon3bbid $p |- ( ph -> ( -. ps <-> A =/= B ) ) $=
      ( wne wn wceq bicomd necon3abid ) ACDFBGABCDABCDHEIJI $.
  $}

  ${
    necon3bid.1 $e |- ( ph -> ( A = B <-> C = D ) ) $.
    $( Deduction from equality to inequality.  (Contributed by NM,
       23-Feb-2005.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    necon3bid $p |- ( ph -> ( A =/= B <-> C =/= D ) ) $=
      ( wne wceq wn df-ne necon3bbid bitrid ) BCGBCHZIADEGBCJAMDEFKL $.
  $}

  ${
    necon3ad.1 $e |- ( ph -> ( ps -> A = B ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       2-Apr-2007.)  (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
    necon3ad $p |- ( ph -> ( A =/= B -> -. ps ) ) $=
      ( wne wceq wn df-ne con3d biimtrid ) CDFCDGZHABHCDIABLEJK $.
  $}

  ${
    necon3bd.1 $e |- ( ph -> ( A = B -> ps ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       2-Apr-2007.)  (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
    necon3bd $p |- ( ph -> ( -. ps -> A =/= B ) ) $=
      ( wn wceq wne con3d df-ne imbitrrdi ) ABFCDGZFCDHALBEICDJK $.
  $}

  ${
    necon3d.1 $e |- ( ph -> ( A = B -> C = D ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       10-Jun-2006.) $)
    necon3d $p |- ( ph -> ( C =/= D -> A =/= B ) ) $=
      ( wne wceq wn necon3ad df-ne imbitrrdi ) ADEGBCHZIBCGAMDEFJBCKL $.
  $}

  $( Characterization of inequality in terms of reversed equality (see
     ~ bicom ).  (Contributed by BJ, 7-Jul-2018.) $)
  nesym $p |- ( A =/= B <-> -. B = A ) $=
    ( wceq eqcom necon3abii ) BACABABDE $.

  ${
    nesymi.1 $e |- A =/= B $.
    $( Inference associated with ~ nesym .  (Contributed by BJ, 7-Jul-2018.) $)
    nesymi $p |- -. B = A $=
      ( wne wceq wn nesym mpbi ) ABDBAEFCABGH $.
  $}

  ${
    nesymir.1 $e |- -. A = B $.
    $( Inference associated with ~ nesym .  (Contributed by BJ, 7-Jul-2018.) $)
    nesymir $p |- B =/= A $=
      ( wne wceq wn nesym mpbir ) BADABEFCBAGH $.
  $}

  ${
    necon3i.1 $e |- ( A = B -> C = D ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       9-Aug-2006.) $)
    necon3i $p |- ( C =/= D -> A =/= B ) $=
      ( wceq wi wne id necon3d ax-mp ) ABFCDFGZCDHABHGELABCDLIJK $.
  $}

  ${
    necon3ai.1 $e |- ( ph -> A = B ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       23-May-2007.)  (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
    necon3ai $p |- ( A =/= B -> -. ph ) $=
      ( wne wceq wn df-ne con3i sylbi ) BCEBCFZGAGBCHAKDIJ $.
  $}

  ${
    necon3bi.1 $e |- ( A = B -> ph ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       1-Jun-2007.)  (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
    necon3bi $p |- ( -. ph -> A =/= B ) $=
      ( wn wceq wne con3i df-ne sylibr ) AEBCFZEBCGKADHBCIJ $.
  $}

  ${
    necon1aidc.1 $e |- ( DECID ph -> ( -. ph -> A = B ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       15-May-2018.) $)
    necon1aidc $p |- ( DECID ph -> ( A =/= B -> ph ) ) $=
      ( wne wceq wn wdc df-ne wi con1dc mpd biimtrid ) BCEBCFZGZAHZABCIPAGNJOAJ
      DANKLM $.
  $}

  ${
    necon1bidc.1 $e |- ( DECID A = B -> ( A =/= B -> ph ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       15-May-2018.) $)
    necon1bidc $p |- ( DECID A = B -> ( -. ph -> A = B ) ) $=
      ( wceq wdc wn wi wne df-ne biimtrrid con1dc mpd ) BCEZFZNGZAHAGNHPBCIOABC
      JDKNALM $.
  $}

  ${
    necon1idc.1 $e |- ( A =/= B -> C = D ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon1idc $p |- ( DECID A = B -> ( C =/= D -> A = B ) ) $=
      ( wceq wn wi wdc wne df-ne sylbir a1i necon1aidc ) ABFZCDOGZCDFZHOIPABJQA
      BKELMN $.
  $}

  ${
    necon2ai.1 $e |- ( A = B -> -. ph ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       16-Jan-2007.)  (Proof rewritten by Jim Kingdon, 16-May-2018.) $)
    necon2ai $p |- ( ph -> A =/= B ) $=
      ( wceq wn wne con2i df-ne sylibr ) ABCEZFBCGKADHBCIJ $.
  $}

  ${
    necon2bi.1 $e |- ( ph -> A =/= B ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       1-Apr-2007.) $)
    necon2bi $p |- ( A = B -> -. ph ) $=
      ( wceq neneqd con2i ) ABCEABCDFG $.
  $}

  ${
    necon2i.1 $e |- ( A = B -> C =/= D ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       18-Mar-2007.) $)
    necon2i $p |- ( C = D -> A =/= B ) $=
      ( wceq neneqd necon2ai ) CDFABABFCDEGH $.
  $}

  ${
    necon2ad.1 $e |- ( ph -> ( A = B -> -. ps ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       19-Apr-2007.)  (Proof rewritten by Jim Kingdon, 16-May-2018.) $)
    necon2ad $p |- ( ph -> ( ps -> A =/= B ) ) $=
      ( wceq wn wne con2d df-ne imbitrrdi ) ABCDFZGCDHALBEICDJK $.
  $}

  ${
    necon2bd.1 $e |- ( ph -> ( ps -> A =/= B ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       13-Apr-2007.) $)
    necon2bd $p |- ( ph -> ( A = B -> -. ps ) ) $=
      ( wceq wne wn df-ne imbitrdi con2d ) ABCDFZABCDGLHECDIJK $.
  $}

  ${
    necon2d.1 $e |- ( ph -> ( A = B -> C =/= D ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       28-Dec-2008.) $)
    necon2d $p |- ( ph -> ( C = D -> A =/= B ) ) $=
      ( wceq wne wn df-ne imbitrdi necon2ad ) ADEGZBCABCGDEHMIFDEJKL $.
  $}

  ${
    necon1abiidc.1 $e |- ( DECID ph -> ( -. ph <-> A = B ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon1abiidc $p |- ( DECID ph -> ( A =/= B <-> ph ) ) $=
      ( wne wceq wn wdc df-ne con1biidc bitrid ) BCEBCFZGAHABCIALDJK $.
  $}

  ${
    necon1bbiidc.1 $e |- ( DECID A = B -> ( A =/= B <-> ph ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon1bbiidc $p |- ( DECID A = B -> ( -. ph <-> A = B ) ) $=
      ( wceq wn wne wdc df-ne bitr3id con1biidc ) BCEZALFBCGLHABCIDJK $.
  $}

  ${
    necon1abiddc.1 $e |- ( ph -> ( DECID ps -> ( -. ps <-> A = B ) ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon1abiddc $p |- ( ph -> ( DECID ps -> ( A =/= B <-> ps ) ) ) $=
      ( wdc wceq wn wb wne con1biddc df-ne bibi1i imbitrrdi ) ABFCDGZHZBICDJZBI
      ABOEKQPBCDLMN $.
  $}

  ${
    necon1bbiddc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B <-> ps ) ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon1bbiddc $p |- ( ph -> ( DECID A = B -> ( -. ps <-> A = B ) ) ) $=
      ( wceq wdc wne wb wn df-ne bibi1i imbitrdi con1biddc ) ACDFZBAOGCDHZBIOJZ
      BIEPQBCDKLMN $.
  $}

  ${
    necon2abiidc.1 $e |- ( DECID ph -> ( A = B <-> -. ph ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon2abiidc $p |- ( DECID ph -> ( ph <-> A =/= B ) ) $=
      ( wdc wne wceq wn bicomd necon1abiidc ) AEZBCFAABCKBCGAHDIJI $.
  $}

  ${
    necon2bbii.1 $e |- ( DECID A = B -> ( ph <-> A =/= B ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon2bbiidc $p |- ( DECID A = B -> ( A = B <-> -. ph ) ) $=
      ( wceq wdc wn wne bicomd necon1bbiidc ) BCEZFZAGKABCLABCHDIJI $.
  $}

  ${
    necon2abiddc.1 $e |- ( ph -> ( DECID ps -> ( A = B <-> -. ps ) ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon2abiddc $p |- ( ph -> ( DECID ps -> ( ps <-> A =/= B ) ) ) $=
      ( wdc wne wb wceq wn bicom imbitrdi necon1abiddc ) ABFZCDGZBHBOHABCDANCDI
      ZBJZHQPHEPQKLMOBKL $.
  $}

  ${
    necon2bbiddc.1 $e |- ( ph -> ( DECID A = B -> ( ps <-> A =/= B ) ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon2bbiddc $p |- ( ph -> ( DECID A = B -> ( A = B <-> -. ps ) ) ) $=
      ( wceq wdc wn wb wne bicom imbitrdi necon1bbiddc ) ACDFZGZBHZNINPIABCDAOB
      CDJZIQBIEBQKLMPNKL $.
  $}

  ${
    necon4aidc.1 $e |- ( DECID A = B -> ( A =/= B -> -. ph ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon4aidc $p |- ( DECID A = B -> ( ph -> A = B ) ) $=
      ( wceq wdc wn wi wne df-ne biimtrrid condc mpd ) BCEZFZNGZAGZHANHPBCIOQBC
      JDKNALM $.
  $}

  ${
    necon4idc.1 $e |- ( DECID A = B -> ( A =/= B -> C =/= D ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       16-May-2018.) $)
    necon4idc $p |- ( DECID A = B -> ( C = D -> A = B ) ) $=
      ( wceq wdc wne wn df-ne imbitrdi necon4aidc ) CDFZABABFGABHCDHMIECDJKL $.
  $}

  ${
    necon4addc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B -> -. ps ) ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       17-May-2018.) $)
    necon4addc $p |- ( ph -> ( DECID A = B -> ( ps -> A = B ) ) ) $=
      ( wceq wdc wne wn wi df-ne imbi1i condc biimtrid sylcom ) ACDFZGZCDHZBIZJ
      ZBPJZETPIZSJQUARUBSCDKLPBMNO $.
  $}

  ${
    necon4bddc.1 $e |- ( ph -> ( DECID ps -> ( -. ps -> A =/= B ) ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       17-May-2018.) $)
    necon4bddc $p |- ( ph -> ( DECID ps -> ( A = B -> ps ) ) ) $=
      ( wdc wn wceq wi wne df-ne syl8ib condc sylcom ) ABFZBGZCDHZGZIQBIAOPCDJR
      ECDKLBQMN $.
  $}

  ${
    necon4ddc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B -> C =/= D ) ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by Jim Kingdon,
       17-May-2018.) $)
    necon4ddc $p |- ( ph -> ( DECID A = B -> ( C = D -> A = B ) ) ) $=
      ( wceq wdc wn wi wne df-ne imbi12i imbitrdi condc sylcom ) ABCGZHZQIZDEGZ
      IZJZTQJARBCKZDEKZJUBFUCSUDUABCLDELMNQTOP $.
  $}

  ${
    necon4abiddc.1 $e |- ( ph -> ( DECID A = B -> ( DECID ps ->
        ( A =/= B <-> -. ps ) ) ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by Jim
       Kingdon, 18-May-2018.) $)
    necon4abiddc $p |- ( ph -> ( DECID A = B -> ( DECID ps ->
        ( A = B <-> ps ) ) ) ) $=
      ( wceq wdc wne wn wb df-ne bibi1i syl8ib con4biddc ) ACDFZBAOGBGCDHZBIZJO
      IZQJEPRQCDKLMN $.
  $}

  ${
    necon4bbiddc.1 $e |- ( ph -> ( DECID ps -> ( DECID A = B ->
        ( -. ps <-> A =/= B ) ) ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by Jim
       Kingdon, 19-May-2018.) $)
    necon4bbiddc $p |- ( ph -> ( DECID ps -> ( DECID A = B ->
        ( ps <-> A = B ) ) ) ) $=
      ( wdc wceq wb wne wn bicom syl8ib com23 necon4abiddc ) ABFZCDGZFZPBHZBPHA
      QORABCDAOQCDIZBJZHZAOQTSHUAETSKLMNMPBKL $.
  $}

  ${
    necon4biddc.1 $e |- ( ph -> ( DECID A = B -> ( DECID C = D ->
        ( A =/= B <-> C =/= D ) ) ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by Jim
       Kingdon, 19-May-2018.) $)
    necon4biddc $p |- ( ph -> ( DECID A = B -> ( DECID C = D ->
        ( A = B <-> C = D ) ) ) ) $=
      ( wceq wdc wne wb wn df-ne bibi2i syl8ib necon4abiddc ) ADEGZBCABCGHPHBCI
      ZDEIZJQPKZJFRSQDELMNO $.
  $}

  ${
    necon1addc.1 $e |- ( ph -> ( DECID ps -> ( -. ps -> A = B ) ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by Jim Kingdon,
       19-May-2018.) $)
    necon1addc $p |- ( ph -> ( DECID ps -> ( A =/= B -> ps ) ) ) $=
      ( wne wceq wn wdc df-ne wi con1dc sylcom syl7bi ) CDFCDGZHZABIZBCDJAQBHOK
      PBKEBOLMN $.
  $}

  ${
    necon1bddc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B -> ps ) ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by Jim Kingdon,
       19-May-2018.) $)
    necon1bddc $p |- ( ph -> ( DECID A = B -> ( -. ps -> A = B ) ) ) $=
      ( wceq wdc wn wi wne df-ne imbi1i imbitrdi con1dc sylcom ) ACDFZGZPHZBIZB
      HPIAQCDJZBISETRBCDKLMPBNO $.
  $}

  ${
    necon1ddc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B -> C = D ) ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by Jim
       Kingdon, 19-May-2018.) $)
    necon1ddc $p |- ( ph -> ( DECID A = B -> ( C =/= D -> A = B ) ) ) $=
      ( wne wceq wn wdc df-ne necon1bddc syl7bi ) DEGDEHZIABCHZJODEKANBCFLM $.
  $}

  ${
    neneqad.1 $e |- ( ph -> -. A = B ) $.
    $( If it is not the case that two classes are equal, they are unequal.
       Converse of ~ neneqd .  One-way deduction form of ~ df-ne .
       (Contributed by David Moews, 28-Feb-2017.) $)
    neneqad $p |- ( ph -> A =/= B ) $=
      ( wceq con2i necon2ai ) ABCABCEDFG $.
  $}

  $( Contraposition law for inequality.  (Contributed by Jim Kingdon,
     19-May-2018.) $)
  nebidc $p |- ( DECID A = B -> ( DECID C = D ->
      ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) ) ) ) $=
    ( wceq wdc wb wne wa id necon3bid wi a1d necon4biddc com3l imp impbid2 ex )
    ABEZFZCDEZFZSUAGZABHCDHGZGTUBIUCUDUCABCDUCJKTUBUDUCLUDTUBUCUDABCDUDUBUDLTUD
    UDUBUDJMMNOPQR $.

  $( Theorem *13.18 in [WhiteheadRussell] p. 178.  (Contributed by Andrew
     Salmon, 3-Jun-2011.) $)
  pm13.18 $p |- ( ( A = B /\ A =/= C ) -> B =/= C ) $=
    ( wceq wne eqeq1 biimprd necon3d imp ) ABDZACEBCEJBCACJACDBCDABCFGHI $.

  $( Theorem *13.181 in [WhiteheadRussell] p. 178.  (Contributed by Andrew
     Salmon, 3-Jun-2011.) $)
  pm13.181 $p |- ( ( A = B /\ B =/= C ) -> A =/= C ) $=
    ( wceq wne eqcom pm13.18 sylanb ) ABDBADBCEACEABFBACGH $.

  ${
    pm2.21ddne.1 $e |- ( ph -> A = B ) $.
    pm2.21ddne.2 $e |- ( ph -> A =/= B ) $.
    $( A contradiction implies anything.  Equality/inequality deduction form.
       (Contributed by David Moews, 28-Feb-2017.) $)
    pm2.21ddne $p |- ( ph -> ps ) $=
      ( wceq neneqd pm2.21dd ) ACDGBEACDFHI $.
  $}

  $( Commutation of inequality.  (Contributed by NM, 14-May-1999.) $)
  necom $p |- ( A =/= B <-> B =/= A ) $=
    ( eqcom necon3bii ) ABBAABCD $.

  ${
    necomi.1 $e |- A =/= B $.
    $( Inference from commutative law for inequality.  (Contributed by NM,
       17-Oct-2012.) $)
    necomi $p |- B =/= A $=
      ( wne necom mpbi ) ABDBADCABEF $.
  $}

  ${
    necomd.1 $e |- ( ph -> A =/= B ) $.
    $( Deduction from commutative law for inequality.  (Contributed by NM,
       12-Feb-2008.) $)
    necomd $p |- ( ph -> B =/= A ) $=
      ( wne necom sylib ) ABCECBEDBCFG $.
  $}

  $( A De Morgan's law for inequality.  (Contributed by NM, 18-May-2007.) $)
  neanior $p |- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) ) $=
    ( wne wa wceq wn wo df-ne anbi12i pm4.56 bitri ) ABEZCDEZFABGZHZCDGZHZFPRIH
    NQOSABJCDJKPRLM $.

  $( A De Morgan's law for inequality.  (Contributed by NM, 30-Sep-2013.)
     (Proof rewritten by Jim Kingdon, 19-May-2018.) $)
  ne3anior $p |- ( ( A =/= B /\ C =/= D /\ E =/= F )
        <-> -. ( A = B \/ C = D \/ E = F ) ) $=
    ( wne w3a wceq wn w3o df-ne 3anbi123i 3ioran bitr4i ) ABGZCDGZEFGZHABIZJZCD
    IZJZEFIZJZHSUAUCKJPTQUBRUDABLCDLEFLMSUAUCNO $.

  ${
    nemtbir.1 $e |- A =/= B $.
    nemtbir.2 $e |- ( ph <-> A = B ) $.
    $( An inference from an inequality, related to modus tollens.  (Contributed
       by NM, 13-Apr-2007.) $)
    nemtbir $p |- -. ph $=
      ( wceq wne wn df-ne mpbi mtbir ) ABCFZBCGLHDBCIJEK $.
  $}

  $( Two classes are different if they don't contain the same element.
     (Contributed by NM, 3-Feb-2012.) $)
  nelne1 $p |- ( ( A e. B /\ -. A e. C ) -> B =/= C ) $=
    ( wcel wn wne wceq eleq2 biimpcd necon3bd imp ) ABDZACDZEBCFLMBCBCGLMBCAHIJ
    K $.

  $( Two classes are different if they don't belong to the same class.
     (Contributed by NM, 25-Jun-2012.) $)
  nelne2 $p |- ( ( A e. C /\ -. B e. C ) -> A =/= B ) $=
    ( wcel wn wne wceq eleq1 biimpcd necon3bd imp ) ACDZBCDZEABFLMABABGLMABCHIJ
    K $.

  $( Two classes are different if they don't belong to the same class.
     (Contributed by Rodolfo Medina, 17-Oct-2010.)  (Proof shortened by AV,
     10-May-2020.) $)
  nelelne $p |- ( -. A e. B -> ( C e. B -> C =/= A ) ) $=
    ( wcel wn wne nelne2 expcom ) CBDABDECAFCABGH $.

  ${
    nfne.1 $e |- F/_ x A $.
    nfne.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for inequality.  (Contributed by NM,
       10-Nov-2007.)  (Revised by Mario Carneiro, 7-Oct-2016.) $)
    nfne $p |- F/ x A =/= B $=
      ( wne wceq wn df-ne nfeq nfn nfxfr ) BCFBCGZHABCIMAABCDEJKL $.
  $}

  ${
    nfned.1 $e |- ( ph -> F/_ x A ) $.
    nfned.2 $e |- ( ph -> F/_ x B ) $.
    $( Bound-variable hypothesis builder for inequality.  (Contributed by NM,
       10-Nov-2007.)  (Revised by Mario Carneiro, 7-Oct-2016.) $)
    nfned $p |- ( ph -> F/ x A =/= B ) $=
      ( wne wceq wn df-ne nfeqd nfnd nfxfrd ) CDGCDHZIABCDJANBABCDEFKLM $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Negated membership
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c e/ $. $( Not an element of (epsilon with slash through it). $)

  $( Extend wff notation to include negated membership. $)
  wnel $a wff A e/ B $.

  $( Define negated membership.  (Contributed by NM, 7-Aug-1994.) $)
  df-nel $a |- ( A e/ B <-> -. A e. B ) $.

  ${
    neli.1 $e |- A e/ B $.
    $( Inference associated with ~ df-nel .  (Contributed by BJ,
       7-Jul-2018.) $)
    neli $p |- -. A e. B $=
      ( wnel wcel wn df-nel mpbi ) ABDABEFCABGH $.
  $}

  ${
    nelir.1 $e |- -. A e. B $.
    $( Inference associated with ~ df-nel .  (Contributed by BJ,
       7-Jul-2018.) $)
    nelir $p |- A e/ B $=
      ( wnel wcel wn df-nel mpbir ) ABDABEFCABGH $.
  $}

  $( Equality theorem for negated membership.  (Contributed by NM,
     20-Nov-1994.) $)
  neleq1 $p |- ( A = B -> ( A e/ C <-> B e/ C ) ) $=
    ( wceq wcel wn wnel eleq1 notbid df-nel 3bitr4g ) ABDZACEZFBCEZFACGBCGLMNAB
    CHIACJBCJK $.

  $( Equality theorem for negated membership.  (Contributed by NM,
     20-Nov-1994.) $)
  neleq2 $p |- ( A = B -> ( C e/ A <-> C e/ B ) ) $=
    ( wceq wcel wn wnel eleq2 notbid df-nel 3bitr4g ) ABDZCAEZFCBEZFCAGCBGLMNAB
    CHICAJCBJK $.

  ${
    neleq12d.1 $e |- ( ph -> A = B ) $.
    neleq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality theorem for negated membership.  (Contributed by FL,
       10-Aug-2016.) $)
    neleq12d $p |- ( ph -> ( A e/ C <-> B e/ D ) ) $=
      ( wnel wceq wb neleq1 syl neleq2 bitrd ) ABDHZCDHZCEHZABCIOPJFBCDKLADEIPQ
      JGDECMLN $.
  $}

  ${
    nfnel.1 $e |- F/_ x A $.
    nfnel.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for negated membership.  (Contributed
       by David Abernethy, 26-Jun-2011.)  (Revised by Mario Carneiro,
       7-Oct-2016.) $)
    nfnel $p |- F/ x A e/ B $=
      ( wnel wcel wn df-nel nfel nfn nfxfr ) BCFBCGZHABCIMAABCDEJKL $.
  $}

  ${
    nfneld.1 $e |- ( ph -> F/_ x A ) $.
    nfneld.2 $e |- ( ph -> F/_ x B ) $.
    $( Bound-variable hypothesis builder for negated membership.  (Contributed
       by David Abernethy, 26-Jun-2011.)  (Revised by Mario Carneiro,
       7-Oct-2016.) $)
    nfneld $p |- ( ph -> F/ x A e/ B ) $=
      ( wnel wcel wn df-nel nfeld nfnd nfxfrd ) CDGCDHZIABCDJANBABCDEFKLM $.
  $}

  $( Two classes are different if they don't contain the same element.
     (Contributed by AV, 28-Jan-2020.) $)
  elnelne1 $p |- ( ( A e. B /\ A e/ C ) -> B =/= C ) $=
    ( wnel wcel wn wne df-nel nelne1 sylan2b ) ACDABEACEFBCGACHABCIJ $.

  $( Two classes are different if they don't belong to the same class.
     (Contributed by AV, 28-Jan-2020.) $)
  elnelne2 $p |- ( ( A e. C /\ B e/ C ) -> A =/= B ) $=
    ( wnel wcel wn wne df-nel nelne2 sylan2b ) BCDACEBCEFABGBCHABCIJ $.

  ${
    nelcon3d.1 $e |- ( ph -> ( A e. B -> C e. D ) ) $.
    $( Contrapositive law deduction for negated membership.  (Contributed by
       AV, 28-Jan-2020.) $)
    nelcon3d $p |- ( ph -> ( C e/ D -> A e/ B ) ) $=
      ( wcel wn wnel con3d df-nel 3imtr4g ) ADEGZHBCGZHDEIBCIANMFJDEKBCKL $.
  $}

  $( A contradiction concerning membership implies anything.  (Contributed by
     Alexander van der Vekens, 25-Jan-2018.) $)
  elnelall $p |- ( A e. B -> ( A e/ B -> ph ) ) $=
    ( wnel wcel wn df-nel pm2.24 biimtrid ) BCDBCEZFJABCGJAHI $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Restricted quantification
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Extend wff notation to include restricted universal quantification. $)
  wral $a wff A. x e. A ph $.

  $( Extend wff notation to include restricted existential quantification. $)
  wrex $a wff E. x e. A ph $.

  $( Extend wff notation to include restricted existential uniqueness. $)
  wreu $a wff E! x e. A ph $.

  $( Extend wff notation to include restricted "at most one". $)
  wrmo $a wff E* x e. A ph $.

  $( Extend class notation to include the restricted class abstraction (class
     builder). $)
  crab $a class { x e. A | ph } $.

  $( Define restricted universal quantification.  Special case of Definition
     4.15(3) of [TakeutiZaring] p. 22.  (Contributed by NM, 19-Aug-1993.) $)
  df-ral $a |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) ) $.

  $( Define restricted existential quantification.  Special case of Definition
     4.15(4) of [TakeutiZaring] p. 22.  (Contributed by NM, 30-Aug-1993.) $)
  df-rex $a |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) ) $.

  $( Define restricted existential uniqueness.  (Contributed by NM,
     22-Nov-1994.) $)
  df-reu $a |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) ) $.

  $( Define restricted "at most one".  (Contributed by NM, 16-Jun-2017.) $)
  df-rmo $a |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) ) $.

  $( Define a restricted class abstraction (class builder), which is the class
     of all ` x ` in ` A ` such that ` ph ` is true.  Definition of
     [TakeutiZaring] p. 20.  (Contributed by NM, 22-Nov-1994.) $)
  df-rab $a |- { x e. A | ph } = { x | ( x e. A /\ ph ) } $.

  $( Relationship between restricted universal and existential quantifiers.
     (Contributed by NM, 21-Jan-1997.) $)
  ralnex $p |- ( A. x e. A -. ph <-> -. E. x e. A ph ) $=
    ( wn wral cv wcel wi wal wrex df-ral wa wex alinexa df-rex xchbinxr bitri )
    ADZBCEBFCGZRHBIZABCJZDRBCKTSALBMUASABNABCOPQ $.

  $( Relationship between restricted universal and existential quantifiers.  In
     classical logic this would be a biconditional.  (Contributed by Jim
     Kingdon, 17-Aug-2018.) $)
  rexnalim $p |- ( E. x e. A -. ph -> -. A. x e. A ph ) $=
    ( wn wrex cv wcel wa wex wral df-rex wi wal exanaliim df-ral sylnibr sylbi
    ) ADZBCEBFCGZRHBIZABCJZDRBCKTSALBMUASABNABCOPQ $.

  $( The double negation of a universal quantification implies the universal
     quantification of the double negation.  Restricted quantifier version of
     ~ nnal .  (Contributed by Jim Kingdon, 1-Aug-2024.) $)
  nnral $p |- ( -. -. A. x e. A ph -> A. x e. A -. -. ph ) $=
    ( wral wn wrex rexnalim con3i ralnex sylibr ) ABCDEZEAEZBCFZELEBCDMKABCGHLB
    CIJ $.

  $( Relationship between restricted universal and existential quantifiers.
     (Contributed by Jim Kingdon, 29-Jun-2022.) $)
  dfrex2dc $p |- ( DECID E. x e. A ph
      -> ( E. x e. A ph <-> -. A. x e. A -. ph ) ) $=
    ( wrex wdc cv wcel wa wex wn wal wral wb df-rex dcbii dfexdc sylbi wi imnan
    df-ral albii bitri notbii 3bitr4g ) ABCDZEZBFCGZAHZBIZUHJZBKZJZUEAJZBCLZJUF
    UIEUIULMUEUIABCNZOUHBPQUOUNUKUNUGUMRZBKUKUMBCTUPUJBUGASUAUBUCUD $.

  $( Relationship between restricted universal and existential quantifiers.
     (Contributed by Jim Kingdon, 17-Aug-2018.) $)
  ralexim $p |- ( A. x e. A ph -> -. E. x e. A -. ph ) $=
    ( wn wrex wral rexnalim con2i ) ADBCEABCFABCGH $.

  $( Relationship between restricted universal and existential quantifiers.
     (Contributed by Jim Kingdon, 17-Aug-2018.) $)
  rexalim $p |- ( E. x e. A ph -> -. A. x e. A -. ph ) $=
    ( wn wral wrex ralnex biimpi con2i ) ADBCEZABCFZJKDABCGHI $.

  ${
    ralbida.1 $e |- F/ x ph $.
    ralbida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  (Contributed by NM, 6-Oct-2003.) $)
    ralbida $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
      ( cv wcel wi wal wral pm5.74da albid df-ral 3bitr4g ) ADHEIZBJZDKQCJZDKBD
      ELCDELARSDFAQBCGMNBDEOCDEOP $.

    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 6-Oct-2003.) $)
    rexbida $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
      ( cv wcel wa wex wrex pm5.32da exbid df-rex 3bitr4g ) ADHEIZBJZDKQCJZDKBD
      ELCDELARSDFAQBCGMNBDEOCDEOP $.
  $}

  ${
    $d x ph $.
    ralbidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  (Contributed by NM, 4-Mar-1997.) $)
    ralbidva $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
      ( nfv ralbida ) ABCDEADGFH $.

    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 9-Mar-1997.) $)
    rexbidva $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
      ( nfv rexbida ) ABCDEADGFH $.
  $}

  ${
    ralbid.1 $e |- F/ x ph $.
    ralbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  (Contributed by NM, 27-Jun-1998.) $)
    ralbid $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
      ( wb cv wcel adantr ralbida ) ABCDEFABCHDIEJGKL $.

    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 27-Jun-1998.) $)
    rexbid $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
      ( wb cv wcel adantr rexbida ) ABCDEFABCHDIEJGKL $.
  $}

  ${
    $d x ph $.
    ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  (Contributed by NM, 20-Nov-1994.) $)
    ralbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
      ( nfv ralbid ) ABCDEADGFH $.

    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 20-Nov-1994.) $)
    rexbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
      ( nfv rexbid ) ABCDEADGFH $.
  $}

  ${
    $d x ph $.
    ralbidv2.1 $e |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  (Contributed by NM, 6-Apr-1997.) $)
    ralbidv2 $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
      ( cv wcel wi wal wral albidv df-ral 3bitr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
      AQRDGMBDENCDFNO $.
  $}

  ${
    $d x ph $.
    rexbidv2.1 $e |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 22-May-1999.) $)
    rexbidv2 $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
      ( cv wcel wa wex wrex exbidv df-rex 3bitr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
      AQRDGMBDENCDFNO $.
  $}

  ${
    ralbid2.nf $e |- F/ x ph $.
    ralbid2.1 $e |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  (Contributed by BJ, 14-Jul-2024.) $)
    ralbid2 $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
      ( cv wcel wi wal wral albid df-ral 3bitr4g ) ADIZEJBKZDLQFJCKZDLBDEMCDFMA
      RSDGHNBDEOCDFOP $.
  $}

  ${
    rexbid2.nf $e |- F/ x ph $.
    rexbid2.1 $e |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by BJ, 14-Jul-2024.) $)
    rexbid2 $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
      ( cv wcel wa wex wrex exbid df-rex 3bitr4g ) ADIZEJBKZDLQFJCKZDLBDEMCDFMA
      RSDGHNBDEOCDFOP $.
  $}

  ${
    ralbii.1 $e |- ( ph <-> ps ) $.
    $( Inference adding restricted universal quantifier to both sides of an
       equivalence.  (Contributed by NM, 23-Nov-1994.)  (Revised by Mario
       Carneiro, 17-Oct-2016.) $)
    ralbii $p |- ( A. x e. A ph <-> A. x e. A ps ) $=
      ( wral wb wtru a1i ralbidv mptru ) ACDFBCDFGHABCDABGHEIJK $.

    $( Inference adding restricted existential quantifier to both sides of an
       equivalence.  (Contributed by NM, 23-Nov-1994.)  (Revised by Mario
       Carneiro, 17-Oct-2016.) $)
    rexbii $p |- ( E. x e. A ph <-> E. x e. A ps ) $=
      ( wrex wb wtru a1i rexbidv mptru ) ACDFBCDFGHABCDABGHEIJK $.

    $( Inference adding two restricted universal quantifiers to both sides of
       an equivalence.  (Contributed by NM, 1-Aug-2004.) $)
    2ralbii $p |- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps ) $=
      ( wral ralbii ) ADFHBDFHCEABDFGII $.

    $( Inference adding two restricted existential quantifiers to both sides of
       an equivalence.  (Contributed by NM, 11-Nov-1995.) $)
    2rexbii $p |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps ) $=
      ( wrex rexbii ) ADFHBDFHCEABDFGII $.
  $}

  ${
    ralbii2.1 $e |- ( ( x e. A -> ph ) <-> ( x e. B -> ps ) ) $.
    $( Inference adding different restricted universal quantifiers to each side
       of an equivalence.  (Contributed by NM, 15-Aug-2005.) $)
    ralbii2 $p |- ( A. x e. A ph <-> A. x e. B ps ) $=
      ( cv wcel wi wal wral albii df-ral 3bitr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
      CFLACDMBCEMN $.
  $}

  ${
    rexbii2.1 $e |- ( ( x e. A /\ ph ) <-> ( x e. B /\ ps ) ) $.
    $( Inference adding different restricted existential quantifiers to each
       side of an equivalence.  (Contributed by NM, 4-Feb-2004.) $)
    rexbii2 $p |- ( E. x e. A ph <-> E. x e. B ps ) $=
      ( cv wcel wa wex wrex exbii df-rex 3bitr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
      CFLACDMBCEMN $.
  $}

  ${
    raleqbii.1 $e |- A = B $.
    raleqbii.2 $e |- ( ps <-> ch ) $.
    $( Equality deduction for restricted universal quantifier, changing both
       formula and quantifier domain.  Inference form.  (Contributed by David
       Moews, 1-May-2017.) $)
    raleqbii $p |- ( A. x e. A ps <-> A. x e. B ch ) $=
      ( cv wcel eleq2i imbi12i ralbii2 ) ABCDECHZDIMEIABDEMFJGKL $.

    $( Equality deduction for restricted existential quantifier, changing both
       formula and quantifier domain.  Inference form.  (Contributed by David
       Moews, 1-May-2017.) $)
    rexeqbii $p |- ( E. x e. A ps <-> E. x e. B ch ) $=
      ( cv wcel eleq2i anbi12i rexbii2 ) ABCDECHZDIMEIABDEMFJGKL $.
  $}

  ${
    ralbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
    $( Inference adding restricted universal quantifier to both sides of an
       equivalence.  (Contributed by NM, 26-Nov-2000.) $)
    ralbiia $p |- ( A. x e. A ph <-> A. x e. A ps ) $=
      ( cv wcel pm5.74i ralbii2 ) ABCDDCFDGABEHI $.

    $( Inference adding restricted existential quantifier to both sides of an
       equivalence.  (Contributed by NM, 26-Oct-1999.) $)
    rexbiia $p |- ( E. x e. A ph <-> E. x e. A ps ) $=
      ( cv wcel pm5.32i rexbii2 ) ABCDDCFDGABEHI $.
  $}

  ${
    $d x y $.  $d y A $.
    2rexbiia.1 $e |- ( ( x e. A /\ y e. B ) -> ( ph <-> ps ) ) $.
    $( Inference adding two restricted existential quantifiers to both sides of
       an equivalence.  (Contributed by NM, 1-Aug-2004.) $)
    2rexbiia $p |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps ) $=
      ( wrex cv wcel rexbidva rexbiia ) ADFHBDFHCECIEJABDFGKL $.
  $}

  ${
    $d x y $.
    r2alf.1 $e |- F/_ y A $.
    $( Double restricted universal quantification.  (Contributed by Mario
       Carneiro, 14-Oct-2016.) $)
    r2alf $p |- ( A. x e. A A. y e. B ph <->
               A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
      ( wral cv wi wal wa df-ral nfcri 19.21 impexp albii imbi2i 3bitr4i bitr4i
      wcel ) ACEGZBDGBHDTZUAIZBJUBCHETZKAIZCJZBJUABDLUFUCBUBUDAIZIZCJUBUGCJZIUF
      UCUBUGCCBDFMNUEUHCUBUDAOPUAUIUBACELQRPS $.

    $( Double restricted existential quantification.  (Contributed by Mario
       Carneiro, 14-Oct-2016.) $)
    r2exf $p |- ( E. x e. A E. y e. B ph <->
               E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
      ( wrex cv wcel wex df-rex nfcri 19.42 anass exbii anbi2i 3bitr4i bitr4i
      wa ) ACEGZBDGBHDIZTSZBJUACHEIZSASZCJZBJTBDKUEUBBUAUCASZSZCJUAUFCJZSUEUBUA
      UFCCBDFLMUDUGCUAUCANOTUHUAACEKPQOR $.
  $}

  ${
    $d x y $.  $d y A $.
    $( Double restricted universal quantification.  (Contributed by NM,
       19-Nov-1995.) $)
    r2al $p |- ( A. x e. A A. y e. B ph <->
               A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
      ( nfcv r2alf ) ABCDECDFG $.

    $( Double restricted existential quantification.  (Contributed by NM,
       11-Nov-1995.) $)
    r2ex $p |- ( E. x e. A E. y e. B ph <->
               E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
      ( nfcv r2exf ) ABCDECDFG $.
  $}

  ${
    $d x y $.  $d y A $.
    2ralbida.1 $e |- F/ x ph $.
    2ralbida.2 $e |- F/ y ph $.
    2ralbida.3 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  (Contributed by NM, 24-Feb-2004.) $)
    2ralbida $p |- ( ph ->
                     ( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
      ( wral cv wcel wa nfv nfan wb anassrs ralbida ) ABEGKCEGKDFHADLFMZNBCEGAT
      EITEOPATELGMBCQJRSS $.
  $}

  ${
    $d x y ph $.  $d y A $.
    2ralbidva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifiers (deduction
       form).  (Contributed by NM, 4-Mar-1997.) $)
    2ralbidva $p |- ( ph ->
                     ( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
      ( nfv 2ralbida ) ABCDEFGADIAEIHJ $.

    $( Formula-building rule for restricted existential quantifiers (deduction
       form).  (Contributed by NM, 15-Dec-2004.) $)
    2rexbidva $p |- ( ph ->
                    ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) ) $=
      ( wrex cv wcel wa wb anassrs rexbidva ) ABEGICEGIDFADJFKZLBCEGAPEJGKBCMHN
      OO $.
  $}

  ${
    $d x ph $.  $d y ph $.
    2ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifiers (deduction
       form).  (Contributed by NM, 28-Jan-2006.)  (Revised by Szymon
       Jaroszewicz, 16-Mar-2007.) $)
    2ralbidv $p |- ( ph ->
                     ( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
      ( wral ralbidv ) ABEGICEGIDFABCEGHJJ $.

    $( Formula-building rule for restricted existential quantifiers (deduction
       form).  (Contributed by NM, 28-Jan-2006.) $)
    2rexbidv $p |- ( ph ->
                    ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) ) $=
      ( wrex rexbidv ) ABEGICEGIDFABCEGHJJ $.

    $( Formula-building rule for restricted quantifiers (deduction form).
       (Contributed by NM, 28-Jan-2006.) $)
    rexralbidv $p |- ( ph ->
                    ( E. x e. A A. y e. B ps <-> E. x e. A A. y e. B ch ) ) $=
      ( wral ralbidv rexbidv ) ABEGICEGIDFABCEGHJK $.
  $}

  $( A transformation of restricted quantifiers and logical connectives.
     (Contributed by NM, 4-Sep-2005.) $)
  ralinexa $p |- ( A. x e. A ( ph -> -. ps ) <-> -. E. x e. A ( ph /\ ps ) ) $=
    ( wn wi wral wa wrex imnan ralbii ralnex bitri ) ABEFZCDGABHZEZCDGOCDIENPCD
    ABJKOCDLM $.

  ${
    $d x A $.  $d x B $.
    $( Two ways to say " ` A ` belongs to ` B ` ".  (Contributed by NM,
       22-Nov-1994.) $)
    risset $p |- ( A e. B <-> E. x e. B x = A ) $=
      ( cv wcel wceq wa wex wrex exancom df-rex df-clel 3bitr4ri ) ADZCEZNBFZGA
      HPOGAHPACIBCEOPAJPACKABCLM $.
  $}

  ${
    hbral.1 $e |- ( y e. A -> A. x y e. A ) $.
    hbral.2 $e |- ( ph -> A. x ph ) $.
    $( Bound-variable hypothesis builder for restricted quantification.
       (Contributed by NM, 1-Sep-1999.)  (Revised by David Abernethy,
       13-Dec-2009.) $)
    hbral $p |- ( A. y e. A ph -> A. x A. y e. A ph ) $=
      ( wral cv wcel wi wal df-ral hbim hbal hbxfrbi ) ACDGCHDIZAJZCKBACDLQBCPA
      BEFMNO $.
  $}

  $( ` x ` is not free in ` A. x e. A ph ` .  (Contributed by NM,
     18-Oct-1996.) $)
  hbra1 $p |- ( A. x e. A ph -> A. x A. x e. A ph ) $=
    ( wral cv wcel wi wal df-ral hba1 hbxfrbi ) ABCDBECFAGZBHBABCILBJK $.

  $( ` x ` is not free in ` A. x e. A ph ` .  (Contributed by NM, 18-Oct-1996.)
     (Revised by Mario Carneiro, 7-Oct-2016.) $)
  nfra1 $p |- F/ x A. x e. A ph $=
    ( wral cv wcel wi wal df-ral nfa1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.

  ${
    $d x y $.
    nfraldw.1 $e |- F/ y ph $.
    nfraldw.2 $e |- ( ph -> F/_ x A ) $.
    nfraldw.3 $e |- ( ph -> F/ x ps ) $.
    $( Not-free for restricted universal quantification where ` x ` and ` y `
       are distinct.  See ~ nfraldya for a version with ` y ` and ` A `
       distinct instead.  (Contributed by NM, 15-Feb-2013.)  (Revised by GG,
       10-Jan-2024.) $)
    nfraldw $p |- ( ph -> F/ x A. y e. A ps ) $=
      ( wral cv wcel wi wal df-ral nfcvd nfeld nfimd nfald nfxfrd ) BDEIDJZEKZB
      LZDMACBDENAUBCDFAUABCACTEACTOGPHQRS $.
  $}

  ${
    $d x y $.
    nfraldxy.2 $e |- F/ y ph $.
    nfraldxy.3 $e |- ( ph -> F/_ x A ) $.
    nfraldxy.4 $e |- ( ph -> F/ x ps ) $.
    $( Old name for ~ nfraldw .  (Contributed by Jim Kingdon, 29-May-2018.)
       (New usage is discouraged.) $)
    nfraldxy $p |- ( ph -> F/ x A. y e. A ps ) $=
      ( wral cv wcel wi wal df-ral wnfc nfcv a1i nfeld nfimd nfald nfxfrd ) BDE
      IDJZEKZBLZDMACBDENAUDCDFAUCBCACUBECUBOACUBPQGRHSTUA $.

    $( Not-free for restricted existential quantification where ` x ` and ` y `
       are distinct.  See ~ nfrexdya for a version with ` y ` and ` A `
       distinct instead.  (Contributed by Jim Kingdon, 30-May-2018.) $)
    nfrexdxy $p |- ( ph -> F/ x E. y e. A ps ) $=
      ( wrex cv wcel wa wex df-rex wnfc nfcv a1i nfeld nfand nfexd nfxfrd ) BDE
      IDJZEKZBLZDMACBDENAUDCDFAUCBCACUBECUBOACUBPQGRHSTUA $.
  $}

  ${
    $d y A $.  $d x z $.  $d y z $.  $d A z $.  $d ph z $.  $d ps z $.
    nfraldya.2 $e |- F/ y ph $.
    nfraldya.3 $e |- ( ph -> F/_ x A ) $.
    nfraldya.4 $e |- ( ph -> F/ x ps ) $.
    $( Not-free for restricted universal quantification where ` y ` and ` A `
       are distinct.  See ~ nfraldxy for a version with ` x ` and ` y `
       distinct instead.  (Contributed by Jim Kingdon, 30-May-2018.) $)
    nfraldya $p |- ( ph -> F/ x A. y e. A ps ) $=
      ( vz wral cv wcel wi wal df-ral wsb sbim clelsb1 nfv nfxfrd bitri 3bitr4i
      imbi1i albii sb8 nfsbd nfraldxy ) BDEJDKELZBMZDNZACBDEOUJBDIPZIEJZACUIDIP
      ZINIKELZUKMZINUJULUMUOIUMUHDIPZUKMUOUHBDIQUPUNUKDIERUCUAUDUIDIUIISUEUKIEO
      UBAUKCIEAISGABDICFHUFUGTT $.

    $( Not-free for restricted existential quantification where ` y ` and ` A `
       are distinct.  See ~ nfrexdxy for a version with ` x ` and ` y `
       distinct instead.  (Contributed by Jim Kingdon, 30-May-2018.) $)
    nfrexdya $p |- ( ph -> F/ x E. y e. A ps ) $=
      ( vz wrex cv wcel wa wex df-rex wsb sban clelsb1 nfv nfxfrd bitri 3bitr4i
      anbi1i exbii sb8e nfsbd nfrexdxy ) BDEJDKELZBMZDNZACBDEOUJBDIPZIEJZACUIDI
      PZINIKELZUKMZINUJULUMUOIUMUHDIPZUKMUOUHBDIQUPUNUKDIERUCUAUDUIDIUIISUEUKIE
      OUBAUKCIEAISGABDICFHUFUGTT $.
  $}

  ${
    $d x y $.
    nfralw.1 $e |- F/_ x A $.
    nfralw.2 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for restricted quantification.  See
       ~ nfralya for a version with ` y ` and ` A ` distinct instead of ` x `
       and ` y ` .  (Contributed by NM, 1-Sep-1999.)  (Revised by GG,
       10-Jan-2024.) $)
    nfralw $p |- F/ x A. y e. A ph $=
      ( wral wnf wtru nftru wnfc a1i nfraldw mptru ) ACDGBHIABCDCJBDKIELABHIFLM
      N $.
  $}

  ${
    $d x y $.
    nfralxy.1 $e |- F/_ x A $.
    nfralxy.2 $e |- F/ x ph $.
    $( Old name for ~ nfralw .  (Contributed by Jim Kingdon, 30-May-2018.)
       (New usage is discouraged.) $)
    nfralxy $p |- F/ x A. y e. A ph $=
      ( wral wnf wtru nftru wnfc a1i nfraldxy mptru ) ACDGBHIABCDCJBDKIELABHIFL
      MN $.

    $( Not-free for restricted existential quantification where ` x ` and ` y `
       are distinct.  See ~ nfrexya for a version with ` y ` and ` A ` distinct
       instead.  (Contributed by Jim Kingdon, 30-May-2018.) $)
    nfrexw $p |- F/ x E. y e. A ph $=
      ( wrex wnf wtru nftru wnfc a1i nfrexdxy mptru ) ACDGBHIABCDCJBDKIELABHIFL
      MN $.
  $}

  ${
    $d y A $.
    nfralya.1 $e |- F/_ x A $.
    nfralya.2 $e |- F/ x ph $.
    $( Not-free for restricted universal quantification where ` y ` and ` A `
       are distinct.  See ~ nfralxy for a version with ` x ` and ` y ` distinct
       instead.  (Contributed by Jim Kingdon, 3-Jun-2018.) $)
    nfralya $p |- F/ x A. y e. A ph $=
      ( wral wnf wtru nftru wnfc a1i nfraldya mptru ) ACDGBHIABCDCJBDKIELABHIFL
      MN $.

    $( Not-free for restricted existential quantification where ` y ` and ` A `
       are distinct.  See ~ nfrexw for a version with ` x ` and ` y ` distinct
       instead.  (Contributed by Jim Kingdon, 3-Jun-2018.) $)
    nfrexya $p |- F/ x E. y e. A ph $=
      ( wrex wnf wtru nftru wnfc a1i nfrexdya mptru ) ACDGBHIABCDCJBDKIELABHIFL
      MN $.
  $}

  ${
    $d x y $.  $d A y $.
    $( Not-free given two restricted quantifiers.  (Contributed by Jim Kingdon,
       20-Aug-2018.) $)
    nfra2xy $p |- F/ y A. x e. A A. y e. B ph $=
      ( wral nfcv nfra1 nfralxy ) ACEFCBDCDGACEHI $.
  $}

  $( ` x ` is not free in ` E. x e. A ph ` .  (Contributed by NM, 19-Mar-1997.)
     (Revised by Mario Carneiro, 7-Oct-2016.) $)
  nfre1 $p |- F/ x E. x e. A ph $=
    ( wrex cv wcel wa wex df-rex nfe1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.

  ${
    $d x y z $.  $d y z A $.  $d z B $.
    $( Triple restricted universal quantification.  (Contributed by NM,
       19-Nov-1995.) $)
    r3al $p |- ( A. x e. A A. y e. B A. z e. C ph <->
               A. x A. y A. z ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) ) $=
      ( cv wcel wa wi wal wral w3a df-ral r2al ralbii bitri albii 19.21v 3anass
      imbi1i impexp 3bitr4i ) CHFIZDHGIZJZAKZDLZCLZBEMBHEIZUJKZBLADGMCFMZBEMUKU
      EUFNZAKZDLZCLZBLUJBEOUMUJBEACDFGPQUQULBUQUKUIKZCLULUPURCUPUKUHKZDLURUOUSD
      UOUKUGJZAKUSUNUTAUKUEUFUAUBUKUGAUCRSUKUHDTRSUKUICTRSUD $.
  $}

  $( Universal quantification implies restricted quantification.  (Contributed
     by NM, 20-Oct-2006.) $)
  alral $p |- ( A. x ph -> A. x e. A ph ) $=
    ( wal cv wcel wi wral ax-1 alimi df-ral sylibr ) ABDBECFZAGZBDABCHANBAMIJAB
    CKL $.

  $( Restricted existence implies existence.  (Contributed by NM,
     11-Nov-1995.) $)
  rexex $p |- ( E. x e. A ph -> E. x ph ) $=
    ( wrex cv wcel wa wex df-rex simpr eximi sylbi ) ABCDBECFZAGZBHABHABCINABMA
    JKL $.

  $( Restricted specialization.  (Contributed by NM, 17-Oct-1996.) $)
  rsp $p |- ( A. x e. A ph -> ( x e. A -> ph ) ) $=
    ( wral cv wcel wi wal df-ral sp sylbi ) ABCDBECFAGZBHLABCILBJK $.

  $( Restricted specialization.  (Contributed by Glauco Siliprandi,
     11-Dec-2019.) $)
  rspa $p |- ( ( A. x e. A ph /\ x e. A ) -> ph ) $=
    ( wral cv wcel rsp imp ) ABCDBECFAABCGH $.

  $( Restricted specialization.  (Contributed by NM, 12-Oct-1999.) $)
  rspe $p |- ( ( x e. A /\ ph ) -> E. x e. A ph ) $=
    ( cv wcel wa wex wrex 19.8a df-rex sylibr ) BDCEAFZLBGABCHLBIABCJK $.

  $( Restricted specialization.  (Contributed by NM, 11-Feb-1997.) $)
  rsp2 $p |- ( A. x e. A A. y e. B ph -> ( ( x e. A /\ y e. B ) -> ph ) ) $=
    ( wral cv wcel wi rsp syl6 impd ) ACEFZBDFZBGDHZCGEHZANOMPAIMBDJACEJKL $.

  $( Restricted specialization.  (Contributed by FL, 4-Jun-2012.) $)
  rsp2e $p |- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph ) $=
    ( cv wcel w3a wrex wa wex simp1 rspe 3adant1 19.8a syl2anc df-rex sylibr )
    BFDGZCFEGZAHZSACEIZJZBKZUBBDIUASUBUDSTALTAUBSACEMNUCBOPUBBDQR $.

  ${
    rspec.1 $e |- A. x e. A ph $.
    $( Specialization rule for restricted quantification.  (Contributed by NM,
       19-Nov-1994.) $)
    rspec $p |- ( x e. A -> ph ) $=
      ( wral cv wcel wi rsp ax-mp ) ABCEBFCGAHDABCIJ $.
  $}

  ${
    rgen.1 $e |- ( x e. A -> ph ) $.
    $( Generalization rule for restricted quantification.  (Contributed by NM,
       19-Nov-1994.) $)
    rgen $p |- A. x e. A ph $=
      ( wral cv wcel wi df-ral mpgbir ) ABCEBFCGAHBABCIDJ $.
  $}

  ${
    $d y z A $.  $d x z $.
    rgen2a.1 $e |- ( ( x e. A /\ y e. A ) -> ph ) $.
    $( Generalization rule for restricted quantification.  Note that ` x ` and
       ` y ` are not required to be disjoint.  This proof illustrates the use
       of ~ dvelim .  Usage of ~ rgen2 instead is highly encouraged.
       (Contributed by NM, 23-Nov-1994.)  (Proof rewritten by Jim Kingdon,
       1-Jun-2018.)  (New usage is discouraged.) $)
    rgen2a $p |- A. x e. A A. y e. A ph $=
      ( vz wral cv wcel wi wal wceq wnf wo nfv eleq1 dvelimor ex biimtrdi alimi
      pm2.43d a1d nfr syl6 jaoi ax-mp df-ral sylibr rgen ) ACDGZBDBHZDIZCHZDIZA
      JZCKZUJUMUKLZCKZULCMZNULUPJZFHZDIZULCBFVBCOVAUKDPQURUTUSURUPULUQUOCUQUNAU
      QUNULUOUMUKDPULUNAERZSUATUBUSULULCKUPULCUCULUOCVCTUDUEUFACDUGUHUI $.
  $}

  ${
    rgenw.1 $e |- ph $.
    $( Generalization rule for restricted quantification.  (Contributed by NM,
       18-Jun-2014.) $)
    rgenw $p |- A. x e. A ph $=
      ( cv wcel a1i rgen ) ABCABECFDGH $.

    $( Generalization rule for restricted quantification.  Note that ` x ` and
       ` y ` needn't be distinct.  (Contributed by NM, 18-Jun-2014.) $)
    rgen2w $p |- A. x e. A A. y e. B ph $=
      ( wral rgenw ) ACEGBDACEFHH $.
  $}

  ${
    mprg.1 $e |- ( A. x e. A ph -> ps ) $.
    mprg.2 $e |- ( x e. A -> ph ) $.
    $( Modus ponens combined with restricted generalization.  (Contributed by
       NM, 10-Aug-2004.) $)
    mprg $p |- ps $=
      ( wral rgen ax-mp ) ACDGBACDFHEI $.
  $}

  ${
    mprgbir.1 $e |- ( ph <-> A. x e. A ps ) $.
    mprgbir.2 $e |- ( x e. A -> ps ) $.
    $( Modus ponens on biconditional combined with restricted generalization.
       (Contributed by NM, 21-Mar-2004.) $)
    mprgbir $p |- ph $=
      ( wral rgen mpbir ) ABCDGBCDFHEI $.
  $}

  $( Distribution of restricted quantification over implication.  (Contributed
     by NM, 9-Feb-1997.) $)
  ralim $p |- ( A. x e. A ( ph -> ps ) ->
               ( A. x e. A ph -> A. x e. A ps ) ) $=
    ( wi wral cv wcel wal df-ral ax-2 al2imi sylbi 3imtr4g ) ABEZCDFZCGDHZAEZCI
    ZQBEZCIZACDFBCDFPQOEZCISUAEOCDJUBRTCQABKLMACDJBCDJN $.

  ${
    ralimi2.1 $e |- ( ( x e. A -> ph ) -> ( x e. B -> ps ) ) $.
    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 22-Feb-2004.) $)
    ralimi2 $p |- ( A. x e. A ph -> A. x e. B ps ) $=
      ( cv wcel wi wal wral alimi df-ral 3imtr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
      CFLACDMBCEMN $.
  $}

  ${
    ralimia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 19-Jul-1996.) $)
    ralimia $p |- ( A. x e. A ph -> A. x e. A ps ) $=
      ( cv wcel a2i ralimi2 ) ABCDDCFDGABEHI $.
  $}

  ${
    ralimiaa.1 $e |- ( ( x e. A /\ ph ) -> ps ) $.
    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 4-Aug-2007.) $)
    ralimiaa $p |- ( A. x e. A ph -> A. x e. A ps ) $=
      ( cv wcel ex ralimia ) ABCDCFDGABEHI $.
  $}

  ${
    ralimi.1 $e |- ( ph -> ps ) $.
    $( Inference quantifying both antecedent and consequent, with strong
       hypothesis.  (Contributed by NM, 4-Mar-1997.) $)
    ralimi $p |- ( A. x e. A ph -> A. x e. A ps ) $=
      ( wi cv wcel a1i ralimia ) ABCDABFCGDHEIJ $.

    $( Inference quantifying both antecedent and consequent two times, with
       strong hypothesis.  (Contributed by AV, 3-Dec-2021.) $)
    2ralimi $p |- ( A. x e. A A. y e. B ph -> A. x e. A A. y e. B ps ) $=
      ( wral ralimi ) ADFHBDFHCEABDFGII $.
  $}

  ${
    ral2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference quantifying antecedent, nested antecedent, and consequent,
       with a strong hypothesis.  (Contributed by NM, 19-Dec-2006.) $)
    ral2imi $p |- ( A. x e. A ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
      ( wral wi ralimi ralim syl ) ADEGBCHZDEGBDEGCDEGHALDEFIBCDEJK $.
  $}

  ${
    ralimdaa.1 $e |- F/ x ph $.
    ralimdaa.2 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
    $( Deduction quantifying both antecedent and consequent, based on Theorem
       19.20 of [Margaris] p. 90.  (Contributed by NM, 22-Sep-2003.) $)
    ralimdaa $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
      ( cv wcel wi wal wral ex a2d alimd df-ral 3imtr4g ) ADHEIZBJZDKRCJZDKBDEL
      CDELASTDFARBCARBCJGMNOBDEPCDEPQ $.
  $}

  ${
    $d x ph $.
    ralimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
    $( Deduction quantifying both antecedent and consequent, based on Theorem
       19.20 of [Margaris] p. 90.  (Contributed by NM, 22-May-1999.) $)
    ralimdva $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
      ( nfv ralimdaa ) ABCDEADGFH $.
  $}

  ${
    $d x ph $.
    ralimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction quantifying both antecedent and consequent, based on Theorem
       19.20 of [Margaris] p. 90.  (Contributed by NM, 8-Oct-2003.) $)
    ralimdv $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
      ( wi cv wcel adantr ralimdva ) ABCDEABCGDHEIFJK $.
  $}

  ${
    $d y A $.  $d x y ph $.
    ralimdvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) ) $.
    $( Deduction doubly quantifying both antecedent and consequent, based on
       Theorem 19.20 of [Margaris] p. 90 ( ~ alim ).  (Contributed by AV,
       27-Nov-2019.) $)
    ralimdvva $p |- ( ph
                   -> ( A. x e. A A. y e. B ps -> A. x e. A A. y e. B ch ) ) $=
      ( wral cv wcel wa wi anassrs ralimdva ) ABEGICEGIDFADJFKZLBCEGAPEJGKBCMHN
      OO $.
  $}

  ${
    $d x ph $.
    ralimdv2.1 $e |- ( ph -> ( ( x e. A -> ps ) -> ( x e. B -> ch ) ) ) $.
    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 1-Feb-2005.) $)
    ralimdv2 $p |- ( ph -> ( A. x e. A ps -> A. x e. B ch ) ) $=
      ( cv wcel wi wal wral alimdv df-ral 3imtr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
      AQRDGMBDENCDFNO $.
  $}

  ${
    ralrimi.1 $e |- F/ x ph $.
    ralrimi.2 $e |- ( ph -> ( x e. A -> ps ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by NM, 10-Oct-1999.) $)
    ralrimi $p |- ( ph -> A. x e. A ps ) $=
      ( cv wcel wi wal wral alrimi df-ral sylibr ) ACGDHBIZCJBCDKAOCEFLBCDMN $.
  $}

  ${
    $d x ph $.
    ralrimiv.1 $e |- ( ph -> ( x e. A -> ps ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 22-Nov-1994.) $)
    ralrimiv $p |- ( ph -> A. x e. A ps ) $=
      ( nfv ralrimi ) ABCDACFEG $.
  $}

  ${
    $d x ph $.
    ralrimiva.1 $e |- ( ( ph /\ x e. A ) -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 2-Jan-2006.) $)
    ralrimiva $p |- ( ph -> A. x e. A ps ) $=
      ( cv wcel ex ralrimiv ) ABCDACFDGBEHI $.
  $}

  ${
    $d x ph $.
    ralrimivw.1 $e |- ( ph -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 18-Jun-2014.) $)
    ralrimivw $p |- ( ph -> A. x e. A ps ) $=
      ( cv wcel a1d ralrimiv ) ABCDABCFDGEHI $.
  $}

  $( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed
     theorem version).  (Contributed by NM, 1-Mar-2008.) $)
  r19.21t $p |- ( F/ x ph ->
           ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) ) $=
    ( wnf cv wcel wi wal wral bi2.04 albii 19.21t bitrid df-ral imbi2i 3bitr4g
    ) ACEZCFDGZABHZHZCIZASBHZCIZHZTCDJABCDJZHUBAUCHZCIRUEUAUGCSABKLAUCCMNTCDOUF
    UDABCDOPQ $.

  ${
    r19.21.1 $e |- F/ x ph $.
    $( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers.
       (Contributed by Scott Fenton, 30-Mar-2011.) $)
    r19.21 $p |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) $=
      ( wnf wi wral wb r19.21t ax-mp ) ACFABGCDHABCDHGIEABCDJK $.
  $}

  ${
    $d x ph $.
    $( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers.
       (Contributed by NM, 15-Oct-2003.)  (Proof shortened by Andrew Salmon,
       30-May-2011.) $)
    r19.21v $p |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) $=
      ( nfv r19.21 ) ABCDACEF $.
  $}

  ${
    ralrimd.1 $e |- F/ x ph $.
    ralrimd.2 $e |- F/ x ps $.
    ralrimd.3 $e |- ( ph -> ( ps -> ( x e. A -> ch ) ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 16-Feb-2004.) $)
    ralrimd $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
      ( cv wcel wi wal wral alrimd df-ral imbitrrdi ) ABDIEJCKZDLCDEMABQDFGHNCD
      EOP $.
  $}

  ${
    $d x ph $.  $d x ps $.
    ralrimdv.1 $e |- ( ph -> ( ps -> ( x e. A -> ch ) ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 27-May-1998.) $)
    ralrimdv $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
      ( nfv ralrimd ) ABCDEADGBDGFH $.
  $}

  ${
    $d x ph $.  $d x ps $.
    ralrimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 2-Feb-2008.) $)
    ralrimdva $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
      ( cv wcel wi ex com23 ralrimdv ) ABCDEADGEHZBCAMBCIFJKL $.
  $}

  ${
    $d x y ph $.  $d y A $.
    ralrimivv.1 $e |- ( ph -> ( ( x e. A /\ y e. B ) -> ps ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version with double quantification.)  (Contributed by NM,
       24-Jul-2004.) $)
    ralrimivv $p |- ( ph -> A. x e. A A. y e. B ps ) $=
      ( wral cv wcel expd ralrimdv ralrimiv ) ABDFHCEACIEJZBDFANDIFJBGKLM $.
  $}

  ${
    $d ph x y $.  $d A y $.
    ralrimivva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version with double quantification.)  (Contributed by Jeff
       Madsen, 19-Jun-2011.) $)
    ralrimivva $p |- ( ph -> A. x e. A A. y e. B ps ) $=
      ( cv wcel wa ex ralrimivv ) ABCDEFACHEIDHFIJBGKL $.
  $}

  ${
    $d ph x y z $.  $d A y z $.  $d B z $.
    ralrimivvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B /\ z e. C ) ) -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version with triple quantification.)  (Contributed by Mario
       Carneiro, 9-Jul-2014.) $)
    ralrimivvva $p |- ( ph -> A. x e. A A. y e. B A. z e. C ps ) $=
      ( wral cv wcel wa 3anassrs ralrimiva ) ABEHJZDGJCFACKFLZMZPDGRDKGLZMBEHAQ
      SEKHLBINOOO $.
  $}

  ${
    $d x y ph $.  $d x y ps $.  $d y A $.
    ralrimdvv.1 $e |- ( ph -> ( ps -> ( ( x e. A /\ y e. B ) -> ch ) ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version with double quantification.)  (Contributed by NM,
       1-Jun-2005.) $)
    ralrimdvv $p |- ( ph -> ( ps -> A. x e. A A. y e. B ch ) ) $=
      ( wral wa cv wcel wi imp ralrimivv ex ) ABCEGIDFIABJCDEFGABDKFLEKGLJCMHNO
      P $.
  $}

  ${
    $d x y ph $.  $d x y ps $.  $d y A $.
    ralrimdvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version with double quantification.)  (Contributed by NM,
       2-Feb-2008.) $)
    ralrimdvva $p |- ( ph -> ( ps -> A. x e. A A. y e. B ch ) ) $=
      ( cv wcel wa wi ex com23 ralrimdvv ) ABCDEFGADIFJEIGJKZBCAPBCLHMNO $.
  $}

  ${
    $d x y $.  $d y A $.
    rgen2.1 $e |- ( ( x e. A /\ y e. B ) -> ph ) $.
    $( Generalization rule for restricted quantification.  (Contributed by NM,
       30-May-1999.) $)
    rgen2 $p |- A. x e. A A. y e. B ph $=
      ( wral cv wcel ralrimiva rgen ) ACEGBDBHDIACEFJK $.
  $}

  ${
    $d y z A $.  $d z B $.  $d x y z $.
    rgen3.1 $e |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) $.
    $( Generalization rule for restricted quantification.  (Contributed by NM,
       12-Jan-2008.) $)
    rgen3 $p |- A. x e. A A. y e. B A. z e. C ph $=
      ( wral cv wcel wa 3expa ralrimiva rgen2 ) ADGIBCEFBJEKZCJFKZLADGPQDJGKAHM
      NO $.
  $}

  ${
    r19.21bi.1 $e |- ( ph -> A. x e. A ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 20-Nov-1994.) $)
    r19.21bi $p |- ( ( ph /\ x e. A ) -> ps ) $=
      ( cv wcel wi wral wal df-ral sylib 19.21bi imp ) ACFDGZBAOBHZCABCDIPCJEBC
      DKLMN $.
  $}

  ${
    rspec2.1 $e |- A. x e. A A. y e. B ph $.
    $( Specialization rule for restricted quantification.  (Contributed by NM,
       20-Nov-1994.) $)
    rspec2 $p |- ( ( x e. A /\ y e. B ) -> ph ) $=
      ( cv wcel wral rspec r19.21bi ) BGDHACEACEIBDFJK $.
  $}

  ${
    rspec3.1 $e |- A. x e. A A. y e. B A. z e. C ph $.
    $( Specialization rule for restricted quantification.  (Contributed by NM,
       20-Nov-1994.) $)
    rspec3 $p |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) $=
      ( cv wcel wa wral rspec2 r19.21bi 3impa ) BIEJZCIFJZDIGJAPQKADGADGLBCEFHM
      NO $.
  $}

  ${
    r19.21be.1 $e |- ( ph -> A. x e. A ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 21-Nov-1994.) $)
    r19.21be $p |- A. x e. A ( ph -> ps ) $=
      ( wi cv wcel r19.21bi expcom rgen ) ABFCDACGDHBABCDEIJK $.
  $}

  ${
    nrex.1 $e |- ( x e. A -> -. ps ) $.
    $( Inference adding restricted existential quantifier to negated wff.
       (Contributed by NM, 16-Oct-2003.) $)
    nrex $p |- -. E. x e. A ps $=
      ( wn wral wrex rgen ralnex mpbi ) AEZBCFABCGEKBCDHABCIJ $.
  $}

  ${
    $d x ph $.
    nrexdv.1 $e |- ( ( ph /\ x e. A ) -> -. ps ) $.
    $( Deduction adding restricted existential quantifier to negated wff.
       (Contributed by NM, 16-Oct-2003.) $)
    nrexdv $p |- ( ph -> -. E. x e. A ps ) $=
      ( wn wral wrex ralrimiva ralnex sylib ) ABFZCDGBCDHFALCDEIBCDJK $.
  $}

  $( Theorem 19.22 of [Margaris] p. 90.  (Restricted quantifier version.)
     (Contributed by NM, 22-Nov-1994.)  (Proof shortened by Andrew Salmon,
     30-May-2011.) $)
  rexim $p |- ( A. x e. A ( ph -> ps ) ->
               ( E. x e. A ph -> E. x e. A ps ) ) $=
    ( wi wral cv wcel wa wex wrex wal df-ral simpl a1i pm3.31 jcad alimi df-rex
    sylbi exim syl 3imtr4g ) ABEZCDFZCGDHZAIZCJZUFBIZCJZACDKBCDKUEUGUIEZCLZUHUJ
    EUEUFUDEZCLULUDCDMUMUKCUMUGUFBUGUFEUMUFANOUFABPQRTUGUICUAUBACDSBCDSUC $.

  ${
    reximia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 10-Feb-1997.) $)
    reximia $p |- ( E. x e. A ph -> E. x e. A ps ) $=
      ( wi wrex rexim mprg ) ABFACDGBCDGFCDABCDHEI $.
  $}

  ${
    reximi2.1 $e |- ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) ) $.
    $( Inference quantifying both antecedent and consequent, based on Theorem
       19.22 of [Margaris] p. 90.  (Contributed by NM, 8-Nov-2004.) $)
    reximi2 $p |- ( E. x e. A ph -> E. x e. B ps ) $=
      ( cv wcel wa wex wrex eximi df-rex 3imtr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
      CFLACDMBCEMN $.
  $}

  ${
    reximi.1 $e |- ( ph -> ps ) $.
    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 18-Oct-1996.) $)
    reximi $p |- ( E. x e. A ph -> E. x e. A ps ) $=
      ( wi cv wcel a1i reximia ) ABCDABFCGDHEIJ $.
  $}

  ${
    reximdai.1 $e |- F/ x ph $.
    reximdai.2 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 31-Aug-1999.) $)
    reximdai $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
      ( wi wral wrex ralrimi rexim syl ) ABCHZDEIBDEJCDEJHANDEFGKBCDELM $.
  $}

  ${
    $d x ph $.
    reximdv2.1 $e |- ( ph -> ( ( x e. A /\ ps ) -> ( x e. B /\ ch ) ) ) $.
    $( Deduction quantifying both antecedent and consequent, based on Theorem
       19.22 of [Margaris] p. 90.  (Contributed by NM, 17-Sep-2003.) $)
    reximdv2 $p |- ( ph -> ( E. x e. A ps -> E. x e. B ch ) ) $=
      ( cv wcel wa wex wrex eximdv df-rex 3imtr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
      AQRDGMBDENCDFNO $.
  $}

  ${
    $d x ph $.
    reximdvai.1 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
    $( Deduction quantifying both antecedent and consequent, based on Theorem
       19.22 of [Margaris] p. 90.  (Contributed by NM, 14-Nov-2002.) $)
    reximdvai $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
      ( nfv reximdai ) ABCDEADGFH $.
  $}

  ${
    $d x ph $.
    reximdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Restricted
       quantifier version with strong hypothesis.)  (Contributed by NM,
       24-Jun-1998.) $)
    reximdv $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
      ( wi cv wcel a1d reximdvai ) ABCDEABCGDHEIFJK $.
  $}

  ${
    $d x ph $.
    reximdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
    $( Deduction quantifying both antecedent and consequent, based on Theorem
       19.22 of [Margaris] p. 90.  (Contributed by NM, 22-May-1999.) $)
    reximdva $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
      ( cv wcel wi ex reximdvai ) ABCDEADGEHBCIFJK $.
  $}

  ${
    $d x ph $.
    reximddva.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch ) $.
    reximddva.2 $e |- ( ph -> E. x e. A ps ) $.
    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Contributed by
       Thierry Arnoux, 7-Dec-2016.) $)
    reximddv $p |- ( ph -> E. x e. A ch ) $=
      ( wrex cv wcel expr reximdva mpd ) ABDEHCDEHGABCDEADIEJBCFKLM $.
  $}

  ${
    $d x ph $.
    reximssdv.1 $e |- ( ph -> E. x e. B ps ) $.
    reximssdv.2 $e |- ( ( ph /\ ( x e. B /\ ps ) ) -> x e. A ) $.
    reximssdv.3 $e |- ( ( ph /\ ( x e. B /\ ps ) ) -> ch ) $.
    $( Derivation of a restricted existential quantification over a subset (the
       second hypothesis implies ` A C_ B ` ), deduction form.  (Contributed by
       AV, 21-Aug-2022.) $)
    reximssdv $p |- ( ph -> E. x e. A ch ) $=
      ( wrex cv wcel wa jca ex reximdv2 mpd ) ABDFJCDEJGABCDFEADKZFLBMZRELZCMAS
      MTCHINOPQ $.
  $}

  ${
    $d A y $.  $d ph x y $.
    reximddv2.1 $e |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch ) $.
    reximddv2.2 $e |- ( ph -> E. x e. A E. y e. B ps ) $.
    $( Double deduction from Theorem 19.22 of [Margaris] p. 90.  (Contributed
       by Thierry Arnoux, 15-Dec-2019.) $)
    reximddv2 $p |- ( ph -> E. x e. A E. y e. B ch ) $=
      ( wrex cv wcel wa ex reximdva impr reximddv ) ABEGJZCEGJZDFADKFLZRSATMZBC
      EGUAEKGLMBCHNOPIQ $.
  $}

  $( A transformation of restricted quantifiers and logical connectives.
     (Contributed by NM, 4-Sep-2005.)  (Revised by Jim Kingdon,
     18-Jan-2026.) $)
  rexanaliim $p |- ( E. x e. A ( ph /\ -. ps )
      -> -. A. x e. A ( ph -> ps ) ) $=
    ( wn wa wrex wi wral annimim reximi rexnalim syl ) ABEFZCDGABHZEZCDGOCDIENP
    CDABJKOCDLM $.

  ${
    $d x y $.  $d y A $.  $d x B $.
    $( Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers.
       (Contributed by NM, 15-Oct-2003.)  (Proof shortened by Andrew Salmon,
       30-May-2011.) $)
    r19.12 $p |- ( E. x e. A A. y e. B ph -> A. y e. B E. x e. A ph ) $=
      ( wral wrex nfcv nfra1 nfrexw wcel ax-1 ralrimi rsp com12 reximdv ralimia
      cv syl ) ACEFZBDGZUACEFABDGZCEFUAUACETCBDCDHACEIJUACREKZLMUAUBCEUCTABDTUC
      AACENOPQS $.
  $}

  $( Closed theorem form of ~ r19.23 .  (Contributed by NM, 4-Mar-2013.)
     (Revised by Mario Carneiro, 8-Oct-2016.) $)
  r19.23t $p |- ( F/ x ps ->
    ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) ) $=
    ( wnf cv wcel wa wal wex wral wrex 19.23t df-ral impexp albii bitr4i df-rex
    wi imbi1i 3bitr4g ) BCECFDGZAHZBSZCIZUCCJZBSABSZCDKZACDLZBSUCBCMUHUBUGSZCIU
    EUGCDNUDUJCUBABOPQUIUFBACDRTUA $.

  ${
    r19.23.1 $e |- F/ x ps $.
    $( Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers.
       (Contributed by NM, 22-Oct-2010.)  (Proof shortened by Mario Carneiro,
       8-Oct-2016.) $)
    r19.23 $p |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) $=
      ( wnf wi wral wrex wb r19.23t ax-mp ) BCFABGCDHACDIBGJEABCDKL $.
  $}

  ${
    $d x ps $.
    $( Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers.
       (Contributed by NM, 31-Aug-1999.) $)
    r19.23v $p |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) $=
      ( nfv r19.23 ) ABCDBCEF $.
  $}

  ${
    rexlimi.1 $e |- F/ x ps $.
    rexlimi.2 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 30-Nov-2003.)  (Proof
       shortened by Andrew Salmon, 30-May-2011.) $)
    rexlimi $p |- ( E. x e. A ph -> ps ) $=
      ( wi wral wrex rgen r19.23 mpbi ) ABGZCDHACDIBGMCDFJABCDEKL $.
  $}

  ${
    $d x ps $.
    rexlimiv.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 20-Nov-1994.) $)
    rexlimiv $p |- ( E. x e. A ph -> ps ) $=
      ( nfv rexlimi ) ABCDBCFEG $.
  $}

  ${
    $d x ps $.
    rexlimiva.1 $e |- ( ( x e. A /\ ph ) -> ps ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by NM, 18-Dec-2006.) $)
    rexlimiva $p |- ( E. x e. A ph -> ps ) $=
      ( cv wcel ex rexlimiv ) ABCDCFDGABEHI $.
  $}

  ${
    $d ps x $.
    rexlimivw.1 $e |- ( ph -> ps ) $.
    $( Weaker version of ~ rexlimiv .  (Contributed by FL, 19-Sep-2011.) $)
    rexlimivw $p |- ( E. x e. A ph -> ps ) $=
      ( wi cv wcel a1i rexlimiv ) ABCDABFCGDHEIJ $.
  $}

  ${
    rexlimd.1 $e |- F/ x ph $.
    rexlimd.2 $e |- F/ x ch $.
    rexlimd.3 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
    $( Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by NM, 27-May-1998.)  (Proof shortened by Andrew
       Salmon, 30-May-2011.) $)
    rexlimd $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( wi wral wrex ralrimi r19.23 sylib ) ABCIZDEJBDEKCIAODEFHLBCDEGMN $.
  $}

  ${
    rexlimd2.1 $e |- F/ x ph $.
    rexlimd2.2 $e |- ( ph -> F/ x ch ) $.
    rexlimd2.3 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
    $( Version of ~ rexlimd with deduction version of second hypothesis.
       (Contributed by NM, 21-Jul-2013.)  (Revised by Mario Carneiro,
       8-Oct-2016.) $)
    rexlimd2 $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( wi wral wrex ralrimi wnf wb r19.23t syl mpbid ) ABCIZDEJZBDEKCIZARDEFHL
      ACDMSTNGBCDEOPQ $.
  $}

  ${
    $d x ph $.  $d x ch $.
    rexlimdv.1 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by NM, 14-Nov-2002.)  (Proof shortened by Eric
       Schmidt, 22-Dec-2006.) $)
    rexlimdv $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( nfv rexlimd ) ABCDEADGCDGFH $.
  $}

  ${
    $d x ph $.  $d x ch $.
    rexlimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by NM, 20-Jan-2007.) $)
    rexlimdva $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( cv wcel wi ex rexlimdv ) ABCDEADGEHBCIFJK $.
  $}

  ${
    $d x ph $.  $d x ch $.
    rexlimdvaa.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by Mario Carneiro, 15-Jun-2016.) $)
    rexlimdvaa $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( cv wcel expr rexlimdva ) ABCDEADGEHBCFIJ $.
  $}

  ${
    $d x ph $.  $d x ch $.
    rexlimdv3a.1 $e |- ( ( ph /\ x e. A /\ ps ) -> ch ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  Frequently-used variant of ~ rexlimdv .  (Contributed by NM,
       7-Jun-2015.) $)
    rexlimdv3a $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( cv wcel 3exp rexlimdv ) ABCDEADGEHBCFIJ $.
  $}

  ${
    $d ch x $.  $d ph x $.
    rexlimdva2.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by Glauco Siliprandi, 2-Jan-2022.) $)
    rexlimdva2 $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( cv wcel exp31 rexlimdv ) ABCDEADGEHBCFIJ $.
  $}

  ${
    $d x ph $.  $d x ch $.
    rexlimdvw.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by NM, 18-Jun-2014.) $)
    rexlimdvw $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( wi cv wcel a1d rexlimdv ) ABCDEABCGDHEIFJK $.
  $}

  ${
    $d x ph $.  $d x ch $.
    rexlimddv.1 $e |- ( ph -> E. x e. A ps ) $.
    rexlimddv.2 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch ) $.
    $( Restricted existential elimination rule of natural deduction.
       (Contributed by Mario Carneiro, 15-Jun-2016.) $)
    rexlimddv $p |- ( ph -> ch ) $=
      ( wrex rexlimdvaa mpd ) ABDEHCFABCDEGIJ $.
  $}

  ${
    $d x y ps $.  $d y A $.
    rexlimivv.1 $e |- ( ( x e. A /\ y e. B ) -> ( ph -> ps ) ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by NM, 17-Feb-2004.) $)
    rexlimivv $p |- ( E. x e. A E. y e. B ph -> ps ) $=
      ( wrex cv wcel rexlimdva rexlimiv ) ADFHBCECIEJABDFGKL $.
  $}

  ${
    $d x y ph $.  $d x y ch $.  $d y A $.
    rexlimdvv.1 $e |- ( ph -> ( ( x e. A /\ y e. B ) -> ( ps -> ch ) ) ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 22-Jul-2004.) $)
    rexlimdvv $p |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) ) $=
      ( wrex cv wcel wa wi expdimp rexlimdv rexlimdva ) ABEGICDFADJFKZLBCEGAQEJ
      GKBCMHNOP $.
  $}

  ${
    $d x y ph $.  $d x y ch $.  $d y A $.
    rexlimdvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90.  (Restricted
       quantifier version.)  (Contributed by NM, 18-Jun-2014.) $)
    rexlimdvva $p |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) ) $=
      ( cv wcel wa wi ex rexlimdvv ) ABCDEFGADIFJEIGJKBCLHMN $.
  $}

  $( Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers.
     (Contributed by NM, 28-Jan-1997.)  (Proof shortened by Andrew Salmon,
     30-May-2011.) $)
  r19.26 $p |- ( A. x e. A ( ph /\ ps ) <->
               ( A. x e. A ph /\ A. x e. A ps ) ) $=
    ( wa wral simpl ralimi simpr jca pm3.2 ral2imi imp impbii ) ABEZCDFZACDFZBC
    DFZEPQROACDABGHOBCDABIHJQRPABOCDABKLMN $.

  ${
    $d x ps $.
    $( Restricted quantitifer version of one direction of ~ 19.27 .  (The other
       direction holds when ` A ` is inhabited, see ~ r19.27mv .)  (Contributed
       by NM, 3-Jun-2004.)  (Proof shortened by Andrew Salmon, 30-May-2011.)
       (Proof shortened by Wolf Lammen, 17-Jun-2023.) $)
    r19.27v $p |- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) ) $=
      ( wral wa id ralrimivw anim2i r19.26 sylibr ) ACDEZBFLBCDEZFABFCDEBMLBBCD
      BGHIABCDJK $.
  $}

  ${
    $d x ph $.
    $( Restricted quantifier version of one direction of ~ 19.28 .  (The other
       direction holds when ` A ` is inhabited, see ~ r19.28mv .)  (Contributed
       by NM, 2-Apr-2004.)  (Proof shortened by Wolf Lammen, 17-Jun-2023.) $)
    r19.28v $p |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) ) $=
      ( wral wa id ralrimivw anim1i r19.26 sylibr ) ABCDEZFACDEZLFABFCDEAMLAACD
      AGHIABCDJK $.
  $}

  $( Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers.
     (Contributed by NM, 10-Aug-2004.) $)
  r19.26-2 $p |- ( A. x e. A A. y e. B ( ph /\ ps ) <->
               ( A. x e. A A. y e. B ph /\ A. x e. A A. y e. B ps ) ) $=
    ( wa wral r19.26 ralbii bitri ) ABGDFHZCEHADFHZBDFHZGZCEHMCEHNCEHGLOCEABDFI
    JMNCEIK $.

  $( Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers.
     (Contributed by FL, 22-Nov-2010.) $)
  r19.26-3 $p |- ( A. x e. A ( ph /\ ps /\ ch ) <->
               ( A. x e. A ph /\ A. x e. A ps /\ A. x e. A ch ) ) $=
    ( w3a wral wa df-3an ralbii r19.26 anbi1i bitr4i 3bitri ) ABCFZDEGABHZCHZDE
    GPDEGZCDEGZHZADEGZBDEGZSFZOQDEABCIJPCDEKTUAUBHZSHUCRUDSABDEKLUAUBSIMN $.

  $( Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers.  (Contributed by
     NM, 22-Feb-2004.) $)
  r19.26m $p |- ( A. x ( ( x e. A -> ph ) /\ ( x e. B -> ps ) ) <->
               ( A. x e. A ph /\ A. x e. B ps ) ) $=
    ( cv wcel wi wa wal wral 19.26 df-ral anbi12i bitr4i ) CFZDGAHZPEGBHZICJQCJ
    ZRCJZIACDKZBCEKZIQRCLUASUBTACDMBCEMNO $.

  $( Distribute a restricted universal quantifier over a biconditional.
     Theorem 19.15 of [Margaris] p. 90 with restricted quantification.
     (Contributed by NM, 6-Oct-2003.) $)
  ralbi $p |- ( A. x e. A ( ph <-> ps ) ->
               ( A. x e. A ph <-> A. x e. A ps ) ) $=
    ( wb wral nfra1 cv wcel rsp imp ralbida ) ABEZCDFZABCDMCDGNCHDIMMCDJKL $.

  $( Distribute a restricted existential quantifier over a biconditional.
     Theorem 19.18 of [Margaris] p. 90 with restricted quantification.
     (Contributed by Jim Kingdon, 21-Jan-2019.) $)
  rexbi $p |- ( A. x e. A ( ph <-> ps ) ->
               ( E. x e. A ph <-> E. x e. A ps ) ) $=
    ( wb wral nfra1 cv wcel rsp imp rexbida ) ABEZCDFZABCDMCDGNCHDIMMCDJKL $.

  $( Split a biconditional and distribute quantifier.  (Contributed by NM,
     3-Jun-2012.) $)
  ralbiim $p |- ( A. x e. A ( ph <-> ps ) <->
             ( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) ) $=
    ( wb wral wi wa dfbi2 ralbii r19.26 bitri ) ABEZCDFABGZBAGZHZCDFNCDFOCDFHMP
    CDABIJNOCDKL $.

  ${
    $d x ps $.
    $( Restricted version of one direction of Theorem 19.27 of [Margaris]
       p. 90.  (The other direction doesn't hold when ` A ` is empty.)
       (Contributed by NM, 3-Jun-2004.)  (Proof shortened by Andrew Salmon,
       30-May-2011.) $)
    r19.27av $p |- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) ) $=
      ( wral wa cv wcel ax-1 ralrimiv anim2i r19.26 sylibr ) ACDEZBFNBCDEZFABFC
      DEBONBBCDBCGDHIJKABCDLM $.
  $}

  ${
    $d x ph $.
    $( Restricted version of one direction of Theorem 19.28 of [Margaris]
       p. 90.  (The other direction doesn't hold when ` A ` is empty.)
       (Contributed by NM, 2-Apr-2004.) $)
    r19.28av $p |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) ) $=
      ( wral wa r19.27av ancom ralbii 3imtr4i ) BCDEZAFBAFZCDEAKFABFZCDEBACDGAK
      HMLCDABHIJ $.
  $}

  $( Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers.
     (Contributed by NM, 31-Aug-1999.)  (Proof shortened by Andrew Salmon,
     30-May-2011.) $)
  r19.29 $p |- ( ( A. x e. A ph /\ E. x e. A ps ) ->
               E. x e. A ( ph /\ ps ) ) $=
    ( wral wrex wa wi pm3.2 ralimi rexim syl imp ) ACDEZBCDFZABGZCDFZNBPHZCDEOQ
    HARCDABIJBPCDKLM $.

  $( Variation of Theorem 19.29 of [Margaris] p. 90 with restricted
     quantifiers.  (Contributed by NM, 31-Aug-1999.) $)
  r19.29r $p |- ( ( E. x e. A ph /\ A. x e. A ps ) ->
                E. x e. A ( ph /\ ps ) ) $=
    ( wral wrex wa r19.29 ancom rexbii 3imtr4i ) BCDEZACDFZGBAGZCDFMLGABGZCDFBA
    CDHMLIONCDABIJK $.

  $( Relationship between two restricted universal and existential quantifiers.
     (Contributed by Glauco Siliprandi, 11-Dec-2019.)  (Proof shortened by Wolf
     Lammen, 18-May-2023.) $)
  ralnex2 $p |- ( A. x e. A A. y e. B -. ph <-> -. E. x e. A E. y e. B ph ) $=
    ( wn wral wrex ralnex ralbii bitri ) AFCEGZBDGACEHZFZBDGMBDHFLNBDACEIJMBDIK
    $.

  ${
    r19.29af2.p $e |- F/ x ph $.
    r19.29af2.c $e |- F/ x ch $.
    r19.29af2.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
    r19.29af2.2 $e |- ( ph -> E. x e. A ps ) $.
    $( A commonly used pattern based on ~ r19.29 .  (Contributed by Thierry
       Arnoux, 17-Dec-2017.) $)
    r19.29af2 $p |- ( ph -> ch ) $=
      ( wrex wi wral wa cv wcel exp31 ralrimi jca r19.29r pm3.35 rexlimi 3syl
      a1i ) ABDEJZBCKZDELZMBUEMZDEJCAUDUFIAUEDEFADNEOZBCHPQRBUEDESUGCDEGUGCKUHB
      CTUCUAUB $.
  $}

  ${
    $d x ch $.
    r19.29af.0 $e |- F/ x ph $.
    r19.29af.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
    r19.29af.2 $e |- ( ph -> E. x e. A ps ) $.
    $( A commonly used pattern based on ~ r19.29 .  (Contributed by Thierry
       Arnoux, 29-Nov-2017.) $)
    r19.29af $p |- ( ph -> ch ) $=
      ( nfv r19.29af2 ) ABCDEFCDIGHJ $.
  $}

  ${
    $d x ch $.  $d x ph $.
    r19.29an.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
    $( A commonly used pattern based on ~ r19.29 .  (Contributed by Thierry
       Arnoux, 29-Dec-2019.) $)
    r19.29an $p |- ( ( ph /\ E. x e. A ps ) -> ch ) $=
      ( wrex wa nfv nfre1 nfan cv wcel adantllr simpr r19.29af ) ABDEGZHBCDEAQD
      ADIBDEJKADLEMBCQFNAQOP $.
  $}

  ${
    $d x ch $.  $d x ph $.
    r19.29a.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
    r19.29a.2 $e |- ( ph -> E. x e. A ps ) $.
    $( A commonly used pattern based on ~ r19.29 .  (Contributed by Thierry
       Arnoux, 22-Nov-2017.) $)
    r19.29a $p |- ( ph -> ch ) $=
      ( nfv r19.29af ) ABCDEADHFGI $.
  $}

  ${
    r19.29d2r.1 $e |- ( ph -> A. x e. A A. y e. B ps ) $.
    r19.29d2r.2 $e |- ( ph -> E. x e. A E. y e. B ch ) $.
    $( Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers,
       deduction version.  (Contributed by Thierry Arnoux, 30-Jan-2017.) $)
    r19.29d2r $p |- ( ph -> E. x e. A E. y e. B ( ps /\ ch ) ) $=
      ( wral wrex wa r19.29 syl2anc reximi syl ) ABEGJZCEGKZLZDFKZBCLEGKZDFKAQD
      FJRDFKTHIQRDFMNSUADFBCEGMOP $.
  $}

  ${
    $d y A $.  $d x y ch $.  $d x y ph $.
    r19.29vva.1 $e |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch ) $.
    r19.29vva.2 $e |- ( ph -> E. x e. A E. y e. B ps ) $.
    $( A commonly used pattern based on ~ r19.29 , version with two restricted
       quantifiers.  (Contributed by Thierry Arnoux, 26-Nov-2017.) $)
    r19.29vva $p |- ( ph -> ch ) $=
      ( wi wa wrex wral cv wcel ex ralrimiva r19.29d2r pm3.35 rexlimivw ancoms
      syl ) ABCJZBKZEGLZDFLCAUCBDEFGAUCEGMDFADNFOKZUCEGUFENGOKBCHPQQIRUECDFUDCE
      GBUCCBCSUATTUB $.
  $}

  ${
    r19.32r.1 $e |- F/ x ph $.
    $( One direction of Theorem 19.32 of [Margaris] p. 90 with restricted
       quantifiers.  For decidable propositions this is an equivalence.
       (Contributed by Jim Kingdon, 19-Aug-2018.) $)
    r19.32r $p |- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) ) $=
      ( wral wo cv wcel wal orc a1d alrimi df-ral olc imim2i alimi sylbi jaoi
      wi sylibr ) ABCDFZGCHDIZABGZTZCJZUDCDFAUFUBAUECEAUDUCABKLMUBUCBTZCJUFBCDN
      UGUECBUDUCBAOPQRSUDCDNUA $.
  $}

  $( Restricted quantifier version of ~ 19.30dc .  (Contributed by Scott
     Fenton, 25-Feb-2011.)  (Proof shortened by Wolf Lammen, 18-Jun-2023.) $)
  r19.30dc $p |- ( ( A. x e. A ( ph \/ ps ) /\ DECID E. x e. A ps )
      -> ( A. x e. A ph \/ E. x e. A ps ) ) $=
    ( wo wral wdc wa wn wi ralnex pm2.53 orcoms ral2imi biimtrrid adantr dfordc
    wrex wb adantl mpbird orcomd ) ABEZCDFZBCDRZGZHZUEACDFZUGUEUHEZUEIZUHJZUDUK
    UFUJBIZCDFUDUHBCDKUCULACDBAULAJBALMNOPUFUIUKSUDUEUHQTUAUB $.

  ${
    $d x ph $.
    $( One direction of Theorem 19.32 of [Margaris] p. 90 with restricted
       quantifiers.  For decidable propositions this is an equivalence, as seen
       at ~ r19.32vdc .  (Contributed by Jim Kingdon, 19-Aug-2018.) $)
    r19.32vr $p |- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) ) $=
      ( nfv r19.32r ) ABCDACEF $.

    $( Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where
       ` ph ` is decidable.  (Contributed by Jim Kingdon, 4-Jun-2018.) $)
    r19.32vdc $p |- ( DECID ph ->
        ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) ) $=
      ( wdc wn wi wral wo wb r19.21v a1i dfordc ralbidv 3bitr4d ) AEZAFZBGZCDHZ
      QBCDHZGZABIZCDHATISUAJPQBCDKLPUBRCDABMNATMO $.
  $}

  $( Restricted quantifier version of ~ 19.35-1 .  (Contributed by Jim Kingdon,
     4-Jun-2018.) $)
  r19.35-1 $p |- ( E. x e. A ( ph -> ps ) ->
      ( A. x e. A ph -> E. x e. A ps ) ) $=
    ( wral wi wrex wa r19.29 pm3.35 reximi syl expcom ) ACDEZABFZCDGZBCDGZNPHAO
    HZCDGQAOCDIRBCDABJKLM $.

  ${
    $d x ps $.
    $( One direction of a restricted quantifier version of Theorem 19.36 of
       [Margaris] p. 90.  In classical logic, the converse would hold if ` A `
       has at least one element, but in intuitionistic logic, that is not a
       sufficient condition.  (Contributed by NM, 22-Oct-2003.) $)
    r19.36av $p |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) ) $=
      ( wi wrex wral r19.35-1 cv wcel idd rexlimiv imim2i syl ) ABECDFACDGZBCDF
      ZEOBEABCDHPBOBBCDCIDJBKLMN $.
  $}

  ${
    r19.37.1 $e |- F/ x ph $.
    $( Restricted version of one direction of Theorem 19.37 of [Margaris]
       p. 90.  In classical logic the converse would hold if ` A ` has at least
       one element, but that is not sufficient in intuitionistic logic.
       (Contributed by FL, 13-May-2012.)  (Revised by Mario Carneiro,
       11-Dec-2016.) $)
    r19.37 $p |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) ) $=
      ( wral wi wrex cv wcel ax-1 ralrimi r19.35-1 syl5 ) AACDFABGCDHBCDHAACDEA
      CIDJKLABCDMN $.
  $}

  ${
    $d x ph $.
    $( Restricted version of one direction of Theorem 19.37 of [Margaris]
       p. 90.  (Contributed by NM, 2-Apr-2004.) $)
    r19.37av $p |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) ) $=
      ( nfv r19.37 ) ABCDACEF $.
  $}

  $( Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90.
     (Contributed by NM, 2-Apr-2004.) $)
  r19.40 $p |- ( E. x e. A ( ph /\ ps ) ->
               ( E. x e. A ph /\ E. x e. A ps ) ) $=
    ( wa wrex simpl reximi simpr jca ) ABEZCDFACDFBCDFKACDABGHKBCDABIHJ $.

  ${
    r19.41.1 $e |- F/ x ps $.
    $( Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
       (Contributed by NM, 1-Nov-2010.) $)
    r19.41 $p |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) ) $=
      ( cv wcel wa wex wrex anass exbii 19.41 bitr3i df-rex anbi1i 3bitr4i ) CF
      DGZABHZHZCIZRAHZCIZBHZSCDJACDJZBHUAUBBHZCIUDUFTCRABKLUBBCEMNSCDOUEUCBACDO
      PQ $.
  $}

  ${
    $d x ps $.
    $( Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
       (Contributed by NM, 17-Dec-2003.) $)
    r19.41v $p |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) ) $=
      ( nfv r19.41 ) ABCDBCEF $.
  $}

  ${
    $d x ph $.
    $( Restricted version of Theorem 19.42 of [Margaris] p. 90.  (Contributed
       by NM, 27-May-1998.) $)
    r19.42v $p |- ( E. x e. A ( ph /\ ps ) <-> ( ph /\ E. x e. A ps ) ) $=
      ( wa wrex r19.41v ancom rexbii 3bitr4i ) BAEZCDFBCDFZAEABEZCDFALEBACDGMKC
      DABHIALHJ $.
  $}

  $( Restricted version of Theorem 19.43 of [Margaris] p. 90.  (Contributed by
     NM, 27-May-1998.)  (Proof rewritten by Jim Kingdon, 5-Jun-2018.) $)
  r19.43 $p |- ( E. x e. A ( ph \/ ps ) <->
               ( E. x e. A ph \/ E. x e. A ps ) ) $=
    ( wo wrex cv wcel wa wex df-rex andi exbii bitri 19.43 orbi12i bitr4i ) ABE
    ZCDFZCGDHZAIZCJZTBIZCJZEZACDFZBCDFZESUAUCEZCJZUESTRIZCJUIRCDKUJUHCTABLMNUAU
    CCONUFUBUGUDACDKBCDKPQ $.

  ${
    $d x ps $.
    $( One direction of a restricted quantifier version of Theorem 19.44 of
       [Margaris] p. 90.  The other direction doesn't hold when ` A ` is empty.
       (Contributed by NM, 2-Apr-2004.) $)
    r19.44av $p |- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ ps ) ) $=
      ( wo wrex r19.43 cv wcel idd rexlimiv orim2i sylbi ) ABECDFACDFZBCDFZENBE
      ABCDGOBNBBCDCHDIBJKLM $.
  $}

  ${
    $d x ph $.
    $( Restricted version of one direction of Theorem 19.45 of [Margaris]
       p. 90.  (The other direction doesn't hold when ` A ` is empty.)
       (Contributed by NM, 2-Apr-2004.) $)
    r19.45av $p |- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) ) $=
      ( wo wrex r19.43 cv wcel idd rexlimiv orim1i sylbi ) ABECDFACDFZBCDFZEAOE
      ABCDGNAOAACDCHDIAJKLM $.
  $}

  ${
    $d x y $.
    ralcomf.1 $e |- F/_ y A $.
    ralcomf.2 $e |- F/_ x B $.
    $( Commutation of restricted quantifiers.  (Contributed by Mario Carneiro,
       14-Oct-2016.) $)
    ralcomf $p |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph ) $=
      ( cv wcel wa wi wal wral ancomsimp 2albii alcom bitri r2alf 3bitr4i ) BHD
      IZCHEIZJAKZCLBLZUATJAKZBLCLZACEMBDMABDMCEMUCUDCLBLUEUBUDBCTUAANOUDBCPQABC
      DEFRACBEDGRS $.

    $( Commutation of restricted quantifiers.  (Contributed by Mario Carneiro,
       14-Oct-2016.) $)
    rexcomf $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $=
      ( cv wcel wa wex wrex ancom anbi1i 2exbii excom bitri r2exf 3bitr4i ) BHD
      IZCHEIZJZAJZCKBKZUATJZAJZBKCKZACELBDLABDLCELUDUFCKBKUGUCUFBCUBUEATUAMNOUF
      BCPQABCDEFRACBEDGRS $.
  $}

  ${
    $d x y $.  $d x B $.  $d y A $.
    $( Commutation of restricted quantifiers.  (Contributed by NM,
       13-Oct-1999.)  (Revised by Mario Carneiro, 14-Oct-2016.) $)
    ralcom $p |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph ) $=
      ( nfcv ralcomf ) ABCDECDFBEFG $.

    $( Commutation of restricted quantifiers.  (Contributed by NM,
       19-Nov-1995.)  (Revised by Mario Carneiro, 14-Oct-2016.) $)
    rexcom $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $=
      ( nfcv rexcomf ) ABCDECDFBEFG $.
  $}

  ${
    $d z A $.  $d z B $.  $d x C $.  $d y C $.  $d x z $.  $d y z $.
    $( Rotate three restricted universal quantifiers.  (Contributed by AV,
       3-Dec-2021.) $)
    ralrot3 $p |- ( A. x e. A A. y e. B A. z e. C ph
                    <-> A. z e. C A. x e. A A. y e. B ph ) $=
      ( wral ralcom ralbii bitri ) ADGHCFHZBEHACFHZDGHZBEHMBEHDGHLNBEACDFGIJMBD
      EGIK $.
  $}

  ${
    $d y z A $.  $d x z B $.  $d x y C $.
    $( Swap 1st and 3rd restricted existential quantifiers.  (Contributed by
       NM, 8-Apr-2015.) $)
    rexcom13 $p |- ( E. x e. A E. y e. B E. z e. C ph
         <-> E. z e. C E. y e. B E. x e. A ph ) $=
      ( wrex rexcom rexbii 3bitri ) ADGHZCFHBEHLBEHZCFHABEHZDGHZCFHNCFHDGHLBCEF
      IMOCFABDEGIJNCDFGIK $.
  $}

  ${
    $d w z A $.  $d w z B $.  $d w x y C $.  $d x y z D $.
    $( Rotate existential restricted quantifiers twice.  (Contributed by NM,
       8-Apr-2015.) $)
    rexrot4 $p |- ( E. x e. A E. y e. B E. z e. C E. w e. D ph
        <-> E. z e. C E. w e. D E. x e. A E. y e. B ph ) $=
      ( wrex rexcom13 rexbii bitri ) AEIJDHJCGJZBFJACGJZDHJEIJZBFJOBFJEIJDHJNPB
      FACDEGHIKLOBEDFIHKM $.
  $}

  $( A commutative law for restricted quantifiers that swaps the domain of the
     restriction.  (Contributed by NM, 22-Feb-2004.) $)
  ralcom3 $p |- ( A. x e. A ( x e. B -> ph ) <->
                  A. x e. B ( x e. A -> ph ) ) $=
    ( cv wcel wi wral pm2.04 ralimi2 impbii ) BEZDFZAGZBCHLCFZAGZBDHNPBCDOMAIJP
    NBDCMOAIJK $.

  ${
    $d y A $.  $d x B $.  $d x y $.
    reean.1 $e |- F/ y ph $.
    reean.2 $e |- F/ x ps $.
    $( Rearrange existential quantifiers.  (Contributed by NM, 27-Oct-2010.)
       (Proof shortened by Andrew Salmon, 30-May-2011.) $)
    reean $p |- ( E. x e. A E. y e. B ( ph /\ ps ) <->
                 ( E. x e. A ph /\ E. y e. B ps ) ) $=
      ( cv wcel wa wex wrex an4 2exbii nfv nfan eean bitri df-rex r2ex anbi12i
      3bitr4i ) CIEJZDIFJZKABKZKZDLCLZUDAKZCLZUEBKZDLZKZUFDFMCEMACEMZBDFMZKUHUI
      UKKZDLCLUMUGUPCDUDUEABNOUIUKCDUDADUDDPGQUEBCUECPHQRSUFCDEFUAUNUJUOULACETB
      DFTUBUC $.
  $}

  ${
    $d y ph $.  $d x ps $.  $d x y $.  $d y A $.  $d x B $.
    $( Rearrange existential quantifiers.  (Contributed by NM, 9-May-1999.) $)
    reeanv $p |- ( E. x e. A E. y e. B ( ph /\ ps ) <->
                 ( E. x e. A ph /\ E. y e. B ps ) ) $=
      ( nfv reean ) ABCDEFADGBCGH $.
  $}

  ${
    $d ph y z $.  $d ps x z $.  $d ch x y $.  $d A y $.  $d B x z $.
    $d C x y $.
    $( Rearrange three existential quantifiers.  (Contributed by Jeff Madsen,
       11-Jun-2010.) $)
    3reeanv $p |- ( E. x e. A E. y e. B E. z e. C ( ph /\ ps /\ ch )
                      <-> ( E. x e. A ph /\ E. y e. B ps /\ E. z e. C ch ) ) $=
      ( wa wrex w3a r19.41v reeanv anbi1i bitri df-3an 2rexbii rexbii 3bitr4i )
      ABJZEHKZCFIKZJZDGKZADGKZBEHKZJZUCJZABCLZFIKEHKZDGKUFUGUCLUEUBDGKZUCJUIUBU
      CDGMULUHUCABDEGHNOPUKUDDGUKUACJZFIKEHKUDUJUMEFHIABCQRUACEFHINPSUFUGUCQT
      $.
  $}

  $( ` x ` is not free in ` E! x e. A ph ` .  (Contributed by NM,
     19-Mar-1997.) $)
  nfreu1 $p |- F/ x E! x e. A ph $=
    ( wreu cv wcel wa weu df-reu nfeu1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.

  $( ` x ` is not free in ` E* x e. A ph ` .  (Contributed by NM,
     16-Jun-2017.) $)
  nfrmo1 $p |- F/ x E* x e. A ph $=
    ( wrmo cv wcel wa wmo df-rmo nfmo1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.

  ${
    $d x y $.
    nfreudxy.1 $e |- F/ y ph $.
    nfreudxy.2 $e |- ( ph -> F/_ x A ) $.
    nfreudxy.3 $e |- ( ph -> F/ x ps ) $.
    $( Not-free deduction for restricted uniqueness.  This is a version where
       ` x ` and ` y ` are distinct.  (Contributed by Jim Kingdon,
       6-Jun-2018.) $)
    nfreudxy $p |- ( ph -> F/ x E! y e. A ps ) $=
      ( cv wcel wa weu wnf wreu wnfc nfcv a1i nfeld nfand nfeud df-reu sylibr
      nfbii ) ADIZEJZBKZDLZCMBDENZCMAUFCDFAUEBCACUDECUDOACUDPQGRHSTUHUGCBDEUAUC
      UB $.
  $}

  ${
    $d x y $.
    nfreuw.1 $e |- F/_ x A $.
    nfreuw.2 $e |- F/ x ph $.
    $( Not-free for restricted uniqueness.  This is a version where ` x ` and
       ` y ` are distinct.  (Contributed by Jim Kingdon, 6-Jun-2018.) $)
    nfreuw $p |- F/ x E! y e. A ph $=
      ( wreu wnf wtru nftru wnfc a1i nfreudxy mptru ) ACDGBHIABCDCJBDKIELABHIFL
      MN $.
  $}

  $( An "identity" law of concretion for restricted abstraction.  Special case
     of Definition 2.1 of [Quine] p. 16.  (Contributed by NM, 9-Oct-2003.) $)
  rabid $p |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) ) $=
    ( cv wcel wa crab df-rab abeq2i ) BDCEAFBABCGABCHI $.

  ${
    reqabi.1 $e |- A = { x e. B | ph } $.
    $( Inference from equality of a class variable and a restricted class
       abstraction.  (Contributed by NM, 16-Feb-2004.) $)
    reqabi $p |- ( x e. A <-> ( x e. B /\ ph ) ) $=
      ( cv wcel crab wa eleq2i rabid bitri ) BFZCGMABDHZGMDGAICNMEJABDKL $.
  $}

  ${
    $d x A $.
    $( An "identity" law for restricted class abstraction.  (Contributed by NM,
       9-Oct-2003.)  (Proof shortened by Andrew Salmon, 30-May-2011.) $)
    rabid2 $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
      ( cv wcel wa cab wceq wi wal crab abeq2 pm4.71 albii bitr4i df-rab eqeq2i
      wral wb df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCRUEUBUCSZBJUGUCB
      CLUFUIBUBAMNOUHUDCABCPQABCTUA $.
  $}

  $( Equivalent wff's correspond to equal restricted class abstractions.
     Closed theorem form of ~ rabbidva .  (Contributed by NM, 25-Nov-2013.) $)
  rabbi $p |- ( A. x e. A ( ps <-> ch )
       <-> { x e. A | ps } = { x e. A | ch } ) $=
    ( cv wcel wa wb wal cab wceq wral abbibcom df-ral pm5.32 albii bitri df-rab
    crab wi eqeq12i 3bitr4i ) CEDFZAGZUCBGZHZCIZUDCJZUECJZKABHZCDLZACDSZBCDSZKU
    DUECMUKUCUJTZCIUGUJCDNUNUFCUCABOPQULUHUMUIACDRBCDRUAUB $.

  $( Swap with a membership relation in a restricted class abstraction.
     (Contributed by NM, 4-Jul-2005.) $)
  rabswap $p |- { x e. A | x e. B } = { x e. B | x e. A } $=
    ( cv wcel wa cab crab ancom abbii df-rab 3eqtr4i ) ADZBEZMCEZFZAGONFZAGOABH
    NACHPQANOIJOABKNACKL $.

  $( The abstraction variable in a restricted class abstraction isn't free.
     (Contributed by NM, 19-Mar-1997.) $)
  nfrab1 $p |- F/_ x { x e. A | ph } $=
    ( crab cv wcel wa cab df-rab nfab1 nfcxfr ) BABCDBECFAGZBHABCILBJK $.

  ${
    $d x y $.
    nfrabw.1 $e |- F/ x ph $.
    nfrabw.2 $e |- F/_ x A $.
    $( A variable not free in a wff remains so in a restricted class
       abstraction.  (Contributed by Jim Kingdon, 19-Jul-2018.) $)
    nfrabw $p |- F/_ x { y e. A | ph } $=
      ( crab cv wcel wa cab df-rab nfcri nfan nfab nfcxfr ) BACDGCHDIZAJZCKACDL
      RBCQABBCDFMENOP $.
  $}

  ${
    reubida.1 $e |- F/ x ph $.
    reubida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by Mario Carneiro, 19-Nov-2016.) $)
    reubida $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
      ( cv wcel wa weu wreu pm5.32da eubid df-reu 3bitr4g ) ADHEIZBJZDKQCJZDKBD
      ELCDELARSDFAQBCGMNBDEOCDEOP $.
  $}

  ${
    $d x y A $.
    cbvrmow.1 $e |- F/ y ph $.
    cbvrmow.2 $e |- F/ x ps $.
    cbvrmow.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a restricted at-most-one quantifier using
       implicit substitution.  Version of ~ cbvrmo with a disjoint variable
       condition.  (Contributed by NM, 16-Jun-2017.)  (Revised by GG,
       23-May-2024.) $)
    cbvrmow $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
      ( cv wcel wa wmo wrmo nfv nfan weq eleq1w anbi12d cbvmow df-rmo 3bitr4i )
      CIEJZAKZCLDIEJZBKZDLACEMBDEMUCUECDUBADUBDNFOUDBCUDCNGOCDPUBUDABCDEQHRSACE
      TBDETUA $.
  $}

  ${
    $d x ph $.
    reubidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 13-Nov-2004.) $)
    reubidva $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
      ( nfv reubida ) ABCDEADGFH $.
  $}

  ${
    $d x ph $.
    reubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 17-Oct-1996.) $)
    reubidv $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
      ( wb cv wcel adantr reubidva ) ABCDEABCGDHEIFJK $.
  $}

  ${
    reubiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
    $( Formula-building rule for restricted existential quantifier (inference
       form).  (Contributed by NM, 14-Nov-2004.) $)
    reubiia $p |- ( E! x e. A ph <-> E! x e. A ps ) $=
      ( cv wcel wa weu wreu pm5.32i eubii df-reu 3bitr4i ) CFDGZAHZCIOBHZCIACDJ
      BCDJPQCOABEKLACDMBCDMN $.
  $}

  ${
    reubii.1 $e |- ( ph <-> ps ) $.
    $( Formula-building rule for restricted existential quantifier (inference
       form).  (Contributed by NM, 22-Oct-1999.) $)
    reubii $p |- ( E! x e. A ph <-> E! x e. A ps ) $=
      ( wb cv wcel a1i reubiia ) ABCDABFCGDHEIJ $.
  $}

  ${
    rmobida.1 $e |- F/ x ph $.
    rmobida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 16-Jun-2017.) $)
    rmobida $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
      ( cv wcel wa wmo wrmo pm5.32da mobid df-rmo 3bitr4g ) ADHEIZBJZDKQCJZDKBD
      ELCDELARSDFAQBCGMNBDEOCDEOP $.
  $}

  ${
    $d x ph $.
    rmobidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 16-Jun-2017.) $)
    rmobidva $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
      ( nfv rmobida ) ABCDEADGFH $.
  $}

  ${
    $d x ph $.
    rmobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 16-Jun-2017.) $)
    rmobidv $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
      ( wb cv wcel adantr rmobidva ) ABCDEABCGDHEIFJK $.
  $}

  ${
    rmobiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
    $( Formula-building rule for restricted existential quantifier (inference
       form).  (Contributed by NM, 16-Jun-2017.) $)
    rmobiia $p |- ( E* x e. A ph <-> E* x e. A ps ) $=
      ( cv wcel wa wmo wrmo pm5.32i mobii df-rmo 3bitr4i ) CFDGZAHZCIOBHZCIACDJ
      BCDJPQCOABEKLACDMBCDMN $.
  $}

  ${
    rmobii.1 $e |- ( ph <-> ps ) $.
    $( Formula-building rule for restricted existential quantifier (inference
       form).  (Contributed by NM, 16-Jun-2017.) $)
    rmobii $p |- ( E* x e. A ph <-> E* x e. A ps ) $=
      ( wb cv wcel a1i rmobiia ) ABCDABFCGDHEIJ $.
  $}

  ${
    raleq1f.1 $e |- F/_ x A $.
    raleq1f.2 $e |- F/_ x B $.
    $( Equality theorem for restricted universal quantifier, with
       bound-variable hypotheses instead of distinct variable restrictions.
       (Contributed by NM, 7-Mar-2004.)  (Revised by Andrew Salmon,
       11-Jul-2011.) $)
    raleqf $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $=
      ( wceq cv wcel wi wal wral nfeq eleq2 imbi1d albid df-ral 3bitr4g ) CDGZB
      HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.

    $( Equality theorem for restricted existential quantifier, with
       bound-variable hypotheses instead of distinct variable restrictions.
       (Contributed by NM, 9-Oct-2003.)  (Revised by Andrew Salmon,
       11-Jul-2011.) $)
    rexeqf $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
      ( wceq cv wcel wa wex wrex nfeq eleq2 anbi1d exbid df-rex 3bitr4g ) CDGZB
      HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.

    $( Equality theorem for restricted unique existential quantifier, with
       bound-variable hypotheses instead of distinct variable restrictions.
       (Contributed by NM, 5-Apr-2004.)  (Revised by Andrew Salmon,
       11-Jul-2011.) $)
    reueq1f $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $=
      ( wceq cv wcel wa weu wreu nfeq eleq2 anbi1d eubid df-reu 3bitr4g ) CDGZB
      HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.

    $( Equality theorem for restricted at-most-one quantifier, with
       bound-variable hypotheses instead of distinct variable restrictions.
       (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
    rmoeq1f $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $=
      ( wceq cv wcel wa wmo wrmo nfeq eleq2 anbi1d mobid df-rmo 3bitr4g ) CDGZB
      HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Equality theorem for restricted universal quantifier.  (Contributed by
       NM, 16-Nov-1995.) $)
    raleq $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $=
      ( nfcv raleqf ) ABCDBCEBDEF $.

    $( Equality theorem for restricted existential quantifier.  (Contributed by
       NM, 29-Oct-1995.) $)
    rexeq $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
      ( nfcv rexeqf ) ABCDBCEBDEF $.

    $( Equality theorem for restricted unique existential quantifier.
       (Contributed by NM, 5-Apr-2004.) $)
    reueq1 $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $=
      ( nfcv reueq1f ) ABCDBCEBDEF $.

    $( Equality theorem for restricted at-most-one quantifier.  (Contributed by
       Alexander van der Vekens, 17-Jun-2017.) $)
    rmoeq1 $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $=
      ( nfcv rmoeq1f ) ABCDBCEBDEF $.
  $}

  ${
    $d A x $.  $d B x $.
    raleq1i.1 $e |- A = B $.
    $( Equality inference for restricted universal qualifier.  (Contributed by
       Paul Chapman, 22-Jun-2011.) $)
    raleqi $p |- ( A. x e. A ph <-> A. x e. B ph ) $=
      ( wceq wral wb raleq ax-mp ) CDFABCGABDGHEABCDIJ $.

    $( Equality inference for restricted existential qualifier.  (Contributed
       by Mario Carneiro, 23-Apr-2015.) $)
    rexeqi $p |- ( E. x e. A ph <-> E. x e. B ph ) $=
      ( wceq wrex wb rexeq ax-mp ) CDFABCGABDGHEABCDIJ $.
  $}

  ${
    $d x A $.  $d x B $.
    raleq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for restricted universal quantifier.  (Contributed by
       NM, 13-Nov-2005.) $)
    raleqdv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) ) $=
      ( wceq wral wb raleq syl ) ADEGBCDHBCEHIFBCDEJK $.

    $( Equality deduction for restricted existential quantifier.  (Contributed
       by NM, 14-Jan-2007.) $)
    rexeqdv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) ) $=
      ( wceq wrex wb rexeq syl ) ADEGBCDHBCEHIFBCDEJK $.
  $}

  ${
    $d A x $.  $d B x $.
    raleqtrdv.1 $e |- ( ph -> A. x e. A ps ) $.
    raleqtrdv.2 $e |- ( ph -> A = B ) $.
    $( Substitution of equal classes into a restricted universal quantifier.
       (Contributed by Matthew House, 21-Jul-2025.) $)
    raleqtrdv $p |- ( ph -> A. x e. B ps ) $=
      ( wral raleqdv mpbid ) ABCDHBCEHFABCDEGIJ $.
  $}

  ${
    $d A x $.  $d B x $.
    rexeqtrdv.1 $e |- ( ph -> E. x e. A ps ) $.
    rexeqtrdv.2 $e |- ( ph -> A = B ) $.
    $( Substitution of equal classes into a restricted existential quantifier.
       (Contributed by Matthew House, 21-Jul-2025.) $)
    rexeqtrdv $p |- ( ph -> E. x e. B ps ) $=
      ( wrex rexeqdv mpbid ) ABCDHBCEHFABCDEGIJ $.
  $}

  ${
    $d A x $.  $d B x $.
    raleqtrrdv.1 $e |- ( ph -> A. x e. A ps ) $.
    raleqtrrdv.2 $e |- ( ph -> B = A ) $.
    $( Substitution of equal classes into a restricted universal quantifier.
       (Contributed by Matthew House, 21-Jul-2025.) $)
    raleqtrrdv $p |- ( ph -> A. x e. B ps ) $=
      ( wral raleqdv mpbird ) ABCEHBCDHFABCEDGIJ $.
  $}

  ${
    $d A x $.  $d B x $.
    rexeqtrrdv.1 $e |- ( ph -> E. x e. A ps ) $.
    rexeqtrrdv.2 $e |- ( ph -> B = A ) $.
    $( Substitution of equal classes into a restricted existential quantifier.
       (Contributed by Matthew House, 21-Jul-2025.) $)
    rexeqtrrdv $p |- ( ph -> E. x e. B ps ) $=
      ( wrex rexeqdv mpbird ) ABCEHBCDHFABCEDGIJ $.
  $}

  ${
    $d x A $.  $d x B $.
    raleqd.1 $e |- ( A = B -> ( ph <-> ps ) ) $.
    $( Equality deduction for restricted universal quantifier.  (Contributed by
       NM, 16-Nov-1995.) $)
    raleqbi1dv $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ps ) ) $=
      ( wceq wral raleq ralbidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.

    $( Equality deduction for restricted existential quantifier.  (Contributed
       by NM, 18-Mar-1997.) $)
    rexeqbi1dv $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) ) $=
      ( wceq wrex rexeq rexbidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.

    $( Equality deduction for restricted unique existential quantifier.
       (Contributed by NM, 5-Apr-2004.) $)
    reueqd $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) ) $=
      ( wceq wreu reueq1 reubidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.

    $( Equality deduction for restricted at-most-one quantifier.  (Contributed
       by Alexander van der Vekens, 17-Jun-2017.) $)
    rmoeqd $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) ) $=
      ( wceq wrmo rmoeq1 rmobidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    raleqbidv.1 $e |- ( ph -> A = B ) $.
    raleqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equality deduction for restricted universal quantifier.  (Contributed by
       NM, 6-Nov-2007.) $)
    raleqbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
      ( wral raleqdv ralbidv bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.

    $( Equality deduction for restricted universal quantifier.  (Contributed by
       NM, 6-Nov-2007.) $)
    rexeqbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
      ( wrex rexeqdv rexbidv bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    raleqbidva.1 $e |- ( ph -> A = B ) $.
    raleqbidva.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Equality deduction for restricted universal quantifier.  (Contributed by
       Mario Carneiro, 5-Jan-2017.) $)
    raleqbidva $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
      ( wral ralbidva raleqdv bitrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.

    $( Equality deduction for restricted universal quantifier.  (Contributed by
       Mario Carneiro, 5-Jan-2017.) $)
    rexeqbidva $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
      ( wrex rexbidva rexeqdv bitrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.
  $}

  $( Unrestricted "at most one" implies restricted "at most one".  (Contributed
     by NM, 16-Jun-2017.) $)
  mormo $p |- ( E* x ph -> E* x e. A ph ) $=
    ( wmo cv wcel wa wrmo moan df-rmo sylibr ) ABDBECFZAGBDABCHALBIABCJK $.

  $( Restricted uniqueness in terms of "at most one".  (Contributed by NM,
     23-May-1999.)  (Revised by NM, 16-Jun-2017.) $)
  reu5 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) ) $=
    ( cv wcel wa weu wex wmo wreu wrex eu5 df-reu df-rex df-rmo anbi12i 3bitr4i
    wrmo ) BDCEAFZBGSBHZSBIZFABCJABCKZABCRZFSBLABCMUBTUCUAABCNABCOPQ $.

  $( Restricted unique existence implies restricted existence.  (Contributed by
     NM, 19-Aug-1999.) $)
  reurex $p |- ( E! x e. A ph -> E. x e. A ph ) $=
    ( wreu wrex wrmo reu5 simplbi ) ABCDABCEABCFABCGH $.

  $( Restricted existential uniqueness implies restricted "at most one."
     (Contributed by NM, 16-Jun-2017.) $)
  reurmo $p |- ( E! x e. A ph -> E* x e. A ph ) $=
    ( wreu wrex wrmo reu5 simprbi ) ABCDABCEABCFABCGH $.

  $( Restricted "at most one" in term of uniqueness.  (Contributed by NM,
     16-Jun-2017.) $)
  rmo5 $p |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) ) $=
    ( cv wcel wa wmo wex weu wrmo wrex wreu df-mo df-rmo df-rex imbi12i 3bitr4i
    wi df-reu ) BDCEAFZBGTBHZTBIZRABCJABCKZABCLZRTBMABCNUCUAUDUBABCOABCSPQ $.

  $( Nonexistence implies restricted "at most one".  (Contributed by NM,
     17-Jun-2017.) $)
  nrexrmo $p |- ( -. E. x e. A ph -> E* x e. A ph ) $=
    ( wrex wn wreu wi wrmo pm2.21 rmo5 sylibr ) ABCDZELABCFZGABCHLMIABCJK $.

  ${
    $d x y $.
    cbvralfw.1 $e |- F/_ x A $.
    cbvralfw.2 $e |- F/_ y A $.
    cbvralfw.3 $e |- F/ y ph $.
    cbvralfw.4 $e |- F/ x ps $.
    cbvralfw.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvralf with a disjoint variable condition.  Although we
       don't do so yet, we expect this disjoint variable condition will allow
       us to remove reliance on ~ ax-i12 and ~ ax-bndl in the proof.
       (Contributed by NM, 7-Mar-2004.)  (Revised by GG, 23-May-2024.) $)
    cbvralfw $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( cv wcel wi wal wral nfcri nfim weq eleq1w df-ral imbi12d cbvalv1
      3bitr4i ) CKELZAMZCNDKELZBMZDNACEOBDEOUEUGCDUDADDCEGPHQUFBCCDEFPIQCDRUDUF
      ABCDESJUAUBACETBDETUC $.
  $}

  ${
    $d x y $.
    cbvrexfw.1 $e |- F/_ x A $.
    cbvrexfw.2 $e |- F/_ y A $.
    cbvrexfw.3 $e |- F/ y ph $.
    cbvrexfw.4 $e |- F/ x ps $.
    cbvrexfw.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvrexf with a disjoint variable condition.  Although we
       don't do so yet, we expect this disjoint variable condition will allow
       us to remove reliance on ~ ax-i12 and ~ ax-bndl in the proof.
       (Contributed by FL, 27-Apr-2008.)  (Revised by GG, 10-Jan-2024.) $)
    cbvrexfw $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( cv wcel wa wex wrex nfcri nfan weq eleq1w df-rex anbi12d cbvexv1
      3bitr4i ) CKELZAMZCNDKELZBMZDNACEOBDEOUEUGCDUDADDCEGPHQUFBCCDEFPIQCDRUDUF
      ABCDESJUAUBACETBDETUC $.
  $}

  ${
    $d x z $.  $d y z $.  $d z A $.  $d z ps $.  $d z ph $.
    cbvralf.1 $e |- F/_ x A $.
    cbvralf.2 $e |- F/_ y A $.
    cbvralf.3 $e |- F/ y ph $.
    cbvralf.4 $e |- F/ x ps $.
    cbvralf.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 7-Mar-2004.)  (Revised by Mario Carneiro,
       9-Oct-2016.) $)
    cbvralf $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( vz cv wcel wi wal wral wsb nfv nfcri nfim nfs1v weq eleq1 sbequ12 cbval
      imbi12d nfsb sbequ sbie bitrdi bitri df-ral 3bitr4i ) CLZEMZANZCOZDLZEMZB
      NZDOZACEPBDEPUQKLZEMZACKQZNZKOVAUPVECKUPKRVCVDCCKEFSACKUATCKUBUOVCAVDUNVB
      EUCACKUDUFUEVEUTKDVCVDDDKEGSACKDHUGTUTKRKDUBZVCUSVDBVBUREUCVFVDACDQBAKDCU
      HABCDIJUIUJUFUEUKACEULBDEULUM $.

    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by FL, 27-Apr-2008.)  (Revised by Mario Carneiro,
       9-Oct-2016.)  (Proof rewritten by Jim Kingdon, 10-Jun-2018.) $)
    cbvrexf $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( vz cv wcel wa wex wrex wsb nfv nfcri nfan nfs1v weq eleq1 sbequ12 cbvex
      anbi12d nfsb sbequ sbie bitrdi bitri df-rex 3bitr4i ) CLZEMZANZCOZDLZEMZB
      NZDOZACEPBDEPUQKLZEMZACKQZNZKOVAUPVECKUPKRVCVDCCKEFSACKUATCKUBUOVCAVDUNVB
      EUCACKUDUFUEVEUTKDVCVDDDKEGSACKDHUGTUTKRKDUBZVCUSVDBVBUREUCVFVDACDQBAKDCU
      HABCDIJUIUJUFUEUKACEULBDEULUM $.
  $}

  ${
    $d x y A $.
    cbvralw.1 $e |- F/ y ph $.
    cbvralw.2 $e |- F/ x ps $.
    cbvralw.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvral with a disjoint variable condition.  Although we
       don't do so yet, we expect this disjoint variable condition will allow
       us to remove reliance on ~ ax-i12 and ~ ax-bndl in the proof.
       (Contributed by NM, 31-Jul-2003.)  (Revised by GG, 10-Jan-2024.) $)
    cbvralw $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( nfcv cbvralfw ) ABCDECEIDEIFGHJ $.

    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvrexfw with more disjoint variable conditions.  Although
       we don't do so yet, we expect the disjoint variable conditions will
       allow us to remove reliance on ~ ax-i12 and ~ ax-bndl in the proof.
       (Contributed by NM, 31-Jul-2003.)  (Revised by GG, 10-Jan-2024.) $)
    cbvrexw $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( nfcv cbvrexfw ) ABCDECEIDEIFGHJ $.
  $}

  ${
    $d A x y $.
    cbvreuw.1 $e |- F/ y ph $.
    cbvreuw.2 $e |- F/ x ps $.
    cbvreuw.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a restricted unique existential quantifier
       using implicit substitution.  Version of ~ cbvreu with a disjoint
       variable condition.  (Contributed by Mario Carneiro, 15-Oct-2016.)
       (Revised by GG, 10-Jan-2024.)  (Revised by Wolf Lammen, 10-Dec-2024.) $)
    cbvreuw $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
      ( wrex wrmo wa wreu cbvrexw cbvrmow anbi12i reu5 3bitr4i ) ACEIZACEJZKBDE
      IZBDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $.
  $}

  ${
    $d x z A $.  $d y z A $.  $d z ph $.  $d z ps $.
    cbvral.1 $e |- F/ y ph $.
    cbvral.2 $e |- F/ x ps $.
    cbvral.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 31-Jul-2003.) $)
    cbvral $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( nfcv cbvralf ) ABCDECEIDEIFGHJ $.

    $( Rule used to change bound variables, using implicit substitution.
       (Contributed by NM, 31-Jul-2003.)  (Proof shortened by Andrew Salmon,
       8-Jun-2011.) $)
    cbvrex $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( nfcv cbvrexf ) ABCDECEIDEIFGHJ $.

    $( Change the bound variable of a restricted unique existential quantifier
       using implicit substitution.  (Contributed by Mario Carneiro,
       15-Oct-2016.) $)
    cbvreu $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
      ( vz cv wcel wa weu wreu wsb nfv sb8eu sban eubii df-reu anbi1i nfsb nfan
      clelsb1 weq eleq1 sbequ sbie bitrdi anbi12d cbveu bitri 3bitri 3bitr4i )
      CJEKZALZCMZDJZEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZVAUPCIUPIPQVBVEI
      UOACIRSVFIJZEKZVDLZIMVAVEVIIVCVHVDCIEUDUASVIUTIDVHVDDVHDPACIDFUBUCUTIPIDU
      EZVHUSVDBVGUREUFVJVDACDOBAIDCUGABCDGHUHUIUJUKULUMACETBDETUN $.

    $( Change the bound variable of restricted "at most one" using implicit
       substitution.  (Contributed by NM, 16-Jun-2017.) $)
    cbvrmo $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
      ( wrex wreu wi wrmo cbvrex cbvreu imbi12i rmo5 3bitr4i ) ACEIZACEJZKBDEIZ
      BDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $.
  $}

  ${
    $d x A $.  $d y A $.  $d y ph $.  $d x ps $.
    cbvralv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a restricted universal quantifier using
       implicit substitution.  (Contributed by NM, 28-Jan-1997.) $)
    cbvralv $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( nfv cbvral ) ABCDEADGBCGFH $.

    $( Change the bound variable of a restricted existential quantifier using
       implicit substitution.  (Contributed by NM, 2-Jun-1998.) $)
    cbvrexv $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( nfv cbvrex ) ABCDEADGBCGFH $.

    $( Change the bound variable of a restricted unique existential quantifier
       using implicit substitution.  (Contributed by NM, 5-Apr-2004.)  (Revised
       by Mario Carneiro, 15-Oct-2016.) $)
    cbvreuv $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
      ( nfv cbvreu ) ABCDEADGBCGFH $.

    $( Change the bound variable of a restricted at-most-one quantifier using
       implicit substitution.  (Contributed by Alexander van der Vekens,
       17-Jun-2017.) $)
    cbvrmov $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
      ( nfv cbvrmo ) ABCDEADGBCGFH $.
  $}

  ${
    $d x y A $.  $d y ph $.  $d x ps $.
    cbvralvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ cbvralv with a disjoint variable condition.  (Contributed
       by GG, 10-Jan-2024.)  Reduce axiom usage.  (Revised by GG,
       25-Aug-2024.) $)
    cbvralvw $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( cv wcel wi wal wral weq eleq1w imbi12d cbvalvw df-ral 3bitr4i ) CGEHZAI
      ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $.
    $( $j usage 'cbvralvw' avoids 'ax-io' 'ax-7' 'ax-10' 'ax-11' 'ax-i12'
       'ax-bndl' 'ax-i5r' 'ax-ext'; $)

    $( Version of ~ cbvrexv with a disjoint variable condition.  (Contributed
       by GG, 10-Jan-2024.)  Reduce axiom usage.  (Revised by GG,
       25-Aug-2024.) $)
    cbvrexvw $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( cv wcel wa wex wrex weq eleq1w anbi12d cbvexvw df-rex 3bitr4i ) CGEHZAI
      ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $.
    $( $j usage 'cbvralvw' avoids 'ax-io' 'ax-7' 'ax-10' 'ax-11' 'ax-i12'
       'ax-bndl' 'ax-i5r' 'ax-ext'; $)

    $d x y z A $.  $d ph z $.  $d ps z $.
    $( Version of ~ cbvreuv with a disjoint variable condition.  (Contributed
       by GG, 10-Jan-2024.)  Reduce axiom usage.  (Revised by GG,
       25-Aug-2024.) $)
    cbvreuvw $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
      ( vz cv wcel wa weu wreu weq wb wal wex eleq1w df-eu 3bitr4ri df-reu
      anbi12d equequ1 bibi12d cbvalvw exbii ) DHEIZBJZDKZCHEIZAJZCKZBDELACELUJC
      GMZNZCOZGPUGDGMZNZDOZGPUKUHUNUQGUMUPCDCDMZUJUGULUOURUIUFABCDEQFUACDGUBUCU
      DUEUJCGRUGDGRSBDETACETS $.
    $( $j usage 'cbvralvw' avoids 'ax-io' 'ax-7' 'ax-10' 'ax-11' 'ax-i12'
       'ax-bndl' 'ax-i5r' 'ax-ext'; $)
  $}

  ${
    $d A y $.  $d ps y $.  $d B x $.  $d ch x $.  $d x ph y $.
    cbvraldva2.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
    cbvraldva2.2 $e |- ( ( ph /\ x = y ) -> A = B ) $.
    $( Rule used to change the bound variable in a restricted universal
       quantifier with implicit substitution which also changes the quantifier
       domain.  Deduction form.  (Contributed by David Moews, 1-May-2017.) $)
    cbvraldva2 $p |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) ) $=
      ( cv wcel wi wal wral weq wa simpr eleq12d imbi12d df-ral cbvaldva
      3bitr4g ) ADJZFKZBLZDMEJZGKZCLZEMBDFNCEGNAUEUHDEADEOZPZUDUGBCUJUCUFFGAUIQ
      IRHSUABDFTCEGTUB $.

    $( Rule used to change the bound variable in a restricted existential
       quantifier with implicit substitution which also changes the quantifier
       domain.  Deduction form.  (Contributed by David Moews, 1-May-2017.) $)
    cbvrexdva2 $p |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) ) $=
      ( cv wcel wa wex wrex weq simpr eleq12d anbi12d cbvexdva df-rex 3bitr4g )
      ADJZFKZBLZDMEJZGKZCLZEMBDFNCEGNAUDUGDEADEOZLZUCUFBCUIUBUEFGAUHPIQHRSBDFTC
      EGTUA $.
  $}

  ${
    $d ps y $.  $d ch x $.  $d A x y $.  $d x ph y $.
    cbvraldva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
    $( Rule used to change the bound variable in a restricted universal
       quantifier with implicit substitution.  Deduction form.  (Contributed by
       David Moews, 1-May-2017.) $)
    cbvraldva $p |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) ) $=
      ( weq wa eqidd cbvraldva2 ) ABCDEFFGADEHIFJK $.

    $( Rule used to change the bound variable in a restricted existential
       quantifier with implicit substitution.  Deduction form.  (Contributed by
       David Moews, 1-May-2017.) $)
    cbvrexdva $p |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) ) $=
      ( weq wa eqidd cbvrexdva2 ) ABCDEFFGADEHIFJK $.
  $}

  ${
    $d x z $.  $d w y $.  $d x A $.  $d z A $.  $d x y B $.  $d z y B $.
    $d w B $.  $d z ph $.  $d y ps $.  $d x ch $.  $d w ch $.
    cbvral2vw.1 $e |- ( x = z -> ( ph <-> ch ) ) $.
    cbvral2vw.2 $e |- ( y = w -> ( ch <-> ps ) ) $.
    $( Change bound variables of double restricted universal quantification,
       using implicit substitution.  Version of ~ cbvral2v with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       NM, 10-Aug-2004.)  (Revised by GG, 10-Jan-2024.) $)
    cbvral2vw $p |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps ) $=
      ( wral weq ralbidv cbvralvw ralbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHD
      FMACEIJNOSTFHCBEGIKOPQ $.
    $( $j usage 'cbvral2vw' avoids 'ax-io' 'ax-7' 'ax-10' 'ax-11' 'ax-i12'
       'ax-bndl' 'ax-i5r' 'ax-ext'; $)
  $}

  ${
    $d x z $.  $d w y $.  $d A x $.  $d A z $.  $d B w $.  $d B x y $.
    $d B z y $.  $d ch w $.  $d ch x $.  $d ph z $.  $d ps y $.
    cbvrex2vw.1 $e |- ( x = z -> ( ph <-> ch ) ) $.
    cbvrex2vw.2 $e |- ( y = w -> ( ch <-> ps ) ) $.
    $( Change bound variables of double restricted universal quantification,
       using implicit substitution.  Version of ~ cbvrex2v with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       FL, 2-Jul-2012.)  (Revised by GG, 10-Jan-2024.) $)
    cbvrex2vw $p |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps ) $=
      ( wrex weq rexbidv cbvrexvw rexbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHD
      FMACEIJNOSTFHCBEGIKOPQ $.
    $( $j usage 'cbvrex2vw' avoids 'ax-io' 'ax-7' 'ax-10' 'ax-11' 'ax-i12'
       'ax-bndl' 'ax-i5r' 'ax-ext'; $)
  $}

  ${
    $d x A $.  $d z A $.  $d x y B $.  $d z y B $.  $d w B $.  $d z ph $.
    $d y ps $.  $d x ch $.  $d w ch $.
    cbvral2v.1 $e |- ( x = z -> ( ph <-> ch ) ) $.
    cbvral2v.2 $e |- ( y = w -> ( ch <-> ps ) ) $.
    $( Change bound variables of double restricted universal quantification,
       using implicit substitution.  (Contributed by NM, 10-Aug-2004.) $)
    cbvral2v $p |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps ) $=
      ( wral weq ralbidv cbvralv ralbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHDF
      MACEIJNOSTFHCBEGIKOPQ $.
  $}

  ${
    $d A x $.  $d A z $.  $d B w $.  $d B x y $.  $d B z y $.  $d ch w $.
    $d ch x $.  $d ph z $.  $d ps y $.
    cbvrex2v.1 $e |- ( x = z -> ( ph <-> ch ) ) $.
    cbvrex2v.2 $e |- ( y = w -> ( ch <-> ps ) ) $.
    $( Change bound variables of double restricted universal quantification,
       using implicit substitution.  (Contributed by FL, 2-Jul-2012.) $)
    cbvrex2v $p |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps ) $=
      ( wrex weq rexbidv cbvrexv rexbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHDF
      MACEIJNOSTFHCBEGIKOPQ $.
  $}

  ${
    $d w ph $.  $d z ps $.  $d x ch $.  $d v ch $.  $d y u th $.  $d x A $.
    $d w A $.  $d x y B $.  $d w y B $.  $d v B $.  $d x y z C $.
    $d w y z C $.  $d v z C $.  $d z y C $.  $d z C $.  $d u C $.
    cbvral3v.1 $e |- ( x = w -> ( ph <-> ch ) ) $.
    cbvral3v.2 $e |- ( y = v -> ( ch <-> th ) ) $.
    cbvral3v.3 $e |- ( z = u -> ( th <-> ps ) ) $.
    $( Change bound variables of triple restricted universal quantification,
       using implicit substitution.  (Contributed by NM, 10-May-2005.) $)
    cbvral3v $p |- ( A. x e. A A. y e. B A. z e. C ph <->
                     A. w e. A A. v e. B A. u e. C ps ) $=
      ( wral weq 2ralbidv cbvralv cbvral2v ralbii bitri ) AGMQFLQZEKQCGMQFLQZHK
      QBJMQILQZHKQUDUEEHKEHRACFGLMNSTUEUFHKCBDFGIJLMOPUAUBUC $.
  $}

  ${
    $d z x A $.  $d y A $.  $d z y ph $.
    $( Change bound variable by using a substitution.  (Contributed by NM,
       20-Nov-2005.)  (Revised by Andrew Salmon, 11-Jul-2011.) $)
    cbvralsv $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $=
      ( vz wral wsb nfv nfs1v sbequ12 cbvral nfsb sbequ bitri ) ABDFABEGZEDFABC
      GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $.
  $}

  ${
    $d z x A $.  $d y z ph $.  $d y A $.
    $( Change bound variable by using a substitution.  (Contributed by NM,
       2-Mar-2008.)  (Revised by Andrew Salmon, 11-Jul-2011.) $)
    cbvrexsv $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $=
      ( vz wrex wsb nfv nfs1v sbequ12 cbvrex nfsb sbequ bitri ) ABDFABEGZEDFABC
      GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $.
  $}

  ${
    $d x y z $.  $d x z ps $.  $d y z ph $.
    sbralie.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Implicit to explicit substitution that swaps variables in a quantified
       expression.  (Contributed by NM, 5-Sep-2004.) $)
    sbralie $p |- ( A. x e. y ph <-> [ y / x ] A. y e. x ps ) $=
      ( vz cv wral wsb cbvralsv sbbii nfv raleq sbie sbco2 wb weq equcoms bitri
      bicomd ralbii 3bitrri ) BDCGZHZCDIBDFIZFUCHZCDIUEFDGZHZACUGHZUDUFCDBDFUCJ
      KUFUHCDUHCLUEFUCUGMNUHUEFCIZCUGHUIUEFCUGJUJACUGUJBDCIABDCFBFLOBADCADLBAPC
      DCDQABETRNSUASUB $.
  $}

  ${
    $d x ph $.
    rabbidva2.1 $e |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) ) $.
    $( Equivalent wff's yield equal restricted class abstractions.
       (Contributed by Thierry Arnoux, 4-Feb-2017.) $)
    rabbidva2 $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
      ( cv wcel wa cab crab abbidv df-rab 3eqtr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
      AQRDGMBDENCDFNO $.
  $}

  ${
    rabbia2.1 $e |- ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) $.
    $( Equivalent wff's yield equal restricted class abstractions.
       (Contributed by Glauco Siliprandi, 26-Jun-2021.) $)
    rabbia2 $p |- { x e. A | ps } = { x e. B | ch } $=
      ( crab wceq wtru cv wcel wa wb a1i rabbidva2 mptru ) ACDGBCEGHIABCDECJZDK
      ALQEKBLMIFNOP $.
  $}

  ${
    rabbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
    $( Equivalent wff's yield equal restricted class abstractions (inference
       form).  (Contributed by NM, 22-May-1999.) $)
    rabbiia $p |- { x e. A | ph } = { x e. A | ps } $=
      ( cv wcel wa cab crab pm5.32i abbii df-rab 3eqtr4i ) CFDGZAHZCIOBHZCIACDJ
      BCDJPQCOABEKLACDMBCDMN $.
  $}

  ${
    rabbii.1 $e |- ( ph <-> ps ) $.
    $( Equivalent wff's correspond to equal restricted class abstractions.
       Inference form of ~ rabbidv .  (Contributed by Peter Mazsa,
       1-Nov-2019.) $)
    rabbii $p |- { x e. A | ph } = { x e. A | ps } $=
      ( wb cv wcel a1i rabbiia ) ABCDABFCGDHEIJ $.
  $}

  ${
    $d x ph $.
    rabbidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal restricted class abstractions (deduction
       form).  (Contributed by NM, 28-Nov-2003.) $)
    rabbidva $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
      ( wb wral crab wceq ralrimiva rabbi sylib ) ABCGZDEHBDEICDEIJANDEFKBCDELM
      $.
  $}

  ${
    $d x ph $.
    rabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal restricted class abstractions (deduction
       form).  (Contributed by NM, 10-Feb-1995.) $)
    rabbidv $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
      ( wb cv wcel adantr rabbidva ) ABCDEABCGDHEIFJK $.
  $}

  ${
    rabeqf.1 $e |- F/_ x A $.
    rabeqf.2 $e |- F/_ x B $.
    $( Equality theorem for restricted class abstractions, with bound-variable
       hypotheses instead of distinct variable restrictions.  (Contributed by
       NM, 7-Mar-2004.) $)
    rabeqf $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $=
      ( wceq cv wcel wa cab crab nfeq eleq2 anbi1d abbid df-rab 3eqtr4g ) CDGZB
      HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.

    rabeqif.3 $e |- A = B $.
    $( Equality theorem for restricted class abstractions.  Inference form of
       ~ rabeqf .  (Contributed by Glauco Siliprandi, 26-Jun-2021.) $)
    rabeqif $p |- { x e. A | ph } = { x e. B | ph } $=
      ( wceq crab rabeqf ax-mp ) CDHABCIABDIHGABCDEFJK $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Equality theorem for restricted class abstractions.  (Contributed by NM,
       15-Oct-2003.) $)
    rabeq $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $=
      ( nfcv rabeqf ) ABCDBCEBDEF $.

    ${
      rabeqi.1 $e |- A = B $.
      $( Equality theorem for restricted class abstractions.  Inference form of
         ~ rabeq .  (Contributed by Glauco Siliprandi, 26-Jun-2021.) $)
      rabeqi $p |- { x e. A | ph } = { x e. B | ph } $=
        ( nfcv rabeqif ) ABCDBCFBDFEG $.
    $}

    rabeqdv.1 $e |- ( ph -> A = B ) $.
    $( Equality of restricted class abstractions.  Deduction form of ~ rabeq .
       (Contributed by Glauco Siliprandi, 5-Apr-2020.) $)
    rabeqdv $p |- ( ph -> { x e. A | ps } = { x e. B | ps } ) $=
      ( wceq crab rabeq syl ) ADEGBCDHBCEHGFBCDEIJ $.
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.
    rabeqbidv.1 $e |- ( ph -> A = B ) $.
    rabeqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equality of restricted class abstractions.  (Contributed by Jeff Madsen,
       1-Dec-2009.) $)
    rabeqbidv $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
      ( crab wceq rabeq syl rabbidv eqtrd ) ABDEIZBDFIZCDFIAEFJOPJGBDEFKLABCDFH
      MN $.
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.
    rabeqbidva.1 $e |- ( ph -> A = B ) $.
    rabeqbidva.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Equality of restricted class abstractions.  (Contributed by Mario
       Carneiro, 26-Jan-2017.) $)
    rabeqbidva $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
      ( crab rabbidva wceq rabeq syl eqtrd ) ABDEICDEIZCDFIZABCDEHJAEFKOPKGCDEF
      LMN $.
  $}

  ${
    rabeq2i.1 $e |- A = { x e. B | ph } $.
    $( Inference from equality of a class variable and a restricted class
       abstraction.  (Contributed by NM, 16-Feb-2004.) $)
    rabeq2i $p |- ( x e. A <-> ( x e. B /\ ph ) ) $=
      ( cv wcel crab wa eleq2i rabid bitri ) BFZCGMABDHZGMDGAICNMEJABDKL $.
  $}

  ${
    $d x z $.  $d y z $.  $d A z $.  $d ph z $.  $d ps z $.
    cbvrab.1 $e |- F/_ x A $.
    cbvrab.2 $e |- F/_ y A $.
    cbvrab.3 $e |- F/ y ph $.
    cbvrab.4 $e |- F/ x ps $.
    cbvrab.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule to change the bound variable in a restricted class abstraction,
       using implicit substitution.  This version has bound-variable hypotheses
       in place of distinct variable conditions.  (Contributed by Andrew
       Salmon, 11-Jul-2011.)  (Revised by Mario Carneiro, 9-Oct-2016.) $)
    cbvrab $p |- { x e. A | ph } = { y e. A | ps } $=
      ( vz cv wcel wa cab crab wsb nfv nfcri nfan nfs1v weq eleq1 sbequ12 cbvab
      anbi12d nfsb sbequ sbie bitrdi eqtri df-rab 3eqtr4i ) CLZEMZANZCOZDLZEMZB
      NZDOZACEPBDEPUQKLZEMZACKQZNZKOVAUPVECKUPKRVCVDCCKEFSACKUATCKUBUOVCAVDUNVB
      EUCACKUDUFUEVEUTKDVCVDDDKEGSACKDHUGTUTKRKDUBZVCUSVDBVBUREUCVFVDACDQBAKDCU
      HABCDIJUIUJUFUEUKACEULBDEULUM $.
  $}

  ${
    $d x y A $.  $d y ph $.  $d x ps $.
    cbvrabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule to change the bound variable in a restricted class abstraction,
       using implicit substitution.  (Contributed by NM, 26-May-1999.) $)
    cbvrabv $p |- { x e. A | ph } = { y e. A | ps } $=
      ( nfcv nfv cbvrab ) ABCDECEGDEGADHBCHFI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The universal class
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare the symbol for the universal class. $)
  $c _V $. $( Letter V (for the universal class) $)

  $( Extend class notation to include the universal class symbol. $)
  cvv $a class _V $.

  ${
    $d z x $.  $d z y $.
    $( Soundness justification theorem for ~ df-v .  (Contributed by Rodolfo
       Medina, 27-Apr-2010.) $)
    vjust $p |- { x | x = x } = { y | y = y } $=
      ( vz weq cab wsb cv wcel equid sbt 2th df-clab 3bitr4i eqriv ) CAADZAEZBB
      DZBEZOACFZQBCFZCGZPHUARHSTOACAIJQBCBIJKOCALQCBLMN $.
  $}

  $( Define the universal class.  Definition 5.20 of [TakeutiZaring] p. 21.
     Also Definition 2.9 of [Quine] p. 19.  (Contributed by NM, 5-Aug-1993.) $)
  df-v $a |- _V = { x | x = x } $.

  $( All setvar variables are sets (see ~ isset ).  Theorem 6.8 of [Quine]
     p. 43.  (Contributed by NM, 5-Aug-1993.) $)
  vex $p |- x e. _V $=
    ( cv cvv wcel weq equid df-v abeq2i mpbir ) ABCDAAEZAFJACAGHI $.

  ${
    elv.1 $e |- ( x e. _V -> ph ) $.
    $( Technical lemma used to shorten proofs.  If a proposition is implied by
       ` x e. _V ` (which is true, see ~ vex ), then it is true.  (Contributed
       by Peter Mazsa, 13-Oct-2018.) $)
    elv $p |- ph $=
      ( cv cvv wcel vex ax-mp ) BDEFABGCH $.
  $}

  ${
    elvd.1 $e |- ( ( ph /\ x e. _V ) -> ps ) $.
    $( Technical lemma used to shorten proofs.  If a proposition is implied by
       ` x e. _V ` (which is true, see ~ vex ) and another antecedent, then it
       is implied by the other antecedent.  (Contributed by Peter Mazsa,
       23-Oct-2018.) $)
    elvd $p |- ( ph -> ps ) $=
      ( cv cvv wcel vex mpan2 ) ACEFGBCHDI $.
  $}

  ${
    el2v.1 $e |- ( ( x e. _V /\ y e. _V ) -> ph ) $.
    $( If a proposition is implied by ` x e. _V ` and ` y e. _V ` (which is
       true, see ~ vex ), then it is true.  (Contributed by Peter Mazsa,
       13-Oct-2018.) $)
    el2v $p |- ph $=
      ( cv cvv wcel vex mp2an ) BEFGCEFGABHCHDI $.
  $}

  ${
    $d x A $.
    $( Two ways to say " ` A ` is a set":  A class ` A ` is a member of the
       universal class ` _V ` (see ~ df-v ) if and only if the class ` A `
       exists (i.e. there exists some set ` x ` equal to class ` A ` ).
       Theorem 6.9 of [Quine] p. 43. _Notational convention_:  We will use the
       notational device " ` A e. _V ` " to mean " ` A ` is a set" very
       frequently, for example in ~ uniex .  Note the when ` A ` is not a set,
       it is called a proper class.  In some theorems, such as ~ uniexg , in
       order to shorten certain proofs we use the more general antecedent
       ` A e. V ` instead of ` A e. _V ` to mean " ` A ` is a set."

       Note that a constant is implicitly considered distinct from all
       variables.  This is why ` _V ` is not included in the distinct variable
       list, even though ~ df-clel requires that the expression substituted for
       ` B ` not contain ` x ` .  (Also, the Metamath spec does not allow
       constants in the distinct variable list.)  (Contributed by NM,
       26-May-1993.) $)
    isset $p |- ( A e. _V <-> E. x x = A ) $=
      ( cvv wcel cv wceq wa wex df-clel vex biantru exbii bitr4i ) BCDAEZBFZNCD
      ZGZAHOAHABCIOQAPOAJKLM $.
  $}

  ${
    $d A y $.  $d x y $.
    issetf.1 $e |- F/_ x A $.
    $( A version of isset that does not require x and A to be distinct.
       (Contributed by Andrew Salmon, 6-Jun-2011.)  (Revised by Mario Carneiro,
       10-Oct-2016.) $)
    issetf $p |- ( A e. _V <-> E. x x = A ) $=
      ( vy cvv wcel cv wceq wex isset nfeq2 nfv eqeq1 cbvex bitri ) BEFDGZBHZDI
      AGZBHZAIDBJQSDAAPBCKSDLPRBMNO $.
  $}

  ${
    $d x A $.
    isseti.1 $e |- A e. _V $.
    $( A way to say " ` A ` is a set" (inference form).  (Contributed by NM,
       5-Aug-1993.) $)
    isseti $p |- E. x x = A $=
      ( cvv wcel cv wceq wex isset mpbi ) BDEAFBGAHCABIJ $.
  $}

  ${
    $d x A $.
    issetri.1 $e |- E. x x = A $.
    $( A way to say " ` A ` is a set" (inference form).  (Contributed by NM,
       5-Aug-1993.) $)
    issetri $p |- A e. _V $=
      ( cvv wcel cv wceq wex isset mpbir ) BDEAFBGAHCABIJ $.
  $}

  $( A class equal to a variable is a set.  Note the absence of disjoint
     variable condition, contrary to ~ isset and ~ issetri .  (Contributed by
     BJ, 27-Apr-2019.) $)
  eqvisset $p |- ( x = A -> A e. _V ) $=
    ( cv wceq cvv wcel vex eleq1 mpbii ) ACZBDJEFBEFAGJBEHI $.

  ${
    $d x A $.  $d x B $.
    $( If a class is a member of another class, then it is a set.  Theorem 6.12
       of [Quine] p. 44.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by
       Andrew Salmon, 8-Jun-2011.) $)
    elex $p |- ( A e. B -> A e. _V ) $=
      ( vx cv wceq wcel wa wex cvv exsimpl df-clel isset 3imtr4i ) CDZAEZNBFZGC
      HOCHABFAIFOPCJCABKCALM $.
  $}

  ${
    elisseti.1 $e |- A e. B $.
    $( If a class is a member of another class, it is a set.  (Contributed by
       NM, 11-Jun-1994.) $)
    elexi $p |- A e. _V $=
      ( wcel cvv elex ax-mp ) ABDAEDCABFG $.
  $}

  ${
    elexd.1 $e |- ( ph -> A e. V ) $.
    $( If a class is a member of another class, it is a set.  (Contributed by
       Glauco Siliprandi, 11-Oct-2020.) $)
    elexd $p |- ( ph -> A e. _V ) $=
      ( wcel cvv elex syl ) ABCEBFEDBCGH $.
  $}

  ${
    $d x A $.
    $( An element of a class exists.  (Contributed by NM, 1-May-1995.) $)
    elisset $p |- ( A e. V -> E. x x = A ) $=
      ( wcel cvv cv wceq wex elex isset sylib ) BCDBEDAFBGAHBCIABJK $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( If two classes each contain another class, then both contain some set.
       (Contributed by Alan Sare, 24-Oct-2011.) $)
    elex22 $p |- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) ) $=
      ( wcel wa cv wceq wi wal eleq1a anim12ii alrimiv elisset adantr exim sylc
      wex ) BCEZBDEZFZAGZBHZUBCEZUBDEZFZIZAJUCARZUFARUAUGASUCUDTUEBCUBKBDUBKLMS
      UHTABCNOUCUFAPQ $.
  $}

  ${
    $d x A $.  $d x B $.
    $( If a class contains another class, then it contains some set.
       (Contributed by Alan Sare, 25-Sep-2011.) $)
    elex2 $p |- ( A e. B -> E. x x e. B ) $=
      ( wcel cv wceq wi wal wex eleq1a alrimiv elisset exim sylc ) BCDZAEZBFZPC
      DZGZAHQAIRAIOSABCPJKABCLQRAMN $.
  $}

  $( A universal quantifier restricted to the universe is unrestricted.
     (Contributed by NM, 26-Mar-2004.) $)
  ralv $p |- ( A. x e. _V ph <-> A. x ph ) $=
    ( cvv wral cv wcel wi wal df-ral vex a1bi albii bitr4i ) ABCDBECFZAGZBHABHA
    BCIAOBNABJKLM $.

  $( An existential quantifier restricted to the universe is unrestricted.
     (Contributed by NM, 26-Mar-2004.) $)
  rexv $p |- ( E. x e. _V ph <-> E. x ph ) $=
    ( cvv wrex cv wcel wa wex df-rex vex biantrur exbii bitr4i ) ABCDBECFZAGZBH
    ABHABCIAOBNABJKLM $.

  $( A unique existential quantifier restricted to the universe is
     unrestricted.  (Contributed by NM, 1-Nov-2010.) $)
  reuv $p |- ( E! x e. _V ph <-> E! x ph ) $=
    ( cvv wreu cv wcel wa weu df-reu vex biantrur eubii bitr4i ) ABCDBECFZAGZBH
    ABHABCIAOBNABJKLM $.

  $( An at-most-one quantifier restricted to the universe is unrestricted.
     (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
  rmov $p |- ( E* x e. _V ph <-> E* x ph ) $=
    ( cvv wrmo cv wcel wa wmo df-rmo vex biantrur mobii bitr4i ) ABCDBECFZAGZBH
    ABHABCIAOBNABJKLM $.

  $( A class abstraction restricted to the universe is unrestricted.
     (Contributed by NM, 27-Dec-2004.)  (Proof shortened by Andrew Salmon,
     8-Jun-2011.) $)
  rabab $p |- { x e. _V | ph } = { x | ph } $=
    ( cvv crab cv wcel wa cab df-rab vex biantrur abbii eqtr4i ) ABCDBECFZAGZBH
    ABHABCIAOBNABJKLM $.

  ${
    $d x y $.  $d y A $.
    $( Commutation of restricted and unrestricted universal quantifiers.
       (Contributed by NM, 26-Mar-2004.)  (Proof shortened by Andrew Salmon,
       8-Jun-2011.) $)
    ralcom4 $p |- ( A. x e. A A. y ph <-> A. y A. x e. A ph ) $=
      ( cvv wral wal ralcom ralv ralbii 3bitr3i ) ACEFZBDFABDFZCEFACGZBDFMCGABC
      DEHLNBDACIJMCIK $.

    $( Commutation of restricted and unrestricted existential quantifiers.
       (Contributed by NM, 12-Apr-2004.)  (Proof shortened by Andrew Salmon,
       8-Jun-2011.) $)
    rexcom4 $p |- ( E. x e. A E. y ph <-> E. y E. x e. A ph ) $=
      ( cvv wrex wex rexcom rexv rexbii 3bitr3i ) ACEFZBDFABDFZCEFACGZBDFMCGABC
      DEHLNBDACIJMCIK $.
  $}

  ${
    $d A x $.  $d x y $.  $d ph x $.
    $( Specialized existential commutation lemma.  (Contributed by Jeff Madsen,
       1-Jun-2011.) $)
    rexcom4a $p |- ( E. x E. y e. A ( ph /\ ps )
                          <-> E. y e. A ( ph /\ E. x ps ) ) $=
      ( wa wrex wex rexcom4 19.42v rexbii bitr3i ) ABFZDEGCHMCHZDEGABCHFZDEGMDC
      EINODEABCJKL $.

    $d B x $.
    rexcom4b.1 $e |- B e. _V $.
    $( Specialized existential commutation lemma.  (Contributed by Jeff Madsen,
       1-Jun-2011.) $)
    rexcom4b $p |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph ) $=
      ( cv wceq wa wrex wex rexcom4a isseti biantru rexbii bitr4i ) ABGEHZICDJB
      KAQBKZIZCDJACDJAQBCDLASCDRABEFMNOP $.
  $}

  ${
    $d x A $.
    $( Closed theorem version of ~ ceqsalg .  (Contributed by NM, 28-Feb-2013.)
       (Revised by Mario Carneiro, 10-Oct-2016.) $)
    ceqsalt $p |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V )
         -> ( A. x ( x = A -> ph ) <-> ps ) ) $=
      ( wnf cv wceq wb wi wal wcel w3a elisset 3ad2ant3 biimp 3ad2ant2 3ad2ant1
      wex imim3i al2imi 19.23t sylibd biimpr imim2i com23 alimi 19.21t impbid
      mpid mpbid ) BCFZCGDHZABIZJZCKZDELZMZUMAJZCKZBURUTUMCSZBUQULVAUPCDENOURUT
      UMBJZCKZVABJZUPULUTVCJUQUOUSVBCUNABUMABPTUAQULUPVCVDIUQUMBCUBRUCUJURBUSJZ
      CKZBUTJZUPULVFUQUOVECUOUMBAUNBAJUMABUDUEUFUGQULUPVFVGIUQBUSCUHRUKUI $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Restricted quantifier version of ~ ceqsalt .  (Contributed by NM,
       28-Feb-2013.)  (Revised by Mario Carneiro, 10-Oct-2016.) $)
    ceqsralt $p |- ( ( F/ x ps
              /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B )
         -> ( A. x e. B ( x = A -> ph ) <-> ps ) ) $=
      ( wnf cv wceq wb wi wal wcel w3a wral df-ral eleq1 pm5.32ri imbi1i impexp
      wa 3bitr3i albii a1i bitrid 19.21v bitrdi biimt 3ad2ant3 ceqsalt 3bitr2d
      ) BCFZCGZDHZABIJCKZDELZMZUMAJZCENZUOUQCKZJZUSBUPURUOUQJZCKZUTURULELZUQJZC
      KZUPVBUQCEOVEVBIUPVDVACVCUMTZAJUOUMTZAJVDVAVFVGAUMVCUOULDEPQRVCUMASUOUMAS
      UAUBUCUDUOUQCUEUFUOUKUSUTIUNUOUSUGUHABCDEUIUJ $.
  $}

  ${
    $d x A $.
    ceqsalg.1 $e |- F/ x ps $.
    ceqsalg.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( A representation of explicit substitution of a class for a variable,
       inferred from an implicit substitution hypothesis.  (Contributed by NM,
       29-Oct-2003.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    ceqsalg $p |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) ) $=
      ( wcel cv wceq wi wal wex elisset nfa1 biimpd a2i sps exlimd syl5com
      biimprcd alrimi impbid1 ) DEHZCIDJZAKZCLZBUDUECMUGBCDENUGUEBCUFCOFUFUEBKC
      UEABUEABGPQRSTBUFCFUEABGUAUBUC $.
  $}

  ${
    $d x A $.
    ceqsal.1 $e |- F/ x ps $.
    ceqsal.2 $e |- A e. _V $.
    ceqsal.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( A representation of explicit substitution of a class for a variable,
       inferred from an implicit substitution hypothesis.  (Contributed by NM,
       18-Aug-1993.) $)
    ceqsal $p |- ( A. x ( x = A -> ph ) <-> ps ) $=
      ( cvv wcel cv wceq wi wal wb ceqsalg ax-mp ) DHICJDKALCMBNFABCDHEGOP $.
  $}

  ${
    $d x A $.  $d x ps $.
    ceqsalv.1 $e |- A e. _V $.
    ceqsalv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( A representation of explicit substitution of a class for a variable,
       inferred from an implicit substitution hypothesis.  (Contributed by NM,
       18-Aug-1993.) $)
    ceqsalv $p |- ( A. x ( x = A -> ph ) <-> ps ) $=
      ( nfv ceqsal ) ABCDBCGEFH $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    ceqsralv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Restricted quantifier version of ~ ceqsalv .  (Contributed by NM,
       21-Jun-2013.) $)
    ceqsralv $p |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) ) $=
      ( wnf cv wceq wb wi wal wcel wral nfv ax-gen ceqsralt mp3an12 ) BCGCHDIZA
      BJKZCLDEMSAKCENBJBCOTCFPABCDEQR $.
  $}

  ${
    $d x ps $.
    gencl.1 $e |- ( th <-> E. x ( ch /\ A = B ) ) $.
    gencl.2 $e |- ( A = B -> ( ph <-> ps ) ) $.
    gencl.3 $e |- ( ch -> ph ) $.
    $( Implicit substitution for class with embedded variable.  (Contributed by
       NM, 17-May-1996.) $)
    gencl $p |- ( th -> ps ) $=
      ( wceq wa wex imbitrid impcom exlimiv sylbi ) DCFGKZLZEMBHSBERCBCARBJINOP
      Q $.
  $}

  ${
    $d x y $.  $d x R $.  $d x ps $.  $d y C $.  $d y S $.  $d y ch $.
    2gencl.1 $e |- ( C e. S <-> E. x e. R A = C ) $.
    2gencl.2 $e |- ( D e. S <-> E. y e. R B = D ) $.
    2gencl.3 $e |- ( A = C -> ( ph <-> ps ) ) $.
    2gencl.4 $e |- ( B = D -> ( ps <-> ch ) ) $.
    2gencl.5 $e |- ( ( x e. R /\ y e. R ) -> ph ) $.
    $( Implicit substitution for class with embedded variable.  (Contributed by
       NM, 17-May-1996.) $)
    2gencl $p |- ( ( C e. S /\ D e. S ) -> ch ) $=
      ( wcel wi cv wceq wrex wa wex df-rex bitri imbi2d ex gencl com12 impcom )
      IKQZHKQZCULBRULCRESJQZUKEGIUKGITZEJUAUMUNUBEUCMUNEJUDUEUNBCULOUFULUMBUMAR
      UMBRDSJQZULDFHULFHTZDJUAUOUPUBDUCLUPDJUDUEUPABUMNUFUOUMAPUGUHUIUHUJ $.
  $}

  ${
    $d x y z $.  $d y z D $.  $d z F $.  $d x y R $.  $d y z S $.  $d x ps $.
    $d y ch $.  $d z th $.
    3gencl.1 $e |- ( D e. S <-> E. x e. R A = D ) $.
    3gencl.2 $e |- ( F e. S <-> E. y e. R B = F ) $.
    3gencl.3 $e |- ( G e. S <-> E. z e. R C = G ) $.
    3gencl.4 $e |- ( A = D -> ( ph <-> ps ) ) $.
    3gencl.5 $e |- ( B = F -> ( ps <-> ch ) ) $.
    3gencl.6 $e |- ( C = G -> ( ch <-> th ) ) $.
    3gencl.7 $e |- ( ( x e. R /\ y e. R /\ z e. R ) -> ph ) $.
    $( Implicit substitution for class with embedded variable.  (Contributed by
       NM, 17-May-1996.) $)
    3gencl $p |- ( ( D e. S /\ F e. S /\ G e. S ) -> th ) $=
      ( wcel wa wi wceq wrex wex df-rex bitri imbi2d 3expia 2gencl com12 3impia
      cv gencl ) KMUCZNMUCZOMUCZDUTURUSUDZDVACUEVADUEGUPLUCZUTGJOUTJOUFZGLUGVBV
      CUDGUHRVCGLUIUJVCCDVAUAUKVAVBCVBAUEVBBUEVBCUEEFHIKNLMPQHKUFABVBSUKINUFBCV
      BTUKEUPLUCFUPLUCVBAUBULUMUNUQUNUO $.
  $}

  ${
    $d x A $.  $d x ps $.
    cgsexg.1 $e |- ( x = A -> ch ) $.
    cgsexg.2 $e |- ( ch -> ( ph <-> ps ) ) $.
    $( Implicit substitution inference for general classes.  (Contributed by
       NM, 26-Aug-2007.) $)
    cgsexg $p |- ( A e. V ->
                     ( E. x ( ch /\ ph ) <-> ps ) ) $=
      ( wcel wa wex biimpa exlimiv cv wceq elisset eximi syl biimprcd ancld
      eximdv syl5com impbid2 ) EFIZCAJZDKZBUEBDCABHLMUDCDKZBUFUDDNEOZDKUGDEFPUH
      CDGQRBCUEDBCACABHSTUAUBUC $.
  $}

  ${
    $d x y ps $.  $d x y A $.  $d x y B $.
    cgsex2g.1 $e |- ( ( x = A /\ y = B ) -> ch ) $.
    cgsex2g.2 $e |- ( ch -> ( ph <-> ps ) ) $.
    $( Implicit substitution inference for general classes.  (Contributed by
       NM, 26-Jul-1995.) $)
    cgsex2g $p |- ( ( A e. V /\ B e. W ) ->
                     ( E. x E. y ( ch /\ ph ) <-> ps ) ) $=
      ( wcel wa wex biimpa exlimivv cv wceq elisset anim12i eeanv sylibr 2eximi
      syl biimprcd ancld 2eximdv syl5com impbid2 ) FHLZGILZMZCAMZENDNZBUMBDECAB
      KOPULCENDNZBUNULDQFRZEQGRZMZENDNZUOULUPDNZUQENZMUSUJUTUKVADFHSEGISTUPUQDE
      UAUBURCDEJUCUDBCUMDEBCACABKUEUFUGUHUI $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.  $d x y z w C $.  $d x y z w D $.
    $d x y z w ps $.
    cgsex4g.1 $e |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch ) $.
    cgsex4g.2 $e |- ( ch -> ( ph <-> ps ) ) $.
    $( An implicit substitution inference for 4 general classes.  (Contributed
       by NM, 5-Aug-1995.) $)
    cgsex4g $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) ->
                    ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) ) $=
      ( wcel wa wex cv wceq biimpa exlimivv elisset anim12i eeanv sylibr ee4anv
      2eximi syl biimprcd ancld 2eximdv syl5com impbid2 ) HLPZIMPZQZJLPZKMPZQZQ
      ZCAQZGRFRZERDRZBVCBDEVBBFGCABOUAUBUBVACGRFRZERDRZBVDVADSHTZESITZQZFSJTZGS
      KTZQZQZGRFRZERDRZVFVAVIERDRZVLGRFRZQVOUQVPUTVQUQVGDRZVHERZQVPUOVRUPVSDHLU
      CEIMUCUDVGVHDEUEUFUTVJFRZVKGRZQVQURVTUSWAFJLUCGKMUCUDVJVKFGUEUFUDVIVLDEFG
      UGUFVNVEDEVMCFGNUHUHUIBVEVCDEBCVBFGBCACABOUJUKULULUMUN $.
  $}

  ${
    $d x A $.
    ceqsex.1 $e |- F/ x ps $.
    ceqsex.2 $e |- A e. _V $.
    ceqsex.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Elimination of an existential quantifier, using implicit substitution.
       (Contributed by NM, 2-Mar-1995.)  (Revised by Mario Carneiro,
       10-Oct-2016.) $)
    ceqsex $p |- ( E. x ( x = A /\ ph ) <-> ps ) $=
      ( cv wceq wa wex biimpa exlimi wal biimprcd alrimi isseti exintr mpisyl
      wi impbii ) CHDIZAJZCKZBUCBCEUBABGLMBUBATZCNUBCKUDBUECEUBABGOPCDFQUBACRSU
      A $.
  $}

  ${
    $d x A $.  $d x ps $.
    ceqsexv.1 $e |- A e. _V $.
    ceqsexv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Elimination of an existential quantifier, using implicit substitution.
       (Contributed by NM, 2-Mar-1995.) $)
    ceqsexv $p |- ( E. x ( x = A /\ ph ) <-> ps ) $=
      ( nfv ceqsex ) ABCDBCGEFH $.
  $}

  ${
    $d x A $.
    ceqsexv2d.1 $e |- A e. _V $.
    ceqsexv2d.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsexv2d.3 $e |- ps $.
    $( Elimination of an existential quantifier, using implicit substitution.
       (Contributed by Thierry Arnoux, 10-Sep-2016.)  Shorten, reduce dv
       conditions.  (Revised by Wolf Lammen, 5-Jun-2025.)  (Proof shortened by
       SN, 5-Jun-2025.) $)
    ceqsexv2d $p |- E. x ph $=
      ( cv wceq isseti mpbiri eximii ) CHDIZACCDEJMABGFKL $.
  $}

  ${
    $d x y A $.  $d x y B $.
    ceqsex2.1 $e |- F/ x ps $.
    ceqsex2.2 $e |- F/ y ch $.
    ceqsex2.3 $e |- A e. _V $.
    ceqsex2.4 $e |- B e. _V $.
    ceqsex2.5 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsex2.6 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( Elimination of two existential quantifiers, using implicit substitution.
       (Contributed by Scott Fenton, 7-Jun-2006.) $)
    ceqsex2 $p |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch ) $=
      ( cv wceq w3a wex wa exbii ceqsex 3anass 19.42v nfan anbi2d exbidv 3bitri
      bitri nfv nfex ) DNFOZENGOZAPZEQZDQUJUKARZEQZRZDQUKBRZEQZCUMUPDUMUJUNRZEQ
      UPULUSEUJUKAUASUJUNEUBUGSUOURDFUQDEUKBDUKDUHHUCUIJUJUNUQEUJABUKLUDUETBCEG
      IKMTUF $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x ps $.  $d y ch $.
    ceqsex2v.1 $e |- A e. _V $.
    ceqsex2v.2 $e |- B e. _V $.
    ceqsex2v.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsex2v.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( Elimination of two existential quantifiers, using implicit substitution.
       (Contributed by Scott Fenton, 7-Jun-2006.) $)
    ceqsex2v $p |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch ) $=
      ( nfv ceqsex2 ) ABCDEFGBDLCELHIJKM $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x ps $.  $d y ch $.
    $d z th $.
    ceqsex3v.1 $e |- A e. _V $.
    ceqsex3v.2 $e |- B e. _V $.
    ceqsex3v.3 $e |- C e. _V $.
    ceqsex3v.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsex3v.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
    ceqsex3v.6 $e |- ( z = C -> ( ch <-> th ) ) $.
    $( Elimination of three existential quantifiers, using implicit
       substitution.  (Contributed by NM, 16-Aug-2011.) $)
    ceqsex3v $p |- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph )
                 <-> th ) $=
      ( cv wceq wa wex anass 3anass anbi1i df-3an anbi2i 3bitr4i 2exbii 19.42vv
      w3a bitri exbii 3anbi3d 2exbidv ceqsexv ceqsex2v ) EQHRZFQIRZGQJRZUIZASZG
      TFTZETUPUQURAUIZGTFTZSZETZDVAVDEVAUPVBSZGTFTVDUTVFFGUPUQURSZSZASUPVGASZSU
      TVFUPVGAUAUSVHAUPUQURUBUCVBVIUPUQURAUDUEUFUGUPVBFGUHUJUKVEUQURBUIZGTFTZDV
      CVKEHKUPVBVJFGUPABUQURNULUMUNBCDFGIJLMOPUOUJUJ $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.  $d x y z w C $.  $d x y z w D $.
    $d x ps $.  $d y ch $.  $d z th $.  $d w ta $.
    ceqsex4v.1 $e |- A e. _V $.
    ceqsex4v.2 $e |- B e. _V $.
    ceqsex4v.3 $e |- C e. _V $.
    ceqsex4v.4 $e |- D e. _V $.
    ceqsex4v.7 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsex4v.8 $e |- ( y = B -> ( ps <-> ch ) ) $.
    ceqsex4v.9 $e |- ( z = C -> ( ch <-> th ) ) $.
    ceqsex4v.10 $e |- ( w = D -> ( th <-> ta ) ) $.
    $( Elimination of four existential quantifiers, using implicit
       substitution.  (Contributed by NM, 23-Sep-2011.) $)
    ceqsex4v $p |- ( E. x E. y E. z E. w
          ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ta ) $=
      ( wceq w3a wex 19.42vv 3anass df-3an anbi2i bitr4i 2exbii 3bitr4i 3anbi3d
      cv wa 2exbidv ceqsex2v 3bitri ) FUMJUBZGUMKUBZUNZHUMLUBZIUMMUBZUNZAUCZIUD
      HUDZGUDFUDURUSVAVBAUCZIUDHUDZUCZGUDFUDVAVBCUCZIUDHUDZEVEVHFGUTVFUNZIUDHUD
      UTVGUNVEVHUTVFHIUEVDVKHIVDUTVCAUNZUNVKUTVCAUFVFVLUTVAVBAUGUHUIUJURUSVGUGU
      KUJVGVAVBBUCZIUDHUDVJFGJKNOURVFVMHIURABVAVBRULUOUSVMVIHIUSBCVAVBSULUOUPCD
      EHILMPQTUAUPUQ $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u D $.  $d x y z w v u E $.  $d x y z w v u F $.  $d x ps $.
    $d y ch $.  $d z th $.  $d w ta $.  $d v et $.  $d u ze $.
    ceqsex6v.1 $e |- A e. _V $.
    ceqsex6v.2 $e |- B e. _V $.
    ceqsex6v.3 $e |- C e. _V $.
    ceqsex6v.4 $e |- D e. _V $.
    ceqsex6v.5 $e |- E e. _V $.
    ceqsex6v.6 $e |- F e. _V $.
    ceqsex6v.7 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsex6v.8 $e |- ( y = B -> ( ps <-> ch ) ) $.
    ceqsex6v.9 $e |- ( z = C -> ( ch <-> th ) ) $.
    ceqsex6v.10 $e |- ( w = D -> ( th <-> ta ) ) $.
    ceqsex6v.11 $e |- ( v = E -> ( ta <-> et ) ) $.
    ceqsex6v.12 $e |- ( u = F -> ( et <-> ze ) ) $.
    $( Elimination of six existential quantifiers, using implicit substitution.
       (Contributed by NM, 21-Sep-2011.) $)
    ceqsex6v $p |- ( E. x E. y E. z E. w E. v E. u
          ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph )
                 <-> ze ) $=
      ( cv wceq w3a wex wa 3anass 3exbii 19.42vvv bitri anbi2d 3exbidv ceqsex3v
      ) HULNUMZIULOUMZJULPUMZUNZKULQUMLULRUMMULSUMUNZAUNZMUOLUOKUOZJUOIUOHUOVGV
      HAUPZMUOLUOKUOZUPZJUOIUOHUOZGVJVMHIJVJVGVKUPZMUOLUOKUOVMVIVOKLMVGVHAUQURV
      GVKKLMUSUTURVNVHDUPZMUOLUOKUOZGVLVHBUPZMUOLUOKUOVHCUPZMUOLUOKUOVQHIJNOPTU
      AUBVDVKVRKLMVDABVHUFVAVBVEVRVSKLMVEBCVHUGVAVBVFVSVPKLMVFCDVHUHVAVBVCDEFGK
      LMQRSUCUDUEUIUJUKVCUTUT $.
  $}

  ${
    $d x y z w v u t s A $.  $d x y z w v u t s B $.  $d x y z w v u t s C $.
    $d x y z w v u t s D $.  $d x y z w v u t s E $.  $d x y z w v u t s F $.
    $d x y z w v u t s G $.  $d x y z w v u t s H $.  $d x ps $.  $d y ch $.
    $d z th $.  $d w ta $.  $d v et $.  $d u ze $.  $d t si $.  $d s rh $.
    ceqsex8v.1 $e |- A e. _V $.
    ceqsex8v.2 $e |- B e. _V $.
    ceqsex8v.3 $e |- C e. _V $.
    ceqsex8v.4 $e |- D e. _V $.
    ceqsex8v.5 $e |- E e. _V $.
    ceqsex8v.6 $e |- F e. _V $.
    ceqsex8v.7 $e |- G e. _V $.
    ceqsex8v.8 $e |- H e. _V $.
    ceqsex8v.9 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsex8v.10 $e |- ( y = B -> ( ps <-> ch ) ) $.
    ceqsex8v.11 $e |- ( z = C -> ( ch <-> th ) ) $.
    ceqsex8v.12 $e |- ( w = D -> ( th <-> ta ) ) $.
    ceqsex8v.13 $e |- ( v = E -> ( ta <-> et ) ) $.
    ceqsex8v.14 $e |- ( u = F -> ( et <-> ze ) ) $.
    ceqsex8v.15 $e |- ( t = G -> ( ze <-> si ) ) $.
    ceqsex8v.16 $e |- ( s = H -> ( si <-> rh ) ) $.
    $( Elimination of eight existential quantifiers, using implicit
       substitution.  (Contributed by NM, 23-Sep-2011.) $)
    ceqsex8v $p |- ( E. x E. y E. z E. w E. v E. u E. t E. s
              ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) )
         /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> rh ) $=
      ( cv wceq wa w3a wex 19.42vvvv 3anass df-3an anbi2i bitr4i 2exbii 3bitr4i
      3anbi3d 4exbidv ceqsex4v bitri ) JVBQVCZKVBRVCZVDZLVBSVCZMVBTVCZVDZVDZNVB
      UAVCOVBUBVCVDZPVBUCVCUEVBUDVCVDZVDZAVEZUEVFPVFZOVFNVFZMVFLVFZKVFJVFVTWCWE
      WFAVEZUEVFPVFOVFNVFZVEZMVFLVFZKVFJVFZIWKWOJKWJWNLMWDWLVDZUEVFPVFZOVFNVFWD
      WMVDWJWNWDWLOPUENVGWIWRNOWHWQPUEWHWDWGAVDZVDWQWDWGAVHWLWSWDWEWFAVIVJVKVLV
      LVTWCWMVIVMVLVLWPWEWFEVEZUEVFPVFOVFNVFZIWMWEWFBVEZUEVFPVFOVFNVFWEWFCVEZUE
      VFPVFOVFNVFWEWFDVEZUEVFPVFOVFNVFXAJKLMQRSTUFUGUHUIVRWLXBNOPUEVRABWEWFUNVN
      VOVSXBXCNOPUEVSBCWEWFUOVNVOWAXCXDNOPUEWACDWEWFUPVNVOWBXDWTNOPUEWBDEWEWFUQ
      VNVOVPEFGHINOPUEUAUBUCUDUJUKULUMURUSUTVAVPVQVQ $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x th $.  $d y ch $.  $d y A $.
    gencbvex.1 $e |- A e. _V $.
    gencbvex.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
    gencbvex.3 $e |- ( A = y -> ( ch <-> th ) ) $.
    gencbvex.4 $e |- ( th <-> E. x ( ch /\ A = y ) ) $.
    $( Change of bound variable using implicit substitution.  (Contributed by
       NM, 17-May-1996.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    gencbvex $p |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) ) $=
      ( cv wceq wa wex excom wb anbi12d bicomd exbii eqcoms ceqsexv simpr eqcom
      19.41v biimpi adantl eximi sylbi adantr ancri impbii bitri 3bitr3i ) FLZG
      MZDBNZNZFOZEOUREOZFOCANZEOUQFOUREFPUSVAEUQVAFGHUQVAQGUOGUOMZVAUQVBCDABJIR
      SUAUBTUTUQFUTUPEOZUQNZUQUPUQEUEVDUQVCUQUCUQVCDVCBDCVBNZEOVCKVEUPEVBUPCVBU
      PGUOUDUFUGUHUIUJUKULUMTUN $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x th $.  $d y ch $.  $d y A $.
    gencbvex2.1 $e |- A e. _V $.
    gencbvex2.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
    gencbvex2.3 $e |- ( A = y -> ( ch <-> th ) ) $.
    gencbvex2.4 $e |- ( th -> E. x ( ch /\ A = y ) ) $.
    $( Restatement of ~ gencbvex with weaker hypotheses.  (Contributed by Jeff
       Hankins, 6-Dec-2006.) $)
    gencbvex2 $p |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) ) $=
      ( cv wceq wa wex biimpac exlimiv impbii gencbvex ) ABCDEFGHIJDCGFLMZNZEOK
      UADETCDJPQRS $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x th $.  $d y ch $.  $d y A $.
    gencbval.1 $e |- A e. _V $.
    gencbval.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
    gencbval.3 $e |- ( A = y -> ( ch <-> th ) ) $.
    gencbval.4 $e |- ( th <-> E. x ( ch /\ A = y ) ) $.
    $( Change of bound variable using implicit substitution.  (Contributed by
       NM, 17-May-1996.)  (Proof rewritten by Jim Kingdon, 20-Jun-2018.) $)
    gencbval $p |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) ) $=
      ( cv wceq wi wal alcom wb imbi12d albii wex bicomd eqcoms 19.23v wa eqcom
      ceqsalv biimpi adantl eximi sylbi pm2.04 mpdi ax-1 impbii bitri 3bitr3i )
      FLZGMZDBNZNZFOZEOUTEOZFOCANZEOUSFOUTEFPVAVCEUSVCFGHUSVCQGUQGUQMZVCUSVDCDA
      BJIRUAUBUFSVBUSFVBURETZUSNZUSURUSEUCVFUSVFDVEBDCVDUDZETVEKVGUREVDURCVDURG
      UQUEUGUHUIUJVEDBUKULUSVEUMUNUOSUP $.
  $}

  ${
    $d A x $.  $d x y $.
    sbhypf.1 $e |- F/ x ps $.
    sbhypf.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Introduce an explicit substitution into an implicit substitution
       hypothesis.  See also csbhypf .  (Contributed by Raph Levien,
       10-Apr-2004.) $)
    sbhypf $p |- ( y = A -> ( [ y / x ] ph <-> ps ) ) $=
      ( cv wceq wa wex wsb wb vex eqeq1 ceqsexv nfs1v nfbi sbequ12 bicomd
      sylan9bb exlimi sylbir ) DHZEIZCHZUDIZUFEIZJZCKACDLZBMZUHUECUDDNUFUDEOPUI
      UKCUJBCACDQFRUGUJAUHBUGAUJACDSTGUAUBUC $.
  $}

  ${
    $d z A $.  $d x z $.
    $( Closed theorem form of ~ vtoclgf .  (Contributed by NM, 17-Feb-2013.)
       (Revised by Mario Carneiro, 12-Oct-2016.) $)
    vtoclgft $p |- ( ( ( F/_ x A /\ F/ x ps )
                  /\ ( A. x ( x = A -> ( ph <-> ps ) )
                     /\ A. x ph ) /\ A e. V ) -> ps ) $=
      ( vz wcel wnfc wnf wa cv wceq wb wi wal cvv elex w3a wex mpbid elisset id
      3ad2ant3 nfnfc1 nfcvd nfeqd eqeq1 a1i cbvexd ad2antrr 3adant3 biimp com23
      imim2i imp alanimi 3ad2ant2 simp1r 19.23t syl mpd syl3an3 ) DEGCDHZBCIZJZ
      CKZDLZABMZNZCOACOJZDPGZBDEQVEVJVKRZVGCSZBVLFKZDLZFSZVMVKVEVPVJFDPUAUCVEVJ
      VPVMMZVKVCVQVDVJVCVOVGFCCDUDVCCVNDVCCVNUEVCUBUFVNVFLVOVGMNVCVNVFDUGUHUIUJ
      UKTVLVGBNZCOZVMBNZVJVEVSVKVIAVRCVIAVRVIVGABVHABNVGABULUNUMUOUPUQVLVDVSVTM
      VCVDVJVKURVGBCUSUTTVAVB $.
  $}

  ${
    vtocld.1 $e |- ( ph -> A e. V ) $.
    vtocld.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    vtocld.3 $e |- ( ph -> ps ) $.
    ${
      vtocldf.4 $e |- F/ x ph $.
      vtocldf.5 $e |- ( ph -> F/_ x A ) $.
      vtocldf.6 $e |- ( ph -> F/ x ch ) $.
      $( Implicit substitution of a class for a setvar variable.  (Contributed
         by Mario Carneiro, 15-Oct-2016.) $)
      vtocldf $p |- ( ph -> ch ) $=
        ( wnfc wnf cv wceq wb wi wal alrimi wcel ex vtoclgft syl221anc ) ADEMCD
        NDOEPZBCQZRZDSBDSEFUACKLAUGDJAUEUFHUBTABDJITGBCDEFUCUD $.
    $}

    $d x A $.  $d x ph $.  $d x ch $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       Mario Carneiro, 15-Oct-2016.) $)
    vtocld $p |- ( ph -> ch ) $=
      ( nfv nfcvd nfvd vtocldf ) ABCDEFGHIADJADEKACDLM $.
  $}

  ${
    $d x A $.
    vtoclf.1 $e |- F/ x ps $.
    vtoclf.2 $e |- A e. _V $.
    vtoclf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtoclf.4 $e |- ph $.
    $( Implicit substitution of a class for a setvar variable.  This is a
       generalization of ~ chvar .  (Contributed by NM, 30-Aug-1993.) $)
    vtoclf $p |- ps $=
      ( cv wceq wi isseti biimpd eximii 19.36i mpg ) ABCABCECIDJZABKCCDFLQABGMN
      OHP $.
  $}

  ${
    $d x A $.  $d x ps $.
    vtocl.1 $e |- A e. _V $.
    vtocl.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl.3 $e |- ph $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 30-Aug-1993.) $)
    vtocl $p |- ps $=
      ( nfv vtoclf ) ABCDBCHEFGI $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ps $.
    vtocl2.1 $e |- A e. _V $.
    vtocl2.2 $e |- B e. _V $.
    vtocl2.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    vtocl2.4 $e |- ph $.
    $( Implicit substitution of classes for setvar variables.  (Contributed by
       NM, 26-Jul-1995.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    vtocl2 $p |- ps $=
      ( wal wi wex cv wceq isseti wa eeanv biimpd 2eximi mp2an 19.36-1 19.36aiv
      sylbir nfv eximii ax-gen mpg ) ADKZBCUIBCABLZDMZUIBLCCNEOZCMZDNFOZDMZUKCM
      ZCEGPDFHPUMUOQULUNQZDMCMUPULUNCDRUQUJCDUQABISTUDUAABDBDUEUBUFUCADJUGUH $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z ps $.
    vtocl3.1 $e |- A e. _V $.
    vtocl3.2 $e |- B e. _V $.
    vtocl3.3 $e |- C e. _V $.
    vtocl3.4 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
    vtocl3.5 $e |- ph $.
    $( Implicit substitution of classes for setvar variables.  (Contributed by
       NM, 3-Jun-1995.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    vtocl3 $p |- ps $=
      ( wal wi wex cv wceq isseti w3a eeeanv biimpd eximi 2eximi sylbir 19.36-1
      mp3an nfv ax-mp eximii 19.36aiv gen2 mpg ) AENZDNZBCUOBCUNBOZDPZUOBOCABOZ
      EPZDPCPZUQCPCQFRZCPZDQGRZDPZEQHRZEPZUTCFISDGJSEHKSVBVDVFTVAVCVETZEPZDPCPU
      TVAVCVECDEUAVHUSCDVGUREVGABLUBUCUDUEUGUSUPCDABEBEUHUFUDUIUNBDBDUHUFUJUKAD
      EMULUM $.
  $}

  ${
    $d x A $.  $d x ch $.  $d x th $.
    vtoclb.1 $e |- A e. _V $.
    vtoclb.2 $e |- ( x = A -> ( ph <-> ch ) ) $.
    vtoclb.3 $e |- ( x = A -> ( ps <-> th ) ) $.
    vtoclb.4 $e |- ( ph <-> ps ) $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 23-Dec-1993.) $)
    vtoclb $p |- ( ch <-> th ) $=
      ( wb cv wceq bibi12d vtocl ) ABKCDKEFGELFMACBDHINJO $.
  $}

  ${
    vtoclgf.1 $e |- F/_ x A $.
    vtoclgf.2 $e |- F/ x ps $.
    vtoclgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtoclgf.4 $e |- ph $.
    $( Implicit substitution of a class for a setvar variable, with
       bound-variable hypotheses in place of distinct variable restrictions.
       (Contributed by NM, 21-Sep-2003.)  (Proof shortened by Mario Carneiro,
       10-Oct-2016.) $)
    vtoclgf $p |- ( A e. V -> ps ) $=
      ( wcel cvv elex cv wceq wex issetf mpbii exlimi sylbi syl ) DEJDKJZBDELUA
      CMDNZCOBCDFPUBBCGUBABIHQRST $.
  $}

  ${
    $d x A $.
    vtoclg1f.nf $e |- F/ x ps $.
    vtoclg1f.maj $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtoclg1f.min $e |- ph $.
    $( Version of ~ vtoclgf with one nonfreeness hypothesis replaced with a
       disjoint variable condition, thus avoiding dependency on ~ ax-11 and
       ~ ax-13 .  (Contributed by BJ, 1-May-2019.) $)
    vtoclg1f $p |- ( A e. V -> ps ) $=
      ( wcel cvv elex cv wceq wex isset mpbii exlimi sylbi syl ) DEIDJIZBDEKTCL
      DMZCNBCDOUABCFUAABHGPQRS $.
  $}

  ${
    $d x A $.  $d x ps $.
    vtoclg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtoclg.2 $e |- ph $.
    $( Implicit substitution of a class expression for a setvar variable.
       (Contributed by NM, 17-Apr-1995.) $)
    vtoclg $p |- ( A e. V -> ps ) $=
      ( nfcv nfv vtoclgf ) ABCDECDHBCIFGJ $.
  $}

  ${
    $d x A $.  $d x ch $.  $d x th $.
    vtoclbg.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
    vtoclbg.2 $e |- ( x = A -> ( ps <-> th ) ) $.
    vtoclbg.3 $e |- ( ph <-> ps ) $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 29-Apr-1994.) $)
    vtoclbg $p |- ( A e. V -> ( ch <-> th ) ) $=
      ( wb cv wceq bibi12d vtoclg ) ABKCDKEFGELFMACBDHINJO $.
  $}

  ${
    vtocl2gf.1 $e |- F/_ x A $.
    vtocl2gf.2 $e |- F/_ y A $.
    vtocl2gf.3 $e |- F/_ y B $.
    vtocl2gf.4 $e |- F/ x ps $.
    vtocl2gf.5 $e |- F/ y ch $.
    vtocl2gf.6 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl2gf.7 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl2gf.8 $e |- ph $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 25-Apr-1995.) $)
    vtocl2gf $p |- ( ( A e. V /\ B e. W ) -> ch ) $=
      ( wcel cvv wi elex nfel1 nfim cv wceq imbi2d vtoclgf mpan9 ) FHRFSRZGIRCF
      HUAUIBTUICTEGILUICEEFSKUBNUCEUDGUEBCUIPUFABDFSJMOQUGUGUH $.
  $}

  ${
    vtocl3gf.a $e |- F/_ x A $.
    vtocl3gf.b $e |- F/_ y A $.
    vtocl3gf.c $e |- F/_ z A $.
    vtocl3gf.d $e |- F/_ y B $.
    vtocl3gf.e $e |- F/_ z B $.
    vtocl3gf.f $e |- F/_ z C $.
    vtocl3gf.1 $e |- F/ x ps $.
    vtocl3gf.2 $e |- F/ y ch $.
    vtocl3gf.3 $e |- F/ z th $.
    vtocl3gf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl3gf.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl3gf.6 $e |- ( z = C -> ( ch <-> th ) ) $.
    vtocl3gf.7 $e |- ph $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 10-Aug-2013.)  (Revised by Mario Carneiro, 10-Oct-2016.) $)
    vtocl3gf $p |- ( ( A e. V /\ B e. W /\ C e. X ) -> th ) $=
      ( wcel cvv wa elex wi nfel1 nfim wceq imbi2d vtoclgf vtocl2gf mpan9 3impb
      cv ) HKUGZILUGZJMUGZDVAHUHUGZVBVCUIDHKUJVDBUKVDCUKVDDUKFGIJLMQRSVDCFFHUHO
      ULUAUMVDDGGHUHPULUBUMFUTIUNBCVDUDUOGUTJUNCDVDUEUOABEHUHNTUCUFUPUQURUS $.
  $}

  ${
    $d x A $.  $d y A $.  $d y B $.  $d x ps $.  $d y ch $.
    vtocl2g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl2g.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl2g.3 $e |- ph $.
    $( Implicit substitution of 2 classes for 2 setvar variables.  (Contributed
       by NM, 25-Apr-1995.) $)
    vtocl2g $p |- ( ( A e. V /\ B e. W ) -> ch ) $=
      ( nfcv nfv vtocl2gf ) ABCDEFGHIDFMEFMEGMBDNCENJKLO $.
  $}

  ${
    $d x B $.
    vtoclgaf.1 $e |- F/_ x A $.
    vtoclgaf.2 $e |- F/ x ps $.
    vtoclgaf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtoclgaf.4 $e |- ( x e. B -> ph ) $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 17-Feb-2006.)  (Revised by Mario Carneiro, 10-Oct-2016.) $)
    vtoclgaf $p |- ( A e. B -> ps ) $=
      ( wcel cv wi nfel1 nfim wceq eleq1 imbi12d vtoclgf pm2.43i ) DEJZBCKZEJZA
      LTBLCDEFTBCCDEFMGNUADOUBTABUADEPHQIRS $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    vtoclga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtoclga.2 $e |- ( x e. B -> ph ) $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 20-Aug-1995.) $)
    vtoclga $p |- ( A e. B -> ps ) $=
      ( nfcv nfv vtoclgaf ) ABCDECDHBCIFGJ $.
  $}

  ${
    $d x y C $.  $d x y D $.
    vtocl2gaf.a $e |- F/_ x A $.
    vtocl2gaf.b $e |- F/_ y A $.
    vtocl2gaf.c $e |- F/_ y B $.
    vtocl2gaf.1 $e |- F/ x ps $.
    vtocl2gaf.2 $e |- F/ y ch $.
    vtocl2gaf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl2gaf.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl2gaf.5 $e |- ( ( x e. C /\ y e. D ) -> ph ) $.
    $( Implicit substitution of 2 classes for 2 setvar variables.  (Contributed
       by NM, 10-Aug-2013.) $)
    vtocl2gaf $p |- ( ( A e. C /\ B e. D ) -> ch ) $=
      ( wcel wa wi cv nfel1 nfan nfim wceq eleq1 anbi1d imbi12d anbi2d vtocl2gf
      nfv pm2.43i ) FHRZGIRZSZCDUAZHRZEUAZIRZSZATUMUSSZBTUOCTDEFGHIJKLVABDUMUSD
      DFHJUBUSDUKUCMUDUOCEUMUNEEFHKUBEGILUBUCNUDUPFUEZUTVAABVBUQUMUSUPFHUFUGOUH
      URGUEZVAUOBCVCUSUNUMURGIUFUIPUHQUJUL $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.  $d x y D $.  $d x ps $.  $d y ch $.
    vtocl2ga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl2ga.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl2ga.3 $e |- ( ( x e. C /\ y e. D ) -> ph ) $.
    $( Implicit substitution of 2 classes for 2 setvar variables.  (Contributed
       by NM, 20-Aug-1995.) $)
    vtocl2ga $p |- ( ( A e. C /\ B e. D ) -> ch ) $=
      ( nfcv nfv vtocl2gaf ) ABCDEFGHIDFMEFMEGMBDNCENJKLO $.
  $}

  ${
    $d x y z R $.  $d x y z S $.  $d x y z T $.
    vtocl3gaf.a $e |- F/_ x A $.
    vtocl3gaf.b $e |- F/_ y A $.
    vtocl3gaf.c $e |- F/_ z A $.
    vtocl3gaf.d $e |- F/_ y B $.
    vtocl3gaf.e $e |- F/_ z B $.
    vtocl3gaf.f $e |- F/_ z C $.
    vtocl3gaf.1 $e |- F/ x ps $.
    vtocl3gaf.2 $e |- F/ y ch $.
    vtocl3gaf.3 $e |- F/ z th $.
    vtocl3gaf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl3gaf.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl3gaf.6 $e |- ( z = C -> ( ch <-> th ) ) $.
    vtocl3gaf.7 $e |- ( ( x e. R /\ y e. S /\ z e. T ) -> ph ) $.
    $( Implicit substitution of 3 classes for 3 setvar variables.  (Contributed
       by NM, 10-Aug-2013.)  (Revised by Mario Carneiro, 11-Oct-2016.) $)
    vtocl3gaf $p |- ( ( A e. R /\ B e. S /\ C e. T ) -> th ) $=
      ( wcel w3a cv nfel1 nf3an nfim wceq eleq1 3anbi1d imbi12d 3anbi2d 3anbi3d
      wi nfv vtocl3gf pm2.43i ) HKUGZILUGZJMUGZUHZDEUIZKUGZFUIZLUGZGUIZMUGZUHZA
      USVCVJVLUHZBUSVCVDVLUHZCUSVFDUSEFGHIJKLMNOPQRSVNBEVCVJVLEEHKNUJVJEUTVLEUT
      UKTULVOCFVCVDVLFFHKOUJFILQUJVLFUTUKUAULVFDGVCVDVEGGHKPUJGILRUJGJMSUJUKUBU
      LVGHUMZVMVNABVPVHVCVJVLVGHKUNUOUCUPVIIUMZVNVOBCVQVJVDVCVLVIILUNUQUDUPVKJU
      MZVOVFCDVRVLVEVCVDVKJMUNURUEUPUFVAVB $.
  $}

  ${
    $d x y z A $.  $d y z B $.  $d z C $.  $d x y z D $.  $d x y z R $.
    $d x y z S $.  $d x ps $.  $d y ch $.  $d z th $.
    vtocl3ga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl3ga.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl3ga.3 $e |- ( z = C -> ( ch <-> th ) ) $.
    vtocl3ga.4 $e |- ( ( x e. D /\ y e. R /\ z e. S ) -> ph ) $.
    $( Implicit substitution of 3 classes for 3 setvar variables.  (Contributed
       by NM, 20-Aug-1995.) $)
    vtocl3ga $p |- ( ( A e. D /\ B e. R /\ C e. S ) -> th ) $=
      ( nfcv nfv vtocl3gaf ) ABCDEFGHIJKLMEHRFHRGHRFIRGIRGJRBESCFSDGSNOPQT $.
  $}

  ${
    $d A x $.  $d A y $.  $d B y $.  $d ps x $.  $d ch y $.  $d C z $.
    $d C w $.  $d D w $.  $d A z $.  $d Q z $.  $d B z $.  $d R z $.
    $d rh z $.  $d A w $.  $d Q w $.  $d B w $.  $d R w $.  $d th w $.
    vtocl4g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl4g.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl4g.3 $e |- ( z = C -> ( ch <-> rh ) ) $.
    vtocl4g.4 $e |- ( w = D -> ( rh <-> th ) ) $.
    vtocl4g.5 $e |- ph $.
    $( Implicit substitution of 4 classes for 4 setvar variables.  (Contributed
       by AV, 22-Jan-2019.) $)
    vtocl4g $p |- ( ( ( A e. Q /\ B e. R )
                     /\ ( C e. S /\ D e. T ) ) -> th ) $=
      ( wcel wa wi cv wceq imbi2d vtocl2g impcom ) LPUCMQUCUDJNUCKOUCUDZDUKCUEU
      KEUEUKDUEHILMPQHUFLUGCEUKTUHIUFMUGEDUKUAUHABCFGJKNORSUBUIUIUJ $.
  $}

  ${
    $d w x y z A $.  $d w y z B $.  $d w z C $.  $d w D $.  $d w x y z R $.
    $d w x y z S $.  $d w x y z T $.  $d w x y z Q $.  $d x ps $.  $d z rh $.
    $d w th $.  $d y ch $.
    vtocl4ga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl4ga.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl4ga.3 $e |- ( z = C -> ( ch <-> rh ) ) $.
    vtocl4ga.4 $e |- ( w = D -> ( rh <-> th ) ) $.
    vtocl4ga.5 $e |- ( ( ( x e. Q /\ y e. R )
                         /\ ( z e. S /\ w e. T ) ) -> ph ) $.
    $( Implicit substitution of 4 classes for 4 setvar variables.  (Contributed
       by AV, 22-Jan-2019.)  (Proof shortened by Wolf Lammen, 31-May-2025.) $)
    vtocl4ga $p |- ( ( ( A e. Q /\ B e. R )
                       /\ ( C e. S /\ D e. T ) ) -> th ) $=
      ( wcel wa wi cv wceq imbi2d ex vtocl2ga com12 impcom ) LPUCMQUCUDJNUCKOUC
      UDZDUMCUEUMEUEUMDUEHILMPQHUFZLUGCEUMTUHIUFZMUGEDUMUAUHUMUNPUCUOQUCUDZCUPA
      UEUPBUEUPCUEFGJKNOFUFZJUGABUPRUHGUFZKUGBCUPSUHUQNUCUROUCUDUPAUBUIUJUKUJUL
      $.
  $}

  ${
    $d x A $.  $d x ph $.
    vtocleg.1 $e |- ( x = A -> ph ) $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 10-Jan-2004.) $)
    vtocleg $p |- ( A e. V -> ph ) $=
      ( wcel cv wceq wex elisset exlimiv syl ) CDFBGCHZBIABCDJMABEKL $.
  $}

  ${
    $d x A $.
    $( Implicit substitution of a class for a setvar variable.  (Closed theorem
       version of ~ vtoclef .)  (Contributed by NM, 7-Nov-2005.)  (Revised by
       Mario Carneiro, 11-Oct-2016.) $)
    vtoclegft $p |- ( ( A e. B /\ F/ x ph /\
                   A. x ( x = A -> ph ) ) -> ph ) $=
      ( wcel wnf cv wceq wi wal w3a wex elisset mpan9 3adant2 wb 19.9t 3ad2ant2
      exim mpbid ) CDEZABFZBGCHZAIBJZKABLZAUAUDUEUBUAUCBLUDUEBCDMUCABSNOUBUAUEA
      PUDABQRT $.
  $}

  ${
    $d x A $.
    vtoclef.1 $e |- F/ x ph $.
    vtoclef.2 $e |- A e. _V $.
    vtoclef.3 $e |- ( x = A -> ph ) $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 18-Aug-1993.) $)
    vtoclef $p |- ph $=
      ( cv wceq wex isseti exlimi ax-mp ) BGCHZBIABCEJMABDFKL $.
  $}

  ${
    $d x A $.  $d x ph $.
    vtocle.1 $e |- A e. _V $.
    vtocle.2 $e |- ( x = A -> ph ) $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 9-Sep-1993.) $)
    vtocle $p |- ph $=
      ( cvv wcel vtocleg ax-mp ) CFGADABCFEHI $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    vtoclri.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtoclri.2 $e |- A. x e. B ph $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 21-Nov-1994.) $)
    vtoclri $p |- ( A e. B -> ps ) $=
      ( rspec vtoclga ) ABCDEFACEGHI $.
  $}

  ${
    spcimgft.1 $e |- F/ x ps $.
    spcimgft.2 $e |- F/_ x A $.
    $( A closed version of ~ spcimgf .  (Contributed by Mario Carneiro,
       4-Jan-2017.) $)
    spcimgft $p |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B ->
                      ( A. x ph -> ps ) ) ) $=
      ( wcel cvv cv wceq wi wal elex wex issetf exim biimtrid 19.36-1 syl6 syl5
      ) DEHDIHZCJDKZABLZLCMZACMBLZDENUEUBUDCOZUFUBUCCOUEUGCDGPUCUDCQRABCFSTUA
      $.

    $( A closed version of ~ spcgf .  (Contributed by Andrew Salmon,
       6-Jun-2011.)  (Revised by Mario Carneiro, 4-Jan-2017.) $)
    spcgft $p |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B ->
                      ( A. x ph -> ps ) ) ) $=
      ( cv wceq wb wi wal wcel biimp imim2i alimi spcimgft syl ) CHDIZABJZKZCLS
      ABKZKZCLDEMACLBKKUAUCCTUBSABNOPABCDEFGQR $.

    $( A closed version of ~ spcimegf .  (Contributed by Mario Carneiro,
       4-Jan-2017.) $)
    spcimegft $p |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. B ->
                      ( ps -> E. x ph ) ) ) $=
      ( wcel cvv cv wceq wi wal wex elex issetf exim biimtrid 19.37-1 syl6 syl5
      ) DEHDIHZCJDKZBALZLCMZBACNLZDEOUEUBUDCNZUFUBUCCNUEUGCDGPUCUDCQRBACFSTUA
      $.

    $( A closed version of ~ spcegf .  (Contributed by Jim Kingdon,
       22-Jun-2018.) $)
    spcegft $p |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B ->
                      ( ps -> E. x ph ) ) ) $=
      ( cv wceq wb wi wal wcel wex biimpr imim2i alimi spcimegft syl ) CHDIZABJ
      ZKZCLTBAKZKZCLDEMBACNKKUBUDCUAUCTABOPQABCDEFGRS $.
  $}

  ${
    spcimgf.1 $e |- F/_ x A $.
    spcimgf.2 $e |- F/ x ps $.
    ${
      spcimgf.3 $e |- ( x = A -> ( ph -> ps ) ) $.
      $( Rule of specialization, using implicit substitution.  Compare Theorem
         7.3 of [Quine] p. 44.  (Contributed by Mario Carneiro, 4-Jan-2017.) $)
      spcimgf $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
        ( cv wceq wi wcel wal spcimgft mpg ) CIDJABKKDELACMBKKCABCDEGFNHO $.
    $}

    spcimegf.3 $e |- ( x = A -> ( ps -> ph ) ) $.
    $( Existential specialization, using implicit substitution.  (Contributed
       by Mario Carneiro, 4-Jan-2017.) $)
    spcimegf $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
      ( cv wceq wi wcel wex spcimegft mpg ) CIDJBAKKDELBACMKKCABCDEGFNHO $.
  $}

  ${
    spcgf.1 $e |- F/_ x A $.
    spcgf.2 $e |- F/ x ps $.
    spcgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Rule of specialization, using implicit substitution.  Compare Theorem
       7.3 of [Quine] p. 44.  (Contributed by NM, 2-Feb-1997.)  (Revised by
       Andrew Salmon, 12-Aug-2011.) $)
    spcgf $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
      ( cv wceq wb wi wcel wal spcgft mpg ) CIDJABKLDEMACNBLLCABCDEGFOHP $.

    $( Existential specialization, using implicit substitution.  (Contributed
       by NM, 2-Feb-1997.) $)
    spcegf $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
      ( cv wceq wb wi wcel wex spcegft mpg ) CIDJABKLDEMBACNLLCABCDEGFOHP $.
  $}

  ${
    $d x A $.  $d x ph $.  $d x ch $.
    spcimdv.1 $e |- ( ph -> A e. B ) $.
    ${
      spcimdv.2 $e |- ( ( ph /\ x = A ) -> ( ps -> ch ) ) $.
      $( Restricted specialization, using implicit substitution.  (Contributed
         by Mario Carneiro, 4-Jan-2017.) $)
      spcimdv $p |- ( ph -> ( A. x ps -> ch ) ) $=
        ( cv wceq wi wal wcel ex alrimiv nfv nfcv spcimgft sylc ) ADIEJZBCKZKZD
        LEFMBDLCKAUBDATUAHNOGBCDEFCDPDEQRS $.
    $}

    ${
      spcdv.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
      $( Rule of specialization, using implicit substitution.  Analogous to
         ~ rspcdv .  (Contributed by David Moews, 1-May-2017.) $)
      spcdv $p |- ( ph -> ( A. x ps -> ch ) ) $=
        ( cv wceq wa biimpd spcimdv ) ABCDEFGADIEJKBCHLM $.
    $}

    spcimedv.2 $e |- ( ( ph /\ x = A ) -> ( ch -> ps ) ) $.
    $( Restricted existential specialization, using implicit substitution.
       (Contributed by Mario Carneiro, 4-Jan-2017.) $)
    spcimedv $p |- ( ph -> ( ch -> E. x ps ) ) $=
      ( cv wceq wi wal wcel wex ex alrimiv nfv nfcv spcimegft sylc ) ADIEJZCBKZ
      KZDLEFMCBDNKAUCDAUAUBHOPGBCDEFCDQDERST $.
  $}

  ${
    $d x ps $.  $d x A $.
    spcgv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Rule of specialization, using implicit substitution.  Compare Theorem
       7.3 of [Quine] p. 44.  (Contributed by NM, 22-Jun-1994.) $)
    spcgv $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
      ( nfcv nfv spcgf ) ABCDECDGBCHFI $.

    $( Existential specialization, using implicit substitution.  (Contributed
       by NM, 14-Aug-1994.) $)
    spcegv $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
      ( nfcv nfv spcegf ) ABCDECDGBCHFI $.
  $}

  ${
    $d X x $.  $d ch x $.
    spcedv.1 $e |- ( ph -> X e. _V ) $.
    spcedv.2 $e |- ( ph -> ch ) $.
    spcedv.3 $e |- ( x = X -> ( ps <-> ch ) ) $.
    $( Existential specialization, using implicit substitution, deduction
       version.  (Contributed by RP, 12-Aug-2020.) $)
    spcedv $p |- ( ph -> E. x ps ) $=
      ( cvv wcel wex spcegv sylc ) AEIJCBDKFGBCDEIHLM $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ps $.
    spc2egv.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( Existential specialization with 2 quantifiers, using implicit
       substitution.  (Contributed by NM, 3-Aug-1995.) $)
    spc2egv $p |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ph ) ) $=
      ( wcel wa cv wceq wex elisset anim12i eeanv sylibr biimprcd 2eximdv
      syl5com ) EGJZFHJZKZCLEMZDLFMZKZDNCNZBADNCNUDUECNZUFDNZKUHUBUIUCUJCEGODFH
      OPUEUFCDQRBUGACDUGABISTUA $.

    $( Specialization with 2 quantifiers, using implicit substitution.
       (Contributed by NM, 27-Apr-2004.) $)
    spc2gv $p |- ( ( A e. V /\ B e. W ) -> ( A. x A. y ph -> ps ) ) $=
      ( wcel wa cv wceq wex wal elisset anim12i wi exim 19.9v eeanv sylibr 3syl
      biimpcd 2alimi alimi bitri imbitrdi syl5com ) EGJZFHJZKZCLEMZDLFMZKZDNZCN
      ZADOCOZBULUMCNZUNDNZKUQUJUSUKUTCEGPDFHPQUMUNCDUAUBURUQBDNZCNZBURUOBRZDOZC
      OUPVARZCOUQVBRAVCCDUOABIUDUEVDVECUOBDSUFUPVACSUCVBVABVACTBDTUGUHUI $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z ps $.
    spc3egv.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
    $( Existential specialization with 3 quantifiers, using implicit
       substitution.  (Contributed by NM, 12-May-2008.) $)
    spc3egv $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
               ( ps -> E. x E. y E. z ph ) ) $=
      ( wcel w3a cv wceq wex elisset 3anim123i eeeanv biimprcd 2eximdv syl5com
      sylibr eximdv ) FIMZGJMZHKMZNZCOFPZDOGPZEOHPZNZEQZDQCQZBAEQZDQCQUIUJCQZUK
      DQZULEQZNUOUFUQUGURUHUSCFIRDGJREHKRSUJUKULCDETUDBUNUPCDBUMAEUMABLUAUEUBUC
      $.

    $( Specialization with 3 quantifiers, using implicit substitution.
       (Contributed by NM, 12-May-2008.) $)
    spc3gv $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
               ( A. x A. y A. z ph -> ps ) ) $=
      ( wcel w3a cv wceq wex wal elisset wi 3anim123i eeeanv sylibr 2alimi exim
      biimpcd alimi syl 3syl 19.9v 3bitri imbitrdi syl5com ) FIMZGJMZHKMZNZCOFP
      ZDOGPZEOHPZNZEQZDQZCQZAERDRZCRZBUQURCQZUSDQZUTEQZNVDUNVGUOVHUPVICFISDGJSE
      HKSUAURUSUTCDEUBUCVFVDBEQZDQZCQZBVFVBVJTZDRZCRZVCVKTZCRVDVLTVFVABTZERZDRZ
      CRVOVEVSCAVQDEVAABLUFUDUGVRVMCDVABEUEUDUHVNVPCVBVJDUEUGVCVKCUEUIVLVKVJBVK
      CUJVJDUJBEUJUKULUM $.
  $}

  ${
    $d x A $.  $d x ps $.
    spcv.1 $e |- A e. _V $.
    spcv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Rule of specialization, using implicit substitution.  (Contributed by
       NM, 22-Jun-1994.) $)
    spcv $p |- ( A. x ph -> ps ) $=
      ( cvv wcel wal wi spcgv ax-mp ) DGHACIBJEABCDGFKL $.

    $( Existential specialization, using implicit substitution.  (Contributed
       by NM, 31-Dec-1993.)  (Proof shortened by Eric Schmidt, 22-Dec-2006.) $)
    spcev $p |- ( ps -> E. x ph ) $=
      ( cvv wcel wex wi spcegv ax-mp ) DGHBACIJEABCDGFKL $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ps $.
    spc2ev.1 $e |- A e. _V $.
    spc2ev.2 $e |- B e. _V $.
    spc2ev.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( Existential specialization, using implicit substitution.  (Contributed
       by NM, 3-Aug-1995.) $)
    spc2ev $p |- ( ps -> E. x E. y ph ) $=
      ( cvv wcel wex wi spc2egv mp2an ) EJKFJKBADLCLMGHABCDEFJJINO $.
  $}

  ${
    $d x A $.  $d x B $.
    rspct.1 $e |- F/ x ps $.
    $( A closed version of ~ rspc .  (Contributed by Andrew Salmon,
       6-Jun-2011.) $)
    rspct $p |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B ->
                   ( A. x e. B ph -> ps ) ) ) $=
      ( cv wceq wb wi wal wcel wral df-ral wa eleq1 adantr simpr imbi12d ex a2i
      alimi nfv nfim nfcv spcgft syl syl7bi com34 pm2.43d ) CGZDHZABIZJZCKZDELZ
      ACEMZBJUOUPUQUPBUQUKELZAJZCKZUOUPUPBJZACENUOULUSVAIZJZCKUPUTVAJJUNVCCULUM
      VBULUMVBULUMOURUPABULURUPIUMUKDEPQULUMRSTUAUBUSVACDEUPBCUPCUCFUDCDUEUFUGU
      HUIUJ $.
  $}

  ${
    $d x A $.  $d x B $.
    rspc.1 $e |- F/ x ps $.
    rspc.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Restricted specialization, using implicit substitution.  (Contributed by
       NM, 19-Apr-2005.)  (Revised by Mario Carneiro, 11-Oct-2016.) $)
    rspc $p |- ( A e. B -> ( A. x e. B ph -> ps ) ) $=
      ( wral cv wcel wi wal df-ral nfcv nfv nfim wceq eleq1 imbi12d spcgf
      pm2.43a biimtrid ) ACEHCIZEJZAKZCLZDEJZBACEMUFUGBUEUGBKCDECDNUGBCUGCOFPUC
      DQUDUGABUCDERGSTUAUB $.

    $( Restricted existential specialization, using implicit substitution.
       (Contributed by NM, 26-May-1998.)  (Revised by Mario Carneiro,
       11-Oct-2016.) $)
    rspce $p |- ( ( A e. B /\ ps ) -> E. x e. B ph ) $=
      ( wcel wa cv wex wrex nfcv nfv nfan wceq eleq1 anbi12d spcegf anabsi5
      df-rex sylibr ) DEHZBIZCJZEHZAIZCKZACELUCBUHUGUDCDECDMUCBCUCCNFOUEDPUFUCA
      BUEDEQGRSTACEUAUB $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    rspcv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Restricted specialization, using implicit substitution.  (Contributed by
       NM, 26-May-1998.) $)
    rspcv $p |- ( A e. B -> ( A. x e. B ph -> ps ) ) $=
      ( nfv rspc ) ABCDEBCGFH $.

    $( Restricted specialization, using implicit substitution.  (Contributed by
       NM, 2-Feb-2006.) $)
    rspccv $p |- ( A. x e. B ph -> ( A e. B -> ps ) ) $=
      ( wcel wral rspcv com12 ) DEGACEHBABCDEFIJ $.

    $( Restricted specialization, using implicit substitution.  (Contributed by
       NM, 13-Sep-2005.) $)
    rspcva $p |- ( ( A e. B /\ A. x e. B ph ) -> ps ) $=
      ( wcel wral rspcv imp ) DEGACEHBABCDEFIJ $.

    $( Restricted specialization, using implicit substitution.  (Contributed by
       NM, 26-Jul-2006.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    rspccva $p |- ( ( A. x e. B ph /\ A e. B ) -> ps ) $=
      ( wcel wral rspcv impcom ) DEGACEHBABCDEFIJ $.

    $( Restricted existential specialization, using implicit substitution.
       (Contributed by NM, 26-May-1998.) $)
    rspcev $p |- ( ( A e. B /\ ps ) -> E. x e. B ph ) $=
      ( nfv rspce ) ABCDEBCGFH $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.  $d x ch $.
    rspcimdv.1 $e |- ( ph -> A e. B ) $.
    ${
      rspcimdv.2 $e |- ( ( ph /\ x = A ) -> ( ps -> ch ) ) $.
      $( Restricted specialization, using implicit substitution.  (Contributed
         by Mario Carneiro, 4-Jan-2017.) $)
      rspcimdv $p |- ( ph -> ( A. x e. B ps -> ch ) ) $=
        ( wral cv wcel wi wal df-ral wceq wa simpr eleq1d biimprd imim12d mpid
        spcimdv biimtrid ) BDFIDJZFKZBLZDMZACBDFNAUGEFKZCGAUFUHCLDEFGAUDEOZPZUH
        UEBCUJUEUHUJUDEFAUIQRSHTUBUAUC $.
    $}

    rspcimedv.2 $e |- ( ( ph /\ x = A ) -> ( ch -> ps ) ) $.
    $( Restricted existential specialization, using implicit substitution.
       (Contributed by Mario Carneiro, 4-Jan-2017.) $)
    rspcimedv $p |- ( ph -> ( ch -> E. x e. B ps ) ) $=
      ( cv wcel wa wex wrex wceq simpr eleq1d biimprd anim12d spcimedv mpand
      df-rex imbitrrdi ) ACDIZFJZBKZDLZBDFMAEFJZCUFGAUEUGCKDEFGAUCENZKZUGUDCBUI
      UDUGUIUCEFAUHOPQHRSTBDFUAUB $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.  $d x ch $.
    rspcdv.1 $e |- ( ph -> A e. B ) $.
    rspcdv.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Restricted specialization, using implicit substitution.  (Contributed by
       NM, 17-Feb-2007.)  (Revised by Mario Carneiro, 4-Jan-2017.) $)
    rspcdv $p |- ( ph -> ( A. x e. B ps -> ch ) ) $=
      ( cv wceq wa biimpd rspcimdv ) ABCDEFGADIEJKBCHLM $.

    $( Restricted existential specialization, using implicit substitution.
       (Contributed by FL, 17-Apr-2007.)  (Revised by Mario Carneiro,
       4-Jan-2017.) $)
    rspcedv $p |- ( ph -> ( ch -> E. x e. B ps ) ) $=
      ( cv wceq wa biimprd rspcimedv ) ABCDEFGADIEJKBCHLM $.
  $}

  ${
    $d A x $.  $d C x $.  $d ch x $.
    rspcdva.1 $e |- ( x = C -> ( ps <-> ch ) ) $.
    rspcdva.2 $e |- ( ph -> A. x e. A ps ) $.
    rspcdva.3 $e |- ( ph -> C e. A ) $.
    $( Restricted specialization, using implicit substitution.  (Contributed by
       Thierry Arnoux, 21-Jun-2020.) $)
    rspcdva $p |- ( ph -> ch ) $=
      ( wcel wral rspcv sylc ) AFEJBDEKCIHBCDFEGLM $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.  $d x ch $.
    rspcedvd.1 $e |- ( ph -> A e. B ) $.
    rspcedvd.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    rspcedvd.3 $e |- ( ph -> ch ) $.
    $( Restricted existential specialization, using implicit substitution.
       Variant of ~ rspcedv .  (Contributed by AV, 27-Nov-2019.) $)
    rspcedvd $p |- ( ph -> E. x e. B ps ) $=
      ( wrex rspcedv mpd ) ACBDFJIABCDEFGHKL $.
  $}

  ${
    $d A x $.  $d B x $.  $d ch x $.
    rspcedvdw.s $e |- ( x = A -> ( ps <-> ch ) ) $.
    rspcedvdw.1 $e |- ( ph -> A e. B ) $.
    rspcedvdw.2 $e |- ( ph -> ch ) $.
    $( Version of ~ rspcedvd where the implicit substitution hypothesis does
       not have an antecedent, which also avoids a disjoint variable condition
       on ` ph , x ` .  (Contributed by SN, 20-Aug-2024.) $)
    rspcedvdw $p |- ( ph -> E. x e. B ps ) $=
      ( wcel wrex rspcev syl2anc ) AEFJCBDFKHIBCDEFGLM $.
  $}

  ${
    $d x ph $.  $d x B $.  $d x A $.
    rspcime.1 $e |- ( ( ph /\ x = A ) -> ps ) $.
    rspcime.2 $e |- ( ph -> A e. B ) $.
    $( Prove a restricted existential.  (Contributed by Rohan Ridenour,
       3-Aug-2023.) $)
    rspcime $p |- ( ph -> E. x e. B ps ) $=
      ( cv wceq wa simpl 2thd id rspcedvd ) ABACDEGACHDIZJBAFAOKLAMN $.
  $}

  ${
    $d x y A $.  $d x B $.  $d x C $.  $d x ps $.  $d x ch $.
    rspceaimv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Restricted existential specialization of a universally quantified
       implication.  (Contributed by BJ, 24-Aug-2022.) $)
    rspceaimv $p |- ( ( A e. B /\ A. y e. C ( ps -> ch ) ) ->
                                          E. x e. B A. y e. C ( ph -> ch ) ) $=
      ( wi wral cv wceq imbi1d ralbidv rspcev ) ACJZEHKBCJZEHKDFGDLFMZQREHSABCI
      NOP $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    rspcedeqvd.1 $e |- ( ph -> A e. B ) $.
    rspcedeqvd.2 $e |- ( ( ph /\ x = A ) -> C = D ) $.
    ${
      $d x D $.
      $( Restricted existential specialization, using implicit substitution.
         Variant of ~ rspcedvd for equations, in which the left hand side
         depends on the quantified variable.  (Contributed by AV,
         24-Dec-2019.) $)
      rspcedeq1vd $p |- ( ph -> E. x e. B C = D ) $=
        ( wceq cv wa eqeq1d eqidd rspcedvd ) AEFIFFIBCDGABJCIKEFFHLAFMN $.
    $}

    $d x C $.
    $( Restricted existential specialization, using implicit substitution.
       Variant of ~ rspcedvd for equations, in which the right hand side
       depends on the quantified variable.  (Contributed by AV,
       24-Dec-2019.) $)
    rspcedeq2vd $p |- ( ph -> E. x e. B C = D ) $=
      ( wceq cv wa eqcomd eqeq2d eqidd rspcedvd ) AEFIEEIBCDGABJCIKZFEEPEFHLMAE
      NO $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x C $.  $d x y D $.
    rspc2.1 $e |- F/ x ch $.
    rspc2.2 $e |- F/ y ps $.
    rspc2.3 $e |- ( x = A -> ( ph <-> ch ) ) $.
    rspc2.4 $e |- ( y = B -> ( ch <-> ps ) ) $.
    $( 2-variable restricted specialization, using implicit substitution.
       (Contributed by NM, 9-Nov-2012.) $)
    rspc2 $p |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph ->
                  ps ) ) $=
      ( wcel wral nfcv nfralxy cv wceq rspc ralbidv sylan9 ) FHNAEIOZDHOCEIOZGI
      NBUCUDDFHCDEIDIPJQDRFSACEILUATCBEGIKMTUB $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y V $.  $d x y W $.  $d x y ps $.
    rspc2gv.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( Restricted specialization with two quantifiers, using implicit
       substitution.  (Contributed by BJ, 2-Dec-2021.) $)
    rspc2gv $p |- ( ( A e. V /\ B e. W ) ->
                                          ( A. x e. V A. y e. W ph -> ps ) ) $=
      ( wral cv wcel wi wal wa df-ral albii wceq eleq1 biimtrid imbi2i bi2anan9
      19.21v bicomi impexp imbi12d bitr3id spc2gv pm2.43a ) ADHJZCGJCKZGLZUJMZC
      NZEGLZFHLZOZBUJCGPUNULDKZHLZAMZDNZMZCNZUQBUMVBCUJVAULADHPUAQVCULUTMZDNZCN
      ZUQBVBVECVEVBULUTDUCUDQVFUQBVDUQBMZCDEFGHVDULUSOZAMUKERZURFRZOZVGULUSAUEV
      KVHUQABVIULUOVJUSUPUKEGSURFHSUBIUFUGUHUITTT $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x C $.  $d x y D $.  $d x ch $.  $d y ps $.
    rspc2v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
    rspc2v.2 $e |- ( y = B -> ( ch <-> ps ) ) $.
    $( 2-variable restricted specialization, using implicit substitution.
       (Contributed by NM, 13-Sep-1999.) $)
    rspc2v $p |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph ->
                  ps ) ) $=
      ( nfv rspc2 ) ABCDEFGHICDLBELJKM $.

    $( 2-variable restricted specialization, using implicit substitution.
       (Contributed by NM, 18-Jun-2014.) $)
    rspc2va $p |- ( ( ( A e. C /\ B e. D ) /\ A. x e. C A. y e. D ph ) ->
                  ps ) $=
      ( wcel wa wral rspc2v imp ) FHLGILMAEINDHNBABCDEFGHIJKOP $.

    $( 2-variable restricted existential specialization, using implicit
       substitution.  (Contributed by NM, 16-Oct-1999.) $)
    rspc2ev $p |- ( ( A e. C /\ B e. D /\ ps ) -> E. x e. C E. y e. D ph ) $=
      ( wcel w3a wrex wa rspcev anim2i 3impb cv wceq rexbidv syl ) FHLZGILZBMUC
      CEINZOZAEINZDHNUCUDBUFUDBOUEUCCBEGIKPQRUGUEDFHDSFTACEIJUAPUB $.
  $}

  ${
    $d z ps $.  $d x ch $.  $d y th $.  $d x y z A $.  $d y z B $.  $d z C $.
    $d x R $.  $d x y S $.  $d x y z T $.
    rspc3v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
    rspc3v.2 $e |- ( y = B -> ( ch <-> th ) ) $.
    rspc3v.3 $e |- ( z = C -> ( th <-> ps ) ) $.
    $( 3-variable restricted specialization, using implicit substitution.
       (Contributed by NM, 10-May-2005.) $)
    rspc3v $p |- ( ( A e. R /\ B e. S /\ C e. T ) ->
                  ( A. x e. R A. y e. S A. z e. T ph -> ps ) ) $=
      ( wcel wral cv wceq wi wa ralbidv rspc2v rspcv sylan9 3impa ) HKQZILQZJMQ
      ZAGMRZFLREKRZBUAUHUIUBULDGMRZUJBUKUMCGMREFHIKLESHTACGMNUCFSITCDGMOUCUDDBG
      JMPUEUFUG $.

    $( 3-variable restricted existentional specialization, using implicit
       substitution.  (Contributed by NM, 25-Jul-2012.) $)
    rspc3ev $p |- ( ( ( A e. R /\ B e. S /\ C e. T ) /\ ps ) ->
                  E. x e. R E. y e. S E. z e. T ph ) $=
      ( wcel wrex cv wceq w3a wa simpl1 simpl2 rspcev 3ad2antl3 rexbidv rspc2ev
      syl3anc ) HKQZILQZJMQZUABUBUJUKDGMRZAGMRZFLREKRUJUKULBUCUJUKULBUDULUJBUMU
      KDBGJMPUEUFUNUMCGMREFHIKLESHTACGMNUGFSITCDGMOUGUHUI $.
  $}

  ${
    $d A x $.  $d B x $.  $d D x $.  $d E x $.
    rspceeqv.1 $e |- ( x = A -> C = D ) $.
    $( Restricted existential specialization in an equality, using implicit
       substitution.  (Contributed by BJ, 2-Sep-2022.) $)
    rspceeqv $p |- ( ( A e. B /\ E = D ) -> E. x e. B E = C ) $=
      ( wceq cv eqeq2d rspcev ) FDHFEHABCAIBHDEFGJK $.
  $}

  ${
    $d x A $.  $d x B $.
    eqvinc.1 $e |- A e. _V $.
    $( A variable introduction law for class equality.  (Contributed by NM,
       14-Apr-1995.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    eqvinc $p |- ( A = B <-> E. x ( x = A /\ x = B ) ) $=
      ( wceq cv wa wex wi isseti ax-1 eqtr jca eximi pm3.43 mp2b 19.37aiv eqtr2
      ex exlimiv impbii ) BCEZAFZBEZUCCEZGZAHUBUFAUDAHUBUDIZUBUEIZGZAHUBUFIZAHA
      BDJUDUIAUDUGUHUDUBKUDUBUEUCBCLSMNUIUJAUBUDUEONPQUFUBAUCBCRTUA $.
  $}

  ${
    $d x A $.  $d x B $.
    $( A variable introduction law for class equality, deduction version.
       (Contributed by Thierry Arnoux, 2-Mar-2017.) $)
    eqvincg $p |- ( A e. V -> ( A = B <-> E. x ( x = A /\ x = B ) ) ) $=
      ( wcel wceq cv wa wex wi elisset ax-1 eqtr ex jca pm3.43 3syl nfv 19.37-1
      eximi syl eqtr2 exlimiv impbid1 ) BDEZBCFZAGZBFZUGCFZHZAIZUEUFUJJZAIZUFUK
      JUEUHAIUFUHJZUFUIJZHZAIUMABDKUHUPAUHUNUOUHUFLUHUFUIUGBCMNOTUPULAUFUHUIPTQ
      UFUJAUFARSUAUJUFAUGBCUBUCUD $.
  $}

  ${
    $d A y $.  $d B y $.  $d x y $.
    eqvincf.1 $e |- F/_ x A $.
    eqvincf.2 $e |- F/_ x B $.
    eqvincf.3 $e |- A e. _V $.
    $( A variable introduction law for class equality, using bound-variable
       hypotheses instead of distinct variable conditions.  (Contributed by NM,
       14-Sep-2003.) $)
    eqvincf $p |- ( A = B <-> E. x ( x = A /\ x = B ) ) $=
      ( vy wceq cv wa wex eqvinc nfeq2 nfan nfv eqeq1 anbi12d cbvex bitri ) BCH
      GIZBHZTCHZJZGKAIZBHZUDCHZJZAKGBCFLUCUGGAUAUBAATBDMATCEMNUGGOTUDHUAUEUBUFT
      UDBPTUDCPQRS $.
  $}

  ${
    $d x A y $.  $d ph y $.
    alexeq.1 $e |- A e. _V $.
    $( Two ways to express substitution of ` A ` for ` x ` in ` ph ` .
       (Contributed by NM, 2-Mar-1995.) $)
    alexeq $p |- ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) $=
      ( vy cv wceq wa wex wi wal anbi1d exbidv imbi1d albidv sb56 vtoclb bicomi
      eqeq2 ) BFZCGZAHZBIZUAAJZBKZTEFZGZAHZBIUGAJZBKUCUEECDUFCGZUHUBBUJUGUAAUFC
      TSZLMUJUIUDBUJUGUAAUKNOABEPQR $.
  $}

  ${
    $d x A y $.  $d ph y $.
    $( Equality implies equivalence with substitution.  (Contributed by NM,
       2-Mar-1995.) $)
    ceqex $p |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) $=
      ( vy cvv wcel cv wceq wa wex wb 19.8a isset sylibr wi eqeq2 anbi1d exbidv
      bibi2d imbi12d ex wal vex alexeq sp com12 biimtrrid impbid vtoclg mpcom )
      CEFZBGZCHZAUMAIZBJZKZUMUMBJUKUMBLBCMNULDGZHZAURAIZBJZKZOUMUPODCEUQCHZURUM
      VAUPUQCULPZVBUTUOAVBUSUNBVBURUMAVCQRSTURAUTURAUTUSBLUAUTURAOZBUBZURAABUQD
      UCUDVEURAVDBUEUFUGUHUIUJ $.
  $}

  ${
    $d x A $.
    ceqsexg.1 $e |- F/ x ps $.
    ceqsexg.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( A representation of explicit substitution of a class for a variable,
       inferred from an implicit substitution hypothesis.  (Contributed by NM,
       11-Oct-2004.) $)
    ceqsexg $p |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) ) $=
      ( wb cv wceq wa wex nfcv nfe1 nfbi ceqex bibi12d biid vtoclgf ) AAHCIDJZA
      KZCLZBHCDECDMUBBCUACNFOTAUBABACDPGQARS $.
  $}

  ${
    $d x A $.  $d x ps $.
    ceqsexgv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Elimination of an existential quantifier, using implicit substitution.
       (Contributed by NM, 29-Dec-1996.) $)
    ceqsexgv $p |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) ) $=
      ( nfv ceqsexg ) ABCDEBCGFH $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    ceqsrexv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Elimination of a restricted existential quantifier, using implicit
       substitution.  (Contributed by NM, 30-Apr-2004.) $)
    ceqsrexv $p |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) ) $=
      ( cv wceq wa wrex wcel wex df-rex an12 exbii bitr4i eleq1 anbi12d bianabs
      ceqsexgv bitrid ) CGZDHZAIZCEJZUCUBEKZAIZIZCLZDEKZBUEUFUDIZCLUIUDCEMUHUKC
      UCUFANOPUJUIBUGUJBICDEUCUFUJABUBDEQFRTSUA $.

    $( Elimination of a restricted existential quantifier, using implicit
       substitution.  (Contributed by Mario Carneiro, 14-Mar-2014.) $)
    ceqsrexbv $p |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) ) $=
      ( wcel cv wceq wa wrex r19.42v eleq1 adantr pm5.32ri bicomi baib ceqsrexv
      wb rexbiia pm5.32i 3bitr3i ) DEGZCHZDIZAJZJZCEKUCUFCEKZJUHUCBJUCUFCELUGUF
      CEUGUDEGZUFUIUFJUGUFUIUCUEUIUCSAUDDEMNOPQTUCUHBABCDEFRUAUB $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x C $.  $d x y D $.  $d x ps $.  $d y ch $.
    ceqsrex2v.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsrex2v.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( Elimination of a restricted existential quantifier, using implicit
       substitution.  (Contributed by NM, 29-Oct-2005.) $)
    ceqsrex2v $p |- ( ( A e. C /\ B e. D ) ->
      ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) ) $=
      ( wcel cv wceq wa wrex anass rexbii r19.42v ceqsrexv bitri anbi2d rexbidv
      bitrid sylan9bb ) FHLZDMFNZEMGNZOAOZEIPZDHPZUHBOZEIPZGILCUKUGUHAOZEIPZOZD
      HPUFUMUJUPDHUJUGUNOZEIPUPUIUQEIUGUHAQRUGUNEISUARUOUMDFHUGUNULEIUGABUHJUBU
      CTUDBCEGIKTUE $.
  $}

  ${
    $d x A $.  $d x B $.
    clel2.1 $e |- A e. _V $.
    $( An alternate definition of class membership when the class is a set.
       (Contributed by NM, 18-Aug-1993.) $)
    clel2 $p |- ( A e. B <-> A. x ( x = A -> x e. B ) ) $=
      ( cv wceq wcel wi wal eleq1 ceqsalv bicomi ) AEZBFMCGZHAIBCGZNOABDMBCJKL
      $.
  $}

  ${
    $d x A $.  $d x B $.
    $( An alternate definition of class membership when the class is a set.
       (Contributed by NM, 13-Aug-2005.) $)
    clel3g $p |- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) ) $=
      ( wcel cv wceq wa wex eleq2 ceqsexgv bicomd ) CDEAFZCGBMEZHAIBCEZNOACDMCB
      JKL $.
  $}

  ${
    $d x A $.  $d x B $.
    clel3.1 $e |- B e. _V $.
    $( An alternate definition of class membership when the class is a set.
       (Contributed by NM, 18-Aug-1993.) $)
    clel3 $p |- ( A e. B <-> E. x ( x = B /\ A e. x ) ) $=
      ( cvv wcel cv wceq wa wex wb clel3g ax-mp ) CEFBCFAGZCHBNFIAJKDABCELM $.
  $}

  ${
    $d x A $.  $d x B $.
    clel4.1 $e |- B e. _V $.
    $( An alternate definition of class membership when the class is a set.
       (Contributed by NM, 18-Aug-1993.) $)
    clel4 $p |- ( A e. B <-> A. x ( x = B -> A e. x ) ) $=
      ( cv wceq wcel wi wal eleq2 ceqsalv bicomi ) AEZCFBMGZHAIBCGZNOACDMCBJKL
      $.
  $}

  ${
    $d A x $.  $d X x $.
    $( Alternate definition of class membership: a class ` X ` is an element of
       another class ` A ` iff there is an element of ` A ` equal to ` X ` .
       (Contributed by AV, 13-Nov-2020.)  (Revised by Steven Nguyen,
       19-May-2023.) $)
    clel5 $p |- ( X e. A <-> E. x e. A X = x ) $=
      ( wcel cv wceq wrex risset eqcom rexbii bitri ) CBDAEZCFZABGCLFZABGACBHMN
      ABLCIJK $.
  $}

  ${
    $d y A z $.  $d y B z $.
    $( Compare theorem *13.183 in [WhiteheadRussell] p. 178.  Only ` A ` is
       required to be a set.  (Contributed by Andrew Salmon, 3-Jun-2011.) $)
    pm13.183 $p |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) ) $=
      ( vy cv wceq wal eqeq1 eqeq2 bibi1d albidv alrimiv wsb stdpc4 sbbi bibi2i
      wb eqsb1 sylbi equsb1 biimp mpi syl impbii vtoclbg ) EFZCGZAFZUGGZUICGZRZ
      AHZBCGUIBGZUKRZAHEBDUGBCIUGBGZULUOAUPUJUNUKUGBUIJKLUHUMUHULAUGCUIJMUMULAE
      NZUHULAEOUQUJAENZUKAENZRZUHUJUKAEPUTURUHRZUHUSUHURAECSQVAURUHAEUAURUHUBUC
      TTUDUEUF $.
  $}

  ${
    $d y A $.  $d x y $.  $d y ph $.
    $( Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89.
       (Contributed by NM, 25-Oct-2012.) $)
    rr19.3v $p |- ( A. x e. A A. y e. A ph <-> A. x e. A ph ) $=
      ( wral cv weq biidd rspcv ralimia wcel ax-1 ralrimiv ralimi impbii ) ACDE
      ZBDEABDEPABDAACBFDCBGAHIJAPBDAACDACFDKLMNO $.

    $( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90.
       (Contributed by NM, 29-Oct-2012.) $)
    rr19.28v $p |- ( A. x e. A A. y e. A ( ph /\ ps )
                      <-> A. x e. A ( ph /\ A. y e. A ps ) ) $=
      ( wa wral cv wcel simpl ralimi weq biidd rspcv syl5 wi simpr jcad ralimia
      a1i r19.28av impbii ) ABFZDEGZCEGABDEGZFZCEGUDUFCECHZEIZUDAUEUDADEGUHAUCA
      DEABJKAADUGEDCLAMNOUDUEPUHUCBDEABQKTRSUFUDCEABDEUAKUB $.
  $}

  ${
    $d x A $.  $d x ps $.
    $( Membership in a class abstraction, using implicit substitution.  (Closed
       theorem version of ~ elabg .)  (Contributed by NM, 7-Nov-2005.)  (Proof
       shortened by Andrew Salmon, 8-Jun-2011.) $)
    elabgt $p |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) ->
                 ( A e. { x | ph } <-> ps ) ) $=
      ( cv wceq wb wal wcel cab abid eleq1 bitr3id bibi1d biimpd a2i alimi nfcv
      wi nfab1 nfel2 nfv nfbi pm5.5 spcgf imp sylan2 ) CFZDGZABHZTZCIDEJZUJDACK
      ZJZBHZTZCIZUPULUQCUJUKUPUJUKUPUJAUOBAUIUNJUJUOACLUIDUNMNOPQRUMURUPUQUPCDE
      CDSUOBCCDUNACUAUBBCUCUDUJUPUEUFUGUH $.
  $}

  ${
    elabgf.1 $e |- F/_ x A $.
    elabgf.2 $e |- F/ x ps $.
    elabgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a class abstraction, using implicit substitution.  Compare
       Theorem 6.13 of [Quine] p. 44.  This version has bound-variable
       hypotheses in place of distinct variable restrictions.  (Contributed by
       NM, 21-Sep-2003.)  (Revised by Mario Carneiro, 12-Oct-2016.) $)
    elabgf $p |- ( A e. B -> ( A e. { x | ph } <-> ps ) ) $=
      ( cv cab wcel wb nfab1 nfel nfbi wceq eleq1 bibi12d abid vtoclgf ) CIZACJ
      ZKZALDUBKZBLCDEFUDBCCDUBFACMNGOUADPUCUDABUADUBQHRACST $.
  $}

  ${
    $d x A $.
    elabf.1 $e |- F/ x ps $.
    elabf.2 $e |- A e. _V $.
    elabf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a class abstraction, using implicit substitution.
       (Contributed by NM, 1-Aug-1994.)  (Revised by Mario Carneiro,
       12-Oct-2016.) $)
    elabf $p |- ( A e. { x | ph } <-> ps ) $=
      ( cvv wcel cab wb nfcv elabgf ax-mp ) DHIDACJIBKFABCDHCDLEGMN $.
  $}

  ${
    $d x ps $.  $d x A $.
    elab.1 $e |- A e. _V $.
    elab.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a class abstraction, using implicit substitution.  Compare
       Theorem 6.13 of [Quine] p. 44.  (Contributed by NM, 1-Aug-1994.) $)
    elab $p |- ( A e. { x | ph } <-> ps ) $=
      ( nfv elabf ) ABCDBCGEFH $.
  $}

  ${
    elab.xex $e |- ( ph -> X e. _V ) $.
    elab.xmaj $e |- ( ph -> ch ) $.
    elab.xsub $e |- ( x = X -> ( ps <-> ch ) ) $.
    $d x ch $.  $d x X $.
    $( Explicit demonstration the class ` { x | ps } ` is not empty by the
       example ` X ` .  (Contributed by RP, 12-Aug-2020.) $)
    elabd $p |- ( ph -> E. x ps ) $=
      ( cvv wcel wex spcegv sylc ) AEIJCBDKFGBCDEIHLM $.
  $}

  ${
    $d x ps $.  $d x A $.
    elabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a class abstraction, using implicit substitution.  Compare
       Theorem 6.13 of [Quine] p. 44.  (Contributed by NM, 14-Apr-1995.) $)
    elabg $p |- ( A e. V -> ( A e. { x | ph } <-> ps ) ) $=
      ( nfcv nfv elabgf ) ABCDECDGBCHFI $.
  $}

  ${
    $d x ps $.  $d x A $.
    elab2g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    elab2g.2 $e |- B = { x | ph } $.
    $( Membership in a class abstraction, using implicit substitution.
       (Contributed by NM, 13-Sep-1995.) $)
    elab2g $p |- ( A e. V -> ( A e. B <-> ps ) ) $=
      ( wcel cab eleq2i elabg bitrid ) DEIDACJZIDFIBENDHKABCDFGLM $.
  $}

  ${
    $d x ps $.  $d x A $.
    elab2.1 $e |- A e. _V $.
    elab2.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    elab2.3 $e |- B = { x | ph } $.
    $( Membership in a class abstraction, using implicit substitution.
       (Contributed by NM, 13-Sep-1995.) $)
    elab2 $p |- ( A e. B <-> ps ) $=
      ( cvv wcel wb elab2g ax-mp ) DIJDEJBKFABCDEIGHLM $.
  $}

  ${
    $d x ps $.  $d x A $.
    elab4g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    elab4g.2 $e |- B = { x | ph } $.
    $( Membership in a class abstraction, using implicit substitution.
       (Contributed by NM, 17-Oct-2012.) $)
    elab4g $p |- ( A e. B <-> ( A e. _V /\ ps ) ) $=
      ( wcel cvv elex elab2g biadan2 ) DEHDIHBDEJABCDEIFGKL $.
  $}

  ${
    elab3gf.1 $e |- F/_ x A $.
    elab3gf.2 $e |- F/ x ps $.
    elab3gf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a class abstraction, with a weaker antecedent than
       ~ elabgf .  (Contributed by NM, 6-Sep-2011.) $)
    elab3gf $p |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) ) $=
      ( wcel wi cab elabgf ibi wb imim2i biimpr syli impbid2 ) BDEIZJZDACKZIZBU
      BBABCDUAFGHLMBTUBBNZUBSUCBABCDEFGHLOUBBPQR $.
  $}

  ${
    $d x ps $.  $d x A $.
    elab3g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a class abstraction, with a weaker antecedent than
       ~ elabg .  (Contributed by NM, 29-Aug-2006.) $)
    elab3g $p |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) ) $=
      ( nfcv nfv elab3gf ) ABCDECDGBCHFI $.
  $}

  ${
    $d x ps $.  $d x A $.
    elab3.1 $e |- ( ps -> A e. _V ) $.
    elab3.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a class abstraction using implicit substitution.
       (Contributed by NM, 10-Nov-2000.) $)
    elab3 $p |- ( A e. { x | ph } <-> ps ) $=
      ( cvv wcel wi cab wb elab3g ax-mp ) BDGHIDACJHBKEABCDGFLM $.
  $}

  ${
    $d A x $.  $d V x $.
    $( Implication for the membership in a restricted class abstraction.
       (Contributed by Alexander van der Vekens, 31-Dec-2017.) $)
    elrabi $p |- ( A e. { x e. V | ph } -> A e. V ) $=
      ( wcel cv wa cab crab wceq wex clelab eleq1 anbi1d simprbda exlimiv sylbi
      df-rab eleq2s ) CDEZCBFZDEZAGZBHZABDICUDEUACJZUCGZBKTUCBCLUFTBUEUCTAUEUBT
      AUACDMNOPQABDRS $.
  $}

  ${
    elrabf.1 $e |- F/_ x A $.
    elrabf.2 $e |- F/_ x B $.
    elrabf.3 $e |- F/ x ps $.
    elrabf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a restricted class abstraction, using implicit
       substitution.  This version has bound-variable hypotheses in place of
       distinct variable restrictions.  (Contributed by NM, 21-Sep-2003.) $)
    elrabf $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
      ( crab wcel cvv wa elex adantr cv cab df-rab eleq2i nfel nfan wceq elabgf
      eleq1 anbi12d bitrid pm5.21nii ) DACEJZKZDLKZDEKZBMZDUHNUKUJBDENOUIDCPZEK
      ZAMZCQZKUJULUHUPDACERSUOULCDLFUKBCCDEFGTHUAUMDUBUNUKABUMDEUDIUEUCUFUG $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    $( Membership in a restricted class abstraction, using implicit
       substitution.  (Closed theorem version of ~ elrab3 .)  (Contributed by
       Thierry Arnoux, 31-Aug-2017.) $)
    elrab3t $p |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) ->
                                           ( A e. { x e. B | ph } <-> ps ) ) $=
      ( cv wceq wb wi wal wcel wa cab crab simpr nfa1 nfv nfan simpl 19.21bi
      eleq1 biimparc biantrurd bibi1d adantl mpbid alrimi elabgt syl2anc df-rab
      pm5.74da eleq2i bibi1i sylibr ) CFZDGZABHZIZCJZDEKZLZDUOEKZALZCMZKZBHZDAC
      ENZKZBHVAUTUPVCBHZIZCJVFUSUTOVAVJCUSUTCURCPUTCQRVAURVJVAURCUSUTSTUTURVJHU
      SUTUPUQVIUTUPLZAVCBVKVBAUPVBUTUODEUAUBUCUDUKUEUFUGVCBCDEUHUIVHVEBVGVDDACE
      UJULUMUN $.
  $}

  ${
    $d x ps $.  $d x A $.  $d x B $.
    elrab.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a restricted class abstraction, using implicit
       substitution.  (Contributed by NM, 21-May-1999.) $)
    elrab $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
      ( nfcv nfv elrabf ) ABCDECDGCEGBCHFI $.

    $( Membership in a restricted class abstraction, using implicit
       substitution.  (Contributed by NM, 5-Oct-2006.) $)
    elrab3 $p |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) ) $=
      ( crab wcel elrab baib ) DACEGHDEHBABCDEFIJ $.
  $}

  ${
    $d A x $.  $d B x $.  $d ch x $.
    elrabd.1 $e |- ( x = A -> ( ps <-> ch ) ) $.
    elrabd.2 $e |- ( ph -> A e. B ) $.
    elrabd.3 $e |- ( ph -> ch ) $.
    $( Membership in a restricted class abstraction, using implicit
       substitution.  Deduction version of ~ elrab .  (Contributed by Glauco
       Siliprandi, 23-Oct-2021.) $)
    elrabd $p |- ( ph -> A e. { x e. B | ps } ) $=
      ( wcel wa crab jca elrab sylibr ) AEFJZCKEBDFLJAPCHIMBCDEFGNO $.
  $}

  ${
    $d x ps $.  $d x A $.  $d x B $.
    elrab2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    elrab2.2 $e |- C = { x e. B | ph } $.
    $( Membership in a class abstraction, using implicit substitution.
       (Contributed by NM, 2-Nov-2006.) $)
    elrab2 $p |- ( A e. C <-> ( A e. B /\ ps ) ) $=
      ( wcel crab wa eleq2i elrab bitri ) DFIDACEJZIDEIBKFODHLABCDEGMN $.
  $}

  ${
    $d x y $.  $d y A $.  $d y ps $.
    ralab.1 $e |- ( y = x -> ( ph <-> ps ) ) $.
    $( Universal quantification over a class abstraction.  (Contributed by Jeff
       Madsen, 10-Jun-2010.) $)
    ralab $p |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) ) $=
      ( cab wral cv wcel wi wal df-ral vex elab imbi1i albii bitri ) CDAEGZHDIZ
      SJZCKZDLBCKZDLCDSMUBUCDUABCABETDNFOPQR $.

    $( Universal quantification over a restricted class abstraction.
       (Contributed by Jeff Madsen, 10-Jun-2010.) $)
    ralrab $p |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) ) $=
      ( wi crab cv wcel wa elrab imbi1i impexp bitri ralbii2 ) CBCHZDAEFIZFDJZS
      KZCHTFKZBLZCHUBRHUAUCCABETFGMNUBBCOPQ $.

    $( Existential quantification over a class abstraction.  (Contributed by
       Mario Carneiro, 23-Jan-2014.)  (Revised by Mario Carneiro,
       3-Sep-2015.) $)
    rexab $p |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) ) $=
      ( cab wrex cv wcel wa wex df-rex vex elab anbi1i exbii bitri ) CDAEGZHDIZ
      SJZCKZDLBCKZDLCDSMUBUCDUABCABETDNFOPQR $.

    $( Existential quantification over a class abstraction.  (Contributed by
       Jeff Madsen, 17-Jun-2011.)  (Revised by Mario Carneiro, 3-Sep-2015.) $)
    rexrab $p |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) ) $=
      ( wa crab cv wcel elrab anbi1i anass bitri rexbii2 ) CBCHZDAEFIZFDJZRKZCH
      SFKZBHZCHUAQHTUBCABESFGLMUABCNOP $.
  $}

  ${
    $d x y $.  $d x A $.  $d x ch $.  $d x ph $.  $d y ps $.
    ralab2.1 $e |- ( x = y -> ( ps <-> ch ) ) $.
    $( Universal quantification over a class abstraction.  (Contributed by
       Mario Carneiro, 3-Sep-2015.) $)
    ralab2 $p |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) ) $=
      ( cab wral cv wcel wi wal df-ral nfsab1 nfv nfim weq eleq1 abid bitrdi
      imbi12d cbval bitri ) BDAEGZHDIZUDJZBKZDLACKZELBDUDMUGUHDEUFBEAEDNBEOPUHD
      ODEQZUFABCUIUFEIZUDJAUEUJUDRAESTFUAUBUC $.

    $( Universal quantification over a restricted class abstraction.
       (Contributed by Mario Carneiro, 3-Sep-2015.) $)
    ralrab2 $p |- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) ) $=
      ( crab wral cv wcel wa cab wi wal df-rab raleqi ralab2 impexp albii
      df-ral bitr4i 3bitri ) BDAEFHZIBDEJFKZALZEMZIUFCNZEOZACNZEFIZBDUDUGAEFPQU
      FBCDEGRUIUEUJNZEOUKUHULEUEACSTUJEFUAUBUC $.

    $( Existential quantification over a class abstraction.  (Contributed by
       Mario Carneiro, 3-Sep-2015.) $)
    rexab2 $p |- ( E. x e. { y | ph } ps <-> E. y ( ph /\ ch ) ) $=
      ( cab wrex cv wcel wa wex df-rex nfsab1 nfv nfan weq eleq1 abid bitrdi
      anbi12d cbvex bitri ) BDAEGZHDIZUDJZBKZDLACKZELBDUDMUGUHDEUFBEAEDNBEOPUHD
      ODEQZUFABCUIUFEIZUDJAUEUJUDRAESTFUAUBUC $.

    $( Existential quantification over a class abstraction.  (Contributed by
       Mario Carneiro, 3-Sep-2015.) $)
    rexrab2 $p |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) ) $=
      ( crab wrex cv wcel wa cab wex df-rab rexeqi rexab2 anass exbii df-rex
      bitr4i 3bitri ) BDAEFHZIBDEJFKZALZEMZIUECLZENZACLZEFIZBDUCUFAEFOPUEBCDEGQ
      UHUDUILZENUJUGUKEUDACRSUIEFTUAUB $.
  $}

  ${
    $d x z $.  $d z A $.
    $( Identity used to create closed-form versions of bound-variable
       hypothesis builders for class expressions.  (Contributed by NM,
       10-Nov-2005.)  (Proof shortened by Mario Carneiro, 12-Oct-2016.) $)
    abidnf $p |- ( F/_ x A -> { z | A. x z e. A } = A ) $=
      ( wnfc cv wcel wal sp nfcr nfrd impbid2 abbi1dv ) ACDZBECFZAGZBCMONNAHMNA
      ABCIJKL $.
  $}

  ${
    $d x z $.  $d z A $.
    dedhb.1 $e |- ( A = { z | A. x z e. A } -> ( ph <-> ps ) ) $.
    dedhb.2 $e |- ps $.
    $( A deduction theorem for converting the inference ` |- F/_ x A ` =>
       ` |- ph ` into a closed theorem.  Use ~ nfa1 and ~ nfab to eliminate the
       hypothesis of the substitution instance ` ps ` of the inference.  For
       converting the inference form into a deduction form, ~ abidnf is useful.
       (Contributed by NM, 8-Dec-2006.) $)
    dedhb $p |- ( F/_ x A -> ph ) $=
      ( wnfc cv wcel wal cab wceq wb abidnf eqcomd syl mpbiri ) CEHZABGSEDIEJCK
      DLZMABNSTECDEOPFQR $.
  $}

  ${
    $d y ph $.  $d x y ps $.  $d x y A $.
    eqeu.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( A condition which implies existential uniqueness.  (Contributed by Jeff
       Hankins, 8-Sep-2009.) $)
    eqeu $p |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph ) $=
      ( vy wcel cv wceq wi wal w3a wex weu spcegv imp 3adant3 eqeq2 imbi2d nfv
      albidv 3adant2 eu3 sylanbrc ) DEHZBACIZDJZKZCLZMACNZAUGGIZJZKZCLZGNZACOUF
      BUKUJUFBUKABCDEFPQRUFUJUPBUFUJUPUOUJGDEULDJZUNUICUQUMUHAULDUGSTUBPQUCACGA
      GUAUDUE $.
  $}

  ${
    $d x y A $.
    $( Equality has existential uniqueness.  (Contributed by NM,
       25-Nov-1994.) $)
    eueq $p |- ( A e. _V <-> E! x x = A ) $=
      ( vy cv wceq wex wa wi wal cvv wcel weu eqtr3 biantru isset eqeq1 3bitr4i
      gen2 eu4 ) ADZBEZAFZUBUACDZBEZGTUCEHZCIAIZGBJKUAALUFUBUEACTUCBMRNABOUAUDA
      CTUCBPSQ $.
  $}

  ${
    $d x A $.
    eueq1.1 $e |- A e. _V $.
    $( Equality has existential uniqueness.  (Contributed by NM,
       5-Apr-1995.) $)
    eueq1 $p |- E! x x = A $=
      ( cvv wcel cv wceq weu eueq mpbi ) BDEAFBGAHCABIJ $.
  $}

  ${
    $d x ph $.  $d x A $.  $d x B $.
    eueq2dc.1 $e |- A e. _V $.
    eueq2dc.2 $e |- B e. _V $.
    $( Equality has existential uniqueness (split into 2 cases).  (Contributed
       by NM, 5-Apr-1995.) $)
    eueq2dc $p |- ( DECID ph ->
        E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) $=
      ( wdc wn wo wceq weu eueq1 euanv biimpri mpan2 euorv bianfd eubidv mpbid
      wa cv df-dc notnot syl2anc orcom orbi2d bitrid mpdan id orbi1d jaoi sylbi
      ) AGAAHZIABUAZCJZTZUMUNDJZTZIZBKZAUBAUTUMAUMUPIZBKZUTAUMHUPBKZVBAUCZAUOBK
      ZVCBCELVCAVETAUOBMNOUMUPBPUDAVAUSBVAUPUMIAUSUMUPUEAUMURUPAUMUQVDQUFUGRSUM
      AURIZBKZUTUMURBKZVGUMUQBKZVHBDFLVHUMVITUMUQBMNOAURBPUHUMVFUSBUMAUPURUMAUO
      UMUIQUJRSUKUL $.
  $}

  ${
    $d x ph $.  $d x ps $.  $d x A $.  $d x B $.  $d x C $.
    eueq3dc.1 $e |- A e. _V $.
    eueq3dc.2 $e |- B e. _V $.
    eueq3dc.3 $e |- C e. _V $.
    eueq3dc.4 $e |- -. ( ph /\ ps ) $.
    $( Equality has existential uniqueness (split into 3 cases).  (Contributed
       by NM, 5-Apr-1995.)  (Proof shortened by Mario Carneiro,
       28-Sep-2015.) $)
    eueq3dc $p |- ( DECID ph -> ( DECID ps ->
        E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B )
               \/ ( ps /\ x = C ) ) ) ) $=
      ( wdc wo wceq wa wn w3o weu eueq1 con2i jaoi cv dcor df-dc ibar wb pm2.45
      imnani bianfd orbi12d mtbid biorf bitrd 3orrot df-3or bitri bitr4di mpbii
      syl eubidv adantr pm2.46 simpl orim12i con3i 3orcomb sylbi syl6 ) AKBKABL
      ZKZACUAZDMZNZVHOZVJEMZNZBVJFMZNZPZCQZABUBVIVHVMLVSVHUCVHVSVMAVSBAVKCQVSCD
      GRAVKVRCAVKVOVQLZVLLZVRAVKVLWAAVKUDAVTOVLWAUEAVMBLZVTWBAVMAOBABUFZABABJUG
      ZSTSAVMVOBVQAVMVNVMAWCSUHABVPWDUHUIUJVTVLUKURULVRVOVQVLPWAVLVOVQUMVOVQVLU
      NUOUPUSUQBVPCQVSCFIRBVPVRCBVPVLVOLZVQLZVRBVPVQWFBVPUDBWEOVQWFUEWEBVLBOZVO
      AWGVKWDUTVMWGVNABVAUTTSWEVQUKURULVLVOVQUNUPUSUQTVMVNCQVSCEHRVMVNVRCVMVNVL
      VQLZVOLZVRVMVNVOWIVMVNUDVMWHOVOWIUEWHVHVLAVQBAVKVBBVPVBVCVDWHVOUKURULVRVL
      VQVOPWIVLVOVQVEVLVQVOUNUOUPUSUQTVFVG $.
  $}

  ${
    $d x A $.
    $( There is at most one set equal to a class.  (Contributed by NM,
       8-Mar-1995.) $)
    moeq $p |- E* x x = A $=
      ( cv wceq wmo wex weu wi cvv wcel isset eueq bitr3i biimpi df-mo mpbir )
      ACBDZAEQAFZQAGZHRSRBIJSABKABLMNQAOP $.
  $}

  ${
    $d x ph $.  $d x ps $.  $d x A $.  $d x B $.  $d x C $.
    moeq3dc.1 $e |- A e. _V $.
    moeq3dc.2 $e |- B e. _V $.
    moeq3dc.3 $e |- C e. _V $.
    moeq3dc.4 $e |- -. ( ph /\ ps ) $.
    $( "At most one" property of equality (split into 3 cases).  (Contributed
       by Jim Kingdon, 7-Jul-2018.) $)
    moeq3dc $p |- ( DECID ph -> ( DECID ps ->
        E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B )
               \/ ( ps /\ x = C ) ) ) ) $=
      ( wdc cv wceq wa wo wn w3o weu wmo eueq3dc eumo syl6 ) AKBKACLZDMNABOPUCE
      MNBUCFMNQZCRUDCSABCDEFGHIJTUDCUAUB $.
  $}

  ${
    $d x y A $.
    $( "At most one" remains true after substitution.  (Contributed by Jim
       Kingdon, 18-Jan-2019.) $)
    mosubt $p |- ( A. y E* x ph -> E* x E. y ( y = A /\ ph ) ) $=
      ( wmo wal cv wceq wex wa wi weu cvv wcel eueq isset bitr3i euexex sylanbr
      nfv expcom moanimv sylibr simpl eximi ancri moimi syl ) ABECFZCGDHZCIZUJA
      JZCIZJZBEZUMBEZUIUKUPKUOUKUIUPUKUJCLZUIUPUQDMNUKCDOCDPQUJACBUJBTRSUAUKUMB
      UBUCUMUNBUMUKULUJCUJAUDUEUFUGUH $.
  $}

  ${
    $d x y A $.
    mosub.1 $e |- E* x ph $.
    $( "At most one" remains true after substitution.  (Contributed by NM,
       9-Mar-1995.) $)
    mosub $p |- E* x E. y ( y = A /\ ph ) $=
      ( wmo cv wceq wa wex mosubt mpg ) ABFCGDHAICJBFCABCDKEL $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( Theorem for inferring "at most one".  (Contributed by NM,
       17-Oct-1996.) $)
    mo2icl $p |- ( A. x ( ph -> x = A ) -> E* x ph ) $=
      ( vy cv wceq wal wex wmo cvv wcel nfa1 vex eleq1 mpbii imim2i eximd 19.9v
      wi sps imbitrdi eqeq2 imbi2d albidv imbi1d mo2r 19.23bi vtoclg com12 syld
      nfv moabs sylibr ) ABEZCFZSZBGZABHZABIZSUSUQURCJKZUSUQURUTBHUTUQAUTBUPBLU
      PAUTSBUOUTAUOUNJKUTBMUNCJNOPTQUTBRUAUTUQUSAUNDEZFZSZBGZUSSUQUSSDCJVACFZVD
      UQUSVEVCUPBVEVBUOAVACUNUBUCUDUEVDUSDABDADUKUFUGUHUIUJABULUM $.
  $}

  ${
    $d x y A $.  $d y ph $.  $d x y ps $.
    moi2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Consequence of "at most one".  (Contributed by NM, 2-Jan-2015.) $)
    mob2 $p |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) ) $=
      ( vy wcel wmo w3a cv wceq simp3 syl5ibcom wi wa wsb wal nfs1v sbequ12 nfv
      mo4f sylbi sbhypf anbi2d eqeq2 imbi12d spcgv syl5 imp expd 3impia impbid
      sp ) DEHZACIZAJZCKZDLZBUQAUSBUOUPAMFNUOUPABUSOUOUPPABUSUOUPABPZUSOZUPAACG
      QZPZURGKZLZOZGRZUOVAUPVGCRVGAVBCGACGSACGTUBVGCUNUCVFVAGDEVDDLZVCUTVEUSVHV
      BBAABCGDBCUAFUDUEVDDURUFUGUHUIUJUKULUM $.

    $( Consequence of "at most one".  (Contributed by NM, 29-Jun-2008.) $)
    moi2 $p |- ( ( ( A e. B /\ E* x ph ) /\ ( ph /\ ps ) ) -> x = A ) $=
      ( wcel wmo wa cv wceq wb mob2 3expa biimprd impr ) DEGZACHZIZABCJDKZSAITB
      QRATBLABCDEFMNOP $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ch $.  $d x ps $.
    moi.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    moi.2 $e |- ( x = B -> ( ph <-> ch ) ) $.
    $( Equality implied by "at most one".  (Contributed by NM, 18-Feb-2006.) $)
    mob $p |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) ->
                ( A = B <-> ch ) ) $=
      ( wcel wa wmo wceq wb wi cvv elex w3a nfv cv nfcv nfmo1 nf3an nfim bibi1d
      3anbi3d eqeq1 imbi12d mob2 vtoclgf com12 3expib syl com3r imp 3impib ) EG
      KZFHKZLADMZBEFNZCOZURUSUTBLZVBPUSVCURVBUSFQKZVCURVBPZPFHRVDUTBVEURVDUTBSZ
      VBVDUTASZDUAZFNZCOZPVFVBPDEGDEUBVFVBDVDUTBDVDDTADUCBDTUDVBDTUEVHENZVGVFVJ
      VBVKABVDUTIUGVKVIVACVHEFUHUFUIACDFQJUJUKULUMUNUOUPUQ $.

    $( Equality implied by "at most one".  (Contributed by NM, 18-Feb-2006.) $)
    moi $p |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) ->
              A = B ) $=
      ( wcel wa wmo wceq wi w3a mob biimprd 3expia impd 3impia ) EGKFHKLZADMZBC
      LEFNZUBUCLBCUDUBUCBCUDOUBUCBPUDCABCDEFGHIJQRSTUA $.
  $}

  ${
    $d B x $.  $d A x $.  $d ps x $.
    morex.1 $e |- B e. _V $.
    morex.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
    $( Derive membership from uniqueness.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    morex $p |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) ) $=
      ( wmo wrex wcel wi cv wa wex df-rex exancom bitri wal nfmo1 nfe1 nfan syl
      mopick alrimi wceq eleq1 imbi12d spcv sylan2b ancoms ) ACHZACDIZBEDJZKZUL
      UKACLZDJZMZCNZUNULUPAMCNURACDOUPACPQUKURMZAUPKZCRUNUSUTCUKURCACSUQCTUAAUP
      CUCUDUTUNCEFUOEUEABUPUMGUOEDUFUGUHUBUIUJ $.
  $}

  ${
    $d x ph $.  $d x A $.
    euxfr2dc.1 $e |- A e. _V $.
    euxfr2dc.2 $e |- E* y x = A $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by NM,
       14-Nov-2004.) $)
    euxfr2dc $p |- ( DECID E. y E. x ( x = A /\ ph ) ->
        ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) ) $=
      ( cv wa wex wdc weu wmo wal wi moani mobii mpbi ax-gen 2euswapdc mpi wceq
      ancom excom dcbii sylbi moeq impbid biidd ceqsexv eubii bitrdi ) BGDUAZAH
      ZBIZCIZJZUMCIZBKZUNCKZACKUPURUSUPUMCLZBMZURUSNZUTBAULHZCLUTULACFOVCUMCAUL
      UBZPQRUPUQBIZJVAVBNUOVEUMCBUCUDUMBCSUETUPUMBLZCMUSURNVFCVCBLVFULABBDUFOVC
      UMBVDPQRUMCBSTUGUNACAABDEULAUHUIUJUK $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x A $.
    euxfrdc.1 $e |- A e. _V $.
    euxfrdc.2 $e |- E! y x = A $.
    euxfrdc.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by NM,
       14-Nov-2004.) $)
    euxfrdc $p |- ( DECID E. y E. x ( x = A /\ ps ) ->
        ( E! x ph <-> E! y ps ) ) $=
      ( weu cv wceq wa wex wdc euex ax-mp biantrur 19.41v pm5.32i exbii 3bitr2i
      eubii eumoi euxfr2dc bitrid ) ACICJEKZBLZDMZCIUGCMDMNBDIAUHCAUFDMZALUFALZ
      DMUHUIAUFDIUIGUFDOPQUFADRUJUGDUFABHSTUAUBBCDEFUFDGUCUDUE $.
  $}

  ${
    $d y z w ph $.  $d x z ps $.  $d y z w A $.  $d x z B $.  $d x y w $.
    euind.1 $e |- B e. _V $.
    euind.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    euind.3 $e |- ( x = y -> A = B ) $.
    $( Existential uniqueness via an indirect equality.  (Contributed by NM,
       11-Oct-2010.) $)
    euind $p |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph )
                 -> E! z A. x ( ph -> z = A ) ) $=
      ( vw wa wceq wi wal wex cv exbii bitri imim2i weu cbvexv biantrur 3bitr2i
      isseti 19.41v excom wb eqeq2 biimpr an31 imbi1i impexp 3bitr3i syl 2alimi
      sylib 19.23v albii 19.21v eximdv biimtrid imp pm4.24 biimpi eqtr3 alanimi
      anim12 syl56 com12 syl5 alrimivv adantl eqeq1 imbi2d albidv eu4 sylanbrc
      ) ABLZFGMZNZDOCOZACPZLAEQZFMZNZCOZEPZWGAKQZFMZNZCOZLZWDWIMZNZKOEOZWGEUAWB
      WCWHWCWDGMZBLZDPZEPZWBWHWCBDPZWTABCDIUBXAWQEPZBLZDPWREPZDPWTBXCDXBBEGHUEU
      CRXDXCDWQBEUFRWRDEUGUDSWBWSWGEWBWRWFNZDOZCOZWSWGNZWAXECDWAVSWEWQUHZNZXEVT
      XIVSFGWDUITXJVSWQWENZNZXEXIXKVSWEWQUJTVSWQLZWENWRALZWENXLXEXMXNWEABWQUKUL
      VSWQWEUMWRAWEUMUNUQUOUPXGWSWFNZCOXHXFXOCWRWFDURUSWSWFCUTSUQVAVBVCWCWPWBWC
      WOEKWMAWNNZCOZWCWNWFWKXPCAAALZWFWKLWEWJLWNAXRAVDVEAWEAWJVHWDWIFVFVIVGXQWC
      WNXQWCWNNAWNCURVEVJVKVLVMWGWLEKWNWFWKCWNWEWJAWDWIFVNVOVPVQVR $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y ph $.  $d x ps $.
    $( A way to express restricted uniqueness.  (Contributed by NM,
       22-Nov-1994.) $)
    reu2 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
               A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
      ( cv wcel wa weu wex wsb weq wi wal wreu wral df-ral impexp albii 3bitr4i
      nfv wrex eu2 df-reu df-rex 19.21v nfs1v nfan eleq1 sbequ12 anbi12d anbi2i
      sbie an4 bitri imbi1i 3bitri imbi2i bitr4i anbi12i ) BEZDFZAGZBHVBBIZVBVB
      BCJZGZBCKZLZCMZBMZGABDNABDUAZAABCJZGZVFLZCDOZBDOZGVBBCVBCTUBABDUCVJVCVOVI
      ABDUDVOVAVNLZBMVIVNBDPVHVPBVACEZDFZVMLZLZCMVAVSCMZLVHVPVAVSCUEVGVTCVGVAVR
      GZVLGZVFLWBVMLVTVEWCVFVEVBVRVKGZGWCVDWDVBVBWDBCVRVKBVRBTABCUFUGVFVAVRAVKU
      TVQDUHABCUIUJULUKVAAVRVKUMUNUOWBVLVFQVAVRVMQUPRVNWAVAVMCDPUQSRURUSS $.

    $( A way to express restricted uniqueness.  (Contributed by NM,
       20-Oct-2006.) $)
    reu6 $p |- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) ) $=
      ( cv wcel wa weq wb wal wex biimp biimpr adantr impbid imim2i imp 3bitr4i
      wi ex wreu weu wral wrex df-reu wsb eleq1 sbequ12 anbi12d equequ1 bibi12d
      19.28v equid tbt simpl biimtrdi spimv expdimp simpr syl6 sps jca a5i impd
      sylbir adantl eleq1a com23 adantll alimi impbii df-ral anbi2i exbii df-eu
      jcai df-rex bitri ) ABDUABEZDFZAGZBUBZABCHZIZBDUCZCDUDZABDUEWAWCIZBJZCKCE
      ZDFZWEGZCKWBWFWHWKCWJVTWDSZGZBJZWJWLBJZGWHWKWJWLBULWHWNWGWMBWHWJWLWGWJBCW
      CWGWJABCUFZGZCCHZIZWJWCWAWQWCWRWCVTWJAWPVSWIDUGABCUHUIBCCUJUKWSWQWJWRWQCU
      MUNWJWPUOVEUPUQWGWLBWGVTWDWGVTGAWCWGVTAWCWAWCLURWGWCASZVTWGWCWAAWAWCMVTAU
      SUTNOTVAVBVCWMWGBWMWAWCWLWAWCSWJWLVTAWCWDAWCSVTAWCLPVDVFWMWCWAWMWCGVTAWMW
      CVTWJWCVTSWLWIDVSVGNQWLWCVTASZWJWLWCXAWLVTWCAWDWTVTAWCMPVHQVIVPTOVJVKWEWO
      WJWDBDVLVMRVNWABCVOWECDVQRVR $.

    $( A way to express restricted uniqueness.  (Contributed by NM,
       24-Oct-2006.) $)
    reu3 $p |- ( E! x e. A ph <->
               ( E. x e. A ph /\ E. y e. A A. x e. A ( ph -> x = y ) ) ) $=
      ( wreu wrex weq wi wral wa reurex wb reu6 biimp ralimi reximi jca wex wal
      sylbi rexex anim2i cv wcel weu nfv eu3 df-reu df-rex df-ral impexp bitr4i
      albii exbii anbi12i 3bitr4i sylibr impbii ) ABDEZABDFZABCGZHZBDIZCDFZJZUS
      UTVDABDKUSAVALZBDIZCDFVDABCDMVGVCCDVFVBBDAVANOPTQVEUTVCCRZJZUSVDVHUTVCCDU
      AUBBUCDUDZAJZBUEVKBRZVKVAHZBSZCRZJUSVIVKBCVKCUFUGABDUHUTVLVHVOABDUIVCVNCV
      CVJVBHZBSVNVBBDUJVMVPBVJAVAUKUMULUNUOUPUQUR $.

    $( A condition which implies existential uniqueness.  (Contributed by Mario
       Carneiro, 2-Oct-2015.) $)
    reu6i $p |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph ) $=
      ( vy wcel cv wceq wb wral wa wrex wreu eqeq2 bibi2d ralbidv rspcev sylibr
      reu6 ) DCFABGZDHZIZBCJZKATEGZHZIZBCJZECLABCMUGUCEDCUDDHZUFUBBCUHUEUAAUDDT
      NOPQABECSR $.

    eqreu.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
    $( A condition which implies existential uniqueness.  (Contributed by Mario
       Carneiro, 2-Oct-2015.) $)
    eqreu $p |- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) ->
      E! x e. A ph ) $=
      ( wcel cv wceq wi wral wreu wa wb ralbiim ceqsralv anbi2d bitrid reu6i ex
      sylbird 3impib 3com23 ) EDGZACHEIZJCDKZBACDLZUDUFBUGUDUFBMZAUENCDKZUGUIUF
      UEAJCDKZMUDUHAUECDOUDUJBUFABCEDFPQRUDUIUGACDESTUAUBUC $.
  $}

  ${
    $d x y z A $.  $d y z ph $.  $d x z ps $.
    rmo4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Restricted "at most one" using implicit substitution.  (Contributed by
       NM, 24-Oct-2006.)  (Revised by NM, 16-Jun-2017.) $)
    rmo4 $p |- ( E* x e. A ph <->
               A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) $=
      ( wrmo cv wcel wa wmo weq wi wral df-rmo wal bitri impexp albii df-ral
      an4 ancom anbi1i imbi1i 3bitri r19.21v 3bitr2i eleq1 anbi12d mo4 3bitr4i
      ) ACEGCHZEIZAJZCKZABJZCDLZMZDENZCENZACEOUNDHZEIZBJZJZUQMZDPZCPUMUSMZCPUOU
      TVFVGCVFVBUMURMZMZDPVHDENVGVEVIDVEVBUMJZUPJZUQMVJURMVIVDVKUQVDUMVBJZUPJVK
      UMAVBBUAVLVJUPUMVBUBUCQUDVJUPUQRVBUMURRUESVHDETUMURDEUFUGSUNVCCDUQUMVBABU
      LVAEUHFUIUJUSCETUKQ $.

    $( Restricted uniqueness using implicit substitution.  (Contributed by NM,
       23-Nov-1994.) $)
    reu4 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
             A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) ) $=
      ( wreu wrex wrmo wa weq wi wral reu5 rmo4 anbi2i bitri ) ACEGACEHZACEIZJR
      ABJCDKLDEMCEMZJACENSTRABCDEFOPQ $.

    $( Restricted uniqueness using implicit substitution.  (Contributed by NM,
       24-Oct-2006.) $)
    reu7 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
             E. x e. A A. y e. A ( ps -> x = y ) ) ) $=
      ( vz wreu wrex weq wi wral wa equequ1 equcom bitrdi imbi12d cbvralv bitri
      reu3 rexbii imbi2d ralbidv cbvrexv anbi2i ) ACEHACEIZACGJZKZCELZGEIZMUFBC
      DJZKZDELZCEIZMACGETUJUNUFUJBGDJZKZDELZGEIUNUIUQGEUHUPCDEUKABUGUOFUKUGDGJU
      OCDGNDGOPQRUAUQUMGCEGCJZUPULDEURUOUKBGCDNUBUCUDSUES $.

    $( Restricted uniqueness using implicit substitution.  (Contributed by NM,
       24-Oct-2006.) $)
    reu8 $p |- ( E! x e. A ph <-> E. x e. A ( ph /\
                A. y e. A ( ps -> x = y ) ) ) $=
      ( wreu weq wb wral wrex wi wa cbvreuv reu6 cv wcel ralbii wal bitrid a1i
      dfbi2 r19.26 ancom equcom imbi2i biimt df-ral bi2.04 albii imbi12d bicomd
      vex eleq1 equcoms ceqsalv 3bitrri bitrdi anbi12d bitr4id rexbiia 3bitri )
      ACEGBDEGBDCHZIZDEJZCEKABCDHZLZDEJZMZCEKABCDEFNBDCEOVEVICEVEBVCLZVCBLZMZDE
      JZCPZEQZVIVDVLDEBVCUBRVOVMVJDEJZVKDEJZMZVIVJVKDEUCVIVHAMVOVRAVHUDVOVHVPAV
      QVHVPIVOVGVJDEVFVCBCDUEUFRUAVOAVOALZVQVOAUGVQDPZEQZVKLZDSVCWABLZLZDSVSVKD
      EUHWBWDDWAVCBUIUJWCVSDVNCUMWCVSICDVFVSWCVFVOWAABVNVTEUNFUKULUOUPUQURUSTUT
      TVAVB $.
  $}

  ${
    $d x y $.
    rmo3f.1 $e |- F/_ x A $.
    rmo3f.2 $e |- F/_ y A $.
    rmo3f.3 $e |- F/ y ph $.
    $( Restricted "at most one" using explicit substitution.  (Contributed by
       NM, 4-Nov-2012.)  (Revised by NM, 16-Jun-2017.)  (Revised by Thierry
       Arnoux, 8-Oct-2017.) $)
    rmo3f $p |- ( E* x e. A ph <->
               A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
      ( cv wcel wa wsb wi wral wal anbi1i bitri 3bitri impexp albii df-ral wrmo
      wmo weq df-rmo sban clelsb1f anbi2i an4 ancom imbi1i nfcri r19.21 3bitr2i
      nfan mo3 3bitr4i ) ABDUABHDIZAJZBUBZAABCKZJZBCUCZLZCDMZBDMZABDUDURURBCKZJ
      ZVBLZCNZBNUQVDLZBNUSVEVIVJBVICHDIZUQVCLZLZCNVLCDMVJVHVMCVHVKUQJZVAJZVBLVN
      VCLVMVGVOVBVGURVKUTJZJUQVKJZVAJVOVFVPURVFUQBCKZUTJVPUQABCUEVRVKUTBCDEUFOP
      UGUQAVKUTUHVQVNVAUQVKUIOQUJVNVAVBRVKUQVCRQSVLCDTUQVCCDCBDFUKZULUMSURBCUQA
      CVSGUNUOVDBDTUPP $.
  $}

  ${
    $d x y $.  $d y ph $.
    rmo4f.1 $e |- F/_ x A $.
    rmo4f.2 $e |- F/_ y A $.
    rmo4f.3 $e |- F/ x ps $.
    rmo4f.4 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Restricted "at most one" using implicit substitution.  (Contributed by
       NM, 24-Oct-2006.)  (Revised by Thierry Arnoux, 11-Oct-2016.)  (Revised
       by Thierry Arnoux, 8-Mar-2017.)  (Revised by Thierry Arnoux,
       8-Oct-2017.) $)
    rmo4f $p |- ( E* x e. A ph <->
               A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) $=
      ( wrmo wsb wa weq wi wral nfv rmo3f sbie anbi2i imbi1i 2ralbii bitri ) AC
      EJAACDKZLZCDMZNZDEOCEOABLZUENZDEOCEOACDEFGADPQUFUHCDEEUDUGUEUCBAABCDHIRST
      UAUB $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Equality has existential uniqueness.  (Contributed by Mario Carneiro,
       1-Sep-2015.) $)
    reueq $p |- ( B e. A <-> E! x e. A x = B ) $=
      ( wcel cv wceq wrex wreu risset wrmo wmo moeq mormo ax-mp mpbiran2 bitr4i
      reu5 ) CBDAECFZABGZRABHZACBITSRABJZRAKUAACLRABMNRABQOP $.
  $}

  $( Restricted "at most one" still holds when a conjunct is added.
     (Contributed by NM, 16-Jun-2017.) $)
  rmoan $p |- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) ) $=
    ( cv wcel wa wmo wrmo moan an12 mobii sylib df-rmo 3imtr4i ) CEDFZAGZCHZPBA
    GZGZCHZACDISCDIRBQGZCHUAQBCJUBTCBPAKLMACDNSCDNO $.

  $( Restricted "at most one" is preserved through implication (note wff
     reversal).  (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
  rmoim $p |- ( A. x e. A ( ph -> ps )
        -> ( E* x e. A ps -> E* x e. A ph ) ) $=
    ( wi wral cv wcel wa wal wrmo df-ral imdistan albii wmo moim df-rmo 3imtr4g
    bitri sylbi ) ABEZCDFZCGDHZAIZUCBIZEZCJZBCDKZACDKZEUBUCUAEZCJUGUACDLUJUFCUC
    ABMNSUGUECOUDCOUHUIUDUECPBCDQACDQRT $.

  ${
    rmoimia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Restricted "at most one" is preserved through implication (note wff
       reversal).  (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
    rmoimia $p |- ( E* x e. A ps -> E* x e. A ph ) $=
      ( wi wrmo rmoim mprg ) ABFBCDGACDGFCDABCDHEI $.
  $}

  ${
    rmoimi2.1 $e |- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) ) $.
    $( Restricted "at most one" is preserved through implication (note wff
       reversal).  (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
    rmoimi2 $p |- ( E* x e. B ps -> E* x e. A ph ) $=
      ( cv wcel wa wmo wrmo wi wal moim ax-mp df-rmo 3imtr4i ) CGZEHBIZCJZRDHAI
      ZCJZBCEKACDKUASLCMTUBLFUASCNOBCEPACDPQ $.
  $}

  ${
    $d x y A $.  $d x B $.
    $( A condition allowing swap of uniqueness and existential quantifiers.
       (Contributed by Thierry Arnoux, 7-Apr-2017.)  (Revised by NM,
       16-Jun-2017.) $)
    2reuswapdc $p |- ( DECID E. x E. y ( x e. A /\ ( y e. B /\ ph ) ) ->
        ( A. x e. A E* y e. B ph ->
          ( E! x e. A E. y e. B ph -> E! y e. B E. x e. A ph ) ) ) $=
      ( wral cv wcel wa wmo wex wrex wreu wal weu df-reu r19.42v df-rex bitri
      wi wrmo df-rmo ralbii df-ral moanimv albii bitr4i bitr3i an12 exbii eubii
      wdc 2euswapdc syl7bi imbi2i imbitrrdi biimtrid ) ACEUAZBDFCGEHZAIZCJZBDFZ
      BGDHZUTIZCKZBKULZACELZBDMZABDLZCEMZTZURVABDACEUBUCVBVDCJZBNZVFVKVBVCVATZB
      NVMVABDUDVLVNBVCUTCUEUFUGVFVMVHVDBKZCOZTVKVHVEBOZVFVMVPVHVCVGIZBOVQVGBDPV
      RVEBVRUSVCAIZIZCKZVEVRVSCELWAVCACEQVSCERUHVTVDCUSVCAUIUJSUKSVDBCUMUNVJVPV
      HVJUSVIIZCOVPVICEPWBVOCWBUTBDLVOUSABDQUTBDRUHUKSUOUPUQUQ $.
  $}

  ${
    $d w y z A $.  $d x z B $.  $d w x y z C $.  $d w y z ph $.  $d x z ps $.
    reuind.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    reuind.2 $e |- ( x = y -> A = B ) $.
    $( Existential uniqueness via an indirect equality.  (Contributed by NM,
       16-Oct-2010.) $)
    reuind $p |- ( ( A. x A. y ( ( ( A e. C /\ ph ) /\ ( B e. C /\ ps ) )
         -> A = B ) /\ E. x ( A e. C /\ ph ) )
                 -> E! z e. C A. x ( ( A e. C /\ ph ) -> z = A ) ) $=
      ( vw wcel wa wceq wi wal wex cv wrex bitri wral wreu eleq1d anbi12d exbii
      cbvexv r19.41v rexcom4 risset anbi1i 3bitr4ri wb eqeq2 imim2i biimpr an31
      imbi1i impexp 3bitr3i sylib syl 2alimi 19.23v an12 adantr pm5.32ri bitr4i
      eleq1 19.42v albii 19.21v expd reximdvai biimtrid imp pm4.24 biimpi eqtr3
      anim12 syl56 alanimi com12 syl5 ralrimivv adantl eqeq1 imbi2d albidv reu4
      a1d sylanbrc ) FHLZAMZGHLZBMZMZFGNZOZDPCPZWMCQZMWMERZFNZOZCPZEHSZXDWMKRZF
      NZOZCPZMZXAXFNZOZKHUAEHUAZXDEHUBWSWTXEWTXAGNZBMZDQZEHSZWSXEWTWODQZXQWMWOC
      DCRDRNZWLWNABXSFGHJUCIUDUFXOEHSZDQXNEHSZBMZDQXQXRXTYBDXNBEHUGUEXOEDHUHWOY
      BDWNYABEGHUIUJUEUKTWSXPXDEHWSXAHLZXPXDWSXNWOMZXCOZDPZCPZYCXPMZXDOZWRYECDW
      RWPXBXNULZOZYEWQYJWPFGXAUMUNYKWPXNXBOZOZYEYJYLWPXBXNUOUNWPXNMZXBOYDWMMZXB
      OYMYEYNYOXBWMWOXNUPUQWPXNXBURYDWMXBURUSUTVAVBYGYHXCOZCPYIYFYPCYFYDDQZXCOY
      PYDXCDVCYQYHXCYQYCXOMZDQYHYDYRDYDWNXOMYRXNWNBVDXOYCWNXNYCWNULBXAGHVHVEVFV
      GUEYCXODVITUQTVJYHXCCVKTUTVLVMVNVOWTXMWSWTXLEKHHWTXLYCXFHLMXJWMXKOZCPZWTX
      KXCXHYSCWMWMWMMZXCXHMXBXGMXKWMUUAWMVPVQWMXBWMXGVSXAXFFVRVTWAYTWTXKYTWTXKO
      WMXKCVCVQWBWCWJWDWEXDXIEKHXKXCXHCXKXBXGWMXAXFFWFWGWHWIWK $.
  $}

  ${
    $d y A $.  $d x B $.  $d x y $.
    $( Double restricted quantification with "at most one," analogous to
       ~ 2moex .  (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
    2rmorex $p |- ( E* x e. A E. y e. B ph -> A. y e. B E* x e. A ph ) $=
      ( wrex wrmo cv wcel wa wmo wal wral wex df-rex mobii df-rmo albii sylib
      wi anbi2i 19.42v 3bitr4i 2moex sylbi moanimv df-ral imbi2i bitri sylibr
      an12 ) ACEFZBDGZCHEIZBHDIZAJZBKZTZCLZABDGZCEMZUMUNUPJZBKZCLZUSUMUOUNAJZJZ
      BKZCLZVDUMVFCNZBKZVHUOULJZBKUOVECNZJZBKUMVJVKVMBULVLUOACEOUAPULBDQVIVMBUO
      VECUBPUCVFBCUDUEVGVCCVFVBBUOUNAUKPRSVCURCUNUPBUFRSVAUNUTTZCLUSUTCEUGVNURC
      UTUQUNABDQUHRUIUJ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    nelrdva.1 $e |- ( ( ph /\ x e. A ) -> x =/= B ) $.
    $( Deduce negative membership from an implication.  (Contributed by Thierry
       Arnoux, 27-Nov-2017.) $)
    nelrdva $p |- ( ph -> -. B e. A ) $=
      ( wcel wceq wa eqidd wne cv wi eleq1 anbi2d imbi12d vtoclg anabsi7 neneqd
      neeq1 pm2.65da ) ADCFZDDGAUAHZDIUBDDAUADDJZABKZCFZHZUDDJZLUBUCLBDCUDDGZUF
      UBUGUCUHUEUAAUDDCMNUDDDSOEPQRT $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Conditional equality (experimental)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This is a very useless definition, which "abbreviates" ` ( x = y -> ph ) ` as
  ` CondEq ( x = y -> ph ) ` . What this display hides, though, is that the
  first expression, even though it has a shorter constant string, is actually
  much more complicated in its parse tree: it is parsed as
  (wi (wceq (cv vx) (cv vy)) wph), while the ` CondEq ` version is parsed as
  (wcdeq vx vy wph).  It also allows us to give a name to the specific ternary
  operation ` ( x = y -> ph ) ` .

  This is all used as part of a metatheorem: we want to say that
  ` |- ( x = y -> ( ph ( x ) <-> ph ( y ) ) ) ` and
  ` |- ( x = y -> A ( x ) = A ( y ) ) ` are provable, for any expressions
  ` ph ( x ) ` or ` A ( x ) ` in the language.  The proof is by induction, so
  the base case is each of the primitives, which is why you will see a theorem
  for each of the set.mm primitive operations.

  The metatheorem comes with a disjoint variables condition: every variable in
  ` ph ( x ) ` is assumed disjoint from ` x ` except ` x ` itself.  For such a
  proof by induction, we must consider each of the possible forms of
  ` ph ( x ) ` .  If it is a variable other than ` x ` , then we have
  ` CondEq ( x = y -> A = A ) ` or ` CondEq ( x = y -> ( ph <-> ph ) ) ` ,
  which is provable by ~ cdeqth and reflexivity.  Since we are only working
  with class and wff expressions, it can't be ` x ` itself in set.mm, but if it
  was we'd have to also prove ` CondEq ( x = y -> x = y ) ` (where _set_
  equality is being used on the right).

  Otherwise, it is a primitive operation applied to smaller expressions.  In
  these cases, for each setvar variable parameter to the operation, we must
  consider if it is equal to ` x ` or not, which yields 2^n proof obligations.
  Luckily, all primitive operations in set.mm have either zero or one set
  variable, so we only need to prove one statement for the non-set constructors
  (like implication) and two for the constructors taking a set (the forall and
  the class builder).

  In each of the primitive proofs, we are allowed to assume that ` y ` is
  disjoint from ` ph ( x ) ` and vice versa, because this is maintained through
  the induction.  This is how we satisfy the disjoint variable conditions of
  ~ cdeqab1 and ~ cdeqab .

$)

  $c CondEq $. $( conditional equality $)

  $( Extend wff notation to include conditional equality.  This is a technical
     device used in the proof that ` F/ ` is the not-free predicate, and that
     definitions are conservative as a result. $)
  wcdeq $a wff CondEq ( x = y -> ph ) $.

  $( Define conditional equality.  All the notation to the left of the ` <-> `
     is fake; the parentheses and arrows are all part of the notation, which
     could equally well be written ` CondEq x y ph ` .  On the right side is
     the actual implication arrow.  The reason for this definition is to
     "flatten" the structure on the right side (whose tree structure is
     something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy
     wph).  (Contributed by Mario Carneiro, 11-Aug-2016.) $)
  df-cdeq $a |- ( CondEq ( x = y -> ph ) <-> ( x = y -> ph ) ) $.

  ${
    cdeqi.1 $e |- ( x = y -> ph ) $.
    $( Deduce conditional equality.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    cdeqi $p |- CondEq ( x = y -> ph ) $=
      ( wcdeq weq wi df-cdeq mpbir ) ABCEBCFAGDABCHI $.
  $}

  ${
    cdeqri.1 $e |- CondEq ( x = y -> ph ) $.
    $( Property of conditional equality.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    cdeqri $p |- ( x = y -> ph ) $=
      ( wcdeq weq wi df-cdeq mpbi ) ABCEBCFAGDABCHI $.
  $}

  ${
    cdeqth.1 $e |- ph $.
    $( Deduce conditional equality from a theorem.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    cdeqth $p |- CondEq ( x = y -> ph ) $=
      ( weq a1i cdeqi ) ABCABCEDFG $.
  $}

  ${
    cdeqnot.1 $e |- CondEq ( x = y -> ( ph <-> ps ) ) $.
    $( Distribute conditional equality over negation.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    cdeqnot $p |- CondEq ( x = y -> ( -. ph <-> -. ps ) ) $=
      ( wn wb weq cdeqri notbid cdeqi ) AFBFGCDCDHABABGCDEIJK $.

    ${
      $d x z $.  $d y z $.
      $( Distribute conditional equality over quantification.  (Contributed by
         Mario Carneiro, 11-Aug-2016.) $)
      cdeqal $p |- CondEq ( x = y -> ( A. z ph <-> A. z ps ) ) $=
        ( wal wb weq cdeqri albidv cdeqi ) AEGBEGHCDCDIABEABHCDFJKL $.

      $( Distribute conditional equality over abstraction.  (Contributed by
         Mario Carneiro, 11-Aug-2016.) $)
      cdeqab $p |- CondEq ( x = y -> { z | ph } = { z | ps } ) $=
        ( cab wceq weq wb cdeqri abbidv cdeqi ) AEGBEGHCDCDIABEABJCDFKLM $.
    $}

    ${
      $d x ps $.  $d y ph $.
      $( Distribute conditional equality over quantification.  (Contributed by
         Mario Carneiro, 11-Aug-2016.) $)
      cdeqal1 $p |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) ) $=
        ( wal wb cdeqri cbvalv cdeqth ) ACFBDFGCDABCDABGCDEHIJ $.

      $( Distribute conditional equality over abstraction.  (Contributed by
         Mario Carneiro, 11-Aug-2016.) $)
      cdeqab1 $p |- CondEq ( x = y -> { x | ph } = { y | ps } ) $=
        ( cab wceq wb cdeqri cbvabv cdeqth ) ACFBDFGCDABCDABHCDEIJK $.
    $}

    cdeqim.1 $e |- CondEq ( x = y -> ( ch <-> th ) ) $.
    $( Distribute conditional equality over implication.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    cdeqim $p |- CondEq ( x = y -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) $=
      ( wi wb weq cdeqri imbi12d cdeqi ) ACIBDIJEFEFKABCDABJEFGLCDJEFHLMN $.
  $}

  $( Conditional equality for set-to-class promotion.  (Contributed by Mario
     Carneiro, 11-Aug-2016.) $)
  cdeqcv $p |- CondEq ( x = y -> x = y ) $=
    ( weq id cdeqi ) ABCZABFDE $.

  ${
    cdeqeq.1 $e |- CondEq ( x = y -> A = B ) $.
    cdeqeq.2 $e |- CondEq ( x = y -> C = D ) $.
    $( Distribute conditional equality over equality.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    cdeqeq $p |- CondEq ( x = y -> ( A = C <-> B = D ) ) $=
      ( wceq wb weq cdeqri eqeq12d cdeqi ) CEIDFIJABABKCDEFCDIABGLEFIABHLMN $.

    $( Distribute conditional equality over elementhood.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) $)
    cdeqel $p |- CondEq ( x = y -> ( A e. C <-> B e. D ) ) $=
      ( wcel wb weq wceq cdeqri eleq12d cdeqi ) CEIDFIJABABKCDEFCDLABGMEFLABHMN
      O $.
  $}

  ${
    $d x ps $.  $d y ph $.
    nfcdeq.1 $e |- F/ x ph $.
    nfcdeq.2 $e |- CondEq ( x = y -> ( ph <-> ps ) ) $.
    $( If we have a conditional equality proof, where ` ph ` is ` ph ( x ) `
       and ` ps ` is ` ph ( y ) ` , and ` ph ( x ) ` in fact does not have
       ` x ` free in it according to ` F/ ` , then ` ph ( x ) <-> ph ( y ) `
       unconditionally.  This proves that ` F/ x ph ` is actually a not-free
       predicate.  (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nfcdeq $p |- ( ph <-> ps ) $=
      ( wsb sbf nfv wb cdeqri sbie bitr3i ) AACDGBACDEHABCDBCIABJCDFKLM $.
  $}

  ${
    $d x z B $.  $d y z A $.
    nfccdeq.1 $e |- F/_ x A $.
    nfccdeq.2 $e |- CondEq ( x = y -> A = B ) $.
    $( Variation of ~ nfcdeq for classes.  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nfccdeq $p |- A = B $=
      ( vz cv wcel nfcri weq equid cdeqth cdeqel nfcdeq eqriv ) GCDGHZCIQDIABAG
      CEJABQQCDGGKABGLMFNOP $.
  $}

  $( Let the computer know the theorems to look for to prove the metatheorem $)
  $( $j
    condequality 'wcdeq' from 'cdeqth';
    condcongruence 'cdeqnot' 'cdeqim' 'cdeqal1' 'cdeqal' 'cdeqcv' 'cdeqeq'
      'cdeqel' 'cdeqab1' 'cdeqab';
    notfree 'wnf' from 'nfcdeq';
    notfree 'wnfc' from 'nfccdeq';
  $)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Russell's Paradox
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y $.
    $( Russell's Paradox.  Proposition 4.14 of [TakeutiZaring] p. 14.

       In the late 1800s, Frege's Axiom of (unrestricted) Comprehension,
       expressed in our notation as ` A e. _V ` , asserted that any collection
       of sets ` A ` is a set i.e. belongs to the universe ` _V ` of all sets.
       In particular, by substituting ` { x | x e/ x } ` (the "Russell class")
       for ` A ` , it asserted ` { x | x e/ x } e. _V ` , meaning that the
       "collection of all sets which are not members of themselves" is a set.
       However, here we prove ` { x | x e/ x } e/ _V ` .  This contradiction
       was discovered by Russell in 1901 (published in 1903), invalidating the
       Comprehension Axiom and leading to the collapse of Frege's system.

       In 1908, Zermelo rectified this fatal flaw by replacing Comprehension
       with a weaker Subset (or Separation) Axiom asserting that ` A ` is a set
       only when it is smaller than some other set ` B ` .  The intuitionistic
       set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla]
       p.  "Axioms of CZF and IZF", which we include as ~ ax-sep .
       (Contributed by NM, 7-Aug-1994.) $)
    ru $p |- { x | x e/ x } e/ _V $=
      ( vy cv wnel cab cvv wcel wn wceq wex wel wal pm5.19 eleq1 df-nel eleq12d
      wb id notbid mtbir bitrid bibi12d spv mto abeq2 nex isset mpbir ) ACZUIDZ
      AEZFDUKFGZHULBCZUKIZBJUNBUNABKZUJQZALZUQBBKZURHZQZURMUPUTABUIUMIZUOURUJUS
      UIUMUMNUJAAKZHVAUSUIUIOVAVBURVAUIUMUIUMVARZVCPSUAUBUCUDUJAUMUETUFBUKUGTUK
      FOUH $.
    $( $j usage 'ru' avoids 'ax-setind'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Proper substitution of classes for sets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c [. $.
  $c ]. $.

  $( Extend wff notation to include the proper substitution of a class for a
     set.  Read this notation as "the proper substitution of class ` A ` for
     setvar variable ` x ` in wff ` ph ` ". $)
  wsbc $a wff [. A / x ]. ph $.

  $( Define the proper substitution of a class for a set.

     When ` A ` is a proper class, our definition evaluates to false.  This is
     somewhat arbitrary: we could have, instead, chosen the conclusion of
     ~ sbc6 for our definition, which always evaluates to true for proper
     classes.

     Our definition also does not produce the same results as discussed in the
     proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does
     hold, as shown by ~ dfsbcq below).  Unfortunately, Quine's definition
     requires a recursive syntactical breakdown of ` ph ` , and it does not
     seem possible to express it with a single closed formula.

     If we did not want to commit to any specific proper class behavior, we
     could use this definition _only_ to prove Theorem ~ dfsbcq , which holds
     for both our definition and Quine's, and from which we can derive a weaker
     version of ~ df-sbc in the form of ~ sbc8g .  However, the behavior of
     Quine's definition at proper classes is similarly arbitrary, and for
     practical reasons (to avoid having to prove sethood of ` A ` in every use
     of this definition) we allow direct reference to ~ df-sbc and assert that
     ` [. A / x ]. ph ` is always false when ` A ` is a proper class.

     The related definition df-csb defines proper substitution into a class
     variable (as opposed to a wff variable).  (Contributed by NM,
     14-Apr-1995.)  (Revised by NM, 25-Dec-2016.) $)
  df-sbc $a |- ( [. A / x ]. ph <-> A e. { x | ph } ) $.

  $( This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
     under both our definition and Quine's, provides us with a weak definition
     of the proper substitution of a class for a set.  Since our ~ df-sbc does
     not result in the same behavior as Quine's for proper classes, if we
     wished to avoid conflict with Quine's definition we could start with this
     theorem and ~ dfsbcq2 instead of ~ df-sbc .  ( ~ dfsbcq2 is needed because
     unlike Quine we do not overload the ~ df-sb syntax.)  As a consequence of
     these theorems, we can derive ~ sbc8g , which is a weaker version of
     ~ df-sbc that leaves substitution undefined when ` A ` is a proper class.

     However, it is often a nuisance to have to prove the sethood hypothesis of
     ~ sbc8g , so we will allow direct use of ~ df-sbc .  Proper substiution
     with a proper class is rarely needed, and when it is, we can simply use
     the expansion of Quine's definition.  (Contributed by NM, 14-Apr-1995.) $)
  dfsbcq $p |- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) ) $=
    ( wceq cab wcel wsbc eleq1 df-sbc 3bitr4g ) CDECABFZGDLGABCHABDHCDLIABCJABD
    JK $.

  $( This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
     under both our definition and Quine's, relates logic substitution ~ df-sb
     and substitution for class variables ~ df-sbc .  Unlike Quine, we use a
     different syntax for each in order to avoid overloading it.  See remarks
     in ~ dfsbcq .  (Contributed by NM, 31-Dec-2016.) $)
  dfsbcq2 $p |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) ) $=
    ( cv wceq cab wcel wsb wsbc eleq1 df-clab df-sbc bicomi 3bitr3g ) CEZDFPABG
    ZHDQHZABCIABDJZPDQKACBLSRABDMNO $.

  $( Show that ~ df-sb and ~ df-sbc are equivalent when the class term ` A ` in
     ~ df-sbc is a setvar variable.  This theorem lets us reuse theorems based
     on ~ df-sb for proofs involving ~ df-sbc .  (Contributed by NM,
     31-Dec-2016.)  (Proof modification is discouraged.) $)
  sbsbc $p |- ( [ y / x ] ph <-> [. y / x ]. ph ) $=
    ( weq wsb cv wsbc wb eqid dfsbcq2 ax-mp ) CCDABCEABCFZGHLIABCLJK $.

  ${
    sbceq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for class substitution.  (Contributed by Mario
       Carneiro, 9-Feb-2017.)  (Revised by NM, 30-Jun-2018.) $)
    sbceq1d $p |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ps ) ) $=
      ( wceq wsbc wb dfsbcq syl ) ADEGBCDHBCEHIFBCDEJK $.

    sbceq1dd.2 $e |- ( ph -> [. A / x ]. ps ) $.
    $( Equality theorem for class substitution.  (Contributed by Mario
       Carneiro, 9-Feb-2017.)  (Revised by NM, 30-Jun-2018.) $)
    sbceq1dd $p |- ( ph -> [. B / x ]. ps ) $=
      ( wsbc sbceq1d mpbid ) ABCDHBCEHGABCDEFIJ $.
  $}

  ${
    $d x ph $.
    sbceqbid.1 $e |- ( ph -> A = B ) $.
    sbceqbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equality theorem for class substitution.  (Contributed by Thierry
       Arnoux, 4-Sep-2018.) $)
    sbceqbid $p |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) ) $=
      ( cab wcel wsbc abbidv eleq12d df-sbc 3bitr4g ) AEBDIZJFCDIZJBDEKCDFKAEFP
      QGABCDHLMBDENCDFNO $.
  $}

  ${
    $d y A $.  $d y ph $.  $d x y $.
    $( This is the closest we can get to ~ df-sbc if we start from ~ dfsbcq
       (see its comments) and ~ dfsbcq2 .  (Contributed by NM, 18-Nov-2008.)
       (Proof shortened by Andrew Salmon, 29-Jun-2011.)
       (Proof modification is discouraged.) $)
    sbc8g $p |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) ) $=
      ( vy cv wsbc cab wcel dfsbcq eleq1 wsb df-clab weq wb equid dfsbcq2 ax-mp
      bitr2i vtoclbg ) ABEFZGZUAABHZIZABCGCUCIECDABUACJUACUCKUDABELZUBAEBMEENUE
      UBOEPABEUAQRST $.
  $}

  $( By our definition of proper substitution, it can only be true if the
     substituted expression is a set.  (Contributed by Mario Carneiro,
     13-Oct-2016.) $)
  sbcex $p |- ( [. A / x ]. ph -> A e. _V ) $=
    ( wsbc cab wcel cvv df-sbc elex sylbi ) ABCDCABEZFCGFABCHCKIJ $.

  $( Equality theorem for class substitution.  Class version of ~ sbequ12 .
     (Contributed by NM, 26-Sep-2003.) $)
  sbceq1a $p |- ( x = A -> ( ph <-> [. A / x ]. ph ) ) $=
    ( wsb cv wceq wsbc sbid dfsbcq2 bitr3id ) AABBDBECFABCGABHABBCIJ $.

  $( Equality theorem for class substitution.  Class version of ~ sbequ12r .
     (Contributed by NM, 4-Jan-2017.) $)
  sbceq2a $p |- ( A = x -> ( [. A / x ]. ph <-> ph ) ) $=
    ( cv wceq wsbc wb sbceq1a eqcoms bicomd ) CBDZEAABCFZALGKCABCHIJ $.

  ${
    $d ph y $.  $d A y $.  $d x y $.
    $( Specialization: if a formula is true for all sets, it is true for any
       class which is a set.  Similar to Theorem 6.11 of [Quine] p. 44.  See
       also ~ stdpc4 and ~ rspsbc .  (Contributed by NM, 16-Jan-2004.) $)
    spsbc $p |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) ) $=
      ( vy wal wsbc wi cv wceq wsb stdpc4 sbsbc sylib dfsbcq imbitrid vtocleg )
      ABFZABCGZHECDRABEIZGZTCJSRABEKUAABELABEMNABTCOPQ $.

    spsbcd.1 $e |- ( ph -> A e. V ) $.
    spsbcd.2 $e |- ( ph -> A. x ps ) $.
    $( Specialization: if a formula is true for all sets, it is true for any
       class which is a set.  Similar to Theorem 6.11 of [Quine] p. 44.  See
       also ~ stdpc4 and ~ rspsbc .  (Contributed by Mario Carneiro,
       9-Feb-2017.) $)
    spsbcd $p |- ( ph -> [. A / x ]. ps ) $=
      ( wcel wal wsbc spsbc sylc ) ADEHBCIBCDJFGBCDEKL $.
  $}

  ${
    sbcth.1 $e |- ph $.
    $( A substitution into a theorem remains true (when ` A ` is a set).
       (Contributed by NM, 5-Nov-2005.) $)
    sbcth $p |- ( A e. V -> [. A / x ]. ph ) $=
      ( wcel wal wsbc ax-gen spsbc mpi ) CDFABGABCHABEIABCDJK $.
  $}

  ${
    $d x ph $.
    sbcthdv.1 $e |- ( ph -> ps ) $.
    $( Deduction version of ~ sbcth .  (Contributed by NM, 30-Nov-2005.)
       (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    sbcthdv $p |- ( ( ph /\ A e. V ) -> [. A / x ]. ps ) $=
      ( wal wcel wsbc alrimiv spsbc mpan9 ) ABCGDEHBCDIABCFJBCDEKL $.
  $}

  $( An identity theorem for substitution.  See ~ sbid .  (Contributed by Mario
     Carneiro, 18-Feb-2017.) $)
  sbcid $p |- ( [. x / x ]. ph <-> ph )
    $=
    ( cv wsbc wsb sbsbc sbid bitr3i ) ABBCDABBEAABBFABGH $.

  ${
    nfsbc1d.2 $e |- ( ph -> F/_ x A ) $.
    $( Deduction version of ~ nfsbc1 .  (Contributed by NM, 23-May-2006.)
       (Revised by Mario Carneiro, 12-Oct-2016.) $)
    nfsbc1d $p |- ( ph -> F/ x [. A / x ]. ps ) $=
      ( wsbc cab wcel df-sbc wnfc nfab1 a1i nfeld nfxfrd ) BCDFDBCGZHACBCDIACDO
      ECOJABCKLMN $.
  $}

  ${
    nfsbc1.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for class substitution.  (Contributed
       by Mario Carneiro, 12-Oct-2016.) $)
    nfsbc1 $p |- F/ x [. A / x ]. ph $=
      ( wsbc wnf wtru wnfc a1i nfsbc1d mptru ) ABCEBFGABCBCHGDIJK $.
  $}

  ${
    $d x A $.
    $( Bound-variable hypothesis builder for class substitution.  (Contributed
       by Mario Carneiro, 12-Oct-2016.) $)
    nfsbc1v $p |- F/ x [. A / x ]. ph $=
      ( nfcv nfsbc1 ) ABCBCDE $.
  $}

  ${
    nfsbcd.1 $e |- F/ y ph $.
    nfsbcd.2 $e |- ( ph -> F/_ x A ) $.
    nfsbcd.3 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfsbc .  (Contributed by NM, 23-Nov-2005.)
       (Revised by Mario Carneiro, 12-Oct-2016.) $)
    nfsbcd $p |- ( ph -> F/ x [. A / y ]. ps ) $=
      ( wsbc cab wcel df-sbc nfabd nfeld nfxfrd ) BDEIEBDJZKACBDELACEPGABCDFHMN
      O $.
  $}

  ${
    nfsbc.1 $e |- F/_ x A $.
    nfsbc.2 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for class substitution.  (Contributed
       by NM, 7-Sep-2014.)  (Revised by Mario Carneiro, 12-Oct-2016.) $)
    nfsbc $p |- F/ x [. A / y ]. ph $=
      ( wsbc wnf wtru nftru wnfc a1i nfsbcd mptru ) ACDGBHIABCDCJBDKIELABHIFLMN
      $.
  $}

  ${
    $d x z $.  $d z A $.  $d y z ph $.
    $( A composition law for class substitution.  (Contributed by NM,
       26-Sep-2003.)  (Revised by Mario Carneiro, 13-Oct-2016.) $)
    sbcco $p |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) $=
      ( vz cv wsbc cvv wcel sbcex dfsbcq wsb sbsbc sbbii sbco2 3bitr3ri vtoclbg
      nfv bitri pm5.21nii ) ABCFGZCDGZDHIABDGZUACDJABDJUACEFZGZABUDGZUBUCEDHUAC
      UDDKABUDDKUEABELZUFABCLZCELUACELUGUEUHUACEABCMNABECACROUACEMPABEMSQT $.
  $}

  ${
    $d x y $.  $d y ph $.  $d A y $.
    sbcco2.1 $e |- ( x = y -> A = B ) $.
    $( A composition law for class substitution.  Importantly, ` x ` may occur
       free in the class expression substituted for ` A ` .  (Contributed by
       NM, 5-Sep-2004.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    sbcco2 $p |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph ) $=
      ( wsbc cv wsb sbsbc nfv weq wceq wb equcoms dfsbcq bicomd syl sbie bitr3i
      ) ABEGZCBHGUACBIABDGZUACBJUAUBCBUBCKCBLDEMZUAUBNUCBCFOUCUBUAABDEPQRST $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( An equivalence for class substitution.  (Contributed by NM,
       23-Aug-1993.)  (Revised by Mario Carneiro, 12-Oct-2016.) $)
    sbc5 $p |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) $=
      ( vy wsbc cvv wcel cv wceq wex sbcex exsimpl isset sylibr wsb weq dfsbcq2
      wa eqeq2 anbi1d exbidv sb5 vtoclbg pm5.21nii ) ABCEZCFGZBHZCIZARZBJZABCKU
      JUHBJUFUHABLBCMNABDOBDPZARZBJUEUJDCFABDCQDHZCIZULUIBUNUKUHAUMCUGSTUAABDUB
      UCUD $.
  $}

  ${
    $d x A $.
    $( An equivalence for class substitution.  (Contributed by NM,
       11-Oct-2004.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
    sbc6g $p |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) ) $=
      ( wcel wsbc cv wceq wa wex wi wal sbc5 nfe1 ceqex ceqsalg bitr4id ) CDEAB
      CFBGCHZAIZBJZRAKBLABCMATBCDSBNABCOPQ $.
  $}

  ${
    $d x A $.
    sbc6.1 $e |- A e. _V $.
    $( An equivalence for class substitution.  (Contributed by NM,
       23-Aug-1993.)  (Proof shortened by Eric Schmidt, 17-Jan-2007.) $)
    sbc6 $p |- ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) $=
      ( cvv wcel wsbc cv wceq wi wal wb sbc6g ax-mp ) CEFABCGBHCIAJBKLDABCEMN
      $.
  $}

  ${
    $d y A $.  $d y ph $.  $d x y $.
    $( An equivalence for class substitution in the spirit of ~ df-clab .  Note
       that ` x ` and ` A ` don't have to be distinct.  (Contributed by NM,
       18-Nov-2008.)  (Revised by Mario Carneiro, 13-Oct-2016.) $)
    sbc7 $p |- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) ) $=
      ( wsbc cv wceq wa wex sbcco sbc5 bitr3i ) ABDEABCFZEZCDEMDGNHCIABCDJNCDKL
      $.
  $}

  ${
    $d x y $.
    cbvsbcw.1 $e |- F/ y ph $.
    cbvsbcw.2 $e |- F/ x ps $.
    cbvsbcw.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ cbvsbc with a disjoint variable condition.  (Contributed by
       GG, 10-Jan-2024.) $)
    cbvsbcw $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
      ( cab wcel wsbc cbvabw eleq2i df-sbc 3bitr4i ) EACIZJEBDIZJACEKBDEKPQEABC
      DFGHLMACENBDENO $.
  $}

  ${
    cbvsbc.1 $e |- F/ y ph $.
    cbvsbc.2 $e |- F/ x ps $.
    cbvsbc.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variables in a wff substitution.  (Contributed by Jeff
       Hankins, 19-Sep-2009.)  (Proof shortened by Andrew Salmon,
       8-Jun-2011.) $)
    cbvsbc $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
      ( cab wcel wsbc cbvab eleq2i df-sbc 3bitr4i ) EACIZJEBDIZJACEKBDEKPQEABCD
      FGHLMACENBDENO $.
  $}

  ${
    $d y ph $.  $d x ps $.
    cbvsbcv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a class substitution using implicit
       substitution.  (Contributed by NM, 30-Sep-2008.)  (Revised by Mario
       Carneiro, 13-Oct-2016.) $)
    cbvsbcv $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
      ( nfv cbvsbc ) ABCDEADGBCGFH $.
  $}

  ${
    $d x A $.
    $( Conversion of implicit substitution to explicit class substitution,
       using a bound-variable hypothesis instead of distinct variables.
       (Closed theorem version of ~ sbciegf .)  (Contributed by NM,
       10-Nov-2005.)  (Revised by Mario Carneiro, 13-Oct-2016.) $)
    sbciegft $p |- ( ( A e. V /\ F/ x ps /\
            A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) ) $=
      ( wcel wnf cv wceq wb wi wal w3a wsbc imim2i alimi biimpa sylan2 3adant1
      wa wex sbc5 biimp impd 19.23t biimtrid biimpr com23 19.21t sbc6g 3ad2ant1
      sylibrd impbid ) DEFZBCGZCHDIZABJZKZCLZMZACDNZBVAUPATZCUAZUTBACDUBUOUSVCB
      KZUNUSUOVBBKZCLZVDURVECURUPABUQABKUPABUCOUDPUOVFVDVBBCUEQRSUFUTBUPAKZCLZV
      AUOUSBVHKZUNUSUOBVGKZCLZVIURVJCURUPBAUQBAKUPABUGOUHPUOVKVIBVGCUIQRSUNUOVA
      VHJUSACDEUJUKULUM $.
  $}

  ${
    $d x A $.
    sbciegf.1 $e |- F/ x ps $.
    sbciegf.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       (Contributed by NM, 14-Dec-2005.)  (Revised by Mario Carneiro,
       13-Oct-2016.) $)
    sbciegf $p |- ( A e. V -> ( [. A / x ]. ph <-> ps ) ) $=
      ( wcel wnf cv wceq wb wi wal wsbc ax-gen sbciegft mp3an23 ) DEHBCICJDKABL
      MZCNACDOBLFSCGPABCDEQR $.
  $}

  ${
    $d x A $.  $d x ps $.
    sbcieg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       (Contributed by NM, 10-Nov-2005.) $)
    sbcieg $p |- ( A e. V -> ( [. A / x ]. ph <-> ps ) ) $=
      ( nfv sbciegf ) ABCDEBCGFH $.
  $}

  ${
    $d x y $.  $d A y $.  $d ch y $.  $d ph y $.  $d ps x $.
    sbcie2g.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    sbcie2g.2 $e |- ( y = A -> ( ps <-> ch ) ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       This version of ~ sbcie avoids a disjointness condition on ` x ` and
       ` A ` by substituting twice.  (Contributed by Mario Carneiro,
       15-Oct-2016.) $)
    sbcie2g $p |- ( A e. V -> ( [. A / x ]. ph <-> ch ) ) $=
      ( cv wsbc dfsbcq wsb sbsbc nfv sbie bitr3i vtoclbg ) ADEJZKZBADFKCEFGADSF
      LITADEMBADENABDEBDOHPQR $.
  $}

  ${
    $d x A $.  $d x ps $.
    sbcie.1 $e |- A e. _V $.
    sbcie.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       (Contributed by NM, 4-Sep-2004.) $)
    sbcie $p |- ( [. A / x ]. ph <-> ps ) $=
      ( cvv wcel wsbc wb sbcieg ax-mp ) DGHACDIBJEABCDGFKL $.
  $}

  ${
    $d x A $.
    sbcied.1 $e |- ( ph -> A e. V ) $.
    sbcied.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    ${
      sbciedf.3 $e |- F/ x ph $.
      sbciedf.4 $e |- ( ph -> F/ x ch ) $.
      $( Conversion of implicit substitution to explicit class substitution,
         deduction form.  (Contributed by NM, 29-Dec-2014.) $)
      sbciedf $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
        ( wcel wnf cv wceq wb wi wal wsbc ex alrimi sbciegft syl3anc ) AEFKCDLD
        MENZBCOZPZDQBDERCOGJAUEDIAUCUDHSTBCDEFUAUB $.
    $}

    $d x ph $.  $d x ch $.
    $( Conversion of implicit substitution to explicit class substitution,
       deduction form.  (Contributed by NM, 13-Dec-2014.) $)
    sbcied $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
      ( nfv nfvd sbciedf ) ABCDEFGHADIACDJK $.
  $}

  ${
    $d x A $.  $d x ph $.  $d x ch $.
    sbcied2.1 $e |- ( ph -> A e. V ) $.
    sbcied2.2 $e |- ( ph -> A = B ) $.
    sbcied2.3 $e |- ( ( ph /\ x = B ) -> ( ps <-> ch ) ) $.
    $( Conversion of implicit substitution to explicit class substitution,
       deduction form.  (Contributed by NM, 13-Dec-2014.) $)
    sbcied2 $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
      ( cv wceq wb id sylan9eqr syldan sbcied ) ABCDEGHADKZELZRFLBCMSAREFSNIOJP
      Q $.
  $}

  ${
    $d y A $.  $d y B $.  $d y ph $.  $d x y $.
    elrabsf.1 $e |- F/_ x B $.
    $( Membership in a restricted class abstraction, expressed with explicit
       class substitution.  (The variation ~ elrabf has implicit substitution).
       The hypothesis specifies that ` x ` must not be a free variable in
       ` B ` .  (Contributed by NM, 30-Sep-2003.)  (Proof shortened by Mario
       Carneiro, 13-Oct-2016.) $)
    elrabsf $p |- ( A e. { x e. B | ph }
          <-> ( A e. B /\ [. A / x ]. ph ) ) $=
      ( vy cv wsbc crab dfsbcq nfcv nfv nfsbc1v sbceq1a cbvrab elrab2 ) ABFGZHZ
      ABCHFCDABDIABQCJARBFDEFDKAFLABQMABQNOP $.
  $}

  ${
    $d x y B $.  $d y A $.
    $( Substitution for the left-hand side in an equality.  Class version of
       ~ eqsb1 .  (Contributed by Andrew Salmon, 29-Jun-2011.) $)
    eqsbc1 $p |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) ) $=
      ( vy cv wceq wsbc dfsbcq eqeq1 wsb sbsbc eqsb1 bitr3i vtoclbg ) AFCGZAEFZ
      HZQCGZPABHBCGEBDPAQBIQBCJRPAEKSPAELAECMNO $.
  $}

  ${
    $d x y $.  $d y A $.  $d y ph $.  $d y ps $.
    $( Move negation in and out of class substitution.  (Contributed by NM,
       16-Jan-2004.) $)
    sbcng $p |- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) ) $=
      ( vy wn wsb wsbc dfsbcq2 cv wceq notbid sbn vtoclbg ) AFZBEGABEGZFOBCHABC
      HZFECDOBECIEJCKPQABECILABEMN $.

    $( Distribution of class substitution over implication.  (Contributed by
       NM, 16-Jan-2004.) $)
    sbcimg $p |- ( A e. V ->
     ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) ) $=
      ( vy wi wsb wsbc dfsbcq2 cv wceq imbi12d sbim vtoclbg ) ABGZCFHACFHZBCFHZ
      GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.

    $( Distribution of class substitution over conjunction.  (Contributed by
       NM, 31-Dec-2016.) $)
    sbcan $p |- ( [. A / x ]. ( ph /\ ps )
        <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) $=
      ( vy wa wsbc cvv wcel sbcex adantl dfsbcq2 cv wceq anbi12d sban pm5.21nii
      wsb vtoclbg ) ABFZCDGZDHIZACDGZBCDGZFZTCDJUDUBUCBCDJKTCERACERZBCERZFUAUEE
      DHTCEDLEMDNUFUCUGUDACEDLBCEDLOABCEPSQ $.

    $( Distribution of class substitution over conjunction.  (Contributed by
       NM, 21-May-2004.) $)
    sbcang $p |- ( A e. V ->
     ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) ) $=
      ( vy wa wsb wsbc dfsbcq2 cv wceq anbi12d sban vtoclbg ) ABGZCFHACFHZBCFHZ
      GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.

    $( Distribution of class substitution over disjunction.  (Contributed by
       NM, 31-Dec-2016.) $)
    sbcor $p |- ( [. A / x ]. ( ph \/ ps )
         <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) $=
      ( vy wo wsbc cvv wcel sbcex jaoi wsb dfsbcq2 cv wceq orbi12d sbor vtoclbg
      pm5.21nii ) ABFZCDGZDHIZACDGZBCDGZFZTCDJUCUBUDACDJBCDJKTCELACELZBCELZFUAU
      EEDHTCEDMENDOUFUCUGUDACEDMBCEDMPABCEQRS $.

    $( Distribution of class substitution over disjunction.  (Contributed by
       NM, 21-May-2004.) $)
    sbcorg $p |- ( A e. V ->
     ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) ) $=
      ( vy wo wsb wsbc dfsbcq2 cv wceq orbi12d sbor vtoclbg ) ABGZCFHACFHZBCFHZ
      GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.

    $( Distribution of class substitution over biconditional.  (Contributed by
       Raph Levien, 10-Apr-2004.) $)
    sbcbig $p |- ( A e. V ->
   ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) ) $=
      ( vy wb wsb wsbc dfsbcq2 cv wceq bibi12d sbbi vtoclbg ) ABGZCFHACFHZBCFHZ
      GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
  $}

  $( Move negation in and out of class substitution.  One direction of ~ sbcng
     that holds for proper classes.  (Contributed by NM, 17-Aug-2018.) $)
  sbcn1 $p |- ( [. A / x ]. -. ph -> -. [. A / x ]. ph ) $=
    ( cvv wcel wn wsbc sbcex sbcng biimpd mpcom ) CDEZAFZBCGZABCGFZMBCHLNOABCDI
    JK $.

  $( Distribution of class substitution over implication.  One direction of
     ~ sbcimg that holds for proper classes.  (Contributed by NM,
     17-Aug-2018.) $)
  sbcim1 $p |- ( [. A / x ]. ( ph -> ps )
          -> ( [. A / x ]. ph -> [. A / x ]. ps ) ) $=
    ( cvv wcel wi wsbc sbcex sbcimg biimpd mpcom ) DEFZABGZCDHZACDHBCDHGZNCDIMO
    PABCDEJKL $.

  $( Distribution of class substitution over biconditional.  One direction of
     ~ sbcbig that holds for proper classes.  (Contributed by NM,
     17-Aug-2018.) $)
  sbcbi1 $p |- ( [. A / x ]. ( ph <-> ps )
       -> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) $=
    ( cvv wcel wb wsbc sbcex sbcbig biimpd mpcom ) DEFZABGZCDHZACDHBCDHGZNCDIMO
    PABCDEJKL $.

  $( Substituting into equivalent wff's gives equivalent results.  (Contributed
     by Giovanni Mascellani, 9-Apr-2018.) $)
  sbcbi2 $p |- ( A. x ( ph <-> ps ) ->
       ( [. A / x ]. ph <-> [. A / x ]. ps ) ) $=
    ( wb wal cab wcel wsbc wceq abbibcom eleq2 sylbi df-sbc 3bitr4g ) ABECFZDAC
    GZHZDBCGZHZACDIBCDIPQSJRTEABCKQSDLMACDNBCDNO $.

  ${
    $d x z A $.  $d x y z $.  $d z ph $.
    $( Move universal quantifier in and out of class substitution.
       (Contributed by NM, 31-Dec-2016.) $)
    sbcal $p |- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) $=
      ( vz wal wsbc cvv wcel sbcex sps dfsbcq2 cv wceq albidv vtoclbg pm5.21nii
      wsb sbal ) ABFZCDGZDHIZACDGZBFZTCDJUCUBBACDJKTCERACERZBFUAUDEDHTCEDLEMDNU
      EUCBACEDLOABCESPQ $.

    $( Move universal quantifier in and out of class substitution.
       (Contributed by NM, 16-Jan-2004.) $)
    sbcalg $p |- ( A e. V
        -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) ) $=
      ( vz wal wsb wsbc dfsbcq2 cv wceq albidv sbal vtoclbg ) ABGZCFHACFHZBGPCD
      IACDIZBGFDEPCFDJFKDLQRBACFDJMABCFNO $.

    $( Move existential quantifier in and out of class substitution.
       (Contributed by NM, 21-May-2004.) $)
    sbcex2 $p |- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) $=
      ( vz wex wsbc cvv wcel sbcex exlimiv wsb dfsbcq2 wceq exbidv sbex vtoclbg
      cv pm5.21nii ) ABFZCDGZDHIZACDGZBFZTCDJUCUBBACDJKTCELACELZBFUAUDEDHTCEDME
      RDNUEUCBACEDMOABCEPQS $.

    $( Move existential quantifier in and out of class substitution.
       (Contributed by NM, 21-May-2004.) $)
    sbcexg $p |- ( A e. V
         -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) ) $=
      ( vz wex wsb wsbc dfsbcq2 cv wceq exbidv sbex vtoclbg ) ABGZCFHACFHZBGPCD
      IACDIZBGFDEPCFDJFKDLQRBACFDJMABCFNO $.
  $}

  ${
    $d x B $.  $d x A $.
    $( A variation of extensionality for classes.  (Contributed by Andrew
       Salmon, 28-Jun-2011.) $)
    sbceqal $p |- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) ) $=
      ( wcel cv wceq wi wal wsbc spsbc sbcimg wb eqsbc1 mpbiri pm5.5 syl 3bitrd
      eqid sylibd ) BDEZAFZBGZUBCGZHZAIUEABJZBCGZUEABDKUAUFUCABJZUDABJZHZUIUGUC
      UDABDLUAUHUJUIMUAUHBBGBSABBDNOUHUIPQABCDNRT $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Theorem *14.121 in [WhiteheadRussell] p. 185.  (Contributed by Andrew
       Salmon, 28-Jun-2011.)  (Proof shortened by Wolf Lammen, 9-May-2013.) $)
    sbeqalb $p |- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <->
      x = B ) ) -> A = B ) ) $=
      ( cv wceq wb wal wa wi wcel bibi1 biimpa biimpd alanimi sbceqal syl5 ) AB
      FZCGZHZBIASDGZHZBIJTUBKZBICELCDGUAUCUDBUAUCJTUBUAUCTUBHATUBMNOPBCDEQR $.
  $}

  ${
    sbcbid.1 $e |- F/ x ph $.
    sbcbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building deduction for class substitution.  (Contributed by NM,
       29-Dec-2014.) $)
    sbcbid $p |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) ) $=
      ( cab wcel wsbc abbid eleq2d df-sbc 3bitr4g ) AEBDHZIECDHZIBDEJCDEJAOPEAB
      CDFGKLBDEMCDEMN $.
  $}

  ${
    $d x ph $.
    sbcbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building deduction for class substitution.  (Contributed by NM,
       29-Dec-2014.) $)
    sbcbidv $p |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) ) $=
      ( nfv sbcbid ) ABCDEADGFH $.
  $}

  ${
    sbcbii.1 $e |- ( ph <-> ps ) $.
    $( Formula-building inference for class substitution.  (Contributed by NM,
       11-Nov-2005.) $)
    sbcbii $p |- ( [. A / x ]. ph <-> [. A / x ]. ps ) $=
      ( wsbc wb wtru a1i sbcbidv mptru ) ACDFBCDFGHABCDABGHEIJK $.
  $}

  ${
    $d x B $.
    $( Substitution for the right-hand side in an equality.  (Contributed by
       Alan Sare, 24-Oct-2011.)  (Proof shortened by JJ, 7-Jul-2021.) $)
    eqsbc2 $p |- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) ) $=
      ( wcel cv wceq wsbc eqsbc1 eqcom sbcbii 3bitr4g ) BDEAFZCGZABHBCGCMGZABHC
      BGABCDIONABCMJKCBJL $.
  $}

  $( Distribution of class substitution over triple conjunction.  (Contributed
     by NM, 14-Dec-2006.)  (Revised by NM, 17-Aug-2018.) $)
  sbc3an $p |- ( [. A / x ]. ( ph /\ ps /\ ch ) <->
              ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) ) $=
    ( w3a wsbc wa df-3an sbcbii sbcan anbi1i 3bitri bitr4i ) ABCFZDEGZADEGZBDEG
    ZHZCDEGZHZQRTFPABHZCHZDEGUBDEGZTHUAOUCDEABCIJUBCDEKUDSTABDEKLMQRTIN $.

  ${
    $d y A $.  $d x y B $.
    $( Class substitution into a membership relation.  (Contributed by NM,
       17-Aug-2018.) $)
    sbcel1v $p |- ( [. A / x ]. x e. B <-> A e. B ) $=
      ( vy wcel wsbc cvv sbcex elex wsb dfsbcq2 eleq1 clelsb1 vtoclbg pm5.21nii
      cv ) APCEZABFZBGEBCEZQABHBCIQADJDPZCERSDBGQADBKTBCLADCMNO $.
  $}

  ${
    $d y B $.  $d x y A $.
    $( Class substitution into a membership relation.  (Contributed by NM,
       17-Nov-2006.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    sbcel2gv $p |- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) ) $=
      ( vy cv wcel eleq2 sbcie2g ) BAFZGBEFZGBCGAECDJKBHKCBHI $.

    $( Class substitution into a membership relation.  One direction of
       ~ sbcel2gv that holds for proper classes.  (Contributed by NM,
       17-Aug-2018.) $)
    sbcel21v $p |- ( [. B / x ]. A e. x -> A e. B ) $=
      ( cvv wcel cv wsbc sbcex sbcel2gv biimpd mpcom ) CDEZBAFEZACGZBCEZMACHLNO
      ABCDIJK $.
  $}

  ${
    $d x ph $.
    sbcimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Substitution analogue of Theorem 19.20 of [Margaris] p. 90 ( ~ alim ).
       (Contributed by NM, 11-Nov-2005.)  (Revised by NM, 17-Aug-2018.)  (Proof
       shortened by JJ, 7-Jul-2021.) $)
    sbcimdv $p |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) $=
      ( wsbc cvv wcel sbcex wi wal alrimiv spsbc sbcim1 syl56 com3l mpdi ) ABDE
      GZEHIZCDEGZBDEJTASUAABCKZDLTUBDEGSUAKAUBDFMUBDEHNBCDEOPQR $.
  $}

  ${
    $d x y $.  $d y A $.  $d y ph $.
    $( Substitution for a variable not free in a wff does not affect it.
       (Contributed by Mario Carneiro, 14-Oct-2016.) $)
    sbctt $p |- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) ) $=
      ( vy wcel wnf wsbc wb wsb wi wceq dfsbcq2 bibi1d imbi2d sbft vtoclg imp
      cv ) CDFABGZABCHZAIZTABEJZAIZKTUBKECDESCLZUDUBTUEUCUAAABECMNOABEPQR $.
  $}

  ${
    sbcgf.1 $e |- F/ x ph $.
    $( Substitution for a variable not free in a wff does not affect it.
       (Contributed by NM, 11-Oct-2004.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    sbcgf $p |- ( A e. V -> ( [. A / x ]. ph <-> ph ) ) $=
      ( wcel wnf wsbc wb sbctt mpan2 ) CDFABGABCHAIEABCDJK $.

    $( Substitution for a variable not free in antecedent affects only the
       consequent.  (Contributed by NM, 11-Oct-2004.) $)
    sbc19.21g $p |- ( A e. V ->
            ( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) ) $=
      ( wcel wi wsbc sbcimg sbcgf imbi1d bitrd ) DEGZABHCDIACDIZBCDIZHAPHABCDEJ
      NOAPACDEFKLM $.
  $}

  ${
    $d x ph $.
    $( Substitution for a variable not occurring in a wff does not affect it.
       Distinct variable form of ~ sbcgf .  (Contributed by Alan Sare,
       10-Nov-2012.) $)
    sbcg $p |- ( A e. V -> ( [. A / x ]. ph <-> ph ) ) $=
      ( nfv sbcgf ) ABCDABEF $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x V $.  $d y W $.
    sbc2iegf.1 $e |- F/ x ps $.
    sbc2iegf.2 $e |- F/ y ps $.
    sbc2iegf.3 $e |- F/ x B e. W $.
    sbc2iegf.4 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       (Contributed by Mario Carneiro, 19-Dec-2013.) $)
    sbc2iegf $p |- ( ( A e. V /\ B e. W ) ->
            ( [. A / x ]. [. B / y ]. ph <-> ps ) ) $=
      ( wcel wa simpl cv wceq wb adantll nfv wsbc wnf a1i sbciedf nfan ) EGMZFH
      MZNZADFUAZBCEGUFUGOUGCPEQZUIBRUFUGUJNZABDFHUGUJOUJDPFQABRUGLSUKDTBDUBUKJU
      CUDSUFUGCUFCTKUEBCUBUHIUCUD $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y ps $.
    sbc2ie.1 $e |- A e. _V $.
    sbc2ie.2 $e |- B e. _V $.
    sbc2ie.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       (Contributed by NM, 16-Dec-2008.)  (Revised by Mario Carneiro,
       19-Dec-2013.) $)
    sbc2ie $p |- ( [. A / x ]. [. B / y ]. ph <-> ps ) $=
      ( cvv wcel wsbc wb nfv nfth sbc2iegf mp2an ) EJKFJKZADFLCELBMGHABCDEFJJBC
      NBDNRCHOIPQ $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y ph $.  $d x y ch $.
    sbc2iedv.1 $e |- A e. _V $.
    sbc2iedv.2 $e |- B e. _V $.
    sbc2iedv.3 $e |- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       (Contributed by NM, 16-Dec-2008.)  (Proof shortened by Mario Carneiro,
       18-Oct-2016.) $)
    sbc2iedv $p |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) ) $=
      ( wsbc cvv wcel a1i cv wceq wa wb impl sbcied ) ABEGKCDFLFLMAHNADOFPZQZBC
      EGLGLMUBINAUAEOGPBCRJSTT $.
  $}

  ${
    $d x y z A $.  $d y z B $.  $d z C $.  $d x y z ps $.
    sbc3ie.1 $e |- A e. _V $.
    sbc3ie.2 $e |- B e. _V $.
    sbc3ie.3 $e |- C e. _V $.
    sbc3ie.4 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       (Contributed by Mario Carneiro, 19-Jun-2014.)  (Revised by Mario
       Carneiro, 29-Dec-2014.) $)
    sbc3ie $p |- ( [. A / x ]. [. B / y ]. [. C / z ]. ph <-> ps ) $=
      ( wsbc cv wceq wa cvv wcel a1i wb 3expa sbcied sbc2ie ) AEHMBCDFGIJCNFOZD
      NGOZPZABEHQHQRUFKSUDUEENHOABTLUAUBUC $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Lemma for ~ sbccom .  (Contributed by NM, 14-Nov-2005.)  (Revised by
       Mario Carneiro, 18-Oct-2016.) $)
    sbccomlem $p |- ( [. A / x ]. [. B / y ]. ph
         <-> [. B / y ]. [. A / x ]. ph ) $=
      ( cv wceq wa wex wsbc excom exdistr an12 exbii bitri 3bitr3i sbc5 3bitr4i
      19.42v sbcbii ) CFEGZAHZCIZBDJZBFDGZAHZBIZCEJZACEJZBDJABDJZCEJUEUCHBIZUAU
      GHZCIZUDUHUEUBHZCIBIUNBIZCIUKUMUNBCKUEUBBCLUOULCUOUAUFHZBIULUNUPBUEUAAMNU
      AUFBSONPUCBDQUGCEQRUIUCBDACEQTUJUGCEABDQTR $.
  $}

  ${
    $d w y z A $.  $d w x z B $.  $d w z ph $.  $d x y $.
    $( Commutative law for double class substitution.  (Contributed by NM,
       15-Nov-2005.)  (Proof shortened by Mario Carneiro, 18-Oct-2016.) $)
    sbccom $p |- ( [. A / x ]. [. B / y ]. ph
          <-> [. B / y ]. [. A / x ]. ph ) $=
      ( vw vz cv wsbc sbccomlem sbcbii bitri 3bitr3i sbcco ) ACFHZIZFEIZBDIZABG
      HZIZGDIZCEIZACEIZBDIABDIZCEIQBSIZGDIZUACOIZFEIZRUBTCOIZFEIZGDIUIGDIZFEIUF
      UHUIGFDEJUJUEGDUJPBSIZFEIUEUIULFEACBOSJKPFBESJLKUKUGFETGCDOJKMQBGDNUACFEN
      MQUCBDACFENKUAUDCEABGDNKM $.
  $}

  ${
    $d x y z $.  $d A z $.  $d B x z $.  $d V z $.  $d ph z $.
    $( Interchange class substitution and restricted quantifier.  (Contributed
       by NM, 1-Mar-2008.)  (Revised by David Abernethy, 22-Feb-2010.) $)
    sbcralt $p |- ( ( A e. V /\ F/_ y A ) ->
           ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) ) $=
      ( vz wral wsbc cv wcel wnfc wa sbcco simpl wsb wceq sbsbc nfcv wb nfralxy
      nfs1v weq sbequ12 ralbidv sbie bitr3i nfnfc1 nfcvd id nfeqd nfan1 dfsbcq2
      adantl ralbid adantll bitrid sbcied bitr3id ) ACEHZBDIUTBGJZIZGDIDFKZCDLZ
      MZABDIZCEHZUTBGDNVEVBVGGDFVCVDOVBABGPZCEHZVEVADQZMVGVBUTBGPVIUTBGRUTVIBGV
      HBCEBESABGUBUABGUCAVHCEABGUDUEUFUGVDVJVIVGTVCVDVJMVHVFCEVDVJCCDUHVDCVADVD
      CVAUIVDUJUKULVJVHVFTVDABGDUMUNUOUPUQURUS $.

    $( Interchange class substitution and restricted existential quantifier.
       (Contributed by NM, 1-Mar-2008.)  (Proof shortened by Mario Carneiro,
       13-Oct-2016.) $)
    sbcrext $p |- ( F/_ y A ->
        ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) ) $=
      ( vz wnfc cvv wcel wrex wsbc wi sbcex a1i nfcvd cv wb wa wsb wceq id 2a1i
      nfnfc1 nfeld rexlimd2 sbcco simpl sbsbc nfcv nfs1v nfrexw sbequ12 rexbidv
      bitr3i nfeqd nfan1 dfsbcq2 adantl rexbid adantll bitrid bitr3id pm5.21ndd
      sbie sbcied expcom ) CDGZDHIZACEJZBDKZABDKZCEJZVJVHLVGVIBDMNVGVKVHCECDUCZ
      VGCDHVGUAZVGCHOUDVGCPEIVKVHLABDMUBUEVHVGVJVLQVJVIBFPZKZFDKVHVGRZVLVIBFDUF
      VQVPVLFDHVHVGUGVPABFSZCEJZVQVODTZRVLVPVIBFSVSVIBFUHVIVSBFVRBCEBEUIABFUJUK
      BPVOTAVRCEABFULUMVDUNVGVTVSVLQVHVGVTRVRVKCEVGVTCVMVGCVODVGCVOOVNUOUPVTVRV
      KQVGABFDUQURUSUTVAVEVBVFVC $.
  $}

  ${
    $d y z A $.  $d x B $.  $d x y z $.  $d ph z $.  $d B z $.
    $( Interchange class substitution and restricted quantifier.  (Contributed
       by NM, 15-Nov-2005.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    sbcralg $p |- ( A e. V ->
                 ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) ) $=
      ( vz wral wsb wsbc dfsbcq2 cv wceq ralbidv nfcv nfs1v nfralxy weq sbequ12
      sbie vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGLDMUCUECEABGDKNUBU
      DBGUCBCEBEOABGPQBGRAUCCEABGSNTUA $.

    $( Interchange class substitution and restricted existential quantifier.
       (Contributed by NM, 15-Nov-2005.)  (Revised by NM, 18-Aug-2018.) $)
    sbcrex $p |- ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) $=
      ( wnfc wrex wsbc wb nfcv sbcrext ax-mp ) CDFACEGBDHABDHCEGICDJABCDEKL $.

    $( Interchange class substitution and restricted unique existential
       quantifier.  (Contributed by NM, 24-Feb-2013.) $)
    sbcreug $p |- ( A e. V ->
                 ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) ) $=
      ( vz wreu wsb wsbc dfsbcq2 cv wceq reubidv nfcv nfs1v nfreuw sbequ12 sbie
      weq vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGLDMUCUECEABGDKNUBUD
      BGUCBCEBEOABGPQBGTAUCCEABGRNSUA $.
  $}

  ${
    $d w x y A $.  $d w ph $.  $d w ps $.  $d y ch $.
    reu8nf.1 $e |- F/ x ps $.
    reu8nf.2 $e |- F/ x ch $.
    reu8nf.3 $e |- ( x = w -> ( ph <-> ch ) ) $.
    reu8nf.4 $e |- ( w = y -> ( ch <-> ps ) ) $.
    $( Restricted uniqueness using implicit substitution.  This version of
       ~ reu8 uses a nonfreeness hypothesis for ` x ` and ` ps ` instead of
       distinct variable conditions.  (Contributed by AV, 21-Jan-2022.) $)
    reu8nf $p |- ( E! x e. A ph <-> E. x e. A ( ph /\
                A. y e. A ( ps -> x = y ) ) ) $=
      ( wreu weq wi wral wa wrex nfv cbvreuw reu8 nfcv nfim nfralw nfan equcoms
      wb bicomd equequ1 imbi2d ralbidv anbi12d cbvrexw 3bitri ) ADGLCFGLCBFEMZN
      ZEGOZPZFGQABDEMZNZEGOZPZDGQACDFGAFRIJSCBFEGKTUQVAFDGCUPDIUODEGDGUABUNDHUN
      DRUBUCUDVAFRFDMZCAUPUTCAUFDFDFMACJUGUEVBUOUSEGVBUNURBFDEUHUIUJUKULUM $.
  $}

  ${
    $d y w A $.  $d w B $.  $d w ph $.  $d x y $.  $d w x $.
    sbcabel.1 $e |- F/_ x B $.
    $( Interchange class substitution and class abstraction.  (Contributed by
       NM, 5-Nov-2005.) $)
    sbcabel $p |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <->
                  { y | [. A / x ]. ph } e. B ) ) $=
      ( vw wcel cvv cab wsbc wb cv wceq wa wex wal bitrd abeq2 elex sbcexg sbcg
      sbcang sbcalg sbcbig bibi1d albidv sbcbii 3bitr4g nfcri sbcgf anbi12d syl
      exbidv df-clel ) DFIDJIZACKZEIZBDLZABDLZCKZEIZMDFUAUQHNZUROZVDEIZPZHQZBDL
      ZVDVBOZVFPZHQZUTVCUQVIVGBDLZHQVLVGHBDJUBUQVMVKHUQVMVEBDLZVFBDLZPVKVEVFBDJ
      UDUQVNVJVOVFUQCNVDIZAMZCRZBDLZVPVAMZCRZVNVJUQVSVQBDLZCRWAVQCBDJUEUQWBVTCU
      QWBVPBDLZVAMVTVPABDJUFUQWCVPVAVPBDJUCUGSUHSVEVRBDACVDTUIVACVDTUJVFBDJBHEG
      UKULUMSUOSUSVHBDHUREUPUIHVBEUPUJUN $.
  $}

  ${
    $d y A $.  $d x y B $.  $d y ph $.
    $( Restricted quantifier version of Axiom 4 of [Mendelson] p. 69.  This
       provides an axiom for a predicate calculus for a restricted domain.
       This theorem generalizes the unrestricted ~ stdpc4 and ~ spsbc .  See
       also ~ rspsbca and rspcsbela .  (Contributed by NM, 17-Nov-2006.)
       (Proof shortened by Mario Carneiro, 13-Oct-2016.) $)
    rspsbc $p |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) ) $=
      ( vy wral wsb wcel wsbc cbvralsv dfsbcq2 rspcv biimtrid ) ABDFABEGZEDFCDH
      ABCIZABEDJNOECDABECKLM $.

    $( Restricted quantifier version of Axiom 4 of [Mendelson] p. 69.
       (Contributed by NM, 14-Dec-2005.) $)
    rspsbca $p |- ( ( A e. B /\ A. x e. B ph ) -> [. A / x ]. ph ) $=
      ( wcel wral wsbc rspsbc imp ) CDEABDFABCGABCDHI $.

    $( Existence form of ~ rspsbca .  (Contributed by NM, 29-Feb-2008.)  (Proof
       shortened by Mario Carneiro, 13-Oct-2016.) $)
    rspesbca $p |- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph ) $=
      ( vy wcel wsbc wa wsb wrex dfsbcq2 rspcev cbvrexsv sylibr ) CDFABCGZHABEI
      ZEDJABDJPOECDABECKLABEDMN $.

    $( Existence form of ~ spsbc .  (Contributed by Mario Carneiro,
       18-Nov-2016.) $)
    spesbc $p |- ( [. A / x ]. ph -> E. x ph ) $=
      ( wsbc cvv wrex wex wcel sbcex rspesbca mpancom rexv sylib ) ABCDZABEFZAB
      GCEHNOABCIABCEJKABLM $.

    spesbcd.1 $e |- ( ph -> [. A / x ]. ps ) $.
    $( form of ~ spsbc .  (Contributed by Mario Carneiro, 9-Feb-2017.) $)
    spesbcd $p |- ( ph -> E. x ps ) $=
      ( wsbc wex spesbc syl ) ABCDFBCGEBCDHI $.
  $}

  ${
    $d x B $.
    sbcth2.1 $e |- ( x e. B -> ph ) $.
    $( A substitution into a theorem.  (Contributed by NM, 1-Mar-2008.)  (Proof
       shortened by Mario Carneiro, 13-Oct-2016.) $)
    sbcth2 $p |- ( A e. B -> [. A / x ]. ph ) $=
      ( wcel wral wsbc rgen rspsbc mpi ) CDFABDGABCHABDEIABCDJK $.
  $}

  ${
    ra5.1 $e |- F/ x ph $.
    $( Restricted quantifier version of Axiom 5 of [Mendelson] p. 69.  This is
       an axiom of a predicate calculus for a restricted domain.  Compare the
       unrestricted ~ stdpc5 .  (Contributed by NM, 16-Jan-2004.) $)
    ra5 $p |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) ) $=
      ( wi wral cv wcel wal df-ral bi2.04 albii bitri stdpc5 sylbi imbitrrdi )
      ABFZCDGZACHDIZBFZCJZBCDGSAUAFZCJZAUBFSTRFZCJUDRCDKUEUCCTABLMNAUACEOPBCDKQ
      $.
  $}

  ${
    $d x y A $.
    rmo2.1 $e |- F/ y ph $.
    $( Condition implying restricted at-most-one quantifier.  (Contributed by
       Jim Kingdon, 14-Jul-2018.) $)
    rmo2ilem $p |- ( E. y A. x e. A ( ph -> x = y ) -> E* x e. A ph ) $=
      ( cv wceq wi wral wex wcel wal wrmo impexp albii df-ral bitr4i exbii wmo
      wa nfv nfan mo2r df-rmo sylibr sylbir ) ABFZCFGZHZBDIZCJUGDKZATZUHHZBLZCJ
      ZABDMZUNUJCUNUKUIHZBLUJUMUQBUKAUHNOUIBDPQRUOULBSUPULBCUKACUKCUAEUBUCABDUD
      UEUF $.

    $( Condition implying restricted at-most-one quantifier.  (Contributed by
       NM, 17-Jun-2017.) $)
    rmo2i $p |- ( E. y e. A A. x e. A ( ph -> x = y ) -> E* x e. A ph ) $=
      ( cv wceq wi wral wrex wex wrmo rexex rmo2ilem syl ) ABFCFGHBDIZCDJPCKABD
      LPCDMABCDENO $.

    $( Restricted at-most-one quantifier using explicit substitution.
       (Contributed by NM, 4-Nov-2012.)  (Revised by NM, 16-Jun-2017.) $)
    rmo3 $p |- ( E* x e. A ph <->
               A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
      ( wrmo cv wcel wa wmo wsb wi wral anbi1i bitri 3bitri impexp albii df-ral
      wal weq df-rmo sban clelsb1 anbi2i an4 ancom r19.21v 3bitr2i nfv nfan mo3
      imbi1i 3bitr4i ) ABDFBGDHZAIZBJZAABCKZIZBCUAZLZCDMZBDMZABDUBUPUPBCKZIZUTL
      ZCTZBTUOVBLZBTUQVCVGVHBVGCGDHZUOVALZLZCTVJCDMVHVFVKCVFVIUOIZUSIZUTLVLVALV
      KVEVMUTVEUPVIURIZIUOVIIZUSIVMVDVNUPVDUOBCKZURIVNUOABCUCVPVIURBCDUDNOUEUOA
      VIURUFVOVLUSUOVIUGNPUMVLUSUTQVIUOVAQPRVJCDSUOVACDUHUIRUPBCUOACUOCUJEUKULV
      BBDSUNO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ps $.  $d x ch $.
    rmoi.b $e |- ( x = B -> ( ph <-> ps ) ) $.
    rmoi.c $e |- ( x = C -> ( ph <-> ch ) ) $.
    $( Consequence of "at most one", using implicit substitution.  (Contributed
       by NM, 2-Jan-2015.)  (Revised by NM, 16-Jun-2017.) $)
    rmob $p |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) ->
        ( B = C <-> ( C e. A /\ ch ) ) ) $=
      ( wrmo cv wcel wa wmo wceq wb df-rmo simprl eleq1 anbi12d syl5ibcom simpl
      wi a1i simplrl simpr simpll simplrr mob syl212anc ex pm5.21ndd sylanb ) A
      DEJDKZELZAMZDNZFELZBMZFGOZGELZCMZPZADEQUQUSMZVAUTVBVDURUTVAUQURBRFGESUAVB
      VAUCVDVACUBUDVDVAVCVDVAMURVAUQURBVCUQURBVAUEZVDVAUFUQUSVAUGVEUQURBVAUHUPU
      SVBDFGEEUNFOUOURABUNFESHTUNGOUOVAACUNGESITUIUJUKULUM $.

    $( Consequence of "at most one", using implicit substitution.  (Contributed
       by NM, 4-Nov-2012.)  (Revised by NM, 16-Jun-2017.) $)
    rmoi $p |- ( ( E* x e. A ph
          /\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C ) $=
      ( wrmo wcel wa wceq rmob biimp3ar ) ADEJFEKBLFGMGEKCLABCDEFGHINO $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Proper substitution of classes for sets into classes
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c [_ $.  $( Underlined left bracket $)
  $c ]_ $.  $( Underlined right bracket $)

  $( Extend class notation to include the proper substitution of a class for a
     set into another class. $)
  csb $a class [_ A / x ]_ B $.

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( Define the proper substitution of a class for a set into another class.
       The underlined brackets distinguish it from the substitution into a wff,
       ~ wsbc , to prevent ambiguity.  Theorem ~ sbcel1g shows an example of
       how ambiguity could arise if we didn't use distinguished brackets.
       Theorem ~ sbccsbg recreates substitution into a wff from this
       definition.  (Contributed by NM, 10-Nov-2005.) $)
    df-csb $a |- [_ A / x ]_ B = { y | [. A / x ]. y e. B } $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y $.
    $( Alternate expression for the proper substitution into a class, without
       referencing substitution into a wff.  Note that ` x ` can be free in
       ` B ` but cannot occur in ` A ` .  (Contributed by NM, 2-Dec-2013.) $)
    csb2 $p |- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) } $=
      ( csb cv wcel wsbc cab wceq wa wex df-csb sbc5 abbii eqtri ) ACDEBFDGZACH
      ZBIAFCJQKALZBIABCDMRSBQACNOP $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.
    $( Analog of ~ dfsbcq for proper substitution into a class.  (Contributed
       by NM, 10-Nov-2005.) $)
    csbeq1 $p |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C ) $=
      ( vy wceq cv wcel wsbc cab csb dfsbcq abbidv df-csb 3eqtr4g ) BCFZEGDHZAB
      IZEJQACIZEJABDKACDKPRSEQABCLMAEBDNAECDNO $.
  $}

  ${
    $d x y z $.  $d z A $.  $d z C $.  $d z D $.
    cbvcsbw.1 $e |- F/_ y C $.
    cbvcsbw.2 $e |- F/_ x D $.
    cbvcsbw.3 $e |- ( x = y -> C = D ) $.
    $( Version of ~ cbvcsb with a disjoint variable condition.  (Contributed by
       GG, 10-Jan-2024.) $)
    cbvcsbw $p |- [_ A / x ]_ C = [_ A / y ]_ D $=
      ( vz cv wcel wsbc cab csb nfcri weq eleq2d cbvsbcw abbii df-csb 3eqtr4i )
      IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE
      TUA $.
  $}

  ${
    $d x z $.  $d y z $.  $d z A $.  $d z C $.  $d z D $.
    cbvcsb.1 $e |- F/_ y C $.
    cbvcsb.2 $e |- F/_ x D $.
    cbvcsb.3 $e |- ( x = y -> C = D ) $.
    $( Change bound variables in a class substitution.  Interestingly, this
       does not require any bound variable conditions on ` A ` .  (Contributed
       by Jeff Hankins, 13-Sep-2009.)  (Revised by Mario Carneiro,
       11-Dec-2016.) $)
    cbvcsb $p |- [_ A / x ]_ C = [_ A / y ]_ D $=
      ( vz cv wcel wsbc cab csb nfcri weq eleq2d cbvsbc abbii df-csb 3eqtr4i )
      IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE
      TUA $.
  $}

  ${
    $d x y $.  $d y B $.  $d x C $.
    cbvcsbv.1 $e |- ( x = y -> B = C ) $.
    $( Change the bound variable of a proper substitution into a class using
       implicit substitution.  (Contributed by NM, 30-Sep-2008.)  (Revised by
       Mario Carneiro, 13-Oct-2016.) $)
    cbvcsbv $p |- [_ A / x ]_ B = [_ A / y ]_ C $=
      ( nfcv cbvcsb ) ABCDEBDGAEGFH $.
  $}

  ${
    csbeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for proper substitution into a class.  (Contributed
       by NM, 3-Dec-2005.) $)
    csbeq1d $p |- ( ph -> [_ A / x ]_ C = [_ B / x ]_ C ) $=
      ( wceq csb csbeq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
  $}

  ${
    $d x y $.  $d y A $.
    $( Analog of ~ sbid for proper substitution into a class.  (Contributed by
       NM, 10-Nov-2005.) $)
    csbid $p |- [_ x / x ]_ A = A $=
      ( vy cv csb wcel wsbc cab df-csb sbcid abbii abid2 3eqtri ) AADZBECDBFZAN
      GZCHOCHBACNBIPOCOAJKCBLM $.
  $}

  $( Equality theorem for proper substitution into a class.  (Contributed by
     NM, 10-Nov-2005.) $)
  csbeq1a $p |- ( x = A -> B = [_ A / x ]_ B ) $=
    ( cv wceq csb csbid csbeq1 eqtr3id ) ADZBECAJCFABCFACGAJBCHI $.

  ${
    $d z A $.  $d y z B $.  $d x z $.
    $( Composition law for chained substitutions into a class.

       Use the weaker ~ csbcow when possible.  (Contributed by NM,
       10-Nov-2005.)  (New usage is discouraged.) $)
    csbco $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $=
      ( vz cv csb wcel wsbc cab df-csb abeq2i sbcbii sbcco bitri abbii 3eqtr4i
      ) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM
      UCABCNOPBECTKAECDKQ $.
  $}

  ${
    $d z A $.  $d x y z w $.  $d y z w B $.
    $( Composition law for chained substitutions into a class.  Version of
       ~ csbco with a disjoint variable condition, which requires fewer axioms.
       (Contributed by NM, 10-Nov-2005.)  (Revised by GG, 25-Aug-2024.) $)
    csbcow $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $=
      ( vz vw cv csb wcel wsbc cab df-csb weq wal wsb sb6 df-clab df-sbc bitri
      wi abeq2i sbcbii equequ2 imbi1d albidv sbiev bitr4i 3bitr4i eqriv 3eqtr4i
      nfv eleq2i abbii ) EGZABGZDHZIZBCJZEKUNDIZACJZEKBCUPHACDHURUTEURUSAUOJZBC
      JZUTUQVABCVAEUPAEUODLUAUBCABMZUSTZANZBKZIZCUSAKZIVBUTVFVHCFVFVHVEBFOZUSAF
      OZFGZVFIVKVHIVIAFMZUSTZANZVJVEVNBFVNBUKBFMZVDVMAVOVCVLUSBFAUCUDUEUFUSAFPU
      GVEFBQUSFAQUHUIULVBCVABKZIVGVABCRVPVFCVAVEBVAUOVHIZVEUSAUORVQUSABOVEUSBAQ
      USABPSSUMULSUSACRUHSUMBECUPLAECDLUJ $.
    $( $j usage 'csbcow' avoids 'ax-io' 'ax-10' 'ax-i12' 'ax-bndl'; $)
  $}

  ${
    $d y A $.  $d y B $.  $d y V $.  $d x y $.
    $( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
       free).  (Contributed by Mario Carneiro, 14-Oct-2016.) $)
    csbtt $p |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B ) $=
      ( vy wcel wnfc wa csb cv wsbc cab df-csb wnf wb nfcr sbctt sylan2 abbi1dv
      eqtrid ) BDFZACGZHZABCIEJCFZABKZELCAEBCMUCUEECUBUAUDANUEUDOAECPUDABDQRST
      $.
  $}

  ${
    csbconstgf.1 $e |- F/_ x B $.
    $( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
       free).  (Contributed by NM, 10-Nov-2005.) $)
    csbconstgf $p |- ( A e. V -> [_ A / x ]_ B = B ) $=
      ( wcel wnfc csb wceq csbtt mpan2 ) BDFACGABCHCIEABCDJK $.
  $}

  ${
    $d B x $.
    $( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
       free). ~ csbconstgf with distinct variable requirement.  (Contributed by
       Alan Sare, 22-Jul-2012.) $)
    csbconstg $p |- ( A e. V -> [_ A / x ]_ B = B ) $=
      ( nfcv csbconstgf ) ABCDACEF $.
  $}

  ${
    $d x y z $.  $d y z A $.  $d y z B $.  $d y z C $.
    $( Distribute proper substitution through a membership relation.
       (Contributed by NM, 10-Nov-2005.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    sbcel12g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
                   [_ A / x ]_ B e. [_ A / x ]_ C ) ) $=
      ( vy vz wcel wsbc cv cab csb wsb dfsbcq2 abbidv eleq12d nfs1v nfab df-csb
      sbab wceq nfel weq sbie vtoclbg eleq12i bitr4di ) BEHCDHZABIZFJZCHZABIZFK
      ZUJDHZABIZFKZHZABCLZABDLZHUHAGMUKAGMZFKZUNAGMZFKZHZUIUQGBEUHAGBNGJBUAZVAU
      MVCUPVEUTULFUKAGBNOVEVBUOFUNAGBNOPUHVDAGAVAVCUTAFUKAGQRVBAFUNAGQRUBAGUCCV
      ADVCAGFCTAGFDTPUDUEURUMUSUPAFBCSAFBDSUFUG $.

    $( Distribute proper substitution through an equality relation.
       (Contributed by NM, 10-Nov-2005.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    sbceqg $p |- ( A e. V -> ( [. A / x ]. B = C <->
                   [_ A / x ]_ B = [_ A / x ]_ C ) ) $=
      ( vy vz wcel wceq wsbc cab csb wsb dfsbcq2 abbidv eqeq12d nfs1v nfab sbab
      cv nfeq weq sbie vtoclbg df-csb eqeq12i bitr4di ) BEHCDIZABJZFTZCHZABJZFK
      ZUJDHZABJZFKZIZABCLZABDLZIUHAGMUKAGMZFKZUNAGMZFKZIZUIUQGBEUHAGBNGTBIZVAUM
      VCUPVEUTULFUKAGBNOVEVBUOFUNAGBNOPUHVDAGAVAVCUTAFUKAGQRVBAFUNAGQRUAAGUBCVA
      DVCAGFCSAGFDSPUCUDURUMUSUPAFBCUEAFBDUEUFUG $.
  $}

  $( Distribute proper substitution through negated membership.  (Contributed
     by Andrew Salmon, 18-Jun-2011.) $)
  sbcnel12g $p |- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/
                    [_ A / x ]_ C ) ) $=
    ( wcel wnel wsbc wn csb wb df-nel sbcbii a1i sbcel12g notbid bitr4di 3bitrd
    sbcng ) BEFZCDGZABHZCDFZIZABHZUCABHZIZABCJZABDJZGZUBUEKTUAUDABCDLMNUCABESTU
    GUHUIFZIUJTUFUKABCDEOPUHUILQR $.

  $( Distribute proper substitution through an inequality.  (Contributed by
     Andrew Salmon, 18-Jun-2011.) $)
  sbcne12g $p |- ( A e. V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/=
                   [_ A / x ]_ C ) ) $=
    ( wcel wne wsbc csb wb wceq sbceqg notbid df-ne sbcbii sbcng bitrid bibi12d
    wn a1i mpbird ) BEFZCDGZABHZABCIZABDIZGZJCDKZABHZSZUEUFKZSZJUBUIUKABCDELMUB
    UDUJUGULUDUHSZABHUBUJUCUMABCDNOUHABEPQUGULJUBUEUFNTRUA $.

  ${
    $d x C $.
    $( Move proper substitution in and out of a membership relation.  Note that
       the scope of ` [. A / x ]. ` is the wff ` B e. C ` , whereas the scope
       of ` [_ A / x ]_ ` is the class ` B ` .  (Contributed by NM,
       10-Nov-2005.) $)
    sbcel1g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
                    [_ A / x ]_ B e. C ) ) $=
      ( wcel wsbc csb sbcel12g csbconstg eleq2d bitrd ) BEFZCDFABGABCHZABDHZFND
      FABCDEIMODNABDEJKL $.

    $( Move proper substitution to first argument of an equality.  (Contributed
       by NM, 30-Nov-2005.) $)
    sbceq1g $p |- ( A e. V -> ( [. A / x ]. B = C <->
                    [_ A / x ]_ B = C ) ) $=
      ( wcel wceq wsbc csb sbceqg csbconstg eqeq2d bitrd ) BEFZCDGABHABCIZABDIZ
      GODGABCDEJNPDOABDEKLM $.
  $}

  ${
    $d x B $.
    $( Move proper substitution in and out of a membership relation.
       (Contributed by NM, 14-Nov-2005.) $)
    sbcel2g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
                    B e. [_ A / x ]_ C ) ) $=
      ( wcel wsbc csb sbcel12g csbconstg eleq1d bitrd ) BEFZCDFABGABCHZABDHZFCO
      FABCDEIMNCOABCEJKL $.

    $( Move proper substitution to second argument of an equality.
       (Contributed by NM, 30-Nov-2005.) $)
    sbceq2g $p |- ( A e. V -> ( [. A / x ]. B = C <->
                    B = [_ A / x ]_ C ) ) $=
      ( wcel wceq wsbc csb sbceqg csbconstg eqeq1d bitrd ) BEFZCDGABHABCIZABDIZ
      GCPGABCDEJNOCPABCEKLM $.
  $}

  ${
    $d y z A $.  $d x z B $.  $d z C $.  $d x y $.
    $( Commutative law for double substitution into a class.  (Contributed by
       NM, 14-Nov-2005.) $)
    csbcomg $p |- ( ( A e. V /\ B e. W ) ->
                 [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C ) $=
      ( vz wcel cvv csb wceq elex wsbc wb sbcel2g sbcbidv adantl adantr 3bitr3d
      wa cv sbccom a1i eqrdv syl2an ) CFICJIZDJIZACBDEKZKZBDACEKZKZLDGICFMDGMUG
      UHUAZHUJULUMHUBZUIIZACNZUNUKIZBDNZUNUJIZUNULIZUMUNEIZBDNZACNZVAACNZBDNZUP
      URVCVEOUMVAABCDUCUDUHVCUPOUGUHVBUOACBDUNEJPQRUGVEUROUHUGVDUQBDACUNEJPQSTU
      GUPUSOUHACUNUIJPSUHURUTOUGBDUNUKJPRTUEUF $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.
    $( Substituting into equivalent classes gives equivalent results.
       (Contributed by Giovanni Mascellani, 9-Apr-2018.) $)
    csbeq2 $p |- ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) $=
      ( vy wceq wal cv wcel cab csb wb eleq2 alimi sbcbi2 abbidv df-csb 3eqtr4g
      wsbc syl ) CDFZAGZEHZCIZABSZEJUCDIZABSZEJABCKABDKUBUEUGEUBUDUFLZAGUEUGLUA
      UHACDUCMNUDUFABOTPAEBCQAEBDQR $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.  $d y ph $.
    csbeq2d.1 $e |- F/ x ph $.
    csbeq2d.2 $e |- ( ph -> B = C ) $.
    $( Formula-building deduction for class substitution.  (Contributed by NM,
       22-Nov-2005.)  (Revised by Mario Carneiro, 1-Sep-2015.) $)
    csbeq2d $p |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C ) $=
      ( vy cv wcel wsbc cab csb eleq2d sbcbid abbidv df-csb 3eqtr4g ) AHIZDJZBC
      KZHLSEJZBCKZHLBCDMBCEMAUAUCHATUBBCFADESGNOPBHCDQBHCEQR $.
  $}

  ${
    $d x ph $.
    csbeq2dv.1 $e |- ( ph -> B = C ) $.
    $( Formula-building deduction for class substitution.  (Contributed by NM,
       10-Nov-2005.)  (Revised by Mario Carneiro, 1-Sep-2015.) $)
    csbeq2dv $p |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C ) $=
      ( nfv csbeq2d ) ABCDEABGFH $.
  $}

  ${
    csbeq2i.1 $e |- B = C $.
    $( Formula-building inference for class substitution.  (Contributed by NM,
       10-Nov-2005.)  (Revised by Mario Carneiro, 1-Sep-2015.) $)
    csbeq2i $p |- [_ A / x ]_ B = [_ A / x ]_ C $=
      ( csb wceq wtru a1i csbeq2dv mptru ) ABCFABDFGHABCDCDGHEIJK $.
  $}

  ${
    $d y z A $.  $d x y z $.
    $( The proper substitution of a class for setvar variable results in the
       class (if the class exists).  (Contributed by NM, 10-Nov-2005.) $)
    csbvarg $p |- ( A e. V -> [_ A / x ]_ x = A ) $=
      ( vz vy wcel cvv cv csb wceq elex wsbc cab df-csb sbcel2gv abbi1dv eqtrid
      vex ax-mp csbeq2i csbco 3eqtr3i syl ) BCFBGFZABAHZIZBJBCKUDUFDHZEHZFEBLZD
      MZBEBAUHUEIZIEBUHIUFUJEBUKUHUHGFZUKUHJERULUKUGUEFAUHLZDMUHADUHUENULUMDUHA
      UGUHGOPQSTAEBUEUAEDBUHNUBUDUIDBEUGBGOPQUC $.
  $}

  ${
    $d x y $.
    $( Substitution into a wff expressed in terms of substitution into a class.
       (Contributed by NM, 15-Aug-2007.) $)
    sbccsbg $p |- ( A e. V ->
                 ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } ) ) $=
      ( wsbc cv cab wcel csb abid sbcbii sbcel2g bitr3id ) ABDFCGZACHZIZBDFDEIO
      BDPJIQABDACKLBDOPEMN $.
  $}

  $( Substitution into a wff expressed in using substitution into a class.
     (Contributed by NM, 27-Nov-2005.) $)
  sbccsb2g $p |- ( A e. V ->
               ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) ) $=
    ( wsbc cv cab wcel csb abid sbcbii sbcel12g csbvarg eleq1d bitrd bitr3id )
    ABCEBFZABGZHZBCEZCDHZCBCRIZHZSABCABJKUATBCQIZUBHUCBCQRDLUAUDCUBBCDMNOP $.

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y ph $.
    nfcsb1d.1 $e |- ( ph -> F/_ x A ) $.
    $( Bound-variable hypothesis builder for substitution into a class.
       (Contributed by Mario Carneiro, 12-Oct-2016.) $)
    nfcsb1d $p |- ( ph -> F/_ x [_ A / x ]_ B ) $=
      ( vy csb cv wcel wsbc cab df-csb nfv nfsbc1d nfabd nfcxfrd ) ABBCDGFHDIZB
      CJZFKBFCDLARBFAFMAQBCENOP $.
  $}

  ${
    nfcsb1.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for substitution into a class.
       (Contributed by Mario Carneiro, 12-Oct-2016.) $)
    nfcsb1 $p |- F/_ x [_ A / x ]_ B $=
      ( csb wnfc wtru a1i nfcsb1d mptru ) AABCEFGABCABFGDHIJ $.
  $}

  ${
    $d x A $.
    $( Bound-variable hypothesis builder for substitution into a class.
       (Contributed by NM, 17-Aug-2006.)  (Revised by Mario Carneiro,
       12-Oct-2016.) $)
    nfcsb1v $p |- F/_ x [_ A / x ]_ B $=
      ( nfcv nfcsb1 ) ABCABDE $.
  $}

  ${
    $d x y $.
    nfsbcdw.1 $e |- F/ y ph $.
    nfsbcdw.2 $e |- ( ph -> F/_ x A ) $.
    nfsbcdw.3 $e |- ( ph -> F/ x ps ) $.
    $( Version of ~ nfsbcd with a disjoint variable condition.  (Contributed by
       NM, 23-Nov-2005.)  (Revised by GG, 10-Jan-2024.) $)
    nfsbcdw $p |- ( ph -> F/ x [. A / y ]. ps ) $=
      ( wsbc cab wcel df-sbc nfabdw nfeld nfxfrd ) BDEIEBDJZKACBDELACEPGABCDFHM
      NO $.
  $}

  ${
    $d x y $.
    nfsbcw.1 $e |- F/_ x A $.
    nfsbcw.2 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for class substitution.  Version of
       ~ nfsbc with a disjoint variable condition, which in the future may make
       it possible to reduce axiom usage.  (Contributed by NM, 7-Sep-2014.)
       (Revised by GG, 10-Jan-2024.) $)
    nfsbcw $p |- F/ x [. A / y ]. ph $=
      ( wsbc wnf wtru nftru wnfc a1i nfsbcdw mptru ) ACDGBHIABCDCJBDKIELABHIFLM
      N $.
  $}

  ${
    $d x z $.  $d y z $.  $d z A $.  $d z B $.  $d z ph $.
    nfcsbd.1 $e |- F/ y ph $.
    nfcsbd.2 $e |- ( ph -> F/_ x A ) $.
    nfcsbd.3 $e |- ( ph -> F/_ x B ) $.
    $( Deduction version of ~ nfcsb .  (Contributed by NM, 21-Nov-2005.)
       (Revised by Mario Carneiro, 12-Oct-2016.) $)
    nfcsbd $p |- ( ph -> F/_ x [_ A / y ]_ B ) $=
      ( vz csb cv wcel wsbc cab df-csb nfv nfcrd nfsbcd nfabd nfcxfrd ) ABCDEJI
      KELZCDMZINCIDEOAUBBIAIPAUABCDFGABIEHQRST $.
  $}

  ${
    $d x y z $.  $d A z $.  $d B z $.
    nfcsbw.1 $e |- F/_ x A $.
    nfcsbw.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for substitution into a class.
       Version of ~ nfcsb with a disjoint variable condition.  (Contributed by
       Mario Carneiro, 12-Oct-2016.)  (Revised by GG, 10-Jan-2024.) $)
    nfcsbw $p |- F/_ x [_ A / y ]_ B $=
      ( vz csb wnfc wtru cv wcel wsbc cab df-csb nftru a1i nfcrd nfsbcdw nfabdw
      nfcxfrd mptru ) ABCDHZIJAUCGKDLZBCMZGNBGCDOJUEAGGPJUDABCBPACIJEQJAGDADIJF
      QRSTUAUB $.
  $}

  ${
    nfcsb.1 $e |- F/_ x A $.
    nfcsb.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for substitution into a class.
       (Contributed by Mario Carneiro, 12-Oct-2016.) $)
    nfcsb $p |- F/_ x [_ A / y ]_ B $=
      ( csb wnfc wtru nftru a1i nfcsbd mptru ) ABCDGHIABCDBJACHIEKADHIFKLM $.
  $}

  ${
    $d x y $.
    csbhypf.1 $e |- F/_ x A $.
    csbhypf.2 $e |- F/_ x C $.
    csbhypf.3 $e |- ( x = A -> B = C ) $.
    $( Introduce an explicit substitution into an implicit substitution
       hypothesis.  See ~ sbhypf for class substitution version.  (Contributed
       by NM, 19-Dec-2008.) $)
    csbhypf $p |- ( y = A -> [_ y / x ]_ B = C ) $=
      ( cv wceq wi csb nfeq2 nfcsb1v nfeq nfim eqeq1 csbeq1a eqeq1d imbi12d
      chvar ) AIZCJZDEJZKBIZCJZAUEDLZEJZKABUFUHAAUECFMAUGEAUEDNGOPUBUEJZUCUFUDU
      HUBUECQUIDUGEAUEDRSTHUA $.
  $}

  ${
    $d x A $.
    $( Conversion of implicit substitution to explicit substitution into a
       class.  (Closed theorem version of ~ csbiegf .)  (Contributed by NM,
       11-Nov-2005.) $)
    csbiebt $p |- ( ( A e. V /\ F/_ x C ) ->
                 ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) ) $=
      ( wcel cvv wnfc cv wceq wi wal csb wb elex wa wsbc adantl a1i nfeqd spsbc
      adantr simpl biimt csbeq1a eqeq1d bitr3d nfv nfnfc1 nfcsb1v simpr sbciedf
      nfan sylibd id nfan1 biimprcd alrimi ex impbid sylan ) BEFBGFZADHZAIBJZCD
      JZKZALZABCMZDJZNBEOVBVCPZVGVIVJVGVFABQZVIVBVGVKKVCVFABGUAUBVJVFVIABGVBVCU
      CVDVFVINVJVDVEVFVIVDVEUDVDCVHDABCUEUFZUGRVBVCAVBAUHADUIZUMVJAVHDAVHHZVJAB
      CUJZSVBVCUKTULUNVCVIVGKVBVCVIVGVCVIPVFAVCVIAVMVCAVHDVNVCVOSVCUOTUPVIVFVCV
      DVEVIVLUQRURUSRUTVA $.

    csbiedf.1 $e |- F/ x ph $.
    csbiedf.2 $e |- ( ph -> F/_ x C ) $.
    csbiedf.3 $e |- ( ph -> A e. V ) $.
    csbiedf.4 $e |- ( ( ph /\ x = A ) -> B = C ) $.
    $( Conversion of implicit substitution to explicit substitution into a
       class.  (Contributed by Mario Carneiro, 13-Oct-2016.) $)
    csbiedf $p |- ( ph -> [_ A / x ]_ B = C ) $=
      ( cv wceq wi wal csb ex alrimi wcel wnfc wb csbiebt syl2anc mpbid ) ABKCL
      ZDELZMZBNZBCDOELZAUFBGAUDUEJPQACFRBESUGUHTIHBCDEFUAUBUC $.
  $}

  ${
    $d x A $.
    csbieb.1 $e |- A e. _V $.
    csbieb.2 $e |- F/_ x C $.
    $( Bidirectional conversion between an implicit class substitution
       hypothesis ` x = A -> B = C ` and its explicit substitution equivalent.
       (Contributed by NM, 2-Mar-2008.) $)
    csbieb $p |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) $=
      ( cvv wcel wnfc cv wceq wi wal csb wb csbiebt mp2an ) BGHADIAJBKCDKLAMABC
      NDKOEFABCDGPQ $.
  $}

  ${
    $d a x A $.  $d a B $.  $d a C $.
    csbiebg.2 $e |- F/_ x C $.
    $( Bidirectional conversion between an implicit class substitution
       hypothesis ` x = A -> B = C ` and its explicit substitution equivalent.
       (Contributed by NM, 24-Mar-2013.)  (Revised by Mario Carneiro,
       11-Dec-2016.) $)
    csbiebg $p |- ( A e. V ->
          ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) ) $=
      ( va cv wceq wal csb eqeq2 imbi1d albidv csbeq1 eqeq1d vex csbieb vtoclbg
      wi ) AHZGHZIZCDIZTZAJAUBCKZDIUABIZUDTZAJABCKZDIGBEUBBIZUEUHAUJUCUGUDUBBUA
      LMNUJUFUIDAUBBCOPAUBCDGQFRS $.
  $}

  ${
    $d x A $.  $d x V $.
    csbiegf.1 $e |- ( A e. V -> F/_ x C ) $.
    csbiegf.2 $e |- ( x = A -> B = C ) $.
    $( Conversion of implicit substitution to explicit substitution into a
       class.  (Contributed by NM, 11-Nov-2005.)  (Revised by Mario Carneiro,
       13-Oct-2016.) $)
    csbiegf $p |- ( A e. V -> [_ A / x ]_ B = C ) $=
      ( wcel cv wceq wi wal csb ax-gen wnfc wb csbiebt mpdan mpbii ) BEHZAIBJCD
      JKZALZABCMDJZUAAGNTADOUBUCPFABCDEQRS $.
  $}

  ${
    $d x A $.
    csbief.1 $e |- A e. _V $.
    csbief.2 $e |- F/_ x C $.
    csbief.3 $e |- ( x = A -> B = C ) $.
    $( Conversion of implicit substitution to explicit substitution into a
       class.  (Contributed by NM, 26-Nov-2005.)  (Revised by Mario Carneiro,
       13-Oct-2016.) $)
    csbief $p |- [_ A / x ]_ B = C $=
      ( cvv wcel csb wceq wnfc a1i csbiegf ax-mp ) BHIZABCJDKEABCDHADLPFMGNO $.
  $}

  ${
    $d x A $.  $d x C $.
    csbie.1 $e |- A e. _V $.
    csbie.2 $e |- ( x = A -> B = C ) $.
    $( Conversion of implicit substitution to explicit substitution into a
       class.  (Contributed by AV, 2-Dec-2019.) $)
    csbie $p |- [_ A / x ]_ B = C $=
      ( nfcv csbief ) ABCDEADGFH $.
  $}

  ${
    $d x A $.  $d x C $.  $d x ph $.
    csbied.1 $e |- ( ph -> A e. V ) $.
    csbied.2 $e |- ( ( ph /\ x = A ) -> B = C ) $.
    $( Conversion of implicit substitution to explicit substitution into a
       class.  (Contributed by Mario Carneiro, 2-Dec-2014.)  (Revised by Mario
       Carneiro, 13-Oct-2016.) $)
    csbied $p |- ( ph -> [_ A / x ]_ B = C ) $=
      ( nfv nfcvd csbiedf ) ABCDEFABIABEJGHK $.
  $}

  ${
    $d x A $.  $d x ph $.  $d x D $.
    csbied2.1 $e |- ( ph -> A e. V ) $.
    csbied2.2 $e |- ( ph -> A = B ) $.
    csbied2.3 $e |- ( ( ph /\ x = B ) -> C = D ) $.
    $( Conversion of implicit substitution to explicit class substitution,
       deduction form.  (Contributed by Mario Carneiro, 2-Jan-2017.) $)
    csbied2 $p |- ( ph -> [_ A / x ]_ C = D ) $=
      ( cv wceq id sylan9eqr syldan csbied ) ABCEFGHABKZCLZQDLEFLRAQCDRMINJOP
      $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y D $.
    csbie2t.1 $e |- A e. _V $.
    csbie2t.2 $e |- B e. _V $.
    $( Conversion of implicit substitution to explicit substitution into a
       class (closed form of ~ csbie2 ).  (Contributed by NM, 3-Sep-2007.)
       (Revised by Mario Carneiro, 13-Oct-2016.) $)
    csbie2t $p |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) ->
                  [_ A / x ]_ [_ B / y ]_ C = D ) $=
      ( cv wceq wa wi wal csb cvv nfa1 nfcvd wcel a1i csbiedf nfa2 nfv nfan sps
      sp impl ) AICJZBIDJZKEFJZLZBMZAMZACBDENFOUKAPULAFQCORULGSULUGKZBDEFOULUGB
      UJBAUAUGBUBUCUMBFQDORUMHSULUGUHUIUKUJAUJBUEUDUFTT $.

    csbie2.3 $e |- ( ( x = A /\ y = B ) -> C = D ) $.
    $( Conversion of implicit substitution to explicit substitution into a
       class.  (Contributed by NM, 27-Aug-2007.) $)
    csbie2 $p |- [_ A / x ]_ [_ B / y ]_ C = D $=
      ( cv wceq wa wi wal csb gen2 csbie2t ax-mp ) AJCKBJDKLEFKMZBNANACBDEOOFKS
      ABIPABCDEFGHQR $.
  $}

  ${
    $d x y z $.  $d A y z $.  $d B y z $.  $d C x $.  $d D y z $.  $d V z $.
    csbie2g.1 $e |- ( x = y -> B = C ) $.
    csbie2g.2 $e |- ( y = A -> C = D ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       This version of ~ sbcie avoids a disjointness condition on ` x ` and
       ` A ` by substituting twice.  (Contributed by Mario Carneiro,
       11-Nov-2016.) $)
    csbie2g $p |- ( A e. V -> [_ A / x ]_ B = D ) $=
      ( vz wcel csb cv wsbc cab df-csb wceq eleq2d sbcie2g abbi1dv eqtrid ) CGK
      ZACDLJMZDKZACNZJOFAJCDPUBUEJFUDUCEKUCFKABCGAMBMZQDEUCHRUFCQEFUCIRSTUA $.
  $}

  ${
    $d x z $.  $d y z $.  $d z A $.  $d z B $.  $d z C $.  $d z ph $.
    $( Nest the composition of two substitutions.  (Contributed by Mario
       Carneiro, 11-Nov-2016.) $)
    sbcnestgf $p |- ( ( A e. V /\ A. y F/ x ph ) ->
         ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $=
      ( vz wcel wnf wal wsbc csb wb cv wi wceq dfsbcq syl cvv a1i csbeq1 imbi2d
      bibi12d vex csbeq1a adantl nfnf1 nfal nfa1 wnfc nfcsb1v sp nfsbcd sbciedf
      vtoclg imp ) DFHABIZCJZACEKZBDKZACBDELZKZMZURUSBGNZKZACBVDELZKZMZOURVCOGD
      FVDDPZVHVCURVIVEUTVGVBUSBVDDQVIVFVAPVGVBMBVDDEUAACVFVAQRUCUBURUSVGBVDSVDS
      HURGUDTBNVDPZUSVGMZURVJEVFPVKBVDEUEACEVFQRUFUQBCABUGUHURABCVFUQCUIBVFUJUR
      BVDEUKTUQCULUMUNUOUP $.

    $( Nest the composition of two substitutions.  (Contributed by NM,
       23-Nov-2005.)  (Proof shortened by Mario Carneiro, 10-Nov-2016.) $)
    csbnestgf $p |- ( ( A e. V /\ A. y F/_ x C ) ->
         [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
      ( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb abeq2i wb nfcr
      sbcbii wnf alimi sbcnestgf sylan2 bitrid abbidv sylan 3eqtr4g ) CFHZAEIZB
      JZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVBG
      URUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUDZBJVGVBUAUMVHBAGEUBUEUTABCDPU
      FUGUHUIUJAGCUPSBGVAESUK $.

    $d x ph $.
    $( Nest the composition of two substitutions.  (Contributed by NM,
       27-Nov-2005.)  (Proof shortened by Mario Carneiro, 11-Nov-2016.) $)
    sbcnestg $p |- ( A e. V ->
          ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $=
      ( wcel wnf wal wsbc csb wb nfv ax-gen sbcnestgf mpan2 ) DFGABHZCIACEJBDJA
      CBDEKJLQCABMNABCDEFOP $.

    $d x C $.
    $( Nest the composition of two substitutions.  (Contributed by NM,
       23-Nov-2005.)  (Proof shortened by Mario Carneiro, 10-Nov-2016.) $)
    csbnestg $p |- ( A e. V ->
          [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
      ( wcel wnfc wal csb wceq nfcv ax-gen csbnestgf mpan2 ) CFGAEHZBIACBDEJJBA
      CDJEJKPBAELMABCDEFNO $.
  $}

  ${
    $d x y $.  $d y C $.
    $( Nest the composition of two substitutions.  (Contributed by NM,
       23-May-2006.)  (Proof shortened by Mario Carneiro, 11-Nov-2016.) $)
    csbnest1g $p |- ( A e. V ->
         [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C ) $=
      ( vy wcel cv csb wnfc wceq nfcsb1v ax-gen csbnestgf mpan2 csbeq2i 3eqtr3g
      wal csbco ) BEGZABFCAFHZDIZIZIZFABCIZUBIZABACDIZIAUEDITAUBJZFRUDUFKUHFAUA
      DLMAFBCUBENOABUCUGAFCDSPAFUEDSQ $.
  $}

  ${
    $d x A $.
    $( Idempotent law for class substitutions.  (Contributed by NM,
       1-Mar-2008.) $)
    csbidmg $p |- ( A e. V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B ) $=
      ( wcel cvv csb wceq elex csbnest1g csbconstg csbeq1d eqtrd syl ) BDEBFEZA
      BABCGZGZPHBDIOQAABBGZCGPABBCFJOARBCABBFKLMN $.
  $}

  ${
    $d x A $.  $d x ph $.  $d x C $.  $d x D $.
    sbcco3g.1 $e |- ( x = A -> B = C ) $.
    $( Composition of two substitutions.  (Contributed by NM, 27-Nov-2005.)
       (Revised by Mario Carneiro, 11-Nov-2016.) $)
    sbcco3g $p |- ( A e. V ->
         ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) ) $=
      ( wcel wsbc csb sbcnestg cvv wceq wb elex nfcvd csbiegf dfsbcq 3syl bitrd
      ) DGIZACEJBDJACBDEKZJZACFJZABCDEGLUBDMIZUCFNUDUEODGPBDEFMUFBFQHRACUCFSTUA
      $.

    $( Composition of two class substitutions.  (Contributed by NM,
       27-Nov-2005.)  (Revised by Mario Carneiro, 11-Nov-2016.) $)
    csbco3g $p |- ( A e. V ->
                 [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D ) $=
      ( wcel csb csbnestg cvv wceq elex nfcvd csbiegf syl csbeq1d eqtrd ) CGIZA
      CBDFJJBACDJZFJBEFJABCDFGKTBUAEFTCLIZUAEMCGNACDELUBAEOHPQRS $.
  $}

  ${
    $d x B $.  $d x D $.
    $( Special case related to ~ rspsbc .  (Contributed by NM, 10-Dec-2005.)
       (Proof shortened by Eric Schmidt, 17-Jan-2007.) $)
    rspcsbela $p |- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D ) $=
      ( wcel wral csb wsbc rspsbc sbcel1g sylibd imp ) BCFZDEFZACGZABDHEFZNPOAB
      IQOABCJABDECKLM $.
  $}

  ${
    $d w x y z $.  $d w y z A $.
    $( Two ways of expressing " ` x ` is (effectively) not free in ` A ` ."
       (Contributed by Mario Carneiro, 14-Oct-2016.) $)
    sbnfc2 $p |- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A ) $=
      ( vw cv csb wceq wal cvv wcel vex csbtt mpan wsb wsbc sbsbc sbcel2g ax-mp
      wb eqtr4d alrimivv nfv wnf eleq2 bitri 3bitr4g 2alimi sbnf2 sylibr impbii
      wnfc nfcd ) ADULZABFZDGZACFZDGZHZCIBIZUNUSBCUNUPDURUOJKZUNUPDHBLZAUODJMNU
      QJKZUNURDHCLZAUQDJMNUAUBUTAEDUTEUCUTEFZDKZABOZVFACOZTZCIBIVFAUDUSVIBCUSVE
      UPKZVEURKZVGVHUPURVEUEVGVFAUOPZVJVFABQVAVLVJTVBAUOVEDJRSUFVHVFAUQPZVKVFAC
      QVCVMVKTVDAUQVEDJRSUFUGUHVFABCUIUJUMUK $.
  $}

  ${
    $d y z A $.  $d z ph $.  $d x y z $.  $d V z $.
    $( Move substitution into a class abstraction.  (Contributed by NM,
       13-Dec-2005.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
    csbabg $p |- ( A e. V ->
                 [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } ) $=
      ( vz wcel cab csb wsbc cv sbccom wsb df-clab sbsbc sbcbii 3bitr4i sbcel2g
      bitri bitr2id eqrdv ) DEGZFBDACHZIZABDJZCHZFKZUFGZUGUCGZBDJZUBUGUDGUECUGJ
      ZACUGJZBDJUHUJACBUGDLUHUECFMUKUEFCNUECFOSUIULBDUIACFMULAFCNACFOSPQBDUGUCE
      RTUA $.
  $}

  ${
    $d x v z $.  $d y v z $.  $d A v z $.  $d B v z $.  $d ph v z $.
    $d ps v z $.
    cbvralcsf.1 $e |- F/_ y A $.
    cbvralcsf.2 $e |- F/_ x B $.
    cbvralcsf.3 $e |- F/ y ph $.
    cbvralcsf.4 $e |- F/ x ps $.
    cbvralcsf.5 $e |- ( x = y -> A = B ) $.
    cbvralcsf.6 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( A more general version of ~ cbvralf that doesn't require ` A ` and ` B `
       to be distinct from ` x ` or ` y ` .  Changes bound variables using
       implicit substitution.  (Contributed by Andrew Salmon, 13-Jul-2011.) $)
    cbvralcsf $p |- ( A. x e. A ph <-> A. y e. B ps ) $=
      ( vz vv cv wcel wi wal wsbc nfcri wral csb nfv nfcsb1v nfsbc1v id csbeq1a
      nfim weq eleq12d sbceq1a imbi12d cbval nfcv nfcsb nfsbc csbeq1 cab df-csb
      eleq2d sbie sbsbc bitr3i abbi2i eqtr4i eqtrdi dfsbcq bitrdi bitri 3bitr4i
      wsb df-ral ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACWASZQZMRVT
      VOWECMVOMUCWCWDCCMWBCWAEUDTACWAUEUHCMUIZVNWCAWDWFVMWAEWBWFUFCWAEUGUJACWAU
      KULUMWEVSMDWCWDDDMWBDCWAEDWAUNZGUOTADCWAWGIUPUHVSMUCMDUIZWCVRWDBWHWAVQWBF
      WHUFWHWBCVQEUBZFCWAVQEUQWINOZEPZCVQSZNURFCNVQEUSWLNFWJFPZWKCDVKWLWKWMCDCN
      FHTCDUIEFWJKUTVAWKCDVBVCVDVEVFUJWHWDACVQSZBACWAVQVGWNACDVKBACDVBABCDJLVAV
      CVHULUMVIACEVLBDFVLVJ $.

    $( A more general version of ~ cbvrexf that has no distinct variable
       restrictions.  Changes bound variables using implicit substitution.
       (Contributed by Andrew Salmon, 13-Jul-2011.)  (Proof shortened by Mario
       Carneiro, 7-Dec-2014.) $)
    cbvrexcsf $p |- ( E. x e. A ph <-> E. y e. B ps ) $=
      ( vz vv cv wcel wa wex wsbc nfcri wrex csb nfv nfcsb1v nfsbc1v id csbeq1a
      nfan weq eleq12d sbceq1a anbi12d cbvex nfcv nfcsb nfsbc csbeq1 cab df-csb
      eleq2d sbie sbsbc bitr3i abbi2i eqtr4i eqtrdi dfsbcq bitrdi bitri 3bitr4i
      wsb df-rex ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACWASZQZMRVT
      VOWECMVOMUCWCWDCCMWBCWAEUDTACWAUEUHCMUIZVNWCAWDWFVMWAEWBWFUFCWAEUGUJACWAU
      KULUMWEVSMDWCWDDDMWBDCWAEDWAUNZGUOTADCWAWGIUPUHVSMUCMDUIZWCVRWDBWHWAVQWBF
      WHUFWHWBCVQEUBZFCWAVQEUQWINOZEPZCVQSZNURFCNVQEUSWLNFWJFPZWKCDVKWLWKWMCDCN
      FHTCDUIEFWJKUTVAWKCDVBVCVDVEVFUJWHWDACVQSZBACWAVQVGWNACDVKBACDVBABCDJLVAV
      CVHULUMVIACEVLBDFVLVJ $.

    $( A more general version of ~ cbvreuv that has no distinct variable
       rextrictions.  Changes bound variables using implicit substitution.
       (Contributed by Andrew Salmon, 13-Jul-2011.) $)
    cbvreucsf $p |- ( E! x e. A ph <-> E! y e. B ps ) $=
      ( vz vv cv wcel wa weu wsb nfcri wreu csb nfv nfcsb1v nfan weq id csbeq1a
      nfs1v eleq12d sbequ12 anbi12d cbveu nfcv nfcsb nfsb csbeq1 cab wsbc sbsbc
      abbii eleq2d sbie bicomi abbi2i df-csb 3eqtr4ri eqtrdi sbequ bitrdi bitri
      df-reu 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACMSZQZM
      RWAVPWFCMVPMUCWDWECCMWCCWBEUDTACMUIUECMUFZVOWDAWEWGVNWBEWCWGUGCWBEUHUJACM
      UKULUMWFVTMDWDWEDDMWCDCWBEDWBUNGUOTACMDIUPUEVTMUCMDUFZWDVSWEBWHWBVRWCFWHU
      GWHWCCVREUBZFCWBVREUQNOZEPZCDSZNURWKCVRUSZNURFWIWLWMNWKCDUTVAWLNFWLWJFPZW
      KWNCDCNFHTCDUFEFWJKVBVCVDVECNVREVFVGVHUJWHWEACDSBAMDCVIABCDJLVCVJULUMVKAC
      EVLBDFVLVM $.

    $( A more general version of ~ cbvrab with no distinct variable
       restrictions.  (Contributed by Andrew Salmon, 13-Jul-2011.) $)
    cbvrabcsf $p |- { x e. A | ph } = { y e. B | ps } $=
      ( vz vv cv wcel wa cab wsb nfcri crab csb nfv nfcsb1v nfan weq id csbeq1a
      nfs1v eleq12d sbequ12 anbi12d cbvab nfcv nfcsb csbeq1 df-csb eleq2d sbsbc
      nfsb wsbc bitr3i abbi2i eqtr4i eqtrdi sbequ bitrdi eqtri df-rab 3eqtr4i
      sbie ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVOMOZCVTEUBZPZACMSZQZMRVSVNWDCMV
      NMUCWBWCCCMWACVTEUDTACMUIUECMUFZVMWBAWCWEVLVTEWAWEUGCVTEUHUJACMUKULUMWDVR
      MDWBWCDDMWADCVTEDVTUNGUOTACMDIUTUEVRMUCMDUFZWBVQWCBWFVTVPWAFWFUGWFWACVPEU
      BZFCVTVPEUPWGNOZEPZCVPVAZNRFCNVPEUQWJNFWHFPZWICDSWJWIWKCDCNFHTCDUFEFWHKUR
      VKWICDUSVBVCVDVEUJWFWCACDSBAMDCVFABCDJLVKVGULUMVHACEVIBDFVIVJ $.
  $}

  ${
    $d A y $.  $d ps y $.  $d B x $.  $d ch x $.
    cbvralv2.1 $e |- ( x = y -> ( ps <-> ch ) ) $.
    cbvralv2.2 $e |- ( x = y -> A = B ) $.
    $( Rule used to change the bound variable in a restricted universal
       quantifier with implicit substitution which also changes the quantifier
       domain.  (Contributed by David Moews, 1-May-2017.) $)
    cbvralv2 $p |- ( A. x e. A ps <-> A. y e. B ch ) $=
      ( nfcv nfv cbvralcsf ) ABCDEFDEICFIADJBCJHGK $.

    $( Rule used to change the bound variable in a restricted existential
       quantifier with implicit substitution which also changes the quantifier
       domain.  (Contributed by David Moews, 1-May-2017.) $)
    cbvrexv2 $p |- ( E. x e. A ps <-> E. y e. B ch ) $=
      ( nfcv nfv cbvrexcsf ) ABCDEFDEICFIADJBCJHGK $.
  $}

  ${
    $d A x y $.  $d B y $.  $d C x $.  $d D y $.  $d E x $.  $d ph x $.
    $d ch x $.  $d ps y $.
    rspc2vd.a $e |- ( x = A -> ( th <-> ch ) ) $.
    rspc2vd.b $e |- ( y = B -> ( ch <-> ps ) ) $.
    rspc2vd.c $e |- ( ph -> A e. C ) $.
    rspc2vd.d $e |- ( ( ph /\ x = A ) -> D = E ) $.
    rspc2vd.e $e |- ( ph -> B e. E ) $.
    $( Deduction version of 2-variable restricted specialization, using
       implicit substitution.  Notice that the class ` D ` for the second set
       variable ` y ` may depend on the first set variable ` x ` .
       (Contributed by AV, 29-Mar-2021.) $)
    rspc2vd $p |- ( ph -> ( A. x e. C A. y e. D th -> ps ) ) $=
      ( csb wcel wral csbied eleqtrrd nfcsb1v nfv nfralw wceq csbeq1a raleqbidv
      wi cv rspc syl rspcv sylsyld ) AHEGJQZRDFJSZEISZCFUNSZBAHKUNPAEGJKINOTUAA
      GIRUPUQUHNUOUQEGICEFUNEGJUBCEUCUDEUIGUEDCFJUNEGJUFLUGUJUKCBFHUNMULUM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Define basic set operations and relations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new symbols. $)
  $c \ $. $( Backslash (difference) $)
  $c u. $. $( Cup (union) $)
  $c i^i $. $( Cap (intersection) $)
  $c C_ $. $( Subclass or subset symbol $)

  $( Extend class notation to include class difference (read:  " ` A ` minus
     ` B ` "). $)
  cdif $a class ( A \ B ) $.

  $( Extend class notation to include union of two classes (read:  " ` A `
     union ` B ` "). $)
  cun $a class ( A u. B ) $.

  $( Extend class notation to include the intersection of two classes (read:
      " ` A ` intersect ` B ` "). $)
  cin $a class ( A i^i B ) $.

  $( Extend wff notation to include the subclass relation.  This is
     read " ` A ` is a subclass of ` B ` " or " ` B ` includes ` A ` ".  When
     ` A ` exists as a set, it is also read " ` A ` is a subset of ` B ` ". $)
  wss $a wff A C_ B $.

  ${
    $d x A $.  $d x B $.  $d y A $.  $d y B $.  $d z x $.  $d z y $.  $d z A $.
    $d z B $.
    $( Soundness justification theorem for ~ df-dif .  (Contributed by Rodolfo
       Medina, 27-Apr-2010.)  (Proof shortened by Andrew Salmon,
       9-Jul-2011.) $)
    difjust $p |- { x | ( x e. A /\ -. x e. B ) }
                  = { y | ( y e. A /\ -. y e. B ) } $=
      ( vz cv wcel wn wa cab weq eleq1 notbid anbi12d cbvabv eqtri ) AFZCGZQDGZ
      HZIZAJEFZCGZUBDGZHZIZEJBFZCGZUGDGZHZIZBJUAUFAEAEKZRUCTUEQUBCLULSUDQUBDLMN
      OUFUKEBEBKZUCUHUEUJUBUGCLUMUDUIUBUGDLMNOP $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Define class difference, also called relative complement.  Definition
       5.12 of [TakeutiZaring] p. 20.  Contrast this operation with union
       ` ( A u. B ) ` ( ~ df-un ) and intersection ` ( A i^i B ) ` ( ~ df-in ).
       Several notations are used in the literature; we chose the ` \ `
       convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the
       more common minus sign to reserve the latter for later use in, e.g.,
       arithmetic.  We will use the terminology " ` A ` excludes ` B ` " to
       mean ` A \ B ` .  We will use " ` B ` is removed from ` A ` " to mean
       ` A \ { B } ` i.e. the removal of an element or equivalently the
       exclusion of a singleton.  (Contributed by NM, 29-Apr-1994.) $)
    df-dif $a |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) } $.
  $}

  ${
    $d x A $.  $d x B $.  $d y A $.  $d y B $.  $d z x $.  $d z y $.  $d z A $.
    $d z B $.
    $( Soundness justification theorem for ~ df-un .  (Contributed by Rodolfo
       Medina, 28-Apr-2010.)  (Proof shortened by Andrew Salmon,
       9-Jul-2011.) $)
    unjust $p |- { x | ( x e. A \/ x e. B ) } = { y | ( y e. A \/ y e. B ) } $=
      ( vz cv wcel wo cab weq eleq1 orbi12d cbvabv eqtri ) AFZCGZODGZHZAIEFZCGZ
      SDGZHZEIBFZCGZUCDGZHZBIRUBAEAEJPTQUAOSCKOSDKLMUBUFEBEBJTUDUAUESUCCKSUCDKL
      MN $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Define the union of two classes.  Definition 5.6 of [TakeutiZaring]
       p. 16.  Contrast this operation with difference ` ( A \ B ) `
       ( ~ df-dif ) and intersection ` ( A i^i B ) ` ( ~ df-in ).  (Contributed
       by NM, 23-Aug-1993.) $)
    df-un $a |- ( A u. B ) = { x | ( x e. A \/ x e. B ) } $.
  $}

  ${
    $d x A $.  $d x B $.  $d y A $.  $d y B $.  $d z x $.  $d z y $.  $d z A $.
    $d z B $.
    $( Soundness justification theorem for ~ df-in .  (Contributed by Rodolfo
       Medina, 28-Apr-2010.)  (Proof shortened by Andrew Salmon,
       9-Jul-2011.) $)
    injust $p |- { x | ( x e. A /\ x e. B ) }
                  = { y | ( y e. A /\ y e. B ) } $=
      ( vz cv wcel wa cab weq eleq1 anbi12d cbvabv eqtri ) AFZCGZODGZHZAIEFZCGZ
      SDGZHZEIBFZCGZUCDGZHZBIRUBAEAEJPTQUAOSCKOSDKLMUBUFEBEBJTUDUAUESUCCKSUCDKL
      MN $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Define the intersection of two classes.  Definition 5.6 of
       [TakeutiZaring] p. 16.  Contrast this operation with union
       ` ( A u. B ) ` ( ~ df-un ) and difference ` ( A \ B ) ` ( ~ df-dif ).
       (Contributed by NM, 29-Apr-1994.) $)
    df-in $a |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) } $.

    $( Alternate definition for the intersection of two classes.  (Contributed
       by NM, 6-Jul-2005.) $)
    dfin5 $p |- ( A i^i B ) = { x e. A | x e. B } $=
      ( cin cv wcel wa cab crab df-in df-rab eqtr4i ) BCDAEZBFMCFZGAHNABIABCJNA
      BKL $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of class difference.  (Contributed by NM,
       25-Mar-2004.) $)
    dfdif2 $p |- ( A \ B ) = { x e. A | -. x e. B } $=
      ( cdif cv wcel wn wa cab crab df-dif df-rab eqtr4i ) BCDAEZBFNCFGZHAIOABJ
      ABCKOABLM $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Expansion of membership in a class difference.  (Contributed by NM,
       29-Apr-1994.) $)
    eldif $p |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) ) $=
      ( vx cdif wcel cvv wn wa elex adantr cv wceq notbid anbi12d df-dif elab2g
      eleq1 pm5.21nii ) ABCEZFAGFZABFZACFZHZIZATJUBUAUDABJKDLZBFZUFCFZHZIUEDATG
      UFAMZUGUBUIUDUFABRUJUHUCUFACRNODBCPQS $.
  $}

  ${
    eldifd.1 $e |- ( ph -> A e. B ) $.
    eldifd.2 $e |- ( ph -> -. A e. C ) $.
    $( If a class is in one class and not another, it is also in their
       difference.  One-way deduction form of ~ eldif .  (Contributed by David
       Moews, 1-May-2017.) $)
    eldifd $p |- ( ph -> A e. ( B \ C ) ) $=
      ( wcel wn cdif eldif sylanbrc ) ABCGBDGHBCDIGEFBCDJK $.
  $}

  ${
    eldifad.1 $e |- ( ph -> A e. ( B \ C ) ) $.
    $( If a class is in the difference of two classes, it is also in the
       minuend.  One-way deduction form of ~ eldif .  (Contributed by David
       Moews, 1-May-2017.) $)
    eldifad $p |- ( ph -> A e. B ) $=
      ( wcel wn cdif wa eldif sylib simpld ) ABCFZBDFGZABCDHFMNIEBCDJKL $.
  $}

  ${
    eldifbd.1 $e |- ( ph -> A e. ( B \ C ) ) $.
    $( If a class is in the difference of two classes, it is not in the
       subtrahend.  One-way deduction form of ~ eldif .  (Contributed by David
       Moews, 1-May-2017.) $)
    eldifbd $p |- ( ph -> -. A e. C ) $=
      ( wcel wn cdif wa eldif sylib simprd ) ABCFZBDFGZABCDHFMNIEBCDJKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Subclasses and subsets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Define the subclass relationship.  Exercise 9 of [TakeutiZaring] p. 18.
     Note that ` A C_ A ` (proved in ~ ssid ).  For a more traditional
     definition, but requiring a dummy variable, see ~ ssalel .  Other possible
     definitions are given by ~ dfss3 , ~ ssequn1 , ~ ssequn2 , and ~ sseqin2 .
     (Contributed by NM, 27-Apr-1994.) $)
  df-ss $a |- ( A C_ B <-> ( A i^i B ) = A ) $.

  $( Variant of subclass definition ~ df-ss .  (Contributed by NM,
     3-Sep-2004.) $)
  dfss $p |- ( A C_ B <-> A = ( A i^i B ) ) $=
    ( wss cin wceq df-ss eqcom bitri ) ABCABDZAEAIEABFIAGH $.

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of the subclass relationship between two classes.
       Definition 5.9 of [TakeutiZaring] p. 17.  (Contributed by NM,
       8-Jan-2002.) $)
    ssalel $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
      ( wss cv wcel wa wb wal wi cin wceq dfss df-in eqeq2i abeq2 3bitri pm4.71
      cab albii bitr4i ) BCDZAEZBFZUDUCCFZGZHZAIZUDUEJZAIUBBBCKZLBUFASZLUHBCMUJ
      UKBABCNOUFABPQUIUGAUDUERTUA $.

    $( Alternate definition of subclass relationship.  (Contributed by NM,
       14-Oct-1999.) $)
    dfss3 $p |- ( A C_ B <-> A. x e. A x e. B ) $=
      ( wss cv wcel wi wal wral ssalel df-ral bitr4i ) BCDAEZBFMCFZGAHNABIABCJN
      ABKL $.
  $}

  ${
    $( Alternate definition of the subclass relationship between two classes.
       Exercise 9 of [TakeutiZaring] p. 18.  This is another name for ~ df-ss
       which is more consistent with the naming in the Metamath Proof Explorer.
       (Contributed by NM, 27-Apr-1994.) $)
    dfss2 $p |- ( A C_ B <-> ( A i^i B ) = A ) $=
      ( df-ss ) ABC $.
  $}

  ${
    $d z A $.  $d z B $.  $d x z $.
    dfss2f.1 $e |- F/_ x A $.
    dfss2f.2 $e |- F/_ x B $.
    $( Equivalence for subclass relation, using bound-variable hypotheses
       instead of distinct variable conditions.  (Contributed by NM,
       3-Jul-1994.)  (Revised by Andrew Salmon, 27-Aug-2011.) $)
    dfssf $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
      ( vz wss cv wcel wal ssalel nfcri nfim nfv weq eleq1 imbi12d cbval bitri
      wi ) BCGFHZBIZUACIZTZFJAHZBIZUECIZTZAJFBCKUDUHFAUBUCAAFBDLAFCELMUHFNFAOUB
      UFUCUGUAUEBPUAUECPQRS $.

    $( Equivalence for subclass relation, using bound-variable hypotheses
       instead of distinct variable conditions.  (Contributed by NM,
       3-Jul-1994.)  (Revised by Andrew Salmon, 27-Aug-2011.) $)
    dfss2f $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
      ( vz wss cv wcel wal ssalel nfcri nfim nfv weq eleq1 imbi12d cbval bitri
      wi ) BCGFHZBIZUACIZTZFJAHZBIZUECIZTZAJFBCKUDUHFAUBUCAAFBDLAFCELMUHFNFAOUB
      UFUCUGUAUEBPUAUECPQRS $.

    $( Equivalence for subclass relation, using bound-variable hypotheses
       instead of distinct variable conditions.  (Contributed by NM,
       20-Mar-2004.) $)
    dfss3f $p |- ( A C_ B <-> A. x e. A x e. B ) $=
      ( wss cv wcel wi wal wral dfss2f df-ral bitr4i ) BCFAGZBHOCHZIAJPABKABCDE
      LPABMN $.

    $( If ` x ` is not free in ` A ` and ` B ` , it is not free in ` A C_ B ` .
       (Contributed by NM, 27-Dec-1996.) $)
    nfss $p |- F/ x A C_ B $=
      ( wss cv wcel wral dfss3f nfra1 nfxfr ) BCFAGCHZABIAABCDEJMABKL $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Membership relationships follow from a subclass relationship.
       (Contributed by NM, 5-Aug-1993.) $)
    ssel $p |- ( A C_ B -> ( C e. A -> C e. B ) ) $=
      ( vx wss cv wceq wcel wex wal ssalel biimpi 19.21bi anim2d eximdv df-clel
      wa wi 3imtr4g ) ABEZDFZCGZUAAHZQZDIUBUABHZQZDICAHCBHTUDUFDTUCUEUBTUCUERZD
      TUGDJDABKLMNODCAPDCBPS $.
  $}

  $( Membership relationships follow from a subclass relationship.
     (Contributed by NM, 7-Jun-2004.) $)
  ssel2 $p |- ( ( A C_ B /\ C e. A ) -> C e. B ) $=
    ( wss wcel ssel imp ) ABDCAECBEABCFG $.

  ${
    sseli.1 $e |- A C_ B $.
    $( Membership inference from subclass relationship.  (Contributed by NM,
       5-Aug-1993.) $)
    sseli $p |- ( C e. A -> C e. B ) $=
      ( wss wcel wi ssel ax-mp ) ABECAFCBFGDABCHI $.

    ${
      sselii.2 $e |- C e. A $.
      $( Membership inference from subclass relationship.  (Contributed by NM,
         31-May-1999.) $)
      sselii $p |- C e. B $=
        ( wcel sseli ax-mp ) CAFCBFEABCDGH $.
    $}

    ${
      sselid.2 $e |- ( ph -> C e. A ) $.
      $( Membership inference from subclass relationship.  (Contributed by NM,
         25-Jun-2014.) $)
      sselid $p |- ( ph -> C e. B ) $=
        ( wcel sseli syl ) ADBGDCGFBCDEHI $.
    $}
  $}

  ${
    sseld.1 $e |- ( ph -> A C_ B ) $.
    $( Membership deduction from subclass relationship.  (Contributed by NM,
       15-Nov-1995.) $)
    sseld $p |- ( ph -> ( C e. A -> C e. B ) ) $=
      ( wss wcel wi ssel syl ) ABCFDBGDCGHEBCDIJ $.

    $( Membership deduction from subclass relationship.  (Contributed by NM,
       26-Jun-2014.) $)
    sselda $p |- ( ( ph /\ C e. A ) -> C e. B ) $=
      ( wcel sseld imp ) ADBFDCFABCDEGH $.

    ${
      sseldd.2 $e |- ( ph -> C e. A ) $.
      $( Membership inference from subclass relationship.  (Contributed by NM,
         14-Dec-2004.) $)
      sseldd $p |- ( ph -> C e. B ) $=
        ( wcel sseld mpd ) ADBGDCGFABCDEHI $.
    $}
  $}

  ${
    ssneld.1 $e |- ( ph -> A C_ B ) $.
    $( If a class is not in another class, it is also not in a subclass of that
       class.  Deduction form.  (Contributed by David Moews, 1-May-2017.) $)
    ssneld $p |- ( ph -> ( -. C e. B -> -. C e. A ) ) $=
      ( wcel sseld con3d ) ADBFDCFABCDEGH $.

    ssneldd.2 $e |- ( ph -> -. C e. B ) $.
    $( If an element is not in a class, it is also not in a subclass of that
       class.  Deduction form.  (Contributed by David Moews, 1-May-2017.) $)
    ssneldd $p |- ( ph -> -. C e. A ) $=
      ( wcel wn ssneld mpd ) ADCGHDBGHFABCDEIJ $.
  $}

  ${
    $d x A $.  $d x B $.
    ssriv.1 $e |- ( x e. A -> x e. B ) $.
    $( Inference based on subclass definition.  (Contributed by NM,
       5-Aug-1993.) $)
    ssriv $p |- A C_ B $=
      ( wss cv wcel wi ssalel mpgbir ) BCEAFZBGKCGHAABCIDJ $.
  $}

  ${
    ssrd.0 $e |- F/ x ph $.
    ssrd.1 $e |- F/_ x A $.
    ssrd.2 $e |- F/_ x B $.
    ssrd.3 $e |- ( ph -> ( x e. A -> x e. B ) ) $.
    $( Deduction based on subclass definition.  (Contributed by Thierry Arnoux,
       8-Mar-2017.) $)
    ssrd $p |- ( ph -> A C_ B ) $=
      ( cv wcel wi wal wss alrimi dfss2f sylibr ) ABIZCJQDJKZBLCDMARBEHNBCDFGOP
      $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    ssrdv.1 $e |- ( ph -> ( x e. A -> x e. B ) ) $.
    $( Deduction based on subclass definition.  (Contributed by NM,
       15-Nov-1995.) $)
    ssrdv $p |- ( ph -> A C_ B ) $=
      ( cv wcel wi wal wss alrimiv ssalel sylibr ) ABFZCGNDGHZBICDJAOBEKBCDLM
      $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Transitivity of subclasses.  Exercise 5 of [TakeutiZaring] p. 17.
       (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
       14-Jun-2011.) $)
    sstr2 $p |- ( A C_ B -> ( B C_ C -> A C_ C ) ) $=
      ( vx wss cv wcel wi wal ssel imim1d alimdv ssalel 3imtr4g ) ABEZDFZBGZPCG
      ZHZDIPAGZRHZDIBCEACEOSUADOTQRABPJKLDBCMDACMN $.
  $}

  $( Transitivity of subclasses.  Theorem 6 of [Suppes] p. 23.  (Contributed by
     NM, 5-Sep-2003.) $)
  sstr $p |- ( ( A C_ B /\ B C_ C ) -> A C_ C ) $=
    ( wss sstr2 imp ) ABDBCDACDABCEF $.

  ${
    sstri.1 $e |- A C_ B $.
    sstri.2 $e |- B C_ C $.
    $( Subclass transitivity inference.  (Contributed by NM, 5-May-2000.) $)
    sstri $p |- A C_ C $=
      ( wss sstr2 mp2 ) ABFBCFACFDEABCGH $.
  $}

  ${
    sstrd.1 $e |- ( ph -> A C_ B ) $.
    sstrd.2 $e |- ( ph -> B C_ C ) $.
    $( Subclass transitivity deduction.  (Contributed by NM, 2-Jun-2004.) $)
    sstrd $p |- ( ph -> A C_ C ) $=
      ( wss sstr syl2anc ) ABCGCDGBDGEFBCDHI $.
  $}

  ${
    sstrid.1 $e |- A C_ B $.
    sstrid.2 $e |- ( ph -> B C_ C ) $.
    $( Subclass transitivity deduction.  (Contributed by NM, 6-Feb-2014.) $)
    sstrid $p |- ( ph -> A C_ C ) $=
      ( wss a1i sstrd ) ABCDBCGAEHFI $.
  $}

  ${
    sstrdi.1 $e |- ( ph -> A C_ B ) $.
    sstrdi.2 $e |- B C_ C $.
    $( Subclass transitivity deduction.  (Contributed by Jonathan Ben-Naim,
       3-Jun-2011.) $)
    sstrdi $p |- ( ph -> A C_ C ) $=
      ( wss a1i sstrd ) ABCDECDGAFHI $.
  $}

  ${
    sylan9ss.1 $e |- ( ph -> A C_ B ) $.
    sylan9ss.2 $e |- ( ps -> B C_ C ) $.
    $( A subclass transitivity deduction.  (Contributed by NM, 27-Sep-2004.)
       (Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
    sylan9ss $p |- ( ( ph /\ ps ) -> A C_ C ) $=
      ( wss sstr syl2an ) ACDHDEHCEHBFGCDEIJ $.
  $}

  ${
    sylan9ssr.1 $e |- ( ph -> A C_ B ) $.
    sylan9ssr.2 $e |- ( ps -> B C_ C ) $.
    $( A subclass transitivity deduction.  (Contributed by NM, 27-Sep-2004.) $)
    sylan9ssr $p |- ( ( ps /\ ph ) -> A C_ C ) $=
      ( wss sylan9ss ancoms ) ABCEHABCDEFGIJ $.
  $}

  ${
    $d x A $.  $d x B $.
    $( The subclass relationship is antisymmetric.  Compare Theorem 4 of
       [Suppes] p. 22.  (Contributed by NM, 5-Aug-1993.) $)
    eqss $p |- ( A = B <-> ( A C_ B /\ B C_ A ) ) $=
      ( vx cv wcel wb wal wi wa wceq wss albiim dfcleq ssalel anbi12i 3bitr4i )
      CDZAEZQBEZFCGRSHCGZSRHCGZIABJABKZBAKZIRSCLCABMUBTUCUACABNCBANOP $.
  $}

  ${
    eqssi.1 $e |- A C_ B $.
    eqssi.2 $e |- B C_ A $.
    $( Infer equality from two subclass relationships.  Compare Theorem 4 of
       [Suppes] p. 22.  (Contributed by NM, 9-Sep-1993.) $)
    eqssi $p |- A = B $=
      ( wceq wss eqss mpbir2an ) ABEABFBAFCDABGH $.
  $}

  ${
    eqssd.1 $e |- ( ph -> A C_ B ) $.
    eqssd.2 $e |- ( ph -> B C_ A ) $.
    $( Equality deduction from two subclass relationships.  Compare Theorem 4
       of [Suppes] p. 22.  (Contributed by NM, 27-Jun-2004.) $)
    eqssd $p |- ( ph -> A = B ) $=
      ( wss wceq eqss sylanbrc ) ABCFCBFBCGDEBCHI $.
  $}

  ${
    eqrd.0 $e |- F/ x ph $.
    eqrd.1 $e |- F/_ x A $.
    eqrd.2 $e |- F/_ x B $.
    eqrd.3 $e |- ( ph -> ( x e. A <-> x e. B ) ) $.
    $( Deduce equality of classes from equivalence of membership.  (Contributed
       by Thierry Arnoux, 21-Mar-2017.) $)
    eqrd $p |- ( ph -> A = B ) $=
      ( cv wcel biimpd ssrd biimprd eqssd ) ACDABCDEFGABIZCJZODJZHKLABDCEGFAPQH
      MLN $.
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.
    eqelssd.1 $e |- ( ph -> A C_ B ) $.
    eqelssd.2 $e |- ( ( ph /\ x e. B ) -> x e. A ) $.
    $( Equality deduction from subclass relationship and membership.
       (Contributed by AV, 21-Aug-2022.) $)
    eqelssd $p |- ( ph -> A = B ) $=
      ( cv wcel ex ssrdv eqssd ) ACDEABDCABGZDHLCHFIJK $.
  $}

  ${
    $d A x $.
    $( Any class is a subclass of itself.  Exercise 10 of [TakeutiZaring]
       p. 18.  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew
       Salmon, 14-Jun-2011.) $)
    ssid $p |- A C_ A $=
      ( vx cv wcel id ssriv ) BAABCADEF $.
  $}

  $( Weakening of ~ ssid .  (Contributed by BJ, 1-Sep-2022.) $)
  ssidd $p |- ( ph -> A C_ A ) $=
    ( wss ssid a1i ) BBCABDE $.

  ${
    $d A x $.
    $( Any class is a subclass of the universal class.  Dual of ~ 0ss .
       (Contributed by NM, 31-Oct-1995.) $)
    ssv $p |- A C_ _V $=
      ( vx cvv cv elex ssriv ) BACBDAEF $.
  $}

  $( Equality theorem for subclasses.  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Andrew Salmon, 21-Jun-2011.) $)
  sseq1 $p |- ( A = B -> ( A C_ C <-> B C_ C ) ) $=
    ( wceq wss wa wb eqss wi sstr2 adantl adantr impbid sylbi ) ABDABEZBAEZFZAC
    EZBCEZGABHQRSPRSIOBACJKOSRIPABCJLMN $.

  $( Equality theorem for the subclass relationship.  (Contributed by NM,
     25-Jun-1998.) $)
  sseq2 $p |- ( A = B -> ( C C_ A <-> C C_ B ) ) $=
    ( wss wa wi wceq wb sstr2 com12 anim12i eqss dfbi2 3imtr4i ) ABDZBADZECADZC
    BDZFZRQFZEABGQRHOSPTQORCABIJRPQCBAIJKABLQRMN $.

  $( Equality theorem for the subclass relationship.  (Contributed by NM,
     31-May-1999.) $)
  sseq12 $p |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) ) $=
    ( wceq wss sseq1 sseq2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.

  ${
    sseq1i.1 $e |- A = B $.
    $( An equality inference for the subclass relationship.  (Contributed by
       NM, 18-Aug-1993.) $)
    sseq1i $p |- ( A C_ C <-> B C_ C ) $=
      ( wceq wss wb sseq1 ax-mp ) ABEACFBCFGDABCHI $.

    $( An equality inference for the subclass relationship.  (Contributed by
       NM, 30-Aug-1993.) $)
    sseq2i $p |- ( C C_ A <-> C C_ B ) $=
      ( wceq wss wb sseq2 ax-mp ) ABECAFCBFGDABCHI $.

    ${
      sseq12i.2 $e |- C = D $.
      $( An equality inference for the subclass relationship.  (Contributed by
         NM, 31-May-1999.)  (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
      sseq12i $p |- ( A C_ C <-> B C_ D ) $=
        ( wceq wss wb sseq12 mp2an ) ABGCDGACHBDHIEFABCDJK $.
    $}
  $}

  ${
    sseq1d.1 $e |- ( ph -> A = B ) $.
    $( An equality deduction for the subclass relationship.  (Contributed by
       NM, 14-Aug-1994.) $)
    sseq1d $p |- ( ph -> ( A C_ C <-> B C_ C ) ) $=
      ( wceq wss wb sseq1 syl ) ABCFBDGCDGHEBCDIJ $.

    $( An equality deduction for the subclass relationship.  (Contributed by
       NM, 14-Aug-1994.) $)
    sseq2d $p |- ( ph -> ( C C_ A <-> C C_ B ) ) $=
      ( wceq wss wb sseq2 syl ) ABCFDBGDCGHEBCDIJ $.

    ${
      sseq12d.2 $e |- ( ph -> C = D ) $.
      $( An equality deduction for the subclass relationship.  (Contributed by
         NM, 31-May-1999.) $)
      sseq12d $p |- ( ph -> ( A C_ C <-> B C_ D ) ) $=
        ( wss sseq1d sseq2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
    $}
  $}

  ${
    eqsstr.1 $e |- A = B $.
    eqsstr.2 $e |- B C_ C $.
    $( Substitution of equality into a subclass relationship.  (Contributed by
       NM, 16-Jul-1995.) $)
    eqsstri $p |- A C_ C $=
      ( wss sseq1i mpbir ) ACFBCFEABCDGH $.
  $}

  ${
    eqsstr3.1 $e |- B = A $.
    eqsstr3.2 $e |- B C_ C $.
    $( Substitution of equality into a subclass relationship.  (Contributed by
       NM, 19-Oct-1999.) $)
    eqsstrri $p |- A C_ C $=
      ( eqcomi eqsstri ) ABCBADFEG $.
  $}

  ${
    sseqtr.1 $e |- A C_ B $.
    sseqtr.2 $e |- B = C $.
    $( Substitution of equality into a subclass relationship.  (Contributed by
       NM, 28-Jul-1995.) $)
    sseqtri $p |- A C_ C $=
      ( wss sseq2i mpbi ) ABFACFDBCAEGH $.
  $}

  ${
    sseqtrri.1 $e |- A C_ B $.
    sseqtrri.2 $e |- C = B $.
    $( Substitution of equality into a subclass relationship.  (Contributed by
       NM, 4-Apr-1995.) $)
    sseqtrri $p |- A C_ C $=
      ( eqcomi sseqtri ) ABCDCBEFG $.
  $}

  ${
    eqsstrd.1 $e |- ( ph -> A = B ) $.
    eqsstrd.2 $e |- ( ph -> B C_ C ) $.
    $( Substitution of equality into a subclass relationship.  (Contributed by
       NM, 25-Apr-2004.) $)
    eqsstrd $p |- ( ph -> A C_ C ) $=
      ( wss sseq1d mpbird ) ABDGCDGFABCDEHI $.
  $}

  ${
    eqsstrrd.1 $e |- ( ph -> B = A ) $.
    eqsstrrd.2 $e |- ( ph -> B C_ C ) $.
    $( Substitution of equality into a subclass relationship.  (Contributed by
       NM, 25-Apr-2004.) $)
    eqsstrrd $p |- ( ph -> A C_ C ) $=
      ( eqcomd eqsstrd ) ABCDACBEGFH $.
  $}

  ${
    sseqtrd.1 $e |- ( ph -> A C_ B ) $.
    sseqtrd.2 $e |- ( ph -> B = C ) $.
    $( Substitution of equality into a subclass relationship.  (Contributed by
       NM, 25-Apr-2004.) $)
    sseqtrd $p |- ( ph -> A C_ C ) $=
      ( wss sseq2d mpbid ) ABCGBDGEACDBFHI $.
  $}

  ${
    sseqtrrd.1 $e |- ( ph -> A C_ B ) $.
    sseqtrrd.2 $e |- ( ph -> C = B ) $.
    $( Substitution of equality into a subclass relationship.  (Contributed by
       NM, 25-Apr-2004.) $)
    sseqtrrd $p |- ( ph -> A C_ C ) $=
      ( eqcomd sseqtrd ) ABCDEADCFGH $.
  $}

  ${
    3sstr3.1 $e |- A C_ B $.
    3sstr3.2 $e |- A = C $.
    3sstr3.3 $e |- B = D $.
    $( Substitution of equality in both sides of a subclass relationship.
       (Contributed by NM, 13-Jan-1996.)  (Proof shortened by Eric Schmidt,
       26-Jan-2007.) $)
    3sstr3i $p |- C C_ D $=
      ( wss sseq12i mpbi ) ABHCDHEACBDFGIJ $.
  $}

  ${
    3sstr4.1 $e |- A C_ B $.
    3sstr4.2 $e |- C = A $.
    3sstr4.3 $e |- D = B $.
    $( Substitution of equality in both sides of a subclass relationship.
       (Contributed by NM, 13-Jan-1996.)  (Proof shortened by Eric Schmidt,
       26-Jan-2007.) $)
    3sstr4i $p |- C C_ D $=
      ( wss sseq12i mpbir ) CDHABHECADBFGIJ $.
  $}

  ${
    3sstr3g.1 $e |- ( ph -> A C_ B ) $.
    3sstr3g.2 $e |- A = C $.
    3sstr3g.3 $e |- B = D $.
    $( Substitution of equality into both sides of a subclass relationship.
       (Contributed by NM, 1-Oct-2000.) $)
    3sstr3g $p |- ( ph -> C C_ D ) $=
      ( wss sseq12i sylib ) ABCIDEIFBDCEGHJK $.
  $}

  ${
    3sstr4g.1 $e |- ( ph -> A C_ B ) $.
    3sstr4g.2 $e |- C = A $.
    3sstr4g.3 $e |- D = B $.
    $( Substitution of equality into both sides of a subclass relationship.
       (Contributed by NM, 16-Aug-1994.)  (Proof shortened by Eric Schmidt,
       26-Jan-2007.) $)
    3sstr4g $p |- ( ph -> C C_ D ) $=
      ( wss sseq12i sylibr ) ABCIDEIFDBECGHJK $.
  $}

  ${
    3sstr3d.1 $e |- ( ph -> A C_ B ) $.
    3sstr3d.2 $e |- ( ph -> A = C ) $.
    3sstr3d.3 $e |- ( ph -> B = D ) $.
    $( Substitution of equality into both sides of a subclass relationship.
       (Contributed by NM, 1-Oct-2000.) $)
    3sstr3d $p |- ( ph -> C C_ D ) $=
      ( wss sseq12d mpbid ) ABCIDEIFABDCEGHJK $.
  $}

  ${
    3sstr4d.1 $e |- ( ph -> A C_ B ) $.
    3sstr4d.2 $e |- ( ph -> C = A ) $.
    3sstr4d.3 $e |- ( ph -> D = B ) $.
    $( Substitution of equality into both sides of a subclass relationship.
       (Contributed by NM, 30-Nov-1995.)  (Proof shortened by Eric Schmidt,
       26-Jan-2007.) $)
    3sstr4d $p |- ( ph -> C C_ D ) $=
      ( wss sseq12d mpbird ) ADEIBCIFADBECGHJK $.
  $}

  ${
    eqsstrid.1 $e |- A = B $.
    eqsstrid.2 $e |- ( ph -> B C_ C ) $.
    $( B chained subclass and equality deduction.  (Contributed by NM,
       25-Apr-2004.) $)
    eqsstrid $p |- ( ph -> A C_ C ) $=
      ( wss sseq1i sylibr ) ACDGBDGFBCDEHI $.
  $}

  ${
    eqsstrrid.1 $e |- B = A $.
    eqsstrrid.2 $e |- ( ph -> B C_ C ) $.
    $( B chained subclass and equality deduction.  (Contributed by NM,
       25-Apr-2004.) $)
    eqsstrrid $p |- ( ph -> A C_ C ) $=
      ( eqcomi eqsstrid ) ABCDCBEGFH $.
  $}

  ${
    sseqtrdi.1 $e |- ( ph -> A C_ B ) $.
    sseqtrdi.2 $e |- B = C $.
    $( A chained subclass and equality deduction.  (Contributed by NM,
       25-Apr-2004.) $)
    sseqtrdi $p |- ( ph -> A C_ C ) $=
      ( wss sseq2i sylib ) ABCGBDGECDBFHI $.
  $}

  ${
    sseqtrrdi.1 $e |- ( ph -> A C_ B ) $.
    sseqtrrdi.2 $e |- C = B $.
    $( A chained subclass and equality deduction.  (Contributed by NM,
       25-Apr-2004.) $)
    sseqtrrdi $p |- ( ph -> A C_ C ) $=
      ( eqcomi sseqtrdi ) ABCDEDCFGH $.
  $}

  ${
    sseqtrid.1 $e |- B C_ A $.
    sseqtrid.2 $e |- ( ph -> A = C ) $.
    $( Subclass transitivity deduction.  (Contributed by Jonathan Ben-Naim,
       3-Jun-2011.) $)
    sseqtrid $p |- ( ph -> B C_ C ) $=
      ( wceq wss sseq2 biimpa sylancl ) ABDGZCBHZCDHZFELMNBDCIJK $.
  $}

  ${
    sseqtrrid.1 $e |- B C_ A $.
    sseqtrrid.2 $e |- ( ph -> C = A ) $.
    $( Subclass transitivity deduction.  (Contributed by Jonathan Ben-Naim,
       3-Jun-2011.) $)
    sseqtrrid $p |- ( ph -> B C_ C ) $=
      ( wss a1i sseqtrrd ) ACBDCBGAEHFI $.
  $}

  ${
    eqsstrdi.1 $e |- ( ph -> A = B ) $.
    eqsstrdi.2 $e |- B C_ C $.
    $( A chained subclass and equality deduction.  (Contributed by Mario
       Carneiro, 2-Jan-2017.) $)
    eqsstrdi $p |- ( ph -> A C_ C ) $=
      ( wss a1i eqsstrd ) ABCDECDGAFHI $.
  $}

  ${
    eqsstrrdi.1 $e |- ( ph -> B = A ) $.
    eqsstrrdi.2 $e |- B C_ C $.
    $( A chained subclass and equality deduction.  (Contributed by Mario
       Carneiro, 2-Jan-2017.) $)
    eqsstrrdi $p |- ( ph -> A C_ C ) $=
      ( eqcomd eqsstrdi ) ABCDACBEGFH $.
  $}

  $( Equality implies the subclass relation.  (Contributed by NM, 5-Aug-1993.)
     (Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
  eqimss $p |- ( A = B -> A C_ B ) $=
    ( wceq wss eqss simplbi ) ABCABDBADABEF $.

  $( Equality implies the subclass relation.  (Contributed by NM,
     23-Nov-2003.) $)
  eqimss2 $p |- ( B = A -> A C_ B ) $=
    ( wss eqimss eqcoms ) ABCABABDE $.

  ${
    eqimssi.1 $e |- A = B $.
    $( Infer subclass relationship from equality.  (Contributed by NM,
       6-Jan-2007.) $)
    eqimssi $p |- A C_ B $=
      ( ssid sseqtri ) AABADCE $.

    $( Infer subclass relationship from equality.  (Contributed by NM,
       7-Jan-2007.) $)
    eqimss2i $p |- B C_ A $=
      ( ssid sseqtrri ) BBABDCE $.
  $}

  $( Two classes are different if they don't include the same class.
     (Contributed by NM, 23-Apr-2015.) $)
  nssne1 $p |- ( ( A C_ B /\ -. A C_ C ) -> B =/= C ) $=
    ( wss wn wne wceq sseq2 biimpcd necon3bd imp ) ABDZACDZEBCFLMBCBCGLMBCAHIJK
    $.

  $( Two classes are different if they are not subclasses of the same class.
     (Contributed by NM, 23-Apr-2015.) $)
  nssne2 $p |- ( ( A C_ C /\ -. B C_ C ) -> A =/= B ) $=
    ( wss wn wne wceq sseq1 biimpcd necon3bd imp ) ACDZBCDZEABFLMABABGLMABCHIJK
    $.

  ${
    $d x A $.  $d x B $.
    $( Negation of subclass relationship.  One direction of Exercise 13 of
       [TakeutiZaring] p. 18.  (Contributed by Jim Kingdon, 15-Jul-2018.) $)
    nssr $p |- ( E. x ( x e. A /\ -. x e. B ) -> -. A C_ B ) $=
      ( cv wcel wn wa wex wi wal wss exanaliim ssalel sylnibr ) ADZBEZOCEZFGAHP
      QIAJBCKPQALABCMN $.
  $}

  $( Demonstrate by witnesses that two classes lack a subclass relation.
     (Contributed by Stefan O'Rear, 5-Feb-2015.) $)
  nelss $p |- ( ( A e. B /\ -. A e. C ) -> -. B C_ C ) $=
    ( wcel wss ssel com12 con3dimp ) ABDZBCEZACDZJIKBCAFGH $.

  ${
    ssrexf.1 $e |- F/_ x A $.
    ssrexf.2 $e |- F/_ x B $.
    $( Restricted existential quantification follows from a subclass
       relationship.  (Contributed by Glauco Siliprandi, 20-Apr-2017.) $)
    ssrexf $p |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) ) $=
      ( wss cv wcel wa wex wrex nfss ssel anim1d eximd df-rex 3imtr4g ) CDGZBHZ
      CIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.

    $( "At most one" existential quantification restricted to a subclass.
       (Contributed by Thierry Arnoux, 8-Oct-2017.) $)
    ssrmof $p |- ( A C_ B -> ( E* x e. B ph -> E* x e. A ph ) ) $=
      ( wss cv wcel wa wmo wrmo wi wal dfss2f biimpi pm3.45 alimi moim df-rmo
      3syl 3imtr4g ) CDGZBHZDIZAJZBKZUDCIZAJZBKZABDLABCLUCUHUEMZBNZUIUFMZBNUGUJ
      MUCULBCDEFOPUKUMBUHUEAQRUIUFBSUAABDTABCTUB $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Quantification restricted to a subclass.  (Contributed by NM,
       11-Mar-2006.) $)
    ssralv $p |- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) ) $=
      ( wss cv wcel ssel imim1d ralimdv2 ) CDEZAABDCKBFZCGLDGACDLHIJ $.

    $( Existential quantification restricted to a subclass.  (Contributed by
       NM, 11-Jan-2007.) $)
    ssrexv $p |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) ) $=
      ( wss cv wcel ssel anim1d reximdv2 ) CDEZAABCDKBFZCGLDGACDLHIJ $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Restricted universal quantification on a subset in terms of superset.
       (Contributed by Stefan O'Rear, 3-Apr-2015.) $)
    ralss $p |- ( A C_ B -> ( A. x e. A ph <->
          A. x e. B ( x e. A -> ph ) ) ) $=
      ( wss cv wcel wi wa ssel pm4.71rd imbi1d impexp bitrdi ralbidv2 ) CDEZABF
      ZCGZAHZBCDPSQDGZRIZAHTSHPRUAAPRTCDQJKLTRAMNO $.

    $( Restricted existential quantification on a subset in terms of superset.
       (Contributed by Stefan O'Rear, 3-Apr-2015.) $)
    rexss $p |- ( A C_ B -> ( E. x e. A ph <->
          E. x e. B ( x e. A /\ ph ) ) ) $=
      ( wss cv wcel wa ssel pm4.71rd anbi1d anass bitrdi rexbidv2 ) CDEZABFZCGZ
      AHZBCDORPDGZQHZAHSRHOQTAOQSCDPIJKSQALMN $.
  $}

  $( Class abstractions in a subclass relationship.  (Contributed by NM,
     3-Jul-1994.) $)
  ss2ab $p |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) ) $=
    ( cab wss cv wcel wi wal nfab1 dfss2f abid imbi12i albii bitri ) ACDZBCDZEC
    FZPGZRQGZHZCIABHZCICPQACJBCJKUAUBCSATBACLBCLMNO $.

  ${
    $d x A $.
    $( Class abstraction in a subclass relationship.  (Contributed by NM,
       16-Aug-2006.) $)
    abss $p |- ( { x | ph } C_ A <-> A. x ( ph -> x e. A ) ) $=
      ( cab wss cv wcel wi wal abid2 sseq2i ss2ab bitr3i ) ABDZCENBFCGZBDZEAOHB
      IPCNBCJKAOBLM $.

    $( Subclass of a class abstraction.  (Contributed by NM, 16-Aug-2006.) $)
    ssab $p |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) ) $=
      ( cab wss cv wcel wi wal abid2 sseq1i ss2ab bitr3i ) CABDZEBFCGZBDZNEOAHB
      IPCNBCJKOABLM $.

    $( The relation for a subclass of a class abstraction is equivalent to
       restricted quantification.  (Contributed by NM, 6-Sep-2006.) $)
    ssabral $p |- ( A C_ { x | ph } <-> A. x e. A ph ) $=
      ( cab wss cv wcel wi wal wral ssab df-ral bitr4i ) CABDEBFCGAHBIABCJABCKA
      BCLM $.
  $}

  ${
    ss2abi.1 $e |- ( ph -> ps ) $.
    $( Inference of abstraction subclass from implication.  (Contributed by NM,
       31-Mar-1995.) $)
    ss2abi $p |- { x | ph } C_ { x | ps } $=
      ( cab wss wi ss2ab mpgbir ) ACEBCEFABGCABCHDI $.
  $}

  ${
    $d x ph $.
    ss2abdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction of abstraction subclass from implication.  (Contributed by NM,
       29-Jul-2011.) $)
    ss2abdv $p |- ( ph -> { x | ps } C_ { x | ch } ) $=
      ( wi wal cab wss alrimiv ss2ab sylibr ) ABCFZDGBDHCDHIAMDEJBCDKL $.
  $}

  ${
    $d x ph $.  $d x A $.
    abssdv.1 $e |- ( ph -> ( ps -> x e. A ) ) $.
    $( Deduction of abstraction subclass from implication.  (Contributed by NM,
       20-Jan-2006.) $)
    abssdv $p |- ( ph -> { x | ps } C_ A ) $=
      ( cv wcel wi wal cab wss alrimiv abss sylibr ) ABCFDGHZCIBCJDKAOCELBCDMN
      $.
  $}

  ${
    $d x A $.
    abssi.1 $e |- ( ph -> x e. A ) $.
    $( Inference of abstraction subclass from implication.  (Contributed by NM,
       20-Jan-2006.) $)
    abssi $p |- { x | ph } C_ A $=
      ( cab cv wcel ss2abi abid2 sseqtri ) ABEBFCGZBECAKBDHBCIJ $.
  $}

  $( Restricted abstraction classes in a subclass relationship.  (Contributed
     by NM, 30-May-1999.) $)
  ss2rab $p |- ( { x e. A | ph } C_ { x e. A | ps } <->
               A. x e. A ( ph -> ps ) ) $=
    ( crab wss cv wcel wa cab wi wal df-rab sseq12i ss2ab df-ral imdistan albii
    wral bitr2i 3bitri ) ACDEZBCDEZFCGDHZAIZCJZUDBIZCJZFUEUGKZCLZABKZCDSZUBUFUC
    UHACDMBCDMNUEUGCOULUDUKKZCLUJUKCDPUMUICUDABQRTUA $.

  ${
    $d x B $.
    $( Restricted class abstraction in a subclass relationship.  (Contributed
       by NM, 16-Aug-2006.) $)
    rabss $p |- ( { x e. A | ph } C_ B <-> A. x e. A ( ph -> x e. B ) ) $=
      ( crab wss cv wcel wa cab wal wral df-rab sseq1i abss impexp albii df-ral
      wi bitr4i 3bitri ) ABCEZDFBGZCHZAIZBJZDFUEUCDHZSZBKZAUGSZBCLZUBUFDABCMNUE
      BDOUIUDUJSZBKUKUHULBUDAUGPQUJBCRTUA $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Subclass of a restricted class abstraction.  (Contributed by NM,
       16-Aug-2006.) $)
    ssrab $p |- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) ) $=
      ( crab wss cv wcel wa cab wal wral df-rab sseq2i ssab dfss3 anbi1i r19.26
      wi df-ral 3bitr2ri 3bitri ) DABCEZFDBGZCHZAIZBJZFUDDHUFSBKZDCFZABDLZIZUCU
      GDABCMNUFBDOUKUEBDLZUJIUFBDLUHUIULUJBDCPQUEABDRUFBDTUAUB $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    ssrabdv.1 $e |- ( ph -> B C_ A ) $.
    ssrabdv.2 $e |- ( ( ph /\ x e. B ) -> ps ) $.
    $( Subclass of a restricted class abstraction (deduction form).
       (Contributed by NM, 31-Aug-2006.) $)
    ssrabdv $p |- ( ph -> B C_ { x e. A | ps } ) $=
      ( wss wral crab ralrimiva ssrab sylanbrc ) AEDHBCEIEBCDJHFABCEGKBCDELM $.
  $}

  ${
    $d x B $.  $d x ph $.
    rabssdv.1 $e |- ( ( ph /\ x e. A /\ ps ) -> x e. B ) $.
    $( Subclass of a restricted class abstraction (deduction form).
       (Contributed by NM, 2-Feb-2015.) $)
    rabssdv $p |- ( ph -> { x e. A | ps } C_ B ) $=
      ( cv wcel wi wral crab wss 3exp ralrimiv rabss sylibr ) ABCGZEHZIZCDJBCDK
      ELASCDAQDHBRFMNBCDEOP $.
  $}

  ${
    $d x ph $.
    ss2rabdv.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
    $( Deduction of restricted abstraction subclass from implication.
       (Contributed by NM, 30-May-2006.) $)
    ss2rabdv $p |- ( ph -> { x e. A | ps } C_ { x e. A | ch } ) $=
      ( wi wral crab wss ralrimiva ss2rab sylibr ) ABCGZDEHBDEICDEIJANDEFKBCDEL
      M $.
  $}

  ${
    ss2rabi.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Inference of restricted abstraction subclass from implication.
       (Contributed by NM, 14-Oct-1999.) $)
    ss2rabi $p |- { x e. A | ph } C_ { x e. A | ps } $=
      ( crab wss wi ss2rab mprgbir ) ACDFBCDFGABHCDABCDIEJ $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Subclass law for restricted abstraction.  (Contributed by NM,
       18-Dec-2004.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    rabss2 $p |- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } ) $=
      ( wss cv wcel wa cab crab wi wal pm3.45 alimi ssalel ss2ab 3imtr4i df-rab
      3sstr4g ) CDEZBFZCGZAHZBIZUADGZAHZBIZABCJABDJUBUEKZBLUCUFKZBLTUDUGEUHUIBU
      BUEAMNBCDOUCUFBPQABCRABDRS $.

    $( Subclass relation for the restriction of a class abstraction.  Prefer
       using the more natural statement ~ ssrab2 .  (Contributed by NM,
       31-Mar-1995.) $)
    ssab2 $p |- { x | ( x e. A /\ ph ) } C_ A $=
      ( cv wcel wa simpl abssi ) BDCEZAFBCIAGH $.

    $( Subclass relation for a restricted class.  (Contributed by NM,
       19-Mar-1997.) $)
    ssrab2 $p |- { x e. A | ph } C_ A $=
      ( crab cv wcel wa cab df-rab ssab2 eqsstri ) ABCDBECFAGBHCABCIABCJK $.
  $}

  ${
    $d A x $.
    ssrab3.1 $e |- B = { x e. A | ph } $.
    $( Subclass relation for a restricted class abstraction.  (Contributed by
       Jonathan Ben-Naim, 3-Jun-2011.) $)
    ssrab3 $p |- B C_ A $=
      ( crab ssrab2 eqsstri ) DABCFCEABCGH $.
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.
    rabssrabd.1 $e |- ( ph -> A C_ B ) $.
    rabssrabd.2 $e |- ( ( ph /\ ps /\ x e. A ) -> ch ) $.
    $( Subclass of a restricted class abstraction.  (Contributed by AV,
       4-Jun-2022.) $)
    rabssrabd $p |- ( ph -> { x e. A | ps } C_ { x e. B | ch } ) $=
      ( crab cv wcel wa w3a 3anan32 sylbir ex ss2rabdv wss rabss2 syl sstrd ) A
      BDEICDEIZCDFIZABCDEADJEKZLZBCUEBLABUDMCABUDNHOPQAEFRUBUCRGCDEFSTUA $.
  $}

  ${
    $d V x $.
    $( If the restricting class of a restricted class abstraction is a subset
       of this restricted class abstraction, it is equal to this restricted
       class abstraction.  (Contributed by Alexander van der Vekens,
       31-Dec-2017.) $)
    ssrabeq $p |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } ) $=
      ( crab wss wa wceq ssrab2 biantru eqss bitr4i ) CABCDZEZMLCEZFCLGNMABCHIC
      LJK $.
  $}

  $( A restricted class is a subclass of the corresponding unrestricted class.
     (Contributed by Mario Carneiro, 23-Dec-2016.) $)
  rabssab $p |- { x e. A | ph } C_ { x | ph } $=
    ( crab cv wcel wa cab df-rab simpr ss2abi eqsstri ) ABCDBECFZAGZBHABHABCINA
    BMAJKL $.

  ${
    $d x y $.  $d y z A $.  $d y z B $.  $d x z C $.
    $( A subset relationship useful for converting union to indexed union using
       dfiun2 or dfiun2g and intersection to indexed intersection using
       dfiin2 .  (Contributed by NM, 5-Oct-2006.)  (Proof shortened by Mario
       Carneiro, 26-Sep-2015.) $)
    uniiunlem $p |- ( A. x e. A B e. D ->
                     ( A. x e. A B e. C <-> { y | E. x e. A y = B } C_ C ) ) $=
      ( vz cv wceq wrex cab wss wcel wi wal wral eqeq1 rexbidv cbvabv wb sseq1i
      r19.23v albii ralcom4 abss 3bitr4i bitr4i nfv eleq1 ceqsalg ralbi bitr2id
      ralimi syl ) BHZDIZACJZBKZELZGHZDIZUTEMZNZGOZACPZDFMZACPZDEMZACPZUSVAACJZ
      GKZELZVEURVKEUQVJBGUOUTIUPVAACUOUTDQRSUAVCACPZGOVJVBNZGOVEVLVMVNGVAVBACUB
      UCVCAGCUDVJGEUEUFUGVGVDVHTZACPVEVITVFVOACVBVHGDFVHGUHUTDEUIUJUMVDVHACUKUN
      UL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The difference, union, and intersection of two classes
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  The difference of two classes
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d A x $.  $d B x y $.
    $( Alternate definition of class difference.  Definition of relative set
       complement in Section 2.3 of [Pierik], p. 10.  (Contributed by BJ and
       Jim Kingdon, 16-Jun-2022.) $)
    dfdif3 $p |- ( A \ B ) = { x e. A | A. y e. B x =/= y } $=
      ( cdif cv wcel wn crab wne wral dfdif2 wi wal weq wex a9ev bitr4i 3bitri
      wa biantrur 19.41v equcom imbi1i eleq1w notbid pm5.74i con2b df-ne bicomi
      sb56 imbi2i bitri albii df-ral rabbii eqtri ) CDEAFZDGZHZACIURBFZJZBDKZAC
      IACDLUTVCACUTVADGZVBMZBNZVCUTBAOZUTTBPZVGUTMZBNVFUTVGBPZUTTVHVJUTBAQUAVGU
      TBUBRUTBAUKVIVEBVIABOZUTMZVEVGVKUTBAUCUDVLVKVDHZMVDVKHZMVEVKUTVMVKUSVDABD
      UEUFUGVKVDUHVNVBVDVBVNURVAUIUJULSUMUNSVBBDUORUPUQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Equality theorem for class difference.  (Contributed by NM,
       10-Feb-1997.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    difeq1 $p |- ( A = B -> ( A \ C ) = ( B \ C ) ) $=
      ( vx wceq cv wcel wn crab cdif rabeq dfdif2 3eqtr4g ) ABEDFCGHZDAINDBIACJ
      BCJNDABKDACLDBCLM $.

    $( Equality theorem for class difference.  (Contributed by NM,
       10-Feb-1997.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    difeq2 $p |- ( A = B -> ( C \ A ) = ( C \ B ) ) $=
      ( vx wceq cv wcel wn crab cdif eleq2 notbid rabbidv dfdif2 3eqtr4g ) ABEZ
      DFZAGZHZDCIQBGZHZDCICAJCBJPSUADCPRTABQKLMDCANDCBNO $.
  $}

  $( Equality theorem for class difference.  (Contributed by FL,
     31-Aug-2009.) $)
  difeq12 $p |- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) ) $=
    ( wceq cdif difeq1 difeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.

  ${
    difeq1i.1 $e |- A = B $.
    $( Inference adding difference to the right in a class equality.
       (Contributed by NM, 15-Nov-2002.) $)
    difeq1i $p |- ( A \ C ) = ( B \ C ) $=
      ( wceq cdif difeq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Inference adding difference to the left in a class equality.
       (Contributed by NM, 15-Nov-2002.) $)
    difeq2i $p |- ( C \ A ) = ( C \ B ) $=
      ( wceq cdif difeq2 ax-mp ) ABECAFCBFEDABCGH $.

    ${
      difeq12i.2 $e |- C = D $.
      $( Equality inference for class difference.  (Contributed by NM,
         29-Aug-2004.) $)
      difeq12i $p |- ( A \ C ) = ( B \ D ) $=
        ( cdif difeq1i difeq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
    $}
  $}

  ${
    difeq1d.1 $e |- ( ph -> A = B ) $.
    $( Deduction adding difference to the right in a class equality.
       (Contributed by NM, 15-Nov-2002.) $)
    difeq1d $p |- ( ph -> ( A \ C ) = ( B \ C ) ) $=
      ( wceq cdif difeq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Deduction adding difference to the left in a class equality.
       (Contributed by NM, 15-Nov-2002.) $)
    difeq2d $p |- ( ph -> ( C \ A ) = ( C \ B ) ) $=
      ( wceq cdif difeq2 syl ) ABCFDBGDCGFEBCDHI $.
  $}

  ${
    difeq12d.1 $e |- ( ph -> A = B ) $.
    difeq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality deduction for class difference.  (Contributed by FL,
       29-May-2014.) $)
    difeq12d $p |- ( ph -> ( A \ C ) = ( B \ D ) ) $=
      ( cdif difeq1d difeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    difeqri.1 $e |- ( ( x e. A /\ -. x e. B ) <-> x e. C ) $.
    $( Inference from membership to difference.  (Contributed by NM,
       17-May-1998.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    difeqri $p |- ( A \ B ) = C $=
      ( cdif cv wcel wn wa eldif bitri eqriv ) ABCFZDAGZNHOBHOCHIJODHOBCKELM $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    nfdif.1 $e |- F/_ x A $.
    nfdif.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for class difference.  (Contributed by
       NM, 3-Dec-2003.)  (Revised by Mario Carneiro, 13-Oct-2016.) $)
    nfdif $p |- F/_ x ( A \ B ) $=
      ( vy cdif cv wcel wn crab dfdif2 nfcri nfn nfrabw nfcxfr ) ABCGFHCIZJZFBK
      FBCLRAFBQAAFCEMNDOP $.
  $}

  $( Implication of membership in a class difference.  (Contributed by NM,
     29-Apr-1994.) $)
  eldifi $p |- ( A e. ( B \ C ) -> A e. B ) $=
    ( cdif wcel wn eldif simplbi ) ABCDEABEACEFABCGH $.

  $( Implication of membership in a class difference.  (Contributed by NM,
     3-May-1994.) $)
  eldifn $p |- ( A e. ( B \ C ) -> -. A e. C ) $=
    ( cdif wcel wn eldif simprbi ) ABCDEABEACEFABCGH $.

  $( A set does not belong to a class excluding it.  (Contributed by NM,
     27-Jun-1994.) $)
  elndif $p |- ( A e. B -> -. A e. ( C \ B ) ) $=
    ( cdif wcel eldifn con2i ) ACBDEABEACBFG $.

  ${
    $d x A $.  $d x B $.
    $( Double class difference.  Exercise 11 of [TakeutiZaring] p. 22.
       (Contributed by NM, 17-May-1998.) $)
    difdif $p |- ( A \ ( B \ A ) ) = A $=
      ( vx cdif cv wcel wn wa simpl pm4.45im imanim eldif sylnibr anim2i impbii
      wi sylbi difeqri ) CABADZACEZAFZTSFZGZHZUAUAUCIUAUATBFZUAPZHUDUAUEJUFUCUA
      UFUEUAGHUBUEUAKTBALMNQOR $.

    $( Subclass relationship for class difference.  Exercise 14 of
       [TakeutiZaring] p. 22.  (Contributed by NM, 29-Apr-1994.) $)
    difss $p |- ( A \ B ) C_ A $=
      ( vx cdif cv eldifi ssriv ) CABDACEABFG $.
  $}

  $( A difference of two classes is contained in the minuend.  Deduction form
     of ~ difss .  (Contributed by David Moews, 1-May-2017.) $)
  difssd $p |- ( ph -> ( A \ B ) C_ A ) $=
    ( cdif wss difss a1i ) BCDBEABCFG $.

  $( If a class is contained in a difference, it is contained in the minuend.
     (Contributed by David Moews, 1-May-2017.) $)
  difss2 $p |- ( A C_ ( B \ C ) -> A C_ B ) $=
    ( cdif wss id difss sstrdi ) ABCDZEZAIBJFBCGH $.

  ${
    difss2d.1 $e |- ( ph -> A C_ ( B \ C ) ) $.
    $( If a class is contained in a difference, it is contained in the minuend.
       Deduction form of ~ difss2 .  (Contributed by David Moews,
       1-May-2017.) $)
    difss2d $p |- ( ph -> A C_ B ) $=
      ( cdif wss difss2 syl ) ABCDFGBCGEBCDHI $.
  $}

  $( Preservation of a subclass relationship by class difference.  (Contributed
     by NM, 15-Feb-2007.) $)
  ssdifss $p |- ( A C_ B -> ( A \ C ) C_ B ) $=
    ( cdif wss difss sstr mpan ) ACDZAEABEIBEACFIABGH $.

  ${
    $d x A $.
    ddifnel.1 $e |- ( -. x e. ( _V \ A ) -> x e. A ) $.
    $( Double complement under universal class.  The hypothesis corresponds to
       stability of membership in ` A ` , which is weaker than decidability
       (see ~ dcstab ).  Actually, the conclusion is a characterization of
       stability of membership in a class (see ~ ddifstab ) .  Exercise 4.10(s)
       of [Mendelson] p. 231, but with an additional hypothesis.  For a version
       without a hypothesis, but which only states that ` A ` is a subset of
       ` _V \ ( _V \ A ) ` , see ~ ddifss .  (Contributed by Jim Kingdon,
       21-Jul-2018.) $)
    ddifnel $p |- ( _V \ ( _V \ A ) ) = A $=
      ( cvv cdif cv wcel wn wa adantl elndif vex jctil impbii difeqri ) ADDBEZB
      AFZDGZQPGHZIQBGZSTRCJTSRQBDKALMNO $.
  $}

  ${
    $d x A $.
    $( A class is equal to its double complement if and only if it is stable
       (that is, membership in it is a stable property).  (Contributed by BJ,
       12-Dec-2021.) $)
    ddifstab $p |- ( ( _V \ ( _V \ A ) ) = A <-> A. x STAB x e. A ) $=
      ( cvv cdif wceq cv wb wal wstab dfcleq wn wi wa eldif vex biantrur bitr4i
      wcel notbii bitri 3bitr2i bibi1i biimp id notnot impbid1 impbii df-stab
      albii ) CCBDZDZBEAFZUKRZULBRZGZAHUNIZAHAUKBJUOUPAUOUNKZKZUNLZUPUOURUNGZUS
      UMURUNUMULCRZULUJRZKZMVCURULCUJNVAVCAOZPVBUQVBVAUQMUQULCBNVAUQVDPQSUAUBUT
      USURUNUCUSURUNUSUDUNUEUFUGTUNUHQUIT $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Contraposition law for subsets.  (Contributed by NM, 22-Mar-1998.) $)
    ssconb $p |- ( ( A C_ C /\ B C_ C ) ->
                  ( A C_ ( C \ B ) <-> B C_ ( C \ A ) ) ) $=
      ( vx wss wa cv wcel cdif wi wal wn wb ssel pm5.1 jcab eldif imbi2i ssalel
      3bitr4g syl2an con2b a1i anbi12d albidv ) ACEZBCEZFZDGZAHZUICBIZHZJZDKUIB
      HZUICAIZHZJZDKAUKEBUOEUHUMUQDUHUJUICHZUNLZFZJZUNURUJLZFZJZUMUQUHUJURJZUJU
      SJZFUNURJZUNVBJZFVAVDUHVEVGVFVHUFVEVGVEVGMUGACUINBCUINVEVGOUAVFVHMUHUJUNU
      BUCUDUJURUSPUNURVBPTULUTUJUICBQRUPVCUNUICAQRTUEDAUKSDBUOST $.

    $( Contraposition law for subsets.  Exercise 15 of [TakeutiZaring] p. 22.
       (Contributed by NM, 22-Mar-1998.) $)
    sscon $p |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) ) $=
      ( vx wss cdif cv wcel wn wa ssel con3d anim2d eldif 3imtr4g ssrdv ) ABEZD
      CBFZCAFZQDGZCHZTBHZIZJUATAHZIZJTRHTSHQUCUEUAQUDUBABTKLMTCBNTCANOP $.

    $( Difference law for subsets.  (Contributed by NM, 28-May-1998.) $)
    ssdif $p |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) ) $=
      ( vx wss cdif cv wcel wn wa ssel anim1d eldif 3imtr4g ssrdv ) ABEZDACFZBC
      FZPDGZAHZSCHIZJSBHZUAJSQHSRHPTUBUAABSKLSACMSBCMNO $.
  $}

  ${
    ssdifd.1 $e |- ( ph -> A C_ B ) $.
    $( If ` A ` is contained in ` B ` , then ` ( A \ C ) ` is contained in
       ` ( B \ C ) ` .  Deduction form of ~ ssdif .  (Contributed by David
       Moews, 1-May-2017.) $)
    ssdifd $p |- ( ph -> ( A \ C ) C_ ( B \ C ) ) $=
      ( wss cdif ssdif syl ) ABCFBDGCDGFEBCDHI $.

    $( If ` A ` is contained in ` B ` , then ` ( C \ B ) ` is contained in
       ` ( C \ A ) ` .  Deduction form of ~ sscon .  (Contributed by David
       Moews, 1-May-2017.) $)
    sscond $p |- ( ph -> ( C \ B ) C_ ( C \ A ) ) $=
      ( wss cdif sscon syl ) ABCFDCGDBGFEBCDHI $.

    $( If ` A ` is contained in ` B ` , then ` ( A \ C ) ` is also contained in
       ` B ` .  Deduction form of ~ ssdifss .  (Contributed by David Moews,
       1-May-2017.) $)
    ssdifssd $p |- ( ph -> ( A \ C ) C_ B ) $=
      ( wss cdif ssdifss syl ) ABCFBDGCFEBCDHI $.

    ssdif2d.2 $e |- ( ph -> C C_ D ) $.
    $( If ` A ` is contained in ` B ` and ` C ` is contained in ` D ` , then
       ` ( A \ D ) ` is contained in ` ( B \ C ) ` .  Deduction form.
       (Contributed by David Moews, 1-May-2017.) $)
    ssdif2d $p |- ( ph -> ( A \ D ) C_ ( B \ C ) ) $=
      ( cdif sscond ssdifd sstrd ) ABEHBDHCDHADEBGIABCDFJK $.
  $}

  $( Restricted universal quantification on a class difference in terms of an
     implication.  (Contributed by Alexander van der Vekens, 3-Jan-2018.) $)
  raldifb $p |- ( A. x e. A ( x e/ B -> ph ) <-> A. x e. ( A \ B ) ph ) $=
    ( cv wnel wi cdif wcel wa impexp bicomi wn df-nel anbi2i eldif bitri imbi1i
    ralbii2 ) BEZDFZAGZABCCDHZTCIZUBGZUDUAJZAGZTUCIZAGUGUEUDUAAKLUFUHAUFUDTDIMZ
    JZUHUAUIUDTDNOUHUJTCDPLQRQS $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  The union of two classes
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Expansion of membership in class union.  Theorem 12 of [Suppes] p. 25.
       (Contributed by NM, 7-Aug-1994.) $)
    elun $p |- ( A e. ( B u. C ) <-> ( A e. B \/ A e. C ) ) $=
      ( vx cun wcel cvv wo elex jaoi wceq eleq1 orbi12d df-un elab2g pm5.21nii
      cv ) ABCEZFAGFZABFZACFZHZARITSUAABIACIJDQZBFZUCCFZHUBDARGUCAKUDTUEUAUCABL
      UCACLMDBCNOP $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    uneqri.1 $e |- ( ( x e. A \/ x e. B ) <-> x e. C ) $.
    $( Inference from membership to union.  (Contributed by NM, 5-Aug-1993.) $)
    uneqri $p |- ( A u. B ) = C $=
      ( cun cv wcel wo elun bitri eqriv ) ABCFZDAGZMHNBHNCHINDHNBCJEKL $.
  $}

  ${
    $d x A $.
    $( Idempotent law for union of classes.  Theorem 23 of [Suppes] p. 27.
       (Contributed by NM, 5-Aug-1993.) $)
    unidm $p |- ( A u. A ) = A $=
      ( vx cv wcel oridm uneqri ) BAAABCADEF $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Commutative law for union of classes.  Exercise 6 of [TakeutiZaring]
       p. 17.  (Contributed by NM, 25-Jun-1998.)  (Proof shortened by Andrew
       Salmon, 26-Jun-2011.) $)
    uncom $p |- ( A u. B ) = ( B u. A ) $=
      ( vx cun cv wcel wo orcom elun bitr4i uneqri ) CABBADZCEZAFZMBFZGONGMLFNO
      HMBAIJK $.
  $}

  $( If a class equals the union of two other classes, then it equals the union
     of those two classes commuted.  (Contributed by Alan Sare,
     18-Feb-2012.) $)
  equncom $p |- ( A = ( B u. C ) <-> A = ( C u. B ) ) $=
    ( cun uncom eqeq2i ) BCDCBDABCEF $.

  ${
    equncomi.1 $e |- A = ( B u. C ) $.
    $( Inference form of ~ equncom .  (Contributed by Alan Sare,
       18-Feb-2012.) $)
    equncomi $p |- A = ( C u. B ) $=
      ( cun wceq equncom mpbi ) ABCEFACBEFDABCGH $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Equality theorem for union of two classes.  (Contributed by NM,
       5-Aug-1993.) $)
    uneq1 $p |- ( A = B -> ( A u. C ) = ( B u. C ) ) $=
      ( vx wceq cun cv wcel wo eleq2 orbi1d elun 3bitr4g eqrdv ) ABEZDACFZBCFZO
      DGZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
  $}

  $( Equality theorem for the union of two classes.  (Contributed by NM,
     5-Aug-1993.) $)
  uneq2 $p |- ( A = B -> ( C u. A ) = ( C u. B ) ) $=
    ( wceq cun uneq1 uncom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.

  $( Equality theorem for union of two classes.  (Contributed by NM,
     29-Mar-1998.) $)
  uneq12 $p |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) ) $=
    ( wceq cun uneq1 uneq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.

  ${
    uneq1i.1 $e |- A = B $.
    $( Inference adding union to the right in a class equality.  (Contributed
       by NM, 30-Aug-1993.) $)
    uneq1i $p |- ( A u. C ) = ( B u. C ) $=
      ( wceq cun uneq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Inference adding union to the left in a class equality.  (Contributed by
       NM, 30-Aug-1993.) $)
    uneq2i $p |- ( C u. A ) = ( C u. B ) $=
      ( wceq cun uneq2 ax-mp ) ABECAFCBFEDABCGH $.

    ${
      uneq12i.2 $e |- C = D $.
      $( Equality inference for union of two classes.  (Contributed by NM,
         12-Aug-2004.)  (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
      uneq12i $p |- ( A u. C ) = ( B u. D ) $=
        ( wceq cun uneq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
    $}
  $}

  ${
    uneq1d.1 $e |- ( ph -> A = B ) $.
    $( Deduction adding union to the right in a class equality.  (Contributed
       by NM, 29-Mar-1998.) $)
    uneq1d $p |- ( ph -> ( A u. C ) = ( B u. C ) ) $=
      ( wceq cun uneq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Deduction adding union to the left in a class equality.  (Contributed by
       NM, 29-Mar-1998.) $)
    uneq2d $p |- ( ph -> ( C u. A ) = ( C u. B ) ) $=
      ( wceq cun uneq2 syl ) ABCFDBGDCGFEBCDHI $.

    ${
      uneq12d.2 $e |- ( ph -> C = D ) $.
      $( Equality deduction for union of two classes.  (Contributed by NM,
         29-Sep-2004.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
      uneq12d $p |- ( ph -> ( A u. C ) = ( B u. D ) ) $=
        ( wceq cun uneq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
    $}
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    nfun.1 $e |- F/_ x A $.
    nfun.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for the union of classes.
       (Contributed by NM, 15-Sep-2003.)  (Revised by Mario Carneiro,
       14-Oct-2016.) $)
    nfun $p |- F/_ x ( A u. B ) $=
      ( vy cun cv wcel wo cab df-un nfcri nfor nfab nfcxfr ) ABCGFHZBIZQCIZJZFK
      FBCLTAFRSAAFBDMAFCEMNOP $.
  $}

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Associative law for union of classes.  Exercise 8 of [TakeutiZaring]
       p. 17.  (Contributed by NM, 3-May-1994.)  (Proof shortened by Andrew
       Salmon, 26-Jun-2011.) $)
    unass $p |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) ) $=
      ( vx cun cv wcel wo elun orbi2i orbi1i orass bitr2i 3bitrri uneqri ) DABE
      ZCABCEZEZDFZRGSAGZSQGZHTSBGZSCGZHZHZSPGZUCHZSAQIUAUDTSBCIJUGTUBHZUCHUEUFU
      HUCSABIKTUBUCLMNO $.
  $}

  $( A rearrangement of union.  (Contributed by NM, 12-Aug-2004.) $)
  un12 $p |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) ) $=
    ( cun uncom uneq1i unass 3eqtr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH
    $.

  $( A rearrangement of union.  (Contributed by NM, 12-Aug-2004.)  (Proof
     shortened by Andrew Salmon, 26-Jun-2011.) $)
  un23 $p |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B ) $=
    ( cun unass un12 uncom 3eqtri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.

  $( A rearrangement of the union of 4 classes.  (Contributed by NM,
     12-Aug-2004.) $)
  un4 $p |- ( ( A u. B ) u. ( C u. D ) ) =
            ( ( A u. C ) u. ( B u. D ) ) $=
    ( cun un12 uneq2i unass 3eqtr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHA
    CLHI $.

  $( Union distributes over itself.  (Contributed by NM, 17-Aug-2004.) $)
  unundi $p |- ( A u. ( B u. C ) ) = ( ( A u. B ) u. ( A u. C ) ) $=
    ( cun unidm uneq1i un4 eqtr3i ) AADZBCDZDAJDABDACDDIAJAEFAABCGH $.

  $( Union distributes over itself.  (Contributed by NM, 17-Aug-2004.) $)
  unundir $p |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. ( B u. C ) ) $=
    ( cun unidm uneq2i un4 eqtr3i ) ABDZCCDZDICDACDBCDDJCICEFABCCGH $.

  ${
    $d x A $.  $d x B $.
    $( Subclass relationship for union of classes.  Theorem 25 of [Suppes]
       p. 27.  (Contributed by NM, 5-Aug-1993.) $)
    ssun1 $p |- A C_ ( A u. B ) $=
      ( vx cun cv wcel wo orc elun sylibr ssriv ) CAABDZCEZAFZNMBFZGMLFNOHMABIJ
      K $.
  $}

  $( Subclass relationship for union of classes.  (Contributed by NM,
     30-Aug-1993.) $)
  ssun2 $p |- A C_ ( B u. A ) $=
    ( cun ssun1 uncom sseqtri ) AABCBACABDABEF $.

  $( Subclass law for union of classes.  (Contributed by NM, 5-Aug-1993.) $)
  ssun3 $p |- ( A C_ B -> A C_ ( B u. C ) ) $=
    ( wss cun ssun1 sstr2 mpi ) ABDBBCEZDAIDBCFABIGH $.

  $( Subclass law for union of classes.  (Contributed by NM, 14-Aug-1994.) $)
  ssun4 $p |- ( A C_ B -> A C_ ( C u. B ) ) $=
    ( wss cun ssun2 sstr2 mpi ) ABDBCBEZDAIDBCFABIGH $.

  $( Membership law for union of classes.  (Contributed by NM, 5-Aug-1993.) $)
  elun1 $p |- ( A e. B -> A e. ( B u. C ) ) $=
    ( cun ssun1 sseli ) BBCDABCEF $.

  $( Membership law for union of classes.  (Contributed by NM, 30-Aug-1993.) $)
  elun2 $p |- ( A e. B -> A e. ( C u. B ) ) $=
    ( cun ssun2 sseli ) BCBDABCEF $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Subclass law for union of classes.  (Contributed by NM, 14-Oct-1999.)
       (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    unss1 $p |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) ) $=
      ( vx wss cun cv wcel wo ssel orim1d elun 3imtr4g ssrdv ) ABEZDACFZBCFZODG
      ZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.

    $( A relationship between subclass and union.  Theorem 26 of [Suppes]
       p. 27.  (Contributed by NM, 30-Aug-1993.)  (Proof shortened by Andrew
       Salmon, 26-Jun-2011.) $)
    ssequn1 $p |- ( A C_ B <-> ( A u. B ) = B ) $=
      ( vx cv wcel wi wal cun wb wceq wo bicom pm4.72 elun bibi1i 3bitr4i albii
      wss ssalel dfcleq ) CDZAEZUABEZFZCGUAABHZEZUCIZCGABRUEBJUDUGCUCUBUCKZIUHU
      CIUDUGUCUHLUBUCMUFUHUCUAABNOPQCABSCUEBTP $.
  $}

  $( Subclass law for union of classes.  Exercise 7 of [TakeutiZaring] p. 18.
     (Contributed by NM, 14-Oct-1999.) $)
  unss2 $p |- ( A C_ B -> ( C u. A ) C_ ( C u. B ) ) $=
    ( wss cun unss1 uncom 3sstr4g ) ABDACEBCECAECBEABCFCAGCBGH $.

  $( Subclass law for union of classes.  (Contributed by NM, 2-Jun-2004.) $)
  unss12 $p |- ( ( A C_ B /\ C C_ D ) -> ( A u. C ) C_ ( B u. D ) ) $=
    ( wss cun unss1 unss2 sylan9ss ) ABECDEACFBCFBDFABCGCDBHI $.

  $( A relationship between subclass and union.  (Contributed by NM,
     13-Jun-1994.) $)
  ssequn2 $p |- ( A C_ B <-> ( B u. A ) = B ) $=
    ( wss cun wceq ssequn1 uncom eqeq1i bitri ) ABCABDZBEBADZBEABFJKBABGHI $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( The union of two subclasses is a subclass.  Theorem 27 of [Suppes] p. 27
       and its converse.  (Contributed by NM, 11-Jun-2004.) $)
    unss $p |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C ) $=
      ( vx cun wss cv wcel wi wa ssalel 19.26 wo elun imbi1i jaob bitri anbi12i
      wal albii 3bitr4i bitr2i ) ABEZCFDGZUCHZUDCHZIZDSZACFZBCFZJZDUCCKUDAHZUFI
      ZUDBHZUFIZJZDSUMDSZUODSZJUHUKUMUODLUGUPDUGULUNMZUFIUPUEUSUFUDABNOULUFUNPQ
      TUIUQUJURDACKDBCKRUAUB $.
  $}

  ${
    unssi.1 $e |- A C_ C $.
    unssi.2 $e |- B C_ C $.
    $( An inference showing the union of two subclasses is a subclass.
       (Contributed by Raph Levien, 10-Dec-2002.) $)
    unssi $p |- ( A u. B ) C_ C $=
      ( wss wa cun pm3.2i unss mpbi ) ACFZBCFZGABHCFLMDEIABCJK $.
  $}

  ${
    unssd.1 $e |- ( ph -> A C_ C ) $.
    unssd.2 $e |- ( ph -> B C_ C ) $.
    $( A deduction showing the union of two subclasses is a subclass.
       (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
    unssd $p |- ( ph -> ( A u. B ) C_ C ) $=
      ( wss cun wa unss biimpi syl2anc ) ABDGZCDGZBCHDGZEFMNIOBCDJKL $.
  $}

  ${
    unssad.1 $e |- ( ph -> ( A u. B ) C_ C ) $.
    $( If ` ( A u. B ) ` is contained in ` C ` , so is ` A ` .  One-way
       deduction form of ~ unss .  Partial converse of ~ unssd .  (Contributed
       by David Moews, 1-May-2017.) $)
    unssad $p |- ( ph -> A C_ C ) $=
      ( wss cun wa unss sylibr simpld ) ABDFZCDFZABCGDFLMHEBCDIJK $.

    $( If ` ( A u. B ) ` is contained in ` C ` , so is ` B ` .  One-way
       deduction form of ~ unss .  Partial converse of ~ unssd .  (Contributed
       by David Moews, 1-May-2017.) $)
    unssbd $p |- ( ph -> B C_ C ) $=
      ( wss cun wa unss sylibr simprd ) ABDFZCDFZABCGDFLMHEBCDIJK $.
  $}

  $( A condition that implies inclusion in the union of two classes.
     (Contributed by NM, 23-Nov-2003.) $)
  ssun $p |- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) ) $=
    ( wss cun ssun3 ssun4 jaoi ) ABDABCEDACDABCFACBGH $.

  $( Restricted existential quantification over union.  (Contributed by Jeff
     Madsen, 5-Jan-2011.) $)
  rexun $p |- ( E. x e. ( A u. B ) ph <->
                          ( E. x e. A ph \/ E. x e. B ph ) ) $=
    ( cun wrex cv wcel wa wo df-rex 19.43 elun anbi1i andir bitri exbii orbi12i
    wex 3bitr4i ) ABCDEZFBGZUAHZAIZBSZABCFZABDFZJZABUAKUBCHZAIZUBDHZAIZJZBSUJBS
    ZULBSZJUEUHUJULBLUDUMBUDUIUKJZAIUMUCUPAUBCDMNUIUKAOPQUFUNUGUOABCKABDKRTP $.

  $( Restricted quantification over a union.  (Contributed by Scott Fenton,
     12-Apr-2011.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
  ralunb $p |- ( A. x e. ( A u. B ) ph <->
                 ( A. x e. A ph /\ A. x e. B ph ) ) $=
    ( cv cun wcel wi wal wa wral wo elun imbi1i jaob bitri albii df-ral anbi12i
    19.26 3bitr4i ) BEZCDFZGZAHZBIZUBCGZAHZBIZUBDGZAHZBIZJZABUCKABCKZABDKZJUFUH
    UKJZBIUMUEUPBUEUGUJLZAHUPUDUQAUBCDMNUGAUJOPQUHUKBTPABUCRUNUIUOULABCRABDRSUA
    $.

  $( Restricted quantification over union.  (Contributed by Jeff Madsen,
     2-Sep-2009.) $)
  ralun $p |- ( ( A. x e. A ph /\ A. x e. B ph ) -> A. x e. ( A u. B ) ph ) $=
    ( cun wral wa ralunb biimpri ) ABCDEFABCFABDFGABCDHI $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  The intersection of two classes
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Expansion of membership in an intersection of two classes.  Theorem 12
       of [Suppes] p. 25.  (Contributed by NM, 29-Apr-1994.) $)
    elin $p |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) ) $=
      ( vx cin wcel wa elex adantl cv wceq eleq1 anbi12d df-in elab2g pm5.21nii
      cvv ) ABCEZFAQFZABFZACFZGZARHUASTACHIDJZBFZUCCFZGUBDARQUCAKUDTUEUAUCABLUC
      ACLMDBCNOP $.
  $}

  ${
    elini.1 $e |- A e. B $.
    elini.2 $e |- A e. C $.
    $( Membership in an intersection of two classes.  (Contributed by Glauco
       Siliprandi, 17-Aug-2020.) $)
    elini $p |- A e. ( B i^i C ) $=
      ( cin wcel elin mpbir2an ) ABCFGABGACGDEABCHI $.
  $}

  ${
    elind.1 $e |- ( ph -> X e. A ) $.
    elind.2 $e |- ( ph -> X e. B ) $.
    $( Deduce membership in an intersection of two classes.  (Contributed by
       Jonathan Ben-Naim, 3-Jun-2011.) $)
    elind $p |- ( ph -> X e. ( A i^i B ) ) $=
      ( wcel cin elin sylanbrc ) ADBGDCGDBCHGEFDBCIJ $.
  $}

  $( Membership in an intersection implies membership in the first set.
     (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
  elinel1 $p |- ( A e. ( B i^i C ) -> A e. B ) $=
    ( cin wcel elin simplbi ) ABCDEABEACEABCFG $.

  $( Membership in an intersection implies membership in the second set.
     (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
  elinel2 $p |- ( A e. ( B i^i C ) -> A e. C ) $=
    ( cin wcel elin simprbi ) ABCDEABEACEABCFG $.

  ${
    elin2.x $e |- X = ( B i^i C ) $.
    $( Membership in a class defined as an intersection.  (Contributed by
       Stefan O'Rear, 29-Mar-2015.) $)
    elin2 $p |- ( A e. X <-> ( A e. B /\ A e. C ) ) $=
      ( wcel cin wa eleq2i elin bitri ) ADFABCGZFABFACFHDLAEIABCJK $.
  $}

  ${
    elin1d.1 $e |- ( ph -> X e. ( A i^i B ) ) $.
    $( Elementhood in the first set of an intersection - deduction version.
       (Contributed by Thierry Arnoux, 3-May-2020.) $)
    elin1d $p |- ( ph -> X e. A ) $=
      ( cin wcel elinel1 syl ) ADBCFGDBGEDBCHI $.

    $( Elementhood in the first set of an intersection - deduction version.
       (Contributed by Thierry Arnoux, 3-May-2020.) $)
    elin2d $p |- ( ph -> X e. B ) $=
      ( cin wcel elinel2 syl ) ADBCFGDCGEDBCHI $.
  $}

  ${
    elin3.x $e |- X = ( ( B i^i C ) i^i D ) $.
    $( Membership in a class defined as a ternary intersection.  (Contributed
       by Stefan O'Rear, 29-Mar-2015.) $)
    elin3 $p |- ( A e. X <-> ( A e. B /\ A e. C /\ A e. D ) ) $=
      ( cin wcel wa w3a elin anbi1i elin2 df-3an 3bitr4i ) ABCGZHZADHZIABHZACHZ
      IZRIAEHSTRJQUARABCKLAPDEFMSTRNO $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Commutative law for intersection of classes.  Exercise 7 of
       [TakeutiZaring] p. 17.  (Contributed by NM, 5-Aug-1993.) $)
    incom $p |- ( A i^i B ) = ( B i^i A ) $=
      ( vx cin cv wcel wa ancom elin 3bitr4i eqriv ) CABDZBADZCEZAFZNBFZGPOGNLF
      NMFOPHNABINBAIJK $.
  $}

  $( Two ways of expressing that two classes have a given intersection.  This
     is often used when that given intersection is the empty set, in which case
     the statement displays two ways of expressing that two classes are
     disjoint (when ` C = (/) ` : ` ( ( A i^i B ) = (/) <-> `
     ` ( B i^i A ) = (/) ) ` ).  (Contributed by Peter Mazsa, 22-Mar-2017.) $)
  ineqcom $p |- ( ( A i^i B ) = C <-> ( B i^i A ) = C ) $=
    ( cin incom eqeq1i ) ABDBADCABEF $.

  ${
    ineqcomi.1 $e |- ( A i^i B ) = C $.
    $( Two ways of expressing that two classes have a given intersection.
       Inference form of ~ ineqcom .  Disjointness inference when ` C = (/) ` .
       (Contributed by Peter Mazsa, 26-Mar-2017.)  (Proof shortened by SN,
       20-Sep-2024.) $)
    ineqcomi $p |- ( B i^i A ) = C $=
      ( cin incom eqtri ) BAEABECBAFDG $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    ineqri.1 $e |- ( ( x e. A /\ x e. B ) <-> x e. C ) $.
    $( Inference from membership to intersection.  (Contributed by NM,
       5-Aug-1993.) $)
    ineqri $p |- ( A i^i B ) = C $=
      ( cin cv wcel wa elin bitri eqriv ) ABCFZDAGZMHNBHNCHINDHNBCJEKL $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Equality theorem for intersection of two classes.  (Contributed by NM,
       14-Dec-1993.) $)
    ineq1 $p |- ( A = B -> ( A i^i C ) = ( B i^i C ) ) $=
      ( vx wceq cin cv wcel wa eleq2 anbi1d elin 3bitr4g eqrdv ) ABEZDACFZBCFZO
      DGZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
  $}

  $( Equality theorem for intersection of two classes.  (Contributed by NM,
     26-Dec-1993.) $)
  ineq2 $p |- ( A = B -> ( C i^i A ) = ( C i^i B ) ) $=
    ( wceq cin ineq1 incom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.

  $( Equality theorem for intersection of two classes.  (Contributed by NM,
     8-May-1994.) $)
  ineq12 $p |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) ) $=
    ( wceq cin ineq1 ineq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.

  ${
    ineq1i.1 $e |- A = B $.
    $( Equality inference for intersection of two classes.  (Contributed by NM,
       26-Dec-1993.) $)
    ineq1i $p |- ( A i^i C ) = ( B i^i C ) $=
      ( wceq cin ineq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality inference for intersection of two classes.  (Contributed by NM,
       26-Dec-1993.) $)
    ineq2i $p |- ( C i^i A ) = ( C i^i B ) $=
      ( wceq cin ineq2 ax-mp ) ABECAFCBFEDABCGH $.

    ${
      ineq12i.2 $e |- C = D $.
      $( Equality inference for intersection of two classes.  (Contributed by
         NM, 24-Jun-2004.)  (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
      ineq12i $p |- ( A i^i C ) = ( B i^i D ) $=
        ( wceq cin ineq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
    $}
  $}

  ${
    ineq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for intersection of two classes.  (Contributed by NM,
       10-Apr-1994.) $)
    ineq1d $p |- ( ph -> ( A i^i C ) = ( B i^i C ) ) $=
      ( wceq cin ineq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Equality deduction for intersection of two classes.  (Contributed by NM,
       10-Apr-1994.) $)
    ineq2d $p |- ( ph -> ( C i^i A ) = ( C i^i B ) ) $=
      ( wceq cin ineq2 syl ) ABCFDBGDCGFEBCDHI $.

    ${
      ineq12d.2 $e |- ( ph -> C = D ) $.
      $( Equality deduction for intersection of two classes.  (Contributed by
         NM, 24-Jun-2004.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
      ineq12d $p |- ( ph -> ( A i^i C ) = ( B i^i D ) ) $=
        ( wceq cin ineq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
    $}

    ${
      ineqan12d.2 $e |- ( ps -> C = D ) $.
      $( Equality deduction for intersection of two classes.  (Contributed by
         NM, 7-Feb-2007.) $)
      ineqan12d $p |- ( ( ph /\ ps ) -> ( A i^i C ) = ( B i^i D ) ) $=
        ( wceq cin ineq12 syl2an ) ACDIEFICEJDFJIBGHCDEFKL $.
    $}
  $}

  $( A frequently-used variant of subclass definition ~ df-ss .  (Contributed
     by NM, 10-Jan-2015.) $)
  dfss1 $p |- ( A C_ B <-> ( B i^i A ) = A ) $=
    ( wss cin wceq df-ss incom eqeq1i bitri ) ABCABDZAEBADZAEABFJKAABGHI $.

  $( Another definition of subclasshood.  Similar to ~ df-ss , ~ dfss , and
     ~ dfss1 .  (Contributed by David Moews, 1-May-2017.) $)
  dfss5 $p |- ( A C_ B <-> A = ( B i^i A ) ) $=
    ( wss cin wceq dfss1 eqcom bitri ) ABCBADZAEAIEABFIAGH $.

  ${
    $d x y $.  $d y A $.  $d y B $.
    nfin.1 $e |- F/_ x A $.
    nfin.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for the intersection of classes.
       (Contributed by NM, 15-Sep-2003.)  (Revised by Mario Carneiro,
       14-Oct-2016.) $)
    nfin $p |- F/_ x ( A i^i B ) $=
      ( vy cin cv wcel crab dfin5 nfcri nfrabw nfcxfr ) ABCGFHCIZFBJFBCKOAFBAFC
      ELDMN $.
  $}

  ${
    $d A y $.  $d C y $.  $d D y $.  $d x y $.
    $( Distribute proper substitution through an intersection relation.
       (Contributed by Alan Sare, 22-Jul-2012.) $)
    csbing $p |- ( A e. B -> [_ A / x ]_ ( C i^i D ) =
                  ( [_ A / x ]_ C i^i [_ A / x ]_ D ) ) $=
      ( vy cv cin csb wceq csbeq1 ineq12d eqeq12d nfcsb1v csbeq1a csbief vtoclg
      vex nfin ) AFGZDEHZIZATDIZATEIZHZJABUAIZABDIZABEIZHZJFBCTBJZUBUFUEUIATBUA
      KUJUCUGUDUHATBDKATBEKLMATUAUEFRAUCUDATDNATENSAGTJDUCEUDATDOATEOLPQ $.
  $}

  ${
    $d x ph $.  $d x A $.  $d x B $.
    rabbi2dva.1 $e |- ( ( ph /\ x e. A ) -> ( x e. B <-> ps ) ) $.
    $( Deduction from a wff to a restricted class abstraction.  (Contributed by
       NM, 14-Jan-2014.) $)
    rabbi2dva $p |- ( ph -> ( A i^i B ) = { x e. A | ps } ) $=
      ( cin cv wcel crab dfin5 rabbidva eqtrid ) ADEGCHEIZCDJBCDJCDEKANBCDFLM
      $.
  $}

  ${
    $d x A $.
    $( Idempotent law for intersection of classes.  Theorem 15 of [Suppes]
       p. 26.  (Contributed by NM, 5-Aug-1993.) $)
    inidm $p |- ( A i^i A ) = A $=
      ( vx cv wcel anidm ineqri ) BAAABCADEF $.
  $}

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Associative law for intersection of classes.  Exercise 9 of
       [TakeutiZaring] p. 17.  (Contributed by NM, 3-May-1994.) $)
    inass $p |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) ) $=
      ( vx cin cv wcel wa anass elin anbi2i bitr4i anbi1i 3bitr4i ineqri ) DABE
      ZCABCEZEZDFZAGZSBGZHZSCGZHZTSQGZHZSPGZUCHSRGUDTUAUCHZHUFTUAUCIUEUHTSBCJKL
      UGUBUCSABJMSAQJNO $.
  $}

  $( A rearrangement of intersection.  (Contributed by NM, 21-Apr-2001.) $)
  in12 $p |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) ) $=
    ( cin incom ineq1i inass 3eqtr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH
    $.

  $( A rearrangement of intersection.  (Contributed by NM, 21-Apr-2001.)
     (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
  in32 $p |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i B ) $=
    ( cin inass in12 incom 3eqtri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.

  $( A rearrangement of intersection.  (Contributed by NM, 27-Aug-2012.) $)
  in13 $p |- ( A i^i ( B i^i C ) ) = ( C i^i ( B i^i A ) ) $=
    ( cin in32 incom 3eqtr4i ) BCDZADBADZCDAHDCIDBCAEAHFCIFG $.

  $( A rearrangement of intersection.  (Contributed by NM, 27-Aug-2012.) $)
  in31 $p |- ( ( A i^i B ) i^i C ) = ( ( C i^i B ) i^i A ) $=
    ( cin in12 incom 3eqtr4i ) CABDZDACBDZDHCDIADCABEHCFIAFG $.

  $( Rotate the intersection of 3 classes.  (Contributed by NM,
     27-Aug-2012.) $)
  inrot $p |- ( ( A i^i B ) i^i C ) = ( ( C i^i A ) i^i B ) $=
    ( cin in31 in32 eqtri ) ABDCDCBDADCADBDABCECBAFG $.

  $( Rearrangement of intersection of 4 classes.  (Contributed by NM,
     21-Apr-2001.) $)
  in4 $p |- ( ( A i^i B ) i^i ( C i^i D ) ) =
            ( ( A i^i C ) i^i ( B i^i D ) ) $=
    ( cin in12 ineq2i inass 3eqtr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHA
    CLHI $.

  $( Intersection distributes over itself.  (Contributed by NM, 6-May-1994.) $)
  inindi $p |- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) ) $=
    ( cin inidm ineq1i in4 eqtr3i ) AADZBCDZDAJDABDACDDIAJAEFAABCGH $.

  $( Intersection distributes over itself.  (Contributed by NM,
     17-Aug-2004.) $)
  inindir $p |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i ( B i^i C ) ) $=
    ( cin inidm ineq2i in4 eqtr3i ) ABDZCCDZDICDACDBCDDJCICEFABCCGH $.

  $( A relationship between subclass and intersection.  Similar to Exercise 9
     of [TakeutiZaring] p. 18.  (Contributed by NM, 17-May-1994.) $)
  sseqin2 $p |- ( A C_ B <-> ( B i^i A ) = A ) $=
    ( dfss1 ) ABC $.

  ${
    $d x A $.  $d x B $.
    $( The intersection of two classes is a subset of one of them.  Part of
       Exercise 12 of [TakeutiZaring] p. 18.  (Contributed by NM,
       27-Apr-1994.) $)
    inss1 $p |- ( A i^i B ) C_ A $=
      ( vx cin cv wcel elin simplbi ssriv ) CABDZACEZJFKAFKBFKABGHI $.
  $}

  $( The intersection of two classes is a subset of one of them.  Part of
     Exercise 12 of [TakeutiZaring] p. 18.  (Contributed by NM,
     27-Apr-1994.) $)
  inss2 $p |- ( A i^i B ) C_ B $=
    ( cin incom inss1 eqsstrri ) ABCBACBBADBAEF $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Subclass of intersection.  Theorem 2.8(vii) of [Monk1] p. 26.
       (Contributed by NM, 15-Jun-2004.)  (Proof shortened by Andrew Salmon,
       26-Jun-2011.) $)
    ssin $p |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) ) $=
      ( vx cv wcel wi wal cin wss elin imbi2i albii jcab 3bitrri ssalel anbi12i
      wa 19.26 3bitr4i ) DEZAFZUABFZGZDHZUBUACFZGZDHZRZUBUABCIZFZGZDHZABJZACJZR
      AUJJUMUBUCUFRZGZDHUDUGRZDHUIULUQDUKUPUBUABCKLMUQURDUBUCUFNMUDUGDSOUNUEUOU
      HDABPDACPQDAUJPT $.
  $}

  ${
    ssini.1 $e |- A C_ B $.
    ssini.2 $e |- A C_ C $.
    $( An inference showing that a subclass of two classes is a subclass of
       their intersection.  (Contributed by NM, 24-Nov-2003.) $)
    ssini $p |- A C_ ( B i^i C ) $=
      ( wss wa cin pm3.2i ssin mpbi ) ABFZACFZGABCHFLMDEIABCJK $.
  $}

  ${
    ssind.1 $e |- ( ph -> A C_ B ) $.
    ssind.2 $e |- ( ph -> A C_ C ) $.
    $( A deduction showing that a subclass of two classes is a subclass of
       their intersection.  (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
    ssind $p |- ( ph -> A C_ ( B i^i C ) ) $=
      ( wss cin wa ssin biimpi syl2anc ) ABCGZBDGZBCDHGZEFMNIOBCDJKL $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Add right intersection to subclass relation.  (Contributed by NM,
       16-Aug-1994.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    ssrin $p |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) ) $=
      ( vx wss cin cv wcel wa ssel anim1d elin 3imtr4g ssrdv ) ABEZDACFZBCFZODG
      ZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.

    $( Add left intersection to subclass relation.  (Contributed by NM,
       19-Oct-1999.) $)
    sslin $p |- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) ) $=
      ( wss cin ssrin incom 3sstr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
  $}

  ${
    ssrind.1 $e |- ( ph -> A C_ B ) $.
    $( Add right intersection to subclass relation.  (Contributed by Glauco
       Siliprandi, 2-Jan-2022.) $)
    ssrind $p |- ( ph -> ( A i^i C ) C_ ( B i^i C ) ) $=
      ( wss cin ssrin syl ) ABCFBDGCDGFEBCDHI $.
  $}

  $( Intersection of subclasses.  (Contributed by NM, 5-May-2000.) $)
  ss2in $p |- ( ( A C_ B /\ C C_ D ) -> ( A i^i C ) C_ ( B i^i D ) ) $=
    ( wss cin ssrin sslin sylan9ss ) ABECDEACFBCFBDFABCGCDBHI $.

  $( Intersection preserves subclass relationship.  (Contributed by NM,
     14-Sep-1999.) $)
  ssinss1 $p |- ( A C_ C -> ( A i^i B ) C_ C ) $=
    ( cin wss wi inss1 sstr2 ax-mp ) ABDZAEACEJCEFABGJACHI $.

  $( Inclusion of an intersection of two classes.  (Contributed by NM,
     30-Oct-2014.) $)
  inss $p |- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C ) $=
    ( wss cin ssinss1 incom eqsstrid jaoi ) ACDABEZCDBCDZABCFKJBAECABGBACFHI $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Combinations of difference, union, and intersection of two classes
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( Absorption law for union.  (Contributed by NM, 16-Apr-2006.) $)
  unabs $p |- ( A u. ( A i^i B ) ) = A $=
    ( cin wss cun wceq inss1 ssequn2 mpbi ) ABCZADAJEAFABGJAHI $.

  $( Absorption law for intersection.  (Contributed by NM, 16-Apr-2006.) $)
  inabs $p |- ( A i^i ( A u. B ) ) = A $=
    ( cun wss cin wceq ssun1 df-ss mpbi ) AABCZDAJEAFABGAJHI $.

  ${
    $d x y A $.  $d y B $.
    $( Subclass defined in terms of class difference.  (Contributed by NM,
       22-Mar-1998.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    dfss4st $p |- ( A. x STAB x e. A
        -> ( A C_ B <-> ( B \ ( B \ A ) ) = A ) ) $=
      ( vy cv wcel wstab wal wss cdif wceq wb eleq1w stbid nfcv wn eldif bitrid
      wa bitr4id cbvalv cin sseqin2 nfa1 notbii anbi2i elin bitri imanst anbi2d
      wi abai sps eqrd eqeq1d sylbi ) AEZBFZGZAHDEZBFZGZDHZBCIZCCBJZJZBKZLUSVBA
      DUQUTKURVAADBMNUAVCVDCBUBZBKVGBCUCVCVFVHBVCDVFVHVBDUDDVFODVHOVBUTVFFZUTVH
      FZLDVIUTCFZUTVEFZPZSZVBVJUTCVEQVBVNVKVKVAPSZPZSZVJVMVPVKVLVOUTCBQUEUFVJVK
      VKVAUKZSZVBVQVJVKVASVSUTCBUGVKVAULUHVBVRVPVKVKVAUIUJRTRUMUNUOTUP $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Double complement and subset.  Similar to ~ ddifss but inside a class
       ` B ` instead of the universal class ` _V ` .  In classical logic the
       subset operation on the right hand side could be an equality (that is,
       ` A C_ B <-> ( B \ ( B \ A ) ) = A ` ).  (Contributed by Jim Kingdon,
       24-Jul-2018.) $)
    ssddif $p |- ( A C_ B <-> A C_ ( B \ ( B \ A ) ) ) $=
      ( vx cv wcel wi wal cdif wss wa wn simpr con2i anim2i eldif notbii anbi2i
      ancr bitri ssalel sylibr syl6 eldifi imim2i impbii albii 3bitr4i ) CDZAEZ
      UHBEZFZCGUIUHBBAHZHZEZFZCGABIAUMIUKUOCUKUOUKUIUJUIJZUNUIUJRUPUJUJUIKZJZKZ
      JZUNUIUSUJURUIUJUQLMNUNUJUHULEZKZJUTUHBULOVBUSUJVAURUHBAOPQSUAUBUNUJUIUHB
      ULUCUDUEUFCABTCAUMTUG $.

    $( Union of two classes and class difference.  In classical logic this
       would be an equality.  (Contributed by Jim Kingdon, 24-Jul-2018.) $)
    unssdif $p |- ( A u. B ) C_ ( _V \ ( ( _V \ A ) \ B ) ) $=
      ( vx cun cvv cdif cv wcel wo wn wa vex eldif mpbiran anbi1i ioran 3bitr4i
      biimpi con2i elun 3imtr4i ssriv ) CABDZEEAFZBFZFZCGZAHZUGBHZIZUGUEHZJZUGU
      CHUGUFHZUKUJUKUJJZUGUDHZUIJZKUHJZUPKUKUNUOUQUPUOUGEHZUQCLZUGEAMNOUGUDBMUH
      UIPQRSUGABTUMURULUSUGEUEMNUAUB $.

    $( Intersection of two classes and class difference.  In classical logic
       this would be an equality.  (Contributed by Jim Kingdon,
       24-Jul-2018.) $)
    inssdif $p |- ( A i^i B ) C_ ( A \ ( _V \ B ) ) $=
      ( vx cin cvv cdif cv wcel wa wn elndif anim2i elin eldif 3imtr4i ssriv )
      CABDZAEBFZFZCGZAHZTBHZIUATRHJZITQHTSHUBUCUATBEKLTABMTARNOP $.

    $( Difference with intersection.  Theorem 33 of [Suppes] p. 29.
       (Contributed by NM, 31-Mar-1998.)  (Proof shortened by Andrew Salmon,
       26-Jun-2011.) $)
    difin $p |- ( A \ ( A i^i B ) ) = ( A \ B ) $=
      ( vx cin cdif cv wcel wa wn wfal wi ax-in2 expd dfnot imbitrrdi imdistani
      com12 simpr con3i eldif anim2i impbii notbii anbi2i bitri 3bitr4i eqriv
      elin ) CAABDZEZABEZCFZAGZUMULBGZHZIZHZUMUNIZHZULUJGZULUKGUQUSUMUPURUPUMUR
      UPUMUNJKURUPUMUNJUOJLMUNNOQPURUPUMUOUNUMUNRSUAUBUTUMULUIGZIZHUQULAUITVBUP
      UMVAUOULABUHUCUDUEULABTUFUG $.
  $}

  $( Double complement under universal class.  In classical logic (or given an
     additional hypothesis, as in ~ ddifnel ), this is equality rather than
     subset.  (Contributed by Jim Kingdon, 24-Jul-2018.) $)
  ddifss $p |- A C_ ( _V \ ( _V \ A ) ) $=
    ( cvv wss cdif ssv ssddif mpbi ) ABCABBADDCAEABFG $.

  ${
    $d x A $.  $d x B $.
    $( Union as a subset of class complement and intersection (De Morgan's
       law).  One direction of the definition of union in [Mendelson] p. 231.
       This would be an equality, rather than subset, in classical logic.
       (Contributed by Jim Kingdon, 25-Jul-2018.) $)
    unssin $p |- ( A u. B ) C_ ( _V \ ( ( _V \ A ) i^i ( _V \ B ) ) ) $=
      ( vx cun cvv cdif cin cv wcel wo wn wa oranim eldifn anim12i nsyl sylnibr
      elin elun vex eldif mpbiran 3imtr4i ssriv ) CABDZEEAFZEBFZGZFZCHZAIZUJBIZ
      JZUJUHIZKZUJUEIUJUIIZUMUJUFIZUJUGIZLZUNUMUKKZULKZLUSUKULMUQUTURVAUJEANUJE
      BNOPUJUFUGRQUJABSUPUJEIUOCTUJEUHUAUBUCUD $.

    $( Intersection in terms of class difference and union (De Morgan's law).
       Similar to Exercise 4.10(n) of [Mendelson] p. 231.  This would be an
       equality, rather than subset, in classical logic.  (Contributed by Jim
       Kingdon, 25-Jul-2018.) $)
    inssun $p |- ( A i^i B ) C_ ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) $=
      ( vx cin cvv cdif cv wcel wa wn wo pm3.1 eldifn orim12i nsyl elun sylnibr
      cun elin vex eldif mpbiran 3imtr4i ssriv ) CABDZEEAFZEBFZRZFZCGZAHZUJBHZI
      ZUJUHHZJZUJUEHUJUIHZUMUJUFHZUJUGHZKZUNUMUKJZULJZKUSUKULLUQUTURVAUJEAMUJEB
      MNOUJUFUGPQUJABSUPUJEHUOCTUJEUHUAUBUCUD $.
  $}

  $( Intersection of two classes and class difference.  In classical logic,
     such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather
     than subset.  (Contributed by Jim Kingdon, 26-Jul-2018.) $)
  inssddif $p |- ( A i^i B ) C_ ( A \ ( A \ B ) ) $=
    ( cin cdif wss inss1 ssddif mpbi difin difeq2i sseqtri ) ABCZAALDZDZAABDZDL
    AELNEABFLAGHMOAABIJK $.

  ${
    $d x A $.  $d x B $.
    $( Intersection with universal complement.  Remark in [Stoll] p. 20.
       (Contributed by NM, 17-Aug-2004.) $)
    invdif $p |- ( A i^i ( _V \ B ) ) = ( A \ B ) $=
      ( vx cvv cdif cin cv wcel wa vex eldif mpbiran anbi2i elin 3bitr4i eqriv
      wn ) CADBEZFZABEZCGZAHZUARHZIUBUABHQZIUASHUATHUCUDUBUCUADHUDCJUADBKLMUAAR
      NUAABKOP $.

    $( Intersection with class difference.  Theorem 34 of [Suppes] p. 29.
       (Contributed by NM, 17-Aug-2004.) $)
    indif $p |- ( A i^i ( A \ B ) ) = ( A \ B ) $=
      ( vx cdif cin cv wcel wn wa anabs5 elin eldif anbi2i bitri 3bitr4i eqriv
      ) CAABDZEZQCFZAGZTSBGHZIZIZUBSRGZSQGZTUAJUDTUEIUCSAQKUEUBTSABLZMNUFOP $.
  $}

  $( Bring an intersection in and out of a class difference.  (Contributed by
     Jeff Hankins, 15-Jul-2009.) $)
  indif2 $p |- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C ) $=
    ( cin cvv cdif inass invdif ineq2i 3eqtr3ri ) ABDZECFZDABLDZDKCFABCFZDABLGK
    CHMNABCHIJ $.

  $( Bring an intersection in and out of a class difference.  (Contributed by
     Mario Carneiro, 15-May-2015.) $)
  indif1 $p |- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C ) $=
    ( cdif cin indif2 incom difeq1i 3eqtr3i ) BACDZEBAEZCDJBEABEZCDBACFBJGKLCBA
    GHI $.

  $( Commutation law for intersection and difference.  (Contributed by Scott
     Fenton, 18-Feb-2013.) $)
  indifcom $p |- ( A i^i ( B \ C ) ) = ( B i^i ( A \ C ) ) $=
    ( cin cdif incom difeq1i indif2 3eqtr4i ) ABDZCEBADZCEABCEDBACEDJKCABFGABCH
    BACHI $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Distributive law for intersection over union.  Exercise 10 of
       [TakeutiZaring] p. 17.  (Contributed by NM, 30-Sep-2002.)  (Proof
       shortened by Andrew Salmon, 26-Jun-2011.) $)
    indi $p |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) ) $=
      ( vx cun cin cv wcel wo wa andi elin orbi12i bitr4i anbi2i 3bitr4i ineqri
      elun ) DABCEZABFZACFZEZDGZAHZUCBHZUCCHZIZJZUCTHZUCUAHZIZUDUCSHZJUCUBHUHUD
      UEJZUDUFJZIUKUDUEUFKUIUMUJUNUCABLUCACLMNULUGUDUCBCROUCTUARPQ $.

    $( Distributive law for union over intersection.  Exercise 11 of
       [TakeutiZaring] p. 17.  (Contributed by NM, 30-Sep-2002.)  (Proof
       shortened by Andrew Salmon, 26-Jun-2011.) $)
    undi $p |- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) ) $=
      ( vx cin cv wcel wo wa elin orbi2i ordi elun anbi12i bitr2i 3bitri uneqri
      cun ) DABCEZABRZACRZEZDFZAGZUCSGZHUDUCBGZUCCGZIZHUDUFHZUDUGHZIZUCUBGZUEUH
      UDUCBCJKUDUFUGLULUCTGZUCUAGZIUKUCTUAJUMUIUNUJUCABMUCACMNOPQ $.
  $}

  $( Distributive law for intersection over union.  Theorem 28 of [Suppes]
     p. 27.  (Contributed by NM, 30-Sep-2002.) $)
  indir $p |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) ) $=
    ( cun cin indi incom uneq12i 3eqtr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMK
    NLACGBCGHI $.

  $( Distributive law for union over intersection.  Theorem 29 of [Suppes]
     p. 27.  (Contributed by NM, 30-Sep-2002.) $)
  undir $p |- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) ) $=
    ( cin cun undi uncom ineq12i 3eqtr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMK
    NLACGBCGHI $.

  $( Equality of union and intersection is equivalent to equality of the
     arguments.  (Contributed by NM, 16-Apr-2006.)  (Proof shortened by Andrew
     Salmon, 26-Jun-2011.) $)
  uneqin $p |- ( ( A u. B ) = ( A i^i B ) <-> A = B ) $=
    ( cun cin wceq wss wa eqimss unss ssin sstr sylbir simpl anim12i syl sylibr
    eqss unidm inidm eqtr4i uneq2 ineq2 3eqtr3a impbii ) ABCZABDZEZABEZUGABFZBA
    FZGZUHUGUEUFFZUKUEUFHULAUFFZBUFFZGUKABUFIUMUIUNUJUMAAFUIGUIAABJAABKLUNUJBBF
    ZGUJBABJUJUOMLNLOABQPUHAACZAADZUEUFUPAUQARASTABAUAABAUBUCUD $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Distributive law for class difference.  Theorem 39 of [Suppes] p. 29.
       (Contributed by NM, 17-Aug-2004.) $)
    difundi $p |- ( A \ ( B u. C ) ) = ( ( A \ B ) i^i ( A \ C ) ) $=
      ( vx cun cdif cin cv wcel wa wn eldif anbi12i elin wo notbii anbi2i bitri
      elun ioran anandi 3bitr4ri eqriv ) DABCEZFZABFZACFZGZDHZUFIZUIUGIZJUIAIZU
      IBIZKZJZULUICIZKZJZJZUIUHIUIUEIZUJUOUKURUIABLUIACLMUIUFUGNUTULUNUQJZJZUSU
      TULUMUPOZKZJZVBUTULUIUDIZKZJVEUIAUDLVGVDULVFVCUIBCSPQRVDVAULUMUPTQRULUNUQ
      UARUBUC $.
  $}

  $( Distributive law for class difference.  (Contributed by NM,
     17-Aug-2004.) $)
  difundir $p |- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) ) $=
    ( cun cvv cdif cin indir invdif uneq12i 3eqtr3i ) ABDZECFZGAMGZBMGZDLCFACFZ
    BCFZDABMHLCINPOQACIBCIJK $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Distributive law for class difference.  In classical logic, for example,
       theorem 40 of [Suppes] p. 29, this is an equality instead of subset.
       (Contributed by Jim Kingdon, 26-Jul-2018.) $)
    difindiss $p |- ( ( A \ B ) u. ( A \ C ) ) C_ ( A \ ( B i^i C ) ) $=
      ( vx cdif cun cin cv wcel wo elun wa wn orbi12i andi bitr4i pm3.14 anim2i
      eldif sylbi elin notbii anbi2i bitr2i sylib ssriv ) DABEZACEZFZABCGZEZDHZ
      UIIULUGIZULUHIZJZULUKIZULUGUHKUOULAIZULBIZULCIZLZMZLZUPUOUQURMZUSMZJZLZVB
      UOUQVCLZUQVDLZJVFUMVGUNVHULABSULACSNUQVCVDOPVEVAUQURUSQRTUPUQULUJIZMZLVBU
      LAUJSVJVAUQVIUTULBCUAUBUCUDUETUF $.
  $}

  $( Distributive law for class difference.  (Contributed by NM,
     17-Aug-2004.) $)
  difindir $p |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) ) $=
    ( cin cvv cdif inindir invdif ineq12i 3eqtr3i ) ABDZECFZDALDZBLDZDKCFACFZBC
    FZDABLGKCHMONPACHBCHIJ $.

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Distribute intersection over difference.  (Contributed by Scott Fenton,
       14-Apr-2011.) $)
    indifdir $p |- ( ( A \ B ) i^i C ) = ( ( A i^i C ) \ ( B i^i C ) ) $=
      ( vx cdif cv wcel wn wa elin notbii anbi12i eldif anbi1i bitri an32 simpl
      cin con3i wfal anim2i ax-in2 expcomd impcom dfnot sylibr adantll 3bitr4ri
      wi jca impbii eqriv ) DABEZCRZACRZBCRZEZDFZUOGZURUPGZHZIURAGZURCGZIZURBGZ
      VCIZHZIZURUQGURUNGZUSVDVAVGURACJUTVFURBCJKLURUOUPMVIVBVEHZIZVCIZVHVIURUMG
      ZVCIVLURUMCJVMVKVCURABMNOVLVDVJIZVHVBVJVCPVNVHVJVGVDVFVEVEVCQSUAVHVDVJVDV
      GQVCVGVJVBVCVGIVETUIZVJVGVCVOVGVEVCTVFTUBUCUDVEUEUFUGUJUKOOUHUL $.
  $}

  $( Set difference with a set difference.  In classical logic this would be
     equality rather than subset.  (Contributed by Jim Kingdon,
     27-Jul-2018.) $)
  difdif2ss $p |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B \ C ) ) $=
    ( cin cun cvv wss inssdif unss2 ax-mp difindiss sstri invdif eqcomi difeq2i
    cdif sseqtrri ) ABPZACDZEZABFCPZDZPZABCPZPTRAUAPZEZUCSUEGTUFGACHSUERIJABUAK
    LUDUBAUBUDBCMNOQ $.

  $( De Morgan's law for union.  Theorem 5.2(13) of [Stoll] p. 19.
     (Contributed by NM, 18-Aug-2004.) $)
  undm $p |- ( _V \ ( A u. B ) ) = ( ( _V \ A ) i^i ( _V \ B ) ) $=
    ( cvv difundi ) CABD $.

  $( De Morgan's law for intersection.  In classical logic, this would be
     equality rather than subset, as in Theorem 5.2(13') of [Stoll] p. 19.
     (Contributed by Jim Kingdon, 27-Jul-2018.) $)
  indmss $p |- ( ( _V \ A ) u. ( _V \ B ) ) C_ ( _V \ ( A i^i B ) ) $=
    ( cvv difindiss ) CABD $.

  $( A relationship involving double difference and union.  (Contributed by NM,
     29-Aug-2004.) $)
  difun1 $p |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C ) $=
    ( cvv cdif cin cun inass invdif eqtr3i undm ineq2i difeq1i ) ADBEZFZCEZABCG
    ZEZABEZCEANDCEZFZFZPROTFUBPANTHOCIJADQEZFUBRUCUAABCKLAQIJJOSCABIMJ $.

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( A subset relationship involving class union and class difference.  In
       classical logic, this would be equality rather than subset, as in the
       first equality of Exercise 13 of [TakeutiZaring] p. 22.  (Contributed by
       Jim Kingdon, 28-Jul-2018.) $)
    undif3ss $p |- ( A u. ( B \ C ) ) C_ ( ( A u. B ) \ ( C \ A ) ) $=
      ( vx cdif cun cv wcel wo wn wa elun eldif orbi2i orc olc jca anim12i jaoi
      simpl orcd adantr adantl ccase impbii 3bitri biimpri pm4.53r sylibr sylbi
      sylnibr ssriv ) DABCEZFZABFZCAEZEZDGZUNHZURAHZURBHZIZURCHZJZUTIZKZURUQHZU
      SUTURUMHZIUTVAVDKZIZVFURAUMLVHVIUTURBCMNVJVFUTVFVIUTVBVEUTVAOUTVDPQVAVBVD
      VEVAUTPVDUTORSUTVDVAUTVJUTVDKUTVIUTVDTUAVIUTPUTVJUTUTVIOZUBUTVJVAVKUCUDUE
      UFVFURUOHZURUPHZJZKVGVBVLVEVNVLVBURABLUGVEVCUTJKVMVCUTUHURCAMUKRURUOUPMUI
      UJUL $.

    $( Represent a set difference as an intersection with a larger difference.
       (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    difin2 $p |- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) ) $=
      ( vx wss cdif cin cv wcel wn ssel pm4.71d anbi1d eldif anbi1i ancom anass
      wa elin bitr4i 3bitri 3bitr4g eqrdv ) ACEZDABFZCBFZAGZUDDHZAIZUHBIJZRUIUH
      CIZRZUJRZUHUEIUHUGIZUDUIULUJUDUIUKACUHKLMUHABNUNUHUFIZUIRUKUJRZUIRZUMUHUF
      ASUOUPUIUHCBNOUQUIUPRUMUPUIPUIUKUJQTUAUBUC $.
  $}

  $( Swap second and third argument of double difference.  (Contributed by NM,
     18-Aug-2004.) $)
  dif32 $p |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B ) $=
    ( cun cdif uncom difeq2i difun1 3eqtr3i ) ABCDZEACBDZEABECEACEBEJKABCFGABCH
    ACBHI $.

  $( Absorption-like law for class difference: you can remove a class only
     once.  (Contributed by FL, 2-Aug-2009.) $)
  difabs $p |- ( ( A \ B ) \ B ) = ( A \ B ) $=
    ( cun cdif difun1 unidm difeq2i eqtr3i ) ABBCZDABDZBDJABBEIBABFGH $.

  $( Two ways to express symmetric difference.  This theorem shows the
     equivalence of the definition of symmetric difference in [Stoll] p. 13 and
     the restated definition in Example 4.1 of [Stoll] p. 262.  (Contributed by
     NM, 17-Aug-2004.) $)
  symdif1 $p |- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) ) $=
    ( cun cin cdif difundir difin incom difeq2i eqtri uneq12i eqtr2i ) ABCABDZE
    AMEZBMEZCABEZBAEZCABMFNPOQABGOBBADZEQMRBABHIBAGJKL $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Class abstractions with difference, union, and intersection of two classes
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)
  ${
    $d x A $.  $d x B $.
    $( Expressing symmetric difference with exclusive-or or two differences.
       (Contributed by Jim Kingdon, 28-Jul-2018.) $)
    symdifxor $p |- ( ( A \ B ) u. ( B \ A ) ) =
                  { x | ( x e. A \/_ x e. B ) } $=
      ( cv wcel wxo cdif cun wo wn eldif orbi12i elun excxor ancom orbi2i bitri
      wa 3bitr4i abbi2i ) ADZBEZUACEZFZABCGZCBGZHZUAUEEZUAUFEZIUBUCJRZUCUBJZRZI
      ZUAUGEUDUHUJUIULUABCKUACBKLUAUEUFMUDUJUKUCRZIUMUBUCNUNULUJUKUCOPQST $.
  $}

  ${
    $d x y $.  $d ph y $.  $d ps y $.
    $( Union of two class abstractions.  (Contributed by NM, 29-Sep-2002.)
       (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    unab $p |- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) } $=
      ( vy cab wo wsb cv wcel sbor df-clab orbi12i 3bitr4ri uneqri ) DACEZBCEZA
      BFZCEZQCDGACDGZBCDGZFDHZRIUAOIZUAPIZFABCDJQDCKUBSUCTADCKBDCKLMN $.

    $( Intersection of two class abstractions.  (Contributed by NM,
       29-Sep-2002.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    inab $p |- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) } $=
      ( vy cab wa wsb cv wcel sban df-clab anbi12i 3bitr4ri ineqri ) DACEZBCEZA
      BFZCEZQCDGACDGZBCDGZFDHZRIUAOIZUAPIZFABCDJQDCKUBSUCTADCKBDCKLMN $.

    $( Difference of two class abstractions.  (Contributed by NM, 23-Oct-2004.)
       (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    difab $p |- ( { x | ph } \ { x | ps } ) = { x | ( ph /\ -. ps ) } $=
      ( vy cab wn wcel wsb df-clab sban bicomi xchbinxr anbi12i 3bitrri difeqri
      wa cv sbn ) DACEZBCEZABFZPZCEZDQZUCGUBCDHACDHZUACDHZPUDSGZUDTGZFZPUBDCIAU
      ACDJUEUGUFUIUGUEADCIKUFBCDHUHBCDRBDCILMNO $.
  $}

  ${
    $( A class abstraction with a conjunction is a subset of the class
       abstraction with the left conjunct only.  (Contributed by AV,
       7-Aug-2024.)  (Proof shortened by SN, 22-Aug-2024.) $)
    abanssl $p |- { f | ( ph /\ ps ) } C_ { f | ph } $=
      ( wa simpl ss2abi ) ABDACABEF $.

    $( A class abstraction with a conjunction is a subset of the class
       abstraction with the right conjunct only.  (Contributed by AV,
       7-Aug-2024.)  (Proof shortened by SN, 22-Aug-2024.) $)
    abanssr $p |- { f | ( ph /\ ps ) } C_ { f | ps } $=
      ( wa simpr ss2abi ) ABDBCABEF $.
  $}

  $( A class builder defined by a negation.  (Contributed by FL,
     18-Sep-2010.) $)
  notab $p |- { x | -. ph } = ( _V \ { x | ph } ) $=
    ( cv cvv wcel wn wa cab cdif crab df-rab rabab eqtr3i difab abid2 difeq1i )
    BCDEZAFZGBHZRBHZDABHZIZRBDJSTRBDKRBLMQBHZUAISUBQABNUCDUABDOPMM $.

  $( Union of two restricted class abstractions.  (Contributed by NM,
     25-Mar-2004.) $)
  unrab $p |- ( { x e. A | ph } u. { x e. A | ps } ) =
              { x e. A | ( ph \/ ps ) } $=
    ( crab cun cv wcel wa cab wo df-rab uneq12i unab andi abbii eqtr4i ) ACDEZB
    CDEZFCGDHZAIZCJZTBIZCJZFZABKZCDEZRUBSUDACDLBCDLMUGTUFIZCJZUEUFCDLUEUAUCKZCJ
    UIUAUCCNUHUJCTABOPQQQ $.

  $( Intersection of two restricted class abstractions.  (Contributed by NM,
     1-Sep-2006.) $)
  inrab $p |- ( { x e. A | ph } i^i { x e. A | ps } ) =
              { x e. A | ( ph /\ ps ) } $=
    ( crab cin cv wcel wa cab df-rab ineq12i inab anandi abbii eqtr4i ) ACDEZBC
    DEZFCGDHZAIZCJZSBIZCJZFZABIZCDEZQUARUCACDKBCDKLUFSUEIZCJZUDUECDKUDTUBIZCJUH
    TUBCMUGUICSABNOPPP $.

  ${
    $d x B $.
    $( Intersection with a restricted class abstraction.  (Contributed by NM,
       19-Nov-2007.) $)
    inrab2 $p |- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph } $=
      ( crab cin cv wcel cab df-rab abid2 eqcomi ineq12i inab elin anbi1i bitri
      wa an32 eqtr4i abbii ) ABCEZDFBGZCHZARZBIZUCDHZBIZFZABCDFZEZUBUFDUHABCJUH
      DBDKLMUKUCUJHZARZBIZUIABUJJUIUEUGRZBIUNUEUGBNUMUOBUMUDUGRZARUOULUPAUCCDOP
      UDUGASQUATTT $.
  $}

  $( Difference of two restricted class abstractions.  (Contributed by NM,
     23-Oct-2004.) $)
  difrab $p |- ( { x e. A | ph } \ { x e. A | ps } ) =
              { x e. A | ( ph /\ -. ps ) } $=
    ( crab cdif cv wcel wa wn df-rab difeq12i difab anass simpr con3i anim2i wi
    cab eqtr4i pm3.2 adantr con3d imdistani impbii bitr3i abbii ) ACDEZBCDEZFCG
    DHZAIZCSZUJBIZCSZFZABJZIZCDEZUHULUIUNACDKBCDKLURUJUQIZCSZUOUQCDKUOUKUMJZIZC
    SUTUKUMCMUSVBCUSUKUPIZVBUJAUPNVCVBUPVAUKUMBUJBOPQUKVAUPUKBUMUJBUMRAUJBUAUBU
    CUDUEUFUGTTT $.

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of restricted class abstraction.  (Contributed by
       NM, 20-Sep-2003.) $)
    dfrab2 $p |- { x e. A | ph } = ( { x | ph } i^i A ) $=
      ( crab cv wcel wa cab cin df-rab inab abid2 ineq1i eqtr3i incom 3eqtri )
      ABCDBECFZAGBHZCABHZIZSCIABCJQBHZSIRTQABKUACSBCLMNCSOP $.

    $( Alternate definition of restricted class abstraction.  (Contributed by
       Mario Carneiro, 8-Sep-2013.) $)
    dfrab3 $p |- { x e. A | ph } = ( A i^i { x | ph } ) $=
      ( crab cv wcel wa cab cin df-rab inab abid2 ineq1i 3eqtr2i ) ABCDBECFZAGB
      HOBHZABHZICQIABCJOABKPCQBCLMN $.

    $( Complementation of restricted class abstractions.  (Contributed by Mario
       Carneiro, 3-Sep-2015.) $)
    notrab $p |- ( A \ { x e. A | ph } ) = { x e. A | -. ph } $=
      ( cv wcel cab cdif wn crab difab cin difin dfrab3 difeq2i difeq1i 3eqtr4i
      wa abid2 df-rab ) BDCEZBFZABFZGZTAHZQBFCABCIZGZUDBCITABJCCUBKZGCUBGUFUCCU
      BLUEUGCABCMNUACUBBCROPUDBCSP $.

    $( Restricted class abstraction with a common superset.  (Contributed by
       Stefan O'Rear, 12-Sep-2015.)  (Proof shortened by Mario Carneiro,
       8-Nov-2015.) $)
    dfrab3ss $p |- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) ) $=
      ( wss cab crab wceq df-ss ineq1 eqcomd sylbi dfrab3 ineq2i eqtr4i 3eqtr4g
      cin inass ) CDEZCABFZQZCDQZTQZABCGCABDGZQZSUBCHZUAUCHCDIUFUCUAUBCTJKLABCM
      UECDTQZQUCUDUGCABDMNCDTROP $.
  $}

  $( Abstraction restricted to a union.  (Contributed by Stefan O'Rear,
     5-Feb-2015.) $)
  rabun2 $p |- { x e. ( A u. B ) | ph } =
      ( { x e. A | ph } u. { x e. B | ph } ) $=
    ( cun crab cv wcel wa cab df-rab uneq12i elun anbi1i andir bitri abbii unab
    wo eqtr4i ) ABCDEZFBGZUAHZAIZBJZABCFZABDFZEZABUAKUHUBCHZAIZBJZUBDHZAIZBJZEZ
    UEUFUKUGUNABCKABDKLUEUJUMSZBJUOUDUPBUDUIULSZAIUPUCUQAUBCDMNUIULAOPQUJUMBRTT
    T $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Restricted uniqueness with difference, union, and intersection
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d x A $.  $d x B $.
    $( Transfer uniqueness to a smaller subclass.  (Contributed by NM,
       20-Oct-2005.) $)
    reuss2 $p |- ( ( ( A C_ B /\ A. x e. A ( ph -> ps ) ) /\
                 ( E. x e. A ph /\ E! x e. B ps ) ) -> E! x e. A ph ) $=
      ( wrex wreu wa wss wi wral wcel wex weu df-rex df-reu anbi12i wal sylan2b
      cv wmo df-ral ssel anim12 sylan exp4b com23 a2d alimdv imp euimmo syl eu5
      imp4a simplbi2 syl9 imp32 sylibr ) ACDFZBCEGZHDEIZABJZCDKZHZCTZDLZAHZCMZV
      EELZBHZCNZHZACDGZUSVHUTVKACDOBCEPQVDVLHVGCNZVMVDVHVKVNVDVKVGCUAZVHVNVDVGV
      JJZCRZVKVOJVCVAVFVBJZCRZVQVBCDUBVAVSVQVAVRVPCVAVRVFAVJVAVFVBAVJJZVAVBVFVT
      VAVBVFAVJVAVFVIJVBVPDEVEUCVFVIABUDUEUFUGUHUNUIUJSVGVJCUKULVNVHVOVGCUMUOUP
      UQACDPURS $.

    $( Transfer uniqueness to a smaller subclass.  (Contributed by NM,
       21-Aug-1999.) $)
    reuss $p |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) ->
                E! x e. A ph ) $=
      ( wss wrex wreu wi wral wa cv wcel idd rgen reuss2 mpanl2 3impb ) CDEZABC
      FZABDGZABCGZRAAHZBCISTJUAUBBCBKCLAMNAABCDOPQ $.

    $( Transfer uniqueness to a smaller class.  (Contributed by NM,
       21-Oct-2005.) $)
    reuun1 $p |- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) )
                 -> E! x e. A ph ) $=
      ( cun wss wo wi wral wrex wreu wa ssun1 orc rgenw reuss2 mpanl12 ) DDEFZG
      AABHZIZCDJACDKTCSLMACDLDENUACDABOPATCDSQR $.

    $( Transfer uniqueness to a smaller or larger class.  (Contributed by NM,
       21-Oct-2005.) $)
    reuun2 $p |- ( -. E. x e. B ph ->
             ( E! x e. ( A u. B ) ph <-> E! x e. A ph ) ) $=
      ( wrex wn cv wcel wa wo weu cun wreu wex df-rex euor2 sylnbi df-reu bitri
      wb elun anbi1i andir orcom eubii 3bitr4g ) ABDEZFBGZDHZAIZUHCHZAIZJZBKZUL
      BKZABCDLZMZABCMUGUJBNUNUOTABDOUJULBPQUQUHUPHZAIZBKUNABUPRUSUMBUSUKUIJZAIZ
      UMURUTAUHCDUAUBVAULUJJUMUKUIAUCULUJUDSSUESABCRUF $.

    $( Restricted uniqueness "picks" a member of a subclass.  (Contributed by
       NM, 21-Aug-1999.) $)
    reupick $p |- ( ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) /\ ph ) ->
                  ( x e. A <-> x e. B ) ) $=
      ( wss wrex wreu wa cv wcel wi ssel ad2antrr wex weu df-rex df-reu anbi12i
      ancrd anim1d an32 imbitrdi eximdv eupick syl9 com23 imp32 sylan2b expcomd
      ex imp impbid ) CDEZABCFZABDGZHZHZAHBIZCJZURDJZUMUSUTKUPACDURLZMUQAUTUSKU
      QUTAUSUPUMUSAHZBNZUTAHZBOZHVDUSKZUNVCUOVEABCPABDQRUMVCVEVFUMVEVCVFUMVCVDU
      SHZBNZVEVFUMVBVGBUMVBUTUSHZAHVGUMUSVIAUMUSUTVASTUTUSAUAUBUCVEVHVFVDUSBUDU
      JUEUFUGUHUIUKUL $.

    $( Restricted uniqueness "picks" a member of a subclass.  (Contributed by
       Mario Carneiro, 19-Nov-2016.) $)
    reupick3 $p  |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) ->
       ( ph -> ps ) ) $=
      ( wreu wa wrex cv wcel wi weu wex df-reu df-rex anass exbii bitr4i eupick
      syl2anb expd 3impia ) ACDEZABFZCDGZCHDIZABJUBUDFUEABUBUEAFZCKUFBFZCLZUFBJ
      UDACDMUDUEUCFZCLUHUCCDNUGUICUEABOPQUFBCRSTUA $.

    $( Restricted uniqueness "picks" a member of a subclass.  (Contributed by
       Mario Carneiro, 15-Dec-2013.)  (Proof shortened by Mario Carneiro,
       19-Nov-2016.) $)
    reupick2 $p |- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\
                       E! x e. A ph ) /\ x e. A ) -> ( ph <-> ps ) ) $=
      ( wi wral wrex wreu w3a cv wcel ancr ralimi rexim syl reupick3 3exp com12
      wa syl6 3imp1 rsp 3ad2ant1 imp impbid ) BAEZCDFZBCDGZACDHZIZCJDKZSABUGUHU
      IUKABEZUGUHABSZCDGZUIUKULEZEUGBUMEZCDFUHUNEUFUPCDBALMBUMCDNOUIUNUOUIUNUKU
      LABCDPQRTUAUJUKUFUGUHUKUFEUIUFCDUBUCUDUE $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The empty set
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare the symbol for the empty or null set. $)
  $c (/) $. $( empty set $)

  $( Extend class notation to include the empty set. $)
  c0 $a class (/) $.

  $( Define the empty set.  Special case of Exercise 4.10(o) of [Mendelson]
     p. 231.  For a more traditional definition, but requiring a dummy
     variable, see ~ dfnul2 .  (Contributed by NM, 5-Aug-1993.) $)
  df-nul $a |- (/) = ( _V \ _V ) $.

  $( Alternate definition of the empty class/set.  (Contributed by BJ,
     30-Nov-2019.)  Avoid ~ ax-13 , ~ df-clel .  (Revised by GG, 3-Sep-2024.)
     Prove directly from definition to allow shortening ~ dfnul2 .  (Revised by
     BJ, 23-Sep-2024.) $)
  dfnul4 $p |- (/) = { x | F. } $=
    ( c0 cvv cdif cv wcel wn cab wfal df-nul df-dif pm3.24 bifal abbii 3eqtri
    wa ) BCCDAECFZQGPZAHIAHJACCKRIARQLMNO $.
  $( $j usage 'dfnul4' avoids 'df-clel' 'ax-10' 'ax-i12' 'ax-bndl' 'ax-16'
     'ax-13'; $)

  $( Alternate definition of the empty set.  Definition 5.14 of [TakeutiZaring]
     p. 20.  (Contributed by NM, 26-Dec-1996.)  Reduce axiom dependencies.
     (Revised by Steven Nguyen, 3-May-2023.)  (Proof shortened by BJ,
     23-Sep-2024.) $)
  dfnul2 $p |- (/) = { x | -. x = x } $=
    ( c0 wfal cab weq wn dfnul4 equid notnoti bifal abbii eqtr4i ) BCADAAEZFZAD
    AGNCANMAHIJKL $.
  $( $j usage 'dfnul2' avoids 'df-clel' 'ax-10' 'ax-i12' 'ax-bndl' 'ax-16'
     'ax-13'; $)

  $( Alternate definition of the empty set.  (Contributed by NM, 25-Mar-2004.)
     (Proof shortened by BJ, 23-Sep-2024.) $)
  dfnul3 $p |- (/) = { x e. A | -. x e. A } $=
    ( wfal cab cv wcel wn wa crab fal pm3.24 2false abbii dfnul4 df-rab 3eqtr4i
    c0 ) CADAEBFZRGZHZADQSABICTACTJRKLMANSABOP $.
  $( $j usage 'dfnul3' avoids 'df-clel' 'ax-10' 'ax-i12' 'ax-bndl' 'ax-16'
     'ax-13'; $)

  $( The empty set has no elements.  Theorem 6.14 of [Quine] p. 44.
     (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Mario Carneiro,
     1-Sep-2015.) $)
  noel $p |- -. A e. (/) $=
    ( c0 wcel cvv cdif eldifi eldifn pm2.65i df-nul eleq2i mtbir ) ABCADDEZCZMA
    DCADDFADDGHBLAIJK $.

  $( The empty set has no elements.  (Contributed by Peter Mazsa,
     4-Jan-2018.) $)
  nel02 $p |- ( A = (/) -> -. B e. A ) $=
    ( c0 wceq wcel noel eleq2 mtbiri ) ACDBAEBCEBFACBGH $.

  $( If a set has elements, it is not empty.  A set with elements is also
     inhabited, see ~ elex2 .  (Contributed by NM, 31-Dec-1993.) $)
  n0i $p |- ( B e. A -> -. A = (/) ) $=
    ( c0 wceq wcel noel eleq2 mtbiri con2i ) ACDZBAEZJKBCEBFACBGHI $.

  $( If a set has elements, it is not empty.  A set with elements is also
     inhabited, see ~ elex2 .  (Contributed by NM, 31-Dec-1993.) $)
  ne0i $p |- ( B e. A -> A =/= (/) ) $=
    ( wcel c0 n0i neneqad ) BACADABEF $.

  ${
    ne0d.1 $e |- ( ph -> B e. A ) $.
    $( Deduction form of ~ ne0i .  If a class has elements, then it is
       nonempty.  (Contributed by Glauco Siliprandi, 23-Oct-2021.) $)
    ne0d $p |- ( ph -> A =/= (/) ) $=
      ( wcel c0 wne ne0i syl ) ACBEBFGDBCHI $.
  $}

  ${
    n0ii.1 $e |- A e. B $.
    $( If a class has elements, then it is not empty.  Inference associated
       with ~ n0i .  (Contributed by BJ, 15-Jul-2021.) $)
    n0ii $p |- -. B = (/) $=
      ( wcel c0 wceq wn n0i ax-mp ) ABDBEFGCBAHI $.

    $( If a class has elements, then it is nonempty.  Inference associated with
       ~ ne0i .  (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
    ne0ii $p |- B =/= (/) $=
      ( wcel c0 wne ne0i ax-mp ) ABDBEFCBAGH $.
  $}

  $( The universal class is not equal to the empty set.  (Contributed by NM,
     11-Sep-2008.) $)
  vn0 $p |- _V =/= (/) $=
    ( vx cv cvv wcel c0 wne vex ne0i ax-mp ) ABZCDCEFAGCJHI $.

  $( The universal class is inhabited.  (Contributed by Jim Kingdon,
     17-Dec-2018.) $)
  vn0m $p |- E. x x e. _V $=
    ( cv cvv wcel wex vex 19.8a ax-mp ) ABCDZIAEAFIAGH $.

  ${
    n0rf.1 $e |- F/_ x A $.
    $( An inhabited class is nonempty.  Following the Definition of [Bauer],
       p. 483, we call a class ` A ` nonempty if ` A =/= (/) ` and inhabited if
       it has at least one element.  In classical logic these two concepts are
       equivalent, for example see Proposition 5.17(1) of [TakeutiZaring]
       p. 20.  This version of ~ n0r requires only that ` x ` not be free in,
       rather than not occur in, ` A ` .  (Contributed by Jim Kingdon,
       31-Jul-2018.) $)
    n0rf $p |- ( E. x x e. A -> A =/= (/) ) $=
      ( cv wcel wex wn wal c0 wne exalim wceq wb nfcv cleqf noel nbn necon3abii
      albii bitr4i sylibr ) ADZBEZAFUCGZAHZGBIJUCAKUEBIBILUCUBIEZMZAHUEABICAINO
      UDUGAUFUCUBPQSTRUA $.
  $}

  ${
    $d x A $.
    $( An inhabited class is nonempty.  See ~ n0rf for more discussion.
       (Contributed by Jim Kingdon, 31-Jul-2018.) $)
    n0r $p |- ( E. x x e. A -> A =/= (/) ) $=
      ( nfcv n0rf ) ABABCD $.

    $( An inhabited class is nonempty.  See ~ n0rf for more discussion.
       (Contributed by Jim Kingdon, 31-Jul-2018.) $)
    neq0r $p |- ( E. x x e. A -> -. A = (/) ) $=
      ( cv wcel wex c0 n0r neneqd ) ACBDAEBFABGH $.
  $}

  ${
    $d x A $.  $d x ph $.
    reximdva0m.1 $e |- ( ( ph /\ x e. A ) -> ps ) $.
    $( Restricted existence deduced from inhabited class.  (Contributed by Jim
       Kingdon, 31-Jul-2018.) $)
    reximdva0m $p |- ( ( ph /\ E. x x e. A ) -> E. x e. A ps ) $=
      ( cv wcel wex wa wrex ex ancld eximdv imp df-rex sylibr ) ACFDGZCHZIQBIZC
      HZBCDJARTAQSCAQBAQBEKLMNBCDOP $.
  $}

  ${
    $d A x $.
    $( A case of equivalence of "at most one" and "only one".  If a class is
       inhabited, that class having at most one element is equivalent to it
       having only one element.  (Contributed by Jim Kingdon, 31-Jul-2018.) $)
    n0mmoeu $p |- ( E. x x e. A -> ( E* x x e. A <-> E! x x e. A ) ) $=
      ( cv wcel exmoeu2 ) ACBDAE $.
  $}

  $( Vacuous existential quantification is false.  (Contributed by NM,
     15-Oct-2003.) $)
  rex0 $p |- -. E. x e. (/) ph $=
    ( c0 cv wcel wn noel pm2.21i nrex ) ABCBDZCEAFJGHI $.

  ${
    $d x A $.
    $( The empty set has no elements.  Theorem 2 of [Suppes] p. 22.
       (Contributed by NM, 29-Aug-1993.) $)
    eq0 $p |- ( A = (/) <-> A. x -. x e. A ) $=
      ( c0 wceq cv wcel wb wal wn nfcv cleqf noel nbn albii bitr4i ) BCDAEZBFZP
      CFZGZAHQIZAHABCABJACJKTSARQPLMNO $.

    $( The universe contains every set.  (Contributed by NM, 11-Sep-2006.) $)
    eqv $p |- ( A = _V <-> A. x x e. A ) $=
      ( cvv wceq cv wcel wb wal dfcleq vex tbt albii bitr4i ) BCDAEZBFZNCFZGZAH
      OAHABCIOQAPOAJKLM $.

    $( A class is not inhabited if and only if it is empty.  (Contributed by
       Jim Kingdon, 1-Jul-2022.) $)
    notm0 $p |- ( -. E. x x e. A <-> A = (/) ) $=
      ( c0 wceq cv wcel wn wal wex eq0 alnex bitr2i ) BCDAEBFZGAHMAIGABJMAKL $.
  $}

  ${
    $d x A $.
    nel0.1 $e |- -. x e. A $.
    $( From the general negation of membership in ` A ` , infer that ` A ` is
       the empty set.  (Contributed by BJ, 6-Oct-2018.) $)
    nel0 $p |- A = (/) $=
      ( c0 wceq cv wcel wn eq0 mpgbir ) BDEAFBGHAABICJ $.
  $}

  ${
    $d x A $.  $d x y $.
    $( Membership of the empty set in another class.  (Contributed by NM,
       29-Jun-2004.) $)
    0el $p |- ( (/) e. A <-> E. x e. A A. y -. y e. x ) $=
      ( c0 wcel cv wceq wrex wn wal risset eq0 rexbii bitri ) DCEAFZDGZACHBFOEI
      BJZACHADCKPQACBOLMN $.
  $}

  ${
    $d x ph $.
    $( The class builder of a decidable proposition not containing the
       abstraction variable is either the universal class or the empty set.
       (Contributed by Jim Kingdon, 1-Aug-2018.) $)
    abvor0dc $p |- ( DECID ph -> ( { x | ph } = _V \/ { x | ph } = (/) ) ) $=
      ( wdc wn wo cab cvv wceq c0 df-dc cv wcel id vex a1i 2thd abbi1dv 2falsed
      noel orim12i sylbi ) ACAADZEABFZGHZUCIHZEAJAUDUBUEAABGAABKZGLZAMUGABNOPQU
      BABIUBAUFILZUBMUHDUBUFSORQTUA $.
  $}

  $( Nonempty class abstraction.  (Contributed by Jim Kingdon, 1-Aug-2018.) $)
  abn0r $p |- ( E. x ph -> { x | ph } =/= (/) ) $=
    ( wex cv cab wcel c0 wne abid exbii nfab1 n0rf sylbir ) ABCBDABEZFZBCNGHOAB
    ABIJBNABKLM $.

  ${
    $d ph y $.  $d x y $.
    $( Inhabited class abstraction.  (Contributed by Jim Kingdon,
       8-Jul-2022.) $)
    abn0m $p |- ( E. y y e. { x | ph } <-> E. x ph ) $=
      ( cv cab wcel wex nfv nfsab1 eleq1w cbvex abid exbii bitr3i ) CDABEZFZCGB
      DOFZBGABGQPBCQCHABCIBCOJKQABABLMN $.
  $}

  $( Nonempty restricted class abstraction.  (Contributed by Jim Kingdon,
     1-Aug-2018.) $)
  rabn0r $p |- ( E. x e. A ph -> { x e. A | ph } =/= (/) ) $=
    ( cv wcel wa wex cab c0 wne wrex crab abn0r df-rex df-rab neeq1i 3imtr4i )
    BDCEAFZBGRBHZIJABCKABCLZIJRBMABCNTSIABCOPQ $.

  ${
    $d x y $.  $d A y $.  $d ph y $.
    $( Inhabited restricted class abstraction.  (Contributed by Jim Kingdon,
       18-Sep-2018.) $)
    rabn0m $p |- ( E. y y e. { x e. A | ph } <-> E. x e. A ph ) $=
      ( wrex cv wcel wa wex crab df-rex rabid exbii nfv cab df-rab eleq2i nfxfr
      nfsab1 eleq1 cbvex 3bitr2ri ) ABDEBFZDGAHZBIUCABDJZGZBICFZUEGZCIABDKUFUDB
      ABDLMUFUHBCUFCNUHUGUDBOZGBUEUIUGABDPQUDBCSRUCUGUETUAUB $.
  $}

  ${
    $d x y $.  $d ph y $.  $d ps x $.
    ab0w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( The class of sets verifying a property is the empty class if and only if
       that property is a contradiction.  (Contributed by GG, 3-Oct-2024.) $)
    ab0w $p |- ( { x | ph } = (/) <-> A. y -. ps ) $=
      ( cab c0 wceq wfal wn wal dfnul4 eqeq2i cv wcel wsb df-clab bitri albii
      wb sbv bibi2i eqabcbw nbfal 3bitr4i ) ACFZGHUFICFZHZBJZDKZGUGUFCLMBDNUGOZ
      TZDKBITZDKUHUJULUMDUKIBUKICDPIIDCQICDUARUBSABCDUGEUCUIUMDBUDSUER $.
    $( $j usage 'ab0w' avoids 'df-clel' 'ax-13'; $)
  $}

  $( Any restricted class abstraction restricted to the empty set is empty.
     (Contributed by NM, 15-Oct-2003.)  (Proof shortened by Andrew Salmon,
     26-Jun-2011.) $)
  rab0 $p |- { x e. (/) | ph } = (/) $=
    ( cv c0 wcel wa wceq wn crab noel intnanr equid notnoti 2false abbii df-rab
    cab dfnul2 3eqtr4i ) BCZDEZAFZBQTTGZHZBQABDIDUBUDBUBUDUAATJKUCBLMNOABDPBRS
    $.

  ${
    $d x y $.  $d A y $.  $d ph y $.

    $( Condition for a restricted class abstraction to be empty.  (Contributed
       by Jeff Madsen, 7-Jun-2010.) $)
    rabeq0 $p |- ( { x e. A | ph } = (/) <-> A. x e. A -. ph ) $=
      ( vy cv wcel wn wi wa wral crab c0 wceq imnan albii df-ral bitri 3bitr4ri
      wal wsb sbn nfv sb8 eq0 cab df-rab eleq2i df-clab notbii ) BECFZAGZHZBSUJ
      AIZGZBSZUKBCJABCKZLMZULUNBUJANOUKBCPUNBDTZDSUMBDTZGZDSZUOUQURUTDUMBDUAOUN
      BDUNDUBUCUQDEZUPFZGZDSVADUPUDVDUTDVCUSVCVBUMBUEZFUSUPVEVBABCUFUGUMDBUHQUI
      OQRR $.

    $( Condition for a class abstraction to be empty.  (Contributed by Jim
       Kingdon, 12-Aug-2018.) $)
    abeq0 $p |- ( { x | ph } = (/) <-> A. x -. ph ) $=
      ( vy wn wsb wal cab c0 wceq sbn albii nfv sb8 cv eq0 df-clab notbii bitri
      wcel 3bitr4ri ) ADZBCEZCFABCEZDZCFZUABFABGZHIZUBUDCABCJKUABCUACLMUGCNUFSZ
      DZCFUECUFOUIUDCUHUCACBPQKRT $.
  $}

  ${
    $d A x $.
    $( Law of excluded middle given decidability, in terms of restricted class
       abstractions.  (Contributed by Jeff Madsen, 20-Jun-2011.)  (Revised by
       Jim Kingdon, 17-Jun-2026.) $)
    rabxmdc $p |- ( A. x e. A DECID ph ->
        A = ( { x e. A | ph } u. { x e. A | -. ph } ) ) $=
      ( wdc wral wn wo crab cun wceq exmiddc ralimi rabid2 sylibr unrab eqtr4di
      ) ADZBCEZCAAFZGZBCHZABCHSBCHIRTBCECUAJQTBCAKLTBCMNASBCOP $.

    $( Law of noncontradiction, in terms of restricted class abstractions.
       (Contributed by Jeff Madsen, 20-Jun-2011.) $)
    rabnc $p |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/) $=
      ( crab wn cin wa c0 inrab wceq rabeq0 cv wcel pm3.24 a1i mprgbir eqtri )
      ABCDAEZBCDFARGZBCDZHARBCITHJSEZBCSBCKUABLCMANOPQ $.
  $}

  ${
    $d x A $.
    $( The union of a class with the empty set is itself.  Dual of ~ inv1 .
       Theorem 24 of [Suppes] p. 27.  (Contributed by NM, 5-Aug-1993.) $)
    un0 $p |- ( A u. (/) ) = A $=
      ( vx c0 cv wcel wo noel biorfi bicomi uneqri ) BACABDZAEZLKCEZFMLKGHIJ $.

    $( The intersection of a class with the empty set is the empty set.  Dual
       of ~ unv .  Commuted form of ~ 0in .  Theorem 16 of [Suppes] p. 26.
       (Contributed by NM, 5-Aug-1993.) $)
    in0 $p |- ( A i^i (/) ) = (/) $=
      ( vx c0 cv wcel wa noel bianfi bicomi ineqri ) BACCBDZCEZKAEZLFLMKGHIJ $.
  $}

  $( The intersection of the empty set with a class is the empty set.  Commuted
     form of ~ in0 .  (Contributed by Glauco Siliprandi, 17-Aug-2020.) $)
  0in $p |- ( (/) i^i A ) = (/) $=
    ( c0 cin incom in0 eqtri ) BACABCBBADAEF $.

  $( The intersection of a class with the universal class is itself.  Dual of
     ~ un0 .  Exercise 4.10(k) of [Mendelson] p. 231.  (Contributed by NM,
     17-May-1998.) $)
  inv1 $p |- ( A i^i _V ) = A $=
    ( cvv cin inss1 ssid ssv ssini eqssi ) ABCAABDAABAEAFGH $.

  $( The union of a class with the universal class is the universal class.
     Dual of ~ in0 .  Exercise 4.10(l) of [Mendelson] p. 231.  (Contributed by
     NM, 17-May-1998.) $)
  unv $p |- ( A u. _V ) = _V $=
    ( cvv cun ssv ssun2 eqssi ) ABCZBGDBAEF $.

  ${
    $d A x $.
    $( The empty set is a subset of any class.  Dual of ~ ssv .  Part of
       Exercise 1 of [TakeutiZaring] p. 22.  (Contributed by NM,
       5-Aug-1993.) $)
    0ss $p |- (/) C_ A $=
      ( vx c0 cv wcel noel pm2.21i ssriv ) BCABDZCEIAEIFGH $.
  $}

  $( Any subset of the empty set is empty.  Dual of ~ vss .  Theorem 5 of
     [Suppes] p. 23 and its converse.  (Contributed by NM, 17-Sep-2003.) $)
  ss0b $p |- ( A C_ (/) <-> A = (/) ) $=
    ( c0 wceq wss 0ss eqss mpbiran2 bicomi ) ABCZABDZIJBADAEABFGH $.

  $( Any subset of the empty set is empty.  Theorem 5 of [Suppes] p. 23.
     (Contributed by NM, 13-Aug-1994.) $)
  ss0 $p |- ( A C_ (/) -> A = (/) ) $=
    ( c0 wss wceq ss0b biimpi ) ABCABDAEF $.

  $( The only subclass of the empty class is itself.  (Contributed by NM,
     7-Mar-2007.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.)  Strengthen
     ~ sseq0 to a biconditional.  (Revised by BJ, 19-Jul-2026.) $)
  sseq0b $p |- ( A = (/) -> ( B C_ A <-> B = (/) ) ) $=
    ( c0 wceq wss sseq2 ss0b bitrdi ) ACDBAEBCEBCDACBFBGH $.

  $( A subclass of an empty class is empty.  (Contributed by NM, 7-Mar-2007.)
     (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
  sseq0 $p |- ( ( A C_ B /\ B = (/) ) -> A = (/) ) $=
    ( c0 wceq wss sseq0b biimpac ) BCDABEACDBAFG $.

  $( A class with a nonempty subclass is nonempty.  (Contributed by NM,
     17-Feb-2007.) $)
  ssn0 $p |- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) ) $=
    ( wss c0 wne wceq sseq0 ex necon3d imp ) ABCZADEBDEKBDADKBDFADFABGHIJ $.

  $( Two classes are both empty if and only if their union is empty.  Dual of
     ~ vvin .  (Contributed by NM, 11-Aug-2004.) $)
  un00 $p |- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) ) $=
    ( c0 wceq wa cun uneq12 un0 eqtrdi wss ssun1 sseq2 mpbii sylib ssun2 impbii
    ss0b jca ) ACDZBCDZEZABFZCDZUAUBCCFCACBCGCHIUCSTUCACJZSUCAUBJUDABKUBCALMAQN
    UCBCJZTUCBUBJUEBAOUBCBLMBQNRP $.

  $( Only the universal class has the universal class as a subclass.  Dual of
     ~ ss0b .  (Contributed by NM, 17-Sep-2003.)  (Proof shortened by Andrew
     Salmon, 26-Jun-2011.) $)
  vss $p |- ( _V C_ A <-> A = _V ) $=
    ( cvv wss wa wceq ssv biantrur eqss bitr4i ) BACZABCZJDABEKJAFGABHI $.

  $( Two classes are both the universal class if and only if their intersection
     is the universal class.  Dual of ~ un00 .  (Contributed by BJ,
     12-Jul-2026.) $)
  vvin $p |- ( ( A = _V /\ B = _V ) <-> ( A i^i B ) = _V ) $=
    ( cvv wceq cin ineq12 inv1 eqtrdi wss inss1 sseq1 mpbii vss sylib inss2 jca
    wa impbii ) ACDZBCDZQZABEZCDZUAUBCCECACBCFCGHUCSTUCCAIZSUCUBAIUDABJUBCAKLAM
    NUCCBIZTUCUBBIUEABOUBCBKLBMNPR $.

  ${
    abf.1 $e |- -. ph $.
    $( A class builder with a false argument is empty.  (Contributed by NM,
       20-Jan-2012.) $)
    abf $p |- { x | ph } = (/) $=
      ( cab c0 wss wceq cv wcel pm2.21i abssi ss0 ax-mp ) ABDZEFNEGABEABHEICJKN
      LM $.
  $}

  ${
    $d x A $.  $d x ph $.
    eq0rdv.1 $e |- ( ph -> -. x e. A ) $.
    $( Deduction for equality to the empty set.  (Contributed by NM,
       11-Jul-2014.) $)
    eq0rdv $p |- ( ph -> A = (/) ) $=
      ( c0 wss wceq cv wcel pm2.21d ssrdv ss0 syl ) ACEFCEGABCEABHZCINEIDJKCLM
      $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    $( The proper substitution of a proper class for a set into a class results
       in the empty set.  (Contributed by NM, 17-Aug-2018.) $)
    csbprc $p |- ( -. A e. _V -> [_ A / x ]_ B = (/) ) $=
      ( vy cvv wcel wn csb cv wsbc cab c0 df-csb wfal sbcex con3i pm2.21d falim
      impbid1 abbidv fal abf eqtrdi eqtrid ) BEFZGZABCHDICFZABJZDKZLADBCMUFUIND
      KLUFUHNDUFUHNUFUHNUHUEUGABOPQUHRSTNDUAUBUCUD $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Two ways of saying that two classes are disjoint (have no members in
       common).  (Contributed by NM, 17-Feb-2004.) $)
    disj $p |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B ) $=
      ( cin c0 wceq cv wcel wn wi wal wral wa cab df-in eqeq1i abeq1 imnan noel
      wb nbn bitr2i albii 3bitri df-ral bitr4i ) BCDZEFZAGZBHZUICHZIZJZAKZULABL
      UHUJUKMZANZEFUOUIEHZTZAKUNUGUPEABCOPUOAEQURUMAUMUOIURUJUKRUQUOUISUAUBUCUD
      ULABUEUF $.

    $( Two ways of saying that two classes are disjoint.  (Contributed by Jeff
       Madsen, 19-Jun-2011.) $)
    disjr $p |- ( ( A i^i B ) = (/) <-> A. x e. B -. x e. A ) $=
      ( cin c0 wceq cv wcel wn wral incom eqeq1i disj bitri ) BCDZEFCBDZEFAGBHI
      ACJOPEBCKLACBMN $.

    $( Two ways of saying that two classes are disjoint (have no members in
       common).  (Contributed by NM, 19-Aug-1993.) $)
    disj1 $p |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) ) $=
      ( cin c0 wceq cv wcel wn wral wi wal disj df-ral bitri ) BCDEFAGZCHIZABJP
      BHQKALABCMQABNO $.

    $( Two ways of saying that two classes are disjoint, using the complement
       of ` B ` relative to a universe ` C ` .  (Contributed by NM,
       15-Feb-2007.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    reldisj $p |- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) ) $=
      ( vx wss cv wcel wn wi wal cdif cin c0 wceq wb ssalel pm5.44 eldif imbi2i
      wa bitr4di sps sylbi albidv disj1 3bitr4g ) ACEZDFZAGZUHBGHZIZDJUIUHCBKZG
      ZIZDJABLMNAULEUGUKUNDUGUIUHCGZIZDJUKUNOZDACPUPUQDUPUKUIUOUJTZIUNUIUOUJQUM
      URUIUHCBRSUAUBUCUDDABUEDAULPUF $.

    $( Two ways of saying that two classes are disjoint.  (Contributed by NM,
       19-May-1998.) $)
    disj3 $p |- ( ( A i^i B ) = (/) <-> A = ( A \ B ) ) $=
      ( vx cv wcel wn wi wal cdif wb cin c0 wa pm4.71 eldif bibi2i bitr4i albii
      wceq disj1 dfcleq 3bitr4i ) CDZAEZUCBEFZGZCHUDUCABIZEZJZCHABKLSAUGSUFUICU
      FUDUDUEMZJUIUDUENUHUJUDUCABOPQRCABTCAUGUAUB $.

    $( Members of disjoint sets are not equal.  (Contributed by NM,
       28-Mar-2007.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    disjne $p |- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D ) $=
      ( vx cin c0 wceq wcel wne cv wn wral wi disj eleq1 notbid rspccva eleq1a
      wa necon3bd syl5com sylanb 3impia ) ABFGHZCAIZDBIZCDJZUEEKZBIZLZEAMZUFUGU
      HNEABOULUFTCBIZLZUGUHUKUNECAUICHUJUMUICBPQRUGUMCDDBCSUAUBUCUD $.
  $}

  $( A set can't belong to both members of disjoint classes.  (Contributed by
     NM, 28-Feb-2015.) $)
  disjel $p |- ( ( ( A i^i B ) = (/) /\ C e. A ) -> -. C e. B ) $=
    ( cin c0 wceq wcel wn cdif wi disj3 eleq2 eldifn biimtrdi sylbi imp ) ABDEF
    ZCAGZCBGHZQAABIZFZRSJABKUARCTGSATCLCABMNOP $.

  $( Two ways of saying that two classes are disjoint.  (Contributed by NM,
     17-May-1998.) $)
  disj2 $p |- ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) ) $=
    ( cvv wss cin c0 wceq cdif wb ssv reldisj ax-mp ) ACDABEFGACBHDIAJABCKL $.

  $( Intersection with a subclass of a disjoint class.  (Contributed by FL,
     24-Jan-2007.) $)
  ssdisj $p |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) = (/) ) $=
    ( wss cin c0 wceq wa ss0b wi ssrin sstr2 syl biimtrrid imp ss0 ) ABDZBCEZFG
    ZHACEZFDZTFGQSUASRFDZQUARIQTRDUBUAJABCKTRFLMNOTPM $.

  $( The union of disjoint classes is disjoint.  (Contributed by NM,
     26-Sep-2004.) $)
  undisj1 $p |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <->
               ( ( A u. B ) i^i C ) = (/) ) $=
    ( cin c0 wceq wa cun un00 indir eqeq1i bitr4i ) ACDZEFBCDZEFGMNHZEFABHCDZEF
    MNIPOEABCJKL $.

  $( The union of disjoint classes is disjoint.  (Contributed by NM,
     13-Sep-2004.) $)
  undisj2 $p |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <->
               ( A i^i ( B u. C ) ) = (/) ) $=
    ( cin c0 wceq wa cun un00 indi eqeq1i bitr4i ) ABDZEFACDZEFGMNHZEFABCHDZEFM
    NIPOEABCJKL $.

  $( Subclass implies empty intersection with difference from the universal
     class.  (Contributed by NM, 17-Sep-2003.) $)
  ssindif0im $p |- ( A C_ B -> ( A i^i ( _V \ B ) ) = (/) ) $=
    ( wss cvv cdif cin c0 wceq ddifss sstr mpan2 disj2 sylibr ) ABCZADDBEZEZCZA
    OFGHNBPCQBIABPJKAOLM $.

  $( The intersection of classes with a common member is nonempty.
     (Contributed by NM, 7-Apr-1994.) $)
  inelcm $p |- ( ( A e. B /\ A e. C ) -> ( B i^i C ) =/= (/) ) $=
    ( wcel wa cin c0 wne elin ne0i sylbir ) ABDACDEABCFZDLGHABCILAJK $.

  $( A minimum element of a class has no elements in common with the class.
     (Contributed by NM, 22-Jun-1994.) $)
  minel $p |- ( ( A e. B /\ ( C i^i B ) = (/) ) -> -. A e. C ) $=
    ( cin c0 wceq wcel wn wa wi inelcm necon2bi imnan sylibr con2d impcom ) CBD
    ZEFZABGZACGZHRTSRTSIZHTSHJUAQEACBKLTSMNOP $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Distribute union over difference.  (Contributed by NM, 17-May-1998.)
       (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    undif4 $p |- ( ( A i^i C ) = (/) ->
                 ( A u. ( B \ C ) ) = ( ( A u. B ) \ C ) ) $=
      ( vx cv wcel wn wi wal cdif cun wb cin c0 wceq wo wa eldif elun 3bitr4g
      pm2.621 olc impbid1 anbi2d orbi2i bitri anbi1i alimi disj1 dfcleq 3imtr4i
      ordi ) DEZAFZUMCFGZHZDIUMABCJZKZFZUMABKZCJZFZLZDIACMNOURVAOUPVCDUPUNUMUQF
      ZPZUMUTFZUOQZUSVBUPUNUMBFZPZUNUOPZQZVIUOQVEVGUPVJUOVIUPVJUOUNUOUAUOUNUBUC
      UDVEUNVHUOQZPVKVDVLUNUMBCRUEUNVHUOULUFVFVIUOUMABSUGTUMAUQSUMUTCRTUHDACUID
      URVAUJUK $.

    $( Subset relation for disjoint classes.  (Contributed by NM, 25-Oct-2005.)
       (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    disjssun $p |- ( ( A i^i B ) = (/) -> ( A C_ ( B u. C ) <-> A C_ C ) ) $=
      ( cin c0 wceq cun wss indi equncomi uneq2 un0 eqtrdi eqtrid df-ss 3bitr4g
      eqeq1d ) ABDZEFZABCGZDZAFACDZAFATHACHSUAUBASUAUBRGZUBUARUBABCIJSUCUBEGUBR
      EUBKUBLMNQATOACOP $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Subclass implies empty difference.  One direction of Exercise 7 of
       [TakeutiZaring] p. 22.  In classical logic this would be an equivalence.
       (Contributed by Jim Kingdon, 2-Aug-2018.) $)
    ssdif0im $p |- ( A C_ B -> ( A \ B ) = (/) ) $=
      ( vx cv wcel wi wal cdif wn wss c0 wceq imanim eldif sylnibr alimi ssalel
      wa eq0 3imtr4i ) CDZAEZUABEZFZCGUAABHZEZIZCGABJUEKLUDUGCUDUBUCIRUFUBUCMUA
      ABNOPCABQCUEST $.
  $}

  $( Universal class equality in terms of empty difference.  (Contributed by
     Jim Kingdon, 3-Aug-2018.) $)
  vdif0im $p |- ( A = _V -> ( _V \ A ) = (/) ) $=
    ( cvv wceq wss cdif c0 vss ssdif0im sylbir ) ABCBADBAEFCAGBAHI $.

  ${
    $d V x $.
    $( If the difference between the restricting class of a restricted class
       abstraction and the restricted class abstraction is empty, the
       restricting class is equal to this restricted class abstraction.
       (Contributed by Jim Kingdon, 3-Aug-2018.) $)
    difrab0eqim $p |-
        ( V = { x e. V | ph } -> ( V \ { x e. V | ph } ) = (/) ) $=
      ( crab wceq wss cdif c0 ssrabeq ssdif0im sylbir ) CABCDZECLFCLGHEABCICLJK
      $.
  $}

  $( Intersection, subclass, and difference relationship.  The converse holds
     in classical logic but not in intuitionistic logic.  (Contributed by Jim
     Kingdon, 3-Aug-2018.)  (Proof shortened by BJ, 18-Jul-2026.) $)
  inssdif0im $p |- ( ( A i^i B ) C_ C -> ( A i^i ( B \ C ) ) = (/) ) $=
    ( cin wss cdif c0 indif2 ssdif0im eqtrid ) ABDZCEABCFDKCFGABCHKCIJ $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Obsolete version of ~ inssdif0im as of 18-Jul-2026.  (Contributed by Jim
       Kingdon, 3-Aug-2018.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    inssdif0imOLD $p |- ( ( A i^i B ) C_ C -> ( A i^i ( B \ C ) ) = (/) ) $=
      ( vx cv cin wcel wi wal cdif wn c0 wceq wa elin imbi1i imanim sylbi eldif
      wss anbi2i anass 3bitr4ri sylnib alimi ssalel eq0 3imtr4i ) DEZABFZGZUICG
      ZHZDIUIABCJZFZGZKZDIUJCTUOLMUMUQDUMUIAGZUIBGZNZULKZNZUPUMUTULHVBKUKUTULUI
      ABOPUTULQRURUIUNGZNURUSVANZNUPVBVCVDURUIBCSUAUIAUNOURUSVAUBUCUDUEDUJCUFDU
      OUGUH $.
  $}

  $( The difference between a class and itself is the empty set.  Proposition
     5.15 of [TakeutiZaring] p. 20.  Also Theorem 32 of [Suppes] p. 28.
     (Contributed by NM, 22-Apr-2004.) $)
  difid $p |- ( A \ A ) = (/) $=
    ( wss cdif c0 wceq ssid ssdif0im ax-mp ) AABAACDEAFAAGH $.

  ${
    $d x A $.
    $( The difference between a class and itself is the empty set.  Proposition
       5.15 of [TakeutiZaring] p. 20.  Also Theorem 32 of [Suppes] p. 28.
       Alternate proof of ~ difid .  (Contributed by David Abernethy,
       17-Jun-2012.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    difidALT $p |- ( A \ A ) = (/) $=
      ( vx cdif cv wcel wn crab c0 dfdif2 dfnul3 eqtr4i ) AACBDAEFBAGHBAAIBAJK
      $.
  $}

  $( The difference between a class and the empty set.  Part of Exercise 4.4 of
     [Stoll] p. 16.  (Contributed by NM, 17-Aug-2004.) $)
  dif0 $p |- ( A \ (/) ) = A $=
    ( cdif c0 difid difeq2i difdif eqtr3i ) AAABZBACBAHCAADEAAFG $.

  $( The difference between the empty set and a class.  Part of Exercise 4.4 of
     [Stoll] p. 16.  (Contributed by NM, 17-Aug-2004.) $)
  0dif $p |- ( (/) \ A ) = (/) $=
    ( c0 cdif wss wceq difss ss0 ax-mp ) BACZBDIBEBAFIGH $.

  $( A class and its relative complement are disjoint.  (Contributed by NM,
     24-Mar-1998.)  Generalize from ~ disjdif .  (Revised by BJ,
     19-Jul-2026.) $)
  disjdifg $p |- ( A C_ B -> ( A i^i ( C \ B ) ) = (/) ) $=
    ( wss cin cdif c0 wceq ssinss1 inssdif0im syl ) ABDACEBDACBFEGHACBIACBJK $.

  $( A class and its relative complement are disjoint.  Theorem 38 of [Suppes]
     p. 29.  (Contributed by NM, 24-Mar-1998.) $)
  disjdif $p |- ( A i^i ( B \ A ) ) = (/) $=
    ( wss cdif cin c0 wceq ssid disjdifg ax-mp ) AACABADEFGAHAABIJ $.

  $( A class and its relative complement are disjoint.  Commuted form of
     ~ disjdif .  (Contributed by Thierry Arnoux, 29-Nov-2023.) $)
  disjdifr $p |- ( ( B \ A ) i^i A ) = (/) $=
    ( cdif c0 disjdif ineqcomi ) ABACDABEF $.

  $( The difference of a class from its intersection is empty.  Theorem 37 of
     [Suppes] p. 29.  (Contributed by NM, 17-Aug-2004.)  (Proof shortened by
     Andrew Salmon, 26-Jun-2011.) $)
  difin0 $p |- ( ( A i^i B ) \ B ) = (/) $=
    ( cin wss cdif c0 wceq inss2 ssdif0im ax-mp ) ABCZBDKBEFGABHKBIJ $.

  $( Absorption of difference by union.  In classical logic, as Theorem 35 of
     [Suppes] p. 29, this would be equality rather than subset.  (Contributed
     by Jim Kingdon, 4-Aug-2018.) $)
  undif1ss $p |- ( ( A \ B ) u. B ) C_ ( A u. B ) $=
    ( cdif wss cun difss unss1 ax-mp ) ABCZADIBEABEDABFIABGH $.

  $( Absorption of difference by union.  In classical logic, as in Part of
     proof of Corollary 6K of [Enderton] p. 144, this would be equality rather
     than subset.  (Contributed by Jim Kingdon, 4-Aug-2018.) $)
  undif2ss $p |- ( A u. ( B \ A ) ) C_ ( A u. B ) $=
    ( cdif cun undif1ss uncom 3sstr4i ) BACZADBADAHDABDBAEAHFABFG $.

  $( Absorption of difference by union.  (Contributed by NM, 18-Aug-2013.) $)
  undifabs $p |- ( A u. ( A \ B ) ) = A $=
    ( cdif cun ssid difss unssi ssun1 eqssi ) AABCZDAAJAAEABFGAJHI $.

  $( The intersection and class difference of a class with another class are
     contained in the original class.  In classical logic we'd be able to make
     a stronger statement: that everything in the original class is in the
     intersection or the difference (that is, this theorem would be equality
     rather than subset).  (Contributed by Jim Kingdon, 4-Aug-2018.) $)
  inundifss $p |- ( ( A i^i B ) u. ( A \ B ) ) C_ A $=
    ( cin cdif inss1 difss unssi ) ABCABDAABEABFG $.

  $( The difference of a class and a class disjoint from it is the original
     class.  (Contributed by BJ, 21-Apr-2019.) $)
  disjdif2 $p |- ( ( A i^i B ) = (/) -> ( A \ B ) = A ) $=
    ( cin c0 wceq cdif difeq2 difin dif0 3eqtr3g ) ABCZDEAKFADFABFAKDAGABHAIJ
    $.

  $( Absorption of union by difference.  Theorem 36 of [Suppes] p. 29.
     (Contributed by NM, 19-May-1998.) $)
  difun2 $p |- ( ( A u. B ) \ B ) = ( A \ B ) $=
    ( cun cdif c0 difundir difid uneq2i un0 3eqtri ) ABCBDABDZBBDZCKECKABBFLEKB
    GHKIJ $.

  $( Union of complementary parts into whole.  (Contributed by Jim Kingdon,
     4-Aug-2018.) $)
  undifss $p |- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B ) $=
    ( wss cdif cun wa difss jctr unss sylib ssun1 sstr mpan impbii ) ABCZABADZE
    ZBCZOOPBCZFROSBAGHAPBIJAQCROAPKAQBLMN $.

  $( A subset of a difference does not intersect the subtrahend.  (Contributed
     by Jeff Hankins, 1-Sep-2013.)  (Proof shortened by Mario Carneiro,
     24-Aug-2015.) $)
  ssdifin0 $p |- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) ) $=
    ( cdif wss cin c0 wceq ssrin incom disjdif eqtri sseq0 sylancl ) ABCDZEACFZ
    OCFZEQGHPGHAOCIQCOFGOCJCBKLPQMN $.

  $( A class is a subclass of itself subtracted from another iff it is the
     empty set.  (Contributed by Steve Rodriguez, 20-Nov-2015.) $)
  ssdifeq0 $p |- ( A C_ ( B \ A ) <-> A = (/) ) $=
    ( cdif wss wceq cin inidm ssdifin0 eqtr3id 0ss difeq2 sseq12d mpbiri impbii
    c0 id ) ABACZDZAOEZRAAAFOAGABAHISROBOCZDTJSAOQTSPAOBKLMN $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( A consequence of inclusion in the union of two classes.  In classical
       logic this would be a biconditional.  (Contributed by Jim Kingdon,
       4-Aug-2018.) $)
    ssundifim $p |- ( A C_ ( B u. C ) -> ( A \ B ) C_ C ) $=
      ( vx cv wcel cun wi wal cdif wss wo wn wa pm5.6r elun imbi2i eldif ssalel
      3imtr4i imbi1i alimi ) DEZAFZUCBCGZFZHZDIUCABJZFZUCCFZHZDIAUEKUHCKUGUKDUD
      UCBFZUJLZHUDULMNZUJHUGUKUDULUJOUFUMUDUCBCPQUIUNUJUCABRUATUBDAUESDUHCST $.
  $}

  $( Distributive law for class difference.  In classical logic, as in Exercise
     4.8 of [Stoll] p. 16, this would be equality rather than subset.
     (Contributed by Jim Kingdon, 4-Aug-2018.) $)
  difdifdirss $p |- ( ( A \ B ) \ C ) C_ ( ( A \ C ) \ ( B \ C ) ) $=
    ( cdif cvv cun cin dif32 invdif eqtr4i un0 indi disjdif incom eqtr3i uneq2i
    c0 wss ax-mp sseqtri ddifss unss2 indmss difeq2i sstri sslin eqsstri ) ABDC
    DZACDZEBDZCFZGZUIBCDZDZUHUIUJGZQFZULUHUOUPUHUIBDUOABCHUIBIJUOKJULUOUICGZFUP
    UIUJCLQUQUOCUIGQUQCAMCUINOPJJULUIEUMDZGZUNUKURRULUSRUKUJEECDZDZFZURCVARUKVB
    RCUACVAUJUBSVBEBUTGZDURBUTUCVCUMEBCIUDTUEUKURUIUFSUIUMITUG $.

  $( Two ways that ` A ` and ` B ` can "partition" ` C ` (when ` A ` and ` B `
     don't overlap and ` A ` is a part of ` C ` ).  In classical logic, the
     second implication would be a biconditional.  (Contributed by Jim Kingdon,
     4-Aug-2018.) $)
  uneqdifeqim $p |- ( ( A C_ C /\ ( A i^i B ) = (/) )
     -> ( ( A u. B ) = C -> ( C \ A ) = B ) ) $=
    ( cin c0 wceq cun cdif wi wss uncom eqcomd difeq1 difun2 incom eqeq1i disj3
    wa eqtr bitri expcom eqcoms sylbi syl5com sylancl syl mpan com12 adantl ) A
    BDZEFZABGZCFZCAHZBFZIACJUMUKUOBAGZULFZUMUKUOIZBAKUQUMRZCUPFZURUSUPCUPULCSLU
    TUNUPAHZFZVABAHZFZURCUPAMBANVBVDRUNVCFZUKUOUNVAVCSUKBVCFZVEUOIZUKBADZEFVFUJ
    VHEABOPBAQTVGVCBVEVCBFUOUNVCBSUAUBUCUDUEUFUGUHUI $.

  ${
    $d x A $.  $d y A $.  $d x z $.  $d z y $.  $d z A $.
    $( Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare
       ~ 19.2 ).  The restricted version is valid only when the domain of
       quantification is inhabited.  (Contributed by Jim Kingdon, 5-Aug-2018.)
       (Revised by Jim Kingdon, 7-Apr-2023.) $)
    r19.2m $p |- ( ( E. y y e. A /\ A. x e. A ph ) -> E. x e. A ph ) $=
      ( vz cv wcel wex wral wrex eleq1w cbvexv bitri wa wal df-ral exintr sylbi
      wi df-rex imbitrrdi impcom sylanbr ) CFDGZCHZBFDGZBHZABDIZABDJZUGEFDGZEHU
      EUFUJBEBEDKLUJUDECECDKLMUHUGUIUHUGUFANBHZUIUHUFASBOUGUKSABDPUFABQRABDTUAU
      BUC $.
  $}

  ${
    $d x A $.
    $( Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare
       ~ 19.2 ).  The restricted version is valid only when the domain of
       quantification is inhabited.  (Contributed by Jim Kingdon, 5-Aug-2018.)
       Obsolete version of ~ r19.2m as of 7-Apr-2023.
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    r19.2mOLD $p |- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph ) $=
      ( wral cv wcel wex wrex wa wi df-ral exintr sylbi df-rex imbitrrdi impcom
      wal ) ABCDZBECFZBGZABCHZRTSAIBGZUARSAJBQTUBJABCKSABLMABCNOP $.
  $}

  ${
    $d a x A $.  $d a y A $.
    r19.3rm.1 $e |- F/ x ph $.
    $( Restricted quantification of wff not containing quantified variable.
       (Contributed by Jim Kingdon, 19-Dec-2018.) $)
    r19.3rm $p |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) ) $=
      ( va cv wcel wex wral wb eleq1 cbvexv wi biimt df-ral 19.23 bitri bitr4di
      wal sylbi sylbir ) CGZDHZCIFGZDHZFIZAABDJZKZUFUDFCUEUCDLMUGBGZDHZBIZUIUFU
      KFBUEUJDLMULAULANZUHULAOUHUKANBTUMABDPUKABEQRSUAUB $.
  $}

  ${
    $d x A $.
    r19.28m.1 $e |- F/ x ph $.
    $( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90.  It
       is valid only when the domain of quantification is inhabited.
       (Contributed by Jim Kingdon, 5-Aug-2018.) $)
    r19.28m $p |- ( E. x x e. A ->
                   ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) ) $=
      ( cv wcel wex wa wral r19.26 r19.3rm anbi1d bitr4id ) CFDGCHZABICDJACDJZB
      CDJZIAQIABCDKOAPQACCDELMN $.
  $}

  ${
    $d x A $.  $d y A $.  $d x ph $.
    $( Restricted quantification of wff not containing quantified variable.
       (Contributed by Jim Kingdon, 6-Aug-2018.) $)
    r19.3rmv $p |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) ) $=
      ( nfv r19.3rm ) ABCDABEF $.
  $}

  ${
    $d x a A $.  $d y a A $.  $d x ph $.
    $( Restricted quantification of wff not containing quantified variable.
       (Contributed by Jim Kingdon, 5-Aug-2018.) $)
    r19.9rmv $p |- ( E. y y e. A -> ( ph <-> E. x e. A ph ) ) $=
      ( va cv wcel wex wrex wb eleq1 cbvexv wa df-rex 19.41v bitri baibr sylbir
      sylbi ) CFZDGZCHEFZDGZEHZAABDIZJZUCUAECUBTDKLUDBFZDGZBHZUFUCUHEBUBUGDKLUE
      UIAUEUHAMBHUIAMABDNUHABOPQSR $.
  $}

  ${
    $d x A $.  $d x ph $.
    $( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90.  It
       is valid only when the domain of quantification is inhabited.
       (Contributed by Jim Kingdon, 6-Aug-2018.) $)
    r19.28mv $p |- ( E. x x e. A ->
                   ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) ) $=
      ( nfv r19.28m ) ABCDACEF $.

    $( Restricted version of Theorem 19.45 of [Margaris] p. 90.  (Contributed
       by NM, 27-May-1998.) $)
    r19.45mv $p |- ( E. x x e. A ->
                   ( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) ) $=
      ( cv wcel wex wo wrex r19.43 r19.9rmv orbi1d bitr4id ) CEDFCGZABHCDIACDIZ
      BCDIZHAPHABCDJNAOPACCDKLM $.
  $}

  ${
    $d A x $.  $d A y $.  $d ps x $.
    $( Restricted version of Theorem 19.44 of [Margaris] p. 90.  (Contributed
       by NM, 27-May-1998.) $)
    r19.44mv $p |- ( E. y y e. A ->
                   ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ ps ) ) ) $=
      ( cv wcel wex wo wrex r19.43 r19.9rmv orbi2d bitr4id ) DFEGDHZABICEJACEJZ
      BCEJZIPBIABCEKOBQPBCDELMN $.
  $}

  ${
    $d x A $.
    r19.27m.1 $e |- F/ x ps $.
    $( Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90.  It
       is valid only when the domain of quantification is inhabited.
       (Contributed by Jim Kingdon, 5-Aug-2018.) $)
    r19.27m $p |- ( E. x x e. A ->
                   ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) ) $=
      ( cv wcel wex wa wral r19.26 r19.3rm anbi2d bitr4id ) CFDGCHZABICDJACDJZB
      CDJZIPBIABCDKOBQPBCCDELMN $.
  $}

  ${
    $d x A $.  $d x ps $.
    $( Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90.  It
       is valid only when the domain of quantification is inhabited.
       (Contributed by Jim Kingdon, 5-Aug-2018.) $)
    r19.27mv $p |- ( E. x x e. A ->
                   ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) ) $=
      ( nfv r19.27m ) ABCDBCEF $.
  $}

  ${
    $d x A $.
    $( Vacuous quantification is always true.  (Contributed by NM,
       11-Mar-1997.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    rzal $p |- ( A = (/) -> A. x e. A ph ) $=
      ( c0 wceq cv wcel ne0i necon2bi pm2.21d ralrimiv ) CDEZABCLBFZCGZANCDCMHI
      JK $.

    $( Restricted existential quantification implies its restriction is
       nonempty (it is also inhabited as shown in ~ rexm ).  (Contributed by
       Szymon Jaroszewicz, 3-Apr-2007.) $)
    rexn0 $p |- ( E. x e. A ph -> A =/= (/) ) $=
      ( c0 wne cv wcel ne0i a1d rexlimiv ) ACDEZBCBFZCGKACLHIJ $.

    $( Restricted existential quantification implies its restriction is
       inhabited.  (Contributed by Jim Kingdon, 16-Oct-2018.) $)
    rexm $p |- ( E. x e. A ph -> E. x x e. A ) $=
      ( wrex cv wcel wa wex df-rex simpl eximi sylbi ) ABCDBECFZAGZBHMBHABCINMB
      MAJKL $.

    $( Idempotent law for restricted quantifier.  (Contributed by NM,
       28-Mar-1997.) $)
    ralidm $p |- ( A. x e. A A. x e. A ph <-> A. x e. A ph ) $=
      ( wral nfra1 cv wcel wa anidm rsp2 biimtrrid ralrimi wal wex 19.23 sylibr
      wi ax-1 df-ral impbii ) ABCDZBCDZUAUBABCUABCEBFCGZUCUCHUBAUCIABBCCJKLUAUC
      UAQBMZUBUAUCBNZUAQUDUAUERUCUABABCEOPUABCSPT $.
  $}

  $( Vacuous universal quantification is always true.  (Contributed by NM,
     20-Oct-2005.) $)
  ral0 $p |- A. x e. (/) ph $=
    ( c0 cv wcel noel pm2.21i rgen ) ABCBDZCEAIFGH $.

  ${
    $d x A $.
    ralf0.1 $e |- -. ph $.
    $( The quantification of a falsehood is vacuous when true.  (Contributed by
       NM, 26-Nov-2005.) $)
    ralf0 $p |- ( A. x e. A ph <-> A = (/) ) $=
      ( wral c0 wceq cv wcel wi wal wn con3 mpi alimi df-ral eq0 3imtr4i impbii
      rzal ) ABCEZCFGZBHCIZAJZBKUCLZBKUAUBUDUEBUDALUEDUCAMNOABCPBCQRABCTS $.
  $}

  $( Inhabited classes and restricted quantification.  (Contributed by Jim
     Kingdon, 6-Aug-2018.) $)
  ralm $p |- ( ( E. x x e. A -> A. x e. A ph ) <-> A. x e. A ph ) $=
    ( cv wcel wex wral wi wal df-ral imbi2i 19.38 sylbi pm2.43 alimi syl sylibr
    ax-1 impbii ) BDCEZBFZABCGZHZUBUCTAHZBIZUBUCTUDHZBIZUEUCUAUEHUGUBUEUAABCJZK
    TUDBLMUFUDBTANOPUHQUBUARS $.

  ${
    $d x y A $.
    raaan.1 $e |- F/ y ph $.
    raaan.2 $e |- F/ x ps $.
    $( Special case of ~ raaan where ` A ` is inhabited.  (Contributed by Jim
       Kingdon, 6-Aug-2018.) $)
    raaanlem $p |- ( E. x x e. A ->
      ( A. x e. A A. y e. A ( ph /\ ps ) <->
         ( A. x e. A ph /\ A. y e. A ps ) ) ) $=
      ( cv wcel wex wa wral wb eleq1 cbvexv r19.28m ralbidv sylbi nfcv nfralxy
      r19.27m bitrd ) CHZEIZCJZABKDELZCELZABDELZKZCELZACELUHKUEDHZEIZDJZUGUJMUD
      ULCDUCUKENOUMUFUICEABDEFPQRAUHCEBCDECESGTUAUB $.

    $( Rearrange restricted quantifiers.  (Contributed by NM, 26-Oct-2010.) $)
    raaan $p |- ( A. x e. A A. y e. A ( ph /\ ps ) <->
                  ( A. x e. A ph /\ A. y e. A ps ) ) $=
      ( cv wcel wex wa wral wi raaanlem pm5.74i ralm jcab eleq1 cbvexv bitri
      imbi1i anbi12i 3bitr3i ) CHZEIZCJZABKDELZCELZMUFACELZBDELZKZMZUHUKUFUHUKA
      BCDEFGNOUGCEPULUFUIMZUFUJMZKUKUFUIUJQUMUIUNUJACEPUNDHZEIZDJZUJMUJUFUQUJUE
      UPCDUDUOERSUABDEPTUBTUC $.
  $}

  ${
    $d y ph $.  $d x ps $.  $d x y A $.
    $( Rearrange restricted quantifiers.  (Contributed by NM, 11-Mar-1997.) $)
    raaanv $p |- ( A. x e. A A. y e. A ( ph /\ ps ) <->
                  ( A. x e. A ph /\ A. y e. A ps ) ) $=
      ( nfv raaan ) ABCDEADFBCFG $.
  $}

  ${
    $d z y $.  $d z x A $.
    $( Set substitution into the first argument of a subset relation.
       (Contributed by Rodolfo Medina, 7-Jul-2010.)  (Proof shortened by Mario
       Carneiro, 14-Nov-2016.) $)
    sbss $p |- ( [ y / x ] x C_ A <-> y C_ A ) $=
      ( vz cv wss wsb vex sbequ sseq1 nfv sbie vtoclb ) AEZCFZADGDEZCFZOABGBEZC
      FDRBHODBAIPRCJOQADQAKNPCJLM $.
  $}

  ${
    $d A y $.  $d B y $.  $d C y $.  $d V y $.  $d x y $.
    $( Distribute proper substitution through a subclass relation.
       (Contributed by Alan Sare, 22-Jul-2012.)  (Proof shortened by Alexander
       van der Vekens, 23-Jul-2017.) $)
    sbcssg $p |- ( A e. V -> ( [. A / x ]. B C_ C <->
      [_ A / x ]_ B C_ [_ A / x ]_ C ) ) $=
      ( vy wcel cv wal wsbc csb wss sbcalg sbcimg sbcel2g imbi12d albidv ssalel
      wi bitrd sbcbii 3bitr4g ) BEGZFHZCGZUDDGZSZFIZABJZUDABCKZGZUDABDKZGZSZFIZ
      CDLZABJUJULLUCUIUGABJZFIUOUGFABEMUCUQUNFUCUQUEABJZUFABJZSUNUEUFABENUCURUK
      USUMABUDCEOABUDDEOPTQTUPUHABFCDRUAFUJULRUB $.
  $}

  ${
    dcun.a $e |- ( ph -> DECID C e. A ) $.
    dcun.b $e |- ( ph -> DECID C e. B ) $.
    $( The union of two decidable classes is decidable.  (Contributed by Jim
       Kingdon, 5-Oct-2022.)  (Revised by Jim Kingdon, 13-Oct-2025.) $)
    dcun $p |- ( ph -> DECID C e. ( A u. B ) ) $=
      ( wcel cun wdc wn wa elun1 adantl orcd df-dc sylibr exmiddc syl mpjaodan
      wo elun2 simplr simpr ioran sylanbrc elun sylnibr olcd adantr ) ADBGZDBCH
      GZIZUJJZAUJKZUKUKJZTZULUNUKUOUJUKADBCLMNUKOZPAUMKZDCGZULUSJZURUSKZUPULVAU
      KUOUSUKURDCBUAMNUQPURUTKZUPULVBUOUKVBUJUSTZUKVBUMUTVCJAUMUTUBURUTUCUJUSUD
      UEDBCUFUGUHUQPAUSUTTZUMAUSIVDFUSQRUISAUJIUJUMTEUJQRS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Conditional operator
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constant symbols. $)
  $c if $.  $( Conditional operator (was "ded" for "deduction class"). $)

  $( Extend class notation to include the conditional operator.  See ~ df-if
     for a description.  (In older databases this was denoted "ded".) $)
  cif $a class if ( ph , A , B ) $.

  ${
    $d x ph $.  $d x A $.  $d x B $.
    $( Define the conditional operator.  Read ` if ( ph , A , B ) ` as "if
       ` ph ` then ` A ` else ` B ` ".  See ~ iftrue and ~ iffalse for its
       values.  In mathematical literature, this operator is rarely defined
       formally but is implicit in informal definitions such as "let f(x)=0 if
       x=0 and 1/x otherwise."

       In the absence of excluded middle, this will tend to be useful where
       ` ph ` is decidable (in the sense of ~ df-dc ).  (Contributed by NM,
       15-May-1999.) $)
    df-if $a |- if ( ph , A , B ) =
                 { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } $.
  $}

  ${
    $d x ph $.  $d x A $.  $d x B $.  $d x C $.
    $( An alternate definition of the conditional operator ~ df-if as a simple
       class abstraction.  (Contributed by Mario Carneiro, 8-Sep-2013.) $)
    dfif6 $p |- if ( ph , A , B ) =
                 ( { x e. A | ph } u. { x e. B | -. ph } ) $=
      ( cv wcel wa cab wn cun wo crab cif unab df-rab uneq12i df-if 3eqtr4ri )
      BEZCFAGZBHZSDFAIZGZBHZJTUCKBHABCLZUBBDLZJACDMTUCBNUEUAUFUDABCOUBBDOPABCDQ
      R $.
  $}

  ${
    $d A x $.  $d ph x $.
    $( Expression of a conditional class as a class abstraction when the False
       alternative is the empty class: in that case, the conditional class is
       the extension, in the True alternative, of the condition.  (Contributed
       by BJ, 16-Aug-2024.) $)
    if0ab $p |- if ( ph , A , (/) ) = { x e. A | ph } $=
      ( c0 cif crab wn cun dfif6 rab0 uneq2i un0 3eqtri ) ACDEABCFZAGZBDFZHNDHN
      ABCDIPDNOBJKNLM $.

    $( A conditional class with the False alternative being sent to the empty
       class is included in the class corresponding to the True alternative.
       (Contributed by BJ, 5-May-2026.) $)
    if0ss $p |- if ( ph , A , (/) ) C_ A $=
      ( vx c0 cif if0ab ssrab3 ) ACBABDEACBFG $.
  $}

  ${
    $d x ph $.  $d x A $.  $d x B $.  $d x C $.
    $( Equality theorem for conditional operator.  (Contributed by NM,
       1-Sep-2004.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    ifeq1 $p |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) ) $=
      ( vx wceq crab wn cun cif rabeq uneq1d dfif6 3eqtr4g ) BCFZAEBGZAHEDGZIAE
      CGZQIABDJACDJOPRQAEBCKLAEBDMAECDMN $.

    $( Equality theorem for conditional operator.  (Contributed by NM,
       1-Sep-2004.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    ifeq2 $p |- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) ) $=
      ( vx wceq crab wn cun cif rabeq uneq2d dfif6 3eqtr4g ) BCFZAEDGZAHZEBGZIP
      QECGZIADBJADCJORSPQEBCKLAEDBMAEDCMN $.

    $( Value of the conditional operator when its first argument is true.
       (Contributed by NM, 15-May-1999.)  (Proof shortened by Andrew Salmon,
       26-Jun-2011.) $)
    iftrue $p |- ( ph -> if ( ph , A , B ) = A ) $=
      ( vx cif cv wcel wa wn wo cab df-if dedlema abbi2dv eqtr4id ) AABCEDFZBGZ
      AHPCGZAIHJZDKBADBCLASDBAQRMNO $.
  $}

  ${
    iftruei.1 $e |- ph $.
    $( Inference associated with ~ iftrue .  (Contributed by BJ,
       7-Oct-2018.) $)
    iftruei $p |- if ( ph , A , B ) = A $=
      ( cif wceq iftrue ax-mp ) AABCEBFDABCGH $.
  $}

  ${
    iftrued.1 $e |- ( ph -> ch ) $.
    $( Value of the conditional operator when its first argument is true.
       (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
    iftrued $p |- ( ph -> if ( ch , A , B ) = A ) $=
      ( cif wceq iftrue syl ) ABBCDFCGEBCDHI $.
  $}

  ${
    $d x ph $.  $d x A $.  $d x B $.
    $( Value of the conditional operator when its first argument is false.
       (Contributed by NM, 14-Aug-1999.) $)
    iffalse $p |- ( -. ph -> if ( ph , A , B ) = B ) $=
      ( vx wn cif cv wcel wa wo cab df-if dedlemb abbi2dv eqtr4id ) AEZABCFDGZB
      HZAIQCHZPIJZDKCADBCLPTDCARSMNO $.
  $}

  ${
    iffalsei.1 $e |- -. ph $.
    $( Inference associated with ~ iffalse .  (Contributed by BJ,
       7-Oct-2018.) $)
    iffalsei $p |- if ( ph , A , B ) = B $=
      ( wn cif wceq iffalse ax-mp ) AEABCFCGDABCHI $.
  $}

  ${
    iffalsed.1 $e |- ( ph -> -. ch ) $.
    $( Value of the conditional operator when its first argument is false.
       (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
    iffalsed $p |- ( ph -> if ( ch , A , B ) = B ) $=
      ( wn cif wceq iffalse syl ) ABFBCDGDHEBCDIJ $.
  $}

  $( When values are unequal, but an "if" condition checks if they are equal,
     then the "false" branch results.  This is a simple utility to provide a
     slight shortening and simplification of proofs versus applying ~ iffalse
     directly in this case.  (Contributed by David A. Wheeler, 15-May-2015.) $)
  ifnefalse $p |- ( A =/= B -> if ( A = B , C , D ) = D ) $=
    ( wne wceq wn cif df-ne iffalse sylbi ) ABEABFZGLCDHDFABILCDJK $.

  ${
    $d A x $.  $d B x $.  $d C x $.  $d ph x $.
    $( Membership in a conditional operator.  (Contributed by NM,
       14-Feb-2005.) $)
    elif $p |- ( A e. if ( ph , B , C ) <->
        ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) ) $=
      ( vx cif wcel cvv wa wn wo elex adantl jaoi cv eleq1 anbi1d orbi12d ancom
      wceq df-if elab2g orbi12i bitrdi pm5.21nii ) BACDFZGZBHGZABCGZIZAJZBDGZIZ
      KZBUFLUJUHUMUIUHABCLMULUHUKBDLMNUHUGUIAIZULUKIZKZUNEOZCGZAIZURDGZUKIZKUQE
      BUFHURBTZUTUOVBUPVCUSUIAURBCPQVCVAULUKURBDPQRAECDUAUBUOUJUPUMUIASULUKSUCU
      DUE $.
  $}

  ${
    ifsbdc.1 $e |- ( if ( ph , A , B ) = A -> C = D ) $.
    ifsbdc.2 $e |- ( if ( ph , A , B ) = B -> C = E ) $.
    $( Distribute a function over an if-clause.  (Contributed by Jim Kingdon,
       1-Jan-2022.) $)
    ifsbdc $p |- ( DECID ph -> C = if ( ph , D , E ) ) $=
      ( wdc wn wo cif wceq exmiddc iftrue syl eqtr4d iffalse jaoi ) AIAAJZKDAEF
      LZMZANAUBTADEUAAABCLZBMDEMABCOGPAEFOQTDFUATUCCMDFMABCRHPAEFRQSP $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y ph $.
    dfif3.1 $e |- C = { x | ph } $.
    $( Alternate definition of the conditional operator ~ df-if .  Note that
       ` ph ` is independent of ` x ` i.e. a constant true or false.
       (Contributed by NM, 25-Aug-2013.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    dfif3 $p |- if ( ph , A , B )
                  = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) ) $=
      ( vy cif crab wn cun cin cvv cdif dfif6 cab weq ineq2i dfrab3 eqtr4i
      biidd cbvabv eqtri notab difeq2i eqtr2i uneq12i ) ACDHAGCIZAJZGDIZKCELZDM
      ENZLZKAGCDOUKUHUMUJUKCAGPZLUHEUNCEABPUNFAABGBGQAUAUBUCZRAGCSTUJDUIGPZLUMU
      IGDSUPULDUPMUNNULAGUDEUNMUOUETRUFUGT $.
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.
    $( A conditional class is included in the union of its two alternatives.
       (Contributed by BJ, 15-Aug-2024.) $)
    ifssun $p |- if ( ph , A , B ) C_ ( A u. B ) $=
      ( vx cif crab wn cun dfif6 wss ssrab2 unss12 mp2an eqsstri ) ABCEADBFZAGZ
      DCFZHZBCHZADBCIOBJQCJRSJADBKPDCKOBQCLMN $.
  $}

  ${
    $( A conditional class whose two alternatives are equal is included in that
       alternative.  With excluded middle, we can prove it is equal to it.
       (Contributed by BJ, 15-Aug-2024.) $)
    ifidss $p |- if ( ph , A , A ) C_ A $=
      ( cif cun ifssun unidm sseqtri ) ABBCBBDBABBEBFG $.
  $}

  $( Equality theorem for conditional operators.  (Contributed by NM,
     1-Sep-2004.) $)
  ifeq12 $p |- ( ( A = B /\ C = D ) ->
                if ( ph , A , C ) = if ( ph , B , D ) ) $=
    ( wceq cif ifeq1 ifeq2 sylan9eq ) BCFDEFABDGACDGACEGABCDHADECIJ $.

  ${
    ifeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for conditional operator.  (Contributed by NM,
       16-Feb-2005.) $)
    ifeq1d $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) ) $=
      ( wceq cif ifeq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.

    $( Equality deduction for conditional operator.  (Contributed by NM,
       16-Feb-2005.) $)
    ifeq2d $p |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) ) $=
      ( wceq cif ifeq2 syl ) ACDGBECHBEDHGFBCDEIJ $.

    ifeq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality deduction for conditional operator.  (Contributed by NM,
       24-Mar-2015.) $)
    ifeq12d $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) ) $=
      ( cif ifeq1d ifeq2d eqtrd ) ABCEIBDEIBDFIABCDEGJABEFDHKL $.
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.  $d ps x $.
    $( Equivalence theorem for conditional operators.  (Contributed by Raph
       Levien, 15-Jan-2004.) $)
    ifbi $p |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) ) $=
      ( vx wb cv wcel wa wn wo cab cif anbi2 notbi anbi2d orbi12d df-if 3eqtr4g
      abbidv ) ABFZEGZCHZAIZUBDHZAJZIZKZELUCBIZUEBJZIZKZELACDMBCDMUAUHULEUAUDUI
      UGUKABUCNUAUFUJUEABOPQTAECDRBECDRS $.
  $}

  ${
    ifbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalence deduction for conditional operators.  (Contributed by NM,
       18-Apr-2005.) $)
    ifbid $p |- ( ph -> if ( ps , A , B ) = if ( ch , A , B ) ) $=
      ( wb cif wceq ifbi syl ) ABCGBDEHCDEHIFBCDEJK $.
  $}

  ${
    ifbieq1d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    ifbieq1d.2 $e |- ( ph -> A = B ) $.
    $( Equivalence/equality deduction for conditional operators.  (Contributed
       by JJ, 25-Sep-2018.) $)
    ifbieq1d $p |- ( ph -> if ( ps , A , C ) = if ( ch , B , C ) ) $=
      ( cif ifbid ifeq1d eqtrd ) ABDFICDFICEFIABCDFGJACDEFHKL $.
  $}

  ${
    ifbieq2i.1 $e |- ( ph <-> ps ) $.
    ifbieq2i.2 $e |- A = B $.
    $( Equivalence/equality inference for conditional operators.  (Contributed
       by Paul Chapman, 22-Jun-2011.) $)
    ifbieq2i $p |- if ( ph , C , A ) = if ( ps , C , B ) $=
      ( cif wb wceq ifbi ax-mp ifeq2 eqtri ) AECHZBECHZBEDHZABIOPJFABECKLCDJPQJ
      GBCDEMLN $.
  $}

  ${
    ifbieq2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    ifbieq2d.2 $e |- ( ph -> A = B ) $.
    $( Equivalence/equality deduction for conditional operators.  (Contributed
       by Paul Chapman, 22-Jun-2011.) $)
    ifbieq2d $p |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) ) $=
      ( cif ifbid ifeq2d eqtrd ) ABFDICFDICFEIABCFDGJACDEFHKL $.
  $}

  ${
    ifbieq12i.1 $e |- ( ph <-> ps ) $.
    ifbieq12i.2 $e |- A = C $.
    ifbieq12i.3 $e |- B = D $.
    $( Equivalence deduction for conditional operators.  (Contributed by NM,
       18-Mar-2013.) $)
    ifbieq12i $p |- if ( ph , A , B ) = if ( ps , C , D ) $=
      ( cif wceq ifeq1 ax-mp ifbieq2i eqtri ) ACDJZAEDJZBEFJCEKPQKHACEDLMABDFEG
      INO $.
  $}

  ${
    ifbieq12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    ifbieq12d.2 $e |- ( ph -> A = C ) $.
    ifbieq12d.3 $e |- ( ph -> B = D ) $.
    $( Equivalence deduction for conditional operators.  (Contributed by Jeff
       Madsen, 2-Sep-2009.) $)
    ifbieq12d $p |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) ) $=
      ( cif ifbid ifeq12d eqtrd ) ABDEKCDEKCFGKABCDEHLACDFEGIJMN $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y ph $.  $d y ps $.
    nfifd.2 $e |- ( ph -> F/ x ps ) $.
    nfifd.3 $e |- ( ph -> F/_ x A ) $.
    nfifd.4 $e |- ( ph -> F/_ x B ) $.
    $( Deduction version of ~ nfif .  (Contributed by NM, 15-Feb-2013.)
       (Revised by Mario Carneiro, 13-Oct-2016.) $)
    nfifd $p |- ( ph -> F/_ x if ( ps , A , B ) ) $=
      ( vy cif cv wcel wa wn wo cab df-if nfv nfcrd nfand nford nfabd nfcxfrd
      nfnd ) ACBDEJIKZDLZBMZUEELZBNZMZOZIPBIDEQAUKCIAIRAUGUJCAUFBCACIDGSFTAUHUI
      CACIEHSABCFUDTUAUBUC $.
  $}

  ${
    nfif.1 $e |- F/ x ph $.
    nfif.2 $e |- F/_ x A $.
    nfif.3 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for a conditional operator.
       (Contributed by NM, 16-Feb-2005.)  (Proof shortened by Andrew Salmon,
       26-Jun-2011.) $)
    nfif $p |- F/_ x if ( ph , A , B ) $=
      ( cif wnfc wtru wnf a1i nfifd mptru ) BACDHIJABCDABKJELBCIJFLBDIJGLMN $.
  $}

  ${
    ifcldadc.1 $e |- ( ( ph /\ ps ) -> A e. C ) $.
    ifcldadc.2 $e |- ( ( ph /\ -. ps ) -> B e. C ) $.
    ifcldadc.dc $e |- ( ph -> DECID ps ) $.
    $( Conditional closure.  (Contributed by Jim Kingdon, 11-Jan-2022.) $)
    ifcldadc $p |- ( ph -> if ( ps , A , B ) e. C ) $=
      ( cif wcel wn wa wceq iftrue adantl eqeltrd iffalse wdc wo exmiddc syl
      mpjaodan ) ABBCDIZEJBKZABLUCCEBUCCMABCDNOFPAUDLUCDEUDUCDMABCDQOGPABRBUDSH
      BTUAUB $.
  $}

  ${
    ifeq1dadc.1 $e |- ( ( ph /\ ps ) -> A = B ) $.
    ifeq1dadc.dc $e |- ( ph -> DECID ps ) $.
    $( Conditional equality.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    ifeq1dadc $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) ) $=
      ( cif wceq wn wa ifeq1d iffalse eqtr4d adantl wdc wo exmiddc syl mpjaodan
      ) ABBCEHZBDEHZIZBJZABKBCDEFLUDUCAUDUAEUBBCEMBDEMNOABPBUDQGBRST $.
  $}

  ${
    ifeq2da.1 $e |- ( ( ph /\ -. ps ) -> A = B ) $.
    ifeq2dadc.dc $e |- ( ph -> DECID ps ) $.
    $( Conditional equality.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    ifeq2dadc $p |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) ) $=
      ( cif wceq wn wa simpr iftrued eqtr4d ifeq2d wdc wo exmiddc syl mpjaodan
      ) ABBECHZBEDHZIBJZABKZUAEUBUDBECABLZMUDBEDUEMNAUCKBCDEFOABPBUCQGBRST $.
  $}

  ${
    ifeqda.1 $e |- ( ( ph /\ ps ) -> A = C ) $.
    ifeqda.2 $e |- ( ( ph /\ -. ps ) -> B = C ) $.
    ifeqdadc.dc $e |- ( ph -> DECID ps ) $.
    $( Separation of the values of the conditional operator.  (Contributed by
       Alexander van der Vekens, 13-Apr-2018.) $)
    ifeqdadc $p |- ( ph -> if ( ps , A , B ) = C ) $=
      ( cif wceq wn wa iftrue adantl eqtrd iffalse wdc wo exmiddc syl mpjaodan
      ) ABBCDIZEJBKZABLUBCEBUBCJABCDMNFOAUCLUBDEUCUBDJABCDPNGOABQBUCRHBSTUA $.
  $}

  ${
    ifbothdc.1 $e |- ( A = if ( ph , A , B ) -> ( ps <-> th ) ) $.
    ifbothdc.2 $e |- ( B = if ( ph , A , B ) -> ( ch <-> th ) ) $.

    ${
      ifbothdadc.3 $e |- ( ( et /\ ph ) -> ps ) $.
      ifbothdadc.4 $e |- ( ( et /\ -. ph ) -> ch ) $.
      ifbothdadc.dc $e |- ( et -> DECID ph ) $.
      $( A formula ` th ` containing a decidable conditional operator is true
         when both of its cases are true.  (Contributed by Jim Kingdon,
         3-Jun-2022.) $)
      ifbothdadc $p |- ( et -> th ) $=
        ( wn wa wb wceq eqcomd syl adantl mpbid cif iftrue iffalse wdc mpjaodan
        wo exmiddc ) EADAMZEANBDJABDOZEAFAFGUAZPUIAUJFAFGUBQHRSTEUHNCDKUHCDOZEU
        HGUJPUKUHUJGAFGUCQIRSTEAUDAUHUFLAUGRUE $.
    $}

    $( A wff ` th ` containing a conditional operator is true when both of its
       cases are true.  (Contributed by Jim Kingdon, 8-Aug-2021.) $)
    ifbothdc $p |- ( ( ps /\ ch /\ DECID ph ) -> th ) $=
      ( wdc w3a wn wi cif wceq wb iftrue eqcomd syl biimpcd 3ad2ant1 iffalse wo
      3ad2ant2 exmiddc 3ad2ant3 mpjaod ) BCAIZJADAKZBCADLUGABDAEAEFMZNBDOAUIEAE
      FPQGRSTCBUHDLUGUHCDUHFUINCDOUHUIFAEFUAQHRSUCUGBAUHUBCAUDUEUF $.
  $}

  $( Identical true and false arguments in the conditional operator.
     (Contributed by NM, 18-Apr-2005.) $)
  ifiddc $p |- ( DECID ph -> if ( ph , A , A ) = A ) $=
    ( wdc wn wo cif wceq exmiddc iftrue iffalse jaoi syl ) ACAADZEABBFBGZAHANMA
    BBIABBJKL $.

  $( Expansion of an equality with a conditional operator.  (Contributed by Jim
     Kingdon, 28-Jul-2022.) $)
  eqifdc $p |- ( DECID ph -> ( A = if ( ph , B , C ) <->
             ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) ) $=
    ( wdc cif wceq wa wn exmiddc simpr simpl iftrued eqtrd jca iffalsed orim12d
    wo ex eqtr4d syl5com jaoi impbid1 ) AEZBACDFZGZABCGZHZAIZBDGZHZRZUDAUIRUFUL
    AJUFAUHUIUKUFAUHUFAHZAUGUFAKZUMBUECUFALUMACDUNMNOSUFUIUKUFUIHZUIUJUFUIKZUOB
    UEDUFUILUOACDUPPNOSQUAUHUFUKUHBCUEAUGKUHACDAUGLMTUKBDUEUIUJKUKACDUIUJLPTUBU
    C $.

  ${
    ifcldcd.a $e |- ( ph -> A e. C ) $.
    ifcldcd.b $e |- ( ph -> B e. C ) $.
    ifcldcd.dc $e |- ( ph -> DECID ps ) $.
    $( Membership (closure) of a conditional operator, deduction form.
       (Contributed by Jim Kingdon, 8-Aug-2021.) $)
    ifcldcd $p |- ( ph -> if ( ps , A , B ) e. C ) $=
      ( cif wcel wn wa wceq iftrue adantl adantr eqeltrd iffalse wdc wo df-dc
      sylib mpjaodan ) ABBCDIZEJBKZABLUDCEBUDCMABCDNOACEJBFPQAUELUDDEUEUDDMABCD
      ROADEJUEGPQABSBUETHBUAUBUC $.
  $}

  $( Negating the first argument swaps the last two arguments of a conditional
     operator.  (Contributed by NM, 21-Jun-2007.) $)
  ifnotdc $p |- ( DECID ph -> if ( -. ph , A , B ) = if ( ph , B , A ) ) $=
    ( wdc wn wo cif wceq df-dc notnot iffalsed iftrue eqtr4d iffalse jaoi sylbi
    ) ADAAEZFQBCGZACBGZHZAIATQARCSAQBCAJKACBLMQRBSQBCLACBNMOP $.

  ${
    2if2.1 $e |- ( ( ph /\ ps ) -> D = A ) $.
    2if2.2 $e |- ( ( ph /\ -. ps /\ th ) -> D = B ) $.
    2if2.3 $e |- ( ( ph /\ -. ps /\ -. th ) -> D = C ) $.
    2if2dc.ps $e |- ( ph -> DECID ps ) $.
    2if2dc.th $e |- ( ( ph /\ -. ps ) -> DECID th ) $.
    $( Resolve two nested conditionals.  (Contributed by Alexander van der
       Vekens, 27-Mar-2018.) $)
    2if2dc $p |- ( ph -> D = if ( ps , A , if ( th , B , C ) ) ) $=
      ( cif wceq wn wa iftrue adantl eqtr4d 3expa iffalse eqcomd wdc wo exmiddc
      eqtrd syl mpjaodan ) ABGBDCEFMZMZNBOZABPGDUJHBUJDNABDUIQRSAUKPZGUIUJULCGU
      INCOZULCPGEUIAUKCGENITCUIENULCEFQRSULUMPGFUIAUKUMGFNJTUMFUINULUMUIFCEFUAU
      BRUFULCUCCUMUDLCUEUGUHUKUJUINABDUIUARSABUCBUKUDKBUEUGUH $.
  $}

  $( Rewrite a conjunction in a conditional as two nested conditionals.
     (Contributed by Mario Carneiro, 28-Jul-2014.) $)
  ifandc $p |- ( DECID ph
      -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) ) $=
    ( wdc wn wo wa cif wceq df-dc iftrue ibar ifbid eqtr2d simpl con3i iffalsed
    iffalse eqtr4d jaoi sylbi ) AEAAFZGABHZCDIZABCDIZDIZJZAKAUHUCAUGUFUEAUFDLAB
    UDCDABMNOUCUEDUGUCUDCDUDAABPQRAUFDSTUAUB $.

  $( Rewrite a disjunction in a conditional as two nested conditionals.
     (Contributed by Mario Carneiro, 28-Jul-2014.) $)
  ifordc $p |- ( DECID ph
      -> if ( ( ph \/ ps ) , A , B ) = if ( ph , A , if ( ps , A , B ) ) ) $=
    ( wdc wn wo wceq exmiddc iftrue orcs eqtr4d iffalse biorf ifbid eqtr2d jaoi
    cif syl ) AEAAFZGABGZCDRZACBCDRZRZHZAIAUETAUBCUDABUBCHUACDJKACUCJLTUDUCUBAC
    UCMTBUACDABNOPQS $.

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ph $.
    $( If a conditional class is inhabited, then the condition is decidable.
       This shows that conditionals are not very useful unless one can prove
       the condition decidable.  (Contributed by BJ, 24-Sep-2022.) $)
    ifmdc $p |- ( A e. if ( ph , B , C ) -> DECID ph ) $=
      ( vx cif wcel wn wo cv wi wceq eleq1 imbi1d wa df-if abeq2i simpr orim12i
      wdc sylbi vtoclg pm2.43i df-dc sylibr ) BACDFZGZAAHZIZATUGUIEJZUFGZUIKUGU
      IKEBUFUJBLUKUGUIUJBUFMNUKUJCGZAOZUJDGZUHOZIZUIUPEUFAECDPQUMAUOUHULARUNUHR
      SUAUBUCAUDUE $.
  $}

  $( Deduce truth from a conditional operator value.  (Contributed by Thierry
     Arnoux, 20-Feb-2025.) $)
  ifnetruedc $p |- ( ( DECID ph /\ A =/= B /\ if ( ph , A , B ) = A )
      -> ph ) $=
    ( wdc wne cif wceq w3a simp1 wn iffalse adantl simpl3 simpl2 eqnetrd neneqd
    wa condandc mpcom ) ADZTBCEZABCFZBGZHZATUAUCIUDAUBCGZAJZUEUDABCKLUDUFQZUBCU
    GUBBCTUAUCUFMTUAUCUFNOPRS $.

  $( Deduce falsehood from a conditional operator value.  (Contributed by
     Thierry Arnoux, 20-Feb-2025.) $)
  ifnefals $p |- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph ) $=
    ( wne cif wceq iftrue adantl simplr simpll necomd eqnetrd neneqd pm2.65da
    wa ) BCDZABCEZCFZOZAQBFZATSABCGHSAOZQBUAQCBPRAIUABCPRAJKLMN $.

  $( The converse of ~ ifbi holds if the two values are not equal.
     (Contributed by Thierry Arnoux, 20-Feb-2025.) $)
  ifnebibdc $p |- ( ( DECID ph /\ DECID ps /\ A =/= B ) ->
               ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ph <-> ps ) ) ) $=
    ( wdc wne w3a cif wceq wb wa wn wo eqifdc 3ad2ant2 adantld simpl jca2 syl6
    wi ifnetruedc 3expia 3adant2 pm5.1 ifnefals ex 3ad2ant3 pm5.21 jaod impbid1
    sylbid ifbi ) AEZBEZCDFZGZACDHZBCDHIZABJZUPURBUQCIZKZBLZUQDIZKZMZUSUNUMURVE
    JUOBUQCDNOUPVAUSVDUPVAABKUSUPVAABUPUTABUMUOUTATUNUMUOUTAACDUAUBUCPBUTQRABUD
    SUPVDALZVBKUSUPVDVFVBUPVCVFVBUOUMVCVFTUNUOVCVFACDUEUFUGPVBVCQRABUHSUIUKABCD
    ULUJ $.

  ${
    $d a x $.  $d x C $.  $d x X $.  $d x Y $.  $d x V $.  $d x W $.
    $d x ps $.  $d x th $.
    ifeqeqx.1 $e |- ( x = X -> A = C ) $.
    ifeqeqx.2 $e |- ( x = Y -> B = a ) $.
    ifeqeqx.3 $e |- ( x = X -> ( ch <-> th ) ) $.
    ifeqeqx.4 $e |- ( x = Y -> ( ch <-> ps ) ) $.
    ifeqeqx.5 $e |- ( ph -> a = C ) $.
    ifeqeqx.6 $e |- ( ( ph /\ ps ) -> th ) $.
    ifeqeqx.y $e |- ( ph -> Y e. V ) $.
    ifeqeqx.x $e |- ( ph -> X e. W ) $.
    ifeqeqxdc.dc $e |- ( ph -> DECID ps ) $.
    $( An equality theorem tailored for ~ ballotfilemsf1o .  (Contributed by
       Thierry Arnoux, 14-Apr-2017.) $)
    ifeqeqxdc $p |- ( ( ph /\ x = if ( ps , X , Y ) )
                                                  -> a = if ( ch , A , B ) ) $=
      ( cv wceq cif wa eqeq2 csb simplr wsbc simpll simpr sbceq1a biimpd dfsbcq
      sylc wi csbeq1 eqeq2d imbi12d wcel nfcvd csbiegf syl eqtr4d adantr eqcomd
      a1d wn pm3.24 wb sbcieg anbi1d mtbiri pm2.21d anass1rs ifbothdadc csbeq1a
      imp ex biimprd notbid nsyld anim2d mtoi expdimp wdc adantlr iftrued eqtrd
      wo mpbird orcd df-dc sylibr ad2antrr iffalsed dcbid exmiddc mpjaodan ) CM
      UCZFUDZXAGUDZXACFGUEZUDAEUCZBKLUEZUDZUFZFGFXDXAUGGXDXAUGXHCUFZXGXAEXFFUHZ
      UDZXBAXGCUIZXIACEXFUJZXKAXGCUKXIXGCXMXLXHCULXGCXMCEXFUMZUNUPBCEKUJZXAEKFU
      HZUDZUQCELUJZXAELFUHZUDZUQXMXKUQAKLKXFUDZXOXMXQXKCEKXFUOZYAXPXJXAEKXFFURU
      SUTLXFUDZXRXMXTXKCELXFUOZYCXSXJXAELXFFURUSUTABUFZXQXOYEXPXAAXPXAUDBAXPHXA
      AKJVAZXPHUDUAEKFHJYFEHVBNVCVDRVEVFVGVHABVIZUFZXRXTAXRYGXTAXRYGUFZXTAYIXTA
      YIBYGUFZBVJZAXRBYGALIVAZXRBVKTCBELIQVLVDVMVNVOVSVPVTUBVQUPXGXBXKXGFXJXAEX
      FFVRUSWAUPXHCVIZUFZXGXAEXFGUHZUDZXCAXGYMUIZYNAXMVIZYPAXGYMUKYNXGYMYRYQXHY
      MULXGYMYRXGCXMXNWBUNUPBXOVIZXAEKGUHZUDZUQXRVIZXAELGUHZUDZUQYRYPUQAKLYAYSY
      RUUAYPYAXOXMYBWBYAYTYOXAEKXFGURUSUTYCUUBYRUUDYPYCXRXMYDWBYCUUCYOXAELXFGUR
      USUTABYSUUAABYSUFZUUAAUUEYJYKAYSYGBAYSDBAYSDVIAXODAYFXODVKUACDEKJPVLVDWBU
      NABDSVTWCWDWEVOWFYHUUDUUBYHUUCXAAUUCXAUDZYGAYLUUFTELGXAIYLEXAVBOVCVDVFVGV
      HUBVQUPXGXCYPXGGYOXAEXFGVRUSWAUPXHBCWGZYGXHBUFZCYMWKUUGUUHCYMUUHCDABDXGSW
      HUUHXEKUDCDVKUUHXEXFKAXGBUIUUHBKLXHBULWIWJPVDWLWMCWNWOXHYGUFZUUGBWGZAUUJX
      GYGUBWPUUICBUUIXELUDCBVKUUIXEXFLAXGYGUIUUIBKLXHYGULWQWJQVDWRWLABYGWKZXGAU
      UJUUKUBBWSVDVFWTVQ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Power classes
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare the symbol for power class. $)
  $c ~P $.  $( Calligraphic P $)

  $( Extend class notation to include power class.  (The tilde in the Metamath
     token is meant to suggest the calligraphic font of the P.) $)
  cpw $a class ~P A $.

  ${
    $d x A $.  $d y A $.  $d z A $.  $d z x $.  $d z y $.
    $( Soundness justification theorem for ~ df-pw .  (Contributed by Rodolfo
       Medina, 28-Apr-2010.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    pwjust $p |- { x | x C_ A } = { y | y C_ A } $=
      ( vz cv wss cab sseq1 cbvabv eqtri ) AEZCFZAGDEZCFZDGBEZCFZBGLNADKMCHINPD
      BMOCHIJ $.
  $}

  ${
    $d x A $.
    $( Define power class.  Definition 5.10 of [TakeutiZaring] p. 17, but we
       also let it apply to proper classes, i.e. those that are not members of
       ` _V ` .  When applied to a set, this produces its power set.  A power
       set of S is the set of all subsets of S, including the empty set and S
       itself.  For example, if ` A ` is { 3 , 5 , 7 }, then ` ~P A ` is {
       (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 ,
       5 , 7 } }.  We will later introduce the Axiom of Power Sets.  Still
       later we will prove that the size of the power set of a finite set is 2
       raised to the power of the size of the set.  (Contributed by NM,
       5-Aug-1993.) $)
    df-pw $a |- ~P A = { x | x C_ A } $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Equality theorem for power class.  (Contributed by NM, 5-Aug-1993.) $)
    pweq $p |- ( A = B -> ~P A = ~P B ) $=
      ( vx wceq cv wss cab cpw sseq2 abbidv df-pw 3eqtr4g ) ABDZCEZAFZCGNBFZCGA
      HBHMOPCABNIJCAKCBKL $.
  $}

  ${
    pweqi.1 $e |- A = B $.
    $( Equality inference for power class.  (Contributed by NM,
       27-Nov-2013.) $)
    pweqi $p |- ~P A = ~P B $=
      ( wceq cpw pweq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    pweqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for power class.  (Contributed by NM,
       27-Nov-2013.) $)
    pweqd $p |- ( ph -> ~P A = ~P B ) $=
      ( wceq cpw pweq syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    $d A x $.  $d B x $.
    ${
      elpw.1 $e |- A e. _V $.
      $( Membership in a power class.  Theorem 86 of [Suppes] p. 47.
         (Contributed by NM, 31-Dec-1993.) $)
      elpw $p |- ( A e. ~P B <-> A C_ B ) $=
        ( vx cv wss cpw sseq1 df-pw elab2 ) DEZBFABFDABGCKABHDBIJ $.
    $}

    $( Setvar variable membership in a power class (common case).  See ~ elpw .
       (Contributed by David A. Wheeler, 8-Dec-2018.) $)
    velpw $p |- ( x e. ~P A <-> x C_ A ) $=
      ( cv vex elpw ) ACBADE $.

    $( Membership in a power class.  Theorem 86 of [Suppes] p. 47.
       (Contributed by NM, 6-Aug-2000.) $)
    elpwg $p |- ( A e. V -> ( A e. ~P B <-> A C_ B ) ) $=
      ( vx cv cpw wcel wss eleq1 sseq1 vex elpw vtoclbg ) DEZBFZGNBHAOGABHDACNA
      OINABJNBDKLM $.
  $}

  ${
    elpwd.1 $e |- ( ph -> A e. V ) $.
    elpwd.2 $e |- ( ph -> A C_ B ) $.
    $( Membership in a power class.  (Contributed by Glauco Siliprandi,
       11-Oct-2020.) $)
    elpwd $p |- ( ph -> A e. ~P B ) $=
      ( cpw wcel wss wb elpwg syl mpbird ) ABCGHZBCIZFABDHNOJEBCDKLM $.
  $}

  $( Subset relation implied by membership in a power class.  (Contributed by
     NM, 17-Feb-2007.) $)
  elpwi $p |- ( A e. ~P B -> A C_ B ) $=
    ( cpw wcel wss elpwg ibi ) ABCZDABEABHFG $.

  $( Characterization of the elements of a power class.  (Contributed by BJ,
     29-Apr-2021.) $)
  elpwb $p |- ( A e. ~P B <-> ( A e. _V /\ A C_ B ) ) $=
    ( cpw wcel cvv wss elex elpwg biadan2 ) ABCZDAEDABFAJGABEHI $.

  ${
    elpwid.1 $e |- ( ph -> A e. ~P B ) $.
    $( An element of a power class is a subclass.  Deduction form of ~ elpwi .
       (Contributed by David Moews, 1-May-2017.) $)
    elpwid $p |- ( ph -> A C_ B ) $=
      ( cpw wcel wss elpwi syl ) ABCEFBCGDBCHI $.
  $}

  $( If ` A ` belongs to a part of ` C ` then ` A ` belongs to ` C ` .
     (Contributed by FL, 3-Aug-2009.) $)
  elelpwi $p |- ( ( A e. B /\ B e. ~P C ) -> A e. C ) $=
    ( cpw wcel elpwi sseld impcom ) BCDEZABEACEIBCABCFGH $.

  ${
    $d A x $.  $d B x $.
    $( The powerclass preserves inclusion.  See ~ sspwb for the biconditional
       version.  (Contributed by NM, 13-Oct-1996.)  Extract forward implication
       of ~ sspwb since it requires fewer axioms.  (Revised by BJ,
       13-Apr-2024.) $)
    sspw $p |- ( A C_ B -> ~P A C_ ~P B ) $=
      ( vx wss cpw cv wcel sstr2 com12 velpw 3imtr4g ssrdv ) ABDZCAEZBEZMCFZADZ
      PBDZPNGPOGQMRPABHICAJCBJKL $.
  $}

  ${
    sspwi.1 $e |- A C_ B $.
    $( The powerclass preserves inclusion (inference form).  (Contributed by
       BJ, 13-Apr-2024.) $)
    sspwi $p |- ~P A C_ ~P B $=
      ( wss cpw sspw ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    sspwd.1 $e |- ( ph -> A C_ B ) $.
    $( The powerclass preserves inclusion (deduction form).  (Contributed by
       BJ, 13-Apr-2024.) $)
    sspwd $p |- ( ph -> ~P A C_ ~P B ) $=
      ( wss cpw sspw syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    $d y A $.  $d x y $.
    nfpw.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for power class.  (Contributed by NM,
       28-Oct-2003.)  (Revised by Mario Carneiro, 13-Oct-2016.) $)
    nfpw $p |- F/_ x ~P A $=
      ( vy cpw cv wss cab df-pw nfcv nfss nfab nfcxfr ) ABEDFZBGZDHDBIOADANBANJ
      CKLM $.
  $}

  $( Membership of the original in a power set.  (Contributed by Stefan O'Rear,
     1-Feb-2015.) $)
  pwidg $p |- ( A e. V -> A e. ~P A ) $=
    ( wcel cpw wss ssid elpwg mpbiri ) ABCAADCAAEAFAABGH $.

  ${
    pwid.1 $e |- A e. _V $.
    $( A set is a member of its power class.  Theorem 87 of [Suppes] p. 47.
       (Contributed by NM, 5-Aug-1993.) $)
    pwid $p |- A e. ~P A $=
      ( cvv wcel cpw pwidg ax-mp ) ACDAAEDBACFG $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Subclass relationship for power class.  (Contributed by NM,
       21-Jun-2009.) $)
    pwss $p |- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) ) $=
      ( cpw wss cv wcel wi wal ssalel df-pw abeq2i imbi1i albii bitri ) BDZCEAF
      ZPGZQCGZHZAIQBEZSHZAIAPCJTUBARUASUAAPABKLMNO $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Unordered and ordered pairs
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new symbols needed. $)
  $c <. $.  $( Bracket (the period distinguishes it from 'less than') $)
  $c >. $.  $( Bracket (the period distinguishes it from 'greater than') $)

  $( Extend class notation to include singleton. $)
  csn $a class { A } $.
  $( Extend class notation to include unordered pair. $)
  cpr $a class { A , B } $.
  $( Extend class notation to include unordered triplet. $)
  ctp $a class { A , B , C } $.
  $( Extend class notation to include ordered pair. $)
  cop $a class <. A , B >. $.
  $( Extend class notation to include ordered triple. $)
  cotp $a class <. A , B , C >. $.

  ${
    $d x A $.  $d y A $.  $d z x $.  $d z y $.  $d z A $.
    $( Soundness justification theorem for ~ df-sn .  (Contributed by Rodolfo
       Medina, 28-Apr-2010.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    snjust $p |- { x | x = A } = { y | y = A } $=
      ( vz cv wceq cab eqeq1 cbvabv eqtri ) AEZCFZAGDEZCFZDGBEZCFZBGLNADKMCHINP
      DBMOCHIJ $.
  $}

  ${
    $d x A $.
    $( Define the singleton of a class.  Definition 7.1 of [Quine] p. 48.  For
       convenience, it is well-defined for proper classes, i.e., those that are
       not elements of ` _V ` , although it is not very meaningful in this
       case.  For an alternate definition see ~ dfsn2 .  (Contributed by NM,
       5-Aug-1993.) $)
    df-sn $a |- { A } = { x | x = A } $.
  $}

  $( Define unordered pair of classes.  Definition 7.1 of [Quine] p. 48.  They
     are unordered, so ` { A , B } = { B , A } ` as proven by ~ prcom .  For a
     more traditional definition, but requiring a dummy variable, see ~ dfpr2 .
     (Contributed by NM, 5-Aug-1993.) $)
  df-pr $a |- { A , B } = ( { A } u. { B } ) $.

  $( Define unordered triple of classes.  Definition of [Enderton] p. 19.
     (Contributed by NM, 9-Apr-1994.) $)
  df-tp $a |- { A , B , C } = ( { A , B } u. { C } ) $.

  ${
    $d x A $.  $d x B $.
    $( Definition of an ordered pair, equivalent to Kuratowski's definition
       ` { { A } , { A , B } } ` when the arguments are sets.  Since the
       behavior of Kuratowski definition is not very useful for proper classes,
       we define it to be empty in this case (see ~ opprc1 and ~ opprc2 ).  For
       Kuratowski's actual definition when the arguments are sets, see ~ dfop .

       Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally
       as ` <. A , B >. = { { A } , { A , B } } ` , which has different
       behavior from our ~ df-op when the arguments are proper classes.
       Ordinarily this difference is not important, since neither definition is
       meaningful in that case.  Our ~ df-op was chosen because it often makes
       proofs shorter by eliminating unnecessary sethood hypotheses.

       There are other ways to define ordered pairs.  The basic requirement is
       that two ordered pairs are equal iff their respective members are equal.
       In 1914 Norbert Wiener gave the first successful definition
       ` <. A , B >. `_2 ` = { { { A } , (/) } , { { B } } } ` .  This was
       simplified by Kazimierz Kuratowski in 1921 to our present definition.
       An even simpler definition is ` <. A , B >. `_3
       ` = { A , { A , B } } ` , but it requires the Axiom of Regularity for
       its justification and is not commonly used.  Finally, an ordered pair of
       real numbers can be represented by a complex number.  (Contributed by
       NM, 28-May-1995.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    df-op $a |- <. A , B >. = { x |
      ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } $.
  $}

  $( Define ordered triple of classes.  Definition of ordered triple in [Stoll]
     p. 25.  (Contributed by NM, 3-Apr-2015.) $)
  df-ot $a |- <. A , B , C >. = <. <. A , B >. , C >. $.

  ${
    $d x A $.  $d x B $.
    $( Equality theorem for singletons.  Part of Exercise 4 of [TakeutiZaring]
       p. 15.  (Contributed by NM, 5-Aug-1993.) $)
    sneq $p |- ( A = B -> { A } = { B } ) $=
      ( vx wceq cv cab csn eqeq2 abbidv df-sn 3eqtr4g ) ABDZCEZADZCFMBDZCFAGBGL
      NOCABMHICAJCBJK $.
  $}

  ${
    sneqi.1 $e |- A = B $.
    $( Equality inference for singletons.  (Contributed by NM, 22-Jan-2004.) $)
    sneqi $p |- { A } = { B } $=
      ( wceq csn sneq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    sneqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for singletons.  (Contributed by NM, 22-Jan-2004.) $)
    sneqd $p |- ( ph -> { A } = { B } ) $=
      ( wceq csn sneq syl ) ABCEBFCFEDBCGH $.
  $}

  $( Alternate definition of singleton.  Definition 5.1 of [TakeutiZaring]
     p. 15.  (Contributed by NM, 24-Apr-1994.) $)
  dfsn2 $p |- { A } = { A , A } $=
    ( cpr csn cun df-pr unidm eqtr2i ) AABACZHDHAAEHFG $.

  ${
    $d A x $.  $d B x $.
    $( There is exactly one element in a singleton.  Exercise 2 of
       [TakeutiZaring] p. 15 (generalized).  (Contributed by NM, 13-Sep-1995.)
       (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    elsng $p |- ( A e. V -> ( A e. { B } <-> A = B ) ) $=
      ( vx cv wceq csn eqeq1 df-sn elab2g ) DEZBFABFDABGCKABHDBIJ $.
  $}

  ${
    elsn.1 $e |- A e. _V $.
    $( There is exactly one element in a singleton.  Exercise 2 of
       [TakeutiZaring] p. 15.  (Contributed by NM, 13-Sep-1995.) $)
    elsn $p |- ( A e. { B } <-> A = B ) $=
      ( cvv wcel csn wceq wb elsng ax-mp ) ADEABFEABGHCABDIJ $.
  $}

  $( There is only one element in a singleton.  Exercise 2 of [TakeutiZaring]
     p. 15.  (Contributed by NM, 21-Jun-1993.) $)
  velsn $p |- ( x e. { A } <-> x = A ) $=
    ( cv vex elsn ) ACBADE $.

  $( There is only one element in a singleton.  (Contributed by NM,
     5-Jun-1994.) $)
  elsni $p |- ( A e. { B } -> A = B ) $=
    ( csn wcel wceq elsng ibi ) ABCZDABEABHFG $.

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of unordered pair.  Definition 5.1 of
       [TakeutiZaring] p. 15.  (Contributed by NM, 24-Apr-1994.) $)
    dfpr2 $p |- { A , B } = { x | ( x = A \/ x = B ) } $=
      ( cpr csn cun cv wceq wo df-pr wcel elun velsn orbi12i bitri abbi2i eqtri
      cab ) BCDBEZCEZFZAGZBHZUBCHZIZARBCJUEAUAUBUAKUBSKZUBTKZIUEUBSTLUFUCUGUDAB
      MACMNOPQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( A member of an unordered pair of classes is one or the other of them.
       Exercise 1 of [TakeutiZaring] p. 15, generalized.  (Contributed by NM,
       13-Sep-1995.) $)
    elprg $p |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) ) $=
      ( vx cv wceq wo cpr eqeq1 orbi12d dfpr2 elab2g ) EFZBGZNCGZHABGZACGZHEABC
      IDNAGOQPRNABJNACJKEBCLM $.
  $}

  ${
    elpr.1 $e |- A e. _V $.
    $( A member of an unordered pair of classes is one or the other of them.
       Exercise 1 of [TakeutiZaring] p. 15.  (Contributed by NM,
       13-Sep-1995.) $)
    elpr $p |- ( A e. { B , C } <-> ( A = B \/ A = C ) ) $=
      ( cvv wcel cpr wceq wo wb elprg ax-mp ) AEFABCGFABHACHIJDABCEKL $.
  $}

  ${
    elpr2.1 $e |- B e. _V $.
    elpr2.2 $e |- C e. _V $.
    $( A member of an unordered pair of classes is one or the other of them.
       Exercise 1 of [TakeutiZaring] p. 15.  (Contributed by NM,
       14-Oct-2005.) $)
    elpr2 $p |- ( A e. { B , C } <-> ( A = B \/ A = C ) ) $=
      ( cpr wcel wceq wo elprg ibi cvv wb eleq1 mpbiri jaoi syl ibir impbii ) A
      BCFZGZABHZACHZIZUAUDABCTJKUDUAUDALGZUAUDMUBUEUCUBUEBLGDABLNOUCUECLGEACLNO
      PABCLJQRS $.
  $}

  $( If a class is an element of a pair, then it is one of the two paired
     elements.  (Contributed by Scott Fenton, 1-Apr-2011.) $)
  elpri $p |- ( A e. { B , C } -> ( A = B \/ A = C ) ) $=
    ( cpr wcel wceq wo elprg ibi ) ABCDZEABFACFGABCJHI $.

  ${
    nelpri.1 $e |- A =/= B $.
    nelpri.2 $e |- A =/= C $.
    $( If an element doesn't match the items in an unordered pair, it is not in
       the unordered pair.  (Contributed by David A. Wheeler, 10-May-2015.) $)
    nelpri $p |- -. A e. { B , C } $=
      ( wne cpr wcel wn wa wceq wo neanior elpri con3i sylbi mp2an ) ABFZACFZAB
      CGHZIZDERSJABKACKLZIUAABACMTUBABCNOPQ $.
  $}

  ${
    prneli.1 $e |- A =/= B $.
    prneli.2 $e |- A =/= C $.
    $( If an element doesn't match the items in an unordered pair, it is not in
       the unordered pair, using ` e/ ` .  (Contributed by David A. Wheeler,
       10-May-2015.) $)
    prneli $p |- A e/ { B , C } $=
      ( cpr nelpri nelir ) ABCFABCDEGH $.
  $}

  ${
    nelprd.1 $e |- ( ph -> A =/= B ) $.
    nelprd.2 $e |- ( ph -> A =/= C ) $.
    $( If an element doesn't match the items in an unordered pair, it is not in
       the unordered pair, deduction version.  (Contributed by Alexander van
       der Vekens, 25-Jan-2018.) $)
    nelprd $p |- ( ph -> -. A e. { B , C } ) $=
      ( wne cpr wcel wn wa wceq wo neanior elpri con3i sylbi syl2anc ) ABCGZBDG
      ZBCDHIZJZEFSTKBCLBDLMZJUBBCBDNUAUCBCDOPQR $.
  $}

  $( Membership in a set with two elements removed.  Similar to ~ eldifsn and
     ~ eldiftp .  (Contributed by Mario Carneiro, 18-Jul-2017.) $)
  eldifpr $p |- ( A e. ( B \ { C , D } ) <->
                ( A e. B /\ A =/= C /\ A =/= D ) ) $=
    ( wcel cpr wn wa wne cdif w3a wo elprg notbid neanior bitr4di pm5.32i eldif
    wceq 3anass 3bitr4i ) ABEZACDFZEZGZHUBACIZADIZHZHABUCJEUBUFUGKUBUEUHUBUEACS
    ADSLZGUHUBUDUIACDBMNACADOPQABUCRUBUFUGTUA $.

  $( Restricted existential quantification over a set with two elements
     removed.  (Contributed by Alexander van der Vekens, 7-Feb-2018.) $)
  rexdifpr $p |- ( E. x e. ( A \ { B , C } ) ph
      <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) ) $=
    ( cv wne w3a cpr cdif wcel eldifpr 3anass bitri anbi1i df-3an bicomi anbi2i
    wa anass rexbii2 ) ABFZDGZUBEGZAHZBCDEIJZCUBUFKZASUBCKZUCUDSZSZASZUHUESZUGU
    JAUGUHUCUDHUJUBCDELUHUCUDMNOUKUHUIASZSULUHUIATUMUEUHUEUMUCUDAPQRNNUA $.

  $( A set is a member of its singleton.  Part of Theorem 7.6 of [Quine] p. 49.
     (Contributed by NM, 28-Oct-2003.) $)
  snidg $p |- ( A e. V -> A e. { A } ) $=
    ( wcel csn wceq eqid elsng mpbiri ) ABCAADCAAEAFAABGH $.

  $( A class is a set iff it is a member of its singleton.  (Contributed by NM,
     5-Apr-2004.) $)
  snidb $p |- ( A e. _V <-> A e. { A } ) $=
    ( cvv wcel csn snidg elex impbii ) ABCAADZCABEAHFG $.

  ${
    snid.1 $e |- A e. _V $.
    $( A set is a member of its singleton.  Part of Theorem 7.6 of [Quine]
       p. 49.  (Contributed by NM, 31-Dec-1993.) $)
    snid $p |- A e. { A } $=
      ( cvv wcel csn snidb mpbi ) ACDAAEDBAFG $.
  $}

  $( A setvar variable is a member of its singleton (common case).
     (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  vsnid $p |- x e. { x } $=
    ( cv vex snid ) ABACD $.

  $( There is only one element in a singleton.  Exercise 2 of [TakeutiZaring]
     p. 15.  This variation requires only that ` B ` , rather than ` A ` , be a
     set.  (Contributed by NM, 28-Oct-2003.) $)
  elsn2g $p |- ( B e. V -> ( A e. { B } <-> A = B ) ) $=
    ( wcel csn wceq elsni snidg eleq1 syl5ibrcom impbid2 ) BCDZABEZDZABFZABGLNO
    BMDBCHABMIJK $.

  ${
    elsn2.1 $e |- B e. _V $.
    $( There is only one element in a singleton.  Exercise 2 of [TakeutiZaring]
       p. 15.  This variation requires only that ` B ` , rather than ` A ` , be
       a set.  (Contributed by NM, 12-Jun-1994.) $)
    elsn2 $p |- ( A e. { B } <-> A = B ) $=
      ( cvv wcel csn wceq wb elsn2g ax-mp ) BDEABFEABGHCABDIJ $.
  $}

  $( If a class is not equal to the class in a singleton, then it is not in the
     singleton.  (Contributed by Glauco Siliprandi, 17-Aug-2020.)  (Proof
     shortened by BJ, 4-May-2021.) $)
  nelsn $p |- ( A =/= B -> -. A e. { B } ) $=
    ( csn wcel elsni necon3ai ) ABCDABABEF $.

  ${
    $d x A $.
    $( A singleton has at most one element.  This works whether ` A ` is a
       proper class or not, and in that sense can be seen as encompassing both
       ~ snmg and ~ snprc .  (Contributed by Jim Kingdon, 30-Aug-2018.) $)
    mosn $p |- E* x x e. { A } $=
      ( cv csn wcel wmo wceq moeq velsn mobii mpbir ) ACZBDEZAFLBGZAFABHMNAABIJ
      K $.
  $}

  ${
    $d A x $.  $d ps x $.
    $( Substitution expressed in terms of quantification over a singleton.
       (Contributed by NM, 14-Dec-2005.)  (Revised by Mario Carneiro,
       23-Apr-2015.) $)
    ralsnsg $p |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) ) $=
      ( wcel csn wral cv wceq wi wal wsbc df-ral velsn imbi1i albii bitri sbc6g
      bitr4id ) CDEABCFZGZBHZCIZAJZBKZABCLUAUBTEZAJZBKUEABTMUGUDBUFUCABCNOPQABC
      DRS $.

    $( Substitution expressed in terms of quantification over a singleton.
       (Contributed by Mario Carneiro, 23-Apr-2015.) $)
    ralsns $p |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) ) $=
      ( wcel csn wral cv wceq wi wal wsbc df-ral velsn imbi1i albii bitri sbc6g
      bitr4id ) CDEABCFZGZBHZCIZAJZBKZABCLUAUBTEZAJZBKUEABTMUGUDBUFUCABCNOPQABC
      DRS $.

    $( Restricted existential quantification over a singleton.  (Contributed by
       Mario Carneiro, 23-Apr-2015.)  (Revised by NM, 22-Aug-2018.) $)
    rexsns $p |- ( E. x e. { A } ph <-> [. A / x ]. ph ) $=
      ( cv csn wcel wex wceq wrex wsbc velsn anbi1i exbii df-rex sbc5 3bitr4i
      wa ) BDZCEZFZAQZBGRCHZAQZBGABSIABCJUAUCBTUBABCKLMABSNABCOP $.

    ralsng.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Substitution expressed in terms of quantification over a singleton.
       (Contributed by NM, 14-Dec-2005.)  (Revised by Mario Carneiro,
       23-Apr-2015.) $)
    ralsng $p |- ( A e. V -> ( A. x e. { A } ph <-> ps ) ) $=
      ( wcel csn wral wsbc ralsns sbcieg bitrd ) DEGACDHIACDJBACDEKABCDEFLM $.

    $( Restricted existential quantification over a singleton.  (Contributed by
       NM, 29-Jan-2012.) $)
    rexsng $p |- ( A e. V -> ( E. x e. { A } ph <-> ps ) ) $=
      ( csn wrex wsbc wcel rexsns sbcieg bitrid ) ACDGHACDIDEJBACDKABCDEFLM $.
  $}

  $( There is a set being the element of a singleton if and only if there is an
     element of the singleton.  (Contributed by Alexander van der Vekens,
     1-Jan-2018.) $)
  exsnrex $p |- ( E. x M = { x } <-> E. x e. M M = { x } ) $=
    ( cv csn wceq wex wcel wa wrex vex snid eleq2 mpbiri pm4.71ri df-rex bitr4i
    exbii ) BACZDZEZAFRBGZTHZAFTABITUBATUATUARSGRAJKBSRLMNQTABOP $.

  ${
    $d A x $.  $d ps x $.
    ralsn.1 $e |- A e. _V $.
    ralsn.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Convert a quantification over a singleton to a substitution.
       (Contributed by NM, 27-Apr-2009.) $)
    ralsn $p |- ( A. x e. { A } ph <-> ps ) $=
      ( cvv wcel csn wral wb ralsng ax-mp ) DGHACDIJBKEABCDGFLM $.

    $( Restricted existential quantification over a singleton.  (Contributed by
       Jeff Madsen, 5-Jan-2011.) $)
    rexsn $p |- ( E. x e. { A } ph <-> ps ) $=
      ( cvv wcel csn wrex wb rexsng ax-mp ) DGHACDIJBKEABCDGFLM $.
  $}

  $( Members of an unordered triple of classes.  (Contributed by FL,
     2-Feb-2014.)  (Proof shortened by Mario Carneiro, 11-Feb-2015.) $)
  eltpg $p |- ( A e. V -> ( A e. { B , C , D } <->
     ( A = B \/ A = C \/ A = D ) ) ) $=
    ( wcel cpr csn wceq ctp w3o elprg elsng orbi12d cun df-tp eleq2i elun bitri
    wo df-3or 3bitr4g ) AEFZABCGZFZADHZFZTZABIZACIZTZADIZTABCDJZFZUIUJULKUCUEUK
    UGULABCELADEMNUNAUDUFOZFUHUMUOABCDPQAUDUFRSUIUJULUAUB $.

  $( Membership in a set with three elements removed.  Similar to ~ eldifsn and
     ~ eldifpr .  (Contributed by David A. Wheeler, 22-Jul-2017.) $)
  eldiftp $p |- ( A e. ( B \ { C , D , E } ) <->
                ( A e. B /\ ( A =/= C /\ A =/= D /\ A =/= E ) ) ) $=
    ( ctp cdif wcel wn wa wne w3a eldif wceq w3o eltpg ne3anior bitr4di pm5.32i
    notbid bitri ) ABCDEFZGHABHZAUBHZIZJUCACKADKAEKLZJABUBMUCUEUFUCUEACNADNAENO
    ZIUFUCUDUGACDEBPTACADAEQRSUA $.

  $( A member of an unordered triple of classes is one of them.  (Contributed
     by Mario Carneiro, 11-Feb-2015.) $)
  eltpi $p |- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) ) $=
    ( ctp wcel wceq w3o eltpg ibi ) ABCDEZFABGACGADGHABCDKIJ $.

  ${
    eltp.1 $e |- A e. _V $.
    $( A member of an unordered triple of classes is one of them.  Special case
       of Exercise 1 of [TakeutiZaring] p. 17.  (Contributed by NM,
       8-Apr-1994.)  (Revised by Mario Carneiro, 11-Feb-2015.) $)
    eltp $p |- ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) $=
      ( cvv wcel ctp wceq w3o wb eltpg ax-mp ) AFGABCDHGABIACIADIJKEABCDFLM $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Alternate definition of unordered triple of classes.  Special case of
       Definition 5.3 of [TakeutiZaring] p. 16.  (Contributed by NM,
       8-Apr-1994.) $)
    dftp2 $p |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) } $=
      ( cv wceq w3o ctp vex eltp abbi2i ) AEZBFLCFLDFGABCDHLBCDAIJK $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    nfpr.1 $e |- F/_ x A $.
    nfpr.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for unordered pairs.  (Contributed by
       NM, 14-Nov-1995.) $)
    nfpr $p |- F/_ x { A , B } $=
      ( vy cpr cv wceq wo cab dfpr2 nfeq2 nfor nfab nfcxfr ) ABCGFHZBIZQCIZJZFK
      FBCLTAFRSAAQBDMAQCEMNOP $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ps $.  $d x ch $.  $d x th $.
    ralprg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ralprg.2 $e |- ( x = B -> ( ph <-> ch ) ) $.
    $( Convert a quantification over a pair to a conjunction.  (Contributed by
       NM, 17-Sep-2011.)  (Revised by Mario Carneiro, 23-Apr-2015.) $)
    ralprg $p |- ( ( A e. V /\ B e. W ) ->
      ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) ) $=
      ( cpr wral csn wa wcel cun df-pr raleqi ralunb ralsng bi2anan9 bitrid
      bitri ) ADEFKZLZADEMZLZADFMZLZNZEGOZFHOZNBCNUEADUFUHPZLUJADUDUMEFQRADUFUH
      SUCUKUGBULUICABDEGITACDFHJTUAUB $.

    $( Convert a quantification over a pair to a disjunction.  (Contributed by
       NM, 17-Sep-2011.)  (Revised by Mario Carneiro, 23-Apr-2015.) $)
    rexprg $p |- ( ( A e. V /\ B e. W ) ->
      ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) ) $=
      ( cpr wrex csn wo wcel wa cun df-pr rexeqi rexsng orbi1d orbi2d sylan9bb
      rexun bitri bitrid ) ADEFKZLZADEMZLZADFMZLZNZEGOZFHOZPBCNZUHADUIUKQZLUMAD
      UGUQEFRSADUIUKUDUEUNUMBULNUOUPUNUJBULABDEGITUAUOULCBACDFHJTUBUCUF $.

    raltpg.3 $e |- ( x = C -> ( ph <-> th ) ) $.
    $( Convert a quantification over a triple to a conjunction.  (Contributed
       by NM, 17-Sep-2011.)  (Revised by Mario Carneiro, 23-Apr-2015.) $)
    raltpg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
      ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) ) $=
      ( wcel w3a cpr wral csn wa ctp wb ralprg ralsng bi2anan9 3impa cun raleqi
      df-tp ralunb bitri df-3an 3bitr4g ) FIOZGJOZHKOZPAEFGQZRZAEHSZRZTZBCTZDTZ
      AEFGHUAZRZBCDPUNUOUPVAVCUBUNUOTURVBUPUTDABCEFGIJLMUCADEHKNUDUEUFVEAEUQUSU
      GZRVAAEVDVFFGHUIUHAEUQUSUJUKBCDULUM $.

    $( Convert a quantification over a triple to a disjunction.  (Contributed
       by Mario Carneiro, 23-Apr-2015.) $)
    rextpg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
      ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) ) $=
      ( wcel w3a cpr wrex csn wo ctp wb wa rexprg orbi1d rexsng orbi2d sylan9bb
      w3o 3impa cun df-tp rexeqi rexun bitri df-3or 3bitr4g ) FIOZGJOZHKOZPAEFG
      QZRZAEHSZRZTZBCTZDTZAEFGHUAZRZBCDUIURUSUTVEVGUBURUSUCZVEVFVDTUTVGVJVBVFVD
      ABCEFGIJLMUDUEUTVDDVFADEHKNUFUGUHUJVIAEVAVCUKZRVEAEVHVKFGHULUMAEVAVCUNUOB
      CDUPUQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.  $d x ch $.
    ralpr.1 $e |- A e. _V $.
    ralpr.2 $e |- B e. _V $.
    ralpr.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ralpr.4 $e |- ( x = B -> ( ph <-> ch ) ) $.
    $( Convert a quantification over a pair to a conjunction.  (Contributed by
       NM, 3-Jun-2007.)  (Revised by Mario Carneiro, 23-Apr-2015.) $)
    ralpr $p |- ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) $=
      ( cvv wcel cpr wral wa wb ralprg mp2an ) EKLFKLADEFMNBCOPGHABCDEFKKIJQR
      $.

    $( Convert an existential quantification over a pair to a disjunction.
       (Contributed by NM, 3-Jun-2007.)  (Revised by Mario Carneiro,
       23-Apr-2015.) $)
    rexpr $p |- ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) $=
      ( cvv wcel cpr wrex wo wb rexprg mp2an ) EKLFKLADEFMNBCOPGHABCDEFKKIJQR
      $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ps $.  $d x ch $.  $d x th $.
    raltp.1 $e |- A e. _V $.
    raltp.2 $e |- B e. _V $.
    raltp.3 $e |- C e. _V $.
    raltp.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
    raltp.5 $e |- ( x = B -> ( ph <-> ch ) ) $.
    raltp.6 $e |- ( x = C -> ( ph <-> th ) ) $.
    $( Convert a quantification over a triple to a conjunction.  (Contributed
       by NM, 13-Sep-2011.)  (Revised by Mario Carneiro, 23-Apr-2015.) $)
    raltp $p |- ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) $=
      ( cvv wcel ctp wral w3a wb raltpg mp3an ) FOPGOPHOPAEFGHQRBCDSTIJKABCDEFG
      HOOOLMNUAUB $.

    $( Convert a quantification over a triple to a disjunction.  (Contributed
       by Mario Carneiro, 23-Apr-2015.) $)
    rextp $p |- ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) $=
      ( cvv wcel ctp wrex w3o wb rextpg mp3an ) FOPGOPHOPAEFGHQRBCDSTIJKABCDEFG
      HOOOLMNUAUB $.
  $}

  ${
    $d x A $.
    $( TODO - make obsolete; use ralsnsSBC instead - also,
       shorten posn with ralsn or ralsng $)
    $( Substitution expressed in terms of quantification over a singleton.
       (Contributed by NM, 14-Dec-2005.)  (Revised by Mario Carneiro,
       23-Apr-2015.) $)
    sbcsng $p |- ( A e. V -> ( [. A / x ]. ph <-> A. x e. { A } ph ) ) $=
      ( wcel csn wral wsbc ralsns bicomd ) CDEABCFGABCHABCDIJ $.
  $}

  ${
    nfsn.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for singletons.  (Contributed by NM,
       14-Nov-1995.) $)
    nfsn $p |- F/_ x { A } $=
      ( csn cpr dfsn2 nfpr nfcxfr ) ABDBBEBFABBCCGH $.
  $}

  ${
    $d A y $.  $d B y $.  $d V y $.  $d x y $.
    $( Distribute proper substitution through the singleton of a class.
       (Contributed by Alan Sare, 10-Nov-2012.) $)
    csbsng $p |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } ) $=
      ( vy wcel wceq cab csb csn wsbc csbabg sbceq2g abbidv eqtrd df-sn csbeq2i
      cv 3eqtr4g ) BDFZABERZCGZEHZIZUAABCIZGZEHZABCJZIUEJTUDUBABKZEHUGUBAEBDLTU
      IUFEABUACDMNOABUHUCECPQEUEPS $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Intersection with the singleton of a non-member is disjoint.
       (Contributed by NM, 22-May-1998.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.)  (Proof shortened by Wolf Lammen, 30-Sep-2014.) $)
    disjsn $p |- ( ( A i^i { B } ) = (/) <-> -. B e. A ) $=
      ( vx csn cin c0 wceq cv wcel wn wi wal wa disj1 con2b velsn imbi1i 3bitri
      imnan albii wex alnex df-clel xchbinxr ) ABDZEFGCHZAIZUFUEIZJKZCLUFBGZUGM
      ZJZCLZBAIZJCAUENUIULCUIUHUGJZKUJUOKULUGUHOUHUJUOCBPQUJUGSRTUMUKCUAUNUKCUB
      CBAUCUDR $.
  $}

  $( Intersection of distinct singletons is disjoint.  (Contributed by NM,
     25-May-1998.) $)
  disjsn2 $p |- ( A =/= B -> ( { A } i^i { B } ) = (/) ) $=
    ( wne csn wcel wn cin c0 wceq elsni eqcomd necon3ai disjsn sylibr ) ABCBADZ
    EZFOBDGHIPABPBABAJKLOBMN $.

  $( The intersection of distinct unordered pairs is disjoint.  (Contributed by
     Alexander van der Vekens, 11-Nov-2017.) $)
  disjpr2 $p |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) )
                  -> ( { A , B } i^i { C , D } ) = (/) ) $=
    ( wne wa cpr cin csn c0 wceq df-pr ineq1i indir eqtri disjsn2 adantr adantl
    cun eqtrid a1i ineq2d indi jca un00 sylib uneq12d un0 eqtrdi eqtrd ) ACEZBC
    EZFZADEZBDEZFZFZABGZCDGZHURCIZDIZSZHZJUQUSVBURUSVBKUQCDLUAUBUQVCURUTHZURVAH
    ZSZJURUTVAUCUQVFJJSJUQVDJVEJUQVDAIZUTHZBIZUTHZSZJVDVGVISZUTHVKURVLUTABLZMVG
    VIUTNOUQVHJKZVJJKZFVKJKUQVNVOUMVNUPUKVNULACPQQUMVOUPULVOUKBCPRQUDVHVJUEUFTU
    QVEVGVAHZVIVAHZSZJVEVLVAHVRURVLVAVMMVGVIVANOUQVPJKZVQJKZFVRJKUQVSVTUPVSUMUN
    VSUOADPQRUPVTUMUOVTUNBDPRRUDVPVQUEUFTUGJUHUITUJ $.

  ${
    $d x A $.
    $( The singleton of a proper class (one that doesn't exist) is the empty
       set.  Theorem 7.2 of [Quine] p. 48.  (Contributed by NM, 5-Aug-1993.) $)
    snprc $p |- ( -. A e. _V <-> { A } = (/) ) $=
      ( vx cv csn wcel wex wn wceq cvv velsn exbii notbii wal alnex bitri isset
      c0 eq0 3bitr4ri ) BCZADZEZBFZGZTAHZBFZGUAQHZAIEZGUCUFUBUEBBAJKLUGUBGBMUDB
      UARUBBNOUHUFBAPLS $.
  $}

  ${
    $d x y A $.  $d x B $.
    $( Special case of ~ r19.12 where its converse holds.  (Contributed by NM,
       19-May-2008.)  (Revised by Mario Carneiro, 23-Apr-2015.)  (Revised by
       BJ, 20-Dec-2021.) $)
    r19.12sn $p |- ( A e. V ->
             ( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph ) ) $=
      ( wcel wral wsbc csn wrex sbcralg rexsns ralbii 3bitr4g ) DFGACEHZBDIABDI
      ZCEHPBDJZKABRKZCEHABCDEFLPBDMSQCEABDMNO $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Condition where a restricted class abstraction is a singleton.
       (Contributed by NM, 28-May-2006.) $)
    rabsn $p |- ( B e. A -> { x e. A | x = B } = { B } ) $=
      ( wcel cv wceq wa cab crab csn eleq1 pm5.32ri abbidv df-rab df-sn 3eqtr4g
      baib ) CBDZAEZBDZSCFZGZAHUAAHUAABICJRUBUAAUBRUAUATRSCBKLQMUAABNACOP $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( A restricted class abstraction restricted to a singleton is either the
       empty set or the singleton itself.  (Contributed by AV, 21-Jul-2019.) $)
    rabsnifsb $p |- { x e. { A } | ph }
                    = if ( [. A / x ]. ph , { A } , (/) ) $=
      ( vy cv csn wcel wa cab wsbc c0 wn wo wi syl imdistani nfan eleq1w anbi1d
      nfv crab cif wceq elsni sbceq1a biimpd orcd biimprd pm2.21i adantr impbii
      noel jaoi abbii nfsbc1v nfn weq orbi12d cbvabw eqtri df-rab df-if 3eqtr4i
      nfor ) BEZCFZGZAHZBIZDEZVFGZABCJZHZVJKGZVLLZHZMZDIZABVFUAVLVFKUBVIVGVLHZV
      EKGZVOHZMZBIVRVHWBBVHWBVHVSWAVGAVLVGVECUCZAVLNVECUDZWCAVLABCUEZUFOPUGVSVH
      WAVGVLAVGWCVLANWDWCAVLWEUHOPVTVHVOVTVHVEULUIUJUMUKUNWBVQBDWBDTVMVPBVKVLBV
      KBTABCUOZQVNVOBVNBTVLBWFUPQVDBDUQZVSVMWAVPWGVGVKVLBDVFRSWGVTVNVOBDKRSURUS
      UTABVFVAVLDVFKVBVC $.
  $}

  ${
    $d A x y $.  $d ph y $.  $d ps x y $.
    rabsnif.f $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( A restricted class abstraction restricted to a singleton is either the
       empty set or the singleton itself.  (Contributed by AV, 12-Apr-2019.)
       (Proof shortened by AV, 21-Jul-2019.) $)
    rabsnif $p |- { x e. { A } | ph } = if ( ps , { A } , (/) ) $=
      ( vy csn crab c0 cif cv wcel cvv wceq wex elrabi 19.8ad sylibr wa wn noel
      elsni syl intnan a1i wo elif biimpi ecased simpl2im wsbc rabsnifsb sbcieg
      isset ifbid eqtrid eleq2d pm5.21nii eqriv ) FACDGZHZBUTIJZFKZVALZDMLZVCVB
      LZVDVCDNZFOZVEVDVGFVDVCUTLZVGACVCUTPVCDUBZUCQFDUNZRVFVHVEVFVGFVFBVIVGVFBV
      ISZBTZVCILZSZVOTVFVNVMVCUAUDUEVFVLVOUFBVCUTIUGUHUIVJUJQVKRVEVAVBVCVEVAACD
      UKZUTIJVBACDULVEVPBUTIABCDMEUMUOUPUQURUS $.
  $}

  ${
    $d x A $.
    rabrsndc.1 $e |- A e. _V $.
    rabrsndc.2 $e |- DECID ph $.
    $( A class abstraction over a decidable proposition restricted to a
       singleton is either the empty set or the singleton itself.  (Contributed
       by Jim Kingdon, 8-Aug-2018.) $)
    rabrsndc $p |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) ) $=
      ( csn crab wceq c0 wo wral wsbc cvv ax-mp wb ralsns orbi12i mpbir eqeq1
      wn wcel pm2.1dc sbcth sbcor mpbi rabeq0 eqcom rabid2 bitri orbi12d mpbiri
      wdc ) DABCGZHZIZDJIZDUNIZKUOJIZUOUNIZKZVAAUAZBUNLZABUNLZKZVEVBBCMZABCMZKZ
      VBAKZBCMZVHCNUBZVJEVIBCNAUMVIFAUCOUDOVBABCUEUFVCVFVDVGVKVCVFPEVBBCNQOVKVD
      VGPEABCNQORSUSVCUTVDABUNUGUTUNUOIVDUOUNUHABUNUIUJRSUPUQUSURUTDUOJTDUOUNTU
      KUL $.
  $}

  ${
    $d x y $.  $d y ph $.  $d y A $.
    $( Another way to express existential uniqueness of a wff: its class
       abstraction is a singleton.  (Contributed by Mario Carneiro,
       14-Nov-2016.) $)
    euabsn2 $p |- ( E! x ph <-> E. y { x | ph } = { y } ) $=
      ( weu cv wceq wb wal wex cab csn df-eu wcel abeq1 velsn albii bitri exbii
      bibi2i bitr4i ) ABDABEZCEZFZGZBHZCIABJUBKZFZCIABCLUGUECUGAUAUFMZGZBHUEABU
      FNUIUDBUHUCABUBOSPQRT $.

    $( Another way to express existential uniqueness of a wff: its class
       abstraction is a singleton.  (Contributed by NM, 22-Feb-2004.) $)
    euabsn $p |- ( E! x ph <-> E. x { x | ph } = { x } ) $=
      ( vy weu cab cv csn wceq wex euabsn2 nfab1 nfeq1 sneq eqeq2d cbvex bitr4i
      nfv ) ABDABEZCFZGZHZCIRBFZGZHZBIABCJUDUABCUDCQBRTABKLUBSHUCTRUBSMNOP $.

    $( A way to express restricted existential uniqueness of a wff: its
       restricted class abstraction is a singleton.  (Contributed by NM,
       30-May-2006.)  (Proof shortened by Mario Carneiro, 14-Nov-2016.) $)
    reusn $p |- ( E! x e. A ph <-> E. y { x e. A | ph } = { y } ) $=
      ( cv wcel wa weu cab csn wceq wex wreu euabsn2 df-reu df-rab eqeq1i exbii
      crab 3bitr4i ) BEDFAGZBHUABIZCEJZKZCLABDMABDSZUCKZCLUABCNABDOUFUDCUEUBUCA
      BDPQRT $.

    $( Restricted existential uniqueness determined by a singleton.
       (Contributed by NM, 29-May-2006.) $)
    absneu $p |- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph ) $=
      ( vy wcel cab csn wceq wa wex weu sneq eqeq2d spcegv imp euabsn2 sylibr
      cv ) CDFZABGZCHZIZJUAESZHZIZEKZABLTUCUGUFUCECDUDCIUEUBUAUDCMNOPABEQR $.

    $( Restricted existential uniqueness determined by a singleton.
       (Contributed by NM, 29-May-2006.)  (Revised by Mario Carneiro,
       23-Dec-2016.) $)
    rabsneu $p |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x e. B ph ) $=
      ( wcel crab csn wceq wa cv weu df-rab eqeq1i absneu sylan2b df-reu sylibr
      wreu cab ) CEFZABDGZCHZIZJBKDFAJZBLZABDSUDUAUEBTZUCIUFUBUGUCABDMNUEBCEOPA
      BDQR $.
  $}

  ${
    $d x A $.
    $( Two ways to express " ` A ` is a singleton".  (Contributed by NM,
       30-Oct-2010.) $)
    eusn $p |- ( E! x x e. A <-> E. x A = { x } ) $=
      ( cv wcel weu cab csn wceq wex euabsn abid2 eqeq1i exbii bitri ) ACZBDZAE
      PAFZOGZHZAIBRHZAIPAJSTAQBRABKLMN $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    rabsnt.1 $e |- B e. _V $.
    rabsnt.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
    $( Truth implied by equality of a restricted class abstraction and a
       singleton.  (Contributed by NM, 29-May-2006.)  (Proof shortened by Mario
       Carneiro, 23-Dec-2016.) $)
    rabsnt $p |- ( { x e. A | ph } = { B } -> ps ) $=
      ( crab csn wceq wcel snid id eleqtrrid elrab simprbi syl ) ACDHZEIZJZERKZ
      BTESREFLTMNUAEDKBABCEDGOPQ $.
  $}

  $( Commutative law for unordered pairs.  (Contributed by NM, 5-Aug-1993.) $)
  prcom $p |- { A , B } = { B , A } $=
    ( csn cun cpr uncom df-pr 3eqtr4i ) ACZBCZDJIDABEBAEIJFABGBAGH $.

  $( Equality theorem for unordered pairs.  (Contributed by NM,
     29-Mar-1998.) $)
  preq1 $p |- ( A = B -> { A , C } = { B , C } ) $=
    ( wceq csn cun cpr sneq uneq1d df-pr 3eqtr4g ) ABDZAEZCEZFBEZNFACGBCGLMONAB
    HIACJBCJK $.

  $( Equality theorem for unordered pairs.  (Contributed by NM, 5-Aug-1993.) $)
  preq2 $p |- ( A = B -> { C , A } = { C , B } ) $=
    ( wceq cpr preq1 prcom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.

  $( Equality theorem for unordered pairs.  (Contributed by NM,
     19-Oct-2012.) $)
  preq12 $p |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } ) $=
    ( wceq cpr preq1 preq2 sylan9eq ) ACEBDEABFCBFCDFACBGBDCHI $.

  ${
    preq1i.1 $e |- A = B $.
    $( Equality inference for unordered pairs.  (Contributed by NM,
       19-Oct-2012.) $)
    preq1i $p |- { A , C } = { B , C } $=
      ( wceq cpr preq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality inference for unordered pairs.  (Contributed by NM,
       19-Oct-2012.) $)
    preq2i $p |- { C , A } = { C , B } $=
      ( wceq cpr preq2 ax-mp ) ABECAFCBFEDABCGH $.

    ${
      preq12i.2 $e |- C = D $.
      $( Equality inference for unordered pairs.  (Contributed by NM,
         19-Oct-2012.) $)
      preq12i $p |- { A , C } = { B , D } $=
        ( wceq cpr preq12 mp2an ) ABGCDGACHBDHGEFACBDIJ $.
    $}
  $}

  ${
    preq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for unordered pairs.  (Contributed by NM,
       19-Oct-2012.) $)
    preq1d $p |- ( ph -> { A , C } = { B , C } ) $=
      ( wceq cpr preq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Equality deduction for unordered pairs.  (Contributed by NM,
       19-Oct-2012.) $)
    preq2d $p |- ( ph -> { C , A } = { C , B } ) $=
      ( wceq cpr preq2 syl ) ABCFDBGDCGFEBCDHI $.

    preq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality deduction for unordered pairs.  (Contributed by NM,
       19-Oct-2012.) $)
    preq12d $p |- ( ph -> { A , C } = { B , D } ) $=
      ( wceq cpr preq12 syl2anc ) ABCHDEHBDICEIHFGBDCEJK $.
  $}

  $( Equality theorem for unordered triples.  (Contributed by NM,
     13-Sep-2011.) $)
  tpeq1 $p |- ( A = B -> { A , C , D } = { B , C , D } ) $=
    ( wceq cpr csn cun ctp preq1 uneq1d df-tp 3eqtr4g ) ABEZACFZDGZHBCFZPHACDIB
    CDINOQPABCJKACDLBCDLM $.

  $( Equality theorem for unordered triples.  (Contributed by NM,
     13-Sep-2011.) $)
  tpeq2 $p |- ( A = B -> { C , A , D } = { C , B , D } ) $=
    ( wceq cpr csn cun ctp preq2 uneq1d df-tp 3eqtr4g ) ABEZCAFZDGZHCBFZPHCADIC
    BDINOQPABCJKCADLCBDLM $.

  $( Equality theorem for unordered triples.  (Contributed by NM,
     13-Sep-2011.) $)
  tpeq3 $p |- ( A = B -> { C , D , A } = { C , D , B } ) $=
    ( wceq cpr csn cun ctp sneq uneq2d df-tp 3eqtr4g ) ABEZCDFZAGZHOBGZHCDAICDB
    INPQOABJKCDALCDBLM $.

  ${
    tpeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for unordered triples.  (Contributed by NM,
       22-Jun-2014.) $)
    tpeq1d $p |- ( ph -> { A , C , D } = { B , C , D } ) $=
      ( wceq ctp tpeq1 syl ) ABCGBDEHCDEHGFBCDEIJ $.

    $( Equality theorem for unordered triples.  (Contributed by NM,
       22-Jun-2014.) $)
    tpeq2d $p |- ( ph -> { C , A , D } = { C , B , D } ) $=
      ( wceq ctp tpeq2 syl ) ABCGDBEHDCEHGFBCDEIJ $.

    $( Equality theorem for unordered triples.  (Contributed by NM,
       22-Jun-2014.) $)
    tpeq3d $p |- ( ph -> { C , D , A } = { C , D , B } ) $=
      ( wceq ctp tpeq3 syl ) ABCGDEBHDECHGFBCDEIJ $.

    tpeq123d.2 $e |- ( ph -> C = D ) $.
    tpeq123d.3 $e |- ( ph -> E = F ) $.
    $( Equality theorem for unordered triples.  (Contributed by NM,
       22-Jun-2014.) $)
    tpeq123d $p |- ( ph -> { A , C , E } = { B , D , F } ) $=
      ( ctp tpeq1d tpeq2d tpeq3d 3eqtrd ) ABDFKCDFKCEFKCEGKABCDFHLADECFIMAFGCEJ
      NO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Rotation of the elements of an unordered triple.  (Contributed by Alan
       Sare, 24-Oct-2011.) $)
    tprot $p |- { A , B , C } = { B , C , A } $=
      ( vx cv wceq w3o cab ctp 3orrot abbii dftp2 3eqtr4i ) DEZAFZNBFZNCFZGZDHP
      QOGZDHABCIBCAIRSDOPQJKDABCLDBCALM $.
  $}

  $( Swap 1st and 2nd members of an undordered triple.  (Contributed by NM,
     22-May-2015.) $)
  tpcoma $p |- { A , B , C } = { B , A , C } $=
    ( cpr csn cun ctp prcom uneq1i df-tp 3eqtr4i ) ABDZCEZFBADZMFABCGBACGLNMABH
    IABCJBACJK $.

  $( Swap 2nd and 3rd members of an undordered triple.  (Contributed by NM,
     22-May-2015.) $)
  tpcomb $p |- { A , B , C } = { A , C , B } $=
    ( ctp tpcoma tprot 3eqtr4i ) BCADCBADABCDACBDBCAEABCFACBFG $.

  $( Split off the first element of an unordered triple.  (Contributed by Mario
     Carneiro, 5-Jan-2016.) $)
  tpass $p |- { A , B , C } = ( { A } u. { B , C } ) $=
    ( ctp cpr csn cun df-tp tprot uncom 3eqtr4i ) BCADBCEZAFZGABCDMLGBCAHABCIML
    JK $.

  $( Two ways to write an unordered quadruple.  (Contributed by Mario Carneiro,
     5-Jan-2016.) $)
  qdass $p |- ( { A , B } u. { C , D } ) = ( { A , B , C } u. { D } ) $=
    ( cpr csn cun ctp unass df-tp uneq1i df-pr uneq2i 3eqtr4ri ) ABEZCFZGZDFZGO
    PRGZGABCHZRGOCDEZGOPRITQRABCJKUASOCDLMN $.

  $( Two ways to write an unordered quadruple.  (Contributed by Mario Carneiro,
     5-Jan-2016.) $)
  qdassr $p |- ( { A , B } u. { C , D } ) = ( { A } u. { B , C , D } ) $=
    ( csn cun cpr ctp unass df-pr uneq1i tpass uneq2i 3eqtr4i ) AEZBEZFZCDGZFOP
    RFZFABGZRFOBCDHZFOPRITQRABJKUASOBCDLMN $.

  $( Unordered triple ` { A , A , B } ` is just an overlong way to write
     ` { A , B } ` .  (Contributed by David A. Wheeler, 10-May-2015.) $)
  tpidm12 $p |- { A , A , B } = { A , B } $=
    ( csn cun cpr ctp dfsn2 uneq1i df-pr df-tp 3eqtr4ri ) ACZBCZDAAEZMDABEAABFL
    NMAGHABIAABJK $.

  $( Unordered triple ` { A , B , A } ` is just an overlong way to write
     ` { A , B } ` .  (Contributed by David A. Wheeler, 10-May-2015.) $)
  tpidm13 $p |- { A , B , A } = { A , B } $=
    ( ctp cpr tprot tpidm12 eqtr3i ) AABCABACABDAABEABFG $.

  $( Unordered triple ` { A , B , B } ` is just an overlong way to write
     ` { A , B } ` .  (Contributed by David A. Wheeler, 10-May-2015.) $)
  tpidm23 $p |- { A , B , B } = { A , B } $=
    ( ctp cpr tprot tpidm12 prcom 3eqtri ) ABBCBBACBADABDABBEBAFBAGH $.

  $( Unordered triple ` { A , A , A } ` is just an overlong way to write
     ` { A } ` .  (Contributed by David A. Wheeler, 10-May-2015.) $)
  tpidm $p |- { A , A , A } = { A } $=
    ( ctp cpr csn tpidm12 dfsn2 eqtr4i ) AAABAACADAAEAFG $.

  $( An unordered triple is an unordered pair if one of its elements is
     identical with another element.  (Contributed by Alexander van der Vekens,
     6-Oct-2017.) $)
  tppreq3 $p |- ( B = C -> { A , B , C } = { A , B } ) $=
    ( wceq ctp cpr tpeq3 eqcoms tpidm23 eqtrdi ) BCDABCEZABBEZABFKLDCBCBABGHABI
    J $.

  $( An unordered pair contains its first member.  Part of Theorem 7.6 of
     [Quine] p. 49.  (Contributed by Stefan Allan, 8-Nov-2008.) $)
  prid1g $p |- ( A e. V -> A e. { A , B } ) $=
    ( wcel cpr wceq wo eqid orci elprg mpbiri ) ACDAABEDAAFZABFZGLMAHIAABCJK $.

  $( An unordered pair contains its second member.  Part of Theorem 7.6 of
     [Quine] p. 49.  (Contributed by Stefan Allan, 8-Nov-2008.) $)
  prid2g $p |- ( B e. V -> B e. { A , B } ) $=
    ( wcel cpr prid1g prcom eleqtrdi ) BCDBBAEABEBACFBAGH $.

  ${
    prid1.1 $e |- A e. _V $.
    $( An unordered pair contains its first member.  Part of Theorem 7.6 of
       [Quine] p. 49.  (Contributed by NM, 5-Aug-1993.) $)
    prid1 $p |- A e. { A , B } $=
      ( cvv wcel cpr prid1g ax-mp ) ADEAABFECABDGH $.
  $}

  ${
    prid2.1 $e |- B e. _V $.
    $( An unordered pair contains its second member.  Part of Theorem 7.6 of
       [Quine] p. 49.  (Contributed by NM, 5-Aug-1993.) $)
    prid2 $p |- B e. { A , B } $=
      ( cpr prid1 prcom eleqtri ) BBADABDBACEBAFG $.
  $}

  $( Membership of a conditional operator in an unordered pair.  (Contributed
     by NM, 17-Jun-2007.) $)
  ifprdc $p |- ( ( A e. C /\ B e. D /\ DECID ph )
      -> if ( ph , A , B ) e. { A , B } ) $=
    ( wcel wdc w3a cpr prid1g 3ad2ant1 prid2g 3ad2ant2 simp3 ifcldcd ) BDFZCEFZ
    AGZHABCBCIZPQBSFRBCDJKQPCSFRBCELMPQRNO $.

  $( An unordered pair is a singleton or a subset of itself.  This theorem is
     helpful to convert theorems about walks in arbitrary graphs into theorems
     about walks in pseudographs.  (Contributed by AV, 27-Feb-2021.) $)
  ifpprsnssdc $p |- ( ( P = { A , B } /\ DECID A = B )
                    -> if- ( A = B , P = { A } , { A , B } C_ P ) ) $=
    ( cpr wceq wdc wa csn wss wif wi preq2 eqtr4di eqcoms eqeq2d biimpcd adantr
    wn dfsn2 eqimss2 a1d wb dfifp2dc adantl mpbir2and ) CABDZEZABEZFZGUHCAHZEZU
    FCIZJZUHUKKZUHRZULKZUGUNUIUHUGUKUHUFUJCUFUJEBABAEUFAADUJBAALASMNOPQUGUPUIUG
    ULUOUFCTUAQUIUMUNUPGUBUGUHUKULUCUDUE $.

  $( A proper class vanishes in an unordered pair.  (Contributed by NM,
     5-Aug-1993.) $)
  prprc1 $p |- ( -. A e. _V -> { A , B } = { B } ) $=
    ( cvv wcel wn csn c0 wceq cpr snprc cun uneq1 df-pr uncom un0 3eqtr4g sylbi
    eqtr2i ) ACDEAFZGHZABIZBFZHAJTSUBKGUBKZUAUBSGUBLABMUCUBGKUBGUBNUBORPQ $.

  $( A proper class vanishes in an unordered pair.  (Contributed by NM,
     22-Mar-2006.) $)
  prprc2 $p |- ( -. B e. _V -> { A , B } = { A } ) $=
    ( cvv wcel wn cpr csn prcom prprc1 eqtrid ) BCDEABFBAFAGABHBAIJ $.

  $( An unordered pair containing two proper classes is the empty set.
     (Contributed by NM, 22-Mar-2006.) $)
  prprc $p |- ( ( -. A e. _V /\ -. B e. _V ) -> { A , B } = (/) ) $=
    ( cvv wcel wn cpr csn c0 prprc1 wceq snprc biimpi sylan9eq ) ACDEBCDEZABFBG
    ZHABINOHJBKLM $.

  ${
    tpid1.1 $e |- A e. _V $.
    $( One of the three elements of an unordered triple.  (Contributed by NM,
       7-Apr-1994.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    tpid1 $p |- A e. { A , B , C } $=
      ( ctp wcel wceq w3o eqid 3mix1i eltp mpbir ) AABCEFAAGZABGZACGZHMNOAIJAAB
      CDKL $.
  $}

  $( Closed theorem form of ~ tpid1 .  (Contributed by Glauco Siliprandi,
     23-Oct-2021.) $)
  tpid1g $p |- ( A e. B -> A e. { A , C , D } ) $=
    ( wcel ctp wceq w3o eqid 3mix1i eltpg mpbiri ) ABEAACDFEAAGZACGZADGZHMNOAIJ
    AACDBKL $.

  ${
    tpid2.1 $e |- B e. _V $.
    $( One of the three elements of an unordered triple.  (Contributed by NM,
       7-Apr-1994.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    tpid2 $p |- B e. { A , B , C } $=
      ( ctp wcel wceq w3o eqid 3mix2i eltp mpbir ) BABCEFBAGZBBGZBCGZHNMOBIJBAB
      CDKL $.
  $}

  $( Closed theorem form of ~ tpid2 .  (Contributed by Glauco Siliprandi,
     23-Oct-2021.) $)
  tpid2g $p |- ( A e. B -> A e. { C , A , D } ) $=
    ( wcel ctp wceq w3o eqid 3mix2i eltpg mpbiri ) ABEACADFEACGZAAGZADGZHNMOAIJ
    ACADBKL $.

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x D $.
    $( Closed theorem form of ~ tpid3 .  (Contributed by Alan Sare,
       24-Oct-2011.) $)
    tpid3g $p |- ( A e. B -> A e. { C , D , A } ) $=
      ( vx wcel cv wceq wex ctp elisset w3o cab 3mix3 a1i abid imbitrrdi eleq2i
      wi dftp2 eleq1 mpbidi exlimdv mpd ) ABFZEGZAHZEIACDAJZFZEABKUEUGUIEUGUFUH
      FZUIUEUEUGUFUFCHZUFDHZUGLZEMZFZUJUEUGUMUOUGUMSUEUGUKULNOUMEPQUHUNUFECDATR
      QUFAUHUAUBUCUD $.
  $}

  ${
    tpid3.1 $e |- C e. _V $.
    $( One of the three elements of an unordered triple.  (Contributed by NM,
       7-Apr-1994.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    tpid3 $p |- C e. { A , B , C } $=
      ( ctp wcel wceq w3o eqid 3mix3i eltp mpbir ) CABCEFCAGZCBGZCCGZHOMNCIJCAB
      CDKL $.
  $}

  $( The singleton of a set is not empty.  It is also inhabited as shown at
     ~ snmg .  (Contributed by NM, 14-Dec-2008.) $)
  snnzg $p |- ( A e. V -> { A } =/= (/) ) $=
    ( wcel csn c0 wne snidg ne0i syl ) ABCAADZCJEFABGJAHI $.

  ${
    $d x A $.
    $( The singleton of a set is inhabited.  (Contributed by Jim Kingdon,
       11-Aug-2018.) $)
    snmg $p |- ( A e. V -> E. x x e. { A } ) $=
      ( wcel csn cv wex snidg elex2 syl ) BCDBBEZDAFKDAGBCHABKIJ $.
  $}

  ${
    $d x A $.
    snnz.1 $e |- A e. _V $.
    $( The singleton of a set is not empty.  It is also inhabited as shown at
       ~ snm .  (Contributed by NM, 10-Apr-1994.) $)
    snnz $p |- { A } =/= (/) $=
      ( cvv wcel csn c0 wne snnzg ax-mp ) ACDAEFGBACHI $.

    $( The singleton of a set is inhabited.  (Contributed by Jim Kingdon,
       11-Aug-2018.) $)
    snm $p |- E. x x e. { A } $=
      ( cvv wcel cv csn wex snmg ax-mp ) BDEAFBGEAHCABDIJ $.
  $}

  ${
    $d A x $.
    $( A singleton is inhabited iff its argument is a set.  (Contributed by
       Scott Fenton, 8-May-2018.)  (Revised by Jim Kingdon, 29-Dec-2025.) $)
    snmb $p |- ( A e. _V <-> E. x x e. { A } ) $=
      ( cvv wcel cv wceq wex csn isset velsn exbii bitr4i ) BCDAEZBFZAGMBHDZAGA
      BIONAABJKL $.
  $}

  ${
    $d x A $.  $d x B $.
    $( A pair containing a set is inhabited.  (Contributed by Jim Kingdon,
       21-Sep-2018.) $)
    prmg $p |- ( A e. V -> E. x x e. { A , B } ) $=
      ( wcel cv csn wex cpr snmg wceq wo orc velsn vex elpr 3imtr4i eximi syl )
      BDEAFZBGEZAHTBCIEZAHABDJUAUBATBKZUCTCKZLUAUBUCUDMABNTBCAOPQRS $.
  $}

  ${
    $d x A $.  $d x B $.
    prnz.1 $e |- A e. _V $.
    $( A pair containing a set is not empty.  It is also inhabited (see
       ~ prm ).  (Contributed by NM, 9-Apr-1994.) $)
    prnz $p |- { A , B } =/= (/) $=
      ( cpr wcel c0 wne prid1 ne0i ax-mp ) AABDZEKFGABCHKAIJ $.

    $( A pair containing a set is inhabited.  (Contributed by Jim Kingdon,
       21-Sep-2018.) $)
    prm $p |- E. x x e. { A , B } $=
      ( cvv wcel cv cpr wex prmg ax-mp ) BEFAGBCHFAIDABCEJK $.
  $}

  ${
    $d x A $.  $d x B $.
    $( A pair containing a set is not empty.  It is also inhabited (see
       ~ prmg ).  (Contributed by FL, 19-Sep-2011.) $)
    prnzg $p |- ( A e. V -> { A , B } =/= (/) ) $=
      ( vx cv cpr c0 wne wceq preq1 neeq1d vex prnz vtoclg ) DEZBFZGHABFZGHDACO
      AIPQGOABJKOBDLMN $.
  $}

  ${
    tpnz.1 $e |- A e. _V $.
    $( A triplet containing a set is not empty.  (Contributed by NM,
       10-Apr-1994.) $)
    tpnz $p |- { A , B , C } =/= (/) $=
      ( ctp wcel c0 wne tpid1 ne0i ax-mp ) AABCEZFLGHABCDILAJK $.
  $}

  ${
    $d A x $.  $d B x $.
    snssOLD.1 $e |- A e. _V $.
    $( Obsolete version of ~ snss as of 1-Jan-2025.  (Contributed by NM,
       5-Aug-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    snssOLD $p |- ( A e. B <-> { A } C_ B ) $=
      ( vx cv csn wcel wi wal wceq wss velsn imbi1i albii ssalel clel2 3bitr4ri
      ) DEZAFZGZRBGZHZDIRAJZUAHZDISBKABGUBUDDTUCUADALMNDSBODABCPQ $.
  $}

  $( Membership in a set with an element removed.  (Contributed by NM,
     10-Oct-2007.) $)
  eldifsn $p |- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) ) $=
    ( csn cdif wcel wn wa wne eldif elsng necon3bbid pm5.32i bitri ) ABCDZEFABF
    ZAOFZGZHPACIZHABOJPRSPQACACBKLMN $.

  $( Subset of a set with an element removed.  (Contributed by Emmett Weisz,
     7-Jul-2021.)  (Proof shortened by JJ, 31-May-2022.) $)
  ssdifsn $p |- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ -. C e. A ) ) $=
    ( csn cdif wss cin c0 wceq wcel difss2 reldisj bicomd biadan2 disjsn anbi2i
    wa wn bitri ) ABCDZEFZABFZATGHIZQUBCAJRZQUAUBUCABTKUBUCUAATBLMNUCUDUBACOPS
    $.

  $( Membership in a set with an element removed.  (Contributed by NM,
     10-Mar-2015.) $)
  eldifsni $p |- ( A e. ( B \ { C } ) -> A =/= C ) $=
    ( csn cdif wcel wne eldifsn simprbi ) ABCDEFABFACGABCHI $.

  $( ` A ` is not in ` ( B \ { A } ) ` .  (Contributed by David Moews,
     1-May-2017.) $)
  neldifsn $p |- -. A e. ( B \ { A } ) $=
    ( csn cdif wcel wne neirr eldifsni mto ) ABACDEAAFAGABAHI $.

  $( ` A ` is not in ` ( B \ { A } ) ` .  Deduction form.  (Contributed by
     David Moews, 1-May-2017.) $)
  neldifsnd $p |- ( ph -> -. A e. ( B \ { A } ) ) $=
    ( csn cdif wcel wn neldifsn a1i ) BCBDEFGABCHI $.

  $( Restricted existential quantification over a set with an element removed.
     (Contributed by NM, 4-Feb-2015.) $)
  rexdifsn $p |- ( E. x e. ( A \ { B } ) ph
      <-> E. x e. A ( x =/= B /\ ph ) ) $=
    ( cv wne wa csn cdif wcel eldifsn anbi1i anass bitri rexbii2 ) ABEZDFZAGZBC
    DHIZCPSJZAGPCJZQGZAGUARGTUBAPCDKLUAQAMNO $.

  $( A set is an element of the universal class excluding a singleton iff it is
     not the singleton element.  (Contributed by AV, 7-Apr-2019.) $)
  eldifvsn $p |- ( A e. V -> ( A e. ( _V \ { B } ) <-> A =/= B ) ) $=
    ( wcel cvv csn cdif wne wa eldifsn elex biantrurd bitr4id ) ACDZAEBFGDAEDZA
    BHZIPAEBJNOPACKLM $.

  ${
    $d x A $.  $d x B $.
    $( Characterization of the inclusion of a singleton in a class.
       (Contributed by BJ, 1-Jan-2025.) $)
    snssb $p |- ( { A } C_ B <-> ( A e. _V -> A e. B ) ) $=
      ( vx csn wss cv wcel wal wceq cvv ssalel velsn imbi1i albii eleq1 pm5.74i
      wi wex 19.23v 3bitri isset bicomi ) ADZBECFZUCGZUDBGZQZCHUDAIZUFQZCHZAJGZ
      ABGZQZCUCBKUGUICUEUHUFCALMNUJUHULQZCHUHCRZULQUMUIUNCUHUFULUDABOPNUHULCSUO
      UKULUKUOCAUAUBMTT $.
  $}

  $( The singleton formed on a set is included in a class if and only if the
     set is an element of that class.  Theorem 7.4 of [Quine] p. 49.
     (Contributed by NM, 22-Jul-2001.)  (Proof shortened by BJ, 1-Jan-2025.) $)
  snssg $p |- ( A e. V -> ( A e. B <-> { A } C_ B ) ) $=
    ( cvv wcel wi csn wss wb snssb bicomi elex imbibi mpsyl ) ADEZABEZFZAGBHZIA
    CEOPRIRQABJKACLOPRMN $.

  ${
    snss.1 $e |- A e. _V $.
    $( The singleton of an element of a class is a subset of the class
       (inference form of ~ snssg ).  Theorem 7.4 of [Quine] p. 49.
       (Contributed by NM, 21-Jun-1993.)  (Proof shortened by BJ,
       1-Jan-2025.) $)
    snss $p |- ( A e. B <-> { A } C_ B ) $=
      ( cvv wcel csn wss wb snssg ax-mp ) ADEABEAFBGHCABDIJ $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Obsolete version of ~ snssgOLD as of 1-Jan-2025.  (Contributed by NM,
       22-Jul-2001.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    snssgOLD $p |- ( A e. V -> ( A e. B <-> { A } C_ B ) ) $=
      ( vx cv wcel csn wss eleq1 wceq sneq sseq1d vex snss vtoclbg ) DEZBFPGZBH
      ABFAGZBHDACPABIPAJQRBPAKLPBDMNO $.

    $( An element not in a set can be removed without affecting the set.
       (Contributed by NM, 16-Mar-2006.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    difsn $p |- ( -. A e. B -> ( B \ { A } ) = B ) $=
      ( vx wcel wn csn cdif cv wne wa eldifsn simpl wceq eleq1 biimpcd necon3bd
      com12 ancld impbid2 bitrid eqrdv ) ABDZEZCBAFGZBCHZUDDUEBDZUEAIZJZUCUFUEB
      AKUCUHUFUFUGLUCUFUGUFUCUGUFUBUEAUEAMUFUBUEABNOPQRSTUA $.

    $( Removal of a singleton from an unordered pair.  (Contributed by NM,
       16-Mar-2006.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    difprsnss $p |- ( { A , B } \ { A } ) C_ { B } $=
      ( vx cpr csn cdif cv wcel wn wa wceq vex elpr velsn notbii biorf biimparc
      wo syl2anb eldif 3imtr4i ssriv ) CABDZAEZFZBEZCGZUCHZUGUDHZIZJUGBKZUGUEHU
      GUFHUHUGAKZUKRZULIZUKUJUGABCLMUIULCANOUNUKUMULUKPQSUGUCUDTCBNUAUB $.
  $}

  $( Removal of a singleton from an unordered pair.  (Contributed by Thierry
     Arnoux, 4-Feb-2017.) $)
  difprsn1 $p |- ( A =/= B -> ( { A , B } \ { A } ) = { B } ) $=
    ( wne cpr csn cdif wceq necom cun df-pr equncomi difeq1i difun2 cin disjsn2
    eqtri c0 disj3 sylib eqtr4id sylbir ) ABCBACZABDZAEZFZBEZGBAHUBUEUFUDFZUFUE
    UFUDIZUDFUGUCUHUDUCUDUFABJKLUFUDMPUBUFUDNQGUFUGGBAOUFUDRSTUA $.

  $( Removal of a singleton from an unordered pair.  (Contributed by Alexander
     van der Vekens, 5-Oct-2017.) $)
  difprsn2 $p |- ( A =/= B -> ( { A , B } \ { B } ) = { A } ) $=
    ( wne cpr csn cdif prcom difeq1i wceq necom difprsn1 sylbi eqtrid ) ABCZABD
    ZBEZFBADZPFZAEZOQPABGHNBACRSIABJBAKLM $.

  $( Removal of a singleton from an unordered triple.  (Contributed by
     Alexander van der Vekens, 5-Oct-2017.) $)
  diftpsn3 $p |- ( ( A =/= C /\ B =/= C )
                    -> ( { A , B , C } \ { C } ) = { A , B } ) $=
    ( wne wa ctp csn cdif cpr cun a1i c0 cin necom disjsn2 sylbi uneq12d eqtrdi
    wceq 3eqtrd df-tp difeq1d df-pr ineq1d incom indi eqtri adantr adantl unidm
    difundir disj3 sylib eqcomd difid un0 ) ACDZBCDZEZABCFZCGZHABIZVAJZVAHZVBVA
    HZVAVAHZJZVBUSUTVCVAUTVCSUSABCUAKUBVDVGSUSVBVAVAUKKUSVGVBLJVBUSVEVBVFLUSVBV
    EUSVBVAMZLSVBVESUSVHAGZBGZJZVAMZVAVIMZVAVJMZJZLUSVBVKVAVBVKSUSABUCKUDVLVOSU
    SVLVAVKMVOVKVAUEVAVIVJUFUGKUSVOLLJLUSVMLVNLUQVMLSZURUQCADVPACNCAOPUHURVNLSZ
    UQURCBDVQBCNCBOPUIQLUJRTVBVAULUMUNVFLSUSVAUOKQVBUPRT $.

  $( Removing two elements as pair of elements corresponds to removing each of
     the two elements as singletons.  (Contributed by Alexander van der Vekens,
     13-Jul-2018.) $)
  difpr $p |- ( A \ { B , C } ) = ( ( A \ { B } ) \ { C } ) $=
    ( cpr cdif csn cun df-pr difeq2i difun1 eqtri ) ABCDZEABFZCFZGZEAMENELOABCH
    IAMNJK $.

  $( ` ( B \ { A } ) ` equals ` B ` if and only if ` A ` is not a member of
     ` B ` .  Generalization of ~ difsn .  (Contributed by David Moews,
     1-May-2017.) $)
  difsnb $p |- ( -. A e. B <-> ( B \ { A } ) = B ) $=
    ( wcel wn csn cdif wceq difsn neldifsnd nelne1 mpdan necomd necon2bi impbii
    wne ) ABCZDBAEFZBGABHPQBPBQPAQCDBQOPABIABQJKLMN $.

  $( The singleton of an element of a class is a subset of the class.
     (Contributed by NM, 6-Jun-1994.) $)
  snssi $p |- ( A e. B -> { A } C_ B ) $=
    ( wcel csn wss snssg ibi ) ABCADBEABBFG $.

  ${
    snssd.1 $e |- ( ph -> A e. B ) $.
    $( The singleton of an element of a class is a subset of the class
       (deduction form).  (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
    snssd $p |- ( ph -> { A } C_ B ) $=
      ( wcel csn wss wb snssg syl mpbid ) ABCEZBFCGZDALLMHDBCCIJK $.
  $}

  $( If we remove a single element from a class then put it back in, we end up
     with a subset of the original class.  If equality is decidable, we can
     replace subset with equality as seen in ~ nndifsnid .  (Contributed by Jim
     Kingdon, 10-Aug-2018.) $)
  difsnss $p |- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A ) $=
    ( wcel csn cdif cun uncom wss snssi undifss sylib eqsstrid ) BACZABDZEZNFNO
    FZAONGMNAHPAHBAINAJKL $.

  $( Note that ` x ` is a dummy variable in the proof below. $)
  $( Compute the power set of the empty set.  Theorem 89 of [Suppes] p. 47.
     (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew Salmon,
     29-Jun-2011.) $)
  pw0 $p |- ~P (/) = { (/) } $=
    ( vx cv c0 wss cab wceq cpw csn ss0b abbii df-pw df-sn 3eqtr4i ) ABZCDZAENC
    FZAECGCHOPANIJACKACLM $.

  $( A singleton is a subset of an unordered pair containing its member.
     (Contributed by NM, 27-Aug-2004.) $)
  snsspr1 $p |- { A } C_ { A , B } $=
    ( csn cun cpr ssun1 df-pr sseqtrri ) ACZIBCZDABEIJFABGH $.

  $( A singleton is a subset of an unordered pair containing its member.
     (Contributed by NM, 2-May-2009.) $)
  snsspr2 $p |- { B } C_ { A , B } $=
    ( csn cun cpr ssun2 df-pr sseqtrri ) BCZACZIDABEIJFABGH $.

  $( A singleton is a subset of an unordered triple containing its member.
     (Contributed by NM, 9-Oct-2013.) $)
  snsstp1 $p |- { A } C_ { A , B , C } $=
    ( csn cpr cun ctp snsspr1 ssun1 sstri df-tp sseqtrri ) ADZABEZCDZFZABCGMNPA
    BHNOIJABCKL $.

  $( A singleton is a subset of an unordered triple containing its member.
     (Contributed by NM, 9-Oct-2013.) $)
  snsstp2 $p |- { B } C_ { A , B , C } $=
    ( csn cpr cun ctp snsspr2 ssun1 sstri df-tp sseqtrri ) BDZABEZCDZFZABCGMNPA
    BHNOIJABCKL $.

  $( A singleton is a subset of an unordered triple containing its member.
     (Contributed by NM, 9-Oct-2013.) $)
  snsstp3 $p |- { C } C_ { A , B , C } $=
    ( csn cpr cun ctp ssun2 df-tp sseqtrri ) CDZABEZKFABCGKLHABCIJ $.

  $( A pair is a subset of an unordered triple containing its members.
     (Contributed by Jim Kingdon, 11-Aug-2018.) $)
  prsstp12 $p |- { A , B } C_ { A , B , C } $=
    ( cpr csn cun ctp ssun1 df-tp sseqtrri ) ABDZKCEZFABCGKLHABCIJ $.

  $( A pair is a subset of an unordered triple containing its members.
     (Contributed by Jim Kingdon, 11-Aug-2018.) $)
  prsstp13 $p |- { A , C } C_ { A , B , C } $=
    ( cpr ctp prsstp12 tpcomb sseqtrri ) ACDACBEABCEACBFABCGH $.

  $( A pair is a subset of an unordered triple containing its members.
     (Contributed by Jim Kingdon, 11-Aug-2018.) $)
  prsstp23 $p |- { B , C } C_ { A , B , C } $=
    ( cpr ctp prsstp12 tprot sseqtrri ) BCDBCAEABCEBCAFABCGH $.

  ${
    prss.1 $e |- A e. _V $.
    prss.2 $e |- B e. _V $.
    $( A pair of elements of a class is a subset of the class.  Theorem 7.5 of
       [Quine] p. 49.  (Contributed by NM, 30-May-1994.)  (Proof shortened by
       Andrew Salmon, 29-Jun-2011.) $)
    prss $p |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) $=
      ( csn wss wa cun wcel cpr unss snss anbi12i df-pr sseq1i 3bitr4i ) AFZCGZ
      BFZCGZHRTIZCGACJZBCJZHABKZCGRTCLUCSUDUAACDMBCEMNUEUBCABOPQ $.
  $}

  $( A pair of elements of a class is a subset of the class.  Theorem 7.5 of
     [Quine] p. 49.  (Contributed by NM, 22-Mar-2006.)  (Proof shortened by
     Andrew Salmon, 29-Jun-2011.) $)
  prssg $p |- ( ( A e. V /\ B e. W )
              -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) ) $=
    ( wcel wa csn wss cpr snssg bi2anan9 cun unss df-pr sseq1i bitr4i bitrdi )
    ADFZBEFZGACFZBCFZGAHZCIZBHZCIZGZABJZCIZSUAUDTUBUFACDKBCEKLUGUCUEMZCIUIUCUEC
    NUHUJCABOPQR $.

  $( A pair of elements of a class is a subset of the class.  (Contributed by
     NM, 16-Jan-2015.) $)
  prssi $p |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C ) $=
    ( wcel wa cpr wss prssg ibi ) ACDBCDEABFCGABCCCHI $.

  ${
    prssd.1 $e |- ( ph -> A e. C ) $.
    prssd.2 $e |- ( ph -> B e. C ) $.
    $( Deduction version of ~ prssi :  A pair of elements of a class is a
       subset of the class.  (Contributed by Glauco Siliprandi,
       17-Aug-2020.) $)
    prssd $p |- ( ph -> { A , B } C_ C ) $=
      ( wcel cpr wss prssi syl2anc ) ABDGCDGBCHDIEFBCDJK $.
  $}

  $( An unordered pair belongs to the power class of a class iff each member
     belongs to the class.  (Contributed by Thierry Arnoux, 3-Oct-2016.)
     (Revised by NM, 18-Jan-2018.) $)
  prsspwg $p |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ ~P C
      <-> ( A C_ C /\ B C_ C ) ) ) $=
    ( wcel wa cpw cpr wss prssg elpwg bi2anan9 bitr3d ) ADFZBEFZGACHZFZBQFZGABI
    QJACJZBCJZGABQDEKORTPSUAACDLBCELMN $.

  $( A pair as subset of a pair.  (Contributed by AV, 26-Oct-2020.) $)
  ssprss $p |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ { C , D }
                        <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) ) $=
    ( wcel wa cpr wss wceq wo prssg elprg bi2anan9 bitr3d ) AEGZBFGZHACDIZGZBSG
    ZHABISJACKADKLZBCKBDKLZHABSEFMQTUBRUAUCACDENBCDFNOP $.

  $( A proper pair is a subset of a pair iff it is equal to the superset.
     (Contributed by AV, 26-Oct-2020.) $)
  ssprsseq $p |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D }
                                               <-> { A , B } = { C , D } ) ) $=
    ( wcel wne w3a cpr wceq wo wa wi eqtr3 syl11 3ad2ant3 com12 preq12 a1d wss
    wb ssprss 3adant3 eqneqall prcom eqtrdi ccase sylbid eqimss impbid1 ) AEGZB
    FGZABHZIZABJZCDJZUAZUPUQKZUOURACKZADKZLBCKZBDKZLMZUSULUMURVDUBUNABCDEFUCUDV
    DUOUSUTVBVAVCUOUSNUOUTVBMZUSUNULVEUSNUMABKZUNUSVEUSABUEZABCOPQRVAVBMZUSUOVH
    UPDCJUQABDCSDCUFUGTUTVCMUSUOABCDSTUOVAVCMZUSUNULVIUSNUMVFUNUSVIVGABDOPQRUHR
    UIUPUQUJUK $.

  ${
    $d x A $.  $d x B $.
    $( Empty set and the singleton itself are subsets of a singleton.
       Concerning the converse, see ~ exmidsssn .  (Contributed by Jim Kingdon,
       10-Aug-2018.) $)
    sssnr $p |- ( ( A = (/) \/ A = { B } ) -> A C_ { B } ) $=
      ( c0 wceq csn wss 0ss sseq1 mpbiri eqimss jaoi ) ACDZABEZFZAMDLNCMFMGACMH
      IAMJK $.

    $( The inhabited subset of a singleton.  (Contributed by Jim Kingdon,
       10-Aug-2018.) $)
    sssnm $p |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) ) $=
      ( cv wcel wex csn wss wceq wa wb ssel elsni syl6 eleq1 ibd exlimdv anc2li
      snssi eqss imbitrrdi com12 eqimss impbid1 ) ADZBEZAFZBCGZHZBUHIZUIUGUJUIU
      GUIUHBHZJUJUIUGUKUIUGCBEZUKUIUFULAUIUFULUIUFUECIZUFULKUIUFUEUHEUMBUHUELUE
      CMNUECBONPQCBSNRBUHTUAUBBUHUCUD $.

    $( Two ways to express that an inhabited set equals a singleton.
       (Contributed by Jim Kingdon, 11-Aug-2018.) $)
    eqsnm $p |- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) ) $=
      ( cv wcel wex csn wss wceq wral sssnm dfss3 velsn ralbii bitri bitr3di )
      ADZBEAFBCGZHZBRIQCIZABJZABCKSQREZABJUAABRLUBTABACMNOP $.
  $}

  $( The subsets of a pair.  (Contributed by Jim Kingdon, 11-Aug-2018.) $)
  ssprr $p |- (
      ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) )
      -> A C_ { B , C } ) $=
    ( c0 wceq csn wo cpr wss 0ss sseq1 mpbiri snsspr1 jaoi snsspr2 eqimss ) ADE
    ZABFZEZGABCHZIZACFZEZATEZGQUASQUADTITJADTKLSUARTIBCMARTKLNUCUAUDUCUAUBTIBCO
    AUBTKLATPNN $.

  $( The subsets of a triple.  (Contributed by Jim Kingdon, 11-Aug-2018.) $)
  sstpr $p |- (
      ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/
        ( ( A = { D } \/ A = { B , D } ) \/
          ( A = { C , D } \/ A = { B , C , D } ) ) )
      -> A C_ { B , C , D } ) $=
    ( c0 csn wo cpr ctp wss ssprr prsstp12 sstrdi snsstp3 sseq1 mpbiri prsstp13
    wceq jaoi prsstp23 eqimss ) AERABFRGACFRABCHZRGGZABCDIZJZADFZRZABDHZRZGZACD
    HZRZAUDRZGZGUCAUBUDABCKBCDLMUJUEUNUGUEUIUGUEUFUDJBCDNAUFUDOPUIUEUHUDJBCDQAU
    HUDOPSULUEUMULUEUKUDJBCDTAUKUDOPAUDUASSS $.

  ${
    tpss.1 $e |- A e. _V $.
    tpss.2 $e |- B e. _V $.
    tpss.3 $e |- C e. _V $.
    $( A triplet of elements of a class is a subset of the class.  (Contributed
       by NM, 9-Apr-1994.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    tpss $p |- ( ( A e. D /\ B e. D /\ C e. D ) <-> { A , B , C } C_ D ) $=
      ( cpr wss csn wa cun wcel w3a ctp unss df-3an prss snss anbi12i 3bitr4i
      bitri df-tp sseq1i ) ABHZDIZCJZDIZKZUEUGLZDIADMZBDMZCDMZNZABCOZDIUEUGDPUN
      UKULKZUMKUIUKULUMQUPUFUMUHABDEFRCDGSTUBUOUJDABCUCUDUA $.
  $}

  $( A triple of elements of a class is a subset of the class.  (Contributed by
     Alexander van der Vekens, 1-Feb-2018.) $)
  tpssi $p |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D ) $=
    ( w3a ctp cpr csn cun df-tp wss prssi 3adant3 snssi 3ad2ant3 unssd eqsstrid
    wcel ) ADRZBDRZCDRZEZABCFABGZCHZIDABCJUBUCUDDSTUCDKUAABDLMUASUDDKTCDNOPQ $.

  ${
    $d x A $.  $d x B $.
    sneqr.1 $e |- A e. _V $.
    $( If the singletons of two sets are equal, the two sets are equal.  Part
       of Exercise 4 of [TakeutiZaring] p. 15.  (Contributed by NM,
       27-Aug-1993.) $)
    sneqr $p |- ( { A } = { B } -> A = B ) $=
      ( csn wceq wcel snid eleq2 mpbii elsn sylib ) ADZBDZEZAMFZABENALFOACGLMAH
      IABCJK $.

    $( If a singleton is a subset of another, their members are equal.
       (Contributed by NM, 28-May-2006.) $)
    snsssn $p |- ( { A } C_ { B } -> A = B ) $=
      ( vx csn wss cv wceq wi wal wcel ssalel velsn imbi12i albii bitri sbceqal
      cvv ax-mp sylbi ) AEZBEZFZDGZAHZUDBHZIZDJZABHZUCUDUAKZUDUBKZIZDJUHDUAUBLU
      LUGDUJUEUKUFDAMDBMNOPARKUHUIICDABRQST $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Closed form of ~ sneqr .  (Contributed by Scott Fenton, 1-Apr-2011.) $)
    sneqrg $p |- ( A e. V -> ( { A } = { B } -> A = B ) ) $=
      ( vx cv csn wceq wi sneq eqeq1d eqeq1 imbi12d vex sneqr vtoclg ) DEZFZBFZ
      GZPBGZHAFZRGZABGZHDACPAGZSUBTUCUDQUARPAIJPABKLPBDMNO $.
  $}

  $( Two singletons of sets are equal iff their elements are equal.
     (Contributed by Scott Fenton, 16-Apr-2012.) $)
  sneqbg $p |- ( A e. V -> ( { A } = { B } <-> A = B ) ) $=
    ( wcel csn wceq sneqrg sneq impbid1 ) ACDAEBEFABFABCGABHI $.

  ${
    $d x A $.
    $( The singleton of a class is a subset of its power class.  (Contributed
       by NM, 5-Aug-1993.) $)
    snsspw $p |- { A } C_ ~P A $=
      ( vx csn cpw cv wceq wss wcel eqimss velsn df-pw abeq2i 3imtr4i ssriv ) B
      ACZADZBEZAFQAGZQOHQPHQAIBAJRBPBAKLMN $.
  $}

  ${
    prsspw.1 $e |- A e. _V $.
    prsspw.2 $e |- B e. _V $.
    $( An unordered pair belongs to the power class of a class iff each member
       belongs to the class.  (Contributed by NM, 10-Dec-2003.)  (Proof
       shortened by Andrew Salmon, 26-Jun-2011.) $)
    prsspw $p |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) $=
      ( cpr cpw wss wcel wa prss elpw anbi12i bitr3i ) ABFCGZHAOIZBOIZJACHZBCHZ
      JABODEKPRQSACDLBCELMN $.
  $}

  $( Reverse equality lemma for unordered pairs.  If two unordered pairs have
     the same second element, the first elements are equal.  Closed form of
     ~ preqr1 .  (Contributed by Jim Kingdon, 21-Sep-2018.) $)
  preqr1g $p |- ( ( A e. _V /\ B e. _V ) ->
      ( { A , C } = { B , C } -> A = B ) ) $=
    ( cvv wcel wa cpr wceq wo wi prid1g eleq2 syl5ibcom elprg sylibd adantr imp
    syl5ibrcom adantl eqcom eqeq2 oplem1 ex ) ADEZBDEZFZACGZBCGZHZABHZUFUIFUJAC
    HZBAHZBCHZUFUIUJUKIZUDUIUNJUEUDUIAUHEZUNUDAUGEUIUOACDKUGUHALMABCDNOPQUFUIUL
    UMIZUEUIUPJUDUEUIBUGEZUPUEUQUIBUHEBCDKUGUHBLRBACDNOSQABTACBUAUBUC $.

  $( Reverse equality lemma for unordered pairs.  If two unordered pairs have
     the same second element, the second elements are equal.  Closed form of
     ~ preqr2 .  (Contributed by Jim Kingdon, 21-Sep-2018.) $)
  preqr2g $p |- ( ( A e. _V /\ B e. _V ) ->
      ( { C , A } = { C , B } -> A = B ) ) $=
    ( cpr wceq cvv wcel wa prcom eqeq12i preqr1g biimtrid ) CADZCBDZEACDZBCDZEA
    FGBFGHABEMONPCAICBIJABCKL $.

  ${
    preqr1.1 $e |- A e. _V $.
    preqr1.2 $e |- B e. _V $.
    $( Reverse equality lemma for unordered pairs.  If two unordered pairs have
       the same second element, the first elements are equal.  (Contributed by
       NM, 18-Oct-1995.) $)
    preqr1 $p |- ( { A , C } = { B , C } -> A = B ) $=
      ( cpr wceq wcel wo prid1 eleq2 mpbii elpr sylib mpbiri eqcom eqeq2 oplem1
      ) ACFZBCFZGZABGZACGZBAGZBCGZUAATHZUBUCIUAASHUFACDJSTAKLABCDMNUABSHZUDUEIU
      AUGBTHBCEJSTBKOBACEMNABPACBQR $.
  $}

  ${
    preqr2.1 $e |- A e. _V $.
    preqr2.2 $e |- B e. _V $.
    $( Reverse equality lemma for unordered pairs.  If two unordered pairs have
       the same first element, the second elements are equal.  (Contributed by
       NM, 5-Aug-1993.) $)
    preqr2 $p |- ( { C , A } = { C , B } -> A = B ) $=
      ( cpr wceq prcom eqeq12i preqr1 sylbi ) CAFZCBFZGACFZBCFZGABGLNMOCAHCBHIA
      BCDEJK $.
  $}

  ${
    preq12b.1 $e |- A e. _V $.
    preq12b.2 $e |- B e. _V $.
    preq12b.3 $e |- C e. _V $.
    preq12b.4 $e |- D e. _V $.
    $( Equality relationship for two unordered pairs.  (Contributed by NM,
       17-Oct-1996.) $)
    preq12b $p |- ( { A , B } = { C , D } <->
                   ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) $=
      ( cpr wceq wa wo wcel preq1 eqeq1d preqr2 biimtrdi com12 ancld prcom elpr
      prid1 eleq2 mpbii sylib wi eqeq2i orim12d mpd preq12 eqtrdi sylan9eq jaoi
      sylbi impbii ) ABIZCDIZJZACJZBDJZKZADJZBCJZKZLZURUSVBLZVEURAUQMZVFURAUPMV
      GABEUBUPUQAUCUDACDEUAUEURUSVAVBVDURUSUTUSURUTUSURCBIZUQJUTUSUPVHUQACBNOBD
      CFHPQRSURVBVCURUPDCIZJZVBVCUFUQVIUPCDTUGVBVJVCVBVJDBIZVIJVCVBUPVKVIADBNZO
      BCDFGPQRUNSUHUIVAURVDABCDUJVBVCUPBDIZUQVBUPVKVMVLDBTUKBCDNULUMUO $.

    $( Equality of two unordered pairs.  (Contributed by NM, 17-Oct-1996.) $)
    prel12 $p |- ( -. A = B -> ( { A , B } = { C , D } <->
                   ( A e. { C , D } /\ B e. { C , D } ) ) ) $=
      ( wceq wn cpr wcel wa eleq2 mpbii wo wi elpr eqeq2 notbid prid1 prid2 jca
      orel2 biimtrdi com3l imp ancrd orel1 orim12d orcom bitri preq12b biimtrid
      3imtr4g ex impd impbid2 ) ABIZJZABKZCDKZIZAVBLZBVBLZMVCVDVEVCAVALVDABEUAV
      AVBANOVCBVALVEABFUBVAVBBNOUCUTVDVEVCVDACIZADIZPZUTVEVCQZACDERUTVHVIUTVHMZ
      BDIZBCIZPZVFVKMZVGVLMZPVEVCVJVKVNVLVOVJVKVFUTVHVKVFQVKUTVHVFVKUTVGJVHVFQV
      KUSVGBDASTVGVFUDUEUFUGUHVJVLVGUTVHVLVGQVLUTVHVGVLUTVFJVHVGQVLUSVFBCASTVFV
      GUIUEUFUGUHUJVEVLVKPVMBCDFRVLVKUKULABCDEFGHUMUOUPUNUQUR $.

    $( A way to represent ordered pairs using unordered pairs with distinct
       members.  (Contributed by NM, 27-Mar-2007.) $)
    opthpr $p |- ( A =/= D ->
                 ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) ) $=
      ( cpr wceq wa wo wne preq12b idd wn wi df-ne pm2.21 sylbi impd orc bitrid
      jaod impbid1 ) ABICDIJACJBDJKZADJZBCJZKZLZADMZUFABCDEFGHNUKUJUFUKUFUFUIUK
      UFOUKUGUHUFUKUGPUGUHUFQZQADRUGULSTUAUDUFUIUBUEUC $.
  $}

  ${
    $d A x y z w $.  $d B x y z w $.  $d C x y z w $.  $d D x y z w $.
    $d V x y z w $.  $d W x y z w $.  $d X x y z w $.  $d Y x y z w $.
    $( Closed form of ~ preq12b .  (Contributed by Scott Fenton,
       28-Mar-2014.) $)
    preq12bg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) ->
       ( { A , B } = { C , D } <->
         ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) ) $=
      ( vx vy vz vw wcel wa cpr wceq wo wb wi cv weq preq1 eqeq1d eqeq1 orbi12d
      anbi1d bibi12d imbi2d preq2 anbi2d eqeq2 w3a vex preq12b vtoclbg vtocl3ga
      eqeq2d a1i 3expa impr ) AEMZBFMZNCGMZDHMZABOZCDOZPZACPZBDPZNZADPZBCPZNZQZ
      RZVAVBVCVDVOSZVDITZJTZOZKTZDOZPZIKUAZVRDPZNZVQDPZJKUAZNZQZRZSZVDAVROZWAPZ
      AVTPZWDNZVKWGNZQZRZSVDVEWAPZWNVINZVKBVTPZNZQZRZSVPIJKABCEFGVQAPZWJWRVDXEW
      BWMWIWQXEVSWLWAVQAVRUBUCXEWEWOWHWPXEWCWNWDVQAVTUDUFXEWFVKWGVQADUDUFUEUGUH
      VRBPZWRXDVDXFWMWSWQXCXFWLVEWAVRBAUIUCXFWOWTWPXBXFWDVIWNVRBDUDUJXFWGXAVKVR
      BVTUDUJUEUGUHVTCPZXDVOVDXGWSVGXCVNXGWAVFVEVTCDUBUQXGWTVJXBVMXGWNVHVIVTCAU
      KUFXGXAVLVKVTCBUKUJUEUGUHWKVQEMVRFMVTGMULVSVTLTZOZPWCJLUAZNZILUAZWGNZQWBW
      ILDHXHDPZXIWAVSXHDVTUIUQXNXKWEXMWHXNXJWDWCXHDVRUKUJXNXLWFWGXHDVQUKUFUEVQV
      RVTXHIUMJUMKUMLUMUNUOURUPUSUT $.
  $}

  $( Two pairs are not equal if at least one element of the first pair is not
     contained in the second pair.  (Contributed by Alexander van der Vekens,
     13-Aug-2017.) $)
  prneimg $p |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) )
                  -> ( ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) )
                       -> { A , B } =/= { C , D } ) ) $=
    ( wcel wa wne wo cpr wceq wn preq12bg oranim df-ne anbi12i sylnibr ad2antll
    orddi simpll pm1.4 jca sylbi biimtrdi anim12i syl6 pm4.56 imbitrdi necon2ad
    ) AEIBFIJCGIDHIJJZACKZADKZJZBCKZBDKZJZLZABMZCDMZUMVAVBNZUPOZUSOZJZUTOUMVCAC
    NZADNZLZBCNZBDNZLZJZVFUMVCVGVKJVHVJJLZVMABCDEFGHPVNVIVGVJLZJZVKVHLZVKVJLZJZ
    JZVMVGVKVHVJUBVTVIVLVIVOVSUCVRVLVPVQVKVJUDUAUEUFUGVIVDVLVEVIVGOZVHOZJUPVGVH
    QUNWAUOWBACRADRSTVLVJOZVKOZJUSVJVKQUQWCURWDBCRBDRSTUHUIUPUSUJUKUL $.

  ${
    preqsn.1 $e |- A e. _V $.
    preqsn.2 $e |- B e. _V $.
    preqsn.3 $e |- C e. _V $.
    $( Equivalence for a pair equal to a singleton.  (Contributed by NM,
       3-Jun-2008.) $)
    preqsn $p |- ( { A , B } = { C } <-> ( A = B /\ B = C ) ) $=
      ( cpr csn wceq wa dfsn2 eqeq2i preq12b oridm eqtr3 simpr jca eqtr bitri
      wo impbii ) ABGZCHZIUBCCGZIZABIZBCIZJZUCUDUBCKLUEACIZUGJZUJTZUHABCCDEFFMU
      KUJUHUJNUJUHUJUFUGABCOUIUGPQUHUIUGABCRUFUGPQUASSS $.
  $}

  ${
    $d A b $.  $d V b $.  $d X b $.  $d Y b $.
    $( For an element ` A ` of an unordered pair which is a subset of a given
       set ` V ` , there is another (maybe the same) element ` b ` of the given
       set ` V ` being an element of the unordered pair.  (Contributed by AV,
       5-Dec-2020.) $)
    elpr2elpr $p |- ( ( X e. V /\ Y e. V /\ A e. { X , Y } )
                      -> E. b e. V { X , Y } = { A , b } ) $=
      ( wcel cpr cv wceq wrex wo wa wi simprr preq12 eqcomd adantlr rspcedeq2vd
      ex simprl prcom eqtr2di jaoi elpri syl11 3impia ) CBFZDBFZACDGZFZUIAEHZGZ
      IZEBJZACIZADIZKUGUHLZUNUJUOUQUNMUPUOUQUNUOUQLEDBUIULUOUGUHNUOUKDIZUMUQUOU
      RLULUIAUKCDOPQRSUPUQUNUPUQLECBUIULUPUGUHTUPUKCIZUMUQUPUSLULDCGUIAUKDCODCU
      AUBQRSUCACDUDUEUF $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Value of the ordered pair when the arguments are sets.  (Contributed by
       Mario Carneiro, 26-Apr-2015.) $)
    dfopg $p |- ( ( A e. V /\ B e. W ) ->
      <. A , B >. = { { A } , { A , B } } ) $=
      ( vx wcel cvv cop csn cpr wceq elex wa cab w3a df-3an baibr abbidv abid2
      cv df-op eqcomi 3eqtr3g eqcomd syl2an ) ACFAGFZBGFZABHZAIABJJZKBDFACLBDLU
      FUGMZUIUHUJETUIFZENUFUGUKOZENZUIUHUJUKULEULUJUKUFUGUKPQREUISUHUMEABUAUBUC
      UDUE $.
  $}

  ${
    dfop.1 $e |- A e. _V $.
    dfop.2 $e |- B e. _V $.
    $( Value of an ordered pair when the arguments are sets, with the
       conclusion corresponding to Kuratowski's original definition.
       (Contributed by NM, 25-Jun-1998.) $)
    dfop $p |- <. A , B >. = { { A } , { A , B } } $=
      ( cvv wcel cop csn cpr wceq dfopg mp2an ) AEFBEFABGAHABIIJCDABEEKL $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Equality theorem for ordered pairs.  (Contributed by NM, 25-Jun-1998.)
       (Revised by Mario Carneiro, 26-Apr-2015.) $)
    opeq1 $p |- ( A = B -> <. A , C >. = <. B , C >. ) $=
      ( vx wceq cvv wcel cv csn cpr w3a cab cop eleq1 anbi1d preq1 df-3an df-op
      wa sneq preq12d eleq2d anbi12d 3bitr4g abbidv 3eqtr4g ) ABEZAFGZCFGZDHZAI
      ZACJZJZGZKZDLBFGZUIUJBIZBCJZJZGZKZDLACMBCMUGUOVADUGUHUISZUNSUPUISZUTSUOVA
      UGVBVCUNUTUGUHUPUIABFNOUGUMUSUJUGUKUQULURABTABCPUAUBUCUHUIUNQUPUIUTQUDUED
      ACRDBCRUF $.

    $( Equality theorem for ordered pairs.  (Contributed by NM, 25-Jun-1998.)
       (Revised by Mario Carneiro, 26-Apr-2015.) $)
    opeq2 $p |- ( A = B -> <. C , A >. = <. C , B >. ) $=
      ( vx wceq cvv wcel cv csn cpr w3a cab cop eleq1 anbi2d eqidd preq2 df-3an
      wa df-op preq12d eleq2d anbi12d 3bitr4g abbidv 3eqtr4g ) ABEZCFGZAFGZDHZC
      IZCAJZJZGZKZDLUHBFGZUJUKCBJZJZGZKZDLCAMCBMUGUOUTDUGUHUISZUNSUHUPSZUSSUOUT
      UGVAVBUNUSUGUIUPUHABFNOUGUMURUJUGUKUKULUQUGUKPABCQUAUBUCUHUIUNRUHUPUSRUDU
      EDCATDCBTUF $.
  $}

  $( Equality theorem for ordered pairs.  (Contributed by NM, 28-May-1995.) $)
  opeq12 $p |- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. ) $=
    ( wceq cop opeq1 opeq2 sylan9eq ) ACEBDEABFCBFCDFACBGBDCHI $.

  ${
    opeq1i.1 $e |- A = B $.
    $( Equality inference for ordered pairs.  (Contributed by NM,
       16-Dec-2006.) $)
    opeq1i $p |- <. A , C >. = <. B , C >. $=
      ( wceq cop opeq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality inference for ordered pairs.  (Contributed by NM,
       16-Dec-2006.) $)
    opeq2i $p |- <. C , A >. = <. C , B >. $=
      ( wceq cop opeq2 ax-mp ) ABECAFCBFEDABCGH $.

    ${
      opeq12i.2 $e |- C = D $.
      $( Equality inference for ordered pairs.  (Contributed by NM,
         16-Dec-2006.)  (Proof shortened by Eric Schmidt, 4-Apr-2007.) $)
      opeq12i $p |- <. A , C >. = <. B , D >. $=
        ( wceq cop opeq12 mp2an ) ABGCDGACHBDHGEFACBDIJ $.
    $}
  $}

  ${
    opeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for ordered pairs.  (Contributed by NM,
       16-Dec-2006.) $)
    opeq1d $p |- ( ph -> <. A , C >. = <. B , C >. ) $=
      ( wceq cop opeq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Equality deduction for ordered pairs.  (Contributed by NM,
       16-Dec-2006.) $)
    opeq2d $p |- ( ph -> <. C , A >. = <. C , B >. ) $=
      ( wceq cop opeq2 syl ) ABCFDBGDCGFEBCDHI $.

    opeq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality deduction for ordered pairs.  (Contributed by NM, 16-Dec-2006.)
       (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    opeq12d $p |- ( ph -> <. A , C >. = <. B , D >. ) $=
      ( wceq cop opeq12 syl2anc ) ABCHDEHBDICEIHFGBDCEJK $.
  $}

  $( Equality theorem for ordered triples.  (Contributed by NM, 3-Apr-2015.) $)
  oteq1 $p |- ( A = B -> <. A , C , D >. = <. B , C , D >. ) $=
    ( wceq cop cotp opeq1 opeq1d df-ot 3eqtr4g ) ABEZACFZDFBCFZDFACDGBCDGLMNDAB
    CHIACDJBCDJK $.

  $( Equality theorem for ordered triples.  (Contributed by NM, 3-Apr-2015.) $)
  oteq2 $p |- ( A = B -> <. C , A , D >. = <. C , B , D >. ) $=
    ( wceq cop cotp opeq2 opeq1d df-ot 3eqtr4g ) ABEZCAFZDFCBFZDFCADGCBDGLMNDAB
    CHICADJCBDJK $.

  $( Equality theorem for ordered triples.  (Contributed by NM, 3-Apr-2015.) $)
  oteq3 $p |- ( A = B -> <. C , D , A >. = <. C , D , B >. ) $=
    ( wceq cop cotp opeq2 df-ot 3eqtr4g ) ABECDFZAFKBFCDAGCDBGABKHCDAICDBIJ $.

  ${
    oteq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for ordered triples.  (Contributed by Mario Carneiro,
       11-Jan-2017.) $)
    oteq1d $p |- ( ph -> <. A , C , D >. = <. B , C , D >. ) $=
      ( wceq cotp oteq1 syl ) ABCGBDEHCDEHGFBCDEIJ $.

    $( Equality deduction for ordered triples.  (Contributed by Mario Carneiro,
       11-Jan-2017.) $)
    oteq2d $p |- ( ph -> <. C , A , D >. = <. C , B , D >. ) $=
      ( wceq cotp oteq2 syl ) ABCGDBEHDCEHGFBCDEIJ $.

    $( Equality deduction for ordered triples.  (Contributed by Mario Carneiro,
       11-Jan-2017.) $)
    oteq3d $p |- ( ph -> <. C , D , A >. = <. C , D , B >. ) $=
      ( wceq cotp oteq3 syl ) ABCGDEBHDECHGFBCDEIJ $.

    oteq123d.2 $e |- ( ph -> C = D ) $.
    oteq123d.3 $e |- ( ph -> E = F ) $.
    $( Equality deduction for ordered triples.  (Contributed by Mario Carneiro,
       11-Jan-2017.) $)
    oteq123d $p |- ( ph -> <. A , C , E >. = <. B , D , F >. ) $=
      ( cotp oteq1d oteq2d oteq3d 3eqtrd ) ABDFKCDFKCEFKCEGKABCDFHLADECFIMAFGCE
      JNO $.
  $}

  ${
    $d y A $.  $d y B $.  $d y x $.
    nfop.1 $e |- F/_ x A $.
    nfop.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for ordered pairs.  (Contributed by
       NM, 14-Nov-1995.) $)
    nfop $p |- F/_ x <. A , B >. $=
      ( vy cop cvv wcel cv csn cpr w3a cab df-op nfel1 nfsn nfpr nfcri nf3an
      nfab nfcxfr ) ABCGBHIZCHIZFJBKZBCLZLZIZMZFNFBCOUIAFUCUDUHAABHDPACHEPAFUGA
      UEUFABDQABCDERRSTUAUB $.
  $}

  ${
    $d z B $.  $d z A $.  $d x z $.
    nfopd.2 $e |- ( ph -> F/_ x A ) $.
    nfopd.3 $e |- ( ph -> F/_ x B ) $.
    $( Deduction version of bound-variable hypothesis builder ~ nfop .  This
       shows how the deduction version of a not-free theorem such as ~ nfop can
       be created from the corresponding not-free inference theorem.
       (Contributed by NM, 4-Feb-2008.) $)
    nfopd $p |- ( ph -> F/_ x <. A , B >. ) $=
      ( vz cv wcel wal cab cop wnfc nfaba1 nfop wb wa nfnfc1 wceq abidnf adantr
      nfan adantl opeq12d nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUHDIZBJGKZLZMZBC
      DLZMZBUJULUIBGNUKBGNOABCMZBDMZUNUPPEFUQURQZBUMUOUQURBBCRBDRUBUSUJCULDUQUJ
      CSURBGCTUAURULDSUQBGDTUCUDUEUFUG $.
  $}

  ${
    opid.1 $e |- A e. _V $.
    $( The ordered pair ` <. A , A >. ` in Kuratowski's representation.
       (Contributed by FL, 28-Dec-2011.) $)
    opid $p |- <. A , A >. = { { A } } $=
      ( csn cpr cop dfsn2 eqcomi preq2i dfop 3eqtr4i ) ACZAADZDKKDAAEKCLKKKLAFG
      HAABBIKFJ $.
  $}

  ${
    $d B x $.  $d ps x $.
    ralunsn.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
    $( Restricted quantification over the union of a set and a singleton, using
       implicit substitution.  (Contributed by Paul Chapman, 17-Nov-2012.)
       (Revised by Mario Carneiro, 23-Apr-2015.) $)
    ralunsn $p |- ( B e. C -> ( A. x e. ( A u. { B } ) ph <->
                                ( A. x e. A ph /\ ps ) ) ) $=
      ( csn cun wral wa wcel ralunb ralsng anbi2d bitrid ) ACDEHZIJACDJZACQJZKE
      FLZRBKACDQMTSBRABCEFGNOP $.
  $}

  ${
    $d A x $.  $d B x y $.  $d C x $.  $d ch x $.  $d ps y $.  $d th x $.
    2ralunsn.1 $e |- ( x = B -> ( ph <-> ch ) ) $.
    2ralunsn.2 $e |- ( y = B -> ( ph <-> ps ) ) $.
    2ralunsn.3 $e |- ( x = B -> ( ps <-> th ) ) $.
    $( Double restricted quantification over the union of a set and a
       singleton, using implicit substitution.  (Contributed by Paul Chapman,
       17-Nov-2012.) $)
    2ralunsn $p |- ( B e. C ->
                     ( A. x e. ( A u. { B } ) A. y e. ( A u. { B } ) ph <->
                       ( ( A. x e. A A. y e. A ph /\ A. x e. A ps ) /\
                         ( A. y e. A ch /\ th ) ) ) ) $=
      ( wcel csn cun wral wa ralunsn ralbidv cv wceq r19.26 anbi1i bitrdi bitrd
      anbi12d ) HIMZAFGHNOZPZEUHPAFGPZBQZEUHPZUJEGPBEGPQZCFGPZDQZQZUGUIUKEUHABF
      GHIKRSUGULUKEGPZUOQUPUKUOEGHIETHUAZUJUNBDURACFGJSLUFRUQUMUOUJBEGUBUCUDUE
      $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Expansion of an ordered pair when either member is a proper class.
       (Contributed by Mario Carneiro, 26-Apr-2015.) $)
    opprc $p |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) ) $=
      ( vx cvv wcel wa wn cop cv csn cpr w3a cab c0 df-op wal wceq 3simpa con3i
      alrimiv abeq0 sylibr eqtrid ) ADEZBDEZFZGZABHUDUECIAJABKKEZLZCMZNCABOUGUI
      GZCPUJNQUGUKCUIUFUDUEUHRSTUICUAUBUC $.
  $}

  $( Expansion of an ordered pair when the first member is a proper class.  See
     also ~ opprc .  (Contributed by NM, 10-Apr-2004.)  (Revised by Mario
     Carneiro, 26-Apr-2015.) $)
  opprc1 $p |- ( -. A e. _V -> <. A , B >. = (/) ) $=
    ( cvv wcel wn wa cop c0 wceq simpl con3i opprc syl ) ACDZENBCDZFZEABGHIPNNO
    JKABLM $.

  $( Expansion of an ordered pair when the second member is a proper class.
     See also ~ opprc .  (Contributed by NM, 15-Nov-1994.)  (Revised by Mario
     Carneiro, 26-Apr-2015.) $)
  opprc2 $p |- ( -. B e. _V -> <. A , B >. = (/) ) $=
    ( cvv wcel wn wa cop c0 wceq simpr con3i opprc syl ) BCDZEACDZNFZEABGHIPNON
    JKABLM $.

  ${
    $d A x $.  $d A y $.  $d B x $.  $d B y $.  $d C y $.
    $( If an ordered pair has an element, then its arguments are sets.
       (Contributed by Mario Carneiro, 26-Apr-2015.) $)
    oprcl $p |- ( C e. <. A , B >. -> ( A e. _V /\ B e. _V ) ) $=
      ( vy vx cop wcel cv wex cvv wa elex2 wsb csn cpr w3a df-op eleq2i df-clab
      cab bitri 3simpa sbimi sylbi nfv sbf sylib exlimiv syl ) CABFZGDHZUJGZDIA
      JGZBJGZKZDCUJLULUODULUOEDMZUOULUMUNEHANABOOGZPZEDMZUPULUKURETZGUSUJUTUKEA
      BQRURDESUAURUOEDUMUNUQUBUCUDUOEDUOEUEUFUGUHUI $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( The power set of a singleton.  (Contributed by Jim Kingdon,
       12-Aug-2018.) $)
    pwsnss $p |- { (/) , { A } } C_ ~P { A } $=
      ( vx cv c0 wceq csn wo cab wss cpr cpw sssnr ss2abi dfpr2 df-pw 3sstr4i )
      BCZDEQAFZEGZBHQRIZBHDRJRKSTBQALMBDRNBROP $.

    $( Compute the power set of the power set of the empty set.  (See ~ pw0 for
       the power set of the empty set.)  Theorem 90 of [Suppes] p. 48 (but with
       subset in place of equality).  (Contributed by Jim Kingdon,
       12-Aug-2018.) $)
    pwpw0ss $p |- { (/) , { (/) } } C_ ~P { (/) } $=
      ( c0 pwsnss ) AB $.

    $( The power set of an unordered pair.  (Contributed by Jim Kingdon,
       13-Aug-2018.) $)
    pwprss $p |- ( { (/) , { A } } u. { { B } , { A , B } } ) C_
        ~P { A , B } $=
      ( vx c0 csn cpr cun cpw cv wcel wo wss wceq elpr orbi12i ssprr sylbi elun
      vex elpw 3imtr4i ssriv ) CDAEZFZBEZABFZFZGZUFHZCIZUDJZUJUGJZKZUJUFLZUJUHJ
      UJUIJUMUJDMUJUCMKZUJUEMUJUFMKZKUNUKUOULUPUJDUCCSZNUJUEUFUQNOUJABPQUJUDUGR
      UJUFUQTUAUB $.

    $( The power set of an unordered triple.  (Contributed by Jim Kingdon,
       13-Aug-2018.) $)
    pwtpss $p |-
        ( ( { (/) , { A } } u. { { B } , { A , B } } ) u.
          ( { { C } , { A , C } } u. { { B , C } , { A , B , C } } ) )
        C_ ~P { A , B , C } $=
      ( vx c0 csn cpr cun ctp cpw cv wceq wo wcel sstpr elun elpr orbi12i bitri
      wss vex elpw 3imtr4i ssriv ) DEAFZGZBFZABGZGZHZCFZACGZGZBCGZABCIZGZHZHZUO
      JZDKZELUTUELMZUTUGLUTUHLMZMZUTUKLUTULLMZUTUNLUTUOLMZMZMZUTUOTUTURNZUTUSNU
      TABCOVHUTUJNZUTUQNZMVGUTUJUQPVIVCVJVFVIUTUFNZUTUINZMVCUTUFUIPVKVAVLVBUTEU
      EDUAZQUTUGUHVMQRSVJUTUMNZUTUPNZMVFUTUMUPPVNVDVOVEUTUKULVMQUTUNUOVMQRSRSUT
      UOVMUBUCUD $.
  $}

  $( Compute the power set of the power set of the power set of the empty set.
     (See also ~ pw0 and ~ pwpw0ss .)  (Contributed by Jim Kingdon,
     13-Aug-2018.) $)
  pwpwpw0ss $p |-
      ( { (/) , { (/) } } u. { { { (/) } } , { (/) , { (/) } } } )
      C_ ~P { (/) , { (/) } } $=
    ( c0 csn pwprss ) AABC $.

  $( The power class of the universe is the universe.  Exercise 4.12(d) of
     [Mendelson] p. 235.  (Contributed by NM, 14-Sep-2003.) $)
  pwv $p |- ~P _V = _V $=
    ( vx cvv cpw cv wcel wss ssv vex elpw mpbir 2th eqriv ) ABCZBADZMEZNBEONBFN
    GNBAHZIJPKL $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The union of a class
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare class union symbol. $)
  $c U. $.  $( Big cup $)

  $( Extend class notation to include the union of a class.  Read:  "union (of)
     ` A ` ". $)
  cuni $a class U. A $.

  ${
    $d x y A $.
    $( Define the union of a class i.e. the collection of all members of the
       members of the class.  Definition 5.5 of [TakeutiZaring] p. 16.  For
       example, ` U. { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } ` .  This is
       similar to the union of two classes ~ df-un .  (Contributed by NM,
       23-Aug-1993.) $)
    df-uni $a |- U. A = { x | E. y ( x e. y /\ y e. A ) } $.
  $}

  ${
    $d x y A $.
    $( Alternate definition of class union.  (Contributed by NM,
       28-Jun-1998.) $)
    dfuni2 $p |- U. A = { x | E. y e. A x e. y } $=
      ( cuni cv wcel wa wex cab wrex df-uni exancom df-rex bitr4i abbii eqtri )
      CDAEBEZFZQCFZGBHZAIRBCJZAIABCKTUAATSRGBHUARSBLRBCMNOP $.
  $}

  ${
    $d x A y $.  $d x B y $.
    $( Membership in class union.  (Contributed by NM, 22-May-1994.) $)
    eluni $p |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) $=
      ( vy cuni wcel cvv cv wa wex elex adantr exlimiv wceq eleq1 anbi1d exbidv
      df-uni elab2g pm5.21nii ) BCEZFBGFZBAHZFZUCCFZIZAJZBUAKUFUBAUDUBUEBUCKLMD
      HZUCFZUEIZAJUGDBUAGUHBNZUJUFAUKUIUDUEUHBUCOPQDACRST $.

    $( Membership in class union.  Restricted quantifier version.  (Contributed
       by NM, 31-Aug-1999.) $)
    eluni2 $p |- ( A e. U. B <-> E. x e. B A e. x ) $=
      ( cv wcel wa wex cuni wrex exancom eluni df-rex 3bitr4i ) BADZEZNCEZFAGPO
      FAGBCHEOACIOPAJABCKOACLM $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Membership in class union.  (Contributed by NM, 24-Mar-1995.) $)
    elunii $p |- ( ( A e. B /\ B e. C ) -> A e. U. C ) $=
      ( vx wcel wa cv cuni wceq eleq2 eleq1 anbi12d spcegv anabsi7 eluni sylibr
      wex ) ABEZBCEZFZADGZEZUACEZFZDQZACHERSUEUDTDBCUABIUBRUCSUABAJUABCKLMNDACO
      P $.
  $}

  ${
    $d y z A $.  $d x y z $.
    nfuni.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for union.  (Contributed by NM,
       30-Dec-1996.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    nfuni $p |- F/_ x U. A $=
      ( vy vz cuni wel wrex cab dfuni2 nfv nfrexw nfab nfcxfr ) ABFDEGZEBHZDIDE
      BJPADOAEBCOAKLMN $.
  $}

  ${
    $d y z A $.  $d x y z $.  $d y z ph $.
    nfunid.3 $e |- ( ph -> F/_ x A ) $.
    $( Deduction version of ~ nfuni .  (Contributed by NM, 18-Feb-2013.) $)
    nfunid $p |- ( ph -> F/_ x U. A ) $=
      ( vy vz cuni wel wrex cab dfuni2 nfv nfvd nfrexdxy nfabd nfcxfrd ) ABCGEF
      HZFCIZEJEFCKARBEAELAQBFCAFLDAQBMNOP $.
  $}

  ${
    $d A y z $.  $d B y z $.  $d V y z $.  $d x y z $.
    $( Distribute proper substitution through the union of a class.
       (Contributed by Alan Sare, 10-Nov-2012.) $)
    csbunig $p |- ( A e. V -> [_ A / x ]_ U. B = U. [_ A / x ]_ B ) $=
      ( vz vy wcel cv wa wex cab csb cuni wsbc csbabg sbcexg sbcang sbcg df-uni
      bitrd sbcel2g anbi12d exbidv abbidv eqtrd csbeq2i 3eqtr4g ) BDGZABEHFHZGZ
      UICGZIZFJZEKZLZUJUIABCLZGZIZFJZEKZABCMZLUPMUHUOUMABNZEKUTUMAEBDOUHVBUSEUH
      VBULABNZFJUSULFABDPUHVCURFUHVCUJABNZUKABNZIURUJUKABDQUHVDUJVEUQUJABDRABUI
      CDUAUBTUCTUDUEABVAUNEFCSUFEFUPSUG $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Equality theorem for class union.  Exercise 15 of [TakeutiZaring] p. 18.
       (Contributed by NM, 10-Aug-1993.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    unieq $p |- ( A = B -> U. A = U. B ) $=
      ( vy vx wceq wel wrex cab cuni rexeq abbidv dfuni2 3eqtr4g ) ABEZCDFZDAGZ
      CHODBGZCHAIBINPQCODABJKCDALCDBLM $.
  $}

  ${
    unieqi.1 $e |- A = B $.
    $( Inference of equality of two class unions.  (Contributed by NM,
       30-Aug-1993.) $)
    unieqi $p |- U. A = U. B $=
      ( wceq cuni unieq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    unieqd.1 $e |- ( ph -> A = B ) $.
    $( Deduction of equality of two class unions.  (Contributed by NM,
       21-Apr-1995.) $)
    unieqd $p |- ( ph -> U. A = U. B ) $=
      ( wceq cuni unieq syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    $d x A y $.  $d ph y $.
    $( Membership in union of a class abstraction.  (Contributed by NM,
       11-Aug-1994.)  (Revised by Mario Carneiro, 14-Nov-2016.) $)
    eluniab $p |- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) ) $=
      ( vy cab cuni wcel cv wa wex eluni nfv nfsab1 nfan weq eleq2 eleq1 bitrdi
      abid anbi12d cbvex bitri ) CABEZFGCDHZGZUDUCGZIZDJCBHZGZAIZBJDCUCKUGUJDBU
      EUFBUEBLABDMNUJDLDBOZUEUIUFAUDUHCPUKUFUHUCGAUDUHUCQABSRTUAUB $.

    $( Membership in union of a class abstraction.  (Contributed by NM,
       4-Oct-2006.) $)
    elunirab $p |- ( A e. U. { x e. B | ph } <->
                   E. x e. B ( A e. x /\ ph ) ) $=
      ( cv wcel cab cuni wex crab wrex eluniab df-rab unieqi eleq2i df-rex an12
      wa exbii bitri 3bitr4i ) CBEZDFZARZBGZHZFCUBFZUDRZBIZCABDJZHZFUGARZBDKZUD
      BCLUKUFCUJUEABDMNOUMUCULRZBIUIULBDPUNUHBUCUGAQSTUA $.
  $}

  ${
    $d x y A $.  $d x y B $.
    unipr.1 $e |- A e. _V $.
    unipr.2 $e |- B e. _V $.
    $( The union of a pair is the union of its members.  Proposition 5.7 of
       [TakeutiZaring] p. 16.  (Contributed by NM, 23-Aug-1993.) $)
    unipr $p |- U. { A , B } = ( A u. B ) $=
      ( vx vy cv wcel wo cab cpr wa wex cun cuni wceq 19.43 bitri clel3 exancom
      vex elpr anbi2i andi exbii orbi12i 3bitr4ri abbii df-un df-uni 3eqtr4ri )
      EGZAHZULBHZIZEJULFGZHZUPABKZHZLZFMZEJABNUROUOVAEUQUPAPZLZUQUPBPZLZIZFMVCF
      MZVEFMZIVAUOVCVEFQUTVFFUTUQVBVDIZLVFUSVIUQUPABFUAUBUCUQVBVDUDRUEUMVGUNVHU
      MVBUQLFMVGFULACSVBUQFTRUNVDUQLFMVHFULBDSVDUQFTRUFUGUHEABUIEFURUJUK $.
  $}

  ${
    $d x y A $.  $d y B $.
    $( The union of a pair is the union of its members.  Proposition 5.7 of
       [TakeutiZaring] p. 16.  (Contributed by NM, 25-Aug-2006.) $)
    uniprg $p |- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) ) $=
      ( vx vy cv cpr cuni cun wceq preq1 unieqd uneq1 eqeq12d preq2 uneq2 unipr
      vex vtocl2g ) EGZFGZHZIZUAUBJZKAUBHZIZAUBJZKABHZIZABJZKEFABCDUAAKZUDUGUEU
      HULUCUFUAAUBLMUAAUBNOUBBKZUGUJUHUKUMUFUIUBBAPMUBBAQOUAUBESFSRT $.
  $}

  ${
    unisn.1 $e |- A e. _V $.
    $( A set equals the union of its singleton.  Theorem 8.2 of [Quine] p. 53.
       (Contributed by NM, 30-Aug-1993.) $)
    unisn $p |- U. { A } = A $=
      ( csn cuni cpr cun dfsn2 unieqi unipr unidm 3eqtri ) ACZDAAEZDAAFALMAGHAA
      BBIAJK $.
  $}

  ${
    $d x A $.
    $( A set equals the union of its singleton.  Theorem 8.2 of [Quine] p. 53.
       (Contributed by NM, 13-Aug-2002.) $)
    unisng $p |- ( A e. V -> U. { A } = A ) $=
      ( vx cv csn cuni wceq sneq unieqd id eqeq12d vex unisn vtoclg ) CDZEZFZOG
      AEZFZAGCABOAGZQSOATPROAHITJKOCLMN $.
  $}

  ${
    $d x y $.  $d y A $.
    $( An alternate statement of the effective freeness of a class ` A ` , when
       it is a set.  (Contributed by Mario Carneiro, 14-Oct-2016.) $)
    dfnfc2 $p |- ( A. x A e. V -> ( F/_ x A <-> A. y F/ x y = A ) ) $=
      ( wcel wal wnfc cv wceq wnf nfcvd id nfeqd alrimiv csn simpr df-nfc velsn
      wa cuni nfbii albii bitri sylibr nfunid nfa1 nfnf1 nfal unisng sps adantr
      nfan nfceqdf mpbid ex impbid2 ) CDEZAFZACGZBHZCIZAJZBFZUSVBBUSAUTCUSAUTKU
      SLMNURVCUSURVCSZACOZTZGUSVDAVEVDVCAVEGZURVCPVGUTVEEZAJZBFVCABVEQVIVBBVHVA
      ABCRUAUBUCUDUEVDAVFCURVCAUQAUFVBABVAAUGUHULURVFCIZVCUQVJACDUIUJUKUMUNUOUP
      $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The class union of the union of two classes.  Theorem 8.3 of [Quine]
       p. 53.  (Contributed by NM, 20-Aug-1993.) $)
    uniun $p |- U. ( A u. B ) = ( U. A u. U. B ) $=
      ( vx vy cun cuni cv wcel wa wo 19.43 elun anbi2i andi bitri exbii orbi12i
      wex eluni 3bitr4i eqriv ) CABEZFZAFZBFZEZCGZDGZHZUHUBHZIZDRZUGUDHZUGUEHZJ
      ZUGUCHUGUFHUIUHAHZIZUIUHBHZIZJZDRUQDRZUSDRZJULUOUQUSDKUKUTDUKUIUPURJZIUTU
      JVCUIUHABLMUIUPURNOPUMVAUNVBDUGASDUGBSQTDUGUBSUGUDUELTUA $.

    $( The class union of the intersection of two classes.  Exercise 4.12(n) of
       [Mendelson] p. 235.  (Contributed by NM, 4-Dec-2003.)  (Proof shortened
       by Andrew Salmon, 29-Jun-2011.) $)
    uniin $p |- U. ( A i^i B ) C_ ( U. A i^i U. B ) $=
      ( vx vy cin cuni cv wcel wex 19.40 elin anbi2i anandi bitri exbii anbi12i
      wa eluni 3imtr4i ssriv ) CABEZFZAFZBFZEZCGZDGZHZUGUAHZQZDIZUFUCHZUFUDHZQZ
      UFUBHUFUEHUHUGAHZQZUHUGBHZQZQZDIUPDIZURDIZQUKUNUPURDJUJUSDUJUHUOUQQZQUSUI
      VBUHUGABKLUHUOUQMNOULUTUMVADUFARDUFBRPSDUFUARUFUCUDKST $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Subclass relationship for class union.  Theorem 61 of [Suppes] p. 39.
       (Contributed by NM, 22-Mar-1998.)  (Proof shortened by Andrew Salmon,
       29-Jun-2011.) $)
    uniss $p |- ( A C_ B -> U. A C_ U. B ) $=
      ( vx vy wss cuni cv wcel wa wex ssel anim2d eximdv eluni 3imtr4g ssrdv )
      ABEZCAFZBFZQCGZDGZHZUAAHZIZDJUBUABHZIZDJTRHTSHQUDUFDQUCUEUBABUAKLMDTANDTB
      NOP $.

    $( Subclass relationship for class union.  (Contributed by NM,
       24-May-1994.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    ssuni $p |- ( ( A C_ B /\ B e. C ) -> A C_ U. C ) $=
      ( vy vx wcel wss cuni cv wi wal eleq2 imbi1d elunii expcom vtoclga imim2d
      wceq alimdv ssalel 3imtr4g impcom ) BCFZABGZACHZGZUCDIZAFZUGBFZJZDKUHUGUE
      FZJZDKUDUFUCUJULDUCUIUKUHUGEIZFZUKJUIUKJEBCUMBRUNUIUKUMBUGLMUNUMCFUKUGUMC
      NOPQSDABTDAUETUAUB $.
  $}

  ${
    unissi.1 $e |- A C_ B $.
    $( Subclass relationship for subclass union.  Inference form of ~ uniss .
       (Contributed by David Moews, 1-May-2017.) $)
    unissi $p |- U. A C_ U. B $=
      ( wss cuni uniss ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    unissd.1 $e |- ( ph -> A C_ B ) $.
    $( Subclass relationship for subclass union.  Deduction form of ~ uniss .
       (Contributed by David Moews, 1-May-2017.) $)
    unissd $p |- ( ph -> U. A C_ U. B ) $=
      ( wss cuni uniss syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    $d x y A $.
    $( The union of a set is empty iff the set is included in the singleton of
       the empty set.  (Contributed by NM, 12-Sep-2004.) $)
    uni0b $p |- ( U. A = (/) <-> A C_ { (/) } ) $=
      ( vx vy cv c0 wceq wral wcel wn wal csn wss cuni eq0 ralbii ralcom4 bitri
      dfss3 velsn albii wrex eluni2 notbii ralnex 3bitr4i 3bitr4ri ) BDZEFZBAGZ
      CDZUGHZIZBAGZCJZAEKZLZAMZEFZUIULCJZBAGUNUHUSBACUGNOULBCAPQUPUGUOHZBAGUIBA
      UORUTUHBABESOQUJUQHZIZCJUKBAUAZIZCJURUNVBVDCVAVCBUJAUBUCTCUQNUMVDCUKBAUDT
      UEUF $.

    $( The union of a set is empty iff all of its members are empty.
       (Contributed by NM, 16-Aug-2006.) $)
    uni0c $p |- ( U. A = (/) <-> A. x e. A x = (/) ) $=
      ( cuni c0 wceq csn wss cv wcel wral uni0b dfss3 velsn ralbii 3bitri ) BCD
      EBDFZGAHZPIZABJQDEZABJBKABPLRSABADMNO $.
  $}

  $( The union of the empty set is the empty set.  Theorem 8.7 of [Quine]
     p. 54.  (Reproved without relying on ax-nul by Eric Schmidt.)
     (Contributed by NM, 16-Sep-1993.)  (Revised by Eric Schmidt,
     4-Apr-2007.) $)
  uni0 $p |- U. (/) = (/) $=
    ( c0 cuni wceq csn wss 0ss uni0b mpbir ) ABACAADZEIFAGH $.

  $( An element of a class is a subclass of its union.  Theorem 8.6 of [Quine]
     p. 54.  Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40.
     (Contributed by NM, 6-Jun-1994.) $)
  elssuni $p |- ( A e. B -> A C_ U. B ) $=
    ( wss wcel cuni ssid ssuni mpan ) AACABDABECAFAABGH $.

  $( Condition turning a subclass relationship for union into an equality.
     (Contributed by NM, 18-Jul-2006.) $)
  unissel $p |- ( ( U. A C_ B /\ B e. A ) -> U. A = B ) $=
    ( cuni wss wcel wa simpl elssuni adantl eqssd ) ACZBDZBAEZFKBLMGMBKDLBAHIJ
    $.

  ${
    $d x y A $.  $d x y B $.
    $( Relationship involving membership, subset, and union.  Exercise 5 of
       [Enderton] p. 26 and its converse.  (Contributed by NM, 20-Sep-2003.) $)
    unissb $p |- ( U. A C_ B <-> A. x e. A x C_ B ) $=
      ( vy cv cuni wcel wi wal wss wral wa wex eluni imbi1i 19.23v albii ssalel
      bitri 3bitr4i bitr4i alcom 19.21v impexp bi2.04 imbi2i df-ral ) DEZBFZGZU
      HCGZHZDIZAEZBGZUNCJZHZAIZUICJUPABKUMUHUNGZUOLZUKHZAIZDIZURULVBDULUTAMZUKH
      VBUJVDUKAUHBNOUTUKAPUAQVCVADIZAIURVADAUBVEUQAUOUSUKHZHZDIUOVFDIZHVEUQUOVF
      DUCVAVGDVAUSUOUKHHVGUSUOUKUDUSUOUKUESQUPVHUODUNCRUFTQSSDUICRUPABUGT $.
  $}

  ${
    $d x A $.  $d x y B $.
    $( A subclass condition on the members of two classes that implies a
       subclass relation on their unions.  Proposition 8.6 of [TakeutiZaring]
       p. 59.  (Contributed by NM, 22-Mar-2004.) $)
    uniss2 $p |- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B ) $=
      ( cv wss wrex wral cuni wcel ssuni expcom rexlimiv ralimi unissb sylibr )
      AEZBEZFZBDGZACHQDIZFZACHCIUAFTUBACSUBBDSRDJUBQRDKLMNACUAOP $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( If the difference ` A \ B ` contains the largest members of ` A ` , then
       the union of the difference is the union of ` A ` .  (Contributed by NM,
       22-Mar-2004.) $)
    unidif $p |- ( A. x e. A E. y e. ( A \ B ) x C_ y ->
               U. ( A \ B ) = U. A ) $=
      ( cv wss cdif wrex wral cuni wceq uniss2 difss unissi jctil eqss sylibr
      wa ) AEBEFBCDGZHACIZSJZCJZFZUBUAFZRUAUBKTUDUCABCSLSCCDMNOUAUBPQ $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Relationship implying union.  (Contributed by NM, 10-Nov-1999.) $)
    ssunieq $p |- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B ) $=
      ( wcel cv wss wral cuni wceq elssuni unissb biimpri anim12i eqss sylibr
      wa ) BCDZAEBFACGZPBCHZFZSBFZPBSIQTRUABCJUARACBKLMBSNO $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Any member of a class is the largest of those members that it includes.
       (Contributed by NM, 13-Aug-2002.) $)
    unimax $p |- ( A e. B -> U. { x e. B | x C_ A } = A ) $=
      ( vy wcel cv wss crab wral cuni wceq ssid sseq1 elrab3 elrab simprbi rgen
      mpbiri wa ssunieq eqcomd sylancl ) BCEZBAFZBGZACHZEZDFZBGZDUFIZUFJZBKUCUG
      BBGZBLUEULABCUDBBMNRUIDUFUHUFEUHCEUIUEUIAUHCUDUHBMOPQUGUJSBUKDBUFTUAUB $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The intersection of a class
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare class intersection symbol. $)
  $c |^| $.  $( Big cap $)

  $( Extend class notation to include the intersection of a class.  Read:
     "intersection (of) ` A ` ". $)
  cint $a class |^| A $.

  ${
    $d x y A $.
    $( Define the intersection of a class.  Definition 7.35 of [TakeutiZaring]
       p. 44.  For example, ` |^| { { 1 , 3 } , { 1 , 8 } } = { 1 } ` .
       Compare this with the intersection of two classes, ~ df-in .
       (Contributed by NM, 18-Aug-1993.) $)
    df-int $a |- |^| A = { x | A. y ( y e. A -> x e. y ) } $.
  $}

  ${
    $d x y A $.
    $( Alternate definition of class intersection.  (Contributed by NM,
       28-Jun-1998.) $)
    dfint2 $p |- |^| A = { x | A. y e. A x e. y } $=
      ( cint cv wcel wi wal cab wral df-int df-ral abbii eqtr4i ) CDBEZCFAEOFZG
      BHZAIPBCJZAIABCKRQAPBCLMN $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Equality law for intersection.  (Contributed by NM, 13-Sep-1999.) $)
    inteq $p |- ( A = B -> |^| A = |^| B ) $=
      ( vx vy wceq wel wral cab cint raleq abbidv dfint2 3eqtr4g ) ABEZCDFZDAGZ
      CHODBGZCHAIBINPQCODABJKCDALCDBLM $.
  $}

  ${
    inteqi.1 $e |- A = B $.
    $( Equality inference for class intersection.  (Contributed by NM,
       2-Sep-2003.) $)
    inteqi $p |- |^| A = |^| B $=
      ( wceq cint inteq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    inteqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for class intersection.  (Contributed by NM,
       2-Sep-2003.) $)
    inteqd $p |- ( ph -> |^| A = |^| B ) $=
      ( wceq cint inteq syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    $d x A y $.  $d x B y $.
    elint.1 $e |- A e. _V $.
    $( Membership in class intersection.  (Contributed by NM, 21-May-1994.) $)
    elint $p |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) ) $=
      ( vy cv wcel wi wal cint wceq eleq1 imbi2d albidv df-int elab2 ) AFZCGZEF
      ZQGZHZAIRBQGZHZAIEBCJDSBKZUAUCAUDTUBRSBQLMNEACOP $.
  $}

  ${
    $d x A $.  $d x B $.
    elint2.1 $e |- A e. _V $.
    $( Membership in class intersection.  (Contributed by NM, 14-Oct-1999.) $)
    elint2 $p |- ( A e. |^| B <-> A. x e. B A e. x ) $=
      ( cint wcel cv wi wal wral elint df-ral bitr4i ) BCEFAGZCFBNFZHAIOACJABCD
      KOACLM $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Membership in class intersection, with the sethood requirement expressed
       as an antecedent.  (Contributed by NM, 20-Nov-2003.) $)
    elintg $p |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) ) $=
      ( vy cv cint wcel wral eleq1 wceq ralbidv vex elint2 vtoclbg ) EFZCGZHPAF
      ZHZACIBQHBRHZACIEBDPBQJPBKSTACPBRJLAPCEMNO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Membership in class intersection.  (Contributed by NM, 14-Oct-1999.)
       (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
    elinti $p |- ( A e. |^| B -> ( C e. B -> A e. C ) ) $=
      ( vx cint wcel wi cv wral elintg eleq2 rspccv biimtrdi pm2.43i ) ABEZFZCB
      FACFZGZPPADHZFZDBIRDABOJTQDCBSCAKLMN $.
  $}

  ${
    $d y z A $.  $d x y z $.
    nfint.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for intersection.  (Contributed by NM,
       2-Feb-1997.)  (Proof shortened by Andrew Salmon, 12-Aug-2011.) $)
    nfint $p |- F/_ x |^| A $=
      ( vy vz cint wel wral cab dfint2 nfv nfralxy nfab nfcxfr ) ABFDEGZEBHZDID
      EBJPADOAEBCOAKLMN $.
  $}

  ${
    $d A x y $.  $d ph y $.
    inteqab.1 $e |- A e. _V $.
    $( Membership in the intersection of a class abstraction.  (Contributed by
       NM, 30-Aug-1993.) $)
    elintab $p |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) $=
      ( vy cab cint wcel cv wi wal elint nfsab1 nfv nfim weq eleq1 bitrdi eleq2
      abid imbi12d cbval bitri ) CABFZGHEIZUDHZCUEHZJZEKACBIZHZJZBKECUDDLUHUKEB
      UFUGBABEMUGBNOUKENEBPZUFAUGUJULUFUIUDHAUEUIUDQABTRUEUICSUAUBUC $.

    $( Membership in the intersection of a class abstraction.  (Contributed by
       NM, 17-Oct-1999.) $)
    elintrab $p |- ( A e. |^| { x e. B | ph } <->
                 A. x e. B ( ph -> A e. x ) ) $=
      ( cv wcel wa cab cint wi wal crab wral elintab impexp albii df-rab inteqi
      bitri eleq2i df-ral 3bitr4i ) CBFZDGZAHZBIZJZGZUEACUDGZKZKZBLZCABDMZJZGUK
      BDNUIUFUJKZBLUMUFBCEOUPULBUEAUJPQTUOUHCUNUGABDRSUAUKBDUBUC $.
  $}

  ${
    $d x y A $.  $d y B $.  $d y ph $.
    $( Membership in the intersection of a class abstraction.  (Contributed by
       NM, 17-Feb-2007.) $)
    elintrabg $p |- ( A e. V -> ( A e. |^| { x e. B | ph } <->
                 A. x e. B ( ph -> A e. x ) ) ) $=
      ( vy cv crab cint wcel wi wral eleq1 wceq imbi2d ralbidv elintrab vtoclbg
      vex ) FGZABDHIZJATBGZJZKZBDLCUAJACUBJZKZBDLFCETCUAMTCNZUDUFBDUGUCUEATCUBM
      OPABTDFSQR $.

    $( The intersection of the empty set is the universal class.  Exercise 2 of
       [TakeutiZaring] p. 44.  (Contributed by NM, 18-Aug-1993.) $)
    int0 $p |- |^| (/) = _V $=
      ( vy vx cv c0 wcel wi wal cab weq cint cvv pm2.21i ax-gen equid 2th abbii
      noel df-int df-v 3eqtr4i ) ACZDEZBCUAEZFZAGZBHBBIZBHDJKUEUFBUEUFUDAUBUCUA
      QLMBNOPBADRBST $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y ph $.
    $( An element of a class includes the intersection of the class.  Exercise
       4 of [TakeutiZaring] p. 44 (with correction), generalized to classes.
       (Contributed by NM, 18-Nov-1995.) $)
    intss1 $p |- ( A e. B -> |^| B C_ A ) $=
      ( vx vy wcel cint cv wal vex elint wceq eleq1 eleq2 imbi12d spcgv pm2.43a
      wi biimtrid ssrdv ) ABEZCBFZACGZUAEDGZBEZUBUCEZQZDHZTUBAEZDUBBCIJUGTUHUFT
      UHQDABUCAKUDTUEUHUCABLUCAUBMNOPRS $.

    $( Subclass of a class intersection.  Theorem 5.11(viii) of [Monk1] p. 52
       and its converse.  (Contributed by NM, 14-Oct-1999.) $)
    ssint $p |- ( A C_ |^| B <-> A. x e. B A C_ x ) $=
      ( vy cint wss cv wcel wral dfss3 vex elint2 ralbii ralcom bitr4i 3bitri )
      BCEZFDGZQHZDBIRAGZHZACIZDBIZBTFZACIZDBQJSUBDBARCDKLMUCUADBIZACIUEUADABCNU
      DUFACDBTJMOP $.

    $( Subclass of the intersection of a class abstraction.  (Contributed by
       NM, 31-Jul-2006.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
    ssintab $p |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) ) $=
      ( vy cab cint wss cv wral wi wal ssint sseq2 ralab2 bitri ) CABEZFGCDHZGZ
      DPIACBHZGZJBKDCPLARTDBQSCMNO $.

    $( Subclass of the least upper bound.  (Contributed by NM, 8-Aug-2000.) $)
    ssintub $p |- A C_ |^| { x e. B | A C_ x } $=
      ( vy cv wss crab cint ssint wcel sseq2 elrab simprbi mprgbir ) BBAEZFZACG
      ZHFBDEZFZDQDBQIRQJRCJSPSARCORBKLMN $.

    $( Subclass of the minimum value of class of supersets.  (Contributed by
       NM, 10-Aug-2006.) $)
    ssmin $p |- A C_ |^| { x | ( A C_ x /\ ph ) } $=
      ( cv wss wa cab cint wi ssintab simpl mpgbir ) CCBDEZAFZBGHENMIBNBCJMAKL
      $.

    $( Any member of a class is the smallest of those members that include it.
       (Contributed by NM, 13-Aug-2002.)  (Proof shortened by Andrew Salmon,
       9-Jul-2011.) $)
    intmin $p |- ( A e. B -> |^| { x e. B | A C_ x } = A ) $=
      ( vy wcel cv wss crab cint wi wral elintrab ssid wceq sseq2 eleq2 imbi12d
      vex rspcv mpii biimtrid ssrdv ssintub a1i eqssd ) BCEZBAFZGZACHIZBUFDUIBD
      FZUIEUHUJUGEZJZACKZUFUJBEZUHAUJCDRLUFUMBBGZUNBMULUOUNJABCUGBNUHUOUKUNUGBB
      OUGBUJPQSTUAUBBUIGUFABCUCUDUE $.

    $( Intersection of subclasses.  (Contributed by NM, 14-Oct-1999.) $)
    intss $p |- ( A C_ B -> |^| B C_ |^| A ) $=
      ( vy vx cv wcel wi wal cint wss imim1 al2imi elint 3imtr4g alrimiv ssalel
      vex 3imtr4i ) CEZAFZSBFZGZCHZDEZBIZFZUDAIZFZGZDHABJUEUGJUCUIDUCUAUDSFZGZC
      HTUJGZCHUFUHUBUKULCTUAUJKLCUDBDQZMCUDAUMMNOCABPDUEUGPR $.

    $( The intersection of an inhabited set is a subclass of its union.
       (Contributed by NM, 29-Jul-2006.) $)
    intssunim $p |- ( E. x x e. A -> |^| A C_ U. A ) $=
      ( vy cv wcel wex cint cuni wral r19.2m ex vex elint2 eluni2 3imtr4g ssrdv
      wrex ) ADZBEAFZCBGZBHZSCDZREZABIZUCABQZUBTEUBUAESUDUEUCAABJKAUBBCLMAUBBNO
      P $.
  $}

  ${
    $d x A $.
    $( Subclass of the intersection of a restricted class builder.
       (Contributed by NM, 30-Jan-2015.) $)
    ssintrab $p |- ( A C_ |^| { x e. B | ph }
           <-> A. x e. B ( ph -> A C_ x ) ) $=
      ( crab cint wss cv wcel wa cab wral df-rab inteqi sseq2i wal impexp albii
      wi ssintab df-ral 3bitr4i bitri ) CABDEZFZGCBHZDIZAJZBKZFZGZACUFGZSZBDLZU
      EUJCUDUIABDMNOUHULSZBPUGUMSZBPUKUNUOUPBUGAULQRUHBCTUMBDUAUBUC $.
  $}

  ${
    $d x A $.
    $( Subclass relationship for intersection and union.  (Contributed by Jim
       Kingdon, 14-Aug-2018.) $)
    intssuni2m $p |- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B ) $=
      ( cv wcel wex wss cint cuni intssunim uniss sylan9ssr ) ADBEAFBCGBHBICIAB
      JBCKL $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    intminss.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Under subset ordering, the intersection of a restricted class
       abstraction is less than or equal to any of its members.  (Contributed
       by NM, 7-Sep-2013.) $)
    intminss $p |- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A ) $=
      ( wcel wa crab cint wss elrab intss1 sylbir ) DEGBHDACEIZGOJDKABCDEFLDOMN
      $.
  $}

  ${
    $d x A $.
    intmin2.1 $e |- A e. _V $.
    $( Any set is the smallest of all sets that include it.  (Contributed by
       NM, 20-Sep-2003.) $)
    intmin2 $p |- |^| { x | A C_ x } = A $=
      ( cv wss cvv crab cint cab rabab inteqi wcel wceq intmin ax-mp eqtr3i ) B
      ADEZAFGZHZQAIZHBRTQAJKBFLSBMCABFNOP $.
  $}

  ${
    $d x A $.  $d x ps $.
    intmin3.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    intmin3.3 $e |- ps $.
    $( Under subset ordering, the intersection of a class abstraction is less
       than or equal to any of its members.  (Contributed by NM,
       3-Jul-2005.) $)
    intmin3 $p |- ( A e. V -> |^| { x | ph } C_ A ) $=
      ( wcel cab cint wss elabg mpbiri intss1 syl ) DEHZDACIZHZQJDKPRBGABCDEFLM
      DQNO $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( Elimination of a conjunct in a class intersection.  (Contributed by NM,
       31-Jul-2006.) $)
    intmin4 $p |- ( A C_ |^| { x | ph } ->
                  |^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } ) $=
      ( vy cab cint wss cv wa wcel wi wal wb ssintab simpr impbid2 imbi1d alimi
      ancr elintab albi syl sylbi vex 3bitr4g eqrdv ) CABEFZGZDCBHZGZAIZBEFZUGU
      HUKDHZUIJZKZBLZAUNKZBLZUMULJUMUGJUHAUJKZBLZUPURMZABCNUTUOUQMZBLVAUSVBBUSU
      KAUNUSUKAUJAOAUJSPQRUOUQBUAUBUCUKBUMDUDZTABUMVCTUEUF $.
  $}

  ${
    $d x z A $.  $d x z ph $.  $d x y z $.
    intab.1 $e |- A e. _V $.
    intab.2 $e |- { x | E. y ( ph /\ x = A ) } e. _V $.
    $( The intersection of a special case of a class abstraction. ` y ` may be
       free in ` ph ` and ` A ` , which can be thought of a ` ph ( y ) ` and
       ` A ( y ) ` .  (Contributed by NM, 28-Jul-2006.)  (Proof shortened by
       Mario Carneiro, 14-Nov-2016.) $)
    intab $p |- |^| { x | A. y ( ph -> A e. x ) } =
                { x | E. y ( ph /\ x = A ) } $=
      ( vz cv wcel wi wal cab cint wceq wa wex wss ex alrimiv sylibr cvv anbi2d
      eqeq1 exbidv cbvabv eqeltri nfe1 nfab nfeq2 eleq2 imbi2d albid elab 19.8a
      wsbc sbc6 df-sbc sylib mpgbir intss1 19.29r simplr pm3.35 adantlr eqeltrd
      ax-mp exlimiv syl vex elintab abssi eqssi eqtri ) ADBHZIZJZCKZBLZMZAGHZDN
      ZOZCPZGLZAVNDNZOZCPZBLZVSWDWDVRIZVSWDQWIADWDIZJZCVQWKCKBWDWDWHUAWCWGGBVTV
      NNZWBWFCWLWAWEAVTVNDUCUBUDUEZFUFVNWDNZVPWKCCVNWDWCCGWBCUGUHUIWNVOWJAVNWDD
      UJUKULUMAWCGDUOZWJAWAWCJZGKWOAWPGAWAWCWBCUNRSWCGDEUPTWCGDUQURUSWDVRUTVFWC
      GVSWCVQVTVNIZJZBKVTVSIWCWRBWCVQWQWCVQOWBVPOZCPWQWBVPCVAWSWQCWSVTDVNAWAVPV
      BAVPVOWAAVOVCVDVEVGVHRSVQBVTGVIVJTVKVLWMVM $.
  $}

  $( The intersection of a class containing the empty set is empty.
     (Contributed by NM, 24-Apr-2004.) $)
  int0el $p |- ( (/) e. A -> |^| A = (/) ) $=
    ( c0 wcel cint intss1 wss 0ss a1i eqssd ) BACZADZBBAEBKFJKGHI $.

  ${
    $d x y A $.  $d x y B $.
    $( The class intersection of the union of two classes.  Theorem 78 of
       [Suppes] p. 42.  (Contributed by NM, 22-Sep-2002.) $)
    intun $p |- |^| ( A u. B ) = ( |^| A i^i |^| B ) $=
      ( vx vy cun cint cin cv wcel wi wal wa 19.26 elun imbi1i jaob bitri elint
      wo 3bitr4i albii vex anbi12i elin eqriv ) CABEZFZAFZBFZGZDHZUFIZCHZUKIZJZ
      DKZUMUHIZUMUIIZLZUMUGIUMUJIUKAIZUNJZUKBIZUNJZLZDKVADKZVCDKZLUPUSVAVCDMUOV
      DDUOUTVBSZUNJVDULVGUNUKABNOUTUNVBPQUAUQVEURVFDUMACUBZRDUMBVHRUCTDUMUFVHRU
      MUHUIUDTUE $.
  $}

  ${
    $d x y A $.  $d x y B $.
    intpr.1 $e |- A e. _V $.
    intpr.2 $e |- B e. _V $.
    $( The intersection of a pair is the intersection of its members.  Theorem
       71 of [Suppes] p. 42.  (Contributed by NM, 14-Oct-1999.) $)
    intpr $p |- |^| { A , B } = ( A i^i B ) $=
      ( vx vy cpr cint cin cv wcel wi wal wa wceq 19.26 wo vex clel4 3bitr4i
      elpr imbi1i jaob bitri albii anbi12i elint elin eqriv ) EABGZHZABIZFJZUJK
      ZEJZUMKZLZFMZUOAKZUOBKZNZUOUKKUOULKUMAOZUPLZUMBOZUPLZNZFMVCFMZVEFMZNURVAV
      CVEFPUQVFFUQVBVDQZUPLVFUNVIUPUMABFRUAUBVBUPVDUCUDUEUSVGUTVHFUOACSFUOBDSUF
      TFUOUJERUGUOABUHTUI $.
  $}

  ${
    $d x y A $.  $d y B $.
    $( The intersection of a pair is the intersection of its members.  Closed
       form of ~ intpr .  Theorem 71 of [Suppes] p. 42.  (Contributed by FL,
       27-Apr-2008.) $)
    intprg $p |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) ) $=
      ( vx vy cv cpr cint cin wceq preq1 inteqd ineq1 eqeq12d preq2 ineq2 intpr
      vex vtocl2g ) EGZFGZHZIZUAUBJZKAUBHZIZAUBJZKABHZIZABJZKEFABCDUAAKZUDUGUEU
      HULUCUFUAAUBLMUAAUBNOUBBKZUGUJUHUKUMUFUIUBBAPMUBBAQOUAUBESFSRT $.
  $}

  $( Intersection of a singleton.  (Contributed by Stefan O'Rear,
     22-Feb-2015.) $)
  intsng $p |- ( A e. V -> |^| { A } = A ) $=
    ( wcel csn cint cpr dfsn2 inteqi cin wceq intprg anidms inidm eqtrdi eqtrid
    ) ABCZADZEAAFZEZAQRAGHPSAAIZAPSTJAABBKLAMNO $.

  ${
    intsn.1 $e |- A e. _V $.
    $( The intersection of a singleton is its member.  Theorem 70 of [Suppes]
       p. 41.  (Contributed by NM, 29-Sep-2002.) $)
    intsn $p |- |^| { A } = A $=
      ( cvv wcel csn cint wceq intsng ax-mp ) ACDAEFAGBACHI $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( The union and intersection of a singleton are equal.  See also ~ eusn .
       (Contributed by Jim Kingdon, 14-Aug-2018.) $)
    uniintsnr $p |- ( E. x A = { x } -> U. A = |^| A ) $=
      ( csn wceq cuni cint vex unisn unieq inteq intsn eqtrdi 3eqtr4a exlimiv
      cv ) BAOZCZDZBEZBFZDARQEPSTPAGZHBQIRTQFPBQJPUAKLMN $.

    $( The union and the intersection of a class abstraction are equal if there
       is a unique satisfying value of ` ph ( x ) ` .  (Contributed by Jim
       Kingdon, 14-Aug-2018.) $)
    uniintabim $p |- ( E! x ph -> U. { x | ph } = |^| { x | ph } ) $=
      ( vy weu cab cv csn wceq wex cuni cint euabsn2 uniintsnr sylbi ) ABDABEZC
      FGHCIOJOKHABCLCOMN $.
  $}

  ${
    intunsn.1 $e |- B e. _V $.
    $( Theorem joining a singleton to an intersection.  (Contributed by NM,
       29-Sep-2002.) $)
    intunsn $p |- |^| ( A u. { B } ) = ( |^| A i^i B ) $=
      ( csn cun cint cin intun intsn ineq2i eqtri ) ABDZEFAFZLFZGMBGALHNBMBCIJK
      $.
  $}

  $( Relative intersection of an empty set.  (Contributed by Stefan O'Rear,
     3-Apr-2015.) $)
  rint0 $p |- ( X = (/) -> ( A i^i |^| X ) = A ) $=
    ( c0 wceq cint cin inteq ineq2d cvv int0 ineq2i inv1 eqtri eqtrdi ) BCDZABE
    ZFACEZFZAOPQABCGHRAIFAQIAJKALMN $.

  ${
    $d B y $.  $d X y $.
    $( Membership in a restricted intersection.  (Contributed by Stefan O'Rear,
       3-Apr-2015.) $)
    elrint $p |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) ) $=
      ( cint cin wcel wa cv wral elin elintg pm5.32i bitri ) DBCEZFGDBGZDOGZHPD
      AIGACJZHDBOKPQRADCBLMN $.

    $( Membership in a restricted intersection.  (Contributed by Stefan O'Rear,
       3-Apr-2015.) $)
    elrint2 $p |- ( X e. A -> ( X e. ( A i^i |^| B ) <->
          A. y e. B X e. y ) ) $=
      ( cint cin wcel cv wral elrint baib ) DBCEFGDBGDAHGACIABCDJK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Indexed union and intersection
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c U_ $. $( Underlined big cup. $)
  $c |^|_ $. $( Underlined big cap. $)

  $( Extend class notation to include indexed union.  Note:  Historically
     (prior to 21-Oct-2005), set.mm used the notation ` U. x e. A B ` , with
     the same union symbol as ~ cuni .  While that syntax was unambiguous, it
     did not allow for LALR parsing of the syntax constructions in set.mm.  The
     new syntax uses as distinguished symbol ` U_ ` instead of ` U. ` and does
     allow LALR parsing.  Thanks to Peter Backes for suggesting this change. $)
  ciun $a class U_ x e. A B $.

  $( Extend class notation to include indexed intersection.  Note:
     Historically (prior to 21-Oct-2005), set.mm used the notation
     ` |^| x e. A B ` , with the same intersection symbol as ~ cint .  Although
     that syntax was unambiguous, it did not allow for LALR parsing of the
     syntax constructions in set.mm.  The new syntax uses a distinguished
     symbol ` |^|_ ` instead of ` |^| ` and does allow LALR parsing.  Thanks to
     Peter Backes for suggesting this change. $)
  ciin $a class |^|_ x e. A B $.

  ${
    $d x y $.  $d y A $.  $d y B $.
    $( Define indexed union.  Definition indexed union in [Stoll] p. 45.  In
       most applications, ` A ` is independent of ` x ` (although this is not
       required by the definition), and ` B ` depends on ` x ` i.e. can be read
       informally as ` B ( x ) ` .  We call ` x ` the index, ` A ` the index
       set, and ` B ` the indexed set.  In most books, ` x e. A ` is written as
       a subscript or underneath a union symbol ` U. ` .  We use a special
       union symbol ` U_ ` to make it easier to distinguish from plain class
       union.  In many theorems, you will see that ` x ` and ` A ` are in the
       same disjoint variable group (meaning ` A ` cannot depend on ` x ` ) and
       that ` B ` and ` x ` do not share a disjoint variable group (meaning
       that can be thought of as ` B ( x ) ` i.e. can be substituted with a
       class expression containing ` x ` ).  An alternate definition tying
       indexed union to ordinary union is ~ dfiun2 .  Theorem ~ uniiun provides
       a definition of ordinary union in terms of indexed union.  (Contributed
       by NM, 27-Jun-1998.) $)
    df-iun $a |- U_ x e. A B = { y | E. x e. A y e. B } $.

    $( Define indexed intersection.  Definition of [Stoll] p. 45.  See the
       remarks for its sibling operation of indexed union ~ df-iun .  An
       alternate definition tying indexed intersection to ordinary intersection
       is ~ dfiin2 .  Theorem ~ intiin provides a definition of ordinary
       intersection in terms of indexed intersection.  (Contributed by NM,
       27-Jun-1998.) $)
    df-iin $a |- |^|_ x e. A B = { y | A. x e. A y e. B } $.
  $}

  ${
    $d x y A $.  $d y B $.  $d y C $.
    $( Membership in indexed union.  (Contributed by NM, 3-Sep-2003.) $)
    eliun $p |- ( A e. U_ x e. B C <-> E. x e. B A e. C ) $=
      ( vy ciun wcel wrex elex rexlimivw cv wceq eleq1 rexbidv df-iun pm5.21nii
      cvv elab2g ) BACDFZGBQGZBDGZACHZBSIUATACBDIJEKZDGZACHUBEBSQUCBLUDUAACUCBD
      MNAECDORP $.

    $( Membership in indexed intersection.  (Contributed by NM, 3-Sep-2003.) $)
    eliin $p |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) ) $=
      ( vy cv wcel wral ciin wceq eleq1 ralbidv df-iin elab2g ) FGZDHZACIBDHZAC
      IFBACDJEPBKQRACPBDLMAFCDNO $.
  $}

  ${
    $d y z A $.  $d x z B $.  $d z C $.  $d x y $.
    $( Commutation of indexed unions.  (Contributed by NM, 18-Dec-2008.) $)
    iuncom $p |- U_ x e. A U_ y e. B C = U_ y e. B U_ x e. A C $=
      ( vz ciun cv wcel wrex rexcom eliun rexbii 3bitr4i eqriv ) FACBDEGZGZBDAC
      EGZGZFHZPIZACJZTRIZBDJZTQITSITEIZBDJZACJUEACJZBDJUBUDUEABCDKUAUFACBTDELMU
      CUGBDATCELMNATCPLBTDRLNO $.
  $}

  ${
    $d y z A $.  $d y z B $.  $d x y z $.
    $( Commutation of union with indexed union.  (Contributed by Mario
       Carneiro, 18-Jan-2014.) $)
    iuncom4 $p |- U_ x e. A U. B = U. U_ x e. A B $=
      ( vy vz cuni ciun cv wcel wa wex df-rex rexbii rexcom4 bitri exbii eluni2
      wrex eliun 3bitr4i r19.41v anbi1i eqriv ) DABCFZGZABCGZFZDHZUDIZABRZUHEHZ
      IZEUFRZUHUEIUHUGIULECRZABRZUKCIZABRZULJZEKZUJUMUOUPULJZABRZEKZUSUOUTEKZAB
      RVBUNVCABULECLMUTAEBNOVAUREUPULABUAPOUIUNABEUHCQMUMUKUFIZULJZEKUSULEUFLVE
      UREVDUQULAUKBCSUBPOTAUHBUDSEUHUFQTUC $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Indexed union of a constant class, i.e. where ` B ` does not depend on
       ` x ` .  (Contributed by Jim Kingdon, 15-Aug-2018.) $)
    iunconstm $p |- ( E. x x e. A -> U_ x e. A B = B ) $=
      ( vy cv wcel wex ciun wrex eliun r19.9rmv bitr4id eqrdv ) AEBFAGZDABCHZCN
      DEZOFPCFZABIQAPBCJQAABKLM $.
  $}

  ${
    $d x z A $.  $d x z B $.  $d y z A $.
    $( Indexed intersection of a constant class, i.e. where ` B ` does not
       depend on ` x ` .  (Contributed by Jim Kingdon, 19-Dec-2018.) $)
    iinconstm $p |- ( E. y y e. A -> |^|_ x e. A B = B ) $=
      ( vz cv wcel wex ciin wral cvv wb vex eliin ax-mp r19.3rmv bitr4id eqrdv
      ) BFCGBHZEACDIZDSEFZTGZUADGZACJZUCUAKGUBUDLEMAUACDKNOUCABCPQR $.
  $}

  ${
    $d x y $.  $d y z A $.  $d x z B $.  $d z C $.
    $( Law combining indexed union with indexed intersection.  Eq. 14 in
       [KuratowskiMostowski] p. 109.  This theorem also appears as the last
       example at ~ http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29 .
       (Contributed by NM, 17-Aug-2004.)  (Proof shortened by Andrew Salmon,
       25-Jul-2011.) $)
    iuniin $p |- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C $=
      ( vz ciin ciun cv wcel wrex wral r19.12 cvv vex eliin ax-mp eliun 3imtr4i
      wb rexbii ralbii ssriv ) FACBDEGZHZBDACEHZGZFIZUDJZACKZUHUFJZBDLZUHUEJUHU
      GJZUHEJZBDLZACKUNACKZBDLUJULUNABCDMUIUOACUHNJZUIUOTFOZBUHDENPQUAUKUPBDAUH
      CERUBSAUHCUDRUQUMULTURBUHDUFNPQSUC $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.
    $( Subclass theorem for indexed union.  (Contributed by NM, 10-Dec-2004.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    iunss1 $p |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C ) $=
      ( vy wss ciun cv wcel wrex ssrexv eliun 3imtr4g ssrdv ) BCFZEABDGZACDGZOE
      HZDIZABJSACJRPIRQISABCKARBDLARCDLMN $.

    $( Subclass theorem for indexed union.  (Contributed by NM,
       24-Jan-2012.) $)
    iinss1 $p |- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C ) $=
      ( vy wss ciin cv wcel wral ssralv cvv wb vex eliin ax-mp 3imtr4g ssrdv )
      BCFZEACDGZABDGZSEHZDIZACJZUCABJZUBTIZUBUAIZUCABCKUBLIZUFUDMENZAUBCDLOPUHU
      GUEMUIAUBBDLOPQR $.

    $( Equality theorem for indexed union.  (Contributed by NM,
       27-Jun-1998.) $)
    iuneq1 $p |- ( A = B -> U_ x e. A C = U_ x e. B C ) $=
      ( wss wa ciun wceq iunss1 anim12i eqss 3imtr4i ) BCEZCBEZFABDGZACDGZEZPOE
      ZFBCHOPHMQNRABCDIACBDIJBCKOPKL $.

    $( Equality theorem for restricted existential quantifier.  (Contributed by
       NM, 27-Jun-1998.) $)
    iineq1 $p |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C ) $=
      ( vy wceq cv wcel wral cab ciin raleq abbidv df-iin 3eqtr4g ) BCFZEGDHZAB
      IZEJQACIZEJABDKACDKPRSEQABCLMAEBDNAECDNO $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.
    $( Subclass theorem for indexed union.  (Contributed by NM, 26-Nov-2003.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    ss2iun $p |- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C ) $=
      ( vy wss wral ciun cv wcel wrex ssel ralimi rexim syl eliun 3imtr4g ssrdv
      wi ) CDFZABGZEABCHZABDHZUAEIZCJZABKZUDDJZABKZUDUBJUDUCJUAUEUGSZABGUFUHSTU
      IABCDUDLMUEUGABNOAUDBCPAUDBDPQR $.

    $( Equality theorem for indexed union.  (Contributed by NM,
       22-Oct-2003.) $)
    iuneq2 $p |- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C ) $=
      ( wss wral wa ciun wceq ss2iun anim12i eqss ralbii r19.26 bitri 3imtr4i )
      CDEZABFZDCEZABFZGZABCHZABDHZEZUCUBEZGCDIZABFZUBUCIRUDTUEABCDJABDCJKUGQSGZ
      ABFUAUFUHABCDLMQSABNOUBUCLP $.

    $( Equality theorem for indexed intersection.  (Contributed by NM,
       22-Oct-2003.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    iineq2 $p |- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C ) $=
      ( vy wceq wral cv wcel cab ciin wb eleq2 ralimi syl abbidv df-iin 3eqtr4g
      ralbi ) CDFZABGZEHZCIZABGZEJUBDIZABGZEJABCKABDKUAUDUFEUAUCUELZABGUDUFLTUG
      ABCDUBMNUCUEABSOPAEBCQAEBDQR $.
  $}

  ${
    iuneq2i.1 $e |- ( x e. A -> B = C ) $.
    $( Equality inference for indexed union.  (Contributed by NM,
       22-Oct-2003.) $)
    iuneq2i $p |- U_ x e. A B = U_ x e. A C $=
      ( wceq ciun iuneq2 mprg ) CDFABCGABDGFABABCDHEI $.

    $( Equality inference for indexed intersection.  (Contributed by NM,
       22-Oct-2003.) $)
    iineq2i $p |- |^|_ x e. A B = |^|_ x e. A C $=
      ( wceq ciin iineq2 mprg ) CDFABCGABDGFABABCDHEI $.
  $}

  ${
    iineq2d.1 $e |- F/ x ph $.
    iineq2d.2 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
    $( Equality deduction for indexed intersection.  (Contributed by NM,
       7-Dec-2011.) $)
    iineq2d $p |- ( ph -> |^|_ x e. A B = |^|_ x e. A C ) $=
      ( wceq wral ciin cv wcel ex ralrimi iineq2 syl ) ADEHZBCIBCDJBCEJHAQBCFAB
      KCLQGMNBCDEOP $.
  $}

  ${
    $d x ph $.
    iuneq2dv.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
    $( Equality deduction for indexed union.  (Contributed by NM,
       3-Aug-2004.) $)
    iuneq2dv $p |- ( ph -> U_ x e. A B = U_ x e. A C ) $=
      ( wceq wral ciun ralrimiva iuneq2 syl ) ADEGZBCHBCDIBCEIGAMBCFJBCDEKL $.

    $( Equality deduction for indexed intersection.  (Contributed by NM,
       3-Aug-2004.) $)
    iineq2dv $p |- ( ph -> |^|_ x e. A B = |^|_ x e. A C ) $=
      ( nfv iineq2d ) ABCDEABGFH $.
  $}

  ${
    $d x A $.  $d x B $.
    iuneq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for indexed union, deduction version.  (Contributed by
       Drahflow, 22-Oct-2015.) $)
    iuneq1d $p |- ( ph -> U_ x e. A C = U_ x e. B C ) $=
      ( wceq ciun iuneq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.

    ${
      $d x ph $.
      iuneq12d.2 $e |- ( ph -> C = D ) $.
      $( Equality deduction for indexed union, deduction version.  (Contributed
         by Drahflow, 22-Oct-2015.) $)
      iuneq12d $p |- ( ph -> U_ x e. A C = U_ x e. B D ) $=
        ( ciun iuneq1d wceq cv wcel adantr iuneq2dv eqtrd ) ABCEIBDEIBDFIABCDEG
        JABDEFAEFKBLDMHNOP $.
    $}
  $}

  ${
    $d x ph $.  $d x A $.
    iuneq2d.2 $e |- ( ph -> B = C ) $.
    $( Equality deduction for indexed union.  (Contributed by Drahflow,
       22-Oct-2015.) $)
    iuneq2d $p |- ( ph -> U_ x e. A B = U_ x e. A C ) $=
      ( wceq cv wcel adantr iuneq2dv ) ABCDEADEGBHCIFJK $.
  $}

  ${
    $d x y $.  $d z A $.  $d z B $.  $d x z $.  $d y z $.
    nfiunxy.1 $e |- F/_ y A $.
    nfiunxy.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for indexed union.  (Contributed by
       Mario Carneiro, 25-Jan-2014.) $)
    nfiunxy $p |- F/_ y U_ x e. A B $=
      ( vz ciun cv wcel wrex cab df-iun nfcri nfrexw nfab nfcxfr ) BACDHGIDJZAC
      KZGLAGCDMSBGRBACEBGDFNOPQ $.

    $( Bound-variable hypothesis builder for indexed intersection.
       (Contributed by Mario Carneiro, 25-Jan-2014.) $)
    nfiinxy $p |- F/_ y |^|_ x e. A B $=
      ( vz ciin cv wcel wral cab df-iin nfcri nfralxy nfab nfcxfr ) BACDHGIDJZA
      CKZGLAGCDMSBGRBACEBGDFNOPQ $.
  $}

  ${
    $d x A $.  $d z A $.  $d z B $.  $d x z $.  $d y z $.
    nfiunya.1 $e |- F/_ y A $.
    nfiunya.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for indexed union.  (Contributed by
       Mario Carneiro, 25-Jan-2014.) $)
    nfiunya $p |- F/_ y U_ x e. A B $=
      ( vz ciun cv wcel wrex cab df-iun nfcri nfrexya nfab nfcxfr ) BACDHGIDJZA
      CKZGLAGCDMSBGRBACEBGDFNOPQ $.

    $( Bound-variable hypothesis builder for indexed intersection.
       (Contributed by Mario Carneiro, 25-Jan-2014.) $)
    nfiinya $p |- F/_ y |^|_ x e. A B $=
      ( vz ciin cv wcel wral cab df-iin nfcri nfralya nfab nfcxfr ) BACDHGIDJZA
      CKZGLAGCDMSBGRBACEBGDFNOPQ $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( Bound-variable hypothesis builder for indexed union.  (Contributed by
       NM, 12-Oct-2003.) $)
    nfiu1 $p |- F/_ x U_ x e. A B $=
      ( vy ciun cv wcel wrex cab df-iun nfre1 nfab nfcxfr ) AABCEDFCGZABHZDIADB
      CJOADNABKLM $.

    $( Bound-variable hypothesis builder for indexed intersection.
       (Contributed by NM, 15-Oct-2003.) $)
    nfii1 $p |- F/_ x |^|_ x e. A B $=
      ( vy ciin cv wcel wral cab df-iin nfra1 nfab nfcxfr ) AABCEDFCGZABHZDIADB
      CJOADNABKLM $.
  $}

  ${
    $d y z w A $.  $d y z w B $.  $d w C z $.  $d w x y z $.
    $( Alternate definition of indexed union when ` B ` is a set.  Definition
       15(a) of [Suppes] p. 44.  (Contributed by NM, 23-Mar-2006.)  (Proof
       shortened by Andrew Salmon, 25-Jul-2011.) $)
    dfiun2g $p |- ( A. x e. A B e. C ->
                  U_ x e. A B = U. { y | E. x e. A y = B } ) $=
      ( vz wcel wral ciun cv wceq wrex cab cuni wa wex nfra1 wb rsp bitrdi syl6
      clel3g rexbida rexcom4 r19.41v exbii exancom bitri eliun eluniab 3bitr4g
      imp eqrdv ) DEGZACHZFACDIZBJZDKZACLZBMNZUOFJZDGZACLZVAUQGZUSOBPZVAUPGVAUT
      GUOVCURVDOZACLZBPZVEUOVCVFBPZACLVHUOVBVIACUNACQUOAJCGZVBVIRZUOVJUNVKUNACS
      BVADEUBUAULUCVFABCUDTVHUSVDOZBPVEVGVLBURVDACUEUFUSVDBUGUHTAVACDUIUSBVAUJU
      KUM $.

    $( Alternate definition of indexed intersection when ` B ` is a set.
       (Contributed by Jeff Hankins, 27-Aug-2009.) $)
    dfiin2g $p |- ( A. x e. A B e. C
               -> |^|_ x e. A B = |^| { y | E. x e. A y = B } ) $=
      ( vw vz wcel wral cv cab wceq wrex wi wal df-ral wb eqeq1 albii bitr4i
      ciin cint eleq2 biimprcd alrimiv eqid imbi12d mpii impbid2 imim2i pm5.74d
      spcgv alimi albi syl sylbi alcom r19.23v vex rexbidv elab imbi1i 3bitr3ri
      19.21v bitrdi bitrid abbidv df-iin df-int 3eqtr4g ) DEHZACIZFJZDHZACIZFKG
      JZBJZDLZACMZBKZHZVMVPHZNZGOZFKACDUAVTUBVLVOWDFVOAJCHZVNNZAOZVLWDVNACPVLWG
      WEVPDLZWBNZGOZNZAOZWDVLWEVKNZAOZWGWLQZVKACPWNWFWKQZAOWOWMWPAWMWEVNWJVKVNW
      JQWEVKVNWJVNWIGWHWBVNVPDVMUCZUDUEVKWJDDLZVNDUFWIWRVNNGDEWHWHWRWBVNVPDDRWQ
      UGULUHUIUJUKUMWFWKAUNUOUPWIACIZGOZWEWINZGOZAOZWDWLWTXAAOZGOXCWSXDGWIACPSX
      AAGUQTWSWCGWSWHACMZWBNWCWHWBACURWAXEWBVSXEBVPGUSVQVPLVRWHACVQVPDRUTVAVBTS
      XBWKAWEWIGVDSVCVEVFVGAFCDVHFGVTVIVJ $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    dfiun2.1 $e |- B e. _V $.
    $( Alternate definition of indexed union when ` B ` is a set.  Definition
       15(a) of [Suppes] p. 44.  (Contributed by NM, 27-Jun-1998.)  (Revised by
       David Abernethy, 19-Jun-2012.) $)
    dfiun2 $p |- U_ x e. A B = U. { y | E. x e. A y = B } $=
      ( cvv wcel ciun cv wceq wrex cab cuni dfiun2g a1i mprg ) DFGZACDHBIDJACKB
      LMJACABCDFNQAICGEOP $.

    $( Alternate definition of indexed intersection when ` B ` is a set.
       Definition 15(b) of [Suppes] p. 44.  (Contributed by NM, 28-Jun-1998.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    dfiin2 $p |- |^|_ x e. A B = |^| { y | E. x e. A y = B } $=
      ( cvv wcel ciin cv wceq wrex cab cint dfiin2g a1i mprg ) DFGZACDHBIDJACKB
      LMJACABCDFNQAICGEOP $.
  $}

  ${
    $d x z $.  $d y z w $.  $d z A $.  $d z w B $.  $d z w C $.
    $( Define double indexed union.  (Contributed by FL, 6-Nov-2013.) $)
    dfiunv2 $p |- U_ x e. A U_ y e. B C =
      { z | E. x e. A E. y e. B z e. C } $=
      ( vw ciun cv wcel wrex cab wceq df-iun a1i iuneq2i vex weq eleq1 rexbidv
      elab rexbii abbii 3eqtri ) ADBEFHZHADGIZFJZBEKZGLZHCIZUIJZADKZCLUJFJZBEKZ
      ADKZCLADUEUIUEUIMAIDJBGEFNOPACDUINULUOCUKUNADUHUNGUJCQGCRUGUMBEUFUJFSTUAU
      BUCUD $.
  $}

  ${
    $d z y A $.  $d z x A $.  $d z B $.  $d z C $.
    cbviun.1 $e |- F/_ y B $.
    cbviun.2 $e |- F/_ x C $.
    cbviun.3 $e |- ( x = y -> B = C ) $.
    $( Rule used to change the bound variables in an indexed union, with the
       substitution specified implicitly by the hypothesis.  (Contributed by
       NM, 26-Mar-2006.)  (Revised by Andrew Salmon, 25-Jul-2011.) $)
    cbviun $p |- U_ x e. A B = U_ y e. A C $=
      ( vz cv wcel wrex cab ciun nfcri weq eleq2d cbvrex abbii df-iun 3eqtr4i )
      IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE
      TUA $.

    $( Change bound variables in an indexed intersection.  (Contributed by Jeff
       Hankins, 26-Aug-2009.)  (Revised by Mario Carneiro, 14-Oct-2016.) $)
    cbviin $p |- |^|_ x e. A B = |^|_ y e. A C $=
      ( vz cv wcel wral cab ciin nfcri weq eleq2d cbvral abbii df-iin 3eqtr4i )
      IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE
      TUA $.
  $}

  ${
    $d x A $.  $d y A $.  $d y B $.  $d x C $.
    cbviunv.1 $e |- ( x = y -> B = C ) $.
    $( Rule used to change the bound variables in an indexed union, with the
       substitution specified implicitly by the hypothesis.  (Contributed by
       NM, 15-Sep-2003.) $)
    cbviunv $p |- U_ x e. A B = U_ y e. A C $=
      ( nfcv cbviun ) ABCDEBDGAEGFH $.

    $( Change bound variables in an indexed intersection.  (Contributed by Jeff
       Hankins, 26-Aug-2009.) $)
    cbviinv $p |- |^|_ x e. A B = |^|_ y e. A C $=
      ( nfcv cbviin ) ABCDEBDGAEGFH $.
  $}

  ${
    $d x y C $.  $d y A $.  $d y B $.
    $( Subset theorem for an indexed union.  (Contributed by NM, 13-Sep-2003.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    iunss $p |- ( U_ x e. A B C_ C <-> A. x e. A B C_ C ) $=
      ( vy ciun wss cv wcel wrex cab wal wral df-iun sseq1i abss ssalel ralcom4
      wi ralbii r19.23v albii 3bitrri 3bitri ) ABCFZDGEHZCIZABJZEKZDGUHUFDIZSZE
      LZCDGZABMZUEUIDAEBCNOUHEDPUNUGUJSZELZABMUOABMZELULUMUPABECDQTUOAEBRUQUKEU
      GUJABUAUBUCUD $.
  $}

  ${
    $d x y C $.  $d y A $.  $d y B $.
    $( Subset implication for an indexed union.  (Contributed by NM,
       3-Sep-2003.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    ssiun $p |- ( E. x e. A C C_ B -> C C_ U_ x e. A B ) $=
      ( vy wss wrex ciun cv wcel ssel reximi r19.37av syl eliun imbitrrdi ssrdv
      wi ) DCFZABGZEDABCHZTEIZDJZUBCJZABGZUBUAJTUCUDRZABGUCUERSUFABDCUBKLUCUDAB
      MNAUBBCOPQ $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( Identity law for subset of an indexed union.  (Contributed by NM,
       12-Oct-2003.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    ssiun2 $p |- ( x e. A -> B C_ U_ x e. A B ) $=
      ( vy cv wcel ciun wrex rspe ex eliun imbitrrdi ssrdv ) AEBFZDCABCGZNDEZCF
      ZQABHZPOFNQRQABIJAPBCKLM $.
  $}

  ${
    $d x A $.  $d x C $.  $d x D $.
    ssiun2s.1 $e |- ( x = C -> B = D ) $.
    $( Subset relationship for an indexed union.  (Contributed by NM,
       26-Oct-2003.) $)
    ssiun2s $p |- ( C e. A -> D C_ U_ x e. A B ) $=
      ( ciun wss nfcv nfiu1 nfss cv wceq sseq1d ssiun2 vtoclgaf ) CABCGZHEQHADB
      ADIAEQAEIABCJKALDMCEQFNABCOP $.
  $}

  ${
    $d x y $.  $d x B $.  $d y C $.  $d x D $.
    $( A subclass condition on the members of two indexed classes ` C ( x ) `
       and ` D ( y ) ` that implies a subclass relation on their indexed
       unions.  Generalization of Proposition 8.6 of [TakeutiZaring] p. 59.
       Compare ~ uniss2 .  (Contributed by NM, 9-Dec-2004.) $)
    iunss2 $p |- ( A. x e. A E. y e. B C C_ D ->
                 U_ x e. A C C_ U_ y e. B D ) $=
      ( wss wrex wral ciun ssiun ralimi iunss sylibr ) EFGBDHZACIEBDFJZGZACIACE
      JPGOQACBDFEKLACEPMN $.
  $}

  ${
    $d C x $.  $d ph x $.
    iunssd.1 $e |- ( ( ph /\ x e. A ) -> B C_ C ) $.
    $( Subset theorem for an indexed union.  (Contributed by Glauco Siliprandi,
       8-Apr-2021.) $)
    iunssd $p |- ( ph -> U_ x e. A B C_ C ) $=
      ( wss wral ciun ralrimiva iunss sylibr ) ADEGZBCHBCDIEGAMBCFJBCDEKL $.
  $}

  ${
    $d y A $.  $d x y $.  $d x B $.
    $( The indexed union of a class abstraction.  (Contributed by NM,
       27-Dec-2004.) $)
    iunab $p |- U_ x e. A { y | ph } = { y | E. x e. A ph } $=
      ( cab ciun wrex wceq cv wcel nfcv nfab1 nfiunxy cleqf abid rexbii 3bitr4i
      wb eliun mpgbir ) BDACEZFZABDGZCEZHCIZUBJZUEUDJZRCCUBUDBCDUACDKACLMUCCLNU
      EUAJZBDGUCUFUGUHABDACOPBUEDUASUCCOQT $.

    $( The indexed union of a restricted class abstraction.  (Contributed by
       NM, 3-Jan-2004.)  (Proof shortened by Mario Carneiro, 14-Nov-2016.) $)
    iunrab $p |- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph } $=
      ( cv wcel cab ciun wrex crab iunab wceq df-rab a1i iuneq2i r19.42v eqtr4i
      wa abbii 3eqtr4i ) BDCFEGZASZCHZIUCBDJZCHZBDACEKZIABDJZCEKZUCBCDLBDUGUDUG
      UDMBFDGACENOPUIUBUHSZCHUFUHCENUEUJCUBABDQTRUA $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.
    iunxdif2.1 $e |- ( x = y -> C = D ) $.
    $( Indexed union with a class difference as its index.  (Contributed by NM,
       10-Dec-2004.) $)
    iunxdif2 $p |- ( A. x e. A E. y e. ( A \ B ) C C_ D ->
                 U_ y e. ( A \ B ) D = U_ x e. A C ) $=
      ( wss cdif wrex wral ciun wceq iunss2 difss iunss1 ax-mp cbviunv sseqtrri
      wa jctil eqss sylibr ) EFHBCDIZJACKZBUDFLZACELZHZUGUFHZTUFUGMUEUIUHABCUDE
      FNUFBCFLZUGUDCHUFUJHCDOBUDCFPQABCEFGRSUAUFUGUBUC $.
  $}

  ${
    $d y A $.  $d y B $.  $d y C $.  $d x y $.
    ssiinf.1 $e |- F/_ x C $.
    $( Subset theorem for an indexed intersection.  (Contributed by FL,
       15-Oct-2012.)  (Proof shortened by Mario Carneiro, 14-Oct-2016.) $)
    ssiinf $p |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B ) $=
      ( vy cv ciin wcel wral wss cvv vex eliin ax-mp ralbii nfcv ralcomf dfss3
      wb bitri 3bitr4i ) FGZABCHZIZFDJZUCCIZFDJZABJZDUDKDCKZABJUFUGABJZFDJUIUEU
      KFDUCLIUEUKTFMAUCBCLNOPUGFADBEFBQRUAFDUDSUJUHABFDCSPUB $.
  $}

  ${
    $d x C $.
    $( Subset theorem for an indexed intersection.  (Contributed by NM,
       15-Oct-2003.) $)
    ssiin $p |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B ) $=
      ( nfcv ssiinf ) ABCDADEF $.
  $}

  ${
    $d x y C $.  $d y A $.  $d y B $.
    $( Subset implication for an indexed intersection.  (Contributed by NM,
       15-Oct-2003.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    iinss $p |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C ) $=
      ( vy wss wrex ciin cv wcel wral cvv wb vex eliin ax-mp wi reximi r19.36av
      ssel syl biimtrid ssrdv ) CDFZABGZEABCHZDEIZUFJZUGCJZABKZUEUGDJZUGLJUHUJM
      ENAUGBCLOPUEUIUKQZABGUJUKQUDULABCDUGTRUIUKABSUAUBUC $.
  $}

  ${
    $d A y $.  $d B y $.  $d x y $.
    $( An indexed intersection is included in any of its members.  (Contributed
       by FL, 15-Oct-2012.) $)
    iinss2 $p |- ( x e. A -> |^|_ x e. A B C_ B ) $=
      ( vy cv wcel ciin wral wi cvv wb vex eliin ax-mp rsp sylbi com12 ssrdv )
      AEBFZDABCGZCDEZTFZSUACFZUBUCABHZSUCIUAJFUBUDKDLAUABCJMNUCABOPQR $.
  $}

  ${
    $d x y A $.
    $( Class union in terms of indexed union.  Definition in [Stoll] p. 43.
       (Contributed by NM, 28-Jun-1998.) $)
    uniiun $p |- U. A = U_ x e. A x $=
      ( vy cuni wel wrex cab cv ciun dfuni2 df-iun eqtr4i ) BDCAEABFCGABAHZICAB
      JACBMKL $.

    $( Class intersection in terms of indexed intersection.  Definition in
       [Stoll] p. 44.  (Contributed by NM, 28-Jun-1998.) $)
    intiin $p |- |^| A = |^|_ x e. A x $=
      ( vy cint wel wral cab cv ciin dfint2 df-iin eqtr4i ) BDCAEABFCGABAHZICAB
      JACBMKL $.

    $( An indexed union of singletons recovers the index set.  (Contributed by
       NM, 6-Sep-2005.) $)
    iunid $p |- U_ x e. A { x } = A $=
      ( vy cv csn ciun wceq cab wcel df-sn equcom abbii eqtri a1i iuneq2i iunab
      wrex risset abid2 3eqtr2i ) ABADZEZFABUACDZGZCHZFZBABUBUEUBUEGUABIUBUCUAG
      ZCHUECUAJUGUDCCAKLMNOUFUDABQZCHUCBIZCHBUDACBPUIUHCAUCBRLCBSTM $.
  $}

  ${
    $d x y $.  $d y A $.
    $( An indexed union of the empty set is empty.  (Contributed by NM,
       26-Mar-2003.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    iun0 $p |- U_ x e. A (/) = (/) $=
      ( vy c0 ciun cv wcel wrex wn noel a1i nrex eliun mtbir 2false eqriv ) CAB
      DEZDCFZQGZRDGZSTABHTABTIAFBGRJZKLARBDMNUAOP $.

    $( An empty indexed union is empty.  (Contributed by NM, 4-Dec-2004.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    0iun $p |- U_ x e. (/) A = (/) $=
      ( vy c0 ciun cv wcel wrex rex0 eliun mtbir noel 2false eqriv ) CADBEZDCFZ
      OGZPDGQPBGZADHRAIAPDBJKPLMN $.

    $( An empty indexed intersection is the universal class.  (Contributed by
       NM, 20-Oct-2005.) $)
    0iin $p |- |^|_ x e. (/) A = _V $=
      ( vy c0 ciin cv wcel wral cab cvv df-iin vex ral0 2th abbi2i eqtr4i ) ADB
      ECFZBGZADHZCIJACDBKSCJQJGSCLRAMNOP $.

    $( Indexed intersection with a universal index class.  (Contributed by NM,
       11-Sep-2008.) $)
    viin $p |- |^|_ x e. _V A = { y | A. x y e. A } $=
      ( cvv ciin cv wcel wral cab wal df-iin ralv abbii eqtri ) ADCEBFCGZADHZBI
      OAJZBIABDCKPQBOALMN $.
  $}

  ${
    $d x y A $.  $d y B $.
    $( There is an inhabited class in an indexed collection ` B ( x ) ` iff the
       indexed union of them is inhabited.  (Contributed by Jim Kingdon,
       16-Aug-2018.) $)
    iunn0m $p |- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B ) $=
      ( cv wcel wex wrex ciun rexcom4 eliun exbii bitr4i ) BEZDFZBGACHOACHZBGNA
      CDIFZBGOABCJQPBANCDKLM $.
  $}

  ${
    $d y A $.  $d x y $.
    $( Indexed intersection of a class builder.  (Contributed by NM,
       6-Dec-2011.) $)
    iinab $p |- |^|_ x e. A { y | ph } = { y | A. x e. A ph } $=
      ( cab ciin wral wceq cv wcel nfcv nfab1 nfiinxy cleqf abid ralbii cvv vex
      wb eliin ax-mp 3bitr4i mpgbir ) BDACEZFZABDGZCEZHCIZUEJZUHUGJZSCCUEUGBCDU
      DCDKACLMUFCLNUHUDJZBDGZUFUIUJUKABDACOPUHQJUIULSCRBUHDUDQTUAUFCOUBUC $.

    $d x A $.  $d x B $.
    $( Indexed intersection of a restricted class builder.  (Contributed by Jim
       Kingdon, 16-Aug-2018.) $)
    iinrabm $p |- ( E. x x e. A
          -> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } ) $=
      ( cv wcel wex wa wral cab crab ciin r19.28mv abbidv wceq df-rab a1i iinab
      iineq2i eqtri 3eqtr4g ) BFDGZBHZCFEGZAIZBDJZCKZUEABDJZIZCKBDACELZMZUICELU
      DUGUJCUEABDNOULBDUFCKZMUHBDUKUMUKUMPUCACEQRTUFBCDSUAUICEQUB $.
  $}

  ${
    $d y A $.  $d x y B $.  $d y C $.
    $( Indexed union of intersection.  Generalization of half of theorem
       "Distributive laws" in [Enderton] p. 30.  Use ~ uniiun to recover
       Enderton's theorem.  (Contributed by NM, 26-Mar-2004.) $)
    iunin2 $p |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C ) $=
      ( vy cin ciun cv wcel wrex r19.42v elin rexbii eliun anbi2i 3bitr4i eqriv
      wa ) EABCDFZGZCABDGZFZEHZSIZABJZUCCIZUCUAIZRZUCTIUCUBIUFUCDIZRZABJUFUIABJ
      ZRUEUHUFUIABKUDUJABUCCDLMUGUKUFAUCBDNOPAUCBSNUCCUALPQ $.

    $( Indexed union of intersection.  Generalization of half of theorem
       "Distributive laws" in [Enderton] p. 30.  Use ~ uniiun to recover
       Enderton's theorem.  (Contributed by Mario Carneiro, 30-Aug-2015.) $)
    iunin1 $p |- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B ) $=
      ( cin ciun iunin2 wceq cv wcel incom a1i iuneq2i 3eqtr4i ) ABCDEZFCABDFZE
      ABDCEZFPCEABCDGABQOQOHAIBJDCKLMPCKN $.

    $( Indexed union of class difference.  Compare to theorem "De Morgan's
       laws" in [Enderton] p. 31.  (Contributed by Jim Kingdon,
       17-Aug-2018.) $)
    iundif2ss $p |- U_ x e. A ( B \ C ) C_ ( B \ |^|_ x e. A C ) $=
      ( vy cdif ciun ciin cv wcel wrex wn wa eldif rexbii r19.42v wral rexnalim
      bitri cvv wb vex eliin ax-mp sylnibr anim2i sylbi eliun 3imtr4i ssriv ) E
      ABCDFZGZCABDHZFZEIZUKJZABKZUOCJZUOUMJZLZMZUOULJUOUNJUQURUODJZLZABKZMZVAUQ
      URVCMZABKVEUPVFABUOCDNOURVCABPSVDUTURVDVBABQZUSVBABRUOTJUSVGUAEUBAUOBDTUC
      UDUEUFUGAUOBUKUHUOCUMNUIUJ $.
  $}

  ${
    $d x B $.  $d y C $.  $d x D $.  $d x y $.
    $( Rearrange indexed unions over intersection.  (Contributed by NM,
       18-Dec-2008.) $)
    2iunin $p |- U_ x e. A U_ y e. B ( C i^i D )
        = ( U_ x e. A C i^i U_ y e. B D ) $=
      ( cin ciun wceq cv wcel iunin2 a1i iuneq2i iunin1 eqtri ) ACBDEFGHZHACEBD
      FHZGZHACEHRGACQSQSIAJCKBDEFLMNACREOP $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.
    $( Indexed intersection of class difference.  Compare to Theorem "De
       Morgan's laws" in [Enderton] p. 31.  (Contributed by Jim Kingdon,
       17-Aug-2018.) $)
    iindif2m $p |- ( E. x x e. A ->
                 |^|_ x e. A ( B \ C ) = ( B \ U_ x e. A C ) ) $=
      ( vy cv wcel wex cdif ciin ciun wral wn r19.28mv eldif bicomi ralbii wrex
      wa cvv ralnex eliun xchbinxr anbi2i 3bitr3g vex eliin ax-mp 3bitr4g eqrdv
      wb ) AFBGAHZEABCDIZJZCABDKZIZULEFZUMGZABLZUQCGZUQUOGZMZSZUQUNGZUQUPGULUTU
      QDGZMZSZABLUTVFABLZSUSVCUTVFABNVGURABURVGUQCDOPQVHVBUTVHVEABRVAVEABUAAUQB
      DUBUCUDUEUQTGVDUSUKEUFAUQBUMTUGUHUQCUOOUIUJ $.

    $( Indexed intersection of intersection.  Compare to Theorem "Distributive
       laws" in [Enderton] p. 30.  (Contributed by Jim Kingdon,
       17-Aug-2018.) $)
    iinin2m $p |- ( E. x x e. A ->
      |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) ) $=
      ( vy cv wcel wex cin ciin wral wa r19.28mv elin ralbii cvv wb eliin ax-mp
      3bitr4g vex anbi2i eqrdv ) AFBGAHZEABCDIZJZCABDJZIZUDEFZUEGZABKZUICGZUIUG
      GZLZUIUFGZUIUHGUDULUIDGZLZABKULUPABKZLUKUNULUPABMUJUQABUICDNOUMURULUIPGZU
      MURQEUAZAUIBDPRSUBTUSUOUKQUTAUIBUEPRSUICUGNTUC $.

    $( Indexed intersection of intersection.  Compare to Theorem "Distributive
       laws" in [Enderton] p. 30.  (Contributed by Jim Kingdon,
       17-Aug-2018.) $)
    iinin1m $p |- ( E. x x e. A ->
      |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) ) $=
      ( cv wcel wex cin ciin iinin2m wceq incom a1i iineq2i 3eqtr4g ) AEBFZAGAB
      CDHZICABDIZHABDCHZIRCHABCDJABSQSQKPDCLMNRCLO $.
  $}

  ${
    $d A x $.  $d X x $.  $d B x $.
    $( Elementhood in a relative intersection.  (Contributed by Mario Carneiro,
       30-Dec-2016.) $)
    elriin $p |- ( B e. ( A i^i |^|_ x e. X S ) <->
      ( B e. A /\ A. x e. X B e. S ) ) $=
      ( ciin cin wcel wa wral elin eliin pm5.32i bitri ) CBAEDFZGHCBHZCOHZIPCDH
      AEJZICBOKPQRACEDBLMN $.

    $( Relative intersection of an empty family.  (Contributed by Stefan
       O'Rear, 3-Apr-2015.) $)
    riin0 $p |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A ) $=
      ( c0 wceq ciin cin iineq1 ineq2d cvv 0iin ineq2i inv1 eqtri eqtrdi ) DEFZ
      BADCGZHBAECGZHZBQRSBADECIJTBKHBSKBACLMBNOP $.

    $( Relative intersection of an inhabited family.  (Contributed by Jim
       Kingdon, 19-Aug-2018.) $)
    riinm $p |- ( ( A. x e. X S C_ A /\ E. x x e. X ) ->
        ( A i^i |^|_ x e. X S ) = |^|_ x e. X S ) $=
      ( wss wral cv wcel wex ciin cin incom wceq wrex r19.2m ancoms iinss df-ss
      wa syl sylib eqtrid ) CBEZADFZAGDHAIZSZBADCJZKUGBKZUGBUGLUFUGBEZUHUGMUFUC
      ADNZUIUEUDUJUCAADOPADCBQTUGBRUAUB $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.  $d y V $.
    iinxsng.1 $e |- ( x = A -> B = C ) $.
    $( A singleton index picks out an instance of an indexed intersection's
       argument.  (Contributed by NM, 15-Jan-2012.)  (Proof shortened by Mario
       Carneiro, 17-Nov-2016.) $)
    iinxsng $p |- ( A e. V -> |^|_ x e. { A } B = C ) $=
      ( vy wcel csn ciin cv wral cab df-iin wceq eleq2d ralsng abbi1dv eqtrid )
      BEHZABIZCJGKZCHZAUALZGMDAGUACNTUDGDUCUBDHABEAKBOCDUBFPQRS $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x y D $.  $d x y E $.  $d y V $.
    $d y W $.
    iinxprg.1 $e |- ( x = A -> C = D ) $.
    iinxprg.2 $e |- ( x = B -> C = E ) $.
    $( Indexed intersection with an unordered pair index.  (Contributed by NM,
       25-Jan-2012.) $)
    iinxprg $p |- ( ( A e. V /\ B e. W )
        -> |^|_ x e. { A , B } C = ( D i^i E ) ) $=
      ( vy wcel wa cv cpr wral cab ciin wceq eleq2d ralprg abbidv df-in 3eqtr4g
      cin df-iin ) BGLCHLMZKNZDLZABCOZPZKQUHELZUHFLZMZKQAUJDREFUEUGUKUNKUIULUMA
      BCGHANZBSDEUHITUOCSDFUHJTUAUBAKUJDUFKEFUCUD $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.  $d y V $.
    iunxsng.1 $e |- ( x = A -> B = C ) $.
    $( A singleton index picks out an instance of an indexed union's argument.
       (Contributed by Mario Carneiro, 25-Jun-2016.) $)
    iunxsng $p |- ( A e. V -> U_ x e. { A } B = C ) $=
      ( vy wcel csn ciun cv wrex eliun wceq eleq2d rexsng bitrid eqrdv ) BEHZGA
      BIZCJZDGKZUAHUBCHZATLSUBDHZAUBTCMUCUDABEAKBNCDUBFOPQR $.
  $}

  ${
    $d x A $.  $d x C $.
    iunxsn.1 $e |- A e. _V $.
    iunxsn.2 $e |- ( x = A -> B = C ) $.
    $( A singleton index picks out an instance of an indexed union's argument.
       (Contributed by NM, 26-Mar-2004.)  (Proof shortened by Mario Carneiro,
       25-Jun-2016.) $)
    iunxsn $p |- U_ x e. { A } B = C $=
      ( cvv wcel csn ciun wceq iunxsng ax-mp ) BGHABICJDKEABCDGFLM $.
  $}

  ${
    $d A x y $.  $d B y $.  $d C y $.  $d V y $.
    iunxsngf.1 $e |- F/_ x C $.
    iunxsngf.2 $e |- ( x = A -> B = C ) $.
    $( A singleton index picks out an instance of an indexed union's argument.
       (Contributed by Mario Carneiro, 25-Jun-2016.)  (Revised by Thierry
       Arnoux, 2-May-2020.) $)
    iunxsngf $p |- ( A e. V -> U_ x e. { A } B = C ) $=
      ( vy wcel csn ciun cv wrex eliun wsbc rexsns nfcri wceq eleq2d bitrid
      sbciegf eqrdv ) BEIZHABJZCKZDHLZUEIUFCIZAUDMZUCUFDIZAUFUDCNUHUGABOUCUIUGA
      BPUGUIABEAHDFQALBRCDUFGSUATTUB $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.
    $( Separate a union in an indexed union.  (Contributed by NM, 27-Dec-2004.)
       (Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
    iunun $p |- U_ x e. A ( B u. C ) = ( U_ x e. A B u. U_ x e. A C ) $=
      ( vy cun ciun cv wcel wrex r19.43 elun rexbii eliun orbi12i 3bitr4i eqriv
      wo ) EABCDFZGZABCGZABDGZFZEHZSIZABJZUDUAIZUDUBIZRZUDTIUDUCIUDCIZUDDIZRZAB
      JUJABJZUKABJZRUFUIUJUKABKUEULABUDCDLMUGUMUHUNAUDBCNAUDBDNOPAUDBSNUDUAUBLP
      Q $.

    $( Separate a union in the index of an indexed union.  (Contributed by NM,
       26-Mar-2004.)  (Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
    iunxun $p |- U_ x e. ( A u. B ) C = ( U_ x e. A C u. U_ x e. B C ) $=
      ( vy cun ciun cv wcel wrex rexun eliun orbi12i bitr4i elun 3bitr4i eqriv
      wo ) EABCFZDGZABDGZACDGZFZEHZDIZASJZUDUAIZUDUBIZRZUDTIUDUCIUFUEABJZUEACJZ
      RUIUEABCKUGUJUHUKAUDBDLAUDCDLMNAUDSDLUDUAUBOPQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x D $.  $d x E $.
    iunxprg.1 $e |- ( x = A -> C = D ) $.
    iunxprg.2 $e |- ( x = B -> C = E ) $.
    $( A pair index picks out two instances of an indexed union's argument.
       (Contributed by Alexander van der Vekens, 2-Feb-2018.) $)
    iunxprg $p |- ( ( A e. V /\ B e. W )
                    -> U_ x e. { A , B } C = ( D u. E ) ) $=
      ( wcel wa cpr ciun csn cun wceq df-pr iuneq1 iunxsng iunxun adantr adantl
      ax-mp eqtri uneq12d eqtrid ) BGKZCHKZLZABCMZDNZABOZDNZACOZDNZPZEFPULAUMUO
      PZDNZUQUKURQULUSQBCRAUKURDSUDAUMUODUAUEUJUNEUPFUHUNEQUIABDEGITUBUIUPFQUHA
      CDFHJTUCUFUG $.
  $}

  ${
    $d x y z $.  $d x z A $.  $d z B $.  $d y z C $.
    $( Separate an indexed union in the index of an indexed union.
       (Contributed by Mario Carneiro, 5-Dec-2016.) $)
    iunxiun $p |- U_ x e. U_ y e. A B C = U_ y e. A U_ x e. B C $=
      ( vz ciun cv wcel wa wex eliun anbi1i r19.41v bitr4i exbii rexcom4 df-rex
      wrex 3bitr4i bitri rexbii eqriv ) FABCDGZEGZBCADEGZGZFHZEIZAUDSZUHUFIZBCS
      ZUHUEIUHUGIAHZUDIZUIJZAKZUMDIZUIJZAKZBCSZUJULUPURBCSZAKUTUOVAAUOUQBCSZUIJ
      VAUNVBUIBUMCDLMUQUIBCNOPURBACQOUIAUDRUKUSBCUKUIADSUSAUHDELUIADRUAUBTAUHUD
      ELBUHCUFLTUC $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( A relationship involving union and indexed intersection.  Exercise 23 of
       [Enderton] p. 33 but with equality changed to subset.  (Contributed by
       Jim Kingdon, 19-Aug-2018.) $)
    iinuniss $p |- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x ) $=
      ( vy cv wcel cint wo cab cun wral ciin r19.32vr elint2 orbi2i elun ralbii
      vex 3imtr4i ss2abi df-un df-iin 3sstr4i ) DEZBFZUDCGZFZHZDIUDBAEZJZFZACKZ
      DIBUFJACUJLUHULDUEUDUIFZACKZHUEUMHZACKUHULUEUMACMUGUNUEAUDCDRNOUKUOACUDBU
      IPQSTDBUFUAADCUJUBUC $.

    $( A relationship involving union and indexed union.  Exercise 25 of
       [Enderton] p. 33 but with biconditional changed to implication.
       (Contributed by Jim Kingdon, 19-Aug-2018.) $)
    iununir $p |- ( ( A u. U. B ) = U_ x e. B ( A u. x ) ->
        ( B = (/) -> A = (/) ) ) $=
      ( c0 wceq cuni cun cv ciun unieq eqtrdi uneq2d un0 iuneq1 eqeq12d biimpcd
      uni0 0iun ) CDEZBCFZGZACBAHGZIZEBDESUABUCDSUABDGBSTDBSTDFDCDJQKLBMKSUCADU
      BIDACDUBNAUBRKOP $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Subclass relationship for power class and union.  (Contributed by NM,
       18-Jul-2006.) $)
    sspwuni $p |- ( A C_ ~P B <-> U. A C_ B ) $=
      ( vx cv cpw wcel wral wss cuni vex elpw ralbii dfss3 unissb 3bitr4i ) CDZ
      BEZFZCAGPBHZCAGAQHAIBHRSCAPBCJKLCAQMCABNO $.

    $( Two ways to express a collection of subclasses.  (Contributed by NM,
       19-Jul-2006.) $)
    pwssb $p |- ( A C_ ~P B <-> A. x e. A x C_ B ) $=
      ( cpw wss cuni cv wral sspwuni unissb bitri ) BCDEBFCEAGCEABHBCIABCJK $.
  $}

  $( Characterization of the elements of a double power class: they are exactly
     the sets whose union is included in that class.  (Contributed by BJ,
     29-Apr-2021.) $)
  elpwpw $p |- ( A e. ~P ~P B <-> ( A e. _V /\ U. A C_ B ) ) $=
    ( cpw wcel cvv wss wa cuni elpwb sspwuni anbi2i bitri ) ABCZCDAEDZAMFZGNAHB
    FZGAMIOPNABJKL $.

  ${
    $d A x $.
    $( The double power class written as a class abstraction: the class of sets
       whose union is included in the given class.  (Contributed by BJ,
       29-Apr-2021.) $)
    pwpwab $p |- ~P ~P A = { x | U. x C_ A } $=
      ( cv cuni wss cpw wcel cvv vex elpwpw mpbiran abbi2i ) ACZDBEZABFFZMOGMHG
      NAIMBJKL $.
  $}

  ${
    $d A x $.
    $( The class of sets whose union is equal to a given class is included in
       the double power class of that class.  (Contributed by BJ,
       29-Apr-2021.) $)
    pwpwssunieq $p |- { x | U. x = A } C_ ~P ~P A $=
      ( cv cuni wceq cab wss cpw eqimss ss2abi pwpwab sseqtrri ) ACDZBEZAFMBGZA
      FBHHNOAMBIJABKL $.
  $}

  $( Relationship for power class and union.  (Contributed by NM,
     18-Jul-2006.) $)
  elpwuni $p |- ( B e. A -> ( A C_ ~P B <-> U. A = B ) ) $=
    ( cpw wss cuni wcel wceq sspwuni unissel expcom eqimss impbid1 bitrid ) ABC
    DAEZBDZBAFZNBGZABHPOQOPQABIJNBKLM $.

  ${
    $d x y A $.
    $( The power class of an intersection in terms of indexed intersection.
       Exercise 24(a) of [Enderton] p. 33.  (Contributed by NM,
       29-Nov-2003.) $)
    iinpw $p |- ~P |^| A = |^|_ x e. A ~P x $=
      ( vy cint cpw cv ciin wss wcel wral ssint vex elpw ralbii bitr4i wb eliin
      cvv ax-mp 3bitr4i eqriv ) CBDZEZABAFZEZGZCFZUBHZUGUEIZABJZUGUCIUGUFIZUHUG
      UDHZABJUJAUGBKUIULABUGUDCLZMNOUGUBUMMUGRIUKUJPUMAUGBUERQSTUA $.

    $( Inclusion of an indexed union of a power class in the power class of the
       union of its index.  Part of Exercise 24(b) of [Enderton] p. 33.
       (Contributed by NM, 25-Nov-2003.) $)
    iunpwss $p |- U_ x e. A ~P x C_ ~P U. A $=
      ( vy cpw ciun cuni wss wrex wcel ssiun eliun vex elpw rexbii bitri uniiun
      cv sseq2i 3imtr4i ssriv ) CABAQZDZEZBFZDZCQZUAGZABHZUFABUAEZGZUFUCIZUFUEI
      ZABUAUFJUKUFUBIZABHUHAUFBUBKUMUGABUFUACLZMNOULUFUDGUJUFUDUNMUDUIUFABPROST
      $.
  $}

  ${
    $d x X $.
    $( Relative intersection of an inhabited class.  (Contributed by Jim
       Kingdon, 19-Aug-2018.) $)
    rintm $p |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X ) $=
      ( cpw wss cv wcel wex wa cint cin incom wceq cuni intssuni2m ssid sspwuni
      mpbi sstrdi df-ss sylib eqtrid ) CBDZEAFCGAHIZBCJZKUEBKZUEBUELUDUEBEUFUEM
      UDUEUCNZBACUCOUCUCEUGBEUCPUCBQRSUEBTUAUB $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Disjointness
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c Disj_ $.

  $( Extend wff notation to include the statement that a family of classes
     ` B ( x ) ` , for ` x e. A ` , is a disjoint family. $)
  wdisj $a wff Disj_ x e. A B $.

  ${
    $d x y $.  $d y A $.  $d y B $.
    $( A collection of classes ` B ( x ) ` is disjoint when for each element
       ` y ` , it is in ` B ( x ) ` for at most one ` x ` .  (Contributed by
       Mario Carneiro, 14-Nov-2016.)  (Revised by NM, 16-Jun-2017.) $)
    df-disj $a |- ( Disj_ x e. A B <-> A. y E* x e. A y e. B ) $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    $( Alternate definition for disjoint classes.  (Contributed by NM,
       17-Jun-2017.) $)
    dfdisj2 $p |- ( Disj_ x e. A B <-> A. y E* x ( x e. A /\ y e. B ) ) $=
      ( wdisj cv wcel wrmo wal wa wmo df-disj df-rmo albii bitri ) ACDEBFDGZACH
      ZBIAFCGPJAKZBIABCDLQRBPACMNO $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.
    $( If each element of a collection is contained in a disjoint collection,
       the original collection is also disjoint.  (Contributed by Mario
       Carneiro, 14-Nov-2016.) $)
    disjss2 $p |- ( A. x e. A B C_ C ->
      ( Disj_ x e. A C -> Disj_ x e. A B ) ) $=
      ( vy wss wral cv wcel wrmo wal wdisj ssel ralimi rmoim syl alimdv df-disj
      wi 3imtr4g ) CDFZABGZEHZDIZABJZEKUCCIZABJZEKABDLABCLUBUEUGEUBUFUDSZABGUEU
      GSUAUHABCDUCMNUFUDABOPQAEBDRAEBCRT $.
  $}

  $( Equality theorem for disjoint collection.  (Contributed by Mario Carneiro,
     14-Nov-2016.) $)
  disjeq2 $p |- ( A. x e. A B = C ->
    ( Disj_ x e. A B <-> Disj_ x e. A C ) ) $=
    ( wceq wral wdisj wss wi eqimss2 ralimi disjss2 syl eqimss impbid ) CDEZABF
    ZABCGZABDGZQDCHZABFRSIPTABDCJKABDCLMQCDHZABFSRIPUAABCDNKABCDLMO $.

  ${
    $d x ph $.
    disjeq2dv.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
    $( Equality deduction for disjoint collection.  (Contributed by Mario
       Carneiro, 14-Nov-2016.) $)
    disjeq2dv $p |- ( ph -> ( Disj_ x e. A B <-> Disj_ x e. A C ) ) $=
      ( wceq wral wdisj wb ralrimiva disjeq2 syl ) ADEGZBCHBCDIBCEIJANBCFKBCDEL
      M $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.
    $( A subset of a disjoint collection is disjoint.  (Contributed by Mario
       Carneiro, 14-Nov-2016.) $)
    disjss1 $p |- ( A C_ B -> ( Disj_ x e. B C -> Disj_ x e. A C ) ) $=
      ( vy wss cv wcel wa wmo wal wdisj wi ssel anim1d alrimiv moim syl dfdisj2
      alimdv 3imtr4g ) BCFZAGZCHZEGDHZIZAJZEKUCBHZUEIZAJZEKACDLABDLUBUGUJEUBUIU
      FMZAKUGUJMUBUKAUBUHUDUEBCUCNOPUIUFAQRTAECDSAEBDSUA $.

    $( Equality theorem for disjoint collection.  (Contributed by Mario
       Carneiro, 14-Nov-2016.) $)
    disjeq1 $p |- ( A = B -> ( Disj_ x e. A C <-> Disj_ x e. B C ) ) $=
      ( wceq wdisj wss wi eqimss2 disjss1 syl eqimss impbid ) BCEZABDFZACDFZNCB
      GOPHCBIACBDJKNBCGPOHBCLABCDJKM $.

    disjeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for disjoint collection.  (Contributed by Mario
       Carneiro, 14-Nov-2016.) $)
    disjeq1d $p |- ( ph -> ( Disj_ x e. A C <-> Disj_ x e. B C ) ) $=
      ( wceq wdisj wb disjeq1 syl ) ACDGBCEHBDEHIFBCDEJK $.

    $d x ph $.
    disjeq12d.1 $e |- ( ph -> C = D ) $.
    $( Equality theorem for disjoint collection.  (Contributed by Mario
       Carneiro, 14-Nov-2016.) $)
    disjeq12d $p |- ( ph -> ( Disj_ x e. A C <-> Disj_ x e. B D ) ) $=
      ( wdisj disjeq1d wceq cv wcel adantr disjeq2dv bitrd ) ABCEIBDEIBDFIABCDE
      GJABDEFAEFKBLDMHNOP $.
  $}

  ${
    $d x y z A $.  $d z B $.  $d z C $.
    cbvdisj.1 $e |- F/_ y B $.
    cbvdisj.2 $e |- F/_ x C $.
    cbvdisj.3 $e |- ( x = y -> B = C ) $.
    $( Change bound variables in a disjoint collection.  (Contributed by Mario
       Carneiro, 14-Nov-2016.) $)
    cbvdisj $p |- ( Disj_ x e. A B <-> Disj_ y e. A C ) $=
      ( vz cv wcel wrmo wal wdisj nfcri weq eleq2d cbvrmo albii df-disj 3bitr4i
      ) IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBI
      CETUA $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x C $.
    cbvdisjv.1 $e |- ( x = y -> B = C ) $.
    $( Change bound variables in a disjoint collection.  (Contributed by Mario
       Carneiro, 11-Dec-2016.) $)
    cbvdisjv $p |- ( Disj_ x e. A B <-> Disj_ y e. A C ) $=
      ( nfcv cbvdisj ) ABCDEBDGAEGFH $.
  $}

  ${
    $d z A $.  $d z B $.  $d x y z $.
    nfdisjv.1 $e |- F/_ y A $.
    nfdisjv.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for disjoint collection.  (Contributed
       by Jim Kingdon, 19-Aug-2018.) $)
    nfdisjv $p |- F/ y Disj_ x e. A B $=
      ( vz wdisj cv wcel wa wmo wal dfdisj2 nfcv nfel nfcri nfan nfmo nfal
      nfxfr ) ACDHAIZCJZGIDJZKZALZGMBAGCDNUFBGUEBAUCUDBBUBCBUBOEPBGDFQRSTUA $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( Bound-variable hypothesis builder for disjoint collection.  (Contributed
       by Mario Carneiro, 14-Nov-2016.) $)
    nfdisj1 $p |- F/ x Disj_ x e. A B $=
      ( vy wdisj cv wcel wrmo wal df-disj nfrmo1 nfal nfxfr ) ABCEDFCGZABHZDIAA
      DBCJOADNABKLM $.
  $}

  ${
    $d i j x A $.  $d j x B $.  $d i x C $.
    disjnim.1 $e |- ( i = j -> B = C ) $.
    $( If a collection ` B ( i ) ` for ` i e. A ` is disjoint, then pairs are
       disjoint.  (Contributed by Mario Carneiro, 26-Mar-2015.)  (Revised by
       Jim Kingdon, 6-Oct-2022.) $)
    disjnim $p |- ( Disj_ i e. A B ->
      A. i e. A A. j e. A ( i =/= j -> ( B i^i C ) = (/) ) ) $=
      ( vx wdisj cv wcel wrmo wal wceq wi wral ralcom4 wex wn ralimi sylbi rmo4
      wne cin c0 df-disj eleq2d albii bitr4i 19.23v biimpi necon3ad notm0 exbii
      wa elin notbii bitr3i imbitrrdi sylbir ) DABHGIZBJZDAKZGLZDIZEIZUBZBCUCZU
      DMZNZEAOZDAOZDGABUEVCVAUTCJZUNZVDVEMZNZEAOZGLZDAOZVKVCVPDAOZGLVRVBVSGVAVL
      DEAVNBCUTFUFUAUGVPDGAPUHVQVJDAVQVOGLZEAOVJVOEGAPVTVIEAVTVFVMGQZRZVHVTWAVD
      VEVTWAVNNVMVNGUIUJUKVHUTVGJZGQZRWBGVGULWDWAWCVMGUTBCUOUMUPUQURSUSSTT $.
  $}

  ${
    $d i j x A $.  $d i j B $.
    $( If a collection ` B ( i ) ` for ` i e. A ` is disjoint, then pairs are
       disjoint.  (Contributed by Mario Carneiro, 14-Nov-2016.)  (Revised by
       Jim Kingdon, 7-Oct-2022.) $)
    disjnims $p |- ( Disj_ x e. A B -> A. i e. A A. j e. A
        ( i =/= j -> ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) ) $=
      ( wdisj cv csb wne c0 wceq wi wral nfcsb1v csbeq1a cbvdisj csbeq1 disjnim
      cin nfcv sylbi ) ABCFDBADGZCHZFUBEGZIUCAUDCHZSJKLEBMDBMADBCUCDCTAUBCNAUBC
      OPBUCUEDEAUBUDCQRUA $.
  $}

  ${
    $d x y z A $.  $d y z B $.  $d x y z C $.  $d x z D $.  $d x y z X $.
    $d x z Y $.
    disji.1 $e |- ( x = X -> B = C ) $.
    disji.2 $e |- ( x = Y -> B = D ) $.
    $( Property of a disjoint collection: if ` B ( X ) = C ` and
       ` B ( Y ) = D ` , and ` X =/= Y ` , then ` C ` and ` D ` are disjoint.
       (Contributed by Mario Carneiro, 14-Nov-2016.) $)
    disji2 $p |- ( ( Disj_ x e. A B /\ ( X e. A /\ Y e. A ) /\
      X =/= Y ) -> ( C i^i D ) = (/) ) $=
      ( vy vz wcel wne cin c0 wceq cv csb wi nfcv wdisj wa wral disjnims ineq1d
      neeq1 csbhypf eqeq1d imbi12d neeq2 ineq2d rspc2v mpan9 3impia ) ABCUAZFBL
      GBLUBZFGMZDENZOPZUOJQZKQZMZAUTCRZAVACRZNZOPZSZKBUCJBUCUPUQUSSZABCJKUDVGVH
      FVAMZDVDNZOPZSJKFGBBUTFPZVBVIVFVKUTFVAUFVLVEVJOVLVCDVDAJFCDAFTADTHUGUEUHU
      IVAGPZVIUQVKUSVAGFUJVMVJUROVMVDEDAKGCEAGTAETIUGUKUHUIULUMUN $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d x C $.
    $( If there is a function ` C ( y ) ` such that ` C ( y ) = x ` for all
       ` y e. B ( x ) ` , then the sets ` B ( x ) ` for distinct ` x e. A ` are
       disjoint.  (Contributed by Mario Carneiro, 10-Dec-2016.) $)
    invdisj $p |- ( A. x e. A A. y e. B C = x -> Disj_ x e. A B ) $=
      ( cv wceq wral wcel wa wmo wal wdisj nfra2xy wi df-ral rsp eqcom imbitrdi
      imim2i impd alimi sylbi mo2icl syl alrimi dfdisj2 sylibr ) EAFZGZBDHZACHZ
      UICIZBFDIZJZAKZBLACDMULUPBUJABCDNULUOUIEGZOZALZUPULUMUKOZALUSUKACPUTURAUT
      UMUNUQUKUNUQOUMUKUNUJUQUJBDQEUIRSTUAUBUCUOAEUDUEUFABCDUGUH $.
  $}

  ${
    $d A z $.  $d B x z $.  $d C y z $.  $d x y z $.
    $( The restricted class abstractions ` { x e. B | C = y } ` for distinct
       ` y e. A ` are disjoint.  (Contributed by AV, 6-May-2020.)  (Proof
       shortened by GG, 26-Jan-2024.) $)
    invdisjrab $p |- Disj_ y e. A { x e. B | C = y } $=
      ( vz cv csb wceq crab wral wdisj wcel wa nfcsb1v nfeq1 weq csbeq1a eqeq1d
      nfcv elrabf simprr sylan2b rgen2 invdisj ax-mp ) AFGZEHZBGZIZFEUIIZADJZKB
      CKBCULLUJBFCULUGULMUICMZUGDMZUJNUJUKUJAUGDAUGTADTAUHUIAUGEOPAFQEUHUIAUGER
      SUAUMUNUJUBUCUDBFCULUHUEUF $.
  $}

  ${
    $d A u v w x y $.  $d B u v w y $.  $d C u v x y $.  $d D u v x y $.
    $( A disjoint collection yields disjoint indexed unions for disjoint index
       sets.  (Contributed by Mario Carneiro, 26-Mar-2015.)  (Revised by Mario
       Carneiro, 14-Nov-2016.) $)
    disjiun $p |- ( ( Disj_ x e. A B /\
      ( C C_ A /\ D C_ A /\ ( C i^i D ) = (/) ) ) ->
        ( U_ x e. C B i^i U_ x e. D B ) = (/) ) $=
      ( vy vw vu vv wss cin c0 wa cv wcel wi wrex weq nfv simprl wdisj wceq w3a
      ciun wn wal wrmo df-disj wne elin eliun anbi12i wsb wral rmo3 csb nfcsb1v
      bitri nfcri csbeq1a eleq2d cbvrex simplrr simplrl ad3antrrr sseldd simprr
      sylib jca simp-4l sbie sylibr nfs1v nfan nfim anbi1d equequ1 sbequ anbi2d
      imbi12d equequ2 rspc2 syl3c eqeltrd inelcm syl2anc rexlimddv exp31 impcom
      sylbi biimtrid necon2bd impancom 3impa alimdv eq0 ) ABCUAZDBJZEBJZDEKZLUB
      ZUCZMFNZADCUDZAECUDZKZOZUEZFUFZXFLUBXBWQXIWQXCCOZABUGZFUFXBXIAFBCUHXBXKXH
      FWRWSXAXKXHPWRWSMZXKXAXHXLXKMZXGWTLXGXJADQZXJAEQZMZXMWTLUIZXGXCXDOZXCXEOZ
      MXPXCXDXEUJXRXNXSXOAXCDCUKAXCECUKULURXKXLXPXQPZXKXJXJAGUMZMZAGRZPZGBUNABU
      NZXLXTPXJAGBXJGSUOYEXLXPXQYEXLMZXPMZXCAHNZCUPZOZXQHDYGXNYJHDQYFXNXOTXJYJA
      HDXJHSAFYIAYHCUQUSZAHRZCYIXCAYHCUTVAZVBVHYGYHDOZYJMZMZXCAINZCUPZOZXQIEYPX
      OYSIEQYFXNXOYOVCXJYSAIEXJISAFYRAYQCUQUSZAIRCYRXCAYQCUTVAZVBVHYPYQEOZYSMZM
      ZYNYHEOXQYGYNYJUUCVDZUUDYHYQEUUDYHBOZYQBOZMYEYJXJAIUMZMZHIRZUUDUUFUUGUUDD
      BYHYFWRXPYOUUCYEWRWSTVEUUEVFUUDEBYQYFWSXPYOUUCYEWRWSVGVEYPUUBYSTZVFVIYEXL
      XPYOUUCVJUUDYJUUHYGYNYJUUCVCUUDYSUUHYPUUBYSVGXJYSAIYTUUAVKVLVIYDUUIUUJPZY
      JYAMZHGRZPAGYHYQBBUUMUUNAYJYAAYKXJAGVMVNUUNASVOUULGSYLYBUUMYCUUNYLXJYJYAY
      MVPAHGVQVTGIRZUUMUUIUUNUUJUUOYAUUHYJXJGIAVRVSGIHWAVTWBWCUUKWDYHDEWEWFWGWG
      WHWJWIWKWLWMWNWOWKWIFXFWPVL $.
  $}

  ${
    $d x y $.  $d y A $.
    $( Any collection of singletons is disjoint.  (Contributed by Mario
       Carneiro, 14-Nov-2016.) $)
    sndisj $p |- Disj_ x e. A { x } $=
      ( vy cv csn wdisj wcel wa wmo dfdisj2 moeq simpr velsn sylib eqcomd moimi
      weq ax-mp mpgbir ) ABADZEZFTBGZCDZUAGZHZAIZCACBUAJACQZAIUFAUCKUEUGAUEUCTU
      EUDCAQUBUDLCTMNOPRS $.
  $}

  $( Any collection of empty sets is disjoint.  (Contributed by Mario Carneiro,
     14-Nov-2016.) $)
  0disj $p |- Disj_ x e. A (/) $=
    ( c0 cv csn wss wral wdisj 0ss rgenw sndisj disjss2 mp2 ) CADEZFZABGABNHABC
    HOABNIJABKABCNLM $.

  ${
    $d x y A $.  $d y B $.
    $( A singleton collection is disjoint.  (Contributed by Mario Carneiro,
       14-Nov-2016.) $)
    disjxsn $p |- Disj_ x e. { A } B $=
      ( vy csn wdisj cv wcel wa wmo dfdisj2 wceq moeq elsni adantr moimi mpgbir
      ax-mp ) ABEZCFAGZSHZDGCHZIZAJZDADSCKTBLZAJUDABMUCUEAUAUEUBTBNOPRQ $.

    $( An empty collection is disjoint.  (Contributed by Mario Carneiro,
       14-Nov-2016.) $)
    disjx0 $p |- Disj_ x e. (/) B $=
      ( c0 csn wss wdisj 0ss disjxsn disjss1 mp2 ) CCDZEAKBFACBFKGACBHACKBIJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Binary relations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Extend wff notation to include the general binary relation predicate.
     Note that the syntax is simply three class symbols in a row.  Since binary
     relations are the only possible wff expressions consisting of three class
     expressions in a row, the syntax is unambiguous. $)
  wbr $a wff A R B $.

  $( Define a general binary relation.  Note that the syntax is simply three
     class symbols in a row.  Definition 6.18 of [TakeutiZaring] p. 29
     generalized to arbitrary classes.  This definition of relations is
     well-defined, although not very meaningful, when classes ` A ` and/or
     ` B ` are proper classes (i.e. are not sets).  On the other hand, we often
     find uses for this definition when ` R ` is a proper class (see for
     example ~ iprc ).  (Contributed by NM, 31-Dec-1993.) $)
  df-br $a |- ( A R B <-> <. A , B >. e. R ) $.

  $( Equality theorem for binary relations.  (Contributed by NM,
     4-Jun-1995.) $)
  breq $p |- ( R = S -> ( A R B <-> A S B ) ) $=
    ( wceq cop wcel wbr eleq2 df-br 3bitr4g ) CDEABFZCGLDGABCHABDHCDLIABCJABDJK
    $.

  $( Equality theorem for a binary relation.  (Contributed by NM,
     31-Dec-1993.) $)
  breq1 $p |- ( A = B -> ( A R C <-> B R C ) ) $=
    ( wceq cop wcel wbr opeq1 eleq1d df-br 3bitr4g ) ABEZACFZDGBCFZDGACDHBCDHMN
    ODABCIJACDKBCDKL $.

  $( Equality theorem for a binary relation.  (Contributed by NM,
     31-Dec-1993.) $)
  breq2 $p |- ( A = B -> ( C R A <-> C R B ) ) $=
    ( wceq cop wcel wbr opeq2 eleq1d df-br 3bitr4g ) ABEZCAFZDGCBFZDGCADHCBDHMN
    ODABCIJCADKCBDKL $.

  $( Equality theorem for a binary relation.  (Contributed by NM,
     8-Feb-1996.) $)
  breq12 $p |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) ) $=
    ( wceq wbr breq1 breq2 sylan9bb ) ABFACEGBCEGCDFBDEGABCEHCDBEIJ $.

  ${
    breqi.1 $e |- R = S $.
    $( Equality inference for binary relations.  (Contributed by NM,
       19-Feb-2005.) $)
    breqi $p |- ( A R B <-> A S B ) $=
      ( wceq wbr wb breq ax-mp ) CDFABCGABDGHEABCDIJ $.
  $}

  ${
    breq1i.1 $e |- A = B $.
    $( Equality inference for a binary relation.  (Contributed by NM,
       8-Feb-1996.) $)
    breq1i $p |- ( A R C <-> B R C ) $=
      ( wceq wbr wb breq1 ax-mp ) ABFACDGBCDGHEABCDIJ $.

    $( Equality inference for a binary relation.  (Contributed by NM,
       8-Feb-1996.) $)
    breq2i $p |- ( C R A <-> C R B ) $=
      ( wceq wbr wb breq2 ax-mp ) ABFCADGCBDGHEABCDIJ $.

    ${
      breq12i.2 $e |- C = D $.
      $( Equality inference for a binary relation.  (Contributed by NM,
         8-Feb-1996.)  (Proof shortened by Eric Schmidt, 4-Apr-2007.) $)
      breq12i $p |- ( A R C <-> B R D ) $=
        ( wceq wbr wb breq12 mp2an ) ABHCDHACEIBDEIJFGABCDEKL $.
    $}
  $}

  ${
    breq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for a binary relation.  (Contributed by NM,
       8-Feb-1996.) $)
    breq1d $p |- ( ph -> ( A R C <-> B R C ) ) $=
      ( wceq wbr wb breq1 syl ) ABCGBDEHCDEHIFBCDEJK $.

    $( Equality deduction for a binary relation.  (Contributed by NM,
       29-Oct-2011.) $)
    breqd $p |- ( ph -> ( C A D <-> C B D ) ) $=
      ( wceq wbr wb breq syl ) ABCGDEBHDECHIFDEBCJK $.

    $( Equality deduction for a binary relation.  (Contributed by NM,
       8-Feb-1996.) $)
    breq2d $p |- ( ph -> ( C R A <-> C R B ) ) $=
      ( wceq wbr wb breq2 syl ) ABCGDBEHDCEHIFBCDEJK $.

    ${
      breq12d.2 $e |- ( ph -> C = D ) $.
      $( Equality deduction for a binary relation.  (Contributed by NM,
         8-Feb-1996.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
      breq12d $p |- ( ph -> ( A R C <-> B R D ) ) $=
        ( wceq wbr wb breq12 syl2anc ) ABCIDEIBDFJCEFJKGHBCDEFLM $.
    $}

    ${
      breq123d.2 $e |- ( ph -> R = S ) $.
      breq123d.3 $e |- ( ph -> C = D ) $.
      $( Equality deduction for a binary relation.  (Contributed by NM,
         29-Oct-2011.) $)
      breq123d $p |- ( ph -> ( A R C <-> B S D ) ) $=
        ( wbr breq12d breqd bitrd ) ABDFKCEFKCEGKABCDEFHJLAFGCEIMN $.
    $}

    ${
      breqdi.1 $e |- ( ph -> C A D ) $.
      $( Equality deduction for a binary relation.  (Contributed by Thierry
         Arnoux, 5-Oct-2020.) $)
      breqdi $p |- ( ph -> C B D ) $=
        ( wbr breqd mpbid ) ADEBHDECHGABCDEFIJ $.
    $}

    ${
      breqan12i.2 $e |- ( ps -> C = D ) $.
      $( Equality deduction for a binary relation.  (Contributed by NM,
         8-Feb-1996.) $)
      breqan12d $p |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) ) $=
        ( wceq wbr wb breq12 syl2an ) ACDJEFJCEGKDFGKLBHICDEFGMN $.

      $( Equality deduction for a binary relation.  (Contributed by NM,
         8-Feb-1996.) $)
      breqan12rd $p |- ( ( ps /\ ph ) -> ( A R C <-> B R D ) ) $=
        ( wbr wb breqan12d ancoms ) ABCEGJDFGJKABCDEFGHILM $.
    $}
  $}

  ${
    eqnbrtrd.1 $e |- ( ph -> A = B ) $.
    eqnbrtrd.2 $e |- ( ph -> -. B R C ) $.
    $( Substitution of equal classes into the negation of a binary relation.
       (Contributed by Glauco Siliprandi, 3-Jan-2021.) $)
    eqnbrtrd $p |- ( ph -> -. A R C ) $=
      ( wbr breq1d mtbird ) ABDEHCDEHGABCDEFIJ $.
  $}

  $( Two classes are different if they don't have the same relationship to a
     third class.  (Contributed by NM, 3-Jun-2012.) $)
  nbrne1 $p |- ( ( A R B /\ -. A R C ) -> B =/= C ) $=
    ( wbr wn wne wceq breq2 biimpcd necon3bd imp ) ABDEZACDEZFBCGMNBCBCHMNBCADI
    JKL $.

  $( Two classes are different if they don't have the same relationship to a
     third class.  (Contributed by NM, 3-Jun-2012.) $)
  nbrne2 $p |- ( ( A R C /\ -. B R C ) -> A =/= B ) $=
    ( wbr wn wne wceq breq1 biimpcd necon3bd imp ) ACDEZBCDEZFABGMNABABHMNABCDI
    JKL $.

  ${
    eqbrtr.1 $e |- A = B $.
    eqbrtr.2 $e |- B R C $.
    $( Substitution of equal classes into a binary relation.  (Contributed by
       NM, 5-Aug-1993.) $)
    eqbrtri $p |- A R C $=
      ( wbr breq1i mpbir ) ACDGBCDGFABCDEHI $.
  $}

  ${
    eqbrtrd.1 $e |- ( ph -> A = B ) $.
    eqbrtrd.2 $e |- ( ph -> B R C ) $.
    $( Substitution of equal classes into a binary relation.  (Contributed by
       NM, 8-Oct-1999.) $)
    eqbrtrd $p |- ( ph -> A R C ) $=
      ( wbr breq1d mpbird ) ABDEHCDEHGABCDEFIJ $.
  $}

  ${
    eqbrtrr.1 $e |- A = B $.
    eqbrtrr.2 $e |- A R C $.
    $( Substitution of equal classes into a binary relation.  (Contributed by
       NM, 5-Aug-1993.) $)
    eqbrtrri $p |- B R C $=
      ( eqcomi eqbrtri ) BACDABEGFH $.
  $}

  ${
    eqbrtrrd.1 $e |- ( ph -> A = B ) $.
    eqbrtrrd.2 $e |- ( ph -> A R C ) $.
    $( Substitution of equal classes into a binary relation.  (Contributed by
       NM, 24-Oct-1999.) $)
    eqbrtrrd $p |- ( ph -> B R C ) $=
      ( eqcomd eqbrtrd ) ACBDEABCFHGI $.
  $}

  ${
    breqtr.1 $e |- A R B $.
    breqtr.2 $e |- B = C $.
    $( Substitution of equal classes into a binary relation.  (Contributed by
       NM, 5-Aug-1993.) $)
    breqtri $p |- A R C $=
      ( wbr breq2i mpbi ) ABDGACDGEBCADFHI $.
  $}

  ${
    breqtrd.1 $e |- ( ph -> A R B ) $.
    breqtrd.2 $e |- ( ph -> B = C ) $.
    $( Substitution of equal classes into a binary relation.  (Contributed by
       NM, 24-Oct-1999.) $)
    breqtrd $p |- ( ph -> A R C ) $=
      ( wbr breq2d mpbid ) ABCEHBDEHFACDBEGIJ $.
  $}

  ${
    breqtrr.1 $e |- A R B $.
    breqtrr.2 $e |- C = B $.
    $( Substitution of equal classes into a binary relation.  (Contributed by
       NM, 5-Aug-1993.) $)
    breqtrri $p |- A R C $=
      ( eqcomi breqtri ) ABCDECBFGH $.
  $}

  ${
    breqtrrd.1 $e |- ( ph -> A R B ) $.
    breqtrrd.2 $e |- ( ph -> C = B ) $.
    $( Substitution of equal classes into a binary relation.  (Contributed by
       NM, 24-Oct-1999.) $)
    breqtrrd $p |- ( ph -> A R C ) $=
      ( eqcomd breqtrd ) ABCDEFADCGHI $.
  $}

  ${
    3brtr3.1 $e |- A R B $.
    3brtr3.2 $e |- A = C $.
    3brtr3.3 $e |- B = D $.
    $( Substitution of equality into both sides of a binary relation.
       (Contributed by NM, 11-Aug-1999.) $)
    3brtr3i $p |- C R D $=
      ( eqbrtrri breqtri ) CBDEACBEGFIHJ $.
  $}

  ${
    3brtr4.1 $e |- A R B $.
    3brtr4.2 $e |- C = A $.
    3brtr4.3 $e |- D = B $.
    $( Substitution of equality into both sides of a binary relation.
       (Contributed by NM, 11-Aug-1999.) $)
    3brtr4i $p |- C R D $=
      ( eqbrtri breqtrri ) CBDECABEGFIHJ $.
  $}

  ${
    3brtr3d.1 $e |- ( ph -> A R B ) $.
    3brtr3d.2 $e |- ( ph -> A = C ) $.
    3brtr3d.3 $e |- ( ph -> B = D ) $.
    $( Substitution of equality into both sides of a binary relation.
       (Contributed by NM, 18-Oct-1999.) $)
    3brtr3d $p |- ( ph -> C R D ) $=
      ( wbr breq12d mpbid ) ABCFJDEFJGABDCEFHIKL $.
  $}

  ${
    3brtr4d.1 $e |- ( ph -> A R B ) $.
    3brtr4d.2 $e |- ( ph -> C = A ) $.
    3brtr4d.3 $e |- ( ph -> D = B ) $.
    $( Substitution of equality into both sides of a binary relation.
       (Contributed by NM, 21-Feb-2005.) $)
    3brtr4d $p |- ( ph -> C R D ) $=
      ( wbr breq12d mpbird ) ADEFJBCFJGADBECFHIKL $.
  $}

  ${
    3brtr3g.1 $e |- ( ph -> A R B ) $.
    3brtr3g.2 $e |- A = C $.
    3brtr3g.3 $e |- B = D $.
    $( Substitution of equality into both sides of a binary relation.
       (Contributed by NM, 16-Jan-1997.) $)
    3brtr3g $p |- ( ph -> C R D ) $=
      ( wbr breq12i sylib ) ABCFJDEFJGBDCEFHIKL $.
  $}

  ${
    3brtr4g.1 $e |- ( ph -> A R B ) $.
    3brtr4g.2 $e |- C = A $.
    3brtr4g.3 $e |- D = B $.
    $( Substitution of equality into both sides of a binary relation.
       (Contributed by NM, 16-Jan-1997.) $)
    3brtr4g $p |- ( ph -> C R D ) $=
      ( wbr breq12i sylibr ) ABCFJDEFJGDBECFHIKL $.
  $}

  ${
    eqbrtrid.1 $e |- A = B $.
    eqbrtrid.2 $e |- ( ph -> B R C ) $.
    $( B chained equality inference for a binary relation.  (Contributed by NM,
       11-Oct-1999.) $)
    eqbrtrid $p |- ( ph -> A R C ) $=
      ( eqid 3brtr4g ) ACDBDEGFDHI $.
  $}

  ${
    eqbrtrrid.1 $e |- B = A $.
    eqbrtrrid.2 $e |- ( ph -> B R C ) $.
    $( B chained equality inference for a binary relation.  (Contributed by NM,
       17-Sep-2004.) $)
    eqbrtrrid $p |- ( ph -> A R C ) $=
      ( eqid 3brtr3g ) ACDBDEGFDHI $.
  $}

  ${
    breqtrid.1 $e |- A R B $.
    breqtrid.2 $e |- ( ph -> B = C ) $.
    $( B chained equality inference for a binary relation.  (Contributed by NM,
       11-Oct-1999.) $)
    breqtrid $p |- ( ph -> A R C ) $=
      ( wbr a1i breqtrd ) ABCDEBCEHAFIGJ $.
  $}

  ${
    breqtrrid.1 $e |- A R B $.
    breqtrrid.2 $e |- ( ph -> C = B ) $.
    $( B chained equality inference for a binary relation.  (Contributed by NM,
       24-Apr-2005.) $)
    breqtrrid $p |- ( ph -> A R C ) $=
      ( eqcomd breqtrid ) ABCDEFADCGHI $.
  $}

  ${
    eqbrtrdi.1 $e |- ( ph -> A = B ) $.
    eqbrtrdi.2 $e |- B R C $.
    $( A chained equality inference for a binary relation.  (Contributed by NM,
       12-Oct-1999.) $)
    eqbrtrdi $p |- ( ph -> A R C ) $=
      ( wbr breq1d mpbiri ) ABDEHCDEHGABCDEFIJ $.
  $}

  ${
    eqbrtrrdi.1 $e |- ( ph -> B = A ) $.
    eqbrtrrdi.2 $e |- B R C $.
    $( A chained equality inference for a binary relation.  (Contributed by NM,
       4-Jan-2006.) $)
    eqbrtrrdi $p |- ( ph -> A R C ) $=
      ( eqcomd eqbrtrdi ) ABCDEACBFHGI $.
  $}

  ${
    breqtrdi.1 $e |- ( ph -> A R B ) $.
    breqtrdi.2 $e |- B = C $.
    $( A chained equality inference for a binary relation.  (Contributed by NM,
       11-Oct-1999.) $)
    breqtrdi $p |- ( ph -> A R C ) $=
      ( eqid 3brtr3g ) ABCBDEFBHGI $.
  $}

  ${
    breqtrrdi.1 $e |- ( ph -> A R B ) $.
    breqtrrdi.2 $e |- C = B $.
    $( A chained equality inference for a binary relation.  (Contributed by NM,
       24-Apr-2005.) $)
    breqtrrdi $p |- ( ph -> A R C ) $=
      ( eqcomi breqtrdi ) ABCDEFDCGHI $.
  $}

  ${
    ssbrd.1 $e |- ( ph -> A C_ B ) $.
    $( Deduction from a subclass relationship of binary relations.
       (Contributed by NM, 30-Apr-2004.) $)
    ssbrd $p |- ( ph -> ( C A D -> C B D ) ) $=
      ( cop wcel wbr sseld df-br 3imtr4g ) ADEGZBHMCHDEBIDECIABCMFJDEBKDECKL $.
  $}

  $( Implication from a subclass relationship of binary relations.
     (Contributed by Peter Mazsa, 11-Nov-2019.) $)
  ssbr $p |- ( A C_ B -> ( C A D -> C B D ) ) $=
    ( wss id ssbrd ) ABEZABCDHFG $.

  ${
    ssbri.1 $e |- A C_ B $.
    $( Inference from a subclass relationship of binary relations.
       (Contributed by NM, 28-Mar-2007.)  (Revised by Mario Carneiro,
       8-Feb-2015.) $)
    ssbri $p |- ( C A D -> C B D ) $=
      ( wbr wi wtru wss a1i ssbrd mptru ) CDAFCDBFGHABCDABIHEJKL $.
  $}

  ${
    nfbrd.2 $e |- ( ph -> F/_ x A ) $.
    nfbrd.3 $e |- ( ph -> F/_ x R ) $.
    nfbrd.4 $e |- ( ph -> F/_ x B ) $.
    $( Deduction version of bound-variable hypothesis builder ~ nfbr .
       (Contributed by NM, 13-Dec-2005.)  (Revised by Mario Carneiro,
       14-Oct-2016.) $)
    nfbrd $p |- ( ph -> F/ x A R B ) $=
      ( wbr cop wcel df-br nfopd nfeld nfxfrd ) CDEICDJZEKABCDELABPEABCDFHMGNO
      $.
  $}

  ${
    nfbr.1 $e |- F/_ x A $.
    nfbr.2 $e |- F/_ x R $.
    nfbr.3 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for binary relation.  (Contributed by
       NM, 1-Sep-1999.)  (Revised by Mario Carneiro, 14-Oct-2016.) $)
    nfbr $p |- F/ x A R B $=
      ( wbr wnf wtru wnfc a1i nfbrd mptru ) BCDHAIJABCDABKJELADKJFLACKJGLMN $.
  $}

  ${
    $d x y $.  $d y z A $.  $d y z R $.
    $( Relationship between a binary relation and a class abstraction.
       (Contributed by Andrew Salmon, 8-Jul-2011.) $)
    brab1 $p |- ( x R A <-> x e. { z | z R A } ) $=
      ( vy cv wbr wsbc cab wcel cvv wb vex breq1 sbcie2g ax-mp df-sbc bitr3i )
      AFZCDGZBFZCDGZBSHZSUBBIJSKJUCTLAMUBEFZCDGTBESKUAUDCDNUDSCDNOPUBBSQR $.
  $}

  $( The empty binary relation never holds.  (Contributed by NM,
     23-Aug-2018.) $)
  br0 $p |- -. A (/) B $=
    ( c0 wbr cop wcel noel df-br mtbir ) ABCDABEZCFJGABCHI $.

  $( If two sets are in a binary relation, the relation cannot be empty.  In
     fact, the relation is also inhabited, as seen at ~ brm .  (Contributed by
     Alexander van der Vekens, 7-Jul-2018.) $)
  brne0 $p |- ( A R B -> R =/= (/) ) $=
    ( wbr cop wcel c0 wne df-br ne0i sylbi ) ABCDABEZCFCGHABCICLJK $.

  ${
    $d A x $.  $d B x $.  $d R x $.

    $( If two sets are in a binary relation, the relation is inhabited.
       (Contributed by Jim Kingdon, 31-Dec-2023.) $)
    brm $p |- ( A R B -> E. x x e. R ) $=
      ( wbr cop wcel cv wex df-br elex2 sylbi ) BCDEBCFZDGAHDGAIBCDJAMDKL $.
  $}

  $( The union of two binary relations.  (Contributed by NM, 21-Dec-2008.) $)
  brun $p |- ( A ( R u. S ) B <-> ( A R B \/ A S B ) ) $=
    ( cop cun wcel wo wbr elun df-br orbi12i 3bitr4i ) ABEZCDFZGNCGZNDGZHABOIAB
    CIZABDIZHNCDJABOKRPSQABCKABDKLM $.

  $( The intersection of two relations.  (Contributed by FL, 7-Oct-2008.) $)
  brin $p |- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) ) $=
    ( cop cin wcel wa wbr elin df-br anbi12i 3bitr4i ) ABEZCDFZGNCGZNDGZHABOIAB
    CIZABDIZHNCDJABOKRPSQABCKABDKLM $.

  $( The difference of two binary relations.  (Contributed by Scott Fenton,
     11-Apr-2011.) $)
  brdif $p |- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) ) $=
    ( cop cdif wcel wn wa wbr eldif df-br notbii anbi12i 3bitr4i ) ABEZCDFZGPCG
    ZPDGZHZIABQJABCJZABDJZHZIPCDKABQLUARUCTABCLUBSABDLMNO $.

  ${
    $d y A $.  $d y B $.  $d y C $.  $d y D $.  $d y R $.  $d x y $.
    $( Move substitution in and out of a binary relation.  (Contributed by NM,
       13-Dec-2005.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
    sbcbrg $p |- ( A e. D -> ( [. A / x ]. B R C <->
           [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) ) $=
      ( vy wbr wsb cv csb wsbc dfsbcq2 wceq csbeq1 breq123d nfcsb1v weq csbeq1a
      nfbr sbie vtoclbg ) CDFHZAGIAGJZCKZAUDDKZAUDFKZHZUCABLABCKZABDKZABFKZHGBE
      UCAGBMUDBNUEUIUFUJUGUKAUDBCOAUDBFOAUDBDOPUCUHAGAUEUFUGAUDCQAUDFQAUDDQTAGR
      CUEDUFFUGAUDCSAUDFSAUDDSPUAUB $.
  $}

  ${
    $d x R $.
    $( Move substitution in and out of a binary relation.  (Contributed by NM,
       13-Dec-2005.) $)
    sbcbr12g $p |- ( A e. D ->
                 ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) ) $=
      ( wcel wbr wsbc csb sbcbrg csbconstg breqd bitrd ) BEGZCDFHABIABCJZABDJZA
      BFJZHPQFHABCDEFKORFPQABFELMN $.
  $}

  ${
    $d x C $.  $d x R $.
    $( Move substitution in and out of a binary relation.  (Contributed by NM,
       13-Dec-2005.) $)
    sbcbr1g $p |- ( A e. D ->
                  ( [. A / x ]. B R C <-> [_ A / x ]_ B R C ) ) $=
      ( wcel wbr wsbc csb sbcbr12g csbconstg breq2d bitrd ) BEGZCDFHABIABCJZABD
      JZFHPDFHABCDEFKOQDPFABDELMN $.
  $}

  ${
    $d x B $.  $d x R $.
    $( Move substitution in and out of a binary relation.  (Contributed by NM,
       13-Dec-2005.) $)
    sbcbr2g $p |- ( A e. D ->
                  ( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) ) $=
      ( wcel wbr wsbc csb sbcbr12g csbconstg breq1d bitrd ) BEGZCDFHABIABCJZABD
      JZFHCQFHABCDEFKOPCQFABCELMN $.
  $}

  ${
    $d A x $.  $d B x y $.  $d R x $.  $d X x $.  $d Y x $.
    $( Restricted existential specialization with a restricted universal
       quantifier over a relation, closed form.  (Contributed by AV,
       20-Aug-2022.) $)
    brralrspcev $p |- ( ( B e. X /\ A. y e. Y A R B )
                        -> E. x e. X A. y e. Y A R x ) $=
      ( cv wbr wral wceq breq2 ralbidv rspcev ) CAHZEIZBGJCDEIZBGJADFODKPQBGODC
      ELMN $.

    $d ph x $.  $d ps x $.
    $( Restricted existential specialization with a restricted universal
       quantifier over an implication with a relation in the antecedent, closed
       form.  (Contributed by AV, 20-Aug-2022.) $)
    brimralrspcev $p |- ( ( B e. X /\ A. y e. Y ( ( ph /\ A R B ) -> ps ) )
                          -> E. x e. X A. y e. Y ( ( ph /\ A R x ) -> ps ) ) $=
      ( cv wbr wa wceq breq2 anbi2d rspceaimv ) AECJZGKZLAEFGKZLBCDFHIQFMRSAQFE
      GNOP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordered-pair class abstractions (class builders)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c |-> $.  $( Maps-to symbol $)

  $( Extend class notation to include ordered-pair class abstraction (class
     builder). $)
  copab $a class { <. x , y >. | ph } $.

  $( Extend the definition of a class to include maps-to notation for defining
     a function via a rule. $)
  cmpt $a class ( x e. A |-> B ) $.

  ${
    $d x z $.  $d y z $.  $d z ph $.
    $( Define the class abstraction of a collection of ordered pairs.
       Definition 3.3 of [Monk1] p. 34.  Usually ` x ` and ` y ` are distinct,
       although the definition doesn't strictly require it.  The brace notation
       is called "class abstraction" by Quine; it is also (more commonly)
       called a "class builder" in the literature.  (Contributed by NM,
       4-Jul-1994.) $)
    df-opab $a |- { <. x , y >. | ph } =
                  { z | E. x E. y ( z = <. x , y >. /\ ph ) } $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    $( Define maps-to notation for defining a function via a rule.  Read as
       "the function defined by the map from ` x ` (in ` A ` ) to
       ` B ( x ) ` ".  The class expression ` B ` is the value of the function
       at ` x ` and normally contains the variable ` x ` .  Similar to the
       definition of mapping in [ChoquetDD] p. 2.  (Contributed by NM,
       17-Feb-2008.) $)
    df-mpt $a |- ( x e. A |-> B ) =
                    { <. x , y >. | ( x e. A /\ y = B ) } $.
  $}

  ${
    $d x z R $.  $d y z R $.
    $( The collection of ordered pairs in a class is a subclass of it.
       (Contributed by NM, 27-Dec-1996.)  (Proof shortened by Andrew Salmon,
       9-Jul-2011.) $)
    opabss $p |- { <. x , y >. | x R y } C_ R $=
      ( vz cv wbr copab cop wceq wa wex cab df-opab df-br eleq1 biimpar sylan2b
      wcel exlimivv abssi eqsstri ) AEZBEZCFZABGDEZUBUCHZIZUDJZBKAKZDLCUDABDMUI
      DCUHUECRZABUDUGUFCRZUJUBUCCNUGUJUKUEUFCOPQSTUA $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.  $d z ps $.  $d z ch $.
    opabbid.1 $e |- F/ x ph $.
    opabbid.2 $e |- F/ y ph $.
    opabbid.3 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal ordered-pair class abstractions (deduction
       form).  (Contributed by NM, 21-Feb-2004.)  (Proof shortened by Andrew
       Salmon, 9-Jul-2011.) $)
    opabbid $p |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } ) $=
      ( vz cv cop wceq wa wex cab copab anbi2d exbid abbidv df-opab 3eqtr4g ) A
      IJDJEJKLZBMZENZDNZIOUBCMZENZDNZIOBDEPCDEPAUEUHIAUDUGDFAUCUFEGABCUBHQRRSBD
      EITCDEITUA $.
  $}

  ${
    $d x ph $.  $d y ph $.
    opabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal ordered-pair class abstractions (deduction
       form).  (Contributed by NM, 15-May-1995.) $)
    opabbidv $p |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } ) $=
      ( nfv opabbid ) ABCDEADGAEGFH $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.  $d z ps $.
    opabbii.1 $e |- ( ph <-> ps ) $.
    $( Equivalent wff's yield equal class abstractions.  (Contributed by NM,
       15-May-1995.) $)
    opabbii $p |- { <. x , y >. | ph } = { <. x , y >. | ps } $=
      ( vz cv wceq copab eqid wb a1i opabbidv ax-mp ) FGZOHZACDIBCDIHOJPABCDABK
      PELMN $.
  $}

  ${
    $d x z w $.  $d y z w $.  $d ph w $.
    nfopab.1 $e |- F/ z ph $.
    $( Bound-variable hypothesis builder for class abstraction.  (Contributed
       by NM, 1-Sep-1999.)  Remove disjoint variable conditions.  (Revised by
       Andrew Salmon, 11-Jul-2011.) $)
    nfopab $p |- F/_ z { <. x , y >. | ph } $=
      ( vw copab cv cop wceq wa wex cab df-opab nfv nfan nfex nfab nfcxfr ) DAB
      CGFHBHCHIJZAKZCLZBLZFMABCFNUCDFUBDBUADCTADTDOEPQQRS $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    $( The first abstraction variable in an ordered-pair class abstraction
       (class builder) is effectively not free.  (Contributed by NM,
       16-May-1995.)  (Revised by Mario Carneiro, 14-Oct-2016.) $)
    nfopab1 $p |- F/_ x { <. x , y >. | ph } $=
      ( vz copab cv cop wceq wa wex cab df-opab nfe1 nfab nfcxfr ) BABCEDFBFCFG
      HAICJZBJZDKABCDLQBDPBMNO $.

    $( The second abstraction variable in an ordered-pair class abstraction
       (class builder) is effectively not free.  (Contributed by NM,
       16-May-1995.)  (Revised by Mario Carneiro, 14-Oct-2016.) $)
    nfopab2 $p |- F/_ y { <. x , y >. | ph } $=
      ( vz copab cv cop wceq wa wex cab df-opab nfe1 nfex nfab nfcxfr ) CABCEDF
      BFCFGHAIZCJZBJZDKABCDLSCDRCBQCMNOP $.
  $}

  ${
    $d x y z w v $.  $d v ph $.  $d v ps $.
    cbvopab.1 $e |- F/ z ph $.
    cbvopab.2 $e |- F/ w ph $.
    cbvopab.3 $e |- F/ x ps $.
    cbvopab.4 $e |- F/ y ps $.
    cbvopab.5 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables in an ordered-pair class
       abstraction, using implicit substitution.  (Contributed by NM,
       14-Sep-2003.) $)
    cbvopab $p |- { <. x , y >. | ph } = { <. z , w >. | ps } $=
      ( vv cv cop wceq wa wex cab nfv nfan opeq12 eqeq2d anbi12d cbvex2 df-opab
      copab abbii 3eqtr4i ) LMZCMZDMZNZOZAPZDQCQZLRUIEMZFMZNZOZBPZFQEQZLRACDUFB
      EFUFUOVALUNUTCDEFUMAEUMESGTUMAFUMFSHTUSBCUSCSITUSBDUSDSJTUJUPOUKUQOPZUMUS
      ABVBULURUIUJUKUPUQUAUBKUCUDUGACDLUEBEFLUEUH $.
  $}

  ${
    $d x y z w $.  $d z w ph $.  $d x y ps $.
    cbvopabv.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables in an ordered-pair class
       abstraction, using implicit substitution.  (Contributed by NM,
       15-Oct-1996.) $)
    cbvopabv $p |- { <. x , y >. | ph } = { <. z , w >. | ps } $=
      ( nfv cbvopab ) ABCDEFAEHAFHBCHBDHGI $.
  $}

  ${
    $d v w x y $.  $d v w y z $.  $d v w ph $.  $d v w ps $.
    cbvopab1.1 $e |- F/ z ph $.
    cbvopab1.2 $e |- F/ x ps $.
    cbvopab1.3 $e |- ( x = z -> ( ph <-> ps ) ) $.
    $( Change first bound variable in an ordered-pair class abstraction, using
       explicit substitution.  (Contributed by NM, 6-Oct-2004.)  (Revised by
       Mario Carneiro, 14-Oct-2016.) $)
    cbvopab1 $p |- { <. x , y >. | ph } = { <. z , y >. | ps } $=
      ( vw vv cv cop wceq wa wex cab copab wsb nfv nfan nfs1v nfex opeq1 eqeq2d
      sbequ12 anbi12d exbidv cbvex nfsb sbequ sbie bitrdi bitri df-opab 3eqtr4i
      abbii ) IKZCKZDKZLZMZANZDOZCOZIPUQEKZUSLZMZBNZDOZEOZIPACDQBEDQVDVJIVDUQJK
      ZUSLZMZACJRZNZDOZJOVJVCVPCJVCJSVOCDVMVNCVMCSACJUATUBURVKMZVBVODVQVAVMAVNV
      QUTVLUQURVKUSUCUDACJUEUFUGUHVPVIJEVOEDVMVNEVMESACJEFUITUBVIJSVKVEMZVOVHDV
      RVMVGVNBVRVLVFUQVKVEUSUCUDVRVNACERBAJECUJABCEGHUKULUFUGUHUMUPACDIUNBEDIUN
      UO $.
  $}

  ${
    $d w x y z $.  $d w ph $.  $d w ps $.
    cbvopab2.1 $e |- F/ z ph $.
    cbvopab2.2 $e |- F/ y ps $.
    cbvopab2.3 $e |- ( y = z -> ( ph <-> ps ) ) $.
    $( Change second bound variable in an ordered-pair class abstraction, using
       explicit substitution.  (Contributed by NM, 22-Aug-2013.) $)
    cbvopab2 $p |- { <. x , y >. | ph } = { <. x , z >. | ps } $=
      ( vw cv cop wceq wa wex cab copab nfv nfan opeq2 df-opab anbi12d 3eqtr4i
      eqeq2d cbvex exbii abbii ) IJZCJZDJZKZLZAMZDNZCNZIOUGUHEJZKZLZBMZENZCNZIO
      ACDPBCEPUNUTIUMUSCULURDEUKAEUKEQFRUQBDUQDQGRUIUOLZUKUQABVAUJUPUGUIUOUHSUC
      HUAUDUEUFACDITBCEITUB $.
  $}

  ${
    $d x y z w $.  $d z w ph $.
    $( Change first bound variable in an ordered-pair class abstraction, using
       explicit substitution.  (Contributed by NM, 31-Jul-2003.) $)
    cbvopab1s $p |- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph } $=
      ( vw cv cop wceq wa wex cab wsb copab nfv nfs1v nfan opeq1 eqeq2d df-opab
      nfex sbequ12 anbi12d exbidv cbvex abbii 3eqtr4i ) EFZBFZCFZGZHZAIZCJZBJZE
      KUGDFZUIGZHZABDLZIZCJZDJZEKABCMURDCMUNVAEUMUTBDUMDNUSBCUQURBUQBNABDOPTUHU
      OHZULUSCVBUKUQAURVBUJUPUGUHUOUIQRABDUAUBUCUDUEABCESURDCESUF $.
  $}

  ${
    $d x y $.  $d y z $.  $d z ph $.  $d x ps $.
    cbvopab1v.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
    $( Rule used to change the first bound variable in an ordered pair
       abstraction, using implicit substitution.  (Contributed by NM,
       31-Jul-2003.)  (Proof shortened by Eric Schmidt, 4-Apr-2007.) $)
    cbvopab1v $p |- { <. x , y >. | ph } = { <. z , y >. | ps } $=
      ( nfv cbvopab1 ) ABCDEAEGBCGFH $.
  $}

  ${
    $d x y z w $.  $d z w ph $.  $d y w ps $.
    cbvopab2v.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
    $( Rule used to change the second bound variable in an ordered pair
       abstraction, using implicit substitution.  (Contributed by NM,
       2-Sep-1999.) $)
    cbvopab2v $p |- { <. x , y >. | ph } = { <. x , z >. | ps } $=
      ( vw cv cop wceq wex cab copab opeq2 eqeq2d anbi12d cbvexv exbii df-opab
      wa abbii 3eqtr4i ) GHZCHZDHZIZJZATZDKZCKZGLUCUDEHZIZJZBTZEKZCKZGLACDMBCEM
      UJUPGUIUOCUHUNDEUEUKJZUGUMABUQUFULUCUEUKUDNOFPQRUAACDGSBCEGSUB $.
  $}

  ${
    $d w y z A $.  $d w ph $.  $d w x y z $.
    $( Move substitution into a class abstraction.  (Contributed by NM,
       6-Aug-2007.)  (Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
    csbopabg $p |- ( A e. V -> [_ A / x ]_ { <. y , z >. | ph } =
               { <. y , z >. | [. A / x ]. ph } ) $=
      ( vw cv copab csb wsb wceq wsbc csbeq1 dfsbcq2 opabbidv eqeq12d vex nfs1v
      nfopab sbequ12 csbief vtoclg ) BGHZACDIZJZABGKZCDIZLBEUEJZABEMZCDIZLGEFUD
      ELZUFUIUHUKBUDEUENULUGUJCDABGEOPQBUDUEUHGRUGCDBABGSTBHUDLAUGCDABGUAPUBUC
      $.
  $}

  ${
    $d x z $.  $d y z $.  $d ph z $.  $d ps z $.
    $( Union of two ordered pair class abstractions.  (Contributed by NM,
       30-Sep-2002.) $)
    unopab $p |- ( { <. x , y >. | ph } u. { <. x , y >. | ps } ) =
               { <. x , y >. | ( ph \/ ps ) } $=
      ( vz cv cop wceq wa wex cab wo copab unab 19.43 andi exbii bitr2i df-opab
      cun bitr3i abbii eqtri uneq12i 3eqtr4i ) EFCFDFGHZAIZDJZCJZEKZUFBIZDJZCJZ
      EKZTZUFABLZIZDJZCJZEKZACDMZBCDMZTUPCDMUOUIUMLZEKUTUIUMENVCUSEVCUHULLZCJUS
      UHULCOVDURCURUGUKLZDJVDUQVEDUFABPQUGUKDORQUAUBUCVAUJVBUNACDESBCDESUDUPCDE
      SUE $.
  $}

  ${
    $d x y ph $.  $d y A $.  $d y B $.  $d y C $.  $d y D $.
    $( An equality theorem for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.) $)
    mpteq12f $p |- ( ( A. x A = C /\ A. x e. A B = D ) ->
                    ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
      ( vy wceq wal wral wa cv wcel copab cmpt nfa1 nfra1 nfan nfv rsp df-mpt
      imp eqeq2d pm5.32da sp eleq2d anbi1d sylan9bbr opabbid 3eqtr4g ) BDGZAHZC
      EGZABIZJZAKZBLZFKZCGZJZAFMUODLZUQEGZJZAFMABCNADENUNUSVBAFUKUMAUJAOULABPQU
      NFRUMUSUPVAJUKVBUMUPURVAUMUPJCEUQUMUPULULABSUAUBUCUKUPUTVAUKBDUOUJAUDUEUF
      UGUHAFBCTAFDETUI $.

    mpteq12dv.1 $e |- ( ph -> A = C ) $.
    ${
      mpteq12dva.2 $e |- ( ( ph /\ x e. A ) -> B = D ) $.
      $( An equality inference for the maps-to notation.  (Contributed by Mario
         Carneiro, 26-Jan-2017.) $)
      mpteq12dva $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
        ( wceq wal wral cmpt alrimiv ralrimiva mpteq12f syl2anc ) ACEIZBJDFIZBC
        KBCDLBEFLIAQBGMARBCHNBCDEFOP $.
    $}

    mpteq12dv.2 $e |- ( ph -> B = D ) $.
    $( An equality inference for the maps-to notation.  (Contributed by NM,
       24-Aug-2011.)  (Revised by Mario Carneiro, 16-Dec-2013.) $)
    mpteq12dv $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
      ( wceq cv wcel adantr mpteq12dva ) ABCDEFGADFIBJCKHLM $.
  $}

  ${
    $d x A $.  $d x C $.
    $( An equality theorem for the maps-to notation.  (Contributed by NM,
       16-Dec-2013.) $)
    mpteq12 $p |- ( ( A = C /\ A. x e. A B = D ) ->
                    ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
      ( wceq wal wral cmpt ax-17 mpteq12f sylan ) BDFZMAGCEFABHABCIADEIFMAJABCD
      EKL $.
  $}

  ${
    $d x A $.  $d x B $.
    $( An equality theorem for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.) $)
    mpteq1 $p |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) ) $=
      ( wceq wral cmpt cv wcel eqidd rgen mpteq12 mpan2 ) BCEDDEZABFABDGACDGENA
      BAHBIDJKABDCDLM $.

    mpteq1d.1 $e |- ( ph -> A = B ) $.
    $( An equality theorem for the maps-to notation.  (Contributed by Mario
       Carneiro, 11-Jun-2016.) $)
    mpteq1d $p |- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) ) $=
      ( wceq cmpt mpteq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
  $}

  ${
    mpteq2ia.1 $e |- ( x e. A -> B = C ) $.
    $( An equality inference for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.) $)
    mpteq2ia $p |- ( x e. A |-> B ) = ( x e. A |-> C ) $=
      ( wceq wal wral cmpt eqid ax-gen rgen mpteq12f mp2an ) BBFZAGCDFZABHABCIA
      BDIFOABJKPABELABCBDMN $.
  $}

  ${
    mpteq2i.1 $e |- B = C $.
    $( An equality inference for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.) $)
    mpteq2i $p |- ( x e. A |-> B ) = ( x e. A |-> C ) $=
      ( wceq cv wcel a1i mpteq2ia ) ABCDCDFAGBHEIJ $.
  $}

  ${
    mpteq12i.1 $e |- A = C $.
    mpteq12i.2 $e |- B = D $.
    $( An equality inference for the maps-to notation.  (Contributed by Scott
       Fenton, 27-Oct-2010.)  (Revised by Mario Carneiro, 16-Dec-2013.) $)
    mpteq12i $p |- ( x e. A |-> B ) = ( x e. C |-> D ) $=
      ( cmpt wceq wtru a1i mpteq12dv mptru ) ABCHADEHIJABCDEBDIJFKCEIJGKLM $.
  $}

  ${
    mpteq2da.1 $e |- F/ x ph $.
    mpteq2da.2 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
    $( Slightly more general equality inference for the maps-to notation.
       (Contributed by FL, 14-Sep-2013.)  (Revised by Mario Carneiro,
       16-Dec-2013.) $)
    mpteq2da $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
      ( wceq wal wral cmpt eqid ax-gen cv wcel ex ralrimi mpteq12f sylancr ) AC
      CHZBIDEHZBCJBCDKBCEKHTBCLMAUABCFABNCOUAGPQBCDCERS $.
  $}

  ${
    $d x ph $.
    mpteq2dva.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
    $( Slightly more general equality inference for the maps-to notation.
       (Contributed by Scott Fenton, 25-Apr-2012.) $)
    mpteq2dva $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
      ( nfv mpteq2da ) ABCDEABGFH $.
  $}

  ${
    $d x ph $.
    mpteq2dv.1 $e |- ( ph -> B = C ) $.
    $( An equality inference for the maps-to notation.  (Contributed by Mario
       Carneiro, 23-Aug-2014.) $)
    mpteq2dv $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
      ( wceq cv wcel adantr mpteq2dva ) ABCDEADEGBHCIFJK $.
  $}

  ${
    $d z A $.  $d z B $.  $d x y z $.
    nfmpt.1 $e |- F/_ x A $.
    nfmpt.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for the maps-to notation.
       (Contributed by NM, 20-Feb-2013.) $)
    nfmpt $p |- F/_ x ( y e. A |-> B ) $=
      ( vz cmpt cv wcel wceq wa copab df-mpt nfcri nfeq2 nfan nfopab nfcxfr ) A
      BCDHBICJZGIZDKZLZBGMBGCDNUCBGATUBAABCEOAUADFPQRS $.
  $}

  ${
    $d A z $.  $d B z $.  $d x z $.
    $( Bound-variable hypothesis builder for the maps-to notation.
       (Contributed by FL, 17-Feb-2008.) $)
    nfmpt1 $p |- F/_ x ( x e. A |-> B ) $=
      ( vz cmpt cv wcel wceq wa copab df-mpt nfopab1 nfcxfr ) AABCEAFBGDFCHIZAD
      JADBCKNADLM $.
  $}

  ${
    $d w x y z $.  $d w z A $.  $d w z A $.  $d w z B $.  $d w z C $.
    cbvmptf.1 $e |- F/_ x A $.
    cbvmptf.2 $e |- F/_ y A $.
    cbvmptf.3 $e |- F/_ y B $.
    cbvmptf.4 $e |- F/_ x C $.
    cbvmptf.5 $e |- ( x = y -> B = C ) $.
    $( Rule to change the bound variable in a maps-to function, using implicit
       substitution.  This version has bound-variable hypotheses in place of
       distinct variable conditions.  (Contributed by Thierry Arnoux,
       9-Mar-2017.) $)
    cbvmptf $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( vz vw cv wcel wceq wa copab cmpt nfv weq wsb nfcri nfs1v eleq1w sbequ12
      nfan anbi12d cbvopab1 nfeq2 nfsb eqeq2d sbhypf eqtri df-mpt 3eqtr4i ) AMC
      NZKMZDOZPZAKQZBMZCNZUQEOZPZBKQZACDRBCERUTLMCNZURALUAZPZLKQVEUSVHAKLUSLSVF
      VGAALCFUBURALUCUFALTUPVFURVGALCUDURALUEUGUHVHVDLKBVFVGBBLCGUBURALBBUQDHUI
      UJUFVDLSLBTVFVBVGVCLBCUDURVCALVAAUQEIUIABTDEUQJUKULUGUHUMAKCDUNBKCEUNUO
      $.
  $}

  ${
    $d w z x A $.  $d w z y A $.  $d w z B $.  $d w z C $.
    cbvmpt.1 $e |- F/_ y B $.
    cbvmpt.2 $e |- F/_ x C $.
    cbvmpt.3 $e |- ( x = y -> B = C ) $.
    $( Rule to change the bound variable in a maps-to function, using implicit
       substitution.  This version has bound-variable hypotheses in place of
       distinct variable conditions.  (Contributed by NM, 11-Sep-2011.) $)
    cbvmpt $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( vz vw cv wcel wceq wa copab cmpt wsb nfv nfan weq nfs1v sbequ12 anbi12d
      eleq1 cbvopab1 nfeq2 nfsb sbequ eqeq2d sbie bitrdi eqtri df-mpt 3eqtr4i )
      AKZCLZIKZDMZNZAIOZBKZCLZUQEMZNZBIOZACDPBCEPUTJKZCLZURAJQZNZJIOVEUSVIAIJUS
      JRVGVHAVGARURAJUASAJTUPVGURVHUOVFCUDURAJUBUCUEVIVDJIBVGVHBVGBRURAJBBUQDFU
      FUGSVDJRJBTZVGVBVHVCVFVACUDVJVHURABQVCURJBAUHURVCABAUQEGUFABTDEUQHUIUJUKU
      CUEULAICDUMBICEUMUN $.
  $}

  ${
    $d A x $.  $d A y $.  $d B y $.  $d C x $.
    cbvmptv.1 $e |- ( x = y -> B = C ) $.
    $( Rule to change the bound variable in a maps-to function, using implicit
       substitution.  (Contributed by Mario Carneiro, 19-Feb-2013.) $)
    cbvmptv $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( nfcv cbvmpt ) ABCDEBDGAEGFH $.
  $}

  ${
    $d x y $.  $d y B $.
    $( Function with universal domain in maps-to notation.  (Contributed by NM,
       16-Aug-2013.) $)
    mptv $p |- ( x e. _V |-> B ) = { <. x , y >. | y = B } $=
      ( cvv cmpt cv wcel wceq wa copab df-mpt vex biantrur opabbii eqtr4i ) ADC
      EAFDGZBFCHZIZABJQABJABDCKQRABPQALMNO $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Transitive classes
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare a new symbol. $)
  $c Tr $. $( Transitive predicate (read:  "the following class is
              transitive") $)

  $( Extend wff notation to include transitive classes.  Notation from
     [TakeutiZaring] p. 35. $)
  wtr $a wff Tr A $.

  $( Define the transitive class predicate.  Definition of [Enderton] p. 71
     extended to arbitrary classes.  For alternate definitions, see ~ dftr2
     (which is suggestive of the word "transitive"), ~ dftr3 , ~ dftr4 , and
     ~ dftr5 .  The term "complete" is used instead of "transitive" in
     Definition 3 of [Suppes] p. 130.  (Contributed by NM, 29-Aug-1993.) $)
  df-tr $a |- ( Tr A <-> U. A C_ A ) $.

  ${
    $d x y A $.
    $( An alternate way of defining a transitive class.  Exercise 7 of
       [TakeutiZaring] p. 40.  (Contributed by NM, 24-Apr-1994.) $)
    dftr2 $p |- ( Tr A <-> A. x A. y ( ( x e. y /\ y e. A ) -> x e. A ) ) $=
      ( cuni wss cv wcel wi wal wtr ssalel df-tr wex 19.23v eluni imbi1i bitr4i
      wa albii 3bitr4i ) CDZCEAFZUAGZUBCGZHZAICJUBBFZGUFCGRZUDHBIZAIAUACKCLUHUE
      AUHUGBMZUDHUEUGUDBNUCUIUDBUBCOPQST $.

    $( An alternate way of defining a transitive class.  (Contributed by NM,
       20-Mar-2004.) $)
    dftr5 $p |- ( Tr A <-> A. x e. A A. y e. x y e. A ) $=
      ( wtr cv wcel wa wi wal wral dftr2 alcom impexp albii df-ral bitr4i bitri
      r19.21v ) CDBEZAEZFZTCFZGSCFZHZAIBIZUCBTJZACJZBACKUEUDBIZAIZUGUDBALUIUBUF
      HZAIUGUHUJAUHUBUCHZBTJZUJUHUAUKHZBIULUDUMBUAUBUCMNUKBTOPUBUCBTRQNUFACOPQQ
      $.

    $( An alternate way of defining a transitive class.  Definition 7.1 of
       [TakeutiZaring] p. 35.  (Contributed by NM, 29-Aug-1993.) $)
    dftr3 $p |- ( Tr A <-> A. x e. A x C_ A ) $=
      ( vy wtr cv wcel wral wss dftr5 dfss3 ralbii bitr4i ) BDCEBFCAEZGZABGMBHZ
      ABGACBIONABCMBJKL $.
  $}

  $( An alternate way of defining a transitive class.  Definition of [Enderton]
     p. 71.  (Contributed by NM, 29-Aug-1993.) $)
  dftr4 $p |- ( Tr A <-> A C_ ~P A ) $=
    ( wtr cuni wss cpw df-tr sspwuni bitr4i ) ABACADAAEDAFAAGH $.

  $( Equality theorem for the transitive class predicate.  (Contributed by NM,
     17-Sep-1993.) $)
  treq $p |- ( A = B -> ( Tr A <-> Tr B ) ) $=
    ( wceq cuni wss wtr unieq sseq1d sseq2 bitrd df-tr 3bitr4g ) ABCZADZAEZBDZB
    EZAFBFMOPAEQMNPAABGHABPIJAKBKL $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( In a transitive class, the membership relation is transitive.
       (Contributed by NM, 19-Apr-1994.)  (Proof shortened by Andrew Salmon,
       9-Jul-2011.) $)
    trel $p |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) ) $=
      ( vy vx wtr cv wcel wa wi wal dftr2 eleq12 wb eleq1 adantl anbi12d adantr
      wceq imbi12d spc2gv pm2.43b sylbi ) AFDGZEGZHZUEAHZIZUDAHZJZEKDKZBCHZCAHZ
      IZBAHZJZDEALUKUNUOUJUPDEBCCAUDBSZUECSZIZUHUNUIUOUSUFULUGUMUDBUECMURUGUMNU
      QUECAOPQUQUIUONURUDBAORTUAUBUC $.
  $}

  $( In a transitive class, the membership relation is transitive.
     (Contributed by NM, 19-Apr-1994.) $)
  trel3 $p |- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> B e. A ) ) $=
    ( wtr wcel w3a wa 3anass trel anim2d biimtrid syld ) AEZBCFZCDFZDAFZGZOCAFZ
    HZBAFROPQHZHNTOPQINUASOACDJKLABCJM $.

  ${
    $d x A $.  $d x B $.
    $( An element of a transitive class is a subset of the class.  (Contributed
       by NM, 7-Aug-1994.) $)
    trss $p |- ( Tr A -> ( B e. A -> B C_ A ) ) $=
      ( vx wtr wcel wss cv wceq eleq1 sseq1 imbi12d imbi2d wral dftr3 rsp sylbi
      wi vtoclg pm2.43b ) ADZBAEZBAFZTCGZAEZUCAFZQZQTUAUBQZQCBAUCBHZUFUGTUHUDUA
      UEUBUCBAIUCBAJKLTUECAMUFCANUECAOPRS $.
  $}

  ${
    $d x A $.  $d x B $.
    $( The intersection of transitive classes is transitive.  (Contributed by
       NM, 9-May-1994.) $)
    trin $p |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) ) $=
      ( vx wtr wa cv cin wss wral wcel elin im2anan9 biimtrid imbitrdi ralrimiv
      trss ssin dftr3 sylibr ) ADZBDZEZCFZABGZHZCUDIUDDUBUECUDUBUCUDJZUCAHZUCBH
      ZEZUEUFUCAJZUCBJZEUBUIUCABKTUJUGUAUKUHAUCPBUCPLMUCABQNOCUDRS $.
  $}

  $( The empty set is transitive.  (Contributed by NM, 16-Sep-1993.) $)
  tr0 $p |- Tr (/) $=
    ( c0 wtr cpw wss 0ss dftr4 mpbir ) ABAACZDHEAFG $.

  $( The universe is transitive.  (Contributed by NM, 14-Sep-2003.) $)
  trv $p |- Tr _V $=
    ( cvv wtr cuni wss ssv df-tr mpbir ) ABACZADHEAFG $.

  ${
    $d x y A $.  $d y B $.
    $( The indexed union of a class of transitive sets is transitive.
       (Contributed by Mario Carneiro, 16-Nov-2014.) $)
    triun $p |- ( A. x e. A Tr B -> Tr U_ x e. A B ) $=
      ( vy wtr wral cv ciun wss wcel wrex eliun r19.29 nfcv nfiu1 nfss trss imp
      wa ssiun2 sstr2 syl5com syl5 rexlimi syl sylan2b ralrimiva dftr3 sylibr )
      CEZABFZDGZABCHZIZDUMFUMEUKUNDUMULUMJUKULCJZABKZUNAULBCLUKUPSUJUOSZABKUNUJ
      UOABMUQUNABAULUMAULNABCOPUQULCIZAGBJZUNUJUOURCULQRUSCUMIURUNABCTULCUMUAUB
      UCUDUEUFUGDUMUHUI $.

    $( The union of a class of transitive sets is transitive.  Exercise 5(a) of
       [Enderton] p. 73.  (Contributed by Scott Fenton, 21-Feb-2011.)  (Proof
       shortened by Mario Carneiro, 26-Apr-2014.) $)
    truni $p |- ( A. x e. A Tr x -> Tr U. A ) $=
      ( cv wtr wral ciun cuni triun wceq wb uniiun treq ax-mp sylibr ) ACZDABEA
      BOFZDZBGZDZABOHRPISQJABKRPLMN $.

    $( The intersection of a class of transitive sets is transitive.  Exercise
       5(b) of [Enderton] p. 73.  (Contributed by Scott Fenton,
       25-Feb-2011.) $)
    trint $p |- ( A. x e. A Tr x -> Tr |^| A ) $=
      ( vy cv wtr wral wel wss wi cint dftr3 ralbii biimpi df-ral ralcom4 bitri
      wal sylib ralim alimi syl wcel vex elint2 ssint imbi12i albii sylibr ) AD
      ZEZABFZCAGZABFZCDZUIHZABFZIZCQZBJZEZUKULUOIZABFZCQZURUKUOCUIFZABFZVCUKVEU
      JVDABCUIKLMVEVACQZABFVCVDVFABUOCUINLVAACBOPRVBUQCULUOABSTUAUTUNUSHZCUSFZU
      RCUSKVHUNUSUBZVGIZCQURVGCUSNVJUQCVIUMVGUPAUNBCUCUDAUNBUEUFUGPPUH $.

    $( Any inhabited transitive class includes its intersection.  Similar to
       Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include
       the inhabitedness hypothesis).  (Contributed by Jim Kingdon,
       22-Aug-2018.) $)
    trintssm $p |- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A ) $=
      ( cv wcel wex wtr cint wss intss1 trss com12 sstr2 sylsyld exlimiv impcom
      wi ) ACZBDZAEBFZBGZBHZRSUAPARTQHSQBHZUAQBISRUBBQJKTQBLMNO $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  IZF Set Theory - add the Axioms of Collection and Separation
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Collection
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y a b $.
    ax-coll.1 $e |- F/ b ph $.
    $( Axiom of Collection.  Axiom 7 of [Crosilla], p.  "Axioms of CZF and IZF"
       (with unnecessary quantifier removed).  It is similar to ~ bnd but uses
       a freeness hypothesis in place of one of the distinct variable
       conditions.  (Contributed by Jim Kingdon, 23-Aug-2018.) $)
    ax-coll $a |- ( A. x e. a E. y ph -> E. b A. x e. a E. y e. b ph ) $.

    $( Axiom of Replacement.  Axiom 7' of [Crosilla], p.  "Axioms of CZF and
       IZF" (with unnecessary quantifier removed).  In our context this is not
       an axiom, but a theorem proved from ~ ax-coll .  It is identical to
       ~ zfrep6 except for the choice of a freeness hypothesis rather than a
       disjoint variable condition between ` b ` and ` ph ` .  (Contributed by
       Jim Kingdon, 23-Aug-2018.) $)
    repizf $p |- ( A. x e. a E! y ph -> E. b A. x e. a E. y e. b ph ) $=
      ( weu cv wral wex wrex euex ralimi ax-coll syl ) ACGZBDHZIACJZBQIACEHKBQI
      EJPRBQACLMABCDEFNO $.
  $}

  ${
    $d ph w $.  $d x y z w $.
    $( A version of the Axiom of Replacement.  Normally ` ph ` would have free
       variables ` x ` and ` y ` .  Axiom 6 of [Kunen] p. 12.  The Separation
       Scheme ~ ax-sep cannot be derived from this version and must be stated
       as a separate axiom in an axiom system (such as Kunen's) that uses this
       version.  (Contributed by NM, 10-Oct-2003.) $)
    zfrep6 $p |- ( A. x e. z E! y ph -> E. w A. x e. z E. y e. w ph ) $=
      ( nfv repizf ) ABCDEAEFG $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Separation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.  $d y z ph $.
    $( The Axiom of Separation of IZF set theory.  Axiom 6 of [Crosilla], p.
       "Axioms of CZF and IZF" (with unnecessary quantifier removed, and with a
       ` F/ y ph ` condition replaced by a disjoint variable condition between
       ` y ` and ` ph ` ).

       The Separation Scheme is a weak form of Frege's Axiom of Comprehension,
       conditioning it (with ` x e. z ` ) so that it asserts the existence of a
       collection only if it is smaller than some other collection ` z ` that
       already exists.  This prevents Russell's paradox ~ ru .  In some texts,
       this scheme is called "Aussonderung" or the Subset Axiom.

       The variable ` x ` can occur in the formula ` ph ` , which in textbooks
       is often written ` ph ( x ) ` .  To specify this in the Metamath
       language, we _omit_ the distinct variable condition ($d) that ` x ` not
       occur in ` ph ` .

       For a version using a class variable, see ~ sepg , which requires the
       axiom of extensionality as well as the axiom scheme of separation for
       its derivation.

       If we omit the requirement that ` y ` not occur in ` ph ` , we can
       derive a contradiction (the ax-sep in set.mm).  (Contributed by NM,
       11-Sep-2006.) $)
    ax-sep $a |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) $.
  $}

  ${
    $d x y z w $.  $d y ph w $.  $d z w $.
    $( A less restrictive version of the Separation Scheme ~ ax-sep , where
       variables ` x ` and ` z ` can both appear free in the wff ` ph ` , which
       can therefore be thought of as ` ph ( x , z ) ` .  This version was
       derived from the more restrictive ~ ax-sep with no additional set theory
       axioms.  (Contributed by NM, 10-Dec-2006.)  (Proof shortened by Mario
       Carneiro, 17-Nov-2016.) $)
    axsepg $p |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) $=
      ( vw wel wa wb wal wex cv eleq2 anbi1d anabs5 bitrdi bibi2d albidv exbidv
      weq ax-sep chvarv ) BCFZBEFZBDFZAGZGZHZBIZCJUBUEHZBIZCJEDEDSZUHUJCUKUGUIB
      UKUFUEUBUKUFUDUEGUEUKUCUDUEEKDKBKLMUDANOPQRUEBCETUA $.
  $}

  ${
    $d A x y z $.  $d ph y z $.
    $( Version of the axiom of separation where the "containing set" is a class
       variable and the sethood assumption is in the antecedent.  (Contributed
       by NM, 21-Jun-1993.)  Put ~ sepgi in closed form.  (Revised by BJ,
       2-Jul-2022.) $)
    sepg $p |- ( A e. V -> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) ) $=
      ( vz wel wa wb wal wex wcel wceq eleq2 anbi1d bibi2d albidv exbidv ax-sep
      cv vtoclg ) BCGZBFGZAHZIZBJZCKUBBTZDLZAHZIZBJZCKFDEFTZDMZUFUKCUMUEUJBUMUD
      UIUBUMUCUHAULDUGNOPQRABCFSUA $.
  $}

  ${
    $d A x y $.  $d ph y $.
    sepgi.1 $e |- A e. _V $.
    $( Inference associated with ~ sepg .  (Contributed by NM, 21-Jun-1993.)
       (Revised by BJ, 14-Jul-2026.) $)
    sepgi $p |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) $=
      ( cvv wcel wel cv wa wb wal wex sepg ax-mp ) DFGBCHBIDGAJKBLCMEABCDFNO $.

    $d x y A z $.  $d y ph z $.
    $( Obsolete version of ~ sepgi as of 14-Jul-2026.  (Contributed by NM,
       21-Jun-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    zfausclOLD $p |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) $=
      ( vz cv wcel wa wb wal wex eleq2 anbi1d bibi2d albidv exbidv ax-sep vtocl
      wceq ) BGZCGHZUAFGZHZAIZJZBKZCLUBUADHZAIZJZBKZCLFDEUCDTZUGUKCULUFUJBULUEU
      IUBULUDUHAUCDUAMNOPQABCFRS $.
  $}

  ${
    $d x ph z $.  $d x y z $.
    bm1.3ii.1 $e |- E. x A. y ( ph -> y e. x ) $.
    $( Convert implication to equivalence using the Separation Scheme
       (Aussonderung) ~ ax-sep .  Similar to Theorem 1.3ii of [BellMachover]
       p. 463.  (Contributed by NM, 5-Aug-1993.) $)
    bm1.3ii $p |- E. x A. y ( y e. x <-> ph ) $=
      ( vz wel wi wal wa wex weq elequ2 imbi2d albidv cbvexv mpbi ax-sep pm3.2i
      wb exan 19.42v bimsc1 alanimi eximi sylbir exlimiv ax-mp ) ACEFZGZCHZCBFZ
      UHAISZCHZBJZIZEJUKASZCHZBJZUJUNEUJEJZUNAUKGZCHZBJUSDVAUJBEBEKZUTUICVBUKUH
      ABECLMNOPACBEQRTUOUREUOUJUMIZBJURUJUMBUAVCUQBUIULUPCAUHUKUBUCUDUEUFUG $.
  $}

  ${
    $d x y z $.
    $( Derive a weakened version of ~ ax-i9 , where ` x ` and ` y ` must be
       distinct, from Separation ~ ax-sep and Extensionality ~ ax-ext .  The
       theorem ` -. A. x -. x = y ` also holds ( ~ ax9vsep ), but in
       intuitionistic logic ` E. x x = y ` is stronger.  (Contributed by Jim
       Kingdon, 25-Aug-2018.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    a9evsep $p |- E. x x = y $=
      ( vz cv wcel wceq wi wa wb wal wex ax-sep id biantru bibi2i biimpri alimi
      ax-ext syl eximi ax-mp ) CDZADZEZUBBDZEZUBUBFZUGGZHZIZCJZAKUCUEFZAKUHCABL
      UKULAUKUDUFIZCJULUJUMCUMUJUFUIUDUHUFUGMNOPQABCRSTUA $.

    $( Derive a weakened version of ~ ax-9 , where ` x ` and ` y ` must be
       distinct, from Separation ~ ax-sep and Extensionality ~ ax-ext .  In
       intuitionistic logic ~ a9evsep is stronger and also holds.  (Contributed
       by NM, 12-Nov-2013.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ax9vsep $p |- -. A. x -. x = y $=
      ( cv wceq wex wn wal a9evsep exalim ax-mp ) ACBCDZAEKFAGFABHKAIJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Null Set Axiom
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y $.
    zfnuleu.1 $e |- E. x A. y -. y e. x $.
    $( Show the uniqueness of the empty set (using the Axiom of Extensionality
       via ~ bm1.1 to strengthen the hypothesis in the form of ~ axnul ).
       (Contributed by NM, 22-Dec-2007.) $)
    zfnuleu $p |- E! x A. y -. y e. x $=
      ( wel wn wal weu wfal wb wex nbfal albii exbii mpbi nfv bm1.1 ax-mp eubii
      mpbir ) BADZEZBFZAGTHIZBFZAGZUDAJZUEUBAJUFCUBUDAUAUCBTKLZMNHABHAOPQUBUDAU
      GRS $.
  $}

  ${
    $d x y z $.
    $( The Null Set Axiom of ZF set theory: there exists a set with no
       elements.  Axiom of Empty Set of [Enderton] p. 18.  In some textbooks,
       this is presented as a separate axiom; here we show it can be derived
       from Separation ~ ax-sep .  This version of the Null Set Axiom tells us
       that at least one empty set exists, but does not tell us that it is
       unique - we need the Axiom of Extensionality to do that (see
       ~ zfnuleu ).

       This theorem should not be referenced by any proof.  Instead, use
       ~ ax-nul below so that the uses of the Null Set Axiom can be more easily
       identified.  (Contributed by Jeff Hoffman, 3-Feb-2008.)  (Revised by NM,
       4-Feb-2008.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    axnul $p |- E. x A. y -. y e. x $=
      ( vz cv wn wa wb wal wex ax-sep pm3.24 intnan id mtbiri alimi eximi ax-mp
      wcel ) BDZADRZSCDRZSSRZUBEFZFZGZBHZAITEZBHZAIUCBACJUFUHAUEUGBUETUDUCUAUBK
      LUEMNOPQ $.

    $( The Null Set Axiom of IZF set theory.  It was derived as ~ axnul above
       and is therefore redundant, but we state it as a separate axiom here so
       that its uses can be identified more easily.  Axiom 4 of [Crosilla] p.
       "Axioms of CZF and IZF".  (Contributed by NM, 7-Aug-2003.) $)
    ax-nul $a |- E. x A. y -. y e. x $.

    $( The Null Set Axiom of ZF set theory: the empty set exists.  Corollary
       5.16 of [TakeutiZaring] p. 20.  For the unabbreviated version, see
       ~ ax-nul .  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Andrew
       Salmon, 9-Jul-2011.) $)
    0ex $p |- (/) e. _V $=
      ( vx vy c0 cv wceq wex wel wn wal ax-nul eq0 exbii mpbir issetri ) ACADZC
      EZAFBAGHBIZAFABJPQABOKLMN $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( The existence of proper substitution into a class.  (Contributed by NM,
       10-Nov-2005.) $)
    csbexga $p |- ( ( A e. V /\ A. x B e. W ) -> [_ A / x ]_ B e. _V ) $=
      ( vy wcel wal wa csb wsbc cab cvv df-csb abid2 elex eqeltrid alimi spsbc
      cv syl5 imp wb nfcv sbcabel adantr mpbid ) BDGZCEGZAHZIZABCJFTCGZABKFLZMA
      FBCNUKULFLZMGZABKZUMMGZUHUJUPUJUOAHUHUPUIUOAUIUNCMFCOCEPQRUOABDSUAUBUHUPU
      QUCUJULAFBMDAMUDUEUFUGQ $.
  $}

  ${
    csbexa.1 $e |- A e. _V $.
    csbexa.2 $e |- B e. _V $.
    $( The existence of proper substitution into a class.  (Contributed by NM,
       7-Aug-2007.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
    csbexa $p |- [_ A / x ]_ B e. _V $=
      ( cvv wcel csb wal csbexga mpan mpg ) CFGZABCHFGZABFGMAINDABCFFJKEL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Theorems requiring subset and intersection existence
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.
    $( No set contains all sets.  Theorem 41 of [Suppes] p. 30.  (Contributed
       by NM, 23-Aug-1993.) $)
    nalset $p |- -. E. x A. y y e. x $=
      ( vz wel wn wex wal alexnim wa wb ax-sep weq elequ1 elequ2 notbid anbi12d
      bitrd bibi12d spv pclem6 syl eximii mpg ) BADZEZBFUDBGAFEAUDABHCBDZCADZCC
      DZEZIZJZCGZUEBUICBAKULBBDZUDUMEZIZJZUEUKUPCBCBLZUFUMUJUOCBBMUQUGUDUIUNCBA
      MUQUHUMUQUHBCDUMCBCMCBBNQOPRSUMUDTUAUBUC $.
  $}

  ${
    $d x y $.
    $( The universal class does not exist as a set.  (Contributed by NM,
       4-Jul-2005.) $)
    vnex $p |- -. E. x x = _V $=
      ( vy wel wal wex cv cvv wceq nalset wcel wb vex albii dfcleq bitr4i exbii
      tbt mtbi ) BACZBDZAEAFZGHZAEABITUBATSBFGJZKZBDUBSUDBUCSBLQMBUAGNOPR $.
  $}

  $( The universal class is not a member of itself (and thus is not a set).
     Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not
     depend on the Axiom of Regularity.  (Contributed by NM, 23-Aug-1993.) $)
  vprc $p |- -. _V e. _V $=
    ( vx cvv wcel cv wceq wex vnex isset mtbir ) BBCADBEAFAGABHI $.

  $( The universal class does not belong to any class.  (Contributed by FL,
     31-Dec-2006.) $)
  nvel $p |- -. _V e. A $=
    ( cvv wcel vprc elex mto ) BACBBCDBAEF $.

  ${
    $d A x y $.  $d B x y $.
    inex1.1 $e |- A e. _V $.
    $( Separation Scheme (Aussonderung) using class notation.  Compare Exercise
       4 of [TakeutiZaring] p. 22.  (Contributed by NM, 5-Aug-1993.) $)
    inex1 $p |- ( A i^i B ) e. _V $=
      ( vx vy cin cv wceq wex wcel wa wal sepgi dfcleq bibi2i albii bitri exbii
      wb elin mpbir issetri ) DABFZDGZUCHZDIEGZUDJZUFAJUFBJZKZSZELZDIUHEDACMUEU
      KDUEUGUFUCJZSZELUKEUDUCNUMUJEULUIUGUFABTOPQRUAUB $.
  $}

  ${
    inex2.1 $e |- A e. _V $.
    $( Separation Scheme (Aussonderung) using class notation.  (Contributed by
       NM, 27-Apr-1994.) $)
    inex2 $p |- ( B i^i A ) e. _V $=
      ( cin cvv incom inex1 eqeltri ) BADABDEBAFABCGH $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Closed-form, generalized Separation Scheme.  (Contributed by NM,
       7-Apr-1995.) $)
    inex1g $p |- ( A e. V -> ( A i^i B ) e. _V ) $=
      ( vx cv cin cvv wcel wceq ineq1 eleq1d vex inex1 vtoclg ) DEZBFZGHABFZGHD
      ACOAIPQGOABJKOBDLMN $.
  $}

  ${
    ssex.1 $e |- B e. _V $.
    $( The subset of a set is also a set.  Exercise 3 of [TakeutiZaring] p. 22.
       This is one way to express the Axiom of Separation ~ ax-sep (a.k.a.
       Subset Axiom).  (Contributed by NM, 27-Apr-1994.) $)
    ssex $p |- ( A C_ B -> A e. _V ) $=
      ( wss cin wceq cvv wcel df-ss inex2 eleq1 mpbii sylbi ) ABDABEZAFZAGHZABI
      ONGHPBACJNAGKLM $.
  $}

  ${
    ssexi.1 $e |- B e. _V $.
    ssexi.2 $e |- A C_ B $.
    $( The subset of a set is also a set.  (Contributed by NM, 9-Sep-1993.) $)
    ssexi $p |- A e. _V $=
      ( wss cvv wcel ssex ax-mp ) ABEAFGDABCHI $.
  $}

  ${
    $d x A $.  $d x B $.
    $( The subset of a set is also a set.  Exercise 3 of [TakeutiZaring] p. 22
       (generalized).  (Contributed by NM, 14-Aug-1994.) $)
    ssexg $p |- ( ( A C_ B /\ B e. C ) -> A e. _V ) $=
      ( vx wcel wss cvv cv wi wceq sseq2 imbi1d vex ssex vtoclg impcom ) BCEABF
      ZAGEZADHZFZRIQRIDBCSBJTQRSBAKLASDMNOP $.
  $}

  ${
    ssexd.1 $e |- ( ph -> B e. C ) $.
    ssexd.2 $e |- ( ph -> A C_ B ) $.
    $( A subclass of a set is a set.  Deduction form of ~ ssexg .  (Contributed
       by David Moews, 1-May-2017.) $)
    ssexd $p |- ( ph -> A e. _V ) $=
      ( wss wcel cvv ssexg syl2anc ) ABCGCDHBIHFEBCDJK $.
  $}

  $( The superclass of a proper class is a proper class.  (Contributed by AV,
     27-Dec-2020.) $)
  prcssprc $p |- ( ( A C_ B /\ A e/ _V ) -> B e/ _V ) $=
    ( wss cvv wnel wcel ssexg ex nelcon3d imp ) ABCZADEBDEKBDADKBDFADFABDGHIJ
    $.

  $( Existence of a difference.  (Contributed by NM, 26-May-1998.) $)
  difexg $p |- ( A e. V -> ( A \ B ) e. _V ) $=
    ( cdif wss wcel cvv difss ssexg mpan ) ABDZAEACFKGFABHKACIJ $.

  ${
    difexi.1 $e |- A e. _V $.
    $( Existence of a difference, inference version of ~ difexg .  (Contributed
       by Glauco Siliprandi, 3-Mar-2021.)  (Revised by AV, 26-Mar-2021.) $)
    difexi $p |- ( A \ B ) e. _V $=
      ( cvv wcel cdif difexg ax-mp ) ADEABFDECABDGH $.
  $}

  ${
    difexd.1 $e |- ( ph -> A e. V ) $.
    $( Existence of a difference.  (Contributed by SN, 16-Jul-2024.) $)
    difexd $p |- ( ph -> ( A \ B ) e. _V ) $=
      ( wcel cdif cvv difexg syl ) ABDFBCGHFEBCDIJ $.
  $}

  ${
    $d x A $.
    $( Separation Scheme (Aussonderung) in terms of a class abstraction.
       Prefer using the more natural statement ~ rabexg .  (Contributed by NM,
       8-Jun-1994.)  Put in closed form.  (Revised by BJ, 18-Jul-2026.) $)
    sepab $p |- ( A e. V -> { x | ( x e. A /\ ph ) } e. _V ) $=
      ( wcel cv wa cab id wss ssab2 a1i ssexd ) CDEZBFCEAGBHZCDNIOCJNABCKLM $.
  $}

  ${
    $d x A $.
    $( Separation Scheme in terms of a restricted class abstraction.
       (Contributed by NM, 23-Oct-1999.) $)
    rabexg $p |- ( A e. V -> { x e. A | ph } e. _V ) $=
      ( crab wss wcel cvv ssrab2 ssexg mpan ) ABCEZCFCDGLHGABCILCDJK $.
  $}

  ${
    $d x A $.
    rabex.1 $e |- A e. _V $.
    $( Separation Scheme in terms of a restricted class abstraction.
       (Contributed by NM, 19-Jul-1996.) $)
    rabex $p |- { x e. A | ph } e. _V $=
      ( cvv wcel crab rabexg ax-mp ) CEFABCGEFDABCEHI $.
  $}

  ${
    $d x A $.
    rabexd.1 $e |- B = { x e. A | ps } $.
    rabexd.2 $e |- ( ph -> A e. V ) $.
    $( Separation Scheme in terms of a restricted class abstraction, deduction
       form of ~ rabex2 .  (Contributed by AV, 16-Jul-2019.) $)
    rabexd $p |- ( ph -> B e. _V ) $=
      ( crab cvv wcel rabexg syl eqeltrid ) AEBCDIZJGADFKOJKHBCDFLMN $.
  $}

  ${
    $d x A $.
    rabex2.1 $e |- B = { x e. A | ps } $.
    rabex2.2 $e |- A e. _V $.
    $( Separation Scheme in terms of a restricted class abstraction.
       (Contributed by AV, 16-Jul-2019.)  (Revised by AV, 26-Mar-2021.) $)
    rabex2 $p |- B e. _V $=
      ( cvv wcel id rabexd ax-mp ) CGHZDGHFLABCDGELIJK $.
  $}

  ${
    $d x B $.  $d y A $.
    rab2ex.1 $e |- B = { y e. A | ps } $.
    rab2ex.2 $e |- A e. _V $.
    $( A class abstraction based on a class abstraction based on a set is a
       set.  (Contributed by AV, 16-Jul-2019.)  (Revised by AV,
       26-Mar-2021.) $)
    rab2ex $p |- { x e. B | ph } e. _V $=
      ( rabex2 rabex ) ACFBDEFGHIJ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ps $.
    elssabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Membership in a class abstraction involving a subset.  Unlike ~ elabg ,
       ` A ` does not have to be a set.  (Contributed by NM, 29-Aug-2006.) $)
    elssabg $p |- ( B e. V ->
                  ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) ) $=
      ( wcel wss wa cvv wi cv cab wb ssexg expcom adantrd wceq sseq1 elab3g syl
      anbi12d ) EFHZDEIZBJZDKHZLDCMZEIZAJZCNHUFOUDUEUGBUEUDUGDEFPQRUJUFCDKUHDSU
      IUEABUHDETGUCUAUB $.
  $}

  ${
    $d x A $.
    $( The intersection of an inhabited class exists.  (Contributed by Jim
       Kingdon, 27-Aug-2018.) $)
    inteximm $p |- ( E. x x e. A -> |^| A e. _V ) $=
      ( cv wcel cint cvv wss intss1 vex ssex syl exlimiv ) ACZBDZBEZFDZANOMGPMB
      HOMAIJKL $.
  $}

  $( If the intersection of a class exists, the class is nonempty.
     (Contributed by Jim Kingdon, 27-Aug-2018.) $)
  intexr $p |- ( |^| A e. _V -> A =/= (/) ) $=
    ( cint cvv wcel c0 wceq vprc inteq int0 eqtrdi eleq1d mtbiri necon2ai ) ABZ
    CDZAEAEFZOCCDGPNCCPNEBCAEHIJKLM $.

  $( If a class intersection is the universe, it is not a set.  In classical
     logic this would be an equivalence.  (Contributed by Jim Kingdon,
     27-Aug-2018.) $)
  intnexr $p |- ( |^| A = _V -> -. |^| A e. _V ) $=
    ( cint cvv wceq wcel vprc eleq1 mtbiri ) ABZCDICECCEFICCGH $.

  ${
    $d x y $.  $d ph y $.
    $( The intersection of an inhabited class abstraction exists.  (Contributed
       by Jim Kingdon, 27-Aug-2018.) $)
    intexabim $p |- ( E. x ph -> |^| { x | ph } e. _V ) $=
      ( vy wex cv cab wcel cint cvv abid exbii nfsab1 nfv eleq1 inteximm sylbir
      cbvex ) ABDBEZABFZGZBDZSHIGZTABABJKUACEZSGZCDUBUDTCBABCLTCMUCRSNQCSOPP $.
  $}

  $( The intersection of an inhabited restricted class abstraction exists.
     (Contributed by Jim Kingdon, 27-Aug-2018.) $)
  intexrabim $p |- ( E. x e. A ph -> |^| { x e. A | ph } e. _V ) $=
    ( cv wcel wex cab cint cvv wrex crab intexabim df-rex df-rab inteqi 3imtr4i
    wa eleq1i ) BDCEAQZBFSBGZHZIEABCJABCKZHZIESBLABCMUCUAIUBTABCNORP $.

  ${
    $d A x y $.  $d B y $.  $d x y z $.  $d A z $.  $d B z $.
    $( The existence of an indexed union. ` x ` is normally a free-variable
       parameter in ` B ` , which should be read ` B ( x ) ` .  (Contributed by
       Jim Kingdon, 28-Aug-2018.) $)
    iinexgm $p |- ( ( E. x x e. A /\ A. x e. A B e. C ) ->
        |^|_ x e. A B e. _V ) $=
      ( vy vz cv wcel wex wral wa ciin wceq wrex cab cint cvv wi syl sylib abid
      dfiin2g adantl elisset rgenw r19.2m mpan2 r19.35-1 imp rexcom4 sylibr nfv
      exbii nfsab1 eleq1w cbvex inteximm eqeltrd ) AGBHAIZCDHZABJZKZABCLZEGZCMZ
      ABNZEOZPZQVAVCVHMUSAEBCDUBUCVBFGVGHZFIZVHQHVBVDVGHZEIZVJVBVFEIZVLVBVEEIZA
      BNZVMUSVAVOUSUTVNRZABNZVAVORUSVPABJVQVPABECDUDUEVPAABUFUGUTVNABUHSUIVEAEB
      UJTVKVFEVFEUAUMUKVKVIEFVKFULVFEFUNEFVGUOUPTFVGUQSUR $.
  $}

  ${
    $d A x y z $.  $d B x y z $.
    $( The intersection of a union ` U. A ` with a class ` B ` is equal to the
       union of the intersections of each element of ` A ` with ` B ` .
       (Contributed by FL, 24-Mar-2007.) $)
    inuni $p |- ( U. A i^i B ) = U. { x | E. y e. A x = ( y i^i B ) } $=
      ( vz cuni cin cv wceq wrex cab wcel wex eluni2 anbi1i elin r19.41v bitr4i
      wa bitri ancom exbii rexcom4 vex inex1 eleq2 ceqsexv rexbii 3bitr4i eqriv
      eluniab ) ECFZDGZAHZBHZDGZIZBCJZAKFZEHZUMLZUTUNLZURSZAMZUTUSLUTULLZUTDLZS
      UTUOLZBCJZVFSZVAVDVEVHVFBUTCNOUTULDPVDUQVBSZAMZBCJZVIVDVJBCJZAMVLVCVMAVCU
      RVBSVMVBURUAUQVBBCQRUBVJBACUCRVLVGVFSZBCJVIVKVNBCVKUTUPLZVNVBVOAUPUODBUDU
      EUNUPUTUFUGUTUODPTUHVGVFBCQTTUIURAUTUKRUJ $.
  $}

  $( Membership in a power class.  Theorem 86 of [Suppes] p. 47.  (Contributed
     by NM, 7-Aug-2000.) $)
  elpw2g $p |- ( B e. V -> ( A e. ~P B <-> A C_ B ) ) $=
    ( wcel cpw wss elpwi cvv ssexg elpwg biimparc syldan expcom impbid2 ) BCDZA
    BEDZABFZABGQOPQOAHDZPABCIRPQABHJKLMN $.

  ${
    elpw2.1 $e |- B e. _V $.
    $( Membership in a power class.  Theorem 86 of [Suppes] p. 47.
       (Contributed by NM, 11-Oct-2007.) $)
    elpw2 $p |- ( A e. ~P B <-> A C_ B ) $=
      ( cvv wcel cpw wss wb elpw2g ax-mp ) BDEABFEABGHCABDIJ $.
  $}

  ${
    elpwi2.1 $e |- B e. V $.
    elpwi2.2 $e |- A C_ B $.
    $( Membership in a power class.  (Contributed by Glauco Siliprandi,
       3-Mar-2021.)  (Proof shortened by Wolf Lammen, 26-May-2024.) $)
    elpwi2 $p |- A e. ~P B $=
      ( cpw wcel wss elexi elpw2 mpbir ) ABFGABHEABBCDIJK $.
  $}

  $( A conditional class with the False alternative being sent to the empty
     class is an element of the powerset of the class corresponding to the True
     alternative when that class is a set.  This statement requires fewer
     axioms than the general case ~ ifelpwung .  (Contributed by BJ,
     5-May-2026.) $)
  if0elpw $p |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ) $=
    ( wcel c0 cif cpw wss if0ss elpw2g mpbiri ) BCDABEFZBGDLBHABILBCJK $.

  ${
    $d A x y $.  $d V x y $.
    $( The power set of a set is never a subset.  (Contributed by Stefan
       O'Rear, 22-Feb-2015.) $)
    pwnss $p |- ( A e. V -> -. ~P A C_ A ) $=
      ( vx vy cpw cv wnel crab wcel wn wa wb eleq12 anidms notbid df-nel bitrid
      wss wceq cbvrabv elrab2 pclem6 ax-mp ssel mtoi ssrab2 elpw2g mpbiri nsyl3
      ) AEZARZCFZULGZCAHZUJIZABIZUKUOUNAIZUNUNIZUQURJZKLUQJDFZUTIZJZUSDUNAUNUTU
      NSZVAURVCVAURLUTUNUTUNMNOUMVBCDAUMULULIZJULUTSZVBULULPVEVDVAVEVDVALULUTUL
      UTMNOQTUAURUQUBUCUJAUNUDUEUPUOUNARUMCAUFUNABUGUHUI $.
  $}

  $( No set equals its power set.  The sethood antecedent is necessary; compare
     ~ pwv .  (Contributed by NM, 17-Nov-2008.)  (Proof shortened by Mario
     Carneiro, 23-Dec-2016.) $)
  pwne $p |- ( A e. V -> ~P A =/= A ) $=
    ( wcel cpw wss wn wne pwnss eqimss necon3bi syl ) ABCADZAEZFLAGABHMLALAIJK
    $.

  $( Lemma for ~ repizf2 .  If we have a function-like proposition which
     provides at most one value of ` y ` for each ` x ` in a set ` w ` , we can
     change "at most one" to "exactly one" by restricting the values of ` x `
     to those values for which the proposition provides a value of ` y ` .
     (Contributed by Jim Kingdon, 7-Sep-2018.) $)
  repizf2lem $p |- ( A. x e. w E* y ph <->
      A. x e. { x e. w | E. y ph } E! y ph ) $=
    ( cv wcel wmo wi wal wex wral crab df-mo imbi2i albii df-ral wa rabid bitri
    weu imbi1i impexp 3bitr4i ) BEZDEZFZACGZHZBIUFACJZACTZHZHZBIZUGBUEKUJBUIBUE
    LZKZUHULBUGUKUFACMNOUGBUEPUOUDUNFZUJHZBIUMUJBUNPUQULBUQUFUIQZUJHULUPURUJUIB
    UERUAUFUIUJUBSOSUC $.

  ${
    $d v x y z w $.  $d v ph $.
    repizf2.1 $e |- F/ z ph $.
    $( Replacement.  This version of replacement is stronger than ~ repizf in
       the sense that ` ph ` does not need to map all values of ` x ` in ` w `
       to a value of ` y ` .  The resulting set contains those elements for
       which there is a value of ` y ` and in that sense, this theorem combines
       ~ repizf with ~ ax-sep .  Another variation would be
       ` A. x e. w E* y ph -> { y | E. x ( x e. w /\ ph ) } e. _V ` but we
       don't have a proof of that yet.  (Contributed by Jim Kingdon,
       7-Sep-2018.) $)
    repizf2 $p |- ( A. x e. w E* y ph ->
        E. z A. x e. { x e. w | E. y ph } E. y e. z ph ) $=
      ( vv wmo cv wral wrex wex crab wi vex rabex wceq weu repizf2lem raleqf wa
      nfcv nfrab1 repizf biimtrrdi biimtrid wel cab df-rab nfv nfex nfan nfcxfr
      nfab nfeq2 exbid sylibd vtocle ) ACHBEIZJZACDIKZBACLZBUSMZJZDLZNGVCVBBUSE
      OPGIZVCQZUTVABVFJZDLZVEUTACRZBVCJZVGVIABCESVGVKVJBVFJVIVJBVFVCBVFUBZVBBUS
      UCZTABCGDFUDUEUFVGVHVDDDVFVCDVCBEUGZVBUAZBUHVBBUSUIVODBVNVBDVNDUJADCFUKUL
      UNUMUOVABVFVCVLVMTUPUQUR $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Theorems requiring empty set existence
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x A $.
    $( Equality theorem for classes and sets .  (Contributed by NM,
       13-Dec-2005.)  (Proof shortened by Raph Levien, 30-Jun-2006.) $)
    class2seteq $p |- ( A e. V -> { x e. A | A e. _V } = A ) $=
      ( wcel cvv crab wceq elex wral cv ax-1 ralrimiv rabid2 sylibr eqcomd syl
      ) BCDBEDZQABFZBGBCHQBRQQABIBRGQQABQAJBDKLQABMNOP $.
  $}

  $( Every power class contains the empty set.  (Contributed by NM,
     25-Oct-2007.) $)
  0elpw $p |- (/) e. ~P A $=
    ( c0 cpw wcel wss 0ss 0ex elpw mpbir ) BACDBAEAFBAGHI $.

  $( The empty set and its power set are not equal.  (Contributed by NM,
     23-Dec-1993.) $)
  0nep0 $p |- (/) =/= { (/) } $=
    ( c0 csn 0ex snnz necomi ) ABAACDE $.

  $( Something cannot be equal to both the empty set and the power set of the
     empty set.  (Contributed by NM, 30-Sep-2003.) $)
  0inp0 $p |- ( A = (/) -> -. A = { (/) } ) $=
    ( c0 wceq csn wne 0nep0 neeq1 mpbiri neneqd ) ABCZABDZJAKEBKEFABKGHI $.

  ${
    $d A x y $.
    $( The removal of the empty set from a class does not affect its union.
       (Contributed by NM, 22-Mar-2004.) $)
    unidif0 $p |- U. ( A \ { (/) } ) = U. A $=
      ( vx vy c0 csn cdif cuni cv wcel wn wa wex n0i pm4.71i anbi1i exbii eluni
      wceq anbi2i bitri an32 anass 3bitr2ri eldif velsn notbii 3bitr4i eqriv )
      BADEZFZGZAGZBHZCHZIZUNAIZUNDRZJZKZKZCLZUOUPKZCLUMUKIZUMULIUTVBCVBUOURKZUP
      KVBURKUTUOVDUPUOURUNUMMNOUOUPURUAUOUPURUBUCPVCUOUNUJIZKZCLVACUMUJQVFUTCVE
      USUOVEUPUNUIIZJZKUSUNAUIUDVHURUPVGUQCDUEUFSTSPTCUMAQUGUH $.
  $}

  ${
    $d y A $.  $d x A $.
    $( An indexed intersection of the empty set, with an inhabited index set,
       is empty.  (Contributed by Jim Kingdon, 29-Aug-2018.) $)
    iin0imm $p |- ( E. y y e. A -> |^|_ x e. A (/) = (/) ) $=
      ( c0 iinconstm ) ABCDE $.
  $}

  ${
    $d x A $.
    $( If an indexed intersection of the empty set is empty, the index set is
       nonempty.  (Contributed by Jim Kingdon, 29-Aug-2018.) $)
    iin0r $p |- ( |^|_ x e. A (/) = (/) -> A =/= (/) ) $=
      ( c0 ciin wceq cvv wcel wn 0ex n0i ax-mp 0iin eqeq1i iineq1 eqeq1d mtbiri
      mtbir necon2ai ) ABCDZCEZBCBCEZTACCDZCEZUCFCEZCFGUDHIFCJKUBFCACLMQUASUBCA
      BCCNOPR $.
  $}

  $( The intersection of the universal class is empty.  (Contributed by NM,
     11-Sep-2008.) $)
  intv $p |- |^| _V = (/) $=
    ( c0 cvv wcel cint wceq 0ex int0el ax-mp ) ABCBDAEFBGH $.

  ${
    $d x y A $.  $d y z A $.
    axpweq.1 $e |- A e. _V $.
    $( Two equivalent ways to express the Power Set Axiom.  Note that ~ ax-pow
       is not used by the proof.  (Contributed by NM, 22-Jun-2009.) $)
    axpweq $p |- ( ~P A e. _V
                <-> E. x A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) ) $=
      ( cpw cvv wcel cv wex wel wi wal pwidg wceq pweq eleq2d spcegv wss bitri
      mpd elex exlimiv impbii vex elpw2 pwss ssalel imbi1i albii exbii ) DFZGHZ
      ULAIZFZHZAJZCBKCIDHLCMZBAKZLZBMZAJUMUQUMULULFZHZUQULGNUPVCAULGUNULOUOVBUL
      UNULPQRUAUPUMAULUOUBUCUDUPVAAUPULUNSZVAULUNAUEUFVDBIZDSZUSLZBMVABDUNUGVGU
      TBVFURUSCVEDUHUIUJTTUKT $.
    $( $j usage 'axpweq' avoids 'ax-pow'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Collection principle
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d ph z w $.  $d x y z w $.
    $( A very strong generalization of the Axiom of Replacement (compare
       ~ zfrep6 ).  Its strength lies in the rather profound fact that
       ` ph ( x , y ) ` does not have to be a "function-like" wff, as it does
       in the standard Axiom of Replacement.  This theorem is sometimes called
       the Boundedness Axiom.  In the context of IZF, it is just a slight
       variation of ~ ax-coll .  (Contributed by NM, 17-Oct-2004.) $)
    bnd $p |- ( A. x e. z E. y ph -> E. w A. x e. z E. y e. w ph ) $=
      ( nfv ax-coll ) ABCDEAEFG $.
  $}

  ${
    $d ph z w v $.  $d x z w v A $.  $d x y z w v B $.
    bnd2.1 $e |- A e. _V $.
    $( A variant of the Boundedness Axiom ~ bnd that picks a subset ` z ` out
       of a possibly proper class ` B ` in which a property is true.
       (Contributed by NM, 4-Feb-2004.) $)
    bnd2 $p |- ( A. x e. A E. y e. B ph ->
              E. z ( z C_ B /\ A. x e. A E. y e. z ph ) ) $=
      ( vw vv wrex wral cv wcel wa wex wss df-rex wi wceq raleq bnd vtocl sylbi
      ralbii exbidv imbi12d cin inex1 inss2 sseq1 mpbiri biantrurd rexeq anbi1i
      vex elin anass bitri rexbii2 bitrdi ralbidv bitr3d spcev exlimiv syl ) AC
      FJZBEKZCLZFMZANZCHLZJZBEKZHOZDLZFPZACVOJZBEKZNZDOZVGVJCOZBEKZVNVFWABEACFQ
      UDWABILZKZVLBWCKZHOZRWBVNRIEGWCESZWDWBWFVNWABWCETWGWEVMHVLBWCETUEUFVJBCIH
      UAUBUCVMVTHVSVMDVKFUGZVKFHUOUHVOWHSZVRVSVMWIVPVRWIVPWHFPVKFUIVOWHFUJUKULW
      IVQVLBEWIVQACWHJVLACVOWHUMAVJCWHVKVHWHMZANVHVKMZVINZANWKVJNWJWLAVHVKFUPUN
      WKVIAUQURUSUTVAVBVCVDVE $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  IZF Set Theory - add the Axioms of Power Sets and Pairing
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Power Sets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z w $.
    $( Axiom of Power Sets.  An axiom of Intuitionistic Zermelo-Fraenkel set
       theory.  It states that a set ` y ` exists that includes the power set
       of a given set ` x ` i.e. contains every subset of ` x ` .  This is
       Axiom 8 of [Crosilla] p.  "Axioms of CZF and IZF" except (a) unnecessary
       quantifiers are removed, and (b) Crosilla has a biconditional rather
       than an implication (but the two are equivalent by ~ bm1.3ii ).

       The variant ~ axpow2 uses explicit subset notation.  A version using
       class notation is ~ pwex .  (Contributed by NM, 5-Aug-1993.) $)
    ax-pow $a |- E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y ) $.

    $( Axiom of Power Sets expressed with the fewest number of different
       variables.  (Contributed by NM, 14-Aug-2003.) $)
    zfpow $p |- E. x A. y ( A. x ( x e. y -> x e. z ) -> y e. x ) $=
      ( vw wel wal wex ax-pow weq elequ1 imbi12d cbvalv imbi1i albii exbii mpbi
      wi ) DBEZDCEZQZDFZBAEZQZBFZAGABEZACEZQZAFZUBQZBFZAGCABDHUDUJAUCUIBUAUHUBT
      UGDADAIRUESUFDABJDACJKLMNOP $.

    $( A variant of the Axiom of Power Sets ~ ax-pow using subset notation.
       Problem in {BellMachover] p. 466.  (Contributed by NM, 4-Jun-2006.) $)
    axpow2 $p |- E. y A. z ( z C_ x -> z e. y ) $=
      ( vw cv wss wel wi wal wex ax-pow ssalel imbi1i albii exbii mpbir ) CEZAE
      ZFZCBGZHZCIZBJDCGDAGHDIZTHZCIZBJABCDKUBUEBUAUDCSUCTDQRLMNOP $.

    $( A variant of the Axiom of Power Sets ~ ax-pow .  For any set ` x ` ,
       there exists a set ` y ` whose members are exactly the subsets of ` x `
       i.e. the power set of ` x ` .  Axiom Pow of [BellMachover] p. 466.
       (Contributed by NM, 4-Jun-2006.) $)
    axpow3 $p |- E. y A. z ( z C_ x <-> z e. y ) $=
      ( cv wss wel wb wal wex axpow2 bm1.3ii bicom albii exbii mpbir ) CDADEZCB
      FZGZCHZBIQPGZCHZBIPBCABCJKSUABRTCPQLMNO $.
  $}

  ${
    $d x y z $.
    $( Every set is an element of some other set.  (Contributed by NM,
       4-Jan-2002.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    el $p |- E. y x e. y $=
      ( vz wel wi wal zfpow weq ax-14 alrimiv ax-13 embantd spimv eximii ) BCDB
      ADEZBFZCBDZEZCFABDZBBCAGRSCACAHZPQSTOBCABIJCABKLMN $.
  $}

  ${
    $d x y z $.
    $( Power set axiom: the powerclass of a set is a set.  Axiom 4 of
       [TakeutiZaring] p. 17.  (Contributed by NM, 30-Oct-2003.)  (Proof
       shortened by Andrew Salmon, 25-Jul-2011.)  Revised to prove ~ pwexg from
       ~ vpwex .  (Revised by BJ, 10-Aug-2022.) $)
    vpwex $p |- ~P x e. _V $=
      ( vy vz cv cpw wss cab cvv df-pw wceq wex wel wb wal axpow2 bm1.3ii abeq2
      exbii mpbir issetri eqeltri ) ADZEBDUBFZBGZHBUBICUDCDZUDJZCKBCLUCMBNZCKUC
      CBACBOPUFUGCUCBUEQRSTUA $.
  $}

  ${
    $d x A $.
    $( Power set axiom expressed in class notation, with the sethood
       requirement as an antecedent.  (Contributed by NM, 30-Oct-2003.) $)
    pwexg $p |- ( A e. V -> ~P A e. _V ) $=
      ( vx cv cpw cvv wcel wceq pweq eleq1d vpwex vtoclg ) CDZEZFGAEZFGCABMAHNO
      FMAIJCKL $.
  $}

  ${
    pwexd.1 $e |- ( ph -> A e. V ) $.
    $( Deduction version of the power set axiom.  (Contributed by Glauco
       Siliprandi, 26-Jun-2021.) $)
    pwexd $p |- ( ph -> ~P A e. _V ) $=
      ( wcel cpw cvv pwexg syl ) ABCEBFGEDBCHI $.
  $}

  ${
    $d x A $.
    $( Existence of a class of subsets.  (Contributed by NM, 15-Jul-2006.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    abssexg $p |- ( A e. V -> { x | ( x C_ A /\ ph ) } e. _V ) $=
      ( wcel cpw cvv cv wss wa pwexg df-pw eleq1i simpl ss2abi ssexg mpan sylbi
      cab syl ) CDECFZGEZBHCIZAJZBSZGEZCDKUBUCBSZGEZUFUAUGGBCLMUEUGIUHUFUDUCBUC
      ANOUEUGGPQRT $.
  $}

  ${
    pwex.1 $e |- A e. _V $.
    $( Power set axiom expressed in class notation.  (Contributed by NM,
       21-Jun-1993.) $)
    pwex $p |- ~P A e. _V $=
      ( cvv wcel cpw pwexg ax-mp ) ACDAECDBACFG $.
  $}

  $( A singleton whose element exists is a set.  The ` A e. _V ` case of
     Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power
     Set, and Separation.  Replacement is not needed.  (Contributed by Jim
     Kingdon, 1-Sep-2018.) $)
  snexg $p |- ( A e. V -> { A } e. _V ) $=
    ( wcel cpw cvv csn pwexg wss snsspw ssexg mpan syl ) ABCADZECZAFZECZABGOMHN
    PAIOMEJKL $.

  ${
    snex.1 $e |- A e. _V $.
    $( A singleton whose element exists is a set.  (Contributed by NM,
       7-Aug-1994.)  (Revised by Mario Carneiro, 24-May-2019.) $)
    snex $p |- { A } e. _V $=
      ( cvv wcel csn snexg ax-mp ) ACDAECDBACFG $.
  $}

  $( A singleton whose element is a proper class is a set.  The ` -. A e. _V `
     case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality,
     Power Set, and Separation.  Replacement is not needed.  (Contributed by
     Jim Kingdon, 1-Sep-2018.) $)
  snexprc $p |- ( -. A e. _V -> { A } e. _V ) $=
    ( cvv wcel wn csn c0 wceq snprc biimpi 0ex eqeltrdi ) ABCDZAEZFBLMFGAHIJK
    $.

  $( A singleton is never a proper class.  (Contributed by Mario Carneiro and
     Jim Kingdon, 3-Jul-2022.) $)
  notnotsnex $p |- -. -. { A } e. _V $=
    ( csn cvv wcel wn wi snexg con3i snexprc syl pm2.01 ax-mp ) ABCDZEZNEZFONMN
    ACDZEMPMACGHAIJHNKL $.

  $( The power set of the empty set (the ordinal 1) is a set.  (Contributed by
     NM, 23-Dec-1993.) $)
  p0ex $p |- { (/) } e. _V $=
    ( c0 cpw csn cvv pw0 0ex pwex eqeltrri ) ABACDEAFGH $.

  $( ` { (/) , { (/) } } ` (the ordinal 2) is a set.  (Contributed by NM,
     5-Aug-1993.) $)
  pp0ex $p |- { (/) , { (/) } } e. _V $=
    ( c0 csn cpr cpw p0ex pwex pwpw0ss ssexi ) AABZCIDIEFGH $.

  $( The ordinal number 3 is a set, proved without the Axiom of Union.
     (Contributed by NM, 2-May-2009.) $)
  ord3ex $p |- { (/) , { (/) } , { (/) , { (/) } } } e. _V $=
    ( csn cpr ctp cun cvv df-tp cpw pp0ex pwex pwprss ssexi snsspr2 unss2 ax-mp
    c0 wss eqeltri ) OOAZORBZCSSAZDZEORSFUASRAZSBZDZUDSGSHIORJKTUCPUAUDPUBSLTUC
    SMNKQ $.

  ${
    $d x y z $.
    $( At least two sets exist (or in terms of first-order logic, the universe
       of discourse has two or more objects).  This theorem asserts the
       existence of two sets which do not equal each other; compare with
       ~ dtruex in which we are given a set ` y ` and go from there to a set
       ` x ` which is not equal to it.  (Contributed by Jim Kingdon,
       2-Sep-2018.) $)
    dtruarb $p |- E. x E. y -. x = y $=
      ( vz wel wn wa wex weq el wal ax-nul eximii eeanv mpbir2an nelneq2 2eximi
      sp cv ax-mp ) CADZCBDEZFZBGAGZABHEZBGAGUCTAGUABGCAIUACJUABBCKUACQLTUAABMN
      UBUDABCRARBROPS $.
  $}

  ${
    $d A x $.
    $( A class is a subclass of the power class of its union.  Exercise 6(b) of
       [Enderton] p. 38.  (Contributed by NM, 14-Oct-1996.) $)
    pwuni $p |- A C_ ~P U. A $=
      ( vx cuni cpw cv wcel wss elssuni vex elpw sylibr ssriv ) BAACZDZBEZAFOMG
      ONFOAHOMBIJKL $.
  $}

  ${
    $d ph x y z $.
    undifexmid.1 $e |- ( x C_ y <-> ( x u. ( y \ x ) ) = y ) $.
    $( Union of complementary parts producing the whole and excluded middle.
       Although special cases such as ~ undifss and ~ undifdcss are provable,
       the full statement implies excluded middle as shown here.  (Contributed
       by Jim Kingdon, 16-Jun-2022.) $)
    undifexmid $p |- ( ph \/ -. ph ) $=
      ( vz c0 csn crab wcel cdif wo wn cun wss wceq cv wb p0ex mpbi ax-mp rabex
      snid ssrab2 wa sseq12 simpl simpr difeq12d uneq12d eqeq12d bibi12d vtocl2
      0ex eleqtrri elun biidd elrab3 biimpi eldifn sylnib orim12i ) FAEFGZHZIZF
      VBVCJZIZKZAALZKFVCVEMZIVGFVBVIFUMUBZVCVBNZVIVBOZAEVBUCBPZCPZNZVMVNVMJZMZV
      NOZQVKVLQBCVCVBAEVBRUARVMVCOZVNVBOZUDZVOVKVRVLVMVCVNVBUEWAVQVIVNVBWAVMVCV
      PVEVSVTUFZWAVNVBVMVCVSVTUGZWBUHUIWCUJUKDULSUNFVCVEUOSVDAVFVHVDAFVBIVDAQVJ
      AAEFVBEPFOAUPUQTZURVFVDAFVBVCUSWDUTVAT $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  A notation for excluded middle
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c EXMID $.

  $( Formula for an abbreviation of excluded middle. $)
  wem $a wff EXMID $.

  ${
    $( The expression ` EXMID ` will be used as a readable shorthand for any
       form of the law of the excluded middle; this is a useful shorthand
       largely because it hides statements of the form "for any proposition" in
       a system which can only quantify over sets, not propositions.

       To see how this compares with other ways of expressing excluded middle,
       compare ~ undifexmid with ~ exmidundif .  The former may be more
       recognizable as excluded middle because it is in terms of propositions,
       and the proof may be easier to follow for much the same reason (it just
       has to show ` ph ` and ` -. ph ` in the the relevant parts of the
       proof).  The latter, however, has the key advantage of being able to
       prove both directions of the biconditional.  To state that excluded
       middle implies a proposition is hard to do gracefully without
       ` EXMID ` , because there is no way to write a hypothesis
       ` ph \/ -. ph ` for an arbitrary proposition; instead the hypothesis
       would need to be the particular instance of excluded middle which that
       proof needs.  Or to say it another way, ` EXMID ` implies ` DECID ph `
       by ~ exmidexmid but there is no good way to express the converse.

       This definition and how we use it is easiest to understand (and most
       appropriate to assign the name "excluded middle" to) if we assume
       ~ ax-sep , in which case ` EXMID ` means that all propositions are
       decidable (see ~ exmidexmid and notice that it relies on ~ ax-sep ).  If
       we instead work with ~ ax-bdsep , ` EXMID ` as defined here means that
       all bounded propositions are decidable.

       (Contributed by Mario Carneiro and Jim Kingdon, 18-Jun-2022.) $)
    df-exmid $a |- ( EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) ) $.
  $}

  ${
    $d ph x z $.
    $( EXMID implies that an arbitrary proposition is decidable.  That is,
       EXMID captures the usual meaning of excluded middle when stated in terms
       of propositions.

       To get other propositional statements which are equivalent to excluded
       middle, combine this with ~ notnotrdc , ~ peircedc , or ~ condc .

       (Contributed by Jim Kingdon, 18-Jun-2022.) $)
    exmidexmid $p |- ( EXMID -> DECID ph ) $=
      ( vz vx wem c0 csn crab wcel wdc wss ssrab2 cv wi wal df-exmid p0ex rabex
      wceq sseq1 eleq2 dcbid imbi12d spcv sylbi mpi 0ex biidd elrab dcbii sylib
      snid mpbiran ) DEABEFZGZHZIZAIDUNUMJZUPABUMKDCLZUMJZEURHZIZMZCNUQUPMZCOVB
      VCCUNABUMPQURUNRZUSUQVAUPURUNUMSVDUTUOURUNETUAUBUCUDUEUOAUOEUMHAEUFUKAABE
      UMBLERAUGUHULUIUJ $.
  $}

  ${
    $d A x $.
    $( A subclass of ` { (/) } ` contains the empty set if and only if it
       equals ` { (/) } ` .  (Contributed by BJ and Jim Kingdon,
       9-Aug-2024.) $)
    ss1o0el1 $p |- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) ) $=
      ( vx c0 csn wss wcel wceq cv wex elex2 sssnm syl biimpcd 0ex eleq2 mpbiri
      wb snid impbid1 ) ACDZEZCAFZATGZUBUAUCUBBHAFBIUAUCQBCAJBACKLMUCUBCTFCNRAT
      COPS $.
  $}

  ${
    $d x y z $.
    $( Excluded middle is equivalent to saying any subset of ` { (/) } ` is
       either ` (/) ` or ` { (/) } ` .  (Contributed by BJ and Jim Kingdon,
       18-Jun-2022.) $)
    exmid01 $p |- ( EXMID <-> A. x ( x C_ { (/) }
        -> ( x = (/) \/ x = { (/) } ) ) ) $=
      ( vy wem cv c0 csn wss wcel wdc wi wal wceq wo df-exmid wn df-dc orcom wa
      simpll bitrid simpr sseldd velsn sylib eqeltrrd simplr pm2.65da eq0rdv ex
      noel eleq2 mtbiri impbid1 ss1o0el1 orbi12d pm5.74i albii bitri ) CADZEFZG
      ZEUSHZIZJZAKVAUSELZUSUTLZMZJZAKANVDVHAVAVCVGVCVBVBOZMZVAVGVBPVJVIVBMVAVGV
      BVIQVAVIVEVBVFVAVIVEVAVIVEVAVIRZBUSVKBDZUSHZVBVKVMRZVLEUSVNVLUTHVLELVNUSU
      TVLVAVIVMSVKVMUAZUBBEUCUDVOUEVAVIVMUFUGUHUIVEVBEEHEUJUSEEUKULUMUSUNUOTTUP
      UQUR $.
  $}

  ${
    $d A x $.
    $( A slight strengthening of ~ pwtrufal .  (Contributed by Mario Carneiro
       and Jim Kingdon, 12-Sep-2023.) $)
    pwntru $p |- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) ) $=
      ( vx c0 csn wss wne wa cv wcel wex simpr neneqd simpll simpl sselda elsni
      wn wceq syl eqeltrrd snssd eqssd ex exlimdv mtod notm0 sylib ) ACDZEZAUHF
      ZGZBHZAIZBJZQACRUKUNAUHRZUKAUHUIUJKLUKUMUOBUKUMUOUKUMGZAUHUIUJUMMUPCAUPUL
      CAUPULUHIULCRUKAUHULUIUJNOULCPSUKUMKTUAUBUCUDUEBAUFUG $.
  $}

  ${
    $d ph x $.
    exmid1dc.x $e |- ( ( ph /\ x C_ { (/) } ) -> DECID x = { (/) } ) $.
    $( A convenience theorem for proving that something implies ` EXMID ` .
       Think of this as an alternative to using a proposition, as in proofs
       like ~ undifexmid or ~ ordtriexmid .  In this context ` x = { (/) } `
       can be thought of as "x is true".  (Contributed by Jim Kingdon,
       21-Nov-2023.) $)
    exmid1dc $p |- ( ph -> EXMID ) $=
      ( cv c0 csn wss wceq wo wi wal wem wa wn wdc exmiddc syl wne df-ne ex mpd
      pwntru biimtrrid orim2d adantl orcomd alrimiv exmid01 sylibr ) ABDZEFZGZU
      JEHZUJUKHZIZJZBKLAUPBAULUOAULMZUNUMUQUNUNNZIZUNUMIZUQUNOUSCUNPQULUSUTJAUL
      URUMUNURUJUKRZULUMUJUKSULVAUMUJUBTUCUDUEUAUFTUGBUHUI $.
  $}

  ${
    $d x y $.
    $( Excluded middle is equivalent to any set being empty or inhabited.
       (Contributed by Jim Kingdon, 5-Mar-2023.) $)
    exmidn0m $p |- ( EXMID <-> A. x ( x = (/) \/ E. y y e. x ) ) $=
      ( wem cv c0 wceq wel wex wo wal wn wa simpr olcd notm0 biimpi adantl orcd
      wdc exmidexmid exmiddc syl mpjaodan alrimiv csn wss wi orc a1d sssnm jaoi
      biimpa ex alimi exmid01 sylibr impbii ) CADZEFZBAGBHZIZAJZCVAACUTVAUTKZCU
      TLUTUSCUTMNCVCLUSUTVCUSCVCUSBUROPQRCUTSUTVCIUTTUTUAUBUCUDVBUREUEZUFZUSURV
      DFZIZUGZAJCVAVHAUSVHUTUSVGVEUSVFUHUIUTVEVGUTVELVFUSUTVEVFBUREUJULNUMUKUNA
      UOUPUQ $.
  $}

  ${
    $d x y z $.
    $( Excluded middle is equivalent to the biconditionalized version of
       ~ sssnr for sets.  (Contributed by Jim Kingdon, 5-Mar-2023.) $)
    exmidsssn $p |- ( EXMID <-> A. x A. y
        ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) ) $=
      ( vz wem cv csn wss c0 wo wb wal wel wex wa 0ss sseq1 mpbiri adantl simpr
      wceq orcd sssnm wn neq0r biorf syl bitrd exmidn0m biimpi 19.21bi mpjaodan
      2thd alrimivv wi 0ex sneq sseq2d eqeq2d orbi2d bibi12d spcv alimi exmid01
      biimpd sylibr impbii ) DAEZBEZFZGZVGHTZVGVITZIZJZBKZAKZDVNABDVKVNCALCMZDV
      KNZVJVMVKVJDVKVJHVIGVIOVGHVIPQRVRVKVLDVKSUAULVQVNDVQVJVLVMCVGVHUBVQVKUCVL
      VMJCVGUDVKVLUEUFUGRDVKVQIZADVSAKACUHUIUJUKUMVPVGHFZGZVKVGVTTZIZUNZAKDVOWD
      AVOWAWCVNWAWCJBHUOVHHTZVJWAVMWCWEVIVTVGVHHUPZUQWEVLWBVKWEVIVTVGWFURUSUTVA
      VDVBAVCVEVF $.
  $}

  ${
    $d B u x $.  $d B w $.  $d B x y z $.  $d V x y $.
    $( Excluded middle in terms of subsets of a singleton.  This is similar to
       ~ exmid01 but lets you choose any set as the element of the singleton
       rather than just ` (/) ` .  It is similar to ~ exmidsssn but for a
       particular set ` B ` rather than all sets.  (Contributed by Jim Kingdon,
       29-Jul-2023.) $)
    exmidsssnc $p |- ( B e. V -> ( EXMID
        <-> A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) ) ) $=
      ( vu vy vz vw wcel wem cv csn wss c0 wceq wo wi wal wb spcgv cvv biimtrid
      exmidsssn sneq sseq2d eqeq2d orbi2d bibi12d alimdv biimp syl6 wdc wn crab
      alimi ssrab2 snexg rabexg sseq1 orbi12d imbi12d 3syl mpii wral rabeq0 wex
      eqeq1 r19.3rmv syl bitr4id biimpd snidg adantr simpr eleqtrrd biidd elrab
      snmg wa simprbi orim12d syld orcom imbitrdi df-dc imbitrrdi a1dd df-exmid
      ex alrimdv impbid ) BCHZIAJZBKZLZWLMNZWLWMNZOZPZAQZWKIWNWQRZAQZWSIWLDJZKZ
      LZWOWLXCNZOZRZDQZAQWKXAADUBWKXHWTAXGWTDBCXBBNZXDWNXFWQXIXCWMWLXBBUCZUDXIX
      EWPWOXIXCWMWLXJUEUFUGSUHUAWTWRAWNWQUIUNUJWKWSEJZMKLZMXKHZUKZPZEQIWKWSXOEW
      KWSXNXLWKWSXMXMULZOZXNWKWSXPXMOZXQWKWSXMFWMUMZMNZXSWMNZOZXRWKWSXSWMLZYBXM
      FWMUOWKWMTHXSTHWSYCYBPZPBCUPXMFWMTUQWRYDAXSTWLXSNZWNYCWQYBWLXSWMURYEWOXTW
      PYAWLXSMVFWLXSWMVFUSUTSVAVBWKXTXPYAXMWKXTXPWKXTXPFWMVCZXPXMFWMVDWKGJWMHGV
      EXPYFRGBCVQXPFGWMVGVHVIVJWKYAXMWKYAVRZBXSHZXMYGBWMXSWKBWMHZYABCVKVLWKYAVM
      VNYHYIXMXMXMFBWMFJBNXMVOVPVSVHWHVTWAXPXMWBWCXMWDWEWFWIEWGWEWJ $.
  $}

  ${
    $( Excluded middle is equivalent to decidability of ` (/) ` being an
       element of an arbitrary set.  (Contributed by Jim Kingdon,
       18-Jun-2022.) $)
    exmid0el $p |- ( EXMID <-> A. x DECID (/) e. x ) $=
      ( wem c0 cv wcel wdc wal exmidexmid alrimiv csn wss alimi df-exmid sylibr
      wi ax-1 impbii ) BCADZEZFZAGZBTASHIUARCJKZTOZAGBTUCATUBPLAMNQ $.
  $}

  ${
    $d x y $.
    $( Excluded middle is equivalent to decidability of membership for two
       arbitrary sets.  (Contributed by Jim Kingdon, 18-Jun-2022.) $)
    exmidel $p |- ( EXMID <-> A. x A. y DECID x e. y ) $=
      ( wem cv wcel wdc wal exmidexmid alrimivv c0 wceq eleq1 dcbid albidv spcv
      0ex exmid0el sylibr impbii ) CADZBDZEZFZBGZAGZCUCABUBHIUEJUAEZFZBGZCUDUHA
      JPTJKZUCUGBUIUBUFTJUALMNOBQRS $.
  $}

  ${
    $d x y z $.
    $( Excluded middle is equivalent to every subset having a complement.  That
       is, the union of a subset and its relative complement being the whole
       set.  Although special cases such as ~ undifss and ~ undifdcss are
       provable, the full statement is equivalent to excluded middle as shown
       here.  (Contributed by Jim Kingdon, 18-Jun-2022.) $)
    exmidundif $p |- ( EXMID
        <-> A. x A. y ( x C_ y <-> ( x u. ( y \ x ) ) = y ) ) $=
      ( vz wem cv wss cdif cun wceq wb wal wa adantl wcel wn simpr syl wo cvv
      c0 undifss biimpi elun1 simplr eldifd elun2 wdc exmidexmid exmiddc adantr
      mpjaodan ex ssrdv eqssd ssun1 sseq2 mpbii impbid1 alrimivv csn vex sseq12
      p0ex simpl difeq12d uneq12d eqeq12d bibi12d spc2gv mp2an 0ex eleq2 mpbiri
      wi snid eldifn orim2i elun df-dc 3imtr4i biimtrdi alrimiv df-exmid sylibr
      impbii ) DAEZBEZFZWFWGWFGZHZWGIZJZBKAKZDWLABDWHWKDWHWKDWHLWJWGWHWJWGFZDWH
      WNWFWGUAUBMDWGWJFWHDCWGWJDCEZWGNZWOWJNZDWPLZWOWFNZWQWSOZWSWQWRWOWFWIUCMWR
      WTLZWOWINWQXAWOWGWFDWPWTUDWRWTPUEWOWIWFUFQDWSWTRZWPDWSUGXBWSUHWSUIQUJUKUL
      UMUJUNULWKWFWJFWHWFWIUOWJWGWFUPUQURUSWMWOTUTZFZTWONZUGZVNZCKDWMXGCWMXDWOX
      CWOGZHZXCIZXFWOSNXCSNWMXDXJJZVNCVAVCWLXKABWOXCSSWFWOIZWGXCIZLZWHXDWKXJWFW
      OWGXCVBXNWJXIWGXCXNWFWOWIXHXLXMVDZXNWGXCWFWOXLXMPZXOVEVFXPVGVHVIVJXJTXINZ
      XFXJXQTXCNTVKVOXIXCTVLVMXETXHNZRXEXEOZRXQXFXRXSXETXCWOVPVQTWOXHVRXEVSVTQW
      AWBCWCWDWE $.
  $}

  ${
    $d x y z $.
    $( Excluded middle is equivalent to every subset having a complement.
       Variation of ~ exmidundif with an implication rather than a
       biconditional.  (Contributed by Jim Kingdon, 16-Feb-2023.) $)
    exmidundifim $p |- ( EXMID
        <-> A. x A. y ( x C_ y -> ( x u. ( y \ x ) ) = y ) ) $=
      ( vz wem cv wss cdif cun wceq wi wal wa adantl wcel wn simpr syl wo cvv
      c0 undifss biimpi elun1 simplr eldifd elun2 wdc exmidexmid exmiddc adantr
      mpjaodan ex ssrdv eqssd alrimivv csn p0ex sseq12 difeq12d uneq12d eqeq12d
      vex simpl imbi12d spc2gv mp2an snid eleq2 mpbiri eldifn orim2i elun df-dc
      0ex 3imtr4i syl6 alrimiv df-exmid sylibr impbii ) DAEZBEZFZWAWBWAGZHZWBIZ
      JZBKAKZDWGABDWCWFDWCLWEWBWCWEWBFZDWCWIWAWBUAUBMDWBWEFWCDCWBWEDCEZWBNZWJWE
      NZDWKLZWJWANZWLWNOZWNWLWMWJWAWDUCMWMWOLZWJWDNWLWPWJWBWADWKWOUDWMWOPUEWJWD
      WAUFQDWNWORZWKDWNUGWQWNUHWNUIQUJUKULUMUJUNULUOWHWJTUPZFZTWJNZUGZJZCKDWHXB
      CWHWSWJWRWJGZHZWRIZXAWJSNWRSNWHWSXEJZJCVBUQWGXFABWJWRSSWAWJIZWBWRIZLZWCWS
      WFXEWAWJWBWRURXIWEXDWBWRXIWAWJWDXCXGXHVCZXIWBWRWAWJXGXHPZXJUSUTXKVAVDVEVF
      XETXDNZXAXEXLTWRNTVNVGXDWRTVHVIWTTXCNZRWTWTOZRXLXAXMXNWTTWRWJVJVKTWJXCVLW
      TVMVOQVPVQCVRVSVT $.
  $}

  ${
    $d ph x y $.
    exmid1stab.x $e |- ( ( ph /\ x C_ { (/) } ) -> STAB x = { (/) } ) $.
    $( If every proposition is stable, excluded middle follows.  We are
       thinking of ` x ` as a proposition and ` x = { (/) } ` as " ` x ` is
       true".  (Contributed by Jim Kingdon, 28-Nov-2023.) $)
    exmid1stab $p |- ( ph -> EXMID ) $=
      ( vy cv c0 wss wcel wi wal wa wn wo cdif cun wceq eqtrdi ex adantl wstab
      csn wdc wem 0ex snid nnexmid uneq1 undifabs a1i wne df-ne pwntru sylan2br
      difeq2d dif0 uneq12d uncom un0 eqtri jaod con3d mtoi difss biimpi alrimiv
      unss mpan2 cvv p0ex ssex sseq1 eqeq1 stbid imbi12d spcgv 3syl mpd df-stab
      mpan9 sylib eleqtrrid elun eldifn orim2i syl df-dc sylibr df-exmid ) ADEZ
      FUAZGZFWIHZUBZIZDJUCAWNDAWKWMAWKKZWLWLLZMZWMWOWLFWJWINZHZMZWQWOFWIWROZHWT
      WOFWJXAFUDUEWOXAWJPZLZLZXBWKXDAWKXCWIWJPZXELZMZLXEUFWKXGXBWKXEXBXFXEXBIWK
      XEXAWJWROWJWIWJWRUGWJWIUHQUIWKXFXBWKXFKZXAFWJOZWJXHWIFWRWJXFWKWIWJUJWIFPW
      IWJUKWIULUMZXHWRWJFNWJXHWIFWJXJUNWJUOQUPXIWJFOWJFWJUQWJURUSQRUTVAVBSWOXBT
      ZXDXBIWOXAWJGZXKWKXLAWKWRWJGZXLWJWIVCWKXMKXLWIWRWJVFVDVGZSABEZWJGZXOWJPZT
      ZIZBJZWKXLXKIZAXSBAXPXRCRVEWKXLXAVHHXTYAIXNXAWJVIVJXSYABXAVHXOXAPZXPXLXRX
      KXOXAWJVKYBXQXBXOXAWJVLVMVNVOVPVSVQXBVRVTVQWAFWIWRWBVTWSWPWLFWJWIWCWDWEWL
      WFWGRVEDWHWG $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom of Pairing
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x z w $.  $d y z w $.
    $( The Axiom of Pairing of IZF set theory.  Axiom 2 of [Crosilla] p.
       "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed,
       and (b) Crosilla has a biconditional rather than an implication (but the
       two are equivalent by ~ bm1.3ii ).  (Contributed by NM, 14-Nov-2006.) $)
    ax-pr $a |- E. z A. w ( ( w = x \/ w = y ) -> w e. z ) $.

    $( Derive the abbreviated version of the Axiom of Pairing from ~ ax-pr .
       (Contributed by NM, 14-Nov-2006.) $)
    zfpair2 $p |- { x , y } e. _V $=
      ( vz vw cv cpr wceq wex wcel wo wal ax-pr bm1.3ii dfcleq vex bibi2i albii
      wb elpr bitri exbii mpbir issetri ) CAEZBEZFZCEZUFGZCHDEZUGIZUIUDGUIUEGJZ
      RZDKZCHUKCDABCDLMUHUMCUHUJUIUFIZRZDKUMDUGUFNUOULDUNUKUJUIUDUEDOSPQTUAUBUC
      $.
  $}

  $( A singleton built on a setvar is a set.  (Contributed by BJ,
     15-Jan-2025.) $)
  vsnex $p |- { x } e. _V $=
    ( cv csn cpr cvv dfsn2 zfpair2 eqeltri ) ABZCIIDEIFAAGH $.

  ${
    $d x A $.  $d x y B $.  $d x W $.
    $( The Axiom of Pairing using class variables.  Theorem 7.13 of [Quine]
       p. 51, but restricted to classes which exist.  For proper classes, see
       ~ prprc , ~ prprc1 , and ~ prprc2 .  (Contributed by Jim Kingdon,
       16-Sep-2018.) $)
    prexg $p |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V ) $=
      ( vx vy wcel cpr cvv wi cv wceq preq2 eleq1d zfpair2 vtoclg preq1 vtocleg
      imbitrid imp ) ACGBDGZABHZIGZUAUCJEACUAEKZBHZIGZUDALZUCUDFKZHZIGUFFBDUHBL
      UIUEIUHBUDMNEFOPUGUEUBIUDABQNSRT $.
  $}

  $( A singleton of a set is a member of the powerclass of a class if and only
     if that set is a member of that class.  (Contributed by NM, 1-Apr-1998.)
     Put in closed form and avoid ~ ax-nul .  (Revised by BJ, 17-Jan-2025.) $)
  snelpwg $p |- ( A e. V -> ( A e. B <-> { A } e. ~P B ) ) $=
    ( wcel csn wss cpw snssg cvv wb snexg elpwg syl bitr4d ) ACDZABDAEZBFZPBGDZ
    ABCHOPIDRQJACKPBILMN $.

  $( A singleton of a set belongs to the power class of a class containing the
     set.  (Contributed by Alan Sare, 25-Aug-2011.) $)
  snelpwi $p |- ( A e. B -> { A } e. ~P B ) $=
    ( wcel csn cpw wss snssi cvv wb elex snexg elpwg 3syl mpbird ) ABCZADZBECZP
    BFZABGOAHCPHCQRIABJAHKPBHLMN $.

  ${
    snelpw.1 $e |- A e. _V $.
    $( A singleton of a set belongs to the power class of a class containing
       the set.  (Contributed by NM, 1-Apr-1998.) $)
    snelpw $p |- ( A e. B <-> { A } e. ~P B ) $=
      ( wcel csn wss cpw snss snex elpw bitr4i ) ABDAEZBFLBGDABCHLBACIJK $.
  $}

  $( An unordered pair of two sets is a member of the powerclass of a class if
     and only if the two sets are members of that class.  (Contributed by AV,
     8-Jan-2020.) $)
  prelpw $p |- ( ( A e. V /\ B e. W )
                 -> ( ( A e. C /\ B e. C ) <-> { A , B } e. ~P C ) ) $=
    ( wcel wa cpr wss cpw prssg cvv wb prexg elpwg syl bitr4d ) ADFBEFGZACFBCFG
    ABHZCIZSCJFZABCDEKRSLFUATMABDENSCLOPQ $.

  $( A pair of two sets belongs to the power class of a class containing those
     two sets.  (Contributed by Thierry Arnoux, 10-Mar-2017.) $)
  prelpwi $p |- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C ) $=
    ( wcel wa cpr cpw wss prssi cvv wb prexg elpwg syl mpbird ) ACDBCDEZABFZCGD
    ZQCHZABCIPQJDRSKABCCLQCJMNO $.

  ${
    $d x y z $.
    $( A theorem similar to extensionality, requiring the existence of a
       singleton.  Exercise 8 of [TakeutiZaring] p. 16.  (Contributed by NM,
       10-Aug-1993.) $)
    rext $p |- ( A. z ( x e. z -> y e. z ) -> x = y ) $=
      ( cv wcel wi wal csn wceq vsnid vex snex eleq2 imbi12d spcv velsn equcomi
      mpi sylbi syl ) ADZCDZEZBDZUBEZFZCGZUDUAHZEZUAUDIZUGUAUHEZUIAJUFUKUIFCUHU
      AAKLUBUHIUCUKUEUIUBUHUAMUBUHUDMNORUIUDUAIUJBUAPBAQST $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Classes are subclasses if and only if their power classes are
       subclasses.  Exercise 18 of [TakeutiZaring] p. 18.  (Contributed by NM,
       13-Oct-1996.) $)
    sspwb $p |- ( A C_ B <-> ~P A C_ ~P B ) $=
      ( vx wss cpw wcel sstr2 com12 vex elpw 3imtr4g ssrdv csn ssel snex bitr4i
      cv snss 3imtr3g impbii ) ABDZAEZBEZDZUACUBUCUACQZADZUEBDZUEUBFUEUCFUFUAUG
      UEABGHUEACIZJUEBUHJKLUDCABUDUEMZUBFZUIUCFZUEAFZUEBFZUBUCUINUJUIADULUIAUEU
      HOZJUEAUHRPUKUIBDUMUIBUNJUEBUHRPSLT $.
  $}

  ${
    $d A x y $.
    $( A class equals the union of its power class.  Exercise 6(a) of
       [Enderton] p. 38.  (Contributed by NM, 14-Oct-1996.)  (Proof shortened
       by Alan Sare, 28-Dec-2008.) $)
    unipw $p |- U. ~P A = A $=
      ( vx vy cpw cuni cv wcel wex eluni elelpwi exlimiv sylbi csn snid snelpwi
      wa vex elunii sylancr impbii eqriv ) BADZEZABFZUCGZUDAGZUEUDCFZGUGUBGPZCH
      UFCUDUBIUHUFCUDUGAJKLUFUDUDMZGUIUBGUEUDBQNUDAOUDUIUBRSTUA $.
  $}

  $( Membership of a power class.  Exercise 10 of [Enderton] p. 26.
     (Contributed by NM, 13-Jan-2007.) $)
  pwel $p |- ( A e. B -> ~P A e. ~P ~P U. B ) $=
    ( wcel cpw cuni wss elssuni sspwb sylib cvv wb pwexg elpwg syl mpbird ) ABC
    ZADZBEZDZDCZQSFZPARFUAABGARHIPQJCTUAKABLQSJMNO $.

  $( A class is transitive iff its power class is transitive.  (Contributed by
     Alan Sare, 25-Aug-2011.)  (Revised by Mario Carneiro, 15-Jun-2014.) $)
  pwtr $p |- ( Tr A <-> Tr ~P A ) $=
    ( cpw cuni wss wtr unipw sseq1i df-tr dftr4 3bitr4ri ) ABZCZKDAKDKEAELAKAFG
    KHAIJ $.

  ${
    $d A x $.  $d B x $.
    $( An extensionality-like principle defining subclass in terms of subsets.
       (Contributed by NM, 30-Jun-2004.) $)
    ssextss $p |- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) ) $=
      ( wss cpw cv wcel wi wal sspwb ssalel vex elpw imbi12i albii 3bitri ) BCD
      BEZCEZDAFZQGZSRGZHZAISBDZSCDZHZAIBCJAQRKUBUEATUCUAUDSBALZMSCUFMNOP $.

    $( An extensionality-like principle that uses the subset instead of the
       membership relation: two classes are equal iff they have the same
       subsets.  (Contributed by NM, 30-Jun-2004.) $)
    ssext $p |- ( A = B <-> A. x ( x C_ A <-> x C_ B ) ) $=
      ( wss wa cv wi wal wceq wb ssextss anbi12i eqss albiim 3bitr4i ) BCDZCBDZ
      EAFZBDZRCDZGAHZTSGAHZEBCISTJAHPUAQUBABCKACBKLBCMSTANO $.

    $( Negation of subclass relationship.  Compare ~ nssr .  (Contributed by
       Jim Kingdon, 17-Sep-2018.) $)
    nssssr $p |- ( E. x ( x C_ A /\ -. x C_ B ) -> -. A C_ B ) $=
      ( cv wss wn wa wex wi wal exanaliim ssextss sylnibr ) ADZBEZNCEZFGAHOPIAJ
      BCEOPAKABCLM $.
  $}

  $( Classes are equal if and only if their power classes are equal.  Exercise
     19 of [TakeutiZaring] p. 18.  (Contributed by NM, 13-Oct-1996.) $)
  pweqb $p |- ( A = B <-> ~P A = ~P B ) $=
    ( wss wa cpw wceq sspwb anbi12i eqss 3bitr4i ) ABCZBACZDAEZBEZCZNMCZDABFMNF
    KOLPABGBAGHABIMNIJ $.

  ${
    $d x A $.
    intid.1 $e |- A e. _V $.
    $( The intersection of all sets to which a set belongs is the singleton of
       that set.  (Contributed by NM, 5-Jun-2009.) $)
    intid $p |- |^| { x | A e. x } = { A } $=
      ( cv wcel cab cint csn cvv wss snex eleq2 intmin3 ax-mp wi elintab mpgbir
      snid id snssi eqssi ) BADZEZAFGZBHZUEIEUDUEJBCKUCBUEEAUEIUBUEBLBCRMNBUDEZ
      UEUDJUFUCUCOAUCABCPUCSQBUDTNUA $.
  $}

  ${
    $d y x $.  $d y ph $.
    $( The abstraction of a wff with existential uniqueness exists.
       (Contributed by NM, 25-Nov-1994.) $)
    euabex $p |- ( E! x ph -> { x | ph } e. _V ) $=
      ( vy weu cab cv csn wceq wex cvv wcel euabsn2 vex snex eleq1 mpbiri sylbi
      exlimiv ) ABDABEZCFZGZHZCISJKZABCLUBUCCUBUCUAJKTCMNSUAJOPRQ $.
  $}

  ${
    $d w x y $.  $d w x z $.  $d x y A $.
    $( An inhabited class (even if proper) has an inhabited subset.
       (Contributed by Jim Kingdon, 17-Sep-2018.) $)
    mss $p |- ( E. y y e. A -> E. x ( x C_ A /\ E. z z e. x ) ) $=
      ( vw cv wcel wex wss wa csn vex snss snex wceq sseq1 eleq2 exbidv anbi12d
      snm spcev mpan2 sylbi exlimiv elequ1 cbvexv anbi2i exbii sylibr ) BFZDGZB
      HAFZDIZEFZULGZEHZJZAHZUMCFULGZCHZJZAHUKURBUKUJKZDIZURUJDBLZMVCUNVBGZEHZUR
      EUJVDTUQVCVFJAVBUJVDNULVBOZUMVCUPVFULVBDPVGUOVEEULVBUNQRSUAUBUCUDVAUQAUTU
      PUMUSUOCECEAUEUFUGUHUI $.
  $}

  ${
    $d x y z A w $.  $d y z ph w $.
    $( Restricted existence in a class (even if proper) implies restricted
       existence in a subset.  (Contributed by NM, 23-Aug-2003.) $)
    exss $p |- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) ) $=
      ( vz vw wrex cv wcel wa cab wex wss crab rabn0m df-rab eleq2i sylbi bitri
      wsb exbii bitr3i csn snss ssab2 sstr2 simpr wceq equsb1 velsn sbbii mpbir
      vex mpi jctil df-clab sban 3imtr4i elex2 syl sylib snex sseq1 rexeq spcev
      anbi12d syl2anc exlimiv ) ABDGZEHZBHZDIZAJZBKZIZELZCHZDMZABVQGZJZCLZVIVJA
      BDNZIZELVPABEDOWCVOEWBVNVJABDPQUAUBVOWAEVOVJUCZDMZABWDGZWAVOWDVNMZWEVJVNE
      UMZUDWGVNDMWEABDUEWDVNDUFUNRVOFHABWDNZIFLZWFVOVJWIIZWJVLBETZABETZJZVKWDIZ
      BETZWMJZVOWKWNWMWPWLWMUGWPVKVJUHZBETBEUIWOWRBEBVJUJUKULUOVOVMBETWNVMEBUPV
      LABEUQSWKVJWOAJZBKZIZWQWIWTVJABWDPQXAWSBETWQWSEBUPWOABEUQSSURFVJWIUSUTABF
      WDOVAVTWEWFJCWDVJWHVBVQWDUHVRWEVSWFVQWDDVCABVQWDVDVFVEVGVHR $.
  $}

  $( An ordered pair of sets is a set.  (Contributed by Jim Kingdon,
     11-Jan-2019.) $)
  opexg $p |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V ) $=
    ( wcel wa cop csn cpr cvv dfopg elex snexg syl adantr prexg syl2anc eqeltrd
    syl2an ) ACEZBDEZFZABGAHZABIZIZJABCDKUBUCJEZUDJEZUEJETUFUATAJEZUFACLZAJMNOT
    UHBJEUGUAUIBDLABJJPSUCUDJJPQR $.

  ${
    opex.1 $e |- A e. _V $.
    opex.2 $e |- B e. _V $.
    $( An ordered pair of sets is a set.  (Contributed by Jim Kingdon,
       24-Sep-2018.)  (Revised by Mario Carneiro, 24-May-2019.) $)
    opex $p |- <. A , B >. e. _V $=
      ( cvv wcel cop opexg mp2an ) AEFBEFABGEFCDABEEHI $.
  $}

  $( An ordered triple of sets is a set.  (Contributed by Jim Kingdon,
     19-Sep-2018.) $)
  otexg $p |- ( ( A e. U /\ B e. V /\ C e. W ) -> <. A , B , C >. e. _V ) $=
    ( wcel cotp cvv wa cop df-ot opexg sylan eqeltrid 3impa ) ADGZBEGZCFGZABCHZ
    IGQRJZSJTABKZCKZIABCLUAUBIGSUCIGABDEMUBCIFMNOP $.

  ${
    elop.1 $e |- A e. _V $.
    elop.2 $e |- B e. _V $.
    elop.3 $e |- C e. _V $.
    $( An ordered pair has two elements.  Exercise 3 of [TakeutiZaring] p. 15.
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    elop $p |- ( A e. <. B , C >. <-> ( A = { B } \/ A = { B , C } ) ) $=
      ( cop wcel csn cpr wceq wo dfop eleq2i elpr bitri ) ABCGZHABIZBCJZJZHARKA
      SKLQTABCEFMNARSDOP $.
  $}

  ${
    opi1.1 $e |- A e. _V $.
    opi1.2 $e |- B e. _V $.
    $( One of the two elements in an ordered pair.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    opi1 $p |- { A } e. <. A , B >. $=
      ( csn cpr cop snex prid1 dfop eleqtrri ) AEZLABFZFABGLMACHIABCDJK $.

    $( One of the two elements of an ordered pair.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    opi2 $p |- { A , B } e. <. A , B >. $=
      ( cpr csn cop cvv wcel prexg mp2an prid2 dfop eleqtrri ) ABEZAFZOEABGPOAH
      IBHIOHICDABHHJKLABCDMN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordered pair theorem
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A x $.  $d B x $.

    $( An ordered pair is inhabited iff the arguments are sets.  (Contributed
       by Jim Kingdon, 21-Sep-2018.) $)
    opm $p |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) ) $=
      ( cv cop wcel wex cvv csn cpr w3a wa df-op eleq2i exbii abid bitri 19.42v
      cab df-3an 3bitr4ri 3simpa snexg adantr prmg syl sylanbrc impbii 3bitr2i
      id ) ADZBCEZFZAGZBHFZCHFZUKBIZBCJZJFZKZAGZUOUPUSAGZKZUOUPLZUNUKUTASZFZAGV
      AUMVFAULVEUKABCMNOVFUTAUTAPOQVDUSLZAGVDVBLVAVCVDUSARUTVGAUOUPUSTOUOUPVBTZ
      UAVCVDUOUPVBUBVDVDVBVCVDUJVDUQHFZVBUOVIUPBHUCUDAUQURHUEUFVHUGUHUI $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x D $.
    opth1.1 $e |- A e. _V $.
    opth1.2 $e |- B e. _V $.
    $( An ordered pair is nonempty if the arguments are sets (it is also
       inhabited; see ~ opm ).  (Contributed by Mario Carneiro,
       26-Apr-2015.) $)
    opnzi $p |- <. A , B >. =/= (/) $=
      ( vx cv cop wcel wex c0 wne cvv opm mpbir2an n0r ax-mp ) EFABGZHEIZQJKRAL
      HBLHCDEABMNEQOP $.

    $( Equality of the first members of equal ordered pairs.  (Contributed by
       NM, 28-May-2008.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    opth1 $p |- ( <. A , B >. = <. C , D >. -> A = C ) $=
      ( cop wceq csn cpr wi sneqr a1i wcel cvv wa opi1 id eleqtrid syl oprcl wo
      simpld prid1g eleq2 syl5ibrcom elsni eqcomd syl6 dfopg 3syl eleqtrd elpri
      mpjaod ) ABGZCDGZHZAIZCIZHZACHZURCDJZHZUTVAKUQACELMUQVCCURNZVAUQVDVCCVBNZ
      UQCONZVEUQVFDONZUQURUPNZVFVGPZUQURUOUPABEFQUQRSZCDURUAZTUCCDOUDTURVBCUEUF
      VDCACAUGUHUIUQURUSVBJZNUTVCUBUQURUPVLVJUQVHVIUPVLHVJVKCDOOUJUKULURUSVBUMT
      UN $.

    $( The ordered pair theorem.  If two ordered pairs are equal, their first
       elements are equal and their second elements are equal.  Exercise 6 of
       [TakeutiZaring] p. 16.  Note that ` C ` and ` D ` are not required to be
       sets due our specific ordered pair definition.  (Contributed by NM,
       28-May-1995.) $)
    opth $p |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) $=
      ( vx cop wceq wa cvv wcel cpr csn syl eqtr3d dfopg sylancl wi prexg opth1
      opi1 id eleqtrid oprcl simprd opeq1d simpld preqr2g syl2anc mpd cv eqeq2d
      preq2 eqeq2 imbi12d vex preqr2 vtoclg sylc jca opeq12 impbii ) ABHZCDHZIZ
      ACIZBDIZJVFVGVHABCDEFUAZVFDKLZCBMZCDMZIZVHVFCKLZVJVFANZVELVNVJJZVFVOVDVEA
      BEFUBVFUCZUDCDVOUEOZUFVFCNZVKMZVSVLMZIZVMVFVEVTWAVFCBHZVEVTVFVDWCVEVFACBV
      IUGVQPVFVNBKLZWCVTIVFVNVJVRUHZFCBKKQRPVFVPVEWAIVRCDKKQOPVFVKKLZVLKLZWBVMS
      VFVNWDWFWEFCBKKTRVFVPWGVRCDKKTOVKVLVSUIUJUKVKCGULZMZIZBWHIZSVMVHSGDKWHDIZ
      WJVMWKVHWLWIVLVKWHDCUNUMWHDBUOUPBWHCFGUQURUSUTVAABCDVBVC $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.  $d x y D $.
    $( Ordered pair theorem. ` C ` and ` D ` are not required to be sets under
       our specific ordered pair definition.  (Contributed by NM, 14-Oct-2005.)
       (Revised by Mario Carneiro, 26-Apr-2015.) $)
    opthg $p |- ( ( A e. V /\ B e. W ) ->
                 ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) ) $=
      ( vx vy cv cop wceq wa wb opeq1 eqeq1d eqeq1 anbi1d bibi12d opeq2 vex
      anbi2d opth vtocl2g ) GIZHIZJZCDJZKZUDCKZUEDKZLZMAUEJZUGKZACKZUJLZMABJZUG
      KZUNBDKZLZMGHABEFUDAKZUHUMUKUOUTUFULUGUDAUENOUTUIUNUJUDACPQRUEBKZUMUQUOUS
      VAULUPUGUEBASOVAUJURUNUEBDPUARUDUECDGTHTUBUC $.
  $}

  $( Ordered pair theorem.  (Contributed by NM, 14-Oct-2005.)  (Revised by
     Mario Carneiro, 26-Apr-2015.) $)
  opthg2 $p |- ( ( C e. V /\ D e. W ) ->
               ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) ) $=
    ( wcel wa cop wceq opthg eqcom anbi12i 3bitr4g ) CEGDFGHCDIZABIZJCAJZDBJZHP
    OJACJZBDJZHCDABEFKPOLSQTRACLBDLMN $.

  ${
    opth2.1 $e |- C e. _V $.
    opth2.2 $e |- D e. _V $.
    $( Ordered pair theorem.  (Contributed by NM, 21-Sep-2014.) $)
    opth2 $p |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) $=
      ( cvv wcel cop wceq wa wb opthg2 mp2an ) CGHDGHABICDIJACJBDJKLEFABCDGGMN
      $.
  $}

  ${
    otth.1 $e |- A e. _V $.
    otth.2 $e |- B e. _V $.
    otth.3 $e |- R e. _V $.
    $( Ordered triple theorem, with triple express with ordered pairs.
       (Contributed by NM, 1-May-1995.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    otth2 $p |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <->
                ( A = C /\ B = D /\ R = S ) ) $=
      ( cop wceq wa w3a opth anbi1i opex df-3an 3bitr4i ) ABJZCDJZKZEFKZLACKZBD
      KZLZUBLSEJTFJKUCUDUBMUAUEUBABCDGHNOSETFABGHPINUCUDUBQR $.

    $( Ordered triple theorem.  (Contributed by NM, 25-Sep-2014.)  (Revised by
       Mario Carneiro, 26-Apr-2015.) $)
    otth $p |- ( <. A , B , R >. = <. C , D , S >. <->
      ( A = C /\ B = D /\ R = S ) ) $=
      ( cotp wceq cop w3a df-ot eqeq12i otth2 bitri ) ABEJZCDFJZKABLELZCDLFLZKA
      CKBDKEFKMRTSUAABENCDFNOABCDEFGHIPQ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    eqvinop.1 $e |- B e. _V $.
    eqvinop.2 $e |- C e. _V $.
    $( A variable introduction law for ordered pairs.  Analog of Lemma 15 of
       [Monk2] p. 109.  (Contributed by NM, 28-May-1995.) $)
    eqvinop $p |- ( A = <. B , C >. <-> E. x E. y ( A = <. x , y >. /\
                  <. x , y >. = <. B , C >. ) ) $=
      ( cv cop wceq wa wex opth2 anbi2i ancom anass 3bitri exbii eqeq2d ceqsexv
      19.42v opeq2 opeq1 bitr2i ) CAHZBHZIZJZUGDEIZJZKZBLZALUEDJZCUEEIZJZKZALCU
      IJZULUPAULUMUFEJZUHKZKZBLUMUSBLZKUPUKUTBUKUHUMURKZKVBUHKUTUJVBUHUEUFDEFGM
      NUHVBOUMURUHPQRUMUSBUAVAUOUMUHUOBEGURUGUNCUFEUEUBSTNQRUOUQADFUMUNUICUEDEU
      CSTUD $.
  $}

  ${
    $d x z w A $.  $d y z w A $.  $d z w ph $.
    $( Substitution of class ` A ` for ordered pair ` <. x , y >. ` .
       (Contributed by NM, 27-Dec-1996.)  (Revised by Andrew Salmon,
       11-Jul-2011.) $)
    copsexg $p |- ( A = <. x , y >. ->
                  ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) $=
      ( vz vw cv cop wceq wa wex wb wi vex 19.8a weq biimtrid syl5 weu euequ1
      eqvinop 19.23bi ex opth anbi1i 2exbii nfe1 wal wnf wo dveeq2or nfae anass
      anim2d eximd biidd drex1 sylibd exbii 19.40 19.9t biimpd anim1d syl6 jaoi
      a1i ax-mp exlimi equcom eubii mpbi eupick com12 sylan9 sylbi impbid eqeq1
      mpan anbi1d 2exbidv bibi2d imbi12d mpbiri adantr exlimivv pm2.43i ) DBGZC
      GZHZIZAWJAJZCKBKZLZWJDEGZFGZHZIZWPWIIZJZFKEKWJWMMZEFDWGWHBNCNUAWSWTEFWQWT
      WRWQWTWRAWRAJZCKZBKZLZMWRAXCWRAXCXAXCCXBBOUBUCWREBPZFCPZJZXCAMWNWOWGWHENF
      NUDZXCXGAJZCKZBKZXGAXAXIBCWRXGAXHUEUFXKXEXFAJZCKZJZBKZXGAXJXOBXNBUGCBPCUH
      ZXECUIZUJXJXOMZCBEUKXPXRXQXPXJXNCKXOXPXIXNCCBCULXIXEXLJZXPXNXEXFAUMZXPXLX
      MXEXLXMMXPXLCOVFUNQUOXNXNCBXPXNUPUQURXQXJXNXOXJXSCKZXQXNXIXSCXTUSYAXECKZX
      MJXQXNXEXLCUTXQYBXEXMXQYBXEXECVAVBVCRQXNBOVDVEVGVHXEXOXMXFAXOXEXMXEBSZXOX
      EXMMBEPZBSYCBETYDXEBBEVIVJVKXEXMBVLVRVMXMXFAXFCSZXMXFAMCFPZCSYECFTYFXFCCF
      VIVJVKXFACVLVRVMVNRQVOVPWQWJWRWMXDDWPWIVQZWQWLXCAWQWKXABCWQWJWRAYGVSVTWAW
      BWCWDWEVOWF $.
  $}

  ${
    $d x y ps $.  $d x y A $.  $d x y B $.
    $( Closed theorem form of ~ copsex2g .  (Contributed by NM,
       17-Feb-2013.) $)
    copsex2t $p |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
      /\ ( A e. V /\ B e. W ) ) ->
                  ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) $=
      ( wcel wa cv wceq wb wal wex cop elisset nfe1 nfv nfbi anim12i eeanv nfa1
      wi sylibr nfa2 nfex opeq12 copsexg eqcoms syl adantl sp 19.21bi bitr3d ex
      imp exlimd sylan2 ) EGIZFHIZJZCKZELZDKZFLZJZABMZUDZDNZCNZVGDOZCOZEFPZVCVE
      PZLAJZDOZCOZBMZVBVDCOZVFDOZJVMUTVTVAWACEGQDFHQUAVDVFCDUBUEVKVMVSVKVLVSCVJ
      CUCVRBCVQCRBCSTVKVGVSDVIDCUFVRBDVQDCVPDRUGBDSTVKVGVSVKVGJAVRBVGAVRMZVKVGV
      OVNLWBVCVEEFUHWBVNVOACDVNUIUJUKULVKVGVHVKVIDVJCUMUNUQUOUPURURUQUS $.
  $}

  ${
    $d x y ps $.  $d x y A $.  $d x y B $.
    copsex2g.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( Implicit substitution inference for ordered pairs.  (Contributed by NM,
       28-May-1995.) $)
    copsex2g $p |- ( ( A e. V /\ B e. W ) ->
                  ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) $=
      ( wcel cv wceq wex cop wa wb elisset nfe1 nfv nfbi eeanv nfex copsexg syl
      opeq12 eqcoms bitr3d exlimi sylbir syl2an ) EGJCKZELZCMZDKZFLZDMZEFNZUKUN
      NZLAOZDMZCMZBPZFHJCEGQDFHQUMUPOULUOOZDMZCMVBULUOCDUAVDVBCVABCUTCRBCSTVCVB
      DVABDUTDCUSDRUBBDSTVCAVABVCURUQLAVAPZUKUNEFUEVEUQURACDUQUCUFUDIUGUHUHUIUJ
      $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.  $d x y z w C $.  $d x y z w D $.
    $d x y z w ps $.  $d x y z w R $.  $d x y z w S $.
    copsex4g.1 $e |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) ->
                     ( ph <-> ps ) ) $.
    $( An implicit substitution inference for 2 ordered pairs.  (Contributed by
       NM, 5-Aug-1995.) $)
    copsex4g $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) ->
                      ( E. x E. y E. z E. w ( ( <. A , B >. = <. x , y >. /\
                      <. C , D >. = <. z , w >. ) /\ ph ) <-> ps ) ) $=
      ( wcel wa cop cv wceq wex vex eqcom opth bitri anbi12i anbi1i a1i 4exbidv
      wb id cgsex4g bitrd ) GKNHLNOIKNJLNOOZGHPZCQZDQZPZRZIJPZEQZFQZPZRZOZAOZFS
      ESDSCSUNGRUOHROZUSIRUTJROZOZAOZFSESDSCSBULVDVHCDEFVDVHUHULVCVGAUQVEVBVFUQ
      UPUMRVEUMUPUAUNUOGHCTDTUBUCVBVAURRVFURVAUAUSUTIJETFTUBUCUDUEUFUGABVGCDEFG
      HIJKLVGUIMUJUK $.
  $}

  $( A property of ordered pairs.  (Contributed by Mario Carneiro,
     26-Apr-2015.) $)
  0nelop $p |- -. (/) e. <. A , B >. $=
    ( c0 cop wcel csn cpr wo id cvv wa oprcl dfopg syl eleqtrd elpri wne necomd
    wceq wn simpld snnzg prnzg jca neanior sylib pm2.65i ) CABDZEZCAFZSCABGZSHZ
    UICUJUKGZEULUICUHUMUIIUIAJEZBJEZKUHUMSABCLZABJJMNOCUJUKPNUICUJQZCUKQZKULTUI
    UQURUIUJCUIUNUJCQUIUNUOUPUAZAJUBNRUIUKCUIUNUKCQUSABJUCNRUDCUJCUKUEUFUG $.

  $( An ordered pair is equal to the ordered pair without the empty set.  This
     is because no ordered pair contains the empty set.  (Contributed by AV,
     15-Nov-2021.) $)
  opwo0id $p |- <. X , Y >. = ( <. X , Y >. \ { (/) } ) $=
    ( cop c0 csn cdif cin wceq wcel 0nelop disjsn mpbir disjdif2 ax-mp eqcomi
    wn ) ABCZDEZFZQQRGDHZSQHTDQIPABJQDKLQRMNO $.

  ${
    $d A x $.  $d B x $.  $d C x $.  $d D x $.

    $( Equivalence of existence implied by equality of ordered pairs.
       (Contributed by NM, 28-May-2008.) $)
    opeqex $p |- ( <. A , B >. = <. C , D >. ->
      ( ( A e. _V /\ B e. _V ) <-> ( C e. _V /\ D e. _V ) ) ) $=
      ( vx cop wceq cv wcel wex cvv wa eleq2 exbidv opm 3bitr3g ) ABFZCDFZGZEHZ
      QIZEJTRIZEJAKIBKILCKIDKILSUAUBEQRTMNEABOECDOP $.
  $}

  ${
    opcom.1 $e |- A e. _V $.
    opcom.2 $e |- B e. _V $.
    $( An ordered pair commutes iff its members are equal.  (Contributed by NM,
       28-May-2009.) $)
    opcom $p |- ( <. A , B >. = <. B , A >. <-> A = B ) $=
      ( cop wceq wa opth eqcom anbi2i anidm 3bitri ) ABEBAEFABFZBAFZGMMGMABBACD
      HNMMBAIJMKL $.
  $}

  ${
    $d x y A $.  $d y B $.
    moop2.1 $e |- B e. _V $.
    $( "At most one" property of an ordered pair.  (Contributed by NM,
       11-Apr-2004.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    moop2 $p |- E* x A = <. B , x >. $=
      ( vy cv cop wceq wmo csb wa wal eqtr2 vex opth simprbi syl gen2 nfcsb1v
      wi nfcv nfop nfeq2 csbeq1a id opeq12d eqeq2d mo4f mpbir ) BCAFZGZHZAIULBA
      EFZCJZUMGZHZKZUJUMHZTZELALUSAEUQUKUOHZURBUKUOMUTCUNHURCUJUNUMDANOPQRULUPA
      EABUOAUNUMAUMCSAUMUAUBUCURUKUOBURCUNUJUMAUMCUDURUEUFUGUHUI $.
  $}

  ${
    opeqsn.1 $e |- A e. _V $.
    opeqsn.2 $e |- B e. _V $.
    opeqsn.3 $e |- C e. _V $.
    $( Equivalence for an ordered pair equal to a singleton.  (Contributed by
       NM, 3-Jun-2008.) $)
    opeqsn $p |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) ) $=
      ( cop csn wceq cpr wa dfop eqeq1i snex cvv wcel preqsn eqcom bitri 3bitri
      prexg mp2an anbi2i anidm anbi1i dfsn2 preq2 eqtr2id eqeq1d bitrdi pm5.32i
      ) ABGZCHZIAHZABJZJZUMIUNUOIZUOCIZKZABIZCUNIZKZULUPUMABDELMUNUOCADNAOPBOPU
      OOPDEABOOUAUBFQUSUTURKVBUQUTURUQUOUNIUTBAIZKZUTUNUORABADEDQVDUTUTKUTVCUTU
      TBARUCUTUDSTUEUTURVAUTURUNCIVAUTUOUNCUTUNAAJUOAUFABAUGUHUIUNCRUJUKST $.
  $}

  ${
    opeqpr.1 $e |- A e. _V $.
    opeqpr.2 $e |- B e. _V $.
    opeqpr.3 $e |- C e. _V $.
    opeqpr.4 $e |- D e. _V $.
    $( Equivalence for an ordered pair equal to an unordered pair.
       (Contributed by NM, 3-Jun-2008.) $)
    opeqpr $p |- ( <. A , B >. = { C , D }
  <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) ) $=
      ( cop cpr wceq csn wa wo eqcom dfop eqeq2i snex cvv wcel preq12b 3bitri
      prexg mp2an ) ABIZCDJZKUFUEKUFALZABJZJZKCUGKDUHKMCUHKDUGKMNUEUFOUEUIUFABE
      FPQCDUGUHGHAERASTBSTUHSTEFABSSUCUDUAUB $.
  $}

  ${
    $d a b c x y A $.  $d a b c x y B $.  $d a b c x y C $.  $d a b c x ph $.
    $d y ps $.
    euotd.1 $e |- ( ph -> A e. _V ) $.
    euotd.2 $e |- ( ph -> B e. _V ) $.
    euotd.3 $e |- ( ph -> C e. _V ) $.
    euotd.4 $e |- ( ph -> ( ps <-> ( a = A /\ b = B /\ c = C ) ) ) $.
    $( Prove existential uniqueness for an ordered triple.  (Contributed by
       Mario Carneiro, 20-May-2015.) $)
    euotd $p |- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) ) $=
      ( vy cv wceq wa wex cvv wtru cotp wb wal weu otexg syl3anc w3a biimpa vex
      wcel otth sylibr eqeq2d biimpd impancom expimpd exlimdv exlimdvv wsbc tru
      adantr ad2antrr simpr eqcomd biimpar jca trud 2thd 3anassrs sbcied mpbiri
      spesbcd nfcv nfsbc1v nfex sbceq1a exbidv sylc 2exbidv excom13 sylib eqeq1
      spcegf anbi1d 3exbidv syl5ibrcom impbid alrimiv eqeq2 bibi2d albidv df-eu
      spcegv ) ACOZGOZHOZIOZUAZPZBQZIRZHRGRZWNNOZPZUBZCUCZNRZXBCUDADEFUAZSUJZXB
      WNXHPZUBZCUCZXGADSUJESUJZFSUJZXIJKLDEFSSSUEUFAXKCAXBXJAXAXJGHAWTXJIAWSBXJ
      ABWSXJABQZWSXJXOWRXHWNXOWODPZWPEPZWQFPZUGZWRXHPZABXSMUHWOWPDEWQFGUIHUIIUI
      UKZULUMUNUOUPUQURAXBXJXHWRPZBQZIRHRGRZAYCGRHRZIRZYDAXNYCIFUSZGRZHRZYFLAXM
      YGHEUSZGRZYIKAYJGDAYJGDUSTUTAYJTGDSJAXPQZYGTHESAXMXPKVAYLXQQYCTIFSAXNXPXQ
      LVBAXPXQXRYCTUBAXSQZYCTYMYBBYMWRXHYMXSXTAXSVCYAULVDABXSMVEVFYMVGVHVIVJVJV
      JVKVLYHYKHESHEVMYJHGYGHEVNVOXQYGYJGYGHEVPVQWCVRYEYIIFSIFVMYHIHYGIGYCIFVNV
      OVOXRYCYGHGYCIFVPVSWCVRYCIHGVTWAXJWTYCGHIXJWSYBBWNXHWRWBWDWEWFWGWHXFXLNXH
      SXCXHPZXEXKCYNXDXJXBXCXHWNWIWJWKWMVRXBCNWLUL $.
  $}

  ${
    opthw.1 $e |- A e. _V $.
    opthw.2 $e |- B e. _V $.
    $( The union of an ordered pair.  Theorem 65 of [Suppes] p. 39.
       (Contributed by NM, 17-Aug-2004.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    uniop $p |- U. <. A , B >. = { A , B } $=
      ( cop cuni csn cpr cun dfop unieqi snex cvv prexg mp2an unipr wss snsspr1
      wcel wceq ssequn1 mpbi 3eqtri ) ABEZFAGZABHZHZFUEUFIZUFUDUGABCDJKUEUFACLA
      MSBMSUFMSCDABMMNOPUEUFQUHUFTABRUEUFUAUBUC $.

    $( Ordered pair membership is inherited by class union.  (Contributed by
       NM, 13-May-2008.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    uniopel $p |- ( <. A , B >. e. C -> U. <. A , B >. e. U. C ) $=
      ( cop wcel cuni cpr uniop opi2 eqeltri elssuni sseld mpi ) ABFZCGZPHZPGRC
      HZGRABIPABDEJABDEKLQPSRPCMNO $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordered-pair class abstractions (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x z $.  $d y z $.  $d ph z $.
    $( The law of concretion.  Special case of Theorem 9.5 of [Quine] p. 61.
       (Contributed by NM, 14-Apr-1995.)  (Proof shortened by Andrew Salmon,
       25-Jul-2011.) $)
    opabid $p |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) $=
      ( vz cv cop wceq wa wex copab vex opex copsexg bicomd df-opab elab2 ) DEZ
      BEZCEZFZGZAHCIBIZADTABCJRSBKCKLUAAUBABCQMNABCDOP $.
  $}

  ${
    $d x y z $.  $d ph z $.
    $( The law of concretion.  Special case of Theorem 9.5 of [Quine] p. 61.
       Version of ~ opabid with a disjoint variable condition.  (Contributed by
       NM, 14-Apr-1995.)  (Revised by GG, 26-Jan-2024.) $)
    opabidw $p |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) $=
      ( opabid ) ABCD $.
  $}

  ${
    $d x z A $.  $d y z A $.  $d z ph $.
    $( Membership in a class abstraction of ordered pairs.  (Contributed by NM,
       24-Mar-1998.) $)
    elopab $p |- ( A e. { <. x , y >. | ph } <->
                 E. x E. y ( A = <. x , y >. /\ ph ) ) $=
      ( vz copab wcel cvv cv cop wceq wex elex vex eleq1 mpbiri adantr exlimivv
      wa opex eqeq1 anbi1d 2exbidv df-opab elab2g pm5.21nii ) DABCFZGDHGZDBIZCI
      ZJZKZASZCLBLZDUGMUMUHBCULUHAULUHUKHGUIUJBNCNTDUKHOPQREIZUKKZASZCLBLUNEDUG
      HUODKZUQUMBCURUPULAUODUKUAUBUCABCEUDUEUF $.
  $}

  ${
    $d x y z $.  $d x y w $.
    $( The law of concretion in terms of substitutions.  Less general than
       ~ opelopabsb , but having a much shorter proof.  (Contributed by NM,
       30-Sep-2002.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    opelopabsbALT $p |- ( <. z , w >. e. { <. x , y >. | ph }
                 <-> [ w / y ] [ z / x ] ph ) $=
      ( cv cop wceq wa wex copab wcel wsb excom vex opth equcom anbi12ci anbi1i
      bitri 2exbii elopab 2sb5 3bitr4i ) DFZEFZGZBFZCFZGHZAIZCJBJZUIUFHZUHUEHZI
      ZAIZBJCJZUGABCKLABDMCEMULUKBJCJUQUKBCNUKUPCBUJUOAUJUEUHHZUFUIHZIUOUEUFUHU
      IDOEOPURUNUSUMDBQECQRTSUATABCUGUBACBEDUCUD $.
  $}

  ${
    $d u v x y z w $.  $d u v w z A $.  $d u v w x B $.  $d u v w z ph $.
    $( The law of concretion in terms of substitutions.  (Contributed by NM,
       30-Sep-2002.)  (Revised by Mario Carneiro, 18-Nov-2016.) $)
    opelopabsb $p |- ( <. A , B >. e. { <. x , y >. | ph }
                 <-> [. A / x ]. [. B / y ]. ph ) $=
      ( vv vu vz vw cop wcel cvv wa wsb cv wceq wex nfs1v weq wb copab wsbc vex
      elopab simpl eqcomd opth sylib 2eximi eeanv isset bitr4i sylbi nfv nfsbxy
      anbi12i sbequ12 sylan9bbr cbvopab eleq2s sbcex spesbc exlimiv syl dfsbcq2
      jca opeq1 eleq1d bibi12d opeq2 sbcbidv nfopab1 nfel2 nfopab2 opabid chvar
      nfbi vtocl2g pm5.21nii ) DEJZABCUAZKZDLKZELKZMZACEUBZBDUBZWEVTACFNZBGNZGF
      UAZWAVTWJKVTGOZFOZJZPZWIMZFQGQZWEWIGFVTUDWPWKDPZWLEPZMZFQGQZWEWOWSGFWOWMV
      TPWSWOVTWMWNWIUEUFWKWLDEGUCFUCUGUHUIWTWQGQZWRFQZMWEWQWRGFUJWCXAWDXBGDUKFE
      UKUPULUHUMAWIBCGFAGUNAFUNWHBGRWHBGCACFRUOCFSAWHBGSWIACFUQWHBGUQURUSUTWGWC
      WDWFBDVAWGWFBQWDWFBDVBWFWDBACEVAVCVDVFHOZIOZJZWAKZACINZBHNZTZDXDJZWAKZXGB
      DUBZTWBWGTHIDELLXCDPZXFXKXHXLXMXEXJWAXCDXDVGVHXGBHDVEVIXDEPZXKWBXLWGXNXJV
      TWAXDEDVJVHXNXGWFBDACIEVEVKVIBOZXDJZWAKZXGTZXIBHXFXHBBXEWAABCVLVMXGBHRVQB
      HSZXQXFXGXHXSXPXEWAXOXCXDVGVHXGBHUQVIXOCOZJZWAKZATXRCIXQXGCCXPWAABCVNVMAC
      IRVQCISZYBXQAXGYCYAXPWAXTXDXOVJVHACIUQVIABCVOVPVPVRVS $.

    brabsb.1 $e |- R = { <. x , y >. | ph } $.
    $( The law of concretion in terms of substitutions.  (Contributed by NM,
       17-Mar-2008.) $)
    brabsb $p |- ( A R B <-> [. A / x ]. [. B / y ]. ph ) $=
      ( wbr cop wcel copab wsbc df-br eleq2i opelopabsb 3bitri ) DEFHDEIZFJQABC
      KZJACELBDLDEFMFRQGNABCDEOP $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ch $.
    $( Closed theorem form of ~ opelopab .  (Contributed by NM,
       19-Feb-2013.) $)
    opelopabt $p |- ( ( A. x A. y ( x = A -> ( ph <-> ps ) )
                     /\ A. x A. y ( y = B -> ( ps <-> ch ) )
                     /\ ( A e. V /\ B e. W ) ) ->
                    ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) ) $=
      ( cop copab wcel cv wceq wa wex wb wi wal w3a elopab anim12 2alimi sylbir
      19.26-2 bitr syl6 copsex2t sylan 3impa bitrid ) FGJZADEKLULDMZEMZJNAOEPDP
      ZUMFNZABQZRZESDSZUNGNZBCQZRZESDSZFHLGILOZTCADEULUAUSVCVDUOCQZUSVCOZUPUTOZ
      ACQZRZESDSZVDVEVFURVBOZESDSVJURVBDEUEVKVIDEVKVGUQVAOVHUPUQUTVAUBABCUFUGUC
      UDACDEFGHIUHUIUJUK $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ps $.
    opelopabga.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( The law of concretion.  Theorem 9.5 of [Quine] p. 61.  (Contributed by
       Mario Carneiro, 19-Dec-2013.) $)
    opelopabga $p |- ( ( A e. V /\ B e. W ) ->
                    ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) ) $=
      ( cop copab wcel cv wceq wa wex elopab copsex2g bitrid ) EFJZACDKLTCMDMJN
      AODPCPEGLFHLOBACDTQABCDEFGHIRS $.

    ${
      brabga.2 $e |- R = { <. x , y >. | ph } $.
      $( The law of concretion for a binary relation.  (Contributed by Mario
         Carneiro, 19-Dec-2013.) $)
      brabga $p |- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) ) $=
        ( wbr cop copab wcel wa df-br eleq2i bitri opelopabga bitrid ) EFGLZEFM
        ZACDNZOZEHOFIOPBUBUCGOUEEFGQGUDUCKRSABCDEFHIJTUA $.
    $}

    $d x y C $.  $d x y D $.
    $( Ordered pair membership in an ordered pair class abstraction.
       (Contributed by Mario Carneiro, 19-Dec-2013.) $)
    opelopab2a $p |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e.
                 { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) ) $=
      ( wcel wa cop cv copab wceq eleq1 bi2anan9 anbi12d opelopabga bianabs ) E
      GJZFHJZKZEFLCMZGJZDMZHJZKZAKZCDNJBUIUCBKCDEFGHUDEOZUFFOZKUHUCABUJUEUAUKUG
      UBUDEGPUFFHPQIRST $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ps $.
    opelopaba.1 $e |- A e. _V $.
    opelopaba.2 $e |- B e. _V $.
    opelopaba.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( The law of concretion.  Theorem 9.5 of [Quine] p. 61.  (Contributed by
       Mario Carneiro, 19-Dec-2013.) $)
    opelopaba $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) $=
      ( cvv wcel cop copab wb opelopabga mp2an ) EJKFJKEFLACDMKBNGHABCDEFJJIOP
      $.

    ${
      braba.4 $e |- R = { <. x , y >. | ph } $.
      $( The law of concretion for a binary relation.  (Contributed by NM,
         19-Dec-2013.) $)
      braba $p |- ( A R B <-> ps ) $=
        ( cvv wcel wbr wb brabga mp2an ) ELMFLMEFGNBOHIABCDEFGLLJKPQ $.
    $}
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ch $.
    opelopabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    opelopabg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( The law of concretion.  Theorem 9.5 of [Quine] p. 61.  (Contributed by
       NM, 28-May-1995.)  (Revised by Mario Carneiro, 19-Dec-2013.) $)
    opelopabg $p |- ( ( A e. V /\ B e. W ) ->
                    ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) ) $=
      ( cv wceq sylan9bb opelopabga ) ACDEFGHIDLFMABELGMCJKNO $.

    ${
      brabg.5 $e |- R = { <. x , y >. | ph } $.
      $( The law of concretion for a binary relation.  (Contributed by NM,
         16-Aug-1999.)  (Revised by Mario Carneiro, 19-Dec-2013.) $)
      brabg $p |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) ) $=
        ( cv wceq sylan9bb brabga ) ACDEFGJHIDNFOABENGOCKLPMQ $.
    $}
  $}

  ${
    $d x y A $.  $d x y B $.
    opelopabgf.x $e |- F/ x ps $.
    opelopabgf.y $e |- F/ y ch $.
    opelopabgf.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    opelopabgf.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( The law of concretion.  Theorem 9.5 of [Quine] p. 61.  This version of
       ~ opelopabg uses bound-variable hypotheses in place of distinct variable
       conditions.  (Contributed by Alexander van der Vekens, 8-Jul-2018.) $)
    opelopabgf $p |- ( ( A e. V /\ B e. W )
                       -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) ) $=
      ( cop copab wcel wsbc wa opelopabsb sbciegf nfcv cv wceq sbcbidv sylan9bb
      nfsbcw bitrid ) FGNADEOPAEGQZDFQZFHPZGIPZRCADEFGSUJUIBEGQZUKCUHULDFHBDEGD
      GUAJUFDUBFUCABEGLUDTBCEGIKMTUEUG $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y ch $.
    opelopab2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    opelopab2.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( Ordered pair membership in an ordered pair class abstraction.
       (Contributed by NM, 14-Oct-2007.)  (Revised by Mario Carneiro,
       19-Dec-2013.) $)
    opelopab2 $p |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e.
                 { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ch ) ) $=
      ( cv wceq sylan9bb opelopab2a ) ACDEFGHIDLFMABELGMCJKNO $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ch $.
    opelopab.1 $e |- A e. _V $.
    opelopab.2 $e |- B e. _V $.
    opelopab.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    opelopab.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( The law of concretion.  Theorem 9.5 of [Quine] p. 61.  (Contributed by
       NM, 16-May-1995.) $)
    opelopab $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) $=
      ( cvv wcel cop copab wb opelopabg mp2an ) FLMGLMFGNADEOMCPHIABCDEFGLLJKQR
      $.

    ${
      brab.5 $e |- R = { <. x , y >. | ph } $.
      $( The law of concretion for a binary relation.  (Contributed by NM,
         16-Aug-1999.) $)
      brab $p |- ( A R B <-> ch ) $=
        ( cvv wcel wbr wb brabg mp2an ) FNOGNOFGHPCQIJABCDEFGNNHKLMRS $.
    $}
  $}

  ${
    $d x y A $.  $d x y B $.
    opelopabaf.x $e |- F/ x ps $.
    opelopabaf.y $e |- F/ y ps $.
    opelopabaf.1 $e |- A e. _V $.
    opelopabaf.2 $e |- B e. _V $.
    opelopabaf.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( The law of concretion.  Theorem 9.5 of [Quine] p. 61.  This version of
       ~ opelopab uses bound-variable hypotheses in place of distinct variable
       conditions.  (Contributed by Mario Carneiro, 19-Dec-2013.)  (Proof
       shortened by Mario Carneiro, 18-Nov-2016.) $)
    opelopabaf $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) $=
      ( cop copab wcel wsbc opelopabsb cvv wb nfv sbc2iegf mp2an bitri ) EFLACD
      MNADFOCEOZBACDEFPEQNFQNZUCBRIJABCDEFQQGHUDCSKTUAUB $.
  $}

  ${
    $d x y A $.  $d x y B $.
    opelopabf.x $e |- F/ x ps $.
    opelopabf.y $e |- F/ y ch $.
    opelopabf.1 $e |- A e. _V $.
    opelopabf.2 $e |- B e. _V $.
    opelopabf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    opelopabf.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( The law of concretion.  Theorem 9.5 of [Quine] p. 61.  This version of
       ~ opelopab uses bound-variable hypotheses in place of distinct variable
       conditions.  (Contributed by NM, 19-Dec-2008.) $)
    opelopabf $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) $=
      ( cop wcel wsbc cvv wb sbciegf ax-mp copab opelopabsb nfcv cv wceq 3bitri
      nfsbc sbcbidv ) FGNADEUAOAEGPZDFPZBEGPZCADEFGUBFQOUJUKRJUIUKDFQBDEGDGUCHU
      GDUDFUEABEGLUHSTGQOUKCRKBCEGQIMSTUF $.
  $}

  ${
    $d ph z $.  $d ps z $.  $d x z $.  $d y z $.
    $( Equivalence of ordered pair abstraction subclass and implication.
       (Contributed by NM, 27-Dec-1996.)  (Revised by Mario Carneiro,
       19-May-2013.) $)
    ssopab2 $p |- ( A. x A. y ( ph -> ps ) ->
        { <. x , y >. | ph } C_ { <. x , y >. | ps } ) $=
      ( vz wi wal cv cop wceq wa wex cab copab nfa1 sp anim2d eximd sps df-opab
      ss2abdv 3sstr4g ) ABFZDGZCGZEHCHDHIJZAKZDLZCLZEMUFBKZDLZCLZEMACDNBCDNUEUI
      ULEUEUHUKCUDCOUDUHUKFCUDUGUJDUCDOUDABUFUCDPQRSRUAACDETBCDETUB $.
  $}

  $( Equivalence of ordered pair abstraction subclass and implication.
     (Contributed by NM, 27-Dec-1996.)  (Proof shortened by Mario Carneiro,
     18-Nov-2016.) $)
  ssopab2b $p |- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } <->
             A. x A. y ( ph -> ps ) ) $=
    ( copab wss wi wal nfopab1 nfss nfopab2 cop wcel ssel opabid 3imtr3g alrimi
    cv ssopab2 impbii ) ACDEZBCDEZFZABGZDHZCHUCUECCUAUBACDIBCDIJUCUDDDUAUBACDKB
    CDKJUCCRDRLZUAMUFUBMABUAUBUFNACDOBCDOPQQABCDST $.

  ${
    ssopab2i.1 $e |- ( ph -> ps ) $.
    $( Inference of ordered pair abstraction subclass from implication.
       (Contributed by NM, 5-Apr-1995.) $)
    ssopab2i $p |- { <. x , y >. | ph } C_ { <. x , y >. | ps } $=
      ( wi wal copab wss ssopab2 ax-gen mpg ) ABFZDGACDHBCDHICABCDJMDEKL $.
  $}

  ${
    $d x ph $.  $d y ph $.
    ssopab2dv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference of ordered pair abstraction subclass from implication.
       (Contributed by NM, 19-Jan-2014.)  (Revised by Mario Carneiro,
       24-Jun-2014.) $)
    ssopab2dv $p |- ( ph -> { <. x , y >. | ps } C_ { <. x , y >. | ch } ) $=
      ( wi wal copab wss alrimivv ssopab2 syl ) ABCGZEHDHBDEICDEIJANDEFKBCDELM
      $.
  $}

  $( Equivalence of ordered pair abstraction equality and biconditional.
     (Contributed by Mario Carneiro, 4-Jan-2017.) $)
  eqopab2b $p |- ( { <. x , y >. | ph } = { <. x , y >. | ps }
              <-> A. x A. y ( ph <-> ps ) ) $=
    ( copab wss wa wi wal wceq wb ssopab2b anbi12i eqss 2albiim 3bitr4i ) ACDEZ
    BCDEZFZRQFZGABHDICIZBAHDICIZGQRJABKDICISUATUBABCDLBACDLMQRNABCDOP $.

  ${
    $d z ph $.  $d z x $.  $d z y $.
    $( Inhabited ordered pair class abstraction.  (Contributed by Jim Kingdon,
       29-Sep-2018.) $)
    opabm $p |- ( E. z z e. { <. x , y >. | ph } <-> E. x E. y ph ) $=
      ( cv copab wcel wex cop wceq wa elopab exbii exrot3 isseti 19.41v mpbiran
      vex opex 2exbii 3bitri ) DEZABCFGZDHUBBEZCEZIZJZAKZCHBHZDHUHDHZCHBHACHBHU
      CUIDABCUBLMUHDBCNUJABCUJUGDHADUFUDUEBRCRSOUGADPQTUA $.
  $}

  ${
    $d ph w $.  $d A w x $.  $d A y $.  $d w y z $.  $d x z $.
    $( Move indexed union inside an ordered-pair abstraction.  (Contributed by
       Stefan O'Rear, 20-Feb-2015.) $)
    iunopab $p |- U_ z e. A { <. x , y >. | ph } =
        { <. x , y >. | E. z e. A ph } $=
      ( vw cv copab wcel wrex cab cop wceq wex ciun elopab rexcom4 exbii bitri
      wa rexbii r19.42v abbii df-iun df-opab 3eqtr4i ) FGZABCHZIZDEJZFKUGBGCGLM
      ZADEJZTZCNZBNZFKDEUHOULBCHUJUOFUJUKATZCNZBNZDEJZUOUIURDEABCUGPUAUSUQDEJZB
      NUOUQDBEQUTUNBUTUPDEJZCNUNUPDCEQVAUMCUKADEUBRSRSSUCDFEUHUDULBCFUEUF $.
  $}

  ${
    $d R x $.  $d R y $.
    $( Membership in an ordered-pair class abstraction defined by a binary
       relation.  (Contributed by AV, 16-Feb-2021.)  (Proof shortened by SN,
       11-Dec-2024.) $)
    elopabr $p |- ( A e. { <. x , y >. | x R y } -> A e. R ) $=
      ( cv wbr copab opabss sseli ) AEBEDFABGDCABDHI $.
    $( $j usage 'elopabr' avoids 'ax-sep' 'ax-nul' 'ax-pr'; $)

    $( Membership in an ordered-pair class abstraction defined by a restricted
       binary relation.  (Contributed by AV, 16-Feb-2021.) $)
    elopabran $p |- ( A e. { <. x , y >. | ( x R y /\ ps ) } -> A e. R ) $=
      ( cv wbr wa copab simpl ssopab2i opabss sstri sseli ) BFCFEGZAHZBCIZEDQOB
      CIEPOBCOAJKBCELMN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Power class of union and intersection
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A x $.  $d B x $.
    $( The power class of the intersection of two classes is the intersection
       of their power classes.  Exercise 4.12(j) of [Mendelson] p. 235.
       (Contributed by NM, 23-Nov-2003.) $)
    pwin $p |- ~P ( A i^i B ) = ( ~P A i^i ~P B ) $=
      ( vx cpw cin cv wss wa wcel ssin vex elpw anbi12i 3bitr4i ineqri eqcomi )
      ADZBDZEABEZDZCQRTCFZAGZUABGZHUASGUAQIZUARIZHUATIUAABJUDUBUEUCUAACKZLUABUF
      LMUASUFLNOP $.

    $( The power class of the union of two classes includes the union of their
       power classes.  Exercise 4.12(k) of [Mendelson] p. 235.  (Contributed by
       NM, 23-Nov-2003.) $)
    pwunss $p |- ( ~P A u. ~P B ) C_ ~P ( A u. B ) $=
      ( vx cpw cun cv wss wcel ssun elun vex elpw orbi12i bitri 3imtr4i ssriv
      wo ) CADZBDZEZABEZDZCFZAGZUCBGZQZUCUAGUCTHZUCUBHUCABIUGUCRHZUCSHZQUFUCRSJ
      UHUDUIUEUCACKZLUCBUJLMNUCUAUJLOP $.

    $( The power class of the union of two classes is a subset of the union of
       their power classes, if one class is a subclass of the other.  One
       direction of Exercise 4.12(l) of [Mendelson] p. 235.  (Contributed by
       Jim Kingdon, 30-Sep-2018.) $)
    pwssunim $p |- ( ( A C_ B \/ B C_ A ) ->
               ~P ( A u. B ) C_ ( ~P A u. ~P B ) ) $=
      ( wss wo cun cpw wceq ssequn2 pweq eqimss syl ssequn1 orim12i orcoms ssun
      sylbi ) ABCZBACZDABEZFZAFZCZTBFZCZDZTUAUCECRQUERUBQUDRSAGZUBBAHUFTUAGUBSA
      ITUAJKPQSBGZUDABLUGTUCGUDSBITUCJKPMNTUAUCOK $.
  $}

  $( Break up the power class of a union into a union of smaller classes.
     (Contributed by Jim Kingdon, 30-Sep-2018.) $)
  pwundifss $p |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_
      ~P ( A u. B ) $=
    ( cun cpw cdif undif1ss wss wceq wa pwunss unss simpli ssequn2 mpbi sseqtri
    mpbir ) ABCDZADZERCQRCZQQRFRQGZSQHTBDZQGZTUBIRUACQGABJRUAQKPLRQMNO $.

  $( The power class of the union of two classes equals the union of their
     power classes, iff one class is a subclass of the other.  Part of Exercise
     7(b) of [Enderton] p. 28.  (Contributed by Jim Kingdon, 30-Sep-2018.) $)
  pwunim $p |- ( ( A C_ B \/ B C_ A ) ->
      ~P ( A u. B ) = ( ~P A u. ~P B ) ) $=
    ( wss wo cun cpw wa wceq pwssunim pwunss biantru sylib eqss sylibr ) ABCBAC
    DZABEFZAFBFEZCZQPCZGZPQHORTABISRABJKLPQMN $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Epsilon and identity relations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constant symbols. $)
  $c _E $. $( Letter E (for epsilon relation) $)
  $c _I $.  $( Letter I (for identity relation) $)

  $( Extend class notation to include the epsilon relation. $)
  cep $a class _E $.

  $( Extend the definition of a class to include identity relation. $)
  cid $a class _I $.

  ${
    $d x y $.
    $( Define the epsilon relation.  Similar to Definition 6.22 of
       [TakeutiZaring] p. 30.  The epsilon relation and set membership are the
       same, that is, ` ( A _E B <-> A e. B ) ` when ` B ` is a set by
       ~ epelg .  Thus, 5 ` _E ` { 1 , 5 }.  (Contributed by NM,
       13-Aug-1995.) $)
    df-eprel $a |- _E = { <. x , y >. | x e. y } $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( The epsilon relation and membership are the same.  General version of
       ~ epel .  (Contributed by Scott Fenton, 27-Mar-2011.)  (Revised by Mario
       Carneiro, 28-Apr-2015.) $)
    epelg $p |- ( B e. V -> ( A _E B <-> A e. B ) ) $=
      ( vx vy wcel cvv cep wbr wi cop df-br cv copab wceq wa wex vex sylbi a1i
      elopab pm3.2i opeqex mpbiri simpld adantr exlimivv df-eprel eleq2s eleq12
      elex wb brabga expcom pm5.21ndd ) BCFZAGFZABHIZABFZURUQJUPURABKZHFUQABHLU
      QUTDMZEMZFZDENZHUTVDFUTVAVBKOZVCPZEQDQUQVCDEUTUAVFUQDEVEUQVCVEUQBGFZVEUQV
      GPVAGFZVBGFZPVHVIDRERUBABVAVBUCUDUEUFUGSDEUHZUISTUSUQJUPABUKTUQUPURUSULVC
      USDEABHGCVAAVBBUJVJUMUNUO $.
  $}

  ${
    epelc.1 $e |- B e. _V $.
    $( The epsilon relationship and the membership relation are the same.
       (Contributed by Scott Fenton, 11-Apr-2012.) $)
    epelc $p |- ( A _E B <-> A e. B ) $=
      ( cvv wcel cep wbr wb epelg ax-mp ) BDEABFGABEHCABDIJ $.
  $}

  $( The epsilon relation and the membership relation are the same.
     (Contributed by NM, 13-Aug-1995.) $)
  epel $p |- ( x _E y <-> x e. y ) $=
    ( cv vex epelc ) ACBCBDE $.

  ${
    $d x y $.
    $( Define the identity relation.  Definition 9.15 of [Quine] p. 64.  For
       example, 5 ` _I ` 5 and ` -. ` 4 ` _I ` 5.  (Contributed by NM,
       13-Aug-1995.) $)
    df-id $a |- _I = { <. x , y >. | x = y } $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Partial and total orderings
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  We have not yet defined relations ( ~ df-rel ), but here we introduce a few
  related notions we will use to develop ordinals.  The class variable ` R ` is
  no different from other class variables, but it reminds us that typically it
  represents what we will later call a "relation".

$)

  $( Declare new constant symbols. $)
  $c Po $.  $( Partial ordering predicate symbol (read: 'partial ordering'). $)
  $c Or $.  $( Strict total ordering predicate symbol (read: 'orders'). $)

  $( Extend wff notation to include the strict partial ordering predicate.
     Read:  ' ` R ` is a partial order on ` A ` .' $)
  wpo $a wff R Po A $.

  $( Extend wff notation to include the strict linear ordering predicate.
     Read:  ' ` R ` orders ` A ` .' $)
  wor $a wff R Or A $.

  ${
    $d x y z R $.  $d x y z A $.
    $( Define the strict partial order predicate.  Definition of [Enderton]
       p. 168.  The expression ` R Po A ` means ` R ` is a partial order on
       ` A ` .  (Contributed by NM, 16-Mar-1997.) $)
    df-po $a |- ( R Po A <-> A. x e. A A. y e. A A. z e. A
                ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) $.

    $( Define the strict linear order predicate.  The expression ` R Or A ` is
       true if relationship ` R ` orders ` A ` .  The property
       ` x R y -> ( x R z \/ z R y ) ` is called weak linearity by Proposition
       11.2.3 of [HoTT], p.  (varies).  If we assumed excluded middle, it would
       be equivalent to trichotomy, ` x R y \/ x = y \/ y R x ` .  (Contributed
       by NM, 21-Jan-1996.)  (Revised by Jim Kingdon, 4-Oct-2018.) $)
    df-iso $a |- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A A. z e. A
                ( x R y -> ( x R z \/ z R y ) ) ) ) $.
  $}

  ${
    $d x y z R $.  $d x y z A $.  $d x y z B $.
    $( Subset theorem for the partial ordering predicate.  (Contributed by NM,
       27-Mar-1997.)  (Proof shortened by Mario Carneiro, 18-Nov-2016.) $)
    poss $p |- ( A C_ B -> ( R Po B -> R Po A ) ) $=
      ( vx vy vz wss cv wbr wn wa wi wral wpo ssralv ralimdv syld df-po 3imtr4g
      ) ABGZDHZUACIJUAEHZCIUBFHZCIKUAUCCILKZFBMZEBMZDBMZUDFAMZEAMZDAMZBCNACNTUG
      UFDAMUJUFDABOTUFUIDATUFUEEAMUIUEEABOTUEUHEAUDFABOPQPQDEFBCRDEFACRS $.
  $}

  ${
    $d x y z R $.  $d x y z S $.  $d x y z A $.
    $( Equality theorem for partial ordering predicate.  (Contributed by NM,
       27-Mar-1997.) $)
    poeq1 $p |- ( R = S -> ( R Po A <-> S Po A ) ) $=
      ( vx vy vz wceq cv wbr wn wa wral wpo breq notbid anbi12d imbi12d ralbidv
      wi df-po 2ralbidv 3bitr4g ) BCGZDHZUDBIZJZUDEHZBIZUGFHZBIZKZUDUIBIZSZKZFA
      LZEALDALUDUDCIZJZUDUGCIZUGUICIZKZUDUICIZSZKZFALZEALDALABMACMUCUOVDDEAAUCU
      NVCFAUCUFUQUMVBUCUEUPUDUDBCNOUCUKUTULVAUCUHURUJUSUDUGBCNUGUIBCNPUDUIBCNQP
      RUADEFABTDEFACTUB $.
  $}

  $( Equality theorem for partial ordering predicate.  (Contributed by NM,
     27-Mar-1997.) $)
  poeq2 $p |- ( A = B -> ( R Po A <-> R Po B ) ) $=
    ( wceq wpo wss wi eqimss2 poss syl eqimss impbid ) ABDZACEZBCEZMBAFNOGBAHBA
    CIJMABFONGABKABCIJL $.

  ${
    $d R a b c $.  $d A a b c $.  $d x a b c $.
    nfpo.r $e |- F/_ x R $.
    nfpo.a $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for partial orders.  (Contributed by
       Stefan O'Rear, 20-Jan-2015.) $)
    nfpo $p |- F/ x R Po A $=
      ( va vb vc wpo cv wbr wn wa wi wral df-po nfcv nfbr nfan nfralxy nfn nfim
      nfxfr ) BCIFJZUDCKZLZUDGJZCKZUGHJZCKZMZUDUICKZNZMZHBOZGBOZFBOAFGHBCPUPAFB
      EUOAGBEUNAHBEUFUMAUEAAUDUDCAUDQZDUQRUAUKULAUHUJAAUDUGCUQDAUGQZRAUGUICURDA
      UIQZRSAUDUICUQDUSRUBSTTTUC $.

    $( Bound-variable hypothesis builder for total orders.  (Contributed by
       Stefan O'Rear, 20-Jan-2015.) $)
    nfso $p |- F/ x R Or A $=
      ( va vb vc wor wpo cv wbr wo wi wral wa df-iso nfcv nfbr nfralxy nfxfr
      nfpo nfor nfim nfan ) BCIBCJZFKZGKZCLZUGHKZCLZUJUHCLZMZNZHBOZGBOZFBOZPAFG
      HBCQUFUQAABCDEUBUPAFBEUOAGBEUNAHBEUIUMAAUGUHCAUGRZDAUHRZSUKULAAUGUJCURDAU
      JRZSAUJUHCUTDUSSUCUDTTTUEUA $.
  $}

  ${
    $d x y z R $.  $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.
    $( Properties of partial order relation in class notation.  (Contributed by
       NM, 27-Mar-1997.) $)
    pocl $p |- ( R Po A -> ( ( B e. A /\ C e. A /\ D e. A ) ->
               ( -. B R B /\ ( ( B R C /\ C R D ) -> B R D ) ) ) ) $=
      ( vx vy vz wcel wbr wa wi cv wceq breq1 imbi2d breq2 anbi2d wal wral r3al
      w3a wn id breq12d notbid anbi1d imbi12d anbi12d imbi1d df-po bitri biimpi
      wpo 19.21bbi 19.21bi com12 vtocl3ga ) BAICAIDAIUBAEUNZBBEJZUCZBCEJZCDEJZK
      ZBDEJZLZKZUSFMZVHEJZUCZVHGMZEJZVKHMZEJZKZVHVMEJZLZKZLUSVABVKEJZVNKZBVMEJZ
      LZKZLUSVAVBCVMEJZKZWALZKZLUSVGLFGHBCDAAAVHBNZVRWCUSWHVJVAVQWBWHVIUTWHVHBV
      HBEWHUDZWIUEUFWHVOVTVPWAWHVLVSVNVHBVKEOUGVHBVMEOUHUIPVKCNZWCWGUSWJWBWFVAW
      JVTWEWAWJVSVBVNWDVKCBEQVKCVMEOUIUJRPVMDNZWGVGUSWKWFVFVAWKWEVDWAVEWKWDVCVB
      VMDCEQRVMDBEQUHRPUSVHAIVKAIVMAIUBZVRUSWLVRLZHUSWMHSZFGUSWNGSFSZUSVRHATGAT
      FATWOFGHAEUKVRFGHAAAUAULUMUOUPUQURUQ $.
  $}

  ${
    $d x y z A $.  $d x y z R $.  $d x y z ph $.
    ispod.1 $e |- ( ( ph /\ x e. A ) -> -. x R x ) $.
    ispod.2 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) ->
                    ( ( x R y /\ y R z ) -> x R z ) ) $.
    $( Sufficient conditions for a partial order.  (Contributed by NM,
       9-Jul-2014.) $)
    ispod $p |- ( ph -> R Po A ) $=
      ( cv wbr wn wa wi wral wpo wcel w3a 3ad2antr1 jca ralrimivvva sylibr
      df-po ) ABIZUCFJKZUCCIZFJUEDIZFJLUCUFFJMZLZDENCENBENEFOAUHBCDEEEAUCEPZUEE
      PZUFEPZQLUDUGAUJUIUDUKGRHSTBCDEFUBUA $.
  $}

  ${
    $d x y z A $.  $d x y z ph $.  $d x y z R $.  $d x y z X $.  $d y z Y $.
    $d z Z $.
    swopolem.1 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) ->
                    ( x R y -> ( x R z \/ z R y ) ) ) $.
    $( Perform the substitutions into the strict weak ordering law.
       (Contributed by Mario Carneiro, 31-Dec-2014.) $)
    swopolem $p |- ( ( ph /\ ( X e. A /\ Y e. A /\ Z e. A ) ) ->
                    ( X R Y -> ( X R Z \/ Z R Y ) ) ) $=
      ( cv wbr wo wi wral wcel wceq breq1 imbi12d breq2 w3a ralrimivvva orbi12d
      orbi1d orbi2d imbi2d rspc3v mpan9 ) ABKZCKZFLZUIDKZFLZULUJFLZMZNZDEOCEOBE
      OGEPHEPIEPUAGHFLZGIFLZIHFLZMZNZAUPBCDEEEJUBUPVAGUJFLZGULFLZUNMZNUQVCULHFL
      ZMZNBCDGHIEEEUIGQZUKVBUOVDUIGUJFRVGUMVCUNUIGULFRUDSUJHQZVBUQVDVFUJHGFTVHU
      NVEVCUJHULFTUESULIQZVFUTUQVIVCURVEUSULIGFTULIHFRUCUFUGUH $.
  $}

  ${
    $d x y z A $.  $d x y z R $.  $d x y z ph $.
    swopo.1 $e |- ( ( ph /\ ( y e. A /\ z e. A ) ) ->
                    ( y R z -> -. z R y ) ) $.
    swopo.2 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) ->
                    ( x R y -> ( x R z \/ z R y ) ) ) $.
    $( A strict weak order is a partial order.  (Contributed by Mario Carneiro,
       9-Jul-2014.) $)
    swopo $p |- ( ph -> R Po A ) $=
      ( cv wcel wa wbr wn wi wral weq breq1 breq2 notbid imbi12d ralrimivva w3a
      id ancli rspc2va syl2anr pm2.01d 3adantr1 wo imp orcomd ord expimpd ispod
      sylan2d ) ABCDEFABIZEJZKUPUPFLZUQUQUQKCIZDIZFLZUTUSFLZMZNZDEOCEOURURMZNZA
      UQUQUQUCUDAVDCDEEGUAVDVFUPUTFLZUTUPFLZMZNCDUPUPEECBPZVAVGVCVIUSUPUTFQVJVB
      VHUSUPUTFRSTDBPZVGURVIVEUTUPUPFRVKVHURUTUPUPFQSTUEUFUGAUQUSEJZUTEJZUBKZVA
      VCUPUSFLZVGAVLVMVDUQGUHVNVOVCVGVNVOKZVBVGVPVGVBVNVOVGVBUIHUJUKULUMUOUN $.
  $}

  $( A partial order relation is irreflexive.  (Contributed by NM,
     27-Mar-1997.) $)
  poirr $p |- ( ( R Po A /\ B e. A ) -> -. B R B ) $=
    ( wcel wpo w3a wbr wn wa df-3an anabs1 anidm 3bitrri wi pocl simpld sylan2b
    imp ) BADZACEZSSSFZBBCGZHZUASSIZSIUDSSSSJSSKSLMTUAIUCUBUBIUBNZTUAUCUEIABBBC
    ORPQ $.

  $( A partial order relation is a transitive relation.  (Contributed by NM,
     27-Mar-1997.) $)
  potr $p |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
             ( ( B R C /\ C R D ) -> B R D ) ) $=
    ( wpo wcel w3a wa wbr wn wi pocl imp simprd ) AEFZBAGCAGDAGHZIBBEJKZBCEJCDE
    JIBDEJLZPQRSIABCDEMNO $.

  $( A partial order relation has no 2-cycle loops.  (Contributed by NM,
     27-Mar-1997.) $)
  po2nr $p |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) ->
              -. ( B R C /\ C R B ) ) $=
    ( wpo wcel wa wbr wn poirr adantrr wi potr 3exp2 com34 pm2.43d imp32 mtod )
    ADEZBAFZCAFZGGBCDHCBDHGZBBDHZSTUCIUAABDJKSTUAUBUCLZSTUAUDLSTUATUDSTUATUDABC
    BDMNOPQR $.

  $( A partial order relation has no 3-cycle loops.  (Contributed by NM,
     27-Mar-1997.) $)
  po3nr $p |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
             -. ( B R C /\ C R D /\ D R B ) ) $=
    ( wpo wcel w3a wa wbr wn po2nr 3adantr2 df-3an potr anim1d biimtrid mtod )
    AEFZBAGZCAGZDAGZHIZBCEJZCDEJZDBEJZHZBDEJZUFIZSTUBUIKUAABDELMUGUDUEIZUFIUCUI
    UDUEUFNUCUJUHUFABCDEOPQR $.

  ${
    $d x y z R $.
    $( Any relation is a partial ordering of the empty set.  (Contributed by
       NM, 28-Mar-1997.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    po0 $p |- R Po (/) $=
      ( vx vy vz c0 wpo cv wbr wn wa wi wral ral0 df-po mpbir ) EAFBGZPAHIPCGZA
      HQDGZAHJPRAHKJDELCELZBELSBMBCDEANO $.
  $}

  ${
    $d R v w x y z $.  $d S v w z $.  $d X v w y z $.  $d Y x z $.
    $d A v w x z $.  $d B v w x z $.
    pofun.1 $e |- S = { <. x , y >. | X R Y } $.
    pofun.2 $e |- ( x = y -> X = Y ) $.
    $( A function preserves a partial order relation.  (Contributed by Jeff
       Madsen, 18-Jun-2011.) $)
    pofun $p |- ( ( R Po B /\ A. x e. A X e. B ) -> S Po A ) $=
      ( vv vw vz wcel wa cv wbr csb weq cop wpo wn nfcsb1v nfel1 csbeq1a eleq1d
      wral rspc impcom poirr copab df-br eleq2i nfcv nfbr nfv vex breq1d csbief
      csbeq1 eqtr3id breq2d opelopabf 3bitri sylnibr sylan2 w3a com12 3anim123d
      anassrs wi imp adantll potr anbi12i 3imtr4g adantlr syldan ispod ) DEUAZG
      DNZACUGZOZKLMCFVTWBKPZCNZWDWDFQZUBZWBWEOVTAWDGRZDNZWGWEWBWIWAWIAWDCAWHDAW
      DGUCZUDAKSZGWHDAWDGUEZUFUHZUIVTWIOWHWHEQZWFDWHEUJWFWDWDTZFNWOGHEQZABUKZNW
      NWDWDFULFWQWOIUMWPWHHEQZWNABWDWDAWHHEWJAEUNZAHUNZUOZWNBUPKUQZXBWKGWHHEWLU
      RZBKSZHWHWHEXDHABPZGRZWHAXEGHBUQWTJUSZAXEWDGUTVAVBVCVDVEVFVJWCWELPZCNZMPZ
      CNZVGZWIAXHGRZDNZAXJGRZDNZVGZWDXHFQZXHXJFQZOZWDXJFQZVKZWBXLXQVTWBXLXQWBWE
      WIXIXNXKXPWEWBWIWMVHXIWBXNWAXNAXHCAXMDAXHGUCZUDALSZGXMDAXHGUEZUFUHVHXKWBX
      PWAXPAXJCAXODAXJGUCUDAMSGXODAXJGUEUFUHVHVIVLVMVTXQYBWBVTXQOWHXMEQZXMXOEQZ
      OWHXOEQZXTYADWHXMXOEVNXRYFXSYGXRWDXHTZFNYIWQNYFWDXHFULFWQYIIUMWPWRYFABWDX
      HXAYFBUPXBLUQZXCBLSZHXMWHEYKHXFXMXGAXEXHGUTVAVBVCVDXSXHXJTZFNYLWQNYGXHXJF
      ULFWQYLIUMWPXMHEQYGABXHXJAXMHEYCWSWTUOYGBUPYJMUQZYDGXMHEYEURBMSZHXOXMEYNH
      XFXOXGAXEXJGUTVAZVBVCVDVOYAWDXJTZFNYPWQNYHWDXJFULFWQYPIUMWPWRYHABWDXJXAYH
      BUPXBYMXCYNHXOWHEYOVBVCVDVPVQVRVS $.
  $}

  ${
    $d x y z R $.  $d x y z A $.
    $( A strict linear order is a strict partial order.  (Contributed by NM,
       28-Mar-1997.) $)
    sopo $p |- ( R Or A -> R Po A ) $=
      ( vx vy vz wor wpo cv wbr wo wi wral df-iso simplbi ) ABFABGCHZDHZBIOEHZB
      IQPBIJKEALDALCALCDEABMN $.
  $}

  ${
    $d x y z R $.  $d x y z A $.  $d x y z B $.
    $( Subset theorem for the strict ordering predicate.  (Contributed by NM,
       16-Mar-1997.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    soss $p |- ( A C_ B -> ( R Or B -> R Or A ) ) $=
      ( vx vy vz wpo cv wbr wi wral wor wcel w3a wal ssel r3al 3imtr4g df-iso
      wa wss wo poss 3anim123d imim1d 2alimdv alimdv anim12d ) ABUAZBCGZDHZEHZC
      IUKFHZCIUMULCIUBJZFBKEBKDBKZTACGZUNFAKEAKDAKZTBCLACLUIUJUPUOUQABCUCUIUKBM
      ZULBMZUMBMZNZUNJZFOEOZDOUKAMZULAMZUMAMZNZUNJZFOEOZDOUOUQUIVCVIDUIVBVHEFUI
      VGVAUNUIVDURVEUSVFUTABUKPABULPABUMPUDUEUFUGUNDEFBBBQUNDEFAAAQRUHDEFBCSDEF
      ACSR $.
  $}

  ${
    $d x y z R $.  $d x y z S $.  $d x y z A $.
    $( Equality theorem for the strict ordering predicate.  (Contributed by NM,
       16-Mar-1997.) $)
    soeq1 $p |- ( R = S -> ( R Or A <-> S Or A ) ) $=
      ( vx vy vz wceq wpo cv wbr wo wi wral wa wor poeq1 orbi12d imbi12d df-iso
      breq 2ralbidv ralbidv anbi12d 3bitr4g ) BCGZABHZDIZEIZBJZUGFIZBJZUJUHBJZK
      ZLZFAMEAMZDAMZNACHZUGUHCJZUGUJCJZUJUHCJZKZLZFAMEAMZDAMZNABOACOUEUFUQUPVDA
      BCPUEUOVCDAUEUNVBEFAAUEUIURUMVAUGUHBCTUEUKUSULUTUGUJBCTUJUHBCTQRUAUBUCDEF
      ABSDEFACSUD $.
  $}

  $( Equality theorem for the strict ordering predicate.  (Contributed by NM,
     16-Mar-1997.) $)
  soeq2 $p |- ( A = B -> ( R Or A <-> R Or B ) ) $=
    ( wceq wor wss wa wi wb soss anim12i eqss dfbi2 3imtr4i bicomd ) ABDZBCEZAC
    EZABFZBAFZGQRHZRQHZGPQRISUATUBABCJBACJKABLQRMNO $.

  $( A strict order relation is irreflexive.  (Contributed by NM,
     24-Nov-1995.) $)
  sonr $p |- ( ( R Or A /\ B e. A ) -> -. B R B ) $=
    ( wor wpo wcel wbr wn sopo poirr sylan ) ACDACEBAFBBCGHACIABCJK $.

  $( A strict order relation is a transitive relation.  (Contributed by NM,
     21-Jan-1996.) $)
  sotr $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
             ( ( B R C /\ C R D ) -> B R D ) ) $=
    ( wor wpo wcel w3a wbr wa wi sopo potr sylan ) AEFAEGBAHCAHDAHIBCEJCDEJKBDE
    JLAEMABCDENO $.

  ${
    $d x y z R $.  $d x y z A $.  $d x y z ph $.
    issod.1 $e |- ( ph -> R Po A ) $.
    issod.2 $e |- ( ( ph /\ ( x e. A /\ y e. A ) ) ->
                    ( x R y \/ x = y \/ y R x ) ) $.
    $( An irreflexive, transitive, trichotomous relation is a linear ordering
       (in the sense of ~ df-iso ).  (Contributed by NM, 21-Jan-1996.)
       (Revised by Mario Carneiro, 9-Jul-2014.) $)
    issod $p |- ( ph -> R Or A ) $=
      ( vz wpo cv wbr wo wi wral wor wcel wa w3a syl6 ralrimiva weq w3o 3adant3
      orc a1i simp3r breq1 syl5ibcom simp1 simp2r simp2l simp3l 3jca potr sylan
      olc expcomd imp syl21anc 3jaod mpd 3expa expr anassrs ralcom sylib df-iso
      sylanbrc ) ADEIZBJZHJZEKZVJCJZEKZVMVKEKZLZMZCDNHDNZBDNDEOFAVRBDAVJDPZQZVQ
      HDNZCDNVRVTWACDAVSVMDPZWAAVSWBQZQZVQHDWDVKDPZVLVPAWCWEVLQZVPAWCWFRZVNBCUA
      ZVMVJEKZUBZVPAWCWJWFGUCWGVNVPWHWIVNVPMWGVNVOUDUEWGWHVOVPWGVLWHVOAWCWEVLUF
      ZVJVMVKEUGUHVOVNUPZSWGWIVOVPWGAWBVSWERZVLWIVOMZAWCWFUIWGWBVSWEAVSWBWFUJAV
      SWBWFUKAWCWEVLULUMWKAWMQZVLWNWOWIVLVOAVIWMWIVLQVOMFDVMVJVKEUNUOUQURUSWLSU
      TVAVBVCTVDTVQCHDDVEVFTBHCDEVGVH $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.  $d x y z R $.
    $( A strict order relation satisfies weak linearity.  (Contributed by Jim
       Kingdon, 6-Oct-2018.) $)
    sowlin $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
        ( B R C -> ( B R D \/ D R C ) ) ) $=
      ( vx vy vz wcel w3a wbr wo wi cv wceq breq1 imbi2d breq2 wral wa wor rsp2
      orbi1d imbi12d orbi2d orbi12d wpo df-iso 3anass syl6 impd biimtrid adantl
      rsp sylbi com12 vtocl3ga impcom ) BAICAIDAIJAEUAZBCEKZBDEKZDCEKZLZMZUSFNZ
      GNZEKZVEHNZEKZVHVFEKZLZMZMUSBVFEKZBVHEKZVJLZMZMUSUTVNVHCEKZLZMZMUSVDMFGHB
      CDAAAVEBOZVLVPUSVTVGVMVKVOVEBVFEPVTVIVNVJVEBVHEPUCUDQVFCOZVPVSUSWAVMUTVOV
      RVFCBERWAVJVQVNVFCVHERUEUDQVHDOZVSVDUSWBVRVCUTWBVNVAVQVBVHDBERVHDCEPUFQQU
      SVEAIZVFAIZVHAIZJZVLUSAEUGZVLHASGASZFASZTWFVLMZFGHAEUHWIWJWGWFWCWDWETZTWI
      VLWCWDWEUIWIWCWKVLWIWCWHWKVLMWHFAUNVLGHAAUBUJUKULUMUOUPUQUR $.
  $}

  $( A strict order relation has no 2-cycle loops.  (Contributed by NM,
     21-Jan-1996.) $)
  so2nr $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
              -. ( B R C /\ C R B ) ) $=
    ( wor wpo wcel wa wbr wn sopo po2nr sylan ) ADEADFBAGCAGHBCDICBDIHJADKABCDL
    M $.

  $( A strict order relation has no 3-cycle loops.  (Contributed by NM,
     21-Jan-1996.) $)
  so3nr $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
             -. ( B R C /\ C R D /\ D R B ) ) $=
    ( wor wpo wcel w3a wbr wn sopo po3nr sylan ) AEFAEGBAHCAHDAHIBCEJCDEJDBEJIK
    AELABCDEMN $.

  $( One direction of ~ sotritric holds for all weakly linear orders.
     (Contributed by Jim Kingdon, 28-Sep-2019.) $)
  sotricim $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
      ( B R C -> -. ( B = C \/ C R B ) ) ) $=
    ( wor wcel wa wbr wceq wn wo w3a adantrr 3adant3 wi breq2 biimprcd 3ad2ant3
    sonr mtod 3expia so2nr imnan sylibr jcad ioran imbitrrdi ) ADEZBAFZCAFZGZGZ
    BCDHZBCIZJZCBDHZJZGUNUPKJULUMUOUQUHUKUMUOUHUKUMLUNBBDHZUHUKURJZUMUHUIUSUJAB
    DSMNUMUHUNUROUKUNURUMBCBDPQRTUAULUMUPGJUMUQOABCDUBUMUPUCUDUEUNUPUFUG $.

  ${
    sotritric.or $e |- R Or A $.
    sotritric.tri $e |- ( ( B e. A /\ C e. A ) ->
      ( B R C \/ B = C \/ C R B ) ) $.
    $( A trichotomy relationship, given a trichotomous order.  (Contributed by
       Jim Kingdon, 28-Sep-2019.) $)
    sotritric $p |- ( ( B e. A /\ C e. A ) ->
        ( B R C <-> -. ( B = C \/ C R B ) ) ) $=
      ( wcel wa wbr wceq wo wn wor wi sotricim mpan w3o 3orass ax-1 pm2.24 jaoi
      sylbi syl impbid ) BAGCAGHZBCDIZBCJZCBDIZKZLZADMUEUFUJNEABCDOPUEUFUGUHQZU
      JUFNZFUKUFUIKULUFUGUHRUFULUIUFUJSUIUFTUAUBUCUD $.

    $( A trichotomy relationship, given a trichotomous order.  (Contributed by
       Jim Kingdon, 13-Dec-2019.) $)
    sotritrieq $p |- ( ( B e. A /\ C e. A ) ->
        ( B = C <-> -. ( B R C \/ C R B ) ) ) $=
      ( wcel wa wceq wbr wo wn wi wor sonr mpan breq2 notbid syl5ibcom w3o jcad
      breq1 imbitrrdi adantr 3orrot 3orcomb 3orass 3bitri biimpi orcomd ord syl
      ioran impbid ) BAGZCAGZHZBCIZBCDJZCBDJZKZLZUOURVBMUPUOURUSLZUTLZHVBUOURVC
      VDUOBBDJZLZURVCADNUOVFEABDOPZURVEUSBCBDQRSUOVFURVDVGURVEUTBCBDUBRSUAUSUTU
      MUCUDUQUSURUTTZVBURMFVHVAURVHURVAVHURVAKZVHURUTUSTURUSUTTVIUSURUTUEURUTUS
      UFURUSUTUGUHUIUJUKULUN $.
  $}

  ${
    $d x y z R $.
    $( Any relation is a strict ordering of the empty set.  (Contributed by NM,
       16-Mar-1997.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    so0 $p |- R Or (/) $=
      ( vx vy vz c0 wor wpo cv wbr wo wi wral po0 ral0 df-iso mpbir2an ) EAFEAG
      BHZCHZAIQDHZAISRAIJKDELCELZBELAMTBNBCDEAOP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Founded and set-like relations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constant symbols. $)
  $c FrFor $. $( Well-founded predicate symbol (read: 'well-founded'). $)
  $c Fr $. $( Well-founded predicate symbol (read: 'well-founded'). $)
  $c Se $. $( Set-like predicate symbol (read: 'set-like'). $)
  $c We $. $( Well-ordering predicate symbol (read: 'well-orders') $)

  $( Extend wff notation to include the well-founded predicate. $)
  wfrfor $a wff FrFor R A S $.

  $( Extend wff notation to include the well-founded predicate.  Read:  ' ` R `
     is a well-founded relation on ` A ` .' $)
  wfr $a wff R Fr A $.

  $( Extend wff notation to include the set-like predicate.  Read:  ' ` R ` is
     set-like on ` A ` .' $)
  wse $a wff R Se A $.

  $( Extend wff notation to include the well-ordering predicate.  Read:
      ' ` R ` well-orders ` A ` .' $)
  wwe $a wff R We A $.

  ${
    $d x y R $.  $d x y A $.  $d x y S $.
    $( Define the well-founded relation predicate where ` A ` might be a proper
       class.  By passing in ` S ` we allow it potentially to be a proper class
       rather than a set.  (Contributed by Jim Kingdon and Mario Carneiro,
       22-Sep-2021.) $)
    df-frfor $a |- ( FrFor R A S <->
      ( A. x e. A ( A. y e. A ( y R x -> y e. S ) -> x e. S )
        -> A C_ S ) ) $.
  $}

  ${
    $d s R $.  $d s A $.
    $( Define the well-founded relation predicate.  In the presence of excluded
       middle, there are a variety of equivalent ways to define this.  In our
       case, this definition, in terms of an inductive principle, works better
       than one along the lines of "there is an element which is minimal when A
       is ordered by R".  Because ` s ` is constrained to be a set (not a
       proper class) here, sometimes it may be necessary to use ` FrFor `
       directly rather than via ` Fr ` .  (Contributed by Jim Kingdon and Mario
       Carneiro, 21-Sep-2021.) $)
    df-frind $a |- ( R Fr A <-> A. s FrFor R A s ) $.
  $}

  ${
    $d x y R $.  $d x y A $.
    $( Define the set-like predicate.  (Contributed by Mario Carneiro,
       19-Nov-2014.) $)
    df-se $a |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V ) $.
  $}

  ${
    $d A x y z $.  $d R x y z $.
    $( Define the well-ordering predicate.  It is unusual to define
       "well-ordering" in the absence of excluded middle, but we mean an
       ordering which is like the ordering which we have for ordinals (for
       example, it does not entail trichotomy because ordinals do not have that
       as seen at ~ ordtriexmid ).  Given excluded middle, well-ordering is
       usually defined to require trichotomy (and the definition of ` Fr ` is
       typically also different).  (Contributed by Mario Carneiro and Jim
       Kingdon, 23-Sep-2021.) $)
    df-wetr $a |- ( R We A <-> ( R Fr A /\
      A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) ) $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.  $d x y V $.
    $( The ` R ` -preimage of an element of the base set in a set-like relation
       is a set.  (Contributed by Mario Carneiro, 19-Nov-2014.) $)
    seex $p |- ( ( R Se A /\ B e. A ) -> { x e. A | x R B } e. _V ) $=
      ( vy wse cv wbr crab cvv wcel wral df-se wceq breq2 rabbidv eleq1d sylanb
      rspccva ) BDFAGZEGZDHZABIZJKZEBLCBKTCDHZABIZJKZEABDMUDUGECBUACNZUCUFJUHUB
      UEABUACTDOPQSR $.

    $( Any relation on a set is set-like on it.  (Contributed by Mario
       Carneiro, 22-Jun-2015.) $)
    exse $p |- ( A e. V -> R Se A ) $=
      ( vy vx wcel cv wbr crab cvv wral wse rabexg ralrimivw df-se sylibr ) ACF
      ZDGEGBHZDAIJFZEAKABLQSEARDACMNEDABOP $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.  $d x y S $.
    $( Subset theorem for the set-like predicate.  (Contributed by Mario
       Carneiro, 24-Jun-2015.) $)
    sess1 $p |- ( R C_ S -> ( S Se A -> R Se A ) ) $=
      ( vy vx wss cv wbr crab cvv wcel wral wi simpl ssbrd ss2rabdv ssexg df-se
      wse wa ex syl ralimdv 3imtr4g ) BCFZDGZEGZCHZDAIZJKZEALUFUGBHZDAIZJKZEALA
      CSABSUEUJUMEAUEULUIFZUJUMMUEUKUHDAUEUFAKZTBCUFUGUEUONOPUNUJUMULUIJQUAUBUC
      EDACREDABRUD $.

    $( Subset theorem for the set-like predicate.  (Contributed by Mario
       Carneiro, 24-Jun-2015.) $)
    sess2 $p |- ( A C_ B -> ( R Se B -> R Se A ) ) $=
      ( vy vx wss cv wbr crab cvv wcel wral wse ssralv wi rabss2 ssexg ex df-se
      syl ralimdv syld 3imtr4g ) ABFZDGEGCHZDBIZJKZEBLZUEDAIZJKZEALZBCMACMUDUHU
      GEALUKUGEABNUDUGUJEAUDUIUFFZUGUJOUEDABPULUGUJUIUFJQRTUAUBEDBCSEDACSUC $.
  $}

  $( Equality theorem for the set-like predicate.  (Contributed by Mario
     Carneiro, 24-Jun-2015.) $)
  seeq1 $p |- ( R = S -> ( R Se A <-> S Se A ) ) $=
    ( wceq wse wss wi eqimss2 sess1 syl eqimss impbid ) BCDZABEZACEZMCBFNOGCBHA
    CBIJMBCFONGBCKABCIJL $.

  $( Equality theorem for the set-like predicate.  (Contributed by Mario
     Carneiro, 24-Jun-2015.) $)
  seeq2 $p |- ( A = B -> ( R Se A <-> R Se B ) ) $=
    ( wceq wse wss wi eqimss2 sess2 syl eqimss impbid ) ABDZACEZBCEZMBAFNOGBAHB
    ACIJMABFONGABKABCIJL $.

  ${
    $d R a b $.  $d A a b $.  $d x a b $.
    nfse.r $e |- F/_ x R $.
    nfse.a $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for set-like relations.  (Contributed
       by Mario Carneiro, 24-Jun-2015.)  (Revised by Mario Carneiro,
       14-Oct-2016.) $)
    nfse $p |- F/ x R Se A $=
      ( va vb wse wbr crab cvv wcel wral df-se nfcv nfbr nfrabw nfel1 nfralxy
      cv nfxfr ) BCHFTZGTZCIZFBJZKLZGBMAGFBCNUFAGBEAUEKUDAFBAUBUCCAUBODAUCOPEQR
      SUA $.
  $}

  ${
    $d x y A $.
    $( The epsilon relation is set-like on any class.  (This is the origin of
       the term "set-like": a set-like relation "acts like" the epsilon
       relation of sets and their elements.)  (Contributed by Mario Carneiro,
       22-Jun-2015.) $)
    epse $p |- _E Se A $=
      ( vy vx cep wse cv wbr crab cvv wcel wral cab epel bicomi abbi2i eqeltrri
      vex rabssab ssexi rgenw df-se mpbir ) ADEBFZCFZDGZBAHZIJZCAKUGCAUFUEBLZUD
      UHIUEBUDUEUCUDJBCMNOCQPUEBARSTCBADUAUB $.
  $}

  ${
    $d A x y $.  $d R x y $.  $d S x y $.  $d T x y $.
    $( Equality theorem for the well-founded predicate.  (Contributed by Jim
       Kingdon, 22-Sep-2021.) $)
    frforeq1 $p |- ( R = S -> ( FrFor R A T <-> FrFor S A T ) ) $=
      ( vy vx wceq cv wbr wcel wi wral wss wfrfor breq ralbidv df-frfor 3bitr4g
      imbi1d ) BCGZEHZFHZBIZUADJZKZEALZUBDJZKZFALZADMZKUAUBCIZUDKZEALZUGKZFALZU
      JKABDNACDNTUIUOUJTUHUNFATUFUMUGTUEULEATUCUKUDUAUBBCOSPSPSFEABDQFEACDQR $.
  $}

  ${
    $d s R $.  $d s S $.  $d s A $.
    $( Equality theorem for the well-founded predicate.  (Contributed by NM,
       9-Mar-1997.) $)
    freq1 $p |- ( R = S -> ( R Fr A <-> S Fr A ) ) $=
      ( vs wceq cv wfrfor wal wfr frforeq1 albidv df-frind 3bitr4g ) BCEZABDFZG
      ZDHACOGZDHABIACINPQDABCOJKABDLACDLM $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d R x y $.  $d T x y $.
    $( Equality theorem for the well-founded predicate.  (Contributed by Jim
       Kingdon, 22-Sep-2021.) $)
    frforeq2 $p |- ( A = B -> ( FrFor R A T <-> FrFor R B T ) ) $=
      ( vy vx wceq cv wbr wcel wi wral wss wfrfor raleq imbi1d raleqbi1dv sseq1
      imbi12d df-frfor 3bitr4g ) ABGZEHZFHZCIUCDJKZEALZUDDJZKZFALZADMZKUEEBLZUG
      KZFBLZBDMZKACDNBCDNUBUIUMUJUNUHULFABUBUFUKUGUEEABOPQABDRSFEACDTFEBCDTUA
      $.
  $}

  ${
    $d A s $.  $d B s $.  $d R s $.
    $( Equality theorem for the well-founded predicate.  (Contributed by NM,
       3-Apr-1994.) $)
    freq2 $p |- ( A = B -> ( R Fr A <-> R Fr B ) ) $=
      ( vs wceq cv wfrfor wal wfr frforeq2 albidv df-frind 3bitr4g ) ABEZACDFZG
      ZDHBCOGZDHACIBCINPQDABCOJKACDLBCDLM $.
  $}

  ${
    $d A x y $.  $d R x y $.  $d S x y $.  $d T x y $.
    $( Equality theorem for the well-founded predicate.  (Contributed by Jim
       Kingdon, 22-Sep-2021.) $)
    frforeq3 $p |- ( S = T -> ( FrFor R A S <-> FrFor R A T ) ) $=
      ( vy vx wceq cv wbr wcel wi wss wfrfor eleq2 imbi2d ralbidv imbi12d sseq2
      wral df-frfor 3bitr4g ) CDGZEHZFHZBIZUCCJZKZEASZUDCJZKZFASZACLZKUEUCDJZKZ
      EASZUDDJZKZFASZADLZKABCMABDMUBUKURULUSUBUJUQFAUBUHUOUIUPUBUGUNEAUBUFUMUEC
      DUCNOPCDUDNQPCDARQFEABCTFEABDTUA $.
  $}

  ${
    $d A u v $.  $d R u v $.  $d S u v $.  $d u v x $.
    nffrfor.r $e |- F/_ x R $.
    nffrfor.a $e |- F/_ x A $.
    nffrfor.s $e |- F/_ x S $.
    $( Bound-variable hypothesis builder for well-founded relations.
       (Contributed by Stefan O'Rear, 20-Jan-2015.)  (Revised by Mario
       Carneiro, 14-Oct-2016.) $)
    nffrfor $p |- F/ x FrFor R A S $=
      ( vv vu wfrfor cv wbr wcel wi wral wss nfcv nfcri nfim nfralxy nfbr nfxfr
      df-frfor nfss ) BCDJHKZIKZCLZUEDMZNZHBOZUFDMZNZIBOZBDPZNAIHBCDUCUMUNAULAI
      BFUJUKAUIAHBFUGUHAAUEUFCAUEQEAUFQUAAHDGRSTAIDGRSTABDFGUDSUB $.
  $}

  ${
    $d A s $.  $d R s $.  $d s x $.
    nffr.r $e |- F/_ x R $.
    nffr.a $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for well-founded relations.
       (Contributed by Stefan O'Rear, 20-Jan-2015.)  (Revised by Mario
       Carneiro, 14-Oct-2016.) $)
    nffr $p |- F/ x R Fr A $=
      ( vs wfr cv wfrfor wal df-frind nfcv nffrfor nfal nfxfr ) BCGBCFHZIZFJABC
      FKQAFABCPDEAPLMNO $.
  $}

  ${
    $d A s x y $.  $d B s x y $.  $d R s x y $.  $d V x $.
    $( A well-founded relation is irreflexive.  This is the case where ` A `
       exists.  (Contributed by Jim Kingdon, 21-Sep-2021.) $)
    frirrg $p |- ( ( R Fr A /\ A e. V /\ B e. A ) -> -. B R B ) $=
      ( vy vx vs wcel wbr wss wa simpr neldifsnd cv wi wral simplr wceq imbi12d
      mpd wfr w3a csn cdif simpl3 sseldd pm2.65da ad2antrr breqtrrd breq1 eleq1
      simpll3 rspcdva velsn sylnibr eldifd ex ralrimiva df-frind df-frfor albii
      wal wfrfor bitri biimpi 3ad2ant1 cvv difexg eleq2 ralbidv sseq2 spcgv syl
      imbi2d 3ad2ant2 adantr mtand ) ACUAZADHZBAHZUBZBBCIZAABUCZUDZJZWAWEBWDHZW
      AWEKZAWDBWAWELVRVSVTWEUEUFWGBAMUGWAWBKZENZFNZCIZWIWDHZOZEAPZWJWDHZOZFAPZW
      EWHWPFAWHWJAHZKZWNWOWSWNKZWJAWCWHWRWNQWTWJBRZWJWCHWTXAWFWTXAKZBWJCIZWFXBB
      BWJCWSWBWNXAWAWBWRQUHWTXALUIXBWMXCWFOEABWIBRWKXCWLWFWIBWJCUJWIBWDUKSWSWNX
      AQWSVTWNXAVRVSVTWBWRULUHUMTXBBAMUGFBUNUOUPUQURWAWQWEOZWBWAWKWIGNZHZOZEAPZ
      WJXEHZOZFAPZAXEJZOZGVBZXDVRVSXNVTVRXNVRACXEVCZGVBXNACGUSXOXMGFEACXEUTVAVD
      VEVFVSVRXNXDOZVTVSWDVGHXPAWCDVHXMXDGWDVGXEWDRZXKWQXLWEXQXJWPFAXQXHWNXIWOX
      QXGWMEAXQXFWLWKXEWDWIVIVNVJXEWDWJVISVJXEWDAVKSVLVMVOTVPTVQ $.
  $}

  ${
    $d R s x y $.
    $( Any relation is well-founded on the empty set.  (Contributed by NM,
       17-Sep-1993.) $)
    fr0 $p |- R Fr (/) $=
      ( vs vy vx c0 wfr cv wfrfor df-frind wbr wel wi wral wss 0ss a1i df-frfor
      mpbir mpgbir ) EAFEABGZHZBEABIUACGDGAJCBKLCEMDBKLDEMZETNZLUCUBTOPDCEATQRS
      $.
  $}

  ${
    $d A x y s z $.  $d R x y s z $.  $d ch x $.  $d ph y s z $.  $d ps x z $.
    frind.sb $e |- ( x = y -> ( ph <-> ps ) ) $.
    frind.ind $e |- ( ( ch /\ x e. A ) ->
      ( A. y e. A ( y R x -> ps ) -> ph ) ) $.
    frind.fr $e |- ( ch -> R Fr A ) $.
    frind.a $e |- ( ch -> A e. V ) $.
    $( Induction over a well-founded set.  (Contributed by Jim Kingdon,
       28-Sep-2021.) $)
    frind $p |- ( ( ch /\ x e. A ) -> ph ) $=
      ( vz vs wss wral cv wcel wi sylib crab wa wbr wsb ralrimiva nfv nfs1v weq
      nfim breq2 imbi1d ralbidv sbequ12 imbi12d cbvral wb elrab3 imbi2d ralbiia
      a1i nfcv elrabf baib sylibr wfrfor wal wfr df-frind rabexg frforeq3 spcgv
      cvv 3syl mpd df-frfor ssrab simprd r19.21bi ) CADFCFFOZADFPZCFADFUAZOZVSV
      TUBCEQZMQZGUCZWCWARZSZEFPZWDWARZSZMFPZWBCWEBSZEFPZADMUDZSZMFPZWKCWCDQZGUC
      ZBSZEFPZASZDFPWPCXADFJUEXAWODMFXAMUFWMWNDWMDUFADMUGZUIDMUHZWTWMAWNXCWSWLE
      FXCWRWEBWQWDWCGUJUKULADMUMZUNUOTWJWOMFWDFRZWHWMWIWNWHWMUPXEWGWLEFWCFRWFBW
      EABDWCFIUQURUSUTWIXEWNAWNDWDFDWDVADFVAXBXDVBVCUNUSVDCFGWAVEZWKWBSCFGNQZVE
      ZNVFZXFCFGVGXIKFGNVHTCFHRWAVLRXIXFSLADFHVIXHXFNWAVLFGXGWAVJVKVMVNMEFGWAVO
      TVNADFFVPTVQVR $.
  $}

  $( Irreflexivity of the epsilon relation: a class founded by epsilon is not a
     member of itself.  (Contributed by NM, 18-Apr-1994.)  (Revised by Mario
     Carneiro, 22-Jun-2015.) $)
  efrirr $p |- ( _E Fr A -> -. A e. A ) $=
    ( cep wfr wcel wa wbr wn frirrg 3anidm23 wb epelg adantl mtbid pm2.01da ) A
    BCZAADZOPEAABFZPOPQGAABAHIPQPJOAAAKLMN $.

  $( Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom
     of Regularity is not required due to antecedent ` _E Fr A ` .
     (Contributed by NM, 4-May-1994.) $)
  tz7.2 $p |- ( ( Tr A /\ _E Fr A /\ B e. A ) -> ( B C_ A /\ B =/= A ) ) $=
    ( wtr cep wfr wcel wss wne wa trss wn wceq efrirr eleq1 syl5ibrcom necon2ad
    notbid anim12ii 3impia ) ACZADEZBAFZBAGZBAHZITUBUCUAUDABJUAUBBAUAUBKBALZAAF
    ZKAMUEUBUFBAANQOPRS $.

  ${
    $d a b c x $.  $d a b c A $.  $d a b c R $.
    nfwe.r $e |- F/_ x R $.
    nfwe.a $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for well-orderings.  (Contributed by
       Stefan O'Rear, 20-Jan-2015.)  (Revised by Mario Carneiro,
       14-Oct-2016.) $)
    nfwe $p |- F/ x R We A $=
      ( va vb vc wwe wfr cv wbr wa wi wral df-wetr nfcv nfbr nfan nfralxy nfxfr
      nffr nfim ) BCIBCJZFKZGKZCLZUFHKZCLZMZUEUHCLZNZHBOZGBOZFBOZMAFGHBCPUDUOAA
      BCDEUBUNAFBEUMAGBEULAHBEUJUKAUGUIAAUEUFCAUEQZDAUFQZRAUFUHCUQDAUHQZRSAUEUH
      CUPDURRUCTTTSUA $.
  $}

  ${
    $d A x y z $.  $d R x y z $.  $d S x y z $.
    $( Equality theorem for the well-ordering predicate.  (Contributed by NM,
       9-Mar-1997.) $)
    weeq1 $p |- ( R = S -> ( R We A <-> S We A ) ) $=
      ( vx vy vz wceq wfr cv wbr wa wral wwe freq1 breq anbi12d imbi12d ralbidv
      wi df-wetr 3bitr4g ) BCGZABHZDIZEIZBJZUEFIZBJZKZUDUGBJZSZFALZEALZDALZKACH
      ZUDUECJZUEUGCJZKZUDUGCJZSZFALZEALZDALZKABMACMUBUCUOUNVCABCNUBUMVBDAUBULVA
      EAUBUKUTFAUBUIURUJUSUBUFUPUHUQUDUEBCOUEUGBCOPUDUGBCOQRRRPDEFABTDEFACTUA
      $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d R x y z $.
    $( Equality theorem for the well-ordering predicate.  (Contributed by NM,
       3-Apr-1994.) $)
    weeq2 $p |- ( A = B -> ( R We A <-> R We B ) ) $=
      ( vx vy vz wceq wfr cv wbr wa wral freq2 raleq raleqbi1dv anbi12d df-wetr
      wi wwe 3bitr4g ) ABGZACHZDIZEIZCJUDFIZCJKUCUECJRZFALZEALZDALZKBCHZUFFBLZE
      BLZDBLZKACSBCSUAUBUJUIUMABCMUHULDABUGUKEABUFFABNOOPDEFACQDEFBCQT $.
  $}

  ${
    $d A x y z $.  $d R x y z $.
    $( A well-ordering is well-founded.  (Contributed by NM, 22-Apr-1994.) $)
    wefr $p |- ( R We A -> R Fr A ) $=
      ( vx vy vz wwe wfr cv wbr wa wi wral df-wetr simplbi ) ABFABGCHZDHZBIPEHZ
      BIJOQBIKEALDALCALCDEABMN $.
  $}

  ${
    $d A x y z $.  $d R x y z $.  $d V x y z $.
    $( A well-ordering is a partial ordering.  (Contributed by Jim Kingdon,
       23-Sep-2021.) $)
    wepo $p |- ( ( R We A /\ A e. V ) -> R Po A ) $=
      ( vx vy vz wwe wcel wa cv wbr wfr wefr frirrg syl3an1 3expa wral r19.21bi
      wn anasss w3a wi df-3an df-wetr simprbi adantr sylan2b ispod ) ABGZACHZIZ
      DEFABUIUJDJZAHZULULBKSZUIABLZUJUMUNABMAULBCNOPUMEJZAHZFJZAHZUAUKUMUQIZUSI
      ULUPBKUPURBKIULURBKUBZUMUQUSUCUKUTUSVAUKUTIVAFAUKUMUQVAFAQZUKUMIVBEAUKVBE
      AQZDAUIVCDAQZUJUIUOVDDEFABUDUEUFRRTRTUGUH $.
  $}

  ${
    $d A x y z $.
    $( An epsilon well-ordering is a transitive relation.  (Contributed by NM,
       22-Apr-1994.) $)
    wetrep $p |- ( ( _E We A /\ ( x e. A /\ y e. A /\ z e. A ) ) ->
               ( ( x e. y /\ y e. z ) -> x e. z ) ) $=
      ( cep wwe cv wcel w3a wa wbr wel wi df-3an wral wfr df-wetr r19.21bi epel
      anasss simprbi sylan2b anbi12i 3imtr3g ) DEFZAGZDHZBGZDHZCGZDHZIZJUFUHEKZ
      UHUJEKZJZUFUJEKZABLZBCLZJACLULUEUGUIJZUKJUOUPMZUGUIUKNUEUSUKUTUEUSJUTCDUE
      UGUIUTCDOZUEUGJVABDUEVABDOZADUEDEPVBADOABCDEQUARRTRTUBUMUQUNURABSBCSUCACS
      UD $.
  $}

  ${
    $d R x y z $.
    $( Any relation is a well-ordering of the empty set.  (Contributed by NM,
       16-Mar-1997.) $)
    we0 $p |- R We (/) $=
      ( vx vy vz c0 wwe wfr cv wbr wa wi wral fr0 ral0 df-wetr mpbir2an ) EAFEA
      GBHZCHZAIRDHZAIJQSAIKDELCELZBELAMTBNBCDEAOP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordinals
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c Ord $. $( Ordinal predicate $)
  $c On $. $( The class of ordinal numbers $)
  $c Lim $. $( Limit ordinal predicate $)
  $c suc $. $( Successor function (read:  'successor of') $)

  $( Extend the definition of a wff to include the ordinal predicate. $)
  word $a wff Ord A $.

  $( Extend the definition of a class to include the class of all ordinal
     numbers.  (The 0 in the name prevents creating a file called con.html,
     which causes problems in Windows.) $)
  con0 $a class On $.

  $( Extend the definition of a wff to include the limit ordinal predicate. $)
  wlim $a wff Lim A $.

  $( Extend class notation to include the successor function. $)
  csuc $a class suc A $.

  ${
    $d x A $.
    $( Define the ordinal predicate, which is true for a class that is
       transitive and whose elements are transitive.  Definition of ordinal in
       [Crosilla], p.  "Set-theoretic principles incompatible with
       intuitionistic logic".

       Some sources will define a notation for ordinal order corresponding to
       ` < ` and ` <_ ` but we just use ` e. ` and ` C_ ` respectively.

       (Contributed by Jim Kingdon, 10-Oct-2018.)  Use its alias ~ dford3
       instead for naming consistency with set.mm.
       (New usage is discouraged.) $)
    df-iord $a |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) ) $.
  $}

  ${
    $d x A $.
    $( Alias for ~ df-iord .  Use it instead of ~ df-iord for naming
       consistency with set.mm.  (Contributed by Jim Kingdon, 10-Oct-2018.) $)
    dford3 $p |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) ) $=
      ( df-iord ) ABC $.
  $}

  $( Define the class of all ordinal numbers.  Definition 7.11 of
     [TakeutiZaring] p. 38.  (Contributed by NM, 5-Jun-1994.) $)
  df-on $a |- On = { x | Ord x } $.

  $( Define the limit ordinal predicate, which is true for an ordinal that has
     the empty set as an element and is not a successor (i.e. that is the union
     of itself).  Our definition combines the definition of Lim of
     [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then
     changes ` A =/= (/) ` to ` (/) e. A ` (which would be equivalent given the
     law of the excluded middle, but which is not for us).  (Contributed by Jim
     Kingdon, 11-Nov-2018.)  Use its alias ~ dflim2 instead for naming
     consistency with set.mm.  (New usage is discouraged.) $)
  df-ilim $a |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) ) $.

  $( Alias for ~ df-ilim .  Use it instead of ~ df-ilim for naming consistency
     with set.mm.  (Contributed by NM, 4-Nov-2004.) $)
  dflim2 $p |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) ) $=
    ( df-ilim ) AB $.

  $( Define the successor of a class.  When applied to an ordinal number, the
     successor means the same thing as "plus 1".  Definition 7.22 of
     [TakeutiZaring] p. 41, who use "+ 1" to denote this function.  Our
     definition is a generalization to classes.  Although it is not
     conventional to use it with proper classes, it has no effect on a proper
     class ( ~ sucprc ).  Some authors denote the successor operation with a
     prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134
     and the definition of successor in [Mendelson] p. 246 (who uses the symbol
     "Suc" as a predicate to mean "is a successor ordinal").  The definition of
     successor of [Enderton] p. 68 denotes the operation with a plus-sign
     superscript.  (Contributed by NM, 30-Aug-1993.) $)
  df-suc $a |- suc A = ( A u. { A } ) $.

  ${
    $d x A $.  $d x B $.
    $( Equality theorem for the ordinal predicate.  (Contributed by NM,
       17-Sep-1993.) $)
    ordeq $p |- ( A = B -> ( Ord A <-> Ord B ) ) $=
      ( vx wceq wtr cv wral wa word treq raleq anbi12d dford3 3bitr4g ) ABDZAEZ
      CFEZCAGZHBEZQCBGZHAIBIOPSRTABJQCABKLCAMCBMN $.
  $}

  ${
    $d x A $.
    $( An ordinal number is an ordinal set.  (Contributed by NM,
       5-Jun-1994.) $)
    elong $p |- ( A e. V -> ( A e. On <-> Ord A ) ) $=
      ( vx cv word con0 ordeq df-on elab2g ) CDZEAECAFBJAGCHI $.
  $}

  ${
    elon.1 $e |- A e. _V $.
    $( An ordinal number is an ordinal set.  (Contributed by NM,
       5-Jun-1994.) $)
    elon $p |- ( A e. On <-> Ord A ) $=
      ( cvv wcel con0 word wb elong ax-mp ) ACDAEDAFGBACHI $.
  $}

  $( An ordinal number has the ordinal property.  (Contributed by NM,
     5-Jun-1994.) $)
  eloni $p |- ( A e. On -> Ord A ) $=
    ( con0 wcel word elong ibi ) ABCADABEF $.

  $( An ordinal number is an ordinal set.  (Contributed by NM, 8-Feb-2004.) $)
  elon2 $p |- ( A e. On <-> ( Ord A /\ A e. _V ) ) $=
    ( con0 wcel word cvv wa eloni elex jca elong biimparc impbii ) ABCZADZAECZF
    MNOAGABHIOMNAEJKL $.

  $( Equality theorem for the limit predicate.  (Contributed by NM,
     22-Apr-1994.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
  limeq $p |- ( A = B -> ( Lim A <-> Lim B ) ) $=
    ( wceq word c0 wcel cuni wlim ordeq eleq2 id unieq eqeq12d 3anbi123d dflim2
    w3a 3bitr4g ) ABCZADZEAFZAAGZCZPBDZEBFZBBGZCZPAHBHRSUCTUDUBUFABIABEJRABUAUE
    RKABLMNAOBOQ $.

  ${
    $d x A $.
    $( An ordinal class is transitive.  (Contributed by NM, 3-Apr-1994.) $)
    ordtr $p |- ( Ord A -> Tr A ) $=
      ( vx word wtr cv wral dford3 simplbi ) ACADBEDBAFBAGH $.
  $}

  $( An element of an ordinal class is a subset of it.  (Contributed by NM,
     30-May-1994.) $)
  ordelss $p |- ( ( Ord A /\ B e. A ) -> B C_ A ) $=
    ( word wtr wcel wss ordtr trss imp sylan ) ACADZBAEZBAFZAGKLMABHIJ $.

  ${
    $d A x $.  $d B x $.
    $( A transitive subclass of an ordinal class is ordinal.  (Contributed by
       NM, 29-May-1994.) $)
    trssord $p |- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A ) $=
      ( vx wtr wss word w3a cv wral dford3 simprbi ssralv syl5 imp anim2i 3impb
      wa sylibr ) ADZABEZBFZGSCHDZCAIZQZAFSTUAUDTUAQUCSTUAUCUAUBCBIZTUCUABDUECB
      JKUBCABLMNOPCAJR $.
  $}

  ${
    $d x A $.  $d x B $.
    $( An element of an ordinal class is ordinal.  Proposition 7.6 of
       [TakeutiZaring] p. 36.  (Contributed by NM, 23-Apr-1994.) $)
    ordelord $p |- ( ( Ord A /\ B e. A ) -> Ord B ) $=
      ( vx word wcel cv wa wceq eleq1 anbi2d ordeq imbi12d wtr wss wral simprbi
      wi dford3 r19.21bi ordelss simpl trssord syl3anc vtoclg anabsi7 ) ADZBAEZ
      BDZUFCFZAEZGZUIDZQUFUGGZUHQCBAUIBHZUKUMULUHUNUJUGUFUIBAIJUIBKLUKUIMZUIANU
      FULUFUOCAUFAMUOCAOCARPSAUITUFUJUAUIAUBUCUDUE $.
  $}

  ${
    $d x y $.
    $( The class of all ordinal numbers is transitive.  (Contributed by NM,
       4-May-2009.) $)
    tron $p |- Tr On $=
      ( vx vy con0 wtr cv wss dftr3 wcel word elon ordelord sylanb ex imbitrrdi
      vex ssrdv mprgbir ) CDAEZCFACACGRCHZBRCSBEZRHZTIZTCHSUAUBSRIUAUBRAOJRTKLM
      TBOJNPQ $.
  $}

  $( An element of an ordinal class is an ordinal number.  (Contributed by NM,
     26-Oct-2003.) $)
  ordelon $p |- ( ( Ord A /\ B e. A ) -> B e. On ) $=
    ( word wcel wa con0 ordelord wb elong adantl mpbird ) ACZBADZEBFDZBCZABGMNO
    HLBAIJK $.

  $( An element of an ordinal number is an ordinal number.  Theorem 2.2(iii) of
     [BellMachover] p. 469.  (Contributed by NM, 26-Oct-2003.) $)
  onelon $p |- ( ( A e. On /\ B e. A ) -> B e. On ) $=
    ( con0 wcel word eloni ordelon sylan ) ACDAEBADBCDAFABGH $.

  $( The intersection of two ordinal classes is ordinal.  Proposition 7.9 of
     [TakeutiZaring] p. 37.  (Contributed by NM, 9-May-1994.) $)
  ordin $p |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) ) $=
    ( word cin wtr ordtr trin syl2an wss inss2 trssord mp3an2 sylancom ) ACZBCZ
    ABDZEZPCZNAEBEQOAFBFABGHQPBIORABJPBKLM $.

  $( The intersection of two ordinal numbers is an ordinal number.
     (Contributed by NM, 7-Apr-1995.) $)
  onin $p |- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On ) $=
    ( con0 wcel wa cin word eloni ordin syl2an cvv wb simpl inex1g elong mpbird
    3syl ) ACDZBCDZEZABFZCDZUAGZRAGBGUCSAHBHABIJTRUAKDUBUCLRSMABCNUAKOQP $.

  $( An element of an ordinal number is a subset of the number.  (Contributed
     by NM, 5-Jun-1994.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
  onelss $p |- ( A e. On -> ( B e. A -> B C_ A ) ) $=
    ( con0 wcel word wss wi eloni ordelss ex syl ) ACDAEZBADZBAFZGAHLMNABIJK $.

  $( Transitive law for ordinal classes.  (Contributed by NM, 12-Dec-2004.) $)
  ordtr1 $p |- ( Ord C -> ( ( A e. B /\ B e. C ) -> A e. C ) ) $=
    ( word wtr wcel wa wi ordtr trel syl ) CDCEABFBCFGACFHCICABJK $.

  $( Transitive law for ordinal numbers.  Theorem 7M(b) of [Enderton] p. 192.
     (Contributed by NM, 11-Aug-1994.) $)
  ontr1 $p |- ( C e. On -> ( ( A e. B /\ B e. C ) -> A e. C ) ) $=
    ( con0 wcel word wa wi eloni ordtr1 syl ) CDECFABEBCEGACEHCIABCJK $.

  ${
    $d x ps $.  $d x A $.
    onintss.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( If a property is true for an ordinal number, then the minimum ordinal
       number for which it is true is smaller or equal.  Theorem Schema 61 of
       [Suppes] p. 228.  (Contributed by NM, 3-Oct-2003.) $)
    onintss $p |- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) ) $=
      ( con0 wcel crab cint wss intminss ex ) DFGBACFHIDJABCDFEKL $.
  $}

  $( The empty set is an ordinal class.  (Contributed by NM, 11-May-1994.) $)
  ord0 $p |- Ord (/) $=
    ( vx c0 word wtr cv wral tr0 ral0 dford3 mpbir2an ) BCBDAEDZABFGKAHABIJ $.

  $( The empty set is an ordinal number.  Corollary 7N(b) of [Enderton] p. 193.
     (Contributed by NM, 17-Sep-1993.) $)
  0elon $p |- (/) e. On $=
    ( c0 con0 wcel word ord0 0ex elon mpbir ) ABCADEAFGH $.

  $( The intersection of the class of ordinal numbers is the empty set.
     (Contributed by NM, 20-Oct-2003.) $)
  inton $p |- |^| On = (/) $=
    ( c0 con0 wcel cint wceq 0elon int0el ax-mp ) ABCBDAEFBGH $.

  $( The empty set is not a limit ordinal.  (Contributed by NM, 24-Mar-1995.)
     (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
  nlim0 $p |- -. Lim (/) $=
    ( c0 wlim word wcel cuni wceq w3a noel simp2 mto dflim2 mtbir ) ABACZAADZAA
    EFZGZPNAHMNOIJAKL $.

  $( A limit ordinal is ordinal.  (Contributed by NM, 4-May-1995.) $)
  limord $p |- ( Lim A -> Ord A ) $=
    ( wlim word c0 wcel cuni wceq dflim2 simp1bi ) ABACDAEAAFGAHI $.

  $( A limit ordinal is its own supremum (union).  (Contributed by NM,
     4-May-1995.) $)
  limuni $p |- ( Lim A -> A = U. A ) $=
    ( wlim word c0 wcel cuni wceq dflim2 simp3bi ) ABACDAEAAFGAHI $.

  $( The union of a limit ordinal is a limit ordinal.  (Contributed by NM,
     19-Sep-2006.) $)
  limuni2 $p |- ( Lim A -> Lim U. A ) $=
    ( wlim cuni wceq wb limuni limeq syl ibi ) ABZACZBZJAKDJLEAFAKGHI $.

  $( A limit ordinal contains the empty set.  (Contributed by NM,
     15-May-1994.) $)
  0ellim $p |- ( Lim A -> (/) e. A ) $=
    ( wlim word c0 wcel cuni wceq dflim2 simp2bi ) ABACDAEAAFGAHI $.

  $( A limit ordinal class that is also a set is an ordinal number.
     (Contributed by NM, 26-Apr-2004.) $)
  limelon $p |- ( ( A e. B /\ Lim A ) -> A e. On ) $=
    ( wcel wlim con0 word limord elong imbitrrid imp ) ABCZADZAECZLMKAFAGABHIJ
    $.

  $( The class of all ordinal numbers is not empty.  (Contributed by NM,
     17-Sep-1995.) $)
  onn0 $p |- On =/= (/) $=
    ( c0 con0 wcel wne 0elon ne0i ax-mp ) ABCBADEBAFG $.

  $( The class of all ordinal numbers is inhabited.  (Contributed by Jim
     Kingdon, 6-Mar-2019.) $)
  onm $p |- E. x x e. On $=
    ( cv c0 wceq con0 wcel wa wex 0elon 0ex eleq1 ceqsexv mpbir exsimpr ax-mp )
    ABZCDZPEFZGAHZRAHSCEFZIRTACJPCEKLMQRANO $.

  $( Equality of successors.  (Contributed by NM, 30-Aug-1993.)  (Proof
     shortened by Andrew Salmon, 25-Jul-2011.) $)
  suceq $p |- ( A = B -> suc A = suc B ) $=
    ( wceq csn cun csuc id sneq uneq12d df-suc 3eqtr4g ) ABCZAADZEBBDZEAFBFLABM
    NLGABHIAJBJK $.

  $( Membership in a successor.  This one-way implication does not require that
     either ` A ` or ` B ` be sets.  (Contributed by NM, 6-Jun-1994.) $)
  elsuci $p |- ( A e. suc B -> ( A e. B \/ A = B ) ) $=
    ( csuc wcel csn wo wceq cun df-suc eleq2i elun bitri elsni orim2i sylbi ) A
    BCZDZABDZABEZDZFZRABGZFQABSHZDUAPUCABIJABSKLTUBRABMNO $.

  $( Membership in a successor.  Exercise 5 of [TakeutiZaring] p. 17.
     (Contributed by NM, 15-Sep-1995.) $)
  elsucg $p |- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) ) $=
    ( csuc wcel csn wo wceq cun df-suc eleq2i elun bitri elsng orbi2d bitrid )
    ABDZEZABEZABFZEZGZACEZSABHZGRABTIZEUBQUEABJKABTLMUCUAUDSABCNOP $.

  $( Variant of membership in a successor, requiring that ` B ` rather than
     ` A ` be a set.  (Contributed by NM, 28-Oct-2003.) $)
  elsuc2g $p |- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) ) $=
    ( csuc wcel csn cun wceq wo df-suc eleq2i elun elsn2g orbi2d bitrid ) ABDZE
    ABBFZGZEZBCEZABEZABHZIZPRABJKSUAAQEZITUCABQLTUDUBUAABCMNOO $.

  ${
    elsuc.1 $e |- A e. _V $.
    $( Membership in a successor.  Exercise 5 of [TakeutiZaring] p. 17.
       (Contributed by NM, 15-Sep-2003.) $)
    elsuc $p |- ( A e. suc B <-> ( A e. B \/ A = B ) ) $=
      ( cvv wcel csuc wceq wo wb elsucg ax-mp ) ADEABFEABEABGHICABDJK $.

    $( Membership in a successor.  (Contributed by NM, 15-Sep-2003.) $)
    elsuc2 $p |- ( B e. suc A <-> ( B e. A \/ B = A ) ) $=
      ( cvv wcel csuc wceq wo wb elsuc2g ax-mp ) ADEBAFEBAEBAGHICBADJK $.
  $}

  ${
    nfsuc.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for successor.  (Contributed by NM,
       15-Sep-2003.) $)
    nfsuc $p |- F/_ x suc A $=
      ( csuc csn cun df-suc nfsn nfun nfcxfr ) ABDBBEZFBGABKCABCHIJ $.
  $}

  $( Membership in a successor.  (Contributed by NM, 20-Jun-1998.) $)
  elelsuc $p |- ( A e. B -> A e. suc B ) $=
    ( wcel csuc wceq wo orc elsucg mpbird ) ABCZABDCJABEZFJKGABBHI $.

  ${
    $d x y A $.  $d x B $.
    $( Membership of a successor in another class.  (Contributed by NM,
       29-Jun-2004.) $)
    sucel $p |- ( suc A e. B <->
                E. x e. B A. y ( y e. x <-> ( y e. A \/ y = A ) ) ) $=
      ( csuc wcel cv wceq wrex wo wb wal risset dfcleq elsuc bibi2i albii bitri
      vex rexbii ) CEZDFAGZUAHZADIBGZUBFZUDCFUDCHJZKZBLZADIAUADMUCUHADUCUEUDUAF
      ZKZBLUHBUBUANUJUGBUIUFUEUDCBSOPQRTR $.
  $}

  $( The successor of the empty set.  (Contributed by NM, 1-Feb-2005.) $)
  suc0 $p |- suc (/) = { (/) } $=
    ( c0 csuc csn cun df-suc uncom un0 3eqtri ) ABAACZDIADIAEAIFIGH $.

  $( A proper class is its own successor.  (Contributed by NM, 3-Apr-1995.) $)
  sucprc $p |- ( -. A e. _V -> suc A = A ) $=
    ( cvv wcel wn csuc cun csn df-suc wceq snprc uneq2 sylbi eqtrid un0 eqtrdi
    c0 ) ABCDZAEZAPFZAQRAAGZFZSAHQTPIUASIAJTPAKLMANO $.

  ${
    unisuc.1 $e |- A e. _V $.
    $( A transitive class is equal to the union of its successor.  Combines
       Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73.
       (Contributed by NM, 30-Aug-1993.) $)
    unisuc $p |- ( Tr A <-> U. suc A = A ) $=
      ( cuni wss cun wceq wtr csuc ssequn1 df-tr csn df-suc unieqi uniun uneq2i
      unisn 3eqtri eqeq1i 3bitr4i ) ACZADTAEZAFAGAHZCZAFTAIAJUCUAAUCAAKZEZCTUDC
      ZEUAUBUEALMAUDNUFATABPOQRS $.
  $}

  $( A transitive class is equal to the union of its successor.  Combines
     Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73.
     (Contributed by Jim Kingdon, 18-Aug-2019.) $)
  unisucg $p |- ( A e. V -> ( Tr A <-> U. suc A = A ) ) $=
    ( wcel wtr cuni cun wceq csuc df-tr ssequn1 bitri df-suc unieqi uniun eqtri
    wss csn unisng uneq2d eqtrid eqeq1d bitr4id ) ABCZADZAEZAFZAGZAHZEZAGUDUEAP
    UGAIUEAJKUCUIUFAUCUIUEAQZEZFZUFUIAUJFZEULUHUMALMAUJNOUCUKAUEABRSTUAUB $.

  $( A class is included in its own successor.  Part of Proposition 7.23 of
     [TakeutiZaring] p. 41 (generalized to arbitrary classes).  (Contributed by
     NM, 31-May-1994.) $)
  sssucid $p |- A C_ suc A $=
    ( csn cun csuc ssun1 df-suc sseqtrri ) AAABZCADAHEAFG $.

  $( Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized).
     (Contributed by NM, 25-Mar-1995.)  (Proof shortened by Scott Fenton,
     20-Feb-2012.) $)
  sucidg $p |- ( A e. V -> A e. suc A ) $=
    ( wcel csuc wceq wo eqid olci elsucg mpbiri ) ABCAADCAACZAAEZFLKAGHAABIJ $.

  ${
    sucid.1 $e |- A e. _V $.
    $( A set belongs to its successor.  (Contributed by NM, 22-Jun-1994.)
       (Proof shortened by Alan Sare, 18-Feb-2012.)  (Proof shortened by Scott
       Fenton, 20-Feb-2012.) $)
    sucid $p |- A e. suc A $=
      ( cvv wcel csuc sucidg ax-mp ) ACDAAEDBACFG $.
  $}

  $( No successor is empty.  (Contributed by Jim Kingdon, 14-Oct-2018.) $)
  nsuceq0g $p |- ( A e. V -> suc A =/= (/) ) $=
    ( wcel csuc c0 wceq noel sucidg eleq2 syl5ibcom mtoi neneqad ) ABCZADZEMNEF
    ZAECZAGMANCOPABHNEAIJKL $.

  ${
    eqelsuc.1 $e |- A e. _V $.
    $( A set belongs to the successor of an equal set.  (Contributed by NM,
       18-Aug-1994.) $)
    eqelsuc $p |- ( A = B -> A e. suc B ) $=
      ( wceq csuc sucid suceq eleqtrid ) ABDAAEBEACFABGH $.
  $}

  ${
    $d A x $.  $d C x $.
    iunsuc.1 $e |- A e. _V $.
    iunsuc.2 $e |- ( x = A -> B = C ) $.
    $( Inductive definition for the indexed union at a successor.  (Contributed
       by Mario Carneiro, 4-Feb-2013.)  (Proof shortened by Mario Carneiro,
       18-Nov-2016.) $)
    iunsuc $p |- U_ x e. suc A B = ( U_ x e. A B u. C ) $=
      ( csuc ciun csn cun wceq df-suc iuneq1 ax-mp iunxun iunxsn uneq2i 3eqtri
      ) ABGZCHZABBIZJZCHZABCHZAUACHZJUDDJSUBKTUCKBLASUBCMNABUACOUEDUDABCDEFPQR
      $.
  $}

  ${
    $d y z A $.
    $( The successor of a transitive class is transitive.  (Contributed by Alan
       Sare, 11-Apr-2009.) $)
    suctr $p |- ( Tr A -> Tr suc A ) $=
      ( vz vy wtr cv wcel csuc wa wi wal wceq simpr vex elsuc sylib simpl eleq2
      wo syl6 mpdi syl5ibcom elelsuc trel expd adantrd syl8 jao alrimivv sylibr
      dftr2 ) ADZBEZCEZFZUMAGZFZHZULUOFZIZCJBJUODUKUSBCUKUQUMAFZUMAKZRZURUQUPVB
      UNUPLUMACMNOUKUQVAURIZVBURIZUQVAULAFZURUQUNVAVEUNUPPUMAULQUAULAUBZSUKUQUT
      URIVCVDIUKUQUTVEURUKUNUTVEIUPUKUNUTVEAULUMUCUDUEVFUFUTURVAUGSTTUHBCUOUJUI
      $.
  $}

  $( A set whose successor belongs to a transitive class also belongs.
     (Contributed by NM, 5-Sep-2003.)  (Proof shortened by Andrew Salmon,
     12-Aug-2011.) $)
  trsuc $p |- ( ( Tr A /\ suc B e. A ) -> B e. A ) $=
    ( wtr csuc wcel cvv wss sssucid ssexg mpan sucidg syl ancri trel syl5 imp
    wa ) ACZBDZAEZBAEZTBSEZTQRUATUBTBFEZUBBSGTUCBHBSAIJBFKLMABSNOP $.

  $( A member of the successor of a transitive class is a subclass of it.
     (Contributed by NM, 4-Oct-2003.) $)
  trsucss $p |- ( Tr A -> ( B e. suc A -> B C_ A ) ) $=
    ( csuc wcel wceq wo wtr wss elsuci trss wi eqimss a1i jaod syl5 ) BACDBADZB
    AEZFAGZBAHZBAIRPSQABJQSKRBALMNO $.

  $( A set whose successor is a subset of another class is a member of that
     class.  (Contributed by NM, 16-Sep-1995.) $)
  sucssel $p |- ( A e. V -> ( suc A C_ B -> A e. B ) ) $=
    ( wcel csuc wss sucidg ssel syl5com ) ACDAAEZDJBFABDACGJBAHI $.

  $( An ordinal class includes its union.  (Contributed by NM, 13-Sep-2003.) $)
  orduniss $p |- ( Ord A -> U. A C_ A ) $=
    ( word wtr cuni wss ordtr df-tr sylib ) ABACADAEAFAGH $.

  ${
    on.1 $e |- A e. On $.
    $( An ordinal number is an ordinal class.  (Contributed by NM,
       11-Jun-1994.) $)
    onordi $p |- Ord A $=
      ( con0 wcel word eloni ax-mp ) ACDAEBAFG $.

    $( An ordinal number is a transitive class.  (Contributed by NM,
       11-Jun-1994.) $)
    ontrci $p |- Tr A $=
      ( word wtr onordi ordtr ax-mp ) ACADABEAFG $.

    $( A member of an ordinal number is an ordinal number.  Theorem 7M(a) of
       [Enderton] p. 192.  (Contributed by NM, 11-Jun-1994.) $)
    oneli $p |- ( B e. A -> B e. On ) $=
      ( con0 wcel onelon mpan ) ADEBAEBDECABFG $.

    $( A member of an ordinal number is a subset of it.  (Contributed by NM,
       11-Aug-1994.) $)
    onelssi $p |- ( B e. A -> B C_ A ) $=
      ( con0 wcel wss wi onelss ax-mp ) ADEBAEBAFGCABHI $.

    $( An element of an ordinal number equals the intersection with it.
       (Contributed by NM, 11-Jun-1994.) $)
    onelini $p |- ( B e. A -> B = ( B i^i A ) ) $=
      ( wcel wss cin wceq onelssi dfss sylib ) BADBAEBBAFGABCHBAIJ $.

    $( An ordinal number equals its union with any element.  (Contributed by
       NM, 13-Jun-1994.) $)
    oneluni $p |- ( B e. A -> ( A u. B ) = A ) $=
      ( wcel wss cun wceq onelssi ssequn2 sylib ) BADBAEABFAGABCHBAIJ $.

    $( An ordinal number is equal to the union of its successor.  (Contributed
       by NM, 12-Jun-1994.) $)
    onunisuci $p |- U. suc A = A $=
      ( wtr csuc cuni wceq ontrci con0 elexi unisuc mpbi ) ACADEAFABGAAHBIJK $.
  $}


$(
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  IZF Set Theory - add the Axiom of Union
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$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Union
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d w x y z $.
    $( Axiom of Union.  An axiom of Intuitionistic Zermelo-Fraenkel set theory.
       It states that a set ` y ` exists that includes the union of a given set
       ` x ` i.e. the collection of all members of the members of ` x ` .  The
       variant ~ axun2 states that the union itself exists.  A version with the
       standard abbreviation for union is ~ uniex2 .  A version using class
       notation is ~ uniex .

       This is Axiom 3 of [Crosilla] p.  "Axioms of CZF and IZF", except (a)
       unnecessary quantifiers are removed, (b) Crosilla has a biconditional
       rather than an implication (but the two are equivalent by ~ bm1.3ii ),
       and (c) the order of the conjuncts is swapped (which is equivalent by
       ~ ancom ).

       The union of a class ~ df-uni should not be confused with the union of
       two classes ~ df-un .  Their relationship is shown in ~ unipr .
       (Contributed by NM, 23-Dec-1993.) $)
    ax-un $a |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y ) $.

    $( Axiom of Union expressed with the fewest number of different variables.
       (Contributed by NM, 14-Aug-2003.)  (New usage is discouraged.) $)
    zfun $p |- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x ) $=
      ( vw wel wa wex wi wal ax-un weq elequ2 elequ1 anbi12d cbvexv albii exbii
      imbi1i mpbi ) BDEZDCEZFZDGZBAEZHZBIZAGUDACEZFZAGZUDHZBIZAGCABDJUFUKAUEUJB
      UCUIUDUBUHDADAKTUDUAUGDABLDACMNORPQS $.

    $( A variant of the Axiom of Union ~ ax-un .  For any set ` x ` , there
       exists a set ` y ` whose members are exactly the members of the members
       of ` x ` i.e. the union of ` x ` .  Axiom Union of [BellMachover]
       p. 466.  (Contributed by NM, 4-Jun-2006.) $)
    axun2 $p |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) ) $=
      ( wel wa wex ax-un bm1.3ii ) CDEDAEFDGBCABCDHI $.

    $( The Axiom of Union using the standard abbreviation for union.  Given any
       set ` x ` , its union ` y ` exists.  (Contributed by NM, 4-Jun-2006.)
       (Proof shortened by BJ, 14-Jul-2026.) $)
    uniex2 $p |- E. y y = U. x $=
      ( vz vw wel cv cuni wcel wb wal wceq wex wa axun2 eluni albii exbii mpbir
      bibi2i dfcleq biimpri eximii ) CBEZCFZAFZGZHZIZCJZBFZUFKZBUIBLUCCDEDAEMDL
      ZIZCJZBLABCDNUIUNBUHUMCUGULUCDUDUEOSPQRUKUICUJUFTUAUB $.

    $( Obsolete version of ~ uniex2 as of 14-Jul-2026.  (Contributed by NM,
       4-Jun-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    uniex2OLD $p |- E. y y = U. x $=
      ( vz cv cuni wceq wex wcel wb wi wa zfun eluni imbi1i albii exbii bm1.3ii
      wal mpbir dfcleq ) BDZADZEZFZBGCDZUAHZUEUCHZICRZBGUGBCUGUFJZCRZBGUFUAUBHK
      BGZUFJZCRZBGBCALUJUMBUIULCUGUKUFBUEUBMNOPSQUDUHBCUAUCTPS $.
  $}

  ${
    $d x y A $.
    uniex.1 $e |- A e. _V $.
    $( The Axiom of Union in class notation.  This says that if ` A ` is a set
       i.e. ` A e. _V ` (see ~ isset ), then the union of ` A ` is also a set.
       Same as Axiom 3 of [TakeutiZaring] p. 16.  (Contributed by NM,
       11-Aug-1993.) $)
    uniex $p |- U. A e. _V $=
      ( vx vy cv cuni cvv wcel wceq unieq eleq1d uniex2 issetri vtocl ) CEZFZGH
      AFZGHCABOAIPQGOAJKDPCDLMN $.
  $}

  $( The union of a setvar is a set.  (Contributed by BJ, 3-May-2021.) $)
  vuniex $p |- U. x e. _V $=
    ( cv vex uniex ) ABACD $.

  ${
    $d x A $.
    $( The ZF Axiom of Union in class notation, in the form of a theorem
       instead of an inference.  We use the antecedent ` A e. V ` instead of
       ` A e. _V ` to make the theorem more general and thus shorten some
       proofs; obviously the universal class constant ` _V ` is one possible
       substitution for class variable ` V ` .  (Contributed by NM,
       25-Nov-1994.) $)
    uniexg $p |- ( A e. V -> U. A e. _V ) $=
      ( vx cv cuni cvv wcel wceq unieq eleq1d vex uniex vtoclg ) CDZEZFGAEZFGCA
      BNAHOPFNAIJNCKLM $.
  $}

  ${
    uniexd.1 $e |- ( ph -> A e. V ) $.
    $( Deduction version of the ZF Axiom of Union in class notation.
       (Contributed by Glauco Siliprandi, 26-Jun-2021.) $)
    uniexd $p |- ( ph -> U. A e. _V ) $=
      ( wcel cuni cvv uniexg syl ) ABCEBFGEDBCHI $.
  $}

  ${
    unex.1 $e |- A e. _V $.
    unex.2 $e |- B e. _V $.
    $( The union of two sets is a set.  Corollary 5.8 of [TakeutiZaring] p. 16.
       (Contributed by NM, 1-Jul-1994.) $)
    unex $p |- ( A u. B ) e. _V $=
      ( cpr cuni cun cvv unipr wcel prexg mp2an uniex eqeltrri ) ABEZFABGHABCDI
      OAHJBHJOHJCDABHHKLMN $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Existence of union is equivalent to existence of its components.
       (Contributed by NM, 11-Jun-1998.) $)
    unexb $p |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V ) $=
      ( vx vy cvv wcel wa cun cv wceq uneq1 eleq1d uneq2 vex unex vtocl2g ssun1
      wss ssexg mpan ssun2 jca impbii ) AEFZBEFZGABHZEFZCIZDIZHZEFAUIHZEFUGCDAB
      EEUHAJUJUKEUHAUIKLUIBJUKUFEUIBAMLUHUICNDNOPUGUDUEAUFRUGUDABQAUFESTBUFRUGU
      EBAUABUFESTUBUC $.
  $}

  $( A union of two sets is a set.  Corollary 5.8 of [TakeutiZaring] p. 16.
     (Contributed by NM, 18-Sep-2006.) $)
  unexg $p |- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V ) $=
    ( wcel cvv cun elex wa unexb biimpi syl2an ) ACEAFEZBFEZABGFEZBDEACHBDHMNIO
    ABJKL $.

  $( An unordered triple of classes exists.  (Contributed by NM,
     10-Apr-1994.) $)
  tpexg $p |- ( ( A e. U /\ B e. V /\ C e. W ) -> { A , B , C } e. _V ) $=
    ( wcel w3a ctp cpr csn cun cvv df-tp wa prexg snexg anim12i 3impa unexg syl
    eqeltrid ) ADGZBEGZCFGZHZABCIABJZCKZLZMABCNUFUGMGZUHMGZOZUIMGUCUDUEULUCUDOU
    JUEUKABDEPCFQRSUGUHMMTUAUB $.

  ${
    $d x A $.  $d x B $.
    $( Union of a singleton in the form of a restricted class abstraction.
       (Contributed by NM, 3-Jul-2008.) $)
    unisn3 $p |- ( A e. B -> U. { x e. B | x = A } = A ) $=
      ( wcel cv wceq crab cuni csn rabsn unieqd unisng eqtrd ) BCDZAEBFACGZHBIZ
      HBNOPACBJKBCLM $.
  $}

  ${
    $d A x y $.  $d F y $.
    $( Sufficient condition for a class abstraction to be a proper class.  The
       class ` F ` can be thought of as an expression in ` x ` and the
       abstraction appearing in the statement as the class of values ` F ` as
       ` x ` varies through ` A ` .  Assuming the antecedents, if that class is
       a set, then so is the "domain" ` A ` .  The converse holds without
       antecedent, see ~ abrexexg .  Note that the second antecedent
       ` A. x e. A x e. F ` cannot be translated to ` A C_ F ` since ` F ` may
       depend on ` x ` .  In applications, one may take ` F = { x } ` or
       ` F = ~P x ` (see ~ snnex and ~ pwnex respectively, proved from
       ~ abnex , which is a consequence of ~ abnexg with ` A = _V ` ).
       (Contributed by BJ, 2-Dec-2021.) $)
    abnexg $p |- ( A. x e. A ( F e. V /\ x e. F ) ->
                               ( { y | E. x e. A y = F } e. W -> A e. _V ) ) $=
      ( cv wceq wrex cab wcel cuni cvv wa wral ciun wi ralimi syl wss simpl csn
      uniexg dfiun2g eleq1d biimprd simpr iunid snssi ss2iun eqsstrrid ssexg ex
      3syl syld syl5 ) BGDHACIBJZFKUQLZMKZDEKZAGZDKZNZACOZCMKZUQFUCVDUSACDPZMKZ
      VEVDUTACOZUSVGQVCUTACUTVBUARVHVGUSVHVFURMABCDEUDUEUFSVDVBACOZCVFTZVGVEQVC
      VBACUTVBUGRVICACVAUBZPZVFACUHVIVKDTZACOVLVFTVBVMACVADUIRACVKDUJSUKVJVGVEC
      VFMULUMUNUOUP $.
  $}

  ${
    $d x y $.  $d F y $.
    $( Sufficient condition for a class abstraction to be a proper class.
       Lemma for ~ snnex and ~ pwnex .  See the comment of ~ abnexg .
       (Contributed by BJ, 2-May-2021.) $)
    abnex $p |- ( A. x ( F e. V /\ x e. F ) -> -. { y | E. x y = F } e. _V ) $=
      ( wcel cv wa wal wceq wex cab cvv vprc wral wrex alral rexv bicomi eleq1i
      abbii biimpi abnexg syl2im mtoi ) CDEAFCEGZAHZBFCIZAJZBKZLEZLLEZMUFUEALNU
      JUGALOZBKZLEZUKUEALPUJUNUIUMLUHULBULUHUGAQRTSUAABLCDLUBUCUD $.
  $}

  ${
    $d x y z $.
    $( The class of all singletons is a proper class.  (Contributed by NM,
       10-Oct-2008.)  (Proof shortened by Eric Schmidt, 7-Dec-2008.) $)
    snnex $p |- { x | E. y x = { y } } e/ _V $=
      ( vz cv csn wceq wex cab wcel cuni vprc wa vsnid a9ev sneq equcoms eximii
      cvv vex snex eleq2 eqeq1 exbidv anbi12d spcev mp2an mpbir 2th eqriv mtbir
      eluniab eleq1i uniexg mto nelir ) ADZBDZEZFZBGZAHZRVARIVAJZRIZVCRRIKVBRRC
      VBRCDZVBIZVDRIVEVDUPIZUTLZAGZVDVDEZIZVIURFZBGZVHCMUQVDFVKBBCNVKCBVDUQOPQV
      GVJVLLAVIVDCSZTUPVIFZVFVJUTVLUPVIVDUAVNUSVKBUPVIURUBUCUDUEUFUTAVDUKUGVMUH
      UIULUJVARUMUNUO $.
  $}

  ${
    $d x y $.
    $( The class of all power sets is a proper class.  See also ~ snnex .
       (Contributed by BJ, 2-May-2021.) $)
    pwnex $p |- { x | E. y x = ~P y } e/ _V $=
      ( cv cpw cvv wcel wa wceq wex cab wnel wal abnex df-nel sylibr vpwex pwid
      wn vex pm3.2i mpg ) BCZDZEFZUBUCFZGZACUCHBIAJZEKZBUFBLUGEFRUHBAUCEMUGENOU
      DUEBPUBBSQTUA $.
  $}

  ${
    opeluu.1 $e |- A e. _V $.
    opeluu.2 $e |- B e. _V $.
    $( Each member of an ordered pair belongs to the union of the union of a
       class to which the ordered pair belongs.  Lemma 3D of [Enderton] p. 41.
       (Contributed by NM, 31-Mar-1995.)  (Revised by Mario Carneiro,
       27-Feb-2016.) $)
    opeluu $p |- ( <. A , B >. e. C ->
                 ( A e. U. U. C /\ B e. U. U. C ) ) $=
      ( cop wcel cuni cpr prid1 opi2 elunii mpan sylancr prid2 jca ) ABFZCGZACH
      ZHZGZBTGZRAABIZGUCSGZUAABDJUCQGRUDABDEKUCQCLMZAUCSLNRBUCGUDUBABEOUEBUCSLN
      P $.
  $}

  ${
    $d A x y v z $.  $d A x y u z $.
    $( Expression for double union that moves union into a class builder.
       (Contributed by FL, 28-May-2007.) $)
    uniuni $p |- U. U. A = U. { x | E. y ( x = U. y /\ y e. A ) } $=
      ( vz vu vv wel cv cuni wcel wex cab wceq eluni anbi2i exbii 19.42v bicomi
      wa 3bitri excom anass ancom bitr3i 2exbii exdistr vex uniex eleq2 ceqsexv
      exancom bitri 3bitr2i eqeq1 anbi1d exbidv elab abbii df-uni 3eqtr4i ) DEG
      ZEHZCIZJZSZEKZDLDFGZFHZAHZBHZIZMZVJCJZSZBKZALZJZSZFKZDLVCIVPIVFVSDVFVAEBG
      ZVMSZBKZSZEKZVMDHZVKJZSZBKZVSVEWCEVDWBVABVBCNOPWDVAWASZBKZEKZVMVAVTSZEKZS
      ZBKZWHWCWJEWJWCVAWABQRPWKWIEKBKVMWLSZEKBKWOWIEBUAWIWPBEWIWLVMSWPVAVTVMUBW
      LVMUCUDUEVMWLBEUFTWNWGBWMWFVMWFWMEWEVJNROPTWHVGVHVKMZVMSZSZFKZBKWSBKFKZVS
      WGWTBWGVMVGWQSZFKZSVMXBSZFKWTWFXCVMWFWQVGSFKXCVGWFFVKVJBUGUHVHVKWEUIUJWQV
      GFUKUDOVMXBFQXDWSFXDXBVMSWSVMXBUCVGWQVMUBULPUMPWSBFUAXAVGWRBKZSZFKVSVGWRF
      BUFXFVRFXEVQVGVQXEVOXEAVHFUGVIVHMZVNWRBXGVLWQVMVIVHVKUNUOUPUQROPULTTURDEV
      CUSDFVPUSUT $.
  $}

  ${
    $d x y z $.  $d A y z $.
    $( Two ways to express single-valuedness of a class expression
       ` A ( x ) ` .  (Contributed by NM, 14-Oct-2010.) $)
    eusv1 $p |- ( E! y A. x y = A <-> E. y A. x y = A ) $=
      ( vz cv wceq wal weu wex wa wi sp eqtr3 syl2an gen2 eqeq1 albidv mpbiran2
      eu4 ) BEZCFZAGZBHUBBIUBDEZCFZAGZJTUCFZKZDGBGUGBDUBUAUDUFUEUAALUDALTUCCMNO
      UBUEBDUFUAUDATUCCPQSR $.
  $}

  ${
    $d x y z w $.  $d A y z w $.
    $( Even if ` x ` is free in ` A ` , it is effectively bound when
       ` A ( x ) ` is single-valued.  (Contributed by NM, 14-Oct-2010.)
       (Revised by Mario Carneiro, 14-Oct-2016.) $)
    eusvnf $p |- ( E! y A. x y = A -> F/_ x A ) $=
      ( vz vw cv wceq wal csb cvv wcel wi vex nfcv nfcsb1v nfeq2 csbeq1a eqeq2d
      spcgf ax-mp weu wex wnfc euex eqtr3d alrimivv sbnfc2 sylibr exlimiv syl )
      BFZCGZAHZBUAUMBUBACUCZUMBUDUMUNBUMADFZCIZAEFZCIZGZEHDHUNUMUSDEUMUKUPURUOJ
      KUMUKUPGZLDMULUTAUOJAUONAUKUPAUOCOPAFZUOGCUPUKAUOCQRSTUQJKUMUKURGZLEMULVB
      AUQJAUQNAUKURAUQCOPVAUQGCURUKAUQCQRSTUEUFADECUGUHUIUJ $.

    $( Two ways to say that ` A ( x ) ` is a set expression that does not
       depend on ` x ` .  (Contributed by Mario Carneiro, 18-Nov-2016.) $)
    eusvnfb $p |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) ) $=
      ( cv wceq wal weu wnfc cvv wcel eusvnf wex euex vex eqeltrrdi sps exlimiv
      wa id syl jca isset nfcvd nfeqd nfrd eximdv biimtrid eusv1 sylibr impbii
      imp ) BDZCEZAFZBGZACHZCIJZRZUOUPUQABCKUOUNBLZUQUNBMUNUQBUMUQAUMCULIUMSBNO
      PQTUAURUSUOUPUQUSUQUMBLUPUSBCUBUPUMUNBUPUMAUPAULCUPAULUCUPSUDUEUFUGUKABCU
      HUIUJ $.
  $}

  ${
    $d x y $.  $d A y $.
    $( Two ways to express single-valuedness of a class expression
       ` A ( x ) ` .  (Contributed by NM, 14-Oct-2010.)  (Revised by Mario
       Carneiro, 18-Nov-2016.) $)
    eusv2i $p |- ( E! y A. x y = A -> E! y E. x y = A ) $=
      ( cv wceq wal weu wex nfeu1 wnf nfcvd eusvnf nfeqd nf2 sylib 19.2 impbid1
      wi eubid ibir ) BDZCEZAFZBGZUBAHZBGUDUEUCBUCBIUDUEUCUDUBAJUEUCRUDAUACUDAU
      AKABCLMUBANOUBAAPQST $.
  $}

  ${
    $d x y $.  $d A y $.
    eusv2.1 $e |- A e. _V $.
    $( Two ways to express single-valuedness of a class expression
       ` A ( x ) ` .  (Contributed by Mario Carneiro, 18-Nov-2016.) $)
    eusv2nf $p |- ( E! y E. x y = A <-> F/_ x A ) $=
      ( cv wceq wex weu wnfc wnf wal nfeu1 wi nfe1 nfeu wa isseti alrimi sylibr
      cvv 19.8a ancri eximii eupick mpan2 nf3 wcel wb dfnfc2 mpg eusvnfb eusv2i
      mpbiran2 sylbir impbii ) BECFZAGZBHZACIZURUPAJZBKZUSURUTBUQBLURUQUPMZAKUT
      URVBAUQABUPANOURUQUPPZBGVBUPVCBBCDQUPUQUPAUAUBUCUQUPBUDUERUPAUFSRCTUGZUSV
      AUHAABCTUIDUJSUSUPAKBHZURVEUSVDDABCUKUMABCULUNUO $.

    $( Two ways to express single-valuedness of a class expression
       ` A ( x ) ` .  (Contributed by NM, 15-Oct-2010.)  (Proof shortened by
       Mario Carneiro, 18-Nov-2016.) $)
    eusv2 $p |- ( E! y E. x y = A <-> E! y A. x y = A ) $=
      ( cv wceq wex weu wnfc wal eusv2nf cvv wcel eusvnfb mpbiran2 bitr4i ) BEC
      FZAGBHACIZQAJBHZABCDKSRCLMDABCNOP $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ph $.  $d x y $.
    $( Two ways to express single-valuedness of a class expression
       ` C ( y ) ` .  (Contributed by NM, 16-Dec-2012.)  (Proof shortened by
       Mario Carneiro, 18-Nov-2016.) $)
    reusv1 $p |- ( E. y e. B ph -> ( E! x e. A A. y e. B ( ph -> x = C )
                   <-> E. x e. A A. y e. B ( ph -> x = C ) ) ) $=
      ( wrex cv wceq wi wral wmo wrmo wreu wb nfra1 nfmo wcel wa wal impd com12
      rsp alrimiv moeq moim mpisyl ex rexlimi mormo reu5 rbaib 3syl ) ACEGABHFI
      ZJZCEKZBLZUPBDMZUPBDNZUPBDGZOAUQCEUPCBUOCEPQCHERZAUQVAASZUPUNJZBTUNBLUQVB
      VCBUPVBUNUPVAAUNUOCEUCUAUBUDBFUEUPUNBUFUGUHUIUPBDUJUSUTURUPBDUKULUM $.
  $}

  ${
    $d x y z B $.  $d x z C $.  $d x y D $.  $d x z ph $.  $d x y ps $.
    reusv3.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
    reusv3.2 $e |- ( y = z -> C = D ) $.
    $( Two ways of expressing existential uniqueness via an indirect equality.
       (Contributed by NM, 23-Dec-2012.) $)
    reusv3i $p |- ( E. x e. A A. y e. B ( ph -> x = C )
            -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) ) $=
      ( cv wceq wi wral wa eqeq2d imbi12d cbvralv ralimi biimpi eqtr2 rexlimivw
      raaanv anim12 syl6 sylbir mpdan ) ACLZHMZNZDGOZABPZHIMZNZEGOZDGOZCFULBUII
      MZNZEGOZUQULUTUKUSDEGDLELMZABUJURJVAHIUIKQRSUAULUTPUKUSPZEGOZDGOUQUKUSDEG
      UDVCUPDGVBUOEGVBUMUJURPUNAUJBURUEUIHIUBUFTTUGUHUC $.

    $d x y z A $.
    $( Two ways to express single-valuedness of a class expression
       ` C ( y ) ` .  See ~ reusv1 for the connection to uniqueness.
       (Contributed by NM, 27-Dec-2012.) $)
    reusv3 $p |- ( E. y e. B ( ph /\ C e. A )
          -> ( A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D )
                   <-> E. x e. A A. y e. B ( ph -> x = C ) ) ) $=
      ( wcel wa wrex wceq wi wral cv bitri ralbii eleq1d anbi12d cbvrexv risset
      nfra2xy nfv nfim ralcom impexp bi2.04 r19.21v rsp sylbi com3l imp31 eqeq1
      eqcom bitrdi ralbidv syl5ibrcom reximdv ex com23 biimtrid expimpd rexlimi
      imbi2d reusv3i impbid1 ) AHFLZMZDGNZABMHIOZPZEGQDGQZACRZHOZPZDGQZCFNZVLBI
      FLZMZEGNVOVTPZVKWBDEGDRERZOZABVJWAJWEHIFKUAUBUCWBWCEGVOVTEVNDEGGUEVTEUFUG
      WDGLZBWAWCWAVPIOZCFNZWFBMZWCCIFUDWIVOWHVTWIVOWHVTPWIVOMZWGVSCFWJVSWGAVMPZ
      DGQZWFBVOWLVOWFBWLVOBWLPZEGQZWFWMPVOVNDGQZEGQWNVNDEGGUHWOWMEGWOBWKPZDGQWM
      VNWPDGVNABVMPPWPABVMUIABVMUJSTBWKDGUKSTSWMEGULUMUNUOWGVRWKDGWGVQVMAWGVQIH
      OVMVPIHUPIHUQURVGUSUTVAVBVCVDVEVFUMABCDEFGHIJKVHVI $.
  $}

  ${
    $d x A $.  $d y ph $.  $d x ps $.  $d x y $.
    alxfr.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Transfer universal quantification from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by NM,
       18-Feb-2007.) $)
    alxfr $p |- ( ( A. y A e. B /\ A. x E. y x = A ) ->
                ( A. x ph <-> A. y ps ) ) $=
      ( wcel wal cv wceq wex wa wi spcgv com12 alimdv adantr nfa1 nfv sp exlimd
      syl5ibrcom adantl impbid ) EFHZDIZCJEKZDLZCIZMACIZBDIZUGUKULNUJUKUGULUKUF
      BDUFUKBABCEFGOPQPRUJULUKNUGULUJUKULUIACULUHADBDSADTULAUHBBDUAGUCUBQPUDUE
      $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x C $.  $d x ch $.  $d x y ph $.  $d y ps $.
    ralxfrd.1 $e |- ( ( ph /\ y e. C ) -> A e. B ) $.
    ralxfrd.2 $e |- ( ( ph /\ x e. B ) -> E. y e. C x = A ) $.
    ralxfrd.3 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Transfer universal quantification from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by NM,
       15-Aug-2014.)  (Proof shortened by Mario Carneiro, 19-Nov-2016.) $)
    ralxfrd $p |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) ) $=
      ( wral cv wcel wa wceq wb adantlr ralrimdva wrex rspcdv r19.29 wi biimprd
      expimpd ancomsd ad2antrr rexlimdva syl5 mpan2d impbid ) ABDGLZCEHLZAULCEH
      AEMHNZOBCDFGIADMZFPZBCQUNKRUASAUMBDGAUOGNZOZUMUPEHTZBJUMUSOCUPOZEHTURBCUP
      EHUBURUTBEHAUTBUCUQUNAUPCBAUPCBAUPOBCKUDUEUFUGUHUIUJSUK $.

    $( Transfer universal quantification from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by FL,
       10-Apr-2007.)  (Revised by Mario Carneiro, 15-Aug-2014.) $)
    rexxfrd $p |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) ) $=
      ( wrex wal cv wcel wa nfv wex df-rex rexlimdva 19.3 wceq 19.29 an12 exbii
      wi sylib sylibr sylan2b biimpd expimpd ancomsd reximdv syl5 adantr mpan2d
      biimtrrid wb adantlr rspcedv impbid ) ABDGLZCEHLZABVCDGBBEMZADNZGOZPZVCBE
      BEQUAVGVDVEFUBZEHLZVCJAVDVIPZVCUFVFVJBVHPZEHLZAVCVIVDENHOZVHPZERZVLVHEHSV
      DVOPZVMVKPZERZVLVPBVNPZERVRBVNEUCVSVQEBVMVHUDUEUGVKEHSUHUIAVKCEHAVHBCAVHB
      CAVHPBCKUJUKULUMUNUOUPUQTACVBEHAVMPBCDFGIAVHBCURVMKUSUTTVA $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x C $.  $d x ch $.  $d x y ph $.  $d y ps $.
    ralxfr2d.1 $e |- ( ( ph /\ y e. C ) -> A e. V ) $.
    ralxfr2d.2 $e |- ( ph -> ( x e. B <-> E. y e. C x = A ) ) $.
    ralxfr2d.3 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Transfer universal quantification from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by Mario
       Carneiro, 20-Aug-2014.) $)
    ralxfr2d $p |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) ) $=
      ( cv wcel wa wceq wex elisset syl wi wrex biimprd r19.23v sylibr r19.21bi
      wral eleq1 mpbidi exlimdv mpd biimpa ralxfrd ) ABCDEFGHAEMHNOZDMZFPZDQZFG
      NZUMFINUPJDFIRSUMUOUQDUOUNGNZUQUMAUOURTZEHAUOEHUAZURTUSEHUFAURUTKUBUOUREH
      UCUDUEUNFGUGUHUIUJAURUTKUKLUL $.

    $( Transfer universal quantification from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by Mario
       Carneiro, 20-Aug-2014.)  (Proof shortened by Mario Carneiro,
       19-Nov-2016.) $)
    rexxfr2d $p |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) ) $=
      ( cv wcel wa wceq wex elisset syl wi wrex biimprd r19.23v sylibr r19.21bi
      wral eleq1 mpbidi exlimdv mpd biimpa rexxfrd ) ABCDEFGHAEMHNOZDMZFPZDQZFG
      NZUMFINUPJDFIRSUMUOUQDUOUNGNZUQUMAUOURTZEHAUOEHUAZURTUSEHUFAURUTKUBUOUREH
      UCUDUEUNFGUGUHUIUJAURUTKUKLUL $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x A $.  $d x y B $.  $d x C $.
    ralxfr.1 $e |- ( y e. C -> A e. B ) $.
    ralxfr.2 $e |- ( x e. B -> E. y e. C x = A ) $.
    ralxfr.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Transfer universal quantification from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by NM,
       10-Jun-2005.)  (Revised by Mario Carneiro, 15-Aug-2014.) $)
    ralxfr $p |- ( A. x e. B ph <-> A. y e. C ps ) $=
      ( wral wb wtru cv wcel adantl wceq wrex ralxfrd mptru ) ACFKBDGKLMABCDEFG
      DNGOEFOMHPCNZFOUAEQZDGRMIPUBABLMJPST $.

    $( Transfer universal quantification from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  This proof does not use
       ~ ralxfrd .  (Contributed by NM, 10-Jun-2005.)  (Revised by Mario
       Carneiro, 15-Aug-2014.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ralxfrALT $p |- ( A. x e. B ph <-> A. y e. C ps ) $=
      ( wral cv wcel wi rspcv syl com12 ralrimiv wceq wrex nfv biimprcd rexlimd
      nfra1 rsp syl6 syl5 impbii ) ACFKZBDGKZUIBDGDLGMZUIBUKEFMUIBNHABCEFJOPQRU
      JACFCLZFMULESZDGTUJAIUJUMADGBDGUDADUAUJUKBUMANBDGUEUMABJUBUFUCUGRUH $.

    $( Transfer existence from a variable ` x ` to another variable ` y `
       contained in expression ` A ` .  (Contributed by NM, 10-Jun-2005.)
       (Revised by Mario Carneiro, 15-Aug-2014.) $)
    rexxfr $p |- ( E. x e. B ph <-> E. y e. C ps ) $=
      ( wrex wb wtru cv wcel adantl wceq rexxfrd mptru ) ACFKBDGKLMABCDEFGDNGOE
      FOMHPCNZFOTEQZDGKMIPUAABLMJPRS $.
  $}

  ${
    $d x A $.  $d x y D $.  $d y ph $.  $d y ps $.  $d x ch $.
    rabxfrd.1 $e |- F/_ y B $.
    rabxfrd.2 $e |- F/_ y C $.
    rabxfrd.3 $e |- ( ( ph /\ y e. D ) -> A e. D ) $.
    rabxfrd.4 $e |- ( x = A -> ( ps <-> ch ) ) $.
    rabxfrd.5 $e |- ( y = B -> A = C ) $.
    $( Class builder membership after substituting an expression ` A `
       (containing ` y ` ) for ` x ` in the class expression ` ch ` .
       (Contributed by NM, 16-Jan-2012.) $)
    rabxfrd $p |- ( ( ph /\ B e. D ) ->
                 ( C e. { x e. D | ps } <-> B e. { y e. D | ch } ) ) $=
      ( wcel crab wb wa wi imp cv ibibr sylib anbi1d elrab rabid 3bitr4g eleq2d
      ex rabbidva nfcv nfel1 wceq eleq1d elrabf nfrab1 nfel eleq1 pm5.32 sylibr
      3bitr3g ) AGIOZHBDIPZOZGCEIPZOZQZAVBVDRZVBVFRZQVBVGSAGFVCOZEIPZOGEUAZVEOZ
      EIPZOVHVIAVKVNGAVJVMEIAVLIOZRZFIOZCRVOCRVJVMVPVQVOCAVOVQVOQZAVOVQSVOVRSAV
      OVQLUIVOVQUBUCTUDBCDFIMUECEIUFUGUJUHVJVDEGIJEIUKZEHVCKULVLGUMFHVCNUNUOVMV
      FEGIJVSEGVEJCEIUPUQVLGVEURUOVAVBVDVFUSUTT $.
  $}

  ${
    $d x A $.  $d x y D $.  $d y ph $.  $d x ps $.
    rabxfr.1 $e |- F/_ y B $.
    rabxfr.2 $e |- F/_ y C $.
    rabxfr.3 $e |- ( y e. D -> A e. D ) $.
    rabxfr.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
    rabxfr.5 $e |- ( y = B -> A = C ) $.
    $( Class builder membership after substituting an expression ` A `
       (containing ` y ` ) for ` x ` in the class expression ` ph ` .
       (Contributed by NM, 10-Jun-2005.) $)
    rabxfr $p |- ( B e. D ->
                 ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) ) $=
      ( wtru wcel crab wb tru cv adantl rabxfrd mpan ) NFHOGACHPOFBDHPOQRNABCDE
      FGHIJDSHOEHONKTLMUAUB $.
  $}

  ${
    $d y ph $.  $d y B $.  $d y C $.  $d x y $.
    reuhypd.1 $e |- ( ( ph /\ x e. C ) -> B e. C ) $.
    reuhypd.2 $e |- ( ( ph /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) ) $.
    $( A theorem useful for eliminating restricted existential uniqueness
       hypotheses.  (Contributed by NM, 16-Jan-2012.) $)
    reuhypd $p |- ( ( ph /\ x e. C ) -> E! y e. C x = A ) $=
      ( cv wcel wa wceq weu wreu cvv elex syl eueq sylib eleq1 syl5ibrcom 3expa
      pm4.71rd wb pm5.32da bitr4d eubidv mpbid df-reu sylibr ) ABIZFJZKZCIZFJZU
      KDLZKZCMZUPCFNUMUNELZCMZURUMEOJZUTUMEFJZVAGEFPQCERSUMUSUQCUMUSUOUSKUQUMUS
      UOUMUOUSVBGUNEFTUAUCUMUOUPUSAULUOUPUSUDHUBUEUFUGUHUPCFUIUJ $.
  $}

  ${
    $d y B $.  $d y C $.  $d x y $.
    reuhyp.1 $e |- ( x e. C -> B e. C ) $.
    reuhyp.2 $e |- ( ( x e. C /\ y e. C ) -> ( x = A <-> y = B ) ) $.
    $( A theorem useful for eliminating restricted existential uniqueness
       hypotheses.  (Contributed by NM, 15-Nov-2004.) $)
    reuhyp $p |- ( x e. C -> E! y e. C x = A ) $=
      ( wtru cv wcel wceq wreu tru adantl wb 3adant1 reuhypd mpan ) HAIZEJZSCKZ
      BELMHABCDETDEJHFNTBIZEJUAUBDKOHGPQR $.
  $}

  $( The Axiom of Union and its converse.  A class is a set iff its union is a
     set.  (Contributed by NM, 11-Nov-2003.) $)
  uniexb $p |- ( A e. _V <-> U. A e. _V ) $=
    ( cvv wcel cuni uniexg cpw wss pwuni pwexg ssexg sylancr impbii ) ABCZADZBC
    ZABEOANFZGPBCMAHNBIAPBJKL $.

  $( The Axiom of Power Sets and its converse.  A class is a set iff its power
     class is a set.  (Contributed by NM, 11-Nov-2003.) $)
  pwexb $p |- ( A e. _V <-> ~P A e. _V ) $=
    ( cpw cvv wcel cuni uniexb unipw eleq1i bitr2i ) ABZCDJEZCDACDJFKACAGHI $.

  $( A class belongs to a double power class if and only if its union belongs
     to the power class.  (Contributed by BJ, 22-Jan-2023.) $)
  elpwpwel $p |- ( A e. ~P ~P B <-> U. A e. ~P B ) $=
    ( cvv wcel cuni wss wa cpw uniexb anbi1i elpwpw elpwb 3bitr4i ) ACDZAEZBFZG
    OCDZPGABHZHDORDNQPAIJABKOBLM $.

  $( The union of the universe is the universe.  Exercise 4.12(c) of
     [Mendelson] p. 235.  (Contributed by NM, 14-Sep-2003.) $)
  univ $p |- U. _V = _V $=
    ( cvv cpw cuni pwv unieqi unipw eqtr3i ) ABZCACAHADEAFG $.

  ${
    eldifpw.1 $e |- C e. _V $.
    $( Membership in a power class difference.  (Contributed by NM,
       25-Mar-2007.) $)
    eldifpw $p |- ( ( A e. ~P B /\ -. C C_ B ) ->
                   ( A u. C ) e. ( ~P ( B u. C ) \ ~P B ) ) $=
      ( cpw wcel wss wn wa cun cdif elpwi unss1 cvv unexg mpan2 elpwg imbitrrid
      wb syl mpd unssbd con3i anim12i eldif sylibr ) ABEZFZCBGZHZIACJZBCJZEZFZU
      KUGFZHZIUKUMUGKFUHUNUJUPUHABGZUNABLUQUNUHUKULGZABCMUHUKNFZUNURSUHCNFUSDAC
      UGNOPUKULNQTRUAUOUIUOACBUKBLUBUCUDUKUMUGUEUF $.
  $}

  ${
    op1stb.1 $e |- A e. _V $.
    op1stb.2 $e |- B e. _V $.
    $( Extract the first member of an ordered pair.  Theorem 73 of [Suppes]
       p. 42.  (Contributed by NM, 25-Nov-2003.) $)
    op1stb $p |- |^| |^| <. A , B >. = A $=
      ( cop cint csn cpr dfop inteqi cin snex wcel prexg mp2an intpr wceq eqtri
      cvv wss snsspr1 df-ss mpbi intsn ) ABEZFZFAGZFAUFUGUFUGABHZHZFZUGUEUIABCD
      IJUJUGUHKZUGUGUHACLASMBSMUHSMCDABSSNOPUGUHTUKUGQABUAUGUHUBUCRRJACUDR $.
  $}

  $( Extract the first member of an ordered pair.  Theorem 73 of [Suppes]
     p. 42.  (Contributed by Jim Kingdon, 17-Dec-2018.) $)
  op1stbg $p |- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = A ) $=
    ( wcel wa cop cint csn cpr dfopg inteqd cin cvv snexg prexg intprg syl2an2r
    wceq eqtrd wss snsspr1 df-ss mpbi eqtrdi intsng adantr ) ACEZBDEZFZABGZHZHA
    IZHZAUJULUMUJULUMABJZJZHZUMUJUKUPABCDKLUJUQUMUOMZUMUHUMNEUIUONEUQURSACOABCD
    PUMUONNQRUMUOUAURUMSABUBUMUOUCUDUETLUHUNASUIACUFUGT $.

  ${
    $d x y A $.
    iunpw.1 $e |- A e. _V $.
    $( An indexed union of a power class in terms of the power class of the
       union of its index.  Part of Exercise 24(b) of [Enderton] p. 33.
       (Contributed by NM, 29-Nov-2003.) $)
    iunpw $p |- ( E. x e. A x = U. A <-> ~P U. A = U_ x e. A ~P x ) $=
      ( vy cuni wceq wrex cpw ciun wcel sseq2 biimprcd reximdv com12 ssiun elpw
      cv wss eliun wa uniiun sseqtrrdi impbid1 df-pw abeq2i bitri 3bitr4g eqrdv
      vex rexbii ssid uniex eleq2 bitr3id mpbii sylib elssuni elpwi eqss sylibr
      anim12i ex reximia syl impbii ) AQZBEZFZABGZVGHZABVFHZIZFZVIDVJVLVIDQZVGR
      ZVNVFRZABGZVNVJJVNVLJZVIVOVQVOVIVQVOVHVPABVHVPVOVFVGVNKLMNVQVNABVFIVGABVF
      VNOABUAUBUCVNVGDUIPVRVNVKJZABGVQAVNBVKSVSVPABVPDVKDVFUDUEUJUFUGUHVMVGVKJZ
      ABGZVIVMVGVLJZWAVMVGVGRZWBVGUKWCVGVJJVMWBVGVGBCULPVJVLVGUMUNUOAVGBVKSUPVT
      VHABVFBJZVTVHWDVTTVFVGRZVGVFRZTVHWDWEVTWFVFBUQVGVFURVAVFVGUSUTVBVCVDVE $.
  $}

  $( Existence of a conditional class, quantitative version (closed form).
     (Contributed by BJ, 15-Aug-2024.) $)
  ifelpwung $p |-
              ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. ~P ( A u. B ) ) $=
    ( wcel wa cif cun cpw wss ifssun cvv wb unexg elpw2g syl mpbiri ) BDFCEFGZA
    BCHZBCIZJFZTUAKZABCLSUAMFUBUCNBCDEOTUAMPQR $.

  ${
    ifelpwund.1 $e |- ( ph -> A e. V ) $.
    ifelpwund.2 $e |- ( ph -> B e. W ) $.
    $( Existence of a conditional class, quantitative version (deduction form).
       (Contributed by BJ, 15-Aug-2024.) $)
    ifelpwund $p |- ( ph -> if ( ps , A , B ) e. ~P ( A u. B ) ) $=
      ( wcel cif cun cpw ifelpwung syl2anc ) ACEIDFIBCDJCDKLIGHBCDEFMN $.
  $}

  ${
    ifelpwun.1 $e |- A e. _V $.
    ifelpwun.2 $e |- B e. _V $.
    $( Existence of a conditional class, quantitative version (inference form).
       (Contributed by BJ, 15-Aug-2024.) $)
    ifelpwun $p |- if ( ph , A , B ) e. ~P ( A u. B ) $=
      ( cvv wcel cif cun cpw ifelpwung mp2an ) BFGCFGABCHBCIJGDEABCFFKL $.
  $}

  ${
    ifexd.1 $e |- ( ph -> A e. V ) $.
    ifexd.2 $e |- ( ph -> B e. W ) $.
    $( Existence of a conditional class (deduction form).  (Contributed by BJ,
       15-Aug-2024.) $)
    ifexd $p |- ( ph -> if ( ps , A , B ) e. _V ) $=
      ( cif cun cpw ifelpwund elexd ) ABCDICDJKABCDEFGHLM $.
  $}

  $( Existence of the conditional operator (closed form).  (Contributed by NM,
     21-Mar-2011.)  (Proof shortened by BJ, 1-Sep-2022.) $)
  ifexg $p |- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. _V ) $=
    ( wcel wa simpl simpr ifexd ) BDFZCEFZGABCDEKLHKLIJ $.

  ${
    ifex.1 $e |- A e. _V $.
    ifex.2 $e |- B e. _V $.
    $( Existence of the conditional operator (inference form).  (Contributed by
       NM, 2-Sep-2004.) $)
    ifex $p |- if ( ph , A , B ) e. _V $=
      ( cvv wcel cif ifexg mp2an ) BFGCFGABCHFGDEABCFFIJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordinals (continued)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( The class of all ordinal numbers is ordinal.  Proposition 7.12 of
     [TakeutiZaring] p. 38, but without using the Axiom of Regularity.
     (Contributed by NM, 17-May-1994.) $)
  ordon $p |- Ord On $=
    ( vx con0 word wtr cv wral tron wcel df-on abeq2i ordtr sylbi rgen mpbir2an
    dford3 ) BCBDAEZDZABFGQABPBHPCZQRABAIJPKLMABON $.

  ${
    $d x y A $.
    $( The union of a class of ordinal numbers is ordinal.  Proposition 7.19 of
       [TakeutiZaring] p. 40.  (Contributed by NM, 30-May-1994.)  (Proof
       shortened by Andrew Salmon, 12-Aug-2011.) $)
    ssorduni $p |- ( A C_ On -> Ord U. A ) $=
      ( vx vy con0 wss cuni wtr word cv wral wcel wrex eluni2 wa wi ssel onelss
      syl6 rexlimdv biimtrid anc2r syl ssuni ralrimiv dftr3 sylibr onelon ssrdv
      syl8 ex ordon trssord 3exp mpii sylc ) ADEZAFZGZUQDEZUQHZUPBIZUQEZBUQJURU
      PVBBUQVAUQKZVACIZKZCALZUPVBCVAAMZUPVEVBCAUPVDAKZVEVAVDEZVHNZVBUPVHVEVIOZO
      VHVEVJOOUPVHVDDKZVKADVDPZVDVAQRVHVEVIUAUBVAVDAUCUISTUDBUQUEUFUPBUQDVCVFUP
      VADKZVGUPVEVNCAUPVHVLVEVNOVMVLVEVNVDVAUGUJRSTUHURUSDHZUTUKURUSVOUTUQDULUM
      UNUO $.
  $}

  $( The union of a set of ordinal numbers is an ordinal number.  Theorem 9 of
     [Suppes] p. 132.  (Contributed by NM, 1-Nov-2003.) $)
  ssonuni $p |- ( A e. V -> ( A C_ On -> U. A e. On ) ) $=
    ( con0 wss cuni wcel word ssorduni cvv wb uniexg elong syl imbitrrid ) ACDA
    EZCFZABFZOGZAHQOIFPRJABKOILMN $.

  ${
    ssonuni.1 $e |- A e. _V $.
    $( The union of a set of ordinal numbers is an ordinal number.  Corollary
       7N(d) of [Enderton] p. 193.  (Contributed by NM, 20-Sep-2003.) $)
    ssonunii $p |- ( A C_ On -> U. A e. On ) $=
      ( cvv wcel con0 wss cuni wi ssonuni ax-mp ) ACDAEFAGEDHBACIJ $.
  $}

  $( The union of two ordinal numbers is an ordinal number.  (Contributed by
     Jim Kingdon, 25-Jul-2019.) $)
  onun2 $p |- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On ) $=
    ( con0 wcel wa cpr wss cun prssi cuni wi prexg ssonuni uniprg eleq1d sylibd
    cvv syl mpd ) ACDBCDEZABFZCGZABHZCDZABCITUBUAJZCDZUDTUAQDUBUFKABCCLUAQMRTUE
    UCCABCCNOPS $.

  ${
    onun2i.1 $e |- A e. On $.
    onun2i.2 $e |- B e. On $.
    $( The union of two ordinal numbers is an ordinal number.  (Contributed by
       NM, 13-Jun-1994.)  (Constructive proof by Jim Kingdon, 25-Jul-2019.) $)
    onun2i $p |- ( A u. B ) e. On $=
      ( con0 wcel cun onun2 mp2an ) AEFBEFABGEFCDABHI $.
  $}

  ${
    $d x A $.
    $( Any ordinal class is a subclass of the class of ordinal numbers.
       Corollary 7.15 of [TakeutiZaring] p. 38.  (Contributed by NM,
       18-May-1994.) $)
    ordsson $p |- ( Ord A -> A C_ On ) $=
      ( vx word con0 cv wcel ordelon ex ssrdv ) ACZBADJBEZAFKDFAKGHI $.
  $}

  $( An ordinal number is a subset of the class of ordinal numbers.
     (Contributed by NM, 5-Jun-1994.) $)
  onss $p |- ( A e. On -> A C_ On ) $=
    ( con0 wcel word wss eloni ordsson syl ) ABCADABEAFAGH $.

  $( The union of an ordinal number is an ordinal number.  (Contributed by NM,
     29-Sep-2006.) $)
  onuni $p |- ( A e. On -> U. A e. On ) $=
    ( con0 wcel wss cuni onss ssonuni mpd ) ABCABDAEBCAFABGH $.

  $( The union of an ordinal class is ordinal.  (Contributed by NM,
     12-Sep-2003.) $)
  orduni $p |- ( Ord A -> Ord U. A ) $=
    ( word con0 wss cuni ordsson ssorduni syl ) ABACDAEBAFAGH $.

  ${
    $d x y A $.
    bm2.5ii.1 $e |- A e. _V $.
    $( Problem 2.5(ii) of [BellMachover] p. 471.  (Contributed by NM,
       20-Sep-2003.) $)
    bm2.5ii $p |- ( A C_ On -> U. A = |^| { x e. On | A. y e. A y C_ x } ) $=
      ( con0 wss cuni wcel cv wral crab cint wceq ssonunii intmin wb unissb a1i
      rabbiia inteqi eqtr3di syl ) CEFCGZEHZUCBIAIZFBCJZAEKZLZMCDNUDUCUEFZAEKZL
      UCUHAUCEOUJUGUIUFAEUIUFPUEEHBCUEQRSTUAUB $.
  $}

  $( A successor exists iff its class argument exists.  (Contributed by NM,
     22-Jun-1998.) $)
  sucexb $p |- ( A e. _V <-> suc A e. _V ) $=
    ( cvv wcel csn wa cun csuc unexb snexg pm4.71i df-suc eleq1i 3bitr4i ) ABCZ
    ADZBCZEAOFZBCNAGZBCAOHNPABIJRQBAKLM $.

  $( The successor of a set is a set (generalization).  (Contributed by NM,
     5-Jun-1994.) $)
  sucexg $p |- ( A e. V -> suc A e. _V ) $=
    ( wcel cvv csuc elex sucexb sylib ) ABCADCAEDCABFAGH $.

  ${
    sucex.1 $e |- A e. _V $.
    $( The successor of a set is a set.  (Contributed by NM, 30-Aug-1993.) $)
    sucex $p |- suc A e. _V $=
      ( cvv wcel csuc sucexg ax-mp ) ACDAECDBACFG $.
  $}

  ${
    $d x A $.
    $( The successor of an ordinal class is ordinal.  (Contributed by Jim
       Kingdon, 8-Nov-2018.) $)
    ordsucim $p |- ( Ord A -> Ord suc A ) $=
      ( vx word csuc wtr cv wral ordtr suctr syl wcel wceq wo csn df-suc eleq2i
      cun elun velsn dford3 orbi2i 3bitri wal simprbi df-ral sylib 19.21bi treq
      wi syl5ibrcom jaod biimtrid ralrimiv sylanbrc ) ACZADZEZBFZEZBUPGUPCUOAEZ
      UQAHZAIJUOUSBUPURUPKZURAKZURALZMZUOUSVBURAANZQZKVCURVFKZMVEUPVGURAOPURAVF
      RVHVDVCBASUAUBUOVCUSVDUOVCUSUIZBUOUSBAGZVIBUCUOUTVJBATUDUSBAUEUFUGUOUSVDU
      TVAURAUHUJUKULUMBUPTUN $.
  $}

  $( The successor of an ordinal number is an ordinal number.  Closed form of
     ~ onsuci .  Forward implication of ~ onsucb .  Proposition 7.24 of
     [TakeutiZaring] p. 41.  (Contributed by NM, 6-Jun-1994.) $)
  onsuc $p |- ( A e. On -> suc A e. On ) $=
    ( con0 wcel csuc word eloni ordsucim syl cvv wb sucexg elong mpbird ) ABCZA
    DZBCZOEZNAEQAFAGHNOICPQJABKOILHM $.

  $( The successor of an ordinal class is ordinal.  (Contributed by Jim
     Kingdon, 20-Nov-2018.) $)
  ordsucg $p |- ( A e. _V -> ( Ord A <-> Ord suc A ) ) $=
    ( cvv wcel word csuc ordsucim sucidg ordelord ex syl5com impbid2 ) ABCZADZA
    EZDZAFLANCZOMABGOPMNAHIJK $.

  $( A class is an ordinal number if and only if its successor is an ordinal
     number.  Biconditional form of ~ onsuc .  (Contributed by NM,
     9-Sep-2003.) $)
  onsucb $p |- ( A e. On <-> suc A e. On ) $=
    ( con0 wcel csuc onsuc word eloni wb elex sucexb sylibr elong ordsucg bitrd
    cvv syl mpbird impbii ) ABCZADZBCZAEUASTFZTGUAAOCZSUBHUATOCUCTBIAJKUCSAFUBA
    OLAMNPQR $.

  $( The successor of an element of an ordinal class is a subset of it.
     (Contributed by NM, 21-Jun-1998.) $)
  ordsucss $p |- ( Ord B -> ( A e. B -> suc A C_ B ) ) $=
    ( word wtr wcel csuc wss wi ordtr csn cun trss snssi a1i jcad unss imbitrdi
    wa df-suc sseq1i imbitrrdi syl ) BCBDZABEZAFZBGZHBIUCUDAAJZKZBGZUFUCUDABGZU
    GBGZRUIUCUDUJUKBALUDUKHUCABMNOAUGBPQUEUHBASTUAUB $.

  $( A set belongs to an ordinal iff its successor is a subset of the ordinal.
     Exercise 8 of [TakeutiZaring] p. 42 and its converse.  (Contributed by NM,
     29-Nov-2003.) $)
  ordelsuc $p |- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) ) $=
    ( wcel word wa csuc wss wi ordsucss adantl sucssel adantr impbid ) ACDZBEZF
    ABDZAGBHZPQRIOABJKORQIPABCLMN $.

  ${
    onsucssi.1 $e |- A e. On $.
    onsucssi.2 $e |- B e. On $.
    $( A set belongs to an ordinal number iff its successor is a subset of the
       ordinal number.  Exercise 8 of [TakeutiZaring] p. 42 and its converse.
       (Contributed by NM, 16-Sep-1995.) $)
    onsucssi $p |- ( A e. B <-> suc A C_ B ) $=
      ( con0 wcel word csuc wss wb onordi ordelsuc mp2an ) AEFBGABFAHBIJCBDKABE
      LM $.
  $}

  ${
    $d x A $.
    $( The successor of an ordinal number is the smallest larger ordinal
       number.  (Contributed by NM, 28-Nov-2003.) $)
    onsucmin $p |- ( A e. On -> suc A = |^| { x e. On | A e. x } ) $=
      ( con0 wcel cv crab cint csuc wss word wb ordelsuc sylan2 rabbidva inteqd
      eloni wceq onsucb intmin sylbi eqtr2d ) BCDZBAEZDZACFZGBHZUCIZACFZGZUFUBU
      EUHUBUDUGACUCCDUBUCJUDUGKUCPBUCCLMNOUBUFCDUIUFQBRAUFCSTUA $.
  $}

  $( Membership is inherited by predecessors.  The converse, for all ordinals,
     implies excluded middle, as shown at ~ onsucelsucexmid .  However, the
     converse does hold where ` B ` is a natural number, as seen at
     ~ nnsucelsuc .  (Contributed by Jim Kingdon, 17-Jul-2019.) $)
  onsucelsucr $p |- ( B e. On -> ( suc A e. suc B -> A e. B ) ) $=
    ( csuc wcel con0 cvv wi elex sucexb sylibr wa wceq wo wss onelss eqimss a1i
    jaod adantl wb elsucg sylbi adantr ordelsuc sylan2 3imtr4d impancom mpancom
    word eloni com12 ) ACZBCZDZBEDZABDZAFDZUNUOUPGUNULFDZUQULUMHAIZJUQUOUNUPUQU
    OKULBDZULBLZMZULBNZUNUPUOVBVCGUQUOUTVCVABULOVAVCGUOULBPQRSUQUNVBTZUOUQURVDU
    SULBFUAUBUCUOUQBUIUPVCTBUJABFUDUEUFUGUHUK $.

  $( The subclass relationship between two ordinals is inherited by their
     predecessors.  The converse implies excluded middle, as shown at
     ~ onsucsssucexmid .  (Contributed by Mario Carneiro and Jim Kingdon,
     29-Jul-2019.) $)
  onsucsssucr $p |- ( ( A e. On /\ Ord B ) -> ( suc A C_ suc B -> A C_ B ) ) $=
    ( con0 wcel word wa csuc wss wb ordsucim ordelsuc sylan2 wi wtr trsucss syl
    ordtr adantl sylbird ) ACDZBEZFAGBGZHZAUBDZABHZUATUBEUDUCIBJAUBCKLUAUDUEMZT
    UABNUFBQBAOPRS $.

  $( Successor and union.  The converse (where ` B ` is an ordinal) implies
     excluded middle, as seen at ~ ordsucunielexmid .  (Contributed by Jim
     Kingdon, 2-Aug-2019.) $)
  sucunielr $p |- ( suc A e. B -> A e. U. B ) $=
    ( csuc wcel cuni cvv elex sucexb sylibr sucidg syl elunii mpancom ) AACZDZN
    BDZABEDPAFDZOPNFDQNBGAHIAFJKANBLM $.

  ${
    $d x y A $.
    $( The class of all ordinal numbers is its own union.  Exercise 11 of
       [TakeutiZaring] p. 40.  (Contributed by NM, 12-Nov-2003.) $)
    unon $p |- U. On = On $=
      ( vx vy con0 cuni wcel wrex eluni2 onelon rexlimiva sylbi vex sucid onsuc
      cv csuc elunii sylancr impbii eqriv ) ACDZCANZTEZUACEZUBUABNZEZBCFUCBUACG
      UEUCBCUDUAHIJUCUAUAOZEUFCEUBUAAKLUAMUAUFCPQRS $.

    $( The union of the ordinal subsets of an ordinal number is that number.
       (Contributed by Jim Kingdon, 2-Aug-2019.) $)
    onuniss2 $p |- ( A e. On -> U. { x e. On | x C_ A } = A ) $=
      ( con0 unimax ) ABCD $.
  $}

  $( The class of ordinal numbers is a limit ordinal.  (Contributed by NM,
     24-Mar-1995.) $)
  limon $p |- Lim On $=
    ( con0 wlim word c0 wcel cuni wceq ordon 0elon unon eqcomi dflim2 mpbir3an
    ) ABACDAEAAFZGHINAJKALM $.

  ${
    $d x A $.

    $( An ordinal which contains the successor of each of its members is equal
       to its union.  (Contributed by Jim Kingdon, 14-Nov-2018.) $)
    ordunisuc2r $p |- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) ) $=
      ( word cv csuc wcel wral cuni wss wa wceq wi wal sucid elunii mpan imim2i
      vex alimi df-ral ssalel 3imtr4i a1i orduniss jctird eqss imbitrrdi ) BCZA
      DZEZBFZABGZBBHZIZUMBIZJBUMKUHULUNUOULUNLUHUIBFZUKLZAMUPUIUMFZLZAMULUNUQUS
      AUKURUPUIUJFUKURUIARNUIUJBOPQSUKABTABUMUAUBUCBUDUEBUMUFUG $.
  $}

  ${
    onssi.1 $e |- A e. On $.
    $( An ordinal number is a subset of ` On ` .  (Contributed by NM,
       11-Aug-1994.) $)
    onssi $p |- A C_ On $=
      ( con0 wcel wss onss ax-mp ) ACDACEBAFG $.

    $( The successor of an ordinal number is an ordinal number.  Inference
       associated with ~ onsuc and ~ onsucb .  Corollary 7N(c) of [Enderton]
       p. 193.  (Contributed by NM, 12-Jun-1994.) $)
    onsuci $p |- suc A e. On $=
      ( con0 wcel csuc onsuc ax-mp ) ACDAECDBAFG $.
  $}

  ${
    $d A x $.
    $( The intersection of an inhabited collection of ordinal numbers is an
       ordinal number.  Compare Exercise 6 of [TakeutiZaring] p. 44.
       (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.) $)
    onintonm $p |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On ) $=
      ( con0 wss cv wcel wex wa cint word wtr wral ssel eloni syl syl6 ralrimiv
      ordtr trint cvv nfv nfe1 nfan cuni intssuni2m unon sseqtrdi sseld ralrimi
      adantr dford3 sylanbrc wb inteximm adantl elong mpbird ) BCDZAEZBFZAGZHZB
      IZCFZVCJZVBVCKZUSKZAVCLVEURVFVAURVGABLVFURVGABURUTUSCFZVGBCUSMVHUSJZVGUSN
      ZUSRZOPQABSOUJVBVGAVCURVAAURAUAUTAUBUCVBUSVCFZVIVGVBVLVHVIVBVCCUSVBVCCUDC
      ABCUEUFUGUHVJPVKPUIAVCUKULVBVCTFZVDVEUMVAVMURABUNUOVCTUPOUQ $.
  $}

  ${
    $d ph y $.  $d x y $.
    $( An existence condition which implies an intersection is an ordinal
       number.  (Contributed by Jim Kingdon, 30-Aug-2021.) $)
    onintrab2im $p |- ( E. x e. On ph -> |^| { x e. On | ph } e. On ) $=
      ( vy con0 wrex crab wss cv wcel cint ssrab2 nfrab1 nfcri nfex rabid elex2
      wex wa sylbir ex rexlimi onintonm sylancr ) ABDEABDFZDGCHUDIZCQZUDJDIABDK
      AUFBDUEBCBCUDABDLMNBHZDIZAUFUHARUGUDIUFABDOCUGUDPSTUACUDUBUC $.
  $}

  ${
    $d x y z $.  $d ph y z $.
    $( Lemma for decidability and ordinals.  The set ` { x e. { (/) } | ph } `
       is a way of connecting statements about ordinals (such as trichotomy in
       ~ ordtriexmid or weak linearity in ~ ordsoexmid ) with a proposition
       ` ph ` .  Our lemma states that it is an ordinal number.  (Contributed
       by Jim Kingdon, 28-Jan-2019.) $)
    ordtriexmidlem $p |- { x e. { (/) } | ph } e. On $=
      ( vy vz c0 csn crab con0 wcel word wtr wss cv wa wi wal simpl mpbir ax-mp
      wceq elrabi velsn sylib noel eleq2 mtbiri syl adantl pm2.21dd gen2 ssrab2
      wn dftr2 csuc ord0 ordsucim suc0 ordeq mpbi trssord mp3an p0ex rabex elon
      wb ) ABEFZGZHIVGJZVGKZVGVFLVFJZVHVICMZDMZIZVLVGIZNZVKVGIZOZDPCPVQCDVOVMVP
      VMVNQVNVMULZVMVNVLETZVRVNVLVFIVSABVLVFUADEUBUCVSVMVKEIVKUDVLEVKUEUFUGUHUI
      UJCDVGUMRABVFUKEUNZJZVJEJWAUOEUPSVTVFTWAVJVEUQVTVFURSUSVGVFUTVAVGABVFVBVC
      VDR $.
  $}

  ${
    $d ph x $.
    $( Lemma for decidability and ordinals.  The set ` { x e. { (/) } | ph } `
       is a way of connecting statements about ordinals (such as trichotomy in
       ~ ordtriexmid or weak linearity in ~ ordsoexmid ) with a proposition
       ` ph ` .  Our lemma helps connect that set to excluded middle.
       (Contributed by Jim Kingdon, 28-Jan-2019.) $)
    ordtriexmidlem2 $p |- ( { x e. { (/) } | ph } = (/) -> -. ph ) $=
      ( c0 csn crab wceq wcel noel eleq2 mtbiri wb 0ex snid biidd elrab3 sylnib
      cv ax-mp ) ABCDZEZCFZCTGZAUAUBCCGCHTCCIJCSGUBAKCLMAABCSBQCFANORP $.
  $}

  ${
    $d x y z $.  $d x ph z $.
    ordtriexmid.1 $e |- A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) $.
    $( Ordinal trichotomy implies the law of the excluded middle (that is,
       decidability of an arbitrary proposition).

       This theorem is stated in "Constructive ordinals", [Crosilla], p.
       "Set-theoretic principles incompatible with intuitionistic logic".

       Also see ~ exmidontri which is much the same theorem but
       biconditionalized and using the ` EXMID ` notation.  (Contributed by
       Mario Carneiro and Jim Kingdon, 14-Nov-2018.) $)
    ordtriexmid $p |- ( ph \/ -. ph ) $=
      ( vz wn wo c0 wceq wcel w3o con0 cv eleq1 eleq2 3orbi123d wa wi 0ex ax-mp
      csn crab noel ordtriexmidlem eqeq1 0elon anbi2d eqeq2 imbi12d vtocl mpan2
      rspec2 vtoclga 3orass mpbi mtpor ordtriexmidlem2 snid biidd elrab3 biimpi
      wb orim12i orcom mpbir ) AAFZGVFAGZAEHUAZUBZHIZHVIJZGZVGVIHJZVLVIUCVMVJVK
      KZVMVLGVILJVNAEUDBMZHJZVOHIZHVOJZKZVNBVILVOVIIVPVMVQVJVRVKVOVIHNVOVIHUEVO
      VIHOPVOLJZHLJZVSUFVTCMZLJZQZVOWBJZVOWBIZWBVOJZKZRVTWAQZVSRCHSWBHIZWDWIWHV
      SWJWCWAVTWBHLNUGWJWEVPWFVQWGVRWBHVOOWBHVOUHWBHVONPUIWHBCLLDULUJUKUMTVMVJV
      KUNUOUPVJVFVKAAEUQVKAHVHJVKAVBHSURAAEHVHEMHIAUSUTTVAVCTAVFVDVE $.
  $}

  ${
    $d ph x y z $.
    $( Ordinal trichotomy implies excluded middle.  Closed form of
       ~ ordtriexmid .  (Contributed by Jim Kingdon, 26-Aug-2024.) $)
    ontriexmidim $p |- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x )
        -> DECID ph ) $=
      ( vz cv wcel wceq w3o con0 wral wn wo wdc csn crab eleq1 3orbi123d orcomd
      c0 eleq2 noel a1i wi ordtriexmidlem 0elon eqeq1 eqeq2 rspc2v mp2an 3orass
      sylib ecased ordtriexmidlem2 wb 0ex biidd elrab3 ax-mp biimpi orim12i syl
      snid df-dc sylibr ) BEZCEZFZVEVFGZVFVEFZHZCIJBIJZAAKZLAMVKVLAVKADSNZOZSGZ
      SVNFZLZVLALVKVQVNSFZVRKVKVNUAUBVKVRVQVKVRVOVPHZVRVQLVNIFSIFVKVSUCADUDUEVJ
      VSVNVFFZVNVFGZVFVNFZHBCVNSIIVEVNGVGVTVHWAVIWBVEVNVFPVEVNVFUFVEVNVFTQVFSGV
      TVRWAVOWBVPVFSVNTVFSVNUGVFSVNPQUHUIVRVOVPUJUKRULVOVLVPAADUMVPASVMFVPAUNSU
      OVBAADSVMDESGAUPUQURUSUTVARAVCVD $.
  $}

  ${
    $d ph x y z $.
    ordtri2orexmid.1 $e |- A. x e. On A. y e. On ( x e. y \/ y C_ x ) $.
    $( Ordinal trichotomy implies excluded middle.  (Contributed by Jim
       Kingdon, 31-Jul-2019.) $)
    ordtri2orexmid $p |- ( ph \/ -. ph ) $=
      ( vz wn wo c0 csn crab wcel wss cv con0 wral 0elon wceq orbi12d ax-mp wb
      ordtriexmidlem csuc suc0 eqeltrri eleq1 sseq2 eleq2 sseq1 rspc2va mpanl12
      onsuci elsni ordtriexmidlem2 syl snssg biidd elrab3 biimpi sylbir orim12i
      0ex snid orcom mpbi ) AFZAGZAVEGAEHIZJZVGKZVGVHLZGZVFBMZCMZKZVMVLLZGZCNOB
      NOZVKDVHNKVGNKVQVKAEUAHUBVGNUCHPUKUDVPVKVHVMKZVMVHLZGBCVHVGNNVLVHQVNVRVOV
      SVLVHVMUEVLVHVMUFRVMVGQVRVIVSVJVMVGVHUGVMVGVHUHRUIUJSVIVEVJAVIVHHQVEVHHUL
      AEUMUNVJHVHKZAHNKVTVJTPHVHNUOSVTAHVGKVTATHVAVBAAEHVGEMHQAUPUQSURUSUTSVEAV
      CVD $.
  $}

  $( Version of ~ 2on with the definition of ` 2o ` expanded and expressed in
     terms of ` Ord ` .  (Contributed by Jim Kingdon, 29-Aug-2021.) $)
  2ordpr $p |- Ord { (/) , { (/) } } $=
    ( c0 csuc word csn cpr ord0 ordsucim mp2b wceq cun df-suc suceq ax-mp df-pr
    wb suc0 3eqtr4i ordeq mpbi ) ABZBZCZAADZEZCZACTCUBFAGTGHUAUDIUBUEOUCBZUCUCD
    JUAUDUCKTUCIUAUFIPTUCLMAUCNQUAUDRMS $.

  ${
    $d ph w x y z $.
    ontr2exmid.1 $e |- A. x e. On A. y A. z e. On
      ( ( x C_ y /\ y e. z ) -> x e. z ) $.
    $( An ordinal transitivity law which implies excluded middle.  (Contributed
       by Jim Kingdon, 17-Sep-2021.) $)
    ontr2exmid $p |- ( ph \/ -. ph ) $=
      ( vw wo c0 wceq wcel wss p0ex con0 cv wa wi wral wal eleq1 ax-mp csn crab
      cpr ssrab2 prid2 word 2ordpr pp0ex elon mpbir ordtriexmidlem sseq1 anbi1d
      wn imbi12d ralbidv albidv rspcv mp2 sseq2 anbi12d imbi1d spcv eleq2 mp2an
      anbi2d elpri ordtriexmidlem2 0ex biidd rabsnt orim12i orcom mpbi ) AUNZAG
      ZAVOGAFHUAZUBZHIZVRVQIZGZVPVRHVQUCZJZWAVRVQKZVQWBJZWCAFVQUDHVQLUEWBMJZWDV
      QDNZJZOZVRWGJZPZDMQZWDWEOZWCPZWFWBUFUGWBUHUIUJVRCNZKZWOWGJZOZWJPZDMQZCRZW
      LVRMJBNZWOKZWQOZXBWGJZPZDMQZCRZBMQXAAFUKEXHXABVRMXBVRIZXGWTCXIXFWSDMXIXDW
      RXEWJXIXCWPWQXBVRWOULUMXBVRWGSUOUPUQURUSWTWLCVQLWOVQIZWSWKDMXJWRWIWJXJWPW
      DWQWHWOVQVRUTWOVQWGSVAVBUPVCTWKWNDWBMWGWBIZWIWMWJWCXKWHWEWDWGWBVQVDVFWGWB
      VRVDUOURUSVEVRHVQVGTVSVOVTAAFVHAAFVQHVIFNHIAVJVKVLTVOAVMVN $.
  $}

  ${
    $d ph x y z $.
    $( A set which is ` 2o ` if ` ph ` or ` (/) ` if ` -. ph ` is an ordinal.
       (Contributed by Jim Kingdon, 29-Aug-2021.) $)
    ordtri2or2exmidlem $p |- { x e. { (/) , { (/) } } | ph } e. On $=
      ( vy vz c0 wcel word cv wa wal wceq eleq2 adantl sylib wo vex biidd elrab
      elpr mpbir csn cpr crab con0 wtr wss wi simpll wn mtbiri pm2.21dd biimpac
      noel velsn orc sylibr syl adantlr simprbi ad2antlr sylanbrc mpjaodan gen2
      elrabi dftr2 ssrab2 2ordpr trssord mp3an pp0ex rabex elon ) ABEEUAZUBZUCZ
      UDFVOGZVOUEZVOVNUFVNGVPVQCHZDHZFZVSVOFZIZVRVOFZUGZDJCJWDCDWBVSEKZWCVSVMKZ
      WBWEIVTWCVTWAWEUHWEVTUIWBWEVTVREFVRUMVSEVRLUJMUKWBWFIVRVNFZAWCVTWFWGWAVTW
      FIZVREKZWGWHVRVMFZWIWFVTWJVSVMVRLULCEUNNWIWIVRVMKZOWGWIWKUOVREVMCPSUPUQUR
      WAAVTWFWAVSVNFZAAABVSVNBHZVSKAQRUSUTAABVRVNWMVRKAQRVAWAWEWFOZVTWAWLWNABVS
      VNVDVSEVMDPSNMVBVCCDVOVETABVNVFVGVOVNVHVIVOABVNVJVKVLT $.
  $}

  ${
    $d ph x z $.  $d x y $.
    onsucsssucexmid.1 $e |- A. x e. On A. y e. On
      ( x C_ y -> suc x C_ suc y ) $.
    $( The converse of ~ onsucsssucr implies excluded middle.  (Contributed by
       Mario Carneiro and Jim Kingdon, 29-Jul-2019.) $)
    onsucsssucexmid $p |- ( ph \/ -. ph ) $=
      ( vz wo c0 wcel wceq csuc wss con0 wi cv suceq imbi12d onsuci wa ax-mp wb
      wn crab ssrab2 ordtriexmidlem sseq1 sseq1d suc0 0elon eqeltrri p0ex eleq1
      anbi2d sseq2 sseq2d rspec2 vtocl mpan2 vtoclga word onordi ordelsuc mp2an
      csn mpbir elsucg elsni ordtriexmidlem2 syl 0ex biidd rabsnt orim12i orcom
      mpbi ) AUAZAFZAVOFAEGVCZUBZVQHZVRVQIZFZVPVRVQJZHZWAWCVRJZWBKZVRVQKZWEAEVQ
      UCVRLHZWFWEMZAEUDZBNZVQKZWJJZWBKZMZWHBVRLWJVRIZWKWFWMWEWJVRVQUEWOWLWDWBWJ
      VROUFPWJLHZVQLHZWNGJVQLUGGUHQUIZWPCNZLHZRZWJWSKZWLWSJZKZMZMWPWQRZWNMCVQUJ
      WSVQIZXAXFXEWNXGWTWQWPWSVQLUKULXGXBWKXDWMWSVQWJUMXGXCWBWLWSVQOUNPPXEBCLLD
      UOUPUQURSSWGWBUSWCWETWIWBVQWRQUTVRWBLVAVBVDWGWCWATWIVRVQLVESVNVSVOVTAVSVR
      GIVOVRGVFAEVGVHAAEVQGVIENGIAVJVKVLSVOAVMVN $.
  $}

  ${
    $d ph x y z $.
    $( Lemma for ~ onsucelsucexmid .  (Contributed by Jim Kingdon,
       2-Aug-2019.) $)
    onsucelsucexmidlem1 $p |- (/) e.
        { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } $=
      ( c0 cv wceq csn cpr crab wcel 0ex prid1 eqid eqeq1 orbi1d elrab mpbir2an
      wo orci ) CBDZCEZAQZBCCFZGZHICUCICCEZAQZCUBJKUDACLRUAUEBCUCTTUDASCCMNOP
      $.

    $( Lemma for ~ onsucelsucexmid .  The set
       ` { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ` appears as ` A ` in
       the proof of Theorem 1.3 in [Bauer] p. 483 (see ~ acexmidlema ), and
       similar sets also appear in other proofs that various propositions imply
       excluded middle, for example in ~ ordtriexmidlem .  (Contributed by Jim
       Kingdon, 2-Aug-2019.) $)
    onsucelsucexmidlem $p |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
        e. On $=
      ( vy vz cv c0 wceq wo csn cpr wcel word wa wi eleq2 adantl ex sylib mpbir
      wal crab con0 wtr wss simpll wn noel pm2.21dd biimpac onsucelsucexmidlem1
      mtbiri velsn eqeltrdi adantr elrabi elpr mpjaod gen2 dftr2 ssrab2 trssord
      vex 2ordpr mp3an pp0ex rabex elon ) BEFGAHZBFFIZJZUAZUBKVKLZVKUCZVKVJUDVJ
      LVLVMCEZDEZKZVOVKKZMZVNVKKZNZDTCTVTCDVRVOFGZVSVOVIGZVRWAVSVRWAMVPVSVPVQWA
      UEWAVPUFVRWAVPVNFKVNUGVOFVNOUKPUHQVPWBVSNVQVPWBVSVPWBMZVNFVKWCVNVIKZVNFGW
      BVPWDVOVIVNOUICFULRABUJUMQUNVQWAWBHZVPVQVOVJKWEVHBVOVJUOVOFVIDVBUPRPUQURC
      DVKUSSVHBVJUTVCVKVJVAVDVKVHBVJVEVFVGS $.
  $}

  ${
    $d ph x y z $.
    onsucelsucexmid.1 $e |- A. x e. On A. y e. On
      ( x e. y -> suc x e. suc y ) $.
    $( The converse of ~ onsucelsucr implies excluded middle.  On the other
       hand, if ` y ` is constrained to be a natural number, instead of an
       arbitrary ordinal, then the converse of ~ onsucelsucr does hold, as seen
       at ~ nnsucelsuc .  (Contributed by Jim Kingdon, 2-Aug-2019.) $)
    onsucelsucexmid $p |- ( ph \/ -. ph ) $=
      ( vz c0 csuc cv wceq wo wcel wn con0 wral suceq imbi12d eleq2 ax-mp suc0
      wi csn cpr crab onsucelsucexmidlem1 0elon onsucelsucexmidlem pm3.2i eleq1
      wa eleq1d eleq2d rspc2va mp2an elsuci wb p0ex prid2 eqeltri orbi1d elrab3
      eqeq1 wne cvv 0ex nsuceq0g df-ne mpbi pm2.53 sylbi eqeq1i mtbi elsn mtbir
      mpi mtbii olc sylibr nsyl orim12i ) FGZEHZFIZAJZEFFUAZUBZUCZKZVTWFIZJZAAL
      ZJVTWFGZKZWIFWFKZWLAEUDFMKZWFMKZUIBHZCHZKZWPGZWQGZKZTZCMNBMNWMWLTZWNWOUEA
      EUFUGDXBXCFWQKZVTWTKZTBCFWFMMWPFIZWRXDXAXEWPFWQUHXFWSVTWTWPFOUJPWQWFIZXDW
      MXEWLWQWFFQXGWTWKVTWQWFOUKPULUMRVTWFUNRWGAWHWJWGVTFIZAJZAVTWEKWGXIUOVTWDW
      ESFWDUPUQZURWCXIEVTWEWAVTIWBXHAWAVTFVAUSUTRXIXHLZAVTFVBZXKFVCKXLVDFVCVERV
      TFVFVGZXHAVHVNVIWHWDWFKZAWHWDWFIZXNLVTWDWFSVJXOWDWDKZXNXPWDFIZXHXQXMVTWDF
      SVJVKWDFUPVLVMWDWFWDQVOVIAXQAJZXNAXQVPWDWEKXNXRUOXJWCXREWDWEWAWDIWBXQAWAW
      DFVAUSUTRVQVRVSR $.
  $}

  ${
    $d ph x y a b $.
    ordsucunielexmid.1 $e |- A. x e. On A. y e. On
      ( x e. U. y -> suc x e. y ) $.
    $( The converse of ~ sucunielr (where ` B ` is an ordinal) implies excluded
       middle.  (Contributed by Jim Kingdon, 2-Aug-2019.) $)
    ordsucunielexmid $p |- ( ph \/ -. ph ) $=
      ( va vb cv wcel csuc wi con0 wa cuni wb wtr wceq word eleq2d wral imbi12d
      eloni ordtr syl unisuc sylib adantl onsuc eleq1 suceq unieq eleq2 rspc2va
      vex eleq1d mpan2 sylan2 sylbird rgen2a onsucelsucexmid ) AEFEGZFGZHZUTIZV
      AIZHZJEFKUTKHZVAKHZLVBUTVDMZHZVEVGVIVBNVFVGVHVAUTVGVAOZVHVAPVGVAQVJVAUAVA
      UBUCVAFUMUDUERUFVGVFVDKHZVIVEJZVAUGVFVKLBGZCGZMZHZVMIZVNHZJZCKSBKSVLDVSVL
      UTVOHZVCVNHZJBCUTVDKKVMUTPZVPVTVRWAVMUTVOUHWBVQVCVNVMUTUIUNTVNVDPZVTVIWAV
      EWCVOVHUTVNVDUJRVNVDVCUKTULUOUPUQURUS $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  IZF Set Theory - add the Axiom of Set Induction
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The ZF Axiom of Foundation would imply Excluded Middle
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A y z $.  $d ph x y $.
    regexmidlemm.a $e |-
      A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } $.
    $( Lemma for ~ regexmid . ` A ` is inhabited.  (Contributed by Jim Kingdon,
       3-Sep-2019.) $)
    regexmidlemm $p |- E. y y e. A $=
      ( c0 csn wcel cv wex cpr wceq wa wo p0ex prid2 eqid orci eqeq1 anbi1d
      orbi12d elrab2 mpbir2an elex2 ax-mp ) FGZDHZCIDHCJUGUFFUFKZHUFUFLZUFFLZAM
      ZNZFUFOPUIUKUFQRBIZUFLZUMFLZAMZNULBUFUHDUNUNUIUPUKUMUFUFSUNUOUJAUMUFFSTUA
      EUBUCCUFDUDUE $.

    $( Lemma for ~ regexmid .  If ` A ` has a minimal element, excluded middle
       follows.  (Contributed by Jim Kingdon, 3-Sep-2019.) $)
    regexmidlem1 $p |- ( E. y ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) ->
        ( ph \/ -. ph ) ) $=
      ( cv wcel wn wi wa wo c0 wceq eqeq1 anbi1d orbi12d elrab2 0ex eleq1 eleq2
      wal csn cpr simprbi snid mpbiri notbid imbi12d spcv syl5com prid1 mpbiran
      pm2.46 sylnbi eqid biantrur sylnibr olcd syl6 orc adantl a1d jaoi syl imp
      exlimiv ) CGZEHZDGZVHHZVJEHZIZJZDUBZKAAIZLZCVIVOVQVIVHMUCZNZVHMNZAKZLZVOV
      QJZVIVHMVRUDZHWBBGZVRNZWEMNZAKZLZWBBVHWDEWEVHNZWFVSWHWAWEVHVROWJWGVTAWEVH
      MOPQFRUEVSWCWAVSVOMEHZIZVQVSMVHHZVOWLVSWMMVRHMSUFVHVRMUAUGVNWMWLJDMSVJMNZ
      VKWMVMWLVJMVHTWNVLWKVJMETUHUIUJUKWLVPAWLMMNZAKZAWKMVRNZWPLZWPIWKMWDHWRMVR
      SULWIWRBMWDEWGWFWQWHWPWEMVROWGWGWOAWEMMOPQFRUMWQWPUNUOWOAMUPUQURUSUTWAVQV
      OAVQVTAVPVAVBVCVDVEVFVG $.

    ${
      $d A v $.  $d ph u x $.  $d u v $.
      $( Lemma for ~ reg2exmid .  If ` A ` has a minimal element (expressed by
         ` C_ ` ), excluded middle follows.  (Contributed by Jim Kingdon,
         2-Oct-2021.) $)
      reg2exmidlema $p |- ( E. u e. A A. v e. A u C_ v -> ( ph \/ -. ph ) ) $=
        ( cv wss wral wo wcel wa c0 wceq 0ex sylibr olcd eqeq1 anbi1d orbi12d
        wn csn simplr sseq1 ralbidv adantl mpbid snss ralbii noel cpr eqid jctl
        wb wi prid1 jctil elrab2 eleq2 rspcv syl com12 mtoi simprr orcd simprbi
        adantr mpjaodan rexlimiva ) DGZCGZHZCEIZAAUAZJZDEVJEKZVMLZVJMUBZNZVOVJM
        NZALZVQVSLZVNAWBMVKKZCEIZVNWBVRVKHZCEIZWDWBVMWFVPVMVSUCVSVMWFUNVQVSVLWE
        CEVJVRVKUDUEUFUGWCWECEMVKOUHUIPWDAMMKZMUJAWDWGAMEKZWDWGUOAMMVRUKZKZMVRN
        ZMMNZALZJZLWHAWNWJAWMWKAWLMULUMQMVROUPUQBGZVRNZWOMNZALZJZWNBMWIEWQWPWKW
        RWMWOMVRRWQWQWLAWOMMRSTFURPWCWGCMEVKMMUSUTVAVBVCVAQVQWALAVNVQVTAVDVEVPV
        SWAJZVMVPVJWIKWTWSWTBVJWIEWOVJNZWPVSWRWAWOVJVRRXAWQVTAWOVJMRSTFURVFVGVH
        VI $.
    $}
  $}

  ${
    $d ph x y z w $.
    regexmid.1 $e |- ( E. y y e. x ->
      E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) ) $.
    $( The axiom of foundation implies excluded middle.

       By foundation (or regularity), we mean the principle that every
       inhabited set has an element which is minimal (when arranged by
       ` e. ` ).  The statement of foundation here is taken from Metamath Proof
       Explorer's ax-reg, and is identical (modulo one unnecessary quantifier)
       to the statement of foundation in Theorem "Foundation implies instances
       of EM" of [Crosilla], p.  "Set-theoretic principles incompatible with
       intuitionistic logic".

       For this reason, IZF does not adopt foundation as an axiom and instead
       replaces it with ~ ax-setind .  (Contributed by Jim Kingdon,
       3-Sep-2019.) $)
    regexmid $p |- ( ph \/ -. ph ) $=
      ( vw cv c0 csn wceq wa wo wcel wex wel wn wi wal eleq2 exbidv pp0ex rabex
      crab regexmidlemm notbid imbi2d albidv anbi12d imbi12d vtocl regexmidlem1
      cpr eqid mp2b ) CGZFGZHIZJUPHJAKLZFHUQULZUCZMZCNZVADCOZDGZUTMZPZQZDRZKZCN
      ZAAPLAFCUTUTUMZUDCBOZCNZVLVCDBOZPZQZDRZKZCNZQVBVJQBUTURFUSUAUBBGZUTJZVMVB
      VSVJWAVLVACVTUTUOSZTWAVRVICWAVLVAVQVHWBWAVPVGDWAVOVFVCWAVNVEVTUTVDSUEUFUG
      UHTUIEUJAFCDUTVKUKUN $.
  $}

  ${
    $d ph u w z $.  $d ph u x z $.  $d ph u y z $.  $d x y $.
    reg2exmid.1 $e |- A. z ( E. w w e. z -> E. x e. z A. y e. z x C_ y ) $.
    $( If any inhabited set has a minimal element (when expressed by ` C_ ` ),
       excluded middle follows.  (Contributed by Jim Kingdon, 2-Oct-2021.) $)
    reg2exmid $p |- ( ph \/ -. ph ) $=
      ( vu cv wss c0 csn wceq wa wo wral wrex wcel wex wi ax-mp wn regexmidlemm
      cpr crab eqid wal pp0ex rabex eleq2 exbidv raleq rexeqbi1dv reg2exmidlema
      imbi12d spcv ) BHCHIZCGHZJKZLUQJLAMNZGJURUCZUDZOZBVAPZAAUANEHZVAQZERZVCAG
      EVAVAUEZUBVDDHZQZERZUPCVHOZBVHPZSZDUFVFVCSZFVMVNDVAUSGUTUGUHVHVALZVJVFVLV
      CVOVIVEEVHVAVDUIUJVKVBBVHVAUPCVHVAUKULUNUOTTAGCBVAVGUMT $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Set Induction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d y a $.  $d ph y $.
    $( Axiom of ` e. `-Induction (also known as set induction).  An axiom of
       Intuitionistic Zermelo-Fraenkel set theory.  Axiom 9 of [Crosilla] p.
       "Axioms of CZF and IZF".  This replaces the Axiom of Foundation (also
       called Regularity) from Zermelo-Fraenkel set theory.

       For more on axioms which might be adopted which are incompatible with
       this axiom (that is, Non-wellfounded Set Theory but in the absence of
       excluded middle), see Chapter 20 of [AczelRathjen], p. 183.
       (Contributed by Jim Kingdon, 19-Oct-2018.) $)
    ax-setind $a |- ( A. a ( A. y e. a [ y / a ] ph -> ph ) -> A. a ph ) $.
  $}

  ${
    $d x y S $.
    $( ` e. `-Induction in terms of membership in a class.  (Contributed by
       Mario Carneiro and Jim Kingdon, 22-Oct-2018.) $)
    setindel $p |- ( A. x ( A. y ( y e. x -> y e. S ) -> x e. S ) ->
        S = _V ) $=
      ( cv wcel wi wal cvv wceq wsb wral clelsb1 ralbii df-ral imbi1i ax-setind
      bitri albii sylbir eqv sylibr ) BDZADZEUBCEZFBGZUCCEZFZAGZUFAGZCHIUHUFABJ
      ZBUCKZUFFZAGUIULUGAUKUEUFUKUDBUCKUEUJUDBUCABCLMUDBUCNQORUFBAPSACTUA $.
  $}

  ${
    $d x y A $.
    $( Set (epsilon) induction.  Theorem 5.22 of [TakeutiZaring] p. 21.
       (Contributed by NM, 17-Sep-2003.) $)
    setind $p |- ( A. x ( x C_ A -> x e. A ) -> A = _V ) $=
      ( vy cv wss wcel wi wal cvv wceq ssalel imbi1i albii setindel sylbi ) ADZ
      BEZPBFZGZAHCDZPFTBFGCHZRGZAHBIJSUBAQUARCPBKLMACBNO $.
  $}

  ${
    $d x A $.
    $( Set (epsilon) induction, stated compactly.  Given as a homework problem
       in 1992 by George Boolos (1940-1996).  (Contributed by NM,
       17-Sep-2003.) $)
    setind2 $p |- ( ~P A C_ A -> A = _V ) $=
      ( vx cpw wss cv wcel wi wal cvv wceq pwss setind sylbi ) ACADBEZADNAFGBHA
      IJBAAKBALM $.
  $}

  ${
    $d A x y $.
    $( No class is a member of itself.  Exercise 6 of [TakeutiZaring] p. 22.

       The reason that this theorem is marked as discouraged is a bit subtle.
       If we wanted to reduce usage of ~ ax-setind , we could redefine
       ` Ord A ` ( ~ df-iord ) to also require ` _E Fr A ` ( ~ df-frind ) and
       in that case any theorem related to irreflexivity of ordinals could use
       ~ ordirr (which under that definition would presumably not need
       ~ ax-setind to prove it).  But since ordinals have not yet been defined
       that way, we cannot rely on the "don't add additional axiom use" feature
       of the minimizer to get theorems to use ~ ordirr .  To encourage
       ~ ordirr when possible, we mark this theorem as discouraged.

       (Contributed by NM, 7-Aug-1994.)  (Proof rewritten by Mario Carneiro and
       Jim Kingdon, 26-Nov-2018.)  (New usage is discouraged.) $)
    elirr $p |- -. A e. A $=
      ( vx vy wcel cvv csn cdif cv wal wsb wral wi wn neldifsnd eleq1 spcgv mpd
      wceq sylibr albii wa w3a simp1 imbi12d pm2.43b 3ad2ant2 wb eleq2 3ad2ant3
      imbi1d mpbid 3expia mtod vex eldif mpbiran xchbinx alrimiv df-ral clelsb1
      velsn ex imbi2i bitri imbi1i ax-setind syl pm2.65i ) AADZAEAFZGZDZVIBHZVK
      DZBIZVLVIVNBCJZCVMKZVNLZBIZVOVICHZVMDZVTVKDZLZCIZVNLZBIVSVIWEBVIWDVNVIWDU
      AZVMARZMVNWFWGVLWFAENVIWDWGVLVIWDWGUBZVIVLVIWDWGUCWHAVMDZVLLZVIVLLZWDVIWJ
      WGWDWIVLWCWJCAVMVTARWAWIWBVLVTAVMOVTAVKOUDPUEUFWGVIWJWKUGWDWGWIVIVLVMAAUH
      UJUIUKQULUMVNVMVJDZWGVNVMEDWLMBUNVMEVJUOUPBAVAUQSVBURVRWEBVQWDVNVQWAVPLZC
      IWDVPCVMUSWMWCCVPWBWABCVKUTVCTVDVETSVNCBVFVGVNVLBAAVMAVKOPQVIAENVH $.
  $}

  $( Epsilon irreflexivity of ordinals: no ordinal class is a member of itself.
     Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes.  The
     present proof requires ~ ax-setind .  If in the definition of ordinals
     ~ df-iord , we also required that membership be well-founded on any
     ordinal (see ~ df-frind ), then we could prove ~ ordirr without
     ~ ax-setind .  (Contributed by NM, 2-Jan-1994.) $)
  ordirr $p |- ( Ord A -> -. A e. A ) $=
    ( wcel wn word elirr a1i ) AABCADAEF $.

  ${
    onirri.1 $e |- A e. On $.
    $( An ordinal number is not a member of itself.  Theorem 7M(c) of
       [Enderton] p. 192.  (Contributed by NM, 11-Jun-1994.) $)
    onirri $p |- -. A e. A $=
      ( word wcel wn onordi ordirr ax-mp ) ACAADEABFAGH $.
  $}

  $( A member of an ordinal class is not equal to it.  (Contributed by NM,
     25-May-1998.) $)
  nordeq $p |- ( ( Ord A /\ B e. A ) -> A =/= B ) $=
    ( word wcel wne wn wceq ordirr eleq1 notbid syl5ibcom necon2ad imp ) ACZBAD
    ZABENOABNAADZFABGZOFAHQPOABAIJKLM $.

  $( An ordinal class cannot be an element of one of its members.  Variant of
     first part of Theorem 2.2(vii) of [BellMachover] p. 469.  (Contributed by
     NM, 3-Apr-1994.) $)
  ordn2lp $p |- ( Ord A -> -. ( A e. B /\ B e. A ) ) $=
    ( word wcel wa ordirr wtr wi ordtr trel syl mtod ) ACZABDBADEZAADZAFMAGNOHA
    IAABJKL $.

  $( An ordinal class and its singleton are disjoint.  (Contributed by NM,
     19-May-1998.) $)
  orddisj $p |- ( Ord A -> ( A i^i { A } ) = (/) ) $=
    ( word wcel wn csn cin c0 wceq ordirr disjsn sylibr ) ABAACDAAEFGHAIAAJK $.

  $( Ordinal derived from its successor.  (Contributed by NM, 20-May-1998.) $)
  orddif $p |- ( Ord A -> A = ( suc A \ { A } ) ) $=
    ( word csn cin wceq csuc cdif orddisj disj3 cun df-suc difeq1i difun2 eqtri
    c0 eqeq2i bitr4i sylib ) ABAACZDOEZAAFZSGZEZAHTAASGZEUCASIUBUDAUBASJZSGUDUA
    UESAKLASMNPQR $.

  $( The membership relation is irreflexive: no set is a member of itself.
     Theorem 105 of [Suppes] p. 54.  (Contributed by NM, 19-Aug-1993.) $)
  elirrv $p |- -. x e. x $=
    ( cv elirr ) ABC $.

  ${
    $d A x $.
    $( A class is equal to its successor iff it is a proper class (assuming the
       Axiom of Set Induction).  (Contributed by NM, 9-Jul-2004.) $)
    sucprcreg $p |- ( -. A e. _V <-> suc A = A ) $=
      ( vx cvv wcel wn csuc sucprc cv wi wal elirr nfv eleq1 ceqsalg mtbiri csn
      wceq velsn wss biimtrrid wo olc cun elun df-suc eqeq1i sseq1 sylbi mpbiri
      ssid wb sseld syl5 alrimiv nsyl3 impbii ) ACDZEAFZAQZAGUQBHZAQZUTADZIZBJZ
      USUQVDAADZAKVBVEBACVEBLUTAAMNOUSVCBVAUTAPZDZUSVBBARVGVBVGUAZUSVBVGVBUBVHU
      TAVFUCZDUSVBUTAVFUDUSVIAUTUSVIASZAASZAUJUSVIAQVJVKUKURVIAAUEUFVIAAUGUHUIU
      LTUMTUNUOUP $.
  $}

  $( The Russell class is equal to the universe ` _V ` .  Exercise 5 of
     [TakeutiZaring] p. 22.  (Contributed by Alan Sare, 4-Oct-2008.) $)
  ruv $p |- { x | x e/ x } = _V $=
    ( cvv weq cab cv wnel df-v equid elirrv nelir 2th abbii eqtr2i ) BAACZADAEZ
    OFZADAGNPANPAHOOAIJKLM $.

  $( Alternate proof of Russell's Paradox ~ ru , simplified using (indirectly)
     the Axiom of Set Induction ~ ax-setind .  (Contributed by Alan Sare,
     4-Oct-2008.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ruALT $p |- { x | x e/ x } e/ _V $=
    ( cv wnel cab cvv wcel wn vprc df-nel mpbir wceq wb ruv neleq1 ax-mp ) ABZP
    CADZECZEECZSEEFGHEEIJQEKRSLAMQEENOJ $.

  $( No set contains all ordinal numbers.  Proposition 7.13 of [TakeutiZaring]
     p. 38.  This is also known as the Burali-Forti paradox (remark in
     [Enderton] p. 194).  In 1897, Cesare Burali-Forti noticed that since the
     "set" of all ordinal numbers is an ordinal class ( ~ ordon ), it must be
     both an element of the set of all ordinal numbers yet greater than every
     such element.  ZF set theory resolves this paradox by not allowing the
     class of all ordinal numbers to be a set (so instead it is a proper
     class).  Here we prove the denial of its existence.  (Contributed by NM,
     18-May-1994.) $)
  onprc $p |- -. On e. _V $=
    ( con0 cvv wcel word wn ordon ordirr ax-mp elong mpbiri mto ) ABCZAACZADZME
    FAGHLMNFABIJK $.

  $( The class of all ordinal numbers is its own successor.  (Contributed by
     NM, 12-Sep-2003.) $)
  sucon $p |- suc On = On $=
    ( con0 cvv wcel wn csuc wceq onprc sucprc ax-mp ) ABCDAEAFGAHI $.

  ${
    $d A x y $.  $d B x y $.
    $( No class has 2-cycle membership loops.  Theorem 7X(b) of [Enderton]
       p. 206.  (Contributed by NM, 16-Oct-1996.)  (Proof rewritten by Mario
       Carneiro and Jim Kingdon, 27-Nov-2018.) $)
    en2lp $p |- -. ( A e. B /\ B e. A ) $=
      ( vx vy wcel wa cvv cv wal wi wceq wn elex eldif pm3.4 eleq1 spcgv adantr
      syl mpd cpr cdif wsb wral wo prid2g sylbi com12 mt2d ad2antlr w3a imbi12d
      simp1r pm2.43b 3ad2ant2 wb eleq2 imbi1d 3ad2ant3 mpbid 3expia mtod prid1g
      simp1l sylanbrc vex mpbiran elpr xchbinx sylibr ex alrimiv df-ral clelsb1
      ioran imbi2i albii bitri imbi1i ax-setind pm2.65i ) ABEZBAEZFZAGABUAZUBZE
      ZWDCHZWFEZCIZWGWDWICDUCZDWHUDZWIJZCIZWJWDDHZWHEZWOWFEZJZDIZWIJZCIWNWDWTCW
      DWSWIWDWSFZWHAKZWHBKZUEZLZWIXAXBLXCLXEXAXBBWFEZWCXFLZWBWSWCBGEZXGBAMXHXFB
      WEEZABGUFXFXHXILZXFXHXJFXHXJJBGWENXHXJOUGUHUISUJWDWSXBXFWDWSXBUKZWCXFWBWC
      WSXBUMXKBWHEZXFJZWCXFJZWSWDXMXBWSXLXFWRXMDBWHWOBKWPXLWQXFWOBWHPWOBWFPULQU
      NUOXBWDXMXNUPWSXBXLWCXFWHABUQURUSUTTVAVBXAXCWGWDWGLZWSWBXOWCWBAGEZXOABMXP
      WGAWEEZABGVCWGXPXQLZWGXPXRFXPXRJAGWENXPXROUGUHUISRZRWDWSXCWGWDWSXCUKZWBWG
      WBWCWSXCVDXTAWHEZWGJZWBWGJZWSWDYBXCWSYAWGWRYBDAWHWOAKWPYAWQWGWOAWHPWOAWFP
      ULQUNUOXCWDYBYCUPWSXCYAWBWGWHBAUQURUSUTTVAVBXBXCVOVEWIWHWEEZXDWIWHGEYDLCV
      FZWHGWENVGWHABYEVHVIVJVKVLWMWTCWLWSWIWLWPWKJZDIWSWKDWHVMYFWRDWKWQWPCDWFVN
      VPVQVRVSVQVJWIDCVTSWBWJWGJWCWIWGCABWHAWFPQRTXSWA $.
  $}

  ${
    preleq.1 $e |- A e. _V $.
    preleq.2 $e |- B e. _V $.
    preleq.3 $e |- C e. _V $.
    preleq.4 $e |- D e. _V $.
    $( Equality of two unordered pairs when one member of each pair contains
       the other member.  (Contributed by NM, 16-Oct-1996.) $)
    preleq $p |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) ->
                   ( A = C /\ B = D ) ) $=
      ( wcel wa cpr wceq wn en2lp eleq12 anbi1d mtbiri con2i adantr wo preq12b
      biimpi adantl ecased ) ABIZCDIZJZABKCDKLZJACLBDLJZADLBCLJZUGUJMUHUJUGUJUG
      DCIZUFJDCNUJUEUKUFADBCOPQRSUHUIUJTZUGUHULABCDEFGHUAUBUCUD $.

    $( Theorem for alternate representation of ordered pairs, requiring the
       Axiom of Set Induction ~ ax-setind (via the ~ preleq step).  See ~ df-op
       for a description of other ordered pair representations.  Exercise 34 of
       [Enderton] p. 207.  (Contributed by NM, 16-Oct-1996.) $)
    opthreg $p |- ( { A , { A , B } } = { C , { C , D } } <->
                   ( A = C /\ B = D ) ) $=
      ( cpr wceq wa wcel prid1 cvv prexg mp2an preleq mpanl12 preq1 eqeq1d syl
      preqr2 biimtrdi imdistani adantr preq12 preq2d eqtrd impbii ) AABIZIZCCDI
      ZIZJZACJZBDJZKZUNUOUJULJZKZUQAUJLCULLUNUSABEMCDGMAUJCULEANLBNLUJNLEFABNNO
      PGCNLDNLULNLGHCDNNOPQRUOURUPUOURCBIZULJUPUOUJUTULACBSTBDCFHUBUCUDUAUQUKCU
      JIZUMUOUKVAJUPACUJSUEUQUJULCABCDUFUGUHUI $.
  $}

  $( The successor operation behaves like a one-to-one function (assuming the
     Axiom of Set Induction).  Similar to Exercise 35 of [Enderton] p. 208 and
     its converse.  (Contributed by NM, 25-Oct-2003.) $)
  suc11g $p |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) ) $=
    ( wcel wa csuc wceq w3a wn en2lp wo sucidg syl5ibrcom elsucg sylibd 3adant1
    eleq2 imp syl5ibcom 3adant2 jca eqcom orbi2i anbi1i sylib sylibr ord 3expia
    ordir mpi suceq impbid1 ) ACEZBDEZFAGZBGZHZABHZUNUOURUSUNUOURIZBAEZABEZFZJU
    SBAKUTVCUSUTVAUSLZVBUSLZFZVCUSLUTVABAHZLZVEFVFUTVHVEUOURVHUNUOURVHUOURBUPEZ
    VHUOVIURBUQEBDMUPUQBRNBADOPSQUNURVEUOUNURVEUNURAUQEZVEUNAUPEURVJACMUPUQARTA
    BCOPSUAUBVHVDVEVGUSVABAUCUDUEUFVAVBUSUJUGUHUKUIABULUM $.

  $( The successor operation behaves like a one-to-one function.  Compare
     Exercise 16 of [Enderton] p. 194.  (Contributed by NM, 3-Sep-2003.) $)
  suc11 $p |- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) ) $=
    ( con0 suc11g ) ABCCD $.

  ${
    $d x y $.
    $( At least two sets exist (or in terms of first-order logic, the universe
       of discourse has two or more objects).  Although ~ dtruarb can also be
       summarized as "at least two sets exist", the difference is that
       ~ dtruarb shows the existence of two sets which are not equal to each
       other, but this theorem says that given a specific ` y ` , we can
       construct a set ` x ` which does not equal it.  (Contributed by Jim
       Kingdon, 29-Dec-2018.) $)
    dtruex $p |- E. x -. x = y $=
      ( cv wceq wn wex csn vex snex isseti wa wcel elirrv vsnid mpbiri mto eqtr
      eleq2 ancom mtbi imnani eximii equcom notbii exbii mpbi ) BCZACZDZEZAFUHU
      GDZEZAFUHUGGZDZUJAAUMUGBHIJUNUIUIUNKZUNUIKUOUGUMDZUPUGUGLZBMUPUQUGUMLBNUG
      UMUGROPUGUHUMQPUIUNSTUAUBUJULAUIUKBAUCUDUEUF $.
  $}

  ${
    $d x y $.
    $( At least two sets exist (or in terms of first-order logic, the universe
       of discourse has two or more objects).  If we assumed the law of the
       excluded middle this would be equivalent to ~ dtruex .  (Contributed by
       Jim Kingdon, 29-Dec-2018.) $)
    dtru $p |- -. A. x x = y $=
      ( weq wn wex wal dtruex exnalim ax-mp ) ABCZDAEJAFDABGJAHI $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( Existential uniqueness implies there is a value for which the wff
       argument is false.  (Contributed by Jim Kingdon, 29-Dec-2018.) $)
    eunex $p |- ( E! x ph -> E. x -. ph ) $=
      ( vy weu wex cv wceq wi wal wa wn nfv eu3 dtruex nfa1 con3d eximd exlimiv
      sp mpi adantl sylbi ) ABDABEZABFCFGZHZBIZCEZJAKZBEZABCACLMUGUIUCUFUICUFUD
      KZBEUIBCNUFUJUHBUEBOUFAUDUEBSPQTRUAUB $.
  $}

  ${
    $d x y z w ph $.
    ordsoexmid.1 $e |- _E Or On $.
    $( Weak linearity of ordinals implies the law of the excluded middle (that
       is, decidability of an arbitrary proposition).  (Contributed by Mario
       Carneiro and Jim Kingdon, 29-Jan-2019.) $)
    ordsoexmid $p |- ( ph \/ -. ph ) $=
      ( vw vx vy vz wo c0 wcel con0 wi cv w3a wceq eleq1 imbi12d eleq2 cep wral
      wbr wn csn crab csuc ordtriexmidlem elexi sucid onsuci suc0 0elon 3anbi1d
      eqeltrri orbi1d 3anbi2d orbi2d p0ex 3anbi3d orbi12d imbi2d wpo wor df-iso
      wa mpbi simpri epel orbi12i imbi12i 2ralbii ralbii rspec3 ordtriexmidlem2
      vtocl mp3an sylbi elirr elrabi mto elsuci ord mpi 0ex biidd rabsnt eqcoms
      elsn syl orim12i mp2b orcom ) AUAZAGZAWKGACHUBZUCZWNUDZIZWNWMIZWMWOIZGZWL
      WNWNJACUEZUFZUGWNJIZWOJIZWMJIZWPWSKZWTWNWTUHZHUDWMJUIHUJUHULDLZJIZXCXDMZX
      GWOIZXGWMIZWRGZKZKZXBXCXDMZXEKDWNXAXGWNNZXIXOXMXEXPXHXBXCXDXGWNJOUKXPXJWP
      XLWSXGWNWOOXPXKWQWRXGWNWMOUMPPXHELZJIZXDMZXGXQIZXKWMXQIZGZKZKZXNEWOWOJXFU
      FXQWONZXSXIYCXMYEXRXCXHXDXQWOJOUNYEXTXJYBXLXQWOXGQYEYAWRXKXQWOWMQUOPPXHXR
      FLZJIZMZXTXGYFIZYFXQIZGZKZKYDFWMUPYFWMNZYHXSYLYCYMYGXDXHXRYFWMJOUQYMYKYBX
      TYMYIXKYJYAYFWMXGQYFWMXQOURUSPYLDEFJJJXGXQRTZXGYFRTZYFXQRTZGZKZFJSEJSZDJS
      ZYLFJSEJSZDJSJRUTZYTJRVAUUBYTVCBDEFJRVBVDVEYSUUADJYRYLEFJJYNXTYQYKDEVFYOY
      IYPYJDFVFFEVFVGVHVIVJVDVKVMVMVMVNWQWKWRAWQWNHNWKWNHXAWFACVLVOWRWMWNNZAWRW
      MWNIZUAUUCUUDWMWMIWMVPACWMWMVQVRWRUUDUUCWMWNVSVTWAAWNWMAACWMHWBCLHNAWCWDW
      EWGWHWIWKAWJVD $.
  $}

  ${
    $d A x y $.
    $( The successor of an ordinal class is ordinal.  (Contributed by NM,
       3-Apr-1995.)  (Constructive proof by Mario Carneiro and Jim Kingdon,
       20-Jul-2019.) $)
    ordsuc $p |- ( Ord A <-> Ord suc A ) $=
      ( vy vx word csuc ordsucim wtr cv wcel wa wi wal wceq en2lp eleq1 biimpac
      wn anim2i expr wss mtoi adantl elelsuc adantr ordelss sylan2 pm2.43d impr
      wo sseld elsuci syl ecased ancom2s alrimivv sylibr sssucid trssord mp3an2
      ex dftr2 mpancom impbii ) ADZAEZDZAFAGZVFVDVFBHZCHZIZVIAIZJZVHAIZKZCLBLVG
      VFVNBCVFVLVMVFVKVJVMVFVKVJJZJZVMVHAMZVOVQQVFVOVQVKAVIIZJZVIANVKVJVQVSVJVQ
      JVRVKVQVJVRVHAVIOPRSUAUBVPVHVEIZVMVQUIVFVKVJVTVFVKJVJVTVFVKVJVJVTKVPVIVEV
      HVOVFVIVEIZVIVETVKWAVJVIAUCUDVEVIUEUFUJSUGUHVHAUKULUMUNUTUOBCAVAUPVGAVETV
      FVDAUQAVEURUSVBVC $.
  $}

  $( A successor ordinal is the successor of its union.  (Contributed by NM,
     10-Dec-2004.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  onsucuni2 $p |- ( ( A e. On /\ A = suc B ) -> suc U. A = A ) $=
    ( con0 wcel csuc wceq wa cuni eleq1 biimpac onsucb wtr word eloni ordtr syl
    unisucg mpbid suceq sylbir eqtr4d unieq unieqd eqeq12d anabsi7 adantr eqtrd
    imbitrrid ) ACDZABEZFZGZAHZEZAEZHZAUIUKUNUPFZULUQUKUJHZEZUJEZHZFZULUJCDZVBU
    KUIVCAUJCIJVCUSUJVAVCBCDZUSUJFZBKVDURBFZVEVDBLZVFVDBMVGBNBOPBCQRURBSPTVCUJL
    ZVAUJFVCUJMVHUJNUJOPUJCQRUAPUKUNUSUPVAUKUMURFUNUSFAUJUBUMURSPUKUOUTAUJSUCUD
    UHUEUIUPAFZUKUIALZVIUIAMVJANAOPACQRUFUG $.

  ${
    $d ph x y $.
    0elsucexmid.1 $e |- A. x e. On (/) e. suc x $.
    $( If the successor of any ordinal class contains the empty set, excluded
       middle follows.  (Contributed by Jim Kingdon, 3-Sep-2021.) $)
    0elsucexmid $p |- ( ph \/ -. ph ) $=
      ( vy c0 csn crab wcel wceq wo wn csuc con0 cv ordtriexmidlem suceq eleq2d
      wral 0ex ax-mp rspcv mp2 elsuc mpbi wb snid elrab3 biimpi ordtriexmidlem2
      biidd eqcoms orim12i ) EADEFZGZHZEUNIZJZAAKZJEUNLZHZUQUNMHEBNZLZHZBMRUTAD
      OCVCUTBUNMVAUNIVBUSEVAUNPQUAUBEUNSUCUDUOAUPURUOAEUMHUOAUEESUFAADEUMDNEIAU
      JUGTUHURUNEADUIUKULT $.
  $}

  $( A successor is not a limit ordinal.  (Contributed by NM, 25-Mar-1995.)
     (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  nlimsucg $p |- ( A e. V -> -. Lim suc A ) $=
    ( wcel csuc wlim wn word cuni wceq wa limord ordsuc sylibr limuni jca ordtr
    wtr unisucg biimpa sylan2 eqeq2d ordirr eleq2 notbid syl5ibrcom sucidg syl6
    wi con3i adantl sylbid expimpd syl5 con2d pm2.43i ) ABCZADZEZFUPURUPURAGZUQ
    UQHZIZJUPUPFZURUSVAURUQGUSUQKALMUQNOUPUSVAVBUPUSJZVAUQAIZVBVCUTAUQUSUPAQZUT
    AIZAPUPVEVFABRSTUAUSVDVBUHUPUSVDAUQCZFZVBUSVHVDAACZFAUBVDVGVIUQAAUCUDUEUPVG
    ABUFUIUGUJUKULUMUNUO $.

  ${
    $d x A $.
    $( The collection of ordinals in the power class of an ordinal is a
       superset of its successor.

       We can think of ` ( ~P A i^i On ) ` as another possible definition of
       successor, which would be equivalent to ~ df-suc given excluded middle.
       It is an ordinal, and has some successor-like properties.  For example,
       if ` A e. On ` then both ` U. suc A = A ` ( ~ onunisuci ) and
       ` U. { x e. On | x C_ A } = A ` ( ~ onuniss2 ).

       Constructively ` ( ~P A i^i On ) ` and ` suc A ` cannot be shown to be
       equivalent (as proved at ~ ordpwsucexmid ).  (Contributed by Jim
       Kingdon, 21-Jul-2019.) $)
    ordpwsucss $p |- ( Ord A -> suc A C_ ( ~P A i^i On ) ) $=
      ( vx word csuc cpw con0 cin cv wcel wss wa wi ordsuc ordelon ex sylbi wtr
      ordtr trsucss syl jcad elin velpw anbi2ci bitri imbitrrdi ssrdv ) ACZBADZ
      AEZFGZUHBHZUIIZULFIZULAJZKZULUKIZUHUMUNUOUHUICZUMUNLAMURUMUNUIULNOPUHAQUM
      UOLARAULSTUAUQULUJIZUNKUPULUJFUBUSUOUNBAUCUDUEUFUG $.
  $}

  $( No member of a set of ordinal numbers belongs to its minimum.
     (Contributed by NM, 2-Feb-1997.)  (Constructive proof by Mario Carneiro
     and Jim Kingdon, 21-Jul-2019.) $)
  onnmin $p |- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A ) $=
    ( wcel cint wn con0 wss intss1 elirr ssel mtoi syl adantl ) BACZBADZCZEZAFG
    NOBGZQBAHRPBBCBIOBBJKLM $.

  $( Relationship between subset and elementhood.  In the context of ordinals
     this can be seen as an ordering law.  (Contributed by Jim Kingdon,
     22-Jul-2019.) $)
  ssnel $p |- ( A C_ B -> -. B e. A ) $=
    ( wss wcel elirr ssel mtoi ) ABCBADBBDBEABBFG $.

  ${
    $d ph x z $.
    ordpwsucexmid.1 $e |- A. x e. On suc x = ( ~P x i^i On ) $.
    $( The subset in ~ ordpwsucss cannot be equality.  That is, strengthening
       it to equality implies excluded middle.  (Contributed by Jim Kingdon,
       30-Jul-2019.) $)
    ordpwsucexmid $p |- ( ph \/ -. ph ) $=
      ( vz c0 csn crab wcel wceq wo wn csuc cpw con0 cin 0elpw 0elon elin ax-mp
      cv mpbir2an ordtriexmidlem pweq ineq1d eqeq12d vtoclri eleqtrri elsuci wb
      suceq 0ex snid biidd elrab3 biimpi ordtriexmidlem2 eqcoms orim12i ) EADEF
      ZGZHZEUTIZJZAAKZJEUTLZHVCEUTMZNOZVEEVGHEVFHENHUTPQEVFNRUAUTNHVEVGIZADUBBT
      ZLZVIMZNOZIVHBUTNVIUTIZVJVEVLVGVIUTUJVMVKVFNVIUTUCUDUECUFSUGEUTUHSVAAVBVD
      VAAEUSHVAAUIEUKULAADEUSDTEIAUMUNSUOVDUTEADUPUQURS $.
  $}

  ${
    $d ph x y z $.
    ordtri2or2exmid.1 $e |- A. x e. On A. y e. On ( x C_ y \/ y C_ x ) $.
    $( Ordinal trichotomy implies excluded middle.  (Contributed by Jim
       Kingdon, 29-Aug-2021.) $)
    ordtri2or2exmid $p |- ( ph \/ -. ph ) $=
      ( vz wo c0 wss cv con0 wral wcel 0elon sseq1 sseq2 orbi12d ax-mp wb biidd
      wceq wn csn cpr crab ordtri2or2exmidlem csuc suc0 onsuci eqeltrri rspc2va
      mpanl12 elirr wa simpl simpr p0ex prid2 elrab3 sylibr sseldd ex snssg 0ex
      mtoi prid1 biimpi sylbir orim12i orcom mpbi ) AUAZAFZAVKFAEGGUBZUCZUDZVMH
      ZVMVOHZFZVLBIZCIZHZVTVSHZFZCJKBJKZVRDVOJLVMJLWDVRAEUEGUFVMJUGGMUHUIWCVRVO
      VTHZVTVOHZFBCVOVMJJVSVOTWAWEWBWFVSVOVTNVSVOVTOPVTVMTWEVPWFVQVTVMVOOVTVMVO
      NPUJUKQVPVKVQAVPAVMVMLZVMULVPAWGVPAUMZVOVMVMVPAUNWHAVMVOLZVPAUOVMVNLWIARG
      VMUPUQAAEVMVNEIZVMTASURQUSUTVAVDVQGVOLZAGJLWKVQRMGVOJVBQWKAGVNLWKARGVMVCV
      EAAEGVNWJGTASURQVFVGVHQVKAVIVJ $.
  $}

  ${
    $d ph x y z $.
    $( Ordinal trichotomy implies excluded middle.  Closed form of
       ~ ordtri2or2exmid .  (Contributed by Jim Kingdon, 26-Aug-2024.) $)
    ontri2orexmidim $p |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x )
        -> DECID ph ) $=
      ( vz cv wss wo con0 wral c0 wcel 0elon wceq sseq1 sseq2 orbi12d wb elrab3
      biidd ax-mp wdc csn cpr crab ordtri2or2exmidlem csuc suc0 onsuci eqeltrri
      wn rspc2va mpanl12 onirri wa simpl simpr p0ex prid2 sylibr sseldd ex mtoi
      snssg 0ex prid1 sylbb1 orim12i syl orcomd df-dc ) BEZCEZFZVLVKFZGZCHIBHIZ
      AAUJZGAUAVPVQAVPADJJUBZUCZUDZVRFZVRVTFZGZVQAGVTHKVRHKVPWCADUEJUFVRHUGJLUH
      UIZVOWCVTVLFZVLVTFZGBCVTVRHHVKVTMVMWEVNWFVKVTVLNVKVTVLOPVLVRMWEWAWFWBVLVR
      VTOVLVRVTNPUKULWAVQWBAWAAVRVRKZVRWDUMWAAWGWAAUNZVTVRVRWAAUOWHAVRVTKZWAAUP
      VRVSKWIAQJVRUQURAADVRVSDEZVRMASRTUSUTVAVBJVTKZWBAJHKWKWBQLJVTHVCTJVSKWKAQ
      JVRVDVEAADJVSWJJMASRTVFVGVHVIAVJUS $.
  $}

  ${
    $d ph u v $.  $d x y u v $.
    onintexmid.onint $e |- ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y ) $.
    $( If the intersection (infimum) of an inhabited class of ordinal numbers
       belongs to the class, excluded middle follows.  The hypothesis would be
       provable given excluded middle.  (Contributed by Mario Carneiro and Jim
       Kingdon, 29-Aug-2021.) $)
    onintexmid $p |- ( ph \/ -. ph ) $=
      ( vu vv cv wss wo con0 wcel wa cint wceq wex wi cin eqeq1i dfss1 vex prmg
      cpr prssi adantr zfpair2 sseq1 eleq2 exbidv anbi12d inteq eleq12d imbi12d
      vtocl syl2anc elpri syl incom intpr 3bitr4ri bitr4i sylib ordtri2or2exmid
      id orbi12i rgen2a ) AEFEGZFGZHZVGVFHZIZEFJVFJKZVGJKZLZVFVGUBZMZVFNZVOVGNZ
      IZVJVMVOVNKZVRVMVNJHZBGZVNKZBOZVSVFVGJUCVKWCVLBVFVGJUAUDCGZJHZWAWDKZBOZLZ
      WDMZWDKZPVTWCLZVSPCVNEFUEWDVNNZWHWKWJVSWLWEVTWGWCWDVNJUFWLWFWBBWDVNWAUGUH
      UIWLWIVOWDVNWDVNUJWLVCUKULDUMUNVOVFVGUOUPVPVHVQVIVGVFQZVFNVFVGQZVFNVHVPWM
      WNVFVGVFUQRVFVGSVOWNVFVFVGETFTURZRUSVQWNVGNVIVOWNVGWORVGVFSUTVDVAVEVB $.
  $}

  ${
    $d A s x y $.
    $( The epsilon relation is well-founded on any class.  (Contributed by NM,
       26-Nov-1995.) $)
    zfregfr $p |- _E Fr A $=
      ( vs vy vx cep wfr cv wfrfor df-frind wel wi wral wcel wsb bitr4i clelsb1
      wal bitri ralbii imbi1i wbr bi2.04 albii df-ral sbim imbi12i ralcom3 epel
      wss ax-setind ssalel sylibr sylbir df-frfor mpbir mpgbir ) AEFAEBGZHZBAEB
      IURCGZDGZEUAZCBJZKZCALZDBJZKZDALZAUQUIZKVGUTAMZVEKZDCNZCUTLZVJKZDQZVHVNVL
      VEKZDALZVGVNVIVOKZDQVPVMVQDVLVIVEUBUCVODAUDOVOVFDAVLVDVEVLCDJZVBKZCALZVDV
      LUSAMZVBKZCUTLVTVKWBCUTVKVIDCNZVEDCNZKWBVIVEDCUEWCWAWDVBDCAPDCUQPUFRSVBCU
      TAUGRVCVSCAVAVRVBCDUHTSOTSRVNVJDQVHVJCDUJDAUQUKULUMDCAEUQUNUOUP $.
  $}

  $( Epsilon is well-founded on an ordinal class.  (Contributed by NM,
     22-Apr-1994.) $)
  ordfr $p |- ( Ord A -> _E Fr A ) $=
    ( cep wfr word zfregfr a1i ) ABCADAEF $.

  ${
    $d A x y z $.
    $( Epsilon well-orders every ordinal.  Proposition 7.4 of [TakeutiZaring]
       p. 36.  (Contributed by NM, 3-Apr-1994.) $)
    ordwe $p |- ( Ord A -> _E We A ) $=
      ( vx vy vz word cep wfr cv wbr wa wi wral wwe wcel w3a ordelord 3ad2antr3
      ordfr wel epel ordtr1 anbi12i 3imtr4g syl ralrimivvva df-wetr sylanbrc )
      AEZAFGBHZCHZFIZUJDHZFIZJZUIULFIZKZDALCALBALAFMARUHUPBCDAAAUHUIANZUJANZULA
      NZOJULEZUPUHUQUSUTURAULPQUTBCSZCDSZJBDSUNUOUIUJULUAUKVAUMVBBCTCDTUBBDTUCU
      DUEBCDAFUFUG $.
  $}

  ${
    $d A a b $.  $d A z $.  $d B a b $.  $d B z $.  $d C b $.  $d C z $.
    $d R a b $.  $d R z $.
    wetriext.we $e |- ( ph -> R We A ) $.
    wetriext.a $e |- ( ph -> A e. V ) $.
    wetriext.tri $e |- ( ph ->
      A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) ) $.
    wetriext.b $e |- ( ph -> B e. A ) $.
    wetriext.c $e |- ( ph -> C e. A ) $.
    wetriext.ext $e |- ( ph -> A. z e. A ( z R B <-> z R C ) ) $.
    $( A trichotomous well-order is extensional.  (Contributed by Jim Kingdon,
       26-Sep-2021.) $)
    wetriext $p |- ( ph -> B = C ) $=
      ( wbr wceq cv wb breq1 wa bibi12d rspcdva biimpar wn wfr wcel wefr frirrg
      wwe syl syl3anc adantr pm2.21dd simpr biimpa w3o wi eqeq1 breq2 3orbi123d
      wral eqeq2 rspc2v syl2anc mpd mpjao3dan ) ADEFPZDEQZVIEDFPZAVHUADDFPZVIAV
      KVHABRZDFPZVLEFPZSZVKVHSBCDVLDQVMVKVNVHVLDDFTVLDEFTUBOMUCUDAVKUEZVHACFUFZ
      CGUGZDCUGZVPACFUJVQJCFUHUKZKMCDFGUIULUMUNAVIUOAVJUAEEFPZVIAVJWAAVOVJWASBC
      EVLEQVMVJVNWAVLEDFTVLEEFTUBONUCUPAWAUEZVJAVQVRECUGZWBVTKNCEFGUIULUMUNAHRZ
      IRZFPZWDWEQZWEWDFPZUQZICVBHCVBZVHVIVJUQZLAVSWCWJWKURMNWIWKDWEFPZDWEQZWEDF
      PZUQHIDECCWDDQWFWLWGWMWHWNWDDWEFTWDDWEUSWDDWEFUTVAWEEQWLVHWMVIWNVJWEEDFUT
      WEEDVCWEEDFTVAVDVEVFVG $.
  $}

  ${
    $d A x y z $.  $d B x y z $.
    $( A subset of a set well-ordered by set membership is well-ordered by set
       membership.  (Contributed by Jim Kingdon, 30-Sep-2021.) $)
    wessep $p |- ( ( _E We A /\ B C_ A ) -> _E We B ) $=
      ( vx vy vz cep wwe wss wa cv wbr wral wcel w3a ssel 3anim123d adantl epel
      wi wel wetrep adantlr anbi12i 3imtr4g syl ralrimivvva wfr zfregfr df-wetr
      imdistani mpbiran sylibr ) AFGZBAHZIZCJZDJZFKZUQEJZFKZIZUPUSFKZSZEBLDBLCB
      LZBFGZUOVCCDEBBBUOUPBMZUQBMZUSBMZNZIUOUPAMZUQAMZUSAMZNZIZVCUOVIVMUNVIVMSU
      MUNVFVJVGVKVHVLBAUPOBAUQOBAUSOPQUJVNCDTZDETZIZCETZVAVBUMVMVQVRSUNCDEAUAUB
      URVOUTVPCDRDERUCCERUDUEUFVEBFUGVDBUHCDEBFUIUKUL $.
  $}

  ${
    $d A a b c $.  $d b c x $.  $d ph x $.
    reg3exmidlemwe.a $e |-
      A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } $.
    $( Lemma for ~ reg3exmid .  Our counterexample ` A ` satisfies ` We ` .
       (Contributed by Jim Kingdon, 3-Oct-2021.) $)
    reg3exmidlemwe $p |- _E We A $=
      ( va vb vc cep cv wbr wa wral wcel epel c0 wceq noel eleq2 wo adantr crab
      wwe wfr wi zfregfr w3a anbi12i simpr csn elirr simprr mtbiri simprl nsyl3
      wn cpr elrabi eleq2s elpri syl orcomd 3ad2ant2 ecased elpr sylib 3ad2ant3
      vex 3eltr3d ex mtoi pm2.21dd sylan2b rgen3 df-wetr mpbir2an ) CHUBCHUCEIZ
      FIZHJZVQGIZHJZKZVPVSHJZUDZGCLFCLECLCUEWCEFGCCCVPCMZVQCMZVSCMZUFZWAWBWAWGV
      PVQMZVQVSMZKZWBVRWHVTWIEFNFGNUGWGWJKZWJWBWGWJUHWGWJUOWJWGWJOUIZWLMZWLUJWG
      WJWMWKVQVSWLWLWGWHWIUKZWKVQWLPZVQOPZWPWHWKWPWHVPOMVPQVQOVPRULWGWHWIUMUNWG
      WOWPSZWJWEWDWQWFWEWPWOWEVQOWLUPZMZWPWOSWSVQBIZWLPWTOPAKSZBWRUAZCXABVQWRUQ
      DURVQOWLUSUTVAVBTVCWKVSWLPZVSOPZXDWIWKXDWIVQOMVQQVSOVQRULWNUNWGXCXDSZWJWF
      WDXEWEWFXDXCWFVSWRMZXDXCSXFVSXBCXABVSWRUQDURVSOWLGVGVDVEVAVFTVCVHVIVJTVKV
      LVIVMEFGCHVNVO $.
  $}

  ${
    $d ph u w z $.  $d ph u x y z $.
    reg3exmid.1 $e |- ( ( _E We z /\ E. w w e. z ) ->
        E. x e. z A. y e. z x C_ y ) $.
    $( If any inhabited set satisfying ~ df-wetr for ` _E ` has a minimal
       element, excluded middle follows.  (Contributed by Jim Kingdon,
       3-Oct-2021.) $)
    reg3exmid $p |- ( ph \/ -. ph ) $=
      ( vu cv c0 csn wceq wa wo wcel wex wral wrex cep wwe wi cpr crab wss eqid
      wn regexmidlemm reg3exmidlemwe pp0ex rabex weeq2 eleq2 anbi12d rexeqbi1dv
      exbidv raleq imbi12d vtocl mpan reg2exmidlema mp2b ) EHZGHZIJZKVBIKALMZGI
      VCUAZUBZNZEOZBHCHUCZCVFPZBVFQZAAUEMAGEVFVFUDZUFVFRSZVHVKAGVFVLUGDHZRSZVAV
      NNZEOZLZVICVNPZBVNQZTVMVHLZVKTDVFVDGVEUHUIVNVFKZVRWAVTVKWBVOVMVQVHVNVFRUJ
      WBVPVGEVNVFVAUKUNULVSVJBVNVFVICVNVFUOUMUPFUQURAGCBVFVLUSUT $.
  $}

  ${
    $d ph x y $.
    dcextest.ex $e |- DECID { x | ph } e. _V $.
    $( If it is decidable whether ` { x | ph } ` is a set, then ` -. ph ` is
       decidable (where ` x ` does not occur in ` ph ` ).  From this fact, we
       can deduce (outside the formal system, since we cannot quantify over
       classes) that if it is decidable whether any class is a set, then "weak
       excluded middle" (that is, any negated proposition ` -. ph ` is
       decidable) holds.  (Contributed by Jim Kingdon, 3-Jul-2022.) $)
    dcextest $p |- DECID -. ph $=
      ( vy wn wdc wo cab cvv wcel exmiddc ax-mp vprc weq df-v wb pm5.1im notbii
      wi equid abbidv eqtr2id eleq1d mtbiri con2i c0 cv vex elab biimpri eq0rdv
      biidd 0ex eqeltrdi impbii orbi12i mpbi df-dc mpbir ) AEZFUTUTEZGZABHZIJZV
      DEZGZVBVDFVFCVDKLVDUTVEVAVDUTAVDAVDIIJMAVCIIAIBBNZBHVCBOAVGABVGAVGAPSBTVG
      AQLUAUBUCUDUEUTVCUFIUTDVCDUGZVCJZEUTVIAAABVHDUHBDNAULUIRUJUKUMUNUOZVDUTVJ
      RUPUQUTURUS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Transfinite induction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y A $.
    $( The Principle of Transfinite Induction.  Theorem 7.17 of [TakeutiZaring]
       p. 39.  This principle states that if ` A ` is a class of ordinal
       numbers with the property that every ordinal number included in ` A `
       also belongs to ` A ` , then every ordinal number is in ` A ` .

       (Contributed by NM, 18-Feb-2004.) $)
    tfi $p |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On ) $=
      ( vy con0 wss cv wcel wi wral wa wal df-ral albii bitri ssalel imbi1i spi
      imim1i impexp wsb wceq imdi imbi2i 19.21v bitr4i simpl wtr tron dftr2 jca
      mpbi bi2.04 3imtr3i alimi sylbi adantl sbim clelsb1 imbi12i ralbii sylbir
      ax-setind sylibr syl eqss biimpri syldan ) BDEZAFZBEZVIBGZHZADIZDBEZBDUAZ
      VHVMJCFZVIGZVPDGZVPBGZHZHZCKZVIDGZVKHZHZAKZVNVMWFVHVMWCVQVSHZHZCKZWDHZAKZ
      WFVMWCVJHZWDHZAKZWKVMWCVLHZAKWNVLADLWOWMAWCVJVKUBMNWMWJAWLWIWDWLWCWGCKZHW
      IVJWPWCCVIBOUCWCWGCUDUEPMNWJWEAWBWIWDWAWHCVQVRJZVSHVQWCJZVSHZWAWHWRWQVSWR
      VQVRVQWCUFWRVRHZAWTAKZCDUGXACKUHCADUIUKQQUJRVQVRVSSWSVQWCVSHHWHVQWCVSSVQW
      CVSULNUMUNRUNUOUPWFWDAKZVNWFWDACTZCVIIZWDHZAKXBXEWEAXDWBWDXDVTCVIIWBXCVTC
      VIXCWCACTZVKACTZHVTWCVKACUQXFVRXGVSACDURACBURUSNUTVTCVILNPMWDCAVBVAADBOVC
      VDVOVHVNJBDVEVFVG $.
  $}

  ${
    $d w y z ph $.  $d w x y z $.
    tfis.1 $e |- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) ) $.
    $( Transfinite Induction Schema.  If all ordinal numbers less than a given
       number ` x ` have a property (induction hypothesis), then all ordinal
       numbers have the property (conclusion).  Exercise 25 of [Enderton]
       p. 200.  (Contributed by NM, 1-Aug-1994.)  (Revised by Mario Carneiro,
       20-Nov-2016.) $)
    tfis $p |- ( x e. On -> ph ) $=
      ( vz vw cv con0 wcel crab wss wi wral wceq ssrab2 wa nfcv bitr3id simprbi
      wsb nfrab1 nfss nfcri dfss3 sseq1 rabid eleq1 imbi12d sbequ nfs1v sbequ12
      nfim nfv cbvrab elrab2 ralimi syl5 anc2li vtoclgaf rgen tfi mp2an rabeq2i
      eqcomi ) BGZHIZVFAABHHABHJZHVGHKEGZVGKZVHVGIZLZEHMVGHNABHOVKEHCGZVGIZCVEM
      ZVFAPZLVKBVHHBVHQZVIVJBBVHVGVPABHUAZUBBEVGVQUCULVEVHNZVNVIVOVJVNVEVGKVRVI
      CVEVGUDVEVHVGUERVOVEVGIVRVJABHUFVEVHVGUGRUHVFVNAVNABCTZCVEMVFAVMVSCVEVMVL
      HIVSABFTZVSFVLHVGAFCBUIAVTBFHBHQFHQAFUMABFUJABFUKUNUOSUPDUQURUSUTEVGVAVBV
      DVCS $.
  $}

  ${
    $d y ph $.  $d x y $.
    tfis2f.1 $e |- F/ x ps $.
    tfis2f.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    tfis2f.3 $e |- ( x e. On -> ( A. y e. x ps -> ph ) ) $.
    $( Transfinite Induction Schema, using implicit substitution.  (Contributed
       by NM, 18-Aug-1994.) $)
    tfis2f $p |- ( x e. On -> ph ) $=
      ( wsb cv wral con0 wcel sbie ralbii biimtrid tfis ) ACDACDHZDCIZJBDRJRKLA
      QBDRABCDEFMNGOP $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x y $.
    tfis2.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    tfis2.2 $e |- ( x e. On -> ( A. y e. x ps -> ph ) ) $.
    $( Transfinite Induction Schema, using implicit substitution.  (Contributed
       by NM, 18-Aug-1994.) $)
    tfis2 $p |- ( x e. On -> ph ) $=
      ( nfv tfis2f ) ABCDBCGEFH $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x ch $.  $d x A $.  $d x y $.
    tfis3.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    tfis3.2 $e |- ( x = A -> ( ph <-> ch ) ) $.
    tfis3.3 $e |- ( x e. On -> ( A. y e. x ps -> ph ) ) $.
    $( Transfinite Induction Schema, using implicit substitution.  (Contributed
       by NM, 4-Nov-2003.) $)
    tfis3 $p |- ( A e. On -> ch ) $=
      ( con0 tfis2 vtoclga ) ACDFJHABDEGIKL $.
  $}

  ${
    $d x v w y z T $.  $d v w y z R $.  $d x v w z S $.  $d x v w z ch $.
    $d x v w y z ph $.  $d w y z ps $.  $d x A $.  $d x th $.
    tfisi.a $e |- ( ph -> A e. V ) $.
    tfisi.b $e |- ( ph -> T e. On ) $.
    tfisi.c $e |- ( ( ph /\ ( R e. On /\ R C_ T ) /\
          A. y ( S e. R -> ch ) ) -> ps ) $.
    tfisi.d $e |- ( x = y -> ( ps <-> ch ) ) $.
    tfisi.e $e |- ( x = A -> ( ps <-> th ) ) $.
    tfisi.f $e |- ( x = y -> R = S ) $.
    tfisi.g $e |- ( x = A -> R = T ) $.
    $( A transfinite induction scheme in "implicit" form where the induction is
       done on an object derived from the object of interest.  (Contributed by
       Stefan O'Rear, 24-Aug-2015.) $)
    tfisi $p |- ( ph -> th ) $=
      ( vv wi vz vw wss ssid wceq wa eqid wcel wal con0 weq eqeq2 anbi2d imbi1d
      cv sseq1 imbi12d albidv eqeq1d imbi2d cbvalv bitrdi wral w3a simp3l simp2
      simp1l eqeltrd simp3r eqsstrd csb wsb simpl3l simpl1l simpr simpl2 onelss
      eleqtrd sylc simpl3r sstrd simpl1r rspcdva nfcv csbhypf eqcomd equcoms wb
      eqidd nfv sbhypf bicomd spv mp2and ex alrimiv eleq1d sylib syl121anc 3exp
      tfis3 syl spcgv mpi expd pm2.43i ) AJJUCZDJUDAXGDTAAXGDAJJUEZAXGUFZDTZJUG
      AGKUHHJUEZXIBTZTZEUIZXHXJTZLAJUJUHXNMHUAUOZUEZAXPJUCZUFZBTZTZEUIZIUBUOZUE
      ZAYCJUCZUFZCTZTZFUIZXNUAUBJUAUBUKZYBHYCUEZYFBTZTZEUIYIYJYAYMEYJXQYKXTYLXP
      YCHULYJXSYFBYJXRYEAXPYCJUPUMUNUQURYMYHEFEFUKZYKYDYLYGYNHIYCQUSYNBCYFOUTUQ
      VAVBXPJUEZYAXMEYOXQXKXTXLXPJHULYOXSXIBYOXRXGAXPJJUPUMUNUQURXPUJUHZYIUBXPV
      CZYBYPYQUFZYAEYRXQXSBYRXQXSVDZAHUJUHHJUCIHUHZCTZFUIZBYRXQAXRVEYSHXPUJYRXQ
      XSVFZYPYQXQXSVGVHYSHXPJUUCYRXQAXRVIVJYSESUOHVKZHUHZBESVLZTZSUIUUBYSUUGSYS
      UUEUUFYSUUEUFZAUUDJUCZUUFAXRYRXQUUEVMUUHUUDXPJUUHYPUUDXPUHUUDXPUCYPYQXQXS
      UUEVNUUHUUDHXPYSUUEVOYRXQXSUUEVPVRZXPUUDVQVSAXRYRXQUUEVTWAUUHIUUDUEZAUUIU
      FZCTZTZFUIZUUDUUDUEZUULUUFTZUUHYIUUOUBXPUUDYCUUDUEZYHUUNFUURYDUUKYGUUMYCU
      UDIULUURYFUULCUURYEUUIAYCUUDJUPUMUNUQURYPYQXQXSUUEWBUUJWCUUHUUDWIUUNUUPUU
      QTFSFSUKZUUKUUPUUMUUQUUSIUUDUUDUUKSFSFUKZUUDIESFUOZHIEUVAWDEIWDQWEZWFWGUS
      UUSCUUFUULCUUFWHSFUUTUUFCBCESUVACEWJOWKZWLWGUTUQWMVSWNWOWPUUGUUASFUUTUUEY
      TUUFCUUTUUDIHUVBWQUVCUQVAWRNWSWTWPWOXAXBXMXOEGKEUOGUEZXKXHXLXJUVDHJJRUSUV
      DBDXIPUTUQXCVSXDXEXFXD $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  IZF Set Theory - add the Axiom of Infinity
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Infinity
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y $.
    $( Axiom of Infinity.  Axiom 5 of [Crosilla] p.  "Axioms of CZF and IZF".
       (Contributed by Jim Kingdon, 16-Nov-2018.) $)
    ax-iinf $a |- E. x ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) ) $.

    $( A standard version of the Axiom of Infinity, using definitions to
       abbreviate.  Axiom Inf of [BellMachover] p. 472.  (Contributed by NM,
       30-Aug-1993.) $)
    zfinf2 $p |- E. x ( (/) e. x /\ A. y e. x suc y e. x ) $=
      ( c0 cv wcel csuc wral wa wex wi wal ax-iinf df-ral anbi2i exbii mpbir )
      CADZEZBDZFQEZBQGZHZAIRSQETJBKZHZAIABLUBUDAUAUCRTBQMNOP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The natural numbers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new symbol. $)
  $c _om $. $( Greek letter omega $)

  $( Extend class notation to include the class of natural numbers. $)
  com $a class _om $.

  ${
    $d x y $.
    $( Define the class of natural numbers as the smallest inductive set, which
       is valid provided we assume the Axiom of Infinity.  Definition 6.3 of
       [Eisenberg] p. 82.

       _Note_: the natural numbers ` _om ` are a subset of the ordinal numbers
       ~ df-on .  Later, when we define complex numbers, we will be able to
       also define a subset of the complex numbers ( ~ df-inn ) with analogous
       properties and operations, but they will be different sets.

       We are unable to use the terms finite ordinal and natural number
       interchangeably, as shown at ~ exmidonfin .  (Contributed by NM,
       6-Aug-1994.)  Use its alias ~ dfom3 instead for naming consistency with
       set.mm.  (New usage is discouraged.) $)
    df-iom $a |- _om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } $.
  $}

  ${
    $d x y $.
    $( Alias for ~ df-iom .  Use it instead of ~ df-iom for naming consistency
       with set.mm.  (Contributed by NM, 6-Aug-1994.) $)
    dfom3 $p |- _om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } $=
      ( df-iom ) ABC $.
  $}

  ${
    $d x y $.
    $( The existence of omega (the class of natural numbers).  Axiom 7 of
       [TakeutiZaring] p. 43.  (Contributed by NM, 6-Aug-1994.) $)
    omex $p |- _om e. _V $=
      ( vy vx com cvv wcel c0 cv csuc wral wa cint zfinf2 intexabim ax-mp dfom3
      cab wex eleq1i mpbir ) CDEFAGZEBGHTEBTIJZAPKZDEZUAAQUCABLUAAMNCUBDABORS
      $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Peano's postulates
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.
    $( Zero is a natural number.  One of Peano's five postulates for
       arithmetic.  Proposition 7.30(1) of [TakeutiZaring] p. 42.  (Contributed
       by NM, 15-May-1994.) $)
    peano1 $p |- (/) e. _om $=
      ( vy vx vz c0 cv wcel csuc wral wa cab com wi 0ex elint wsb df-clab simpl
      cint sbimi clelsb2 sylib sylbi mpgbir dfom3 eleqtrri ) DDAEZFZBEGUFFBUFHZ
      IZAJZRZKDUKFCEZUJFZDULFZLCCDUJMNUMUIACOZUNUICAPUOUGACOUNUIUGACUGUHQSACDTU
      AUBUCABUDUE $.
    $( $j usage 'peano1' avoids 'ax-iinf' 'ax-setind'; $)
  $}

  ${
    $d x y z A $.
    $( The successor of any natural number is a natural number.  One of Peano's
       five postulates for arithmetic.  Proposition 7.30(2) of [TakeutiZaring]
       p. 42.  (Contributed by NM, 3-Sep-2003.) $)
    peano2 $p |- ( A e. _om -> suc A e. _om ) $=
      ( vy vx vz cvv wcel com csuc cv wral wa wi imbi12d adantl wsb wal clelsb2
      wb sylib nfv elex c0 cab cint simpl wceq eleq1 suceq eleq1d df-clab simpr
      df-ral sbimi sbim imbi12i bitri sbalv sylbi 19.21bi nfra1 nfan nfsab nfvd
      nfcvd vtocldf ralrimiva ralim elintg sucexg syl imbitrrid mpd dfom3 mpcom
      eleq2i 3imtr4g ) AEFZAGFZAHZGFZAGUAVQAUBBIZFZCIZHZWAFZCWAJZKZBUCZUDZFZVSW
      IFZVRVTVQADIZFZVSWLFZLZDWHJZWJWKLZVQWODWHVQWLWHFZKZWCWLFZWDWLFZLZWOCAEVQW
      RUEWCAUFZXBWORWSXCWTWMXAWNWCAWLUGXCWDVSWLWCAUHUIMNWRXBVQWRXBCWRWGBDOZXBCP
      ZWGDBUJXDWCWAFZWELZCPZBDOXEWGXHBDWGWFXHWBWFUKWECWAULSUMXGXBBDCXGBDOXFBDOZ
      WEBDOZLXBXFWEBDUNXIWTXJXABDWCQBDWDQUOUPUQSURUSNVQWRCVQCTWGCBDWBWFCWBCTWEC
      WAUTVAVBVAWSCAVDWSWOCVCVEVFWPWQVQWMDWHJZWNDWHJZLWMWNDWHVGVQWJXKWKXLDAWHEV
      HVQVSEFWKXLRAEVIDVSWHEVHVJMVKVLGWIABCVMZVOGWIVSXMVOVPVN $.
    $( $j usage 'peano2' avoids 'ax-iinf' 'ax-setind'; $)
  $}

  $( The successor of any natural number is not zero.  One of Peano's five
     postulates for arithmetic.  Proposition 7.30(3) of [TakeutiZaring] p. 42.
     (Contributed by NM, 3-Sep-2003.) $)
  peano3 $p |- ( A e. _om -> suc A =/= (/) ) $=
    ( com nsuceq0g ) ABC $.
  $( $j usage 'peano3' avoids 'ax-iinf' 'ax-setind'; $)

  $( Two natural numbers are equal iff their successors are equal, i.e. the
     successor function is one-to-one.  One of Peano's five postulates for
     arithmetic.  Proposition 7.30(4) of [TakeutiZaring] p. 43.  (Contributed
     by NM, 3-Sep-2003.) $)
  peano4 $p |- ( ( A e. _om /\ B e. _om ) -> ( suc A = suc B <-> A = B ) ) $=
    ( com suc11g ) ABCCD $.
  $( $j usage 'peano4' avoids 'ax-iinf'; $)

  ${
    $d x y A $.
    $( The induction postulate: any class containing zero and closed under the
       successor operation contains all natural numbers.  One of Peano's five
       postulates for arithmetic.  Proposition 7.30(5) of [TakeutiZaring]
       p. 43.  The more traditional statement of mathematical induction as a
       theorem schema, with a basis and an induction step, is derived from this
       theorem as Theorem ~ findes .  (Contributed by NM, 18-Feb-2004.) $)
    peano5 $p |- ( ( (/) e. A /\
                 A. x e. _om ( x e. A -> suc x e. A ) ) -> _om C_ A ) $=
      ( vy c0 wcel cv csuc wi com wral wa cin wss cint elin df-ral sylibr eleq2
      cab wal dfom3 peano1 mpbiran biimpri peano2 adantr a1i jcad alimi imbi12i
      pm3.31 albii 3imtr4i anim12i omex wceq raleqbi1dv anbi12d elab intss1 syl
      inex1 eqsstrid ssid biantrur ssin bitri ) DBEZAFZBEZVIGZBEZHZAIJZKZIIBLZM
      ZIBMZVOIDCFZEZVKVSEZAVSJZKZCSZNZVPCAUAVOVPWDEZWEVPMVODVPEZVKVPEZAVPJZKZWF
      VHWGVNWIWGVHWGDIEVHUBDIBOUCUDVNVIVPEZWHHZATZWIVIIEZVMHZATWNVJKZVKIEZVLKZH
      ZATVNWMWOWSAWOWPWQVLWPWQHWOWNWQVJVIUEUFUGWNVJVLUKUHUIVMAIPWLWSAWKWPWHWRVI
      IBOVKIBOUJULUMWHAVPPQUNWCWJCVPIBUOVBVSVPUPVTWGWBWIVSVPDRWAWHAVSVPVSVPVKRU
      QURUSQVPWDUTVAVCVRIIMZVRKVQWTVRIVDVEIIBVFVGQ $.
    $( $j usage 'peano5' avoids 'ax-setind'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Finite induction (for finite ordinals)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x A $.
    find.1 $e |- ( A C_ _om /\ (/) e. A /\ A. x e. A suc x e. A ) $.
    $( The Principle of Finite Induction (mathematical induction).  Corollary
       7.31 of [TakeutiZaring] p. 43.  The simpler hypothesis shown here was
       suggested in an email from "Colin" on 1-Oct-2001.  The hypothesis states
       that ` A ` is a set of natural numbers, zero belongs to ` A ` , and
       given any member of ` A ` the member's successor also belongs to ` A ` .
       The conclusion is that every natural number is in ` A ` .  (Contributed
       by NM, 22-Feb-2004.)  (Proof shortened by Andrew Salmon,
       27-Aug-2011.) $)
    find $p |- A = _om $=
      ( com wss c0 wcel cv csuc wral simp1i wi wa w3a 3simpc ax-mp df-ral alral
      wal sylbi anim2i peano5 eqssi ) BDBDEZFBGZAHZIBGZABJZCKUEUFBGUGLZADJZMZDB
      EUEUHMZUKUDUEUHNULCUDUEUHOPUHUJUEUHUIASUJUGABQUIADRTUAPABUBPUC $.
  $}

  ${
    $d x y $.  $d x A $.  $d x ps $.  $d x ch $.  $d x th $.  $d x ta $.
    $d y ph $.
    $( Substitutions. $)
    finds.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    finds.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    finds.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    finds.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Basis. $)
    finds.5 $e |- ps $.
    $( Induction step. $)
    finds.6 $e |- ( y e. _om -> ( ch -> th ) ) $.
    $( Principle of Finite Induction (inference schema), using implicit
       substitutions.  The first four hypotheses establish the substitutions we
       need.  The last two are the basis and the induction step.  Theorem
       Schema 22 of [Suppes] p. 136.  This is Metamath 100 proof #74.
       (Contributed by NM, 14-Apr-1995.) $)
    finds $p |- ( A e. _om -> ta ) $=
      ( com wcel cab c0 cv elab csuc wi wral wss 0ex mpbir sucex 3imtr4g peano5
      vex rgen mp2an sseli elabg mpbid ) HOPHAFQZPEOUPHRUPPZGSZUPPZURUAZUPPZUBZ
      GOUCOUPUDUQBMABFRUEITUFVBGOUROPCDUSVANACFURGUJZJTADFUTURVCUGKTUHUKGUPUIUL
      UMAEFHOLUNUO $.
  $}

  ${
    $d x y ta $.  $d x ps $.  $d x ch $.  $d x th $.  $d y ph $.
    $( Substitutions. $)
    finds2.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    finds2.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    finds2.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    $( Basis. $)
    finds2.4 $e |- ( ta -> ps ) $.
    $( Induction step. $)
    finds2.5 $e |- ( y e. _om -> ( ta -> ( ch -> th ) ) ) $.
    $( Principle of Finite Induction (inference schema), using implicit
       substitutions.  The first three hypotheses establish the substitutions
       we need.  The last two are the basis and the induction step.  Theorem
       Schema 22 of [Suppes] p. 136.  (Contributed by NM, 29-Nov-2002.) $)
    finds2 $p |- ( x e. _om -> ( ta -> ph ) ) $=
      ( cv com wcel wi c0 wceq imbi2d elab cab csuc wss 0ex mpbir a2d vex sucex
      wral 3imtr4g rgen peano5 mp2an sseli abid sylib ) FMZNOUQEAPZFUAZOURNUSUQ
      QUSOZGMZUSOZVAUBZUSOZPZGNUINUSUCUTEBPZKURVFFQUDUQQRABEHSTUEVEGNVANOZECPZE
      DPZVBVDVGECDLUFURVHFVAGUGZUQVARACEISTURVIFVCVAVJUHUQVCRADEJSTUJUKGUSULUMU
      NURFUOUP $.
  $}

  ${
    $d x y $.  $d x ps $.  $d x ch $.  $d x th $.  $d y ph $.
    $( Substitutions. $)
    finds1.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    finds1.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    finds1.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    $( Basis. $)
    finds1.4 $e |- ps $.
    $( Induction step. $)
    finds1.5 $e |- ( y e. _om -> ( ch -> th ) ) $.
    $( Principle of Finite Induction (inference schema), using implicit
       substitutions.  The first three hypotheses establish the substitutions
       we need.  The last two are the basis and the induction step.  Theorem
       Schema 22 of [Suppes] p. 136.  (Contributed by NM, 22-Mar-2006.) $)
    finds1 $p |- ( x e. _om -> ph ) $=
      ( cv com wcel c0 wceq eqid a1i wi a1d finds2 mpi ) ELMNOOPZAOQABCDUCEFGHI
      BUCJRFLMNCDSUCKTUAUB $.
  $}

  ${
    $d x y z $.  $d y z ph $.
    findes.1 $e |- [. (/) / x ]. ph $.
    findes.2 $e |- ( x e. _om -> ( ph -> [. suc x / x ]. ph ) ) $.
    $( Finite induction with explicit substitution.  The first hypothesis is
       the basis and the second is the induction step.  Theorem Schema 22 of
       [Suppes] p. 136.  This is an alternative for Metamath 100 proof #74.
       (Contributed by Raph Levien, 9-Jul-2003.) $)
    findes $p |- ( x e. _om -> ph ) $=
      ( vz vy wsb c0 wsbc csuc dfsbcq2 sbequ sbequ12r com wcel nfv nfim imbi12d
      cv wi nfs1v nfsbc1v weq eleq1 sbequ12 wceq suceq dfsbcq syl chvar finds
      wb ) ABEGABHIABFGZABFSZJZIZAEFBSZABEHKAEFBLABEUOKAEBMCUQNOZAABUQJZIZTZTUN
      NOZUMUPTZTBFVBVCBVBBPUMUPBABFUAABUOUBQQBFUCZURVBVAVCUQUNNUDVDAUMUTUPABFUE
      VDUSUOUFUTUPULUQUNUGABUSUOUHUIRRDUJUK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The Natural Numbers (continued)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z A $.
    $( A natural number is either 0 or a successor.  Similar theorems for
       arbitrary sets or real numbers will not be provable (without the law of
       the excluded middle), but equality of natural numbers is decidable.
       (Contributed by NM, 27-May-1998.) $)
    nn0suc $p |- ( A e. _om -> ( A = (/) \/ E. x e. _om A = suc x ) ) $=
      ( vy vz cv c0 wceq csuc com wrex wo eqeq1 rexbidv orbi12d eqid orci suceq
      wcel eqeq2d rspcev mpan2 olcd a1d finds ) CEZFGZUEAEZHZGZAIJZKFFGZFUHGZAI
      JZKDEZFGZUNUHGZAIJZKZUNHZFGZUSUHGZAIJZKZBFGZBUHGZAIJZKCDBUFUFUKUJUMUEFFLU
      FUIULAIUEFUHLMNUEUNGZUFUOUJUQUEUNFLVGUIUPAIUEUNUHLMNUEUSGZUFUTUJVBUEUSFLV
      HUIVAAIUEUSUHLMNUEBGZUFVDUJVFUEBFLVIUIVEAIUEBUHLMNUKUMFOPUNIRZVCURVJVBUTV
      JUSUSGZVBUSOVAVKAUNIUGUNGUHUSUSUGUNQSTUAUBUCUD $.
  $}

  ${
    $d A x y $.
    $( A natural number ordinal is, as a set, included in the set of natural
       number ordinals.  (Contributed by NM, 21-Jun-1998.)  Extract this result
       from the previous proof of ~ elnn .  (Revised by BJ, 7-Aug-2024.) $)
    elomssom $p |- ( A e. _om -> A C_ _om ) $=
      ( vy vx cv com wss c0 csuc sseq1 0ss wcel wa csn cun unss vex snss anbi2i
      df-suc sseq1i 3bitr4i biimpi expcom finds ) BDZEFGEFCDZEFZUFHZEFZAEFBCAUE
      GEIUEUFEIUEUHEIUEAEIEJUGUFEKZUIUGUJLZUIUGUFMZEFZLUFULNZEFUKUIUFULEOUJUMUG
      UFECPQRUHUNEUFSTUAUBUCUD $.
  $}

  $( A member of a natural number is a natural number.  (Contributed by NM,
     21-Jun-1998.) $)
  elnn $p |- ( ( A e. B /\ B e. _om ) -> A e. _om ) $=
    ( com wcel wss elomssom ssel2 ancoms sylan2 ) BCDABDZBCEZACDZBFKJLBCAGHI $.

  ${
    $d x y $.
    $( Omega is ordinal.  Theorem 7.32 of [TakeutiZaring] p. 43.  (Contributed
       by NM, 18-Oct-1995.) $)
    ordom $p |- Ord _om $=
      ( vx vy com word wtr cv wral wcel wa wi wal elnn gen2 dftr2 mpbir c0 csuc
      treq tr0 suctr a1i finds rgen dford3 mpbir2an ) CDCEZAFZEZACGUFUGBFZHUICH
      IUGCHZJZBKAKUKABUGUILMABCNOUHACUIEPEUHUGQZEZUHBAUGUIPRUIUGRZUIULRUNSUHUMJ
      UJUGTUAUBUCACUDUE $.
  $}

  $( Omega is an ordinal number.  (Contributed by Mario Carneiro,
     30-Jan-2013.) $)
  omelon2 $p |- ( _om e. _V -> _om e. On ) $=
    ( com cvv wcel con0 word ordom elong mpbiri ) ABCADCAEFABGH $.

  $( Omega is an ordinal number.  (Contributed by NM, 10-May-1998.)  (Revised
     by Mario Carneiro, 30-Jan-2013.) $)
  omelon $p |- _om e. On $=
    ( com cvv wcel con0 omex omelon2 ax-mp ) ABCADCEFG $.

  $( A natural number is an ordinal number.  (Contributed by NM,
     27-Jun-1994.) $)
  nnon $p |- ( A e. _om -> A e. On ) $=
    ( com omelon oneli ) BACD $.

  ${
    nnoni.1 $e |- A e. _om $.
    $( A natural number is an ordinal number.  (Contributed by NM,
       27-Jun-1994.) $)
    nnoni $p |- A e. On $=
      ( com wcel con0 nnon ax-mp ) ACDAEDBAFG $.
  $}

  $( A natural number is ordinal.  (Contributed by NM, 17-Oct-1995.) $)
  nnord $p |- ( A e. _om -> Ord A ) $=
    ( com wcel con0 word nnon eloni syl ) ABCADCAEAFAGH $.

  $( Omega is a subset of ` On ` .  (Contributed by NM, 13-Jun-1994.) $)
  omsson $p |- _om C_ On $=
    ( vx com con0 cv nnon ssriv ) ABCADEF $.

  ${
    $d x z $.
    $( Omega is a limit ordinal.  Theorem 2.8 of [BellMachover] p. 473.
       (Contributed by NM, 26-Mar-1995.)  (Proof rewritten by Jim Kingdon,
       5-Jan-2019.) $)
    limom $p |- Lim _om $=
      ( vx vz com wlim word c0 wcel cuni wceq ordom peano1 cv wa wex csuc sucex
      wi vex isseti peano2 sucid jctil anbi12d imbitrrid eximii 19.37aiv sylibr
      eleq2 eleq1 eluni ssriv wss orduniss ax-mp eqssi dflim2 mpbir3an ) CDCEZF
      CGCCHZIJKCUSACUSALZCGZUTBLZGZVBCGZMZBNUTUSGVAVEBVBUTOZIZVAVEQBBVFUTARZPSV
      AVEVGUTVFGZVFCGZMVAVJVIUTTUTVHUAUBVGVCVIVDVJVBVFUTUHVBVFCUIUCUDUEUFBUTCUJ
      UGUKURUSCULJCUMUNUOCUPUQ $.
  $}

  $( A class belongs to omega iff its successor does.  (Contributed by NM,
     3-Dec-1995.) $)
  peano2b $p |- ( A e. _om <-> suc A e. _om ) $=
    ( com wcel csuc peano2 cvv elex sucexb sylibr sucidg elnn mpancom impbii
    syl ) ABCZADZBCZAEAPCZQOQAFCZRQPFCSPBGAHIAFJNAPKLM $.

  ${
    $d x A $.
    $( A nonzero natural number is a successor.  (Contributed by NM,
       18-Feb-2004.) $)
    nnsuc $p |- ( ( A e. _om /\ A =/= (/) ) -> E. x e. _om A = suc x ) $=
      ( c0 wne com wcel wceq wn cv csuc wrex df-ne nn0suc ord imp sylan2b ) BCD
      BEFZBCGZHZBAIJGAEKZBCLQSTQRTABMNOP $.
  $}

  ${
    $d A x $.
    $( The successor of the precedessor of a nonzero natural number.
       (Contributed by Jim Kingdon, 31-Jul-2022.) $)
    nnsucpred $p |- ( ( A e. _om /\ A =/= (/) ) -> suc U. A = A ) $=
      ( vx com wcel c0 wne wa cv csuc wceq cuni nnsuc con0 nnon ad2antrr simprr
      onsucuni2 syl2anc rexlimddv ) ACDZAEFZGZABHZIJZAKIAJZBCBALUBUCCDZUDGZGAMD
      ZUDUETUHUAUGANOUBUFUDPAUCQRS $.
  $}

  ${
    $d x y A $.
    $( A natural number is either zero or nonzero.  Decidable equality for
       natural numbers is a special case of the law of the excluded middle
       which holds in most constructive set theories including ours.
       (Contributed by Jim Kingdon, 5-Jan-2019.) $)
    nndceq0 $p |- ( A e. _om -> DECID A = (/) ) $=
      ( vx vy com wcel c0 wceq wn wo wdc cv csuc eqeq1 notbid orbi12d eqid orci
      peano3 neneqd olcd a1d finds df-dc sylibr ) ADEAFGZUEHZIZUEJBKZFGZUIHZIFF
      GZUKHZICKZFGZUNHZIZUMLZFGZURHZIZUGBCAUIUIUKUJULUHFFMZUIUIUKVANOUHUMGZUIUN
      UJUOUHUMFMZVBUIUNVCNOUHUQGZUIURUJUSUHUQFMZVDUIURVENOUHAGZUIUEUJUFUHAFMZVF
      UIUEVGNOUKULFPQUMDEZUTUPVHUSURVHUQFUMRSTUAUBUEUCUD $.

    $( A natural number is either the empty set or has the empty set as an
       element.  (Contributed by Jim Kingdon, 23-Aug-2019.) $)
    0elnn $p |- ( A e. _om -> ( A = (/) \/ (/) e. A ) ) $=
      ( vx vy cv c0 wceq wcel wo csuc eqeq1 eleq2 orbi12d eqid com wi 0ex sucid
      orci suceq a1i eleqtrrid wss sssucid sseld jaod olc syl6 finds ) BDZEFZEU
      IGZHEEFZEEGZHCDZEFZEUNGZHZUNIZEFZEURGZHZAEFZEAGZHBCAUJUJULUKUMUIEEJUIEEKL
      UIUNFUJUOUKUPUIUNEJUIUNEKLUIURFUJUSUKUTUIUREJUIUREKLUIAFUJVBUKVCUIAEJUIAE
      KLULUMEMRUNNGZUQUTVAVDUOUTUPUOUTOVDUOEEIUREPQUNESUATVDUNUREUNURUBVDUNUCTU
      DUEUTUSUFUGUH $.
  $}

  $( A natural number is nonempty iff it contains the empty set.  Although in
     constructive mathematics it is generally more natural to work with
     inhabited sets and ignore the whole concept of nonempty sets, in the
     specific case of natural numbers this theorem may be helpful in converting
     proofs which were written assuming excluded middle.  (Contributed by Jim
     Kingdon, 28-Aug-2019.) $)
  nn0eln0 $p |- ( A e. _om -> ( (/) e. A <-> A =/= (/) ) ) $=
    ( com wcel c0 wceq wo wne wb 0elnn noel eleq2 mtbiri nner 2falsed ne0i 2thd
    id jaoi syl ) ABCADEZDACZFUAADGZHZAITUCUATUAUBTUADDCDJADDKLADMNUAUAUBUAQADO
    PRS $.

  ${
    $d ph w x y z $.
    nnregexmid.1 $e |- ( ( x C_ _om /\ E. y y e. x ) ->
        E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) ) $.
    $( If inhabited sets of natural numbers always have minimal elements,
       excluded middle follows.  The argument is essentially the same as
       ~ regexmid and the larger lesson is that although natural numbers may
       behave "non-constructively" even in a constructive set theory (for
       example see ~ nndceq or ~ nntri3or ), sets of natural numbers are a
       different animal.  (Contributed by Jim Kingdon, 6-Sep-2019.) $)
    nnregexmid $p |- ( ph \/ -. ph ) $=
      ( vw cv c0 wceq wa wo wcel wel wn wi wal wex com wss peano1 csn crab csuc
      cpr ssrab2 suc0 peano2 ax-mp eqeltrri prssi mp2an sstri eqid regexmidlemm
      pp0ex rabex sseq1 eleq2 exbidv anbi12d notbid imbi2d albidv imbi12d vtocl
      regexmidlem1 ) CGZFGZHUAZIVHHIAJKZFHVIUDZUBZLZDCMZDGZVLLZNZOZDPZJZCQZAANK
      VLRSZVMCQZWAVLVKRVJFVKUEHRLZVIRLVKRSTHUCZVIRUFWDWERLTHUGUHUIHVIRUJUKULAFC
      VLVLUMZUNBGZRSZCBMZCQZJZWIVNDBMZNZOZDPZJZCQZOWBWCJZWAOBVLVJFVKUOUPWGVLIZW
      KWRWQWAWSWHWBWJWCWGVLRUQWSWIVMCWGVLVGURZUSUTWSWPVTCWSWIVMWOVSWTWSWNVRDWSW
      MVQVNWSWLVPWGVLVOURVAVBVCUTUSVDEVEUKAFCDVLWFVFUH $.
  $}

  ${
    $d A w k x $.  $d ch w k x $.  $d ph w k y $.  $d ps w k x $.
    $d x y w k $.
    omsinds.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    omsinds.2 $e |- ( x = A -> ( ph <-> ch ) ) $.
    omsinds.3 $e |- ( x e. _om -> ( A. y e. x ps -> ph ) ) $.
    $( Strong (or "total") induction principle over ` _om ` .  (Contributed by
       Scott Fenton, 17-Jul-2015.) $)
    omsinds $p |- ( A e. _om -> ch ) $=
      ( com wcel csuc cv wral c0 wceq suceq raleqdv wi ax-mp csn wsbc ral0 rgen
      vw vk peano1 nfth nfsbc1v nfim raleq sbceq1a imbi12d rspc wb ralsns mpbir
      suc0 raleqi cun simpr peano2 adantr cbvralv sylib nfv mpi sylc syl mpbird
      wa ralun syl2anc df-suc a1i ex finds sucidg rspcdva ) FJKACDFLZFHADUEMZLZ
      NADOLZNZADUFMZLZNZADWFLZNZADVTNUEUFFWAOPADWBWCWAOQRWAWEPADWBWFWAWEQRWAWFP
      ADWBWHWAWFQRWAFPADWBVTWAFQRWDADOUAZNZWKADOUBZBEONZWLBEUCZBEDMZNZASZDJNZWM
      WLSZWQDJIUDZOJKZWRWSSUGWQWSDOJWMWLDWMDWNUHADOUIUJWOOPWPWMAWLBEWOOUKADOULU
      MUNTTTXAWKWLUOUGADOJUPTUQADWCWJURUSUQWEJKZWGWIXBWGVKZWIADWFWFUAZUTZNZXCWG
      ADXDNZXFXBWGVAZXCXGADWFUBZXCWFJKZBEWFNZXIXBXJWGWEVBVCZXCWGXKXHABDEWFGVDVE
      XJWRXKXISZWTWQXMDWFJXKXIDXKDVFADWFUIUJWOWFPWPXKAXIBEWOWFUKADWFULUMUNVGVHX
      CXJXGXIUOXLADWFJUPVIVJADWFXDVLVMXCADWHXEWHXEPXCWFVNVORVJVPVQFJVRVS $.
  $}

  ${
    $d A x $.
    $( The predecessor of a natural number is a natural number.  This theorem
       is most interesting when the natural number is a successor (as seen in
       theorems like ~ onsucuni2 ) but also holds when it is ` (/) ` by
       ~ uni0 .  (Contributed by Jim Kingdon, 31-Jul-2022.) $)
    nnpredcl $p |- ( A e. _om -> U. A e. _om ) $=
      ( vx com wcel c0 wceq cuni cv csuc wrex unieq uni0 peano1 eqeltrdi adantl
      eqeltri wa con0 nnon adantr simpr onsucuni2 ex sylc simpl eqeltrd peano2b
      rexlimdvw sylibr nn0suc mpjaodan ) ACDZAEFZAGZCDZABHZIFZBCJZUMUOULUMUNEGZ
      CAEKUSECLMPNOULURQZUNIZCDUOUTVAACUTARDZURVAAFZULVBURASTULURUAVBUQVCBCVBUQ
      VCAUPUBUCUHUDULURUEUFUNUGUIBAUJUK $.
  $}

  ${
    $( The predecessor (see ~ nnpredcl ) of a nonzero natural number is less
       than (see ~ df-iord ) that number.  (Contributed by Jim Kingdon,
       14-Sep-2024.) $)
    nnpredlt $p |- ( ( A e. _om /\ A =/= (/) ) -> U. A e. A ) $=
      ( com wcel c0 wne cuni csuc nnpredcl adantr sucidg syl nnsucpred eleqtrd
      wa ) ABCZADEZNZAFZRGZAQRBCZRSCOTPAHIRBJKALM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Relations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c X. $. $( Times symbol (cross product symbol) (read: 'cross') $)
  $c `' $. $( Small elevated smiley (converse operation) $)
  $c dom $. $( Domain $)
  $c ran $. $( Range $)
  $c |` $. $( Right hook (restriction symbol) $)
  $c " $. $( Left quote (image symbol) $)
  $c o. $. $( Small circle (composition symbol) $)
  $c Rel $. $( Relation predicate $)

  $( Extend the definition of a class to include the cross product. $)
  cxp $a class ( A X. B ) $.

  $( Extend the definition of a class to include the converse of a class. $)
  ccnv $a class `' A $.

  $( Extend the definition of a class to include the domain of a class. $)
  cdm $a class dom A $.

  $( Extend the definition of a class to include the range of a class. $)
  crn $a class ran A $.

  $( Extend the definition of a class to include the restriction of a class.
     (Read:  The restriction of ` A ` to ` B ` .) $)
  cres $a class ( A |` B ) $.

  $( Extend the definition of a class to include the image of a class.  (Read:
     The image of ` B ` under ` A ` .) $)
  cima $a class ( A " B ) $.

  $( Extend the definition of a class to include the composition of two
     classes.  (Read:  The composition of ` A ` and ` B ` .) $)
  ccom $a class ( A o. B ) $.

  $( Extend the definition of a wff to include the relation predicate.  (Read:
     ` A ` is a relation.) $)
  wrel $a wff Rel A $.

  ${
    $d x y z A $.  $d x y z B $.
    $( Define the Cartesian product of two classes.  This is also sometimes
       called the "cross product" but that term also has other meanings; we
       intentionally choose a less ambiguous term.  Definition 9.11 of [Quine]
       p. 64.  For example, ` ( { 1 , 5 } X. { 2 , 7 } ) = `
       ` ( { <. 1 , 2 >. , <. 1 , 7 >. } u. { <. 5 , 2 >. , <. 5 , 7 >. } ) ` .
       Another example is that the set of rational numbers is defined using the
       Cartesian product as ` ( ZZ X. NN ) ` ; the left- and right-hand sides
       of the Cartesian product represent the top (integer) and bottom
       (natural) numbers of a fraction.  (Contributed by NM, 4-Jul-1994.) $)
    df-xp $a |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) } $.

    $( Define the relation predicate.  Definition 6.4(1) of [TakeutiZaring]
       p. 23.  For alternate definitions, see ~ dfrel2 and ~ dfrel3 .
       (Contributed by NM, 1-Aug-1994.) $)
    df-rel $a |- ( Rel A <-> A C_ ( _V X. _V ) ) $.

    $( Define the converse of a class.  Definition 9.12 of [Quine] p. 64.  The
       converse of a binary relation swaps its arguments, i.e., if ` A e. _V `
       and ` B e. _V ` then ` ( A ``' R B <-> B R A ) ` , as proven in ~ brcnv
       (see ~ df-br and ~ df-rel for more on relations).  For example,
       ` ``' { <. 2 , 6 >. , <. 3 , 9 >. } = { <. 6 , 2 >. , <. 9 , 3 >. } ` .

       We use Quine's breve accent (smile) notation.  Like Quine, we use it as
       a prefix, which eliminates the need for parentheses.  "Converse" is
       Quine's terminology.  Some authors use a "minus one" exponent and call
       it "inverse", especially when the argument is a function, although this
       is not in general a genuine inverse.  (Contributed by NM,
       4-Jul-1994.) $)
    df-cnv $a |- `' A = { <. x , y >. | y A x } $.

    $( Define the composition of two classes.  Definition 6.6(3) of
       [TakeutiZaring] p. 24.  Note that Definition 7 of [Suppes] p. 63
       reverses ` A ` and ` B ` , uses a slash instead of ` o. ` , and calls
       the operation "relative product".  (Contributed by NM, 4-Jul-1994.) $)
    df-co $a |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) } $.

    $( Define the domain of a class.  Definition 3 of [Suppes] p. 59.  For
       example, F = { ` <. ` 2 , 6 ` >. ` , ` <. ` 3 , 9 ` >. ` } ` -> ` dom F
       = { 2 , 3 } .  Contrast with range (defined in ~ df-rn ).  For alternate
       definitions see ~ dfdm2 , ~ dfdm3 , and ~ dfdm4 .  The
       notation " ` dom ` " is used by Enderton; other authors sometimes use
       script D. (Contributed by NM, 1-Aug-1994.) $)
    df-dm $a |- dom A = { x | E. y x A y } $.

    $( Define the range of a class.  For example, F = { ` <. ` 2 , 6 ` >. ` ,
       ` <. ` 3 , 9 ` >. ` } -> ran F = { 6 , 9 } .  Contrast with domain
       (defined in ~ df-dm ).  For alternate definitions, see ~ dfrn2 ,
       ~ dfrn3 , and ~ dfrn4 .  The notation " ` ran ` " is used by Enderton;
       other authors sometimes use script R or script W. (Contributed by NM,
       1-Aug-1994.) $)
    df-rn $a |- ran A = dom `' A $.

    $( Define the restriction of a class.  Definition 6.6(1) of [TakeutiZaring]
       p. 24.  For example,
       ` ( F = { <. 2 , 6 >. , <. 3 , 9 >. } /\ B = { 1 , 2 } ) -> `
       ` ( F |`` B ) = { <. 2 , 6 >. } ` .  We do not introduce a special
       syntax for the corestriction of a class: it will be expressed either as
       the intersection ` ( A i^i ( _V X. B ) ) ` or as the converse of the
       restricted converse.  (Contributed by NM, 2-Aug-1994.) $)
    df-res $a |- ( A |` B ) = ( A i^i ( B X. _V ) ) $.

    $( Define the image of a class (as restricted by another class).
       Definition 6.6(2) of [TakeutiZaring] p. 24.  For example, ( F = { ` <. `
       2 , 6 ` >. ` , ` <. ` 3 , 9 ` >. ` } /\ B = { 1 , 2 } ) -> ( F ` " ` B )
       = { 6 } .  Contrast with restriction ( ~ df-res ) and range ( ~ df-rn ).
       For an alternate definition, see ~ dfima2 .  (Contributed by NM,
       2-Aug-1994.) $)
    df-ima $a |- ( A " B ) = ran ( A |` B ) $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.
    $( Equality theorem for cross product.  (Contributed by NM, 4-Jul-1994.) $)
    xpeq1 $p |- ( A = B -> ( A X. C ) = ( B X. C ) ) $=
      ( vx vy wceq cv wcel wa copab cxp eleq2 anbi1d opabbidv df-xp 3eqtr4g ) A
      BFZDGZAHZEGCHZIZDEJRBHZTIZDEJACKBCKQUAUCDEQSUBTABRLMNDEACODEBCOP $.

    $( Equality theorem for cross product.  (Contributed by NM, 5-Jul-1994.) $)
    xpeq2 $p |- ( A = B -> ( C X. A ) = ( C X. B ) ) $=
      ( vx vy wceq cv wcel wa copab cxp eleq2 anbi2d opabbidv df-xp 3eqtr4g ) A
      BFZDGCHZEGZAHZIZDEJRSBHZIZDEJCAKCBKQUAUCDEQTUBRABSLMNDECAODECBOP $.

    $( Membership in a cross product.  Uses fewer axioms than ~ elxp .
       (Contributed by NM, 4-Jul-1994.) $)
    elxpi $p |- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\
               ( x e. B /\ y e. C ) ) ) $=
      ( vz cv cop wceq wcel wa wex cab cxp eqeq1 anbi1d 2exbidv elabg ibi copab
      df-xp df-opab eqtri eleq2s ) CAGZBGZHZIZUEDJUFEJKZKZBLALZCFGZUGIZUIKZBLAL
      ZFMZDENZCUPJUKUOUKFCUPULCIZUNUJABURUMUHUIULCUGOPQRSUQUIABTUPABDEUAUIABFUB
      UCUD $.

    $( Membership in a cross product.  (Contributed by NM, 4-Jul-1994.) $)
    elxp $p |- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\
               ( x e. B /\ y e. C ) ) ) $=
      ( cxp wcel cv wa copab cop wceq wex df-xp eleq2i elopab bitri ) CDEFZGCAH
      ZDGBHZEGIZABJZGCSTKLUAIBMAMRUBCABDENOUAABCPQ $.

    $( Membership in a cross product.  (Contributed by NM, 23-Feb-2004.) $)
    elxp2 $p |- ( A e. ( B X. C ) <-> E. x e. B E. y e. C A = <. x , y >. ) $=
      ( cv wcel cop wceq wrex wa wex cxp df-rex r19.42v an13 exbii 3bitr3i elxp
      3bitr4ri ) AFZDGZCUABFZHIZBEJZKZALUDUBUCEGZKKZBLZALUEADJCDEMGUFUIAUBUDKZB
      EJUGUJKZBLUFUIUJBENUBUDBEOUKUHBUGUBUDPQRQUEADNABCDEST $.
  $}

  $( Equality theorem for cross product.  (Contributed by FL, 31-Aug-2009.) $)
  xpeq12 $p |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) ) $=
    ( wceq cxp xpeq1 xpeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.

  ${
    xpeq1i.1 $e |- A = B $.
    $( Equality inference for cross product.  (Contributed by NM,
       21-Dec-2008.) $)
    xpeq1i $p |- ( A X. C ) = ( B X. C ) $=
      ( wceq cxp xpeq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality inference for cross product.  (Contributed by NM,
       21-Dec-2008.) $)
    xpeq2i $p |- ( C X. A ) = ( C X. B ) $=
      ( wceq cxp xpeq2 ax-mp ) ABECAFCBFEDABCGH $.
  $}

  ${
    xpeq12i.1 $e |- A = B $.
    xpeq12i.2 $e |- C = D $.
    $( Equality inference for cross product.  (Contributed by FL,
       31-Aug-2009.) $)
    xpeq12i $p |- ( A X. C ) = ( B X. D ) $=
      ( wceq cxp xpeq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
  $}

  ${
    xpeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for cross product.  (Contributed by Jeff Madsen,
       17-Jun-2010.) $)
    xpeq1d $p |- ( ph -> ( A X. C ) = ( B X. C ) ) $=
      ( wceq cxp xpeq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Equality deduction for cross product.  (Contributed by Jeff Madsen,
       17-Jun-2010.) $)
    xpeq2d $p |- ( ph -> ( C X. A ) = ( C X. B ) ) $=
      ( wceq cxp xpeq2 syl ) ABCFDBGDCGFEBCDHI $.

    ${
      xpeq12d.2 $e |- ( ph -> C = D ) $.
      $( Equality deduction for Cartesian product.  (Contributed by NM,
         8-Dec-2013.) $)
      xpeq12d $p |- ( ph -> ( A X. C ) = ( B X. D ) ) $=
        ( wceq cxp xpeq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
    $}

    $( Equality deduction for a Cartesian square, see Wikipedia "Cartesian
       product",
~ https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power .
       (Contributed by AV, 13-Jan-2020.) $)
    sqxpeqd $p |- ( ph -> ( A X. A ) = ( B X. B ) ) $=
      ( xpeq12d ) ABCBCDDE $.
  $}

  ${
    $d y z A $.  $d y z B $.  $d x y z $.
    nfxp.1 $e |- F/_ x A $.
    nfxp.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for cross product.  (Contributed by
       NM, 15-Sep-2003.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nfxp $p |- F/_ x ( A X. B ) $=
      ( vy vz cxp cv wcel wa copab df-xp nfcri nfan nfopab nfcxfr ) ABCHFIBJZGI
      CJZKZFGLFGBCMTFGARSAAFBDNAGCENOPQ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( The empty set is not a member of a cross product.  (Contributed by NM,
       2-May-1996.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    0nelxp $p |- -. (/) e. ( A X. B ) $=
      ( vx vy c0 cxp wcel cv cop wceq wa wex wne wn opnzi simpl eqcomd necon3ai
      vex nex ax-mp elxp mtbir ) EABFGECHZDHZIZJZUDAGUEBGKZKZDLZCLUJCUIDUFEMUIN
      UDUECSDSOUIUFEUIEUFUGUHPQRUATTCDEABUBUC $.

    $( A member of a cross product (ordered pair) doesn't contain the empty
       set.  (Contributed by NM, 15-Dec-2008.) $)
    0nelelxp $p |- ( C e. ( A X. B ) -> -. (/) e. C ) $=
      ( vx vy cxp wcel cv cop wceq wa wex c0 wn elxp 0nelop simpl eleq2d mtbiri
      exlimivv sylbi ) CABFGCDHZEHZIZJZUBAGUCBGKZKZELDLMCGZNZDECABOUGUIDEUGUHMU
      DGUBUCPUGCUDMUEUFQRSTUA $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.
    $( Ordered pair membership in a cross product.  (Contributed by NM,
       15-Nov-1994.)  (Proof shortened by Andrew Salmon, 12-Aug-2011.)
       (Revised by Mario Carneiro, 26-Apr-2015.) $)
    opelxp $p |- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) ) $=
      ( vx vy cop cxp wcel cv wceq wrex wa elxp2 wb opth2 eleq1 bi2anan9 eqeq2d
      vex sylbi biimprcd rexlimivv eqid opeq1 opeq2 rspc2ev mp3an3 impbii bitri
      ) ABGZCDHIUKEJZFJZGZKZFDLECLZACIZBDIZMZEFUKCDNUPUSUOUSEFCDUOUSULCIZUMDIZM
      ZUOAULKZBUMKZMUSVBOABULUMETFTPVCUQUTVDURVAAULCQBUMDQRUAUBUCUQURUKUKKZUPUK
      UDUOVEUKAUMGZKEFABCDULAKUNVFUKULAUMUESUMBKVFUKUKUMBAUFSUGUHUIUJ $.

    $( Binary relation on a cross product.  (Contributed by NM,
       22-Apr-2004.) $)
    brxp $p |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) ) $=
      ( cxp wbr cop wcel wa df-br opelxp bitri ) ABCDEZFABGMHACHBDHIABMJABCDKL
      $.
  $}

  $( Ordered pair membership in a cross product (implication).  (Contributed by
     NM, 28-May-1995.) $)
  opelxpi $p |- ( ( A e. C /\ B e. D ) -> <. A , B >. e. ( C X. D ) ) $=
    ( cop cxp wcel wa opelxp biimpri ) ABECDFGACGBDGHABCDIJ $.

  ${
    opelxpd.1 $e |- ( ph -> A e. C ) $.
    opelxpd.2 $e |- ( ph -> B e. D ) $.
    $( Ordered pair membership in a Cartesian product, deduction form.
       (Contributed by Glauco Siliprandi, 3-Mar-2021.) $)
    opelxpd $p |- ( ph -> <. A , B >. e. ( C X. D ) ) $=
      ( wcel cop cxp opelxpi syl2anc ) ABDHCEHBCIDEJHFGBCDEKL $.
  $}

  $( The first member of an ordered pair of classes in a cross product belongs
     to first cross product argument.  (Contributed by NM, 28-May-2008.)
     (Revised by Mario Carneiro, 26-Apr-2015.) $)
  opelxp1 $p |- ( <. A , B >. e. ( C X. D ) -> A e. C ) $=
    ( cop cxp wcel opelxp simplbi ) ABECDFGACGBDGABCDHI $.

  $( The second member of an ordered pair of classes in a cross product belongs
     to second cross product argument.  (Contributed by Mario Carneiro,
     26-Apr-2015.) $)
  opelxp2 $p |- ( <. A , B >. e. ( C X. D ) -> B e. D ) $=
    ( cop cxp wcel opelxp simprbi ) ABECDFGACGBDGABCDHI $.

  $( The first member of an ordered triple of classes in a cross product
     belongs to first cross product argument.  (Contributed by NM,
     28-May-2008.) $)
  otelxp1 $p |- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T )
          -> A e. R ) $=
    ( cop cxp wcel opelxp1 syl ) ABGZCGDEHZFHILMIADILCMFJABDEJK $.

  ${
    $d A x z $.  $d A y $.  $d B x $.  $d B y z $.  $d ph x $.  $d ph y z $.
    $d ps z $.
    opabssxpd.x $e |- ( ( ph /\ ps ) -> x e. A ) $.
    opabssxpd.y $e |- ( ( ph /\ ps ) -> y e. B ) $.
    $( An ordered-pair class abstraction is a subset of a Cartesian product.
       Formerly part of proof for ~ opabex2 .  (Contributed by AV,
       26-Nov-2021.) $)
    opabssxpd $p |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) ) $=
      ( vz copab cv cop wceq wa wex cab cxp df-opab wcel simprl opelxpd adantrl
      eqeltrd ex exlimdvv abssdv eqsstrid ) ABCDJIKZCKZDKZLZMZBNZDOCOZIPEFQZBCD
      IRAUNIUOAUMUHUOSZCDAUMUPAUMNUHUKUOAULBTABUKUOSULABNUIUJEFGHUAUBUCUDUEUFUG
      $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d y z ph $.  $d x ps $.
    rabxp.1 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
    $( Membership in a class builder restricted to a cross product.
       (Contributed by NM, 20-Feb-2014.) $)
    rabxp $p |- { x e. ( A X. B ) | ph }
             = { <. y , z >. | ( y e. A /\ z e. B /\ ps ) } $=
      ( cv cxp wcel wa cab cop wceq w3a wex crab copab elxp anbi1i anass anbi2d
      19.41vv df-3an bitr4di pm5.32i bitri 2exbii 3bitr2i abbii df-opab 3eqtr4i
      df-rab ) CIZFGJZKZALZCMUODIZEIZNOZUSFKZUTGKZBPZLZEQDQZCMACUPRVDDESURVFCUR
      VAVBVCLZLZEQDQZALVHALZEQDQVFUQVIADEUOFGTUAVHADEUDVJVEDEVJVAVGALZLVEVAVGAU
      BVAVKVDVAVKVGBLVDVAABVGHUCVBVCBUEUFUGUHUIUJUKACUPUNVDDECULUM $.
  $}

  $( A true binary relation on a relation implies the arguments are sets.
     (This is a property of our ordered pair definition.)  (Contributed by
     Mario Carneiro, 26-Apr-2015.) $)
  brrelex12 $p |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) ) $=
    ( wrel wbr wa cvv cxp wcel wss df-rel biimpi ssbrd imp brxp sylib ) CDZABCE
    ZFABGGHZEZAGIBGIFQRTQCSABQCSJCKLMNABGGOP $.

  $( A true binary relation on a relation implies the first argument is a set.
     (This is a property of our ordered pair definition.)  (Contributed by NM,
     18-May-2004.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  brrelex1 $p |- ( ( Rel R /\ A R B ) -> A e. _V ) $=
    ( wrel wbr wa cvv wcel brrelex12 simpld ) CDABCEFAGHBGHABCIJ $.

  $( A true binary relation on a relation implies the first argument is a set.
     (This is a property of our ordered pair definition.)  (Contributed by NM,
     18-May-2004.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  brrelex $p |- ( ( Rel R /\ A R B ) -> A e. _V ) $=
    ( wrel wbr wa cvv wcel brrelex12 simpld ) CDABCEFAGHBGHABCIJ $.

  $( A true binary relation on a relation implies the second argument is a set.
     (This is a property of our ordered pair definition.)  (Contributed by
     Mario Carneiro, 26-Apr-2015.) $)
  brrelex2 $p |- ( ( Rel R /\ A R B ) -> B e. _V ) $=
    ( wrel wbr wa cvv wcel brrelex12 simprd ) CDABCEFAGHBGHABCIJ $.

  ${
    brrelexi.1 $e |- Rel R $.
    $( Two classes that are related by a binary relation are sets.  (An
       artifact of our ordered pair definition.)  (Contributed by BJ,
       3-Oct-2022.) $)
    brrelex12i $p |- ( A R B -> ( A e. _V /\ B e. _V ) ) $=
      ( wrel wbr cvv wcel wa brrelex12 mpan ) CEABCFAGHBGHIDABCJK $.

    $( The first argument of a binary relation exists.  (An artifact of our
       ordered pair definition.)  (Contributed by NM, 4-Jun-1998.) $)
    brrelex1i $p |- ( A R B -> A e. _V ) $=
      ( wrel wbr cvv wcel brrelex1 mpan ) CEABCFAGHDABCIJ $.

    $( The second argument of a binary relation exists.  (An artifact of our
       ordered pair definition.)  (Contributed by Mario Carneiro,
       26-Apr-2015.) $)
    brrelex2i $p |- ( A R B -> B e. _V ) $=
      ( wrel wbr cvv wcel brrelex2 mpan ) CEABCFBGHDABCIJ $.
  $}

  ${
    nprrel.1 $e |- Rel R $.
    nprrel.2 $e |- -. A e. _V $.
    $( No proper class is related to anything via any relation.  (Contributed
       by Roy F. Longton, 30-Jul-2005.) $)
    nprrel $p |- -. A R B $=
      ( wbr cvv wcel brrelex1i mto ) ABCFAGHEABCDIJ $.
  $}

  $( A binary relation does not contain the empty set.  (Contributed by AV,
     15-Nov-2021.) $)
  0nelrel $p |- ( Rel R -> (/) e/ R ) $=
    ( wrel wcel wnel cvv cxp wss df-rel biimpi 0nelxp a1i ssneldd df-nel sylibr
    c0 wn ) ABZOACPOADQAEEFZOQARGAHIORCPQEEJKLOAMN $.

  ${
    $d x y A $.  $d x y B $.
    $( Representation of a constant function using the mapping operation.
       (Note that ` x ` cannot appear free in ` B ` .)  (Contributed by NM,
       12-Oct-1999.)  (Revised by Mario Carneiro, 16-Nov-2013.) $)
    fconstmpt $p |- ( A X. { B } ) = ( x e. A |-> B ) $=
      ( vy cv wcel csn wa copab wceq cxp cmpt velsn anbi2i opabbii df-xp df-mpt
      3eqtr4i ) AEBFZDEZCGZFZHZADISTCJZHZADIBUAKABCLUCUEADUBUDSDCMNOADBUAPADBCQ
      R $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y z C $.  $d x y z R $.
    vtoclr.1 $e |- Rel R $.
    vtoclr.2 $e |- ( ( x R y /\ y R z ) -> x R z ) $.
    $( Variable to class conversion of transitive relation.  (Contributed by
       NM, 9-Jun-1998.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    vtoclr $p |- ( ( A R B /\ B R C ) -> A R C ) $=
      ( wbr wa wi cvv wcel brrelex2i cv wceq breq1 imbi12d breq2 anbi1d anbi12d
      brrelex1i jca imbi2d imbi1d anbi2d vtoclg vtocl2g syl2im imp pm2.43i ) DE
      GJZEFGJZKZDFGJZUMUNUOUPLZUMDMNZEMNZKUNFMNZUQUMURUSDEGHUCDEGHOUDEFGHOUTAPZ
      BPZGJZVBFGJZKZVAFGJZLZLUTDVBGJZVDKZUPLZLUTUQLABDEMMVADQZVGVJUTVKVEVIVFUPV
      KVCVHVDVADVBGRUAVADFGRSUEVBEQZVJUQUTVLVIUOUPVLVHUMVDUNVBEDGTVBEFGRUBUFUEV
      CVBCPZGJZKZVAVMGJZLVGCFMVMFQZVOVEVPVFVQVNVDVCVMFVBGTUGVMFVAGTSIUHUIUJUKUL
      $.
  $}

  $( Ordered pair membership in the universal class of ordered pairs.
     (Contributed by Mario Carneiro, 3-May-2015.) $)
  opelvvg $p |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( _V X. _V ) ) $=
    ( wcel cvv cop cxp elex opelxpi syl2an ) ACEAFEBFEABGFFHEBDEACIBDIABFFJK $.

  ${
    opelvv.1 $e |- A e. _V $.
    opelvv.2 $e |- B e. _V $.
    $( Ordered pair membership in the universal class of ordered pairs.
       (Contributed by NM, 22-Aug-2013.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    opelvv $p |- <. A , B >. e. ( _V X. _V ) $=
      ( cvv wcel cop cxp opelxpi mp2an ) AEFBEFABGEEHFCDABEEIJ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x D $.
    $( Justification theorem for an ordered pair definition that works for any
       classes, including proper classes.  This is a possible definition
       implied by the footnote in [Jech] p. 78, which says, "The sophisticated
       reader will not object to our use of a pair of classes."  (Contributed
       by NM, 28-Sep-2003.) $)
    opthprc $p |- ( ( ( A X. { (/) } ) u. ( B X. { { (/) } } ) ) =
                    ( ( C X. { (/) } ) u. ( D X. { { (/) } } ) )
                  <-> ( A = C /\ B = D ) ) $=
      ( vx c0 csn cxp cun wceq wa wcel wo opelxp mpbiran2 bianfi bitr4i orbi12i
      elun 3bitr4ri cv cop eleq2 0ex snid 0nep0 elsn nemtbir 3bitr4g eqrdv p0ex
      biorfi eqcom bitri wn wb biorf ax-mp jca xpeq1 uneq12 syl2an impbii ) AFG
      ZHZBVDGZHZIZCVDHZDVFHZIZJZACJZBDJZKVLVMVNVLEACVLEUAZFUBZVHLZVPVKLZVOALZVO
      CLZVHVKVPUCVPVELZVPVGLZMVSFVFLZMVQVSWAVSWBWCWAVSFVDLZFUDUEZVOFAVDNOWBVOBL
      ZWCKWCVOFBVFNWCWFWCFVDUFFVDUDUGUHZPQRVPVEVGSWCVSWGULTVPVILZVPVJLZMVTWCMVR
      VTWHVTWIWCWHVTWDWEVOFCVDNOWIVODLZWCKWCVOFDVFNWCWJWGPQRVPVIVJSWCVTWGULTUIU
      JVLEBDVLVOVDUBZVHLZWKVKLZWFWJVHVKWKUCWKVELZWKVGLZMVDVDLZWFMZWLWFWNWPWOWFW
      NVSWPKWPVOVDAVDNWPVSWPFVDUFWPVDFJFVDJVDFUKUGVDFUMUNUHZPQWOWFVDVFLZVDUKUEZ
      VOVDBVFNORWKVEVGSWPUOZWFWQUPWRWPWFUQURTWKVILZWKVJLZMWPWJMZWMWJXBWPXCWJXBV
      TWPKWPVOVDCVDNWPVTWRPQXCWJWSWTVOVDDVFNORWKVIVJSXAWJXDUPWRWPWJUQURTUIUJUSV
      MVEVIJVGVJJVLVNACVDUTBDVFUTVEVIVGVJVAVBVC $.
  $}

  ${
    brel.1 $e |- R C_ ( C X. D ) $.
    $( Two things in a binary relation belong to the relation's domain.
       (Contributed by NM, 17-May-1996.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    brel $p |- ( A R B -> ( A e. C /\ B e. D ) ) $=
      ( wbr cxp wcel wa ssbri brxp sylib ) ABEGABCDHZGACIBDIJENABFKABCDLM $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y ps $.
    brab2a.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    brab2a.2 $e |- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } $.
    $( Ordered pair membership in an ordered pair class abstraction.
       (Contributed by Mario Carneiro, 9-Nov-2015.) $)
    brab2a $p |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) ) $=
      ( wbr wcel wa cv copab cxp simpl ssopab2i df-xp 3sstr4i brel df-br eleq2i
      cop bitri opelopab2a bitrid biadan2 ) EFILZEGMFHMNZBEFGHICOGMDOHMNZANZCDP
      ZULCDPIGHQUMULCDULARSKCDGHTUAUBUJEFUEZUNMZUKBUJUOIMUPEFIUCIUNUOKUDUFABCDE
      FGHJUGUHUI $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Membership in a cross product.  (Contributed by NM, 5-Mar-1995.) $)
    elxp3 $p |- ( A e. ( B X. C ) <->
            E. x E. y ( <. x , y >. = A /\ <. x , y >. e. ( B X. C ) ) ) $=
      ( cxp wcel cv cop wceq wa wex elxp eqcom opelxp anbi12i 2exbii bitr4i ) C
      DEFZGCAHZBHZIZJZTDGUAEGKZKZBLALUBCJZUBSGZKZBLALABCDEMUHUEABUFUCUGUDUBCNTU
      ADEOPQR $.
  $}

  ${
    $d y z A $.  $d y z B $.  $d y z C $.  $d x y z $.
    $( Membership in a union of cross products.  (Contributed by Mario
       Carneiro, 29-Dec-2014.)  (Revised by Mario Carneiro, 1-Jan-2017.) $)
    opeliunxp $p |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <->
                     ( x e. A /\ C e. B ) ) $=
      ( vz vy cv cop csn cxp ciun wcel cvv elex wex wceq eleq2d anbi12d bitri
      wa opexg wsb csb wrex nfv nfs1v nfcv nfcsb1v nfxp nfcri nfan sbequ12 sneq
      df-rex csbeq1a cbvex eleq1 anbi2d exbidv bitrid df-iun elab2g opelxp an12
      xpeq12d anbi2i velsn equcom anbi1i 3bitri sbequ12r equcoms eqcomd ceqsexv
      exbii vex bitrdi pm5.21nii ) AGZDHZABVSIZCJZKZLZVTMLZVSBLZDCLZTZVTWCNVSDB
      CUAWEWDWFAEUBZVTEGZIZAWJCUCZJZLZTZEOZWHFGZWBLZABUDZWPFVTWCMWSWIWQWMLZTZEO
      ZWQVTPZWPWSWFWRTZAOXBWRABUNXDXAAEXDEUEWIWTAWFAEUFAFWMAWKWLAWKUGAWJCUHUIUJ
      UKVSWJPZWFWIWRWTWFAEULXEWBWMWQXEWAWKCWLVSWJUMAWJCUOZVEQRUPSXCXAWOEXCWTWNW
      IWQVTWMUQURUSUTAFBWBVAVBWPWJVSPZWIDWLLZTZTZEOWHWOXJEWOWIVSWKLZXHTZTXKXITX
      JWNXLWIVSDWKWLVCVFWIXKXHVDXKXGXIXKXEXGAWJVGAEVHSVIVJVOXIWHEVSAVPXGWIWFXHW
      GWFEAVKXGWLCDXGCWLCWLPAEXFVLVMQRVNSVQVR $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Distributive law for cross product over union.  Theorem 103 of [Suppes]
       p. 52.  (Contributed by NM, 12-Aug-2004.) $)
    xpundi $p |- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) ) $=
      ( vx vy cun cxp cv wcel wa copab df-xp uneq12i wo elun andi bitri opabbii
      anbi2i eqtr4i unopab ) ABCFZGDHAIZEHZUBIZJZDEKZABGZACGZFZDEAUBLUJUCUDBIZJ
      ZDEKZUCUDCIZJZDEKZFZUGUHUMUIUPDEABLDEACLMUGULUONZDEKUQUFURDEUFUCUKUNNZJUR
      UEUSUCUDBCOSUCUKUNPQRULUODEUATTT $.

    $( Distributive law for cross product over union.  Similar to Theorem 103
       of [Suppes] p. 52.  (Contributed by NM, 30-Sep-2002.) $)
    xpundir $p |- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) ) $=
      ( vx vy cun cxp cv wcel wa copab df-xp uneq12i wo elun anbi1i andir bitri
      opabbii eqtr4i unopab ) ABFZCGDHZUBIZEHCIZJZDEKZACGZBCGZFZDEUBCLUJUCAIZUE
      JZDEKZUCBIZUEJZDEKZFZUGUHUMUIUPDEACLDEBCLMUGULUONZDEKUQUFURDEUFUKUNNZUEJU
      RUDUSUEUCABOPUKUNUEQRSULUODEUATTT $.
  $}

  ${
    $d w y z A $.  $d w y z B $.  $d w x y z C $.
    $( Distributive law for cross product over indexed union.  (Contributed by
       Mario Carneiro, 27-Apr-2014.) $)
    xpiundi $p |- ( C X. U_ x e. A B ) = U_ x e. A ( C X. B ) $=
      ( vz vw vy ciun cxp cv wrex wcel wa wex eliun exbii df-rex rexbii 3bitr4i
      elxp2 cop wceq rexcom anbi1i rexcom4 r19.41v 3bitri eqriv ) EDABCHZIZABDC
      IZHZEJZFJGJZUAUBZGUIKZFDKZUMUKLZABKZUMUJLUMULLUOGCKZABKZFDKUTFDKZABKUQUSU
      TFADBUCUPVAFDUNUILZUOMZGNUNCLZABKZUOMZGNZUPVAVDVGGVCVFUOAUNBCOUDPUOGUIQVA
      VEUOMZGNZABKVIABKZGNVHUTVJABUOGCQRVIAGBUEVKVGGVEUOABUFPUGSRURVBABFGUMDCTR
      SFGUMDUITAUMBUKOSUH $.

    $( Distributive law for cross product over indexed union.  (Contributed by
       Mario Carneiro, 27-Apr-2014.) $)
    xpiundir $p |- ( U_ x e. A B X. C ) = U_ x e. A ( B X. C ) $=
      ( vz vy vw ciun cxp cv cop wrex wcel wa df-rex rexbii eliun elxp2 3bitr4i
      wex wceq rexcom4 anbi1i r19.41v bitr4i exbii 3bitr4ri eqriv ) EABCHZDIZAB
      CDIZHZEJZFJZGJKUAGDLZFUILZUMUKMZABLZUMUJMUMULMUNUIMZUONZFTZUOFCLZABLZUPUR
      UNCMZUONZFTZABLVEABLZFTVCVAVEAFBUBVBVFABUOFCOPUTVGFUTVDABLZUONVGUSVHUOAUN
      BCQUCVDUOABUDUEUFUGUOFUIOUQVBABFGUMCDRPSFGUMUIDRAUMBUKQSUH $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Membership in a union of cross products when the second factor is
       constant.  (Contributed by Mario Carneiro, 29-Dec-2014.) $)
    iunxpconst $p |- U_ x e. A ( { x } X. B ) = ( A X. B ) $=
      ( cv csn ciun cxp xpiundir iunid xpeq1i eqtr3i ) ABADEZFZCGABLCGFBCGABLCH
      MBCABIJK $.
  $}

  $( The cross product of two unions.  (Contributed by NM, 12-Aug-2004.) $)
  xpun $p |- ( ( A u. B ) X. ( C u. D ) ) =
            ( ( ( A X. C ) u. ( A X. D ) ) u. ( ( B X. C ) u. ( B X. D ) ) ) $=
    ( cun cxp xpundi xpundir uneq12i un4 3eqtri ) ABEZCDEFLCFZLDFZEACFZBCFZEZAD
    FZBDFZEZEOREPSEELCDGMQNTABCHABDHIOPRSJK $.

  ${
    $d w x y z A $.
    $( Membership in universal class of ordered pairs.  (Contributed by NM,
       4-Jul-1994.) $)
    elvv $p |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. ) $=
      ( cvv cxp wcel cv cop wceq wa wex elxp vex pm3.2i biantru 2exbii bitr4i )
      CDDEFCAGZBGZHIZRDFZSDFZJZJZBKAKTBKAKABCDDLTUDABUCTUAUBAMBMNOPQ $.

    $( Membership in universal class of ordered triples.  (Contributed by NM,
       17-Dec-2008.) $)
    elvvv $p |- ( A e. ( ( _V X. _V ) X. _V )
                 <-> E. x E. y E. z A = <. <. x , y >. , z >. ) $=
      ( vw cvv cxp wcel cv cop wceq wa wex elxp anass 19.42vv 2exbii vex bitr3i
      bitri ancom biantru elvv anbi2i 3bitr4ri exrot4 excom opex eqeq2d ceqsexv
      opeq1 exbii ) DFFGZFGHDEIZCIZJZKZUNUMHZUOFHZLLZCMEMZDAIZBIZJZUOJZKZCMZBMA
      MZECDUMFNVAUNVDKZUQLZBMAMZCMEMZVHUTVKECUTUQURLZUSLZVKUQURUSOUQVILZBMAMUQV
      IBMAMZLZVKVNUQVIABPVJVOABVIUQUAQVNVMVQUSVMCRUBURVPUQABUNUCUDSUESQVLVJCMEM
      ZBMAMVHVJABECUFVRVGABVRVJEMZCMVGVJECUGVSVFCUQVFEVDVBVCARBRUHVIUPVEDUNVDUO
      UKUIUJULTQSTT $.

    $( An ordered pair contains its union.  (Contributed by NM,
       16-Sep-2006.) $)
    elvvuni $p |- ( A e. ( _V X. _V ) -> U. A e. A ) $=
      ( vx vy cvv cxp wcel cv cop wceq wex cuni elvv cpr vex uniop opi2 eqeltri
      unieq id eleq12d mpbiri exlimivv sylbi ) ADDEFABGZCGZHZIZCJBJAKZAFZBCALUG
      UIBCUGUIUFKZUFFUJUDUEMUFUDUEBNZCNZOUDUEUKULPQUGUHUJAUFAUFRUGSTUAUBUC $.
  $}

  ${
    $d x y z A $.
    $( "At most one" remains true inside ordered pair quantification.
       (Contributed by NM, 28-Aug-2007.) $)
    mosubopt $p |- ( A. y A. z E* x ph ->
                 E* x E. y E. z ( A = <. y , z >. /\ ph ) ) $=
      ( wmo wal cv cop wceq wex wa wi nfa1 nfe1 nfmo nfex copsexg sps exlimd
      mobidv biimpcd moanimv simpl 2eximi ancri moimi sylbir syl ) ABFZDGZCGZEC
      HDHIJZDKZCKZUMALZDKZCKZBFZMZUSULUNUSCUKCNURCBUQCOPUKUNUSMCUKUMUSDUJDNURDB
      UQDCUPDOQPUJUMUSMDUMUJUSUMAURBACDERUAUBSTSTUTUOURLZBFUSUOURBUCURVABURUOUP
      UMCDUMAUDUEUFUGUHUI $.
  $}

  ${
    $d x y z A $.
    mosubop.1 $e |- E* x ph $.
    $( "At most one" remains true inside ordered pair quantification.
       (Contributed by NM, 28-May-1995.) $)
    mosubop $p |- E* x E. y E. z ( A = <. y , z >. /\ ph ) $=
      ( wmo wal cv cop wceq wa wex gen2 mosubopt ax-mp ) ABGZDHCHECIDIJKALDMCMB
      GQCDFNABCDEOP $.
  $}

  $( Intersection of binary relation with Cartesian product.  (Contributed by
     NM, 3-Mar-2007.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  brinxp2 $p |- ( A ( R i^i ( C X. D ) ) B <->
                ( A e. C /\ B e. D /\ A R B ) ) $=
    ( cxp cin wbr wa wcel w3a brin ancom brxp anbi1i df-3an bitr4i 3bitri ) ABE
    CDFZGHABEHZABSHZIUATIZACJZBDJZTKZABESLTUAMUBUCUDIZTIUEUAUFTABCDNOUCUDTPQR
    $.

  $( Intersection of binary relation with Cartesian product.  (Contributed by
     NM, 9-Mar-1997.) $)
  brinxp $p |- ( ( A e. C /\ B e. D ) ->
               ( A R B <-> A ( R i^i ( C X. D ) ) B ) ) $=
    ( cxp cin wbr wcel wa w3a brinxp2 df-3an bitri baibr ) ABECDFGHZACIZBDIZJZA
    BEHZPQRTKSTJABCDELQRTMNO $.

  ${
    $d x y z A $.  $d x y z R $.
    $( Intersection of partial order with cross product of its field.
       (Contributed by Mario Carneiro, 10-Jul-2014.) $)
    poinxp $p |- ( R Po A <-> ( R i^i ( A X. A ) ) Po A ) $=
      ( vx vy vz cv wbr wn wa wi wral cxp cin wpo wcel wb brinxp ralbidva df-po
      anbi12d simpll syl2anc notbid adantll adantlr imbi12d ralbiia 3bitr4i
      adantr ) CFZUJBGZHZUJDFZBGZUMEFZBGZIZUJUOBGZJZIZEAKZDAKZCAKUJUJBAALMZGZHZ
      UJUMVCGZUMUOVCGZIZUJUOVCGZJZIZEAKZDAKZCAKABNAVCNVBVMCAUJAOZVAVLDAVNUMAOZI
      ZUTVKEAVPUOAOZIZULVEUSVJVRUKVDVRVNVNUKVDPVNVOVQUAZVSUJUJAABQUBUCVRUQVHURV
      IVRUNVFUPVGVPUNVFPVQUJUMAABQUIVOVQUPVGPVNUMUOAABQUDTVNVQURVIPVOUJUOAABQUE
      UFTRRUGCDEABSCDEAVCSUH $.

    $( Intersection of linear order with cross product of its field.
       (Contributed by Mario Carneiro, 10-Jul-2014.) $)
    soinxp $p |- ( R Or A <-> ( R i^i ( A X. A ) ) Or A ) $=
      ( vx vy vz wpo cv wbr wo wi wral wa cxp cin wor poinxp wcel brinxp df-iso
      wb 3adant3 3adant2 ancoms 3adant1 orbi12d imbi12d 3expb 2ralbidva ralbiia
      w3a anbi12i 3bitr4i ) ABFZCGZDGZBHZUNEGZBHZUQUOBHZIZJZEAKDAKZCAKZLABAAMNZ
      FZUNUOVDHZUNUQVDHZUQUOVDHZIZJZEAKDAKZCAKZLABOAVDOUMVEVCVLABPVBVKCAUNAQZVA
      VJDEAAVMUOAQZUQAQZVAVJTVMVNVOUJZUPVFUTVIVMVNUPVFTVOUNUOAABRUAVPURVGUSVHVM
      VOURVGTVNUNUQAABRUBVNVOUSVHTZVMVOVNVQUQUOAABRUCUDUEUFUGUHUIUKCDEABSCDEAVD
      SUL $.

    $( Intersection of set-like relation with cross product of its field.
       (Contributed by Mario Carneiro, 22-Jun-2015.) $)
    seinxp $p |- ( R Se A <-> ( R i^i ( A X. A ) ) Se A ) $=
      ( vy vx cv wbr crab cvv wcel cxp cin wse wb brinxp ancoms rabbidva eleq1d
      wral ralbiia df-se 3bitr4i ) CEZDEZBFZCAGZHIZDARUBUCBAAJKZFZCAGZHIZDARABL
      AUGLUFUJDAUCAIZUEUIHUKUDUHCAUBAIUKUDUHMUBUCAABNOPQSDCABTDCAUGTUA $.
  $}

  ${
    $d x y z A $.  $d x y z R $.
    $( Partial ordering of a singleton.  (Contributed by Jim Kingdon,
       5-Dec-2018.) $)
    posng $p |- ( ( Rel R /\ A e. _V ) -> ( R Po { A } <-> -. A R A ) ) $=
      ( vz vy vx csn cv wbr wn wa wi wral cvv wb wceq breq2 anbi2d ralsng bitrd
      ralbidv wpo wrel wcel df-po imbi12d simpl imbitrid biantrud bicomd breq12
      anidms notbid adantl bitrid ) AFZBUACGZUPBHZIZUPDGZBHZUSEGZBHZJZUPVABHZKZ
      JZEUOLZDUOLZCUOLZBUBZAMUCZJAABHZIZCDEUOBUDVKVIVMNVJVKVIURCUOLVMVKVHURCUOV
      KVHURUTUSABHZJZUPABHZKZJZDUOLURVKVGVRDUOVFVREAMVAAOZVEVQURVSVCVOVDVPVSVBV
      NUTVAAUSBPQVAAUPBPUEQRTVRURDAMUSAOZURVRVTVQURVOUTVTVPUTVNUFUSAUPBPUGUHUIR
      STURVMCAMUPAOZUQVLWAUQVLNUPAUPABUJUKULRSUMUN $.

    $( Strict linear ordering on a singleton.  (Contributed by Jim Kingdon,
       5-Dec-2018.) $)
    sosng $p |- ( ( Rel R /\ A e. _V ) -> ( R Or { A } <-> -. A R A ) ) $=
      ( vx vy vz wrel cvv wcel wa csn wor wbr wn wpo sopo cv wral elsni adantl
      wi posng imbitrid biimpar ax-in2 adantr breqan12d imbi1d mpbird ralrimivw
      wo wb ralrimivva df-iso sylanbrc ex impbid ) BFAGHIZAJZBKZAABLZMZUSURBNZU
      QVAURBOABUAZUBUQVAUSUQVAIVBCPZDPZBLZVDEPZBLVGVEBLUJZTZEURQZDURQCURQZUSUQV
      BVAVCUCVAVKUQVAVJCDURURVAVDURHZVEURHZIZIZVIEURVOVIUTVHTZVAVPVNUTVHUDUEVNV
      IVPUKVAVNVFUTVHVLVMVDAVEABVDARVEARUFUGSUHUIULSCDEURBUMUNUOUP $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( An abstraction relation is a subset of a related cross product.
       (Contributed by NM, 16-Jul-1995.) $)
    opabssxp $p |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ph ) }
                   C_ ( A X. B ) $=
      ( cv wcel wa copab cxp simpl ssopab2i df-xp sseqtrri ) BFDGCFEGHZAHZBCIOB
      CIDEJPOBCOAKLBCDEMN $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y ps $.
    brab2ga.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    brab2ga.2 $e |- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } $.
    $( The law of concretion for a binary relation.  See ~ brab2a for alternate
       proof.  TODO: should one of them be deleted?  (Contributed by Mario
       Carneiro, 28-Apr-2015.)  (Proof modification is discouraged.) $)
    brab2ga $p |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) ) $=
      ( wbr wcel wa cv copab cxp opabssxp eqsstri brel eleq2i opelopab2a bitrid
      cop df-br bitri biadan2 ) EFILZEGMFHMNZBEFGHIICOGMDOHMNANCDPZGHQKACDGHRST
      UHEFUDZUJMZUIBUHUKIMULEFIUEIUJUKKUAUFABCDEFGHJUBUCUG $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y ps $.
    optocl.1 $e |- D = ( B X. C ) $.
    optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
    optocl.3 $e |- ( ( x e. B /\ y e. C ) -> ph ) $.
    $( Implicit substitution of class for ordered pair.  (Contributed by NM,
       5-Mar-1995.) $)
    optocl $p |- ( A e. D -> ps ) $=
      ( cxp wcel cv cop wceq wa wex elxp3 sylbi opelxp imbitrid exlimivv eleq2s
      imp ) BEFGLZHEUFMCNZDNZOZEPZUIUFMZQZDRCRBCDEFGSULBCDUJUKBUKAUJBUKUGFMUHGM
      QAUGUHFGUAKTJUBUEUCTIUD $.
  $}

  ${
    $d x y z w A $.  $d z w B $.  $d x y z w C $.  $d x y z w D $.
    $d x y ps $.  $d z w ch $.  $d z w R $.
    2optocl.1 $e |- R = ( C X. D ) $.
    2optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
    2optocl.3 $e |- ( <. z , w >. = B -> ( ps <-> ch ) ) $.
    2optocl.4 $e |- ( ( ( x e. C /\ y e. D ) /\ ( z e. C /\ w e. D ) ) ->
                   ph ) $.
    $( Implicit substitution of classes for ordered pairs.  (Contributed by NM,
       12-Mar-1995.) $)
    2optocl $p |- ( ( A e. R /\ B e. R ) -> ch ) $=
      ( wcel wi cv cop wceq imbi2d wa ex optocl com12 impcom ) ILQHLQZCUHBRUHCR
      FGIJKLMFSZGSZTIUABCUHOUBUHUIJQUJKQUCZBUKARUKBRDEHJKLMDSZESZTHUAABUKNUBULJ
      QUMKQUCUKAPUDUEUFUEUG $.
  $}

  ${
    $d x y z w v u A $.  $d z w v u B $.  $d v u C $.  $d x y z w v u D $.
    $d x y z w v u F $.  $d z w v u R $.  $d x y ps $.  $d z w ch $.
    $d v u th $.
    3optocl.1 $e |- R = ( D X. F ) $.
    3optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
    3optocl.3 $e |- ( <. z , w >. = B -> ( ps <-> ch ) ) $.
    3optocl.4 $e |- ( <. v , u >. = C -> ( ch <-> th ) ) $.
    3optocl.5 $e |- ( ( ( x e. D /\ y e. F ) /\ ( z e. D /\ w e. F )
                    /\ ( v e. D /\ u e. F ) ) -> ph ) $.
    $( Implicit substitution of classes for ordered pairs.  (Contributed by NM,
       12-Mar-1995.) $)
    3optocl $p |- ( ( A e. R /\ B e. R /\ C e. R ) -> th ) $=
      ( wcel wa wi cv cop wceq imbi2d 3expia 2optocl com12 optocl impcom 3impa
      ) KOUBZLOUBZMOUBZDUQUOUPUCZDURCUDURDUDIJMNPOQIUEZJUEZUFMUGCDURTUHURUSNUBU
      TPUBUCZCVAAUDVABUDVACUDEFGHKLNPOQEUEZFUEZUFKUGABVARUHGUEZHUEZUFLUGBCVASUH
      VBNUBVCPUBUCVDNUBVEPUBUCVAAUAUIUJUKULUMUN $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u D $.  $d x y z w v u S $.  $d x y ph $.  $d z w v u ps $.
    opbrop.1 $e |- ( ( ( z = A /\ w = B ) /\ ( v = C /\ u = D ) ) ->
                     ( ph <-> ps ) ) $.
    opbrop.2 $e |- R = { <. x , y >. | ( ( x e. ( S X. S ) /\
                      y e. ( S X. S ) ) /\
                      E. z E. w E. v E. u ( ( x = <. z , w >. /\
                      y = <. v , u >. ) /\ ph ) ) } $.
    $( Ordered pair membership in a relation.  Special case.  (Contributed by
       NM, 5-Aug-1995.) $)
    opbrop $p |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
                   ( <. A , B >. R <. C , D >. <-> ps ) ) $=
      ( wcel wa cv wex cop cxp wbr copsex4g anbi2d cvv opexg eleq1 anbi1d eqeq1
      wceq wb 4exbidv anbi12d brabg syl2an opelxpi anim12i biantrurd 3bitr4d )
      INQJNQRZKNQLNQRZRZIJUAZNNUBZQZKLUAZVEQZRZVDESFSUAZUKZVGGSHSUAZUKZRZARZHTG
      TFTETZRZVIBRVDVGMUCZBVCVPBVIABEFGHIJKLNNOUDUEVAVDUFQVGUFQVRVQULVBIJNNUGKL
      NNUGCSZVEQZDSZVEQZRZVSVJUKZWAVLUKZRZARZHTGTFTETZRVFWBRZVKWERZARZHTGTFTETZ
      RVQCDVDVGUFUFMVSVDUKZWCWIWHWLWMVTVFWBVSVDVEUHUIWMWGWKEFGHWMWFWJAWMWDVKWEV
      SVDVJUJUIUIUMUNWAVGUKZWIVIWLVPWNWBVHVFWAVGVEUHUEWNWKVOEFGHWNWJVNAWNWEVMVK
      WAVGVLUJUEUIUMUNPUOUPVCVIBVAVFVBVHIJNNUQKLNNUQURUSUT $.
  $}

  ${
    $d x y z A $.
    $( The cross product with the empty set is empty.  Part of Theorem 3.13(ii)
       of [Monk1] p. 37.  (Contributed by NM, 4-Jul-1994.) $)
    0xp $p |- ( (/) X. A ) = (/) $=
      ( vz vx vy c0 cxp cv wcel cop wceq wa wex elxp noel simprl mto nex 2false
      bitri eqriv ) BEAFZEBGZUAHUBCGZDGZIJZUCEHZUDAHZKKZDLZCLZUBEHZCDUBEAMUJUKU
      ICUHDUHUFUCNUEUFUGOPQQUBNRST $.
  $}

  ${
    $d A w y z $.  $d B w y z $.  $d C w y z $.  $d D w y z $.  $d w x y z $.
    $( Distribute proper substitution through the cross product of two classes.
       (Contributed by Alan Sare, 10-Nov-2012.) $)
    csbxpg $p |- ( A e. D -> [_ A / x ]_ ( B X. C ) =
                ( [_ A / x ]_ B X. [_ A / x ]_ C ) ) $=
      ( vz vw vy wcel cv wa wex cab csb cxp wsbc sbcexg sbcang sbcel2g bitrd
      wceq csbabg anbi12d exbidv abbidv eqtrd copab df-xp df-opab eqtri csbeq2i
      cop sbcg 3eqtr4g ) BEIZABFJGJZHJZULUAZUPCIZUQDIZKZKZHLZGLZFMZNZURUPABCNZI
      ZUQABDNZIZKZKZHLZGLZFMZABCDOZNVGVIOZUOVFVDABPZFMVOVDAFBEUBUOVRVNFUOVRVCAB
      PZGLVNVCGABEQUOVSVMGUOVSVBABPZHLVMVBHABEQUOVTVLHUOVTURABPZVAABPZKVLURVAAB
      ERUOWAURWBVKURABEUMUOWBUSABPZUTABPZKVKUSUTABERUOWCVHWDVJABUPCESABUQDESUCT
      UCTUDTUDTUEUFABVPVEVPVAGHUGVEGHCDUHVAGHFUIUJUKVQVKGHUGVOGHVGVIUHVKGHFUIUJ
      UN $.
  $}

  $( Equality theorem for the relation predicate.  (Contributed by NM,
     1-Aug-1994.) $)
  releq $p |- ( A = B -> ( Rel A <-> Rel B ) ) $=
    ( wceq cvv cxp wss wrel sseq1 df-rel 3bitr4g ) ABCADDEZFBKFAGBGABKHAIBIJ $.

  ${
    releqi.1 $e |- A = B $.
    $( Equality inference for the relation predicate.  (Contributed by NM,
       8-Dec-2006.) $)
    releqi $p |- ( Rel A <-> Rel B ) $=
      ( wceq wrel wb releq ax-mp ) ABDAEBEFCABGH $.
  $}

  ${
    releqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for the relation predicate.  (Contributed by NM,
       8-Mar-2014.) $)
    releqd $p |- ( ph -> ( Rel A <-> Rel B ) ) $=
      ( wceq wrel wb releq syl ) ABCEBFCFGDBCHI $.
  $}

  ${
    nfrel.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for a relation.  (Contributed by NM,
       31-Jan-2004.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nfrel $p |- F/ x Rel A $=
      ( wrel cvv cxp wss df-rel nfcv nfss nfxfr ) BDBEEFZGABHABLCALIJK $.
  $}

  $( Distribute proper substitution through a relation predicate.  (Contributed
     by Alexander van der Vekens, 23-Jul-2017.) $)
  sbcrel $p |- ( A e. V -> ( [. A / x ]. Rel R <-> Rel [_ A / x ]_ R ) ) $=
    ( wcel cvv cxp wss wsbc sbcssg csbconstg sseq2d bitrd df-rel sbcbii 3bitr4g
    csb wrel ) BDEZCFFGZHZABIZABCQZTHZCRZABIUCRSUBUCABTQZHUDABCTDJSUFTUCABTDKLM
    UEUAABCNOUCNP $.

  $( Subclass theorem for relation predicate.  Theorem 2 of [Suppes] p. 58.
     (Contributed by NM, 15-Aug-1994.) $)
  relss $p |- ( A C_ B -> ( Rel B -> Rel A ) ) $=
    ( wss cvv cxp wrel sstr2 df-rel 3imtr4g ) ABCBDDEZCAJCBFAFABJGBHAHI $.

  ${
    $d x y z A $.  $d x y z B $.
    $( A subclass relationship depends only on a relation's ordered pairs.
       Theorem 3.2(i) of [Monk1] p. 33.  (Contributed by NM, 2-Aug-1994.)
       (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    ssrel $p |- ( Rel A -> ( A C_ B <->
                A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) ) ) $=
      ( vz wrel wss cv cop wcel wi wal ssel alrimivv wceq wex eleq1 imbi12d cvv
      ssalel biimprcd 2alimi 19.23vv sylib com23 a2d alimdv df-rel imbi2i albii
      cxp elvv 3bitri 3imtr4g com12 impbid2 ) CFZCDGZAHBHIZCJZUSDJZKZBLALZURVBA
      BCDUSMNVCUQURVCEHZCJZVDUSOZBPAPZKZELZVEVDDJZKZELUQURVCVHVKEVCVEVGVJVCVGVE
      VJVCVFVKKZBLALVGVKKVBVLABVFVKVBVFVEUTVJVAVDUSCQVDUSDQRUAUBVFVKABUCUDUEUFU
      GUQCSSUKZGVEVDVMJZKZELVICUHECVMTVOVHEVNVGVEABVDULUIUJUMECDTUNUOUP $.

    $( Extensionality principle for relations.  Theorem 3.2(ii) of [Monk1]
       p. 33.  (Contributed by NM, 2-Aug-1994.) $)
    eqrel $p |- ( ( Rel A /\ Rel B ) -> ( A = B <->
                A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) ) $=
      ( wrel wa wss cv cop wcel wi wal wceq ssrel bi2anan9 eqss 2albiim 3bitr4g
      wb ) CEZDEZFCDGZDCGZFAHBHIZCJZUDDJZKBLALZUFUEKBLALZFCDMUEUFSBLALTUBUGUAUC
      UHABCDNABDCNOCDPUEUFABQR $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z R $.  $d x y z S $.
    $( A subclass relationship depends only on a relation's ordered pairs.
       This version of ~ ssrel is restricted to the relation's domain.
       (Contributed by Thierry Arnoux, 25-Jan-2018.) $)
    ssrel2 $p |- ( R C_ ( A X. B ) -> ( R C_ S <-> A. x e. A A. y e. B
      ( <. x , y >. e. R -> <. x , y >. e. S ) ) ) $=
      ( vz cxp wss cv wcel wi wral wrex wal eleq1 ralimi r19.23v bitri ssalel
      cop wa ssel a1d ralrimivv wceq imbi12d biimprcd ralbii sylib com23 alimdv
      a2d elxp2 imbi2i albii 3imtr4g com12 impbid2 ) ECDHZIZEFIZAJZBJZUAZEKZVEF
      KZLZBDMZACMZVBVHABCDVBVHVCCKVDDKUBEFVEUCUDUEVJVAVBVJGJZEKZVKVEUFZBDNZACNZ
      LZGOZVLVKFKZLZGOVAVBVJVPVSGVJVLVOVRVJVOVLVRVJVMVSLZBDMZACMZVOVSLZVIWAACVH
      VTBDVMVSVHVMVLVFVRVGVKVEEPVKVEFPUGUHQQWBVNVSLZACMWCWAWDACVMVSBDRUIVNVSACR
      SUJUKUMULVAVLVKUTKZLZGOVQGEUTTWFVPGWEVOVLABVKCDUNUOUPSGEFTUQURUS $.
  $}

  ${
    $d x y A $.  $d x y B $.
    relssi.1 $e |- Rel A $.
    relssi.2 $e |- ( <. x , y >. e. A -> <. x , y >. e. B ) $.
    $( Inference from subclass principle for relations.  (Contributed by NM,
       31-Mar-1998.) $)
    relssi $p |- A C_ B $=
      ( wss cv cop wcel wi wal wrel wb ssrel ax-mp ax-gen mpgbir ) CDGZAHBHIZCJ
      TDJKZBLZACMSUBALNEABCDOPUABFQR $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ph $.
    relssdv.1 $e |- ( ph -> Rel A ) $.
    relssdv.2 $e |- ( ph -> ( <. x , y >. e. A -> <. x , y >. e. B ) ) $.
    $( Deduction from subclass principle for relations.  (Contributed by NM,
       11-Sep-2004.) $)
    relssdv $p |- ( ph -> A C_ B ) $=
      ( wss cv cop wcel wi wal alrimivv wrel wb ssrel syl mpbird ) ADEHZBICIJZD
      KUAEKLZCMBMZAUBBCGNADOTUCPFBCDEQRS $.
  $}

  ${
    $d x y A $.  $d x y B $.
    eqrelriv.1 $e |- ( <. x , y >. e. A <-> <. x , y >. e. B ) $.
    $( Inference from extensionality principle for relations.  (Contributed by
       FL, 15-Oct-2012.) $)
    eqrelriv $p |- ( ( Rel A /\ Rel B ) -> A = B ) $=
      ( wrel wa wceq cv cop wcel wb wal gen2 eqrel mpbiri ) CFDFGCDHAIBIJZCKQDK
      LZBMAMRABENABCDOP $.
  $}

  ${
    $d x y A $.  $d x y B $.
    eqreliiv.1 $e |- Rel A $.
    eqreliiv.2 $e |- Rel B $.
    eqreliiv.3 $e |- ( <. x , y >. e. A <-> <. x , y >. e. B ) $.
    $( Inference from extensionality principle for relations.  (Contributed by
       NM, 17-Mar-1995.) $)
    eqrelriiv $p |- A = B $=
      ( wrel wceq eqrelriv mp2an ) CHDHCDIEFABCDGJK $.
  $}

  ${
    $d x y A $.  $d x y B $.
    eqbrriv.1 $e |- Rel A $.
    eqbrriv.2 $e |- Rel B $.
    eqbrriv.3 $e |- ( x A y <-> x B y ) $.
    $( Inference from extensionality principle for relations.  (Contributed by
       NM, 12-Dec-2006.) $)
    eqbrriv $p |- A = B $=
      ( cv wbr cop wcel df-br 3bitr3i eqrelriiv ) ABCDEFAHZBHZCIOPDIOPJZCKQDKGO
      PCLOPDLMN $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d ph x $.  $d ph y $.
    eqrelrdv.1 $e |- Rel A $.
    eqrelrdv.2 $e |- Rel B $.
    eqrelrdv.3 $e |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) ) $.
    $( Deduce equality of relations from equivalence of membership.
       (Contributed by Rodolfo Medina, 10-Oct-2010.) $)
    eqrelrdv $p |- ( ph -> A = B ) $=
      ( cv cop wcel wb wal wceq alrimivv wrel eqrel mp2an sylibr ) ABICIJZDKTEK
      LZCMBMZDENZAUABCHODPEPUCUBLFGBCDEQRS $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d ph x $.  $d ph y $.
    eqbrrdv.1 $e |- ( ph -> Rel A ) $.
    eqbrrdv.2 $e |- ( ph -> Rel B ) $.
    eqbrrdv.3 $e |- ( ph -> ( x A y <-> x B y ) ) $.
    $( Deduction from extensionality principle for relations.  (Contributed by
       Mario Carneiro, 3-Jan-2017.) $)
    eqbrrdv $p |- ( ph -> A = B ) $=
      ( wceq cv cop wcel wb wal wbr df-br 3bitr3g alrimivv wrel eqrel syl2anc
      mpbird ) ADEIZBJZCJZKZDLZUFELZMZCNBNZAUIBCAUDUEDOUDUEEOUGUHHUDUEDPUDUEEPQ
      RADSESUCUJMFGBCDETUAUB $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d ph x $.  $d ph y $.
    eqbrrdiv.1 $e |- Rel A $.
    eqbrrdiv.2 $e |- Rel B $.
    eqbrrdiv.3 $e |- ( ph -> ( x A y <-> x B y ) ) $.
    $( Deduction from extensionality principle for relations.  (Contributed by
       Rodolfo Medina, 10-Oct-2010.) $)
    eqbrrdiv $p |- ( ph -> A = B ) $=
      ( cv wbr cop wcel df-br 3bitr3g eqrelrdv ) ABCDEFGABIZCIZDJPQEJPQKZDLRELH
      PQDMPQEMNO $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d ph x $.  $d ph y $.
    eqrelrdv2.1 $e |- ( ph
       -> ( <. x , y >. e. A <-> <. x , y >. e. B ) ) $.
    $( A version of ~ eqrelrdv .  (Contributed by Rodolfo Medina,
       10-Oct-2010.) $)
    eqrelrdv2 $p |- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A = B ) $=
      ( wrel wa wceq cv cop wcel wb wal alrimivv adantl eqrel adantr mpbird ) D
      GEGHZAHDEIZBJCJKZDLUBELMZCNBNZAUDTAUCBCFOPTUAUDMABCDEQRS $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.
    $( A subclass relationship determined by ordered triples.  Use ~ relrelss
       to express the antecedent in terms of the relation predicate.
       (Contributed by NM, 17-Dec-2008.)  (Proof shortened by Andrew Salmon,
       27-Aug-2011.) $)
    ssrelrel $p |- ( A C_ ( ( _V X. _V ) X. _V ) -> ( A C_ B <->
                   A. x A. y A. z ( <. <. x , y >. , z >. e. A
                       -> <. <. x , y >. , z >. e. B ) ) ) $=
      ( vw cvv cxp wss cv cop wcel wi wal ssel alrimiv wex eleq1 sylib ssalel
      alrimivv wceq elvvv imbi12d biimprcd alimi 19.23v 2alimi 19.23vv biimtrid
      com23 a2d alimdv 3imtr4g com12 impbid2 ) DGGHGHZIZDEIZAJBJKCJKZDLZUTELZMZ
      CNZBNANZUSVDABUSVCCDEUTOPUAVEURUSVEFJZDLZVFUQLZMZFNVGVFELZMZFNURUSVEVIVKF
      VEVGVHVJVEVHVGVJVHVFUTUBZCQZBQAQZVEVKABCVFUCVEVMVKMZBNANVNVKMVDVOABVDVLVK
      MZCNVOVCVPCVLVKVCVLVGVAVJVBVFUTDRVFUTERUDUEUFVLVKCUGSUHVMVKABUISUJUKULUMF
      DUQTFDETUNUOUP $.

    $( Extensionality principle for ordered triples, analogous to ~ eqrel .
       Use ~ relrelss to express the antecedent in terms of the relation
       predicate.  (Contributed by NM, 17-Dec-2008.) $)
    eqrelrel $p |- ( ( A u. B ) C_ ( ( _V X. _V ) X. _V ) -> ( A = B <->
                   A. x A. y A. z ( <. <. x , y >. , z >. e. A
                       <-> <. <. x , y >. , z >. e. B ) ) ) $=
      ( cun cvv cxp wss wa wceq cv cop wcel wb wal unss wi ssrelrel bi2anan9
      eqss 2albiim albii 19.26 bitri 3bitr4g sylbir ) DEFGGHGHZIDUHIZEUHIZJZDEK
      ZALBLMCLMZDNZUMENZOCPBPZAPZODEUHQUKDEIZEDIZJUNUORCPBPZAPZUOUNRCPBPZAPZJZU
      LUQUIURVAUJUSVCABCDESABCEDSTDEUAUQUTVBJZAPVDUPVEAUNUOBCUBUCUTVBAUDUEUFUG
      $.
  $}

  ${
    $d x y A $.
    $( A member of a relation is an ordered pair.  (Contributed by NM,
       17-Sep-2006.) $)
    elrel $p |- ( ( Rel R /\ A e. R ) -> E. x E. y A = <. x , y >. ) $=
      ( wrel wcel wa cvv cxp cop wceq wex wss df-rel biimpi sselda elvv sylib
      cv ) DEZCDFGCHHIZFCASBSJKBLALTDUACTDUAMDNOPABCQR $.
  $}

  $( A singleton is a relation iff it is an ordered pair.  (Contributed by NM,
     24-Sep-2013.)  (Revised by BJ, 12-Feb-2022.) $)
  relsng $p |- ( A e. V -> ( Rel { A } <-> A e. ( _V X. _V ) ) ) $=
    ( wcel csn wrel cvv cxp wss df-rel snssg bitr4id ) ABCADZELFFGZHAMCLIAMBJK
    $.

  $( A singleton of an ordered pair is a relation.  (Contributed by NM,
     17-May-1998.)  (Revised by BJ, 12-Feb-2022.) $)
  relsnopg $p |- ( ( A e. V /\ B e. W ) -> Rel { <. A , B >. } ) $=
    ( wcel wa cop csn wrel cvv cxp opelvvg wb opexg relsng syl mpbird ) ACEBDEF
    ZABGZHIZSJJKEZABCDLRSJETUAMABCDNSJOPQ $.

  ${
    relsn.1 $e |- A e. _V $.
    $( A singleton is a relation iff it is an ordered pair.  (Contributed by
       NM, 24-Sep-2013.) $)
    relsn $p |- ( Rel { A } <-> A e. ( _V X. _V ) ) $=
      ( csn wrel cvv cxp wss wcel df-rel snss bitr4i ) ACZDLEEFZGAMHLIAMBJK $.

    relsnop.2 $e |- B e. _V $.
    $( A singleton of an ordered pair is a relation.  (Contributed by NM,
       17-May-1998.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    relsnop $p |- Rel { <. A , B >. } $=
      ( cop csn wrel cvv cxp wcel opelvv opex relsn mpbir ) ABEZFGOHHIJABCDKOAB
      CDLMN $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.
    $( Subset theorem for cross product.  Generalization of Theorem 101 of
       [Suppes] p. 52.  (Contributed by NM, 26-Aug-1995.)  (Proof shortened by
       Andrew Salmon, 27-Aug-2011.) $)
    xpss12 $p |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) ) $=
      ( vx vy wss wa cv wcel copab cxp ssel im2anan9 ssopab2dv df-xp 3sstr4g )
      ABGZCDGZHZEIZAJZFIZCJZHZEFKUABJZUCDJZHZEFKACLBDLTUEUHEFRUBUFSUDUGABUAMCDU
      CMNOEFACPEFBDPQ $.
  $}

  $( A cross product is included in the ordered pair universe.  Exercise 3 of
     [TakeutiZaring] p. 25.  (Contributed by NM, 2-Aug-1994.) $)
  xpss $p |- ( A X. B ) C_ ( _V X. _V ) $=
    ( cvv wss cxp ssv xpss12 mp2an ) ACDBCDABECCEDAFBFACBCGH $.

  $( A cross product is a relation.  Theorem 3.13(i) of [Monk1] p. 37.
     (Contributed by NM, 2-Aug-1994.) $)
  relxp $p |- Rel ( A X. B ) $=
    ( cxp wrel cvv wss xpss df-rel mpbir ) ABCZDJEECFABGJHI $.

  $( Subset relation for cross product.  (Contributed by Jeff Hankins,
     30-Aug-2009.) $)
  xpss1 $p |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) ) $=
    ( wss cxp ssid xpss12 mpan2 ) ABDCCDACEBCEDCFABCCGH $.

  $( Subset relation for cross product.  (Contributed by Jeff Hankins,
     30-Aug-2009.) $)
  xpss2 $p |- ( A C_ B -> ( C X. A ) C_ ( C X. B ) ) $=
    ( wss cxp ssid xpss12 mpan ) CCDABDCAECBEDCFCCABGH $.

  ${
    $d A x y z $.  $d B x y z $.
    $( A cross product is included in the power of the power of the union of
       its arguments.  (Contributed by NM, 13-Sep-2006.) $)
    xpsspw $p |- ( A X. B ) C_ ~P ~P ( A u. B ) $=
      ( vz vx vy cun cpw cv wcel wss wceq wa wex csn cpr snssi sseq1 syl5ibrcom
      vex syl cxp cop elxpi dfop wo ssun3 adantr df-pr ssun4 anim12i unss sylib
      eqsstrid jaod elpr 3imtr4g ssrdv biimpar sylan2 exlimivv sylibr ssriv
      elpw ) CABUAZABFZGZGZCHZVDIZVHVFJZVHVGIVIVHDHZEHZUBZKZVKAIZVLBIZLZLZEMDMV
      JDEVHABUCVRVJDEVQVNVMVFJZVJVQVMVKNZVKVLOZOZVFVKVLDSESUDVQCWBVFVQVHVTKZVHW
      AKZUEVHVEJZVHWBIVHVFIVQWCWEWDVQWEWCVTVEJZVOWFVPVOVTAJWFVKAPVTABUFTZUGVHVT
      VEQRVQWEWDWAVEJVQWAVTVLNZFZVEVKVLUHVQWFWHVEJZLWIVEJVOWFVPWJWGVPWHBJWJVLBP
      WHBAUITUJVTWHVEUKULUMVHWAVEQRUNVHVTWACSZUOVHVEWKVCUPUQUMVNVJVSVHVMVFQURUS
      UTTVHVFWKVCVAVB $.
  $}

  $( The double class union of a cross product is included in the union of its
     arguments.  (Contributed by NM, 16-Sep-2006.) $)
  unixpss $p |- U. U. ( A X. B ) C_ ( A u. B ) $=
    ( cxp cuni cun cpw xpsspw unissi unipw sseqtri ) ABCZDZDABEZFZDMLNLNFZDNKOA
    BGHNIJHMIJ $.

  $( The cross product of two sets is a set.  Proposition 6.2 of
     [TakeutiZaring] p. 23.  (Contributed by NM, 14-Aug-1994.) $)
  xpexg $p |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V ) $=
    ( wcel wa cxp cun cpw wss cvv xpsspw unexg pwexg 3syl ssexg sylancr ) ACEBD
    EFZABGZABHZIZIZJUBKEZSKEABLRTKEUAKEUCABCDMTKNUAKNOSUBKPQ $.

  ${
    xpexd.1 $e |- ( ph -> A e. V ) $.
    xpexd.2 $e |- ( ph -> B e. W ) $.
    $( The Cartesian product of two sets is a set.  (Contributed by Glauco
       Siliprandi, 26-Jun-2021.) $)
    xpexd $p |- ( ph -> ( A X. B ) e. _V ) $=
      ( wcel cxp cvv xpexg syl2anc ) ABDHCEHBCIJHFGBCDEKL $.
  $}

  ${
    xpex.1 $e |- A e. _V $.
    xpex.2 $e |- B e. _V $.
    $( The cross product of two sets is a set.  Proposition 6.2 of
       [TakeutiZaring] p. 23.  (Contributed by NM, 14-Aug-1994.) $)
    xpex $p |- ( A X. B ) e. _V $=
      ( cvv wcel cxp xpexg mp2an ) AEFBEFABGEFCDABEEHI $.
  $}

  ${
    unexd.1 $e |- ( ph -> A e. V ) $.
    unexd.2 $e |- ( ph -> B e. W ) $.
    $( The union of two sets is a set.  (Contributed by SN, 16-Jul-2024.) $)
    unexd $p |- ( ph -> ( A u. B ) e. _V ) $=
      ( wcel cun cvv unexg syl2anc ) ABDHCEHBCIJHFGBCDEKL $.
  $}

  $( The Cartesian square of a set is a set.  (Contributed by AV,
     13-Jan-2020.) $)
  sqxpexg $p |- ( A e. V -> ( A X. A ) e. _V ) $=
    ( wcel cxp cvv xpexg anidms ) ABCAADECAABBFG $.

  $( The union of two relations is a relation.  Compare Exercise 5 of
     [TakeutiZaring] p. 25.  (Contributed by NM, 12-Aug-1994.) $)
  relun $p |- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) ) $=
    ( cvv cxp wss wa cun wrel unss df-rel anbi12i 3bitr4ri ) ACCDZEZBMEZFABGZME
    AHZBHZFPHABMIQNROAJBJKPJL $.

  $( The intersection with a relation is a relation.  (Contributed by NM,
     16-Aug-1994.) $)
  relin1 $p |- ( Rel A -> Rel ( A i^i B ) ) $=
    ( cin wss wrel wi inss1 relss ax-mp ) ABCZADAEJEFABGJAHI $.

  $( The intersection with a relation is a relation.  (Contributed by NM,
     17-Jan-2006.) $)
  relin2 $p |- ( Rel B -> Rel ( A i^i B ) ) $=
    ( cin wss wrel wi inss2 relss ax-mp ) ABCZBDBEJEFABGJBHI $.

  $( A difference cutting down a relation is a relation.  (Contributed by NM,
     31-Mar-1998.) $)
  reldif $p |- ( Rel A -> Rel ( A \ B ) ) $=
    ( cdif wss wrel wi difss relss ax-mp ) ABCZADAEJEFABGJAHI $.

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( An indexed union is a relation iff each member of its indexed family is
       a relation.  (Contributed by NM, 19-Dec-2008.) $)
    reliun $p |- ( Rel U_ x e. A B <-> A. x e. A Rel B ) $=
      ( vy ciun wrel cv wcel wrex cab cvv cxp wss wral df-iun releqi df-rel wal
      wi 3bitri abss ssalel bitri ralbii ralcom4 r19.23v albii bitr4i ) ABCEZFD
      GZCHZABIZDJZFUMKKLZMZCFZABNZUIUMADBCOPUMQUOULUJUNHZSZDRZUQULDUNUAUQUKURSZ
      DRZABNVAABNZDRUTUPVBABUPCUNMVBCQDCUNUBUCUDVAADBUEVCUSDUKURABUFUGTUHT $.
  $}

  $( An indexed intersection is a relation if at least one of the member of the
     indexed family is a relation.  (Contributed by NM, 8-Mar-2014.) $)
  reliin $p |- ( E. x e. A Rel B -> Rel |^|_ x e. A B ) $=
    ( cvv cxp wss wrex ciin wrel iinss df-rel rexbii 3imtr4i ) CDDEZFZABGABCHZN
    FCIZABGPIABCNJQOABCKLPKM $.

  ${
    $d x A $.
    $( The union of a class is a relation iff any member is a relation.
       Exercise 6 of [TakeutiZaring] p. 25 and its converse.  (Contributed by
       NM, 13-Aug-2004.) $)
    reluni $p |- ( Rel U. A <-> A. x e. A Rel x ) $=
      ( cuni wrel cv ciun wral uniiun releqi reliun bitri ) BCZDABAEZFZDMDABGLN
      ABHIABMJK $.

    $( The intersection of a class is a relation if at least one member is a
       relation.  (Contributed by NM, 8-Mar-2014.) $)
    relint $p |- ( E. x e. A Rel x -> Rel |^| A ) $=
      ( cv wrel wrex ciin cint reliin intiin releqi sylibr ) ACZDABEABLFZDBGZDA
      BLHNMABIJK $.
  $}

  $( The empty set is a relation.  (Contributed by NM, 26-Apr-1998.) $)
  rel0 $p |- Rel (/) $=
    ( c0 wrel cvv cxp wss 0ss df-rel mpbir ) ABACCDZEIFAGH $.

  ${
    $d x y $.
    relopabiv.1 $e |- A = { <. x , y >. | ph } $.
    $( A class of ordered pairs is a relation.  For a version without a
       disjoint variable condition, see ~ relopabi .  (Contributed by BJ,
       22-Jul-2023.) $)
    relopabiv $p |- Rel A $=
      ( wrel cvv cxp wss copab cv wcel wa vex pm3.2i a1i ssopab2i df-xp 3sstr4i
      df-rel mpbir ) DFDGGHZIABCJBKGLZCKGLZMZBCJDUBAUEBCUEAUCUDBNCNOPQEBCGGRSDT
      UA $.
  $}

  ${
    $d x y $.
    $( A class of ordered pairs is a relation.  For a version without a
       disjoint variable condition, see ~ relopab .  (Contributed by SN,
       8-Sep-2024.) $)
    relopabv $p |- Rel { <. x , y >. | ph } $=
      ( copab eqid relopabiv ) ABCABCDZGEF $.
  $}

  ${
    $d ph z $.  $d x z $.  $d y z $.
    relopabi.1 $e |- A = { <. x , y >. | ph } $.
    $( A class of ordered pairs is a relation.  (Contributed by Mario Carneiro,
       21-Dec-2013.) $)
    relopabi $p |- Rel A $=
      ( vz wrel cvv cxp wss cv cop wceq wa wex cab copab df-opab wcel vex eqtri
      opelvv eleq1 mpbiri adantr exlimivv abssi eqsstri df-rel mpbir ) DGDHHIZJ
      DFKZBKZCKZLZMZANZCOBOZFPZUKDABCQUSEABCFRUAURFUKUQULUKSZBCUPUTAUPUTUOUKSUM
      UNBTCTUBULUOUKUCUDUEUFUGUHDUIUJ $.
  $}

  $( A class of ordered pairs is a relation.  (Contributed by NM, 8-Mar-1995.)
     (Unnecessary distinct variable restrictions were removed by Alan Sare,
     9-Jul-2013.)  (Proof shortened by Mario Carneiro, 21-Dec-2013.) $)
  relopab $p |- Rel { <. x , y >. | ph } $=
    ( copab eqid relopabi ) ABCABCDZGEF $.

  $( If two classes are in a relationship given by an ordered-pair class
     abstraction, the classes are sets.  (Contributed by Alexander van der
     Vekens, 5-Nov-2017.) $)
  brabv $p |- ( X { <. x , y >. | ph } Y -> ( X e. _V /\ Y e. _V ) ) $=
    ( copab relopab brrelex12i ) DEABCFABCGH $.

  ${
    $d A y $.  $d B y $.  $d x y $.
    $( The maps-to notation always describes a relationship.  (Contributed by
       Scott Fenton, 16-Apr-2012.) $)
    mptrel $p |- Rel ( x e. A |-> B ) $=
      ( vy cv wcel wceq wa cmpt df-mpt relopabi ) AEBFDECGHADABCIADBCJK $.
  $}

  ${
    $d w x y z A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d ph z w $.
    $d ps z w $.
    $( The identity relation is a relation.  Part of Exercise 4.12(p) of
       [Mendelson] p. 235.  (Contributed by NM, 26-Apr-1998.)  (Revised by
       Mario Carneiro, 21-Dec-2013.) $)
    reli $p |- Rel _I $=
      ( vx vy cv wceq cid df-id relopabi ) ACBCDABEABFG $.

    $( The membership relation is a relation.  (Contributed by NM,
       26-Apr-1998.)  (Revised by Mario Carneiro, 21-Dec-2013.) $)
    rele $p |- Rel _E $=
      ( vx vy wel cep df-eprel relopabi ) ABCABDABEF $.

    $( A relation expressed as an ordered pair abstraction.  (Contributed by
       NM, 11-Dec-2006.) $)
    opabid2 $p |- ( Rel A -> { <. x , y >. | <. x , y >. e. A } = A ) $=
      ( vz vw wrel cv cop wcel copab wceq wb wal vex opeq1 eleq1d opelopab gen2
      opeq2 relopab eqrel mpan mpbiri ) CFZAGZBGZHZCIZABJZCKZDGZEGZHZUIIUMCIZLZ
      EMDMZUODEUHUKUFHZCIUNABUKULDNENUEUKKUGUQCUEUKUFOPUFULKUQUMCUFULUKSPQRUIFU
      DUJUPLUHABTDEUICUAUBUC $.

    $( Intersection of two ordered pair class abstractions.  (Contributed by
       NM, 30-Sep-2002.) $)
    inopab $p |- ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) =
               { <. x , y >. | ( ph /\ ps ) } $=
      ( vz vw copab cin wa wrel relopab relin1 ax-mp cv cop wcel wsb sban sbbii
      opelopabsbALT anbi12i 3bitr4ri elin 3bitr4i eqrelriiv ) EFACDGZBCDGZHZABI
      ZCDGZUFJUHJACDKUFUGLMUICDKENFNOZUFPZUKUGPZIZUICEQZDFQZUKUHPUKUJPACEQZBCEQ
      ZIZDFQUQDFQZURDFQZIUPUNUQURDFRUOUSDFABCERSULUTUMVAACDEFTBCDEFTUAUBUKUFUGU
      CUICDEFTUDUE $.

    $( The difference of two ordered-pair abstractions.  (Contributed by Stefan
       O'Rear, 17-Jan-2015.) $)
    difopab $p |- ( { <. x , y >. | ph } \ { <. x , y >. | ps } ) =
        { <. x , y >. | ( ph /\ -. ps ) } $=
      ( vz vw copab wn wa wrel relopab ax-mp cv wcel wsbc sbcbii opelopabsb cvv
      sbcan wb cdif reldif cop vex sbcng notbii anbi12i eldif 3bitr4i eqrelriiv
      3bitr4ri ) EFACDGZBCDGZUAZABHZIZCDGZULJUNJACDKULUMUBLUPCDKEMZFMZUCZULNZUT
      UMNZHZIZUPDUSOZCUROZUTUNNUTUQNADUSOZUODUSOZIZCUROVGCUROZVHCUROZIVFVDVGVHC
      URSVEVICURAUODUSSPVAVJVCVKACDURUSQBDUSOZHZCUROZVLCUROZHZVKVCURRNVNVPTEUDV
      LCURRUELVHVMCURUSRNVHVMTFUDBDUSRUELPVBVOBCDURUSQUFUKUGUKUTULUMUHUPCDURUSQ
      UIUJ $.

    $( The intersection of two cross products.  Exercise 9 of [TakeutiZaring]
       p. 25.  (Contributed by NM, 3-Aug-1994.)  (Proof shortened by Andrew
       Salmon, 27-Aug-2011.) $)
    inxp $p |- ( ( A X. B ) i^i ( C X. D ) ) =
                   ( ( A i^i C ) X. ( B i^i D ) ) $=
      ( vx vy cv wcel wa copab cin cxp inopab elin anbi12i bitr4i opabbii eqtri
      an4 df-xp ineq12i 3eqtr4i ) EGZAHZFGZBHZIZEFJZUCCHZUEDHZIZEFJZKZUCACKZHZU
      EBDKZHZIZEFJZABLZCDLZKUNUPLUMUGUKIZEFJUSUGUKEFMVBUREFVBUDUIIZUFUJIZIURUDU
      FUIUJSUOVCUQVDUCACNUEBDNOPQRUTUHVAULEFABTEFCDTUAEFUNUPTUB $.

    $( Distributive law for cross product over intersection.  Theorem 102 of
       [Suppes] p. 52.  (Contributed by NM, 26-Sep-2004.) $)
    xpindi $p |- ( A X. ( B i^i C ) ) = ( ( A X. B ) i^i ( A X. C ) ) $=
      ( cxp cin inxp inidm xpeq1i eqtr2i ) ABDACDEAAEZBCEZDAKDABACFJAKAGHI $.

    $( Distributive law for cross product over intersection.  Similar to
       Theorem 102 of [Suppes] p. 52.  (Contributed by NM, 26-Sep-2004.) $)
    xpindir $p |- ( ( A i^i B ) X. C ) = ( ( A X. C ) i^i ( B X. C ) ) $=
      ( cxp cin inxp inidm xpeq2i eqtr2i ) ACDBCDEABEZCCEZDJCDACBCFKCJCGHI $.
  $}

  ${
    $d x y z w A $.  $d x y z w C $.  $d z w B $.
    $( Distributive law for cross product over indexed intersection.
       (Contributed by Jim Kingdon, 7-Dec-2018.) $)
    xpiindim $p |- ( E. y y e. A ->
      ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) ) $=
      ( vw vz ciin cxp wrel wa cv wcel wex wral relxp wb cvv eliin opelxp rgenw
      wceq wrex r19.2m mpan2 reliin syl jctil cop eleq1w cbvexv r19.28mv sylbir
      bicomd elv anbi2i ralbii 3bitr4g vex opex ax-mp eqrelrdv2 mpancom ) EACDH
      ZIZJZACEDIZHZJZKBLCMZBNZVEVHUBVKVIVFVKVGJZACUCZVIVKVLACOVMVLACEDPUAVLABCU
      DUEACVGUFUGEVDPUHVKFGVEVHVKFLZEMZGLZVDMZKZVNVPUIZVGMZACOZVSVEMVSVHMZVKVOV
      PDMZACOZKZVOWCKZACOZVRWAVKWGWEVKALCMZANWGWEQWHVJABABCUJUKVOWCACULUMUNVQWD
      VOVQWDQGAVPCDRSUOUPVTWFACVNVPEDTUQURVNVPEVDTVSRMWBWAQVNVPFUSGUSUTAVSCVGRS
      VAURVBVC $.

    $( Distributive law for cross product over relativized indexed
       intersection.  (Contributed by Jim Kingdon, 7-Dec-2018.) $)
    xpriindim $p |- ( E. y y e. A ->
        ( C X. ( D i^i |^|_ x e. A B ) ) =
        ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) ) $=
      ( cv wcel wex ciin cin cxp xpindi xpiindim ineq2d eqtrid ) BGCHBIZEFACDJZ
      KLEFLZERLZKSACEDLJZKEFRMQTUASABCDENOP $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y C $.  $d x y D $.  $d x E $.
    $( Membership in a union of cross products.  Analogue of ~ elxp for
       nonconstant ` B ( x ) ` .  (Contributed by Mario Carneiro,
       29-Dec-2014.) $)
    eliunxp $p |- ( C e. U_ x e. A ( { x } X. B ) <->
      E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) ) $=
      ( cv csn cxp ciun wcel cop wceq wex wa wrel wral relxp rgenw reliun exbii
      mpbir elrel mpan pm4.71ri nfiu1 nfel2 19.41 19.41v eleq1 opeliunxp bitrdi
      pm5.32i bitr3i 3bitr2i ) EACAFZGZDHZIZJZEUOBFZKZLZBMZAMZUSNVCUSNZAMVBUOCJ
      UTDJNZNZBMZAMUSVDUROZUSVDVIUQOZACPVJACUPDQRACUQSUAABEURUBUCUDVCUSAAEURACU
      QUEUFUGVEVHAVEVBUSNZBMVHVBUSBUHVKVGBVBUSVFVBUSVAURJVFEVAURUIACDUTUJUKULTU
      MTUN $.

    $d x A $.
    opeliunxp2.1 $e |- ( x = C -> B = E ) $.
    $( Membership in a union of cross products.  (Contributed by Mario
       Carneiro, 14-Feb-2015.) $)
    opeliunxp2 $p |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <->
      ( C e. A /\ D e. E ) ) $=
      ( cop cv csn cxp ciun wcel cvv wa wbr df-br wrel wral wb reliun brrelex1i
      relxp rgenw mpbir sylbir elex adantr nfcv nfiu1 nfel2 nfv nfbi wceq opeq1
      eleq1d eleq1 eleq2d anbi12d bibi12d opeliunxp vtoclgf pm5.21nii ) DEHZABA
      IZJZCKZLZMZDNMZDBMZEFMZOZVIDEVHPVJDEVHQDEVHVHRVGRZABSVNABVFCUCUDABVGUAUEU
      BUFVKVJVLDBUGUHVEEHZVHMZVEBMZECMZOZTVIVMTADNADUIVIVMAAVDVHABVGUJUKVMAULUM
      VEDUNZVPVIVSVMVTVOVDVHVEDEUOUPVTVQVKVRVLVEDBUQVTCFEGURUSUTABCEVAVBVC $.
  $}

  ${
    $d x y z A $.  $d x z B $.  $d y z ph $.  $d x ps $.
    ralxp.1 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
    $( Write a double restricted quantification as one universal quantifier.
       In this version of ~ ralxp , ` B ( y ) ` is not assumed to be constant.
       (Contributed by Mario Carneiro, 29-Dec-2014.) $)
    raliunxp $p |- ( A. x e. U_ y e. A ( { y } X. B ) ph <->
      A. y e. A A. z e. B ps ) $=
      ( cv csn cxp wcel wi wal wa wral wex albii vex bitri ciun cop wceq imbi1i
      eliunxp 19.23vv bitr4i alrot3 impexp imbi2d ceqsalv 2albii df-ral 3bitr4i
      opex r2al ) CIZDFDIZJGKUAZLZAMZCNZURFLEIZGLOZBMZENDNZACUSPBEGPDFPVBUQURVC
      UBZUCZVDOZAMZENDNZCNZVFVAVKCVAVIEQDQZAMVKUTVMADEFGUQUEUDVIADEUFUGRVLVJCNZ
      ENDNVFVJCDEUHVNVEDEVNVHVDAMZMZCNVEVJVPCVHVDAUIRVOVECVGURVCDSESUOVHABVDHUJ
      UKTULTTACUSUMBDEFGUPUN $.

    $( Write a double restricted quantification as one universal quantifier.
       In this version of ~ rexxp , ` B ( y ) ` is not assumed to be constant.
       (Contributed by Mario Carneiro, 14-Feb-2015.) $)
    rexiunxp $p |- ( E. x e. U_ y e. A ( { y } X. B ) ph <->
      E. y e. A E. z e. B ps ) $=
      ( cv csn cxp ciun wcel wa wex wrex cop exbii vex bitri wceq anbi1i bitr4i
      eliunxp 19.41vv exrot3 anass opex anbi2d ceqsexv 2exbii df-rex 3bitr4i
      r2ex ) CIZDFDIZJGKLZMZANZCOZUPFMEIZGMNZBNZEODOZACUQPBEGPDFPUTUOUPVAQZUAZV
      BNZANZEODOZCOZVDUSVICUSVGEODOZANVIURVKADEFGUOUDUBVGADEUEUCRVJVHCOZEODOVDV
      HCDEUFVLVCDEVLVFVBANZNZCOVCVHVNCVFVBAUGRVMVCCVEUPVADSESUHVFABVBHUIUJTUKTT
      ACUQULBDEFGUNUM $.

    $d y B $.
    $( Universal quantification restricted to a cross product is equivalent to
       a double restricted quantification.  The hypothesis specifies an
       implicit substitution.  (Contributed by NM, 7-Feb-2004.)  (Revised by
       Mario Carneiro, 29-Dec-2014.) $)
    ralxp $p |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps ) $=
      ( cxp wral cv csn ciun iunxpconst raleqi raliunxp bitr3i ) ACFGIZJACDFDKL
      GIMZJBEGJDFJACSRDFGNOABCDEFGHPQ $.

    $( Existential quantification restricted to a cross product is equivalent
       to a double restricted quantification.  (Contributed by NM,
       11-Nov-1995.)  (Revised by Mario Carneiro, 14-Feb-2015.) $)
    rexxp $p |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps ) $=
      ( cxp wrex cv csn ciun iunxpconst rexeqi rexiunxp bitr3i ) ACFGIZJACDFDKL
      GIMZJBEGJDFJACSRDFGNOABCDEFGHPQ $.
  $}

  ${
    $d x A $.
    $( Disjoint union is a subset of a cross product.  (Contributed by Stefan
       O'Rear, 21-Nov-2014.) $)
    djussxp $p |- U_ x e. A ( { x } X. B ) C_ ( A X. _V ) $=
      ( cv csn cxp ciun cvv wss iunss wcel snssi ssv xpss12 sylancl mprgbir ) A
      BADZEZCFZGBHFZISTIZABABSTJQBKRBICHIUAQBLCMRBCHNOP $.
  $}

  ${
    $d u v w x y A $.  $d u v w x y z B $.  $d u v w ph $.  $d u v w ps $.
    ralxpf.1 $e |- F/ y ph $.
    ralxpf.2 $e |- F/ z ph $.
    ralxpf.3 $e |- F/ x ps $.
    ralxpf.4 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
    $( Version of ~ ralxp with bound-variable hypotheses.  (Contributed by NM,
       18-Aug-2006.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    ralxpf $p |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps ) $=
      ( vv vw vu wral wsb cv wceq vex nfsb cxp cbvralsv ralbii nfv nfcv nfralxy
      nfs1v sbequ12 ralbidv cbvral cop wa wex eqvinop nfbi sbhypf opth sylan9bb
      wb sylbi exlimi ralxp 3bitr4ri bitri ) ACFGUAZOACLPZLVEOZBEGOZDFOZACLVEUB
      BDMPZEGOZMFOVJENPZNGOZMFOVIVGVKVMMFVJENGUBUCVHVKDMFVHMUDVJDEGDGUEBDMUGZUF
      DQZMQZRZBVJEGBDMUHZUIUJVFVLLMNFGLQZVPNQZUKZRVSVOEQZUKZRZWCWARZULZEUMZDUMV
      FVLUSZDEVSVPVTMSNSUNWGWHDVFVLDACLDHTVJENDVNTUOWFWHEVFVLEACLEITVJENUGUOWDV
      FBWEVLABCLWCJKUPWEVQWBVTRZULBVLUSVOWBVPVTDSESUQVQBVJWIVLVRVJENUHURUTURVAV
      AUTVBVCVD $.

    $( Version of ~ rexxp with bound-variable hypotheses.  (Contributed by NM,
       19-Dec-2008.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    rexxpf $p |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps ) $=
      ( vv vw vu wrex wsb cv wceq vex nfsb cxp cbvrexsv rexbii nfv nfs1v nfrexw
      nfcv sbequ12 rexbidv cbvrex cop wa wex eqvinop nfbi sbhypf sylan9bb sylbi
      wb opth exlimi rexxp 3bitr4ri bitri ) ACFGUAZOACLPZLVEOZBEGOZDFOZACLVEUBB
      DMPZEGOZMFOVJENPZNGOZMFOVIVGVKVMMFVJENGUBUCVHVKDMFVHMUDVJDEGDGUGBDMUEZUFD
      QZMQZRZBVJEGBDMUHZUIUJVFVLLMNFGLQZVPNQZUKZRVSVOEQZUKZRZWCWARZULZEUMZDUMVF
      VLUSZDEVSVPVTMSNSUNWGWHDVFVLDACLDHTVJENDVNTUOWFWHEVFVLEACLEITVJENUEUOWDVF
      BWEVLABCLWCJKUPWEVQWBVTRZULBVLUSVOWBVPVTDSESUTVQBVJWIVLVRVJENUHUQURUQVAVA
      URVBVCVD $.
  $}

  ${
    $d w x y A $.  $d w x y z B $.  $d w C $.  $d w D $.
    iunxpf.1 $e |- F/_ y C $.
    iunxpf.2 $e |- F/_ z C $.
    iunxpf.3 $e |- F/_ x D $.
    iunxpf.4 $e |- ( x = <. y , z >. -> C = D ) $.
    $( Indexed union on a cross product is equals a double indexed union.  The
       hypothesis specifies an implicit substitution.  (Contributed by NM,
       19-Dec-2008.) $)
    iunxpf $p |- U_ x e. ( A X. B ) C = U_ y e. A U_ z e. B D $=
      ( vw cxp ciun cv wcel wrex nfcri cop eliun wceq eleq2d rexxpf bitri eqriv
      rexbii 3bitr4i ) LADEMZFNZBDCEGNZNZLOZFPZAUHQULGPZCEQZBDQZULUIPULUKPZUMUN
      ABCDEBLFHRCLFIRALGJRAOBOCOSUAFGULKUBUCAULUHFTUQULUJPZBDQUPBULDUJTURUOBDCU
      LEGTUFUDUGUE $.
  $}

  ${
    $d x y A $.  $d x y ph $.
    opabbi2dv.1 $e |- Rel A $.
    opabbi2dv.3 $e |- ( ph -> ( <. x , y >. e. A <-> ps ) ) $.
    $( Deduce equality of a relation and an ordered-pair class builder.
       Compare ~ abbi2dv .  (Contributed by NM, 24-Feb-2014.) $)
    opabbi2dv $p |- ( ph -> A = { <. x , y >. | ps } ) $=
      ( cv cop wcel copab wrel wceq opabid2 ax-mp opabbidv eqtr3id ) AECHDHIEJZ
      CDKZBCDKELSEMFCDENOARBCDGPQ $.
  $}

  ${
    $d v w x y z A $.  $d v w x y z B $.
    relop.1 $e |- A e. _V $.
    relop.2 $e |- B e. _V $.
    $( A necessary and sufficient condition for a Kuratowski ordered pair to be
       a relation.  (Contributed by NM, 3-Jun-2008.)  (Avoid depending on this
       detail.) $)
    relop $p |- ( Rel <. A , B >.
             <-> E. x E. y ( A = { x } /\ B = { x , y } ) ) $=
      ( vz vw vv cop cvv cv wceq wa wex wi wcel bitri weq eqeq2d cxp wss df-rel
      wrel csn cpr wal ssalel vex elop elvv imbi12i jaob albii 19.26 snex eqeq1
      wo eqcom opeqsn bitrdi 2exbidv imbi12d spcv sneq cbvexv a9ev equcom exbii
      mpbi 19.41v mpbiran eqid a1bi 3bitr2ri sylib prexg mp2an mpi opeqpr eqtr2
      idd preqsn simplbi dfsn2 preq2 eqtr2id eqtrid anbi12d biimpd com12 adantr
      syl expd mpd expcom impd jaod biimtrid 2eximdv exlimiv syl2an sylbi simpr
      equid jctl sylibr eqtr4d opeq12 spc2ev adantlr preq12 biimpa dfop eqtr4di
      imp jaodan ex 3imtr4g ssrdv exlimivv impbii ) CDJZUDYCKKUAZUBZCALZUEZMZDY
      FBLZUFZMZNZBOAOZYCUCYEYMYEGLZCUEZMZYNYFYIJZMZBOAOZPZGUGZYNCDUFZMZYSPZGUGZ
      NZYMYEYNYCQZYNYDQZPZGUGZUUFGYCYDUHUUJYTUUDNZGUGUUFUUIUUKGUUIYPUUCURZYSPUU
      KUUGUULUUHYSYNCDGUIEFUJZABYNUKULYPYSUUCUMRUNYTUUDGUORRUUACHLZUEZMZHOZUUBY
      QMZBOAOZYMUUEUUAYOYOMZABSZYHNZBOZAOZPZUUQYTUVEGYOCEUPYPYPUUTYSUVDYNYOYOUQ
      YPYRUVBABYPYRYOYQMZUVBYNYOYQUQUVFYQYOMUVBYOYQUSYFYICAUIZBUIZEUTRVAVBVCVDU
      UQYHAOUVDUVEUUPYHHAHASZUUOYGCUUNYFVETVFUVCYHAUVCUVABOZYHBASZBOUVJBAVGUVKU
      VABBAVHVIVJUVAYHBVKVLVIUUTUVDYOVMVNVOVPUUEUUBUUBMZUUSUUBVMUUDUVLUUSPGUUBC
      KQDKQUUBKQEFCDKKVQVRUUCUUCUVLYSUUSYNUUBUUBUQUUCYRUURABYNUUBYQUQVBVCVDVSUU
      QUUSYMUUPUUSYMPHUUPUURYLABUURYLCYJMZDYGMZNZURZUUPYLUURYQUUBMUVPUUBYQUSYFY
      ICDUVGUVHEFVTRUUPYLYLUVOUUPYLWBUUPUVMUVNYLUVMUUPUVNYLPZUVMUUPNZUVAUVQUVRY
      JUUOMZUVACYJUUOWAUVSUVABHSYFYIUUNUVGUVHHUIWCWDWMUVMUVAUVQPUUPUVAUVMUVQUVA
      UVMUVNYLUVAUVOYLUVAUVMYHUVNYKUVAYJYGCUVAYGYFYFUFZYJYFWEZYFYIYFWFZWGTUVAYG
      YJDUVAYGUVTYJUWAUWBWHTWIWJWNWKWLWOWPWQWRWSWTXAXPXBXCYLYEABYLGYCYDYLUULYNU
      UNILZJZMZIOHOZUUGUUHYLUULUWFYLYPUWFUUCYHYPUWFYKYHYPNZYNYFYFJZMZUWFUWGYNYO
      UWHYHYPXDYHUWHYOMZYPYHAASZYHNUWJYHUWKAXEXFYFYFCUVGUVGEUTXGWLXHUWEUWIHIYFY
      FUVGUVGUVIIASNUWDUWHYNUUNUWCYFYFXITXJWMXKYLUUCNZYRUWFUWLYNYGYJUFZYQYLUUCY
      NUWMMYLUUBUWMYNCDYGYJXLTXMYFYIUVGUVHXNXOUWEYRHIYFYIUVGUVHUVIIBSNUWDYQYNUU
      NUWCYFYIXITXJWMXQXRUUMHIYNUKXSXTYAYBR $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( For sets, the identity relation is the same as equality.  (Contributed
       by NM, 30-Apr-2004.)  (Proof shortened by Andrew Salmon,
       27-Aug-2011.) $)
    ideqg $p |- ( B e. V -> ( A _I B <-> A = B ) ) $=
      ( vx vy wcel cid wbr wceq cvv wa reli brrelex1i adantl simpl jca biimparc
      eleq1 elex cv syl eqeq1 eqeq2 df-id brabg pm5.21nd ) BCFZABGHZABIZAJFZUGK
      UGUHKUJUGUHUJUGABGLMNUGUHOPUGUIKZUJUGUKACFZUJUIULUGABCRQACSUAUGUIOPDTZETZ
      IAUNIUIDEABJCGUMAUNUBUNBAUCDEUDUEUF $.
  $}

  ${
    ideq.1 $e |- B e. _V $.
    $( For sets, the identity relation is the same as equality.  (Contributed
       by NM, 13-Aug-1995.) $)
    ideq $p |- ( A _I B <-> A = B ) $=
      ( cvv wcel cid wbr wceq wb ideqg ax-mp ) BDEABFGABHICABDJK $.
  $}

  $( A set is identical to itself.  (Contributed by NM, 28-May-2008.)  (Proof
     shortened by Andrew Salmon, 27-Aug-2011.) $)
  ididg $p |- ( A e. V -> A _I A ) $=
    ( wcel cid wbr wceq eqid ideqg mpbiri ) ABCAADEAAFAGAABHI $.

  $( Two ways of expressing set existence.  (Contributed by NM, 16-Feb-2008.)
     (Proof shortened by Andrew Salmon, 27-Aug-2011.)  (Revised by Mario
     Carneiro, 26-Apr-2015.) $)
  issetid $p |- ( A e. _V <-> A _I A ) $=
    ( cvv wcel cid wbr ididg reli brrelex1i impbii ) ABCAADEABFAADGHI $.

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.
    $( Subclass theorem for composition.  (Contributed by FL, 30-Dec-2010.) $)
    coss1 $p |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) ) $=
      ( vx vy vz wss cv wbr wa wex copab id ssbrd anim2d eximdv ssopab2dv df-co
      ccom 3sstr4g ) ABGZDHEHZCIZUBFHZAIZJZEKZDFLUCUBUDBIZJZEKZDFLACSBCSUAUGUJD
      FUAUFUIEUAUEUHUCUAABUBUDUAMNOPQDFEACRDFEBCRT $.

    $( Subclass theorem for composition.  (Contributed by NM, 5-Apr-2013.) $)
    coss2 $p |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) ) $=
      ( vx vy vz wss cv wbr wa wex copab id ssbrd anim1d eximdv ssopab2dv df-co
      ccom 3sstr4g ) ABGZDHZEHZAIZUCFHCIZJZEKZDFLUBUCBIZUEJZEKZDFLCASCBSUAUGUJD
      FUAUFUIEUAUDUHUEUAABUBUCUAMNOPQDFECARDFECBRT $.
  $}

  $( Equality theorem for composition of two classes.  (Contributed by NM,
     3-Jan-1997.) $)
  coeq1 $p |- ( A = B -> ( A o. C ) = ( B o. C ) ) $=
    ( wss wa ccom wceq coss1 anim12i eqss 3imtr4i ) ABDZBADZEACFZBCFZDZONDZEABG
    NOGLPMQABCHBACHIABJNOJK $.

  $( Equality theorem for composition of two classes.  (Contributed by NM,
     3-Jan-1997.) $)
  coeq2 $p |- ( A = B -> ( C o. A ) = ( C o. B ) ) $=
    ( wss wa ccom wceq coss2 anim12i eqss 3imtr4i ) ABDZBADZECAFZCBFZDZONDZEABG
    NOGLPMQABCHBACHIABJNOJK $.

  ${
    coeq1i.1 $e |- A = B $.
    $( Equality inference for composition of two classes.  (Contributed by NM,
       16-Nov-2000.) $)
    coeq1i $p |- ( A o. C ) = ( B o. C ) $=
      ( wceq ccom coeq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality inference for composition of two classes.  (Contributed by NM,
       16-Nov-2000.) $)
    coeq2i $p |- ( C o. A ) = ( C o. B ) $=
      ( wceq ccom coeq2 ax-mp ) ABECAFCBFEDABCGH $.
  $}

  ${
    coeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for composition of two classes.  (Contributed by NM,
       16-Nov-2000.) $)
    coeq1d $p |- ( ph -> ( A o. C ) = ( B o. C ) ) $=
      ( wceq ccom coeq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Equality deduction for composition of two classes.  (Contributed by NM,
       16-Nov-2000.) $)
    coeq2d $p |- ( ph -> ( C o. A ) = ( C o. B ) ) $=
      ( wceq ccom coeq2 syl ) ABCFDBGDCGFEBCDHI $.
  $}

  ${
    coeq12i.1 $e |- A = B $.
    coeq12i.2 $e |- C = D $.
    $( Equality inference for composition of two classes.  (Contributed by FL,
       7-Jun-2012.) $)
    coeq12i $p |- ( A o. C ) = ( B o. D ) $=
      ( ccom coeq1i coeq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
  $}

  ${
    coeq12d.1 $e |- ( ph -> A = B ) $.
    coeq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality deduction for composition of two classes.  (Contributed by FL,
       7-Jun-2012.) $)
    coeq12d $p |- ( ph -> ( A o. C ) = ( B o. D ) ) $=
      ( ccom coeq1d coeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  ${
    $d w x y z $.  $d y z w A $.  $d y z w B $.
    nfco.1 $e |- F/_ x A $.
    nfco.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for function value.  (Contributed by
       NM, 1-Sep-1999.) $)
    nfco $p |- F/_ x ( A o. B ) $=
      ( vy vw vz ccom cv wbr wa wex copab df-co nfcv nfbr nfan nfex nfopab
      nfcxfr ) ABCIFJZGJZCKZUCHJZBKZLZGMZFHNFHGBCOUHFHAUGAGUDUFAAUBUCCAUBPEAUCP
      ZQAUCUEBUIDAUEPQRSTUA $.
  $}

  ${
    $d R x y z $.  $d S x y z $.  $d A x y z $.
    $( Elements of a composed relation.  (Contributed by BJ, 10-Jul-2022.) $)
    elco $p |- ( A e. ( R o. S ) <->
                  E. x E. y E. z ( A = <. x , z >. /\ ( x S y /\ y R z ) ) ) $=
      ( ccom wcel cv wbr wa wex copab cop wceq df-co eleq2i elopab exbii bitri
      19.42v bicomi excom ) DEFGZHDAIZBIZFJUFCIZEJKZBLZACMZHZDUEUGNOZUHKZCLBLZA
      LZUDUJDACBEFPQUKULUIKZCLZALUOUIACDRUQUNAUQUMBLZCLUNUPURCURUPULUHBUAUBSUMC
      BUCTSTT $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.
    $( Ordered pair membership in a composition.  (Contributed by NM,
       24-Feb-2015.) $)
    brcog $p |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <->
                   E. x ( A D x /\ x C B ) ) ) $=
      ( vy vz cv wbr wa wex ccom wceq breq1 breq2 bi2anan9 exbidv df-co brabga
      ) HJZAJZEKZUCIJZDKZLZAMBUCEKZUCCDKZLZAMHIBCDENFGUBBOZUECOZLUGUJAUKUDUHULU
      FUIUBBUCEPUECUCDQRSHIADETUA $.

    $( Ordered pair membership in a composition.  (Contributed by NM,
       27-Jan-1997.)  (Revised by Mario Carneiro, 24-Feb-2015.) $)
    opelco2g $p |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. ( C o. D ) <->
                   E. x ( <. A , x >. e. D /\ <. x , B >. e. C ) ) ) $=
      ( wcel wa ccom wbr cv wex cop brcog df-br anbi12i exbii 3bitr3g ) BFHCGHI
      BCDEJZKBALZEKZUACDKZIZAMBCNTHBUANEHZUACNDHZIZAMABCDEFGOBCTPUDUGAUBUEUCUFB
      UAEPUACDPQRS $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x D $.  $d x X $.
    $( Ordered pair membership in a composition.  (Contributed by Thierry
       Arnoux, 14-Jan-2018.) $)
    brcogw $p |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) )
      -> A ( C o. D ) B ) $=
      ( vx wcel w3a wbr wa cv wex ccom simpl1 simpl2 wceq breq2 breq1 3ad2antl3
      anbi12d spcegv imp brcog biimpar syl21anc ) AEJZBFJZGHJZKAGDLZGBCLZMZMUIU
      JAINZDLZUOBCLZMZIOZABCDPLZUIUJUKUNQUIUJUKUNRUKUIUNUSUJUKUNUSURUNIGHUOGSUP
      ULUQUMUOGADTUOGBCUAUCUDUEUBUIUJMUTUSIABCDEFUFUGUH $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d ph x $.  $d ph y $.
    eqbrrdva.1 $e |- ( ph -> A C_ ( C X. D ) ) $.
    eqbrrdva.2 $e |- ( ph -> B C_ ( C X. D ) ) $.
    eqbrrdva.3 $e |- ( ( ph /\ x e. C /\ y e. D ) -> ( x A y <-> x B y ) ) $.
    $( Deduction from extensionality principle for relations, given an
       equivalence only on the relation's domain and range.  (Contributed by
       Thierry Arnoux, 28-Nov-2017.) $)
    eqbrrdva $p |- ( ph -> A = B ) $=
      ( cvv cxp wss wrel sstrdi df-rel sylibr cv wcel wbr xpss wa brxp imbitrdi
      ssbrd wb 3expib pm5.21ndd eqbrrdv ) ABCDEADKKLZMDNADFGLZUJHFGUAZODPQAEUJM
      ENAEUKUJIULOEPQABRZFSZCRZGSZUBZUMUODTZUMUOETZAURUMUOUKTZUQADUKUMUOHUEUMUO
      FGUCZUDAUSUTUQAEUKUMUOIUEVAUDAUNUPURUSUFJUGUHUI $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x D $.
    opelco.1 $e |- A e. _V $.
    opelco.2 $e |- B e. _V $.
    $( Binary relation on a composition.  (Contributed by NM, 21-Sep-2004.)
       (Revised by Mario Carneiro, 24-Feb-2015.) $)
    brco $p |- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) $=
      ( cvv wcel ccom wbr cv wa wex wb brcog mp2an ) BHICHIBCDEJKBALZEKRCDKMANO
      FGABCDEHHPQ $.

    $( Ordered pair membership in a composition.  (Contributed by NM,
       27-Dec-1996.)  (Revised by Mario Carneiro, 24-Feb-2015.) $)
    opelco $p |- ( <. A , B >. e. ( C o. D ) <-> E. x ( A D x /\ x C B ) ) $=
      ( cop ccom wcel wbr cv wa wex df-br brco bitr3i ) BCHDEIZJBCRKBALZEKSCDKM
      ANBCROABCDEFGPQ $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Subset theorem for converse.  (Contributed by NM, 22-Mar-1998.) $)
    cnvss $p |- ( A C_ B -> `' A C_ `' B ) $=
      ( vy vx wss wbr copab ccnv cop wcel ssel 3imtr4g ssopab2dv df-cnv 3sstr4g
      cv df-br ) ABEZCPZDPZAFZDCGSTBFZDCGAHBHRUAUBDCRSTIZAJUCBJUAUBABUCKSTAQSTB
      QLMDCANDCBNO $.
  $}

  $( Equality theorem for converse.  (Contributed by NM, 13-Aug-1995.) $)
  cnveq $p |- ( A = B -> `' A = `' B ) $=
    ( wss wa ccnv wceq cnvss anim12i eqss 3imtr4i ) ABCZBACZDAEZBEZCZNMCZDABFMN
    FKOLPABGBAGHABIMNIJ $.

  ${
    cnveqi.1 $e |- A = B $.
    $( Equality inference for converse.  (Contributed by NM, 23-Dec-2008.) $)
    cnveqi $p |- `' A = `' B $=
      ( wceq ccnv cnveq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    cnveqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for converse.  (Contributed by NM, 6-Dec-2013.) $)
    cnveqd $p |- ( ph -> `' A = `' B ) $=
      ( wceq ccnv cnveq syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    $d x y A $.  $d x y R $.
    $( Membership in a converse.  Equation 5 of [Suppes] p. 62.  (Contributed
       by NM, 24-Mar-1998.) $)
    elcnv $p |- ( A e. `' R <-> E. x E. y ( A = <. x , y >. /\ y R x ) ) $=
      ( ccnv wcel cv wbr copab cop wceq wa wex df-cnv eleq2i elopab bitri ) CDE
      ZFCBGZAGZDHZABIZFCTSJKUALBMAMRUBCABDNOUAABCPQ $.

    $( Membership in a converse.  Equation 5 of [Suppes] p. 62.  (Contributed
       by NM, 11-Aug-2004.) $)
    elcnv2 $p |- ( A e. `' R <->
                 E. x E. y ( A = <. x , y >. /\ <. y , x >. e. R ) ) $=
      ( ccnv wcel cv cop wceq wbr wa wex elcnv df-br anbi2i 2exbii bitri ) CDEF
      CAGZBGZHIZSRDJZKZBLALTSRHDFZKZBLALABCDMUBUDABUAUCTSRDNOPQ $.
  $}

  ${
    $d y z A $.  $d x y z $.
    nfcnv.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for converse.  (Contributed by NM,
       31-Jan-2004.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nfcnv $p |- F/_ x `' A $=
      ( vz vy ccnv cv wbr copab df-cnv nfcv nfbr nfopab nfcxfr ) ABFDGZEGZBHZED
      IEDBJQEDAAOPBAOKCAPKLMN $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.
    $( Ordered-pair membership in converse.  (Contributed by NM, 13-May-1999.)
       (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    opelcnvg $p |- ( ( A e. C /\ B e. D ) ->
                     ( <. A , B >. e. `' R <-> <. B , A >. e. R ) ) $=
      ( vy vx wcel wa ccnv wbr cop cv breq2 breq1 df-cnv brabg df-br 3bitr3g )
      ACHBDHIABEJZKBAEKZABLTHBALEHFMZGMZEKUBAEKUAGFABCDTUCAUBENUBBAEOGFEPQABTRB
      AERS $.
  $}

  $( The converse of a binary relation swaps arguments.  Theorem 11 of [Suppes]
     p. 61.  (Contributed by NM, 10-Oct-2005.) $)
  brcnvg $p |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) ) $=
    ( wcel wa cop ccnv wbr opelcnvg df-br 3bitr4g ) ACFBDFGABHEIZFBAHEFABNJBAEJ
    ABCDEKABNLBAELM $.

  ${
    opelcnv.1 $e |- A e. _V $.
    opelcnv.2 $e |- B e. _V $.
    $( Ordered-pair membership in converse.  (Contributed by NM,
       13-Aug-1995.) $)
    opelcnv $p |- ( <. A , B >. e. `' R <-> <. B , A >. e. R ) $=
      ( cvv wcel cop ccnv wb opelcnvg mp2an ) AFGBFGABHCIGBAHCGJDEABFFCKL $.

    $( The converse of a binary relation swaps arguments.  Theorem 11 of
       [Suppes] p. 61.  (Contributed by NM, 13-Aug-1995.) $)
    brcnv $p |- ( A `' R B <-> B R A ) $=
      ( cvv wcel ccnv wbr wb brcnvg mp2an ) AFGBFGABCHIBACIJDEABFFCKL $.
  $}

  ${
    $d y z A $.  $d y z F $.  $d y z V $.  $d x y z $.
    $( Move class substitution in and out of the converse of a function.
       (Contributed by Thierry Arnoux, 8-Feb-2017.) $)
    csbcnvg $p |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F ) $=
      ( vz vy wcel csb ccnv cv wbr copab wsbc sbcbrg csbconstg breq12d opabbidv
      bitrd csbopabg df-cnv wceq a1i 3eqtr4rd csbeq2i eqtr4di ) BDGZABCHZIZABEJ
      ZFJZCKZFELZHZABCIZHUFUKABMZFELUIUJUGKZFELZUMUHUFUOUPFEUFUOABUIHZABUJHZUGK
      UPABUIUJDCNUFURUIUSUJUGABUIDOABUJDOPRQUKAFEBDSUHUQUAUFFEUGTUBUCABUNULFECT
      UDUE $.
  $}

  ${
    $d x y z A $.  $d x y z B $.
    $( Distributive law of converse over class composition.  Theorem 26 of
       [Suppes] p. 64.  (Contributed by NM, 19-Mar-1998.)  (Proof shortened by
       Andrew Salmon, 27-Aug-2011.) $)
    cnvco $p |- `' ( A o. B ) = ( `' B o. `' A ) $=
      ( vx vy vz cv ccom wbr copab ccnv wa wex exancom brco brcnv anbi12i exbii
      vex 3bitr4i opabbii df-cnv df-co 3eqtr4i ) CFZDFZABGZHZDCIUEEFZAJZHZUHUDB
      JZHZKZELZDCIUFJUKUIGUGUNDCUDUHBHZUHUEAHZKELUPUOKZELUGUNUOUPEMEUDUEABCRZDR
      ZNUMUQEUJUPULUOUEUHAUSERZOUHUDBUTUROPQSTDCUFUADCEUKUIUBUC $.
  $}

  ${
    $d x y z w A $.
    $( The converse of a class union is the (indexed) union of the converses of
       its members.  (Contributed by NM, 11-Aug-2004.) $)
    cnvuni $p |- `' U. A = U_ x e. A `' x $=
      ( vy vz vw cuni ccnv cv ciun wcel wrex cop wa elcnv2 eluni2 anbi2i bitr4i
      wceq wex rexcom4 r19.42v 2exbii rexbii exbii 3bitrri 3bitri eliun eqriv )
      CBFZGZABAHZGZIZCHZUJJZUNULJZABKZUNUMJUOUNDHZEHZLRZUSURLZUIJZMZESDSUTVAUKJ
      ZMZABKZESZDSZUQDEUNUINVCVFDEVCUTVDABKZMVFVBVIUTAVABOPUTVDABUAQUBUQVEESZDS
      ZABKVJABKZDSVHUPVKABDEUNUKNUCVJADBTVLVGDVEAEBTUDUEUFAUNBULUGQUH $.
  $}

  ${
    $d x y A $.
    $( Alternate definition of domain.  Definition 6.5(1) of [TakeutiZaring]
       p. 24.  (Contributed by NM, 28-Dec-1996.) $)
    dfdm3 $p |- dom A = { x | E. y <. x , y >. e. A } $=
      ( cdm cv wbr wex cab cop wcel df-dm df-br exbii abbii eqtri ) CDAEZBEZCFZ
      BGZAHPQICJZBGZAHABCKSUAARTBPQCLMNO $.

    $( Alternate definition of range.  Definition 4 of [Suppes] p. 60.
       (Contributed by NM, 27-Dec-1996.) $)
    dfrn2 $p |- ran A = { y | E. x x A y } $=
      ( crn ccnv cdm cv wbr wex cab df-rn df-dm vex brcnv exbii abbii 3eqtri )
      CDCEZFBGZAGZRHZAIZBJTSCHZAIZBJCKBARLUBUDBUAUCASTCBMAMNOPQ $.

    $( Alternate definition of range.  Definition 6.5(2) of [TakeutiZaring]
       p. 24.  (Contributed by NM, 28-Dec-1996.) $)
    dfrn3 $p |- ran A = { y | E. x <. x , y >. e. A } $=
      ( crn cv wbr wex cab cop wcel dfrn2 df-br exbii abbii eqtri ) CDAEZBEZCFZ
      AGZBHPQICJZAGZBHABCKSUABRTAPQCLMNO $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( Membership in a range.  (Contributed by Scott Fenton, 2-Feb-2011.) $)
    elrn2g $p |- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) ) $=
      ( vy cv cop wcel wex crn wceq opeq2 eleq1d exbidv dfrn3 elab2g ) AFZEFZGZ
      CHZAIQBGZCHZAIEBCJDRBKZTUBAUCSUACRBQLMNAECOP $.

    $( Membership in a range.  (Contributed by Scott Fenton, 2-Feb-2011.) $)
    elrng $p |- ( A e. V -> ( A e. ran B <-> E. x x B A ) ) $=
      ( wcel crn cv cop wex wbr elrn2g df-br exbii bitr4di ) BDEBCFEAGZBHCEZAIO
      BCJZAIABCDKQPAOBCLMN $.
  $}

  ${
    $d A a $.  $d B a $.  $d R a $.  $d Y a $.
    $( If a relation is a subset of a cartesian product, then for each element
       of the range of the relation there is an element of the first set of the
       cartesian product which is related to the element of the range by the
       relation.  (Contributed by AV, 24-Oct-2020.) $)
    ssrelrn $p |- ( ( R C_ ( A X. B ) /\ Y e. ran R ) -> E. a e. A a R Y ) $=
      ( cxp wss crn wcel wa cv wbr wex wrex wi elrng ssbr brxp simplbi syl6 ex
      ancrd adantl eximdv com23 sylbid pm2.43i impcom df-rex sylibr ) CABFZGZDC
      HZIZJEKZAIZUODCLZJZEMZUQEANUNULUSUNULUSOZUNUNUQEMZUTEDCUMPUNULVAUSUNULVAU
      SOUNULJUQUREULUQUROUNULUQUPULUQUODUKLZUPCUKUODQVBUPDBIUODABRSTUBUCUDUAUEU
      FUGUHUQEAUIUJ $.
  $}

  ${
    $d x y A $.
    $( Alternate definition of domain.  (Contributed by NM, 28-Dec-1996.) $)
    dfdm4 $p |- dom A = ran `' A $=
      ( vy vx cv wbr wex cab crn cdm vex brcnv exbii abbii dfrn2 df-dm 3eqtr4ri
      ccnv ) BDZCDZAQZEZBFZCGSRAEZBFZCGTHAIUBUDCUAUCBRSABJCJKLMBCTNCBAOP $.
  $}

  ${
    $d x y w v $.  $d w v A $.
    dfdmf.1 $e |- F/_ x A $.
    dfdmf.2 $e |- F/_ y A $.
    $( Definition of domain, using bound-variable hypotheses instead of
       distinct variable conditions.  (Contributed by NM, 8-Mar-1995.)
       (Revised by Mario Carneiro, 15-Oct-2016.) $)
    dfdmf $p |- dom A = { x | E. y x A y } $=
      ( vw vv cdm cv wbr wex cab df-dm nfcv nfbr nfv breq2 cbvex abbii nfex weq
      breq1 exbidv cbvab 3eqtri ) CHFIZGIZCJZGKZFLUFBIZCJZBKZFLAIZUJCJZBKZALFGC
      MUIULFUHUKGBBUFUGCBUFNEBUGNOUKGPUGUJUFCQRSULUOFAUKABAUFUJCAUFNDAUJNOTUOFP
      FAUAUKUNBUFUMUJCUBUCUDUE $.
  $}

  ${
    $d A w y $.  $d B w y $.  $d V w y $.  $d x w y $.
    $( Distribute proper substitution through the domain of a class.
       (Contributed by Jim Kingdon, 8-Dec-2018.) $)
    csbdmg $p |- ( A e. V -> [_ A / x ]_ dom B = dom [_ A / x ]_ B ) $=
      ( vy vw wcel cop wex cab csb cdm wsbc csbabg sbcex2 sbcel2g exbidv bitrid
      cv dfdm3 abbidv eqtrd csbeq2i 3eqtr4g ) BDGZABESFSHZCGZFIZEJZKZUFABCKZGZF
      IZEJZABCLZKUKLUEUJUHABMZEJUNUHAEBDNUEUPUMEUPUGABMZFIUEUMUGFABOUEUQULFABUF
      CDPQRUAUBABUOUIEFCTUCEFUKTUD $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Domain membership.  Theorem 4 of [Suppes] p. 59.  (Contributed by Mario
       Carneiro, 9-Jul-2014.) $)
    eldmg $p |- ( A e. V -> ( A e. dom B <-> E. y A B y ) ) $=
      ( vx cv wbr wex cdm wceq breq1 exbidv df-dm elab2g ) EFZAFZCGZAHBPCGZAHEB
      CIDOBJQRAOBPCKLEACMN $.

    $( Domain membership.  Theorem 4 of [Suppes] p. 59.  (Contributed by NM,
       27-Jan-1997.)  (Revised by Mario Carneiro, 9-Jul-2014.) $)
    eldm2g $p |- ( A e. V -> ( A e. dom B <-> E. y <. A , y >. e. B ) ) $=
      ( wcel cdm cv wbr wex cop eldmg df-br exbii bitrdi ) BDEBCFEBAGZCHZAIBOJC
      EZAIABCDKPQABOCLMN $.
  $}

  ${
    $d y A $.  $d y B $.
    eldm.1 $e |- A e. _V $.
    $( Membership in a domain.  Theorem 4 of [Suppes] p. 59.  (Contributed by
       NM, 2-Apr-2004.) $)
    eldm $p |- ( A e. dom B <-> E. y A B y ) $=
      ( cvv wcel cdm cv wbr wex wb eldmg ax-mp ) BEFBCGFBAHCIAJKDABCELM $.

    $( Membership in a domain.  Theorem 4 of [Suppes] p. 59.  (Contributed by
       NM, 1-Aug-1994.) $)
    eldm2 $p |- ( A e. dom B <-> E. y <. A , y >. e. B ) $=
      ( cvv wcel cdm cv cop wex wb eldm2g ax-mp ) BEFBCGFBAHICFAJKDABCELM $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Subset theorem for domain.  (Contributed by NM, 11-Aug-1994.) $)
    dmss $p |- ( A C_ B -> dom A C_ dom B ) $=
      ( vx vy wss cdm cv cop wcel wex ssel eximdv vex eldm2 3imtr4g ssrdv ) ABE
      ZCAFZBFZQCGZDGHZAIZDJUABIZDJTRITSIQUBUCDABUAKLDTACMZNDTBUDNOP $.
  $}

  $( Equality theorem for domain.  (Contributed by NM, 11-Aug-1994.) $)
  dmeq $p |- ( A = B -> dom A = dom B ) $=
    ( wss wa cdm wceq dmss anim12i eqss 3imtr4i ) ABCZBACZDAEZBEZCZNMCZDABFMNFK
    OLPABGBAGHABIMNIJ $.

  ${
    dmeqi.1 $e |- A = B $.
    $( Equality inference for domain.  (Contributed by NM, 4-Mar-2004.) $)
    dmeqi $p |- dom A = dom B $=
      ( wceq cdm dmeq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    dmeqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for domain.  (Contributed by NM, 4-Mar-2004.) $)
    dmeqd $p |- ( ph -> dom A = dom B ) $=
      ( wceq cdm dmeq syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    $d y A $.  $d y B $.  $d y C $.
    opeldm.1 $e |- A e. _V $.
    opeldm.2 $e |- B e. _V $.
    $( Membership of first of an ordered pair in a domain.  (Contributed by NM,
       30-Jul-1995.) $)
    opeldm $p |- ( <. A , B >. e. C -> A e. dom C ) $=
      ( vy cop wcel cv wex cdm wceq opeq2 eleq1d spcev eldm2 sylibr ) ABGZCHZAF
      IZGZCHZFJACKHUBSFBETBLUARCTBAMNOFACDPQ $.

    $( Membership of first of a binary relation in a domain.  (Contributed by
       NM, 30-Jul-1995.) $)
    breldm $p |- ( A R B -> A e. dom R ) $=
      ( wbr cop wcel cdm df-br opeldm sylbi ) ABCFABGCHACIHABCJABCDEKL $.
  $}

  ${
    $d y A $.  $d y B $.  $d y C $.
    $( Membership of first of an ordered pair in a domain.  (Contributed by Jim
       Kingdon, 9-Jul-2019.) $)
    opeldmg $p |- ( ( A e. V /\ B e. W ) ->
        ( <. A , B >. e. C -> A e. dom C ) ) $=
      ( vy wcel wa cop cv wex cdm wi wceq opeq2 eleq1d spcegv adantl wb eldm2g
      adantr sylibrd ) ADGZBEGZHABIZCGZAFJZIZCGZFKZACLGZUDUFUJMUCUIUFFBEUGBNUHU
      ECUGBAOPQRUCUKUJSUDFACDTUAUB $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x R $.
    $( Membership of first of a binary relation in a domain.  (Contributed by
       NM, 21-Mar-2007.) $)
    breldmg $p |- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R ) $=
      ( vx wbr w3a cdm cv wex breq2 spcegv imp 3adant1 wb eldmg 3ad2ant1 mpbird
      wcel ) ACTZBDTZABEGZHAEITZAFJZEGZFKZUBUCUGUAUBUCUGUFUCFBDUEBAELMNOUAUBUDU
      GPUCFAECQRS $.

    $( The domain of a union is the union of domains.  Exercise 56(a) of
       [Enderton] p. 65.  (Contributed by NM, 12-Aug-1994.)  (Proof shortened
       by Andrew Salmon, 27-Aug-2011.) $)
    dmun $p |- dom ( A u. B ) = ( dom A u. dom B ) $=
      ( vy vx cv wbr wex cab cun cdm wo unab brun exbii 19.43 abbii eqtri df-dm
      bitr2i uneq12i 3eqtr4ri ) CEZDEZAFZDGZCHZUBUCBFZDGZCHZIZUBUCABIZFZDGZCHZA
      JZBJZIUKJUJUEUHKZCHUNUEUHCLUQUMCUMUDUGKZDGUQULURDUBUCABMNUDUGDOSPQUOUFUPU
      ICDARCDBRTCDUKRUA $.

    $( The domain of an intersection belong to the intersection of domains.
       Theorem 6 of [Suppes] p. 60.  (Contributed by NM, 15-Sep-2004.) $)
    dmin $p |- dom ( A i^i B ) C_ ( dom A i^i dom B ) $=
      ( vx vy cin cdm cv cop wcel wa wex 19.40 eldm2 elin exbii anbi12i 3imtr4i
      vex bitri ssriv ) CABEZFZAFZBFZEZCGZDGHZAIZUGBIZJZDKZUHDKZUIDKZJZUFUBIZUF
      UEIZUHUIDLUOUGUAIZDKUKDUFUACRZMUQUJDUGABNOSUPUFUCIZUFUDIZJUNUFUCUDNUSULUT
      UMDUFAURMDUFBURMPSQT $.
  $}

  ${
    $d x y z $.  $d y z A $.  $d y z B $.
    $( The domain of an indexed union.  (Contributed by Mario Carneiro,
       26-Apr-2016.) $)
    dmiun $p |- dom U_ x e. A B = U_ x e. A dom B $=
      ( vy vz ciun cdm cv cop wcel wex wrex rexcom4 eldm2 rexbii eliun 3bitr4ri
      vex exbii 3bitr4i eqriv ) DABCFZGZABCGZFZDHZEHIZUBJZEKZUFUDJZABLZUFUCJUFU
      EJUGCJZEKZABLULABLZEKUKUIULAEBMUJUMABEUFCDRZNOUHUNEAUGBCPSQEUFUBUONAUFBUD
      PTUA $.

    $d x A $.
    $( The domain of a union.  Part of Exercise 8 of [Enderton] p. 41.
       (Contributed by NM, 3-Feb-2004.) $)
    dmuni $p |- dom U. A = U_ x e. A dom x $=
      ( vy vz cuni cdm cv ciun cop wcel wex wrex excom ancom 19.41v vex 3bitr4i
      wa eldm2 exbii anbi2i bitri eluni df-rex eliun eqriv ) CBEZFZABAGZFZHZCGZ
      DGIZUGJZDKZULUJJZABLZULUHJULUKJUMUIJZUIBJZRZAKZDKZUSUPRZAKZUOUQVBUTDKZAKV
      DUTDAMVEVCAURDKZUSRUSVFRVEVCVFUSNURUSDOUPVFUSDULUICPZSUAQTUBUNVADAUMBUCTU
      PABUDQDULUGVGSAULBUJUEQUF $.
  $}

  ${
    $d x y $.
    $( The domain of a class of ordered pairs.  (Contributed by NM,
       16-May-1995.)  (Revised by Mario Carneiro, 4-Dec-2016.) $)
    dmopab $p |- dom { <. x , y >. | ph } = { x | E. y ph } $=
      ( copab cdm wbr wex cab nfopab1 nfopab2 dfdmf cop wcel df-br opabid bitri
      cv exbii abbii eqtri ) ABCDZEBQZCQZUAFZCGZBHACGZBHBCUAABCIABCJKUEUFBUDACU
      DUBUCLUAMAUBUCUANABCOPRST $.
  $}

  ${
    $d x y A $.
    $( Upper bound for the domain of a restricted class of ordered pairs.
       (Contributed by NM, 31-Jan-2004.) $)
    dmopabss $p |- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A $=
      ( cv wcel wa copab cdm wex cab dmopab 19.42v abbii ssab2 eqsstri ) BEDFZA
      GZBCHIRCJZBKZDRBCLTQACJZGZBKDSUBBQACMNUABDOPP $.
  $}

  ${
    $d x y A $.
    $( The domain of a restricted class of ordered pairs.  (Contributed by NM,
       31-Jan-2004.) $)
    dmopab3 $p |- ( A. x e. A E. y ph <->
               dom { <. x , y >. | ( x e. A /\ ph ) } = A ) $=
      ( wex wral cv wcel wi wal wa wb copab cdm wceq df-ral pm4.71 albii dmopab
      cab 19.42v abbii eqtri eqeq1i eqcom abeq2 3bitr2ri 3bitri ) ACEZBDFBGDHZU
      IIZBJUJUJUIKZLZBJZUJAKZBCMNZDOZUIBDPUKUMBUJUIQRUQULBTZDODUROUNUPURDUPUOCE
      ZBTURUOBCSUSULBUJACUAUBUCUDDURUEULBDUFUGUH $.
  $}

  ${
    $d x y $.
    $( The domain of the empty set is empty.  Part of Theorem 3.8(v) of [Monk1]
       p. 36.  (Contributed by NM, 4-Jul-1994.)  (Proof shortened by Andrew
       Salmon, 27-Aug-2011.) $)
    dm0 $p |- dom (/) = (/) $=
      ( vx vy c0 cdm wceq cv wcel eq0 cop wex noel nex vex eldm2 mtbir mpgbir
      wn ) CDZCEAFZRGZQAARHTSBFIZCGZBJUBBUAKLBSCAMNOP $.

    $( The domain of the identity relation is the universe.  (Contributed by
       NM, 30-Apr-1998.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    dmi $p |- dom _I = _V $=
      ( vx vy cid cdm cvv wceq cv wcel eqv wbr wex a9ev ideq equcom bitri exbii
      vex mpbir eldm mpgbir ) CDZEFAGZUAHZAAUAIUCUBBGZCJZBKZUFUDUBFZBKBALUEUGBU
      EUBUDFUGUBUDBQMABNOPRBUBCAQSRT $.

    $( The domain of the universe is the universe.  (Contributed by NM,
       8-Aug-2003.) $)
    dmv $p |- dom _V = _V $=
      ( cvv cdm ssv cid dmi wss dmss ax-mp eqsstrri eqssi ) ABZAKCADBZKEDAFLKFD
      CDAGHIJ $.
  $}

  ${
    $d x y A $.
    $( An empty domain implies an empty range.  For a similar theorem for
       whether the domain and range are inhabited, see ~ dmmrnm .  (Contributed
       by NM, 21-May-1998.) $)
    dm0rn0 $p |- ( dom A = (/) <-> ran A = (/) ) $=
      ( vx vy cv wbr wex cab c0 wceq wcel wb wal alnex noel albii abeq1 3bitr4i
      wn nbn eqeq1i cdm crn excom xchbinx bitr4i 3bitr3i df-dm dfrn2 ) BDZCDZAE
      ZCFZBGZHIZUKBFZCGZHIZAUAZHIAUBZHIULUIHJZKZBLZUOUJHJZKZCLZUNUQULRZBLZUORZC
      LZVBVEVGUOCFZRVIVGULBFVJULBMUKBCUCUDUOCMUEVFVABUTULUINSOVHVDCVCUOUJNSOUFU
      LBHPUOCHPQURUMHBCAUGTUSUPHBCAUHTQ $.

    $( A relation is empty iff its domain is empty.  For a similar theorem for
       whether the relation and domain are inhabited, see ~ reldmm .
       (Contributed by NM, 15-Sep-2004.) $)
    reldm0 $p |- ( Rel A -> ( A = (/) <-> dom A = (/) ) ) $=
      ( vx vy wrel c0 wceq cv cop wcel wb wal cdm rel0 eqrel mpan2 wn eq0 alnex
      wex albii vex eldm2 xchbinxr noel nbn bitr3i bitr2i bitrdi ) ADZAEFZBGZCG
      HZAIZULEIZJZCKZBKZALZEFZUIEDUJUQJMBCAENOUSUKURIZPZBKUQBURQVAUPBVAUMPZCKZU
      PVCUMCSUTUMCRCUKABUAUBUCVBUOCUNUMULUDUETUFTUGUH $.
  $}

  ${
    $d A x z w $.  $d A y z w $.
    $( A relation is inhabited iff its domain is inhabited.  (Contributed by
       Jim Kingdon, 30-Jan-2026.) $)
    reldmm $p |- ( Rel A -> ( E. x x e. A <-> E. y y e. dom A ) ) $=
      ( vw vz cv wcel wex wrel cdm eleq1w cbvexv wa cop wceq wi elrel eximi nfv
      19.37-1 eleq1 biimpd syl 3syl syldbl2 eldm2 exbii sylibr ex exlimdv elex2
      vex exlimivv sylbi impbid1 bitr3id ) AFCGZAHDFZCGZDHZCIZBFZCJGZBHZUSUQDAD
      ACKLVAUTVDVAUSVDDVAUSVDVAUSMZVBEFNZCGZEHZBHZVDVAUSVIVEURVFOZEHZBHUSVHPZBH
      USVIPBEURCQVKVLBVKUSVGPZEHVLVJVMEVJUSVGURVFCUAUBRUSVGEUSESTUCRUSVHBUSBSTU
      DUEVCVHBEVBCBULUFUGZUHUIUJVDVIUTVNVGUTBEDVFCUKUMUNUOUP $.
  $}

  ${
    $d y z A $.  $d x z A $.
    $( A domain is inhabited if and only if the range is inhabited.
       (Contributed by Jim Kingdon, 15-Dec-2018.) $)
    dmmrnm $p |- ( E. x x e. dom A <-> E. y y e. ran A ) $=
      ( vz cv cdm wcel wex wbr crn cab df-dm exbii abid bitri dfrn2 excom eleq1
      eleq2i cbvexv 3bitr2i ) AEZCFZGZAHZUBDEZCIZDHZAHZUFCJZGZDHZBEZUJGZBHUEUBU
      HAKZGZAHUIUDUPAUCUOUBADCLSMUPUHAUHANMOULUFUGAHZDKZGZDHZUIUKUSDUJURUFADCPS
      MUTUQDHUIUSUQDUQDNMUGDAQOOUKUNDBUFUMUJRTUA $.
  $}

  ${
    $d y z A $.  $d x y z B $.
    $( The domain of a cross product.  Part of Theorem 3.13(x) of [Monk1]
       p. 37.  (Contributed by NM, 28-Jul-1995.)  (Proof shortened by Andrew
       Salmon, 27-Aug-2011.) $)
    dmxpm $p |- ( E. x x e. B -> dom ( A X. B ) = A ) $=
      ( vz vy cv wcel wex cxp cdm wceq eleq1 cbvexv copab df-xp dmeqi ralrimivw
      wa wral id dmopab3 sylib eqtrid sylbi ) AFZCGZAHDFZCGZDHZBCIZJZBKUFUHADUE
      UGCLMUIUKEFBGUHREDNZJZBUJULEDBCOPUIUIEBSUMBKUIUIEBUITQUHEDBUAUBUCUD $.
  $}

  ${
    $d A x y $.
    $( The domain of a square Cartesian product.  (Contributed by NM,
       28-Jul-1995.)  (Revised by Jim Kingdon, 11-Apr-2023.) $)
    dmxpid $p |- dom ( A X. A ) = A $=
      ( vy vx cxp cdm cv wcel wa copab df-xp dmeqi wral wceq elex2 rgen dmopab3
      wex mpbi eqtri ) AADZEBFZAGCFAGZHBCIZEZATUCBCAAJKUBCQZBALUDAMUEBACUAANOUB
      BCAPRS $.
  $}

  $( The domain of the intersection of two square Cartesian products.  Unlike
     ~ dmin , equality holds.  (Contributed by NM, 29-Jan-2008.) $)
  dmxpin $p |- dom ( ( A X. A ) i^i ( B X. B ) ) = ( A i^i B ) $=
    ( cxp cin cdm inxp dmeqi dmxpid eqtri ) AACBBCDZEABDZKCZEKJLAABBFGKHI $.

  $( The Cartesian product of a class with itself is one-to-one.  (Contributed
     by NM, 5-Nov-2006.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  xpid11 $p |- ( ( A X. A ) = ( B X. B ) <-> A = B ) $=
    ( cxp wceq cdm dmeq dmxpid 3eqtr3g xpeq12 anidms impbii ) AACZBBCZDZABDZNLE
    MEABLMFAGBGHONABABIJK $.

  $( The domain of the double converse of a class (which doesn't have to be a
     relation as in ~ dfrel2 ).  (Contributed by NM, 8-Apr-2007.) $)
  dmcnvcnv $p |- dom `' `' A = dom A $=
    ( cdm ccnv crn dfdm4 df-rn eqtr2i ) ABACZDHCBAEHFG $.

  $( The range of the double converse of a class.  (Contributed by NM,
     8-Apr-2007.) $)
  rncnvcnv $p |- ran `' `' A = ran A $=
    ( crn ccnv cdm df-rn dfdm4 eqtr2i ) ABACZDHCBAEHFG $.

  ${
    $d x y A $.  $d x y B $.
    $( The first member of an ordered pair in a relation belongs to the domain
       of the relation.  (Contributed by NM, 28-Jul-2004.) $)
    elreldm $p |- ( ( Rel A /\ B e. A ) -> |^| |^| B e. dom A ) $=
      ( vx vy wrel wcel cint cdm cv cop wceq wex cvv cxp wss wi df-rel ssel vex
      sylbi imbitrdi eleq1 opeldm biimtrdi inteqd op1stb eqtrdi eleq1d exlimivv
      elvv inteq sylibrd syli imp ) AEZBAFZBGZGZAHZFZUPUOBCIZDIZJZKZDLCLZUTUOUP
      BMMNZFZVEUOAVFOUPVGPAQAVFBRTCDBUJUAVDUPUTPCDVDUPVAUSFZUTVDUPVCAFVHBVCAUBV
      AVBACSZDSZUCUDVDURVAUSVDURVCGZGVAVDUQVKBVCUKUEVAVBVIVJUFUGUHULUIUMUN $.
  $}

  $( Equality theorem for range.  (Contributed by NM, 29-Dec-1996.) $)
  rneq $p |- ( A = B -> ran A = ran B ) $=
    ( wceq ccnv cdm crn cnveq dmeqd df-rn 3eqtr4g ) ABCZADZEBDZEAFBFKLMABGHAIBI
    J $.

  ${
    rneqi.1 $e |- A = B $.
    $( Equality inference for range.  (Contributed by NM, 4-Mar-2004.) $)
    rneqi $p |- ran A = ran B $=
      ( wceq crn rneq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    rneqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for range.  (Contributed by NM, 4-Mar-2004.) $)
    rneqd $p |- ( ph -> ran A = ran B ) $=
      ( wceq crn rneq syl ) ABCEBFCFEDBCGH $.
  $}

  $( Subset theorem for range.  (Contributed by NM, 22-Mar-1998.) $)
  rnss $p |- ( A C_ B -> ran A C_ ran B ) $=
    ( wss ccnv cdm crn cnvss dmss syl df-rn 3sstr4g ) ABCZADZEZBDZEZAFBFLMOCNPC
    ABGMOHIAJBJK $.

  $( The second argument of a binary relation belongs to its range.
     (Contributed by NM, 29-Jun-2008.) $)
  brelrng $p |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C ) $=
    ( wcel wbr w3a ccnv cdm crn wb brcnvg ancoms biimp3ar 3com12 syld3an3 df-rn
    breldmg eleqtrrdi ) ADFZBEFZABCGZHBCIZJZCKUAUBUCBAUDGZBUEFZUAUBUFUCUBUAUFUC
    LBAEDCMNOUBUAUFUGBAEDUDSPQCRT $.

  $( Membership of second member of an ordered pair in a range.  (Contributed
     by Jim Kingdon, 26-Jan-2019.) $)
  opelrng $p |- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C ) $=
    ( cop wcel wbr crn df-br brelrng syl3an3br ) ABFCGADGBEGABCHBCIGABCJABCDEKL
    $.

  ${
    brelrn.1 $e |- A e. _V $.
    brelrn.2 $e |- B e. _V $.
    $( The second argument of a binary relation belongs to its range.
       (Contributed by NM, 13-Aug-2004.) $)
    brelrn $p |- ( A C B -> B e. ran C ) $=
      ( cvv wcel wbr crn brelrng mp3an12 ) AFGBFGABCHBCIGDEABCFFJK $.

    $( Membership of second member of an ordered pair in a range.  (Contributed
       by NM, 23-Feb-1997.) $)
    opelrn $p |- ( <. A , B >. e. C -> B e. ran C ) $=
      ( cop wcel wbr crn df-br brelrn sylbir ) ABFCGABCHBCIGABCJABCDEKL $.
  $}

  $( The first argument of a binary relation belongs to its domain.
     (Contributed by NM, 2-Jul-2008.) $)
  releldm $p |- ( ( Rel R /\ A R B ) -> A e. dom R ) $=
    ( wrel wbr wa cvv wcel cdm brrelex brrelex2 simpr breldmg syl3anc ) CDZABCE
    ZFAGHBGHPACIHABCJABCKOPLABGGCMN $.

  $( The second argument of a binary relation belongs to its range.
     (Contributed by NM, 2-Jul-2008.) $)
  relelrn $p |- ( ( Rel R /\ A R B ) -> B e. ran R ) $=
    ( wrel wbr wa cvv wcel crn brrelex brrelex2 simpr brelrng syl3anc ) CDZABCE
    ZFAGHBGHPBCIHABCJABCKOPLABCGGMN $.

  ${
    $d x A $.  $d x R $.
    $( Membership in a domain.  (Contributed by Mario Carneiro, 5-Nov-2015.) $)
    releldmb $p |- ( Rel R -> ( A e. dom R <-> E. x A R x ) ) $=
      ( wrel cdm wcel cv wbr wex eldmg ibi releldm ex exlimdv impbid2 ) CDZBCEZ
      FZBAGZCHZAIZRUAABCQJKPTRAPTRBSCLMNO $.

    $( Membership in a range.  (Contributed by Mario Carneiro, 5-Nov-2015.) $)
    relelrnb $p |- ( Rel R -> ( A e. ran R <-> E. x x R A ) ) $=
      ( wrel crn wcel cv wbr wex elrng ibi relelrn ex exlimdv impbid2 ) CDZBCEZ
      FZAGZBCHZAIZRUAABCQJKPTRAPTRSBCLMNO $.
  $}

  ${
    releldm.1 $e |- Rel R $.
    $( The first argument of a binary relation belongs to its domain.
       (Contributed by NM, 28-Apr-2015.) $)
    releldmi $p |- ( A R B -> A e. dom R ) $=
      ( wrel wbr cdm wcel releldm mpan ) CEABCFACGHDABCIJ $.

    $( The second argument of a binary relation belongs to its range.
       (Contributed by NM, 28-Apr-2015.) $)
    relelrni $p |- ( A R B -> B e. ran R ) $=
      ( wrel wbr crn wcel relelrn mpan ) CEABCFBCGHDABCIJ $.
  $}

  ${
    $d x y w v $.  $d w v A $.
    dfrnf.1 $e |- F/_ x A $.
    dfrnf.2 $e |- F/_ y A $.
    $( Definition of range, using bound-variable hypotheses instead of distinct
       variable conditions.  (Contributed by NM, 14-Aug-1995.)  (Revised by
       Mario Carneiro, 15-Oct-2016.) $)
    dfrnf $p |- ran A = { y | E. x x A y } $=
      ( vv vw crn cv wbr wex cab dfrn2 nfcv nfbr nfv breq1 cbvex abbii nfex weq
      breq2 exbidv cbvab 3eqtri ) CHFIZGIZCJZFKZGLAIZUGCJZAKZGLUJBIZCJZAKZBLFGC
      MUIULGUHUKFAAUFUGCAUFNDAUGNOUKFPUFUJUGCQRSULUOGBUKBABUJUGCBUJNEBUGNOTUOGP
      GBUAUKUNAUGUMUJCUBUCUDUE $.
  $}

  ${
    $d x y A $.  $d x y B $.
    elrn.1 $e |- A e. _V $.
    $( Membership in a range.  (Contributed by NM, 10-Jul-1994.) $)
    elrn2 $p |- ( A e. ran B <-> E. x <. x , A >. e. B ) $=
      ( vy cv cop wcel wex crn wceq opeq2 eleq1d exbidv dfrn3 elab2 ) AFZEFZGZC
      HZAIQBGZCHZAIEBCJDRBKZTUBAUCSUACRBQLMNAECOP $.

    $( Membership in a range.  (Contributed by NM, 2-Apr-2004.) $)
    elrn $p |- ( A e. ran B <-> E. x x B A ) $=
      ( crn wcel cv cop wex wbr elrn2 df-br exbii bitr4i ) BCEFAGZBHCFZAIOBCJZA
      IABCDKQPAOBCLMN $.
  $}

  ${
    $d x y z $.  $d y z A $.
    nfrn.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for domain.  (Contributed by NM,
       30-Jan-2004.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nfdm $p |- F/_ x dom A $=
      ( vy vz cdm cv wbr wex cab df-dm nfcv nfbr nfex nfab nfcxfr ) ABFDGZEGZBH
      ZEIZDJDEBKTADSAEAQRBAQLCARLMNOP $.

    $( Bound-variable hypothesis builder for range.  (Contributed by NM,
       1-Sep-1999.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nfrn $p |- F/_ x ran A $=
      ( crn ccnv cdm df-rn nfcnv nfdm nfcxfr ) ABDBEZFBGAKABCHIJ $.
  $}

  $( Domain of an intersection.  (Contributed by FL, 15-Oct-2012.) $)
  dmiin $p |- dom |^|_ x e. A B C_ |^|_ x e. A dom B $=
    ( ciin cdm wss nfii1 nfdm ssiinf cv wcel iinss2 dmss syl mprgbir ) ABCDZEZA
    BCEZDFQRFZABABRQAPABCGHIAJBKPCFSABCLPCMNO $.

  ${
    $d x y $.
    $( The range of a class of ordered pairs.  (Contributed by NM,
       14-Aug-1995.)  (Revised by Mario Carneiro, 4-Dec-2016.) $)
    rnopab $p |- ran { <. x , y >. | ph } = { y | E. x ph } $=
      ( copab crn wbr wex cab nfopab1 nfopab2 dfrnf cop wcel df-br opabid bitri
      cv exbii abbii eqtri ) ABCDZEBQZCQZUAFZBGZCHABGZCHBCUAABCIABCJKUEUFCUDABU
      DUBUCLUAMAUBUCUANABCOPRST $.
  $}

  ${
    $d y z A $.  $d y z B $.  $d x y z C $.
    rnmpt.1 $e |- F = ( x e. A |-> B ) $.
    $( The range of a function in maps-to notation.  (Contributed by Scott
       Fenton, 21-Mar-2011.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    rnmpt $p |- ran F = { y | E. x e. A y = B } $=
      ( cv wcel wceq wa copab crn wex cab wrex rnopab cmpt df-mpt eqtri rneqi
      df-rex abbii 3eqtr4i ) AGCHBGDIZJZABKZLUEAMZBNELUDACOZBNUEABPEUFEACDQUFFA
      BCDRSTUHUGBUDACUAUBUC $.

    $( The range of a function in maps-to notation.  (Contributed by Mario
       Carneiro, 20-Feb-2015.) $)
    elrnmpt $p |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) ) $=
      ( vy cv wceq wrex crn eqeq1 rexbidv rnmpt elab2g ) HIZCJZABKDCJZABKHDELFQ
      DJRSABQDCMNAHBCEGOP $.

    ${
      $d x A $.  $d x D $.
      elrnmpt1s.1 $e |- ( x = D -> B = C ) $.
      $( Elementhood in an image set.  (Contributed by Mario Carneiro,
         12-Sep-2015.) $)
      elrnmpt1s $p |- ( ( D e. A /\ C e. V ) -> C e. ran F ) $=
        ( wcel wceq wrex crn eqid cv eqeq2d rspcev mpan2 elrnmpt biimparc sylan
        ) EBJZDCKZABLZDGJZDFMJZUBDDKZUDDNUCUGAEBAOEKCDDIPQRUEUFUDABCDFGHSTUA $.
    $}

    $( Elementhood in an image set.  (Contributed by Mario Carneiro,
       31-Aug-2015.) $)
    elrnmpt1 $p |- ( ( x e. A /\ B e. V ) -> B e. ran F ) $=
      ( vz vy wcel cv crn csb wceq wa wex vex wb id csbeq1a nfcsb1v bitr2d wrex
      eleq12d biantrud equcoms spcev df-rex nfv nfcri nfeq2 nfan eqeq2d anbi12d
      cbvex bitri eqeq1 anbi2d exbidv bitrid rnmpt elab2g imbitrrid impcom ) CE
      IZAJZBIZCDKZIZVFVHVDGJZAVIBLZIZCAVICLZMZNZGOZVNVFGVEAPVNVFQAGVEVIMZVFVKVN
      VPVEVIBVJVPRAVIBSUCZVPVMVKAVICSZUDUAUEUFHJZCMZABUBZVOHCVGEWAVKVSVLMZNZGOZ
      VTVOWAVFVTNZAOWDVTABUGWEWCAGWEGUHVKWBAAGVJAVIBTUIAVSVLAVICTUJUKVPVFVKVTWB
      VQVPCVLVSVRULUMUNUOVTWCVNGVTWBVMVKVSCVLUPUQURUSAHBCDFUTVAVBVC $.

    $( Membership in the range of a function.  (Contributed by NM,
       27-Aug-2007.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    elrnmptg $p |- ( A. x e. A B e. V ->
      ( C e. ran F <-> E. x e. A C = B ) ) $=
      ( vy crn wcel cv wceq wrex cab wral rnmpt eleq2i cvv wa syl wi eleq1 elex
      wb r19.29 biimparc rexlimivw ex eqeq1 rexbidv elab3g bitrid ) DEIZJDHKZCL
      ZABMZHNZJZCFJZABOZDCLZABMZUMUQDAHBCEGPQUTVBDRJZUAURVBUDUTVBVCUTVBSUSVASZA
      BMVCUSVAABUEVDVCABVDDFJZVCVAVEUSDCFUBUFDFUCTUGTUHUPVBHDRUNDLUOVAABUNDCUIU
      JUKTUL $.

    elrnmpti.2 $e |- B e. _V $.
    $( Membership in the range of a function.  (Contributed by NM,
       30-Aug-2004.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    elrnmpti $p |- ( C e. ran F <-> E. x e. A C = B ) $=
      ( cvv wcel wral crn wceq wrex wb rgenw elrnmptg ax-mp ) CHIZABJDEKIDCLABM
      NRABGOABCDEHFPQ $.
  $}

  ${
    $d D x $.  $d x A $.  $d x C $.  $d x ph $.
    elrnmptdv.1 $e |- F = ( x e. A |-> B ) $.
    elrnmptdv.2 $e |- ( ph -> C e. A ) $.
    elrnmptdv.3 $e |- ( ph -> D e. V ) $.
    elrnmptdv.4 $e |- ( ( ph /\ x = C ) -> D = B ) $.
    $( Elementhood in the range of a function in maps-to notation, deduction
       form.  (Contributed by Rohan Ridenour, 3-Aug-2023.) $)
    elrnmptdv $p |- ( ph -> D e. ran F ) $=
      ( crn wcel wceq wrex rspcime wb elrnmpt syl mpbird ) AFGMNZFDOZBCPZAUCBEC
      LJQAFHNUBUDRKBCDFGHISTUA $.
  $}

  ${
    $d x C $.
    elrnmpt2d.1 $e |- F = ( x e. A |-> B ) $.
    elrnmpt2d.2 $e |- ( ph -> C e. ran F ) $.
    $( Elementhood in the range of a function in maps-to notation, deduction
       form.  (Contributed by Rohan Ridenour, 3-Aug-2023.) $)
    elrnmpt2d $p |- ( ph -> E. x e. A C = B ) $=
      ( crn wcel wceq wrex elrnmpt ibi syl ) AEFIZJZEDKBCLZHQRBCDEFPGMNO $.
  $}

  $( The range of the empty set is empty.  Part of Theorem 3.8(v) of [Monk1]
     p. 36.  (Contributed by NM, 4-Jul-1994.) $)
  rn0 $p |- ran (/) = (/) $=
    ( c0 cdm wceq crn dm0 dm0rn0 mpbi ) ABACADACEAFG $.

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( Alternate definition of indexed union when ` B ` is a set.  (Contributed
       by Mario Carneiro, 31-Aug-2015.) $)
    dfiun3g $p |- ( A. x e. A B e. C ->
                  U_ x e. A B = U. ran ( x e. A |-> B ) ) $=
      ( vy wcel wral ciun wceq wrex cab cuni cmpt crn dfiun2g eqid rnmpt unieqi
      cv eqtr4di ) CDFABGABCHESCIABJEKZLABCMZNZLAEBCDOUCUAAEBCUBUBPQRT $.

    $( Alternate definition of indexed intersection when ` B ` is a set.
       (Contributed by Mario Carneiro, 31-Aug-2015.) $)
    dfiin3g $p |- ( A. x e. A B e. C
               -> |^|_ x e. A B = |^| ran ( x e. A |-> B ) ) $=
      ( vy wcel wral ciin wceq wrex cab cint cmpt crn dfiin2g eqid rnmpt inteqi
      cv eqtr4di ) CDFABGABCHESCIABJEKZLABCMZNZLAEBCDOUCUAAEBCUBUBPQRT $.
  $}

  ${
    dfiun3.1 $e |- B e. _V $.
    $( Alternate definition of indexed union when ` B ` is a set.  (Contributed
       by Mario Carneiro, 31-Aug-2015.) $)
    dfiun3 $p |- U_ x e. A B = U. ran ( x e. A |-> B ) $=
      ( cvv wcel ciun cmpt crn cuni wceq dfiun3g cv a1i mprg ) CEFZABCGABCHIJKA
      BABCELPAMBFDNO $.

    $( Alternate definition of indexed intersection when ` B ` is a set.
       (Contributed by Mario Carneiro, 31-Aug-2015.) $)
    dfiin3 $p |- |^|_ x e. A B = |^| ran ( x e. A |-> B ) $=
      ( cvv wcel ciin cmpt crn cint wceq dfiin3g cv a1i mprg ) CEFZABCGABCHIJKA
      BABCELPAMBFDNO $.
  $}

  ${
    $d V k $.  $d X k $.
    $( Express a relative indexed intersection as an intersection.
       (Contributed by Stefan O'Rear, 22-Feb-2015.) $)
    riinint $p |- ( ( X e. V /\ A. k e. I S C_ X ) ->
        ( X i^i |^|_ k e. I S ) = |^| ( { X } u. ran ( k e. I |-> S ) ) ) $=
      ( wcel wss wral wa ciin cin cmpt crn cint csn cun cvv wceq ssexg expcom
      ralimdv imp dfiin3g syl ineq2d intun intsng adantr ineq1d eqtrid eqtr4d )
      EDFZAEGZBCHZIZEBCAJZKEBCALMZNZKZEOZUQPNZUOUPUREUOAQFZBCHZUPURRULUNVCULUMV
      BBCUMULVBAEDSTUAUBBCAQUCUDUEUOVAUTNZURKUSUTUQUFUOVDEURULVDERUNEDUGUHUIUJU
      K $.
  $}

  ${
    $d x y A $.
    $( A relation is empty iff its range is empty.  (Contributed by NM,
       15-Sep-2004.) $)
    relrn0 $p |- ( Rel A -> ( A = (/) <-> ran A = (/) ) ) $=
      ( wrel c0 wceq cdm crn reldm0 dm0rn0 bitrdi ) ABACDAECDAFCDAGAHI $.

    $( The domain and range of a class are included in its double union.
       (Contributed by NM, 13-May-2008.) $)
    dmrnssfld $p |- ( dom A u. ran A ) C_ U. U. A $=
      ( vx vy cdm crn cuni cv cop wex vex eldm2 cpr prid1 wss sseld mpi exlimiv
      wcel sylbi ssriv uniop uniopel eqeltrrid elssuni syl elrn2 prid2 unssi )
      ADZAEZAFZFZBUIULBGZUIRUMCGZHZARZCIUMULRZCUMABJZKUPUQCUPUMUMUNLZRUQUMUNURM
      UPUSULUMUPUSUKRUSULNUPUSUOFUKUMUNURCJZUAUMUNAURUTUBUCUSUKUDUEZOPQSTCUJULU
      NUJRUPBIUNULRZBUNAUTUFUPVBBUPUNUSRVBUMUNUTUGUPUSULUNVAOPQSTUH $.
  $}

  $( The domain of a set is a set.  Corollary 6.8(2) of [TakeutiZaring] p. 26.
     (Contributed by NM, 7-Apr-1995.) $)
  dmexg $p |- ( A e. V -> dom A e. _V ) $=
    ( wcel cuni cvv cdm uniexg wss crn ssun1 dmrnssfld sstri ssexg mpan 3syl
    cun ) ABCADZECQDZECZAFZECZABGQEGTRHSUATTAIZPRTUBJAKLTREMNO $.

  $( The range of a set is a set.  Corollary 6.8(3) of [TakeutiZaring] p. 26.
     Similar to Lemma 3D of [Enderton] p. 41.  (Contributed by NM,
     31-Mar-1995.) $)
  rnexg $p |- ( A e. V -> ran A e. _V ) $=
    ( wcel cuni cvv crn uniexg wss cdm ssun2 dmrnssfld sstri ssexg mpan 3syl
    cun ) ABCADZECQDZECZAFZECZABGQEGTRHSUATAIZTPRTUBJAKLTREMNO $.

  ${
    dmexd.1 $e |- ( ph -> A e. V ) $.
    $( The domain of a set is a set.  (Contributed by Glauco Siliprandi,
       26-Jun-2021.) $)
    dmexd $p |- ( ph -> dom A e. _V ) $=
      ( wcel cdm cvv dmexg syl ) ABCEBFGEDBCHI $.
  $}

  ${
    dmex.1 $e |- A e. _V $.
    $( The domain of a set is a set.  Corollary 6.8(2) of [TakeutiZaring]
       p. 26.  (Contributed by NM, 7-Jul-2008.) $)
    dmex $p |- dom A e. _V $=
      ( cvv wcel cdm dmexg ax-mp ) ACDAECDBACFG $.

    $( The range of a set is a set.  Corollary 6.8(3) of [TakeutiZaring] p. 26.
       Similar to Lemma 3D of [Enderton] p. 41.  (Contributed by NM,
       7-Jul-2008.) $)
    rnex $p |- ran A e. _V $=
      ( cvv wcel crn rnexg ax-mp ) ACDAECDBACFG $.
  $}

  $( The identity function is a proper class.  This means, for example, that we
     cannot use it as a member of the class of continuous functions unless it
     is restricted to a set.  (Contributed by NM, 1-Jan-2007.) $)
  iprc $p |- -. _I e. _V $=
    ( cid cvv wcel cdm vprc dmi eleq1i mtbir dmexg mto ) ABCADZBCZLBBCEKBBFGHAB
    IJ $.

  ${
    $d x y z A $.  $d x y z B $.
    $( Domain of a composition.  Theorem 21 of [Suppes] p. 63.  (Contributed by
       NM, 19-Mar-1998.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    dmcoss $p |- dom ( A o. B ) C_ dom B $=
      ( vx vy vz ccom cdm cv cop wcel wex wbr nfe1 wa exsimpl vex opelco cbvexv
      breq2 3imtr4i exlimi eldm2 eldm ssriv ) CABFZGZBGZCHZDHZIUEJZDKUHUIBLZDKZ
      UHUFJUHUGJUJULDUKDMUHEHZBLZUMUIALZNEKUNEKUJULUNUOEOEUHUIABCPZDPQUKUNDEUIU
      MUHBSRTUADUHUEUPUBDUHBUPUCTUD $.
  $}

  $( Range of a composition.  (Contributed by NM, 19-Mar-1998.) $)
  rncoss $p |- ran ( A o. B ) C_ ran A $=
    ( ccnv ccom cdm crn dmcoss df-rn cnvco dmeqi eqtri 3sstr4i ) BCZACZDZEZNEAB
    DZFZAFMNGRQCZEPQHSOABIJKAHL $.

  ${
    $d x y z A $.  $d x y z B $.
    $( Domain of a composition.  (Contributed by NM, 28-May-1998.)  (Proof
       shortened by Andrew Salmon, 27-Aug-2011.) $)
    dmcosseq $p |- ( ran B C_ dom A -> dom ( A o. B ) = dom B ) $=
      ( vx vy vz crn cdm wss ccom dmcoss a1i cv wbr wex wcel wi vex eldm eximdv
      imbitrrdi cop ssel elrn imbi12i 19.8a imim1i pm3.2 sylcom sylbi syl excom
      wa opelco exbii eldm2 3imtr4g ssrdv eqssd ) BFZAGZHZABIZGZBGZVCVDHVAABJKV
      ACVDVCVACLZDLZBMZDNZVEELZUAVBOZENZVEVDOVEVCOVAVHVGVFVIAMZULZDNZENZVKVAVHV
      MENZDNVOVAVGVPDVAVFUSOZVFUTOZPZVGVPPZUSUTVFUBVSVGCNZVLENZPZVTVQWAVRWBCVFB
      DQZUCEVFAWDRUDWCVGWBVPVGWAWBVGCUEUFVGVLVMEVGVLUGSUHUIUJSVMEDUKTVJVNEDVEVI
      ABCQZEQUMUNTDVEBWEREVEVBWEUOUPUQUR $.

    $( Domain of a composition.  (Contributed by NM, 19-Mar-1998.) $)
    dmcoeq $p |- ( dom A = ran B -> dom ( A o. B ) = dom B ) $=
      ( cdm crn wceq wss ccom eqimss2 dmcosseq syl ) ACZBDZELKFABGCBCELKHABIJ
      $.
  $}

  $( Range of a composition.  (Contributed by NM, 19-Mar-1998.) $)
  rncoeq $p |- ( dom A = ran B -> ran ( A o. B ) = ran A ) $=
    ( ccnv cdm crn wceq ccom dmcoeq eqcom df-rn dfdm4 eqeq12i bitri cnvco dmeqi
    eqtri 3imtr4i ) BCZDZACZEZFZRTGZDZTDZFADZBEZFZABGZEZAEZFRTHUHUGUFFUBUFUGIUG
    SUFUABJAKLMUJUDUKUEUJUICZDUDUIJULUCABNOPAJLQ $.

  $( Equality theorem for restrictions.  (Contributed by NM, 7-Aug-1994.) $)
  reseq1 $p |- ( A = B -> ( A |` C ) = ( B |` C ) ) $=
    ( wceq cvv cxp cin cres ineq1 df-res 3eqtr4g ) ABDACEFZGBLGACHBCHABLIACJBCJ
    K $.

  $( Equality theorem for restrictions.  (Contributed by NM, 8-Aug-1994.) $)
  reseq2 $p |- ( A = B -> ( C |` A ) = ( C |` B ) ) $=
    ( wceq cvv cxp cin cres xpeq1 ineq2d df-res 3eqtr4g ) ABDZCAEFZGCBEFZGCAHCB
    HMNOCABEIJCAKCBKL $.

  ${
    reseqi.1 $e |- A = B $.
    $( Equality inference for restrictions.  (Contributed by NM,
       21-Oct-2014.) $)
    reseq1i $p |- ( A |` C ) = ( B |` C ) $=
      ( wceq cres reseq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality inference for restrictions.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    reseq2i $p |- ( C |` A ) = ( C |` B ) $=
      ( wceq cres reseq2 ax-mp ) ABECAFCBFEDABCGH $.

    reseqi.2 $e |- C = D $.
    $( Equality inference for restrictions.  (Contributed by NM,
       21-Oct-2014.) $)
    reseq12i $p |- ( A |` C ) = ( B |` D ) $=
      ( cres reseq1i reseq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
  $}

  ${
    reseqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for restrictions.  (Contributed by NM,
       21-Oct-2014.) $)
    reseq1d $p |- ( ph -> ( A |` C ) = ( B |` C ) ) $=
      ( wceq cres reseq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Equality deduction for restrictions.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    reseq2d $p |- ( ph -> ( C |` A ) = ( C |` B ) ) $=
      ( wceq cres reseq2 syl ) ABCFDBGDCGFEBCDHI $.

    reseqd.2 $e |- ( ph -> C = D ) $.
    $( Equality deduction for restrictions.  (Contributed by NM,
       21-Oct-2014.) $)
    reseq12d $p |- ( ph -> ( A |` C ) = ( B |` D ) ) $=
      ( cres reseq1d reseq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  ${
    nfres.1 $e |- F/_ x A $.
    nfres.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for restriction.  (Contributed by NM,
       15-Sep-2003.)  (Revised by David Abernethy, 19-Jun-2012.) $)
    nfres $p |- F/_ x ( A |` B ) $=
      ( cres cvv cxp cin df-res nfcv nfxp nfin nfcxfr ) ABCFBCGHZIBCJABODACGEAG
      KLMN $.
  $}

  $( Distribute proper substitution through the restriction of a class.
     (Contributed by Alan Sare, 10-Nov-2012.) $)
  csbresg $p |- ( A e. V -> [_ A / x ]_ ( B |` C ) =
                 ( [_ A / x ]_ B |` [_ A / x ]_ C ) ) $=
    ( wcel cvv cxp cin cres csbing csbxpg csbconstg xpeq2d eqtrd ineq2d csbeq2i
    csb df-res 3eqtr4g ) BEFZABCDGHZIZRZABCRZABDRZGHZIZABCDJZRUEUFJUAUDUEABUBRZ
    IUHABECUBKUAUJUGUEUAUJUFABGRZHUGABDGELUAUKGUFABGEMNOPOABUIUCCDSQUEUFST $.

  $( A restriction to the empty set is empty.  (Contributed by NM,
     12-Nov-1994.) $)
  res0 $p |- ( A |` (/) ) = (/) $=
    ( c0 cres cvv cxp cin df-res 0xp ineq2i in0 3eqtri ) ABCABDEZFABFBABGLBADHI
    AJK $.

  ${
    opelres.1 $e |- B e. _V $.
    $( Ordered pair membership in a restriction.  Exercise 13 of
       [TakeutiZaring] p. 25.  (Contributed by NM, 13-Nov-1995.) $)
    opelres $p |- ( <. A , B >. e. ( C |` D ) <->
                    ( <. A , B >. e. C /\ A e. D ) ) $=
      ( cop cres wcel cvv cxp cin wa df-res eleq2i elin opelxp mpbiran2 anbi2i
      3bitri ) ABFZCDGZHTCDIJZKZHTCHZTUBHZLUDADHZLUAUCTCDMNTCUBOUEUFUDUEUFBIHEA
      BDIPQRS $.

    $( Binary relation on a restriction.  (Contributed by NM, 12-Dec-2006.) $)
    brres $p |- ( A ( C |` D ) B <-> ( A C B /\ A e. D ) ) $=
      ( cop cres wcel wa wbr opelres df-br anbi1i 3bitr4i ) ABFZCDGZHOCHZADHZIA
      BPJABCJZRIABCDEKABPLSQRABCLMN $.
  $}

  ${
    $d y A $.  $d y B $.  $d y C $.  $d y D $.
    $( Ordered pair membership in a restriction.  Exercise 13 of
       [TakeutiZaring] p. 25.  (Contributed by NM, 14-Oct-2005.) $)
    opelresg $p |- ( B e. V -> ( <. A , B >. e. ( C |` D ) <->
                    ( <. A , B >. e. C /\ A e. D ) ) ) $=
      ( vy cv cop cres wcel wa wceq opeq2 eleq1d anbi1d vex opelres vtoclbg ) A
      FGZHZCDIZJTCJZADJZKABHZUAJUDCJZUCKFBESBLZTUDUASBAMZNUFUBUEUCUFTUDCUGNOASC
      DFPQR $.

    $( Binary relation on a restriction.  (Contributed by Mario Carneiro,
       4-Nov-2015.) $)
    brresg $p |- ( B e. V -> ( A ( C |` D ) B <-> ( A C B /\ A e. D ) ) ) $=
      ( wcel cop cres wa wbr opelresg df-br anbi1i 3bitr4g ) BEFABGZCDHZFOCFZAD
      FZIABPJABCJZRIABCDEKABPLSQRABCLMN $.
  $}

  ${
    opres.1 $e |- B e. _V $.
    $( Ordered pair membership in a restriction when the first member belongs
       to the restricting class.  (Contributed by NM, 30-Apr-2004.)  (Proof
       shortened by Andrew Salmon, 27-Aug-2011.) $)
    opres $p |- ( A e. D ->
                    ( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) ) $=
      ( cop cres wcel opelres rbaib ) ABFZCDGHKCHADHABCDEIJ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( A restricted identity relation is equivalent to equality in its domain.
       (Contributed by NM, 30-Apr-2004.) $)
    resieq $p |- ( ( B e. A /\ C e. A ) -> ( B ( _I |` A ) C <-> B = C ) ) $=
      ( vx wcel cid cres wbr wceq wb cv wi breq2 eqeq2 bibi12d imbi2d cop opres
      vex df-br ideq bitr3i 3bitr4g vtoclg impcom ) CAEBAEZBCFAGZHZBCIZJZUFBDKZ
      UGHZBUKIZJZLUFUJLDCAUKCIZUNUJUFUOULUHUMUIUKCBUGMUKCBNOPUFBUKQZUGEUPFEZULU
      MBUKFADSZRBUKUGTUMBUKFHUQBUKURUABUKFTUBUCUDUE $.
  $}

  $( ` <. A , A >. ` belongs to a restriction of the identity class iff ` A `
     belongs to the restricting class.  (Contributed by FL, 27-Oct-2008.)
     (Revised by NM, 30-Mar-2016.) $)
  opelresi $p |- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> A e. B ) ) $=
    ( wcel cop cid cres wa opelresg wbr ididg df-br sylib biantrurd bitr4d ) AC
    DZAAEZFBGDQFDZABDZHSAAFBCIPRSPAAFJRACKAAFLMNO $.

  $( The restriction of a restriction.  (Contributed by NM, 27-Mar-2008.) $)
  resres $p |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) ) $=
    ( cres cvv cxp cin df-res ineq1i xpindir ineq2i inass 3eqtr4ri 3eqtri ) ABD
    ZCDOCEFZGABEFZGZPGZABCGZDZOCHORPABHIATEFZGAQPGZGUASUBUCABCEJKATHAQPLMN $.

  $( Distributive law for restriction over union.  Theorem 31 of [Suppes]
     p. 65.  (Contributed by NM, 30-Sep-2002.) $)
  resundi $p |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) ) $=
    ( cun cvv cxp cin cres xpundir ineq2i indi eqtri df-res uneq12i 3eqtr4i ) A
    BCDZEFZGZABEFZGZACEFZGZDZAPHABHZACHZDRASUADZGUCQUFABCEIJASUAKLAPMUDTUEUBABM
    ACMNO $.

  $( Distributive law for restriction over union.  (Contributed by NM,
     23-Sep-2004.) $)
  resundir $p |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) ) $=
    ( cun cvv cxp cin cres indir df-res uneq12i 3eqtr4i ) ABDZCEFZGANGZBNGZDMCH
    ACHZBCHZDABNIMCJQORPACJBCJKL $.

  $( Class restriction distributes over intersection.  (Contributed by FL,
     6-Oct-2008.) $)
  resindi $p |- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) ) $=
    ( cin cvv cxp cres xpindir ineq2i inindi eqtri df-res ineq12i 3eqtr4i ) ABC
    DZEFZDZABEFZDZACEFZDZDZAOGABGZACGZDQARTDZDUBPUEABCEHIARTJKAOLUCSUDUAABLACLM
    N $.

  $( Class restriction distributes over intersection.  (Contributed by NM,
     18-Dec-2008.) $)
  resindir $p |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) ) $=
    ( cin cvv cxp cres inindir df-res ineq12i 3eqtr4i ) ABDZCEFZDAMDZBMDZDLCGAC
    GZBCGZDABMHLCIPNQOACIBCIJK $.

  $( Move intersection into class restriction.  (Contributed by NM,
     18-Dec-2008.) $)
  inres $p |- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C ) $=
    ( cin cvv cxp cres inass df-res ineq2i 3eqtr4ri ) ABDZCEFZDABMDZDLCGABCGZDA
    BMHLCIONABCIJK $.

  $( Commutative law for restriction and difference.  (Contributed by AV,
     7-Jun-2021.) $)
  resdifcom $p |- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B ) $=
    ( cdif cvv cxp cin cres indif1 df-res difeq1i 3eqtr4ri ) ACDZBEFZGANGZCDMBH
    ABHZCDANCIMBJPOCABJKL $.

  ${
    $d x C $.
    $( Distribution of restriction over indexed union.  (Contributed by Mario
       Carneiro, 29-May-2015.) $)
    resiun1 $p |- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C ) $=
      ( cvv cxp cin ciun cres iunin2 wceq cv wcel df-res incom iuneq2i 3eqtr4ri
      eqtri a1i ) ABDEFZCGZHTABCHZGZABCDIZHUBDIZABTCJABUDUAUDUAKALBMUDCTGUACDNC
      TORSPUEUBTGUCUBDNUBTORQ $.

    $( Distribution of restriction over indexed union.  (Contributed by Mario
       Carneiro, 29-May-2015.) $)
    resiun2 $p |- ( C |` U_ x e. A B ) = U_ x e. A ( C |` B ) $=
      ( ciun cres cvv cxp cin df-res wceq cv a1i iuneq2i xpiundir ineq2i iunin2
      wcel eqtr4i ) DABCEZFDTGHZIZABDCFZEZDTJUDABDCGHZIZEZUBABUCUFUCUFKALBRDCJM
      NUBDABUEEZIUGUAUHDABCGOPABDUEQSSS $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The domain of a restriction.  Exercise 14 of [TakeutiZaring] p. 25.
       (Contributed by NM, 1-Aug-1994.) $)
    dmres $p |- dom ( A |` B ) = ( B i^i dom A ) $=
      ( vx vy cdm cin cres cv wcel cop wex wa eldm2 19.41v opelres exbii anbi1i
      vex 3bitr4i bitr2i ineqri incom eqtr3i ) AEZBFABGZEZBUDFCUDBUFCHZUFIUGDHZ
      JZUEIZDKZUGUDIZUGBIZLZDUGUECRZMUIAIZUMLZDKUPDKZUMLUKUNUPUMDNUJUQDUGUHABDR
      OPULURUMDUGAUOMQSTUAUDBUBUC $.
  $}

  $( A domain restricted to a subclass equals the subclass.  (Contributed by
     NM, 2-Mar-1997.) $)
  ssdmres $p |- ( A C_ dom B <-> dom ( B |` A ) = A ) $=
    ( cdm wss cin wceq cres df-ss dmres eqeq1i bitr4i ) ABCZDALEZAFBAGCZAFALHNM
    ABAIJK $.

  $( The domain of a restriction to a set exists.  (Contributed by NM,
     7-Apr-1995.) $)
  dmresexg $p |- ( B e. V -> dom ( A |` B ) e. _V ) $=
    ( wcel cres cdm cin cvv dmres inex1g eqeltrid ) BCDABEFBAFZGHABIBLCJK $.

  $( A class includes its restriction.  Exercise 15 of [TakeutiZaring] p. 25.
     (Contributed by NM, 2-Aug-1994.) $)
  resss $p |- ( A |` B ) C_ A $=
    ( cres cvv cxp cin df-res inss1 eqsstri ) ABCABDEZFAABGAJHI $.

  $( Commutative law for restriction.  (Contributed by NM, 27-Mar-1998.) $)
  rescom $p |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B ) $=
    ( cin cres incom reseq2i resres 3eqtr4i ) ABCDZEACBDZEABECEACEBEJKABCFGABCH
    ACBHI $.

  $( Subclass theorem for restriction.  (Contributed by NM, 16-Aug-1994.) $)
  ssres $p |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) ) $=
    ( wss cvv cxp cin cres ssrin df-res 3sstr4g ) ABDACEFZGBLGACHBCHABLIACJBCJK
    $.

  $( Subclass theorem for restriction.  (Contributed by NM, 22-Mar-1998.)
     (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  ssres2 $p |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) ) $=
    ( wss cvv cxp cin cres xpss1 sslin syl df-res 3sstr4g ) ABDZCAEFZGZCBEFZGZC
    AHCBHNOQDPRDABEIOQCJKCALCBLM $.

  $( A restriction is a relation.  Exercise 12 of [TakeutiZaring] p. 25.
     (Contributed by NM, 2-Aug-1994.)  (Proof shortened by Andrew Salmon,
     27-Aug-2011.) $)
  relres $p |- Rel ( A |` B ) $=
    ( cres cvv cxp wss wrel cin df-res inss2 eqsstri relxp relss mp2 ) ABCZBDEZ
    FPGOGOAPHPABIAPJKBDLOPMN $.

  $( Absorption law for restriction.  Exercise 17 of [TakeutiZaring] p. 25.
     (Contributed by NM, 9-Aug-1994.) $)
  resabs1 $p |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) ) $=
    ( wss cres cin resres wceq sseqin2 reseq2 sylbi eqtrid ) BCDZACEBEACBFZEZAB
    EZACBGMNBHOPHBCINBAJKL $.

  ${
    resabs1d.b $e |- ( ph -> B C_ C ) $.
    $( Absorption law for restriction, deduction form.  (Contributed by Glauco
       Siliprandi, 11-Dec-2019.) $)
    resabs1d $p |- ( ph -> ( ( A |` C ) |` B ) = ( A |` B ) ) $=
      ( wss cres wceq resabs1 syl ) ACDFBDGCGBCGHEBCDIJ $.
  $}

  $( Absorption law for restriction.  (Contributed by NM, 27-Mar-1998.) $)
  resabs2 $p |- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) ) $=
    ( wss cres rescom resabs1 eqtrid ) BCDABEZCEACEBEIABCFABCGH $.

  $( Idempotent law for restriction.  (Contributed by NM, 27-Mar-1998.) $)
  residm $p |- ( ( A |` B ) |` B ) = ( A |` B ) $=
    ( wss cres wceq ssid resabs2 ax-mp ) BBCABDZBDIEBFABBGH $.

  $( A restriction to an image.  (Contributed by NM, 29-Sep-2004.) $)
  resima $p |- ( ( A |` B ) " B ) = ( A " B ) $=
    ( cres crn cima residm rneqi df-ima 3eqtr4i ) ABCZBCZDJDJBEABEKJABFGJBHABHI
    $.

  $( Image under a restricted class.  (Contributed by FL, 31-Aug-2009.) $)
  resima2 $p |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) ) $=
    ( wss cres cima crn df-ima cin resres rneqi wceq df-ss incom reseq2d reseq2
    a1i rneqd eqtr4di eqtrid eqtrd sylbi ) BCDZACEZBFUDBEZGZABFZUDBHUCUFACBIZEZ
    GZUGUEUIACBJKUCBCIZBLZUJUGLBCMULUJAUKEZGZUGULUIUMULUHUKAUHUKLULCBNQORULUNAB
    EZGUGULUMUOUKBAPRABHSUAUBTT $.

  $( Restriction of a constant function (or other cross product).  (Contributed
     by Stefan O'Rear, 24-Jan-2015.) $)
  xpssres $p |- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) ) $=
    ( wss cxp cres cin df-res inxp incom inv1 xpeq12i 3eqtri wceq biimpi xpeq1d
    cvv df-ss eqtrid ) CADZABEZCFZCAGZBEZCBEUBUACQEGACGZBQGZEUDUACHABCQIUEUCUFB
    ACJBKLMTUCCBTUCCNCAROPS $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Membership in a restriction.  (Contributed by Scott Fenton,
       17-Mar-2011.) $)
    elres $p |- ( A e. ( B |` C )
          <-> E. x e. C E. y ( A = <. x , y >. /\ <. x , y >. e. B ) ) $=
      ( cres wcel cv cop wceq wex wrex wrel relres elrel mpan eleq1 biimpd vex
      wa opelres biimpi ancomd syl6com an12 imbitrdi 2eximdv mpd rexcom4 df-rex
      ancld exbii excom 3bitri sylibr simplbi2com biimprd syl9 exlimdv rexlimiv
      impd impbii ) CDEFZGZCAHZBHZIZJZVGDGZTZBKZAELZVDVEEGZVJTZBKAKZVLVDVHBKAKZ
      VOVCMVDVPDENABCVCOPVDVHVNABVDVHVHVMVITZTVNVDVHVQVHVDVGVCGZVQVHVDVRCVGVCQZ
      RVRVIVMVRVIVMTVEVFDEBSUAZUBUCUDUKVHVMVIUEUFUGUHVLVJAELZBKVNAKZBKVOVJABEUI
      WAWBBVJAEUJULVNBAUMUNUOVKVDAEVMVJVDBVMVHVIVDVMVIVRVHVDVRVIVMVTUPVHVDVRVSU
      QURVAUSUTVB $.

    ${
      elsnres.1 $e |- C e. _V $.
      $( Memebership in restriction to a singleton.  (Contributed by Scott
         Fenton, 17-Mar-2011.) $)
      elsnres $p |- ( A e. ( B |` { C } )
            <-> E. y ( A = <. C , y >. /\ <. C , y >. e. B ) ) $=
        ( vx csn cres wcel cv cop wceq wa wex elres rexcom4 opeq1 eqeq2d eleq1d
        wrex anbi12d rexsn exbii 3bitri ) BCDGZHIBFJZAJZKZLZUHCIZMZANFUETUKFUET
        ZANBDUGKZLZUMCIZMZANFABCUEOUKFAUEPULUPAUKUPFDEUFDLZUIUNUJUOUQUHUMBUFDUG
        QZRUQUHUMCURSUAUBUCUD $.
    $}

    $( Simplification law for restriction.  (Contributed by NM,
       16-Aug-1994.) $)
    relssres $p |- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A ) $=
      ( vx vy wrel cdm wss wa cres wceq simpl cv cop wcel vex opeldm ssel ancld
      wi syl5 opelres imbitrrdi adantl relssdv resss jctil eqss sylibr ) AEZAFZ
      BGZHZABIZAGZAUMGZHUMAJULUOUNULCDAUMUIUKKUKCLZDLZMZANZURUMNZSUIUKUSUSUPBNZ
      HUTUKUSVAUSUPUJNUKVAUPUQACODOZPUJBUPQTRUPUQABVBUAUBUCUDABUEUFUMAUGUH $.
  $}

  $( A relation restricted to its domain equals itself.  (Contributed by NM,
     12-Dec-2006.) $)
  resdm $p |- ( Rel A -> ( A |` dom A ) = A ) $=
    ( wrel cdm wss cres wceq ssid relssres mpan2 ) ABACZJDAJEAFJGAJHI $.

  $( The restriction of a set is a set.  (Contributed by NM, 28-Mar-1998.)
     (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  resexg $p |- ( A e. V -> ( A |` B ) e. _V ) $=
    ( cres wss wcel cvv resss ssexg mpan ) ABDZAEACFKGFABHKACIJ $.

  ${
    resex.1 $e |- A e. _V $.
    $( The restriction of a set is a set.  (Contributed by Jeff Madsen,
       19-Jun-2011.) $)
    resex $p |- ( A |` B ) e. _V $=
      ( cvv wcel cres resexg ax-mp ) ADEABFDECABDGH $.
  $}

  $( When restricting a relation, intersecting with the domain of the relation
     has no effect.  (Contributed by FL, 6-Oct-2008.) $)
  resindm $p |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) ) $=
    ( wrel cres cdm cin resdm resindi incom inres inidm reseq1i 3eqtrri 3eqtr4g
    ineq2d ) ACZABDZAAEZDZFQAFZABRFDQPSAQAGOABRHTAQFAAFZBDQQAIAABJUAABAKLMN $.

  $( Restricting a relation to its domain without a set is the same as
     restricting the relation to the universe without this set.  (Contributed
     by AV, 2-Dec-2018.) $)
  resdmdfsn $p |- ( Rel R -> ( R |` ( _V \ { X } ) )
                             = ( R |` ( dom R \ { X } ) ) ) $=
    ( wrel cvv csn cdif cdm cin cres resindm indif1 incom eqtri difeq1i reseq2i
    inv1 eqtr3di ) ACADBEZFZAGZHZIASIATRFZIASJUAUBAUADTHZRFUBDTRKUCTRUCTDHTDTLT
    PMNMOQ $.

  ${
    $d x y A $.
    $( Restriction of a class abstraction of ordered pairs.  (Contributed by
       NM, 5-Nov-2002.) $)
    resopab $p |- ( { <. x , y >. | ph } |` A ) =
                  { <. x , y >. | ( x e. A /\ ph ) } $=
      ( copab cres cvv cxp cin cv wa df-res df-xp biantru opabbii eqtr4i ineq2i
      wcel vex eqtri incom inopab ) ABCEZDFUCDGHZIZBJDRZAKBCEZUCDLUEUFBCEZUCIZU
      GUEUCUHIUIUDUHUCUDUFCJGRZKZBCEUHBCDGMUFUKBCUJUFCSNOPQUCUHUATUFABCUBTT $.

    $( The existence of a restricted identity function, proved without using
       the Axiom of Replacement.  (Contributed by NM, 13-Jan-2007.) $)
    resiexg $p |- ( A e. V -> ( _I |` A ) e. _V ) $=
      ( vx vy wcel cid cres cxp wss cvv relres weq cv wa cop simpr eleq1 biimpa
      jca vex opelres wbr df-br bitr3i anbi1i bitri opelxp 3imtr4i relssi xpexg
      ideq anidms ssexg sylancr ) ABEZFAGZAAHZIUQJEZUPJECDUPUQFAKCDLZCMZAEZNZVA
      DMZAEZNUTVCOZUPEZVEUQEVBVAVDUSVAPUSVAVDUTVCAQRSVFVEFEZVANVBUTVCFADTZUAVGU
      SVAVGUTVCFUBUSUTVCFUCUTVCVHUKUDUEUFUTVCAAUGUHUIUOURAABBUJULUPUQJUMUN $.

    $( A subclass of the identity function is the identity function restricted
       to its domain.  (Contributed by NM, 13-Dec-2003.)  (Proof shortened by
       Andrew Salmon, 27-Aug-2011.) $)
    iss $p |- ( A C_ _I <-> A = ( _I |` dom A ) ) $=
      ( vx vy cid wss cdm cres wceq cv cop wcel wb wal ssel vex opeldm biimtrid
      wa wi wrel a1i jcad wbr df-br ideq bitr3i wex eldm2 opeq2 eleq1d biimprcd
      sylcom exlimdv imbi2d syl5ibcom impd impbid opelres bitr4di alrimivv reli
      relss mpi relres eqrel sylancl mpbird resss sseq1 mpbiri impbii ) ADEZADA
      FZGZHZVLVOBIZCIZJZAKZVRVNKZLZCMBMZVLWABCVLVSVRDKZVPVMKZRZVTVLVSWEVLVSWCWD
      ADVRNZVSWDSVLVPVQABOZCOZPUAUBVLWCWDVSWCVPVQHZVLWDVSSZWCVPVQDUCWIVPVQDUDVP
      VQWHUEUFZVLWDVPVPJZAKZSWIWJWDVSCUGVLWMCVPAWGUHVLVSWMCVLVSWCWMWFWCWIVSWMWK
      WIWMVSWIWLVRAVPVQVPUIUJZUKQULUMQWIWMVSWDWNUNUOQUPUQVPVQDVMWHURUSUTVLATZVN
      TVOWBLVLDTWOVAADVBVCDVMVDBCAVNVEVFVGVOVLVNDEDVMVHAVNDVIVJVK $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.
    $( Restriction of a class abstraction of ordered pairs.  (Contributed by
       NM, 24-Aug-2007.) $)
    resopab2 $p |- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) =
                  { <. x , y >. | ( x e. A /\ ph ) } ) $=
      ( wss cv wcel wa copab cres resopab pm4.71d anbi1d anass bitr2di opabbidv
      ssel eqtrid ) DEFZBGZEHZAIZBCJDKUADHZUCIZBCJUDAIZBCJUCBCDLTUEUFBCTUFUDUBI
      ZAIUETUDUGATUDUBDEUARMNUDUBAOPQS $.

    $( Restriction of the mapping operation.  (Contributed by Mario Carneiro,
       15-Jul-2013.) $)
    resmpt $p |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) ) $=
      ( vy wss cv wcel wceq wa copab cres cmpt resopab2 df-mpt reseq1i 3eqtr4g
      ) CBFAGZBHEGDIZJAEKZCLRCHSJAEKABDMZCLACDMSAECBNUATCAEBDOPAECDOQ $.

    $( Unconditional restriction of the mapping operation.  (Contributed by
       Stefan O'Rear, 24-Jan-2015.)  (Proof shortened by Mario Carneiro,
       22-Mar-2015.) $)
    resmpt3 $p |- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C ) $=
      ( cmpt cres cin resres wss wceq ssid resmpt ax-mp reseq1i inss1 3eqtr3i )
      ABDEZBFZCFQBCGZFZQCFASDEZQBCHRQCBBIRQJBKABBDLMNSBITUAJBCOABSDLMP $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.
    resmptf.a $e |- F/_ x A $.
    resmptf.b $e |- F/_ x B $.
    $( Restriction of the mapping operation.  (Contributed by Thierry Arnoux,
       28-Mar-2017.) $)
    resmptf $p |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) ) $=
      ( vy wss cv cmpt cres resmpt nfcv nfcsb1v csbeq1a cbvmptf reseq1i 3eqtr4g
      csb ) CBHGBAGIZDSZJZCKGCUAJABDJZCKACDJGBCUALUCUBCAGBDUAEGBMGDMZATDNZATDOZ
      PQAGCDUAFGCMUDUEUFPR $.
  $}

  ${
    $d A x $.  $d B x $.
    resmptd.b $e |- ( ph -> B C_ A ) $.
    $( Restriction of the mapping operation, deduction form.  (Contributed by
       Glauco Siliprandi, 11-Dec-2019.) $)
    resmptd $p |- ( ph -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) ) $=
      ( wss cmpt cres wceq resmpt syl ) ADCGBCEHDIBDEHJFBCDEKL $.
  $}

  ${
    $d w x y z A $.  $d w x y z R $.
    $( Alternate definition of the restriction operation.  (Contributed by
       Mario Carneiro, 5-Nov-2013.) $)
    dfres2 $p |- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) } $=
      ( vz vw cres cv wbr wa copab relres relopab cop vex brres df-br ancom weq
      wcel 3bitr3i eleq1 breq1 anbi12d breq2 anbi2d opelopab bitr4i eqrelriiv )
      EFDCGZAHZCTZUKBHZDIZJZABKZDCLUOABMEHZFHZNZUJTZUQCTZUQURDIZJZUSUPTUQURUJIV
      BVAJUTVCUQURDCFOZPUQURUJQVBVARUAUOVAUQUMDIZJVCABUQUREOVDAESULVAUNVEUKUQCU
      BUKUQUMDUCUDBFSVEVBVAUMURUQDUEUFUGUHUI $.
  $}

  ${
    $d A x y $.
    $( The restricted identity relation expressed as an ordered-pair class
       abstraction.  (Contributed by FL, 25-Apr-2012.) $)
    opabresid $p |- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) } $=
      ( cid cres weq copab cv wcel df-id equcom opabbii eqtri reseq1i resopab
      wa ) DCEBAFZABGZCEAHCIQPABGDRCDABFZABGRABJSQABABKLMNQABCOM $.
  $}


  ${
    $d A x y $.
    $( The restricted identity relation expressed in maps-to notation.
       (Contributed by FL, 25-Apr-2012.) $)
    mptresid $p |- ( _I |` A ) = ( x e. A |-> x ) $=
      ( vy cid cres cv wcel weq wa copab cmpt opabresid df-mpt eqtr4i ) DBEAFZB
      GCAHIACJABOKACBLACBOMN $.
  $}

  $( The domain of a restricted identity function.  (Contributed by NM,
     27-Aug-2004.) $)
  dmresi $p |- dom ( _I |` A ) = A $=
    ( cid cdm wss cres wceq cvv ssv dmi sseqtrri ssdmres mpbi ) ABCZDBAECAFAGMA
    HIJABKL $.

  ${
    $d A x y $.
    $( Restriction of the identity to a singleton.  (Contributed by FL,
       2-Aug-2009.)  (Proof shortened by JJ, 25-Aug-2021.)  (Proof shortened by
       Peter Mazsa, 6-Oct-2022.) $)
    restidsing $p |- ( _I |` { A } ) = ( { A } X. { A } ) $=
      ( vx vy cid csn cres cxp relres relxp cv wcel cop wa wceq wbr anbi12i vex
      velsn ideq eqeq1 eqcom bitrdi pm5.32i bitri df-br anbi2i 3bitr2ri opelres
      biancomi opelxp 3bitr4i eqrelriiv ) BCDAEZFZUMUMGZDUMHUMUMIBJZUMKZUPCJZLZ
      DKZMZUQURUMKZMZUSUNKZUSUOKVCUPANZURANZMZUQUPURDOZMZVAUQVEVBVFBARZCARPVIVE
      UPURNZMVGUQVEVHVKVJUPURCQZSPVEVKVFVEVKAURNVFUPAURTAURUAUBUCUDVHUTUQUPURDU
      EUFUGVDUQUTUPURDUMVLUHUIUPURUMUMUJUKUL $.
  $}

  $( TODO : replace uses of ~ resid with set.mm/dfrel3 and delete ~ resid $)
  $( Any relation restricted to the universe is itself.  (Contributed by NM,
     16-Mar-2004.) $)
  resid $p |- ( Rel A -> ( A |` _V ) = A ) $=
    ( wrel cdm cvv wss cres wceq ssv relssres mpan2 ) ABACZDEADFAGKHADIJ $.

  $( Equality theorem for image.  (Contributed by NM, 14-Aug-1994.) $)
  imaeq1 $p |- ( A = B -> ( A " C ) = ( B " C ) ) $=
    ( wceq cres crn cima reseq1 rneqd df-ima 3eqtr4g ) ABDZACEZFBCEZFACGBCGLMNA
    BCHIACJBCJK $.

  $( Equality theorem for image.  (Contributed by NM, 14-Aug-1994.) $)
  imaeq2 $p |- ( A = B -> ( C " A ) = ( C " B ) ) $=
    ( wceq cres crn cima reseq2 rneqd df-ima 3eqtr4g ) ABDZCAEZFCBEZFCAGCBGLMNA
    BCHICAJCBJK $.

  ${
    imaeq1i.1 $e |- A = B $.
    $( Equality theorem for image.  (Contributed by NM, 21-Dec-2008.) $)
    imaeq1i $p |- ( A " C ) = ( B " C ) $=
      ( wceq cima imaeq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality theorem for image.  (Contributed by NM, 21-Dec-2008.) $)
    imaeq2i $p |- ( C " A ) = ( C " B ) $=
      ( wceq cima imaeq2 ax-mp ) ABECAFCBFEDABCGH $.
  $}

  ${
    imaeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for image.  (Contributed by FL, 15-Dec-2006.) $)
    imaeq1d $p |- ( ph -> ( A " C ) = ( B " C ) ) $=
      ( wceq cima imaeq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Equality theorem for image.  (Contributed by FL, 15-Dec-2006.) $)
    imaeq2d $p |- ( ph -> ( C " A ) = ( C " B ) ) $=
      ( wceq cima imaeq2 syl ) ABCFDBGDCGFEBCDHI $.

    imaeq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality theorem for image.  (Contributed by Mario Carneiro,
       4-Dec-2016.) $)
    imaeq12d $p |- ( ph -> ( A " C ) = ( B " D ) ) $=
      ( cima imaeq1d imaeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Alternate definition of image.  Compare definition (d) of [Enderton]
       p. 44.  (Contributed by NM, 19-Apr-2004.)  (Proof shortened by Andrew
       Salmon, 27-Aug-2011.) $)
    dfima2 $p |- ( A " B ) = { y | E. x e. B x A y } $=
      ( cima cres crn cv wbr wex cab wrex df-ima dfrn2 wa vex brres ancom bitri
      wcel exbii df-rex bitr4i abbii 3eqtri ) CDECDFZGAHZBHZUFIZAJZBKUGUHCIZADL
      ZBKCDMABUFNUJULBUJUGDTZUKOZAJULUIUNAUIUKUMOUNUGUHCDBPQUKUMRSUAUKADUBUCUDU
      E $.

    $( Alternate definition of image.  Compare definition (d) of [Enderton]
       p. 44.  (Contributed by NM, 14-Aug-1994.)  (Proof shortened by Andrew
       Salmon, 27-Aug-2011.) $)
    dfima3 $p |- ( A " B ) = { y | E. x ( x e. B /\ <. x , y >. e. A ) } $=
      ( cima cv wbr wrex cab wcel cop wa dfima2 df-br rexbii df-rex bitri abbii
      wex eqtri ) CDEAFZBFZCGZADHZBIUADJUAUBKCJZLASZBIABCDMUDUFBUDUEADHUFUCUEAD
      UAUBCNOUEADPQRT $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Membership in an image.  Theorem 34 of [Suppes] p. 65.  (Contributed by
       NM, 20-Jan-2007.) $)
    elimag $p |- ( A e. V -> ( A e. ( B " C ) <-> E. x e. C x B A ) ) $=
      ( vy cv wbr wrex cima wceq breq2 rexbidv dfima2 elab2g ) AGZFGZCHZADIPBCH
      ZADIFBCDJEQBKRSADQBPCLMAFCDNO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    elima.1 $e |- A e. _V $.
    $( Membership in an image.  Theorem 34 of [Suppes] p. 65.  (Contributed by
       NM, 19-Apr-2004.) $)
    elima $p |- ( A e. ( B " C ) <-> E. x e. C x B A ) $=
      ( cvv wcel cima cv wbr wrex wb elimag ax-mp ) BFGBCDHGAIBCJADKLEABCDFMN
      $.

    $( Membership in an image.  Theorem 34 of [Suppes] p. 65.  (Contributed by
       NM, 11-Aug-2004.) $)
    elima2 $p |- ( A e. ( B " C ) <-> E. x ( x e. C /\ x B A ) ) $=
      ( cima wcel cv wbr wrex wa wex elima df-rex bitri ) BCDFGAHZBCIZADJPDGQKA
      LABCDEMQADNO $.

    $( Membership in an image.  Theorem 34 of [Suppes] p. 65.  (Contributed by
       NM, 14-Aug-1994.) $)
    elima3 $p |- ( A e. ( B " C ) <-> E. x ( x e. C /\ <. x , A >. e. B ) ) $=
      ( cima wcel cv wbr wa wex cop elima2 df-br anbi2i exbii bitri ) BCDFGAHZD
      GZRBCIZJZAKSRBLCGZJZAKABCDEMUAUCATUBSRBCNOPQ $.
  $}

  ${
    nfima.1 $e |- F/_ x A $.
    nfima.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for image.  (Contributed by NM,
       30-Dec-1996.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    nfima $p |- F/_ x ( A " B ) $=
      ( cima cres crn df-ima nfres nfrn nfcxfr ) ABCFBCGZHBCIAMABCDEJKL $.
  $}

  ${
    $d x z $.  $d B z $.  $d A z $.
    nfimad.2 $e |- ( ph -> F/_ x A ) $.
    nfimad.3 $e |- ( ph -> F/_ x B ) $.
    $( Deduction version of bound-variable hypothesis builder ~ nfima .
       (Contributed by FL, 15-Dec-2006.)  (Revised by Mario Carneiro,
       15-Oct-2016.) $)
    nfimad $p |- ( ph -> F/_ x ( A " B ) ) $=
      ( vz cv wcel wal cab cima wnfc nfaba1 nfima wb wa nfnfc1 nfan abidnf
      imaeq1d imaeq2d sylan9eq nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUGDIZBJGKZL
      ZMZBCDLZMZBUIUKUHBGNUJBGNOABCMZBDMZUMUOPEFUPUQQBULUNUPUQBBCRBDRSUPUQULCUK
      LUNUPUICUKBGCTUAUQUKDCBGDTUBUCUDUEUF $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The image of the domain of a class is the range of the class.
       (Contributed by NM, 14-Aug-1994.) $)
    imadmrn $p |- ( A " dom A ) = ran A $=
      ( vx vy cv cdm wcel cop wa wex cab cima crn vex opeldm ancom bitr2i exbii
      pm4.71i abbii dfima3 dfrn3 3eqtr4i ) BDZAEZFZUCCDZGAFZHZBIZCJUGBIZCJAUDKA
      LUIUJCUHUGBUGUGUEHUHUGUEUCUFABMCMNRUGUEOPQSBCAUDTBCAUAUB $.

    $( The image of a class is a subset of its range.  Theorem 3.16(xi) of
       [Monk1] p. 39.  (Contributed by NM, 31-Mar-1995.) $)
    imassrn $p |- ( A " B ) C_ ran A $=
      ( vx vy cv wcel cop wex cab cima crn exsimpr ss2abi dfima3 dfrn3 3sstr4i
      wa ) CEZBFZRDEGAFZQCHZDITCHZDIABJAKUAUBDSTCLMCDABNCDAOP $.
  $}

  ${
    $d A x $.  $d C x $.
    $( Image of a function in maps-to notation.  (Contributed by Glauco
       Siliprandi, 23-Oct-2021.) $)
    mptima $p |- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B ) $=
      ( cmpt cima cres crn cin df-ima resmpt3 rneqi eqtri ) ABCEZDFNDGZHABDICEZ
      HNDJOPABDCKLM $.
  $}

  ${
    $d A x $.  $d C x $.
    mptimass.1 $e |- ( ph -> C C_ A ) $.
    $( Image of a function in maps-to notation for a subset.  (Contributed by
       Glauco Siliprandi, 23-Oct-2021.) $)
    mptimass $p |- ( ph -> ( ( x e. A |-> B ) " C ) = ran ( x e. C |-> B ) ) $=
      ( cmpt cima cin crn mptima wss wceq sseqin2 sylib mpteq1d rneqd eqtrid )
      ABCDGEHBCEIZDGZJBEDGZJBCDEKATUAABSEDAECLSEMFECNOPQR $.
  $}

  $( The image of a set is a set.  Theorem 3.17 of [Monk1] p. 39.  (Contributed
     by NM, 24-Jul-1995.) $)
  imaexg $p |- ( A e. V -> ( A " B ) e. _V ) $=
    ( wcel cima crn wss cvv imassrn rnexg ssexg sylancr ) ACDABEZAFZGNHDMHDABIA
    CJMNHKL $.

  ${
    imaex.1 $e |- A e. _V $.
    $( The image of a set is a set.  Theorem 3.17 of [Monk1] p. 39.
       (Contributed by JJ, 24-Sep-2021.) $)
    imaex $p |- ( A " B ) e. _V $=
      ( cvv wcel cima imaexg ax-mp ) ADEABFDECABDGH $.
  $}

  ${
    $d B x $.
    $( A condition where the converse of ~ xpex holds as well.  Corollary
       6.9(2) in [TakeutiZaring] p. 26.  (Contributed by Andrew Salmon,
       13-Nov-2011.) $)
    xpexcnvm $p |- ( ( E. x x e. B /\ ( A X. B ) e. _V ) -> A e. _V ) $=
      ( cv wcel wex cxp cvv cdm dmexg dmxpm eleq1d imbitrid imp ) ADCEAFZBCGZHE
      ZBHEZQPIZHEORPHJOSBHABCKLMN $.
  $}

  ${
    $d x y A $.
    $( Image under the identity relation.  Theorem 3.16(viii) of [Monk1] p. 38.
       (Contributed by NM, 30-Apr-1998.) $)
    imai $p |- ( _I " A ) = A $=
      ( vx vy cid cima cv wcel cop wex cab dfima3 weq wbr df-br vex ideq bitr3i
      wa anbi2i bitri ancom exbii eleq1 ceqsexv abbii abid2 3eqtri ) DAEBFZAGZU
      HCFZHDGZRZBIZCJUJAGZCJABCDAKUMUNCUMBCLZUIRZBIUNULUPBULUIUORUPUKUOUIUKUHUJ
      DMUOUHUJDNUHUJCOZPQSUIUOUATUBUIUNBUJUQUHUJAUCUDTUECAUFUG $.
  $}

  $( The range of the restricted identity function.  (Contributed by NM,
     27-Aug-2004.) $)
  rnresi $p |- ran ( _I |` A ) = A $=
    ( cid cima cres crn df-ima imai eqtr3i ) BACBADEABAFAGH $.

  $( The image of a restriction of the identity function.  (Contributed by FL,
     31-Dec-2006.) $)
  resiima $p |- ( B C_ A -> ( ( _I |` A ) " B ) = B ) $=
    ( wss cid cres cima crn wceq df-ima a1i resabs1 rneqd rnresi 3eqtrd ) BACZD
    AEZBFZPBEZGZDBEZGZBQSHOPBIJORTDBAKLUABHOBMJN $.

  $( Image of the empty set.  Theorem 3.16(ii) of [Monk1] p. 38.  (Contributed
     by NM, 20-May-1998.) $)
  ima0 $p |- ( A " (/) ) = (/) $=
    ( c0 cima cres crn df-ima res0 rneqi rn0 3eqtri ) ABCABDZEBEBABFKBAGHIJ $.

  $( Image under the empty relation.  (Contributed by FL, 11-Jan-2007.) $)
  0ima $p |- ( (/) " A ) = (/) $=
    ( c0 cima crn imassrn rn0 sseqtri 0ss eqssi ) BACZBJBDBBAEFGJHI $.

  ${
    $d A y $.  $d B y $.  $d C y $.  $d x y $.  $d F y $.
    $( Move class substitution in and out of the image of a function.
       (Contributed by FL, 15-Dec-2006.)  (Proof shortened by Mario Carneiro,
       4-Dec-2016.) $)
    csbima12g $p |- ( A e. C -> [_ A / x ]_ ( F " B ) =
                 ( [_ A / x ]_ F " [_ A / x ]_ B ) ) $=
      ( vy cv cima csb csbeq1 imaeq12d eqeq12d vex nfcsb1v nfima csbeq1a csbief
      wceq vtoclg ) AFGZECHZIZATEIZATCIZHZRABUAIZABEIZABCIZHZRFBDTBRZUBUFUEUIAT
      BUAJUJUCUGUDUHATBEJATBCJKLATUAUEFMAUCUDATENATCNOAGTREUCCUDATEPATCPKQS $.
  $}

  $( A class whose image under another is empty is disjoint with the other's
     domain.  (Contributed by FL, 24-Jan-2007.) $)
  imadisj $p |- ( ( A " B ) = (/) <-> ( dom A i^i B ) = (/) ) $=
    ( cima wceq cres crn cdm cin df-ima eqeq1i dm0rn0 dmres incom eqtri 3bitr2i
    c0 ) ABCZPDABEZFZPDRGZPDAGZBHZPDQSPABIJRKTUBPTBUAHUBABLBUAMNJO $.

  $( A preimage under any class is included in the domain of the class.
     (Contributed by FL, 29-Jan-2007.) $)
  cnvimass $p |- ( `' A " B ) C_ dom A $=
    ( ccnv cima crn cdm imassrn dfdm4 sseqtrri ) ACZBDJEAFJBGAHI $.

  $( The preimage of the range of a class is the domain of the class.
     (Contributed by Jeff Hankins, 15-Jul-2009.) $)
  cnvimarndm $p |- ( `' A " ran A ) = dom A $=
    ( ccnv cdm cima crn imadmrn df-rn imaeq2i dfdm4 3eqtr4i ) ABZKCZDKEKAEZDACK
    FMLKAGHAIJ $.

  ${
    $d x y A $.  $d x B $.  $d x y R $.
    $( The image of a singleton.  (Contributed by NM, 8-May-2005.) $)
    imasng $p |- ( A e. B -> ( R " { A } ) = { y | A R y } ) $=
      ( vx wcel cvv csn cima wbr cab wceq elex wrex dfima2 rexsng abbidv eqtrid
      cv breq1 syl ) BCFBGFZDBHZIZBASZDJZAKZLBCMUBUDESZUEDJZEUCNZAKUGEADUCOUBUJ
      UFAUIUFEBGUHBUEDTPQRUA $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.
    $( Elementhood in the image of a singleton.  (Contributed by Mario
       Carneiro, 3-Nov-2015.) $)
    elrelimasn $p |- ( Rel R -> ( B e. ( R " { A } ) <-> A R B ) ) $=
      ( vy vx wrel cvv wcel csn cima wbr wi cv wceq wex wrex elimag ibi ex wb
      rexm elsni eximi 3syl isset sylibr a1i brrelex1 cab imasng brrelex2 breq2
      eleq2d elab3g syl sylan9bbr pm5.21ndd ) CFZAGHZBCAIZJZHZABCKZVBUSLURVBDMZ
      ANZDOZUSVBVDBCKZDUTPZVDUTHZDOVFVBVHDBCUTVAQRVGDUTUAVIVEDVDAUBUCUDDAUEUFUG
      URVCUSABCUHSURUSVBVCTUSVBBAEMZCKZEUIZHZURVCUSVAVLBEAGCUJUMURVCBGHZLVMVCTU
      RVCVNABCUKSVKVCEBGVJBACULUNUOUPSUQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    elimasn.1 $e |- B e. _V $.
    elimasn.2 $e |- C e. _V $.
    $( Membership in an image of a singleton.  (Contributed by NM,
       15-Mar-2004.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    elimasn $p |- ( C e. ( A " { B } ) <-> <. B , C >. e. A ) $=
      ( vx csn cima wcel wbr cop cv breq2 cvv cab wceq imasng ax-mp elab2 df-br
      bitri ) CABGHZIBCAJZBCKAIBFLZAJZUCFCUBEUDCBAMBNIUBUEFOPDFBNAQRSBCATUA $.
  $}

  ${
    $d A y z $.  $d B y z $.  $d C y z $.
    $( Membership in an image of a singleton.  (Contributed by Raph Levien,
       21-Oct-2006.) $)
    elimasng $p |- ( ( B e. V /\ C e. W ) ->
                   ( C e. ( A " { B } ) <-> <. B , C >. e. A ) ) $=
      ( vz vy cv csn cima wcel cop wceq sneq imaeq2d eleq2d eleq1d bibi12d vex
      wb opeq1 eleq1 opeq2 elimasn vtocl2g ) FHZAGHZIZJZKZUGUFLZAKZTUFABIZJZKZB
      UFLZAKZTCUNKZBCLZAKZTGFBCDEUGBMZUJUOULUQVAUIUNUFVAUHUMAUGBNOPVAUKUPAUGBUF
      UAQRUFCMZUOURUQUTUFCUNUBVBUPUSAUFCBUCQRAUGUFGSFSUDUE $.
  $}

  ${
    $d y F $.  $d x y $.
    $( Two ways to express the class of unique-valued arguments of ` F ` ,
       which is the same as the domain of ` F ` whenever ` F ` is a function.
       The left-hand side of the equality is from Definition 10.2 of [Quine]
       p. 65.  Quine uses the notation "arg ` F ` " for this class (for which
       we have no separate notation).  (Contributed by NM, 8-May-2005.) $)
    args $p |- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y } $=
      ( cv csn cima wceq wex wbr weu cab cvv wcel vex imasng ax-mp eqeq1i exbii
      euabsn bitr4i abbii ) CADZEFZBDZEZGZBHZUBUDCIZBJZAUGUHBKZUEGZBHUIUFUKBUCU
      JUEUBLMUCUJGANBUBLCOPQRUHBSTUA $.
  $}

  ${
    eliniseg.1 $e |- C e. _V $.
    $( Membership in an initial segment.  The idiom ` ( ``' A " { B } ) ` ,
       meaning ` { x | x A B } ` , is used to specify an initial segment in
       (for example) Definition 6.21 of [TakeutiZaring] p. 30.  (Contributed by
       NM, 28-Apr-2004.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    eliniseg $p |- ( B e. V -> ( C e. ( `' A " { B } ) <-> C A B ) ) $=
      ( wcel cvv ccnv csn cima wbr wb wa cop elimasng df-br bitr4di bitrd mpan2
      brcnvg ) BDFZCGFZCAHZBIJFZCBAKZLEUAUBMZUDBCUCKZUEUFUDBCNUCFUGUCBCDGOBCUCP
      QBCDGATRS $.
  $}

  ${
    $d A x $.
    epini.1 $e |- A e. _V $.
    $( Any set is equal to its preimage under the converse epsilon relation.
       (Contributed by Mario Carneiro, 9-Mar-2013.) $)
    epini $p |- ( `' _E " { A } ) = A $=
      ( vx cep ccnv csn cima cv wbr cvv wb vex eliniseg ax-mp epelc bitri eqriv
      wcel ) CDEAFGZACHZSRZTADIZTARAJRUAUBKBDATJCLMNTABOPQ $.
  $}

  ${
    $d x A $.  $d x B $.
    $( An idiom that signifies an initial segment of an ordering, used, for
       example, in Definition 6.21 of [TakeutiZaring] p. 30.  (Contributed by
       NM, 28-Apr-2004.) $)
    iniseg $p |- ( B e. V -> ( `' A " { B } ) = { x | x A B } ) $=
      ( wcel cvv ccnv csn cima cv wbr cab wceq elex vex eliniseg abbi2dv syl )
      CDECFEZBGCHIZAJZCBKZALMCDNSUBATBCUAFAOPQR $.
  $}

  ${
    $d x y A $.  $d x y R $.  $d x V $.
    $( Alternate definition of set-like relation.  (Contributed by Mario
       Carneiro, 23-Jun-2015.) $)
    dfse2 $p |- ( R Se A <-> A. x e. A ( A i^i ( `' R " { x } ) ) e. _V ) $=
      ( vy wse cv wbr crab cvv wcel wral ccnv csn cima cin df-se cab dfrab3 vex
      wceq iniseg ax-mp ineq2i eqtr4i eleq1i ralbii bitri ) BCEDFAFZCGZDBHZIJZA
      BKBCLUHMNZOZIJZABKADBCPUKUNABUJUMIUJBUIDQZOUMUIDBRULUOBUHIJULUOTASDCUHIUA
      UBUCUDUEUFUG $.

    $( Any set relation is set-like.  (Contributed by Mario Carneiro,
       22-Jun-2015.) $)
    exse2 $p |- ( R e. V -> R Se A ) $=
      ( vy vx wcel cv wbr crab cvv wral wse cdm wss wa cab df-rab breldm adantl
      vex abssi eqsstri dmexg ssexg sylancr ralrimivw df-se sylibr ) BCFZDGZEGZ
      BHZDAIZJFZEAKABLUIUNEAUIUMBMZNUOJFUNUMUJAFZULOZDPUOULDAQUQDUOULUJUOFUPUJU
      KBDTETRSUAUBBCUCUMUOJUDUEUFEDABUGUH $.
  $}

  $( Subset theorem for image.  (Contributed by NM, 16-Mar-2004.) $)
  imass1 $p |- ( A C_ B -> ( A " C ) C_ ( B " C ) ) $=
    ( wss cres crn cima ssres rnss syl df-ima 3sstr4g ) ABDZACEZFZBCEZFZACGBCGM
    NPDOQDABCHNPIJACKBCKL $.

  $( Subset theorem for image.  Exercise 22(a) of [Enderton] p. 53.
     (Contributed by NM, 22-Mar-1998.) $)
  imass2 $p |- ( A C_ B -> ( C " A ) C_ ( C " B ) ) $=
    ( wss cres crn cima ssres2 rnss syl df-ima 3sstr4g ) ABDZCAEZFZCBEZFZCAGCBG
    MNPDOQDABCHNPIJCAKCBKL $.

  $( The image of a singleton outside the domain is empty.  (Contributed by NM,
     22-May-1998.) $)
  ndmima $p |- ( -. A e. dom B -> ( B " { A } ) = (/) ) $=
    ( cdm wcel wn csn cima cres crn c0 df-ima wceq cin dmres incom eqtri disjsn
    biimpri eqtrid dm0rn0 sylib ) ABCZDEZBAFZGBUDHZIZJBUDKUCUECZJLUFJLUCUGUBUDM
    ZJUGUDUBMUHBUDNUDUBOPUHJLUCUBAQRSUETUAS $.

  ${
    $d x y A $.
    $( A converse is a relation.  Theorem 12 of [Suppes] p. 62.  (Contributed
       by NM, 29-Oct-1996.) $)
    relcnv $p |- Rel `' A $=
      ( vy vx cv wbr ccnv df-cnv relopabi ) BDCDAECBAFCBAGH $.
  $}

  $( When ` R ` is a relation, the sethood assumptions on ~ brcnv can be
     omitted.  (Contributed by Mario Carneiro, 28-Apr-2015.) $)
  relbrcnvg $p |- ( Rel R -> ( A `' R B <-> B R A ) ) $=
    ( wrel cvv wcel wa ccnv wbr wi relcnv brrelex12 mpan ancomd ex wb pm5.21ndd
    a1i brcnvg ) CDZAEFZBEFZGZABCHZIZBACIZUEUCJTUDDUEUCCKABUDLMRTUFUCTUFGUBUABA
    CLNOUCUEUFPJTABEECSRQ $.

  $( Eliminate the class existence constraint in ~ eliniseg .  (Contributed by
     Mario Carneiro, 5-Dec-2014.)  (Revised by Mario Carneiro, 17-Nov-2015.) $)
  eliniseg2 $p |- ( Rel A -> ( C e. ( `' A " { B } ) <-> C A B ) ) $=
    ( ccnv csn cima wcel wbr wrel wb relcnv elrelimasn ax-mp relbrcnvg bitrid )
    CADZBEFGZBCPHZAICBAHPIQRJAKBCPLMBCANO $.

  ${
    relbrcnv.1 $e |- Rel R $.
    $( When ` R ` is a relation, the sethood assumptions on ~ brcnv can be
       omitted.  (Contributed by Mario Carneiro, 28-Apr-2015.) $)
    relbrcnv $p |- ( A `' R B <-> B R A ) $=
      ( wrel ccnv wbr wb relbrcnvg ax-mp ) CEABCFGBACGHDABCIJ $.
  $}

  ${
    $d x y z R $.
    $( Two ways of saying a relation is transitive.  Definition of transitivity
       in [Schechter] p. 51.  (Contributed by NM, 27-Dec-1996.)  (Proof
       shortened by Andrew Salmon, 27-Aug-2011.) $)
    cotr $p |- ( ( R o. R ) C_ R <->
             A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) ) $=
      ( ccom wss cv cop wcel wi wal wbr wa wrel wb wex df-co vex albii bitri
      relopabi ssrel ax-mp opelco df-br bicomi imbi12i 19.23v bitr4i alcom ) DD
      EZDFZAGZCGZHZUKIZUODIZJZCKZAKZUMBGZDLVAUNDLMZUMUNDLZJZCKBKZAKUKNULUTOVBBP
      ZACUKACBDDQUAACUKDUBUCUSVEAUSVDBKZCKVEURVGCURVFVCJVGUPVFUQVCBUMUNDDARCRUD
      VCUQUMUNDUEUFUGVBVCBUHUISVDCBUJTST $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z R $.  $d x y z S $.  $d z V $.
    $d z W $.
    $( Two ways to state a relation is reflexive.  Adapted from Tarski.
       (Contributed by FL, 15-Jan-2012.)  (Revised by NM, 30-Mar-2016.) $)
    issref $p |- ( ( _I |` A ) C_ R <-> A. x e. A x R x ) $=
      ( vy vz cv wbr wral wcel wi wal cop cid wss df-ral cvv vex opelresi ax-mp
      sylbi cres wb df-br bicomi imbi12i albii ralidm bitri cxp pm2.27 opelresg
      ralv wa weq ideq opeq2 eleq1d biimpcd syl6 biimtrrdi pm2.43i com3r sylbir
      imp biimtrdi ralrimiv sps ralimi eleq1 imbi12d ralxp sylibr relres df-rel
      wceq wrel mpbi sseli ancri pm3.31 syl5 alimi ssalel ssel alrimiv 3bitr2ri
      syl impbii ) AFZWICGZABHWIBIZWJJZAKWIWILZMBUAZIZWMCIZJZAKZWNCNZWJABOWQWLA
      WOWKWPWJWIPIZWOWKUBAQZWIBPRSWJWPWIWICUCUDUEUFWRWSWRDFZWNIZXBCIZJZDKZWSWRW
      QAPHZAPHZXFXHXGWRWQAPUGWQAULUHXHXEDPPUIZHZXFXHWIEFZLZWNIZXLCIZJZEPHZAPHXJ
      XGXPAPXGWTWQJZAKXPWQAPOXQXPAWTXQXPJXAWTXQWQXPWTWQUJWQXOEPXKPIZXMWQXNXRXMX
      LMIZWKUMWQXNJZWIXKMBPUKXSWKXTXSWIXKMGZWKXTJZWIXKMUCYAAEUNZYBWIXKEQUOWKWQY
      CXNWKWQYCXNJZJZWKWKWOYEWIBBRWOWQWPYDWOWPUJYCWPXNYCWMXLCWIXKWIUPUQURUSUTVA
      VBTVCVDVEVBVFUSSVGTVHXEXODAEPPXBXLVOXCXMXDXNXBXLWNVIXBXLCVIVJVKVLXJXBXIIZ
      XEJZDKXFXEDXIOYGXEDXCYFXCUMYGXDXCYFWNXIXBWNVPWNXINMBVMWNVNVQVRVSYFXCXDVTW
      AWBTWGVCDWNCWCVLWSWQAWNCWMWDWEWHWF $.

    $( Two ways of saying a relation is symmetric.  Similar to definition of
       symmetry in [Schechter] p. 51.  (Contributed by NM, 28-Dec-1996.)
       (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    cnvsym $p |- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) ) $=
      ( cv cop ccnv wcel wi wal wss wbr alcom wrel relcnv ssrel ax-mp vex brcnv
      wb df-br bitr3i imbi12i 2albii 3bitr4i ) BDZADZEZCFZGZUGCGZHZAIBIZUKBIAIU
      HCJZUFUECKZUEUFCKZHZBIAIUKBALUHMUMULSCNBAUHCOPUPUKABUNUIUOUJUNUEUFUHKUIUE
      UFCBQAQRUEUFUHTUAUEUFCTUBUCUD $.

    $( Two ways of saying a relation is antisymmetric.  Definition of
       antisymmetry in [Schechter] p. 51.  (Contributed by NM, 9-Sep-2004.)
       (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    intasym $p |- ( ( R i^i `' R ) C_ _I <->
                  A. x A. y ( ( x R y /\ y R x ) -> x = y ) ) $=
      ( ccnv cin cid wss cv cop wcel wi wal wbr wa weq wrel wb df-br vex bitr3i
      relcnv relin2 ssrel mp2b elin brcnv anbi12i bitr4i imbi12i 2albii bitri
      ideq ) CCDZEZFGZAHZBHZIZUNJZURFJZKZBLALZUPUQCMZUQUPCMZNZABOZKZBLALUMPUNPU
      OVBQCUACUMUBABUNFUCUDVAVGABUSVEUTVFUSURCJZURUMJZNVEURCUMUEVCVHVDVIUPUQCRV
      DUPUQUMMVIUPUQCASBSZUFUPUQUMRTUGUHUTUPUQFMVFUPUQFRUPUQVJULTUIUJUK $.

    $( Two ways of saying a relation is antisymmetric and reflexive.
       ` U. U. R ` is the field of a relation by ~ relfld .  (Contributed by
       NM, 6-May-2008.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    asymref $p |- ( ( R i^i `' R ) = ( _I |` U. U. R ) <->
       A. x e. U. U. R A. y ( ( x R y /\ y R x ) <-> x = y ) ) $=
      ( cv wcel cid cuni wb wal wbr wa wceq wi df-br bitr3i bitri 3bitr4i albii
      vex wrel cop ccnv cin cres wral opeluu simpld adantr pm4.71ri bibi1i elin
      sylbi brcnv anbi12i bitr4i opelres ideq pm5.32 19.21v relcnv relin2 ax-mp
      anbi2ci bibi12i relres eqrel mp2an df-ral ) ADZBDZUAZCCUBZUCZEZVKFCGGZUDZ
      EZHZBIZAIZVIVOEZVIVJCJZVJVICJZKZVIVJLZHZBIZMZAIVMVPLZWGAVOUEVSWHAVSWAWFMZ
      BIWHVRWJBWDWAWEKZHWAWDKZWKHVRWJWDWLWKWDWAWBWAWCWBWAVJVOEZWBVKCEZWAWMKVIVJ
      CNZVIVJCASZBSZUFULUGUHUIUJVNWDVQWKVNWNVKVLEZKWDVKCVLUKWBWNWCWRWOWCVIVJVLJ
      WRVIVJCWPWQUMVIVJVLNOUNUOVQVKFEZWAKWKVIVJFVOWQUPWSWEWAWSVIVJFJWEVIVJFNVIV
      JWQUQOVCPVDWAWDWEURQRWAWFBUSPRVMTZVPTWIVTHVLTWTCUTCVLVAVBFVOVEABVMVPVFVGW
      GAVOVHQ $.

    $( Two ways of saying a relation is irreflexive.  Definition of
       irreflexivity in [Schechter] p. 51.  (Contributed by NM, 9-Sep-2004.)
       (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    intirr $p |- ( ( R i^i _I ) = (/) <-> A. x -. x R x ) $=
      ( vy cid cin c0 wceq cv cop wcel cvv cdif wi wal wbr wn wss df-br 3bitr2i
      vex incom eqeq1i disj2 wrel wb reli ssrel ax-mp 3bitri equcom wa biantrur
      ideq opex eldif bitr4i xchnxbir imbi12i 2albii breq2 notbid equsal albii
      nfv ) BDEZFGZAHZCHZIZDJZVIKBLZJZMZCNANZVHVGGZVGVHBOZPZMZCNZANVGVGBOZPZANV
      FDBEZFGDVKQZVNVEWBFBDUAUBDBUCDUDWCVNUEUFACDVKUGUHUIVRVMACVOVJVQVLVOVGVHGV
      GVHDOVJCAUJVGVHCTZUMVGVHDRSVIBJZVLVPWEPZVIKJZWFUKVLWGWFVGVHATWDUNULVIKBUO
      UPVGVHBRUQURUSVSWAAVQWACAWACVDVOVPVTVHVGVGBUTVAVBVCS $.

    $( Two ways of saying that two elements have an upper bound.  (Contributed
       by Mario Carneiro, 3-Nov-2015.) $)
    brcodir $p |- ( ( A e. V /\ B e. W ) ->
      ( A ( `' R o. R ) B <-> E. z ( A R z /\ B R z ) ) ) $=
      ( wcel wa ccnv ccom wbr cv wex brcog wb cvv vex brcnvg mpan anbi2d adantl
      exbidv bitrd ) BEGZCFGZHZBCDIZDJKBALZDKZUHCUGKZHZAMUICUHDKZHZAMABCUGDEFNU
      FUKUMAUEUKUMOUDUEUJULUIUHPGUEUJULOAQUHCPFDRSTUAUBUC $.

    $( Two ways of saying a relation is directed.  (Contributed by Mario
       Carneiro, 22-Nov-2013.) $)
    codir $p |- ( ( A X. B ) C_ ( `' R o. R ) <-> A. x e. A A. y e. B
      E. z ( x R z /\ y R z ) ) $=
      ( cv cop cxp wcel ccnv ccom wi wal wa wbr wral cvv wb vex wex wss brcodir
      opelxp df-br mp2an bitr3i imbi12i 2albii wrel relxp ssrel ax-mp 3bitr4i
      r2al ) AGZBGZHZDEIZJZURFKFLZJZMZBNANZUPDJUQEJOZUPCGZFPUQVFFPOCUAZMZBNANUS
      VAUBZVGBEQADQVCVHABUTVEVBVGUPUQDEUDVBUPUQVAPZVGUPUQVAUEUPRJUQRJVJVGSATBTC
      UPUQFRRUCUFUGUHUIUSUJVIVDSDEUKABUSVAULUMVGABDEUOUN $.

    $( A quantifier-free way of expressing the total order predicate.
       (Contributed by Mario Carneiro, 22-Nov-2013.) $)
    qfto $p |- ( ( A X. B ) C_ ( R u. `' R ) <->
                 A. x e. A A. y e. B ( x R y \/ y R x ) ) $=
      ( cv cop cxp wcel ccnv cun wi wal wa wbr wo wss wral opelxp vex brun wrel
      df-br brcnv orbi2i 3bitr3i imbi12i 2albii relxp ssrel ax-mp r2al 3bitr4i
      wb ) AFZBFZGZCDHZIZUQEEJZKZIZLZBMAMZUOCIUPDINZUOUPEOZUPUOEOZPZLZBMAMURVAQ
      ZVHBDRACRVCVIABUSVEVBVHUOUPCDSUOUPVAOVFUOUPUTOZPVBVHUOUPEUTUAUOUPVAUCVKVG
      VFUOUPEATBTUDUEUFUGUHURUBVJVDUNCDUIABURVAUJUKVHABCDULUM $.

    $( A square cross product ` ( A X. A ) ` is a transitive relation.
       (Contributed by FL, 31-Jul-2009.) $)
    xpidtr $p |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A ) $=
      ( vx vy vz cxp ccom wss cv wbr wa wal wcel simplbi2com adantl sylbi com12
      wi brxp adantr imp ax-gen gen2 cotr mpbir ) AAEZUEFUEGBHZCHZUEIZUGDHZUEIZ
      JUFUIUEIZQZDKZCKBKUMBCULDUHUJUKUHUFALZUGALZJUJUKQZUFUGAARUNUPUOUJUNUKUJUO
      UIALZJUNUKQZUGUIAARUQURUOUKUNUQUFUIAARMNOPSOTUAUBBCDUEUCUD $.

    $( The intersection of two transitive classes is transitive.  (Contributed
       by FL, 31-Jul-2009.) $)
    trin2 $p |- ( ( ( R o. R ) C_ R /\ ( S o. S ) C_ S )
      -> ( ( R i^i S ) o. ( R i^i S ) ) C_ ( R i^i S ) ) $=
      ( vx vy vz ccom wss wa cv cin wbr wal cotr brin simpr simpl com12 alanimi
      wi sylbi anim12d an4s syl2anb imbitrrdi ex imp sylibr ) AAFAGZBBFBGZHCIZD
      IZABJZKZUKEIZULKZHZUJUNULKZSZELZDLZCLZULULFULGUHUIVAUHUJUKAKZUKUNAKZHZUJU
      NAKZSZELZDLZCLZUIVASCDEAMUIVIVAUIUJUKBKZUKUNBKZHZUJUNBKZSZELZDLZCLZVIVASC
      DEBMVQVIVAVPVHUTCVOVGUSDVNVFUREVNVFHZUPVEVMHZUQUPVRVSUMVBVJHVCVKHVRVSSZUO
      UJUKABNUKUNABNVBVCVJVKVTVRVDVLHVSVRVDVEVLVMVNVFOVNVFPUAQUBUCQUJUNABNUDRRR
      UETQTUFCDEULMUG $.

    $( A partial order relation is irreflexive.  (Contributed by Mario
       Carneiro, 2-Nov-2015.) $)
    poirr2 $p |- ( R Po A -> ( R i^i ( _I |` A ) ) = (/) ) $=
      ( vx vy wpo cid cres cin c0 wss wceq wrel relres relin2 cv wcel wbr wa wn
      biimtrid mp1i cop df-br brin bitr3i wi vex brres poirr breq2 sylbi notbid
      ideq syl5ibcom expimpd ancomsd con2d imnan sylib pm2.21d relssdv ss0 syl
      wb ) ABEZBFAGZHZIJVGIKVECDVGIVFLVGLVEFAMBVFNUACOZDOZUBZVGPZVHVIBQZVHVIVFQ
      ZRZVEVJIPZVKVHVIVGQVNVHVIVGUCVHVIBVFUDUEVEVNVOVEVLVMSUFVNSVEVMVLVMVHVIFQZ
      VHAPZRVEVLSZVHVIFADUGZUHVEVQVPVRVEVQVPVRVEVQRVHVHBQZSVPVRAVHBUIVPVTVLVPVH
      VIKVTVLVDVHVIVSUMVHVIVHBUJUKULUNUOUPTUQVLVMURUSUTTVAVGVBVC $.
  $}

  $( The relation induced by a transitive relation on a part of its field is
     transitive.  (Taking the intersection of a relation with a square cross
     product is a way to restrict it to a subset of its field.)  (Contributed
     by FL, 31-Jul-2009.) $)
  trinxp $p |- ( ( R o. R ) C_ R ->
  ( ( R i^i ( A X. A ) ) o. ( R i^i ( A X. A ) ) ) C_ ( R i^i ( A X. A ) ) ) $=
    ( ccom wss cxp cin xpidtr trin2 mpan2 ) BBCBDAAEZJCJDBJFZKCKDAGBJHI $.

  ${
    soi.1 $e |- R Or S $.
    soi.2 $e |- R C_ ( S X. S ) $.
    $( A strict order relation is irreflexive.  (Contributed by NM,
       10-Feb-1996.)  (Revised by Mario Carneiro, 10-May-2013.) $)
    soirri $p |- -. A R A $=
      ( wbr id wor wcel wn brel simpld sonr sylancr pm2.65i ) AABFZPPGPCBHACIZP
      JDPQQAACCBEKLCABMNO $.

    $( A strict order relation is a transitive relation.  (Contributed by NM,
       10-Feb-1996.)  (Revised by Mario Carneiro, 10-May-2013.) $)
    sotri $p |- ( ( A R B /\ B R C ) -> A R C ) $=
      ( wcel wa wbr brel simpld anim12i wi wor w3a sotr mpan 3expb mpcom ) AEHZ
      BEHZCEHZIZIABDJZBCDJZIZACDJZUEUAUFUDUEUAUBABEEDGKLBCEEDGKMUAUBUCUGUHNZEDO
      UAUBUCPUIFEABCDQRST $.

    $( A strict order relation has no 2-cycle loops.  (Contributed by NM,
       10-Feb-1996.)  (Revised by Mario Carneiro, 10-May-2013.) $)
    son2lpi $p |- -. ( A R B /\ B R A ) $=
      ( wbr wa soirri sotri mto ) ABCGBACGHAACGACDEFIABACDEFJK $.

    $( A transitivity relation.  (Read ` -. ` B < A and B < C implies A < C .)
       (Contributed by Mario Carneiro, 10-May-2013.) $)
    sotri2 $p |- ( ( A e. S /\ -. B R A /\ B R C ) -> A R C ) $=
      ( wcel wbr wn w3a simp2 wo wa brel 3ad2ant3 simp1 df-3an sylanbrc simp3
      wor wi sowlin mpan sylc ord mpd ) AEHZBADIZJZBCDIZKZUJACDIZUHUJUKLULUIUMU
      LBEHZCEHZUHKZUKUIUMMZULUNUONZUHUPUKUHURUJBCEEDGOPUHUJUKQUNUOUHRSUHUJUKTED
      UAUPUKUQUBFEBCADUCUDUEUFUG $.

    $( A transitivity relation.  (Read A < B and ` -. ` C < B implies A < C .)
       (Contributed by Mario Carneiro, 10-May-2013.) $)
    sotri3 $p |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C ) $=
      ( wcel wbr wn w3a simp3 wo wa brel 3ad2ant2 simp1 df-3an sylanbrc simp2
      wor wi sowlin mpan sylc ecased ) CEHZABDIZCBDIZJZKZACDIZUIUGUHUJLUKAEHZBE
      HZUGKZUHULUIMZUKUMUNNZUGUOUHUGUQUJABEEDGOPUGUHUJQUMUNUGRSUGUHUJTEDUAUOUHU
      PUBFEABCDUCUDUEUF $.
  $}

  $( Express "less than or equals" for general strict orders.  (Contributed by
     Stefan O'Rear, 17-Jan-2015.) $)
  poleloe $p |- ( B e. V -> ( A ( R u. _I ) B <-> ( A R B \/ A = B ) ) ) $=
    ( cid cun wbr wo wcel wceq brun ideqg orbi2d bitrid ) ABCEFGABCGZABEGZHBDIZ
    OABJZHABCEKQPROABDLMN $.

  $( Transitive law for general strict orders.  (Contributed by Stefan O'Rear,
     17-Jan-2015.) $)
  poltletr $p |- ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) ->
      ( ( A R B /\ B ( R u. _I ) C ) -> A R C ) ) $=
    ( wpo wcel w3a wa wbr cid cun wceq wo poleloe 3ad2ant3 adantl anbi2d com12
    wb wi potr breq2 biimpac a1d jaodan sylbid ) EDFZAEGZBEGZCEGZHZIZABDJZBCDKL
    JZIUNBCDJZBCMZNZIZACDJZUMUOURUNULUOURTZUHUKUIVAUJBCDEOPQRUSUMUTUNUPUMUTUAUQ
    UMUNUPIUTEABCDUBSUNUQIUTUMUQUNUTBCADUCUDUEUFSUG $.

  ${
    $d x y z w $.  $d z w ph $.
    $( The converse of a class abstraction of ordered pairs.  (Contributed by
       NM, 11-Dec-2003.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    cnvopab $p |- `' { <. x , y >. | ph } = { <. y , x >. | ph } $=
      ( vz vw copab ccnv relcnv relopab cop wcel wsb opelopabsbALT sbcom2 bitri
      cv vex opelcnv 3bitr4i eqrelriiv ) DEABCFZGZACBFZUAHACBIEPZDPZJUAKZACDLBE
      LZUEUDJZUBKUHUCKUFABELCDLUGABCEDMABECDNOUEUDUADQEQRACBDEMST $.
  $}

  ${
    $d x y ph $.  $d x C $.  $d x D $.  $d y A $.  $d y B $.
    mptcnv.1 $e |- ( ph -> ( ( x e. A /\ y = B ) <-> ( y e. C /\ x = D ) ) ) $.
    $( The converse of a mapping function.  (Contributed by Thierry Arnoux,
       16-Jan-2017.) $)
    mptcnv $p |- ( ph -> `' ( x e. A |-> B ) = ( y e. C |-> D ) ) $=
      ( cv wcel wceq wa copab cmpt ccnv opabbidv df-mpt cnveqi cnvopab eqtri
      3eqtr4g ) ABIZDJCIZEKLZCBMZUCFJUBGKLZCBMBDENZOZCFGNAUDUFCBHPUHUDBCMZOUEUG
      UIBCDEQRUDBCSTCBFGQUA $.
  $}

  ${
    $d x y $.
    $( The converse of the empty set.  (Contributed by NM, 6-Apr-1998.) $)
    cnv0 $p |- `' (/) = (/) $=
      ( vx vy c0 ccnv relcnv rel0 cv cop wcel vex opelcnv noel 2false eqrelriiv
      bitr4i ) ABCDZCCEFAGZBGZHZPIRQHZCIZSCIZQRCAJBJKUBUASLTLMON $.

    $( The converse of the identity relation.  Theorem 3.7(ii) of [Monk1]
       p. 36.  (Contributed by NM, 26-Apr-1998.)  (Proof shortened by Andrew
       Salmon, 27-Aug-2011.) $)
    cnvi $p |- `' _I = _I $=
      ( vy vx cid wbr copab weq ccnv vex ideq equcom bitri opabbii df-cnv df-id
      cv 3eqtr4i ) AOZBOZCDZBAEBAFZBAECGCSTBASABFTQRBHIABJKLBACMBANP $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The converse of a union is the union of converses.  Theorem 16 of
       [Suppes] p. 62.  (Contributed by NM, 25-Mar-1998.)  (Proof shortened by
       Andrew Salmon, 27-Aug-2011.) $)
    cnvun $p |- `' ( A u. B ) = ( `' A u. `' B ) $=
      ( vy vx cun ccnv cv wbr copab df-cnv unopab brun opabbii eqtr4i uneq12i
      wo ) ABEZFZCGZDGZAHZDCIZSTBHZDCIZEZAFZBFZERSTQHZDCIZUEDCQJUEUAUCPZDCIUIUA
      UCDCKUHUJDCSTABLMNNUFUBUGUDDCAJDCBJON $.

    $( Distributive law for converse over set difference.  (Contributed by
       Mario Carneiro, 26-Jun-2014.) $)
    cnvdif $p |- `' ( A \ B ) = ( `' A \ `' B ) $=
      ( vx vy cdif ccnv relcnv wss wrel difss relss mp2 cv cop wcel wn wa eldif
      vex opelcnv notbii anbi12i bitri 3bitr4i eqrelriiv ) CDABEZFZAFZBFZEZUFGU
      JUHHUHIUJIUHUIJAGUJUHKLDMZCMZNZUFOUMAOZUMBOZPZQZULUKNZUGOURUJOZUMABRULUKU
      FCSZDSZTUSURUHOZURUIOZPZQUQURUHUIRVBUNVDUPULUKAUTVATVCUOULUKBUTVATUAUBUCU
      DUE $.

    $( Distributive law for converse over intersection.  Theorem 15 of [Suppes]
       p. 62.  (Contributed by NM, 25-Mar-1998.)  (Revised by Mario Carneiro,
       26-Jun-2014.) $)
    cnvin $p |- `' ( A i^i B ) = ( `' A i^i `' B ) $=
      ( vy vx cin ccnv cv wbr copab df-cnv inopab brin opabbii eqtr4i ineq12i
      wa ) ABEZFZCGZDGZAHZDCIZSTBHZDCIZEZAFZBFZERSTQHZDCIZUEDCQJUEUAUCPZDCIUIUA
      UCDCKUHUJDCSTABLMNNUFUBUGUDDCAJDCBJON $.
  $}

  $( Distributive law for range over union.  Theorem 8 of [Suppes] p. 60.
     (Contributed by NM, 24-Mar-1998.) $)
  rnun $p |- ran ( A u. B ) = ( ran A u. ran B ) $=
    ( cun ccnv cdm crn cnvun dmeqi dmun eqtri df-rn uneq12i 3eqtr4i ) ABCZDZEZA
    DZEZBDZEZCZNFAFZBFZCPQSCZEUAOUDABGHQSIJNKUBRUCTAKBKLM $.

  $( The range of an intersection belongs the intersection of ranges.  Theorem
     9 of [Suppes] p. 60.  (Contributed by NM, 15-Sep-2004.) $)
  rnin $p |- ran ( A i^i B ) C_ ( ran A i^i ran B ) $=
    ( cin ccnv cdm crn cnvin dmeqi dmin eqsstri df-rn ineq12i 3sstr4i ) ABCZDZE
    ZADZEZBDZEZCZNFAFZBFZCPQSCZEUAOUDABGHQSIJNKUBRUCTAKBKLM $.

  ${
    $d x y z $.  $d y z A $.  $d y z B $.
    $( The range of an indexed union.  (Contributed by Mario Carneiro,
       29-May-2015.) $)
    rniun $p |- ran U_ x e. A B = U_ x e. A ran B $=
      ( vz vy ciun crn cv cop wcel wex wrex rexcom4 elrn2 rexbii eliun 3bitr4ri
      vex exbii 3bitr4i eqriv ) DABCFZGZABCGZFZEHDHZIZUBJZEKZUFUDJZABLZUFUCJUFU
      EJUGCJZEKZABLULABLZEKUKUIULAEBMUJUMABEUFCDRZNOUHUNEAUGBCPSQEUFUBUONAUFBUD
      PTUA $.

    $d x A $.
    $( The range of a union.  Part of Exercise 8 of [Enderton] p. 41.
       (Contributed by NM, 17-Mar-2004.)  (Revised by Mario Carneiro,
       29-May-2015.) $)
    rnuni $p |- ran U. A = U_ x e. A ran x $=
      ( cuni crn cv ciun uniiun rneqi rniun eqtri ) BCZDABAEZFZDABLDFKMABGHABLI
      J $.
  $}

  $( Distributive law for image over union.  Theorem 35 of [Suppes] p. 65.
     (Contributed by NM, 30-Sep-2002.) $)
  imaundi $p |- ( A " ( B u. C ) ) = ( ( A " B ) u. ( A " C ) ) $=
    ( cun cres crn cima resundi rneqi rnun eqtri df-ima uneq12i 3eqtr4i ) ABCDZ
    EZFZABEZFZACEZFZDZAOGABGZACGZDQRTDZFUBPUEABCHIRTJKAOLUCSUDUAABLACLMN $.

  $( The image of a union.  (Contributed by Jeff Hoffman, 17-Feb-2008.) $)
  imaundir $p |- ( ( A u. B ) " C ) = ( ( A " C ) u. ( B " C ) ) $=
    ( cun cima cres crn df-ima resundir rneqi rnun 3eqtri uneq12i eqtr4i ) ABDZ
    CEZACFZGZBCFZGZDZACEZBCEZDPOCFZGQSDZGUAOCHUDUEABCIJQSKLUBRUCTACHBCHMN $.

  ${
    $d x y A $.  $d x y B $.  $d x y R $.
    $( An upper bound for intersection with a domain.  Theorem 40 of [Suppes]
       p. 66, who calls it "somewhat surprising".  (Contributed by NM,
       11-Aug-2004.) $)
    dminss $p |- ( dom R i^i A ) C_ ( `' R " ( R " A ) ) $=
      ( vx vy cdm cin ccnv cima cv wbr wcel wa 19.8a ancoms elima2 sylibr simpl
      wex vex brcnv jca eximi eldm anbi1i elin 19.41v 3bitr4i 3imtr4i ssriv ) C
      BEZAFZBGZBAHZHZCIZDIZBJZUOAKZLZDRZUPUMKZUPUOULJZLZDRUOUKKZUOUNKUSVCDUSVAV
      BUSURUQLZCRZVAURUQVFVECMNCUPBADSZOPUSUQVBUQURQUPUOBVGCSZTPUAUBUOUJKZURLUQ
      DRZURLVDUTVIVJURDUOBVHUCUDUOUJAUEUQURDUFUGDUOULUMVHOUHUI $.

    $( An upper bound for intersection with an image.  Theorem 41 of [Suppes]
       p. 66.  (Contributed by NM, 11-Aug-2004.) $)
    imainss $p |- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) ) $=
      ( vy vx cima cin ccnv cv wcel wbr wa wex brcnv 19.8a sylan2br elin elima2
      vex anbi1i ancoms anim2i simprl anassrs anbi2i bitri sylibr eximi 3bitr4i
      jca 19.41v 3imtr4i ssriv ) DCAFZBGZCACHZBFZGZFZEIZAJZUTDIZCKZLZVBBJZLZEMZ
      UTURJZVCLZEMVBUOJZVBUSJVFVIEVFVAVEVBUTUPKZLZDMZLZVCLZVIVAVCVEVOVAVCVELZLV
      NVCVPVMVAVEVCVMVCVEVKVMVBUTCDSZESZNVLDOPUAUBVAVCVEUCUJUDVHVNVCVHVAUTUQJZL
      VNUTAUQQVSVMVADUTUPBVRRUEUFTUGUHVBUNJZVELVDEMZVELVJVGVTWAVEEVBCAVQRTVBUNB
      QVDVEEUKUIEVBCURVQRULUM $.
  $}

  $( The image of an intersection.  (Contributed by Thierry Arnoux,
     16-Dec-2017.) $)
  inimass $p |- ( ( A i^i B ) " C ) C_ ( ( A " C ) i^i ( B " C ) ) $=
    ( cres cin crn cima rnin df-ima resindir rneqi eqtri ineq12i 3sstr4i ) ACDZ
    BCDZEZFZOFZPFZEABEZCGZACGZBCGZEOPHUBUACDZFRUACIUEQABCJKLUCSUDTACIBCIMN $.

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x V $.
    $( The intersection of the image of singleton.  (Contributed by Thierry
       Arnoux, 16-Dec-2017.) $)
    inimasn $p |- ( C e. V
      -> ( ( A i^i B ) " { C } ) = ( ( A " { C } ) i^i ( B " { C } ) ) ) $=
      ( vx wcel cin csn cima cv wa elin cop a1i cvv vex elimasng mpan2 anbi12d
      wb 3bitr4rd bitr2id eqrdv ) CDFZEABGZCHZIZAUFIZBUFIZGZEJZUJFUKUHFZUKUIFZK
      ZUDUKUGFZUKUHUILUDCUKMZUEFZUPAFZUPBFZKZUOUNUQUTTUDUPABLNUDUKOFZUOUQTEPZUE
      CUKDOQRUDULURUMUSUDVAULURTVBACUKDOQRUDVAUMUSTVBBCUKDOQRSUAUBUC $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The converse of a cross product.  Exercise 11 of [Suppes] p. 67.
       (Contributed by NM, 14-Aug-1999.)  (Proof shortened by Andrew Salmon,
       27-Aug-2011.) $)
    cnvxp $p |- `' ( A X. B ) = ( B X. A ) $=
      ( vy vx cv wcel copab ccnv cxp cnvopab ancom opabbii eqtri cnveqi 3eqtr4i
      wa df-xp ) CEAFZDEBFZPZCDGZHZSRPZDCGZABIZHBAIUBTDCGUDTCDJTUCDCRSKLMUEUACD
      ABQNDCBAQO $.
  $}

  $( The cross product with the empty set is empty.  Part of Theorem 3.13(ii)
     of [Monk1] p. 37.  (Contributed by NM, 12-Apr-2004.) $)
  xp0 $p |- ( A X. (/) ) = (/) $=
    ( c0 cxp ccnv 0xp cnveqi cnvxp cnv0 3eqtr3i ) BACZDBDABCBJBAEFBAGHI $.

  ${
    $d x y z A $.  $d x y z B $.
    $( The cross product of inhabited classes is inhabited.  (Contributed by
       Jim Kingdon, 11-Dec-2018.) $)
    xpmlem $p |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) ) $=
      ( cv wcel wex wa cxp eeanv cop vex opex wceq eleq1 opelxp bitrdi exlimivv
      spcev sylbir elxp simpr 2eximi sylbi exlimiv sylib impbii ) AFZDGZAHBFZEG
      ZBHIZCFZDEJZGZCHZUMUJULIZBHAHZUQUJULABKZURUQABUPURCUIUKLZUIUKAMBMNUNVAOZU
      PVAUOGURUNVAUOPUIUKDEQRTSUAUQUSUMUPUSCUPVBURIZBHAHUSABUNDEUBVCURABVBURUCU
      DUEUFUTUGUH $.
  $}

  ${
    $d x A $.  $d y B $.  $d z A $.  $d z B $.  $d a b w A $.  $d a b w B $.
    $d a x $.  $d b y $.  $d z w $.
    $( The cross product of inhabited classes is inhabited.  (Contributed by
       Jim Kingdon, 13-Dec-2018.) $)
    xpm $p |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) ) $=
      ( va vb vw cv wcel wex wa cxp xpmlem eleq1 cbvexv anbi12i 3bitr3i ) FIZDJ
      ZFKZGIZEJZGKZLHIZDEMZJZHKAIZDJZAKZBIZEJZBKZLCIZUFJZCKFGHDENUAUJUDUMTUIFAS
      UHDOPUCULGBUBUKEOPQUGUOHCUEUNUFOPR $.
  $}

  $( A cross product is empty if at least one member is empty.  (Contributed by
     Jim Kingdon, 12-Dec-2018.) $)
  xpeq0r $p |- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) ) $=
    ( c0 wceq cxp xpeq1 0xp eqtrdi xpeq2 xp0 jaoi ) ACDZABEZCDBCDZLMCBECACBFBGH
    NMACECBCAIAJHK $.

  $( A Cartesian square is empty iff its member is empty.  (Contributed by Jim
     Kingdon, 21-Apr-2023.) $)
  sqxpeq0 $p |- ( ( A X. A ) = (/) <-> A = (/) ) $=
    ( cxp c0 wceq cdm dmeq dmxpid dm0 3eqtr3g xpeq0r orcs impbii ) AABZCDZACDZN
    MECEACMCFAGHIOONAAJKL $.

  $( Cross products with disjoint sets are disjoint.  (Contributed by NM,
     13-Sep-2004.) $)
  xpdisj1 $p |- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) ) $=
    ( cin c0 wceq cxp inxp xpeq1 0xp eqtrdi eqtrid ) ABEZFGZACHBDHENCDEZHZFACBD
    IOQFPHFNFPJPKLM $.

  $( Cross products with disjoint sets are disjoint.  (Contributed by NM,
     13-Sep-2004.) $)
  xpdisj2 $p |- ( ( A i^i B ) = (/) -> ( ( C X. A ) i^i ( D X. B ) ) = (/) ) $=
    ( cin c0 wceq cxp inxp xpeq2 xp0 eqtrdi eqtrid ) ABEZFGZCAHDBHECDEZNHZFCADB
    IOQPFHFNFPJPKLM $.

  $( Cross products with two different singletons are disjoint.  (Contributed
     by NM, 28-Jul-2004.) $)
  xpsndisj $p |- ( B =/= D -> ( ( A X. { B } ) i^i ( C X. { D } ) ) = (/) ) $=
    ( wne csn cin c0 wceq cxp disjsn2 xpdisj2 syl ) BDEBFZDFZGHIANJCOJGHIBDKNOA
    CLM $.

  ${
    $d x A $.  $d y B $.
    $( Disjoint unions with disjoint index sets are disjoint.  (Contributed by
       Stefan O'Rear, 21-Nov-2014.) $)
    djudisj $p |- ( ( A i^i B ) = (/) -> ( U_ x e. A ( { x } X. C ) i^i
            U_ y e. B ( { y } X. D ) ) = (/) ) $=
      ( cin c0 wceq cv csn cxp ciun cvv wss djussxp incom eqtrid ssdisj sylancr
      xpdisj1 ) CDGHIZACAJKELMZCNLZOUDBDBJKFLMZGZHIUCUEGHIACEPUBUFUEUDGZHUDUEQU
      BUEDNLZOUHUDGZHIUGHIBDFPUBUIUDUHGHUHUDQCDNNUARUEUHUDSTRUCUDUEST $.
  $}

  $( A double restriction to disjoint classes is the empty set.  (Contributed
     by NM, 7-Oct-2004.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  resdisj $p |- ( ( A i^i B ) = (/) -> ( ( C |` A ) |` B ) = (/) ) $=
    ( cin c0 wceq cres resres reseq2 res0 eqtrdi eqtrid ) ABDZEFZCAGBGCMGZECABH
    NOCEGEMECICJKL $.

  ${
    $d x A $.
    $( The range of a cross product.  Part of Theorem 3.13(x) of [Monk1] p. 37,
       with nonempty changed to inhabited.  (Contributed by Jim Kingdon,
       12-Dec-2018.) $)
    rnxpm $p |- ( E. x x e. A -> ran ( A X. B ) = B ) $=
      ( cv wcel wex cxp crn cdm ccnv df-rn cnvxp dmeqi eqtri dmxpm eqtrid ) ADB
      EAFBCGZHZCBGZIZCRQJZITQKUASBCLMNACBOP $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The domain of a cross product is a subclass of the first factor.
       (Contributed by NM, 19-Mar-2007.) $)
    dmxpss $p |- dom ( A X. B ) C_ A $=
      ( vx vy cxp cdm cv wcel cop wex vex eldm2 opelxp1 exlimiv sylbi ssriv ) C
      ABEZFZACGZRHSDGZIQHZDJSAHZDSQCKLUAUBDSTABMNOP $.
  $}

  $( The range of a cross product is a subclass of the second factor.
     (Contributed by NM, 16-Jan-2006.)  (Proof shortened by Andrew Salmon,
     27-Aug-2011.) $)
  rnxpss $p |- ran ( A X. B ) C_ B $=
    ( cxp crn ccnv cdm df-rn cnvxp dmeqi dmxpss eqsstri ) ABCZDLEZFZBLGNBACZFBM
    OABHIBAJKK $.

  $( Upper bound for the domain of a binary relation.  (Contributed by BJ,
     10-Jul-2022.) $)
  dmxpss2 $p |- ( R C_ ( A X. B ) -> dom R C_ A ) $=
    ( cxp wss cdm dmss dmxpss sstrdi ) CABDZECFJFACJGABHI $.

  $( Upper bound for the range of a binary relation.  (Contributed by BJ,
     10-Jul-2022.) $)
  rnxpss2 $p |- ( R C_ ( A X. B ) -> ran R C_ B ) $=
    ( cxp wss crn rnss rnxpss sstrdi ) CABDZECFJFBCJGABHI $.

  ${
    $d x y A $.
    $( The range of a square cross product.  (Contributed by FL,
       17-May-2010.) $)
    rnxpid $p |- ran ( A X. A ) = A $=
      ( vx vy cxp crn rnxpss cv wcel cop wex wa opelxp anidm bitri weq wb opeq1
      eleq1d equcoms biimpd spimev sylbir vex elrn2 sylibr ssriv eqssi ) AADZEZ
      AAAFBAUIBGZAHZCGZUJIZUHHZCJZUJUIHUKUJUJIZUHHZUOUQUKUKKUKUJUJAALUKMNUQUNCB
      CBOUQUNUQUNPBCBCOUPUMUHUJULUJQRSTUAUBCUJUHBUCUDUEUFUG $.
  $}

  ${
    $d x a b A $.  $d x a b B $.
    $( A cross-product subclass relationship is equivalent to the relationship
       for its components.  (Contributed by Jim Kingdon, 12-Dec-2018.) $)
    ssxpbm $p |- ( E. x x e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) <->
                ( A C_ C /\ B C_ D ) ) ) $=
      ( va vb cv cxp wcel wex wss cdm wceq adantl sylbir adantr eqsstrrd sstrdi
      wa xpm dmxpm dmss dmxpss crn rnxpm rnss rnxpss jca ex xpss12 impbid1 ) AH
      BCIZJAKZUMDEIZLZBDLZCELZTZUNUPUSUNUPTZUQURUTBUOMZDUTBUMMZVAUNVBBNZUPUNFHB
      JFKZGHCJGKZTZVCFGABCUAZVEVCVDGBCUBOPQUPVBVALUNUMUOUCORDEUDSUTCUOUEZEUTCUM
      UEZVHUNVICNZUPUNVFVJVGVDVJVEFBCUFQPQUPVIVHLUNUMUOUGORDEUHSUIUJBDCEUKUL $.
  $}

  ${
    $d x C $.
    $( Cross product subset cancellation.  (Contributed by Jim Kingdon,
       14-Dec-2018.) $)
    ssxp1 $p |- ( E. x x e. C ->
        ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) ) $=
      ( cv wcel wex cxp wss cdm wceq dmxpm adantr adantl eqsstrrd dmxpss sstrdi
      wa dmss ex xpss1 impbid1 ) AEDFAGZBDHZCDHZIZBCIZUCUFUGUCUFRZBUEJZCUHBUDJZ
      UIUCUJBKUFABDLMUFUJUIIUCUDUESNOCDPQTBCDUAUB $.
  $}

  ${
    $d x C $.
    $( Cross product subset cancellation.  (Contributed by Jim Kingdon,
       14-Dec-2018.) $)
    ssxp2 $p |- ( E. x x e. C ->
        ( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) ) $=
      ( cv wcel wex cxp wss crn wceq rnxpm adantr adantl eqsstrrd rnxpss sstrdi
      wa rnss ex xpss2 impbid1 ) AEDFAGZDBHZDCHZIZBCIZUCUFUGUCUFRZBUEJZCUHBUDJZ
      UIUCUJBKUFADBLMUFUJUIIUCUDUESNODCPQTBCDUAUB $.
  $}

  ${
    $d x A $.  $d y B $.  $d A z $.  $d B z $.  $d C z $.  $d D z $.
    $( The cross product of inhabited classes is one-to-one.  (Contributed by
       Jim Kingdon, 13-Dec-2018.) $)
    xp11m $p |- ( ( E. x x e. A /\ E. y y e. B )
        -> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) ) $=
      ( vz cv wcel wex wa cxp wceq wi xpm anidm wss ssxpbm syl5ibcom eqss eleq2
      exbidv anbi2d bitr3id eqimss eqimss2 anim12d bitr4i imbitrdi sylbid com12
      an4 anbi12i sylbi xpeq12 impbid1 ) AHCIAJBHDIBJKZCDLZEFLZMZCEMZDFMZKZUQGH
      ZURIZGJZUTVCNABGCDOUTVFVCUTVFVFVDUSIZGJZKZVCVFVFVFKUTVIVFPUTVFVHVFUTVEVGG
      URUSVDUAUBUCUDUTVICEQZDFQZKZECQZFDQZKZKZVCUTVFVLVHVOUTURUSQVFVLURUSUEGCDE
      FRSUTUSURQVHVOUSURUFGEFCDRSUGVPVJVMKZVKVNKZKVCVJVKVMVNULVAVQVBVRCETDFTUMU
      HUIUJUKUNCEDFUOUP $.
  $}

  ${
    $d x C $.
    $( Cancellation law for cross-product.  (Contributed by Jim Kingdon,
       14-Dec-2018.) $)
    xpcanm $p |- ( E. x x e. C -> ( ( C X. A ) = ( C X. B ) <-> A = B ) ) $=
      ( cv wcel wex cxp wss wa wceq ssxp2 anbi12d eqss 3bitr4g ) AEDFAGZDBHZDCH
      ZIZRQIZJBCIZCBIZJQRKBCKPSUATUBABCDLACBDLMQRNBCNO $.

    $( Cancellation law for cross-product.  (Contributed by Jim Kingdon,
       14-Dec-2018.) $)
    xpcan2m $p |- ( E. x x e. C -> ( ( A X. C ) = ( B X. C ) <-> A = B ) ) $=
      ( cv wcel wex cxp wss wa wceq ssxp1 anbi12d eqss 3bitr4g ) AEDFAGZBDHZCDH
      ZIZRQIZJBCIZCBIZJQRKBCKPSUATUBABCDLACBDLMQRNBCNO $.
  $}

  ${
    $d x a A $.  $d x b B $.
    $( If a nonempty cross product is a set, so are both of its components.
       (Contributed by Jim Kingdon, 14-Dec-2018.) $)
    xpexr2m $p |- ( ( ( A X. B ) e. C /\ E. x x e. ( A X. B ) ) ->
                 ( A e. _V /\ B e. _V ) ) $=
      ( va vb cv cxp wcel wex wa cvv xpm cdm dmxpm adantl dmexg adantr eqeltrrd
      wceq crn rnxpm rnexg anim12dan ancom2s sylan2br ) AGBCHZIAJUGDIZEGBIEJZFG
      CIFJZKBLIZCLIZKZEFABCMUHUJUIUMUHUJUKUIULUHUJKUGNZBLUJUNBTUHFBCOPUHUNLIUJU
      GDQRSUHUIKUGUAZCLUIUOCTUHEBCUBPUHUOLIUIUGDUCRSUDUEUF $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Subset of the range of a restriction.  (Contributed by NM,
       16-Jan-2006.) $)
    ssrnres $p |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B ) $=
      ( vy vx cxp cin crn wceq wss rnss ax-mp cvv cv wex wa elrn2 bitr2i 3bitri
      wcel cres inss2 rnxpss sstri eqss mpbiran ssv xpss12 mp2an sslin sseqtrri
      ssid df-res sstr mpan2 cop ssel imbitrdi ancrd elin opelxp anbi2i opelres
      vex anbi1i anass exbii 19.41v imbitrrdi ssrdv impbii ) CABFZGZHZBIZBVNJZB
      CAUAZHZJZVOVNBJVPVNVLHZBVMVLJVNVTJCVLUBVMVLKLABUCUDVNBUEUFVPVSVPVNVRJZVSV
      MVQJWAVMCAMFZGZVQVLWBJZVMWCJAAJBMJWDAULBUGAABMUHUIVLWBCUJLCAUMUKVMVQKLBVN
      VRUNUOVSDBVNVSDNZBTZENZWEUPZVQTZEOZWFPZWEVNTZVSWFWJVSWFWEVRTWJBVRWEUQEWEV
      QDVDZQURUSWLWHVMTZEOWIWFPZEOWKEWEVMWMQWNWOEWNWHCTZWHVLTZPWPWGATZWFPZPZWOW
      HCVLUTWQWSWPWGWEABVAVBWOWPWRPZWFPWTWIXAWFWGWECAWMVCVEWPWRWFVFRSVGWIWFEVHS
      VIVJVKR $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.
    $( Range of the intersection with a cross product.  (Contributed by NM,
       17-Jan-2006.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    rninxp $p |- ( ran ( C i^i ( A X. B ) ) = B <->
                 A. y e. B E. x e. A x C y ) $=
      ( cres crn wss wcel wral cxp cin wceq wbr wrex dfss3 ssrnres cima df-ima
      cv eleq2i vex elima bitr3i ralbii 3bitr3i ) DECFGZHBTZUGIZBDJECDKLGDMATUH
      ENACOZBDJBDUGPCDEQUIUJBDUIUHECRZIUJUKUGUHECSUAAUHECBUBUCUDUEUF $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x y C $.
    $( Domain of the intersection with a cross product.  (Contributed by NM,
       17-Jan-2006.) $)
    dminxp $p |- ( dom ( C i^i ( A X. B ) ) = A <->
                 A. x e. A E. y e. B x C y ) $=
      ( cxp cin cdm wceq ccnv crn cv wbr wrex wral dfdm4 cnvin cnvxp eqtri vex
      ineq2i rneqi eqeq1i rninxp brcnv rexbii ralbii 3bitri ) ECDFZGZHZCIEJZDCF
      ZGZKZCIBLZALZULMZBDNZACOUQUPEMZBDNZACOUKUOCUKUJJZKUOUJPVBUNVBULUIJZGUNEUI
      QVCUMULCDRUASUBSUCBADCULUDUSVAACURUTBDUPUQEBTATUEUFUGUH $.
  $}

  $( Image of a relation restricted to a rectangular region.  (Contributed by
     Stefan O'Rear, 19-Feb-2015.) $)
  imainrect $p |- ( ( G i^i ( A X. B ) ) " Y ) =
      ( ( G " ( Y i^i A ) ) i^i B ) $=
    ( cxp cin cres crn cima df-res rneqi df-ima eqtri ineq1i ccnv ineq2i eqtr4i
    cvv cdm 3eqtr4ri cnvin inxp inv1 incom xpeq12i eqtr2i xpindir inass 3eqtr4i
    in32 cnveqi cnvxp dmeqi dmres df-rn ) CABEZFZDGZHUQDREZFZHZUQDICDAFZIZBFZUR
    UTUQDJKUQDLVDCVBREZFZHZBFZVAVCVGBVCCVBGZHVGCVBLVIVFCVBJKMNVFOZBGZSZUTOZSVHV
    AVKVMVFRBEZFZOVJVNOZFZVMVKVFVNUAUTVOCUSFZUPFVRAREZVNFZFZUTVOUPVTVRVTARFZRBF
    ZEUPARRBUBWBAWCBAUCWCBRFBRBUDBUCMUEUFPCUPUSUJVOVRVSFZVNFWAVFWDVNVFCUSVSFZFW
    DVEWECDARUGPCUSVSUHQNVRVSVNUHMUIUKVKVJBREZFVQVJBJVPWFVJRBULPQTUMBVJSZFWGBFV
    LVHBWGUDVJBUNVGWGBVFUONTUTUOTQUI $.

  $( The image by a cross product.  (Contributed by Thierry Arnoux,
     16-Dec-2017.) $)
  xpima1 $p |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) ) $=
    ( cin c0 wceq cxp cima cvv crn cres df-ima df-res rneqi 3eqtri xpeq1 eqtrdi
    inxp 0xp rneq rn0 syl eqtrid ) ACDZEFZABGZCHZUDBIDZGZJZEUGUFCKZJUFCIGDZJUJU
    FCLUKULUFCMNULUIABCIRNOUEUIEFZUJEFUEUIEUHGEUDEUHPUHSQUMUJEJEUIETUAQUBUC $.

  ${
    $d x A $.  $d x C $.
    $( The image by a cross product.  (Contributed by Thierry Arnoux,
       16-Dec-2017.) $)
    xpima2m $p |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B ) $=
      ( cv cin wcel wex cxp cima cvv cres df-ima df-res rneqi inxp 3eqtri rnxpm
      crn eqtrid inv1 eqtrdi ) AEBDFZGAHZBCIZDJZCKFZCUDUFUCUGIZSZUGUFUEDLZSUEDK
      IFZSUIUEDMUJUKUEDNOUKUHBCDKPOQAUCUGRTCUAUB $.
  $}

  ${
    $d A x $.  $d X x $.
    $( The image of a singleton by a cross product.  (Contributed by Thierry
       Arnoux, 14-Jan-2018.) $)
    xpimasn $p |- ( X e. A -> ( ( A X. B ) " { X } ) = B ) $=
      ( vx wcel cv csn cin wex cxp cima wceq snmg wss snssi dfss1 eleq2d exbidv
      sylib mpbird xpima2m syl ) CAEZDFZACGZHZEZDIZABJUEKBLUCUHUDUEEZDIDCAMUCUG
      UIDUCUFUEUDUCUEANUFUELCAOUEAPSQRTDABUEUAUB $.
  $}

  ${
    $d x y R $.
    $( The set of all ordered pairs in a class is the same as the double
       converse.  (Contributed by Mario Carneiro, 16-Aug-2015.) $)
    cnvcnv3 $p |- `' `' R = { <. x , y >. | x R y } $=
      ( ccnv cv wbr copab df-cnv vex brcnv opabbii eqtri ) CDZDBEZAEZMFZABGONCF
      ZABGABMHPQABNOCBIAIJKL $.

    $( Alternate definition of relation.  Exercise 2 of [TakeutiZaring] p. 25.
       (Contributed by NM, 29-Dec-1996.) $)
    dfrel2 $p |- ( Rel R <-> `' `' R = R ) $=
      ( vx vy wrel ccnv wceq relcnv cv cop wcel vex opelcnv bitri eqrelriv mpan
      releq mpbii impbii ) ADZAEZEZAFZUADZSUBTGZBCUAABHZCHZIZUAJUFUEITJUGAJUEUF
      TBKZCKZLUFUEAUIUHLMNOUBUCSUDUAAPQR $.

    $( A relation can be expressed as the set of ordered pairs in it.
       (Contributed by Mario Carneiro, 16-Aug-2015.) $)
    dfrel4v $p |- ( Rel R <-> R = { <. x , y >. | x R y } ) $=
      ( wrel ccnv wceq cv wbr copab dfrel2 eqcom cnvcnv3 eqeq2i 3bitri ) CDCEEZ
      CFCOFCAGBGCHABIZFCJOCKOPCABCLMN $.
  $}

  $( The double converse of a class strips out all elements that are not
     ordered pairs.  (Contributed by NM, 8-Dec-2003.) $)
  cnvcnv $p |- `' `' A = ( A i^i ( _V X. _V ) ) $=
    ( ccnv cvv cxp cin wceq wrel relcnv df-rel mpbi relxp dfrel2 sseqtrri cnvin
    wss dfss cnveqi inss2 mpbir eqtr3i 3eqtr2i ) ABZBZUCCCDZBZBZEZUBUEEZBZAUDEZ
    UCUFOUCUGFUCUDUFUCGUCUDOUBHUCIJUDGUFUDFCCKUDLJMUCUFPJUBUENUJBZBZUIUJUKUHAUD
    NQUJGZULUJFUMUJUDOAUDRUJISUJLJTUA $.

  $( The double converse of a class equals its restriction to the universe.
     (Contributed by NM, 8-Oct-2007.) $)
  cnvcnv2 $p |- `' `' A = ( A |` _V ) $=
    ( ccnv cvv cxp cin cres cnvcnv df-res eqtr4i ) ABBACCDEACFAGACHI $.

  $( The double converse of a class is a subclass.  Exercise 2 of
     [TakeutiZaring] p. 25.  (Contributed by NM, 23-Jul-2004.) $)
  cnvcnvss $p |- `' `' A C_ A $=
    ( ccnv cvv cxp cin cnvcnv inss1 eqsstri ) ABBACCDZEAAFAIGH $.

  $( Equality theorem for converse.  (Contributed by FL, 19-Sep-2011.) $)
  cnveqb $p |- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) ) $=
    ( wrel wa wceq ccnv cnveq wi dfrel2 eqeq2 imbitrrid eqcoms sylbi imbi2d imp
    eqeq1 impbid2 ) ACZBCZDABEZAFZBFZEZABGRSUCTHZRUAFZAESUDHZAIUFAUESUDAUEEZUCU
    EBEZHZSUBFZBEUIBIUIBUJUCUHBUJEUEUJEUAUBGBUJUEJKLMUGTUHUCAUEBPNKLMOQ $.

  $( A relation empty iff its converse is empty.  (Contributed by FL,
     19-Sep-2011.) $)
  cnveq0 $p |- ( Rel A -> ( A = (/) <-> `' A = (/) ) ) $=
    ( c0 ccnv wceq wrel wb cnv0 rel0 cnveqb mpan2 eqeq2 bibi2d imbitrrid eqcoms
    wi ax-mp ) BCZBDAEZABDZACZBDZFZOZGUCBQRUBBQDZSTQDZFZRBEUFHABIJUDUAUESBQTKLM
    NP $.

  $( Alternate definition of relation.  (Contributed by NM, 14-May-2008.) $)
  dfrel3 $p |- ( Rel R <-> ( R |` _V ) = R ) $=
    ( wrel ccnv wceq cvv cres dfrel2 cnvcnv2 eqeq1i bitri ) ABACCZADAEFZADAGKLA
    AHIJ $.

  $( The domain of a universal restriction.  (Contributed by NM,
     14-May-2008.) $)
  dmresv $p |- dom ( A |` _V ) = dom A $=
    ( cvv cres cdm cin dmres incom inv1 3eqtri ) ABCDBADZEJBEJABFBJGJHI $.

  $( The range of a universal restriction.  (Contributed by NM,
     14-May-2008.) $)
  rnresv $p |- ran ( A |` _V ) = ran A $=
    ( ccnv crn cvv cres cnvcnv2 rneqi rncnvcnv eqtr3i ) ABBZCADEZCACJKAFGAHI $.

  $( Range defined in terms of image.  (Contributed by NM, 14-May-2008.) $)
  dfrn4 $p |- ran A = ( A " _V ) $=
    ( cvv cima cres crn df-ima rnresv eqtr2i ) ABCABDEAEABFAGH $.

  ${
    $d A w y $.  $d B w y $.  $d V w y $.  $d x w y $.
    $( Distribute proper substitution through the range of a class.
       (Contributed by Alan Sare, 10-Nov-2012.) $)
    csbrng $p |- ( A e. V -> [_ A / x ]_ ran B = ran [_ A / x ]_ B ) $=
      ( vw vy wcel cv cop wex cab csb crn wsbc csbabg sbcexg exbidv bitrd dfrn3
      sbcel2g abbidv eqtrd csbeq2i 3eqtr4g ) BDGZABEHFHIZCGZEJZFKZLZUFABCLZGZEJ
      ZFKZABCMZLUKMUEUJUHABNZFKUNUHAFBDOUEUPUMFUEUPUGABNZEJUMUGEABDPUEUQULEABUF
      CDTQRUAUBABUOUIEFCSUCEFUKSUD $.
  $}

  $( The restriction of the double converse of a class.  (Contributed by NM,
     8-Apr-2007.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  rescnvcnv $p |- ( `' `' A |` B ) = ( A |` B ) $=
    ( ccnv cres cvv cin cnvcnv2 reseq1i resres wss wceq ssv sseqin2 mpbi 3eqtri
    reseq2i ) ACCZBDAEDZBDAEBFZDABDQRBAGHAEBISBABEJSBKBLBEMNPO $.

  $( The double converse of the restriction of a class.  (Contributed by NM,
     3-Jun-2007.) $)
  cnvcnvres $p |- `' `' ( A |` B ) = ( `' `' A |` B ) $=
    ( cres ccnv wrel wceq relres dfrel2 mpbi rescnvcnv eqtr4i ) ABCZDDZLADDBCLE
    MLFABGLHIABJK $.

  $( The image of the double converse of a class.  (Contributed by NM,
     8-Apr-2007.) $)
  imacnvcnv $p |- ( `' `' A " B ) = ( A " B ) $=
    ( ccnv cres crn cima rescnvcnv rneqi df-ima 3eqtr4i ) ACCZBDZEABDZEKBFABFLM
    ABGHKBIABIJ $.

  ${
    $d x y A $.
    $( The domain of a singleton is inhabited iff the singleton argument is an
       ordered pair.  (Contributed by Jim Kingdon, 15-Dec-2018.) $)
    dmsnm $p |- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } ) $=
      ( vy cvv cxp wcel cop wceq wex csn cdm elvv wbr vex eldm df-br opex exbii
      cv elsn eqcom 3bitri bitr2i bitri ) BDDEFBASZCSZGZHZCIZAIUEBJZKFZAIACBLUI
      UKAUKUEUFUJMZCIUICUEUJANZOULUHCULUGUJFUGBHUHUEUFUJPUGBUEUFUMCNQTUGBUAUBRU
      CRUD $.
  $}

  ${
    $d x A $.
    $( The range of a singleton is inhabited iff the singleton argument is an
       ordered pair.  (Contributed by Jim Kingdon, 15-Dec-2018.) $)
    rnsnm $p |- ( A e. ( _V X. _V ) <-> E. x x e. ran { A } ) $=
      ( cvv cxp wcel cv csn cdm wex crn dmsnm dmmrnm bitri ) BCCDEAFZBGZHEAINOJ
      EAIABKAAOLM $.
  $}

  $( The domain of the singleton of the empty set is empty.  (Contributed by
     NM, 30-Jan-2004.) $)
  dmsn0 $p |- dom { (/) } = (/) $=
    ( vx c0 csn cdm wceq cv wcel wn wal wex cvv cxp 0nelxp dmsnm mtbi alnex eq0
    mpbir ) BCDZBEAFSGZHAIZUATAJZHBKKLGUBKKMABNOTAPRASQR $.

  $( The converse of the singleton of the empty set is empty.  (Contributed by
     Mario Carneiro, 30-Aug-2015.) $)
  cnvsn0 $p |- `' { (/) } = (/) $=
    ( c0 csn ccnv wceq crn dfdm4 dmsn0 eqtr3i wrel wb relcnv relrn0 ax-mp mpbir
    cdm ) ABZCZADZQEZADZPOSAPFGHQIRTJPKQLMN $.

  ${
    $d x A $.
    $( The domain of a singleton is empty if the singleton's argument contains
       the empty set.  (Contributed by NM, 15-Dec-2008.) $)
    dmsn0el $p |- ( (/) e. A -> dom { A } = (/) ) $=
      ( vx c0 wcel cv csn cdm wn wal wceq wex 0nelelxp con2i dmsnm sylnib alnex
      cvv cxp sylibr eq0 ) CADZBEAFGZDZHBIZUBCJUAUCBKZHUDUAAQQRDZUEUFUAQQALMBAN
      OUCBPSBUBTS $.
  $}

  ${
    $d x A $.
    relsn2m.1 $e |- A e. _V $.
    $( A singleton is a relation iff it has an inhabited domain.  (Contributed
       by Jim Kingdon, 16-Dec-2018.) $)
    relsn2m $p |- ( Rel { A } <-> E. x x e. dom { A } ) $=
      ( csn wrel cvv cxp wcel cv cdm wex relsn dmsnm bitri ) BDZEBFFGHAIOJHAKBC
      LABMN $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x V $.
    $( The domain of a singleton of an ordered pair is the singleton of the
       first member.  (Contributed by Mario Carneiro, 26-Apr-2015.) $)
    dmsnopg $p |- ( B e. V -> dom { <. A , B >. } = { A } ) $=
      ( vx vy wcel cop csn cdm cv wceq wex vex opth1 exlimiv opeq1 opeq2 eqeq1d
      spcegv syl5 impbid2 eldm2 opex elsn exbii bitri velsn 3bitr4g eqrdv ) BCF
      ZDABGZHZIZAHZUJDJZEJZGZUKKZELZUOAKZUOUMFZUOUNFUJUSUTURUTEUOUPABDMZEMZNOUT
      UOBGZUKKZUJUSUOABPURVEEBCUPBKUQVDUKUPBUOQRSTUAVAUQULFZELUSEUOULVBUBVFUREU
      QUKUOUPVBVCUCUDUEUFDAUGUHUI $.

    $( The domain of an unordered pair of ordered pairs.  (Contributed by Mario
       Carneiro, 26-Apr-2015.) $)
    dmpropg $p |- ( ( B e. V /\ D e. W ) ->
        dom { <. A , B >. , <. C , D >. } = { A , C } ) $=
      ( wcel wa cop csn cdm cun cpr wceq dmsnopg uneq12 syl2an df-pr dmeqi dmun
      eqtri 3eqtr4g ) BEGZDFGZHABIZJZKZCDIZJZKZLZAJZCJZLZUEUHMZKZACMUCUGULNUJUM
      NUKUNNUDABEOCDFOUGULUJUMPQUPUFUILZKUKUOUQUEUHRSUFUITUAACRUB $.
  $}

  ${
    dmsnop.1 $e |- B e. _V $.
    $( The domain of a singleton of an ordered pair is the singleton of the
       first member.  (Contributed by NM, 30-Jan-2004.)  (Proof shortened by
       Andrew Salmon, 27-Aug-2011.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    dmsnop $p |- dom { <. A , B >. } = { A } $=
      ( cvv wcel cop csn cdm wceq dmsnopg ax-mp ) BDEABFGHAGICABDJK $.

    dmprop.1 $e |- D e. _V $.
    $( The domain of an unordered pair of ordered pairs.  (Contributed by NM,
       13-Sep-2011.) $)
    dmprop $p |- dom { <. A , B >. , <. C , D >. } = { A , C } $=
      ( cvv wcel cop cpr cdm wceq dmpropg mp2an ) BGHDGHABICDIJKACJLEFABCDGGMN
      $.

    dmtpop.1 $e |- F e. _V $.
    $( The domain of an unordered triple of ordered pairs.  (Contributed by NM,
       14-Sep-2011.) $)
    dmtpop $p |- dom { <. A , B >. , <. C , D >. , <. E , F >. }
                        = { A , C , E } $=
      ( cop ctp cdm cpr csn cun df-tp dmeqi dmun dmprop dmsnop uneq12i 3eqtri
      eqtr4i ) ABJZCDJZEFJZKZLZACMZENZOZACEKUHUDUEMZUFNZOZLULLZUMLZOUKUGUNUDUEU
      FPQULUMRUOUIUPUJABCDGHSEFITUAUBACEPUC $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Double converse of a singleton of an ordered pair.  (Unlike ~ cnvsn ,
       this does not need any sethood assumptions on ` A ` and ` B ` .)
       (Contributed by Mario Carneiro, 26-Apr-2015.) $)
    cnvcnvsn $p |- `' `' { <. A , B >. } = `' { <. B , A >. } $=
      ( vy vx cop csn ccnv relcnv cv wcel vex opelcnv wceq wa opth 3bitr4i opex
      ancom elsn bitri eqrelriiv ) CDABEZFZGZGZBAEZFZGZUDHUGHCIZDIZEZUEJUJUIEZU
      DJZUKUHJZUIUJUDCKZDKZLUKUCJZULUGJZUMUNUKUBMZULUFMZUQURUIAMZUJBMZNVBVANUSU
      TVAVBRUIUJABUOUPOUJUIBAUPUOOPUKUBUIUJUOUPQSULUFUJUIUPUOQSPUJUIUCUPUOLUIUJ
      UGUOUPLPTUA $.

    $( The domain of the singleton of the singleton of a singleton.
       (Contributed by Jim Kingdon, 16-Dec-2018.) $)
    dmsnsnsng $p |- ( A e. _V -> dom { { { A } } } = { A } ) $=
      ( vx cop csn cdm wceq cvv vex opid sneq sneqd eqtrid dmeqd eqeq12d dmsnop
      cv vtoclg ) BPZRCZDZEZRDZFADZDZDZEZUCFBAGRAFZUAUFUBUCUGTUEUGSUDUGSUBDUDRB
      HZIUGUBUCRAJZKLKMUINRRUHOQ $.
  $}

  $( The range of a singleton of an ordered pair is the singleton of the second
     member.  (Contributed by NM, 24-Jul-2004.)  (Revised by Mario Carneiro,
     30-Apr-2015.) $)
  rnsnopg $p |- ( A e. V -> ran { <. A , B >. } = { B } ) $=
    ( wcel cop csn crn cdm ccnv df-rn dfdm4 cnvcnvsn dmeqi 3eqtri eqtr4i eqtrid
    dmsnopg ) ACDABEFZGZBAEFZHZBFSRIZHZUARJUATIZGUDIZHUCTKUDJUEUBBALMNOBACQP $.

  $( The range of a pair of ordered pairs is the pair of second members.
     (Contributed by Thierry Arnoux, 3-Jan-2017.) $)
  rnpropg $p |- ( ( A e. V /\ B e. W )
    -> ran { <. A , C >. , <. B , D >. } = { C , D } ) $=
    ( wcel cop cpr crn csn cun df-pr rneqi wceq rnsnopg adantr adantl uneq12d
    wa rnun 3eqtr4g eqtrid ) AEGZBFGZTZACHZBDHZIZJUGKZUHKZLZJZCDIZUIULUGUHMNUFU
    JJZUKJZLCKZDKZLUMUNUFUOUQUPURUDUOUQOUEACEPQUEUPUROUDBDFPRSUJUKUACDMUBUC $.

  ${
    cnvsn.1 $e |- A e. _V $.
    $( The range of a singleton of an ordered pair is the singleton of the
       second member.  (Contributed by NM, 24-Jul-2004.)  (Revised by Mario
       Carneiro, 26-Apr-2015.) $)
    rnsnop $p |- ran { <. A , B >. } = { B } $=
      ( cvv wcel cop csn crn wceq rnsnopg ax-mp ) ADEABFGHBGICABDJK $.

    cnvsn.2 $e |- B e. _V $.
    $( Extract the first member of an ordered pair.  (See ~ op2nda to extract
       the second member and ~ op1stb for an alternate version.)  (Contributed
       by Raph Levien, 4-Dec-2003.) $)
    op1sta $p |- U. dom { <. A , B >. } = A $=
      ( cop csn cdm cuni dmsnop unieqi unisn eqtri ) ABEFGZHAFZHAMNABDIJACKL $.

    $( Converse of a singleton of an ordered pair.  (Contributed by NM,
       11-May-1998.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    cnvsn $p |- `' { <. A , B >. } = { <. B , A >. } $=
      ( cop csn ccnv cnvcnvsn wrel wceq relsnop dfrel2 mpbi eqtr3i ) BAEFZGGZAB
      EFGOBAHOIPOJBADCKOLMN $.

    $( Extract the second member of an ordered pair.  Theorem 5.12(ii) of
       [Monk1] p. 52.  (See ~ op1stb to extract the first member and ~ op2nda
       for an alternate version.)  (Contributed by NM, 25-Nov-2003.) $)
    op2ndb $p |- |^| |^| |^| `' { <. A , B >. } = B $=
      ( cop csn ccnv cint cnvsn inteqi opex intsn eqtri op1stb ) ABEFGZHZHZHBAE
      ZHZHBQSPRPRFZHROTABCDIJRBADCKLMJJBADCNM $.

    $( Extract the second member of an ordered pair.  (See ~ op1sta to extract
       the first member and ~ op2ndb for an alternate version.)  (Contributed
       by NM, 17-Feb-2004.)  (Proof shortened by Andrew Salmon,
       27-Aug-2011.) $)
    op2nda $p |- U. ran { <. A , B >. } = B $=
      ( cop csn crn cuni rnsnop unieqi unisn eqtri ) ABEFGZHBFZHBMNABCIJBDKL $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Converse of a singleton of an ordered pair.  (Contributed by NM,
       23-Jan-2015.) $)
    cnvsng $p |- ( ( A e. V /\ B e. W ) ->
                                      `' { <. A , B >. } = { <. B , A >. } ) $=
      ( vx vy cv cop csn ccnv wceq opeq1 sneqd cnveqd opeq2 eqeq12d vex vtocl2g
      cnvsn ) EGZFGZHZIZJZUATHZIZKAUAHZIZJZUAAHZIZKABHZIZJZBAHZIZKEFABCDTAKZUDU
      IUFUKUQUCUHUQUBUGTAUALMNUQUEUJTAUAOMPUABKZUIUNUKUPURUHUMURUGULUABAOMNURUJ
      UOUABALMPTUAEQFQSR $.

    $( Swap the members of an ordered pair.  (Contributed by Jim Kingdon,
       16-Dec-2018.) $)
    opswapg $p |- ( ( A e. V /\ B e. W ) ->
                                       U. `' { <. A , B >. } = <. B , A >. ) $=
      ( wcel cop csn ccnv cuni cnvsng unieqd cvv wceq elex opexg syl2anr unisng
      wa syl eqtrd ) ACEZBDEZRZABFGHZIBAFZGZIZUEUCUDUFABCDJKUCUELEZUGUEMUBBLEAL
      EUHUABDNACNBALLOPUELQST $.

    $( Membership in a cross product.  This version requires no quantifiers or
       dummy variables.  See also ~ elxp5 .  (Contributed by NM,
       17-Feb-2004.) $)
    elxp4 $p |- ( A e. ( B X. C ) <-> ( A = <. U. dom { A } , U. ran { A } >.
                           /\ ( U. dom { A } e. B /\ U. ran { A } e. C ) ) ) $=
      ( vx vy wcel cvv csn cdm cuni crn cop wceq wa elex eleq1 wex unieqd syl
      wb cxp anim12i opexg adantl adantr mpbird sylan2 cv elxp a1i rneqd op2nda
      sneq vex eqtr2di anbi1i anass bitri exbii snexg rnexg uniexg opeq2 eqeq2d
      pm4.71ri anbi2d anbi12d ceqsexgv bitrid dmeqd dmsnopg unisn eqtrdi eqtr2d
      ex pm4.71rd anbi1d 3bitrd exbidv dmexg opeq1 pm5.21nii ) ABCUAZFZAGFZAAHZ
      IZJZWFKZJZLZMZWHBFZWJCFZNZNZAWCOWOWLWHGFZWJGFZNZWEWMWQWNWRWHBOWJCOUBWLWSN
      WEWKGFZWSWTWLWHWJGGUCUDWLWEWTTWSAWKGPUEUFUGWEWDADUHZEUHZLZMZXABFZXBCFZNZN
      ZEQZDQZXAWHMZAXAWJLZMZXEWNNZNZNZDQZWPWDXJTWEDEABCUIUJWEXIXPDWEXIXOXKXMNZX
      NNZXPXIXBWJMZXHNZEQZWEXOXHYAEXHXTXDNZXGNYAXDYCXGXDXTXDWJXCHZKZJXBXDWIYEXD
      WFYDAXCUMUKRXAXBDUNZEUNULUOVEUPXTXDXGUQURUSWEWRYBXOTWEWIGFZWRWEWFGFZYGAGU
      TZWFGVASWIGVBSZXHXOEWJGXTXDXMXGXNXTXCXLAXBWJXAVCVDXTXFWNXEXBWJCPVFVGVHSVI
      WEXMXRXNWEXMXKWEXMXKWEXMNWHXLHZIZJZXAXMWHYMMWEXMWGYLXMWFYKAXLUMVJRUDWEYMX
      AMXMWEYMXAHZJXAWEYLYNWEWRYLYNMYJXAWJGVKSRXAYFVLVMUEVNVOVPVQXSXPTWEXKXMXNU
      QUJVRVSWEWQXQWPTWEWGGFZWQWEYHYOYIWFGVTSWGGVBSXOWPDWHGXKXMWLXNWOXKXLWKAXAW
      HWJWAVDXKXEWMWNXAWHBPVQVGVHSVRWB $.

    $( Membership in a cross product requiring no quantifiers or dummy
       variables.  Provides a slightly shorter version of ~ elxp4 when the
       double intersection does not create class existence problems (caused by
       ~ int0 ).  (Contributed by NM, 1-Aug-2004.) $)
    elxp5 $p |- ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >.
                              /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) ) $=
      ( vx vy wcel cvv cint csn crn cuni cop wceq wa elex eleq1 adantr wex syl
      wb cxp anim12i opexg adantl mpbird sylan2 cv elxp rneqd unieqd vex op2nda
      eqtr2di pm4.71ri anbi1i anass bitri exbii snexg rnexg uniexg opeq2 eqeq2d
      sneq anbi2d anbi12d ceqsexgv bitrid inteq inteqd op1stbg sylancr pm4.71rd
      eqtr2d anbi1d a1i 3bitrd exbidv eqvisset exlimiv ad2antrl opeq1 pm5.21nii
      ex bitrdi ) ABCUAZFZAGFZAAHZHZAIZJZKZLZMZWJBFZWMCFZNZNZAWFOWRWOWJGFZWMGFZ
      NZWHWPWTWQXAWJBOZWMCOUBWOXBNWHWNGFZXBXDWOWJWMGGUCUDWOWHXDTXBAWNGPQUEUFWHW
      GDUGZWJMZAXEWMLZMZXEBFZWQNZNZNZDRZWSWGAXEEUGZLZMZXIXNCFZNZNZERZDRWHXMDEAB
      CUHWHXTXLDWHXTXKXFXHNZXJNZXLXTXNWMMZXSNZERZWHXKXSYDEXSYCXPNZXRNYDXPYFXRXP
      YCXPWMXOIZJZKXNXPWLYHXPWKYGAXOVDUIUJXEXNDUKZEUKULUMUNUOYCXPXRUPUQURWHXAYE
      XKTWHWLGFZXAWHWKGFYJAGUSWKGUTSWLGVASZXSXKEWMGYCXPXHXRXJYCXOXGAXNWMXEVBVCY
      CXQWQXIXNWMCPVEVFVGSVHWHXHYAXJWHXHXFWHXHXFWHXHNWJXGHZHZXEXHWJYMMWHXHWIYLA
      XGVIVJUDWHYMXEMZXHWHXEGFXAYNYIYKXEWMGGVKVLQVNWDVMVOYBXLTWHXFXHXJUPVPVQVRV
      HXMWTWSXLWTDXFWTXKDWJVSQVTWPWTWOWQXCWAXKWSDWJGXFXHWOXJWRXFXGWNAXEWJWMWBVC
      XFXIWPWQXEWJBPVOVFVGWCWEWC $.
  $}

  ${
    $d s t A $.  $d s t B $.  $d s t F $.
    $( An image under the converse of a restriction.  (Contributed by Jeff
       Hankins, 12-Jul-2009.) $)
    cnvresima $p |- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A ) $=
      ( vt vs cres ccnv cima cin cv cop wa wex vex elima3 anbi1i opelcnv bitr4i
      wcel bitri elin opelres anbi2i anass exbii 19.41v 3bitr4ri eqriv ) DCAFZG
      ZBHZCGZBHZAIZDJZUKSEJZBSZUPUOKZUJSZLZEMZUOUNSZEUOUJBDNZOUOUMSZUOASZLUQURU
      LSZLZEMZVELZVBVAVDVHVEEUOULBVCOPUOUMAUAVAVGVELZEMVIUTVJEUTUQVFVELZLVJUSVK
      UQUSUOUPKZUISZVKUPUOUIENZVCQVMVLCSZVELVKUOUPCAVNUBVFVOVEUPUOCVNVCQPRTUCUQ
      VFVEUDRUEVGVEEUFTUGTUH $.
  $}

  $( A class restricted to its domain equals its double converse.  (Contributed
     by NM, 8-Apr-2007.) $)
  resdm2 $p |- ( A |` dom A ) = `' `' A $=
    ( ccnv cdm cres rescnvcnv wrel relcnv resdm ax-mp dmcnvcnv reseq2i 3eqtr3ri
    wceq ) ABZBZOCZDZAPDOAACZDAPEOFQOMNGOHIPRAAJKL $.

  $( Restriction to the domain of a restriction.  (Contributed by NM,
     8-Apr-2007.) $)
  resdmres $p |- ( A |` dom ( A |` B ) ) = ( A |` B ) $=
    ( cres cdm ccnv cvv cxp in12 df-res resdm2 eqtr3i ineq2i incom 3eqtri dmres
    cin xpeq1i xpindir eqtri 3eqtr4i rescnvcnv ) AABCZDZCZAEEZBCZUBABFGZADZFGZP
    ZPZUEUGPZUDUFUKUGAUIPZPUGUEPULAUGUIHUMUEUGAUHCUMUEAUHIAJKLUGUEMNUDAUCFGZPUK
    AUCIUNUJAUNBUHPZFGUJUCUOFABOQBUHFRSLSUEBITABUAS $.

  $( The image of the domain of a restriction.  (Contributed by NM,
     8-Apr-2007.) $)
  imadmres $p |- ( A " dom ( A |` B ) ) = ( A " B ) $=
    ( cres cdm crn cima resdmres rneqi df-ima 3eqtr4i ) AABCZDZCZEKEALFABFMKABG
    HALIABIJ $.

  ${
    $d x y C $.  $d y A $.  $d y B $.  $d y F $.  $d x V $.
    dmmpo.1 $e |- F = ( x e. A |-> B ) $.
    $( The preimage of a function in maps-to notation.  (Contributed by Stefan
       O'Rear, 25-Jan-2015.) $)
    mptpreima $p |- ( `' F " C ) = { x e. A | B e. C } $=
      ( vy ccnv cima cv wcel wceq wa copab crab eqtri crn wex cab bitri cnvopab
      cmpt df-mpt cnveqi imaeq1i df-ima resopab rneqi ancom anass exbii df-clel
      cres 19.42v bicomi anbi2i abbii rnopab df-rab 3eqtr4i ) EHZDIAJBKZGJZCLZM
      ZGANZDIZCDKZABOZVAVFDVAVEAGNZHVFEVJEABCUBVJFAGBCUCPUDVEAGUAPUEVGVFDUMZQZV
      IVFDUFVLVCDKZVEMZGANZQZVIVKVOVEGADUGUHVNGRZASVBVHMZASVPVIVQVRAVQVBVDVMMZM
      ZGRZVRVNVTGVNVEVMMVTVMVEUIVBVDVMUJTUKWAVBVSGRZMVRVBVSGUNWBVHVBVHWBGCDULUO
      UPTTUQVNGAURVHABUSUTPPP $.

    $( Converse singleton image of a function defined by maps-to.  (Contributed
       by Stefan O'Rear, 25-Jan-2015.) $)
    mptiniseg $p |- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } ) $=
      ( wcel ccnv csn cima crab wceq mptpreima elsn2g rabbidv eqtrid ) DFHZEIDJ
      ZKCSHZABLCDMZABLABCSEGNRTUAABCDFOPQ $.

    $( The domain of the mapping operation in general.  (Contributed by NM,
       16-May-1995.)  (Revised by Mario Carneiro, 22-Mar-2015.) $)
    dmmpt $p |- dom F = { x e. A | B e. _V } $=
      ( cdm ccnv crn cvv cima wcel crab dfdm4 dfrn4 mptpreima 3eqtri ) DFDGZHQI
      JCIKABLDMQNABCIDEOP $.

    $d x A $.
    $( The domain of a mapping is a subset of its base class.  (Contributed by
       Scott Fenton, 17-Jun-2013.) $)
    dmmptss $p |- dom F C_ A $=
      ( cdm cvv wcel crab dmmpt ssrab2 eqsstri ) DFCGHZABIBABCDEJMABKL $.
  $}

  ${
    $d A x $.
    $( The domain of the mapping operation is the stated domain, if the
       function value is always a set.  (Contributed by Mario Carneiro,
       9-Feb-2013.)  (Revised by Mario Carneiro, 14-Sep-2013.) $)
    dmmptg $p |- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A ) $=
      ( wcel wral cmpt cdm cvv crab eqid wceq elex ralimi rabid2 sylibr eqtr4id
      dmmpt ) CDEZABFZABCGZHCIEZABJZBABCUAUAKRTUBABFBUCLSUBABCDMNUBABOPQ $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z C $.
    $( A composition is a relation.  Exercise 24 of [TakeutiZaring] p. 25.
       (Contributed by NM, 26-Jan-1997.) $)
    relco $p |- Rel ( A o. B ) $=
      ( vx vz vy cv wbr wa wex ccom df-co relopabi ) CFDFZBGMEFAGHDICEABJCEDABK
      L $.

    $( Alternate definition of a class composition, using only one bound
       variable.  (Contributed by NM, 19-Dec-2008.) $)
    dfco2 $p |- ( A o. B )
                = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) $=
      ( vy vz ccom cvv ccnv cv csn cima cxp wrel wcel cop wex vex elimasn bitri
      wa ciun relco reliun relxp a1i mprgbir wb opelco2g mp2an wrex rexv opelxp
      eliun opelcnv anbi12i exbii 3bitrri eqrelriiv ) DEBCFZAGCHZAIZJZKZBVBKZLZ
      UAZBCUBVFMVEMZAGAGVEUCVGVAGNVCVDUDUEUFDIZEIZOZUSNZVHVAOCNZVAVIOBNZTZAPZVJ
      VFNZVHGNVIGNVKVOUGDQZEQZAVHVIBCGGUHUIVPVJVENZAGUJVSAPVOAVJGVEUMVSAUKVSVNA
      VSVHVCNZVIVDNZTVNVHVIVCVDULVTVLWAVMVTVAVHOUTNVLUTVAVHAQZVQRVAVHCWBVQUNSBV
      AVIWBVRRUOSUPUQSUR $.

    $( Generalization of ~ dfco2 , where ` C ` can have any value between
       ` dom A i^i ran B ` and ` _V ` .  (Contributed by NM, 21-Dec-2008.)
       (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    dfco2a $p |- ( ( dom A i^i ran B ) C_ C -> ( A o. B )
         = U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) ) $=
      ( vy vz vw cvv cv cima ciun wcel wrex wex wa cop vex sylbi 3bitr4g eliun
      cdm crn cin wss ccom ccnv csn cxp dfco2 wceq wbr wb eliniseg ax-mp brelrn
      elimasn opeldm anim12ci adantl exlimivv elxp elin 3imtr4i pm4.71rd exbidv
      ssel syl5 rexv df-rex eqrdv eqtrid ) BUAZCUBZUCZDUDZBCUEAHCUFAIZUGZJZBVQJ
      ZUHZKZADVTKZABCUIVOEWAWBVOEIZVTLZAHMZWDADMZWCWALWCWBLVOWDANVPDLZWDOZANWEW
      FVOWDWHAVOWDWGWDVPVNLZVOWGWCFIZGIZPUJZWJVRLZWKVSLZOZOZGNFNVPVLLZVPVMLZOZW
      DWIWPWSFGWOWSWLWMWRWNWQWMWJVPCUKZWRVPHLWMWTULAQZCVPWJHFQZUMUNWJVPCXBXAUOR
      WNVPWKPBLWQBVPWKXAGQZUPVPWKBXAXCUQRURUSUTFGWCVRVSVAVPVLVMVBVCVNDVPVFVGVDV
      EWDAVHWDADVISAWCHVTTAWCDVTTSVJVK $.

    $( Class composition distributes over union.  (Contributed by NM,
       21-Dec-2008.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    coundi $p |- ( A o. ( B u. C ) ) = ( ( A o. B ) u. ( A o. C ) ) $=
      ( vx vz vy cv wbr wa wex copab cun ccom wo unopab brun anbi1i andir bitri
      df-co exbii 19.43 bitr2i opabbii eqtri uneq12i 3eqtr4ri ) DGZEGZBHZUIFGAH
      ZIZEJZDFKZUHUICHZUKIZEJZDFKZLZUHUIBCLZHZUKIZEJZDFKZABMZACMZLAUTMUSUMUQNZD
      FKVDUMUQDFOVGVCDFVCULUPNZEJVGVBVHEVBUJUONZUKIVHVAVIUKUHUIBCPQUJUOUKRSUAUL
      UPEUBUCUDUEVEUNVFURDFEABTDFEACTUFDFEAUTTUG $.

    $( Class composition distributes over union.  (Contributed by NM,
       21-Dec-2008.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    coundir $p |- ( ( A u. B ) o. C ) = ( ( A o. C ) u. ( B o. C ) ) $=
      ( vx vy vz cv wbr wa wex copab cun ccom wo unopab brun anbi2i bitri df-co
      andi exbii 19.43 bitr2i opabbii eqtri uneq12i 3eqtr4ri ) DGEGZCHZUHFGZAHZ
      IZEJZDFKZUIUHUJBHZIZEJZDFKZLZUIUHUJABLZHZIZEJZDFKZACMZBCMZLUTCMUSUMUQNZDF
      KVDUMUQDFOVGVCDFVCULUPNZEJVGVBVHEVBUIUKUONZIVHVAVIUIUHUJABPQUIUKUOTRUAULU
      PEUBUCUDUEVEUNVFURDFEACSDFEBCSUFDFEUTCSUG $.

    $( Restricted first member of a class composition.  (Contributed by NM,
       12-Oct-2004.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    cores $p |- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) ) $=
      ( vz vy vx crn wss cv wbr cres wa wex copab ccom wcel wb vex brelrn df-co
      ssel brres rbaib syl56 pm5.32d exbidv opabbidv 3eqtr4g ) BGZCHZDIZEIZBJZU
      LFIZACKZJZLZEMZDFNUMULUNAJZLZEMZDFNUOBOABOUJURVADFUJUQUTEUJUMUPUSUMULUIPU
      JULCPZUPUSQUKULBDRERSUICULUAUPUSVBULUNACFRUBUCUDUEUFUGDFEUOBTDFEABTUH $.

    $( Associative law for the restriction of a composition.  (Contributed by
       NM, 12-Dec-2006.) $)
    resco $p |- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) ) $=
      ( vx vy vz ccom cres relres relco cv wbr wcel wex vex anbi1i 19.41v brres
      wa brco an32 bitr4i exbii 3bitr2i 3bitr4i eqbrriv ) DEABGZCHZABCHZGZUGCIA
      UIJDKZEKZUGLZUKCMZSZUKFKZUILZUPULALZSZFNZUKULUHLUKULUJLUOUKUPBLZURSZFNZUN
      SVBUNSZFNUTUMVCUNFUKULABDOZEOZTPVBUNFQVDUSFVDVAUNSZURSUSVAURUNUAUQVGURUKU
      PBCFORPUBUCUDUKULUGCVFRFUKULAUIVEVFTUEUF $.

    $( Image of the composition of two classes.  (Contributed by Jason
       Orendorff, 12-Dec-2006.) $)
    imaco $p |- ( ( A o. B ) " C ) = ( A " ( B " C ) ) $=
      ( vx vy vz ccom cima cv wbr wrex wcel wa wex df-rex vex elima exbii bitri
      rexcom4 r19.41v brco rexbii anbi1i 3bitr4i 3bitr4ri eqriv ) DABGZCHZABCHZ
      HZEIZDIZAJZEUJKULUJLZUNMZENZUMUKLUMUILZUNEUJOEUMAUJDPZQFIZULBJZUNMZENZFCK
      ZVAFCKZUNMZENZURUQVDVBFCKZENVGVBFECTVHVFEVAUNFCUARSURUTUMUHJZFCKVDFUMUHCU
      SQVIVCFCEUTUMABFPUSUBUCSUPVFEUOVEUNFULBCEPQUDRUEUFUG $.

    $( The range of the composition of two classes.  (Contributed by NM,
       12-Dec-2006.) $)
    rnco $p |- ran ( A o. B ) = ran ( A |` ran B ) $=
      ( vy vx vz ccom crn cres cv wbr wex wcel wa vex exbii excom ancom 3bitr4i
      brco elrn 19.41v anbi2i brres bitr4i 3bitri eqriv ) CABFZGZABGZHZGZDIZCIZ
      UGJZDKZEIZUMUJJZEKZUMUHLUMUKLUOULUPBJZUPUMAJZMZEKZDKVADKZEKURUNVBDEULUMAB
      DNCNZSOVADEPVCUQEVCUTUPUILZMZUQUSDKZUTMUTVGMVCVFVGUTQUSUTDUAVEVGUTDUPBENT
      UBRUPUMAUIVDUCUDOUEDUMUGVDTEUMUJVDTRUF $.
  $}

  $( The range of the composition of two classes.  (Contributed by NM,
     27-Mar-2008.) $)
  rnco2 $p |- ran ( A o. B ) = ( A " ran B ) $=
    ( ccom crn cres cima rnco df-ima eqtr4i ) ABCDABDZEDAJFABGAJHI $.

  $( The domain of a composition.  Exercise 27 of [Enderton] p. 53.
     (Contributed by NM, 4-Feb-2004.) $)
  dmco $p |- dom ( A o. B ) = ( `' B " dom A ) $=
    ( ccom cdm ccnv crn cima dfdm4 cnvco rneqi rnco2 imaeq2i eqtr4i 3eqtri ) AB
    CZDOEZFBEZAEZCZFZQADZGZOHPSABIJTQRFZGUBQRKUAUCQAHLMN $.

  ${
    $d w x y z A $.  $d w y z B $.  $d w y z C $.
    $( Composition with an indexed union.  (Contributed by NM, 21-Dec-2008.) $)
    coiun $p |- ( A o. U_ x e. C B ) = U_ x e. C ( A o. B ) $=
      ( vy vz vw ciun ccom relco wrel cv wcel wbr wa wex wrex cop eliun df-br
      reliun a1i mprgbir rexbii 3bitr4i anbi1i r19.41v bitr4i exbii rexcom4 vex
      opelco bitri eqrelriiv ) EFBADCHZIZADBCIZHZBUOJURKUQKZADADUQUAUSALDMBCJUB
      UCELZGLZUONZVAFLZBNZOZGPZUTVACNZVDOZGPZADQZUTVCRZUPMVKURMZVFVHADQZGPVJVEV
      MGVEVGADQZVDOVMVBVNVDUTVARZUOMVOCMZADQVBVNAVODCSUTVAUOTVGVPADUTVACTUDUEUF
      VGVDADUGUHUIVHAGDUJUHGUTVCBUOEUKZFUKZULVLVKUQMZADQVJAVKDUQSVSVIADGUTVCBCV
      QVRULUDUMUEUN $.
  $}

  $( A composition is not affected by a double converse of its first argument.
     (Contributed by NM, 8-Oct-2007.) $)
  cocnvcnv1 $p |- ( `' `' A o. B ) = ( A o. B ) $=
    ( ccnv ccom cvv cres cnvcnv2 coeq1i crn wss wceq ssv cores ax-mp eqtri ) AC
    CZBDAEFZBDZABDZPQBAGHBIZEJRSKTLABEMNO $.

  $( A composition is not affected by a double converse of its second argument.
     (Contributed by NM, 8-Oct-2007.) $)
  cocnvcnv2 $p |- ( A o. `' `' B ) = ( A o. B ) $=
    ( ccnv ccom cres cnvcnv2 coeq2i resco wrel wceq relco dfrel3 mpbi 3eqtr2i
    cvv ) ABCCZDABOEZDABDZOEZRPQABFGABOHRISRJABKRLMN $.

  $( Absorption of a reverse (preimage) restriction of the second member of a
     class composition.  (Contributed by NM, 11-Dec-2006.) $)
  cores2 $p |- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) ) $=
    ( cdm wss ccnv cres ccom wceq dfdm4 sseq1i cores sylbi cnvco cocnvcnv1 wrel
    crn relco dfrel2 mpbi eqtri 3eqtr4g cnveqd 3eqtr3g ) ADZCEZABFZCGZFZHZFZFZA
    BHZFZFZUJUMUFUKUNUFUHAFZHZUGUPHZUKUNUFUPQZCEUQURIUEUSCAJKUGUPCLMUKUIFUPHUQA
    UINUHUPOUAABNUBUCUJPULUJIAUIRUJSTUMPUOUMIABRUMSTUD $.

  ${
    $d x y z A $.
    $( Composition with the empty set.  Theorem 20 of [Suppes] p. 63.
       (Contributed by NM, 24-Apr-2004.) $)
    co02 $p |- ( A o. (/) ) = (/) $=
      ( vx vy vz c0 ccom relco rel0 cv cop wcel wbr wa noel df-br mtbir intnanr
      wex nex vex opelco 2false eqrelriiv ) BCAEFZEAEGHBIZCIZJZUDKZUGEKUHUEDIZE
      LZUIUFALZMZDRULDUJUKUJUEUIJZEKUMNUEUIEOPQSDUEUFAEBTCTUAPUGNUBUC $.

    $( Composition with the empty set.  (Contributed by NM, 24-Apr-2004.) $)
    co01 $p |- ( (/) o. A ) = (/) $=
      ( c0 ccnv ccom cnv0 cnvco coeq2i co02 3eqtri eqtr4i cnveqi wrel wceq rel0
      dfrel2 mpbi relco 3eqtr3ri ) BCZCZBADZCZCZBUASUBSBUBEUBACZSDUDBDBBAFSBUDE
      GUDHIJKBLTBMNBOPUALUCUAMBAQUAOPR $.

    $( Composition with the identity relation.  Part of Theorem 3.7(i) of
       [Monk1] p. 36.  (Contributed by NM, 22-Apr-2004.) $)
    coi1 $p |- ( Rel A -> ( A o. _I ) = A ) $=
      ( vx vy vz cid ccom wrel wceq relco cv cop wcel wbr wex vex opelco equcom
      wa ideq bitri anbi1i exbii breq1 ceqsexv df-br eqrelriv mpan ) AEFZGAGUHA
      HAEIBCUHABJZCJZKZUHLZUIUJAMZUKALULUIDJZEMZUNUJAMZRZDNZUMDUIUJAEBOZCOPURUN
      UIHZUPRZDNUMUQVADUOUTUPUOUIUNHUTUIUNDOSBDQTUAUBUPUMDUIUSUNUIUJAUCUDTTUIUJ
      AUETUFUG $.

    $( Composition with the identity relation.  Part of Theorem 3.7(i) of
       [Monk1] p. 36.  (Contributed by NM, 22-Apr-2004.) $)
    coi2 $p |- ( Rel A -> ( _I o. A ) = A ) $=
      ( wrel ccnv ccom cnvco wceq relcnv ax-mp cnveqi eqtr3i dfrel2 coeq2 coeq1
      cid coi1 cnvi sylan9eq mpan2 sylbi biimpi 3eqtr3a ) ABZNCZACZCZDZUENADZAU
      DNDZCUFUEUDNEUHUDUDBUHUDFAGUDOHIJUBUEAFZUFUGFZAKZUIUCNFZUJPUIULUFUCADUGUE
      AUCLUCNAMQRSUBUIUKTUA $.
  $}

  $( Composition with a restricted identity relation.  (Contributed by FL,
     19-Jun-2011.)  (Revised by Stefan O'Rear, 7-Mar-2015.) $)
  coires1 $p |- ( A o. ( _I |` B ) ) = ( A |` B ) $=
    ( ccnv cres ccom cocnvcnv1 wrel wceq relcnv coi1 ax-mp eqtr3i reseq1i resco
    cid rescnvcnv ) ACZCZBDZAOBDEZABDAOEZBDSTUARBROEZUARAOFRGUBRHQIRJKLMAOBNLAB
    PL $.

  ${
    $d x y z w A $.  $d x y z w B $.  $d x y z w C $.
    $( Associative law for class composition.  Theorem 27 of [Suppes] p. 64.
       Also Exercise 21 of [Enderton] p. 53.  Interestingly, this law holds for
       any classes whatsoever, not just functions or even relations.
       (Contributed by NM, 27-Jan-1997.) $)
    coass $p |- ( ( A o. B ) o. C ) = ( A o. ( B o. C ) ) $=
      ( vx vy vz vw ccom relco cv wbr wa wex cop wcel brco exbii opelco 3bitr4i
      vex excom anass 2exbii bitr4i anbi2i exdistr anbi1i 19.41v eqrelriiv ) DE
      ABHZCHZABCHZHZUJCIAULIDJZFJZCKZUOGJZBKZUQEJZAKZLZLZGMFMZUPURLZUTLZFMZGMZU
      NUSNZUKOZVHUMOZVCVBFMGMVGVBFGUAVEVBGFUPURUTUBUCUDUPUOUSUJKZLZFMUPVAGMZLZF
      MVIVCVLVNFVKVMUPGUOUSABFTETZPUEQFUNUSUJCDTZVORUPVAFGUFSUNUQULKZUTLZGMVDFM
      ZUTLZGMVJVGVRVTGVQVSUTFUNUQBCVPGTPUGQGUNUSAULVPVORVFVTGVDUTFUHQSSUI $.
  $}

  $( A relation is transitive iff its converse is transitive.  (Contributed by
     FL, 19-Sep-2011.) $)
  relcnvtr $p |- ( Rel R ->
     ( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) ) $=
    ( wrel ccom wss ccnv cnvco cnvss eqsstrrid wceq wi sseq1 dfrel2 coeq1 coeq2
    eqtrd id sseq12d biimpd sylbi com12 biimtrdi mpsyl impbid2 ) ABZAACZADZAEZU
    GCZUGDZUFUHUEEUGAAFUEAGHUIUDUFUHEZUGEZUKCZIZUIUJUKDZUDUFJZUGUGFUHUGGUMUNULU
    KDZUOUJULUKKUDUPUFUDUKAIZUPUFJALUQUPUFUQULUEUKAUQULAUKCUEUKAUKMUKAANOUQPQRS
    TUAUBTUC $.

  ${
    $d x y A $.
    $( A relation is included in the cross product of its domain and range.
       Exercise 4.12(t) of [Mendelson] p. 235.  (Contributed by NM,
       3-Aug-1994.) $)
    relssdmrn $p |- ( Rel A -> A C_ ( dom A X. ran A ) ) $=
      ( vx vy wrel cdm crn cxp id cv cop wi wex 19.8a wa opelxp vex eldm2 elrn2
      wcel anbi12i bitri sylanbrc a1i relssdv ) ADZBCAAEZAFZGZUEHBIZCIZJZASZUKU
      HSZKUEULULCLZULBLZUMULCMULBMUMUIUFSZUJUGSZNUNUONUIUJUFUGOUPUNUQUOCUIABPQB
      UJACPRTUAUBUCUD $.
  $}

  $( The converse is a subset of the cartesian product of range and domain.
     (Contributed by Mario Carneiro, 2-Jan-2017.) $)
  cnvssrndm $p |- `' A C_ ( ran A X. dom A ) $=
    ( ccnv cdm crn cxp wrel relcnv relssdmrn ax-mp df-rn dfdm4 xpeq12i sseqtrri
    wss ) ABZOCZODZEZADZACZEOFORNAGOHISPTQAJAKLM $.

  $( Composition as a subset of the cross product of factors.  (Contributed by
     Mario Carneiro, 12-Jan-2017.) $)
  cossxp $p |- ( A o. B ) C_ ( dom B X. ran A ) $=
    ( ccom cdm crn cxp wrel wss relco relssdmrn ax-mp dmcoss rncoss mp2an sstri
    xpss12 ) ABCZQDZQEZFZBDZAEZFZQGQTHABIQJKRUAHSUBHTUCHABLABMRUASUBPNO $.

  ${
    cossxp2.r $e |- ( ph -> R C_ ( A X. B ) ) $.
    cossxp2.s $e |- ( ph -> S C_ ( B X. C ) ) $.
    $( The composition of two relations is a relation, with bounds on its
       domain and codomain.  (Contributed by BJ, 10-Jul-2022.) $)
    cossxp2 $p |- ( ph -> ( S o. R ) C_ ( A X. C ) ) $=
      ( ccom cdm crn cxp cossxp wss dmxpss2 syl rnxpss2 xpss12 syl2anc sstrid )
      AFEIEJZFKZLZBDLZFEMAUABNZUBDNZUCUDNAEBCLNUEGBCEOPAFCDLNUFHCDFQPUABUBDRST
      $.
  $}

  $( Restricting a relation and a converse relation when they are composed
     together.  (Contributed by BJ, 10-Jul-2022.) $)
  cocnvres $p |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) ) $=
    ( cdm cres ccnv ccom wss wceq resss dmss cores2 rescnvcnv cnveqi coeq2i crn
    ax-mp dfdm4 eqimss2i cores 3eqtr3ri ) BACZDZAEZEBCZDZEZFZUBUCFZUBAUDDZEZFBU
    CFZUBCUDGZUGUHHUBBGULBUAIUBBJPUBUCUDKPUFUJUBUEUIAUDLMNUCOZUAGUHUKHUAUMAQRBU
    CUASPT $.

  $( Upper bound for the composed of a relation and an inverse relation.
     (Contributed by BJ, 10-Jul-2022.) $)
  cocnvss $p
            |- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) ) $=
  ( ccnv ccom cdm cres crn cxp cocnvres cossxp df-rn eqimss2i ssid xpss12 mp2an
  wss sstri eqsstri ) BACDBAEFZABEFZCZDZTGZSGZHZABIUBUAEZUDHZUESUAJUFUCPUDUDPUG
  UEPUCUFTKLUDMUFUCUDUDNOQR $.

  $( Two ways to describe the structure of a two-place operation.  (Contributed
     by NM, 17-Dec-2008.) $)
  relrelss $p |- ( ( Rel A /\ Rel dom A ) <-> A C_ ( ( _V X. _V ) X. _V ) ) $=
    ( vx wrel cdm wa cvv cxp wss df-rel anbi2i crn relssdmrn ssv mpan2 sylan9ss
    xpss12 xpss sstr sylibr dmss cv wcel wex wceq vn0m dmxpm ax-mp sseqtrdi jca
    impbii bitri ) ACZADZCZEULUMFFGZHZEZAUOFGZHZUNUPULUMIJUQUSULUPAUMAKZGZURALU
    PUTFHVAURHUTMUMUOUTFPNOUSULUPUSAUOHZULUSURUOHVBUOFQAURUORNAISUSUMURDZUOAURT
    BUAFUBBUCVCUOUDBUEBUOFUFUGUHUIUJUK $.

  ${
    $d x y A $.  $d x y R $.
    $( The membership relation for a relation is inherited by class union.
       (Contributed by NM, 17-Sep-2006.) $)
    unielrel $p |- ( ( Rel R /\ A e. R ) -> U. A e. U. R ) $=
      ( vx vy wrel wcel wa cv cop wceq wex elrel simpr wi vex uniopel a1i eleq1
      cuni unieq eleq1d 3imtr4d exlimivv sylc ) BEZABFZGACHZDHZIZJZDKCKUFASZBSZ
      FZCDABLUEUFMUJUFUMNCDUJUIBFZUISZULFZUFUMUNUPNUJUGUHBCODOPQAUIBRUJUKUOULAU
      ITUAUBUCUD $.
  $}

  $( The double union of a relation is its field.  (Contributed by NM,
     17-Sep-2006.) $)
  relfld $p |- ( Rel R -> U. U. R = ( dom R u. ran R ) ) $=
    ( wrel cuni cdm crn cun cxp wss relssdmrn 3syl unixpss sstrdi dmrnssfld a1i
    uniss eqssd ) ABZACZCZADZAEZFZQSTUAGZCZCZUBQAUCHRUDHSUEHAIAUCORUDOJTUAKLUBS
    HQAMNP $.

  $( Restriction of a relation to its field.  (Contributed by FL,
     15-Apr-2012.) $)
  relresfld $p |- ( Rel R -> ( R |` U. U. R ) = R ) $=
    ( wrel cuni cres wceq cdm crn wi relfld reseq2d resundi wa eqtr resss resdm
    cun wss ssequn2 uneq1 eqeq2d ex biimtrdi com3r sylbi mpsyl syl5com sylancl
    pm2.43i ) ABZAACCZDZAEZUIUKAAFZAGZPZDZEZUPAUMDZAUNDZPZEZUIULHUIUJUOAAIJAUMU
    NKUQVALUKUTEZUIULUKUPUTMUSAQZUIURAEZVBULHZAUNNAOVCAUSPZAEZVDVEHUSARVDVBVGUL
    VDVBUKVFEZVGULHVDUTVFUKURAUSSTVHVGULUKVFAMUAUBUCUDUEUFUGUH $.

  $( Composition with the identity relation restricted to a relation's field.
     (Contributed by FL, 2-May-2011.) $)
  relcoi2 $p |- ( Rel R -> ( ( _I |` U. U. R ) o. R ) = R ) $=
    ( wrel cid cuni cres ccom crn wss wceq cdm cun dmrnssfld simpr sylbir ax-mp
    wa unss cores mp1i coi2 eqtrd ) ABZCADDZEAFZCAFZAAGZUCHZUDUEIUBAJZUFKUCHZUG
    ALUIUHUCHZUGPUGUHUFUCQUJUGMNOCAUCRSATUA $.

  $( Composition with the identity relation restricted to a relation's field.
     (Contributed by FL, 8-May-2011.) $)
  relcoi1 $p |- ( Rel R -> ( R o. ( _I |` U. U. R ) ) = R ) $=
    ( wrel cid cuni cres ccom cun wceq wi resco reseq1 eqtr eqeq1 imbitrrid syl
    wa ex com3l eqcoms mpcom cdm relfld resundi coeq2 coundi resdm uneq1 uneq2d
    crn coi1 resss ssequn2 mpbi eqtr4id mpsyl syl5com eqtrid mp2b reseq2 coeq2d
    wss mpi eqeq1d eqtrd ) ABZACADDZEZFZACFZAVFAUAZAUIZGZHZVEVHVIHZAUBVEVNVMACV
    LEZFZVIHZVOCVJEZCVKEZGZHVPAVTFZHZVEVQICVJVKUCVOVTAUDVEVQWBWAVIHVEWAAVRFZAVS
    FZGZVIAVRVSUEVEVIVJEZWCHZWEVIHZACVJJVIAHZVEWGWHIZAUJZWIWFAVJEZHZVEWJVIAVJKW
    LAHZVEWMWJIAUFWMWNVEWJWMWNVEWJIZWMWNPWFAHZWOWFWLALWGWPVEWHWPVEWHIZIWCWFWCWF
    HZWPWQWRWPPWCAHZWQWCWFALVIVKEZWDHZWSWEAWDGZHZWQACVKJWCAWDUGVEXAXCWHWIVEXAXC
    WHIZIZWKWIWTAVKEZHZVEXEIVIAVKKXAXGVEXDXGVEXDIZIWDWTWDWTHZXGXHXIXGPZXBAXFGZH
    ZXHXJWDXFAWDWTXFLUHXCXLVEWHXCXLWQXCXLPWEXKHZWQWEXBXKLVEWHXMXKVIHVEXKAVIXFAV
    AXKAHAVKUKXFAULUMWKUNWEXKVIMNOQROQSROTRUOOQSROQRTUPTVBUQVPWAVIMNURVMVHVPVIV
    MVGVOAVFVLCUSUTVCNTWKVD $.

  $( The double union of the converse of a class is its field.  (Contributed by
     NM, 4-Jun-2008.) $)
  unidmrn $p |- U. U. `' A = ( dom A u. ran A ) $=
    ( ccnv cuni crn cdm cun wrel wceq relcnv ax-mp equncomi dfdm4 df-rn uneq12i
    relfld eqtr4i ) ABZCCZQDZQEZFAEZADZFRTSQGRTSFHAIQOJKUASUBTALAMNP $.

  $( if ` R ` is a relation, its double union equals the double union of its
     converse.  (Contributed by FL, 5-Jan-2009.) $)
  relcnvfld $p |- ( Rel R -> U. U. R = U. U. `' R ) $=
    ( wrel cuni cdm crn cun ccnv relfld unidmrn eqtr4di ) ABACCADAEFAGCCAHAIJ
    $.

  $( Alternate definition of domain ~ df-dm that doesn't require dummy
     variables.  (Contributed by NM, 2-Aug-2010.) $)
  dfdm2 $p |- dom A = U. U. ( `' A o. A ) $=
    ( ccnv ccom cuni cdm crn cun cnvco cocnvcnv2 eqtri unieqi eqtr3i wceq df-rn
    unidmrn eqcomi dmcoeq ax-mp rncoeq dfdm4 eqtr4i uneq12i unidm 3eqtrri ) ABZ
    ACZDZDZUFEZUFFZGZAEZULGULUFBZDZDUHUKUNUGUMUFUMUEUEBCUFUEAHUEAIJKKUFOLUIULUJ
    ULUEEZAFZMZUIULMUPUOANPZUEAQRUJUEFZULUQUJUSMURUEASRATUAUBULUCUD $.

  ${
    $d x a A $.  $d x b B $.
    $( The double class union of an inhabited cross product is the union of its
       members.  (Contributed by Jim Kingdon, 18-Dec-2018.) $)
    unixpm $p |- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) ) $=
      ( vb va cv cxp wcel wex cuni cdm crn cun wrel relxp relfld ax-mp wa ancom
      wceq xpm bitri dmxpm rnxpm uneq12 syl2an sylbir eqtrid ) AFBCGZHAIZUIJJZU
      IKZUILZMZBCMZUINUKUNTBCOUIPQUJDFCHDIZEFBHEIZRZUNUOTZURUQUPRUJUPUQSEDABCUA
      UBUPULBTUMCTUSUQDBCUCEBCUDULBUMCUEUFUGUH $.
  $}

  $( The union of an empty cross product is empty.  (Contributed by Jim
     Kingdon, 18-Dec-2018.) $)
  unixp0im $p |- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) ) $=
    ( cxp c0 wceq cuni unieq uni0 eqtrdi ) ABCZDEJFDFDJDGHI $.

  $( The converse of a set is a set.  Corollary 6.8(1) of [TakeutiZaring]
     p. 26.  (Contributed by NM, 17-Mar-1998.) $)
  cnvexg $p |- ( A e. V -> `' A e. _V ) $=
    ( wcel ccnv cdm crn cxp wss cvv wrel relcnv relssdmrn ax-mp df-rn eqeltrrid
    rnexg dfdm4 dmexg xpexg syl2anc ssexg sylancr ) ABCZADZUDEZUDFZGZHZUGICZUDI
    CUDJUHAKUDLMUCUEICUFICUIUCUEAFIANABPOUCUFAEIAQABROUEUFIISTUDUGIUAUB $.

  ${
    cnvex.1 $e |- A e. _V $.
    $( The converse of a set is a set.  Corollary 6.8(1) of [TakeutiZaring]
       p. 26.  (Contributed by NM, 19-Dec-2003.) $)
    cnvex $p |- `' A e. _V $=
      ( cvv wcel ccnv cnvexg ax-mp ) ACDAECDBACFG $.
  $}

  $( A relation is a set iff its converse is a set.  (Contributed by FL,
     3-Mar-2007.) $)
  relcnvexb $p |- ( Rel R -> ( R e. _V <-> `' R e. _V ) ) $=
    ( wrel cvv wcel ccnv cnvexg wceq wi dfrel2 eleq1 imbitrid sylbi impbid2 ) A
    BZACDZAEZCDZACFNPEZAGZQOHAIQRCDSOPCFRACJKLM $.

  ${
    $d x y A $.  $d x y B $.
    $( Restriction of a class to a singleton.  (Contributed by Mario Carneiro,
       28-Dec-2014.) $)
    ressn $p |- ( A |` { B } ) = ( { B } X. ( A " { B } ) ) $=
      ( vx vy csn cres cima cxp relres relxp cv cop wcel wa ancom elimasn elsni
      vex sneqd imaeq2d eleq2d bitr3id pm5.32i opelres opelxp 3bitr4i eqrelriiv
      bitri ) CDABEZFZUIAUIGZHZAUIIUIUKJCKZDKZLZAMZUMUIMZNZUQUNUKMZNZUOUJMUOULM
      URUQUPNUTUPUQOUQUPUSUPUNAUMEZGZMUQUSAUMUNCRDRZPUQVBUKUNUQVAUIAUQUMBUMBQST
      UAUBUCUHUMUNAUIVCUDUMUNUIUKUEUFUG $.
  $}

  ${
    $d A a b x $.  $d B a b $.  $d A a b y $.
    $( The converse of an intersection is the intersection of the converse.
       (Contributed by Jim Kingdon, 18-Dec-2018.) $)
    cnviinm $p |- ( E. y y e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B ) $=
      ( va wcel wex ciin ccnv eleq1w cbvexv wrel relcnv cvv wss wral df-rel cop
      vb cv wceq cxp wrex r19.2m expcom mpbi a1i mprg iinss syl sylibr vex opex
      wi eliin ax-mp opelcnv ralbii bitri 3bitr4i eqrelriv sylancr sylbir sylbi
      wb ) BTCFZBGETZCFZEGZACDHZIZACDIZHZUAZVFVHBEBECJKVIATCFZAGZVNVOVHAEAECJKV
      PVKLVMLZVNVJMVPVMNNUBZOZVQVPVLVROZACUCZVSVTVPWAUNACVPVTACPWAVTAACUDUEVTVO
      VLLVTDMVLQUFUGUHACVLVRUIUJVMQUKESVKVMSTZVGRZVJFZWCDFZACPZVGWBRZVKFWGVMFZW
      CNFWDWFVEWBVGSULZEULZUMAWCCDNUOUPVGWBVJWJWIUQWHWGVLFZACPZWFWGNFWHWLVEVGWB
      WJWIUMAWGCVLNUOUPWKWEACVGWBDWJWIUQURUSUTVAVBVCVD $.
  $}

  ${
    $d x y z w A $.  $d x y z w R $.
    $( The converse of a partial order relation is a partial order relation.
       (Contributed by NM, 15-Jun-2005.) $)
    cnvpom $p |- ( E. x x e. A -> ( R Po A <-> `' R Po A ) ) $=
      ( vw vy vz cv wbr wn wa wral wpo r19.26 ralbidv ralbii 3bitr4g vex ralcom
      wi brcnv wcel wex ralidm r19.3rmv bitr2id anbi1d bitrid id breq12d notbid
      ccnv weq cbvralv anbi12ci imbi12i anbi12i bitr2i bitri bitrdi df-po ) AGB
      UAAUBZDGZVBCHZIZVBEGZCHZVEFGZCHZJZVBVGCHZSZJFBKZEBKDBKZVGVGCUKZHZIZVGVEVN
      HZVEVBVNHZJZVGVBVNHZSZJZDBKZEBKFBKZBCLBVNLVAVLDBKZEBKWCFBKZEBKVMWDVAWEWFE
      BVAWEVDDBKZVKFBKZJZDBKZWFVAVDFBKZWHJZDBKZWGDBKZWHDBKZJZWEWJWMWKDBKZWOJVAW
      PWKWHDBMVAWQWNWOWNWGVAWQVDDBUCVAVDWKDBVDFABUDNUEUFUGVLWLDBVDVKFBMOWGWHDBM
      PWJWBFBKZDBKWFWIWRDBWRVPFBKZWAFBKZJWIVPWAFBMWSWGWTWHVPVDFDBFDULZVOVCVOVGV
      GCHXAVCVGVGCFQZXBTXAVGVBVGVBCXAUHZXCUIUGUJUMWAVKFBVSVIVTVJVQVHVRVFVGVECXB
      EQZTVEVBCXDDQZTUNVGVBCXBXETUOOUPUQOWBDFBBRURUSNVLDEBBRWCFEBBRPDEFBCUTFEDB
      VNUTP $.

    $( The converse of a strict order relation is a strict order relation.
       (Contributed by Jim Kingdon, 19-Dec-2018.) $)
    cnvsom $p |- ( E. x x e. A -> ( R Or A <-> `' R Or A ) ) $=
      ( vy vz cv wcel wex wpo wbr wo wi wral ccnv wor cnvpom vex brcnv df-iso
      wa orbi12i orcom imbi12i ralbii 2ralbii ralcom bitr3i a1i anbi12d 3bitr4g
      wb bitri ) AFZBGAHZBCIZUMDFZCJZUMEFZCJZURUPCJZKZLZEBMZDBMABMZTBCNZIZUPUMV
      EJZUPURVEJZURUMVEJZKZLZEBMZABMDBMZTBCOBVEOUNUOVFVDVMABCPVDVMUKUNVDVLDBMAB
      MVMVLVCADBBVKVBEBVGUQVJVAUPUMCDQZAQZRVJUTUSKVAVHUTVIUSUPURCVNEQZRURUMCVPV
      ORUAUTUSUBULUCUDUEVLADBBUFUGUHUIADEBCSDAEBVESUJ $.
  $}

  $( The composition of two sets is a set.  (Contributed by NM,
     19-Mar-1998.) $)
  coexg $p |- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V ) $=
    ( wcel wa ccom cdm crn cxp wss cvv cossxp dmexg rnexg xpexg syl2anr sylancr
    ssexg ) ACEZBDEZFABGZBHZAIZJZKUELEZUBLEABMUAUCLEUDLEUFTBDNACOUCUDLLPQUBUELS
    R $.

  ${
    coex.1 $e |- A e. _V $.
    coex.2 $e |- B e. _V $.
    $( The composition of two sets is a set.  (Contributed by NM,
       15-Dec-2003.) $)
    coex $p |- ( A o. B ) e. _V $=
      ( cvv wcel ccom coexg mp2an ) AEFBEFABGEFCDABEEHI $.
  $}

  ${
    $d x a c A $.  $d x a c B $.  $d x a c C $.
    $( Composition of two cross products.  (Contributed by Jim Kingdon,
       20-Dec-2018.) $)
    xpcom $p |- ( E. x x e. B -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) ) $=
      ( va vc cv wcel wex cxp wbr wa copab ccom ibar anbi1i brxp anbi12i anandi
      ancom 3bitr4i exbii 19.41v bitr2i bitr2di opabbidv df-co df-xp 3eqtr4g )
      AGZCHZAIZEGZUJBCJZKZUJFGZCDJZKZLZAIZEFMUMBHZUPDHZLZEFMUQUNNBDJULUTVCEFULV
      CULVCLZUTULVCOUTUKVCLZAIVDUSVEAVAUKLZUKVBLZLUKVALZVGLUSVEVFVHVGVAUKTPUOVF
      URVGUMUJBCQUJUPCDQRUKVAVBSUAUBUKVCAUCUDUEUFEFAUQUNUGEFBDUHUI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Definite description binder (inverted iota)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c iota $.
  $( Extend class notation with Russell's definition description binder
     (inverted iota). $)
  cio $a class ( iota x ph ) $.

  ${
    $d w x z $.  $d ph w z $.  $d ph w y $.  $d x y $.
    $( Soundness justification theorem for ~ df-iota .  (Contributed by Andrew
       Salmon, 29-Jun-2011.) $)
    iotajust $p |- U. { y | { x | ph } = { y } } =
       U. { z | { x | ph } = { z } } $=
      ( vw cab cv csn wceq sneq eqeq2d cbvabv eqtri unieqi ) ABFZCGZHZIZCFZODGZ
      HZIZDFZSOEGZHZIZEFUCRUFCEPUDIQUEOPUDJKLUFUBEDUDTIUEUAOUDTJKLMN $.
  $}

  ${
    $d y x $.  $d y ph $.
    $( Define Russell's definition description binder, which can be read as
       "the unique ` x ` such that ` ph ` ," where ` ph ` ordinarily contains
       ` x ` as a free variable.  Our definition is meaningful only when there
       is exactly one ` x ` such that ` ph ` is true (see ~ iotaval );
       otherwise, it evaluates to the empty set (see ~ iotanul ).  Russell used
       the inverted iota symbol ` iota ` to represent the binder.

       Sometimes proofs need to expand an iota-based definition.  That is,
       given "X = the x for which ... x ... x ..." holds, the proof needs to
       get to "...  X ...  X ...".  A general strategy to do this is to use
       ~ iotacl (for unbounded iota).  This can be easier than applying a
       version that applies an explicit substitution, because substituting an
       iota into its own property always has a bound variable clash which must
       be first renamed or else guarded with NF.

       (Contributed by Andrew Salmon, 30-Jun-2011.) $)
    df-iota $a |- ( iota x ph ) = U. { y | { x | ph } = { y } } $.
  $}

  ${
    $d y x $.  $d y ph $.
    $( Alternate definition for descriptions.  Definition 8.18 in [Quine]
       p. 56.  (Contributed by Andrew Salmon, 30-Jun-2011.) $)
    dfiota2 $p |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) } $=
      ( cio cab cv csn wceq cuni wal df-iota df-sn eqeq2i abbibcom bitr4i abbii
      wb unieqi eqtri ) ABDABEZCFZGZHZCEZIABFUAHZQBJZCEZIABCKUDUGUCUFCUCTUEBEZH
      UFUBUHTBUALMAUEBNOPRS $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( Bound-variable hypothesis builder for the ` iota ` class.  (Contributed
       by Andrew Salmon, 11-Jul-2011.)  (Revised by Mario Carneiro,
       15-Oct-2016.) $)
    nfiota1 $p |- F/_ x ( iota x ph ) $=
      ( vy cio weq wb wal cab cuni dfiota2 nfaba1 nfuni nfcxfr ) BABDABCEFZBGCH
      ZIABCJBONBCKLM $.
  $}

  ${
    $d z ps $.  $d z ph $.  $d x y z $.
    nfiotadw.1 $e |- F/ y ph $.
    nfiotadw.2 $e |- ( ph -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for the ` iota ` class.  (Contributed
       by Jim Kingdon, 21-Dec-2018.) $)
    nfiotadw $p |- ( ph -> F/_ x ( iota y ps ) ) $=
      ( vz cio cv wceq wb wal cab cuni dfiota2 nfv wnf nfcv nfeq a1i nfcxfrd
      nfbid nfald nfabd nfunid ) ACBDHBDIZGIZJZKZDLZGMZNBDGOACUKAUJCGAGPAUICDEA
      BUHCFUHCQACUFUGCUFRCUGRSTUBUCUDUEUA $.
  $}

  ${
    $d x y $.
    nfiotaw.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for the ` iota ` class.  (Contributed
       by NM, 23-Aug-2011.) $)
    nfiotaw $p |- F/_ x ( iota y ph ) $=
      ( cio wnfc wtru nftru wnf a1i nfiotadw mptru ) BACEFGABCCHABIGDJKL $.
  $}

  ${
    $d z w x $.  $d z w y $.  $d z w ph $.  $d z w ps $.
    cbviota.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    cbviota.2 $e |- F/ y ph $.
    cbviota.3 $e |- F/ x ps $.
    $( Change bound variables in a description binder.  (Contributed by Andrew
       Salmon, 1-Aug-2011.) $)
    cbviota $p |- ( iota x ph ) = ( iota y ps ) $=
      ( vw vz weq wb wal cab cuni cio wsb nfv nfbi equequ1 bibi12d sbequ12 nfsb
      nfs1v cbval sbequ sbie bitrdi bitri abbii unieqi dfiota2 3eqtr4i ) ACHJZK
      ZCLZHMZNBDHJZKZDLZHMZNACOBDOUPUTUOUSHUOACIPZIHJZKZILUSUNVCCIUNIQVAVBCACIU
      CVBCQRCIJAVAUMVBACIUACIHSTUDVCURIDVAVBDACIDFUBVBDQRURIQIDJZVABVBUQVDVAACD
      PBAIDCUEABCDGEUFUGIDHSTUDUHUIUJACHUKBDHUKUL $.
  $}

  ${
    $d ph y z $.  $d ps x z $.  $d x y $.
    cbviotavw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variables in a description binder.  Version of ~ cbviotav
       with a disjoint variable condition.  (Contributed by Andrew Salmon,
       1-Aug-2011.)  (Revised by GG, 30-Sep-2024.) $)
    cbviotavw $p |- ( iota x ph ) = ( iota y ps ) $=
      ( vz cab cv csn wceq cuni cio cbvabv eqeq1i abbii unieqi df-iota 3eqtr4i
      ) ACGZFHIZJZFGZKBDGZTJZFGZKACLBDLUBUEUAUDFSUCTABCDEMNOPACFQBDFQR $.
  $}

  ${
    $d ph y $.  $d ps x $.
    cbviotav.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variables in a description binder.  (Contributed by Andrew
       Salmon, 1-Aug-2011.) $)
    cbviotav $p |- ( iota x ph ) = ( iota y ps ) $=
      ( nfv cbviota ) ABCDEADFBCFG $.
  $}

  ${
    $d w z ph $.  $d w z x $.  $d w z y $.
    sb8iota.1 $e |- F/ y ph $.
    $( Variable substitution in description binder.  Compare ~ sb8eu .
       (Contributed by NM, 18-Mar-2013.) $)
    sb8iota $p |- ( iota x ph ) = ( iota y [ y / x ] ph ) $=
      ( vz vw weq wal cab cuni wsb cio nfv sb8 sbbi nfsb equsb3 nfxfr dfiota2
      wb nfbi sbequ cbval sblbis albii 3bitri abbii unieqi 3eqtr4i ) ABEGZTZBHZ
      EIZJABCKZCEGZTZCHZEIZJABLUNCLUMURULUQEULUKBFKZFHUKBCKZCHUQUKBFUKFMNUSUTFC
      USABFKZUJBFKZTCAUJBFOVAVBCABFCDPVBFEGZCBFEQVCCMRUARUTFMUKFCBUBUCUTUPCUJUO
      ABCBCEQUDUEUFUGUHABESUNCESUI $.
  $}

  ${
    $d y z $.  $d x z $.  $d ph z $.
    $( Equality theorem for descriptions.  (Contributed by Andrew Salmon,
       30-Jun-2011.) $)
    iotaeq $p |- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) ) $=
      ( vz cv wceq wal cab csn cuni cio wcel drsb1 df-clab 3bitr4g eqrdv eqeq1d
      wsb abbidv df-iota unieqd 3eqtr4g ) BECEFBGZABHZDEZIZFZDHZJACHZUFFZDHZJAB
      KACKUCUHUKUCUGUJDUCUDUIUFUCDUDUIUCABDRACDRUEUDLUEUILABCDMADBNADCNOPQSUAAB
      DTACDTUB $.
  $}

  ${
    $d ph z $.  $d ps z $.  $d x y z $.
    $( Equivalence theorem for descriptions.  (Contributed by Andrew Salmon,
       30-Jun-2011.) $)
    iotabi $p |- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) ) $=
      ( vz wb wal cab cv csn wceq cuni cio abbibcom biimpi eqeq1d abbidv unieqd
      df-iota 3eqtr4g ) ABECFZACGZDHIZJZDGZKBCGZUBJZDGZKACLBCLTUDUGTUCUFDTUAUEU
      BTUAUEJABCMNOPQACDRBCDRS $.

    $( Part of Theorem 8.17 in [Quine] p. 56.  This theorem serves as a lemma
       for the fundamental property of iota.  (Contributed by Andrew Salmon,
       11-Jul-2011.) $)
    uniabio $p |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y ) $=
      ( cv wceq wal cab cuni csn abbibcom biimpi df-sn eqtr4di unieqd vex unisn
      wb eqtrdi ) ABDCDZEZQBFZABGZHSIZHSUAUBUCUAUBTBGZUCUAUBUDEATBJKBSLMNSCOPR
      $.

    $( Theorem 8.19 in [Quine] p. 57.  This theorem is the fundamental property
       of iota.  (Contributed by Andrew Salmon, 11-Jul-2011.) $)
    iotaval $p |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y ) $=
      ( vz weq wb wal cio cab cuni cv dfiota2 wceq cvv wcel vex sbeqalb equcomi
      wa wi ax-mp ex equequ2 equcoms bibi2d biimpd alimdv com12 alrimiv uniabio
      syl6 impbid syl eqtrid ) ABCEZFZBGZABHABDEZFZBGZDIJZCKZABDLUQUTDCEZFZDGVA
      VBMUQVDDUQUTVCUQUTVCVBNOZUQUTSZVCTCPVEVFCDEVCABVBDKNQCDRUKUAUBVCUQUTVCUPU
      SBVCUPUSVCUOURAUOURFCDCDBUCUDUEUFUGUHULUIUTDCUJUMUN $.

    $( Equivalence between two different forms of ` iota ` .  (Contributed by
       Andrew Salmon, 12-Jul-2011.) $)
    iotauni $p |- ( E! x ph -> ( iota x ph ) = U. { x | ph } ) $=
      ( vz weu cv wceq wb wal wex cio cuni df-eu iotaval uniabio eqtr4d exlimiv
      cab sylbi ) ABDABECEZFGBHZCIABJZABQKZFZABCLTUCCTUASUBABCMABCNOPR $.

    $( Equivalence between two different forms of ` iota ` .  (Contributed by
       Mario Carneiro, 24-Dec-2016.) $)
    iotaint $p |- ( E! x ph -> ( iota x ph ) = |^| { x | ph } ) $=
      ( weu cio cab cuni cint iotauni uniintabim eqtrd ) ABCABDABEZFKGABHABIJ
      $.

    $( Property of iota.  (Contributed by NM, 23-Aug-2011.)  (Revised by Mario
       Carneiro, 23-Dec-2016.) $)
    iota1 $p |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) ) $=
      ( vz weu cv wceq wb wal wex cio df-eu iotaval eqeq2d bitr4d eqcom exlimiv
      sp bitrdi sylbi ) ABDABEZCEZFZGZBHZCIAABJZTFZGZABCKUDUGCUDATUEFZUFUDAUBUH
      UCBQUDUEUATABCLMNTUEORPS $.

    $( Theorem 8.22 in [Quine] p. 57.  This theorem is the result if there
       isn't exactly one ` x ` that satisfies ` ph ` .  (Contributed by Andrew
       Salmon, 11-Jul-2011.) $)
    iotanul $p |- ( -. E! x ph -> ( iota x ph ) = (/) ) $=
      ( vz weu wn cio c0 wss wceq weq wb wal wex df-eu cab dfiota2 alnex ax-in2
      cuni wi alimi ss2ab sylibr dfnul2 sseqtrrdi sylbir uni0 sseqtrdi eqsstrid
      unissd sylnbi ss0 syl ) ABDZEABFZGHZUOGIUNABCJKBLZCMZUPABCNUREZUOUQCOZSZG
      ABCPUSVAGSGUSUTGUSUQEZCLZUTGHUQCQVCUTCCJEZCOZGVCUQVDTZCLUTVEHVBVFCUQVDRUA
      UQVDCUBUCCUDUEUFUJUGUHUIUKUOULUM $.
  $}

  ${
    $d ph y $.  $d x y $.
    $( Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition
       added.  This theorem proves the existence of the ` iota ` class under
       our definition.  (Contributed by Jim Kingdon, 21-Dec-2018.) $)
    euiotaex $p |- ( E! x ph -> ( iota x ph ) e. _V ) $=
      ( vy cv wceq wb wal wex cio weu cvv wcel iotaval eqcomd eximi df-eu isset
      3imtr4i ) ABDCDZEFBGZCHSABIZEZCHABJUAKLTUBCTUASABCMNOABCPCUAQR $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( Value of iota based on a proposition which holds only for values which
       are subsets of a given class.  (Contributed by Mario Carneiro and Jim
       Kingdon, 21-Dec-2018.) $)
    iotass $p |- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A ) $=
      ( vy cv wss wal cio cab csn wceq cuni df-iota cpw unieq vex df-pw sspwuni
      wi sylib unisn eqtrdi sseq2i ss2ab bitri biimpri sseq1 syl2anr ex ss2abdv
      biimpa sseqtrrdi eqsstrid ) ABECFZSBGZABHABIZDEZJZKZDIZLZCABDMUOUTCNZFVAC
      FUOUTUQCFZDIVBUOUSVCDUOUSVCUSUPLZUQKZVDCFZVCUOUSVDURLUQUPUROUQDPUAUBUOUPV
      BFZVFVGUOVGUPUNBIZFUOVBVHUPBCQUCAUNBUDUEUFUPCRTVEVFVCVDUQCUGUKUHUIUJDCQUL
      UTCRTUM $.
  $}

  ${
    $d ph z $.  $d x z $.
    $( Existence of the ` iota ` class when all the possible values are
       contained in a set.  (Contributed by Jim Kingdon, 27-May-2025.) $)
    iotaexab $p |- ( { x | ph } e. V -> ( iota x ph ) e. _V ) $=
      ( vz cab wcel cio cuni cvv uniexg wss wal abid elssuni sylbir ax-gen wceq
      cv wi sseq2 nfab1 nfuni nfeq2 imbi2d albid imbi12d iotass vtoclg mpisyl
      ssexd ) ABEZCFZABGZUKHZIUKCJZULUNIFABRZUNKZSZBLZUMUNKZUOURBAUPUKFUQABMUPU
      KNOPAUPDRZKZSZBLZUMVAKZSUSUTSDUNIVAUNQZVDUSVEUTVFVCURBBVAUNBUKABUAUBUCVFV
      BUQAVAUNUPTUDUEVAUNUMTUFABVAUGUHUIUJ $.
  $}

  ${
    $d ph z $.  $d x z $.
    $( Theorem *14.22 in [WhiteheadRussell] p. 190.  (Contributed by Andrew
       Salmon, 12-Jul-2011.) $)
    iota4 $p |- ( E! x ph -> [. ( iota x ph ) / x ]. ph ) $=
      ( vz weu weq wb wal wex cio wsbc df-eu wsb wi biimpr alimi sb2 cv iotaval
      syl wceq eqcomd dfsbcq2 mpbid exlimiv sylbi ) ABDABCEZFZBGZCHABABIZJZABCK
      UHUJCUHABCLZUJUHUFAMZBGUKUGULBAUFNOABCPSUHCQZUITUKUJFUHUIUMABCRUAABCUIUBS
      UCUDUE $.
  $}

  $( Theorem *14.23 in [WhiteheadRussell] p. 191.  (Contributed by Andrew
     Salmon, 12-Jul-2011.) $)
  iota4an $p |- ( E! x ( ph /\ ps )
            -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) $=
    ( wa weu cio wsbc iota4 wi cvv euiotaex simpl sbcth syl wb sbcimg mpbid mpd
    wcel ) ABDZCEZTCTCFZGZACUBGZTCHUATAIZCUBGZUCUDIZUAUBJSZUFTCKZUECUBJABLMNUAU
    HUFUGOUITACUBJPNQR $.

  ${
    $d x y A $.  $d x V $.  $d x ph $.  $d y ps $.
    iota5.1 $e |- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) ) $.
    $( A method for computing iota.  (Contributed by NM, 17-Sep-2013.) $)
    iota5 $p |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A ) $=
      ( vy wcel wa cv wceq wb wal cio alrimiv wi eqeq2 bibi2d albidv imbi12d
      iotaval vtoclg adantl mpd ) ADEHZIZBCJZDKZLZCMZBCNZDKZUFUICFOUEUJULPZABUG
      GJZKZLZCMZUKUNKZPUMGDEUNDKZUQUJURULUSUPUICUSUOUHBUNDUGQRSUNDUKQTBCGUAUBUC
      UD $.
  $}

  ${
    $d x ph $.
    iotabidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building deduction for iota.  (Contributed by NM,
       20-Aug-2011.) $)
    iotabidv $p |- ( ph -> ( iota x ps ) = ( iota x ch ) ) $=
      ( wb wal cio wceq alrimiv iotabi syl ) ABCFZDGBDHCDHIAMDEJBCDKL $.
  $}

  ${
    iotabii.1 $e |- ( ph <-> ps ) $.
    $( Formula-building deduction for iota.  (Contributed by Mario Carneiro,
       2-Oct-2015.) $)
    iotabii $p |- ( iota x ph ) = ( iota x ps ) $=
      ( wb cio wceq iotabi mpg ) ABEACFBCFGCABCHDI $.
  $}

  $( Membership law for descriptions.

     This can useful for expanding an unbounded iota-based definition (see
     ~ df-iota ).

     (Contributed by Andrew Salmon, 1-Aug-2011.) $)
  iotacl $p |- ( E! x ph -> ( iota x ph ) e. { x | ph } ) $=
    ( weu cio wsbc cab wcel iota4 df-sbc sylib ) ABCABABDZEKABFGABHABKIJ $.

  ${
    iota2df.1 $e |- ( ph -> B e. V ) $.
    iota2df.2 $e |- ( ph -> E! x ps ) $.
    iota2df.3 $e |- ( ( ph /\ x = B ) -> ( ps <-> ch ) ) $.
    ${
      iota2df.4 $e |- F/ x ph $.
      iota2df.5 $e |- ( ph -> F/ x ch ) $.
      iota2df.6 $e |- ( ph -> F/_ x B ) $.
      $( A condition that allows us to represent "the unique element such that
         ` ph ` " with a class expression ` A ` .  (Contributed by NM,
         30-Dec-2014.) $)
      iota2df $p |- ( ph -> ( ch <-> ( iota x ps ) = B ) ) $=
        ( cio cv wceq wb wa simpr eqeq2d bibi12d weu iota1 syl wnfc nfiota1 a1i
        nfeqd nfbid vtocldf ) ABBDMZDNZOZPZCUJEOZPDEFGAUKEOZQZBCULUNIUPUKEUJAUO
        RSTABDUAUMHBDUBUCJLACUNDKADUJEDUJUDABDUEUFLUGUHUI $.
    $}

    $d x B $.  $d x ph $.  $d x ch $.
    $( A condition that allows us to represent "the unique element such that
       ` ph ` " with a class expression ` A ` .  (Contributed by NM,
       30-Dec-2014.) $)
    iota2d $p |- ( ph -> ( ch <-> ( iota x ps ) = B ) ) $=
      ( nfv nfvd nfcvd iota2df ) ABCDEFGHIADJACDKADELM $.
  $}

  ${
    $d ph y $.  $d A y $.  $d x y $.
    $( An element of an iota expression.  (Contributed by Jim Kingdon,
       22-Nov-2024.) $)
    eliota $p |- ( A e. ( iota x ph ) <-> E. y ( A e. y /\
        A. x ( ph <-> x = y ) ) ) $=
      ( cio wcel weq wb wal cab cuni cv wa wex dfiota2 eleq2i eluniab bitri ) D
      ABEZFDABCGHBIZCJKZFDCLFTMCNSUADABCOPTCDQR $.
  $}

  ${
    $d A y $.  $d ph y $.  $d x y $.
    $( An inhabited iota expression has a unique value.  (Contributed by Jim
       Kingdon, 22-Nov-2024.) $)
    eliotaeu $p |- ( A e. ( iota x ph ) -> E! x ph ) $=
      ( vy cv wcel weq wb wal wa wex cio weu exsimpr eliota df-eu 3imtr4i ) CDE
      FZABDGHBIZJDKSDKCABLFABMRSDNABDCOABDPQ $.
  $}

  ${
    $d A x $.  $d ps x $.
    iota2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( The unique element such that ` ph ` .  (Contributed by Jeff Madsen,
       1-Jun-2011.)  (Revised by Mario Carneiro, 23-Dec-2016.) $)
    iota2 $p |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) ) $=
      ( wcel cvv weu cio wceq wb elex wa simpl simpr cv adantl nfv nfeu1 nfcvd
      nfan nfvd iota2df sylan ) DEGDHGZACIZBACJDKLDEMUFUGNZABCDHUFUGOUFUGPCQDKA
      BLUHFRUFUGCUFCSACTUBUHBCUCUHCDUAUDUE $.
  $}

  $( A class abstraction with a unique member can be expressed as a singleton.
     (Contributed by Mario Carneiro, 23-Dec-2016.) $)
  sniota $p |- ( E! x ph -> { x | ph } = { ( iota x ph ) } ) $=
    ( weu cv cab wcel cio csn wb wal wceq nfeu1 iota1 eqcom bitrdi abid 3bitr4g
    vex elsn alrimi nfab1 nfiota1 nfsn cleqf sylibr ) ABCZBDZABEZFZUGABGZHZFZIZ
    BJUHUKKUFUMBABLUFAUGUJKZUIULUFAUJUGKUNABMUJUGNOABPUGUJBRSQTBUHUKABUABUJABUB
    UCUDUE $.

  ${
    $d A x z $.  $d A w $.  $d V z $.  $d ph z $.  $d ps x z $.  $d w z $.
    iotam.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Representation of "the unique element such that ` ph ` " with a class
       expression ` A ` which is inhabited (that means that "the unique element
       such that ` ph ` " exists).  (Contributed by AV, 30-Jan-2024.) $)
    iotam $p |- ( ( A e. V /\ E. w w e. A /\ A = ( iota x ph ) ) -> ps ) $=
      ( vz cv wcel wex cio wceq wa wi eleq1w cbvexv simprr eqcomd weu wb simprl
      simpl eleqtrd eliotaeu syl iota2 syl2anc mpbird ex exlimiv 3impib 3com12
      sylbi ) DIEJZDKZEFJZEACLZMZBUPUQUSBUPHIZEJZHKUQUSNZBOZUOVADHDHEPQVAVCHVAV
      BBVAVBNZBUREMZVDEURVAUQUSRZSVDUQACTZBVEUAVAUQUSUBVDUTURJVGVDUTEURVAVBUCVF
      UDACUTUEUFABCEFGUGUHUIUJUKUNULUM $.
  $}

  ${
    $d A y z $.  $d x y z $.  $d ph z $.
    $( Class substitution within a description binder.  (Contributed by Scott
       Fenton, 6-Oct-2017.) $)
    csbiotag $p |- ( A e. V ->
        [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) ) $=
      ( vz cv cio csb wsb wceq wsbc csbeq1 dfsbcq2 iotabidv eqeq12d vex nfiotaw
      nfs1v weq sbequ12 csbief vtoclg ) BFGZACHZIZABFJZCHZKBDUEIZABDLZCHZKFDEUD
      DKZUFUIUHUKBUDDUEMULUGUJCABFDNOPBUDUEUHFQUGBCABFSRBFTAUGCABFUAOUBUC $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Functions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c : $. $( Colon $)
  $c Fun $. $( Function predicate $)
  $c Fn $. $( Function connective $)
  $c --> $. $( Domain-codomain connective $)
  $c -1-1-> $. $( 'One-to-one' domain-codomain connective $)
  $c -onto-> $. $( 'Onto' domain-codomain connective $)
  $c -1-1-onto-> $. $( 'One-to-one' and 'onto' domain-codomain connective $)
  $c ` $. $( Left apostrophe (function value symbol) $)
  $c Isom $. $( Isomorphism $)

  $( Extend the definition of a wff to include the function predicate.  (Read:
     ` A ` is a function.) $)
  wfun $a wff Fun A $.

  $( Extend the definition of a wff to include the function predicate with a
     domain.  (Read: ` A ` is a function on ` B ` .) $)
  wfn $a wff A Fn B $.

  $( Extend the definition of a wff to include the function predicate with
     domain and codomain.  (Read: ` F ` maps ` A ` into ` B ` .) $)
  wf $a wff F : A --> B $.

  $( Extend the definition of a wff to include one-to-one functions.  (Read:
     ` F ` maps ` A ` one-to-one into ` B ` .)  The notation ("1-1" above the
     arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27. $)
  wf1 $a wff F : A -1-1-> B $.

  $( Extend the definition of a wff to include onto functions.  (Read: ` F `
     maps ` A ` onto ` B ` .)  The notation ("onto" below the arrow) is from
     Definition 6.15(4) of [TakeutiZaring] p. 27. $)
  wfo $a wff F : A -onto-> B $.

  $( Extend the definition of a wff to include one-to-one onto functions.
     (Read: ` F ` maps ` A ` one-to-one onto ` B ` .)  The notation ("1-1"
     above the arrow and "onto" below the arrow) is from Definition 6.15(6) of
     [TakeutiZaring] p. 27. $)
  wf1o $a wff F : A -1-1-onto-> B $.

  $( Extend the definition of a class to include the value of a function.
     (Read:  The value of ` F ` at ` A ` , or " ` F ` of ` A ` .") $)
  cfv $a class ( F ` A ) $.

  $( Extend the definition of a wff to include the isomorphism property.
     (Read: ` H ` is an ` R ` , ` S ` isomorphism of ` A ` onto ` B ` .) $)
  wiso $a wff H Isom R , S ( A , B ) $.

  ${
    $d x A $.  $d x F $.
    $( Define predicate that determines if some class ` A ` is a function.
       Definition 10.1 of [Quine] p. 65.  For example, the expression
       ` Fun _I ` is true ( ~ funi ).  This is not the same as defining a
       specific function's mapping, which is typically done using the format of
       ~ cmpt with the maps-to notation (see ~ df-mpt ).  Contrast this
       predicate with the predicates to determine if some class is a function
       with a given domain ( ~ df-fn ), a function with a given domain and
       codomain ( ~ df-f ), a one-to-one function ( ~ df-f1 ), an onto function
       ( ~ df-fo ), or a one-to-one onto function ( ~ df-f1o ).  For alternate
       definitions, see ~ dffun2 , ~ dffun4 , ~ dffun6 , ~ dffun7 , ~ dffun8 ,
       and ~ dffun9 .  (Contributed by NM, 1-Aug-1994.) $)
    df-fun $a |- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) ) $.

    $( Define a function with domain.  Definition 6.15(1) of [TakeutiZaring]
       p. 27.  (Contributed by NM, 1-Aug-1994.) $)
    df-fn $a |- ( A Fn B <-> ( Fun A /\ dom A = B ) ) $.

    $( Define a function (mapping) with domain and codomain.  Definition
       6.15(3) of [TakeutiZaring] p. 27.  (Contributed by NM, 1-Aug-1994.) $)
    df-f $a |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) ) $.

    $( Define a one-to-one function.  Compare Definition 6.15(5) of
       [TakeutiZaring] p. 27.  We use their notation ("1-1" above the arrow).
       (Contributed by NM, 1-Aug-1994.) $)
    df-f1 $a |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) ) $.

    $( Define an onto function.  Definition 6.15(4) of [TakeutiZaring] p. 27.
       We use their notation ("onto" under the arrow).  (Contributed by NM,
       1-Aug-1994.) $)
    df-fo $a |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) ) $.

    $( Define a one-to-one onto function.  Compare Definition 6.15(6) of
       [TakeutiZaring] p. 27.  We use their notation ("1-1" above the arrow and
       "onto" below the arrow).  (Contributed by NM, 1-Aug-1994.) $)
    df-f1o $a |- ( F : A -1-1-onto-> B <->
                ( F : A -1-1-> B /\ F : A -onto-> B ) ) $.

    $( Define the value of a function, ` ( F `` A ) ` , also known as function
       application.  For example, ` ( _I `` (/) ) = (/) ` .  Typically,
       function ` F ` is defined using maps-to notation (see ~ df-mpt ), but
       this is not required.  For example,
       ` F = { <. 2 , 6 >. , <. 3 , 9 >. } -> ( F `` 3 ) = 9 ` .  We will later
       define two-argument functions using ordered pairs as
       ` ( A F B ) = ( F `` <. A , B >. ) ` .  This particular definition is
       quite convenient: it can be applied to any class and evaluates to the
       empty set when it is not meaningful.  The left apostrophe notation
       originated with Peano and was adopted in Definition *30.01 of
       [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and
       Definition 6.11 of [TakeutiZaring] p. 26.  It means the same thing as
       the more familiar ` F ( A ) ` notation for a function's value at ` A ` ,
       i.e., " ` F ` of ` A ` ," but without context-dependent notational
       ambiguity.  (Contributed by NM, 1-Aug-1994.)  Revised to use ` iota ` .
       (Revised by Scott Fenton, 6-Oct-2017.) $)
    df-fv $a |- ( F ` A ) = ( iota x A F x ) $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.  $d x y S $.  $d x y H $.
    $( Define the isomorphism predicate.  We read this as " ` H ` is an ` R ` ,
       ` S ` isomorphism of ` A ` onto ` B ` ".  Normally, ` R ` and ` S ` are
       ordering relations on ` A ` and ` B ` respectively.  Definition 6.28 of
       [TakeutiZaring] p. 32, whose notation is the same as ours except that
       ` R ` and ` S ` are subscripts.  (Contributed by NM, 4-Mar-1997.) $)
    df-isom $a |- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\
                 A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) ) $.
  $}

  ${
    $d x y z A $.
    $( Alternate definition of a function.  (Contributed by NM,
       29-Dec-1996.) $)
    dffun2 $p |- ( Fun A <-> ( Rel A /\
                 A. x A. y A. z ( ( x A y /\ x A z ) -> y = z ) ) ) $=
      ( wfun wrel cid wss wa cv wbr wal wex copab 3bitri vex albii alcom bitri
      wi ccnv ccom df-fun df-id sseq2i df-co sseq1i ssopab2b brcnv anbi1i exbii
      weq imbi1i 19.23v bitr4i anbi2i ) DEDFZDDUAZUBZGHZIUQAJZBJZDKZVACJDKZIZBC
      ULZTZCLZBLALZIDUCUTVIUQUTVBVAURKZVDIZAMZVFTZCLZBLZVHALZBLVIUTUSVFBCNZHVLB
      CNZVQHVOGVQUSBCUDUEUSVRVQBCADURUFUGVLVFBCUHOVNVPBVNVGALZCLVPVMVSCVMVEAMZV
      FTVSVLVTVFVKVEAVJVCVDVBVADBPAPUIUJUKUMVEVFAUNUOQVGCARSQVHBAROUPS $.

    $( Alternate definition of a function.  Definition 6.4(4) of
       [TakeutiZaring] p. 24.  (Contributed by NM, 29-Dec-1996.) $)
    dffun4 $p |- ( Fun A <-> ( Rel A /\
                 A. x A. y A. z ( ( <. x , y >. e. A /\ <. x , z >. e. A )
                 -> y = z ) ) ) $=
      ( wfun wrel cv wbr wa weq wal cop wcel dffun2 df-br anbi12i imbi1i 2albii
      wi albii anbi2i bitri ) DEDFZAGZBGZDHZUDCGZDHZIZBCJZSZCKZBKAKZIUCUDUELDMZ
      UDUGLDMZIZUJSZCKZBKAKZIABCDNUMUSUCULURABUKUQCUIUPUJUFUNUHUOUDUEDOUDUGDOPQ
      TRUAUB $.

    $( A way of proving a relation is a function, analogous to ~ mo2r .
       (Contributed by Jim Kingdon, 27-May-2020.) $)
    dffun5r $p |- ( ( Rel A /\
        A. x E. z A. y ( <. x , y >. e. A -> y = z ) ) -> Fun A ) $=
      ( wrel cv cop wcel weq wi wal wex wa wfun wmo nfv mo2r opeq2 eleq1d mo4
      sylib alimi anim2i dffun4 sylibr ) DEZAFZBFZGZDHZBCIZJBKCLZAKZMUFUJUGCFZG
      ZDHZMUKJCKBKZAKZMDNUMURUFULUQAULUJBOUQUJBCUJCPQUJUPBCUKUIUODUHUNUGRSTUAUB
      UCABCDUDUE $.
  $}

  ${
    $d x y w v u $.  $d A w v u $.
    dffun6f.1 $e |- F/_ x A $.
    dffun6f.2 $e |- F/_ y A $.
    $( Definition of function, using bound-variable hypotheses instead of
       distinct variable conditions.  (Contributed by NM, 9-Mar-1995.)
       (Revised by Mario Carneiro, 15-Oct-2016.) $)
    dffun6f $p |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) ) $=
      ( vw vv vu wfun cv wbr wa weq wal wmo nfcv nfbr nfv breq2 albii wi dffun2
      wrel cbvmo mo4 nfmo breq1 mobidv cbval 3bitr3ri anbi2i bitr4i ) CICUCZFJZ
      GJZCKZUNHJZCKZLGHMUAHNGNZFNZLUMAJZBJZCKZBOZANZLFGHCUBVEUTUMUPGOZFNUNVBCKZ
      BOZFNUTVEVFVHFUPVGGBBUNUOCBUNPEBUOPQVGGRUOVBUNCSUDTVFUSFUPURGHUOUQUNCSUET
      VHVDFAVGABAUNVBCAUNPDAVBPQUFVDFRFAMVGVCBUNVAVBCUGUHUIUJUKUL $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( Alternate definition of a function using "at most one" notation.
       (Contributed by NM, 9-Mar-1995.) $)
    dffun6 $p |- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) ) $=
      ( nfcv dffun6f ) ABCACDBCDE $.

    $( A function has at most one value for each argument.  (Contributed by NM,
       24-May-1998.) $)
    funmo $p |- ( Fun F -> E* y A F y ) $=
      ( vx wfun cv wbr cvv wcel wa wi wal wmo wrel dffun6 simplbi brrelex ancrd
      ex syl alrimiv wceq breq1 mobidv imbi2d simprbi 19.21bi vtoclg com12 moim
      moanimv sylibr sylc ) CEZBAFZCGZBHIZUPJZKZALURAMZUPAMZUNUSAUNUPUQUNCNZUPU
      QKUNVBDFZUOCGZAMZDLZDACOZPVBUPUQBUOCQSTRUAUNUQVAKUTUQUNVAUNVEKUNVAKDBHVCB
      UBZVEVAUNVHVDUPAVCBUOCUCUDUEUNVEDUNVBVFVGUFUGUHUIUQUPAUKULUPURAUJUM $.
  $}

  ${
    $d x y z w $.  $d A w $.
    dffun4f.1 $e |- F/_ x A $.
    dffun4f.2 $e |- F/_ y A $.
    dffun4f.3 $e |- F/_ z A $.
    $( Definition of function like ~ dffun4 but using bound-variable hypotheses
       instead of distinct variable conditions.  (Contributed by Jim Kingdon,
       17-Mar-2019.) $)
    dffun4f $p |- ( Fun A <-> ( Rel A /\
                 A. x A. y A. z ( ( <. x , y >. e. A /\ <. x , z >. e. A )
                 -> y = z ) ) ) $=
      ( vw cv wbr wal wa weq wi cop wcel nfcv nfbr breq2 albii wfun wmo dffun6f
      wrel mo4f nfv nfan nfim anbi2d equequ2 imbi12d cbval bitr4i df-br anbi12i
      anbi2i imbi1i 2albii 3bitri ) DUADUDZAIZBIZDJZBUBZAKZLUTVCVACIZDJZLZBCMZN
      ZCKZBKZAKZLUTVAVBODPZVAVFODPZLZVINZCKBKZAKZLABDEFUCVEVMUTVDVLAVDVCVAHIZDJ
      ZLZBHMZNZHKZBKVLVCWABHBVAVTDBVAQFBVTQRVBVTVADSUEVKWEBVJWDCHVJHUFWBWCCVCWA
      CCVAVBDCVAQZGCVBQRCVAVTDWFGCVTQRUGWCCUFUHCHMZVHWBVIWCWGVGWAVCVFVTVADSUICH
      BUJUKULTUMTUPVMVSUTVLVRAVJVQBCVHVPVIVCVNVGVOVAVBDUNVAVFDUNUOUQURTUPUS $.
  $}

  $( A function is a relation.  (Contributed by NM, 1-Aug-1994.) $)
  funrel $p |- ( Fun A -> Rel A ) $=
    ( wfun wrel ccnv ccom cid wss df-fun simplbi ) ABACAADEFGAHI $.

  $( A function does not contain the empty set.  (Contributed by BJ,
     26-Nov-2021.) $)
  0nelfun $p |- ( Fun R -> (/) e/ R ) $=
    ( wfun wrel c0 wnel funrel 0nelrel syl ) ABACDAEAFAGH $.

  $( Subclass theorem for function predicate.  (Contributed by NM,
     16-Aug-1994.)  (Proof shortened by Mario Carneiro, 24-Jun-2014.) $)
  funss $p |- ( A C_ B -> ( Fun B -> Fun A ) ) $=
    ( wss wrel ccnv ccom cid wa wfun relss wi coss1 cnvss coss2 syl sstrd sstr2
    anim12d df-fun 3imtr4g ) ABCZBDZBBEZFZGCZHADZAAEZFZGCZHBIAIUAUBUFUEUIABJUAU
    HUDCUEUIKUAUHBUGFZUDABUGLUAUGUCCUJUDCABMUGUCBNOPUHUDGQORBSAST $.

  $( Equality theorem for function predicate.  (Contributed by NM,
     16-Aug-1994.) $)
  funeq $p |- ( A = B -> ( Fun A <-> Fun B ) ) $=
    ( wceq wfun wss wi eqimss2 funss syl eqimss impbid ) ABCZADZBDZLBAEMNFBAGBA
    HILABENMFABJABHIK $.

  ${
    funeqi.1 $e |- A = B $.
    $( Equality inference for the function predicate.  (Contributed by Jonathan
       Ben-Naim, 3-Jun-2011.) $)
    funeqi $p |- ( Fun A <-> Fun B ) $=
      ( wceq wfun wb funeq ax-mp ) ABDAEBEFCABGH $.
  $}

  ${
    funeqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for the function predicate.  (Contributed by NM,
       23-Feb-2013.) $)
    funeqd $p |- ( ph -> ( Fun A <-> Fun B ) ) $=
      ( wceq wfun wb funeq syl ) ABCEBFCFGDBCHI $.
  $}

  ${
    nffun.1 $e |- F/_ x F $.
    $( Bound-variable hypothesis builder for a function.  (Contributed by NM,
       30-Jan-2004.) $)
    nffun $p |- F/ x Fun F $=
      ( wfun wrel ccnv ccom cid wa df-fun nfrel nfcnv nfco nfcv nfss nfan nfxfr
      wss ) BDBEZBBFZGZHRZIABJSUBAABCKAUAHABTCABCLMAHNOPQ $.
  $}

  ${
    $d A w y z $.  $d F w y z $.  $d V w y z $.  $d x w y z $.
    $( Distribute proper substitution through the function predicate.
       (Contributed by Alexander van der Vekens, 23-Jul-2017.) $)
    sbcfung $p |- ( A e. V -> ( [. A / x ]. Fun F <-> Fun [_ A / x ]_ F ) ) $=
      ( vw vy vz wrel cv wbr wa wi wal wsbc sbcal csbconstg bitrd bitrid albidv
      csb wcel weq wfun sbcan sbcrel sbcimg sbcbrg breq12d anbi12d sbcg imbi12d
      dffun2 sbcbii 3bitr4g ) BDUAZCHZEIZFIZCJZUQGIZCJZKZFGUBZLZGMZFMZEMZKZABNZ
      ABCTZHZUQURVJJZUQUTVJJZKZVCLZGMZFMZEMZKZCUCZABNVJUCVIUPABNZVGABNZKUOVSUPV
      GABUDUOWAVKWBVRABCDUEWBVFABNZEMUOVRVFEABOUOWCVQEWCVEABNZFMUOVQVEFABOUOWDV
      PFWDVDABNZGMUOVPVDGABOUOWEVOGUOWEVBABNZVCABNZLVOVBVCABDUFUOWFVNWGVCWFUSAB
      NZVAABNZKUOVNUSVAABUDUOWHVLWIVMUOWHABUQTZABURTZVJJVLABUQURDCUGUOWJUQWKURV
      JABUQDPZABURDPUHQUOWIWJABUTTZVJJVMABUQUTDCUGUOWJUQWMUTVJWLABUTDPUHQUIRVCA
      BDUJUKQSRSRSRUIRVTVHABEFGCULUMEFGVJULUN $.
  $}

  ${
    $d y A $.  $d y F $.
    $( There is exactly one value of a function.  (Contributed by NM,
       22-Apr-2004.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
    funeu $p |- ( ( Fun F /\ A F B ) -> E! y A F y ) $=
      ( wfun wbr wa cv wex weu cdm wcel wrel funrel releldm sylan eldmg ibi syl
      wmo wi funmo adantr df-mo sylib mpd ) DEZBCDFZGZBAHDFZAIZUJAJZUIBDKZLZUKU
      GDMUHUNDNBCDOPUNUKABDUMQRSUIUJATZUKULUAUGUOUHABDUBUCUJAUDUEUF $.

    $( There is exactly one value of a function.  (Contributed by NM,
       3-Aug-1994.) $)
    funeu2 $p |- ( ( Fun F /\ <. A , B >. e. F ) -> E! y <. A , y >. e. F ) $=
      ( cop wcel wfun wbr cv weu df-br wa funeu eubii sylib sylan2br ) BCEDFDGZ
      BCDHZBAIZEDFZAJZBCDKQRLBSDHZAJUAABCDMUBTABSDKNOP $.
  $}

  ${
    $d x y A $.
    $( Alternate definition of a function.  One possibility for the definition
       of a function in [Enderton] p. 42.  (Enderton's definition is ambiguous
       because "there is only one" could mean either "there is at most one" or
       "there is exactly one".  However, ~ dffun8 shows that it does not matter
       which meaning we pick.)  (Contributed by NM, 4-Nov-2002.) $)
    dffun7 $p |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) ) $=
      ( wfun wrel cv wbr wmo wal wa cdm wral dffun6 wcel wi wex vex eldm bitr4i
      moabs imbi1i albii df-ral anbi2i bitri ) CDCEZAFZBFCGZBHZAIZJUFUIACKZLZJA
      BCMUJULUFUJUGUKNZUIOZAIULUIUNAUIUHBPZUIOUNUHBTUMUOUIBUGCAQRUASUBUIAUKUCSU
      DUE $.

    $( Alternate definition of a function.  One possibility for the definition
       of a function in [Enderton] p. 42.  Compare ~ dffun7 .  (Contributed by
       NM, 4-Nov-2002.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
    dffun8 $p |- ( Fun A <-> ( Rel A /\ A. x e. dom A E! y x A y ) ) $=
      ( wfun wrel cv wbr wmo cdm wral wa weu dffun7 wex wi wcel df-mo vex eldm
      wb pm5.5 sylbi bitrid ralbiia anbi2i bitri ) CDCEZAFZBFCGZBHZACIZJZKUGUIB
      LZAUKJZKABCMULUNUGUJUMAUKUJUIBNZUMOZUHUKPZUMUIBQUQUOUPUMTBUHCARSUOUMUAUBU
      CUDUEUF $.

    $( Alternate definition of a function.  (Contributed by NM, 28-Mar-2007.)
       (Revised by NM, 16-Jun-2017.) $)
    dffun9 $p |- ( Fun A <->
                 ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) ) $=
      ( wfun wrel cv wbr wmo cdm wral wa wrmo dffun7 wcel brelrn pm4.71ri mobii
      crn vex df-rmo bitr4i ralbii anbi2i bitri ) CDCEZAFZBFZCGZBHZACIZJZKUEUHB
      CRZLZAUJJZKABCMUKUNUEUIUMAUJUIUGULNZUHKZBHUMUHUPBUHUOUFUGCASBSOPQUHBULTUA
      UBUCUD $.
  $}

  $( An equivalence for the function predicate.  (Contributed by NM,
     13-Aug-2004.) $)
  funfn $p |- ( Fun A <-> A Fn dom A ) $=
    ( wfun cdm wceq wa wfn eqid biantru df-fn bitr4i ) ABZKACZLDZEALFMKLGHALIJ
    $.

  ${
    funfnd.1 $e |- ( ph -> Fun A ) $.
    $( A function is a function over its domain.  (Contributed by Glauco
       Siliprandi, 23-Oct-2021.) $)
    funfnd $p |- ( ph -> A Fn dom A ) $=
      ( wfun cdm wfn funfn sylib ) ABDBBEFCBGH $.
  $}

  $( The identity relation is a function.  Part of Theorem 10.4 of [Quine]
     p. 65.  (Contributed by NM, 30-Apr-1998.) $)
  funi $p |- Fun _I $=
    ( cid wfun wrel ccnv ccom wss reli wceq relcnv coi2 ax-mp cnvi eqtri df-fun
    eqimssi mpbir2an ) ABACAADZEZAFGRARQAQCRQHAIQJKLMOANP $.

  $( The universe is not a function.  (Contributed by Raph Levien,
     27-Jan-2004.) $)
  nfunv $p |- -. Fun _V $=
    ( cvv wfun wrel cxp wcel 0nelxp 0ex wss df-rel biimpi sseld mpi mto funrel
    c0 ) ABACZPOAADZEZAAFPOAERGPAQOPAQHAIJKLMANM $.

  ${
    $d t u v w x y z A $.  $d t u v w x y z B $.
    $( A Kuratowski ordered pair is a function only if its components are
       equal.  (Contributed by NM, 5-Jun-2008.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    funopg $p |- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B ) $=
      ( vu vt vx vy vz vw vv wcel cop wceq cv weq wi cpr vex cvv funeqd imbi12d
      wfun opeq1 eqeq1 opeq2 eqeq2 csn wa wex wrel funrel relop sylib opth opid
      preq1i dfop preq2i snex zfpair2 3eqtr4ri eqeq2i bitr3i wal dffun4 simprbi
      prid1 eleq2 mpbiri prid2 jca w3a opeq12 3adant3 eleq1d 3adant2 anbi12d wb
      opex eqeq12 3adant1 spc3gv mp3an syl2im biimtrid dfsn2 preq2 eqeq2d eqtr3
      eqtr2id expcom biimtrdi com13 imp sylcom exlimdvv mpd vtocl2g 3impia ) AC
      LBDLABMZUCZABNZEOZFOZMZUCZEFPZQAXEMZUCZAXENZQXBXCQEFABCDXDANZXGXJXHXKXLXF
      XIXDAXEUDUAXDAXEUEUBXEBNZXJXBXKXCXMXIXAXEBAUFUAXEBAUGUBXGXDGOZUHZNZXEXNHO
      ZRZNZUIZHUJGUJZXHXGXFUKZYAXFULGHXDXEESZFSZUMUNXGXTXHGHXGXTGHPZXHXTXFXNXNM
      ZXNXQMZRZNZXGYEXTXFXOXRMZNYIXDXEXOXRYCYDUOYJYHXFYFXOXRRZRXOUHZYKRYHYJYFYL
      YKXNGSZUPUQYGYKYFXNXQYMHSZURUSXOXRXNYMUTGHVAURVBVCVDXGIOZJOZMZXFLZYOKOZMZ
      XFLZUIZJKPZQZKVEJVEIVEZYIYFXFLZYGXFLZUIZYEXGYBUUEIJKXFVFVGYIUUFUUGYIUUFYF
      YHLYFYGXNXNYMYMVTVHXFYHYFVIVJYIUUGYGYHLYFYGXNXQYMYNVTVKXFYHYGVIVJVLXNTLZU
      UIXQTLUUEUUHYEQZQYMYMYNUUDUUJIJKXNXNXQTTTIGPZJGPZKHPZVMZUUBUUHUUCYEUUNYRU
      UFUUAUUGUUNYQYFXFUUKUULYQYFNUUMYOYPXNXNVNVOVPUUNYTYGXFUUKUUMYTYGNUULYOYSX
      NXQVNVQVPVRUULUUMUUCYEVSUUKYPXNYSXQWAWBUBWCWDWEWFXPXSYEXHQYEXSXPXHYEXSXEX
      ONZXPXHQYEXRXOXEYEXOXNXNRXRXNWGXNXQXNWHWKWIXPUUOXHXDXEXOWJWLWMWNWOWPWQWRW
      SWT $.
  $}

  ${
    $d x y $.
    $( A class of ordered pairs is a function when there is at most one second
       member for each pair.  (Contributed by NM, 16-May-1995.) $)
    funopab $p |- ( Fun { <. x , y >. | ph } <-> A. x E* y ph ) $=
      ( copab wfun wbr wmo wal wrel relopab nfopab1 nfopab2 dffun6f mpbiran cop
      cv wcel df-br opabid bitri mobii albii ) ABCDZEZBPZCPZUCFZCGZBHZACGZBHUDU
      CIUIABCJBCUCABCKABCLMNUHUJBUGACUGUEUFOUCQAUEUFUCRABCSTUAUBT $.
  $}

  ${
    $d x y $.  $d y A $.
    $( A class of ordered pairs of values is a function.  (Contributed by NM,
       14-Nov-1995.) $)
    funopabeq $p |- Fun { <. x , y >. | y = A } $=
      ( cv wceq copab wfun wmo funopab moeq mpgbir ) BDCEZABFGLBHALABIBCJK $.

    $( A class of ordered pairs of values in the form used by ~ df-mpt is a
       function.  (Contributed by NM, 17-Feb-2013.) $)
    funopab4 $p |- Fun { <. x , y >. | ( ph /\ y = A ) } $=
      ( cv wceq wa copab wss wfun simpr ssopab2i funopabeq funss mp2 ) ACEDFZGZ
      BCHZPBCHZISJRJQPBCAPKLBCDMRSNO $.
  $}

  ${
    $d A y $.  $d B y $.  $d x y $.
    $( A function in maps-to notation is a function.  (Contributed by Mario
       Carneiro, 13-Jan-2013.) $)
    funmpt $p |- Fun ( x e. A |-> B ) $=
      ( vy cmpt wfun cv wcel wceq wa copab funopab4 df-mpt funeqi mpbir ) ABCEZ
      FAGBHZDGCIJADKZFQADCLPRADBCMNO $.
  $}

  ${
    funmpt2.1 $e |- F = ( x e. A |-> B ) $.
    $( Functionality of a class given by a maps-to notation.  (Contributed by
       FL, 17-Feb-2008.)  (Revised by Mario Carneiro, 31-May-2014.) $)
    funmpt2 $p |- Fun F $=
      ( wfun cmpt funmpt funeqi mpbir ) DFABCGZFABCHDKEIJ $.
  $}

  ${
    $d x y z F $.  $d x y z G $.
    $( The composition of two functions is a function.  Exercise 29 of
       [TakeutiZaring] p. 25.  (Contributed by NM, 26-Jan-1997.)  (Proof
       shortened by Andrew Salmon, 17-Sep-2011.) $)
    funco $p |- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) ) $=
      ( vx vy vz wfun ccom wrel wbr wmo cdm wral wex wss dmcoss weu sylibr wcel
      wa cv wal funmo alrimiv ralrimivw dffun8 simprbi r19.26 nfv euexex ralimi
      anim12ci ssralv mpsyl copab df-br df-co eleq2i opabid 3bitri mobii ralbii
      syl cop relco jctil dffun7 ) AFZBFZSZABGZHZCTZDTZVJIZDJZCVJKZLZSVJFVIVQVK
      VIVLETZBIZVRVMAIZSEMZDJZCVPLZVQVPBKZNVIWBCWDLZWCABOVIVSEPZVTDJZEUAZSZCWDL
      ZWEVIWFCWDLZWHCWDLZSWJVGWLVHWKVGWHCWDVGWGEDVRAUBUCUDVHBHWKCEBUEUFUKWFWHCW
      DUGQWIWBCWDVSVTEDVSDUHUIUJVBWBCVPWDULUMVOWBCVPVNWADVNVLVMVCZVJRWMWACDUNZR
      WAVLVMVJUOVJWNWMCDEABUPUQWACDURUSUTVAQABVDVECDVJVFQ $.
  $}

  $( A restriction of a function is a function.  Compare Exercise 18 of
     [TakeutiZaring] p. 25.  (Contributed by NM, 16-Aug-1994.) $)
  funres $p |- ( Fun F -> Fun ( F |` A ) ) $=
    ( cres wss wfun wi resss funss ax-mp ) BACZBDBEJEFBAGJBHI $.

  ${
    funresd.1 $e |- ( ph -> Fun F ) $.
    $( A restriction of a function is a function.  (Contributed by Glauco
       Siliprandi, 2-Jan-2022.) $)
    funresd $p |- ( ph -> Fun ( F |` A ) ) $=
      ( wfun cres funres syl ) ACECBFEDBCGH $.
  $}

  ${
    $d x y F $.  $d x y G $.
    $( The restriction of a function to the domain of a subclass equals the
       subclass.  (Contributed by NM, 15-Aug-1994.) $)
    funssres $p |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G ) $=
      ( vx vy wfun wss wa cdm cres wceq cv cop wcel wb wal vex wi wex imp wrel
      opelres ssel opeldm jcad adantl funeu2 eldm2 ancrd eximdv biimtrid eupick
      a1i syl2an exp43 com23 pm2.43d impd impbid bitr4id alrimivv relres funrel
      weu com34 relss mpan9 eqrel sylancr mpbird ) AEZBAFZGZABHZIZBJZCKZDKZLZVN
      MZVRBMZNZDOCOZVLWACDVLVSVRAMZVPVMMZGZVTVPVQAVMDPZUAVLVTWEVKVTWEQVJVKVTWCW
      DBAVRUBZVTWDQVKVPVQBCPZWFUCULUDUEVLWCWDVTVLWCWDVTQVLWCWDWCVTVJVKWCWDWCVTQ
      ZQZQVJWCVKWJVJWCVKWDWIVJWCGWCDVCWCVTGZDRZWIVKWDGDVPVQAUFVKWDWLWDVTDRVKWLD
      VPBWHUGVKVTWKDVKVTWCWGUHUIUJSWCVTDUKUMUNUOSVDUPUQURUSUTVLVNTBTZVOWBNAVMVA
      VJATVKWMAVBBAVEVFCDVNBVGVHVI $.
  $}

  $( Equality of restrictions of a function and a subclass.  (Contributed by
     NM, 16-Aug-1994.) $)
  fun2ssres $p |- ( ( Fun F /\ G C_ F /\ A C_ dom G ) ->
                  ( F |` A ) = ( G |` A ) ) $=
    ( wfun wss cdm cres wceq wa resabs1 eqcomd funssres reseq1d sylan9eqr 3impa
    ) BDZCBEZACFZEZBAGZCAGZHSPQIZTBRGZAGZUASUDTBARJKUBUCCABCLMNO $.

  ${
    $d x y z F $.  $d x y z G $.
    $( The union of functions with disjoint domains is a function.  Theorem 4.6
       of [Monk1] p. 43.  (Contributed by NM, 12-Aug-1994.) $)
    funun $p |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) ->
                Fun ( F u. G ) ) $=
      ( vx vy vz wfun wa cdm wrel cv wi wal anim12i wo wn 19.21bi opeldm dffun4
      wcel vex cin c0 wceq cun cop funrel relun sylibr elun anbi12i anddi bitri
      adantr disj1 biimpi imnan sylib nsyl orel2 syl con2d orel1 orim12d adantl
      biimtrid simprbi 19.21bbi jaao syld alrimiv alrimivv sylanbrc ) AFZBFZGZA
      HZBHZUAUBUCZGZABUDZIZCJZDJZUEZVTSZWBEJZUEZVTSZGZWCWFUCZKZELZDLCLVTFVOWAVR
      VOAIZBIZGWAVMWMVNWNAUFBUFMABUGUHUMVSWLCDVSWKEVSWIWDASZWGASZGZWDBSZWGBSZGZ
      NZWJWIWQWOWSGZNZWRWPGZWTNZNZVSXAWIWOWRNZWPWSNZGXFWEXGWHXHWDABUIWGABUIUJWO
      WRWPWSUKULVRXFXAKVOVRXCWQXEWTVRXBOXCWQKVRWBVPSZWBVQSZGZXBVRXIXJOKZXKOVRXL
      CVRXLCLCVPVQUNUOPZXIXJUPUQWOXIWSXJWBWCACTZDTZQWBWFBXNETZQMURXBWQUSUTVRXDO
      XEWTKVRXJXIGZXDVRXJXIOKXQOVRXIXJXMVAXJXIUPUQWRXJWPXIWBWCBXNXOQWBWFAXNXPQM
      URXDWTVBUTVCVDVEVOXAWJKVRVMWQWJVNWTVMWQWJKZDEVMXRELDLZCVMWMXSCLCDEARVFPVG
      VNWTWJKZDEVNXTELDLZCVNWNYACLCDEBRVFPVGVHUMVIVJVKCDEVTRVL $.
  $}

  ${
    $d x y $.  $d F y $.  $d G y $.
    $( If the union of classes is a function, there is at most one element in
       relation to an arbitrary element regarding one of these classes.
       (Contributed by AV, 18-Jul-2019.) $)
    fununmo $p |- ( Fun ( F u. G ) -> E* y x F y ) $=
      ( cun wfun cv wbr wmo funmo wo orc brun sylibr moimi syl ) CDEZFAGZBGZQHZ
      BIRSCHZBIBRQJUATBUAUARSDHZKTUAUBLRSCDMNOP $.
  $}

  ${
    $d F x y $.  $d G x y $.
    $( If the union of classes is a function, the classes itselves are
       functions.  (Contributed by AV, 18-Jul-2019.) $)
    fununfun $p |- ( Fun ( F u. G ) -> ( Fun F /\ Fun G ) ) $=
      ( vx vy wrel wa cun funrel relun sylib cv wbr wmo fununmo alrimiv anim12i
      wfun wal dffun6 sylibr simpl simpr uncom funeqi sylbi jca mpancom ) AEZBE
      ZFZABGZQZAQZBQZFULUKEUJUKHABIJUJULFZUMUNUOUHCKZDKZALDMZCRZFUMUJUHULUSUHUI
      UAULURCCDABNOPCDASTUOUIUPUQBLDMZCRZFUNUJUIULVAUHUIUBULUTCULBAGZQUTUKVBABU
      CUDCDBANUEOPCDBSTUFUG $.
  $}

  ${
    $d A x y z $.  $d F x y z $.
    $( A function with removed elements is still a function.  (Contributed by
       AV, 7-Jun-2021.) $)
    fundif $p |- ( Fun F -> Fun ( F \ A ) ) $=
      ( vx vy vz wrel cv wbr wa weq wi wal cdif reldif wn brdif pm2.27 ad2ant2r
      wfun dffun2 syl2anb com12 alimi 2alimi anim12i 3imtr4i ) BFZCGZDGZBHZUHEG
      ZBHZIZDEJZKZELZDLCLZIBAMZFZUHUIURHZUHUKURHZIZUNKZELZDLCLZIBSURSUGUSUQVEBA
      NUPVDCDUOVCEVBUOUNUTUJUHUIAHOZIULUHUKAHOZIUOUNKZVAUHUIBAPUHUKBAPUJULVHVFV
      GUMUNQRUAUBUCUDUECDEBTCDEURTUF $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The converse singleton of an ordered pair is a function.  This is
       equivalent to ~ funsn via ~ cnvsn , but stating it this way allows us to
       skip the sethood assumptions on ` A ` and ` B ` .  (Contributed by NM,
       30-Apr-2015.) $)
    funcnvsn $p |- Fun `' { <. A , B >. } $=
      ( vx vy cop csn ccnv wfun wrel cv wbr wmo wal relcnv wceq moeq wcel brcnv
      vex df-br bitri elsni opth1 syl sylbi moimi ax-mp ax-gen dffun6 mpbir2an
      ) ABEZFZGZHUMICJZDJZUMKZDLZCMULNUQCUOAOZDLUQDAPUPURDUPUOUNEZULQZURUPUOUNU
      LKUTUNUOULCSZDSZRUOUNULTUAUTUSUKOURUSUKUBUOUNABVBVAUCUDUEUFUGUHCDUMUIUJ
      $.

    $( A singleton of an ordered pair is a function.  Theorem 10.5 of [Quine]
       p. 65.  (Contributed by NM, 28-Jun-2011.) $)
    funsng $p |- ( ( A e. V /\ B e. W ) -> Fun { <. A , B >. } ) $=
      ( wcel wa cop csn ccnv wfun funcnvsn wceq cnvsng ancoms funeqd mpbii ) AC
      EZBDEZFZBAGHIZJABGHZJBAKSTUARQTUALBADCMNOP $.

    $( Functionality and domain of the singleton of an ordered pair.
       (Contributed by Mario Carneiro, 30-Apr-2015.) $)
    fnsng $p |- ( ( A e. V /\ B e. W ) -> { <. A , B >. } Fn { A } ) $=
      ( wcel wa cop csn wfun cdm wceq wfn funsng dmsnopg adantl df-fn sylanbrc
      ) ACEZBDEZFABGHZITJAHZKZTUALABCDMSUBRABDNOTUAPQ $.

    funsn.1 $e |- A e. _V $.
    funsn.2 $e |- B e. _V $.
    $( A singleton of an ordered pair is a function.  Theorem 10.5 of [Quine]
       p. 65.  (Contributed by NM, 12-Aug-1994.) $)
    funsn $p |- Fun { <. A , B >. } $=
      ( cvv wcel cop csn wfun funsng mp2an ) AEFBEFABGHICDABEEJK $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d V x y z $.  $d W x y z $.
    $( A function based on the singleton of an ordered pair.  Unlike ~ funsng ,
       this holds even if ` A ` or ` B ` is a proper class.  (Contributed by
       Jim Kingdon, 17-Apr-2022.) $)
    funinsn $p |- Fun ( { <. A , B >. } i^i ( V X. W ) ) $=
      ( vx vy vz cop cxp cv wcel wa wal cvv wceq elin simplbi elsni syl vex csn
      cin wfun wrel weq wi wss inss2 xpss sstri df-rel mpbir sylib simprd eqtr3
      opth syl2an gen2 ax-gen dffun4 mpbir2an ) ABHZUAZCDIZUBZUCVEUDZEJZFJZHZVE
      KZVGGJZHZVEKZLFGUEZUFZGMFMZEMVFVENNIZUGVEVDVQVCVDUHCDUIUJVEUKULVPEVOFGVJV
      HBOZVKBOZVNVMVJVGAOZVRVJVIVBOZVTVRLVJVIVCKZWAVJWBVIVDKVIVCVDPQVIVBRSVGVHA
      BETZFTUPUMUNVMVTVSVMVLVBOZVTVSLVMVLVCKZWDVMWEVLVDKVLVCVDPQVLVBRSVGVKABWCG
      TUPUMUNVHVKBUOUQURUSEFGVEUTVA $.
  $}

  $( A set of two pairs is a function if their first members are different.
     (Contributed by FL, 26-Jun-2011.) $)
  funprg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B )
     -> Fun { <. A , C >. , <. B , D >. } ) $=
    ( wcel wa cop csn wfun cdm cin c0 wceq funsng syl2anc dmsnopg simp1l simp2l
    wne w3a cun cpr simp1r simp2r ineq12d disjsn2 3ad2ant3 eqtrd funun syl21anc
    syl df-pr funeqi sylibr ) AEIZBFIZJZCGIZDHIZJZABUCZUDZACKZLZBDKZLZUEZMZVGVI
    UFZMVFVHMZVJMZVHNZVJNZOZPQVLVFUSVBVNUSUTVDVEUAVAVBVCVEUBZACEGRSVFUTVCVOUSUT
    VDVEUGVAVBVCVEUHZBDFHRSVFVRALZBLZOZPVFVPWAVQWBVFVBVPWAQVSACGTUOVFVCVQWBQVTB
    DHTUOUIVEVAWCPQVDABUJUKULVHVJUMUNVMVKVGVIUPUQUR $.

  $( A set of three pairs is a function if their first members are different.
     (Contributed by Alexander van der Vekens, 5-Dec-2017.) $)
  funtpg $p |- ( ( ( X e. U /\ Y e. V /\ Z e. W )
                   /\ ( A e. F /\ B e. G /\ C e. H )
                   /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) )
                 -> Fun { <. X , A >. , <. Y , B >. , <. Z , C >. } ) $=
    ( wcel w3a wne cop wfun wceq syl wn cpr csn cun ctp cdm cin c0 3simpa simp1
    funprg syl3an simp13 simp23 funsng syl2anc 3ad2ant2 dmpropg dmsnopg ineq12d
    wa wo elpri w3o nner eqcoms 3mix2 3mix3 jaoi 3ianorr disjsn sylibr 3ad2ant3
    con2i eqtrd funun syl21anc df-tp funeqi ) JDMZKHMZLIMZNZAEMZBFMZCGMZNZJKOZJ
    LOZKLOZNZNZJAPZKBPZUAZLCPZUBZUCZQZWLWMWOUDZQWKWNQZWPQZWNUEZWPUEZUFZUGRWRWBV
    SVTUTWFWCWDUTZWJWGWTVSVTWAUHWCWDWEUHZWGWHWIUIJKABDHEFUJUKWKWAWEXAVSVTWAWFWJ
    ULWBWCWDWEWJUMZLCIGUNUOWKXDJKUAZLUBZUFZUGWKXBXHXCXIWKXEXBXHRWFWBXEWJXFUPJAK
    BEFUQSWKWEXCXIRXGLCGURSUSWJWBXJUGRZWFWJLXHMZTXKXLWJXLLJRZLKRZVAZWJTZLJKVBXO
    WGTZWHTZWITZVCZXPXMXTXNXMXRXTXRJLJLVDVEXRXQXSVFSXNXSXTXSKLKLVDVEXSXQXRVGSVH
    WGWHWIVISSVMXHLVJVKVLVNWNWPVOVPWSWQWLWMWOVQVRVK $.

  ${
    funpr.1 $e |- A e. _V $.
    funpr.2 $e |- B e. _V $.
    funpr.3 $e |- C e. _V $.
    funpr.4 $e |- D e. _V $.
    $( A function with a domain of two elements.  (Contributed by Jeff Madsen,
       20-Jun-2010.) $)
    funpr $p |- ( A =/= B -> Fun { <. A , C >. , <. B , D >. } ) $=
      ( cvv wcel wa wne cop cpr wfun pm3.2i funprg mp3an12 ) AIJZBIJZKCIJZDIJZK
      ABLACMBDMNOSTEFPUAUBGHPABCDIIIIQR $.
  $}

  ${
    funtp.1 $e |- A e. _V $.
    funtp.2 $e |- B e. _V $.
    funtp.3 $e |- C e. _V $.
    funtp.4 $e |- D e. _V $.
    funtp.5 $e |- E e. _V $.
    funtp.6 $e |- F e. _V $.
    $( A function with a domain of three elements.  (Contributed by NM,
       14-Sep-2011.) $)
    funtp $p |- ( ( A =/= B /\ A =/= C /\ B =/= C )
                    -> Fun { <. A , D >. , <. B , E >. , <. C , F >. } ) $=
      ( wne cop csn wfun wa cin c0 wceq w3a cpr cun ctp funpr funsn jctir df-pr
      cdm dmprop eqtri dmsnop ineq12i disjsn2 anim12i sylib eqtrid funun syl2an
      undisj1 3impb df-tp funeqi sylibr ) ABMZACMZBCMZUAADNZBENZUBZCFNZOZUCZPZV
      HVIVKUDZPVEVFVGVNVEVJPZVLPZQVJUIZVLUIZRZSTVNVFVGQZVEVPVQABDEGHJKUECFILUFU
      GWAVTAOZBOZUCZCOZRZSVRWDVSWEVRABUBWDADBEJKUJABUHUKCFLULUMWAWBWERSTZWCWERS
      TZQWFSTVFWGVGWHACUNBCUNUOWBWCWEUTUPUQVJVLURUSVAVOVMVHVIVKVBVCVD $.
  $}

  ${
    fnsn.1 $e |- A e. _V $.
    fnsn.2 $e |- B e. _V $.
    $( Functionality and domain of the singleton of an ordered pair.
       (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
    fnsn $p |- { <. A , B >. } Fn { A } $=
      ( cvv wcel cop csn wfn fnsng mp2an ) AEFBEFABGHAHICDABEEJK $.
  $}

  $( Function with a domain of two different values.  (Contributed by FL,
     26-Jun-2011.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  fnprg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B )
         -> { <. A , C >. , <. B , D >. } Fn { A , B } ) $=
    ( wcel wa wne w3a cop cpr wfun cdm wceq wfn funprg dmpropg 3ad2ant2 df-fn
    sylanbrc ) AEIBFIJZCGIDHIJZABKZLACMBDMNZOUGPABNZQZUGUHRABCDEFGHSUEUDUIUFACB
    DGHTUAUGUHUBUC $.

  $( Function with a domain of three different values.  (Contributed by
     Alexander van der Vekens, 5-Dec-2017.) $)
  fntpg $p |- ( ( ( X e. U /\ Y e. V /\ Z e. W )
                  /\ ( A e. F /\ B e. G /\ C e. H )
                  /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) )
           -> { <. X , A >. , <. Y , B >. , <. Z , C >. } Fn { X , Y , Z } ) $=
    ( wcel w3a wne cop cdm wceq csn cun ctp wfun wfn funtpg wa dmsnopg 3ad2ant1
    cpr 3ad2ant2 jca uneq12 syl df-pr eqtr4di dmeqi eqeq1i dmun sylibr 3ad2ant3
    bitri uneq12d df-tp eqtri 3eqtr4g df-fn sylanbrc ) JDMKHMLIMNZAEMZBFMZCGMZN
    ZJKOJLOKLONZNZJAPZKBPZLCPZUAZUBVQQZJKLUAZRVQVSUCABCDEFGHIJKLUDVMVNVOUHZQZVP
    SZQZTZJKUHZLSZTVRVSVMWAWEWCWFVMVNSZQZVOSZQZTZWERZWAWERZVMWKJSZKSZTZWEVMWHWN
    RZWJWORZUEZWKWPRVKVGWSVLVKWQWRVHVIWQVJJAEUFUGVIVHWRVJKBFUFUIUJUIWHWNWJWOUKU
    LJKUMUNWMWGWITZQZWERWLWAXAWEVTWTVNVOUMUOUPXAWKWEWGWIUQUPUTURVKVGWCWFRZVLVJV
    HXBVILCGUFUSUIVAVRVTWBTZQWDVQXCVNVOVPVBUOVTWBUQVCJKLVBVDVQVSVEVF $.

  ${
    fntp.1 $e |- A e. _V $.
    fntp.2 $e |- B e. _V $.
    fntp.3 $e |- C e. _V $.
    fntp.4 $e |- D e. _V $.
    fntp.5 $e |- E e. _V $.
    fntp.6 $e |- F e. _V $.
    $( A function with a domain of three elements.  (Contributed by NM,
       14-Sep-2011.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    fntp $p |- ( ( A =/= B /\ A =/= C /\ B =/= C )
          -> { <. A , D >. , <. B , E >. , <. C , F >. } Fn { A , B , C } ) $=
      ( wne w3a cop ctp wfun cdm wceq wfn funtp dmtpop a1i df-fn sylanbrc ) ABM
      ACMBCMNZADOBEOCFOPZQUGRABCPZSZUGUHTABCDEFGHIJKLUAUIUFADBECFJKLUBUCUGUHUDU
      E $.
  $}

  $( The empty set is a function.  Theorem 10.3 of [Quine] p. 65.  (Contributed
     by NM, 7-Apr-1998.) $)
  fun0 $p |- Fun (/) $=
    ( c0 cop csn wss wfun 0ss 0ex funsn funss mp2 ) AAABCZDKEAEKFAAGGHAKIJ $.

  $( The double converse of a function is a function.  (Contributed by NM,
     21-Sep-2004.) $)
  funcnvcnv $p |- ( Fun A -> Fun `' `' A ) $=
    ( ccnv wss wfun wi cnvcnvss funss ax-mp ) ABBZACADIDEAFIAGH $.

  ${
    $d x y A $.
    $( A simpler equivalence for single-rooted (see ~ funcnv ).  (Contributed
       by NM, 9-Aug-2004.) $)
    funcnv2 $p |- ( Fun `' A <-> A. y E* x x A y ) $=
      ( ccnv wfun cv wbr wmo wrel relcnv dffun6 mpbiran brcnv mobii albii bitri
      wal vex ) CDZEZBFZAFZSGZAHZBQZUBUACGZAHZBQTSIUECJBASKLUDUGBUCUFAUAUBCBRAR
      MNOP $.

    $( The converse of a class is a function iff the class is single-rooted,
       which means that for any ` y ` in the range of ` A ` there is at most
       one ` x ` such that ` x A y ` .  Definition of single-rooted in
       [Enderton] p. 43.  See ~ funcnv2 for a simpler version.  (Contributed by
       NM, 13-Aug-2004.) $)
    funcnv $p |- ( Fun `' A <-> A. y e. ran A E* x x A y ) $=
      ( cv wbr wmo wal crn wcel wi ccnv wfun wral wa vex pm4.71ri mobii moanimv
      brelrn bitri albii funcnv2 df-ral 3bitr4i ) ADZBDZCEZAFZBGUFCHZIZUHJZBGCK
      LUHBUIMUHUKBUHUJUGNZAFUKUGULAUGUJUEUFCAOBOSPQUJUGARTUAABCUBUHBUIUCUD $.

    $( A condition showing a class is single-rooted.  (See ~ funcnv ).
       (Contributed by NM, 26-May-2006.) $)
    funcnv3 $p |- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y ) $=
      ( cv wbr wmo crn wral wex ccnv wfun cdm wreu wcel dfrn2 abeq2i biimpi weu
      wa vex biantrurd ralbiia funcnv df-reu breldm pm4.71ri eu5 3bitr2i ralbii
      eubii 3bitr4i ) ADZBDZCEZAFZBCGZHUNAIZUOSZBUPHCJKUNACLZMZBUPHUOURBUPUMUPN
      ZUQUOVAUQUQBUPABCOPQUAUBABCUCUTURBUPUTULUSNZUNSZARUNARURUNAUSUDUNVCAUNVBU
      LUMCATBTUEUFUJUNAUGUHUIUK $.
  $}

  ${
    $d x y z A $.
    $( Another way of expressing that a class is single-rooted.  Counterpart to
       ~ dffun2 .  (Contributed by Jim Kingdon, 24-Dec-2018.) $)
    funcnveq $p |- ( Fun `' A <->
        A. x A. y A. z ( ( x A y /\ z A y ) -> x = z ) ) $=
      ( ccnv wfun cv wbr wa weq wi wal wrel relcnv dffun2 mpbiran alcom anbi12i
      vex brcnv imbi1i 2albii albii 3bitri ) DEZFZBGZAGZUEHZUGCGZUEHZIZACJZKZCL
      ZALBLZUOBLZALUHUGDHZUJUGDHZIZUMKZCLBLZALUFUEMUPDNBACUEOPUOBAQUQVBAUNVABCU
      LUTUMUIURUKUSUGUHDBSZASTUGUJDVCCSTRUAUBUCUD $.
  $}

  ${
    $d f g x y z w v A $.  $d x y B $.  $d x y R $.
    $( The double converse of a class is a function iff the class is
       single-valued.  Each side is equivalent to Definition 6.4(2) of
       [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued.
       Note that ` A ` is not necessarily a function.  (Contributed by NM,
       13-Aug-2004.) $)
    fun2cnv $p |- ( Fun `' `' A <-> A. x E* y x A y ) $=
      ( ccnv wfun cv wbr wmo wal funcnv2 vex brcnv mobii albii bitri ) CDZDEBFZ
      AFZPGZBHZAIRQCGZBHZAIBAPJTUBASUABQRCBKAKLMNO $.

    $( A single-valued relation is a function.  (See ~ fun2cnv for
       "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24.
       (Contributed by NM, 17-Jan-2006.) $)
    svrelfun $p |- ( Fun A <-> ( Rel A /\ Fun `' `' A ) ) $=
      ( vx vy wfun wrel cv wbr wmo wal wa ccnv dffun6 fun2cnv anbi2i bitr4i ) A
      DAEZBFCFAGCHBIZJPAKKDZJBCALRQPBCAMNO $.

    $( Single-rootedness (see ~ funcnv ) of a class cut down by a cross
       product.  (Contributed by NM, 5-Mar-2007.) $)
    fncnv $p |- ( `' ( R i^i ( A X. B ) ) Fn B <->
                  A. y e. B E! x e. A x R y ) $=
      ( cxp cin ccnv wfn wfun cdm wceq wa cv wbr wral wmo wcel wi 3bitr4i df-fn
      crn wreu df-rn eqeq1i anbi2i wrex wrmo rninxp anbi1i funcnv raleq moanimv
      w3a brinxp2 3anan12 bitri mobii df-rmo imbi2i biimt bitr4id bitrdi bitrid
      ralbiia pm5.32i r19.26 ancom reu5 ralbii 3bitr2i ) ECDFGZHZDIVMJZVMKZDLZM
      VNVLUBZDLZMZANZBNZEOZACUCZBDPZVMDUAVRVPVNVQVODVLUDUEUFVRVNMZWBACUGZWBACUH
      ZMZBDPZVSWDVRWGBDPZMWFBDPZWJMWEWIVRWKWJABCDEUIUJVRVNWJVNVTWAVLOZAQZBVQPZV
      RWJABVLUKVRWNWMBDPWJWMBVQDULWMWGBDWADRZWMWOWGSZWGWOVTCRZWBMZMZAQWOWRAQZSW
      MWPWOWRAUMWLWSAWLWQWOWBUNWSVTWACDEUOWQWOWBUPUQURWGWTWOWBACUSUTTWOWGVAVBVE
      VCVDVFWFWGBDVGTVNVRVHWCWHBDWBACVIVJTVK $.

    $( Two ways of stating that ` A ` is one-to-one (but not necessarily a
       function).  Each side is equivalent to Definition 6.4(3) of
       [TakeutiZaring] p. 24, who use the notation "Un_2 (A)" for one-to-one
       (but not necessarily a function).  (Contributed by NM, 17-Jan-2006.) $)
    fun11 $p |- ( ( Fun `' `' A /\ Fun `' A ) <->
         A. x A. y A. z A. w ( ( x A y /\ z A w ) -> ( x = z <-> y = w ) ) ) $=
      ( cv wbr wa weq wi wal ccnv wfun bi2.04 anbi12i 2albii alcom albii 3bitri
      19.26-2 wb dfbi2 imbi2i pm4.76 nfv breq1 anbi1d imbi1d equsal bitri breq2
      3bitr2i bitr2i wmo fun2cnv mo4 funcnv2 alrot4 3bitr4i ) CFZBFZEGZUTDFZEGZ
      HZBDIZJZBKZDKZCKZAFZVCEGZVDHZACIZJZAKZDKCKZHZVKVAEGZVDHZVNVFUAZJZBKAKZDKC
      KZELZLMZWEMZHWBDKCKBKAKWDVHVPHZDKCKVRWCWHCDWCVNVTVFJZJZVFVTVNJZJZHZBKAKWJ
      BKAKZWLBKZAKZHWHWBWMABWBVTVNVFJZVFVNJZHZJVTWQJZVTWRJZHWMWAWSVTVNVFUBUCVTW
      QWRUDWTWJXAWLVTVNVFNVTVFVNNOULPWJWLABTWNVHWPVPWNWJAKZBKVHWJABQXBVGBWIVGAC
      VGAUEVNVTVEVFVNVSVBVDVKUTVAEUFUGUHUIRUJWOVOAWKVOBDVOBUEVFVTVMVNVFVSVLVDVA
      VCVKEUKUGUHUIROSPVHVPCDTUMWFVJWGVQWFVBBUNZCKVGDKBKZCKVJCBEUOXCXDCVBVDBDVA
      VCUTEUKUPRXDVICVGBDQRSWGVLAUNZDKVOCKAKZDKZVQADEUQXEXFDVLVDACVKUTVCEUFUPRX
      GVPCKZDKVQXFXHDVOACQRVPDCQUJSOWBABCDURUS $.

    $( The union of a chain (with respect to inclusion) of functions is a
       function.  (Contributed by NM, 10-Aug-2004.) $)
    fununi $p |- ( A. f e. A ( Fun f /\ A. g e. A ( f C_ g \/ g C_ f ) ) ->
                 Fun U. A ) $=
      ( vx vy vz vw vv cv wfun wss wo wral wa wrel wcel weq wi wal wex cuni cop
      funrel adantr ralimi reluni sylibr r19.28av anim1d dffun4 simprbi 19.21bi
      ssel 19.21bbi syl9r adantl anim2d jaod funeq sseq1 orbi12d anbi12d anbi2d
      sseq2 cbvral2v ralcom orcom bitrid bitri anbi12i anandir 2ralbii r19.26-2
      imp anidm bitr2i 3bitr3i eluni eeanv an4 ancom 2exbii 3bitr2i imbi1i r2al
      19.23v impexp 2albii albii 3bitr2ri 3imtr4i alrimiv alrimivv syl sylanbrc
      ) BIZJZWPCIZKZWRWPKZLZCAMZNZBAMZAUAZOZDIZEIUBZXEPZXGFIUBZXEPZNZEFQZRZFSZE
      SDSZXEJXDWPOZBAMXFXCXQBAWQXQXBWPUCUDUEBAUFUGXDWQXANZCAMZBAMZXPXCXSBAWQXAC
      AUHUEXTXODEXTXNFGIZJZHIZJZNZYAYCKZYCYAKZLZNZHAMZGAMZXHYAPZXJYCPZNZXMRZHAM
      ZGAMZXTXNYJYPGAYIYOHAYEYHYOYEYFYOYGYDYFYORYBYFYNXHYCPZYMNZYDXMYFYLYRYMYAY
      CXHUMUIYDYSXMRZFYDYTFSZDEYDYCOUUAESDSDEFYCUJUKUNULUOUPYBYGYORYDYGYNYLXJYA
      PZNZYBXMYGYMUUBYLYCYAXJUMUQYBUUCXMRZFYBUUDFSZDEYBYAOUUEESDSDEFYAUJUKUNULU
      OUDURVNUEUEXTXTNYBYHNZHAMGAMZYDYHNZHAMGAMZNZXTYKXTUUGXTUUIXRUUFYBYAWRKZWR
      YAKZLZNBCGHAABGQZWQYBXAUUMWPYAUSUUNWSUUKWTUULWPYAWRUTWPYAWRVDVAVBCHQZUUMY
      HYBUUOUUKYFUULYGWRYCYAVDWRYCYAUTVAVCVEXTXRBAMCAMUUIXRBCAAVFXRUUHWQYAWPKZW
      PYAKZLZNCBGHAACGQZXAUURWQXAWTWSLUUSUURWSWTVGUUSWTUUPWSUUQWRYAWPUTWRYAWPVD
      VAVHVCBHQZWQYDUURYHWPYCUSUUTUUPYFUUQYGWPYCYAVDWPYCYAUTVAVBVEVIVJXTVOYKUUF
      UUHNZHAMGAMUUJYIUVAGHAAYBYDYHVKVLUUFUUHGHAAVMVPVQXNYAAPZYCAPZNZYNNZHTZGTZ
      XMRUVFXMRZGSZYQXLUVGXMXLYLUVBNZGTZYMUVCNZHTZNUVJUVLNZHTGTUVGXIUVKXKUVMGXH
      AVRHXJAVRVJUVJUVLGHVSUVNUVEGHUVNYNUVDNUVEYLUVBYMUVCVTYNUVDWAVIWBWCWDUVFXM
      GWFYQUVDYORZHSGSUVEXMRZHSZGSUVIYOGHAAWEUVPUVOGHUVDYNXMWGWHUVQUVHGUVEXMHWF
      WIWJWCWKWLWMWNDEFXEUJWO $.

    $( The union of a chain (with respect to inclusion) of single-rooted sets
       is single-rooted.  (See ~ funcnv for "single-rooted" definition.)
       (Contributed by NM, 11-Aug-2004.) $)
    funcnvuni $p |- ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) )
                    -> Fun `' U. A ) $=
      ( vy vx vz vw vv cv ccnv wfun wss wo wral wa wceq wrex wi wal weq cbvrexv
      cab cuni cnveq eqeq2d wcel funeqd sseq1 sseq2 orbi12d ralbidv rspcv funeq
      anbi12d biimprcd cnvss orim12i wb sseq12 syl5ibrcom expd syl6com rexlimdv
      ancoms com23 alrimdv anim12ii biimtrid alrimiv df-ral eqeq1 rexbidv ralab
      vex elab anbi2i imbi12i albii bitr2i sylib fununi syl cnvuni cnvex dfiun2
      ciun eqtri funeqi sylibr ) BIZJZKZWJCIZLZWMWJLZMZCANZOZBANZDIZEIZJZPZEAQZ
      DUBZUCZKZAUCJZKWSFIZKZXIGIZLZXKXILZMZGXENZOZFXENZXGWSXIXBPZEAQZXJXKXBPZEA
      QZXNRZGSZOZRZFSZXQWSYEFXSXIHIZJZPZHAQWSYDXRYIEHAEHTXBYHXIXAYGUDUEUAWSYIYD
      HAYGAUFWSYHKZYGWMLZWMYGLZMZCANZOZYIYDRWRYOBYGABHTZWLYJWQYNYPWKYHWJYGUDUGY
      PWPYMCAYPWNYKWOYLWJYGWMUHWJYGWMUIUJUKUNULYJYIXJYNYCYIXJYJXIYHUMUOYNYIYBGY
      NYAYIXNYNXTYIXNRZEAXAAUFYNYGXALZXAYGLZMZXTYQRYMYTCXAACETYKYRYLYSWMXAYGUIW
      MXAYGUHUJULYTXTYIXNYTXNXTYIOZYHXBLZXBYHLZMYRUUBYSUUCYGXAUPXAYGUPUQUUAXLUU
      BXMUUCYIXTXLUUBURXIYHXKXBUSVDXKXBXIYHUSUJUTVAVBVCVEVFVGVBVCVHVIXQXIXEUFZX
      PRZFSYFXPFXEVJUUEYEFUUDXSXPYDXDXSDXIFVNDFTXCXREAWTXIXBVKVLVOXOYCXJXDYAXNG
      DDGTXCXTEAWTXKXBVKVLVMVPVQVRVSVTXEFGWAWBXHXFXHEAXBWFXFEAWCEDAXBXAEVNWDWEW
      GWHWI $.

    $( The union of a chain (with respect to inclusion) of one-to-one functions
       is a one-to-one function.  (Contributed by NM, 11-Aug-2004.) $)
    fun11uni $p |- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\
                   A. g e. A ( f C_ g \/ g C_ f ) ) ->
                   ( Fun U. A /\ Fun `' U. A ) ) $=
      ( cv wfun ccnv wa wo wral cuni simpl anim1i ralimi fununi simpr funcnvuni
      wss syl jca ) BDZEZTFEZGZTCDZQUDTQHCAIZGZBAIZAJZEZUHFEZUGUAUEGZBAIUIUFUKB
      AUCUAUEUAUBKLMABCNRUGUBUEGZBAIUJUFULBAUCUBUEUAUBOLMABCPRS $.
  $}

  $( The intersection with a function is a function.  Exercise 14(a) of
     [Enderton] p. 53.  (Contributed by NM, 19-Mar-2004.)  (Proof shortened by
     Andrew Salmon, 17-Sep-2011.) $)
  funin $p |- ( Fun F -> Fun ( F i^i G ) ) $=
    ( cin wss wfun wi inss1 funss ax-mp ) ABCZADAEJEFABGJAHI $.

  $( The restriction of a one-to-one function is one-to-one.  (Contributed by
     NM, 25-Mar-1998.) $)
  funres11 $p |- ( Fun `' F -> Fun `' ( F |` A ) ) $=
    ( cres wss ccnv wfun wi resss cnvss funss mp2b ) BACZBDLEZBEZDNFMFGBAHLBIMN
    JK $.

  $( The converse of a restricted function.  (Contributed by NM,
     27-Mar-1998.) $)
  funcnvres $p |- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) ) $=
    ( ccnv wfun cima cres cdm df-ima df-rn eqtri reseq2i wceq resss cnvss ax-mp
    crn wss funssres mpan2 eqtr2id ) BCZDZUABAEZFUABAFZCZGZFZUEUCUFUAUCUDPUFBAH
    UDIJKUBUEUAQZUGUELUDBQUHBAMUDBNOUAUERST $.

  $( Converse of a restricted identity function.  (Contributed by FL,
     4-Mar-2007.) $)
  cnvresid $p |- `' ( _I |` A ) = ( _I |` A ) $=
    ( cid ccnv wceq wfun cres cnvi eqcomi funi funeq cima funcnvres imai eqtrdi
    mpbii reseq12i mp2b ) BBCZDZREZBAFZCZUADRBGHSBETIBRJOTUBRBAKZFUAABLRBUCAGAM
    PNQ $.

  $( The converse of a restriction of the converse of a function equals the
     function restricted to the image of its converse.  (Contributed by NM,
     4-May-2005.) $)
  funcnvres2 $p |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) ) $=
    ( wfun ccnv cres cima wceq funcnvcnv funcnvres syl wrel funrel dfrel2 sylib
    reseq1d eqtrd ) BCZBDZAEDZRDZRAFZEZBUAEQTCSUBGBHARIJQTBUAQBKTBGBLBMNOP $.

  $( The image of the preimage of a function.  (Contributed by NM,
     25-May-2004.) $)
  funimacnv $p |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) ) $=
    ( wfun ccnv cima crn cin df-ima funcnvres2 rneqd eqtr4id df-rn ineq2i dmres
    cres cdm dfdm4 3eqtr2ri eqtrdi ) BCZBBDZAEZEZUAAOZDZFZABFZGZTUCBUBOZFUFBUBH
    TUEUIABIJKUHAUAPZGUDPUFUGUJABLMUAANUDQRS $.

  $( A kind of contraposition law that infers a subclass of an image from a
     preimage subclass.  (Contributed by NM, 25-May-2004.) $)
  funimass1 $p |- ( ( Fun F /\ A C_ ran F ) ->
                 ( ( `' F " A ) C_ B -> A C_ ( F " B ) ) ) $=
    ( ccnv cima wss wfun crn wa imass2 funimacnv wceq dfss biimpi eqcomd sseq1d
    cin sylan9eq imbitrid ) CDAEZBFCTEZCBEZFCGZACHZFZIZAUBFTBCJUFUAAUBUCUEUAAUD
    QZAACKUEAUGUEAUGLAUDMNORPS $.

  $( A kind of contraposition law that infers an image subclass from a subclass
     of a preimage.  (Contributed by NM, 25-May-2004.) $)
  funimass2 $p |- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B ) $=
    ( ccnv cima wss wfun imass2 crn cin funimacnv sseq2d inss1 mpi biimtrdi imp
    sstr2 sylan2 ) ACDBEZFCGZCAEZCSEZFZUABFZASCHTUCUDTUCUABCIZJZFZUDTUBUFUABCKL
    UGUFBFUDBUEMUAUFBQNOPR $.

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    $( One direction of ~ imadif .  This direction does not require
       ` Fun ``' F ` .  (Contributed by Jim Kingdon, 25-Dec-2018.) $)
    imadiflem $p |- ( ( F " A ) \ ( F " B ) ) C_ ( F " ( A \ B ) ) $=
      ( vx vy cv wbr wrex wn wa cab cdif cima wcel wex df-rex notbii wal sylibr
      dfima2 alnex 19.29r sylan2br simpl simplr wi simpr ancom imnan bitr4i mpd
      sylib jca32 eldif anbi1i anandir bitri eximi syl2anb difeq12i difab eqtri
      syl ss2abi 3sstr4i ) DFZEFCGZDAHZVGDBHZIZJZEKZVGDABLZHZEKCAMZCBMZLZCVMMVK
      VNEVHVFANZVGJZDOZVFBNZVGJZDOZIZVNVJVGDAPVIWCVGDBPQVTWDJZVFVMNZVGJZDOZVNWE
      VSWBIZJZDOZWHWDVTWIDRWKWBDUAVSWIDUBUCWJWGDWJVSWAIZVGJJZWGWJVSWLVGVSWIUDWJ
      VGWLVRVGWIUEZWJWIVGWLUFZVSWIUGWIVGWAJZIWOWBWPWAVGUHQVGWAUIUJULUKWNUMWGVRW
      LJZVGJWMWFWQVGVFABUNUOVRWLVGUPUQSURVCVGDVMPSUSVDVQVHEKZVIEKZLVLVOWRVPWSDE
      CATDECBTUTVHVIEVAVBDECVMTVE $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    $( The image of a difference is the difference of images.  (Contributed by
       NM, 24-May-1998.) $)
    imadif $p |- ( Fun `' F ->
                 ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) ) $=
      ( vx vy cdif cima cv wbr wrex cab wn wa wcel wex exbii wmo df-rex dfima2
      wi ccnv wfun anandir 19.40 sylbi wal nfv nfe1 funmo vex brcnv mobii sylib
      nfan mopick sylan con2d imnan alrimi ex exancom alnex 3imtr3g anim2d syl5
      eldif anbi1i bitri notbii anbi12i 3imtr4g ss2abdv difab eqtri 3sstr4g wss
      difeq12i imadiflem a1i eqssd ) CUAZUBZCABFZGZCAGZCBGZFZWBDHZEHZCIZDWCJZEK
      WJDAJZWJDBJZLZMZEKZWDWGWBWKWOEWBWHANZWHBNZLZMZWJMZDOZWQWJMZDOZWRWJMZDOZLZ
      MZWKWOXBXDWSWJMZDOZMZWBXHXBXCXIMZDOXKXAXLDWQWSWJUCPXCXIDUDUEWBXJXGXDWBWJW
      SMZDOZXELZDUFZXJXGWBXNXPWBXNMZXODWBXNDWBDUGXMDUHUNXQWRWJLTXOXQWJWRWBWJDQZ
      XNWJWSTWBWIWHWAIZDQXRDWIWAUIXSWJDWIWHCEUJDUJUKULUMWJWSDUOUPUQWRWJURUMUSUT
      WJWSDVAXEDVBVCVDVEWKWHWCNZWJMZDOXBWJDWCRYAXADXTWTWJWHABVFVGPVHWLXDWNXGWJD
      ARWMXFWJDBRVIVJVKVLDECWCSWGWLEKZWMEKZFWPWEYBWFYCDECASDECBSVQWLWMEVMVNVOWG
      WDVPWBABCVRVSVT $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    $( One direction of ~ imain .  This direction does not require
       ` Fun ``' F ` .  (Contributed by Jim Kingdon, 25-Dec-2018.) $)
    imainlem $p |- ( F " ( A i^i B ) ) C_ ( ( F " A ) i^i ( F " B ) ) $=
      ( vx vy cv wbr cin wrex cab wa cima wcel df-rex elin anbi1i anandir sylbi
      wex dfima2 bitri exbii 19.40 anbi12i sylibr ss2abi ineq12i eqtri 3sstr4i
      inab ) DFZEFCGZDABHZIZEJULDAIZULDBIZKZEJZCUMLCALZCBLZHZUNUQEUNUKAMZULKZDS
      ZUKBMZULKZDSZKZUQUNUKUMMZULKZDSZVHULDUMNVKVCVFKZDSVHVJVLDVJVBVEKZULKVLVIV
      MULUKABOPVBVEULQUAUBVCVFDUCRRUOVDUPVGULDANULDBNUDUEUFDECUMTVAUOEJZUPEJZHU
      RUSVNUTVODECATDECBTUGUOUPEUJUHUI $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z F $.
    $( The image of an intersection is the intersection of images.
       (Contributed by Paul Chapman, 11-Apr-2009.) $)
    imain $p |- ( Fun `' F ->
                  ( F " ( A i^i B ) ) = ( ( F " A ) i^i ( F " B ) ) ) $=
      ( vx vy vz ccnv cin cima cv wbr wrex cab wcel wex simpr wal df-rex dfima2
      wa wfun wss imainlem a1i eeanv simprll anim12i wi funcnveq biimpi 19.21bi
      weq 19.21bbi imp sylan2 simprrl eqeltrd elin sylanbrc simprlr jca exlimdv
      eximdv biimtrrid anbi12i 3imtr4g ss2abdv ineq12i inab eqtri 3sstr4g eqssd
      ex ) CGUAZCABHZIZCAIZCBIZHZVPVSUBVNABCUCUDVNDJZEJZCKZDALZFJZWACKZFBLZTZEM
      ZWBDVOLZEMVSVPVNWGWIEVNVTANZWBTZDOZWDBNZWETZFOZTZVTVONZWBTZDOZWGWIWPWKWNT
      ZFOZDOVNWSWKWNDFUEVNXAWRDVNWTWRFVNWTWRVNWTTZWQWBXBWJVTBNWQVNWJWBWNUFXBVTW
      DBWTVNWBWETZDFULZWKWBWNWEWJWBPWMWEPUGVNXCXDVNXCXDUHZEFVNXEFQEQZDVNXFDQDEF
      CUIUJUKUMUNUOVNWKWMWEUPUQVTABURUSVNWJWBWNUTVAVMVBVCVDWCWLWFWOWBDARWEFBRVE
      WBDVORVFVGVSWCEMZWFEMZHWHVQXGVRXHDECASFECBSVHWCWFEVIVJDECVOSVKVL $.
  $}

  ${
    $d A b x y z $.  $d B b x y z $.  $d C z $.
    $( Lemma for ~ funimaexg .  It constitutes the interesting part of
       ~ funimaexg , in which ` B C_ dom A ` .  (Contributed by Jim Kingdon,
       27-Dec-2018.) $)
    funimaexglem $p |- ( ( Fun A /\ B e. C /\ B C_ dom A ) ->
        ( A " B ) e. _V ) $=
      ( vz vx vy vb wcel wss w3a cv wex wrex wel wi wal wral wa simprbi exbii
      wfun cdm cima cvv wbr cab wmo dffun7 3ad2ant1 ssralv 3ad2ant3 mpd alrimiv
      wrel wceq weu sseq1 biimpar 3adant1 jca dffun8 adantl ssel adantr sylsyld
      simp1 rsp ralrimiv zfrep6 3syl raleq exbidv 3ad2ant2 mpbid 3com12 vtocleg
      wb 3expib 3impib df-rex exancom bitri sylib 19.29 nfcv nfmo1 nfralxy nfe1
      ralbii r19.26 mopick ralimi sylbir alrimi eximi syl syl2anc r19.23v albii
      nfan abss sylibr dfima2 sseq1i vex ssex exlimiv ) AUAZBCHZBAUBZIZJZABUCZD
      KZIZDLZXMUDHZXLEKZFKAUEZEBMZFUFZXNIZDLZXPXLXTFDNZOZFPZDLZYCXLXSYDOZEBQZFP
      ZDLZYGXLXSFUGZEBQZDPZXSYDRZFLZEBQZDLZYKXLYMDXLYLEXJQZYMXHXIYSXKXHAUNZYSEF
      AUHSUIXKXHYSYMOXIYLEBXJUJUKULUMXLXSFXNMZEBQZDLZYRXIXHXKUUCXIXHXKUUCXHXKRU
      UCOGBCGKZBUOZXHXKUUCXHUUEXKUUCXHUUEXKJZUUAEUUDQZDLZUUCUUFUUDXJIZXHRZXSFUP
      ZEUUDQUUHUUFUUIXHUUEXKUUIXHUUEUUIXKUUDBXJUQURUSXHUUEXKVFUTUUJUUKEUUDUUJUU
      KEXJQZEGNZXRXJHZUUKXHUULUUIXHYTUULEFAVASVBUUIUUMUUNOXHUUDXJXRVCVDUUKEXJVG
      VEVHXSEFGDVIVJUUEXHUUHUUCVQXKUUEUUGUUBDUUAEUUDBVKVLVMVNVOVRVPVSVOUUBYQDUU
      AYPEBUUAYDXSRFLYPXSFXNVTYDXSFWAWBWITWCYNYRRYMYQRZDLYKYMYQDWDUUOYJDUUOYIFY
      MYQFYLFEBFBWEZXSFWFWGYPFEBUUPYOFWHWGWTUUOYLYPRZEBQYIYLYPEBWJUUQYHEBXSYDFW
      KWLWMWNWOWPWQYJYFDYIYEFXSYDEBWRWSTWCYBYFDXTFXNXATXBXOYBDXMYAXNEFABXCXDTXB
      XOXQDXMXNDXEXFXGWP $.
  $}

  $( Axiom of Replacement using abbreviations.  Axiom 39(vi) of [Quine] p. 284.
     Compare Exercise 9 of [TakeutiZaring] p. 29.  (Contributed by NM,
     10-Sep-2006.) $)
  funimaexg $p |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V ) $=
    ( wfun wcel cdm cin cima cvv wrel wceq simpl funrel cres crn resres reseq2i
    wa incom df-ima eqtr4i resdm reseq1d eqtr3id rneqd 3eqtr4g inex1g wss inss2
    3syl funimaexglem mp3an3 sylan2 eqeltrrd ) ADZBCEZRZABAFZGZHZABHZIUQUOAJZUT
    VAKUOUPLAMVBAUSNZOABNZOUTVAVBVCVDVBVCAURNZBNZVDVFAURBGZNVCAURBPUSVGABURSQUA
    VBVEABAUBUCUDUEAUSTABTUFUJUPUOUSIEZUTIEZBURCUGUOVHUSURUHVIBURUIAUSIUKULUMUN
    $.

  ${
    zfrep5.1 $e |- B e. _V $.
    $( The image of a set under any function is also a set.  Equivalent of
       Axiom of Replacement.  Axiom 39(vi) of [Quine] p. 284.  Compare Exercise
       9 of [TakeutiZaring] p. 29.  (Contributed by NM, 17-Nov-2002.) $)
    funimaex $p |- ( Fun A -> ( A " B ) e. _V ) $=
      ( wfun cvv wcel cima funimaexg mpan2 ) ADBEFABGEFCABEHI $.
  $}

  ${
    $d x z A $.  $d b x y z $.  $d z ph $.
    $( Part of a study of the Axiom of Replacement used by the Isabelle prover.
       The object PrimReplace is apparently the image of the function encoded
       by ` ph ( x , y ) ` i.e. the class ` ( { <. x , y >. | ph } " A ) ` .
       If so, we can prove Isabelle's "Axiom of Replacement" conclusion without
       using the Axiom of Replacement, for which I (N. Megill) currently have
       no explanation.  (Contributed by NM, 26-Oct-2006.)  (Proof shortened by
       Mario Carneiro, 4-Dec-2016.) $)
    isarep1 $p |- ( b e. ( { <. x , y >. | ph } " A ) <->
                 E. x e. A [ b / y ] ph ) $=
      ( vz copab cima wcel wbr wrex wsb vex elima cop wsbc df-br sbsbc 3bitri
      cv opelopabsb sbbii bitr2i rexbii nfs1v nfv sbequ12r cbvrex ) ETZABCGZDHI
      FTZUIUJJZFDKACELZBFLZFDKUMBDKFUIUJDEMNULUNFDULUKUIOUJIACUIPZBUKPZUNUKUIUJ
      QABCUKUIUAUNUOBFLUPUMUOBFACERUBUOBFRUCSUDUNUMFBDUMBFUEUMFUFUMFBUGUHS $.
  $}

  ${
    $d w x y A $.  $d y z $.  $d w ph $.  $d z ph $.
    isarep2.1 $e |- A e. _V $.
    isarep2.2 $e |- A. x e. A A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z ) $.
    $( Part of a study of the Axiom of Replacement used by the Isabelle prover.
       In Isabelle, the sethood of PrimReplace is apparently postulated
       implicitly by its type signature " ` [ ` i, ` [ ` i, i ` ] ` => o ` ] `
       => i", which automatically asserts that it is a set without using any
       axioms.  To prove that it is a set in Metamath, we need the hypotheses
       of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement
       in the form ~ funimaex .  (Contributed by NM, 26-Oct-2006.) $)
    isarep2 $p |- E. w w = ( { <. x , y >. | ph } " A ) $=
      ( copab cima cv wcel wa cvv cres resima resopab wmo wi wal imaeq1i eqtr3i
      wfun funopab wsb weq rspec nfv sylibr moanimv mpbir mpgbir funimaex ax-mp
      mo3 eqeltri isseti ) EABCIZFJZUSBKFLZAMZBCIZFJZNURFOZFJUSVCURFPVDVBFABCFQ
      UAUBVBUCZVCNLVEVACRZBVABCUDVFUTACRZSUTAACDUEMCDUFSDTCTZVGVHBFHUGACDADUHUO
      UIUTACUJUKULVBFGUMUNUPUQ $.
  $}

  $( Equality theorem for function predicate with domain.  (Contributed by NM,
     1-Aug-1994.) $)
  fneq1 $p |- ( F = G -> ( F Fn A <-> G Fn A ) ) $=
    ( wceq wfun cdm wa wfn funeq dmeq eqeq1d anbi12d df-fn 3bitr4g ) BCDZBEZBFZ
    ADZGCEZCFZADZGBAHCAHOPSRUABCIOQTABCJKLBAMCAMN $.

  $( Equality theorem for function predicate with domain.  (Contributed by NM,
     1-Aug-1994.) $)
  fneq2 $p |- ( A = B -> ( F Fn A <-> F Fn B ) ) $=
    ( wceq wfun cdm wa wfn eqeq2 anbi2d df-fn 3bitr4g ) ABDZCEZCFZADZGNOBDZGCAH
    CBHMPQNABOIJCAKCBKL $.

  ${
    fneq1d.1 $e |- ( ph -> F = G ) $.
    $( Equality deduction for function predicate with domain.  (Contributed by
       Paul Chapman, 22-Jun-2011.) $)
    fneq1d $p |- ( ph -> ( F Fn A <-> G Fn A ) ) $=
      ( wceq wfn wb fneq1 syl ) ACDFCBGDBGHEBCDIJ $.
  $}

  ${
    fneq2d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for function predicate with domain.  (Contributed by
       Paul Chapman, 22-Jun-2011.) $)
    fneq2d $p |- ( ph -> ( F Fn A <-> F Fn B ) ) $=
      ( wceq wfn wb fneq2 syl ) ABCFDBGDCGHEBCDIJ $.
  $}

  ${
    fneq12d.1 $e |- ( ph -> F = G ) $.
    fneq12d.2 $e |- ( ph -> A = B ) $.
    $( Equality deduction for function predicate with domain.  (Contributed by
       NM, 26-Jun-2011.) $)
    fneq12d $p |- ( ph -> ( F Fn A <-> G Fn B ) ) $=
      ( wfn fneq1d fneq2d bitrd ) ADBHEBHECHABDEFIABCEGJK $.
  $}

  $( Equality theorem for function predicate with domain.  (Contributed by
     Thierry Arnoux, 31-Jan-2017.) $)
  fneq12 $p |- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) ) $=
    ( wceq wa simpl simpr fneq12d ) CDEZABEZFABCDJKGJKHI $.

  ${
    fneq1i.1 $e |- F = G $.
    $( Equality inference for function predicate with domain.  (Contributed by
       Paul Chapman, 22-Jun-2011.) $)
    fneq1i $p |- ( F Fn A <-> G Fn A ) $=
      ( wceq wfn wb fneq1 ax-mp ) BCEBAFCAFGDABCHI $.
  $}

  ${
    fneq2i.1 $e |- A = B $.
    $( Equality inference for function predicate with domain.  (Contributed by
       NM, 4-Sep-2011.) $)
    fneq2i $p |- ( F Fn A <-> F Fn B ) $=
      ( wceq wfn wb fneq2 ax-mp ) ABECAFCBFGDABCHI $.
  $}

  ${
    nffn.1 $e |- F/_ x F $.
    nffn.2 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for a function with domain.
       (Contributed by NM, 30-Jan-2004.) $)
    nffn $p |- F/ x F Fn A $=
      ( wfn wfun cdm wceq wa df-fn nffun nfdm nfeq nfan nfxfr ) CBFCGZCHZBIZJAC
      BKQSAACDLARBACDMENOP $.
  $}

  $( A function with domain is a function.  (Contributed by NM, 1-Aug-1994.) $)
  fnfun $p |- ( F Fn A -> Fun F ) $=
    ( wfn wfun cdm wceq df-fn simplbi ) BACBDBEAFBAGH $.

  $( A function with domain is a relation.  (Contributed by NM, 1-Aug-1994.) $)
  fnrel $p |- ( F Fn A -> Rel F ) $=
    ( wfn wfun wrel fnfun funrel syl ) BACBDBEABFBGH $.

  $( The domain of a function.  (Contributed by NM, 2-Aug-1994.) $)
  fndm $p |- ( F Fn A -> dom F = A ) $=
    ( wfn wfun cdm wceq df-fn simprbi ) BACBDBEAFBAGH $.

  ${
    fndmi.1 $e |- F Fn A $.
    $( The domain of a function.  (Contributed by Wolf Lammen, 1-Jun-2024.) $)
    fndmi $p |- dom F = A $=
      ( wfn cdm wceq fndm ax-mp ) BADBEAFCABGH $.
  $}

  ${
    fndmd.1 $e |- ( ph -> F Fn A ) $.
    $( The domain of a function.  (Contributed by Glauco Siliprandi,
       23-Oct-2021.) $)
    fndmd $p |- ( ph -> dom F = A ) $=
      ( wfn cdm wceq fndm syl ) ACBECFBGDBCHI $.
  $}

  ${
    funfni.1 $e |- ( ( Fun F /\ B e. dom F ) -> ph ) $.
    $( Inference to convert a function and domain antecedent.  (Contributed by
       NM, 22-Apr-2004.) $)
    funfni $p |- ( ( F Fn A /\ B e. A ) -> ph ) $=
      ( wfn wcel wa wfun cdm fnfun adantr fndm eleq2d biimpar syl2anc ) DBFZCBG
      ZHDIZCDJZGZAQSRBDKLQUARQTBCBDMNOEP $.
  $}

  $( A function has a unique domain.  (Contributed by NM, 11-Aug-1994.) $)
  fndmu $p |- ( ( F Fn A /\ F Fn B ) -> A = B ) $=
    ( wfn cdm fndm sylan9req ) CADCBDACEBACFBCFG $.

  $( The first argument of binary relation on a function belongs to the
     function's domain.  (Contributed by NM, 7-May-2004.) $)
  fnbr $p |- ( ( F Fn A /\ B F C ) -> B e. A ) $=
    ( wfn wbr cdm wcel wrel fnrel releldm sylan fndm eleq2d biimpa syldan ) DAE
    ZBCDFZBDGZHZBAHZQDIRTADJBCDKLQTUAQSABADMNOP $.

  $( The first argument of an ordered pair in a function belongs to the
     function's domain.  (Contributed by NM, 8-Aug-1994.) $)
  fnop $p |- ( ( F Fn A /\ <. B , C >. e. F ) -> B e. A ) $=
    ( cop wcel wfn wbr df-br fnbr sylan2br ) BCEDFDAGBCDHBAFBCDIABCDJK $.

  ${
    $d y F $.  $d y B $.
    $( There is exactly one value of a function.  (Contributed by NM,
       22-Apr-2004.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
    fneu $p |- ( ( F Fn A /\ B e. A ) -> E! y B F y ) $=
      ( cv wbr weu wfun cdm wa wmo funmo adantr wex wb eldmg ibi adantl exmoeu2
      wcel syl mpbid funfni ) CAEDFZAGZBCDDHZCDIZTZJZUDAKZUEUFUJUHACDLMUIUDANZU
      JUEOUHUKUFUHUKACDUGPQRUDASUAUBUC $.

    $( There is exactly one value of a function.  (Contributed by NM,
       7-Nov-1995.) $)
    fneu2 $p |- ( ( F Fn A /\ B e. A ) -> E! y <. B , y >. e. F ) $=
      ( wfn wcel wa cv wbr weu cop fneu df-br eubii sylib ) DBECBFGCAHZDIZAJCPK
      DFZAJABCDLQRACPDMNO $.
  $}

  $( The union of two functions with disjoint domains.  (Contributed by NM,
     22-Sep-2004.) $)
  fnun $p |- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) ->
             ( F u. G ) Fn ( A u. B ) ) $=
    ( wfn wa cin c0 wceq cun wfun wi df-fn ineq12 eqeq1d anbi2d funun biimtrrdi
    cdm dmun uneq12 eqtrid jctird imbitrrdi expd impcom an4s syl2anb imp ) CAEZ
    DBEZFABGZHIZCDJZABJZEZUJCKZCSZAIZFDKZDSZBIZFUMUPLZUKCAMDBMUQUTUSVBVCUSVBFZU
    QUTFZVCVDVEUMUPVDVEUMFZUNKZUNSZUOIZFUPVDVFVGVIVDVFVEURVAGZHIZFVGVDVKUMVEVDV
    JULHURAVABNOPCDQRVDVHURVAJUOCDTURAVABUAUBUCUNUOMUDUEUFUGUHUI $.

  ${
    fnunop.x $e |- ( ph -> X e. _V ) $.
    fnunop.y $e |- ( ph -> Y e. _V ) $.
    fnunop.f $e |- ( ph -> F Fn D ) $.
    fnunop.g $e |- G = ( F u. { <. X , Y >. } ) $.
    fnunop.e $e |- E = ( D u. { X } ) $.
    fnunop.d $e |- ( ph -> -. X e. D ) $.
    $( Extension of a function with a new ordered pair.  (Contributed by NM,
       28-Sep-2013.)  (Revised by Mario Carneiro, 30-Apr-2015.) $)
    fnunsn $p |- ( ph -> G Fn E ) $=
      ( cop csn cun wfn cvv wcel sylibr cin c0 fnsng syl2anc wn disjsn syl21anc
      wceq fnun fneq1i fneq2i bitri ) ADFGNOZPZBFOZPZQZECQZADBQUMUOQZBUOUAUBUHZ
      UQJAFRSGRSUSHIFGRRUCUDAFBSUEUTMBFUFTBUODUMUIUGURUNCQUQCEUNKUJCUPUNLUKULT
      $.
  $}

  $( Composition of two functions.  (Contributed by NM, 22-May-2006.) $)
  fnco $p |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B ) $=
    ( wfn crn wss w3a ccom wfun cdm wceq fnfun funco syl2an 3adant3 fndm sseq2d
    wa biimpar dmcosseq syl 3adant2 3ad2ant2 eqtrd df-fn sylanbrc ) CAEZDBEZDFZ
    AGZHZCDIZJZUMKZBLUMBEUHUIUNUKUHCJDJUNUIACMBDMCDNOPULUODKZBUHUKUOUPLZUIUHUKS
    UJCKZGZUQUHUSUKUHURAUJACQRTCDUAUBUCUIUHUPBLUKBDQUDUEUMBUFUG $.

  $( A function does not change when restricted to its domain.  (Contributed by
     NM, 5-Sep-2004.) $)
  fnresdm $p |- ( F Fn A -> ( F |` A ) = F ) $=
    ( wfn wrel cdm wss cres wceq fnrel fndm eqimss syl relssres syl2anc ) BACZB
    DBEZAFZBAGBHABIOPAHQABJPAKLBAMN $.

  $( A function restricted to a class disjoint with its domain is empty.
     (Contributed by NM, 23-Sep-2004.) $)
  fnresdisj $p |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) ) $=
    ( cres c0 wceq cdm wfn wrel wb relres reldm0 ax-mp dmres incom eqtri ineq1d
    cin fndm eqtrid eqeq1d bitr2id ) CBDZEFZUCGZEFZCAHZABRZEFUCIUDUFJCBKUCLMUGU
    EUHEUGUECGZBRZUHUEBUIRUJCBNBUIOPUGUIABACSQTUAUB $.

  $( Membership in two functions restricted by each other's domain.
     (Contributed by NM, 8-Aug-1994.) $)
  2elresin $p |- ( ( F Fn A /\ G Fn B ) ->
                 ( ( <. x , y >. e. F /\ <. x , z >. e. G ) <->
                   ( <. x , y >. e. ( F |` ( A i^i B ) ) /\
                     <. x , z >. e. ( G |` ( A i^i B ) ) ) ) ) $=
    ( wfn wa cv cop wcel cin cres fnop anim12i vex opres resss sseli wi anbi12d
    an4s elin sylibr biimprd syl ex pm2.43d impbid1 ) FDHZGEHZIZAJZBJZKZFLZUNCJ
    ZKZGLZIZUPFDEMZNZLZUSGVBNZLZIZUMVAVGUMVAVAVGUAZUMVAIZUNVBLZVHVIUNDLZUNELZIZ
    VJUKUQULUTVMUKUQIVKULUTIVLDUNUOFOEUNURGOPUCUNDEUDUEVJVGVAVJVDUQVFUTUNUOFVBB
    QRUNURGVBCQRUBUFUGUHUIVDUQVFUTVCFUPFVBSTVEGUSGVBSTPUJ $.

  $( Restriction of a function with a subclass of its domain.  (Contributed by
     NM, 10-Oct-2007.) $)
  fnssresb $p |- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) ) $=
    ( cres wfn wfun cdm wceq wa wss df-fn fnfun funres biantrurd ssdmres sseq2d
    syl fndm bitr3id bitr3d bitrid ) CBDZBEUBFZUBGBHZIZCAEZBAJZUBBKUFUDUEUGUFUC
    UDUFCFUCACLBCMQNUDBCGZJUFUGBCOUFUHABACRPSTUA $.

  $( Restriction of a function with a subclass of its domain.  (Contributed by
     NM, 2-Aug-1994.) $)
  fnssres $p |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B ) $=
    ( wfn cres wss fnssresb biimpar ) CADCBEBDBAFABCGH $.

  ${
    fnssresd.1 $e |- ( ph -> F Fn A ) $.
    fnssresd.2 $e |- ( ph -> B C_ A ) $.
    $( Restriction of a function to a subclass of its domain.  (Contributed by
       Glauco Siliprandi, 5-Feb-2022.) $)
    fnssresd $p |- ( ph -> ( F |` B ) Fn B ) $=
      ( wfn wss cres fnssres syl2anc ) ADBGCBHDCICGEFBCDJK $.
  $}

  $( Restriction of a function's domain with an intersection.  (Contributed by
     NM, 9-Aug-1994.) $)
  fnresin1 $p |- ( F Fn A -> ( F |` ( A i^i B ) ) Fn ( A i^i B ) ) $=
    ( wfn cin wss cres inss1 fnssres mpan2 ) CADABEZAFCKGKDABHAKCIJ $.

  $( Restriction of a function's domain with an intersection.  (Contributed by
     NM, 9-Aug-1994.) $)
  fnresin2 $p |- ( F Fn A -> ( F |` ( B i^i A ) ) Fn ( B i^i A ) ) $=
    ( wfn cin wss cres inss2 fnssres mpan2 ) CADBAEZAFCKGKDBAHAKCIJ $.

  ${
    $d x y A $.  $d x y F $.
    $( An equivalence for functionality of a restriction.  Compare ~ dffun8 .
       (Contributed by Mario Carneiro, 20-May-2015.) $)
    fnres $p |- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y ) $=
      ( cres wfun cdm wa cv wbr wmo wral ancom wal wcel vex mpbiran 3bitr4i wss
      bitri wceq wex wfn wi brres mobii moanimv albii wrel relres dffun6 df-ral
      weu dmres inss1 eqsstri eqss dfss3 elin2 baib eldm bitrdi ralbiia anbi12i
      cin r19.26 df-fn eu5 ralbii ) DCEZFZVJGZCUAZHZAIZBIZDJZBUBZVQBKZHZACLZVJC
      UCVQBUMZACLVSACLZVRACLZHWDWCHVNWAWCWDMVKWCVMWDVOVPVJJZBKZANZVOCOZVSUDZANV
      KWCWFWIAWFWHVQHZBKWIWEWJBWEVQWHHWJVOVPDCBPUEVQWHMTUFWHVQBUGTUHVKVJUIWGDCU
      JABVJUKQVSACULRVMCVLSZWDVMVLCSWKVLCDGZVECDCUNZCWLUOUPVLCUQQWKVOVLOZACLWDA
      CVLURWNVRACWHWNVOWLOZVRWNWHWOVOCWLVLWMUSUTBVODAPVAVBVCTTVDVRVSACVFRVJCVGW
      BVTACVQBVHVIR $.
  $}

  $( Functionality and domain of restricted identity.  (Contributed by NM,
     27-Aug-2004.) $)
  fnresi $p |- ( _I |` A ) Fn A $=
    ( cid cres wfn wfun cdm wceq funi funres ax-mp dmresi df-fn mpbir2an ) BACZ
    ADNEZNFAGBEOHABIJAKNALM $.

  $( The image of a function's domain is its range.  (Contributed by NM,
     4-Nov-2004.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  fnima $p |- ( F Fn A -> ( F " A ) = ran F ) $=
    ( wfn cima cres crn df-ima fnresdm rneqd eqtrid ) BACZBADBAEZFBFBAGKLBABHIJ
    $.

  $( A function with empty domain is empty.  (Contributed by NM, 15-Apr-1998.)
     (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  fn0 $p |- ( F Fn (/) <-> F = (/) ) $=
    ( c0 wfn wceq wrel cdm fnrel fndm reldm0 biimpar syl2anc wfun fun0 mpbir2an
    dm0 df-fn fneq1 mpbiri impbii ) ABCZABDZTAEZAFBDZUABAGBAHUBUAUCAIJKUATBBCZU
    DBLBFBDMOBBPNBABQRS $.

  $( A class that is disjoint with the domain of a function has an empty image
     under the function.  (Contributed by FL, 24-Jan-2007.) $)
  fnimadisj $p |- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( F " C ) = (/) ) $=
    ( wfn cin c0 wceq wa cdm cima fndm ineq1d eqeq1d biimpar imadisj sylibr ) C
    ADZABEZFGZHCIZBEZFGZCBJFGQUBSQUARFQTABACKLMNCBOP $.

  $( Images under a function never map nonempty sets to empty sets.
     (Contributed by Stefan O'Rear, 21-Jan-2015.) $)
  fnimaeq0 $p |- ( ( F Fn A /\ B C_ A ) ->
      ( ( F " B ) = (/) <-> B = (/) ) ) $=
    ( cima c0 wceq cdm cin wfn wa imadisj incom fndm sseq2d biimpar df-ss sylib
    wss eqtrid eqeq1d bitrid ) CBDEFCGZBHZEFCAIZBARZJZBEFCBKUFUCBEUFUCBUBHZBUBB
    LUFBUBRZUGBFUDUHUEUDUBABACMNOBUBPQSTUA $.

  ${
    $d y z A $.  $d y z B $.  $d x y z $.
    $( Alternate definition for the maps-to notation ~ df-mpt .  (Contributed
       by Mario Carneiro, 30-Dec-2016.) $)
    dfmpt3 $p |- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } ) $=
      ( vy vz cmpt cv wcel wceq copab csn cxp ciun df-mpt cop wex anbi2i 2exbii
      wa velsn eliunxp elopab 3bitr4i eqriv eqtr4i ) ABCFAGZBHZDGZCIZSZADJZABUF
      KCKZLMZADBCNEUMUKEGZUFUHOIZUGUHULHZSZSZDPAPUOUJSZDPAPUNUMHUNUKHURUSADUQUJ
      UOUPUIUGDCTQQRADBULUNUAUJADUNUBUCUDUE $.
  $}

  ${
    $d x y A $.
    fnopabg.1 $e |- F = { <. x , y >. | ( x e. A /\ ph ) } $.
    $( Functionality and domain of an ordered-pair class abstraction.
       (Contributed by NM, 30-Jan-2004.)  (Proof shortened by Mario Carneiro,
       4-Dec-2016.) $)
    fnopabg $p |- ( A. x e. A E! y ph <-> F Fn A ) $=
      ( wmo wex wa wral cv wcel copab wfn weu wfun cdm wceq wal 3bitr4i moanimv
      wi albii funopab df-ral 3bitr4ri dmopab3 anbi12i r19.26 df-fn ancom bitri
      eu5 ralbii fneq1i ) ACGZACHZIZBDJZBKDLZAIZBCMZDNZACOZBDJEDNUPBDJZUQBDJZIV
      BPZVBQDRZIUSVCVEVGVFVHVACGZBSUTUPUBZBSVGVEVIVJBUTACUAUCVABCUDUPBDUEUFABCD
      UGUHUPUQBDUIVBDUJTVDURBDVDUQUPIURACUMUQUPUKULUNDEVBFUOT $.
  $}

  ${
    $d x y A $.
    fnopab.1 $e |- ( x e. A -> E! y ph ) $.
    fnopab.2 $e |- F = { <. x , y >. | ( x e. A /\ ph ) } $.
    $( Functionality and domain of an ordered-pair class abstraction.
       (Contributed by NM, 5-Mar-1996.) $)
    fnopab $p |- F Fn A $=
      ( weu wral wfn rgen fnopabg mpbi ) ACHZBDIEDJNBDFKABCDEGLM $.
  $}

  ${
    $d x y A $.  $d y B $.
    mptfng.1 $e |- F = ( x e. A |-> B ) $.
    $( The maps-to notation defines a function with domain.  (Contributed by
       Scott Fenton, 21-Mar-2011.) $)
    mptfng $p |- ( A. x e. A B e. _V <-> F Fn A ) $=
      ( vy cvv wcel wral cv wceq weu wfn eueq ralbii cmpt wa copab df-mpt eqtri
      fnopabg bitri ) CGHZABIFJCKZFLZABIDBMUCUEABFCNOUDAFBDDABCPAJBHUDQAFREAFBC
      STUAUB $.

    $( The maps-to notation defines a function with domain.  (Contributed by
       NM, 9-Apr-2013.) $)
    fnmpt $p |- ( A. x e. A B e. V -> F Fn A ) $=
      ( wcel wral cvv wfn elex ralimi mptfng sylib ) CEGZABHCIGZABHDBJOPABCEKLA
      BCDFMN $.
  $}

  $( A mapping operation with empty domain.  (Contributed by Mario Carneiro,
     28-Dec-2014.) $)
  mpt0 $p |- ( x e. (/) |-> A ) = (/) $=
    ( c0 cmpt wfn wceq cvv wcel wral ral0 eqid fnmpt ax-mp fn0 mpbi ) ACBDZCEZP
    CFBGHZACIQRAJACBPGPKLMPNO $.

  ${
    $d x A $.
    fnmpti.1 $e |- B e. _V $.
    fnmpti.2 $e |- F = ( x e. A |-> B ) $.
    $( Functionality and domain of an ordered-pair class abstraction.
       (Contributed by NM, 29-Jan-2004.)  (Revised by Mario Carneiro,
       31-Aug-2015.) $)
    fnmpti $p |- F Fn A $=
      ( cvv wcel wral wfn rgenw mptfng mpbi ) CGHZABIDBJNABEKABCDFLM $.

    $( Domain of an ordered-pair class abstraction that specifies a function.
       (Contributed by NM, 6-Sep-2005.)  (Revised by Mario Carneiro,
       31-Aug-2015.) $)
    dmmpti $p |- dom F = A $=
      ( wfn cdm wceq fnmpti fndm ax-mp ) DBGDHBIABCDEFJBDKL $.
  $}

  ${
    $d B x $.  $d ph x $.
    dmmptd.a $e |- A = ( x e. B |-> C ) $.
    dmmptd.c $e |- ( ( ph /\ x e. B ) -> C e. V ) $.
    $( The domain of the mapping operation, deduction form.  (Contributed by
       Glauco Siliprandi, 11-Dec-2019.) $)
    dmmptd $p |- ( ph -> dom A = B ) $=
      ( cdm cvv wcel crab dmmpt wral wceq cv wa elexd ralrimiva rabid2 eqtr4id
      sylibr ) ACIEJKZBDLZDBDECGMAUCBDNDUDOAUCBDABPDKQEFHRSUCBDTUBUA $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.
    $( Union of mappings which are mutually compatible.  (Contributed by Mario
       Carneiro, 31-Aug-2015.) $)
    mptun $p |- ( x e. ( A u. B ) |-> C ) =
        ( ( x e. A |-> C ) u. ( x e. B |-> C ) ) $=
      ( vy cmpt cv wcel wceq wa copab df-mpt uneq12i wo elun anbi1i andir bitri
      cun eqtr4i opabbii unopab ) ABCSZDFAGZUCHZEGDIZJZAEKZABDFZACDFZSZAEUCDLUK
      UDBHZUFJZAEKZUDCHZUFJZAEKZSZUHUIUNUJUQAEBDLAECDLMUHUMUPNZAEKURUGUSAEUGULU
      ONZUFJUSUEUTUFUDBCOPULUOUFQRUAUMUPAEUBTTT $.
  $}

  $( Equality theorem for functions.  (Contributed by NM, 1-Aug-1994.) $)
  feq1 $p |- ( F = G -> ( F : A --> B <-> G : A --> B ) ) $=
    ( wceq wfn crn wss wa wf fneq1 rneq sseq1d anbi12d df-f 3bitr4g ) CDEZCAFZC
    GZBHZIDAFZDGZBHZIABCJABDJQRUATUCACDKQSUBBCDLMNABCOABDOP $.

  $( Equality theorem for functions.  (Contributed by NM, 1-Aug-1994.) $)
  feq2 $p |- ( A = B -> ( F : A --> C <-> F : B --> C ) ) $=
    ( wceq wfn crn wss wa wf fneq2 anbi1d df-f 3bitr4g ) ABEZDAFZDGCHZIDBFZQIAC
    DJBCDJOPRQABDKLACDMBCDMN $.

  $( Equality theorem for functions.  (Contributed by NM, 1-Aug-1994.) $)
  feq3 $p |- ( A = B -> ( F : C --> A <-> F : C --> B ) ) $=
    ( wceq wfn crn wss wa wf sseq2 anbi2d df-f 3bitr4g ) ABEZDCFZDGZAHZIPQBHZIC
    ADJCBDJORSPABQKLCADMCBDMN $.

  $( Equality theorem for functions.  (Contributed by FL, 14-Jul-2007.)  (Proof
     shortened by Andrew Salmon, 17-Sep-2011.) $)
  feq23 $p |- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) ) $=
    ( wceq wf feq2 feq3 sylan9bb ) ACFABEGCBEGBDFCDEGACBEHBDCEIJ $.

  ${
    feq1d.1 $e |- ( ph -> F = G ) $.
    $( Equality deduction for functions.  (Contributed by NM, 19-Feb-2008.) $)
    feq1d $p |- ( ph -> ( F : A --> B <-> G : A --> B ) ) $=
      ( wceq wf wb feq1 syl ) ADEGBCDHBCEHIFBCDEJK $.
  $}

  ${
    feq2d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for functions.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    feq2d $p |- ( ph -> ( F : A --> C <-> F : B --> C ) ) $=
      ( wceq wf wb feq2 syl ) ABCGBDEHCDEHIFBCDEJK $.

    $( Equality deduction for functions.  (Contributed by AV, 1-Jan-2020.) $)
    feq3d $p |- ( ph -> ( F : X --> A <-> F : X --> B ) ) $=
      ( wceq wf wb feq3 syl ) ABCGEBDHECDHIFBCEDJK $.
  $}

  ${
    feq12d.1 $e |- ( ph -> F = G ) $.
    feq12d.2 $e |- ( ph -> A = B ) $.
    $( Equality deduction for functions.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    feq12d $p |- ( ph -> ( F : A --> C <-> G : B --> C ) ) $=
      ( wf feq1d feq2d bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.

    feq123d.3 $e |- ( ph -> C = D ) $.
    $( Equality deduction for functions.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    feq123d $p |- ( ph -> ( F : A --> C <-> G : B --> D ) ) $=
      ( wf feq12d wceq wb feq3 syl bitrd ) ABDFKCDGKZCEGKZABCDFGHILADEMRSNJDECG
      OPQ $.
  $}

  $( Equality theorem for functions.  (Contributed by FL, 16-Nov-2008.) $)
  feq123 $p |- ( ( F = G /\ A = C /\ B = D ) ->
    ( F : A --> B <-> G : C --> D ) ) $=
    ( wceq w3a simp1 simp2 simp3 feq123d ) EFGZACGZBDGZHACBDEFMNOIMNOJMNOKL $.

  ${
    feq1i.1 $e |- F = G $.
    $( Equality inference for functions.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    feq1i $p |- ( F : A --> B <-> G : A --> B ) $=
      ( wceq wf wb feq1 ax-mp ) CDFABCGABDGHEABCDIJ $.
  $}

  ${
    feq2i.1 $e |- A = B $.
    $( Equality inference for functions.  (Contributed by NM, 5-Sep-2011.) $)
    feq2i $p |- ( F : A --> C <-> F : B --> C ) $=
      ( wceq wf wb feq2 ax-mp ) ABFACDGBCDGHEABCDIJ $.
  $}

  ${
    feq23i.1 $e |- A = C $.
    feq23i.2 $e |- B = D $.
    $( Equality inference for functions.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    feq23i $p |- ( F : A --> B <-> F : C --> D ) $=
      ( wceq wf wb feq23 mp2an ) ACHBDHABEICDEIJFGABCDEKL $.
  $}

  ${
    feq23d.1 $e |- ( ph -> A = C ) $.
    feq23d.2 $e |- ( ph -> B = D ) $.
    $( Equality deduction for functions.  (Contributed by NM, 8-Jun-2013.) $)
    feq23d $p |- ( ph -> ( F : A --> B <-> F : C --> D ) ) $=
      ( eqidd feq123d ) ABDCEFFAFIGHJ $.
  $}

  ${
    nff.1 $e |- F/_ x F $.
    nff.2 $e |- F/_ x A $.
    nff.3 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for a mapping.  (Contributed by NM,
       29-Jan-2004.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nff $p |- F/ x F : A --> B $=
      ( wf wfn crn wss wa df-f nffn nfrn nfss nfan nfxfr ) BCDHDBIZDJZCKZLABCDM
      SUAAABDEFNATCADEOGPQR $.
  $}

  ${
    $d V x $.  $d X x $.
    $( Distribute proper substitution through the function predicate with a
       domain.  (Contributed by Alexander van der Vekens, 15-Jul-2018.) $)
    sbcfng $p |- ( X e. V -> ( [. X / x ]. F Fn A
                              <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) ) $=
      ( wcel wfn wsbc wfun cdm wa csb wb df-fn a1i sbcbidv sbcfung sbceqg bitrd
      wceq csbdmg eqeq1d anbi12d sbcan 3bitr4g ) EDFZCBGZAEHCIZCJZBTZKZAEHZAECL
      ZAEBLZGZUFUGUKAEUGUKMUFCBNOPUFUHAEHZUJAEHZKUMIZUMJZUNTZKULUOUFUPURUQUTAEC
      DQUFUQAEUILZUNTUTAEUIBDRUFVAUSUNAECDUAUBSUCUHUJAEUDUMUNNUES $.

    $( Distribute proper substitution through the function predicate with
       domain and codomain.  (Contributed by Alexander van der Vekens,
       15-Jul-2018.) $)
    sbcfg $p |- ( X e. V -> ( [. X / x ]. F : A --> B
                 <-> [_ X / x ]_ F : [_ X / x ]_ A --> [_ X / x ]_ B ) ) $=
      ( wcel wf wsbc wfn crn wss wa csb wb df-f a1i sbcbidv sbcfng bitrd sbcssg
      csbrng sseq1d anbi12d sbcan 3bitr4g ) FEGZBCDHZAFIDBJZDKZCLZMZAFIZAFBNZAF
      CNZAFDNZHZUGUHULAFUHULOUGBCDPQRUGUIAFIZUKAFIZMUPUNJZUPKZUOLZMUMUQUGURUTUS
      VBABDEFSUGUSAFUJNZUOLVBAFUJCEUAUGVCVAUOAFDEUBUCTUDUIUKAFUEUNUOUPPUFT $.
  $}

  $( A mapping is a function.  (Contributed by NM, 2-Aug-1994.) $)
  ffn $p |- ( F : A --> B -> F Fn A ) $=
    ( wf wfn crn wss df-f simplbi ) ABCDCAECFBGABCHI $.

  ${
    ffnd.1 $e |- ( ph -> F : A --> B ) $.
    $( A mapping is a function with domain, deduction form.  (Contributed by
       Glauco Siliprandi, 17-Aug-2020.) $)
    ffnd $p |- ( ph -> F Fn A ) $=
      ( wf wfn ffn syl ) ABCDFDBGEBCDHI $.
  $}

  $( Any function is a mapping into ` _V ` .  (Contributed by NM, 31-Oct-1995.)
     (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  dffn2 $p |- ( F Fn A <-> F : A --> _V ) $=
    ( wfn crn cvv wss wa wf ssv biantru df-f bitr4i ) BACZMBDZEFZGAEBHOMNIJAEBK
    L $.

  $( A mapping is a function.  (Contributed by NM, 3-Aug-1994.) $)
  ffun $p |- ( F : A --> B -> Fun F ) $=
    ( wf wfn wfun ffn fnfun syl ) ABCDCAECFABCGACHI $.

  ${
    ffund.1 $e |- ( ph -> F : A --> B ) $.
    $( A mapping is a function, deduction version.  (Contributed by Glauco
       Siliprandi, 3-Mar-2021.) $)
    ffund $p |- ( ph -> Fun F ) $=
      ( wf wfun ffun syl ) ABCDFDGEBCDHI $.
  $}

  $( A mapping is a relation.  (Contributed by NM, 3-Aug-1994.) $)
  frel $p |- ( F : A --> B -> Rel F ) $=
    ( wf wfn wrel ffn fnrel syl ) ABCDCAECFABCGACHI $.

  $( The domain of a mapping.  (Contributed by NM, 2-Aug-1994.) $)
  fdm $p |- ( F : A --> B -> dom F = A ) $=
    ( wf wfn cdm wceq ffn fndm syl ) ABCDCAECFAGABCHACIJ $.

  ${
    fdmd.1 $e |- ( ph -> F : A --> B ) $.
    $( Deduction form of ~ fdm .  The domain of a mapping.  (Contributed by
       Glauco Siliprandi, 26-Jun-2021.) $)
    fdmd $p |- ( ph -> dom F = A ) $=
      ( wf cdm wceq fdm syl ) ABCDFDGBHEBCDIJ $.
  $}

  ${
    fdmi.1 $e |- F : A --> B $.
    $( The domain of a mapping.  (Contributed by NM, 28-Jul-2008.) $)
    fdmi $p |- dom F = A $=
      ( wf cdm wceq fdm ax-mp ) ABCECFAGDABCHI $.
  $}

  $( The range of a mapping.  (Contributed by NM, 3-Aug-1994.) $)
  frn $p |- ( F : A --> B -> ran F C_ B ) $=
    ( wf wfn crn wss df-f simprbi ) ABCDCAECFBGABCHI $.

  ${
    frnd.1 $e |- ( ph -> F : A --> B ) $.
    $( Deduction form of ~ frn .  The range of a mapping.  (Contributed by
       Glauco Siliprandi, 26-Jun-2021.) $)
    frnd $p |- ( ph -> ran F C_ B ) $=
      ( wf crn wss frn syl ) ABCDFDGCHEBCDIJ $.
  $}

  $( A function maps to its range.  (Contributed by NM, 1-Sep-1999.) $)
  dffn3 $p |- ( F Fn A <-> F : A --> ran F ) $=
    ( wfn crn wss wa wf ssid biantru df-f bitr4i ) BACZLBDZMEZFAMBGNLMHIAMBJK
    $.

  $( A function maps to its range.  (Contributed by Glauco Siliprandi,
     3-Mar-2021.) $)
  ffrn $p |- ( F : A --> B -> F : A --> ran F ) $=
    ( wf wfn crn ffn dffn3 sylib ) ABCDCAEACFCDABCGACHI $.

  $( Expanding the codomain of a mapping.  (Contributed by NM, 10-May-1998.)
     (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  fss $p |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C ) $=
    ( wss wf wfn crn wa sstr2 com12 anim2d df-f 3imtr4g impcom ) BCEZABDFZACDFZ
    PDAGZDHZBEZISTCEZIQRPUAUBSUAPUBTBCJKLABDMACDMNO $.

  ${
    fssd.f $e |- ( ph -> F : A --> B ) $.
    fssd.b $e |- ( ph -> B C_ C ) $.
    $( Expanding the codomain of a mapping, deduction form.  (Contributed by
       Glauco Siliprandi, 11-Dec-2019.) $)
    fssd $p |- ( ph -> F : A --> C ) $=
      ( wf wss fss syl2anc ) ABCEHCDIBDEHFGBCDEJK $.
  $}

  ${
    fssdmd.f $e |- ( ph -> F : A --> B ) $.
    fssdmd.d $e |- ( ph -> D C_ dom F ) $.
    $( Expressing that a class is a subclass of the domain of a function
       expressed in maps-to notation, deduction form.  (Contributed by AV,
       21-Aug-2022.) $)
    fssdmd $p |- ( ph -> D C_ A ) $=
      ( cdm fdmd sseqtrd ) ADEHBGABCEFIJ $.
  $}

  ${
    fssdm.d $e |- D C_ dom F $.
    fssdm.f $e |- ( ph -> F : A --> B ) $.
    $( Expressing that a class is a subclass of the domain of a function
       expressed in maps-to notation, semi-deduction form.  (Contributed by AV,
       21-Aug-2022.) $)
    fssdm $p |- ( ph -> D C_ A ) $=
      ( cdm fdmd sseqtrid ) AEHDBFABCEGIJ $.
  $}

  $( The image of a class under a function with domain and codomain is a subset
     of its codomain.  (Contributed by Glauco Siliprandi, 17-Aug-2020.) $)
  fimass $p |- ( F : A --> B -> ( F " X ) C_ B ) $=
    ( wf cima crn imassrn frn sstrid ) ABCECDFCGBCDHABCIJ $.

  ${
    fimassd.1 $e |- ( ph -> F : A --> B ) $.
    $( The image of a class is a subset of its codomain.  (Contributed by
       Glauco Siliprandi, 23-Oct-2021.) $)
    fimassd $p |- ( ph -> ( F " X ) C_ B ) $=
      ( wf cima wss fimass syl ) ABCDGDEHCIFBCDEJK $.
  $}

  $( Composition of two mappings.  (Contributed by NM, 29-Aug-1999.)  (Proof
     shortened by Andrew Salmon, 17-Sep-2011.) $)
  fco $p |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C ) $=
    ( wf wa ccom wfn crn wss df-f wi fnco 3expib adantr rncoss sstr mpan adantl
    jctird imp syl2anb sylibr ) BCDFZABEFZGDEHZAIZUGJZCKZGZACUGFUEDBIZDJZCKZGZE
    AIZEJBKZGZUKUFBCDLABELUOURUKUOURUHUJULURUHMUNULUPUQUHBADENOPUNUJULUIUMKUNUJ
    DEQUIUMCRSTUAUBUCACUGLUD $.

  ${
    fcod.1 $e |- ( ph -> F : B --> C ) $.
    fcod.2 $e |- ( ph -> G : A --> B ) $.
    $( Composition of two mappings.  (Contributed by Glauco Siliprandi,
       26-Jun-2021.) $)
    fcod $p |- ( ph -> ( F o. G ) : A --> C ) $=
      ( wf ccom fco syl2anc ) ACDEIBCFIBDEFJIGHBCDEFKL $.
  $}

  $( Functionality of a composition with weakened out of domain condition on
     the first argument.  (Contributed by Stefan O'Rear, 11-Mar-2015.) $)
  fco2 $p |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) ->
      ( F o. G ) : A --> C ) $=
    ( cres wf wa ccom fco wceq crn wss frn cores syl adantl feq1d mpbid ) BCDBF
    ZGZABEGZHZACTEIZGACDEIZGABCTEJUCACUDUEUBUDUEKZUAUBELBMUFABENDEBOPQRS $.

  $( A mapping is a class of ordered pairs.  (Contributed by NM, 3-Aug-1994.)
     (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  fssxp $p |- ( F : A --> B -> F C_ ( A X. B ) ) $=
    ( wf cdm crn cxp wrel wss frel relssdmrn syl wceq fdm eqimss xpss12 syl2anc
    frn sstrd ) ABCDZCCEZCFZGZABGZTCHCUCIABCJCKLTUAAIZUBBIUCUDITUAAMUEABCNUAAOL
    ABCRUAAUBBPQS $.

  $( A function with bounded domain and codomain is a set.  This version is
     proven without the Axiom of Replacement.  (Contributed by Mario Carneiro,
     24-Jun-2015.) $)
  fex2 $p |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V ) $=
    ( wf wcel w3a cxp cvv xpexg 3adant1 wss fssxp 3ad2ant1 ssexd ) ABCFZADGZBEG
    ZHCABIZJRSTJGQABDEKLQRCTMSABCNOP $.

  $( Two ways of specifying a partial function from ` A ` to ` B ` .
     (Contributed by NM, 13-Nov-2007.) $)
  funssxp $p |- ( ( Fun F /\ F C_ ( A X. B ) ) <->
             ( F : dom F --> B /\ dom F C_ A ) ) $=
    ( wfun cxp wss wa cdm wf wfn funfn biimpi rnss rnxpss sstrdi anim12i sylibr
    crn df-f jca dmss dmxpss adantl ffun adantr fssxp xpss1 sylan9ss impbii ) C
    DZCABEZFZGZCHZBCIZUNAFZGZUMUOUPUMCUNJZCRZBFZGUOUJURULUTUJURCKLULUSUKRBCUKMA
    BNOPUNBCSQULUPUJULUNUKHACUKUAABUBOUCTUQUJULUOUJUPUNBCUDUEUOUPCUNBEUKUNBCUFU
    NABUGUHTUI $.

  $( A mapping is a partial function.  (Contributed by NM, 25-Nov-2007.) $)
  ffdm $p |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) ) $=
    ( wf cdm wss fdm feq2d ibir wceq eqimss syl jca ) ABCDZCEZBCDZOAFZNPNOABCAB
    CGZHINOAJQROAKLM $.

  ${
    ffdmd.1 $e |- ( ph -> F : A --> B ) $.
    $( The domain of a function.  (Contributed by Glauco Siliprandi,
       26-Jun-2021.) $)
    ffdmd $p |- ( ph -> F : dom F --> B ) $=
      ( cdm wf wss wa ffdm syl simpld ) ADFZCDGZMBHZABCDGNOIEBCDJKL $.
  $}

  $( The members of an ordered pair element of a mapping belong to the
     mapping's domain and codomain.  (Contributed by NM, 10-Dec-2003.)
     (Revised by Mario Carneiro, 26-Apr-2015.) $)
  opelf $p |- ( ( F : A --> B /\ <. C , D >. e. F ) ->
              ( C e. A /\ D e. B ) ) $=
    ( wf cop wcel wa cxp fssxp sseld opelxp imbitrdi imp ) ABEFZCDGZEHZCAHDBHIZ
    PRQABJZHSPETQABEKLCDABMNO $.

  $( The union of two functions with disjoint domains.  (Contributed by NM,
     22-Sep-2004.) $)
  fun $p |- ( ( ( F : A --> C /\ G : B --> D ) /\ ( A i^i B ) = (/) ) ->
             ( F u. G ) : ( A u. B ) --> ( C u. D ) ) $=
    ( cin c0 wceq wf wa cun wfn crn wss fnun expcom wi rnun df-f unss12 anim12d
    eqsstrid a1i anbi12i an4 bitri 3imtr4g impcom ) ABGHIZACEJZBDFJZKZABLZCDLZE
    FLZJZUJEAMZFBMZKZENZCOZFNZDOZKZKZUPUNMZUPNZUOOZKUMUQUJUTVGVEVIUTUJVGABEFPQV
    EVIRUJVEVHVAVCLUOEFSVACVCDUAUCUDUBUMURVBKZUSVDKZKVFUKVJULVKACETBDFTUEURVBUS
    VDUFUGUNUOUPTUHUI $.

  $( The union of two functions with disjoint domains.  (Contributed by Mario
     Carneiro, 12-Mar-2015.) $)
  fun2 $p |- ( ( ( F : A --> C /\ G : B --> C ) /\ ( A i^i B ) = (/) ) ->
             ( F u. G ) : ( A u. B ) --> C ) $=
    ( wf wa cin c0 wceq cun fun wb unidm feq3 ax-mp sylib ) ACDFBCEFGABHIJGABKZ
    CCKZDEKZFZRCTFZABCCDELSCJUAUBMCNSCRTOPQ $.

  ${
    fun2d.f $e |- ( ph -> F : A --> C ) $.
    fun2d.g $e |- ( ph -> G : B --> C ) $.
    fun2d.i $e |- ( ph -> ( A i^i B ) = (/) ) $.
    $( The union of functions with disjoint domains is a function, deduction
       version of ~ fun2 .  (Contributed by AV, 11-Oct-2020.)  (Revised by AV,
       24-Oct-2021.) $)
    fun2d $p |- ( ph -> ( F u. G ) : ( A u. B ) --> C ) $=
      ( wf cin c0 wceq cun fun2 syl21anc ) ABDEJCDFJBCKLMBCNDEFNJGHIBCDEFOP $.
  $}

  $( Composition of two functions.  (Contributed by NM, 22-May-2006.) $)
  fnfco $p |- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B ) $=
    ( wf wfn crn wss wa ccom df-f fnco 3expb sylan2b ) BADECAFZDBFZDGAHZICDJBFZ
    BADKOPQRABCDLMN $.

  $( Restriction of a function with a subclass of its domain.  (Contributed by
     NM, 23-Sep-2004.) $)
  fssres $p |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B ) $=
    ( wf wss cres wfn crn df-f fnssres resss rnss ax-mp sstr mpan anim12i an32s
    wa sylanb sylibr ) ABDEZCAFZSDCGZCHZUDIZBFZSZCBUDEUBDAHZDIZBFZSUCUHABDJUIUC
    UKUHUIUCSUEUKUGACDKUFUJFZUKUGUDDFULDCLUDDMNUFUJBOPQRTCBUDJUA $.

  ${
    fssresd.1 $e |- ( ph -> F : A --> B ) $.
    fssresd.2 $e |- ( ph -> C C_ A ) $.
    $( Restriction of a function with a subclass of its domain, deduction form.
       (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
    fssresd $p |- ( ph -> ( F |` C ) : C --> B ) $=
      ( wf wss cres fssres syl2anc ) ABCEHDBIDCEDJHFGBCDEKL $.
  $}

  $( Restriction of a restricted function with a subclass of its domain.
     (Contributed by NM, 21-Jul-2005.) $)
  fssres2 $p |- ( ( ( F |` A ) : A --> B /\ C C_ A ) ->
                ( F |` C ) : C --> B ) $=
    ( cres wf wss wa fssres wb resabs1 feq1d adantl mpbid ) ABDAEZFZCAGZHCBOCEZ
    FZCBDCEZFZABCOIQSUAJPQCBRTDCAKLMN $.

  $( An identity for the mapping relationship under restriction.  (Contributed
     by Scott Fenton, 4-Sep-2011.)  (Proof shortened by Mario Carneiro,
     26-May-2016.) $)
  fresin $p |- ( F : A --> B -> ( F |` X ) : ( A i^i X ) --> B ) $=
    ( wf cin cres wss inss1 fssres mpan2 resres wfn ffn fnresdm reseq1d eqtr3id
    wceq syl feq1d mpbid ) ABCEZADFZBCUCGZEZUCBCDGZEUBUCAHUEADIABUCCJKUBUCBUDUF
    UBUDCAGZDGUFCADLUBUGCDUBCAMUGCRABCNACOSPQTUA $.

  $( If two functions agree on their common domain, their union contains a
     union of three functions with pairwise disjoint domains.  If we assumed
     the law of the excluded middle, this would be equality rather than subset.
     (Contributed by Jim Kingdon, 28-Dec-2018.) $)
  resasplitss $p |- ( ( F Fn A /\ G Fn B /\
      ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) ->
        ( ( F |` ( A i^i B ) ) u.
          ( ( F |` ( A \ B ) ) u. ( G |` ( B \ A ) ) ) )
        C_ ( F u. G )
      ) $=
    ( wfn cin cres wceq w3a cdif cun uneq1i wss inundifss ssres2 ax-mp eqsstrri
    resundi eqsstrrid fnresdm unidm un4 simp3 uneq1d uneq2d incom eqsstri mp2an
    unss12 eqsstrdi uneq12 syl2an 3adant3 sseqtrd ) CAEZDBEZCABFZGZDUQGZHZIZURC
    ABJZGZDBAJZGZKZKZCAGZDBGZKZCDKZVAVGURURKZVFKZVJVLURVFURUALVAVMURVCKZURVEKZK
    ZVJURVCURVEUBVAVPVNUSVEKZKZVJVAVOVQVNVAURUSVEUOUPUTUCUDUEVNVHMVQVIMVRVJMVNC
    UQVBKZGZVHCUQVBRVSAMVTVHMABNVSACOPQVQDUQVDKZGZVIDUQVDRWABMWBVIMWABAFZVDKBUQ
    WCVDABUFLBANUGWABDOPQVNVHVQVIUIUHUJSSUOUPVJVKHZUTUOVHCHVIDHWDUPACTBDTVHCVID
    UKULUMUN $.

  $( From the union of two functions with disjoint domains, either component
     can be recovered by restriction.  (Contributed by Stefan O'Rear,
     9-Oct-2014.)  (Revised by Jim Kingdon, 18-May-2026.) $)
  fresaunres2disj $p |- ( ( F : A --> C /\ G : B --> C /\
      ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G ) $=
    ( wf cin c0 wceq w3a cun cres resundir simp3 wfn ffn 3ad2ant1 fnresdisj syl
    wb mpbid 3ad2ant2 fnresdm uneq12d eqtrid uncom un0 eqtri eqtrdi ) ACDFZBCEF
    ZABGHIZJZDEKBLZHEKZEUMUNDBLZEBLZKUODEBMUMUPHUQEUMULUPHIZUJUKULNUMDAOZULURTU
    JUKUSULACDPQABDRSUAUMEBOZUQEIUKUJUTULBCEPUBBEUCSUDUEUOEHKEHEUFEUGUHUI $.

  $( From the union of two functions with disjoint domains, either component
     can be recovered by restriction.  (Contributed by Mario Carneiro,
     16-Feb-2015.)  (Revised by Jim Kingdon, 18-May-2026.) $)
  fresaunres1disj $p |- ( ( F : A --> C /\ G : B --> C /\
      ( A i^i B ) = (/) ) ->
    ( ( F u. G ) |` A ) = F ) $=
    ( wf cin c0 wceq w3a cres fresaunres2disj 3com12 incom eqeq1i 3anbi3i uncom
    cun reseq1i 3imtr3i ) ACDFZBCEFZBAGZHIZJEDRZAKZDIZUAUBABGZHIZJDERZAKZDIUBUA
    UDUGBACEDLMUDUIUAUBUCUHHBANOPUFUKDUEUJAEDQSOT $.

  $( Composition of a mapping and restricted identity.  (Contributed by NM,
     13-Dec-2003.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  fcoi1 $p |- ( F : A --> B -> ( F o. ( _I |` A ) ) = F ) $=
    ( wf wfn cid cres ccom wceq ffn wfun cdm df-fn wss eqimss ccnv cnvi reseq1i
    wa syl cnveqi eqtr2i coeq2i cores2 eqtrid wrel funrel coi1 sylan9eqr sylbi
    cnvresid ) ABCDCAEZCFAGZHZCIZABCJULCKZCLZAIZSUOCAMURUPUNCFHZCURUQANZUNUSIUQ
    AOUTUNCFPZAGZPZHUSUMVCCVCUMPUMVBUMVAFAQRUAAUKUBUCCFAUDUETUPCUFUSCICUGCUHTUI
    UJT $.

  $( Composition of restricted identity and a mapping.  (Contributed by NM,
     13-Dec-2003.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  fcoi2 $p |- ( F : A --> B -> ( ( _I |` B ) o. F ) = F ) $=
    ( wf wfn crn wss wa cid cres ccom wceq df-f cores wrel fnrel coi2 sylan9eqr
    syl sylbi ) ABCDCAEZCFBGZHIBJCKZCLABCMUBUAUCICKZCICBNUACOUDCLACPCQSRT $.

  ${
    $d y F $.  $d y A $.  $d y B $.  $d y C $.
    $( There is exactly one value of a function in its codomain.  (Contributed
       by NM, 10-Dec-2003.) $)
    feu $p |- ( ( F : A --> B /\ C e. A ) -> E! y e. B <. C , y >. e. F ) $=
      ( wf wcel wa cv cop weu wreu wfn ffn fneu2 sylan wb opelf simprd ex mpbid
      pm4.71rd eubidv adantr df-reu sylibr ) BCEFZDBGZHZAIZCGZDUJJEGZHZAKZULACL
      UIULAKZUNUGEBMUHUOBCENABDEOPUGUOUNQUHUGULUMAUGULUKUGULUKUGULHUHUKBCDUJERS
      TUBUCUDUAULACUEUF $.
  $}

  ${
    $d x y F $.  $d x y A $.  $d x y B $.
    $( The converse of a restriction of a function.  (Contributed by NM,
       26-Mar-1998.) $)
    fcnvres $p |- ( F : A --> B -> `' ( F |` A ) = ( `' F |` B ) ) $=
      ( vy vx wf cres ccnv relcnv cv cop wcel wa ex pm4.71d vex opelcnv opelres
      bitri bitr4di relres opelf simpld simprd anbi1i bitr3d eqrelrdv ) ABCFZDE
      CAGZHZCHZBGZUIIUKBUAUHEJZDJZKZCLZUNUMKZUJLZUQULLZUHUPUPUMALZMZURUHUPUTUHU
      PUTUHUPMZUTUNBLZABUMUNCUBZUCNOURUOUILVAUNUMUIDPZEPZQUMUNCAVERSTUHUPUPVCMZ
      USUHUPVCUHUPVCVBUTVCVDUDNOUSUQUKLZVCMVGUNUMUKBVFRVHUPVCUNUMCVEVFQUESTUFUG
      $.
  $}

  $( The preimage of a class disjoint with a mapping's codomain is empty.
     (Contributed by FL, 24-Jan-2007.) $)
  fimacnvdisj $p |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) ->
    ( `' F " C ) = (/) ) $=
    ( wf cin c0 wceq wa ccnv cdm cima wss crn df-rn frn adantr eqsstrrid ssdisj
    sylancom imadisj sylibr ) ABDEZBCFGHZIZDJZKZCFGHZUFCLGHUCUDUGBMUHUEUGDNZBDO
    UCUIBMUDABDPQRUGBCSTUFCUAUB $.

  ${
    $d x A $.  $d x B $.  $d x F $.
    fintm.1 $e |- E. x x e. B $.
    $( Function into an intersection.  (Contributed by Jim Kingdon,
       28-Dec-2018.) $)
    fintm $p |- ( F : A --> |^| B <-> A. x e. B F : A --> x ) $=
      ( wfn crn cint wss wa cv wral wf ssint anbi2i wcel wex wb r19.28mv df-f
      ax-mp bitr4i ralbii 3bitr4i ) DBFZDGZCHZIZJZUEUFAKZIZJZACLZBUGDMBUJDMZACL
      UIUEUKACLZJZUMUHUOUEAUFCNOUJCPAQUMUPREUEUKACSUAUBBUGDTUNULACBUJDTUCUD $.
  $}

  $( Mapping into an intersection.  (Contributed by NM, 14-Sep-1999.)  (Proof
     shortened by Andrew Salmon, 17-Sep-2011.) $)
  fin $p |- ( F : A --> ( B i^i C ) <-> ( F : A --> B /\ F : A --> C ) ) $=
    ( wfn crn cin wss wa wf ssin anbi2i anandi bitr3i df-f anbi12i 3bitr4i ) DA
    EZDFZBCGZHZIZRSBHZIZRSCHZIZIZATDJABDJZACDJZIUBRUCUEIZIUGUJUARSBCKLRUCUEMNAT
    DOUHUDUIUFABDOACDOPQ $.

  ${
    $d A x $.  $d B x $.
    fabexg.1 $e |- F = { x | ( x : A --> B /\ ph ) } $.
    $( Existence of a set of functions.  (Contributed by Paul Chapman,
       25-Feb-2008.) $)
    fabexg $p |- ( ( A e. C /\ B e. D ) -> F e. _V ) $=
      ( wcel wa cxp cvv cpw xpexg pwexg wss cv cab wf fssxp velpw sylibr anim1i
      ss2abi eqsstri ssab2 sstri ssexg mpan 3syl ) CEIDFIJCDKZLIUKMZLIZGLIZCDEF
      NUKLOGULPUMUNGBQZULIZAJZBRZULGCDUOSZAJZBRURHUTUQBUSUPAUSUOUKPUPCDUOTBUKUA
      UBUCUDUEABULUFUGGULLUHUIUJ $.
  $}

  ${
    $d x A $.  $d x B $.
    fabex.1 $e |- A e. _V $.
    fabex.2 $e |- B e. _V $.
    fabex.3 $e |- F = { x | ( x : A --> B /\ ph ) } $.
    $( Existence of a set of functions.  (Contributed by NM, 3-Dec-2007.) $)
    fabex $p |- F e. _V $=
      ( cvv wcel fabexg mp2an ) CIJDIJEIJFGABCDIIEHKL $.
  $}

  ${
    fndmexd.1 $e |- ( ph -> F e. V ) $.
    fndmexd.2 $e |- ( ph -> F Fn D ) $.
    $( If a function is a set, its domain is a set.  (Contributed by Rohan
       Ridenour, 13-May-2024.) $)
    fndmexd $p |- ( ph -> D e. _V ) $=
      ( cdm cvv fndmd dmexd eqeltrrd ) ACGBHABCFIACDEJK $.
  $}

  $( If a mapping is a set, its domain is a set.  (Contributed by NM,
     27-Aug-2006.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  dmfex $p |- ( ( F e. C /\ F : A --> B ) -> A e. _V ) $=
    ( wf wcel cvv cdm wceq wi fdm dmexg eleq1 imbitrid syl impcom ) ABDEZDCFZAG
    FZQDHZAIZRSJABDKRTGFUASDCLTAGMNOP $.

  $( The empty function.  (Contributed by NM, 14-Aug-1999.) $)
  f0 $p |- (/) : (/) --> A $=
    ( c0 wf wfn crn wss wceq eqid fn0 mpbir rn0 0ss eqsstri df-f mpbir2an ) BAB
    CBBDZBEZAFPBBGBHBIJQBAKALMBABNO $.

  $( A class is a function with empty codomain iff it and its domain are empty.
     (Contributed by NM, 10-Dec-2003.) $)
  f00 $p |- ( F : A --> (/) <-> ( F = (/) /\ A = (/) ) ) $=
    ( c0 wf wceq wa wfn wfun cdm ffun crn wss frn ss0 syl dm0rn0 df-fn sylanbrc
    sylibr fn0 sylib fdm eqtr3d jca f0 feq1 feq2 sylan9bb mpbiri impbii ) ACBDZ
    BCEZACEZFZUKULUMUKBCGZULUKBHBIZCEZUOACBJUKBKZCEZUQUKURCLUSACBMURNOBPSZBCQRB
    TUAUKUPACACBUBUTUCUDUNUKCCCDZCUEULUKACCDUMVAACBCUFACCCUGUHUIUJ $.

  $( A function with empty domain is empty.  (Contributed by Alexander van der
     Vekens, 30-Jun-2018.) $)
  f0bi $p |- ( F : (/) --> X <-> F = (/) ) $=
    ( c0 wf wceq wfn ffn fn0 sylib f0 feq1 mpbiri impbii ) CBADZACEZNACFOCBAGAH
    IONCBCDBJCBACKLM $.

  $( A function is empty iff it has an empty domain.  (Contributed by AV,
     10-Feb-2019.) $)
  f0dom0 $p |- ( F : X --> Y -> ( X = (/) <-> F = (/) ) ) $=
    ( wf wceq feq2 f0bi biimpi biimtrdi com12 feq1 cdm fdm dm0 eqtr3di impbid
    c0 ) BCADZBQEZAQEZSRTSRQCADZTBQCAFUATACGHIJTRSTRBCQDZSBCAQKUBQLBQBCQMNOIJP
    $.

  ${
    $d E y $.  $d Y y $.
    $( If there is no element in the range of a function, its domain must be
       empty.  (Contributed by Alexander van der Vekens, 12-Jul-2018.) $)
    f0rn0 $p |- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> X = (/) ) $=
      ( wf cv crn wcel wrex wn c0 wceq cdm wi fdm wa cin sylbi biimtrrid imp ex
      wss frn wral ralnex disj df-ss incom eqeq1i eqtr2 adantl dm0rn0 sylibr wb
      syl eqeq1 eqcoms adantr mpbird exp32 mpcom ) CDBEZAFBGZHZADIJZCKLZBMZCLZV
      BVEVFNCDBOVHVBVEVFVHVBVEPZPZVFVGKLZVJVCKLZVKVIVLVHVBVEVLVBVCDUBZVEVLNCDBU
      CVEVDJADUDZVMVLVDADUEVNDVCQZKLZVMVLADVCUFVMVCDQZVCLZVPVLNZVCDUGVRVOVCLZVS
      VQVOVCVCDUHUIVTVPVLVOVCKUJUARRSSUOTUKBULUMVHVFVKUNZVIWACVGCVGKUPUQURUSUTV
      AT $.
  $}

  ${
    $d x A $.  $d x B $.
    fconst.1 $e |- B e. _V $.
    $( A cross product with a singleton is a constant function.  (Contributed
       by NM, 14-Aug-1999.)  (Proof shortened by Andrew Salmon,
       17-Sep-2011.) $)
    fconst $p |- ( A X. { B } ) : A --> { B } $=
      ( vx csn cxp wf wfn crn wss fconstmpt fnmpti rnxpss df-f mpbir2an ) ABEZA
      PFZGQAHQIPJDABQCDABKLAPMAPQNO $.
  $}

  ${
    $d x A $.  $d x B $.
    $( A cross product with a singleton is a constant function.  (Contributed
       by NM, 19-Oct-2004.) $)
    fconstg $p |- ( B e. V -> ( A X. { B } ) : A --> { B } ) $=
      ( vx cv csn cxp wf wceq sneq xpeq2d feq1 feq3 sylan9bb syl2anc vex fconst
      wb vtoclg ) ADEZFZAUAGZHZABFZAUDGZHZDBCTBIZUBUEIZUAUDIZUCUFRUGUAUDATBJZKU
      JUHUCAUAUEHUIUFAUAUBUELUAUDAUEMNOATDPQS $.
  $}

  $( A cross product with a singleton is a constant function.  (Contributed by
     NM, 24-Jul-2014.) $)
  fnconstg $p |- ( B e. V -> ( A X. { B } ) Fn A ) $=
    ( wcel csn cxp wf wfn fconstg ffn syl ) BCDABEZALFZGMAHABCIALMJK $.

  $( Constant function with loose range.  (Contributed by Stefan O'Rear,
     1-Feb-2015.) $)
  fconst6g $p |- ( B e. C -> ( A X. { B } ) : A --> C ) $=
    ( wcel csn cxp wf wss fconstg snssi fss syl2anc ) BCDABEZAMFZGMCHACNGABCIBC
    JAMCNKL $.

  ${
    fconst6.1 $e |- B e. C $.
    $( A constant function as a mapping.  (Contributed by Jeff Madsen,
       30-Nov-2009.)  (Revised by Mario Carneiro, 22-Apr-2015.) $)
    fconst6 $p |- ( A X. { B } ) : A --> C $=
      ( wcel csn cxp wf fconst6g ax-mp ) BCEACABFGHDABCIJ $.
  $}

  $( Equality theorem for one-to-one functions.  (Contributed by NM,
     10-Feb-1997.) $)
  f1eq1 $p |- ( F = G -> ( F : A -1-1-> B <-> G : A -1-1-> B ) ) $=
    ( wceq wf ccnv wfun wa wf1 feq1 cnveq funeqd anbi12d df-f1 3bitr4g ) CDEZAB
    CFZCGZHZIABDFZDGZHZIABCJABDJQRUATUCABCDKQSUBCDLMNABCOABDOP $.

  $( Equality theorem for one-to-one functions.  (Contributed by NM,
     10-Feb-1997.) $)
  f1eq2 $p |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) ) $=
    ( wceq wf ccnv wfun wa wf1 feq2 anbi1d df-f1 3bitr4g ) ABEZACDFZDGHZIBCDFZQ
    IACDJBCDJOPRQABCDKLACDMBCDMN $.

  $( Equality theorem for one-to-one functions.  (Contributed by NM,
     10-Feb-1997.) $)
  f1eq3 $p |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) ) $=
    ( wceq wf ccnv wfun wa wf1 feq3 anbi1d df-f1 3bitr4g ) ABEZCADFZDGHZICBDFZQ
    ICADJCBDJOPRQABCDKLCADMCBDMN $.

  ${
    nff1.1 $e |- F/_ x F $.
    nff1.2 $e |- F/_ x A $.
    nff1.3 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for a one-to-one function.
       (Contributed by NM, 16-May-2004.) $)
    nff1 $p |- F/ x F : A -1-1-> B $=
      ( wf1 wf ccnv wfun wa df-f1 nff nfcnv nffun nfan nfxfr ) BCDHBCDIZDJZKZLA
      BCDMSUAAABCDEFGNATADEOPQR $.
  $}

  ${
    $d x y F $.
    $( Alternate definition of a one-to-one function.  (Contributed by NM,
       31-Dec-1996.) $)
    dff12 $p |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. y E* x x F y ) ) $=
      ( wf1 wf ccnv wfun wa cv wbr wmo wal df-f1 funcnv2 anbi2i bitri ) CDEFCDE
      GZEHIZJSAKBKELAMBNZJCDEOTUASABEPQR $.
  $}

  $( A one-to-one mapping is a mapping.  (Contributed by NM, 31-Dec-1996.) $)
  f1f $p |- ( F : A -1-1-> B -> F : A --> B ) $=
    ( wf1 wf ccnv wfun df-f1 simplbi ) ABCDABCECFGABCHI $.

  $( The range of a one-to-one mapping.  (Contributed by BJ, 6-Jul-2022.) $)
  f1rn $p |- ( F : A -1-1-> B -> ran F C_ B ) $=
    ( wf1 wf crn wss f1f frn syl ) ABCDABCECFBGABCHABCIJ $.

  $( A one-to-one mapping is a function on its domain.  (Contributed by NM,
     8-Mar-2014.) $)
  f1fn $p |- ( F : A -1-1-> B -> F Fn A ) $=
    ( wf1 wf wfn f1f ffn syl ) ABCDABCECAFABCGABCHI $.

  $( A one-to-one mapping is a function.  (Contributed by NM, 8-Mar-2014.) $)
  f1fun $p |- ( F : A -1-1-> B -> Fun F ) $=
    ( wf1 wfn wfun f1fn fnfun syl ) ABCDCAECFABCGACHI $.

  $( A one-to-one onto mapping is a relation.  (Contributed by NM,
     8-Mar-2014.) $)
  f1rel $p |- ( F : A -1-1-> B -> Rel F ) $=
    ( wf1 wfn wrel f1fn fnrel syl ) ABCDCAECFABCGACHI $.

  $( The domain of a one-to-one mapping.  (Contributed by NM, 8-Mar-2014.) $)
  f1dm $p |- ( F : A -1-1-> B -> dom F = A ) $=
    ( wf1 wfn cdm wceq f1fn fndm syl ) ABCDCAECFAGABCHACIJ $.

  $( A function that is one-to-one is also one-to-one on some superset of its
     range.  (Contributed by Mario Carneiro, 12-Jan-2013.) $)
  f1ss $p |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C ) $=
    ( wf1 wss wa wf ccnv wfun f1f fss sylan df-f1 simprbi adantr sylanbrc ) ABD
    EZBCFZGACDHZDIJZACDERABDHZSTABDKABCDLMRUASRUBUAABDNOPACDNQ $.

  $( Combine a one-to-one function with a restriction on the domain.
     (Contributed by Stefan O'Rear, 20-Feb-2015.) $)
  f1ssr $p |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C ) $=
    ( wf1 crn wss wa wf ccnv wfun f1fn adantr simpr df-f sylanbrc df-f1 simprbi
    wfn ) ABDEZDFCGZHZACDIZDJKZACDEUBDASZUAUCTUEUAABDLMTUANACDOPTUDUATABDIUDABD
    QRMACDQP $.

  $( If a function is one-to-one from ` A ` to ` B ` and is also a function
     from ` A ` to ` C ` , then it is a one-to-one function from ` A ` to
     ` C ` .  (Contributed by BJ, 4-Jul-2022.) $)
  f1ff1 $p |- ( ( F : A -1-1-> B /\ F : A --> C ) -> F : A -1-1-> C ) $=
    ( wf wf1 crn wss frn f1ssr sylan2 ) ACDEABDFDGCHACDFACDIABCDJK $.

  $( A function that is one-to-one is also one-to-one on any subclass of its
     domain.  (Contributed by Mario Carneiro, 17-Jan-2015.) $)
  f1ssres $p |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-> B ) $=
    ( wf1 wss wa cres wf ccnv wfun f1f fssres sylan simprbi funres11 syl adantr
    df-f1 sylanbrc ) ABDEZCAFZGCBDCHZIZUCJKZCBUCEUAABDIZUBUDABDLABCDMNUAUEUBUAD
    JKZUEUAUFUGABDSOCDPQRCBUCST $.

  $( The restriction of an injective function is injective.  (Contributed by
     AV, 28-Jun-2022.) $)
  f1resf1 $p |- ( ( ( F : A -1-1-> B /\ C C_ A ) /\ ( F |` C ) : C --> D )
                  -> ( F |` C ) : C -1-1-> D ) $=
    ( wf1 wss wa cres wf f1ssres f1ff1 sylan ) ABEFCAGHCBECIZFCDNJCDNFABCEKCBDN
    LM $.

  $( Two ways to express that a set ` A ` (not necessarily a function) is
     one-to-one.  Each side is equivalent to Definition 6.4(3) of
     [TakeutiZaring] p. 24, who use the notation "Un_2 (A)" for one-to-one.  We
     do not introduce a separate notation since we rarely use it.  (Contributed
     by NM, 13-Aug-2004.) $)
  f1cnvcnv $p |- ( `' `' A : dom A -1-1-> _V
             <-> ( Fun `' A /\ Fun `' `' A ) ) $=
    ( cdm cvv ccnv wf1 wf wfun wa df-f1 wfn dffn2 wceq dmcnvcnv mpbiran2 bitr3i
    df-fn wrel relcnv dfrel2 mpbi funeqi anbi12ci bitri ) ABZCADZDZEUDCUFFZUFDZ
    GZHUEGZUFGZHUDCUFIUGUKUIUJUGUFUDJZUKUDUFKULUKUFBUDLAMUFUDPNOUHUEUEQUHUELARU
    ESTUAUBUC $.

  $( Composition of one-to-one functions.  Exercise 30 of [TakeutiZaring]
     p. 25.  (Contributed by NM, 28-May-1998.) $)
  f1co $p |- ( ( F : B -1-1-> C /\ G : A -1-1-> B ) ->
              ( F o. G ) : A -1-1-> C ) $=
    ( wf1 wa ccom wf ccnv wfun df-f1 fco funco cnvco funeqi sylibr anim12i an4s
    ancoms syl2anb ) BCDFZABEFZGACDEHZIZUDJZKZGZACUDFUBBCDIZDJZKZGABEIZEJZKZGUH
    UCBCDLABELUIULUKUNUHUIULGUEUKUNGUGABCDEMUNUKUGUNUKGUMUJHZKUGUMUJNUFUODEOPQT
    RSUAACUDLQ $.

  $( Equality theorem for onto functions.  (Contributed by NM, 1-Aug-1994.) $)
  foeq1 $p |- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) ) $=
    ( wceq wfn crn wa wfo fneq1 rneq eqeq1d anbi12d df-fo 3bitr4g ) CDEZCAFZCGZ
    BEZHDAFZDGZBEZHABCIABDIPQTSUBACDJPRUABCDKLMABCNABDNO $.

  $( Equality theorem for onto functions.  (Contributed by NM, 1-Aug-1994.) $)
  foeq2 $p |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) ) $=
    ( wceq wfn crn wa wfo fneq2 anbi1d df-fo 3bitr4g ) ABEZDAFZDGCEZHDBFZPHACDI
    BCDINOQPABDJKACDLBCDLM $.

  $( Equality theorem for onto functions.  (Contributed by NM, 1-Aug-1994.) $)
  foeq3 $p |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) ) $=
    ( wceq wfn crn wa wfo eqeq2 anbi2d df-fo 3bitr4g ) ABEZDCFZDGZAEZHOPBEZHCAD
    ICBDINQROABPJKCADLCBDLM $.

  ${
    nffo.1 $e |- F/_ x F $.
    nffo.2 $e |- F/_ x A $.
    nffo.3 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for an onto function.  (Contributed by
       NM, 16-May-2004.) $)
    nffo $p |- F/ x F : A -onto-> B $=
      ( wfo wfn crn wceq wa df-fo nffn nfrn nfeq nfan nfxfr ) BCDHDBIZDJZCKZLAB
      CDMSUAAABDEFNATCADEOGPQR $.
  $}

  $( An onto mapping is a mapping.  (Contributed by NM, 3-Aug-1994.) $)
  fof $p |- ( F : A -onto-> B -> F : A --> B ) $=
    ( wfn crn wceq wa wss wfo wf eqimss anim2i df-fo df-f 3imtr4i ) CADZCEZBFZG
    PQBHZGABCIABCJRSPQBKLABCMABCNO $.

  $( An onto mapping is a function.  (Contributed by NM, 29-Mar-2008.) $)
  fofun $p |- ( F : A -onto-> B -> Fun F ) $=
    ( wfo wf wfun fof ffun syl ) ABCDABCECFABCGABCHI $.

  $( An onto mapping is a function on its domain.  (Contributed by NM,
     16-Dec-2008.) $)
  fofn $p |- ( F : A -onto-> B -> F Fn A ) $=
    ( wfo wf wfn fof ffn syl ) ABCDABCECAFABCGABCHI $.

  $( The codomain of an onto function is its range.  (Contributed by NM,
     3-Aug-1994.) $)
  forn $p |- ( F : A -onto-> B -> ran F = B ) $=
    ( wfo wfn crn wceq df-fo simprbi ) ABCDCAECFBGABCHI $.

  $( Alternate definition of an onto function.  (Contributed by NM,
     22-Mar-2006.) $)
  dffo2 $p |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) ) $=
    ( wfo wf crn wceq wa fof forn jca wfn ffn df-fo biimpri sylan impbii ) ABCD
    ZABCEZCFBGZHRSTABCIABCJKSCALZTRABCMRUATHABCNOPQ $.

  $( The image of the domain of an onto function.  (Contributed by NM,
     29-Nov-2002.) $)
  foima $p |- ( F : A -onto-> B -> ( F " A ) = B ) $=
    ( wfo cdm cima crn imadmrn wf wceq fof fdm syl imaeq2d forn 3eqtr3a ) ABCDZ
    CCEZFCGCAFBCHQRACQABCIRAJABCKABCLMNABCOP $.

  $( A function maps onto its range.  (Contributed by NM, 10-May-1998.) $)
  dffn4 $p |- ( F Fn A <-> F : A -onto-> ran F ) $=
    ( wfn crn wceq wa wfo eqid biantru df-fo bitr4i ) BACZLBDZMEZFAMBGNLMHIAMBJ
    K $.

  $( A function maps its domain onto its range.  (Contributed by NM,
     23-Jul-2004.) $)
  funforn $p |- ( Fun A <-> A : dom A -onto-> ran A ) $=
    ( wfun cdm wfn crn wfo funfn dffn4 bitri ) ABAACZDJAEAFAGJAHI $.

  $( An onto function has unique domain and range.  (Contributed by NM,
     5-Nov-2006.) $)
  fodmrnu $p |- ( ( F : A -onto-> B /\ F : C -onto-> D ) ->
                ( A = C /\ B = D ) ) $=
    ( wfo wa wceq wfn fofn fndmu syl2an crn forn sylan9req jca ) ABEFZCDEFZGACH
    ZBDHQEAIECISRABEJCDEJACEKLQRBEMDABENCDENOP $.

  $( A function is a function onto the image of its domain.  (Contributed by
     AV, 1-Dec-2022.) $)
  fimadmfo $p |- ( F : A --> B -> F : A -onto-> ( F " A ) ) $=
    ( wf cdm wceq cima wfo fdm wa crn wfn ffn adantr dffn4 sylib imaeq2 imadmrn
    wb eqtrdi eqcoms adantl foeq3 syl mpbird mpdan ) ABCDZCEZAFZACAGZCHZABCIUGU
    IJZUKACKZCHZULCALZUNUGUOUIABCMNACOPULUJUMFZUKUNSUIUPUGUPAUHAUHFUJCUHGUMAUHC
    QCRTUAUBUJUMACUCUDUEUF $.

  $( Restriction of a function.  (Contributed by NM, 4-Mar-1997.) $)
  fores $p |- ( ( Fun F /\ A C_ dom F ) ->
              ( F |` A ) : A -onto-> ( F " A ) ) $=
    ( wfun cdm wss cres cima wfo funres anim1i wfn wceq df-fn crn df-ima eqcomi
    wa df-fo mpbiran2 ssdmres anbi2i 3bitr4i sylibr ) BCZABDEZQBAFZCZUEQZABAGZU
    FHZUDUGUEABIJUFAKZUGUFDALZQUJUHUFAMUJUKUFNZUILUIUMBAOPAUIUFRSUEULUGABTUAUBU
    C $.

  $( Composition of onto functions.  (Contributed by NM, 22-Mar-2006.) $)
  foco $p |- ( ( F : B -onto-> C /\ G : A -onto-> B ) ->
             ( F o. G ) : A -onto-> C ) $=
    ( wfo wa ccom crn wceq dffo2 fco ad2ant2r cdm fdm eqtr3 sylan rncoeq eqeq1d
    wf biimpar an32s adantrl jca syl2anb sylibr ) BCDFZABEFZGACDEHZTZUIIZCJZGZA
    CUIFUGBCDTZDIZCJZGZABETZEIZBJZGZUMUHBCDKABEKUQVAGUJULUNURUJUPUTABCDELMUQUTU
    LURUNUTUPULUNUTGDNZUSJZUPULUNVBBJUTVCBCDOVBUSBPQVCULUPVCUKUOCDERSUAQUBUCUDU
    EACUIKUF $.

  $( Equality theorem for one-to-one onto functions.  (Contributed by NM,
     10-Feb-1997.) $)
  f1oeq1 $p |- ( F = G -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) ) $=
    ( wceq wf1 wfo wa wf1o f1eq1 foeq1 anbi12d df-f1o 3bitr4g ) CDEZABCFZABCGZH
    ABDFZABDGZHABCIABDIOPRQSABCDJABCDKLABCMABDMN $.

  $( Equality theorem for one-to-one onto functions.  (Contributed by NM,
     10-Feb-1997.) $)
  f1oeq2 $p |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) ) $=
    ( wceq wf1 wfo wa wf1o f1eq2 foeq2 anbi12d df-f1o 3bitr4g ) ABEZACDFZACDGZH
    BCDFZBCDGZHACDIBCDIOPRQSABCDJABCDKLACDMBCDMN $.

  $( Equality theorem for one-to-one onto functions.  (Contributed by NM,
     10-Feb-1997.) $)
  f1oeq3 $p |- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) ) $=
    ( wceq wf1 wfo wa wf1o f1eq3 foeq3 anbi12d df-f1o 3bitr4g ) ABEZCADFZCADGZH
    CBDFZCBDGZHCADICBDIOPRQSABCDJABCDKLCADMCBDMN $.

  $( Equality theorem for one-to-one onto functions.  (Contributed by FL,
     14-Jul-2012.) $)
  f1oeq23 $p |- ( ( A = B /\ C = D ) ->
   ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> D ) ) $=
    ( wceq wf1o f1oeq2 f1oeq3 sylan9bb ) ABFACEGBCEGCDFBDEGABCEHCDBEIJ $.

  ${
    f1eq123d.1 $e |- ( ph -> F = G ) $.
    f1eq123d.2 $e |- ( ph -> A = B ) $.
    f1eq123d.3 $e |- ( ph -> C = D ) $.
    $( Equality deduction for one-to-one functions.  (Contributed by Mario
       Carneiro, 27-Jan-2017.) $)
    f1eq123d $p |- ( ph -> ( F : A -1-1-> C <-> G : B -1-1-> D ) ) $=
      ( wf1 wceq wb f1eq1 syl f1eq2 f1eq3 3bitrd ) ABDFKZBDGKZCDGKZCEGKZAFGLSTM
      HBDFGNOABCLTUAMIBCDGPOADELUAUBMJDECGQOR $.

    $( Equality deduction for onto functions.  (Contributed by Mario Carneiro,
       27-Jan-2017.) $)
    foeq123d $p |- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) ) $=
      ( wfo wceq wb foeq1 syl foeq2 foeq3 3bitrd ) ABDFKZBDGKZCDGKZCEGKZAFGLSTM
      HBDFGNOABCLTUAMIBCDGPOADELUAUBMJDECGQOR $.

    $( Equality deduction for one-to-one onto functions.  (Contributed by Mario
       Carneiro, 27-Jan-2017.) $)
    f1oeq123d $p |- ( ph ->
      ( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) ) $=
      ( wf1o wceq wb f1oeq1 syl f1oeq2 f1oeq3 3bitrd ) ABDFKZBDGKZCDGKZCEGKZAFG
      LSTMHBDFGNOABCLTUAMIBCDGPOADELUAUBMJDECGQOR $.
  $}

  ${
    f1oeq1d.1 $e |- ( ph -> F = G ) $.
    $( Equality deduction for one-to-one onto functions.  (Contributed by
       Glauco Siliprandi, 17-Aug-2020.) $)
    f1oeq1d $p |- ( ph -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) ) $=
      ( wceq wf1o wb f1oeq1 syl ) ADEGBCDHBCEHIFBCDEJK $.
  $}

  ${
    f1oeq2d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for one-to-one onto functions.  (Contributed by
       Glauco Siliprandi, 17-Aug-2020.) $)
    f1oeq2d $p |- ( ph -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) ) $=
      ( wceq wf1o wb f1oeq2 syl ) ABCGBDEHCDEHIFBCDEJK $.
  $}

  ${
    f1oeq3d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for one-to-one onto functions.  (Contributed by
       Glauco Siliprandi, 17-Aug-2020.) $)
    f1oeq3d $p |- ( ph -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) ) $=
      ( wceq wf1o wb f1oeq3 syl ) ABCGDBEHDCEHIFBCDEJK $.
  $}

  ${
    nff1o.1 $e |- F/_ x F $.
    nff1o.2 $e |- F/_ x A $.
    nff1o.3 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for a one-to-one onto function.
       (Contributed by NM, 16-May-2004.) $)
    nff1o $p |- F/ x F : A -1-1-onto-> B $=
      ( wf1o wf1 wfo wa df-f1o nff1 nffo nfan nfxfr ) BCDHBCDIZBCDJZKABCDLQRAAB
      CDEFGMABCDEFGNOP $.
  $}

  $( A one-to-one onto mapping is a one-to-one mapping.  (Contributed by NM,
     12-Dec-2003.) $)
  f1of1 $p |- ( F : A -1-1-onto-> B -> F : A -1-1-> B ) $=
    ( wf1o wf1 wfo df-f1o simplbi ) ABCDABCEABCFABCGH $.

  $( A one-to-one onto mapping is a mapping.  (Contributed by NM,
     12-Dec-2003.) $)
  f1of $p |- ( F : A -1-1-onto-> B -> F : A --> B ) $=
    ( wf1o wf1 wf f1of1 f1f syl ) ABCDABCEABCFABCGABCHI $.

  $( A one-to-one onto mapping is function on its domain.  (Contributed by NM,
     12-Dec-2003.) $)
  f1ofn $p |- ( F : A -1-1-onto-> B -> F Fn A ) $=
    ( wf1o wf wfn f1of ffn syl ) ABCDABCECAFABCGABCHI $.

  $( A one-to-one onto mapping is a function.  (Contributed by NM,
     12-Dec-2003.) $)
  f1ofun $p |- ( F : A -1-1-onto-> B -> Fun F ) $=
    ( wf1o wfn wfun f1ofn fnfun syl ) ABCDCAECFABCGACHI $.

  $( A one-to-one onto mapping is a relation.  (Contributed by NM,
     13-Dec-2003.) $)
  f1orel $p |- ( F : A -1-1-onto-> B -> Rel F ) $=
    ( wf1o wfun wrel f1ofun funrel syl ) ABCDCECFABCGCHI $.

  $( The domain of a one-to-one onto mapping.  (Contributed by NM,
     8-Mar-2014.) $)
  f1odm $p |- ( F : A -1-1-onto-> B -> dom F = A ) $=
    ( wf1o wfn cdm wceq f1ofn fndm syl ) ABCDCAECFAGABCHACIJ $.

  $( Alternate definition of one-to-one onto function.  (Contributed by NM,
     10-Feb-1997.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
  dff1o2 $p |- ( F : A -1-1-onto-> B
        <-> ( F Fn A /\ Fun `' F /\ ran F = B ) ) $=
    ( wf1o wf1 wfo wa wfn ccnv wfun crn w3a df-f1o wf df-f1 df-fo anbi12i anass
    wceq bitri 3anan12 anbi1i eqimss df-f biimpri sylan2 3adant2 ancom 3bitr4ri
    wss pm4.71i ) ABCDABCEZABCFZGZCAHZCIJZCKZBSZLZABCMUNABCNZUPGZUOURGZGZUSULVA
    UMVBABCOABCPQVCUTUPVBGZGZUSUTUPVBRUSUTGVDUTGUSVEUSVDUTUOUPURUAUBUSUTUOURUTU
    PURUOUQBUJZUTUQBUCUTUOVFGABCUDUEUFUGUKUTVDUHUITTT $.

  $( Alternate definition of one-to-one onto function.  (Contributed by NM,
     25-Mar-1998.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
  dff1o3 $p |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) ) $=
    ( wfn ccnv wfun crn wceq w3a wf1o wfo 3anan32 dff1o2 df-fo anbi1i 3bitr4i
    wa ) CADZCEFZCGBHZIRTQZSQABCJABCKZSQRSTLABCMUBUASABCNOP $.

  $( A one-to-one onto function is an onto function.  (Contributed by NM,
     28-Apr-2004.) $)
  f1ofo $p |- ( F : A -1-1-onto-> B -> F : A -onto-> B ) $=
    ( wf1o wfo ccnv wfun dff1o3 simplbi ) ABCDABCECFGABCHI $.

  $( Alternate definition of one-to-one onto function.  (Contributed by NM,
     25-Mar-1998.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
  dff1o4 $p |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) ) $=
    ( wf1o wfn ccnv wfun crn w3a wa dff1o2 3anass cdm df-rn eqeq1i anbi2i df-fn
    wceq bitr4i 3bitri ) ABCDCAEZCFZGZCHZBRZIUAUCUEJZJUAUBBEZJABCKUAUCUELUFUGUA
    UFUCUBMZBRZJUGUEUIUCUDUHBCNOPUBBQSPT $.

  $( Alternate definition of one-to-one onto function.  (Contributed by NM,
     10-Dec-2003.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
  dff1o5 $p |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) ) $=
    ( wf1o wf1 wfo wa crn wceq df-f1o dffo2 f1f biantrurd bitr4id pm5.32i bitri
    wf ) ABCDABCEZABCFZGRCHBIZGABCJRSTRSABCQZTGTABCKRUATABCLMNOP $.

  $( A one-to-one function maps onto its range.  (Contributed by NM,
     13-Aug-2004.) $)
  f1orn $p |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) ) $=
    ( crn wf1o wfn ccnv wfun wceq w3a wa dff1o2 eqid df-3an mpbiran2 bitri ) AB
    CZBDBAEZBFGZPPHZIZQRJZAPBKTUASPLQRSMNO $.

  $( A one-to-one function maps one-to-one onto its range.  (Contributed by NM,
     4-Sep-2004.) $)
  f1f1orn $p |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F ) $=
    ( wf1 wfn ccnv wfun crn wf1o f1fn wf df-f1 simprbi f1orn sylanbrc ) ABCDZCA
    ECFGZACHCIABCJPABCKQABCLMACNO $.

  ${
    $d A f $.  $d B f $.
    f1oabexg.1 $e |- F = { f | ( f : A -1-1-onto-> B /\ ph ) } $.
    $( The class of all 1-1-onto functions mapping one set to another is a set.
       (Contributed by Paul Chapman, 25-Feb-2008.) $)
    f1oabexg $p |- ( ( A e. C /\ B e. D ) -> F e. _V ) $=
      ( wcel wa cv wf1o cab cvv wf wss f1of anim1i ss2abi eqid sylancr eqeltrid
      fabexg ssexg ) BDICEIJZGBCFKZLZAJZFMZNHUEUIBCUFOZAJZFMZPULNIUINIUHUKFUGUJ
      ABCUFQRSAFBCDEULULTUCUIULNUDUAUB $.
  $}

  $( The converse of a one-to-one onto function is also one-to-one onto.
     (Contributed by NM, 11-Feb-1997.)  (Proof shortened by Andrew Salmon,
     22-Oct-2011.) $)
  f1ocnv $p |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A ) $=
    ( wfn ccnv wa wf1o wrel fnrel wceq dfrel2 fneq1 biimprd sylbi anim2i ancoms
    wi mpcom dff1o4 3imtr4i ) CADZCEZBDZFUCUBEZADZFZABCGBAUBGUCUAUFUAUEUCCHZUAU
    EACIUGUDCJZUAUEQCKUHUEUAAUDCLMNROPABCSBAUBST $.

  $( A relation is a one-to-one onto function iff its converse is a one-to-one
     onto function with domain and codomain/range interchanged.  (Contributed
     by NM, 8-Dec-2003.) $)
  f1ocnvb $p |- ( Rel F ->
                ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) ) $=
    ( wrel wf1o ccnv f1ocnv wceq wb dfrel2 f1oeq1 sylbi imbitrid impbid2 ) CDZA
    BCEZBACFZEZABCGRABQFZEZOPBAQGOSCHTPICJABSCKLMN $.

  $( The restriction of a one-to-one function maps one-to-one onto the image.
     (Contributed by NM, 25-Mar-1998.) $)
  f1ores $p |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto->
            ( F " C ) ) $=
    ( wf1 wss wa cres crn wf1o cima f1ssres f1f1orn wceq wb df-ima f1oeq3 ax-mp
    syl sylibr ) ABDECAFGZCDCHZIZUBJZCDCKZUBJZUACBUBEUDABCDLCBUBMSUEUCNUFUDODCP
    UEUCCUBQRT $.

  $( The converse of a one-to-one-onto restricted function.  (Contributed by
     Paul Chapman, 21-Apr-2008.) $)
  f1orescnv $p |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) ->
                    ( `' F |` P ) : P -1-1-onto-> R ) $=
    ( ccnv wfun cres wf1o wa f1ocnv adantl wceq wb cima funcnvres df-ima dff1o5
    crn wf1 simprbi eqtrid reseq2d sylan9eq f1oeq1 syl mpbid ) CDZEZBACBFZGZHZA
    BUHDZGZABUFAFZGZUIULUGBAUHIJUJUKUMKULUNLUGUIUKUFCBMZFUMBCNUIUOAUFUIUOUHQZAC
    BOUIBAUHRUPAKBAUHPSTUAUBABUKUMUCUDUE $.

  $( Preimage of an image.  (Contributed by NM, 30-Sep-2004.) $)
  f1imacnv $p |- ( ( F : A -1-1-> B /\ C C_ A )
                 -> ( `' F " ( F " C ) ) = C ) $=
    ( wf1 wss wa ccnv cima cres resima wfun wceq df-f1 simprbi adantr funcnvres
    wf syl wf1o imaeq1d f1ores f1ocnv cdm crn imadmrn f1odm imaeq2d wfo 3eqtr3a
    f1ofo forn eqtr3d eqtr3id ) ABDEZCAFZGZDHZDCIZIURUSJZUSIZCURUSKUQDCJZHZUSIZ
    VACUQVCUTUSUQURLZVCUTMUOVEUPUOABDRVEABDNOPCDQSUAUQUSCVCTZVDCMUQCUSVBTVFABCD
    UBCUSVBUCSVFVCVCUDZIVCUEZVDCVCUFVFVGUSVCUSCVCUGUHVFUSCVCUIVHCMUSCVCUKUSCVCU
    LSUJSUMUN $.

  $( A reverse version of ~ f1imacnv .  (Contributed by Jeff Hankins,
     16-Jul-2009.) $)
  foimacnv $p |- ( ( F : A -onto-> B /\ C C_ B )
                 -> ( F " ( `' F " C ) ) = C ) $=
    ( wfo wss wa ccnv cima cres resima wfun wceq fofun adantr syl crn cdm df-rn
    eqtr3id funcnvres2 imaeq1d wfn resss cnvss ax-mp cnvcnvss sstri funss mpsyl
    df-ima eqtr2i jctir df-fn sylibr dfdm4 forn sseq2d biimpar sseqtrdi ssdmres
    sylib df-fo sylanbrc foima eqtr3d ) ABDEZCBFZGZDDHZCIZIDVKJZVKIZCDVKKVIVJCJ
    ZHZVKIZVMCVIVOVLVKVIDLZVOVLMVGVQVHABDNZOCDUAPUBVIVKCVOEZVPCMVIVOVKUCZVOQZCM
    VSVIVOLZVORZVKMZGVTVIWBWDVGWBVHVODFVGVQWBVOVJHZDVNVJFVOWEFVJCUDVNVJUEUFDUGU
    HVRVODUIUJOVKVNQWCVJCUKVNSULUMVOVKUNUOVIWAVNRZCVNUPVICVJRZFWFCMVICDQZWGVGCW
    HFVHVGWHBCABDUQURUSDSUTCVJVAVBTVKCVOVCVDVKCVOVEPVFT $.

  $( The union of two onto functions with disjoint domains is an onto function.
     (Contributed by Mario Carneiro, 22-Jun-2016.) $)
  foun $p |- ( ( ( F : A -onto-> B /\ G : C -onto-> D )
            /\ ( A i^i C ) = (/) )
           -> ( F u. G ) : ( A u. C ) -onto-> ( B u. D ) ) $=
    ( wfo wa cin c0 wceq cun wfn crn fofn anim12i fnun sylan rnun forn ad2antrr
    ad2antlr uneq12d eqtrid df-fo sylanbrc ) ABEGZCDFGZHZACIJKZHZEFLZACLZMZULNZ
    BDLZKUMUPULGUIEAMZFCMZHUJUNUGUQUHURABEOCDFOPACEFQRUKUOENZFNZLUPEFSUKUSBUTDU
    GUSBKUHUJABETUAUHUTDKUGUJCDFTUBUCUDUMUPULUEUF $.

  $( The union of two one-to-one onto functions with disjoint domains and
     ranges.  (Contributed by NM, 26-Mar-1998.) $)
  f1oun $p |- ( ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D )
            /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) )
           -> ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) ) $=
    ( wf1o wa cin c0 wceq cun wfn ccnv wi dff1o4 fnun ex cnvun fneq1i imbitrrdi
    sylibr im2anan9 an4s syl2anb imp ) ABEGZCDFGZHZACIJKZBDIJKZHZACLZBDLZEFLZGZ
    UIULUOUMMZUONZUNMZHZUPUGEAMZENZBMZHFCMZFNZDMZHULUTOZUHABEPCDFPVAVDVCVFVGVAV
    DHZUJUQVCVFHZUKUSVHUJUQACEFQRVIUKUSVIUKHVBVELZUNMUSBDVBVEQUNURVJEFSTUBRUCUD
    UEUMUNUOPUAUF $.

  ${
    $d A u v x z $.  $d A u v y $.  $d B u v y $.  $d B u v z $.  $d C u v x $.
    $d D u v $.  $d S u v x $.
    fun11iun.1 $e |- ( x = y -> B = C ) $.
    fun11iun.2 $e |- B e. _V $.
    $( The union of a chain (with respect to inclusion) of one-to-one functions
       is a one-to-one function.  (Contributed by Mario Carneiro, 20-May-2013.)
       (Revised by Mario Carneiro, 24-Jun-2015.) $)
    fun11iun $p |- ( A. x e. A ( B : D -1-1-> S /\
                       A. y e. A ( B C_ C \/ C C_ B ) ) ->
                     U_ x e. A B : U_ x e. A D -1-1-> S ) $=
      ( vz vu vv wss wral wa wfun wceq cv wrex wcel wf1 wo ciun wf ccnv wfn crn
      cdm cab cuni vex eqeq1 rexbidv elab r19.29 nfv nfre1 nfab nfralxy nfan wi
      f1eq1 biimparc df-f1 ffun anim1i syl adantlr eqeq2d cbvrexv sseq12 ancoms
      sylbi orbi12d biimprcd expdimp rexlimivw imp sylan an32s adantlll sylan2b
      wb ralrimiva jca a1i rexlimi fun11uni simpld dfiun2 funeqi sylibr cop wex
      nfra1 eldm2 f1dm eleq2d bitr3id adantr syl6 rexbida eliun rexcom4 3bitr4i
      rsp exbii 3bitr4g eqrdv df-fn sylanbrc rniun ralimi iunss eqsstrid simprd
      f1rn df-f cnveqi ) FGDUAZDEMZEDMZUBZBCNZOZACNZACFUCZGACDUCZUDZYHUEZPZYGGY
      HUAYFYHYGUFZYHUGZGMYIYFYHPZYHUHZYGQYLYFJRZDQZACSZJUIZUJZPZYNYFUUAYTUEZPZY
      FKRZPZUUDUEPZOZUUDLRZMZUUHUUDMZUBZLYSNZOZKYSNUUAUUCOYFUUMKYSUUDYSTYFUUDDQ
      ZACSZUUMYRUUOJUUDKUKZYPUUDQYQUUNACYPUUDDULUMUNYFUUOOYEUUNOZACSUUMYEUUNACU
      OUUQUUMACUUGUULAUUGAUPUUKALYSYRAJYQACUQURUUKAUPUSUTUUQUUMVAARZCTZUUQUUGUU
      LXTUUNUUGYDXTUUNOFGUUDUAZUUGUUNUUTXTFGUUDDVBVCUUTFGUUDUDZUUFOUUGFGUUDVDUV
      AUUEUUFFGUUDVEVFVMVGVHUUQUUKLYSUUHYSTUUQUUHDQZACSZUUKYRUVCJUUHLUKYPUUHQYQ
      UVBACYPUUHDULUMUNUVCUUQUUHEQZBCSZUUKUVBUVDABCUURBRQDEUUHHVIVJYDUUNUVEUUKX
      TYDUVEUUNUUKYDUVEOYCUVDOZBCSZUUNUUKYCUVDBCUOUVGUUNUUKUVFUUNUUKVABCYCUVDUU
      NUUKUVDUUNOZUUKYCUVHUUIYAUUJYBUUNUVDUUIYAWCUUDDUUHEVKVLUUHEUUDDVKVNVOVPVQ
      VRVSVTWAWBWBWDWEWFWGVGWBWDYSKLWHVGZWIYHYTAJCDIWJZWKWLYFKYOYGYFUUDUUHWMZDT
      ZLWNZACSZUUDFTZACSUUDYOTZUUDYGTYFUVMUVOACYEACWOYFUUSUVMUVOWCZYFUUSYEUVQYE
      ACXFXTUVQYDUVMUUDDUHZTXTUVOLUUDDUUPWPXTUVRFUUDFGDWQWRWSWTXAVRXBUVKYHTZLWN
      UVLACSZLWNUVPUVNUVSUVTLAUVKCDXCXGLUUDYHUUPWPUVLALCXDXEAUUDCFXCXHXIYHYGXJX
      KYFYMACDUGZUCZGACDXLYFUWAGMZACNUWBGMYEUWCACXTUWCYDFGDXQWTXMACUWAGXNWLXOYG
      GYHXRXKYFUUCYKYFUUAUUCUVIXPYJUUBYHYTUVJXSWKWLYGGYHVDXK $.
  $}

  $( The restriction of a one-to-one onto function to a difference maps onto
     the difference of the images.  (Contributed by Paul Chapman,
     11-Apr-2009.) $)
  resdif $p |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\
                               ( F |` B ) : B -onto-> D ) ->
                 ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) ) $=
    ( ccnv wfun cres wfo cdif cima wf1o wceq syl wb ax-mp crn df-ima wa forn wf
    w3a cdm wss fofun difss fof fdm sseqtrrid fores syl2anc indif reseq2i eqtri
    cin resres foeq1 rneqi 3eqtr4i foeq3 bitri funres11 biimpri syl2anr 3adant3
    sylib dff1o3 eqtrid anim12i imadif difeq12 sylan9eq sylan2 3impb f1oeq3
    mpbid ) EFGZACEAHZIZBDEBHZIZUBZABJZEWCKZEWCHZLZWCCDJZWELZVQVSWFWAVSWCWDWEIZ
    WEFGZWFVQVSWCVRWCKZVRWCHZIZWIVSVRGWCVRUCZUDWMACVRUEVSAWCWNABUFVSACVRUAWNAMA
    CVRUGACVRUHNUIWCVRUJUKWMWCWKWEIZWIWLWEMWMWOOWLEAWCUOZHWEEAWCUPWPWCEABULUMUN
    ZWCWKWLWEUQPWKWDMWOWIOWLQWEQWKWDWLWEWQURVRWCREWCRUSWKWDWCWEUTPVAVFWCEVBWFWI
    WJSWCWDWEVGVCVDVEWBWDWGMZWFWHOVQVSWAWRVSWASVQEAKZCMZEBKZDMZSZWRVSWTWAXBVSWS
    VRQCEARACVRTVHWAXAVTQDEBRBDVTTVHVIVQXCWDWSXAJWGABEVJWSCXADVKVLVMVNWDWGWCWEV
    ONVP $.

  $( Composition of one-to-one onto functions.  (Contributed by NM,
     19-Mar-1998.) $)
  f1oco $p |- ( ( F : B -1-1-onto-> C /\ G : A -1-1-onto-> B ) ->
              ( F o. G ) : A -1-1-onto-> C ) $=
    ( wf1o wa ccom wf1 wfo df-f1o f1co foco anim12i an4s syl2anb sylibr ) BCDFZ
    ABEFZGACDEHZIZACTJZGZACTFRBCDIZBCDJZGABEIZABEJZGUCSBCDKABEKUDUFUEUGUCUDUFGU
    AUEUGGUBABCDELABCDEMNOPACTKQ $.

  $( The converse of an injective function is bijective.  (Contributed by FL,
     11-Nov-2011.) $)
  f1cnv $p |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A ) $=
    ( wf1 crn wf1o ccnv f1f1orn f1ocnv syl ) ABCDACEZCFKACGFABCHAKCIJ $.

  $( Composition with the converse.  (Contributed by Jeff Madsen,
     2-Sep-2009.) $)
  funcocnv2 $p |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) ) $=
    ( wfun ccnv ccom cid wss crn cres wceq wrel df-fun simprbi cdm dfdm4 dmcoeq
    iss ax-mp df-rn eqtr4i a1i reseq2d eqeq2d bitrid mpbid ) ABZAACZDZEFZUGEAGZ
    HZIZUEAJUHAKLUHUGEUGMZHZIUEUKUGPUEUMUJUGUEULUIEULUIIUEULUFMZUIAMUFGIULUNIAN
    AUFOQARSTUAUBUCUD $.

  $( The composition of an onto function and its converse.  (Contributed by
     Stefan O'Rear, 12-Feb-2015.) $)
  fococnv2 $p |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) ) $=
    ( wfo ccnv ccom cid crn cres wfun wceq fofun funcocnv2 forn reseq2d eqtrd
    syl ) ABCDZCCEFZGCHZIZGBIRCJSUAKABCLCMQRTBGABCNOP $.

  $( The composition of a one-to-one onto function and its converse equals the
     identity relation restricted to the function's range.  (Contributed by NM,
     13-Dec-2003.)  (Proof shortened by Stefan O'Rear, 12-Feb-2015.) $)
  f1ococnv2 $p |- ( F : A -1-1-onto-> B -> ( F o. `' F ) = ( _I |` B ) ) $=
    ( wf1o wfo ccnv ccom cid cres wceq f1ofo fococnv2 syl ) ABCDABCECCFGHBIJABC
    KABCLM $.

  $( Composition of an injective function with its converse.  (Contributed by
     FL, 11-Nov-2011.) $)
  f1cocnv2 $p |- ( F : A -1-1-> B -> ( F o. `' F ) = ( _I |` ran F ) ) $=
    ( wf1 wfun ccnv ccom cid crn cres wceq f1fun funcocnv2 syl ) ABCDCECCFGHCIJ
    KABCLCMN $.

  $( The composition of a one-to-one onto function's converse and itself equals
     the identity relation restricted to the function's domain.  (Contributed
     by NM, 13-Dec-2003.) $)
  f1ococnv1 $p |- ( F : A -1-1-onto-> B -> ( `' F o. F ) = ( _I |` A ) ) $=
    ( wf1o ccnv ccom cres wrel wceq f1orel dfrel2 sylib coeq2d f1ocnv f1ococnv2
    cid syl eqtr3d ) ABCDZCEZTEZFZTCFPAGZSUACTSCHUACIABCJCKLMSBATDUBUCIABCNBATO
    QR $.

  $( Composition of an injective function with its converse.  (Contributed by
     FL, 11-Nov-2011.) $)
  f1cocnv1 $p |- ( F : A -1-1-> B -> ( `' F o. F ) = ( _I |` A ) ) $=
    ( wf1 crn wf1o ccnv ccom cid cres wceq f1f1orn f1ococnv1 syl ) ABCDACEZCFCG
    CHIAJKABCLAOCMN $.

  $( Express a constraint on a composition as a constraint on the composand.
     (Contributed by Stefan O'Rear, 7-Mar-2015.) $)
  funcoeqres $p |- ( ( Fun G /\ ( F o. G ) = H ) ->
      ( F |` ran G ) = ( H o. `' G ) ) $=
    ( wfun ccom wceq crn cres cid funcocnv2 coeq2d coass eqcomi coires1 3eqtr3g
    ccnv coeq1 sylan9req ) BDZABEZCFABGZHZTBPZEZCUCESABUCEZEZAIUAHZEUDUBSUEUGAB
    JKUDUFABUCLMAUANOTCUCQR $.

  $( A subset of an injective function is injective.  (Contributed by AV,
     20-Nov-2020.) $)
  f1ssf1 $p |- ( ( Fun F /\ Fun `' F /\ G C_ F ) -> Fun `' G ) $=
    ( wfun ccnv wss wi cdm cres funssres funres11 cnveq funeqd imbitrrid eqcoms
    wa wceq syl ex com23 3imp ) ACZADCZBAEZBDZCZUAUCUBUEUAUCUBUEFZUAUCOABGZHZBP
    UFABIUFBUHUBUEBUHPZUHDZCUGAJUIUDUJBUHKLMNQRST $.

  ${
    $d x F $.  $d x A $.  $d x B $.
    f11o.1 $e |- F e. _V $.
    $( Relationship between a mapping and an onto mapping.  Figure 38 of
       [Enderton] p. 145.  (Contributed by NM, 10-May-1998.) $)
    ffoss $p |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) ) $=
      ( wf cv wfo wss wa wex crn wfn df-f dffn4 anbi1i bitri rnex wceq foeq3
      sseq1 anbi12d spcev sylbi fof fss sylan exlimiv impbii ) BCDFZBAGZDHZUKCI
      ZJZAKZUJBDLZDHZUPCIZJZUOUJDBMZURJUSBCDNUTUQURBDOPQUNUSAUPDERUKUPSULUQUMUR
      UKUPBDTUKUPCUAUBUCUDUNUJAULBUKDFUMUJBUKDUEBUKCDUFUGUHUI $.

    $( Relationship between one-to-one and one-to-one onto function.
       (Contributed by NM, 4-Apr-1998.) $)
    f11o $p |- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) ) $=
      ( wf ccnv wfun wa cv wfo wss wex wf1 wf1o ffoss anbi1i df-f1 dff1o3 bitri
      an32 exbii 19.41v 3bitr4i ) BCDFZDGHZIBAJZDKZUGCLZIZAMZUFIZBCDNBUGDOZUIIZ
      AMZUEUKUFABCDEPQBCDRUOUJUFIZAMULUNUPAUNUHUFIZUIIUPUMUQUIBUGDSQUHUFUIUATUB
      UJUFAUCTUD $.
  $}

  $( The empty set maps one-to-one into any class.  (Contributed by NM,
     7-Apr-1998.) $)
  f10 $p |- (/) : (/) -1-1-> A $=
    ( c0 wf1 wf ccnv wfun f0 fun0 cnv0 funeqi mpbir df-f1 mpbir2an ) BABCBABDBE
    ZFZAGOBFHNBIJKBABLM $.

  ${
    f10d.f $e |- ( ph -> F = (/) ) $.
    $( The empty set maps one-to-one into any class, deduction version.
       (Contributed by AV, 25-Nov-2020.) $)
    f10d $p |- ( ph -> F : dom F -1-1-> A ) $=
      ( cdm wf1 c0 f10 wceq dm0 f1eq2 ax-mp mpbir dmeqd eqidd f1eq123d mpbiri
      wb ) ACEZBCFGEZBGFZUAGBGFZBHTGIUAUBRJTGBGKLMASTBBCGDACGDNABOPQ $.
  $}

  $( One-to-one onto mapping of the empty set.  (Contributed by NM,
     15-Apr-1998.) $)
  f1o00 $p |- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) ) $=
    ( c0 wf1o wfn ccnv wa wceq dff1o4 fn0 biimpi adantr cnveq cnv0 eqtrdi sylbi
    cdm fneq1d biimpa jca fndm syl dm0 eqtr3di eqid mpbir fneq2 sylan9bb mpbiri
    biimpri impbii bitri ) CABDBCEZBFZAEZGZBCHZACHZGZCABIUPUSUPUQURUMUQUOUMUQBJ
    ZKLUPCQZACUPCAEZVAAHUMUOVBUMAUNCUMUQUNCHUTUQUNCFCBCMNOZPRSACUAUBUCUDTUSUMUO
    UQUMURUMUQUTUJLUSUOCCEZVDCCHCUECJUFUQUOVBURVDUQAUNCVCRACCUGUHUITUKUL $.

  $( Onto mapping of the empty set.  (Contributed by NM, 22-Mar-2006.) $)
  fo00 $p |- ( F : (/) -onto-> A <-> ( F = (/) /\ A = (/) ) ) $=
    ( c0 wfo wf1o wceq wf1 wfn fofn fn0 f10 f1eq1 mpbiri sylbi syl ancri df-f1o
    wa sylibr f1ofo impbii f1o00 bitri ) CABDZCABEZBCFZACFRUDUEUDCABGZUDRUEUDUG
    UDBCHZUGCABIUHUFUGBJUFUGCACGAKCABCLMNOPCABQSCABTUAABUBUC $.

  $( One-to-one onto mapping of the empty set.  (Contributed by NM,
     10-Sep-2004.) $)
  f1o0 $p |- (/) : (/) -1-1-onto-> (/) $=
    ( c0 wf1o wceq eqid f1o00 mpbir2an ) AAABAACZGADZHAAEF $.

  $( A restriction of the identity relation is a one-to-one onto function.
     (Contributed by NM, 30-Apr-1998.)  (Proof shortened by Andrew Salmon,
     22-Oct-2011.) $)
  f1oi $p |- ( _I |` A ) : A -1-1-onto-> A $=
    ( cid cres wf1o wfn ccnv fnresi cnvresid fneq1i mpbir dff1o4 mpbir2an ) AAB
    ACZDMAEZMFZAEZAGZPNQAOMAHIJAAMKL $.

  $( The identity relation is a one-to-one onto function on the universe.
     (Contributed by NM, 16-May-2004.) $)
  f1ovi $p |- _I : _V -1-1-onto-> _V $=
    ( cvv cid cres wf1o f1oi wceq wb wrel reli dfrel3 mpbi f1oeq1 ax-mp ) AABAC
    ZDZAABDZAENBFZOPGBHQIBJKAANBLMK $.

  ${
    f1osn.1 $e |- A e. _V $.
    f1osn.2 $e |- B e. _V $.
    $( A singleton of an ordered pair is one-to-one onto function.
       (Contributed by NM, 18-May-1998.)  (Proof shortened by Andrew Salmon,
       22-Oct-2011.) $)
    f1osn $p |- { <. A , B >. } : { A } -1-1-onto-> { B } $=
      ( csn cop wf1o wfn ccnv fnsn cnvsn fneq1i mpbir dff1o4 mpbir2an ) AEZBEZA
      BFEZGRPHRIZQHZABCDJTBAFEZQHBADCJQSUAABCDKLMPQRNO $.
  $}

  ${
    $d A a b $.  $d B b $.
    $( A singleton of an ordered pair is one-to-one onto function.
       (Contributed by Mario Carneiro, 12-Jan-2013.) $)
    f1osng $p |- ( ( A e. V /\ B e. W ) ->
                   { <. A , B >. } : { A } -1-1-onto-> { B } ) $=
      ( va vb cv csn cop wf1o wceq sneq f1oeq2 syl opeq1 sneqd f1oeq1 bitrd vex
      wb f1oeq3 opeq2 f1osn vtocl2g ) EGZHZFGZHZUEUGIZHZJZAHZUHAUGIZHZJZULBHZAB
      IZHZJZEFABCDUEAKZUKULUHUJJZUOUTUFULKUKVATUEALUFULUHUJMNUTUJUNKVAUOTUTUIUM
      UEAUGOPULUHUJUNQNRUGBKZUOULUPUNJZUSVBUHUPKUOVCTUGBLUHUPULUNUANVBUNURKVCUS
      TVBUMUQUGBAUBPULUPUNURQNRUEUGESFSUCUD $.

    $( A singleton of an ordered pair is a one-to-one function.  (Contributed
       by AV, 17-Apr-2021.) $)
    f1sng $p |- ( ( A e. V /\ B e. W ) ->
                   { <. A , B >. } : { A } -1-1-> W ) $=
      ( wcel wa csn cop wf1 wss wf1o f1osng f1of1 syl snssi adantl f1ss syl2anc
      ) ACEZBDEZFZAGZBGZABHGZIZUCDJZUBDUDIUAUBUCUDKUEABCDLUBUCUDMNTUFSBDOPUBUCD
      UDQR $.
  $}

  ${
    fsnd.a $e |- ( ph -> A e. V ) $.
    fsnd.b $e |- ( ph -> B e. W ) $.
    $( A singleton of an ordered pair is a function.  (Contributed by AV,
       17-Apr-2021.) $)
    fsnd $p |- ( ph -> { <. A , B >. } : { A } --> W ) $=
      ( wcel wa csn cop wf1 wf jca f1sng f1f 3syl ) ABDHZCEHZIBJZEBCKJZLTEUAMAR
      SFGNBCDEOTEUAPQ $.
  $}

  $( An unordered pair of ordered pairs with different elements is a one-to-one
     onto function.  (Contributed by Alexander van der Vekens, 14-Aug-2017.) $)
  f1oprg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) )
      -> ( ( A =/= C /\ B =/= D )
      -> { <. A , B >. , <. C , D >. } : { A , C } -1-1-onto-> { B , D } ) ) $=
    ( wcel wa wne cpr cop wf1o csn cun wceq df-pr eqcomi a1i c0 f1osng ad2antrr
    cin ad2antlr disjsn2 ad2antrl ad2antll f1oun syl22anc f1oeq123d mpbid ex )
    AEIBFIJZCGIDHIJZJZACKZBDKZJZACLZBDLZABMZCDMZLZNZUPUSJZAOZCOZPZBOZDOZPZVBOZV
    COZPZNZVEVFVGVJVMNZVHVKVNNZVGVHUDUAQZVJVKUDUAQZVPUNVQUOUSABEFUBUCUOVRUNUSCD
    GHUBUEUQVSUPURACUFUGURVTUPUQBDUFUHVGVJVHVKVMVNUIUJVFVIUTVLVAVOVDVOVDQVFVDVO
    VBVCRSTVIUTQVFUTVIACRSTVLVAQVFVAVLBDRSTUKULUM $.

  ${
    $d x F $.  $d x A $.
    $( Function value when ` F ` is not a function.  Theorem 6.12(2) of
       [TakeutiZaring] p. 27.  (Contributed by NM, 30-Apr-2004.)  (Proof
       shortened by Mario Carneiro, 31-Aug-2015.) $)
    tz6.12-2 $p |- ( -. E! x A F x -> ( F ` A ) = (/) ) $=
      ( cv wbr weu wn cfv cio c0 df-fv iotanul eqtrid ) BADCEZAFGBCHNAIJABCKNAL
      M $.

    $( The value of a function at a unique point.  (Contributed by Scott
       Fenton, 6-Oct-2017.) $)
    fveu $p |- ( E! x A F x -> ( F ` A ) = U. { x | A F x } ) $=
      ( cv wbr weu cfv cio cab cuni df-fv iotauni eqtrid ) BADCEZAFBCGNAHNAIJAB
      CKNALM $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( If ` A ` is a proper class and ` F ` is any class, then there is no
       unique set which is related to ` A ` through the binary relation ` F ` .
       (Contributed by Scott Fenton, 7-Oct-2017.) $)
    brprcneu $p |- ( -. A e. _V -> -. E! x A F x ) $=
      ( vy wcel wn cv wbr wex wi wa weq cop c0 notbii exbii sylibr opprc1 df-br
      eleq1d cvv wmo weu dtruex equcom mpbir 19.42v anbi12d anidm bitrdi anbi1d
      exbidv imbi12d mpbiri anbi12i anbi1i 3imtr4g eximdv wal exanaliim exnalim
      jctr eximi syl breq2 mo4 syl6 eu5 imnan bitr4i ) BUAEFZBAGZCHZAIZVMAUBZFZ
      JZVMAUCZFZVKVNVMBDGZCHZKZADLZFZKZDIZAIZVPVKVMWFAVKBVLMZCEZWIBVTMZCEZKZWDK
      ZDIZVMWFVKWIWNJNCEZWOWDKZDIZJWOWOWDDIZKWQWOWRWRDALZFZDIDAUDWDWTDWCWSADUEO
      PUFVBWOWDDUGQVKWIWOWNWQVKWHNCBVLRTZVKWMWPDVKWLWOWDVKWLWOWOKWOVKWIWOWKWOXA
      VKWJNCBVTRTUHWOUIUJUKULUMUNBVLCSZWEWMDWBWLWDVMWIWAWKXBBVTCSUOUPPUQURWGWBW
      CJDUSZAUSZFZVPWGXCFZAIXEWFXFAWBWCDUTVCXCAVAVDVOXDVMWAADVLVTBCVEVFOQVGVSVN
      VOKZFVQVRXGVMAVHOVNVOVIVJQ $.
  $}

  ${
    $d x A $.  $d x F $.
    $( A function's value at a proper class is the empty set.  (Contributed by
       NM, 20-May-1998.) $)
    fvprc $p |- ( -. A e. _V -> ( F ` A ) = (/) ) $=
      ( vx cvv wcel wn cv wbr weu cfv c0 wceq brprcneu tz6.12-2 syl ) ADEFACGBH
      CIFABJKLCABMCABNO $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( Alternate definition of function value.  Definition 10.11 of [Quine]
       p. 68.  (Contributed by NM, 30-Apr-2004.)  (Proof shortened by Andrew
       Salmon, 17-Sep-2011.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    fv2 $p |- ( F ` A ) = U. { x | A. y ( A F y <-> y = x ) } $=
      ( cfv cv wbr cio weq wb wal cab cuni df-fv dfiota2 eqtri ) CDECBFDGZBHQBA
      IJBKALMBCDNQBAOP $.
  $}

  ${
    $d F x $.  $d A x $.  $d V x $.
    $( A definition of function value in terms of iota.  (Contributed by Jim
       Kingdon, 29-Dec-2018.) $)
    dffv3g $p |- ( A e. V -> ( F ` A ) = ( iota x x e. ( F " { A } ) ) ) $=
      ( wcel cfv cv wbr cio csn cima df-fv cvv wb vex wa elimasng df-br bitr4di
      cop mpan2 iotabidv eqtr4id ) BDEZBCFBAGZCHZAIUECBJKEZAIABCLUDUGUFAUDUEMEZ
      UGUFNAOUDUHPUGBUETCEUFCBUEDMQBUECRSUAUBUC $.
  $}

  ${
    $d x y A $.  $d x y F $.  $d x y V $.
    $( The previous definition of function value, from before the ` iota `
       operator was introduced.  Although based on the idea embodied by
       Definition 10.2 of [Quine] p. 65 (see ~ args ), this definition
       apparently does not appear in the literature.  (Contributed by NM,
       1-Aug-1994.) $)
    dffv4g $p |- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } ) $=
      ( vy wcel cfv cv csn cima cio wceq cuni dffv3g df-iota abid2 eqeq1i abbii
      cab unieqi eqtri eqtrdi ) BDFBCGEHCBIJZFZEKZUCAHIZLZASZMZEBCDNUEUDESZUFLZ
      ASZMUIUDEAOULUHUKUGAUJUCUFEUCPQRTUAUB $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x y F $.
    $( Membership in a function value.  (Contributed by NM, 30-Apr-2004.) $)
    elfv $p |- ( A e. ( F ` B ) <->
               E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) ) $=
      ( cfv wcel cv wbr weq wb wal cab cuni wa wex fv2 eleq2i eluniab bitri ) C
      DEFZGCDBHEIBAJKBLZAMNZGCAHGUBOAPUAUCCABDEQRUBACST $.
  $}

  ${
    $d x A $.  $d x B $.  $d x F $.  $d x G $.
    $( Equality theorem for function value.  (Contributed by NM,
       29-Dec-1996.) $)
    fveq1 $p |- ( F = G -> ( F ` A ) = ( G ` A ) ) $=
      ( vx wceq cv wbr cio cfv breq iotabidv df-fv 3eqtr4g ) BCEZADFZBGZDHAOCGZ
      DHABIACINPQDAOBCJKDABLDACLM $.

    $( Equality theorem for function value.  (Contributed by NM,
       29-Dec-1996.) $)
    fveq2 $p |- ( A = B -> ( F ` A ) = ( F ` B ) ) $=
      ( vx wceq cv wbr cio cfv breq1 iotabidv df-fv 3eqtr4g ) ABEZADFZCGZDHBOCG
      ZDHACIBCINPQDABOCJKDACLDBCLM $.
  $}

  ${
    fveq1i.1 $e |- F = G $.
    $( Equality inference for function value.  (Contributed by NM,
       2-Sep-2003.) $)
    fveq1i $p |- ( F ` A ) = ( G ` A ) $=
      ( wceq cfv fveq1 ax-mp ) BCEABFACFEDABCGH $.
  $}

  ${
    fveq1d.1 $e |- ( ph -> F = G ) $.
    $( Equality deduction for function value.  (Contributed by NM,
       2-Sep-2003.) $)
    fveq1d $p |- ( ph -> ( F ` A ) = ( G ` A ) ) $=
      ( wceq cfv fveq1 syl ) ACDFBCGBDGFEBCDHI $.
  $}

  ${
    fveq2i.1 $e |- A = B $.
    $( Equality inference for function value.  (Contributed by NM,
       28-Jul-1999.) $)
    fveq2i $p |- ( F ` A ) = ( F ` B ) $=
      ( wceq cfv fveq2 ax-mp ) ABEACFBCFEDABCGH $.
  $}

  ${
    fveq2d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for function value.  (Contributed by NM,
       29-May-1999.) $)
    fveq2d $p |- ( ph -> ( F ` A ) = ( F ` B ) ) $=
      ( wceq cfv fveq2 syl ) ABCFBDGCDGFEBCDHI $.
  $}

  $( Equality theorem for nested function values.  (Contributed by AV,
     14-Aug-2022.) $)
  2fveq3 $p |- ( A = B -> ( F ` ( G ` A ) ) = ( F ` ( G ` B ) ) ) $=
    ( wceq cfv fveq2 fveq2d ) ABEADFBDFCABDGH $.

  ${
    fveq12i.1 $e |- F = G $.
    fveq12i.2 $e |- A = B $.
    $( Equality deduction for function value.  (Contributed by FL,
       27-Jun-2014.) $)
    fveq12i $p |- ( F ` A ) = ( G ` B ) $=
      ( cfv fveq1i fveq2i eqtri ) ACGADGBDGACDEHABDFIJ $.
  $}

  ${
    fveq12d.1 $e |- ( ph -> F = G ) $.
    fveq12d.2 $e |- ( ph -> A = B ) $.
    $( Equality deduction for function value.  (Contributed by FL,
       22-Dec-2008.) $)
    fveq12d $p |- ( ph -> ( F ` A ) = ( G ` B ) ) $=
      ( cfv fveq1d fveq2d eqtrd ) ABDHBEHCEHABDEFIABCEGJK $.
  $}

  ${
    fveqeq2d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for function value.  (Contributed by BJ,
       30-Aug-2022.) $)
    fveqeq2d $p |- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) ) $=
      ( cfv fveq2d eqeq1d ) ABEGCEGDABCEFHI $.
  $}

  $( Equality deduction for function value.  (Contributed by BJ,
     31-Aug-2022.) $)
  fveqeq2 $p |- ( A = B -> ( ( F ` A ) = C <-> ( F ` B ) = C ) ) $=
    ( wceq id fveqeq2d ) ABEZABCDHFG $.

  ${
    $d y F $.  $d y A $.  $d x y $.
    nffv.1 $e |- F/_ x F $.
    nffv.2 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for function value.  (Contributed by
       NM, 14-Nov-1995.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nffv $p |- F/_ x ( F ` A ) $=
      ( vy cfv cv wbr cio df-fv nfcv nfbr nfiotaw nfcxfr ) ABCGBFHZCIZFJFBCKQAF
      ABPCEDAPLMNO $.
  $}

  ${
    $d x C $.
    $( Bound-variable hypothesis builder for mapping, special case.
       (Contributed by Mario Carneiro, 25-Dec-2016.) $)
    nffvmpt1 $p |- F/_ x ( ( x e. A |-> B ) ` C ) $=
      ( cmpt nfmpt1 nfcv nffv ) ADABCEABCFADGH $.
  $}

  ${
    $d z A $.  $d z F $.  $d x z $.
    nffvd.2 $e |- ( ph -> F/_ x F ) $.
    nffvd.3 $e |- ( ph -> F/_ x A ) $.
    $( Deduction version of bound-variable hypothesis builder ~ nffv .
       (Contributed by NM, 10-Nov-2005.)  (Revised by Mario Carneiro,
       15-Oct-2016.) $)
    nffvd $p |- ( ph -> F/_ x ( F ` A ) ) $=
      ( vz cv wcel wal cab cfv wnfc nfaba1 nffv wb wa nfnfc1 wceq abidnf adantr
      nfan adantl fveq12d nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUHDIZBJGKZLZMZBC
      DLZMZBUJULUKBGNUIBGNOABDMZBCMZUNUPPEFUQURQZBUMUOUQURBBDRBCRUBUSUJCULDUQUL
      DSURBGDTUAURUJCSUQBGCTUCUDUEUFUG $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( A function has one value given an argument in its domain.  (Contributed
       by Jim Kingdon, 29-Dec-2018.) $)
    funfveu $p |- ( ( Fun F /\ A e. dom F ) -> E! y A F y ) $=
      ( vx wfun cdm wcel cv wbr weu wa wi wceq eleq1 anbi2d eubidv imbi12d wrel
      breq1 wral dffun8 simprbi r19.21bi vtoclg anabsi7 ) CEZBCFZGZBAHZCIZAJZUF
      DHZUGGZKZULUICIZAJZLUFUHKZUKLDBUGULBMZUNUQUPUKURUMUHUFULBUGNOURUOUJAULBUI
      CSPQUFUPDUGUFCRUPDUGTDACUAUBUCUDUE $.
  $}

  ${
    $d x A $.  $d x B $.  $d x F $.
    $( The value of a function is a subset of ` B ` if every element that could
       be a candidate for the value is a subset of ` B ` .  (Contributed by
       Mario Carneiro, 24-May-2019.) $)
    fvss $p |- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B ) $=
      ( cv wbr wss wi wal cfv cio df-fv iotass eqsstrid ) BAEZDFZOCGHAIBDJPAKCA
      BDLPACMN $.

    $( The result of a function value is always a subset of the union of the
       range, if the input is a set.  (Contributed by Stefan O'Rear,
       2-Nov-2014.)  (Revised by Mario Carneiro, 24-May-2019.) $)
    fvssunirng $p |- ( A e. _V -> ( F ` A ) C_ U. ran F ) $=
      ( vx cvv wcel cv wbr crn cuni wss wi wal cfv vex brelrng 3exp mpi elssuni
      syl6 alrimiv fvss syl ) ADEZACFZBGZUDBHZIZJZKZCLABMUGJUCUICUCUEUDUFEZUHUC
      UDDEZUEUJKCNUCUKUEUJAUDBDDOPQUDUFRSTCAUGBUAUB $.

    $( The result of a function value is always a subset of the union of the
       range, even if it is invalid and thus empty.  (Contributed by Stefan
       O'Rear, 2-Nov-2014.)  (Revised by Mario Carneiro, 24-May-2019.) $)
    relfvssunirn $p |- ( Rel F -> ( F ` A ) C_ U. ran F ) $=
      ( vx wrel cv wbr crn cuni wss wi wal wcel relelrn ex elssuni syl6 alrimiv
      cfv fvss syl ) BDZACEZBFZUBBGZHZIZJZCKABRUEIUAUGCUAUCUBUDLZUFUAUCUHAUBBMN
      UBUDOPQCAUEBST $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( The value of a function exists.  A special case of Corollary 6.13 of
       [TakeutiZaring] p. 27.  (Contributed by Jim Kingdon, 29-Dec-2018.) $)
    funfvex $p |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V ) $=
      ( vy wfun cdm wcel wa cfv wbr cio cvv df-fv weu funfveu euiotaex eqeltrid
      cv syl ) BDABEFGZABHACQBIZCJZKCABLSTCMUAKFCABNTCORP $.

    $( If a function has a set range, then the function value exists
       unconditional on the domain.  (Contributed by Mario Carneiro,
       24-May-2019.) $)
    relrnfvex $p |- ( ( Rel F /\ ran F e. _V ) -> ( F ` A ) e. _V ) $=
      ( wrel cfv crn cuni wss cvv wcel relfvssunirn uniexg ssexg syl2an ) BCABD
      ZBEZFZGPHINHIOHIABJOHKNPHLM $.

    $( Evaluating a set function at a set exists.  (Contributed by Mario
       Carneiro and Jim Kingdon, 28-May-2019.) $)
    fvexg $p |- ( ( F e. V /\ A e. W ) -> ( F ` A ) e. _V ) $=
      ( wcel cfv crn cuni wss cvv elex fvssunirng rnexg uniexg ssexg syl2anr
      syl ) ADEZABFZBGZHZIZUAJEZSJEBCEZRAJEUBADKABLQUDTJEUCBCMTJNQSUAJOP $.

    ${
      fvex.1 $e |- F e. V $.
      fvex.2 $e |- A e. W $.
      $( Evaluating a set function at a set exists.  (Contributed by Mario
         Carneiro and Jim Kingdon, 28-May-2019.) $)
      fvex $p |- ( F ` A ) e. _V $=
        ( wcel cfv cvv fvexg mp2an ) BCGADGABHIGEFABCDJK $.
    $}

    $( If a function is set-like, then the function value exists if the input
       does.  (Contributed by Mario Carneiro, 24-May-2019.) $)
    sefvex $p |- ( ( `' F Se _V /\ A e. _V ) -> ( F ` A ) e. _V ) $=
      ( vy vx cvv ccnv wse wcel wa cfv cv wbr crab cuni wss wi wal w3a vex syl
      a1i simp3 simp2 brcnvg sylancr mpbird breq1 elrab sylanbrc elssuni 3expia
      wb alrimiv fvss seex uniexg ssexg syl2anc ) EBFZGZAEHZIZABJZCKZAUSLZCEMZN
      ZOZVGEHZVCEHVBADKZBLZVJVGOZPZDQVHVBVMDUTVAVKVLUTVAVKRZVJVFHZVLVNVJEHZVJAU
      SLZVOVPVNDSZUAVNVQVKUTVAVKUBVNVPVAVQVKULVRUTVAVKUCVJAEEBUDUEUFVEVQCVJEVDV
      JAUSUGUHUIVJVFUJTUKUMDAVGBUNTVBVFEHVICEAUSUOVFEUPTVCVGEUQUR $.
  $}

  $( Move a conditional outside of a function.  (Contributed by Jim Kingdon,
     1-Jan-2022.) $)
  fvifdc $p |- ( DECID ph
      -> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) ) $=
    ( cif cfv fveq2 ifsbdc ) ABCABCEZDFBDFCDFIBDGICDGH $.

  ${
    $d x y z F $.  $d x y z A $.
    $( Alternate definition of the value of a function.  Definition 6.11 of
       [TakeutiZaring] p. 26.  (Contributed by NM, 30-Apr-2004.)  (Revised by
       Mario Carneiro, 31-Aug-2015.) $)
    fv3 $p |- ( F ` A ) = { x | ( E. y ( x e. y /\ A F y ) /\
              E! y A F y ) } $=
      ( vz cv wcel wbr wa wex weu cfv weq wb wal elfv wi biimpr breq2 sylib vex
      alimi ceqsalv anim2i eximi elequ2 anbi12d cbvexv exsimpr df-eu sylibr jca
      nfeu1 nfa1 nfan nfex nfim biimp ax-14 syl6 com23 impd anc2ri com12 eximdv
      nfv sps biimtrid exlimi imp impbii bitri abbi2i ) AFZBFZGZCVODHZIZBJZVQBK
      ZIZACDLZVNWBGVNEFZGZVQBEMZNZBOZIZEJZWAEBVNCDPWIWAWIVSVTWIWDCWCDHZIZEJVSWH
      WKEWGWJWDWGWEVQQZBOWJWFWLBVQWERUBVQWJBWCEUAVOWCCDSUCTUDUEWKVREBEBMWDVPWJV
      QEBAUFWCVOCDSUGUHTWIWGEJZVTWDWGEUIVQBEUJZUKULVSVTWIVRVTWIQBVTWIBVQBUMWHBE
      WDWGBWDBVFWFBUNUOUPUQVTWMVRWIWNVRWGWHEWGVRWHWGVRWDWFVRWDQBWFVPVQWDWFVQVPW
      DWFVQWEVPWDQVQWEURBEAUSUTVAVBVGVCVDVEVHVIVJVKVLVM $.
  $}

  ${
    $d x F $.  $d x A $.  $d x B $.
    $( The value of a restricted function.  (Contributed by NM, 2-Aug-1994.) $)
    fvres $p |- ( A e. B -> ( ( F |` B ) ` A ) = ( F ` A ) ) $=
      ( vx wcel cv cres wbr cio cfv vex brres rbaib iotabidv df-fv 3eqtr4g ) AB
      EZADFZCBGZHZDIARCHZDIASJACJQTUADTUAQARCBDKLMNDASODACOP $.
  $}

  ${
    fvresd.1 $e |- ( ph -> A e. B ) $.
    $( The value of a restricted function, deduction version of ~ fvres .
       (Contributed by Glauco Siliprandi, 8-Apr-2021.) $)
    fvresd $p |- ( ph -> ( ( F |` B ) ` A ) = ( F ` A ) ) $=
      ( wcel cres cfv wceq fvres syl ) ABCFBDCGHBDHIEBCDJK $.
  $}

  $( The value of a member of the domain of a subclass of a function.
     (Contributed by NM, 15-Aug-1994.) $)
  funssfv $p |- ( ( Fun F /\ G C_ F /\ A e. dom G ) ->
                ( F ` A ) = ( G ` A ) ) $=
    ( wfun wss cdm wcel cfv wceq wa cres fvres eqcomd funssres fveq1d sylan9eqr
    3impa ) BDZCBEZACFZGZABHZACHZIUARSJZUBABTKZHZUCUAUFUBATBLMUDAUECBCNOPQ $.

  ${
    $d y F $.  $d y A $.
    $( Function value.  Theorem 6.12(1) of [TakeutiZaring] p. 27.  (Contributed
       by NM, 30-Apr-2004.) $)
    tz6.12-1 $p |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y ) $=
      ( cv wbr weu wa cfv cio df-fv wceq iota1 biimpac eqtrid ) BADZCEZPAFZGBCH
      PAIZOABCJQPROKPALMN $.

    $( Function value.  Theorem 6.12(1) of [TakeutiZaring] p. 27.  (Contributed
       by NM, 10-Jul-1994.) $)
    tz6.12 $p |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) ->
                 ( F ` A ) = y ) $=
      ( cv cop wcel wbr weu cfv wceq df-br eubii tz6.12-1 syl2anbr ) BADZECFZBO
      CGZQAHBCIOJPAHBOCKZQPARLABCMN $.
  $}

  ${
    $d y z A $.  $d z F $.
    tz6.12f.1 $e |- F/_ y F $.
    $( Function value, using bound-variable hypotheses instead of distinct
       variable conditions.  (Contributed by NM, 30-Aug-1999.) $)
    tz6.12f $p |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) ->
                 ( F ` A ) = y ) $=
      ( vz cv cop wcel weu wa cfv wceq wi opeq2 eleq1d wb nfel2 nfv cbveu a1i
      anbi12d eqeq2 imbi12d tz6.12 chvarv ) BEFZGZCHZUHEIZJZBCKZUFLZMBAFZGZCHZU
      OAIZJZUKUMLZMEAUFUMLZUJUQULURUSUHUOUIUPUSUGUNCUFUMBNOZUIUPPUSUHUOEAAUGCDQ
      UOERUTSTUAUFUMUKUBUCEBCUDUE $.
  $}

  ${
    $d y F $.  $d y A $.
    $( Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.  (Contributed by
       NM, 30-Apr-2004.) $)
    tz6.12c $p |- ( E! y A F y -> ( ( F ` A ) = y <-> A F y ) ) $=
      ( cv wbr weu cfv wceq wex euex wi nfeu1 nfv tz6.12-1 expcom breq2 biimprd
      nfim syli com12 exlimi mpcom syl5ibcom impbid ) BADZCEZAFZBCGZUEHZUFUGBUH
      CEZUIUFUFAIUGUJUFAJUFUGUJKAUGUJAUFALUJAMRUGUFUJUFUGUIUJUFUGUIABCNOZUIUJUF
      UHUEBCPZQSTUAUBULUCUKUD $.
  $}

  ${
    $d x A $.  $d x F $.
    $( The value of a class outside its domain is the empty set.  (Contributed
       by Jim Kingdon, 15-Jan-2019.) $)
    ndmfvg $p |- ( ( A e. _V /\ -. A e. dom F ) -> ( F ` A ) = (/) ) $=
      ( vx cvv wcel cdm wn cfv c0 wceq cv wbr weu euex eldmg imbitrrid tz6.12-2
      wex con3d syl6 imp ) ADEZABFEZGZABHIJZUBUDACKBLZCMZGUEUBUGUCUGUCUBUFCRUFC
      NCABDOPSCABQTUA $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    $( If a function value has a member, the argument belongs to the domain.
       (Contributed by Jim Kingdon, 22-Jan-2019.) $)
    relelfvdm $p |- ( ( Rel F /\ A e. ( F ` B ) ) -> B e. dom F ) $=
      ( vx vy wrel cfv wcel wa cv wbr wex cdm weq wb wal elfv exsimpr wsb syl
      sylbi equsb1 spsbbi mpbiri breq2 sylib eximi anim2i 19.42v sylibr releldm
      nfv sbie exlimiv ) CFZABCGHZIZUOBDJZCKZIZDLZBCMHZUQUOUSDLZIVAUPVCUOUPBEJZ
      CKZEDNZOEPZDLZVCUPAURHZVGIDLVHDEABCQVIVGDRUAVGUSDVGVEEDSZUSVGVJVFEDSEDUBV
      EVFEDUCUDVEUSEDUSEULVDURBCUEUMUFUGTUHUOUSDUIUJUTVBDBURCUKUNT $.
  $}

  ${
    $d A x $.  $d F x $.
    $( The value of a relation outside its domain is the empty set.
       (Contributed by Jim Kingdon, 6-Aug-2026.) $)
    relndmfv $p |- ( ( Rel F /\ -. A e. dom F ) -> ( F ` A ) = (/) ) $=
      ( vx wrel cdm wcel wn wa cfv c0 cvv cv relelfvdm elexd adantlr ex wi noel
      pm2.21i a1i wb wceq simpr simplr ndmfvg syl2anc eleq2d pm5.21ndd eqrdv )
      BDZABEZFGZHZCABIZJUMAKFZCLZUNFZUPJFZUMUQUOUJUQUOULUJUQHAUKUPABMNOPURUOQUM
      URUOUPRSTUMUOUQURUAUMUOHZUNJUPUSUOULUNJUBUMUOUCUJULUOUDABUEUFUGPUHUI $.
  $}

  ${
    $d B k x $.  $d F j k x $.
    $( If a function value has a member, the function is inhabited.
       (Contributed by Jim Kingdon, 14-Jun-2025.) $)
    elfvm $p |- ( A e. ( F ` B ) -> E. j j e. F ) $=
      ( vk vx cfv wcel cv wex wbr weu cio eliotaeu eleq2s euex brm exlimiv 3syl
      df-fv eleq1w cbvexv sylib ) ABDGZHZEIDHZEJZCIDHZCJUEBFIZDKZFLZUJFJUGUKAUJ
      FMUDUJFANFBDTOUJFPUJUGFEBUIDQRSUFUHECECDUAUBUC $.
  $}

  ${
    $d B w $.  $d F w $.
    $( If a function value is inhabited, the function value is a set.
       (Contributed by Jim Kingdon, 30-Jan-2026.) $)
    elfvfvex $p |- ( A e. ( F ` B ) -> ( F ` B ) e. _V ) $=
      ( vw cfv wcel wbr cio cvv df-fv weu eliotaeu eleq2s euiotaex syl eqeltrid
      cv ) ABCEZFZRBDQCGZDHZIDBCJZSTDKZUAIFUCAUARTDALUBMTDNOP $.
  $}

  ${
    $d F w $.  $d X w $.
    $( If a function value is inhabited, the argument is related to the
       function value.  (Contributed by Jim Kingdon, 31-Jan-2026.) $)
    fvmbr $p |- ( A e. ( F ` X ) -> X F ( F ` X ) ) $=
      ( vw cfv wcel wbr cv cio df-fv eqcomi cvv weu wb elfvfvex eliotaeu eleq2s
      wceq breq2 iota2 syl2anc mpbiri ) ACBEZFZCUCBGZCDHZBGZDIZUCRZUCUHDCBJZKUD
      UCLFUGDMZUEUINACBOUKAUHUCUGDAPUJQUGUEDUCLUFUCCBSTUAUB $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.  $d x y z w F $.
    $( The value of a non-member of a restriction is the empty set.
       (Contributed by NM, 13-Nov-1995.) $)
    nfvres $p |- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) ) $=
      ( vz vx vw vy wcel wn cres cfv cv wex cab csn wceq eleq2i bitri sylbi cvv
      wbr weu cuni cio df-fv df-iota eqtri eluni exsimpr wsb df-clab nfv eqeq2d
      wa sneq sbie exbii sylib euabsn2 sylibr euex cop cxp df-br df-res simprbi
      cin elin opelxp1 syl exlimiv 3syl con3i eq0rdv ) ABHZIDACBJZKZDLZVQHZVOVS
      AELZVPUAZEUBZWAEMVOVSWAENZFLZOZPZFMZWBVSWDWCGLZOZPZGNZHZFMZWGVSVRWDHZWLUN
      FMZWMVSVRWKUCZHWOVQWPVRVQWAEUDWPEAVPUEWAEGUFUGQFVRWKUHRWNWLFUISWLWFFWLWJG
      FUJWFWJFGUKWJWFGFWFGULWHWDPWIWEWCWHWDUOUMUPRUQURWAEFUSUTWAEVAWAVOEWAAVTVB
      ZBTVCZHZVOWAWQCWRVGZHZWSWAWQVPHXAAVTVPVDVPWTWQCBVEQRXAWQCHWSWQCWRVHVFSAVT
      BTVIVJVKVLVMVN $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( If the restriction of a class to a singleton is not a function, its
       value is the empty set.  (Contributed by NM, 8-Aug-2010.)  (Proof
       shortened by Andrew Salmon, 22-Oct-2011.) $)
    nfunsn $p |- ( -. Fun ( F |` { A } ) -> ( F ` A ) = (/) ) $=
      ( vy vx csn cres wfun wn cv wbr weu cfv c0 wceq wrel wmo wal wa sylbi syl
      eumo wcel vex brres velsn breq1 biimpac moimi alrimiv relres jctil dffun6
      wb sylibr con3i tz6.12-2 ) BAEZFZGZHACIZBJZCKZHABLMNVBUSVBUROZDIZUTURJZCP
      ZDQZRUSVBVGVCVBVFDVBVACPVFVACUAVEVACVEVDUTBJZVDUQUBZRVAVDUTBUQCUCUDVIVHVA
      VIVDANVHVAUMDAUEVDAUTBUFSUGSUHTUIBUQUJUKDCURULUNUOCABUPT $.
  $}

  ${
    $d x A $.
    $( Function value of the empty set.  (Contributed by Stefan O'Rear,
       26-Nov-2014.) $)
    0fv $p |- ( (/) ` A ) = (/) $=
      ( vx c0 cfv cv wbr cio df-fv weu wn wceq wex cop wcel noel df-br nex euex
      mtbir mto iotanul ax-mp eqtri ) ACDABEZCFZBGZCBACHUEBIZJUFCKUGUEBLUEBUEAU
      DMZCNUHOAUDCPSQUEBRTUEBUAUBUC $.
  $}

  $( A function value of a function value at a proper class is the empty set.
     (Contributed by AV, 8-Apr-2021.) $)
  fv2prc $p |- ( -. A e. _V -> ( ( F ` A ) ` B ) = (/) ) $=
    ( cvv wcel wn cfv c0 fvprc fveq1d 0fv eqtrdi ) ADEFZBACGZGBHGHMBNHACIJBKL
    $.

  ${
    $d y A $.  $d y B $.  $d y C $.  $d y F $.  $d x y $.
    $( Move class substitution in and out of a function value.  (Contributed by
       NM, 11-Nov-2005.) $)
    csbfv12g $p |- ( A e. C ->
                 [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) ) $=
      ( vy wcel wbr cio csb cfv csbiotag sbcbrg csbconstg breq2d bitrd iotabidv
      cv wsbc df-fv eqtrd csbeq2i 3eqtr4g ) BDGZABCFRZEHZFIZJZABCJZUEABEJZHZFIZ
      ABCEKZJUIUJKUDUHUFABSZFIULUFAFBDLUDUNUKFUDUNUIABUEJZUJHUKABCUEDEMUDUOUEUI
      UJABUEDNOPQUAABUMUGFCETUBFUIUJTUC $.
  $}

  ${
    $d x F $.
    $( Move class substitution in and out of a function value.  (Contributed by
       NM, 10-Nov-2005.) $)
    csbfv2g $p |- ( A e. C -> [_ A / x ]_ ( F ` B ) =
                  ( F ` [_ A / x ]_ B ) ) $=
      ( wcel cfv csb csbfv12g csbconstg fveq1d eqtrd ) BDFZABCEGHABCHZABEHZGNEG
      ABCDEIMNOEABEDJKL $.

    $( Substitution for a function value.  (Contributed by NM, 1-Jan-2006.) $)
    csbfvg $p |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) ) $=
      ( wcel cv cfv csb csbfv2g csbvarg fveq2d eqtrd ) BCEZABAFZDGHABNHZDGBDGAB
      NCDIMOBDABCJKL $.
  $}

  ${
    $d y A $.  $d y F $.  $d y B $.
    $( The second argument of a binary relation on a function is the function's
       value.  (Contributed by NM, 30-Apr-2004.)  (Revised by Mario Carneiro,
       28-Apr-2015.) $)
    funbrfv $p |- ( Fun F -> ( A F B -> ( F ` A ) = B ) ) $=
      ( vy wfun wbr cfv wceq cvv wcel wa wrel funrel brrelex2 sylan cv wi breq2
      anbi2d eqeq2 imbi12d weu funeu tz6.12-1 sylan2 anabss7 vtoclg mpcom ex )
      CEZABCFZACGZBHZBIJZUJUKKZUMUJCLUKUNCMABCNOUJADPZCFZKZULUPHZQUOUMQDBIUPBHZ
      URUOUSUMUTUQUKUJUPBACRSUPBULTUAUJUQUSURUQUQDUBUSDAUPCUCDACUDUEUFUGUHUI $.
  $}

  $( The second element in an ordered pair member of a function is the
     function's value.  (Contributed by NM, 19-Jul-1996.) $)
  funopfv $p |- ( Fun F -> ( <. A , B >. e. F -> ( F ` A ) = B ) ) $=
    ( cop wcel wbr wfun cfv wceq df-br funbrfv biimtrrid ) ABDCEABCFCGACHBIABCJ
    ABCKL $.

  ${
    $d x F $.  $d x A $.  $d x B $.  $d x C $.
    $( Equivalence of function value and binary relation.  (Contributed by NM,
       19-Apr-2004.)  (Revised by Mario Carneiro, 28-Apr-2015.) $)
    fnbrfvb $p |- ( ( F Fn A /\ B e. A ) ->
                   ( ( F ` B ) = C <-> B F C ) ) $=
      ( vx wfn wcel wa cfv wceq wbr eqid cvv wb funfvex funfni cv wi breq2 syl
      eqeq2 bibi12d imbi2d fneu tz6.12c vtoclg mpcom mpbii syl5ibcom wfun fnfun
      weu funbrfv adantr impbid ) DAFZBAGZHZBDIZCJZBCDKZURBUSDKZUTVAURUSUSJZVBU
      SLUSMGZURVCVBNZVDABDBDOPURUSEQZJZBVFDKZNZRURVEREUSMVFUSJZVIVEURVJVGVCVHVB
      VFUSUSUAVFUSBDSUBUCURVHEULVIEABDUDEBDUETUFUGUHUSCBDSUIUPVAUTRZUQUPDUJVKAD
      UKBCDUMTUNUO $.
  $}

  $( Equivalence of function value and ordered pair membership.  (Contributed
     by NM, 7-Nov-1995.) $)
  fnopfvb $p |- ( ( F Fn A /\ B e. A ) ->
                 ( ( F ` B ) = C <-> <. B , C >. e. F ) ) $=
    ( wfn wcel wa cfv wceq wbr cop fnbrfvb df-br bitrdi ) DAEBAFGBDHCIBCDJBCKDF
    ABCDLBCDMN $.

  $( Equivalence of function value and binary relation.  (Contributed by NM,
     26-Mar-2006.) $)
  funbrfvb $p |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) ) $=
    ( wfun cdm wfn wcel cfv wceq wbr wb funfn fnbrfvb sylanb ) CDCCEZFAOGACHBIA
    BCJKCLOABCMN $.

  $( Equivalence of function value and ordered pair membership.  Theorem
     4.3(ii) of [Monk1] p. 42.  (Contributed by NM, 26-Jan-1997.) $)
  funopfvb $p |- ( ( Fun F /\ A e. dom F ) ->
                 ( ( F ` A ) = B <-> <. A , B >. e. F ) ) $=
    ( wfun cdm wfn wcel cfv wceq cop wb funfn fnopfvb sylanb ) CDCCEZFAOGACHBIA
    BJCGKCLOABCMN $.

  $( Version of ~ fnbrfvb for functions on Cartesian products: function value
     expressed as a binary relation.  See ~ fnbrovb for the form when ` F ` is
     seen as a binary operation.  (Contributed by BJ, 15-Feb-2022.) $)
  fnbrfvb2 $p |- ( ( F Fn ( V X. W ) /\ ( A e. V /\ B e. W ) ) ->
                           ( ( F ` <. A , B >. ) = C <-> <. A , B >. F C ) ) $=
    ( wcel wa cxp wfn cop cfv wceq wbr wb opelxpi fnbrfvb sylan2 ) AEGBFGHDEFIZ
    JABKZSGTDLCMTCDNOABEFPSTCDQR $.

  ${
    $d A y $.  $d B y $.  $d F y $.  $d X y $.
    $( There is exactly one codomain element for each element of the domain of
       a function.  (Contributed by AV, 20-Apr-2025.) $)
    fdmeu $p |- ( ( F : A --> B /\ X e. A ) -> E! y e. B ( F ` X ) = y ) $=
      ( wf wcel wa cfv cv wceq wreu cop feu wfn wb ffn anim1i adantr fnopfvb
      syl reubidva mpbird ) BCDFZEBGZHZEDIAJZKZACLEUGMDGZACLABCEDNUFUHUIACUFUGC
      GZHDBOZUEHZUHUIPUFULUJUDUKUEBCDQRSBEUGDTUAUBUC $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    $( Function value in terms of a binary relation.  (Contributed by Mario
       Carneiro, 19-Mar-2014.) $)
    funbrfv2b $p |- ( Fun F ->
                       ( A F B <-> ( A e. dom F /\ ( F ` A ) = B ) ) ) $=
      ( wfun wbr cdm wcel wa cfv wceq wrel funrel releldm syl pm4.71rd funbrfvb
      wi ex pm5.32da bitr4d ) CDZABCEZACFGZUBHUCACIBJZHUAUBUCUACKZUBUCQCLUEUBUC
      ABCMRNOUAUCUDUBABCPST $.

    $( Representation of a function in terms of its values.  The converse holds
       given the law of the excluded middle; as it is we have most of the
       converse via ~ funmpt and ~ dmmptss .  (Contributed by Jim Kingdon,
       31-Dec-2018.) $)
    dffn5im $p |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) ) $=
      ( vy wfn cv wcel cfv wceq wa copab cmpt wbr wrel fnrel dfrel4v sylib fnbr
      ex pm4.71rd eqcom fnbrfvb bitrid pm5.32da bitr4d opabbidv df-mpt eqtr4di
      eqtrd ) CBEZCAFZBGZDFZUKCHZIZJZADKZABUNLUJCUKUMCMZADKZUQUJCNCUSIBCOADCPQU
      JURUPADUJURULURJUPUJURULUJURULBUKUMCRSTUJULUOURUOUNUMIUJULJURUMUNUABUKUMC
      UBUCUDUEUFUIADBUNUGUH $.

    $( The range of a function expressed as a collection of the function's
       values.  (Contributed by NM, 20-Oct-2005.)  (Proof shortened by Mario
       Carneiro, 31-Aug-2015.) $)
    fnrnfv $p |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } ) $=
      ( wfn crn cv cfv cmpt wceq wrex cab dffn5im rneqd eqid rnmpt eqtrdi ) DCE
      ZDFACAGDHZIZFBGSJACKBLRDTACDMNABCSTTOPQ $.

    $( A member of a function's range is a value of the function.  (Contributed
       by NM, 31-Oct-1995.) $)
    fvelrnb $p |- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) ) $=
      ( vy cv cfv wceq wrex cvv wcel wi wfn crn wb wa wex df-rex 19.41v simpl
      anim1i ancomd funfvex funfni syl simpr eleq1d adantr mpbid exlimiv sylbir
      sylanb expcom cab fnrnfv eleq2d eqeq1 eqcom bitrdi rexbidv elab3g mpancom
      sylan9bbr ) AFZDGZCHZABIZCJKZLZDBMZCDNZKZVGOVGVJVHVGVDBKZVFPZAQZVJVHVFABR
      VOVJPVNVJPZAQVHVNVJASVPVHAVPVEJKZVHVPVJVMPVQVPVMVJVNVMVJVMVFTUAUBVQBVDDVD
      DUCUDUEVNVQVHOVJVNVECJVMVFUFUGUHUIUJUKULUMVJVLCEFZVEHZABIZEUNZKVIVGVJVKWA
      CAEBDUOUPVTVGECJVRCHZVSVFABWBVSCVEHVFVRCVEUQCVEURUSUTVAVCVB $.

    $( Alternate definition of the image of a function.  (Contributed by Raph
       Levien, 20-Nov-2006.) $)
    dfimafn $p |- ( ( Fun F /\ A C_ dom F ) ->
                  ( F " A ) = { y | E. x e. A ( F ` x ) = y } ) $=
      ( wfun cdm wss wa cima cv wbr wrex cab cfv wceq dfima2 wcel ssel funbrfvb
      wb ex syl9r imp31 rexbidva abbidv eqtr4id ) DEZCDFZGZHZDCIAJZBJZDKZACLZBM
      UKDNULOZACLZBMABDCPUJUPUNBUJUOUMACUGUIUKCQZUOUMTZUIUQUKUHQZUGURCUHUKRUGUS
      URUKULDSUAUBUCUDUEUF $.

    $( Alternate definition of the image of a function as an indexed union of
       singletons of function values.  (Contributed by Raph Levien,
       20-Nov-2006.) $)
    dfimafn2 $p |- ( ( Fun F /\ A C_ dom F ) ->
                   ( F " A ) = U_ x e. A { ( F ` x ) } ) $=
      ( vy wfun cdm wss wa cima cv cfv wceq cab ciun wrex dfimafn iunab eqtr4di
      csn wcel df-sn eqcom abbii eqtri a1i iuneq2i ) CEBCFGHZCBIZABAJZCKZDJZLZD
      MZNZABUJSZNUGUHULABODMUNADBCPULADBQRABUOUMUOUMLUIBTUOUKUJLZDMUMDUJUAUPULD
      UKUJUBUCUDUEUFR $.

    $( Membership relation for the values of a function whose image is a
       subclass.  (Contributed by Raph Levien, 20-Nov-2006.) $)
    funimass4 $p |- ( ( Fun F /\ A C_ dom F ) ->
                    ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) ) $=
      ( vy cima wss cv wcel wi wal wfun cdm wa wral wceq wb wrex bitrid cvv cfv
      ssalel wbr vex elima eqcom ssel funbrfvb ex imp31 rexbidva bitr4id imbi1d
      syl9 r19.23v bitr4di albidv ancoms ralcom4 w3a ssel2 anim2i 3impb funfvex
      nfv eleq1 ceqsalg 3syl 3expa ralbidva bitr3id bitrd ) DBFZCGEHZVMIZVNCIZJ
      ZEKZDLZBDMZGZNZAHZDUAZCIZABOZEVMCUBWBVRVNWDPZVPJZABOZEKZWFWAVSVRWJQWAVSNZ
      VQWIEWKVQWGABRZVPJWIWKVOWLVPWKVOWCVNDUCZABRWLAVNDBEUDUEWKWGWMABWGWDVNPZWK
      WCBIZNWMVNWDUFWAVSWOWNWMQZWAWOWCVTIZVSWPBVTWCUGVSWQWPWCVNDUHUIUNUJSUKULUM
      WGVPABUOUPUQURWJWHEKZABOWBWFWHAEBUSWBWRWEABVSWAWOWRWEQZVSWAWOUTVSWQNZWDTI
      WSVSWAWOWTWAWONWQVSBVTWCVAVBVCWCDVDVPWEEWDTWEEVEVNWDCVFVGVHVIVJVKVLS $.

    $( Function value in an image.  Part of Theorem 4.4(iii) of [Monk1] p. 42.
       (Contributed by NM, 29-Apr-2004.)  (Proof shortened by Andrew Salmon,
       22-Oct-2011.) $)
    fvelima $p |- ( ( Fun F /\ A e. ( F " B ) ) ->
                  E. x e. B ( F ` x ) = A ) $=
      ( wfun cima wcel cv cfv wceq wrex wbr elimag ibi funbrfv reximdv syl5 imp
      ) DEZBDCFZGZAHZDIBJZACKZUAUBBDLZACKZSUDUAUFABDCTMNSUEUCACUBBDOPQR $.
  $}

  ${
    $d A x $.  $d B x $.  $d F x $.  $d Y x $.

    $( A member of a surjective function's codomain is a value of the function.
       (Contributed by Thierry Arnoux, 23-Jan-2020.) $)
    foelcdmi $p |- ( ( F : A -onto-> B /\ Y e. B ) -> E. x e. A ( F ` x ) = Y )
      $=
      ( wfo wcel cv cfv wceq wrex crn forn eleq2d wfn fofn fvelrnb syl bitr3d
      wb biimpa ) BCDFZECGZAHDIEJABKZUBEDLZGZUCUDUBUECEBCDMNUBDBOUFUDTBCDPABEDQ
      RSUA $.
  $}

  ${
    $d x A $.  $d x C $.  $d x F $.
    feqmptd.1 $e |- ( ph -> F : A --> B ) $.
    $( Deduction form of ~ dffn5im .  (Contributed by Mario Carneiro,
       8-Jan-2015.) $)
    feqmptd $p |- ( ph -> F = ( x e. A |-> ( F ` x ) ) ) $=
      ( wfn cv cfv cmpt wceq wf ffn syl dffn5im ) AECGZEBCBHEIJKACDELPFCDEMNBCE
      ON $.

    feqresmpt.2 $e |- ( ph -> C C_ A ) $.
    $( Express a restricted function as a mapping.  (Contributed by Mario
       Carneiro, 18-May-2016.) $)
    feqresmpt $p |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) ) $=
      ( cres cv cfv cmpt wf wss fssres syl2anc feqmptd fvres mpteq2ia eqtrdi )
      AFEIZBEBJZUAKZLBEUBFKZLABEDUAACDFMECNEDUAMGHCDEFOPQBEUCUDUBEFRST $.
  $}

  ${
    $d x z $.  $d x z A $.  $d z F $.
    dffn5imf.1 $e |- F/_ x F $.
    $( Representation of a function in terms of its values.  (Contributed by
       Jim Kingdon, 31-Dec-2018.) $)
    dffn5imf $p |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) ) $=
      ( vz wfn cv cfv cmpt dffn5im nfcv nffv fveq2 cbvmpt eqtrdi ) CBFCEBEGZCHZ
      IABAGZCHZIEBCJEABQSAPCDAPKLESKPRCMNO $.
  $}

  ${
    $d u v A $.  $d x u v B $.  $d x u v C $.  $d x u v F $.
    $( Function value in an image.  (Contributed by NM, 20-Jan-2007.)  (Proof
       shortened by Andrew Salmon, 22-Oct-2011.)  (Revised by David Abernethy,
       17-Dec-2011.) $)
    fvelimab $p |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <->
                  E. x e. B ( F ` x ) = C ) ) $=
      ( vu vv wfn wss wa cima wcel cv cfv wceq wrex cvv eleq1 wb wi elex anim2i
      ssel2 funfvex funfni sylan2 anassrs syl5ibcom rexlimdva imdistani rexbidv
      eqeq2 bibi12d imbi2d wfun cdm fnfun adantr sseq2d biimpar dfimafn syl2anc
      cab fndm abeq2d vtoclg impcom pm5.21nd fveq2 eqeq1d cbvrexv bitrdi ) EBHZ
      CBIZJZDECKZLZFMZENZDOZFCPZAMZENZDOZACPVOVQWAVODQLZJVQWEVODVPUAUBVOWAWEVOV
      TWEFCVOVRCLZJVSQLZVTWEVMVNWFWGVNWFJVMVRBLWGCBVRUCWGBVREVREUDUEUFUGVSDQRUH
      UIUJWEVOVQWASZVOGMZVPLZVSWIOZFCPZSZTVOWHTGDQWIDOZWMWHVOWNWJVQWLWAWIDVPRWN
      WKVTFCWIDVSULUKUMUNVOWLGVPVOEUOZCEUPZIZVPWLGVCOVMWOVNBEUQURVMWQVNVMWPBCBE
      VDUSUTFGCEVAVBVEVFVGVHVTWDFACVRWBOVSWCDVRWBEVIVJVKVL $.
  $}

  $( The value of the identity function.  (Contributed by NM, 1-May-2004.)
     (Revised by Mario Carneiro, 28-Apr-2015.) $)
  fvi $p |- ( A e. V -> ( _I ` A ) = A ) $=
    ( cid wfun wcel wbr cfv wceq funi ididg funbrfv mpsyl ) CDABEAACFACGAHIABJA
    ACKL $.

  ${
    $d x y A $.  $d y B $.  $d x y F $.
    $( The indexed intersection of a function's values is the intersection of
       its range.  (Contributed by NM, 20-Oct-2005.) $)
    fniinfv $p |- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F ) $=
      ( vy wfn cv cfv ciin wceq wrex cab cint crn wcel funfvex funfni ralrimiva
      cvv wral dfiin2g syl fnrnfv inteqd eqtr4d ) CBEZABAFZCGZHZDFUGIABJDKZLZCM
      ZLUEUGRNZABSUHUJIUEULABULBUFCUFCOPQADBUGRTUAUEUKUIADBCUBUCUD $.

    $( Singleton of function value.  (Contributed by NM, 22-May-1998.) $)
    fnsnfv $p |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) ) $=
      ( vy wfn wcel wa cv cfv wceq cab wbr csn cima eqcom fnbrfvb bitrid abbidv
      df-sn a1i imasng adantl 3eqtr4d ) CAEZBAFZGZDHZBCIZJZDKZBUGCLZDKZUHMZCBMN
      ZUFUIUKDUIUHUGJUFUKUGUHOABUGCPQRUMUJJUFDUHSTUEUNULJUDDBACUAUBUC $.
  $}

  $( The image of a pair under a function.  (Contributed by Jeff Madsen,
     6-Jan-2011.) $)
  fnimapr $p |- ( ( F Fn A /\ B e. A /\ C e. A ) ->
                          ( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } ) $=
    ( wfn wcel w3a csn cima cun cfv fnsnfv 3adant3 3adant2 uneq12d eqcomd df-pr
    cpr wceq imaeq2i imaundi eqtri 3eqtr4g ) DAEZBAFZCAFZGZDBHZIZDCHZIZJZBDKZHZ
    CDKZHZJZDBCRZIZUMUORUGUQULUGUNUIUPUKUDUEUNUISUFABDLMUDUFUPUKSUEACDLNOPUSDUH
    UJJZIULURUTDBCQTDUHUJUAUBUMUOQUC $.

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z F $.
    ssimaex.1 $e |- A e. _V $.
    $( The existence of a subimage.  (Contributed by NM, 8-Apr-2007.) $)
    ssimaex $p |- ( ( Fun F /\ B C_ ( F " A ) ) ->
                 E. x ( x C_ A /\ B = ( F " x ) ) ) $=
      ( vy vz vw cima wss cdm cv wceq wa wex cfv wcel wi ex adantr wfun imaeq2i
      cin cres dmres imadmres eqtr3i sseq2i ssrab2 ssel2 adantll fvelima eleq1a
      crab wrex anim2d fveq2 eleq1d elrab imbitrrdi a1i jcad reximdv2 adantl wb
      simpr wfn funfn inss2 sstri fvelimab mpan2 sylbi sylibrd syld adantlr mpd
      eleq1 biimpcd rexlimiv syl6 impbid eqrdv inex1 rabex sseq1 imaeq2 anbi12d
      eqeq2d spcev sylancr inss1 sstr anim1i eximi syl sylan2br ) CDBIZJDUAZCDB
      DKZUCZIZJZALZBJZCDXDIZMZNZAOZXBWRCDDBUDKZIXBWRXJXADDBUEUBDBUFUGUHWSXCNZXD
      XAJZXGNZAOZXIXKFLZDPZCQZFXAUNZXAJZCDXRIZMZXNXQFXAUIZXKGCXTXKGLZCQZYCXTQZX
      KYDYEXKYDNYCXBQZYEXCYDYFWSCXBYCUJUKWSYDYFYERXCWSYDNZYFHLZDPZYCMZHXAUOZYEW
      SYFYKRYDWSYFYKHYCXADULSTYGYKYJHXRUOZYEYDYKYLRWSYDYJYJHXAXRYDYHXAQZYJNZYHX
      RQZYJYDYNYMYICQZNZYOYDYJYPYMYCCYIUMUPXQYPFYHXAXOYHMXPYICXOYHDUQURUSZUTYNY
      JRYDYMYJVFVAVBVCVDWSYEYLVEZYDWSDWTVGZYSDVHYTXRWTJYSXRXAWTYBBWTVIVJHWTXRYC
      DVKVLVMTVNVOVPVQSWSYEYDRXCWSYEYLYDWSYEYLHYCXRDULSYJYDHXRYOYQYJYDRZYRYPUUA
      YMYJYPYDYIYCCVRVSVDVMVTWATWBWCXMXSYANAXRXQFXABWTEWDWEXDXRMZXLXSXGYAXDXRXA
      WFUUBXFXTCXDXRDWGWIWHWJWKXMXHAXLXEXGXLXABJXEBWTWLXDXABWMVLWNWOWPWQ $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d F x y $.
    $( The existence of a subimage.  (Contributed by FL, 15-Apr-2007.) $)
    ssimaexg $p |- ( ( A e. C /\ Fun F /\ B C_ ( F " A ) ) ->
                 E. x ( x C_ A /\ B = ( F " x ) ) ) $=
      ( vy wcel wfun cima wss cv wceq wa wex imaeq2 sseq2d anbi2d sseq2 anbi1d
      wi exbidv imbi12d vex ssimaex vtoclg 3impib ) BDGEHZCEBIZJZAKZBJZCEUJILZM
      ZANZUGCEFKZIZJZMZUJUOJZULMZANZTUGUIMZUNTFBDUOBLZURVBVAUNVCUQUIUGVCUPUHCUO
      BEOPQVCUTUMAVCUSUKULUOBUJRSUAUBAUOCEFUCUDUEUF $.
  $}

  $( A simplified expression for the value of a function when we know it's a
     function.  (Contributed by Jim Kingdon, 1-Jan-2019.) $)
  funfvdm $p |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. ( F " { A } ) ) $=
    ( wfun cdm wcel wa cfv csn cuni cima cvv wceq funfvex unisng syl eqid df-fn
    wfn mpbiran2 fnsnfv sylanbr unieqd eqtr3d ) BCZABDZEZFZABGZHZIZUHBAHJZIUGUH
    KEUJUHLABMUHKNOUGUIUKUDBUERZUFUIUKLULUDUEUELUEPBUEQSUEABTUAUBUC $.

  ${
    $d y A $.  $d y F $.
    $( The value of a function.  Definition of function value in [Enderton]
       p. 43.  (Contributed by Jim Kingdon, 1-Jan-2019.) $)
    funfvdm2 $p |- ( ( Fun F /\ A e. dom F ) ->
        ( F ` A ) = U. { y | A F y } ) $=
      ( wfun cdm wcel wa cfv csn cima cuni wbr cab funfvdm imasng adantl unieqd
      cv wceq eqtrd ) CDZBCEZFZGZBCHCBIJZKBARCLAMZKBCNUDUEUFUCUEUFSUAABUBCOPQT
      $.
  $}

  ${
    $d w A $.  $d w F $.  $d w y $.
    funfvdm2f.1 $e |- F/_ y A $.
    funfvdm2f.2 $e |- F/_ y F $.
    $( The value of a function.  Version of ~ funfvdm2 using a bound-variable
       hypotheses instead of distinct variable conditions.  (Contributed by Jim
       Kingdon, 1-Jan-2019.) $)
    funfvdm2f $p |- ( ( Fun F /\ A e. dom F ) ->
        ( F ` A ) = U. { y | A F y } ) $=
      ( vw wfun cdm wcel wa cfv cv wbr cab cuni funfvdm2 nfcv nfbr nfv breq2
      cbvab unieqi eqtrdi ) CGBCHIJBCKBFLZCMZFNZOBALZCMZANZOFBCPUFUIUEUHFAABUDC
      DEAUDQRUHFSUDUGBCTUAUBUC $.
  $}

  $( The value of a union when the argument is in the first domain.
     (Contributed by Scott Fenton, 29-Jun-2013.) $)
  fvun1 $p |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) ->
    ( ( F u. G ) ` X ) = ( F ` X ) ) $=
    ( wfn cin c0 wceq wcel wa w3a wfun cdm cun cfv fnfun 3ad2ant1 cima cuni csn
    3ad2ant2 ineqan12d eqeq1d biimprd adantrd 3impia simp3r eleq2d mpbird funun
    fndm wb wss ssun1 ax-mp sseli anim12i anasss 3impa funfvdm syl imaundir a1i
    dmss unieqd disjel ndmima 3ad2ant3 uneq2d un0 eqtrdi eqcomd adantrl 3adant2
    wn eqtrd 3eqtrd syl112anc ) CAFZDBFZABGZHIZEAJZKZLZCMZDMZCNZDNZGZHIZEWIJZEC
    DOZPZECPZIVTWAWGWEACQRWAVTWHWEBDQUBVTWAWEWLVTWAKZWCWLWDWQWLWCWQWKWBHVTWAWIA
    WJBACULZBDULUCUDUEUFUGWFWMWDVTWAWCWDUHVTWAWMWDUMWEVTWIAEWRUIRUJWGWHWLWMKZLZ
    WOWNEUAZSZTZCXASZTZWPWTWNMZEWNNZJZKZWOXCIWGWHWSXIWGWHKZWLWMXIXJWLKXFWMXHCDU
    KWIXGECWNUNWIXGUNCDUOCWNVEUPUQURUSUTEWNVAVBWTXCXDDXASZOZTXEWTXBXLXBXLIWTCDX
    AVCVDVFWTXLXDWTXLXDHOXDWTXKHXDWSWGXKHIZWHWSEWJJVPXMWIWJEVGEDVHVBVIVJXDVKVLV
    FVQWGWSXEWPIZWHWGWMXNWLWGWMKWPXEECVAVMVNVOVRVS $.

  $( The value of a union when the argument is in the second domain.
     (Contributed by Scott Fenton, 29-Jun-2013.) $)
  fvun2 $p |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) ->
    ( ( F u. G ) ` X ) = ( G ` X ) ) $=
    ( wfn cin c0 wceq wcel w3a cun cfv uncom fveq1i incom eqeq1i anbi1i fvun1
    wa syl3an3b 3com12 eqtrid ) CAFZDBFZABGZHIZEBJZTZKECDLZMEDCLZMZEDMZEUJUKCDN
    OUEUDUIULUMIZUIUEUDBAGZHIZUHTUNUGUPUHUFUOHABPQRBADCESUAUBUC $.

  ${
    fvun1d.1 $e |- ( ph -> F Fn A ) $.
    fvun1d.2 $e |- ( ph -> G Fn B ) $.
    fvun1d.3 $e |- ( ph -> ( A i^i B ) = (/) ) $.
    fvun1d.4 $e |- ( ph -> X e. A ) $.
    $( The value of a union when the argument is in the first domain, a
       deduction version.  (Contributed by metakunt, 28-May-2024.) $)
    fvun1d $p |- ( ph -> ( ( F u. G ) ` X ) = ( F ` X ) ) $=
      ( wfn cin c0 wceq wcel wa w3a cun cfv jca 3jca fvun1 syl ) ADBKZECKZBCLMN
      ZFBOZPZQFDERSFDSNAUDUEUHGHAUFUGIJTUABCDEFUBUC $.
  $}

  ${
    fvun2d.1 $e |- ( ph -> F Fn A ) $.
    fvun2d.2 $e |- ( ph -> G Fn B ) $.
    fvun2d.3 $e |- ( ph -> ( A i^i B ) = (/) ) $.
    fvun2d.4 $e |- ( ph -> X e. B ) $.
    $( The value of a union when the argument is in the second domain, a
       deduction version.  (Contributed by metakunt, 28-May-2024.) $)
    fvun2d $p |- ( ph -> ( ( F u. G ) ` X ) = ( G ` X ) ) $=
      ( wfn cin c0 wceq wcel wa w3a cun cfv jca 3jca fvun2 syl ) ADBKZECKZBCLMN
      ZFCOZPZQFDERSFESNAUDUEUHGHAUFUGIJTUABCDEFUBUC $.
  $}

  ${
    $d x y A $.  $d x y F $.  $d x y G $.  $d x X $.
    $( Domains of a function composition.  (Contributed by NM, 27-Jan-1997.) $)
    dmfco $p |- ( ( Fun G /\ A e. dom G ) ->
               ( A e. dom ( F o. G ) <-> ( G ` A ) e. dom F ) ) $=
      ( vy vx wfun cdm wcel wa cfv cv cop wex ccom wceq cvv syl exbidv eldm2g
      wb opeq1 eleq1d ceqsexgv eqcom funopfvb bitrid anbi1d bitr3d vex opelco2g
      funfvex mpan2 bitrd adantl 3bitr4rd ) CFZACGZHZIZACJZDKZLZBHZDMZAEKZLCHZV
      EVALZBHZIZEMZDMZUTBGHZABCNZGHZUSVCVJDUSVEUTOZVHIZEMZVCVJUSUTPHZVQVCTACUKZ
      VHVCEUTPVOVGVBBVEUTVAUAUBUCQUSVPVIEUSVOVFVHVOUTVEOUSVFVEUTUDAVECUEUFUGRUH
      RUSVRVLVDTVSDUTBPSQURVNVKTUPURVNAVALVMHZDMVKDAVMUQSURVTVJDURVAPHVTVJTDUIE
      AVABCUQPUJULRUMUNUO $.

    $( Value of a function composition.  Similar to second part of Theorem 3H
       of [Enderton] p. 47.  (Contributed by NM, 9-Oct-2004.)  (Proof shortened
       by Andrew Salmon, 22-Oct-2011.)  (Revised by Stefan O'Rear,
       16-Oct-2014.) $)
    fvco2 $p |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) =
          ( F ` ( G ` X ) ) ) $=
      ( vx wfn wcel wa cv ccom csn cima cio cfv imaco fnsnfv imaeq2d dffv3g cvv
      wceq eqtr4id eleq2d iotabidv adantl funfvex funfni syl 3eqtr4d ) CAFZDAGZ
      HZEIZBCJZDKZLZGZEMZULBDCNZKZLZGZEMZDUMNZURBNZUKUPVAEUKUOUTULUKUOBCUNLZLUT
      BCUNOUKUSVEBADCPQUAUBUCUJVCUQTUIEDUMARUDUKURSGZVDVBTVFADCDCUEUFEURBSRUGUH
      $.
  $}

  $( Value of a function composition.  Similar to Exercise 5 of [TakeutiZaring]
     p. 28.  (Contributed by NM, 22-Apr-2006.)  (Proof shortened by Mario
     Carneiro, 26-Dec-2014.) $)
  fvco $p |- ( ( Fun G /\ A e. dom G ) ->
               ( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) ) $=
    ( wfun cdm wfn wcel ccom cfv wceq funfn fvco2 sylanb ) CDCCEZFANGABCHIACIBI
    JCKNBCALM $.

  $( Value of a function composition.  (Contributed by NM, 3-Jan-2004.)
     (Revised by Mario Carneiro, 26-Dec-2014.) $)
  fvco3 $p |- ( ( G : A --> B /\ C e. A ) ->
             ( ( F o. G ) ` C ) = ( F ` ( G ` C ) ) ) $=
    ( wf wfn wcel ccom cfv wceq ffn fvco2 sylan ) ABEFEAGCAHCDEIJCEJDJKABELADEC
    MN $.

  $( Value of a composition.  (Contributed by BJ, 7-Jul-2022.) $)
  fvco4 $p |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\
                    ( u e. A /\ x = ( K ` u ) ) ) -> ( H ` x ) = ( F ` u ) ) $=
    ( wf ccom wceq wa cv wcel cfv fvco3 ad2ant2r simplr eqcomd fveq1d fveq2
    ad2antll 3eqtr4rd ) CGFHZEFIZDJZKZBLZCMZALZUGFNZJZKZKZUGUDNZUJENZUGDNUIENZU
    CUHUNUOJUEUKCGUGEFOPUMUGDUDUMUDDUCUEULQRSUKUPUOJUFUHUIUJETUAUB $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y ch $.
    fvopab3g.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    fvopab3g.3 $e |- ( y = B -> ( ps <-> ch ) ) $.
    fvopab3g.4 $e |- ( x e. C -> E! y ph ) $.
    fvopab3g.5 $e |- F = { <. x , y >. | ( x e. C /\ ph ) } $.
    $( Value of a function given by ordered-pair class abstraction.
       (Contributed by NM, 6-Mar-1996.)  (Revised by Mario Carneiro,
       28-Apr-2015.) $)
    fvopab3g $p |- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> ch ) ) $=
      ( wcel wa cv wceq wb adantr cop copab cfv anbi12d anbi2d opelopabg fnopab
      eleq1 wfn fnopfvb mpan eleq2i bitrdi ibar 3bitr4d ) FHOZGIOZPFGUAZDQZHOZA
      PZDEUBZOZUPCPZFJUCGRZCVAUPBPVDDEFGHIUSFRUTUPABUSFHUHKUDEQGRBCUPLUEUFUPVEV
      CSUQUPVEURJOZVCJHUIUPVEVFSADEHJMNUGHFGJUJUKJVBURNULUMTUPCVDSUQUPCUNTUO $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y ch $.
    fvopab3ig.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    fvopab3ig.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    fvopab3ig.3 $e |- ( x e. C -> E* y ph ) $.
    fvopab3ig.4 $e |- F = { <. x , y >. | ( x e. C /\ ph ) } $.
    $( Value of a function given by ordered-pair class abstraction.
       (Contributed by NM, 23-Oct-1999.) $)
    fvopab3ig $p |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) ) $=
      ( wcel wa cv cfv wceq wi cop copab eleq1 anbi12d anbi2d opelopabg biimpar
      exp43 pm2.43a imp fveq1i wfun funopab moanimv mpbir mpgbir funopfv eqtrid
      wmo ax-mp syl6 ) FHOZGIOZPZCFGUADQZHOZAPZDEUBZOZFJRZGSVBVCCVITZVCVBVKVBVC
      VBCVIVDVIVBCPZVGVBBPVLDEFGHIVEFSVFVBABVEFHUCKUDEQGSBCVBLUEUFUGUHUIUJVIVJF
      VHRZGFJVHNUKVHULZVIVMGSTVNVGEUSZDVGDEUMVOVFAEUSTMVFAEUNUOUPFGVHUQUTURVA
      $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.  $d x y D $.  $d y F $.
    fvmptss2.1 $e |- ( x = D -> B = C ) $.
    fvmptss2.2 $e |- F = ( x e. A |-> B ) $.
    $( A mapping always evaluates to a subset of the substituted expression in
       the mapping, even if this is a proper class, or we are out of the
       domain.  (Contributed by Mario Carneiro, 13-Feb-2015.)  (Revised by
       Mario Carneiro, 3-Jul-2019.) $)
    fvmptss2 $p |- ( F ` D ) C_ C $=
      ( vy cv wbr wss wi cfv fvss cvv wcel nfcv wceq sylbi funmpt2 funrel ax-mp
      wfun wrel brrelex1i cmpt nfmpt1 nfcxfr nfbr nfv nfim breq1 sseq2d imbi12d
      cop df-br wa copab opabid eqimss adantl df-mpt eqtri eleq2s vtoclgf mpcom
      mpg ) EIJZFKZVIDLZMZEFNDLIIEDFOEPQVJVKEVIFFUDFUEABCFHUAFUBUCUFAJZVIFKZVIC
      LZMVLAEPAERZVJVKAAEVIFVPAFABCUGZHABCUHUIAVIRUJVKAUKULVMESZVNVJVOVKVMEVIFU
      MVRCDVIGUNUOVNVMVIUPZFQVOVMVIFUQVOVSVMBQZVICSZURZAIUSZFVSWCQWBVOWBAIUTWAV
      OVTVICVAVBTFVQWCHAIBCVCVDVETVFVGVH $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x C y $.  $d x D y $.
    fvmptg.1 $e |- ( x = A -> B = C ) $.
    fvmptg.2 $e |- F = ( x e. D |-> B ) $.
    $( Value of a function given in maps-to notation.  (Contributed by NM,
       2-Oct-2007.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    fvmptg $p |- ( ( A e. D /\ C e. R ) -> ( F ` A ) = C ) $=
      ( vy wcel wa wceq cfv eqid cv eqeq2d eqeq1 wmo moeq a1i cmpt copab df-mpt
      eqtri fvopab3ig mpi ) BEKDFKLDDMZBGNDMDOJPZCMZUIDMUHAJBDEFGAPZBMCDUIHQUID
      DRUJJSUKEKZJCTUAGAECUBULUJLAJUCIAJECUDUEUFUG $.

    ${
      fvmpt.3 $e |- C e. _V $.
      $( Value of a function given in maps-to notation.  (Contributed by NM,
         17-Aug-2011.) $)
      fvmpt $p |- ( A e. D -> ( F ` A ) = C ) $=
        ( wcel cvv cfv wceq fvmptg mpan2 ) BEJDKJBFLDMIABCDEKFGHNO $.
    $}
  $}

  ${
    $d y A $.  $d y B $.  $d x y C $.
    fvmpts.1 $e |- F = ( x e. C |-> B ) $.
    $( Value of a function given in maps-to notation, using explicit class
       substitution.  (Contributed by Scott Fenton, 17-Jul-2013.)  (Revised by
       Mario Carneiro, 31-Aug-2015.) $)
    fvmpts $p |- ( ( A e. C /\ [_ A / x ]_ B e. V ) ->
        ( F ` A ) = [_ A / x ]_ B ) $=
      ( vy cv csb csbeq1 cmpt nfcv nfcsb1v csbeq1a cbvmpt eqtri fvmptg ) HBAHIZ
      CJZABCJDFEASBCKEADCLHDTLGAHDCTHCMASCNASCOPQR $.
  $}

  ${
    $d x A $.  $d x C $.  $d x D $.  $d x V $.
    fvmpt3.a $e |- ( x = A -> B = C ) $.
    fvmpt3.b $e |- F = ( x e. D |-> B ) $.
    ${
      fvmpt3.c $e |- ( x e. D -> B e. V ) $.
      $( Value of a function given in maps-to notation, with a slightly
         different sethood condition.  (Contributed by Stefan O'Rear,
         30-Jan-2015.) $)
      fvmpt3 $p |- ( A e. D -> ( F ` A ) = C ) $=
        ( wcel cfv wceq cv eleq1d vtoclga fvmptg mpdan ) BEKDGKZBFLDMCGKSABEANB
        MCDGHOJPABCDEGFHIQR $.
    $}

    fvmpt3i.c $e |- B e. _V $.
    $( Value of a function given in maps-to notation, with a slightly different
       sethood condition.  (Contributed by Mario Carneiro, 11-Sep-2015.) $)
    fvmpt3i $p |- ( A e. D -> ( F ` A ) = C ) $=
      ( cvv wcel cv a1i fvmpt3 ) ABCDEFJGHCJKALEKIMN $.
  $}

  ${
    $d x A $.  $d x C $.  $d x D $.  $d x ph $.
    fvmptd.1 $e |- ( ph -> F = ( x e. D |-> B ) ) $.
    fvmptd.2 $e |- ( ( ph /\ x = A ) -> B = C ) $.
    fvmptd.3 $e |- ( ph -> A e. D ) $.
    fvmptd.4 $e |- ( ph -> C e. V ) $.
    $( Deduction version of ~ fvmpt .  (Contributed by Scott Fenton,
       18-Feb-2013.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    fvmptd $p |- ( ph -> ( F ` A ) = C ) $=
      ( cfv cmpt csb fveq1d wcel wceq csbied eqeltrd eqid fvmpts syl2anc 3eqtrd
      ) ACGMCBFDNZMZBCDOZEACGUEIPACFQUGHQUFUGRKAUGEHABCDEFKJSZLTBCDFUEHUEUAUBUC
      UHUD $.
  $}

  ${
    $d A x $.  $d C x $.  $d D x $.  $d ph x $.
    fvmptd2.1 $e |- F = ( x e. D |-> B ) $.
    fvmptd2.2 $e |- ( ( ph /\ x = A ) -> B = C ) $.
    fvmptd2.3 $e |- ( ph -> A e. D ) $.
    fvmptd2.4 $e |- ( ph -> C e. V ) $.
    $( Deduction version of ~ fvmpt (where the definition of the mapping does
       not depend on the common antecedent ` ph ` ).  (Contributed by Glauco
       Siliprandi, 23-Oct-2021.) $)
    fvmptd2 $p |- ( ph -> ( F ` A ) = C ) $=
      ( cmpt wceq a1i fvmptd ) ABCDEFGHGBFDMNAIOJKLP $.
  $}

  ${
    $d x A $.
    fvmpt2.1 $e |- F = ( x e. A |-> B ) $.
    $( Reverse closure for a mapping:  If the function value of a mapping has a
       member, the argument belongs to the base class of the mapping.
       (Contributed by AV, 4-Apr-2020.)  (Revised by Jim Kingdon,
       27-Mar-2023.) $)
    mptrcl $p |- ( I e. ( F ` X ) -> X e. A ) $=
      ( cfv wcel dmmptss wrel wfun funmpt2 funrel ax-mp relelfvdm mpan sselid
      cdm ) EFDHIZDSZBFABCDGJDKZTFUAIDLUBABCDGMDNOEFDPQR $.

    $d x y A $.  $d y B $.  $d y D $.  $d y F $.
    $( Value of a function given by the maps-to notation.  (Contributed by FL,
       21-Jun-2010.) $)
    fvmpt2 $p |- ( ( x e. A /\ B e. C ) -> ( F ` x ) = B ) $=
      ( vy cv csb weq csbeq1 csbid eqtrdi cmpt nfcv nfcsb1v cbvmpt eqtri fvmptg
      csbeq1a ) GAHZAGHZCIZCBDEGAJUCAUACICAUBUACKACLMEABCNGBUCNFAGBCUCGCOAUBCPA
      UBCTQRS $.

    $d x y C $.
    $( If all the values of the mapping are subsets of a class ` C ` , then so
       is any evaluation of the mapping at a value in the domain of the
       mapping.  (Contributed by Jim Kingdon, 3-Jan-2018.) $)
    fvmptssdm $p |- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C ) $=
      ( vy wcel wss cfv wi cvv cv wceq fveq2 sseq1d imbi2d nfcv nfim cdm nfrab1
      wral crab nfcri nfra1 cmpt nfmpt1 nfcxfr nffv nfss eleq1 imbi12d dmmpt wa
      eleq2i rabeq2i fvmpt2 eqimss syl sylbi adantr dmmptss sseli rsp ex sylbir
      mpan9 sstrd chvar vtoclga eleq2s imp ) EFUAZICDJZABUCZEFKZDJZVPVRLZECMIZA
      BUDZVNVPHNZFKZDJZLZVSHEWAWBEOZWDVRVPWFWCVQDWBEFPQRANZWAIZVPWGFKZDJZLZLWBW
      AIZWELAHWLWEAAHWAVTABUBUEVPWDAVOABUFAWCDAWBFAFABCUGGABCUHUIAWBSUJADSUKTTW
      GWBOZWHWLWKWEWGWBWAULWMWJWDVPWMWIWCDWGWBFPQRUMWHWGVNIZWKVNWAWGABCFGUNZUPW
      NVPWJWNVPUOWICDWNWICJZVPWNWGBIZVTUOZWPVTAVNBWOUQWRWICOWPABCMFGURWICUSUTVA
      VBWNWQVPVOVNBWGABCFGVCVDVOABVEVHVIVFVGVJVKWOVLVM $.

    $d y V $.
    $( Sufficient condition for a maps-to notation to be set-like.
       (Contributed by Mario Carneiro, 3-Jul-2019.) $)
    mptfvex $p |- ( ( A. x B e. V /\ C e. W ) -> ( F ` C ) e. _V ) $=
      ( vy wcel wal wa cfv csb cvv cv cmpt nfel1 wceq eleq1d wss csbeq1 nfcsb1v
      nfcv cbvmpt eqtri fvmptss2 elex alimi cbval sylib spcgv syl5 impcom ssexg
      csbeq1a sylancr ) CFJZAKZDGJZLDEMZADCNZUAVBOJZVAOJIBAIPZCNZVBDEAVDDCUBZEA
      BCQIBVEQHAIBCVEICUDZAVDCUCZAVDCUPZUEUFUGUTUSVCUSVEOJZIKZUTVCUSCOJZAKVKURV
      LACFUHUIVLVJAIICOVGRAVEOVHRAPVDSCVEOVITUJUKVJVCIDGVDDSVEVBOVFTULUMUNVAVBO
      UOUQ $.
  $}

  ${
    $d x A $.
    fvmpt2d.1 $e |- ( ph -> F = ( x e. A |-> B ) ) $.
    fvmpt2d.4 $e |- ( ( ph /\ x e. A ) -> B e. V ) $.
    $( Deduction version of ~ fvmpt2 .  (Contributed by Thierry Arnoux,
       8-Dec-2016.) $)
    fvmpt2d $p   |- ( ( ph /\ x e. A ) -> ( F ` x ) = B ) $=
      ( cv wcel wa cfv cmpt wceq fveq1d adantr simpr eqid fvmpt2 syl2anc eqtrd
      ) ABIZCJZKZUBELZUBBCDMZLZDAUEUGNUCAUBEUFGOPUDUCDFJUGDNAUCQHBCDFUFUFRSTUA
      $.
  $}

  ${
    $d x A $.  $d x D $.  $d x ph $.
    fvmptdf.1 $e |- ( ph -> A e. D ) $.
    fvmptdf.2 $e |- ( ( ph /\ x = A ) -> B e. V ) $.
    fvmptdf.3 $e |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) ) $.
    ${
      fvmptdf.4 $e |- F/_ x F $.
      fvmptdf.5 $e |- F/ x ps $.
      $( Alternate deduction version of ~ fvmpt , suitable for iteration.
         (Contributed by Mario Carneiro, 7-Jan-2017.) $)
      fvmptdf $p |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) ) $=
        ( cv wceq cmpt wi nfv wcel cfv nfmpt1 nfeq nfim cvv wex syl isset sylib
        elex fveq1 simpr fveq2d adantr eqeltrd eqid fvmpt2 eqtr3d eqeq2d sylbid
        wa syl2anc syl5 exlimdd ) ACNZDOZGCFEPZOZBQCACRVGBCCGVFLCFEUAUBMUCADUDS
        ZVECUEADFSZVHIDFUIUFCDUGUHVGDGTZDVFTZOZAVEUTZBDGVFUJVMVLVJEOBVMVKEVJVMV
        DVFTZVKEVMVDDVFAVEUKZULVMVDFSEHSVNEOVMVDDFVOAVIVEIUMUNJCFEHVFVFUOUPVAUQ
        URKUSVBVC $.
    $}

    $d x F $.  $d x ps $.
    $( Alternate deduction version of ~ fvmpt , suitable for iteration.
       (Contributed by Mario Carneiro, 7-Jan-2017.) $)
    fvmptdv $p |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) ) $=
      ( nfcv nfv fvmptdf ) ABCDEFGHIJKCGLBCMN $.
  $}

  ${
    $d x A $.  $d x C $.  $d x D $.  $d x ph $.
    fvmptdv2.1 $e |- ( ph -> A e. D ) $.
    fvmptdv2.2 $e |- ( ( ph /\ x = A ) -> B e. V ) $.
    fvmptdv2.3 $e |- ( ( ph /\ x = A ) -> B = C ) $.
    $( Alternate deduction version of ~ fvmpt , suitable for iteration.
       (Contributed by Mario Carneiro, 7-Jan-2017.) $)
    fvmptdv2 $p |- ( ph -> ( F = ( x e. D |-> B ) -> ( F ` A ) = C ) ) $=
      ( cfv wceq cmpt cvv eqidd cv wcel elex syl wex isset wa eqeltrrd exlimddv
      sylib fvmptd fveq1 eqeq1d syl5ibrcom ) ACGLZEMGBFDNZMZCULLZEMABCDEFULOAUL
      PKIABQCMZEORBACORZUOBUAACFRUPICFSTBCUBUFAUOUCZDEOKUQDHRDORJDHSTUDUEUGUMUK
      UNECGULUHUIUJ $.
  $}

  ${
    $d x A $.
    $( Bidirectional equality theorem for a mapping abstraction.  Equivalent to
       ~ eqfnfv .  (Contributed by Mario Carneiro, 14-Nov-2014.) $)
    mpteqb $p |- ( A. x e. A B e. V ->
      ( ( x e. A |-> B ) = ( x e. A |-> C ) <->
        A. x e. A B = C ) ) $=
      ( wcel wral cvv cmpt wceq wb wfn eqid mptfng wa wi nfmpt1 cfv fvmpt2 syl
      elex ralimi fneq1 3bitr4g biimpd r19.26 nfeq cv simpll ad2ant2lr ad2ant2l
      fveq1d 3eqtr3d exp31 ralrimi ralim biimtrrid expd mpdd com12 mpteq12 mpan
      impbid1 ) CEFZABGCHFZABGZABCIZABDIZJZCDJZABGZKVDVEABCEUAUBVFVIVKVIVFVKVIV
      FDHFZABGZVKVIVFVMVIVGBLVHBLVFVMBVGVHUCABCVGVGMZNABDVHVHMZNUDUEVIVFVMVKVFV
      MOVEVLOZABGZVIVKVEVLABUFVIVPVJPZABGVQVKPVIVRABAVGVHABCQABDQUGVIAUHZBFZVPV
      JVIVTOVPOZVSVGRZVSVHRZCDWAVSVGVHVIVTVPUIULVTVEWBCJVIVLABCHVGVNSUJVTVLWCDJ
      VIVEABDHVHVOSUKUMUNUOVPVJABUPTUQURUSUTBBJVKVIBMABCBDVAVBVCT $.
  $}

  ${
    $d x A $.  $d x C $.  $d x D $.
    $( Closed theorem form of ~ fvmpt .  (Contributed by Scott Fenton,
       21-Feb-2013.)  (Revised by Mario Carneiro, 11-Sep-2015.) $)
    fvmptt $p |- ( ( A. x ( x = A -> B = C ) /\
      F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = C ) $=
      ( cv wceq wi wal cmpt wcel wa w3a cfv simp2 fveq1d wrex cvv elex nfa1 nfv
      risset nfeq1 nfim simprl simplr simprr eqeltrd eqid fvmpt2 syl2anc simpll
      nffvmpt1 fveq2d 3eqtr3d exp43 a2i com23 sps rexlimd syl7 biimtrid 3adant2
      imp32 eqtrd ) AHZBIZCDIZJZAKZFAECLZIZBEMZDGMZNZOZBFPBVMPZDVRBFVMVLVNVQQRV
      LVQVSDIZVNVLVOVPVTVOVIAESZVLVPVTJABEUDVPDTMZVLWAVTDGUAVLVIWBVTJZAEVKAUBWB
      VTAWBAUCAVSDAECBUOUEUFVKVHEMZVIWCJJAVKVIWDWCVIVJWDWCJVIVJWDWBVTVIVJNZWDWB
      NZNZVHVMPZCVSDWGWDCTMWHCIWEWDWBUGWGCDTVIVJWFUHZWEWDWBUIUJAECTVMVMUKULUMWG
      VHBVMVIVJWFUNUPWIUQURUSUTVAVBVCVDVFVEVG $.
  $}

  ${
    $d x D $.
    fvmptf.1 $e |- F/_ x A $.
    fvmptf.2 $e |- F/_ x C $.
    fvmptf.3 $e |- ( x = A -> B = C ) $.
    fvmptf.4 $e |- F = ( x e. D |-> B ) $.
    $( Value of a function given by an ordered-pair class abstraction.  This
       version of ~ fvmptg uses bound-variable hypotheses instead of distinct
       variable conditions.  (Contributed by NM, 8-Nov-2005.)  (Revised by
       Mario Carneiro, 15-Oct-2016.) $)
    fvmptf $p |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C ) $=
      ( wcel cfv wceq cvv elex cv wi nfel1 cmpt nfmpt1 nfcxfr nffv eleq1d fveq2
      nfeq nfim eqeq12d imbi12d fvmpt2 ex vtoclgaf syl5 imp ) BELZDGLZBFMZDNZUP
      DOLZUOURDGPCOLZAQZFMZCNZRUSURRABEHUSURAADOISAUQDABFAFAECTKAECUAUBHUCIUFUG
      VABNZUTUSVCURVDCDOJUDVDVBUQCDVABFUEJUHUIVAELUTVCAECOFKUJUKULUMUN $.
  $}

  ${
    $d A x $.  $d C x $.  $d D x $.
    fvmptd3.1 $e |- F = ( x e. D |-> B ) $.
    fvmptd3.2 $e |- ( x = A -> B = C ) $.
    fvmptd3.3 $e |- ( ph -> A e. D ) $.
    fvmptd3.4 $e |- ( ph -> C e. V ) $.
    $( Deduction version of ~ fvmpt .  (Contributed by Glauco Siliprandi,
       23-Oct-2021.) $)
    fvmptd3 $p |- ( ph -> ( F ` A ) = C ) $=
      ( wcel cfv wceq nfcv fvmptf syl2anc ) ACFMEHMCGNEOKLBCDEFGHBCPBEPJIQR $.
  $}

  ${
    $d ph x $.  $d A x $.  $d C x $.  $d D x $.
    fvmptd4.1 $e |- ( x = A -> B = C ) $.
    fvmptd4.2 $e |- ( ph -> F = ( x e. D |-> B ) ) $.
    fvmptd4.3 $e |- ( ph -> A e. D ) $.
    fvmptd4.4 $e |- ( ph -> C e. V ) $.
    $( Deduction version of ~ fvmpt (where the substitution hypothesis does not
       have the antecedent ` ph ` ).  (Contributed by SN, 26-Jul-2024.) $)
    fvmptd4 $p |- ( ph -> ( F ` A ) = C ) $=
      ( cv wceq adantl fvmptd ) ABCDEFGHJBMCNDENAIOKLP $.
  $}

  ${
    $d M x y $.  $d V x $.  $d X x y $.  $d Y y $.  $d m y $.
    elfvmptrab1.f $e |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } ) $.
    elfvmptrab1.v $e |- ( X e. V -> [_ X / m ]_ M e. _V ) $.
    $( Implications for the value of a function defined by the maps-to notation
       with a class abstraction as a result having an element.  Here, the base
       set of the class abstraction depends on the argument of the function.
       (Contributed by Alexander van der Vekens, 15-Jul-2018.) $)
    elfvmptrab1 $p |- ( Y e. ( F ` X )
                        -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) $=
      ( cdm wcel cfv csb wa crab cvv wceq nfcv wrel cv funmpt2 funrel relelfvdm
      wfun ax-mp mpan wsbc dmmptss sseli rabexg nfsbc1v nfrabw csbeq1 rabeqbidv
      3syl nfcsb sbceq1a fvmptf syl2anc eleq2d elrabi anim12i ex sylbid mpcom )
      HELZMZIHENZMZHGMZIDHFOZMZPZEUAZVKVIEUFVPBGACDBUBZFOZQZEJUCEUDUGIHEUEUHVIV
      KIABHUIZCVMQZMZVOVIVJWAIVIVLWARMZVJWASVHGHBGVSEJUJUKZVIVLVMRMWCWDKVTCVMRU
      LUQBHVSWAGERBHTZVTBCVMABHUMBDHFWEBFTURUNVQHSAVTCVRVMDVQHFUOABHUSUPJUTVAVB
      VIWBVOVIVLWBVNWDVTCIVMVCVDVEVFVG $.
  $}

  ${
    $d M m x y $.  $d V x $.  $d X x y $.  $d Y y $.
    elfvmptrab.f $e |- F = ( x e. V |-> { y e. M | ph } ) $.
    elfvmptrab.v $e |- ( X e. V -> M e. _V ) $.
    $( Implications for the value of a function defined by the maps-to notation
       with a class abstraction as a result having an element.  (Contributed by
       Alexander van der Vekens, 15-Jul-2018.) $)
    elfvmptrab $p |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. M ) ) $=
      ( vm cfv wcel csb wa crab cmpt wceq csbconstg syl cv rabeq mpteq2ia eqtri
      eqcomd cvv eqeltrd elfvmptrab1 eleq2d biimpd imdistani ) HGDLMGFMZHKGENZM
      ZOULHEMZOABCKDEFGHDBFACEPZQBFACKBUAZENZPZQIBFUPUSUQFMZEURRUPUSRUTUREKUQEF
      SUEACEURUBTUCUDULUMEUFKGEFSZJUGUHULUNUOULUNUOULUMEHVAUIUJUKT $.
  $}

  ${
    $d x y A $.
    fvopab4ndm.1 $e |- F = { <. x , y >. | ( x e. A /\ ph ) } $.
    $( Value of a function given by an ordered-pair class abstraction, outside
       of its domain.  (Contributed by NM, 28-Mar-2008.) $)
    fvopab4ndm $p |- ( -. B e. A -> ( F ` B ) = (/) ) $=
      ( cdm wcel cfv c0 wceq cv wa copab dmeqi dmopabss eqsstri sseli wrel mpan
      wn relopabiv relndmfv nsyl5 ) EFHZIZEDIEFJKLZUFDEUFBMDIANZBCOZHDFUJGPABCD
      QRSFTUGUBUHUIBCFGUCEFUDUAUE $.
  $}

  ${
    $d A x y $.  $d B y $.
    fvmptndm.1 $e |- F = ( x e. A |-> B ) $.
    $( Value of a function given by the maps-to notation, outside of its
       domain.  (Contributed by AV, 31-Dec-2020.) $)
    fvmptndm $p |- ( -. X e. A -> ( F ` X ) = (/) ) $=
      ( vy cv wceq cmpt wcel wa copab df-mpt eqtri fvopab4ndm ) GHCIZAGBEDDABCJ
      AHBKQLAGMFAGBCNOP $.
  $}

  ${
    $d A x y $.  $d ps x y $.  $d B y $.  $d C x y $.
    fvopab6.1 $e |- F = { <. x , y >. | ( ph /\ y = B ) } $.
    fvopab6.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    fvopab6.3 $e |- ( x = A -> B = C ) $.
    $( Value of a function given by ordered-pair class abstraction.
       (Contributed by Jeff Madsen, 2-Sep-2009.)  (Revised by Mario Carneiro,
       11-Sep-2015.) $)
    fvopab6 $p |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C ) $=
      ( wcel cfv wceq cvv cv wa copab elex eqeq2d anbi12d iba bicomd moeq moani
      wi wmo a1i vex biantrur opabbii eqtri fvopab3ig sylan 3impia ) EHNZGINZBE
      JOGPZUREQNUSBUTUHEHUAADRZFPZSZBVAGPZSZBCDEGQIJCRZEPZABVBVDLVGFGVAMUBUCVDB
      VEVDBUDUEVCDUIVFQNZVBADDFUFUGUJJVCCDTVHVCSZCDTKVCVICDVHVCCUKULUMUNUOUPUQ
      $.
  $}

  ${
    $d x A $.  $d x F $.  $d x G $.  $d x ph $.
    $( Equality of functions is determined by their values.  Special case of
       Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
       (Contributed by NM, 3-Aug-1994.)  (Proof shortened by Andrew Salmon,
       22-Oct-2011.)  (Proof shortened by Mario Carneiro, 31-Aug-2015.) $)
    eqfnfv $p |- ( ( F Fn A /\ G Fn A ) -> ( F = G <->
                 A. x e. A ( F ` x ) = ( G ` x ) ) ) $=
      ( wfn wa wceq cv cfv cmpt wral dffn5im eqeqan12d wb cvv funfvex ralrimiva
      wcel funfni mpteqb syl adantr bitrd ) CBEZDBEZFCDGABAHZCIZJZABUFDIZJZGZUG
      UIGABKZUDUECUHDUJABCLABDLMUDUKULNZUEUDUGORZABKUMUDUNABUNBUFCUFCPSQABUGUIO
      TUAUBUC $.

    $( Equality of functions is determined by their values.  Exercise 4 of
       [TakeutiZaring] p. 28.  (Contributed by NM, 3-Aug-1994.)  (Revised by
       Mario Carneiro, 31-Aug-2015.) $)
    eqfnfv2 $p |- ( ( F Fn A /\ G Fn B ) -> ( F = G <->
                 ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) ) $=
      ( wfn wa wceq cv cfv wral cdm dmeq fndm eqeqan12d imbitrid pm4.71rd fneq2
      wb biimparc eqfnfv sylan2 anassrs pm5.32da bitrd ) DBFZECFZGZDEHZBCHZUIGU
      JAIZDJUKEJHABKZGUHUIUJUIDLZELZHUHUJDEMUFUGUMBUNCBDNCENOPQUHUJUIULUFUGUJUI
      ULSZUGUJGUFEBFZUOUJUPUGBCERTABDEUAUBUCUDUE $.

    $d x B $.
    $( Derive equality of functions from equality of their values.
       (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    eqfnfv3 $p |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A
                                  ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) ) $=
      ( wfn wa wceq cv cfv wral wss wcel eqfnfv2 ancom bitri anbi1i anass dfss3
      eqss r19.26 bitr4i anbi2i bitrdi ) DBFECFGDEHBCHZAIZDJUFEJHZABKZGZCBLZUFC
      MZUGGABKZGZABCDENUIUJBCLZGZUHGZUMUEUOUHUEUNUJGUOBCTUNUJOPQUPUJUNUHGZGUMUJ
      UNUHRUQULUJUQUKABKZUHGULUNURUHABCSQUKUGABUAUBUCPPUD $.

    eqfnfvd.1 $e |- ( ph -> F Fn A ) $.
    eqfnfvd.2 $e |- ( ph -> G Fn A ) $.
    eqfnfvd.3 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) ) $.
    $( Deduction for equality of functions.  (Contributed by Mario Carneiro,
       24-Jul-2014.) $)
    eqfnfvd $p |- ( ph -> F = G ) $=
      ( wceq cv cfv wral ralrimiva wfn wb eqfnfv syl2anc mpbird ) ADEIZBJZDKTEK
      IZBCLZAUABCHMADCNECNSUBOFGBCDEPQR $.
  $}

  ${
    $d x z A $.  $d z F $.  $d z G $.
    eqfnfv2f.1 $e |- F/_ x F $.
    eqfnfv2f.2 $e |- F/_ x G $.
    $( Equality of functions is determined by their values.  Special case of
       Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
       This version of ~ eqfnfv uses bound-variable hypotheses instead of
       distinct variable conditions.  (Contributed by NM, 29-Jan-2004.) $)
    eqfnfv2f $p |- ( ( F Fn A /\ G Fn A ) -> ( F = G <->
                 A. x e. A ( F ` x ) = ( G ` x ) ) ) $=
      ( vz wfn wa wceq cv cfv wral eqfnfv nfcv nffv nfeq nfv fveq2 eqeq12d
      cbvral bitrdi ) CBHDBHICDJGKZCLZUCDLZJZGBMAKZCLZUGDLZJZABMGBCDNUFUJGABAUD
      UEAUCCEAUCOZPAUCDFUKPQUJGRUCUGJUDUHUEUIUCUGCSUCUGDSTUAUB $.
  $}

  ${
    $d F x $.  $d G x $.
    $( Equality of functions is determined by their values.  (Contributed by
       Scott Fenton, 19-Jun-2011.) $)
    eqfunfv $p |- ( ( Fun F /\ Fun G ) -> ( F = G <->
                    ( dom F = dom G /\
                      A. x e. dom F ( F ` x ) = ( G ` x ) ) ) ) $=
      ( wfun cdm wfn wceq cv cfv wral wa wb funfn eqfnfv2 syl2anb ) BDBBEZFCCEZ
      FBCGPQGAHZBIRCIGAPJKLCDBMCMAPQBCNO $.
  $}

  ${
    $d x B $.  $d x F $.  $d x G $.
    $( Equality of restricted functions is determined by their values.
       (Contributed by NM, 3-Aug-1994.) $)
    fvreseq $p |- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) ->
         ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) ) $=
      ( wfn wa wss cres wceq cv cfv wral wb fnssres anim12i anandirs wcel fvres
      eqfnfv eqeq12d ralbiia bitrdi syl ) DBFZEBFZGCBHZGDCIZCFZECIZCFZGZUHUJJZA
      KZDLZUNELZJZACMZNUEUFUGULUEUGGUIUFUGGUKBCDOBCEOPQULUMUNUHLZUNUJLZJZACMURA
      CUHUJTVAUQACUNCRUSUOUTUPUNCDSUNCESUAUBUCUD $.
  $}

  ${
    $d A a i $.  $d C i $.  $d D a $.  $d M a i $.  $d U a i $.  $d V a i $.
    $d ph a i $.
    fnmptfvd.m $e |- ( ph -> M Fn A ) $.
    fnmptfvd.s $e |- ( i = a -> D = C ) $.
    fnmptfvd.d $e |- ( ( ph /\ i e. A ) -> D e. U ) $.
    fnmptfvd.c $e |- ( ( ph /\ a e. A ) -> C e. V ) $.
    $( A function with a given domain is a mapping defined by its function
       values.  (Contributed by AV, 1-Mar-2019.) $)
    fnmptfvd $p |- ( ph -> ( M = ( a e. A |-> C )
                              <-> A. i e. A ( M ` i ) = D ) ) $=
      ( cmpt wceq cfv wral wfn wcel eqid cv ralrimiva fnmpt syl syl2anc cbvmptv
      wb eqfnfv eqcomi a1i fveq1d eqeq2d ralbidv simpr fvmpt2 ralbidva 3bitrd
      wa ) AGIBCNZOZFUAZGPZVAUSPZOZFBQZVBVAFBDNZPZOZFBQVBDOZFBQAGBRUSBRZUTVEUGJ
      ACHSZIBQVJAVKIBMUBIBCUSHUSTUCUDFBGUSUHUEAVDVHFBAVCVGVBAVAUSVFUSVFOAVFUSFI
      BDCKUFUIUJUKULUMAVHVIFBAVABSZURZVGDVBVMVLDESVGDOAVLUNLFBDEVFVFTUOUEULUPUQ
      $.
  $}

  ${
    $d F x y $.  $d G x y $.  $d A x y $.
    $( Two ways to express the locus of differences between two functions.
       (Contributed by Stefan O'Rear, 17-Jan-2015.) $)
    fndmdif $p |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) =
          { x e. A | ( F ` x ) =/= ( G ` x ) } ) $=
      ( vy wfn wa cdm cv cfv wss wceq wcel wbr wex wn wb eqcom fnbrfvb bitrid
      cdif cin wne crab difss dmss ax-mp fndm adantr sseqtrid dfss1 vex adantll
      sylib eldm necon3abid funfvex funfni adantlr breq2 notbid ceqsexgv anbi1d
      cvv syl brdif bitr4di exbidv 3bitr2rd rabbi2dva eqtr3d ) CBFZDBFZGZBCDUAZ
      HZUBZVPAIZCJZVRDJZUCZABUDVNVPBKVQVPLVNCHZVPBVOCKVPWBKCDUEVOCUFUGVLWBBLVMB
      CUHUIUJVPBUKUNVNWAABVPVRVPMVREIZVONZEOZVNVRBMZGZWAEVRVOAULUOWGWAVRVSDNZPZ
      WCVSLZVRWCDNZPZGZEOZWEWGWHVSVTVMWFVSVTLZWHQVLWOVTVSLVMWFGWHVSVTRBVRVSDSTU
      MUPWGVSVDMZWNWIQVLWFWPVMWPBVRCVRCUQURUSWLWIEVSVDWJWKWHWCVSVRDUTVAVBVEWGWM
      WDEWGWMVRWCCNZWLGWDWGWJWQWLVLWFWJWQQVMWJVSWCLVLWFGWQWCVSRBVRWCCSTUSVCVRWC
      CDVFVGVHVITVJVK $.

    $( The difference set between two functions is commutative.  (Contributed
       by Stefan O'Rear, 17-Jan-2015.) $)
    fndmdifcom $p |- ( ( F Fn A /\ G Fn A ) ->
        dom ( F \ G ) = dom ( G \ F ) ) $=
      ( vx wfn wa cv cfv wne crab cdif cdm wb wcel necom rabbiia fndmdif ancoms
      a1i wceq 3eqtr4a ) BAEZCAEZFDGZBHZUDCHZIZDAJUFUEIZDAJZBCKLCBKLZUGUHDAUGUH
      MUDANUEUFOSPDABCQUCUBUJUITDACBQRUA $.

    $( Two ways to express the locus of equality between two functions.
       (Contributed by Stefan O'Rear, 17-Jan-2015.) $)
    fndmin $p |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) =
          { x e. A | ( F ` x ) = ( G ` x ) } ) $=
      ( vy wfn wa cin cdm cv wcel cfv wceq copab cmpt dffn5im df-mpt eqtrdi wex
      cab crab ineqan12d inopab dmeqd anandi exbii 19.42v bitr3i wb cvv funfvex
      wfun eqeq1 ceqsexgv syl funfni bitrid abbidv dmopab df-rab 3eqtr4g adantr
      pm5.32da eqtrd ) CBFZDBFZGZCDHZIAJZBKZEJZVICLZMZGZVJVKVIDLZMZGZGZAENZIZVL
      VOMZABUAZVGVHVSVGVHVNAENZVQAENZHVSVEVFCWCDWDVECABVLOWCABCPAEBVLQRVFDABVOO
      WDABDPAEBVOQRUBVNVQAEUCRUDVEVTWBMVFVEVRESZATVJWAGZATVTWBVEWEWFAWEVJVMVPGZ
      ESZGZVEWFWEVJWGGZESWIWJVREVJVMVPUEUFVJWGEUGUHVEVJWHWAWHWAUIZBVICCULVICIKG
      VLUJKWKVICUKVPWAEVLUJVKVLVOUMUNUOUPVCUQURVRAEUSWAABUTVAVBVD $.
  $}

  ${
    $d F x $.  $d G x $.  $d A x $.
    $( Two functions are equal iff their equalizer is the whole domain.
       (Contributed by Stefan O'Rear, 7-Mar-2015.) $)
    fneqeql $p |- ( ( F Fn A /\ G Fn A ) ->
        ( F = G <-> dom ( F i^i G ) = A ) ) $=
      ( vx wfn wa wceq cfv crab cin cdm wral eqfnfv eqcom rabid2 bitr4di fndmin
      cv bitri eqeq1d bitr4d ) BAECAEFZBCGZDRZBHUDCHGZDAIZAGZBCJKZAGUBUCUEDALZU
      GDABCMUGAUFGUIUFANUEDAOSPUBUHUFADABCQTUA $.

    $( Two functions are equal iff their equalizer contains the whole domain.
       (Contributed by Stefan O'Rear, 9-Mar-2015.) $)
    fneqeql2 $p |- ( ( F Fn A /\ G Fn A ) ->
        ( F = G <-> A C_ dom ( F i^i G ) ) ) $=
      ( wfn wceq cin cdm wss fneqeql eqss inss1 dmss ax-mp fndm adantr sseqtrid
      wa biantrurd bitr4id bitrd ) BADZCADZQZBCEBCFZGZAEZAUEHZABCIUCUFUEAHZUGQU
      GUEAJUCUHUGUCBGZUEAUDBHUEUIHBCKUDBLMUAUIAEUBABNOPRST $.

    $( Two functions are equal on a subset iff their equalizer contains that
       subset.  (Contributed by Stefan O'Rear, 7-Mar-2015.) $)
    fnreseql $p |- ( ( F Fn A /\ G Fn A /\ X C_ A ) ->
        ( ( F |` X ) = ( G |` X ) <-> X C_ dom ( F i^i G ) ) ) $=
      ( wfn wss w3a cin cdm wb fnssres 3adant2 3adant1 fneqeql syl2anc resindir
      cres wceq dmeqi dmres eqtr3i eqeq1i df-ss bitr4i bitrdi ) BAEZCAEZDAFZGZB
      DQZCDQZRZUJUKHZIZDRZDBCHZIZFZUIUJDEZUKDEZULUOJUFUHUSUGADBKLUGUHUTUFADCKMD
      UJUKNOUODUQHZDRURUNVADUPDQZIUNVAVBUMBCDPSUPDTUAUBDUQUCUDUE $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( The range of a choice function (a function that chooses an element from
       each member of its domain) is included in the union of its domain.
       (Contributed by NM, 31-Aug-1999.) $)
    chfnrn $p |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A ) $=
      ( vy wfn cv cfv wcel wral wa crn cuni wrex wceq fvelrnb biimpd wi biimpcd
      eleq1 ralimi rexim syl sylan9 eluni2 imbitrrdi ssrdv ) CBEZAFZCGZUHHZABIZ
      JZDCKZBLZULDFZUMHZUOUHHZABMZUOUNHUGUPUIUONZABMZUKURUGUPUTABUOCOPUKUSUQQZA
      BIUTURQUJVAABUSUJUQUIUOUHSRTUSUQABUAUBUCAUOBUDUEUF $.
  $}

  $( Ordered pair with function value.  Part of Theorem 4.3(i) of [Monk1]
     p. 41.  (Contributed by NM, 14-Oct-1996.) $)
  funfvop $p |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F ) $=
    ( wfun cdm wcel wa cfv wceq cop eqid funopfvb mpbii ) BCABDEFABGZMHAMIBEMJA
    MBKL $.

  $( Two ways to say that ` A ` is in the domain of ` F ` .  (Contributed by
     Mario Carneiro, 1-May-2014.) $)
  funfvbrb $p |- ( Fun F -> ( A e. dom F <-> A F ( F ` A ) ) ) $=
    ( wfun cdm wcel cfv wbr wa funfvop df-br sylibr wrel funrel releldm impbida
    cop sylan ) BCZABDEZAABFZBGZRSHATPBEUAABIATBJKRBLUASBMATBNQO $.

  $( A member of a preimage is a function value argument.  (Contributed by NM,
     4-May-2007.) $)
  fvimacnvi $p |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B ) $=
    ( wfun ccnv cima wcel wa cfv csn wss snssi funimass2 sylan2 cdm wb cnvimass
    sseli cvv funfvex snssg syl wceq funfn fnsnfv sylanb sseq1d bitrd mpbird
    wfn ) CDZACEBFZGZHZACIZBGZCAJZFZBKZUMUKUQULKUSAULLUQBCMNUNUPUOJZBKZUSUMUKAC
    OZGZUPVAPZULVBACBQRZUKVCHUOSGVDACTUOBSUAUBNUNUTURBUMUKVCUTURUCZVEUKCVBUJVCV
    FCUDVBACUEUFNUGUHUI $.

  $( The argument of a function value belongs to the preimage of any class
     containing the function value.  Raph Levien remarks:  "This proof is
     unsatisfying, because it seems to me that ~ funimass2 could probably be
     strengthened to a biconditional."  (Contributed by Raph Levien,
     20-Nov-2006.) $)
  fvimacnv $p |- ( ( Fun F /\ A e. dom F ) ->
                 ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) ) $=
    ( wfun cdm wcel wa cfv ccnv cima csn cop funfvop cvv wb sylancom mpbird wss
    wi ex funfvex opelcnvg elimasng snssg syl biimtrdi imp sseld mpid fvimacnvi
    imass2 adantr impbid ) CDZACEZFZGZACHZBFZACIZBJZFZUQUSAUTURKZJZFZVBUQVEURAL
    UTFZUQVFAURLCFZACMUNUPURNFZVFVGOACUAZURANUOCUBPQUNUPVHVEVFOVIUTURANUOUCPQUQ
    USVEVBSUQUSGVDVAAUQUSVDVARZUQUSVCBRZVJUQVHUSVKOVIURBNUDUEVCBUTUKUFUGUHTUIUN
    VBUSSUPUNVBUSABCUJTULUM $.

  ${
    $d F x $.  $d A x $.  $d B x $.
    $( A kind of contraposition law that infers an image subclass from a
       subclass of a preimage.  Raph Levien remarks:  "Likely this could be
       proved directly, and ~ fvimacnv would be the special case of ` A ` being
       a singleton, but it works this way round too."  (Contributed by Raph
       Levien, 20-Nov-2006.) $)
    funimass3 $p |- ( ( Fun F /\ A C_ dom F ) ->
                    ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) ) $=
      ( vx wfun cdm wss wa cima cv ccnv wcel wral funimass4 wb ssel fvimacnv ex
      cfv syl9r imp31 ralbidva bitrd dfss3 bitr4di ) CEZACFZGZHZCAIBGZDJZCKBIZL
      ZDAMZAULGUIUJUKCSBLZDAMUNDABCNUIUOUMDAUFUHUKALZUOUMOZUHUPUKUGLZUFUQAUGUKP
      UFURUQUKBCQRTUAUBUCDAULUDUE $.

    $( A subclass of a preimage in terms of function values.  (Contributed by
       NM, 15-May-2007.) $)
    funimass5 $p |- ( ( Fun F /\ A C_ dom F ) ->
                    ( A C_ ( `' F " B ) <-> A. x e. A ( F ` x ) e. B ) ) $=
      ( wfun cdm wss wa cima ccnv cv cfv wcel wral funimass3 funimass4 bitr3d )
      DEBDFGHDBICGBDJCIGAKDLCMABNBCDOABCDPQ $.

    $( Two ways of specifying that a function is constant on a subdomain.
       (Contributed by NM, 8-Mar-2007.) $)
    funconstss $p |- ( ( Fun F /\ A C_ dom F ) ->
                     ( A. x e. A ( F ` x ) = B <-> A C_ ( `' F " { B } ) ) ) $=
      ( wfun cdm wss wa cima csn cv cfv wcel wral ccnv wceq funimass4 funimass3
      cvv wb ssel2 anim2i anassrs funfvex elsng 3syl ralbidva 3bitr3rd ) DEZBDF
      ZGZHZDBICJZGAKZDLZUMMZABNBDOUMIGUOCPZABNABUMDQBUMDRULUPUQABULUNBMZHUIUNUJ
      MZHZUOSMUPUQTUIUKURUTUKURHUSUIBUJUNUAUBUCUNDUDUOCSUEUFUGUH $.
  $}

  $( Membership in the preimage of a set under a function.  (Contributed by
     Jeff Madsen, 2-Sep-2009.) $)
  elpreima $p |- ( F Fn A -> ( B e. ( `' F " C )
                              <-> ( B e. A /\ ( F ` B ) e. C ) ) ) $=
    ( wfn ccnv cima wcel cfv cdm cnvimass sseli fndm eleq2d imbitrid wfun fnfun
    wa fvimacnvi sylan ex jcad wb fvimacnv funfni biimpd expimpd impbid ) DAEZB
    DFCGZHZBAHZBDICHZRUIUKULUMUKBDJZHUIULUJUNBDCKLUIUNABADMNOUIUKUMUIDPUKUMADQB
    CDSTUAUBUIULUMUKUIULRUMUKUMUKUCABDBCDUDUEUFUGUH $.

  $( Membership in the preimage of a singleton, under a function.  (Contributed
     by Mario Carneiro, 12-May-2014.)  (Proof shortened by Mario Carneiro,
     28-Apr-2015.) $)
  fniniseg $p |- ( F Fn A -> ( C e. ( `' F " { B } ) <->
                   ( C e. A /\ ( F ` C ) = B ) ) ) $=
    ( wfn ccnv csn cima wcel cfv wa wceq elpreima wb wfun cdm cvv funfvex elsng
    syl funfni pm5.32da bitrd ) DAEZCDFBGZHICAIZCDJZUEIZKUFUGBLZKACUEDMUDUFUHUI
    UHUINZACDDOCDPIKUGQIUJCDRUGBQSTUAUBUC $.

  ${
    $d x A $.  $d x F $.  $d x B $.
    $( Inverse images under functions expressed as abstractions.  (Contributed
       by Stefan O'Rear, 1-Feb-2015.) $)
    fncnvima2 $p |- ( F Fn A -> ( `' F " B ) = { x e. A | ( F ` x ) e. B } ) $=
      ( wfn ccnv cima cv wcel cfv wa cab crab elpreima abbi2dv df-rab eqtr4di )
      DBEZDFCGZAHZBITDJCIZKZALUAABMRUBASBTCDNOUAABPQ $.

    $( Inverse point images under functions expressed as abstractions.
       (Contributed by Stefan O'Rear, 1-Feb-2015.) $)
    fniniseg2 $p |- ( F Fn A -> ( `' F " { B } ) =
          { x e. A | ( F ` x ) = B } ) $=
      ( wfn ccnv csn cima cv cfv wcel crab wceq fncnvima2 wb cdm wa cvv funfvex
      wfun elsng syl funfni rabbidva eqtrd ) DBEZDFCGZHAIZDJZUGKZABLUICMZABLABU
      GDNUFUJUKABUJUKOZBUHDDTUHDPKQUIRKULUHDSUICRUAUBUCUDUE $.

    $( Support sets of functions expressed as abstractions.  (Contributed by
       Stefan O'Rear, 1-Feb-2015.) $)
    fnniniseg2 $p |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) =
          { x e. A | ( F ` x ) =/= B } ) $=
      ( wfn ccnv cvv csn cdif cima cv cfv wcel wne fncnvima2 wa eldifsn funfvex
      crab funfni biantrurd bitr4id rabbidva eqtrd ) DBEZDFGCHIZJAKZDLZUFMZABSU
      HCNZABSABUFDOUEUIUJABUEUGBMPZUIUHGMZUJPUJUHGCQUKULUJULBUGDUGDRTUAUBUCUD
      $.
  $}

  ${
    $d x F $.  $d x A $.  $d x B $.
    $( Preimage of a union.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    unpreima $p |- ( Fun F -> ( `' F " ( A u. B ) )
                                  = ( ( `' F " A ) u. ( `' F " B ) ) ) $=
      ( vx wfun cdm wfn ccnv cun cima wceq funfn cv wcel cfv wa elpreima anbi2i
      wo elun andi bitri orbi12d bitrid bitr4id bitrd eqrdv sylbi ) CECCFZGZCHZ
      ABIZJZUKAJZUKBJZIZKCLUJDUMUPUJDMZUMNUQUINZUQCOZULNZPZUQUPNZUIUQULCQUJVAUR
      USANZPZURUSBNZPZSZVBVAURVCVESZPVGUTVHURUSABTRURVCVEUAUBVBUQUNNZUQUONZSUJV
      GUQUNUOTUJVIVDVJVFUIUQACQUIUQBCQUCUDUEUFUGUH $.

    $( Preimage of an intersection.  (Contributed by Jeff Madsen, 2-Sep-2009.)
       (Proof shortened by Mario Carneiro, 14-Jun-2016.) $)
    inpreima $p |- ( Fun F -> ( `' F " ( A i^i B ) )
                                  = ( ( `' F " A ) i^i ( `' F " B ) ) ) $=
      ( wfun ccnv cin cima wceq funcnvcnv imain syl ) CDCEZEDLABFGLAGLBGFHCIABL
      JK $.

    $( Preimage of a difference.  (Contributed by Mario Carneiro,
       14-Jun-2016.) $)
    difpreima $p |- ( Fun F -> ( `' F " ( A \ B ) )
                                  = ( ( `' F " A ) \ ( `' F " B ) ) ) $=
      ( wfun ccnv cdif cima wceq funcnvcnv imadif syl ) CDCEZEDLABFGLAGLBGFHCIA
      BLJK $.

    $( The preimage of a restricted function.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    respreima $p |- ( Fun F -> ( `' ( F |` B ) " A )
                                    = ( ( `' F " A ) i^i B ) ) $=
      ( vx wfun cres ccnv cima cin cdm wfn cv wcel wb cfv wa bitri syl elpreima
      elin funfn ancom anbi1i fvres eleq1d adantl pm5.32i a1i an32 bitrdi fnfun
      wceq funres dmres jctir df-fn sylibr anbi1d bitrid 3bitr4d sylbi eqrdv )
      CEZDCBFZGAHZCGAHZBIZVCCCJZKZDLZVEMZVJVGMZNCUAVIVJBVHIZMZVJVDOZAMZPZVJVHMZ
      VJCOZAMZPZVJBMZPZVKVLVIVQVRWBPZVTPZWCVQWENVIVQWDVPPWEVNWDVPVNWBVRPWDVJBVH
      TWBVRUBQUCWDVPVTWBVPVTNVRWBVOVSAVJBCUDUEUFUGQUHVRWBVTUIUJVIVDVMKZVKVQNVIV
      DEZVDJVMULZPWFVIWGWHVIVCWGVHCUKBCUMRCBUNUOVDVMUPUQVMVJAVDSRVLVJVFMZWBPVIW
      CVJVFBTVIWIWAWBVHVJACSURUSUTVAVB $.
  $}

  $( The preimage of the codomain of a mapping is the mapping's domain.
     (Contributed by FL, 25-Jan-2007.) $)
  fimacnv $p |- ( F : A --> B -> ( `' F " B ) = A ) $=
    ( wf ccnv cima crn imassrn cdm dfdm4 fdm ssid eqsstrdi eqsstrrid sstrid wss
    frn wfun wb ffun sseqtrrid funimass3 syl2anc mpbid eqssd ) ABCDZCEZBFZAUFUH
    UGGZAUGBHUFUICIZACJUFUJAAABCKZALZMNOUFCAFZBPZAUHPZUFUMCGBCAHABCQOUFCRAUJPUN
    UOSABCTUFAAUJULUKUAABCUBUCUDUE $.

  $( Ordered pair with function value.  Part of Theorem 4.3(i) of [Monk1]
     p. 41.  (Contributed by NM, 30-Sep-2004.) $)
  fnopfv $p |- ( ( F Fn A /\ B e. A ) -> <. B , ( F ` B ) >. e. F ) $=
    ( cfv cop wcel funfvop funfni ) BBCDECFABCBCGH $.

  ${
    $d x y F $.  $d x A $.
    $( A function's value belongs to its range.  (Contributed by NM,
       14-Oct-1996.) $)
    fvelrn $p |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F ) $=
      ( vx vy wfun cdm wcel cfv crn cv wa wi wceq eleq1 anbi2d fveq2 eleq1d cop
      syl cvv imbi12d wex funfvop vex opeq1 spcev funfvex elrn2g mpbird anabsi7
      wb vtoclg ) BEZABFZGZABHZBIZGZUMCJZUNGZKZUSBHZUQGZLUMUOKZURLCAUNUSAMZVAVD
      VCURVEUTUOUMUSAUNNOVEVBUPUQUSABPQUAVAVCDJZVBRZBGZDUBZVAUSVBRZBGZVIUSBUCVH
      VKDUSCUDVFUSMVGVJBVFUSVBUEQUFSVAVBTGVCVIUKUSBUGDVBBTUHSUIULUJ $.
  $}

  $( A function's value belongs to its range.  (Contributed by NM,
     15-Oct-1996.) $)
  fnfvelrn $p |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F ) $=
    ( cfv crn wcel fvelrn funfni ) BCDCEFABCBCGH $.

  $( A function's value belongs to its codomain.  (Contributed by NM,
     12-Aug-1999.) $)
  ffvelcdm $p |- ( ( F : A --> B /\ C e. A ) -> ( F ` C ) e. B ) $=
    ( wf wcel wa cfv crn wfn ffn fnfvelrn sylan wi frn sseld adantr mpd ) ABDEZ
    CAFZGCDHZDIZFZUABFZSDAJTUCABDKACDLMSUCUDNTSUBBUAABDOPQR $.

  ${
    ffvelcdmi.1 $e |- F : A --> B $.
    $( A function's value belongs to its codomain.  (Contributed by NM,
       6-Apr-2005.) $)
    ffvelcdmi $p |- ( C e. A -> ( F ` C ) e. B ) $=
      ( wf wcel cfv ffvelcdm mpan ) ABDFCAGCDHBGEABCDIJ $.
  $}

  ${
    ffvelcdmd.1 $e |- ( ph -> F : A --> B ) $.
    $( A function's value belongs to its codomain.  (Contributed by Mario
       Carneiro, 29-Dec-2016.) $)
    ffvelcdmda $p |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) $=
      ( wf wcel cfv ffvelcdm sylan ) ABCEGDBHDEICHFBCDEJK $.

    ffvelcdmd.2 $e |- ( ph -> C e. A ) $.
    $( A function's value belongs to its codomain.  (Contributed by Mario
       Carneiro, 29-Dec-2016.) $)
    ffvelcdmd $p |- ( ph -> ( F ` C ) e. B ) $=
      ( wcel cfv ffvelcdmda mpdan ) ADBHDEICHGABCDEFJK $.
  $}

  ${
    $d x y A $.  $d x y F $.  $d x ps $.  $d y ph $.
    rexrn.1 $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $.
    $( Restricted existential quantification over the range of a function.
       (Contributed by Mario Carneiro, 24-Dec-2013.)  (Revised by Mario
       Carneiro, 20-Aug-2014.) $)
    rexrn $p |- ( F Fn A -> ( E. x e. ran F ph <-> E. y e. A ps ) ) $=
      ( wfn cv cfv crn cvv wcel funfvex funfni wceq wrex fvelrnb eqcom rexbii
      bitrdi wb adantl rexxfr2d ) FEHZABCDDIZFJZFKZELUGLMEUFFUFFNOUECIZUHMUGUIP
      ZDEQUIUGPZDEQDEUIFRUJUKDEUGUISTUAUKABUBUEGUCUD $.

    $( Restricted universal quantification over the range of a function.
       (Contributed by Mario Carneiro, 24-Dec-2013.)  (Revised by Mario
       Carneiro, 20-Aug-2014.) $)
    ralrn $p |- ( F Fn A -> ( A. x e. ran F ph <-> A. y e. A ps ) ) $=
      ( wfn cv cfv crn cvv wcel funfvex funfni wceq wrex fvelrnb eqcom rexbii
      bitrdi wb adantl ralxfr2d ) FEHZABCDDIZFJZFKZELUGLMEUFFUFFNOUECIZUHMUGUIP
      ZDEQUIUGPZDEQDEUIFRUJUKDEUGUISTUAUKABUBUEGUCUD $.
  $}

  ${
    $d F x y $.  $d Y x y $.
    $( For any element in the range of a function there is an element in the
       domain of the function for which the function value is the element of
       the range.  (Contributed by Alexander van der Vekens, 8-Dec-2017.) $)
    elrnrexdm $p |- ( Fun F -> ( Y e. ran F
                                 -> E. x e. dom F Y = ( F ` x ) ) ) $=
      ( vy wfun crn wcel cv wceq cfv cdm wa eqidd ancli adantl eqeq2 rspcev syl
      wrex ex wfn wb funfn rexrn sylbi sylibd ) BEZCBFZGZCDHZIZDUHSZCAHBJZIZABK
      ZSZUGUIULUGUILUICCIZLZULUIURUGUIUQUICMNOUKUQDCUHUJCCPQRTUGBUOUAULUPUBBUCU
      KUNDAUOBUJUMCPUDUEUF $.

    $( For any element in the range of a function there is an element in the
       domain of the function for which the function value is the element of
       the range.  (Contributed by Alexander van der Vekens, 17-Dec-2017.) $)
    elrnrexdmb $p |- ( Fun F -> ( Y e. ran F
                                 <-> E. x e. dom F Y = ( F ` x ) ) ) $=
      ( wfun crn wcel cv cfv wceq cdm wrex wfn funfn fvelrnb sylbi eqcom rexbii
      wb bitr4di ) BDZCBEFZAGBHZCIZABJZKZCUBIZAUDKTBUDLUAUERBMAUDCBNOUFUCAUDCUB
      PQS $.

    $( For any element in the domain of a function there is an element in the
       range of the function which is the function value for the element of the
       domain.  (Contributed by Alexander van der Vekens, 8-Dec-2017.) $)
    eldmrexrn $p |- ( Fun F -> ( Y e. dom F
                                 -> E. x e. ran F x = ( F ` Y ) ) ) $=
      ( wfun cdm wcel cv cfv wceq crn wrex wa fvelrn eqid eqeq1 rspcev sylancl
      ex ) BDZCBEFZAGZCBHZIZABJZKZSTLUBUDFUBUBIZUECBMUBNUCUFAUBUDUAUBUBOPQR $.
  $}

  ${
    $d w x z A $.  $d y B $.  $d y ch $.  $d w y z F $.  $d w x z ps $.
    ralrnmpt.1 $e |- F = ( x e. A |-> B ) $.
    ralrnmpt.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( A restricted quantifier over an image set.  (Contributed by Mario
       Carneiro, 20-Aug-2015.) $)
    ralrnmpt $p |- ( A. x e. A B e. V ->
      ( A. y e. ran F ps <-> A. x e. A ch ) ) $=
      ( vw vz wcel wral cv wsbc wb dfsbcq syl nfv crn cfv fnmpt nfsbc1v sbceq1a
      wfn ralrn cbvral bicomi cmpt nfmpt1 nfcxfr nfcv nffv nfsbc weq wceq fveq2
      3bitr3g wa fvmpt2 sbcieg adantl bitrd ralimiaa ralbi ) FHMZCENZADGUAZNZAD
      COZGUBZPZCENZBCENZVHADKOZPZKVINZADLOZGUBZPZLENZVJVNVHGEUFVRWBQCEFGHIUCVQW
      AKLEGADVPVTRUGSVJVRAVQDKVIAKTADVPUDADVPUEUHUIWAVMLCEACDVTCVSGCGCEFUJICEFU
      KULCVSUMUNACTUOVMLTLCUPVTVLUQWAVMQVSVKGURADVTVLRSUHUSVHVMBQZCENVNVOQVGWCC
      EVKEMZVGUTZVMADFPZBWEVLFUQVMWFQCEFHGIVAADVLFRSVGWFBQWDABDFHJVBVCVDVEVMBCE
      VFSVD $.

    $( A restricted quantifier over an image set.  (Contributed by Mario
       Carneiro, 20-Aug-2015.) $)
    rexrnmpt $p |- ( A. x e. A B e. V ->
      ( E. y e. ran F ps <-> E. x e. A ch ) ) $=
      ( vw vz wcel wral wrex cv wsbc wb nfv wa crn cfv wfn fnmpt dfsbcq nfsbc1v
      rexrn syl sbceq1a cbvrex bicomi cmpt nfmpt1 nfcxfr nfcv nfsbc weq sbceq1d
      fveq2 3bitr3g fvmpt2 sbcieg adantl bitrd ralimiaa wi wal wex pm5.32 albii
      nffv exbi sylbi df-ral df-rex bibi12i 3imtr4i ) FHMZCENZADGUAZOZADCPZGUBZ
      QZCEOZBCEOZVSADKPZQZKVTOZADLPZGUBZQZLEOZWAWEVSGEUCWIWMRCEFGHIUDWHWLKLEGAD
      WGWKUEUGUHWAWIAWHDKVTAKSADWGUFADWGUIUJUKWLWDLCEACDWKCWJGCGCEFULICEFUMUNCW
      JUOVKACSUPWDLSLCUQADWKWCWJWBGUSURUJUTVSWDBRZCENZWEWFRZVRWNCEWBEMZVRTZWDAD
      FQZBWRADWCFCEFHGIVAURVRWSBRWQABDFHJVBVCVDVEWQWNVFZCVGZWQWDTZCVHZWQBTZCVHZ
      RZWOWPXAXBXDRZCVGXFWTXGCWQWDBVIVJXBXDCVLVMWNCEVNWEXCWFXEWDCEVOBCEVOVPVQUH
      VD $.
  $}

  $( Alternate definition of a mapping.  (Contributed by NM, 14-Nov-2007.) $)
  dff2 $p |- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) ) $=
    ( wf wfn cxp wss wa ffn fssxp jca crn rnss rnxpss sstrdi anim2i df-f sylibr
    impbii ) ABCDZCAEZCABFZGZHZTUAUCABCIABCJKUDUACLZBGZHTUCUFUAUCUEUBLBCUBMABNO
    PABCQRS $.

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    $( Property of a mapping.  (Contributed by Jim Kingdon, 4-Jan-2019.) $)
    dff3im $p |- ( F : A --> B ->
        ( F C_ ( A X. B ) /\ A. x e. A E! y x F y ) ) $=
      ( wf cxp wss cv wbr weu wral fssxp wcel wa wex adantr syl2anc cvv syl wmo
      cfv cop wfun cdm fdm eleq2d biimpar funfvop df-br sylibr wi funfvex breq2
      ffun spcegv mpd funmo eu5 sylanbrc ralrimiva jca ) CDEFZECDGHAIZBIZEJZBKZ
      ACLCDEMVCVGACVCVDCNZOZVFBPZVFBUAZVGVIVDVDEUBZEJZVJVIVDVLUCENZVMVIEUDZVDEU
      EZNZVNVCVOVHCDEUOZQZVCVQVHVCVPCVDCDEUFUGUHZVDEUIRVDVLEUJUKVIVOVQVMVJULZVS
      VTVOVQOVLSNWAVDEUMVFVMBVLSVEVLVDEUNUPTRUQVCVKVHVCVOVKVRBVDEURTQVFBUSUTVAV
      B $.

    $( Property of a mapping.  (Contributed by Jim Kingdon, 4-Jan-2019.) $)
    dff4im $p |- ( F : A --> B ->
        ( F C_ ( A X. B ) /\ A. x e. A E! y e. B x F y ) ) $=
      ( wf cxp wss cv wbr weu wral wa wreu dff3im wcel cop df-br ssel opelxp2
      syl6 biimtrid pm4.71rd eubidv df-reu bitr4di ralbidv pm5.32i sylib ) CDEF
      ECDGZHZAIZBIZEJZBKZACLZMUKUNBDNZACLZMABCDEOUKUPURUKUOUQACUKUOUMDPZUNMZBKU
      QUKUNUTBUKUNUSUNULUMQZEPZUKUSULUMERUKVBVAUJPUSEUJVASULUMCDTUAUBUCUDUNBDUE
      UFUGUHUI $.

    $( An onto mapping expressed in terms of function values.  (Contributed by
       NM, 29-Oct-2006.) $)
    dffo3 $p |- ( F : A -onto-> B <-> ( F : A --> B /\
                  A. y e. B E. x e. A y = ( F ` x ) ) ) $=
      ( wfo wf crn wceq wa cv cfv wrex wral dffo2 cab wb wcel wal wi wfn fnrnfv
      ffn eqeq1d syl dfbi2 simpr ffvelcdm adantr eqeltrd exp31 rexlimdv bitr4id
      biantrurd albidv abeq1 df-ral 3bitr4g bitrd pm5.32i bitri ) CDEFCDEGZEHZD
      IZJVBBKZAKZELZIZACMZBDNZJCDEOVBVDVJVBVDVIBPZDIZVJVBECUAZVDVLQCDEUCVMVCVKD
      ABCEUBUDUEVBVIVEDRZQZBSVNVITZBSVLVJVBVOVPBVBVOVIVNTZVPJVPVIVNUFVBVQVPVBVH
      VNACVBVFCRZVHVNVBVRJZVHJVEVGDVSVHUGVSVGDRVHCDVFEUHUIUJUKULUNUMUOVIBDUPVIB
      DUQURUSUTVA $.

    $( Alternate definition of an onto mapping.  (Contributed by NM,
       20-Mar-2007.) $)
    dffo4 $p |- ( F : A -onto-> B <->
                ( F : A --> B /\ A. y e. B E. x e. A x F y ) ) $=
      ( wfo wf cv wbr wrex wral wa crn wceq dffo2 simpl wcel wex vex wi bitr3id
      elrn eleq2 biimpar adantll wfn ffn fnbr syl ancrd eximdv df-rex imbitrrdi
      ad2antrr mpd ralrimiva jca sylbi cfv fnbrfvb biimprd eqcom imbitrdi sylan
      ex reximdva ralimdv imdistani dffo3 sylibr impbii ) CDEFZCDEGZAHZBHZEIZAC
      JZBDKZLZVLVMEMZDNZLZVSCDEOWBVMVRVMWAPWBVQBDWBVODQZLVPARZVQWAWCWDVMWAWDWCW
      DVOVTQWAWCAVOEBSUBVTDVOUCUAUDUEVMWDVQTWAWCVMWDVNCQZVPLZARVQVMVPWFAVMVPWEV
      MECUFZVPWETCDEUGZWGVPWECVNVOEUHVEUIUJUKVPACULUMUNUOUPUQURVSVMVOVNEUSZNZAC
      JZBDKZLVLVMVRWLVMVQWKBDVMVPWJACVMWGWEVPWJTWHWGWELZVPWIVONZWJWMWNVPCVNVOEU
      TVAWIVOVBVCVDVFVGVHABCDEVIVJVK $.

    $( Alternate definition of an onto mapping.  (Contributed by NM,
       20-Mar-2007.) $)
    dffo5 $p |- ( F : A -onto-> B <->
                ( F : A --> B /\ A. y e. B E. x x F y ) ) $=
      ( wfo wf cv wbr wrex wral wa wex dffo4 rexex ralimi anim2i wcel wfn wi ex
      ffn fnbr syl ancrd eximdv df-rex imbitrrdi ralimdv imdistani impbii bitri
      ) CDEFCDEGZAHZBHZEIZACJZBDKZLZUMUPAMZBDKZLZABCDENUSVBURVAUMUQUTBDUPACOPQU
      MVAURUMUTUQBDUMUTUNCRZUPLZAMUQUMUPVDAUMUPVCUMECSZUPVCTCDEUBVEUPVCCUNUOEUC
      UAUDUEUFUPACUGUHUIUJUKUL $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d y z C $.  $d y z F $.
    fmpt.1 $e |- F = ( x e. A |-> C ) $.
    $( Functionality of the mapping operation.  (Contributed by Mario Carneiro,
       26-Jul-2013.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    fmpt $p |- ( A. x e. A C e. B <-> F : A --> B ) $=
      ( vy wcel wral wf wfn crn wss fnmpt cv wceq wrex cab rnmpt wa biimparc ex
      r19.29 rexlimivw syl abssdv eqsstrid df-f sylanbrc crab ccnv cima fimacnv
      eleq1 mptpreima eqtr3di rabid2 sylib impbii ) DCHZABIZBCEJZVAEBKELZCMVBAB
      DECFNVAVCGOZDPZABQZGRCAGBDEFSVAVFGCVAVFVDCHZVAVFTUTVETZABQVGUTVEABUCVHVGA
      BVEVGUTVDDCUNUAUDUEUBUFUGBCEUHUIVBBUTABUJZPVAVBEUKCULBVIBCEUMABDCEFUOUPUT
      ABUQURUS $.

    $( Express bijection for a mapping operation.  (Contributed by Mario
       Carneiro, 30-May-2015.)  (Revised by Mario Carneiro, 4-Dec-2016.) $)
    f1ompt $p |- ( F : A -1-1-onto-> B <->
        ( A. x e. A C e. B /\ A. y e. B E! x e. A y = C ) ) $=
      ( vz wa cv wceq wral wcel wfn wbr weu nfcv bitri vex 3bitr4i wf wf1o wreu
      ccnv wb ffn dff1o4 baib syl cres fnres cmpt nfmpt1 nfcxfr nfv breq1 copab
      nfbr df-mpt eqtri breqi cop df-br opabid bitrdi cbveu brcnv df-reu ralbii
      eubii wrel cdm wss relcnv crn df-rn frn eqsstrrid relssres sylancr fneq1d
      bitr3id bitr4d pm5.32i f1of pm4.71ri fmpt anbi1i ) CDFUAZCDFUBZIWIBJZEKZA
      CUCZBDLZIWJEDMACLZWNIWIWJWNWIWJFUDZDNZWNWIFCNZWJWQUECDFUFWJWRWQCDFUGUHUIW
      NWPDUJZDNZWIWQWTWKHJZWPOZHPZBDLWNBHDWPUKXCWMBDXAWKFOZHPAJZCMWLIZAPXCWMXDX
      FHAAXAWKFAXAQAFACEULZGACEUMUNAWKQURXFHUOXAXEKXDXEWKFOZXFXAXEWKFUPXHXEWKXF
      ABUQZOZXFXEWKFXIFXGXIGABCEUSUTVAXJXEWKVBXIMXFXEWKXIVCXFABVDRRVEVFXBXDHWKX
      AFBSHSVGVJWLACVHTVIRWIDWSWPWIWPVKWPVLZDVMWSWPKFVNWIXKFVODFVPCDFVQVRWPDVSV
      TWAWBWCWDWJWICDFWEWFWOWIWNACDEFGWGWHT $.

    fmpti.2 $e |- ( x e. A -> C e. B ) $.
    $( Functionality of the mapping operation.  (Contributed by NM,
       19-Mar-2005.)  (Revised by Mario Carneiro, 1-Sep-2015.) $)
    fmpti $p |- F : A --> B $=
      ( wcel wral wf rgen fmpt mpbi ) DCHZABIBCEJNABGKABCDEFLM $.
  $}

  ${
    $d A x $.  $d C x $.
    fvmptelcdm.1 $e |- ( ph -> ( x e. A |-> B ) : A --> C ) $.
    $( The value of a function at a point of its domain belongs to its
       codomain.  (Contributed by Glauco Siliprandi, 26-Jun-2021.) $)
    fvmptelcdm $p |- ( ( ph /\ x e. A ) -> B e. C ) $=
      ( wcel cmpt wf wral eqid fmpt sylibr r19.21bi ) ADEGZBCACEBCDHZIOBCJFBCED
      PPKLMN $.
  $}

  ${
    $d x A $.  $d x C $.  $d x ph $.
    fmptd.1 $e |- ( ( ph /\ x e. A ) -> B e. C ) $.
    fmptd.2 $e |- F = ( x e. A |-> B ) $.
    $( Domain and codomain of the mapping operation; deduction form.
       (Contributed by Mario Carneiro, 13-Jan-2013.) $)
    fmptd $p |- ( ph -> F : A --> C ) $=
      ( wcel wral wf ralrimiva fmpt sylib ) ADEIZBCJCEFKAOBCGLBCEDFHMN $.
  $}

  ${
    $d x A $.  $d x C $.  $d x ph $.
    fmpttd.1 $e |- ( ( ph /\ x e. A ) -> B e. C ) $.
    $( Version of ~ fmptd with inlined definition.  Domain and codomain of the
       mapping operation; deduction form.  (Contributed by Glauco Siliprandi,
       23-Oct-2021.)  (Proof shortened by BJ, 16-Aug-2022.) $)
    fmpttd $p |- ( ph -> ( x e. A |-> B ) : A --> C ) $=
      ( cmpt eqid fmptd ) ABCDEBCDGZFJHI $.
  $}

  ${
    $d x A $.  $d x C $.  $d x ph $.
    fmpt3d.1 $e |- ( ph -> F = ( x e. A |-> B ) ) $.
    fmpt3d.2 $e |- ( ( ph /\ x e. A ) -> B e. C ) $.
    $( Domain and codomain of the mapping operation; deduction form.
       (Contributed by Thierry Arnoux, 4-Jun-2017.) $)
    fmpt3d $p |- ( ph -> F : A --> C ) $=
      ( wf cmpt fmpttd feq1d mpbird ) ACEFICEBCDJZIABCDEHKACEFNGLM $.
  $}

  ${
    $d A x $.  $d C x $.
    fmptdf.1 $e |- F/ x ph $.
    fmptdf.2 $e |- ( ( ph /\ x e. A ) -> B e. C ) $.
    fmptdf.3 $e |- F = ( x e. A |-> B ) $.
    $( A version of ~ fmptd using bound-variable hypothesis instead of a
       distinct variable condition for ` ph ` .  (Contributed by Glauco
       Siliprandi, 29-Jun-2017.) $)
    fmptdf $p |- ( ph -> F : A --> C ) $=
      ( wcel wral wf cv ex ralrimi fmpt sylib ) ADEJZBCKCEFLARBCGABMCJRHNOBCEDF
      IPQ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    $( A function maps to a class to which all values belong.  (Contributed by
       NM, 3-Dec-2003.) $)
    ffnfv $p |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) $=
      ( vy wf wfn cv cfv wcel wral wa ffn ffvelcdm ralrimiva jca crn simpl wceq
      wss wrex fvelrnb biimpd nfra1 nfv wi rsp eleq1 biimpcd syl6 rexlimd ssrdv
      sylan9 df-f sylanbrc impbii ) BCDFZDBGZAHZDIZCJZABKZLZUQURVBBCDMUQVAABBCU
      SDNOPVCURDQZCTUQURVBRVCEVDCUREHZVDJZUTVESZABUAZVBVECJZURVFVHABVEDUBUCVBVG
      VIABVAABUDVIAUEVBUSBJVAVGVIUFVAABUGVGVAVIUTVECUHUIUJUKUMULBCDUNUOUP $.
  $}

  ${
    $d z A $.  $d z B $.  $d z F $.  $d x z $.
    ffnfvf.1 $e |- F/_ x A $.
    ffnfvf.2 $e |- F/_ x B $.
    ffnfvf.3 $e |- F/_ x F $.
    $( A function maps to a class to which all values belong.  This version of
       ~ ffnfv uses bound-variable hypotheses instead of distinct variable
       conditions.  (Contributed by NM, 28-Sep-2006.) $)
    ffnfvf $p |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) $=
      ( vz wf wfn cv cfv wcel wral wa ffnfv nfcv nffv nfel nfv weq fveq2 eleq1d
      cbvralf anbi2i bitri ) BCDIDBJZHKZDLZCMZHBNZOUGAKZDLZCMZABNZOHBCDPUKUOUGU
      JUNHABHBQEAUICAUHDGAUHQRFSUNHTHAUAUIUMCUHULDUBUCUDUEUF $.
  $}

  ${
    $d x A $.  $d x B $.  $d x F $.
    $( An upper bound for range determined by function values.  (Contributed by
       NM, 8-Oct-2004.) $)
    fnfvrnss $p |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B ) $=
      ( wfn cv cfv wcel wral wa wf crn wss ffnfv frn sylbir ) DBEAFDGCHABIJBCDK
      DLCMABCDNBCDOP $.
  $}

  ${
    $d x A $.  $d x C $.
    rnmptss.1 $e |- F = ( x e. A |-> B ) $.
    $( The range of an operation given by the maps-to notation as a subset.
       (Contributed by Thierry Arnoux, 24-Sep-2017.) $)
    rnmptss $p |- ( A. x e. A B e. C -> ran F C_ C ) $=
      ( wcel wral wf crn wss fmpt frn sylbi ) CDGABHBDEIEJDKABDCEFLBDEMN $.
  $}

  ${
    $d x A $.  $d y A $.  $d y C $.  $d y F $.  $d x ph $.  $d y ph $.
    fmpt2d.2 $e |- ( ( ph /\ x e. A ) -> B e. V ) $.
    fmpt2d.1 $e |- ( ph -> F = ( x e. A |-> B ) ) $.
    fmpt2d.3 $e |- ( ( ph /\ y e. A ) -> ( F ` y ) e. C ) $.
    $( Domain and codomain of the mapping operation; deduction form.
       (Contributed by NM, 27-Dec-2014.) $)
    fmpt2d $p |- ( ph -> F : A --> C ) $=
      ( wfn cv cfv wcel wral wf cmpt ralrimiva eqid fnmpt fneq1d ffnfv sylanbrc
      syl mpbird ) AGDLZCMGNFOZCDPDFGQAUGBDERZDLZAEHOZBDPUJAUKBDISBDEUIHUITUAUE
      ADGUIJUBUFAUHCDKSCDFGUCUD $.
  $}

  ${
    $d x A $.  $d x B $.  $d x F $.
    $( A necessary and sufficient condition for a restricted function.
       (Contributed by Mario Carneiro, 14-Nov-2013.) $)
    ffvresb $p |- ( Fun F -> ( ( F |` A ) : A --> B <->
        A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) ) $=
      ( wfun cres wf cv cdm wcel cfv wa wral fdm cin dmres inss2 adantl wfn wss
      eqsstri eqsstrrdi sselda wceq fvres ffvelcdm eqeltrrd jca ralrimiva simpl
      crn ralimi dfss3 sylibr funfn fnssres sylanb sylan2 simpr eleq1d fnfvrnss
      imbitrrid ralimia syl2anc df-f sylanbrc ex impbid2 ) DEZBCDBFZGZAHZDIZJZV
      LDKZCJZLZABMZVKVQABVKVLBJZLZVNVPVKBVMVLVKBVJIZVMBCVJNWABVMOVMDBPBVMQUAUBU
      CVTVLVJKZVOCVSWBVOUDVKVLBDUEZRBCVLVJUFUGUHUIVIVRVKVIVRLZVJBSZVJUKCTZVKVRV
      IBVMTZWEVRVNABMWGVQVNABVNVPUJULABVMUMUNVIDVMSWGWEDUOVMBDUPUQURZWDWEWBCJZA
      BMZWFWHVRWJVIVQWIABVQWIVSVPVNVPUSVSWBVOCWCUTVBVCRABCVJVAVDBCVJVEVFVGVH $.
  $}

  ${
    $d A x $.  $d ph x $.  $d F x $.  $d Y x $.
    resflem.1 $e |- ( ph -> F : V --> X ) $.
    resflem.2 $e |- ( ph -> A C_ V ) $.
    resflem.3 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) e. Y ) $.
    $( A lemma to bound the range of a restriction.  The conclusion would also
       hold with ` ( X i^i Y ) ` in place of ` Y ` (provided ` x ` does not
       occur in ` X ` ).  If that stronger result is needed, it is however
       simpler to use the instance of ~ resflem where ` ( X i^i Y ) ` is
       substituted for ` Y ` (in both the conclusion and the third hypothesis).
       (Contributed by BJ, 4-Jul-2022.) $)
    resflem $p |- ( ph -> ( F |` A ) : A --> Y ) $=
      ( cres wf cv cdm wcel cfv wa wral sseld syl wceq fdm eleq2d jcad ralrimiv
      sylibrd ex wfun wb ffun ffvresb mpbird ) ACGDCKLZBMZDNZOZUNDPGOZQZBCRZAUR
      BCAUNCOZUPUQAUTUNEOUPACEUNISAUOEUNAEFDLZUOEUAHEFDUBTUCUFAUTUQJUGUDUEADUHZ
      UMUSUIAVAVBHEFDUJTBCGDUKTUL $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x y ph $.  $d y ps $.  $d x ch $.
    f1oresrab.1 $e |- F = ( x e. A |-> C ) $.
    f1oresrab.2 $e |- ( ph -> F : A -1-1-onto-> B ) $.
    f1oresrab.3 $e |- ( ( ph /\ x e. A /\ y = C ) -> ( ch <-> ps ) ) $.
    $( Build a bijection between restricted abstract builders, given a
       bijection between the base classes, deduction version.  (Contributed by
       Thierry Arnoux, 17-Aug-2018.) $)
    f1oresrab $p |- ( ph -> ( F |` { x e. A | ps } ) :
                               { x e. A | ps } -1-1-onto-> { y e. B | ch } ) $=
      ( crab ccnv cres wf1o syl wceq wb wcel wfun f1ofun funcnvcnv 3syl wf1 wss
      cima f1ocnv f1of1 ssrab2 f1ores sylancl mptpreima cv wa wi 3expia alrimiv
      wal wral f1of fmpt sylibr r19.21bi elrab3t syl2anc rabbidva eqtrid f1oeq3
      wf mpbid f1orescnv rescnvcnv f1oeq1 ax-mp sylib ) ABDFMZCEGMZINZNZVQOZPZV
      QVRIVQOZPZAVTUAZVRVQVSVROZPZWBAFGIPZIUAWEKFGIUBIUCUDAVRVSVRUGZWFPZWGAGFVS
      UEZVRGUFWJAGFVSPZWKAWHWLKFGIUHQGFVSUIQCEGUJGFVRVSUKULAWIVQRWJWGSAWIHVRTZD
      FMVQDFHVRIJUMAWMBDFADUNFTZUOZEUNHRZCBSZUPZEUSHGTZWMBSWOWREAWNWPWQLUQURAWS
      DFAFGIVJZWSDFUTAWHWTKFGIVAQDFGHIJVBVCVDCBEHGVEVFVGVHWIVQVRWFVIQVKVQVRVSVL
      VFWAWCRWBWDSIVQVMVQVRWAWCVNVOVP $.
  $}

  ${
    $d u v w x z A $.  $d u x y B $.  $d u w z F $.  $d u w z G $.  $d u y R $.
    $d u w x z ph $.  $d u x S $.  $d u v w y z T $.
    fmptco.1 $e |- ( ( ph /\ x e. A ) -> R e. B ) $.
    fmptco.2 $e |- ( ph -> F = ( x e. A |-> R ) ) $.
    fmptco.3 $e |- ( ph -> G = ( y e. B |-> S ) ) $.
    fmptco.4 $e |- ( y = R -> S = T ) $.
    $( Composition of two functions expressed as ordered-pair class
       abstractions.  If ` F ` has the equation ( x + 2 ) and ` G ` the
       equation ( 3 * z ) then ` ( G o. F ) ` has the equation ( 3 * ( x +
       2 ) ) .  (Contributed by FL, 21-Jun-2012.)  (Revised by Mario Carneiro,
       24-Jul-2014.) $)
    fmptco $p |- ( ph -> ( G o. F ) = ( x e. A |-> T ) ) $=
      ( vu cv wbr wa wcel wceq vz vw vv ccom cmpt relco wfun wrel funmpt funrel
      ax-mp wex csb cop cfv eqid fmptd feq1d mpbird ffun syl funbrfv imp eqcomd
      wf sylan a1d expimpd pm4.71rd exbidv cvv wi exsimpl sylibr a1i cdm adantr
      isset eleq2d biimpar funfvex syl2anc adantrr ex wb breq2 anbi12d ceqsexgv
      fdm breq1 funfvbrb bitr3d fveq1d eqidd breq123d nfcv nfv nffvmpt1 nfcsb1v
      nfbr nfeq2 nfbi nfim fveq2 breq1d csbeq1a eqeq2d bibi12d imbi2d vex simpl
      eleq1d simpr eqeq12d df-mpt brabga sylancl fvmpt2 3bitr4d expcom vtoclgaf
      biantrurd impcom bitrd sylan9bbr pm5.21ndd opelco copab eleq2i nfan eleq1
      pm5.32da eqeq1 anbi2d opelopabf bitri 3bitr4g eqrelrdv ) AUAUBJIUDZBDHUEZ
      JIUFYTUGYTUHBDHUIYTUJUKAUAPZOPZIQZUUBUBPZJQZRZOULZUUADSZUUDBUUAHUMZTZRZUU
      AUUDUNZYSSUULYTSZAUUGUUBUUAIUOZTZUUFRZOULZUUKAUUFUUPOAUUFUUOAUUCUUEUUOAUU
      CRZUUOUUEUURUUNUUBAIUGZUUCUUNUUBTZADEIVEZUUSAUVADEBDFUEZVEABDFEUVBKUVBUPZ
      UQADEIUVBLURUSZDEIUTVAZUUSUUCUUTUUAUUBIVBVCVFVDVGVHVIVJAUUNVKSZUUQUUKUUQU
      VFVLAUUQUUOOULUVFUUOUUFOVMOUUNVRVNVOAUUKUVFAUUHUVFUUJAUUHRUUSUUAIVPZSZUVF
      AUUSUUHUVEVQAUVHUUHAUVGDUUAAUVAUVGDTUVDDEIWIVAVSZVTUUAIWAWBWCWDAUVFUUQUUK
      WEUVFUUQUUAUUNIQZUUNUUDJQZRZAUUKUUFUVLOUUNVKUUOUUCUVJUUEUVKUUBUUNUUAIWFUU
      BUUNUUDJWJWGWHAUVLUUHUUAUVBUOZUUDCEGUEZQZRUUKAUVJUUHUVKUVOAUVHUVJUUHAUUSU
      VHUVJWEUVEUUAIWKVAUVIWLAUUNUVMUUDUUDJUVNAUUAIUVBLWMMAUUDWNWOWGAUUHUVOUUJU
      UHAUVOUUJWEZABPZUVBUOZUUDUVNQZUUDHTZWEZVLAUVPVLBUUADBUUAWPAUVPBABWQUVOUUJ
      BBUVMUUDUVNBDFUUAWRBUVNWPBUUDWPWTBUUDUUIBUUAHWSZXAXBXCUVQUUATZUWAUVPAUWCU
      VSUVOUVTUUJUWCUVRUVMUUDUVNUVQUUAUVBXDXEUWCHUUIUUDBUUAHXFZXGXHXIAUVQDSZUWA
      AUWERZFUUDUVNQZFESZUVTRZUVSUVTUWFUWHUUDVKSUWGUWIWEKUBXJZCPZESZUUBGTZRUWIC
      OFUUDUVNEVKUWKFTZUUBUUDTZRZUWLUWHUWMUVTUWPUWKFEUWNUWOXKXLUWPUUBUUDGHUWNUW
      OXMUWNGHTUWONVQXNWGCOEGXOXPXQUWFUVRFUUDUVNUWFUWEUWHUVRFTAUWEXMKBDFEUVBUVC
      XRWBXEUWFUWHUVTKYBXSXTYAYCYLYDYEWDYFYDOUUAUUDJIUAXJZUWJYGUUMUULUWEUCPZHTZ
      RZBUCYHZSUUKYTUXAUULBUCDHXOYIUWTUUHUWRUUITZRUUKBUCUUAUUDUUHUXBBUUHBWQBUWR
      UUIUWBXAYJUUKUCWQUWQUWJUWCUWEUUHUWSUXBUVQUUADYKUWCHUUIUWRUWDXGWGUWRUUDTUX
      BUUJUUHUWRUUDUUIYMYNYOYPYQYR $.
  $}

  ${
    $d w x y z B $.  $d w y z R $.  $d w x z S $.  $d x z A $.  $d y z T $.
    $d z ph $.
    fmptcof.1 $e |- ( ph -> A. x e. A R e. B ) $.
    fmptcof.2 $e |- ( ph -> F = ( x e. A |-> R ) ) $.
    fmptcof.3 $e |- ( ph -> G = ( y e. B |-> S ) ) $.
    ${
      fmptcof.4 $e |- ( y = R -> S = T ) $.
      $( Version of ~ fmptco where ` ph ` needn't be distinct from ` x ` .
         (Contributed by NM, 27-Dec-2014.) $)
      fmptcof $p |- ( ph -> ( G o. F ) = ( x e. A |-> T ) ) $=
        ( vz vw csb cmpt wceq nfcv ccom cv wcel wral nfcsb1v nfel1 csbeq1a rspc
        eleq1d mpan9 cbvmpt eqtrdi csbeq1 fmptco csbeq1d eqtr4di csbiegf ralimi
        nfcsb eqid nfcvd mpteq12 sylancr syl eqtrd ) AJIUAZBDCFGQZRZBDHRZAVFODC
        BOUBZFQZGQZRVHAOPDEVKCPUBZGQZVLIJAFEUCZBDUDZVJDUCVKEUCZKVOVQBVJDBVKEBVJ
        FUEZUFBUBVJSZFVKEBVJFUGZUIUHUJAIBDFRODVKRLBODFVKOFTVRVTUKULAJCEGRPEVNRM
        CPEGVNPGTCVMGUECVMGUGUKULCVMVKGUMUNBODVGVLOVGTBCVKGVRBGTUSVSCFVKGVTUOUK
        UPAVPVHVISZKVPDDSVGHSZBDUDWADUTVOWBBDCFGHEVOCHVANUQURBDVGDHVBVCVDVE $.
    $}

    $( Composition of two functions expressed as mapping abstractions.
       (Contributed by NM, 22-May-2006.)  (Revised by Mario Carneiro,
       31-Aug-2015.) $)
    fmptcos $p |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) ) $=
      ( vz cv csb cmpt nfcv nfcsb1v csbeq1a cbvmpt eqtrdi csbeq1 fmptcof ) ABMD
      EFCMNZGOZCFGOHIJKAICEGPMEUEPLCMEGUEMGQCUDGRCUDGSTUACUDFGUBUC $.
  $}

  ${
    $d x A $.  $d y B $.  $d x y C $.  $d x y F $.  $d x ph $.
    cofmpt.1 $e |- ( ph -> F : C --> D ) $.
    cofmpt.2 $e |- ( ( ph /\ x e. A ) -> B e. C ) $.
    $( Express composition of a maps-to function with another function in a
       maps-to notation.  (Contributed by Thierry Arnoux, 29-Jun-2017.) $)
    cofmpt $p |- ( ph ->
                     ( F o. ( x e. A |-> B ) ) = ( x e. A |-> ( F ` B ) ) ) $=
      ( vy cv cfv cmpt eqidd feqmptd fveq2 fmptco ) ABJCEDJKZGLDGLBCDMZGIASNAJE
      FGHORDGPQ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x C $.  $d x y D $.  $d x E $.
    $( Express composition of two functions as a maps-to applying both in
       sequence.  (Contributed by Stefan O'Rear, 5-Oct-2014.)  (Proof shortened
       by Mario Carneiro, 27-Dec-2014.) $)
    fcompt $p |- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |->
        ( A ` ( B ` x ) ) ) ) $=
      ( vy wf wa cv cfv wcel ffvelcdm adantll wfn cmpt wceq ffn dffn5im syl
      adantl adantr fveq2 fmptco ) EFBHZDECHZIZAGDEAJZCKZGJZBKZUIBKCBUFUHDLUIEL
      UEDEUHCMNUGCDOZCADUIPQUFULUEDECRUAADCSTUGBEOZBGEUKPQUEUMUFEFBRUBGEBSTUJUI
      BUCUD $.
  $}

  ${
    $d F x y $.  $d I x $.  $d X x y $.  $d Y x y $.
    $( Composition with a constant function.  (Contributed by Stefan O'Rear,
       11-Mar-2015.) $)
    fcoconst $p |- ( ( F Fn X /\ Y e. X ) ->
        ( F o. ( I X. { Y } ) ) = ( I X. { ( F ` Y ) } ) ) $=
      ( vx vy wfn wcel wa csn cxp ccom cfv cmpt simplr wceq fconstmpt a1i cvv
      cv wf simpl dffn2 sylib feqmptd fveq2 fmptco eqtr4di ) ACGZDCHZIZABDJKZLE
      BDAMZNBUMJKUKEFBCDFTZAMUMULAUIUJETBHOULEBDNPUKEBDQRUKFCSAUKUICSAUAUIUJUBC
      AUCUDUEUNDAUFUGEBUMQUH $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    fsn.1 $e |- A e. _V $.
    fsn.2 $e |- B e. _V $.
    $( A function maps a singleton to a singleton iff it is the singleton of an
       ordered pair.  (Contributed by NM, 10-Dec-2003.) $)
    fsn $p |- ( F : { A } --> { B } <-> F = { <. A , B >. } ) $=
      ( vx vy csn cop wceq cv wcel wb wal wa velsn weu eleq1d bitr4i bitr2i feu
      wf opelf anbi12i sylib ex wreu snid mpan2 opeq2 pm5.32i ancom eubii eueq1
      anbi1i biantru euanv df-reu sylibr opeq12 syl5ibrcom impbid vex opex elsn
      3bitr4i opth2 bitrdi alrimivv wrel frel relsnop eqrel sylancl mpbird wf1o
      f1osn f1oeq1 mpbiri f1of syl impbii ) AHZBHZCUBZCABIZHZJZWEWHFKZGKZIZCLZW
      KWGLZMZGNFNZWEWNFGWEWLWIAJZWJBJZOZWMWEWLWRWEWLWRWEWLOWIWCLZWJWDLZOWRWCWDW
      IWJCUCWSWPWTWQFAPGBPZUDUEUFWEWLWRWFCLZWEAWJIZCLZGWDUGZXBWEAWCLXEADUHGWCWD
      ACUAUIXBWQOZGQZWTXDOZGQXBXEXFXHGXHWQXDOZXFWTWQXDXAUOXIWQXBOXFWQXDXBWQXCWF
      CWJBAUJRUKXBWQULSTUMXBXBWQGQZOXGXJXBGBEUNUPXBWQGUQSXDGWDURVFUSWRWKWFCWIWJ
      ABUTRVAVBWMWKWFJWRWKWFWIWJFVCGVCVDVEWIWJABDEVGTVHVIWECVJWGVJWHWOMWCWDCVKA
      BDEVLFGCWGVMVNVOWHWCWDCVPZWEWHXKWCWDWGVPABDEVQWCWDCWGVRVSWCWDCVTWAWB $.
  $}

  ${
    $d A a b $.  $d B b $.  $d F a b $.
    $( A function maps a singleton to a singleton iff it is the singleton of an
       ordered pair.  (Contributed by NM, 26-Oct-2012.) $)
    fsng $p |- ( ( A e. C /\ B e. D ) ->
                 ( F : { A } --> { B } <-> F = { <. A , B >. } ) ) $=
      ( va vb cv csn wf cop wceq wb sneq feq2d opeq1 sneqd eqeq2d bibi12d vex
      feq3 syl opeq2 fsn vtocl2g ) FHZIZGHZIZEJZEUFUHKZIZLZMAIZUIEJZEAUHKZIZLZM
      UNBIZEJZEABKZIZLZMFGABCDUFALZUJUOUMURVDUGUNUIEUFANOVDULUQEVDUKUPUFAUHPQRS
      UHBLZUOUTURVCVEUIUSLUOUTMUHBNUIUSUNEUAUBVEUQVBEVEUPVAUHBAUCQRSUFUHEFTGTUD
      UE $.
  $}

  ${
    fsn2.1 $e |- A e. _V $.
    $( A function that maps a singleton to a class is the singleton of an
       ordered pair.  (Contributed by NM, 19-May-2004.) $)
    fsn2 $p |- ( F : { A } --> B <->
               ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) $=
      ( csn wf cfv cvv wcel cop wceq wa wfn ffn snid funfvex mpan2 syl wb cima
      funfni elex adantr ffvelcdm crn dffn3 biimpi imadmrn fndm imaeq2d eqtr3id
      cdm fnsnfv eqtr4d feq3 mpbid jca wss snssi fss ancoms sylan impbii anbi2d
      fsng mpan bitrid pm5.21nii ) AEZBCFZACGZHIZVKBIZCAVKJEKZLZVJCVIMZVLVIBCNZ
      VPAVIIZVLADOZVLVIACACPUAQRVMVLVNVKBUBUCVJVMVIVKEZCFZLZVLVOVJWBVJVMWAVJVRV
      MVSVIBACUDQVJVPWAVQVPVICUEZCFZWAVPWDVICUFUGVPWCVTKWDWASVPWCCVITZVTVPWCCCU
      LZTWECUHVPWFVICVICUIUJUKVPVRVTWEKVSVIACUMQUNWCVTVICUORUPRUQVMVTBURZWAVJVK
      BUSWAWGVJVIVTBCUTVAVBVCVLWAVNVMAHIVLWAVNSDAVKHHCVEVFVDVGVH $.
  $}

  ${
    $d A a b $.  $d B a b $.  $d F a b $.
    $( A function that maps a singleton to a class is the singleton of an
       ordered pair.  (Contributed by Thierry Arnoux, 11-Jul-2020.) $)
    fsn2g $p |- ( A e. V -> ( F : { A } --> B <->
               ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) ) $=
      ( va cv csn wf cfv wcel cop wceq wa sneq feq2d fveq2 eleq1d opeq12d sneqd
      id eqeq2d anbi12d vex fsn2 vtoclbg ) EFZGZBCHUFCIZBJZCUFUHKZGZLZMAGZBCHAC
      IZBJZCAUNKZGZLZMEADUFALZUGUMBCUFANOUSUIUOULURUSUHUNBUFACPZQUSUKUQCUSUJUPU
      SUFAUHUNUSTUTRSUAUBUFBCEUCUDUE $.
  $}

  $( The cross product of two singletons.  (Contributed by Mario Carneiro,
     30-Apr-2015.) $)
  xpsng $p |- ( ( A e. V /\ B e. W ) ->
    ( { A } X. { B } ) = { <. A , B >. } ) $=
    ( wcel wa csn cxp wf cop wceq fconstg adantl fsng mpbid ) ACEZBDEZFAGZBGZRS
    HZIZTABJGKQUAPRBDLMABCDTNO $.

  ${
    xpsn.1 $e |- A e. _V $.
    xpsn.2 $e |- B e. _V $.
    $( The cross product of two singletons.  (Contributed by NM,
       4-Nov-2006.) $)
    xpsn $p |- ( { A } X. { B } ) = { <. A , B >. } $=
      ( cvv wcel csn cxp cop wceq xpsng mp2an ) AEFBEFAGBGHABIGJCDABEEKL $.
  $}

  ${
    dfmpt.1 $e |- B e. _V $.
    $( Alternate definition for the maps-to notation ~ df-mpt (although it
       requires that ` B ` be a set).  (Contributed by NM, 24-Aug-2010.)
       (Revised by Mario Carneiro, 30-Dec-2016.) $)
    dfmpt $p |- ( x e. A |-> B ) = U_ x e. A { <. x , B >. } $=
      ( cmpt cv csn cxp ciun cop dfmpt3 wceq wcel vex xpsn a1i iuneq2i eqtri )
      ABCEABAFZGCGHZIABSCJGZIABCKABTUATUALSBMSCANDOPQR $.

    $d x y $.  $d y A $.  $d y B $.
    $( A function expressed as the range of another function.  (Contributed by
       Mario Carneiro, 22-Jun-2013.)  (Proof shortened by Mario Carneiro,
       31-Aug-2015.) $)
    fnasrn $p |- ( x e. A |-> B ) = ran ( x e. A |-> <. x , B >. ) $=
      ( vy cmpt cv cop csn ciun crn dfmpt wcel wrex cab wceq rnmpt velsn eqtr4i
      eqid rexbii abbii df-iun ) ABCFABAGCHZIZJZABUDFZKZABCDLUHEGZUEMZABNZEOZUF
      UHUIUDPZABNZEOULAEBUDUGUGTQUKUNEUJUMABEUDRUAUBSAEBUEUCSS $.
  $}

  $( Alternate definition for the maps-to notation ~ df-mpt (which requires
     that ` B ` be a set).  (Contributed by Jim Kingdon, 9-Jan-2019.) $)
  dfmptg $p |- ( A. x e. A B e. V ->
      ( x e. A |-> B ) = U_ x e. A { <. x , B >. } ) $=
    ( wcel wral cmpt cv csn cxp ciun cop dfmpt3 wceq cvv vex mpan ralimi iuneq2
    xpsng syl eqtrid ) CDEZABFZABCGABAHZICIJZKZABUECLIZKZABCMUDUFUHNZABFUGUINUC
    UJABUEOEUCUJAPUECODTQRABUFUHSUAUB $.

  ${
    $d x y $.  $d y A $.  $d y B $.
    $( A function expressed as the range of another function.  (Contributed by
       Jim Kingdon, 9-Jan-2019.) $)
    fnasrng $p |- ( A. x e. A B e. V ->
        ( x e. A |-> B ) = ran ( x e. A |-> <. x , B >. ) ) $=
      ( vy wcel wral cmpt cv cop csn ciun crn dfmptg wrex cab wceq rnmpt eqtr4i
      eqid velsn rexbii abbii df-iun eqtr4di ) CDFABGABCHABAICJZKZLZABUFHZMZABC
      DNUJEIZUGFZABOZEPZUHUJUKUFQZABOZEPUNAEBUFUIUITRUMUPEULUOABEUFUAUBUCSAEBUG
      UDSUE $.
  $}

  ${
    $d F x $.
    $( A function is a union of singletons of ordered pairs indexed by its
       domain.  (Contributed by AV, 18-Sep-2020.) $)
    funiun $p |- ( Fun F -> F = U_ x e. dom F { <. x , ( F ` x ) >. } ) $=
      ( wfun cdm cv cfv cmpt cop csn ciun wfn wceq funfn dffn5im sylbi cvv wcel
      wral funfvex ralrimiva dfmptg syl eqtrd ) BCZBABDZAEZBFZGZAUEUFUGHIJZUDBU
      EKBUHLBMAUEBNOUDUGPQZAUERUHUILUDUJAUEUFBSTAUEUGPUAUBUC $.
  $}

  ${
    $d F a b c d $.  $d X a b c d $.  $d Y a b c d $.
    funopsn.x $e |- X e. _V $.
    funopsn.y $e |- Y e. _V $.
    $( If a function is an ordered pair then it is a singleton of an ordered
       pair.  (Contributed by AV, 20-Sep-2020.)  (Proof shortened by AV,
       15-Jul-2021.)  A function is a class of ordered pairs, so the fact that
       an ordered pair may sometimes be itself a function is an "accident"
       depending on the specific encoding of ordered pairs as classes (in
       set.mm, the Kuratowski encoding).  A more meaningful statement is
       ~ funsng , as ~ relsnopg is to ~ relop .  (New usage is discouraged.) $)
    funopsn $p |- ( ( Fun F /\ F = <. X , Y >. )
                              -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) $=
      ( vb vc vd cv cop wcel wex wceq wa csn cvv cpr vex sylib wfun wi mpbir2an
      opm wrel funrel ad2antrr simpr simplr eleqtrrd syl2anc dfop eqeq2i biimpi
      elrel adantl wo wb dfsn2 simpl funeqd mpbid funopg mp3an12i preq2d eqtrid
      elop eqeq2d orbi2d adantr mpbird eqtr3d snex zfpair2 preqsn simpld simprd
      oridm eqtr2d 3eqtr4a sneqd opid sneqi eqtr4di eqtrd sneq id opeq12d spcev
      anbi12d ex exlimdvv mpd expcom exlimiv ax-mp ) GJZBCKZLZGMZAUAZAWRNZOZBDJ
      ZPZNZAXDXDKZPZNZOZDMZUBZWTBQLZCQLZEFGBCUDUCWSXLGXCWSXKXCWSOZWQHJZIJZKZNZI
      MHMZXKXOAUEZWQALXTXAYAXBWSAUFUGXOWQWRAXCWSUHZXAXBWSUIUJHIWQAUOUKXOXSXKHIX
      OXSXKXOXSOZBXPPZNZAXPXPKZPZNZXKYCYDXPXQRZBYCYDYINZYIBNZYCYDYIRZBPZNYJYKOY
      CWQYLYMXSWQYLNZXOXSYNXRYLWQXPXQHSZISULUMUNUPXOWQYMNZXSXOYPYPUQZYPXOYQYPWQ
      BCRZNZUQZXOWSYTYBWQBCGSEFVGTXCYQYTURWSXCYPYSYPXCYMYRWQXCYMBBRYRBUSXCBCBXM
      XNXCWRUAZBCNEFXCXAUUAXAXBUTXCAWRXAXBUHZVAVBBCQQVCVDVEVFZVHVIVJVKYPVRTVJVL
      YDYIBXPYOVMHIVNEVOTZVPYCYJYKUUDVQVSZYCAYMPZYGXCAUUFNWSXSXCWRYMYRRZAUUFBCE
      FULUUBXCUUFYMYMRUUGYMUSXCYMYRYMUUCVEVFVTUGYCUUFYDPZPYGYCYMUUHYCBYDUUEWAWA
      YFUUHXPYOWBWCWDWEXJYEYHODXPYOXDXPNZXFYEXIYHUUIXEYDBXDXPWFVHUUIXHYGAUUIXGY
      FUUIXDXPXDXPUUIWGZUUJWHWAVHWJWIUKWKWLWMWNWOWP $.

    $( An ordered pair is a function iff it is a singleton of an ordered pair.
       (Contributed by AV, 20-Sep-2020.)  A function is a class of ordered
       pairs, so the fact that an ordered pair may sometimes be itself a
       function is an "accident" depending on the specific encoding of ordered
       pairs as classes (in set.mm, the Kuratowski encoding).  A more
       meaningful statement is ~ funsng , as ~ relsnopg is to ~ relop .
       (New usage is discouraged.) $)
    funop $p |- ( Fun <. X , Y >.
                  <-> E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) ) $=
      ( cop wfun cv csn wceq wa wex funopsn mpan2 vex funsn funeq mpbiri adantl
      eqid exlimiv impbii ) ABFZGZACHZIJZUCUEUEFIZJZKZCLZUDUCUCJUJUCTUCABCDEMNU
      IUDCUHUDUFUHUDUGGUEUECOZUKPUCUGQRSUAUB $.
  $}

  $( Composition of a function with domain and a function as a function with
     domain.  Generalization of ~ fnco .  (Contributed by AV, 17-Sep-2024.) $)
  fncofn $p |- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn ( `' G " A ) ) $=
    ( wfn wfun wa ccom ccnv cima cdm fnfun funco funfnd wceq fndm adantr eqcomd
    sylan imaeq2d dmco eqtr4di fneq2d mpbird ) BADZCEZFZBCGZCHZAIZDUGUGJZDUFUGU
    DBEUEUGEABKBCLRMUFUIUJUGUFUIUHBJZIUJUFAUKUHUFUKAUDUKANUEABOPQSBCTUAUBUC $.

  $( Composition of a function with domain and codomain and a function as a
     function with domain and codomain.  Generalization of ~ fco .
     (Contributed by AV, 18-Sep-2024.) $)
  fcof $p |- ( ( F : A --> B /\ Fun G ) -> ( F o. G ) : ( `' G " A ) --> B ) $=
    ( wf wfun wa ccom ccnv cima wfn crn wss df-f wi fncofn adantr rncoss sstr
    ex mpan adantl jctird imp sylanb sylibr ) ABCEZDFZGCDHZDIAJZKZUILZBMZGZUJBU
    IEUGCAKZCLZBMZGZUHUNABCNURUHUNURUHUKUMUOUHUKOUQUOUHUKACDPTQUQUMUOULUPMUQUMC
    DRULUPBSUAUBUCUDUEUJBUINUF $.

  ${
    $d A x $.  $d B x $.  $d G x $.  $d X x $.  $d Y x $.
    funopdmsn.g $e |- G = <. X , Y >. $.
    funopdmsn.x $e |- X e. V $.
    funopdmsn.y $e |- Y e. W $.
    $( The domain of a function which is an ordered pair is a singleton.
       (Contributed by AV, 15-Nov-2021.)  (Avoid depending on this detail.) $)
    funopdmsn $p |- ( ( Fun G /\ A e. dom G /\ B e. dom G ) -> A = B ) $=
      ( vx wfun cdm wcel wceq csn cop wa elexi eleq2 cv wex funeqi funop eqcomi
      wi bitri eqeq1i vex dmsnop eqtrdi anbi12d elsni eqtr3 syl2an biimtrdi syl
      dmeq sylbi adantl exlimiv 3impib ) CLZACMZNZBVDNZABOZVCFKUAZPZOZFGQZVHVHQ
      PZOZRZKUBZVEVFRZVGUFZVCVKLVOCVKHUCFGKFDISGEJSUDUGVNVQKVMVQVJVMCVLOZVQVKCV
      LCVKHUEUHVRVDVIOZVQVRVDVLMVICVLURVHVHKUIUJUKVSVPAVINZBVINZRVGVSVEVTVFWAVD
      VIATVDVIBTULVTAVHOBVHOVGWAAVHUMBVHUMABVHUNUOUPUQUSUTVAUSVB $.
  $}

  $( If ` A ` is not in ` C ` , then the restriction of a singleton of
     ` <. A , B >. ` to ` C ` is null.  (Contributed by Scott Fenton,
     15-Apr-2011.) $)
  ressnop0 $p |- ( -. A e. C -> ( { <. A , B >. } |` C ) = (/) ) $=
    ( wcel wn cop cvv cxp csn cres c0 wceq opelxp1 con3i cin df-res incom eqtri
    disjsn biimpri eqtrid syl ) ACDZEABFZCGHZDZEZUDIZCJZKLUFUCABCGMNUGUIUEUHOZK
    UIUHUEOUJUHCPUHUEQRUJKLUGUEUDSTUAUB $.

  ${
    fpr.1 $e |- A e. _V $.
    fpr.2 $e |- B e. _V $.
    fpr.3 $e |- C e. _V $.
    fpr.4 $e |- D e. _V $.
    $( A function with a domain of two elements.  (Contributed by Jeff Madsen,
       20-Jun-2010.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
    fpr $p |- ( A =/= B
                -> { <. A , C >. , <. B , D >. } : { A , B } --> { C , D } ) $=
      ( wne cop cpr wfn crn wa jctir sylibr csn cun df-pr rnsnop wss wfun funpr
      wf cdm wceq dmprop df-fn rneqi rnun uneq12i eqtr4i 3eqtri eqimssi df-f )
      ABIZACJZBDJZKZABKZLZUSMZCDKZUAZNUTVCUSUDUPVAVDUPUSUBZUSUEUTUFZNVAUPVEVFAB
      CDEFGHUCACBDGHUGOUSUTUHPVBVCVBUQQZURQZRZMVGMZVHMZRZVCUSVIUQURSUIVGVHUJVLC
      QZDQZRVCVJVMVKVNACETBDFTUKCDSULUMUNOUTVCUSUOP $.
  $}

  $( A function with a domain of two elements.  (Contributed by FL,
     2-Feb-2014.) $)
  fprg $p |- ( ( ( A e. E /\ B e. F ) /\ ( C e. G /\ D e. H ) /\ A =/= B )
                -> { <. A , C >. , <. B , D >. } : { A , B } --> { C , D } ) $=
    ( wcel wa wne cop cpr crn wceq csn cun rnsnopg 3ad2ant1 df-pr w3a wfn fnprg
    wss adantr adantl uneq12d rneqi rnun eqtri 3eqtr4g eqimss syl df-f sylanbrc
    wf ) AEIZBFIZJZCGIDHIJZABKZUAZACLZBDLZMZABMZUBVENZCDMZUDZVFVHVEUPABCDEFGHUC
    VBVGVHOVIVBVCPZNZVDPZNZQZCPZDPZQVGVHVBVKVOVMVPUSUTVKVOOZVAUQVQURACERUESUSUT
    VMVPOZVAURVRUQBDFRUFSUGVGVJVLQZNVNVEVSVCVDTUHVJVLUIUJCDTUKVGVHULUMVFVHVEUNU
    O $.

  $( A function with a domain of three elements.  (Contributed by Alexander van
     der Vekens, 4-Dec-2017.) $)
  ftpg $p |- ( ( ( X e. U /\ Y e. V /\ Z e. W )
                 /\ ( A e. F /\ B e. G /\ C e. H )
                 /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) )
               -> { <. X , A >. , <. Y , B >. , <. Z , C >. } :
                  { X , Y , Z } --> { A , B , C } ) $=
    ( wcel w3a wne cpr wf wceq wa wn csn cun cop ctp cin c0 3simpa simp1 syl3an
    fprg eqidd wb simp3 anim12i 3adant3 syl mpbird df-ne wo elpri eqcom orbi12i
    fsng sylib oranim con2i syl2anb 3adant1 3ad2ant3 disjsn sylibr fun syl21anc
    df-tp feq1i feq23i bitri ) JDMZKHMZLIMZNZAEMZBFMZCGMZNZJKOZJLOZKLOZNZNZJKPZ
    LUAZUBZABPZCUAZUBZJAUCZKBUCZPZLCUCZUAZUBZQZJKLUDZABCUDZWQWRWTUDZQZWJWKWNWSQ
    ZWLWOXAQZWKWLUEUFRZXCWAVRVSSWEWBWCSWIWFXHVRVSVTUGWBWCWDUGWFWGWHUHJKABDHEFUJ
    UIWJXIXAXARZWJXAUKWJVTWDSZXIXKULWAWEXLWIWAVTWEWDVRVSVTUMWBWCWDUMUNUOLCIGXAV
    CUPUQWJLWKMZTZXJWIWAXNWEWGWHXNWFWGJLRZTZKLRZTZXNWHJLURKLURXMXPXRSZXMXOXQUSZ
    XSTXMLJRZLKRZUSXTLJKUTYAXOYBXQLJVALKVAVBVDXOXQVEUPVFVGVHVIWKLVJVKWKWLWNWOWS
    XAVLVMXGXDXEXBQXCXDXEXFXBWQWRWTVNVOXDXEWMWPXBJKLVNABCVNVPVQVK $.

  ${
    ftp.a $e |- A e. _V $.
    ftp.b $e |- B e. _V $.
    ftp.c $e |- C e. _V $.
    ftp.d $e |- X e. _V $.
    ftp.e $e |- Y e. _V $.
    ftp.f $e |- Z e. _V $.
    ftp.g $e |- A =/= B $.
    ftp.h $e |- A =/= C $.
    ftp.i $e |- B =/= C $.
    $( A function with a domain of three elements.  (Contributed by Stefan
       O'Rear, 17-Oct-2014.)  (Proof shortened by Alexander van der Vekens,
       23-Jan-2018.) $)
    ftp $p |- { <. A , X >. , <. B , Y >. , <. C , Z >. } : { A , B , C } -->
        { X , Y , Z } $=
      ( cvv wcel w3a wne ctp cop wf 3pm3.2i ftpg mp3an ) APQZBPQZCPQZRDPQZEPQZF
      PQZRABSZACSZBCSZRABCTDEFTADUABEUACFUATUBUFUGUHGHIUCUIUJUKJKLUCULUMUNMNOUC
      DEFPPPPPPABCUDUE $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x F $.
    $( A function restricted to a singleton.  (Contributed by NM,
       9-Oct-2004.) $)
    fnressn $p |- ( ( F Fn A /\ B e. A ) ->
                  ( F |` { B } ) = { <. B , ( F ` B ) >. } ) $=
      ( vx wcel wfn csn cres cfv cop wceq cv wi sneq reseq2d fveq2 opeq12 mpdan
      wa cvv sneqd eqeq12d imbi2d wss vex snss fnssres sylan2b dffn2 fsn2 bitri
      wf snssi sylan2 vsnid funfvex funfni sylancl biantrurd fvres ax-mp opeq2i
      sneqi eqeq2i bitr3di bitrid mpbid expcom vtoclga impcom ) BAECAFZCBGZHZBB
      CIZJZGZKZVKCDLZGZHZVRVRCIZJZGZKZMVKVQMDBAVRBKZWDVQVKWEVTVMWCVPWEVSVLCVRBN
      OWEWBVOWEWAVNKWBVOKVRBCPVRWABVNQRUAUBUCVKVRAEZWDVKWFSZVTVSFZWDWFVKVSAUDZW
      HVRADUEZUFAVSCUGZUHWHVRVTIZTEZVTVRWLJZGZKZSZWGWDWHVSTVTULWQVSVTUIVRTVTWJU
      JUKWGWPWQWDWGWMWPWGWHVRVSEZWMWFVKWIWHVRAUMWKUNDUOZWMVSVRVTVRVTUPUQURUSWOW
      CVTWNWBWLWAVRWRWLWAKWSVRVSCUTVAVBVCVDVEVFVGVHVIVJ $.

    $( The value of a function restricted to a singleton.  (Contributed by NM,
       9-Oct-2004.) $)
    fressnfv $p |- ( ( F Fn A /\ B e. A ) ->
                 ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) ) $=
      ( vx wcel wfn csn cres wf cfv wb cv wi wceq sneq reseq2 syl wa cop eleq1d
      feq1d feq2 bitrd fveq2 bibi12d imbi2d fnressn vsnid fvres ax-mp sneqi vex
      opeq2i eqeq2i fsn2 iba eleq1i bitr3di bitrid sylbir expcom vtoclga impcom
      ) BAFDAGZBHZCDVFIZJZBDKZCFZLZVEEMZHZCDVMIZJZVLDKZCFZLZNVEVKNEBAVLBOZVRVKV
      EVSVOVHVQVJVSVMVFOZVOVHLVLBPVTVOVMCVGJVHVTVMCVNVGVMVFDQUBVMVFCVGUCUDRVSVP
      VICVLBDUEUAUFUGVEVLAFZVRVEWASVNVLVPTZHZOZVRAVLDUHWDVNVLVLVNKZTZHZOZVRWGWC
      VNWFWBWEVPVLVLVMFWEVPOEUIVLVMDUJUKZUNULUOVOWECFZWHSZWHVQVLCVNEUMUPWHWJWKV
      QWHWJUQWEVPCWIURUSUTVARVBVCVD $.
  $}

  $( The value of a constant function.  (Contributed by NM, 30-May-1999.) $)
  fvconst $p |- ( ( F : A --> { B } /\ C e. A ) -> ( F ` C ) = B ) $=
    ( csn wf wcel wa cfv wceq ffvelcdm elsni syl ) ABEZDFCAGHCDIZNGOBJANCDKOBLM
    $.

  ${
    $d x A $.  $d x B $.
    $( Express a singleton function in maps-to notation.  (Contributed by NM,
       6-Jun-2006.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.)  (Revised
       by Stefan O'Rear, 28-Feb-2015.) $)
    fmptsn $p |- ( ( A e. V /\ B e. W ) ->
        { <. A , B >. } = ( x e. { A } |-> B ) ) $=
      ( wcel wa csn cxp cop cmpt xpsng fconstmpt eqtr3di ) BDFCEFGBHZCHIBCJHAOC
      KBCDELAOCMN $.
  $}

  ${
    $d x A $.  $d x B $.  $d x R $.  $d x S $.
    fmptap.0a $e |- A e. _V $.
    fmptap.0b $e |- B e. _V $.
    fmptap.1 $e |- ( R u. { A } ) = S $.
    fmptap.2 $e |- ( x = A -> C = B ) $.
    $( Append an additional value to a function.  (Contributed by NM,
       6-Jun-2006.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    fmptap $p |- ( ( x e. R |-> C ) u. { <. A , B >. } ) = ( x e. S |-> C ) $=
      ( cmpt cop csn cun cvv wcel wceq fmptsn mp2an cv elsni syl mpteq2ia mptun
      eqtr4i uneq2i mpteq1 ax-mp 3eqtr2i ) AEDKZBCLMZNUJABMZDKZNAEULNZDKZAFDKZU
      KUMUJUKAULCKZUMBOPCOPUKUQQGHABCOORSAULDCATZULPURBQDCQURBUAJUBUCUEUFAEULDU
      DUNFQUOUPQIAUNFDUGUHUI $.
  $}

  ${
    $d x A $.  $d x B $.  $d x R $.  $d x S $.  $d x ph $.
    fmptapd.0a $e |- ( ph -> A e. _V ) $.
    fmptapd.0b $e |- ( ph -> B e. _V ) $.
    fmptapd.1 $e  |- ( ph -> ( R u. { A } ) = S ) $.
    fmptapd.2 $e  |- ( ( ph /\ x = A ) -> C = B ) $.
    $( Append an additional value to a function.  (Contributed by Thierry
       Arnoux, 3-Jan-2017.) $)
    fmptapd $p    |- ( ph ->
      ( ( x e. R |-> C ) u. { <. A , B >. } ) = ( x e. S |-> C ) ) $=
      ( cmpt cop csn cun cvv wcel wceq fmptsn syl2anc cv elsni sylan2 mpteq2dva
      eqtr4d uneq2d mptun a1i mpteq1d 3eqtr2d ) ABFELZCDMNZOUKBCNZELZOZBFUMOZEL
      ZBGELAULUNUKAULBUMDLZUNACPQDPQULURRHIBCDPPSTABUMEDBUAZUMQAUSCREDRUSCUBKUC
      UDUEUFUQUORABFUMEUGUHABUPGEJUIUJ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x D $.  $d x ph $.
    fmptpr.1 $e |- ( ph -> A e. V ) $.
    fmptpr.2 $e |- ( ph -> B e. W ) $.
    fmptpr.3 $e |- ( ph -> C e. X ) $.
    fmptpr.4 $e |- ( ph -> D e. Y ) $.
    fmptpr.5 $e |- ( ( ph /\ x = A ) -> E = C ) $.
    fmptpr.6 $e |- ( ( ph /\ x = B ) -> E = D ) $.
    $( Express a pair function in maps-to notation.  (Contributed by Thierry
       Arnoux, 3-Jan-2017.) $)
    fmptpr $p |- ( ph ->
      { <. A , C >. , <. B , D >. } = ( x e. { A , B } |-> E ) ) $=
      ( cun c0 wcel cop cpr csn cmpt wceq df-pr a1i uneq1i uncom un0 3eqtri cvv
      mpt0 elex syl eqtr3i fmptapd eqtr3id uneq1d eqcomi 3eqtrd ) ACEUAZDFUAZUB
      ZVBUCZVCUCZRZBCUCZGUDZVFRBCDUBZGUDVDVGUEAVBVCUFUGAVEVIVFAVEBSGUDZVERZVIVL
      SVERVESRVEVKSVEBGUMUHSVEUIVEUJUKABCEGSVHACHTCULTLCHUNUOAEJTEULTNEJUNUOSVH
      RZVHUEAVHSRVMVHVHSUIVHUJUPUGPUQURUSABDFGVHVJADITDULTMDIUNUOAFKTFULTOFKUNU
      OVHDUCRZVJUEAVJVNCDUFUTUGQUQVA $.
  $}

  $( The value of a restricted identity function.  (Contributed by NM,
     19-May-2004.) $)
  fvresi $p |- ( B e. A -> ( ( _I |` A ) ` B ) = B ) $=
    ( wcel cid cres cfv fvres fvi eqtrd ) BACBDAEFBDFBBADGBAHI $.

  $( Remove an ordered pair not participating in a function value.
     (Contributed by Jim Kingdon, 7-Jan-2019.) $)
  fvunsng $p |- ( ( D e. V /\ B =/= D )
       -> ( ( A u. { <. B , C >. } ) ` D ) = ( A ` D ) ) $=
    ( wcel wne wa cop csn cun cfv cres wceq snidg fvres syl resundir c0 wn un0
    elsni necon3ai ressnop0 uneq2d eqtrdi eqtrid fveq1d sylan9req adantr eqtrd
    ) DEFZBDGZHDABCIJZKZLZDADJZMZLZDALZULUMUPDUOUQMZLZUSULDUQFZVBUPNDEOZDUQUOPQ
    UMDVAURUMVAURUNUQMZKZURAUNUQRUMVFURSKURUMVESURUMBUQFZTVESNVGBDBDUBUCBCUQUDQ
    UEURUAUFUGUHUIULUSUTNZUMULVCVHVDDUQAPQUJUK $.

  ${
    fvsn.1 $e |- A e. _V $.
    fvsn.2 $e |- B e. _V $.
    $( The value of a singleton of an ordered pair is the second member.
       (Contributed by NM, 12-Aug-1994.) $)
    fvsn $p |- ( { <. A , B >. } ` A ) = B $=
      ( cop csn wfun wcel cfv wceq funsn opex snid funopfv mp2 ) ABEZFZGPQHAQIB
      JABCDKPABCDLMABQNO $.
  $}

  ${
    $d A a b $.  $d B b $.
    $( The value of a singleton of an ordered pair is the second member.
       (Contributed by NM, 26-Oct-2012.) $)
    fvsng $p |- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B ) $=
      ( va vb cv cop csn cfv opeq1 sneqd id fveq12d eqeq1d opeq2 fveq1d eqeq12d
      wceq vex fvsn vtocl2g ) EGZUCFGZHZIZJZUDSAAUDHZIZJZUDSAABHZIZJZBSEFABCDUC
      ASZUGUJUDUNUCAUFUIUNUEUHUCAUDKLUNMNOUDBSZUJUMUDBUOAUIULUOUHUKUDBAPLQUOMRU
      CUDETFTUAUB $.
  $}

  ${
    fvsnun.1 $e |- A e. _V $.
    fvsnun.2 $e |- B e. _V $.
    fvsnun.3 $e |- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) $.
    $( The value of a function with one of its ordered pairs replaced, at the
       replaced ordered pair.  See also ~ fvsnun2 .  (Contributed by NM,
       23-Sep-2007.) $)
    fvsnun1 $p |- ( G ` A ) = B $=
      ( csn cres cfv cop cdif cun c0 cin wceq eqtri ax-mp fvres reseq1i disjdif
      resundir incom resdisj uneq2i un0 fveq1i wcel snid fvsn 3eqtr3i ) AEAIZJZ
      KZAABLIZUMJZKZAEKZBAUNUQUNUPDCUMMZJZNZUMJZUQEVBUMHUAVCUQVAUMJZNZUQUPVAUMU
      CVEUQONUQVDOUQUTUMPZOQVDOQVFUMUTPOUTUMUDUMCUBRUTUMDUESUFUQUGRRRUHAUMUIZUO
      USQAFUJZAUMETSURAUPKZBVGURVIQVHAUMUPTSABFGUKRUL $.

    $( The value of a function with one of its ordered pairs replaced, at
       arguments other than the replaced one.  See also ~ fvsnun1 .
       (Contributed by NM, 23-Sep-2007.) $)
    fvsnun2 $p |- ( D e. ( C \ { A } ) -> ( G ` D ) = ( F ` D ) ) $=
      ( csn cdif wcel cres cfv cop cun c0 wceq 3eqtri fvres reseq1i cin disjdif
      resundir wfn wb fnsn fnresdisj ax-mp mpbi residm uneq12i uncom un0 fveq1i
      3eqtr3a ) DCAJZKZLDFURMZNDEURMZNDFNDENDUSUTUSABOJZUTPZURMVAURMZUTURMZPZUT
      FVBURIUAVAUTURUDVEQUTPUTQPUTVCQVDUTUQURUBQRZVCQRZUQCUCVAUQUEVFVGUFABGHUGU
      QURVAUHUIUJEURUKULQUTUMUTUNSSUODURFTDURETUP $.
  $}

  $( Split a function into a single point and all the rest.  (Contributed by
     Stefan O'Rear, 27-Feb-2015.)  (Revised by Jim Kingdon, 20-Jan-2023.) $)
  fnsnsplitss $p |- ( ( F Fn A /\ X e. A ) ->
      ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F ) $=
    ( wfn wcel csn cdif cun cres cfv cop wss difsnss adantl syl resundi fnressn
    wa ssres2 uneq2d eqtrid wceq fnresdm adantr 3sstr3d ) BADZCAEZRZBACFZGZUIHZ
    IZBAIZBUJIZCCBJKFZHZBUHUKALZULUMLUGUQUFACMNUKABSOUHULUNBUIIZHUPBUJUIPUHURUO
    UNACBQTUAUFUMBUBUGABUCUDUE $.

  $( Adjoining a point to a function gives a function.  (Contributed by Stefan
     O'Rear, 28-Feb-2015.) $)
  fsnunf $p |- ( ( F : S --> T /\ ( X e. V /\ -. X e. S ) /\ Y e. T ) ->
      ( F u. { <. X , Y >. } ) : ( S u. { X } ) --> T ) $=
    ( wf wcel wn wa w3a csn cun cop cin c0 wceq simp1 wf1o syl simp2l simp3 fun
    f1osng syl2anc f1of simp2r disjsn sylibr syl21anc wb snssi 3ad2ant3 ssequn2
    wss sylib feq3 mpbid ) ABCGZEDHZEAHIZJZFBHZKZAELZMZBFLZMZCEFNLZMZGZVFBVJGZV
    DUSVEVGVIGZAVEOPQZVKUSVBVCRVDVEVGVISZVMVDUTVCVOUSUTVAVCUAUSVBVCUBEFDBUDUEVE
    VGVIUFTVDVAVNUSUTVAVCUGAEUHUIAVEBVGCVIUCUJVDVHBQZVKVLUKVDVGBUOZVPVCUSVQVBFB
    ULUMVGBUNUPVHBVFVJUQTUR $.

  $( Recover the added point from a point-added function.  (Contributed by
     Stefan O'Rear, 28-Feb-2015.)  (Revised by NM, 18-May-2017.) $)
  fsnunfv $p |- ( ( X e. V /\ Y e. W /\ -. X e. dom F ) ->
      ( ( F u. { <. X , Y >. } ) ` X ) = Y ) $=
    ( wcel cdm wn w3a cop csn cun cres cfv c0 wceq cin dmres 3adant3 syl disjsn
    incom eqtri biimpri eqtrid 3ad2ant3 wb relres reldm0 ax-mp sylibr wfn fnsng
    wrel fnresdm uneq12d resundir uncom un0 eqtr2i 3eqtr4g snidg 3ad2ant1 fvres
    fveq1d fvsng 3eqtr3d ) DBFZECFZDAGZFHZIZDADEJKZLZDKZMZNZDVMNZDVNNZEVLDVPVMV
    LAVOMZVMVOMZLOVMLZVPVMVLVTOWAVMVLVTGZOPZVTOPZVKVHWDVIVKWCVJVOQZOWCVOVJQWFAV
    ORVOVJUBUCWFOPVKVJDUAUDUEUFVTUNWEWDUGAVOUHVTUIUJUKVLVMVOULZWAVMPVHVIWGVKDEB
    CUMSVOVMUOTUPAVMVOUQWBVMOLVMOVMURVMUSUTVAVEVLDVOFZVQVSPVHVIWHVKDBVBVCDVOVNV
    DTVHVIVREPVKDEBCVFSVG $.

  $( Recover the original function from a point-added function.  (Contributed
     by Stefan O'Rear, 28-Feb-2015.) $)
  fsnunres $p |- ( ( F Fn S /\ -. X e. S ) ->
      ( ( F u. { <. X , Y >. } ) |` S ) = F ) $=
    ( wfn wcel wn wa cres cop csn cun c0 fnresdm adantr ressnop0 adantl uneq12d
    wceq resundir un0 eqcomi 3eqtr4g ) BAEZCAFGZHZBAIZCDJKZAIZLBMLZBUHLAIBUFUGB
    UIMUDUGBSUEABNOUEUIMSUDCDAPQRBUHATUJBBUAUBUC $.

  $( Restricting a function to a domain without one element of the domain of
     the function, and adding a pair of this element and the function value of
     the element results in a subset of the function itself.  (Contributed by
     AV, 2-Dec-2018.)  (Revised by Jim Kingdon, 21-Jan-2023.) $)
  funresdfunsnss $p |- ( ( Fun F /\ X e. dom F ) ->
      ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F ) $=
    ( wfun cdm wcel wa cvv csn cdif cres cfv cop cun wceq wrel funrel resdmdfsn
    syl adantr uneq1d wfn wss funfn fnsnsplitss sylanb eqsstrd ) ACZBADZEZFZAGB
    HZIJZBBAKLHZMAUHUKIJZUMMZAUJULUNUMUGULUNNZUIUGAOUPAPABQRSTUGAUHUAUIUOAUBAUC
    UHABUDUEUF $.

  ${
    fvpr1.1 $e |- A e. _V $.
    fvpr1.2 $e |- C e. _V $.
    $( The value of a function with a domain of two elements.  (Contributed by
       Jeff Madsen, 20-Jun-2010.) $)
    fvpr1 $p |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = C ) $=
      ( wne cop cpr cfv csn cun df-pr fveq1i wceq necom cvv wcel fvunsng mpan
      sylbi eqtrid fvsn eqtrdi ) ABGZAACHZBDHZIZJZAUFKZJZCUEUIAUJUGKLZJZUKAUHUL
      UFUGMNUEBAGZUMUKOZABPAQRUNUOEUJBDAQSTUAUBACEFUCUD $.
  $}

  ${
    fvpr2.1 $e |- B e. _V $.
    fvpr2.2 $e |- D e. _V $.
    $( The value of a function with a domain of two elements.  (Contributed by
       Jeff Madsen, 20-Jun-2010.) $)
    fvpr2 $p |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` B ) = D ) $=
      ( wne cop cpr cfv prcom fveq1i wceq necom fvpr1 sylbi eqtrid ) ABGZBACHZB
      DHZIZJBTSIZJZDBUAUBSTKLRBAGUCDMABNBADCEFOPQ $.
  $}

  $( The value of a function with a domain of (at most) two elements.
     (Contributed by Alexander van der Vekens, 3-Dec-2017.) $)
  fvpr1g $p |- ( ( A e. V /\ C e. W /\ A =/= B )
                 -> ( { <. A , C >. , <. B , D >. } ` A ) = C ) $=
    ( wcel wne w3a cop cpr cfv csn wceq wa cun df-pr fveq1i necom fvunsng fvsng
    sylan2b eqtrid 3adant2 3adant3 eqtrd ) AEGZCFGZABHZIAACJZBDJZKZLZAUJMZLZCUG
    UIUMUONUHUGUIOUMAUNUKMPZLZUOAULUPUJUKQRUIUGBAHUQUONABSUNBDAETUBUCUDUGUHUOCN
    UIACEFUAUEUF $.

  $( The value of a function with a domain of (at most) two elements.
     (Contributed by Alexander van der Vekens, 3-Dec-2017.) $)
  fvpr2g $p |- ( ( B e. V /\ D e. W /\ A =/= B )
                 -> ( { <. A , C >. , <. B , D >. } ` B ) = D ) $=
    ( wcel wne w3a cop cpr cfv csn wceq wa cun prcom df-pr eqtri fveq1i fvunsng
    eqtrid 3adant2 fvsng 3adant3 eqtrd ) BEGZDFGZABHZIBACJZBDJZKZLZBUKMZLZDUGUI
    UMUONUHUGUIOUMBUNUJMPZLUOBULUPULUKUJKUPUJUKQUKUJRSTUNACBEUAUBUCUGUHUODNUIBD
    EFUDUEUF $.

  $( The value of a function with a domain of (at most) three elements.
     (Contributed by Alexander van der Vekens, 4-Dec-2017.) $)
  fvtp1g $p |- ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) )
                -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D ) $=
    ( wcel wa wne cop ctp cfv cpr csn cun df-tp fveq1i wceq necom fvunsng 3expa
    sylan2b ad2ant2rl fvpr1g adantrr eqtrd eqtrid ) AGIZDHIZJZABKZACKZJJZAADLZB
    ELZCFLZMZNAUPUQOZURPQZNZDAUSVAUPUQURRSUOVBAUTNZDUJUNVBVCTZUKUMUNUJCAKVDACUA
    UTCFAGUBUDUEULUMVCDTZUNUJUKUMVEABDEGHUFUCUGUHUI $.

  $( The value of a function with a domain of (at most) three elements.
     (Contributed by Alexander van der Vekens, 4-Dec-2017.) $)
  fvtp2g $p |- ( ( ( B e. V /\ E e. W ) /\ ( A =/= B /\ B =/= C ) )
                -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E ) $=
    ( wcel wa wne cop ctp cfv tprot fveq1i wceq wi necom fvtp1g expcom ancoms
    sylanb impcom eqtrid ) BGIEHIJZABKZBCKZJZJBADLZBELZCFLZMZNBUKULUJMZNZEBUMUN
    UJUKULOPUIUFUOEQZUGBAKZUHUFUPRZABSUHUQURUFUHUQJUPBCAEFDGHTUAUBUCUDUE $.

  $( The value of a function with a domain of (at most) three elements.
     (Contributed by Alexander van der Vekens, 4-Dec-2017.) $)
  fvtp3g $p |- ( ( ( C e. V /\ F e. W ) /\ ( A =/= C /\ B =/= C ) )
                -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` C ) = F ) $=
    ( wcel wa wne cop ctp cfv tprot fveq1i wceq wi necom fvtp2g expcom sylan2b
    ancoms impcom eqtrid ) CGIFHIJZACKZBCKZJZJCADLZBELZCFLZMZNCUKULUJMZNZFCUMUN
    UJUKULOPUIUFUOFQZUHUGUFUPRZUGUHCAKZUQACSUFUHURJUPBCAEFDGHTUAUBUCUDUE $.

  ${
    fvtp1.1 $e |- A e. _V $.
    fvtp1.4 $e |- D e. _V $.
    $( The first value of a function with a domain of three elements.
       (Contributed by NM, 14-Sep-2011.) $)
    fvtp1 $p |- ( ( A =/= B /\ A =/= C )
             -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D ) $=
      ( cvv wcel wne wa cop ctp cfv wceq fvtp1g mpanl12 ) AIJDIJABKACKLAADMBEMC
      FMNODPGHABCDEFIIQR $.
  $}

  ${
    fvtp2.1 $e |- B e. _V $.
    fvtp2.4 $e |- E e. _V $.
    $( The second value of a function with a domain of three elements.
       (Contributed by NM, 14-Sep-2011.) $)
    fvtp2 $p |- ( ( A =/= B /\ B =/= C )
             -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E ) $=
      ( wne wa cop ctp cfv tprot fveq1i wceq necom fvtp1 ancoms sylanb eqtrid )
      ABIZBCIZJBADKZBEKZCFKZLZMBUEUFUDLZMZEBUGUHUDUEUFNOUBBAIZUCUIEPZABQUCUJUKB
      CAEFDGHRSTUA $.
  $}

  ${
    fvtp3.1 $e |- C e. _V $.
    fvtp3.4 $e |- F e. _V $.
    $( The third value of a function with a domain of three elements.
       (Contributed by NM, 14-Sep-2011.) $)
    fvtp3 $p |- ( ( A =/= C /\ B =/= C )
             -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` C ) = F ) $=
      ( wne wa cop ctp cfv tprot fveq1i wceq necom fvtp2 sylan2b ancoms eqtrid
      ) ACIZBCIZJCADKZBEKZCFKZLZMCUEUFUDLZMZFCUGUHUDUEUFNOUCUBUIFPZUBUCCAIUJACQ
      BCAEFDGHRSTUA $.
  $}

  $( The value of a constant function.  (Contributed by NM, 20-Aug-2005.) $)
  fvconst2g $p |- ( ( B e. D /\ C e. A ) -> ( ( A X. { B } ) ` C ) = B ) $=
    ( wcel csn cxp wf cfv wceq fconstg fvconst sylan ) BDEABFZANGZHCAECOIBJABDK
    ABCOLM $.

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x F $.
    $( A constant function expressed as a cross product.  (Contributed by NM,
       27-Nov-2007.) $)
    fconst2g $p |- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) ) $=
      ( vx wcel csn wf cxp wceq wa cv cfv wral fvconst adantlr fvconst2g eqtr4d
      adantll wfn ralrimiva wb ffn fnconstg eqfnfv syl2an mpbird expcom fconstg
      feq1 syl5ibrcom impbid ) BCFZABGZDHZDAUNIZJZUOUMUQUOUMKZUQELZDMZUSUPMZJZE
      ANZURVBEAURUSAFZKUTBVAUOVDUTBJUMABUSDOPUMVDVABJUOABUSCQSRUAUODATUPATUQVCU
      BUMAUNDUCABCUDEADUPUEUFUGUHUMUOUQAUNUPHABCUIAUNDUPUJUKUL $.
  $}

  ${
    fvconst2.1 $e |- B e. _V $.
    $( The value of a constant function.  (Contributed by NM, 16-Apr-2005.) $)
    fvconst2 $p |- ( C e. A -> ( ( A X. { B } ) ` C ) = B ) $=
      ( cvv wcel csn cxp cfv wceq fvconst2g mpan ) BEFCAFCABGHIBJDABCEKL $.

    $( A constant function expressed as a cross product.  (Contributed by NM,
       20-Aug-1999.) $)
    fconst2 $p |- ( F : A --> { B } <-> F = ( A X. { B } ) ) $=
      ( cvv wcel csn wf cxp wceq wb fconst2g ax-mp ) BEFABGZCHCANIJKDABECLM $.
  $}

  ${
    $d x w z A $.  $d x w z B $.  $d x w z F $.  $d y w A $.
    $( A constant function expressed in terms of its functionality, domain, and
       value.  See also ~ fconst2 .  (Contributed by Jim Kingdon,
       8-Jan-2019.) $)
    fconstfvm $p |- ( E. y y e. A -> (
        F : A --> { B } <->
          ( F Fn A /\ A. x e. A ( F ` x ) = B ) ) ) $=
      ( vw vz cv wcel wex csn wf wfn cfv wceq wral wa wrex eqeq1d sylan9bbr ffn
      fvconst ralrimiva jca fvelrnb fveq2 rspccva rexbidva r19.9rmv velsn eqcom
      crn bicomd bitr2i bitrdi eqrdv an32s exp31 imdistand wfo df-fo fof sylbir
      syl6 impbid2 ) BHCIBJZCDKZELZECMZAHZENZDOZACPZQZVHVIVMCVGEUAVHVLACCDVJEUB
      UCUDVFVNVIEULZVGOZQZVHVFVIVMVPVFVIVMVPVFVMVIVPVFVMQZVIQZFVOVGVSFHZVOIZDVT
      OZVTVGIZVIWAGHZENZVTOZGCRZVRWBGCVTEUEVMWGWBGCRZVFWBVMWFWBGCVMWDCIQWEDVTVL
      WEDOAWDCVJWDOVKWEDVJWDEUFSUGSUHVFWBWHWBGBCUIUMTTWCVTDOWBFDUJVTDUKUNUOUPUQ
      URUSVQCVGEUTVHCVGEVACVGEVBVCVDVE $.
  $}

  ${
    $d x A $.  $d y B $.  $d y F $.  $d y A $.  $d y x $.
    $( Two ways to express a constant function.  (Contributed by Jim Kingdon,
       8-Jan-2019.) $)
    fconst3m $p |- ( E. x x e. A -> (
        F : A --> { B } <->
          ( F Fn A /\ A C_ ( `' F " { B } ) ) ) ) $=
      ( vy cv wcel wex csn wf wfn cfv wceq wral wa ccnv cima wss fconstfvm wfun
      cdm wb fnfun fndm eqimss2 syl funconstss syl2anc pm5.32i bitrdi ) AFBGAHB
      CIZDJDBKZEFDLCMEBNZOULBDPUKQRZOEABCDSULUMUNULDTBDUAZRZUMUNUBBDUCULUOBMUPB
      DUDBUOUEUFEBCDUGUHUIUJ $.
  $}

  ${
    $d x A $.
    $( Two ways to express a constant function.  (Contributed by NM,
       8-Mar-2007.) $)
    fconst4m $p |- ( E. x x e. A -> (
        F : A --> { B } <->
          ( F Fn A /\ ( `' F " { B } ) = A ) ) ) $=
      ( cv wcel wex csn wf wfn ccnv cima wss wa wceq fconst3m cdm cnvimass fndm
      sseqtrid biantrurd eqss bitr4di pm5.32i bitrdi ) AEBFAGBCHZDIDBJZBDKUFLZM
      ZNUGUHBOZNABCDPUGUIUJUGUIUHBMZUINUJUGUKUIUGDQUHBDUFRBDSTUAUHBUBUCUDUE $.
  $}

  ${
    $d x A $.  $d x B $.
    $( The restriction of a function to a set exists.  Compare Proposition 6.17
       of [TakeutiZaring] p. 28.  (Contributed by NM, 7-Apr-1995.)  (Revised by
       Mario Carneiro, 22-Jun-2013.) $)
    resfunexg $p |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V ) $=
      ( vx wfun wcel cres cdm cfv cmpt cima cvv wceq wral ralrimiva 3syl adantr
      wa cv sylancr cop crn funres funfvex fnasrng wfn funfn dffn5im syl dmmptg
      sylib vex opexg imaeq2d imadmrn eqtr3di 3eqtr4d funmpt dmresexg funimaexg
      adantl eqeltrd ) AEZBCFZRZABGZDVFHZDSZVHVFIZUAZJZVGKZLVEDVGVIJZVKUBZVFVLV
      CVMVNMZVDVCVFEZVILFZDVGNVOBAUCZVPVQDVGVHVFUDZODVGVILUEPQVEVFVGUFZVFVMMVEV
      PVTVCVPVDVRQVFUGUKDVGVFUHUIVCVLVNMVDVCVKVKHZKVLVNVCWAVGVKVCVPVJLFZDVGNWAV
      GMVRVPWBDVGVPVHVGFRVHLFVQWBDULVSVHVILLUMTODVGVJLUJPUNVKUOUPQUQVEVKEVGLFZV
      LLFDVGVJURVDWCVCABCUSVAVKVGLUTTVB $.
  $}

  $( If the domain of a function is a set, the function is a set.  Theorem
     6.16(1) of [TakeutiZaring] p. 28.  This theorem is derived using the Axiom
     of Replacement in the form of ~ resfunexg .  (Contributed by NM,
     14-Aug-1994.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
  fnex $p |- ( ( F Fn A /\ A e. B ) -> F e. _V ) $=
    ( wfn wcel wrel cdm cres cvv fnrel adantr wfun wceq eleq1a impcom resfunexg
    wa df-fn sylan2 anassrs sylanb resdm eleq1d biimpa syl2anc ) CADZABEZQCFZCC
    GZHZIEZCIEZUFUHUGACJKUFCLZUIAMZQUGUKCARUMUNUGUKUNUGQUMUIBEZUKUGUNUOABUINOCU
    IBPSTUAUHUKULUHUJCICUBUCUDUE $.

  $( The domain of a function is a set iff the function is a set.  (Contributed
     by AV, 8-Aug-2024.) $)
  fndmexb $p |- ( F Fn A -> ( A e. _V <-> F e. _V ) ) $=
    ( wfn cvv wcel fnex ex wa simpr simpl fndmexd impbid ) BACZADEZBDEZMNOADBFG
    MONMOHABDMOIMOJKGL $.

  $( The domain of a function is a set iff the function is a set.  (Contributed
     by AV, 8-Aug-2024.) $)
  fdmexb $p |- ( F : A --> B -> ( A e. _V <-> F e. _V ) ) $=
    ( wf wfn cvv wcel wb ffn fndmexb syl ) ABCDCAEAFGCFGHABCIACJK $.

  $( If the domain of a function exists, so does the function.  Part of Theorem
     4.15(v) of [Monk1] p. 46.  This theorem is derived using the Axiom of
     Replacement in the form of ~ fnex .  (Note:  Any resemblance between
     F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by
     Swedish chefs.)  (Contributed by NM, 11-Nov-1995.) $)
  funex $p |- ( ( Fun F /\ dom F e. B ) -> F e. _V ) $=
    ( wfun cdm wfn wcel cvv funfn fnex sylanb ) BCBBDZEKAFBGFBHKABIJ $.

  ${
    $d x y A $.
    opabex.1 $e |- A e. _V $.
    opabex.2 $e |- ( x e. A -> E* y ph ) $.
    $( Existence of a function expressed as class of ordered pairs.
       (Contributed by NM, 21-Jul-1996.) $)
    opabex $p |- { <. x , y >. | ( x e. A /\ ph ) } e. _V $=
      ( cv wcel wa copab wfun cdm cvv wmo funopab moanimv mpbir mpgbir dmopabss
      wi ssexi funex mp2an ) BGDHZAIZBCJZKZUFLZMHUFMHUGUECNZBUEBCOUIUDACNTFUDAC
      PQRUHDEABCDSUAMUFUBUC $.
  $}

  ${
    $d x A $.
    $( If the domain of a function given by maps-to notation is a set, the
       function is a set.  (Contributed by FL, 6-Jun-2011.)  (Revised by Mario
       Carneiro, 31-Aug-2015.) $)
    mptexg $p |- ( A e. V -> ( x e. A |-> B ) e. _V ) $=
      ( wcel cmpt wfun cdm cvv funmpt wss eqid dmmptss ssexg mpan funex sylancr
      ) BDEZABCFZGSHZIEZSIEABCJTBKRUAABCSSLMTBDNOISPQ $.
  $}

  ${
    $d x A $.
    mptex.1 $e |- A e. _V $.
    $( If the domain of a function given by maps-to notation is a set, the
       function is a set.  (Contributed by NM, 22-Apr-2005.)  (Revised by Mario
       Carneiro, 20-Dec-2013.) $)
    mptex $p |- ( x e. A |-> B ) e. _V $=
      ( cvv wcel cmpt mptexg ax-mp ) BEFABCGEFDABCEHI $.
  $}

  ${
    $d A x $.
    mptexd.1 $e |- ( ph -> A e. V ) $.
    $( If the domain of a function given by maps-to notation is a set, the
       function is a set.  Deduction version of ~ mptexg .  (Contributed by
       Glauco Siliprandi, 24-Dec-2020.) $)
    mptexd $p |- ( ph -> ( x e. A |-> B ) e. _V ) $=
      ( wcel cmpt cvv mptexg syl ) ACEGBCDHIGFBCDEJK $.
  $}

  ${
    $d A w x z $.  $d A x y $.  $d B w z $.  $d C w $.
    $( If a function given by maps-to notation is inhabited, then the class it
       is defined on is inhabited.  (Contributed by Jim Kingdon,
       22-Jul-2026.) $)
    mptmex $p |- ( C e. ( x e. A |-> B ) -> E. y y e. A ) $=
      ( vz vw cmpt wcel cv wex wceq copab elex2 opabm sylib df-mpt eleq2s simpl
      wa exlimiv eximi syl eleq1w cbvexv ) EACDHZIZAJCIZAKZBJCIZBKUGUHFJDLZTZFK
      ZAKZUIUNEULAFMZUFEUOIGJUOIGKUNGEUONULAFGOPAFCDQRUMUHAULUHFUHUKSUAUBUCUHUJ
      ABABCUDUEP $.
  $}

  ${
    $d x y A $.  $d x ph $.
    mptrabex.1 $e |- A e. _V $.
    $( If the domain of a function given by maps-to notation is a class
       abstraction based on a set, the function is a set.  (Contributed by AV,
       16-Jul-2019.)  (Revised by AV, 26-Mar-2021.) $)
    mptrabex $p |- ( x e. { y e. A | ph } |-> B ) e. _V $=
      ( crab rabex mptex ) BACDGEACDFHI $.
  $}

  $( If the domain of a mapping is a set, the function is a set.  (Contributed
     by NM, 3-Oct-1999.) $)
  fex $p |- ( ( F : A --> B /\ A e. C ) -> F e. _V ) $=
    ( wf wfn wcel cvv ffn fnex sylan ) ABDEDAFACGDHGABDIACDJK $.

  ${
    fexd.1 $e |- ( ph -> F : A --> B ) $.
    fexd.2 $e |- ( ph -> A e. C ) $.
    $( If the domain of a mapping is a set, the function is a set.
       (Contributed by Glauco Siliprandi, 26-Jun-2021.) $)
    fexd $p |- ( ph -> F e. _V ) $=
      ( wf wcel cvv fex syl2anc ) ABCEHBDIEJIFGBCDEKL $.
  $}

  ${
    $d f x y A $.  $d f y B $.
    eufnfv.1 $e |- A e. _V $.
    eufnfv.2 $e |- B e. _V $.
    $( A function is uniquely determined by its values.  (Contributed by NM,
       31-Aug-2011.) $)
    eufnfv $p |- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B ) $=
      ( vy cv wfn cfv wceq wral wa weu wb wal wex cmpt mptex cvv eqeq2 pm4.71ri
      bibi2d albidv spcev eqid fnmpti mpbiri dffn5im eqeq1d wcel funfvex funfni
      fneq1 ralrimiva mpteqb syl bitrd pm5.32i bitr2i mpg df-eu mpbir ) DHZBIZA
      HZVDJZCKABLZMZDNVIVDGHZKZOZDPZGQZVIVDABCRZKZOZVNDVMVQDPGVOABCESVJVOKZVLVQ
      DVRVKVPVIVJVOVDUAUCUDUEVPVEVPMVIVPVEVPVEVOBIABCVOFVOUFUGBVDVOUNUHUBVEVPVH
      VEVPABVGRZVOKZVHVEVDVSVOABVDUIUJVEVGTUKZABLVTVHOVEWAABWABVFVDVFVDULUMUOAB
      VGCTUPUQURUSUTVAVIDGVBVC $.
  $}

  $( A function's value in a preimage belongs to the image.  (Contributed by
     NM, 23-Sep-2003.) $)
  funfvima $p |- ( ( Fun F /\ B e. dom F ) -> ( B e. A ->
                 ( F ` B ) e. ( F " A ) ) ) $=
    ( wfun cdm wcel wa cfv cima cres dmres elin2 crn funres fvelrn sylan df-ima
    wi eleq2i fvres eleq1d bitr4id syl5ibrcom biimtrrid expd com12 impd pm2.43b
    ex ) CDZBCEZFZGBAFZBCHZCAIZFZUMUJULUMUPRZUJUMULUQRUJUMULUQUMULGBCAJZEZFZUJU
    QBAUKUSCAKLUJUTUQUJUTGUPUMBURHZURMZFZUJURDUTVCACNBUROPUMUPUNVBFVCUOVBUNCAQS
    UMVAUNVBBACTUAUBUCUIUDUEUFUGUH $.

  $( A function's value in an included preimage belongs to the image.
     (Contributed by NM, 3-Feb-1997.) $)
  funfvima2 $p |- ( ( Fun F /\ A C_ dom F ) -> ( B e. A ->
                  ( F ` B ) e. ( F " A ) ) ) $=
    ( wfun cdm wss wcel cfv cima wi ssel funfvima ex com23 a2d syl5 imp ) CDZAC
    EZFZBAGZBCHCAIGZJZTUABSGZJRUCASBKRUAUDUBRUDUAUBRUDUCABCLMNOPQ $.

  ${
    $d x A $.  $d x F $.  $d x G $.
    $( A class including a function contains the function's value in the image
       of the singleton of the argument.  (Contributed by NM, 23-Mar-2004.) $)
    funfvima3 $p |- ( ( Fun F /\ F C_ G ) -> ( A e. dom F ->
                    ( F ` A ) e. ( G " { A } ) ) ) $=
      ( vx wss wfun cdm wcel cfv csn cima wi wa cop funfvop ssel syl5 wb adantl
      cvv imp cv simpr wceq sneq imaeq2d eleq2d eleq1d bibi12d funfvex elimasng
      opeq1 vex sylancr vtocld mpbird exp32 impcom ) BCEZBFZABGZHZABIZCAJZKZHZL
      USUTVBVFUSUTVBMZMVFAVCNZCHZUSVGVIVGVHBHUSVIABOBCVHPQUAVGVFVIRZUSVGVCCDUBZ
      JZKZHZVKVCNZCHZRZVJDAVAUTVBUCVKAUDZVQVJRVGVRVNVFVPVIVRVMVEVCVRVLVDCVKAUEU
      FUGVRVOVHCVKAVCULUHUISVGVKTHVCTHVQDUMABUJCVKVCTTUKUNUOSUPUQUR $.
  $}

  $( The function value of an operand in a set is contained in the image of
     that set, using the ` Fn ` abbreviation.  (Contributed by Stefan O'Rear,
     10-Mar-2015.) $)
  fnfvima $p |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> ( F ` X ) e. ( F " S ) ) $=
    ( wfn wss wcel w3a wfun cdm wa cima fnfun 3ad2ant1 simp2 wceq fndm sseqtrrd
    cfv jca simp3 funfvima2 sylc ) CAEZBAFZDBGZHZCIZBCJZFZKUFDCSCBLGUGUHUJUDUEU
    HUFACMNUGBAUIUDUEUFOUDUEUIAPUFACQNRTUDUEUFUABDCUBUC $.

  ${
    fnfvimad.1 $e |- ( ph -> F Fn A ) $.
    fnfvimad.2 $e |- ( ph -> B e. A ) $.
    fnfvimad.3 $e |- ( ph -> B e. C ) $.
    $( A function's value belongs to the image.  (Contributed by Glauco
       Siliprandi, 23-Oct-2021.) $)
    fnfvimad $p |- ( ph -> ( F ` B ) e. ( F " C ) ) $=
      ( cin cima cfv wss inss2 imass2 ax-mp wfn wcel inss1 a1i elind fnfvima
      syl3anc sselid ) AEBDIZJZEDJZCEKZUDDLUEUFLBDMUDDENOAEBPUDBLZCUDQUGUEQFUHA
      BDRSABDCGHTBUDECUAUBUC $.
  $}

  ${
    $d x y z $.  $d y z A $.  $d y z F $.
    dfimafnf.1 $e |- F/_ x A $.
    dfimafnf.2 $e |- F/_ x F $.
    $( Alternate definition of the image of a function.  (Contributed by Raph
       Levien, 20-Nov-2006.)  (Revised by Thierry Arnoux, 24-Apr-2017.) $)
    dfimafnf $p |- ( ( Fun F /\ A C_ dom F ) ->
                  ( F " A ) = { y | E. x e. A y = ( F ` x ) } ) $=
      ( vz wfun cdm wss wa cima cv cfv wceq wrex cab wbr wcel nfcv dfima2 eqcom
      wb ssel funbrfvb bitr3id ex syl9r imp31 rexbidva abbidv eqtr4id nfeq2 nfv
      nffv fveq2 eqeq2d cbvrexfw abbii eqtrdi ) DHZCDIZJZKZDCLZBMZGMZDNZOZGCPZB
      QZVFAMZDNZOZACPZBQVDVEVGVFDRZGCPZBQVKGBDCUAVDVJVQBVDVIVPGCVAVCVGCSZVIVPUC
      ZVCVRVGVBSZVAVSCVBVGUDVAVTVSVIVHVFOVAVTKVPVHVFUBVGVFDUEUFUGUHUIUJUKULVJVO
      BVIVNGACGCTEAVFVHAVGDFAVGTUOUMVNGUNVGVLOVHVMVFVGVLDUPUQURUSUT $.
  $}

  ${
    resfvresima.f $e |- ( ph -> Fun F ) $.
    resfvresima.s $e |- ( ph -> S C_ dom F ) $.
    resfvresima.x $e |- ( ph -> X e. S ) $.
    $( The value of the function value of a restriction for a function
       restricted to the image of the restricting subset.  (Contributed by AV,
       6-Mar-2021.) $)
    resfvresima $p |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) )
                              = ( H ` ( F ` X ) ) ) $=
      ( cres cfv cima fvresd fveq2d wfun cdm wss wa wcel jca funfvima2 eqtrd
      sylc ) AECBIJZDCBKZIZJECJZUEJUFDJAUCUFUEAEBCHLMAUFUDDACNZBCOPZQEBRUFUDRAU
      GUHFGSHBECTUBLUA $.
  $}

  ${
    $d A x $.  $d Y x $.  $d F x $.
    $( Given an onto function, an element is in its codomain if and only if it
       is the image of an element of its domain (see ~ foima ).  (Contributed
       by BJ, 6-Jul-2022.) $)
    foima2 $p
            |- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A Y = ( F ` x ) ) ) $=
  ( wfo wcel cima cv cfv wceq wrex foima eqcomd eleq2d wfn wss wb fofn ssid wa
  fvelimab eqcom rexbii bitrdi sylancl bitrd ) BCDFZECGEDBHZGZEAIDJZKZABLZUHCUI
  EUHUICBCDMNOUHDBPZBBQZUJUMRBCDSBTUNUOUAUJUKEKZABLUMABBEDUBUPULABUKEUCUDUEUFUG
  $.
  $}

  ${
    $d A x $.  $d C x $.  $d F x $.
    $( Property of a surjective function.  (Contributed by Jeff Madsen,
       4-Jan-2011.)  (Proof shortened by BJ, 6-Jul-2022.) $)
    foelrn $p
             |- ( ( F : A -onto-> B /\ C e. B ) -> E. x e. A C = ( F ` x ) ) $=
  ( wfo wcel cv cfv wceq wrex foima2 biimpa ) BCEFDCGDAHEIJABKABCEDLM $.
  $}

  ${
    $d F x y z $.  $d G x y z $.  $d A y z $.  $d B x y z $.  $d C x y z $.
    $( If a composition of two functions is surjective, then the function on
       the left is surjective.  (Contributed by Jeff Madsen, 16-Jun-2011.) $)
    foco2 $p |- ( ( F : B --> C /\ G : A --> B /\
                        ( F o. G ) : A -onto-> C ) -> F : B -onto-> C ) $=
      ( vy vx vz wf ccom wfo w3a cv cfv wceq wrex wral wa wcel adantll ffvelcdm
      simp1 foelrn fvco3 fveq2 eqeq2d rspcev syl2anc eqeq1 syl5ibrcom rexlimdva
      rexbidv syl5 impl ralrimiva 3impa dffo3 sylanbrc ) BCDIZABEIZACDEJZKZLUSF
      MZGMZDNZOZGBPZFCQZBCDKUSUTVBUBUSUTVBVHUSUTRZVBRVGFCVIVBVCCSZVGVBVJRVCHMZV
      ANZOZHAPVIVGHACVCVAUCVIVMVGHAVIVKASZRZVGVMVLVEOZGBPZVOVKENZBSZVLVRDNZOZVQ
      UTVNVSUSABVKEUATUTVNWAUSABVKDEUDTVPWAGVRBVDVROVEVTVLVDVRDUEUFUGUHVMVFVPGB
      VCVLVEUIULUJUKUMUNUOUPGFBCDUQUR $.
  $}

  ${
    $d ph y $.  $d ps x $.  $d F x y $.  $d B x y $.  $d A x y $.
    rexima.x $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $.
    $( Existential quantification under an image in terms of the base set.
       (Contributed by Stefan O'Rear, 21-Jan-2015.) $)
    rexima $p |- ( ( F Fn A /\ B C_ A ) ->
        ( E. x e. ( F " B ) ph <-> E. y e. B ps ) ) $=
      ( wfn wss wa cv cfv cima cvv wcel ssel2 funfvex wceq wrex funfni fvelimab
      sylan2 anassrs eqcom rexbii bitrdi wb adantl rexxfr2d ) GEIZFEJZKZABCDDLZ
      GMZGFNZFOUKULUNFPZUOOPZULUQKUKUNEPURFEUNQUREUNGUNGRUAUCUDUMCLZUPPUOUSSZDF
      TUSUOSZDFTDEFUSGUBUTVADFUOUSUEUFUGVAABUHUMHUIUJ $.

    $( Universal quantification under an image in terms of the base set.
       (Contributed by Stefan O'Rear, 21-Jan-2015.) $)
    ralima $p |- ( ( F Fn A /\ B C_ A ) ->
        ( A. x e. ( F " B ) ph <-> A. y e. B ps ) ) $=
      ( wfn wss wa cv cfv cima cvv wcel ssel2 funfvex wceq wrex funfni fvelimab
      sylan2 anassrs eqcom rexbii bitrdi wb adantl ralxfr2d ) GEIZFEJZKZABCDDLZ
      GMZGFNZFOUKULUNFPZUOOPZULUQKUKUNEPURFEUNQUREUNGUNGRUAUCUDUMCLZUPPUOUSSZDF
      TUSUOSZDFTDEFUSGUBUTVADFUOUSUEUFUGVAABUHUMHUIUJ $.
  $}

  ${
    $d A x $.  $d R x $.
    $( TODO:  This is the same as ~ issref (which has a much longer proof).
       Should we replace ~ issref with this one? - NM 9-May-2016.

       Two ways to state a relation is reflexive.  (Adapted from Tarski.)
       (Contributed by FL, 15-Jan-2012.)  (Proof shortened by Mario Carneiro,
       3-Nov-2015.)  (Proof modification is discouraged.) $)
    idref $p |- ( ( _I |` A ) C_ R <-> A. x e. A x R x ) $=
      ( cv cop wcel wral cmpt crn wss wbr cid cres wf eqid fmpt wfn opex fnmpti
      vex df-f bitri df-br ralbii mptresid eqcomi fnasrn eqtr3i sseq1i 3bitr4ri
      mpbiran ) ADZULEZCFZABGZABUMHZIZCJZULULCKZABGLBMZCJUOBCUPNZURABCUMUPUPOZP
      VAUPBQURABUMUPULULATZVCRVBSBCUPUAUKUBUSUNABULULCUCUDUTUQCABULHZUTUQUTVDAB
      UEUFABULVCUGUHUIUJ $.
  $}

  ${
    $d y z B $.  $d x y z A $.
    elabrex.1 $e |- B e. _V $.
    $( Elementhood in an image set.  (Contributed by Mario Carneiro,
       14-Jan-2014.) $)
    elabrex $p |- ( x e. A -> B e. { y | E. x e. A y = B } ) $=
      ( vz cv wcel csb wceq wrex cab wtru tru csbeq1a equcoms trud rspcev mpan2
      2thd eqeq1 rexbidv sylibr nfv nfcsb1v nfeq2 eqeq2d cbvrex abbii eleqtrrdi
      elab ) AGZCHZDBGZAFGZDIZJZFCKZBLZUNDJZACKZBLUMDUPJZFCKZDUSHUMMVCNVBMFULCU
      OULJZVBMVBAFAUODOZPVDQTRSURVCBDEUTUQVBFCUNDUPUAUBUKUCVAURBUTUQAFCUTFUDAUN
      UPAUODUEUFULUOJDUPUNVEUGUHUIUJ $.
  $}

  ${
    $d A x y z $.  $d B y z $.
    $( Elementhood in an image set.  (Contributed by Glauco Siliprandi,
       11-Dec-2019.) $)
    elabrexg $p |- ( ( x e. A /\ B e. V ) -> B e. { y | E. x e. A y = B } ) $=
      ( vz cv wcel csb wceq wrex cab wtru tru csbeq1a equcoms trud 2thd rspcev
      wa mpan2 adantr wb eqeq1 rexbidv elabg adantl mpbird nfcsb1v nfeq2 eqeq2d
      nfv cbvrexw abbii eleqtrrdi ) AGZCHZDEHZTZDBGZAFGZDIZJZFCKZBLZUTDJZACKZBL
      USDVEHZDVBJZFCKZUQVJURUQMVJNVIMFUPCVAUPJZVIMVIAFAVADOZPVKQRSUAUBURVHVJUCU
      QVDVJBDEVFVCVIFCUTDVBUDUEUFUGUHVGVDBVFVCAFCVFFULAUTVBAVADUIUJUPVAJDVBUTVL
      UKUMUNUO $.
  $}

  ${
    $d A y z $.  $d B y z $.  $d C w $.  $d D y $.  $d w x y $.  $d w z y $.
    abrexco.1 $e |- B e. _V $.
    abrexco.2 $e |- ( y = B -> C = D ) $.
    $( Composition of two image maps ` C ( y ) ` and ` B ( w ) ` .
       (Contributed by NM, 27-May-2013.) $)
    abrexco $p |- { x | E. y e. { z | E. w e. A z = B } x = C } =
        { x | E. w e. A x = D } $=
      ( cv wceq wrex cab wa wex wcel df-rex bitr4i bitri vex eqeq1 rexbidv elab
      anbi1i r19.41v exbii rexcom4 eqeq2d ceqsexv rexbii abbii ) AKZGLZBCKZFLZD
      EMZCNZMZUMHLZDEMZAUSBKZFLZUNOZBPZDEMZVAUSVDDEMZBPZVFUSVBURQZUNOZBPVHUNBUR
      RVJVGBVJVCDEMZUNOVGVIVKUNUQVKCVBBUAUOVBLUPVCDEUOVBFUBUCUDUEVCUNDEUFSUGTVD
      DBEUHSVEUTDEUNUTBFIVCGHUMJUIUJUKTUL $.
  $}

  ${
    $d x y z A $.  $d y z B $.  $d y z C $.
    $( The image of an indexed union is the indexed union of the images.
       (Contributed by Mario Carneiro, 18-Jun-2014.) $)
    imaiun $p |- ( A " U_ x e. B C ) = U_ x e. B ( A " C ) $=
      ( vy vz ciun cima cv wcel cop wa wex wrex rexcom4 vex elima3 rexbii eliun
      anbi1i r19.41v bitr4i exbii 3bitr4ri 3bitr4i eqriv ) EBACDGZHZACBDHZGZFIZ
      UGJZUKEIZKBJZLZFMZUMUIJZACNZUMUHJUMUJJUKDJZUNLZFMZACNUTACNZFMURUPUTAFCOUQ
      VAACFUMBDEPZQRUOVBFUOUSACNZUNLVBULVDUNAUKCDSTUSUNACUAUBUCUDFUMBUGVCQAUMCU
      ISUEUF $.
  $}

  ${
    $d x A $.  $d x B $.
    $( The image of a union is the indexed union of the images.  Theorem 3K(a)
       of [Enderton] p. 50.  (Contributed by NM, 9-Aug-2004.)  (Proof shortened
       by Mario Carneiro, 18-Jun-2014.) $)
    imauni $p |- ( A " U. B ) = U_ x e. B ( A " x ) $=
      ( cuni cima cv ciun uniiun imaeq2i imaiun eqtri ) BCDZEBACAFZGZEACBMEGLNB
      ACHIABCMJK $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( The indexed union of a function's values is the union of its range.
       Compare Definition 5.4 of [Monk1] p. 50.  (Contributed by NM,
       27-Sep-2004.) $)
    fniunfv $p |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F ) $=
      ( vy wfn cv cfv ciun wceq wrex cab cuni crn wcel funfvex funfni ralrimiva
      cvv wral dfiun2g syl fnrnfv unieqd eqtr4d ) CBEZABAFZCGZHZDFUGIABJDKZLZCM
      ZLUEUGRNZABSUHUJIUEULABULBUFCUFCOPQADBUGRTUAUEUKUIADBCUBUCUD $.

    $( The indexed union of a function's values is the union of its image under
       the index class.  This theorem is a slight variation of ~ fniunfv .
       (Contributed by Jim Kingdon, 10-Jan-2019.) $)
    funiunfvdm $p |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) ) $=
      ( wfn cv cfv ciun crn cuni cima fniunfv cdm imadmrn imaeq2d eqtr3id eqtrd
      fndm unieqd ) CBDZABAECFGCHZICBJZIABCKSTUASTCCLZJUACMSUBBCBCQNORP $.
  $}

  ${
    $d x z A $.  $d z F $.
    funiunfvf.1 $e |- F/_ x F $.
    $( The indexed union of a function's values is the union of its image under
       the index class.  This version of ~ funiunfvdm uses a bound-variable
       hypothesis in place of a distinct variable condition.  (Contributed by
       Jim Kingdon, 10-Jan-2019.) $)
    funiunfvdmf $p |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) ) $=
      ( vz wfn cv cfv ciun cima cuni nfcv nffv fveq2 cbviun funiunfvdm eqtr3id
      ) CBFABAGZCHZIEBEGZCHZICBJKEABUASATCDATLMESLTRCNOEBCPQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x F $.
    $( Membership in the union of an image of a function.  (Contributed by Jim
       Kingdon, 10-Jan-2019.) $)
    eluniimadm $p |- ( F Fn A ->
                   ( B e. U. ( F " A ) <-> E. x e. A B e. ( F ` x ) ) ) $=
      ( wfn cv cfv ciun wcel cima cuni wrex funiunfvdm eleq2d eliun bitr3di ) D
      BEZCABAFDGZHZICDBJKZICRIABLQSTCABDMNACBROP $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( Membership in the union of the range of a function.  (Contributed by NM,
       24-Sep-2006.) $)
    elunirn $p |- ( Fun F -> ( A e. U. ran F <->
                E. x e. dom F A e. ( F ` x ) ) ) $=
      ( vy crn cuni wcel cv wa wex wfun cfv cdm wrex eluni wceq wfn eleq2 cvv
      wb fvelrnb anbi2d r19.42v bitr4di biimparc reximi biimtrdi exlimdv fvelrn
      funfn sylbi wi funfvex anbi12d spcegv syl mpan2d rexlimdva impbid bitrid
      eleq1 ) BCEZFGBDHZGZVCVBGZIZDJZCKZBAHZCLZGZACMZNZDBVBOVHVGVMVHVFVMDVHVFVD
      VJVCPZIZAVLNZVMVHVFVDVNAVLNZIVPVHVEVQVDVHCVLQVEVQTCUJAVLVCCUAUKUBVDVNAVLU
      CUDVOVKAVLVNVKVDVJVCBRUEUFUGUHVHVKVGAVLVHVIVLGIZVKVJVBGZVGVICUIVRVJSGVKVS
      IZVGULVICUMVFVTDVJSVCVJPVDVKVEVSVCVJBRVCVJVBVAUNUOUPUQURUSUT $.
  $}

  ${
    $d x A $.  $d x I $.  $d x F $.
    $( Membership in a union of some function-defined family of sets.
       (Contributed by Stefan O'Rear, 30-Jan-2015.) $)
    fnunirn $p |- ( F Fn I -> ( A e. U. ran F <->
        E. x e. I A e. ( F ` x ) ) ) $=
      ( wfn crn cuni wcel cv cfv cdm wrex wfun fnfun elunirn fndm rexeqdv bitrd
      wb syl ) CDEZBCFGHZBAICJHZACKZLZUCADLUACMUBUESDCNABCOTUAUCAUDDDCPQR $.
  $}

  ${
    $d x y z A $.  $d z B $.  $d x y z F $.
    $( A one-to-one function in terms of function values.  Compare Theorem
       4.8(iv) of [Monk1] p. 43.  (Contributed by NM, 29-Oct-1996.) $)
    dff13 $p |- ( F : A -1-1-> B <-> ( F : A --> B /\
             A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
      ( vz cv wbr wal wa cfv wceq wi wral wb wcel vex breldm eleq2d bitrid fndm
      wf1 wf wmo dff12 wfn ffn imbitrid anim12d pm4.71rd eqcom fnbrfvb bi2anan9
      cdm anandis pm5.32da bitr4d imbi1d impexp bitrdi albidv 19.21v wex 19.23v
      cvv funfvex funfni eqvincg syl bitr4id adantrr pm5.74da bitrd 2albidv mo4
      breq1 albii alrot3 bitri r2al 3bitr4g pm5.32i ) CDEUBCDEUCZAGZFGZEHZAUDZF
      IZJWCWDEKZBGZEKZLZWDWJLZMZBCNACNZJAFCDEUEWCWHWOWCECUFZWHWOOCDEUGWPWFWJWEE
      HZJZWMMZFIZBIAIZWDCPZWJCPZJZWNMZBIAIWHWOWPWTXEABWPWTXDWEWILZWEWKLZJZWMMZM
      ZFIZXEWPWSXJFWPWSXDXHJZWMMXJWPWRXLWMWPWRXDWRJXLWPWRXDWPWFXBWQXCWFWDEUNZPW
      PXBWDWEEAQFQZRWPXMCWDCEUAZSUHWQWJXMPWPXCWJWEEBQXNRWPXMCWJXOSUHUIUJWPXDXHW
      RWPXBXCXHWROWPXBJZXFWFWPXCJZXGWQXFWIWELXPWFWEWIUKCWDWEEULTXGWKWELXQWQWEWK
      UKCWJWEEULTUMUOUPUQURXDXHWMUSUTVAXKXDXIFIZMWPXEXDXIFVBWPXDXRWNWPXBXRWNOXC
      XPXRXHFVCZWMMWNXHWMFVDXPWLXSWMXPWIVEPZWLXSOXTCWDEWDEVFVGFWIWKVEVHVIURVJVK
      VLTVMVNWHWSBIAIZFIXAWGYAFWFWQABWDWJWEEVPVOVQWSFABVRVSWNABCCVTWAVIWBVS $.
  $}

  ${
    $d A c d $.  $d C c d $.  $d D d $.  $d F c d $.
    $( If the values of a one-to-one function for two arguments are equal, the
       arguments themselves must be equal.  (Contributed by Alexander van der
       Vekens, 12-Nov-2017.) $)
    f1veqaeq $p |- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) ->
                     ( ( F ` C ) = ( F ` D ) -> C = D ) ) $=
      ( vc vd wf1 wcel wa cfv wceq wi wf cv weq wral dff13 fveq2 imbi12d eqeq1d
      eqeq1 eqeq2d eqeq2 rspc2v com12 adantl sylbi imp ) ABEHZCAIDAIJZCEKZDEKZL
      ZCDLZMZUJABENZFOZEKZGOZEKZLZFGPZMZGAQFAQZJUKUPMZFGABERVEVFUQUKVEUPVDUPULV
      ALZCUTLZMFGCDAAURCLZVBVGVCVHVIUSULVAURCESUAURCUTUBTUTDLZVGUNVHUOVJVAUMULU
      TDESUCUTDCUDTUEUFUGUHUI $.
  $}

  ${
    $d x y w v A $.  $d w v B $.  $d w v F $.
    dff13f.1 $e |- F/_ x F $.
    dff13f.2 $e |- F/_ y F $.
    $( A one-to-one function in terms of function values.  Compare Theorem
       4.8(iv) of [Monk1] p. 43.  (Contributed by NM, 31-Jul-2003.) $)
    dff13f $p |- ( F : A -1-1-> B <-> ( F : A --> B /\
             A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
      ( vw vv cv cfv wceq weq wi wral wa nfcv nffv nfeq nfv wf dff13 nfim fveq2
      eqeq2d equequ2 imbi12d cbvral ralbii nfralxy eqeq1d equequ1 ralbidv bitri
      wf1 anbi2i ) CDEUOCDEUAZHJZEKZIJZEKZLZHIMZNZICOZHCOZPUQAJZEKZBJZEKZLZABMZ
      NZBCOZACOZPHICDEUBVFVOUQVFUSVJLZHBMZNZBCOZHCOVOVEVSHCVDVRIBCVBVCBBUSVABUR
      EGBURQRBUTEGBUTQRSVCBTUCVRITIBMZVBVPVCVQVTVAVJUSUTVIEUDUEIBHUFUGUHUIVSVNH
      ACVRABCACQVPVQAAUSVJAUREFAURQRAVIEFAVIQRSVQATUCUJVNHTHAMZVRVMBCWAVPVKVQVL
      WAUSVHVJURVGEUDUKHABULUGUMUHUNUPUN $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.  $d y F $.
    f1mpt.1 $e |- F = ( x e. A |-> C ) $.
    f1mpt.2 $e |- ( x = y -> C = D ) $.
    $( Express injection for a mapping operation.  (Contributed by Mario
       Carneiro, 2-Jan-2017.) $)
    f1mpt $p |- ( F : A -1-1-> B <->
      ( A. x e. A C e. B /\ A. x e. A A. y e. A ( C = D -> x = y ) ) ) $=
      ( wf1 wf cv cfv wceq wi wral wa wcel wb ralbi nfmpt1 nfcxfr dff13f anbi1i
      cmpt nfcv fmpt eleq1d raaanv fvmpt2 fvmptg eqeqan12d an4s imbi1d ralimdva
      cbvralv ex syl6 ralimia syl sylbir sylan2b anidms pm5.32i 3bitr2i ) CDGJC
      DGKZALZGMZBLZGMZNZVGVINZOZBCPZACPZQEDRZACPZVOQVQEFNZVLOZBCPZACPZQABCDGAGA
      CEUEHACEUAUBBGUFUCVQVFVOACDEGHUGUDVQVOWAVQVOWASZVQVQFDRZBCPZWBVPWCABCVLEF
      DIUHUPVQWDQVPWCQZBCPZACPZWBVPWCABCUIWGVNVTSZACPWBWFWHACVGCRZWFVMVSSZBCPWH
      WIWEWJBCWIVICRZQZWEWJWLWEQVKVRVLWIVPWKWCVKVRSWIVPQWKWCQVHEVJFACEDGHUJAVIE
      FCDGIHUKULUMUNUQUOVMVSBCTURUSVNVTACTUTVAVBVCVDVE $.
  $}

  $( Equality of function values for a one-to-one function.  (Contributed by
     NM, 11-Feb-1997.) $)
  f1fveq $p |- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) ->
               ( ( F ` C ) = ( F ` D ) <-> C = D ) ) $=
    ( wf1 wcel wa cfv wceq f1veqaeq fveq2 impbid1 ) ABEFCAGDAGHHCEIDEIJCDJABCDE
    KCDELM $.

  ${
    $d F z $.  $d A z $.  $d Y z $.  $d X z $.  $d B z $.
    $( Membership in the image of a 1-1 map.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    f1elima $p |- ( ( F : A -1-1-> B /\ X e. A /\ Y C_ A )
                                -> ( ( F ` X ) e. ( F " Y ) <-> X e. Y ) ) $=
      ( vz wf1 wcel wss w3a cfv cima cv wceq wrex wb wfn wi wa anassrs fvelimab
      f1fn sylan 3adant2 ssel impac f1fveq ancom2s biimpd biimpcd sylan9 anasss
      eleq1 sylan2 rexlimdva 3impa eqid fveq2 eqeq1d rspcev mpan2 impbid1 bitrd
      ) ABCGZDAHZEAIZJZDCKZCELHZFMZCKZVHNZFEOZDEHZVDVFVIVMPZVEVDCAQVFVOABCUBFAE
      VHCUAUCUDVGVMVNVDVEVFVMVNRVDVESZVFSVLVNFEVPVFVJEHZVLVNRZVFVQSVPVJAHZVQSVR
      VFVQVSEAVJUEUFVPVSVQVRVPVSSVLVJDNZVQVNVDVEVSVLVTRVDVEVSSSVLVTVDVSVEVLVTPA
      BVJDCUGUHUITVTVQVNVJDEUMUJUKULUNTUOUPVNVHVHNZVMVHUQVLWAFDEVTVKVHVHVJDCURU
      SUTVAVBVC $.
  $}

  ${
    $d F a $.  $d A a $.  $d B a $.  $d C a $.  $d D a $.
    $( Taking images under a one-to-one function preserves subsets.
       (Contributed by Stefan O'Rear, 30-Oct-2014.) $)
    f1imass $p |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) ->
      ( ( F " C ) C_ ( F " D ) <-> C C_ D ) ) $=
      ( va wf1 wss wa cima cv wcel wi simplrl sseld wb 3expa f1elima syl3anc ex
      simplr simplll simpr simp1rl simp1rr 3imtr3d pm2.43d ssrdv imass2 impbid1
      cfv syld ) ABEGZCAHZDAHZIZIZECJZEDJZHZCDHZUQUTVAUQUTIZFCDVBFKZCLZVCDLZVBV
      DVCALZVDVEMZVBCAVCUMUNUOUTNOVBVFVGVBVFIZVCEUKZURLZVIUSLZVDVEVHURUSVIUQUTV
      FUAOVHUMVFUNVJVDPUMUPUTVFUBZVBVFUCZUQUTVFUNUNUOUMUTVFUDQABEVCCRSVHUMVFUOV
      KVEPVLVMUQUTVFUOUNUOUMUTVFUEQABEVCDRSUFTULUGUHTCDEUIUJ $.

    $( Taking images under a one-to-one function preserves equality.
       (Contributed by Stefan O'Rear, 30-Oct-2014.) $)
    f1imaeq $p |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) ->
      ( ( F " C ) = ( F " D ) <-> C = D ) ) $=
      ( wf1 wss wa cima wceq f1imass wb ancom2s anbi12d eqss 3bitr4g ) ABEFZCAG
      ZDAGZHHZECIZEDIZGZUBUAGZHCDGZDCGZHUAUBJCDJTUCUEUDUFABCDEKQSRUDUFLABDCEKMN
      UAUBOCDOP $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( A one-to-one onto function in terms of function values.  (Contributed by
       NM, 29-Mar-2008.) $)
    dff1o6 $p |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ ran F = B /\
             A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
      ( wf1o wf1 wfo wa wf cv cfv wceq wi wral wfn crn w3a df-f1o dff13 anbi12i
      df-fo df-3an wss eqimss anim2i df-f sylibr pm4.71ri anbi1i 3bitrri 3bitri
      an32 ) CDEFCDEGZCDEHZICDEJZAKZELBKZELMUQURMNBCOACOZIZECPZEQZDMZIZIZVAVCUS
      RZCDESUNUTUOVDABCDETCDEUBUAVFVDUSIUPVDIZUSIVEVAVCUSUCVDVGUSVDUPVDVAVBDUDZ
      IUPVCVHVAVBDUEUFCDEUGUHUIUJUPVDUSUMUKUL $.
  $}

  $( The converse value of the value of a one-to-one onto function.
     (Contributed by NM, 20-May-2004.) $)
  f1ocnvfv1 $p |- ( ( F : A -1-1-onto-> B /\ C e. A ) ->
                 ( `' F ` ( F ` C ) ) = C ) $=
    ( wf1o wcel wa ccnv ccom cfv cid cres wceq f1ococnv1 fveq1d adantr wf fvco3
    f1of sylan fvresi adantl 3eqtr3d ) ABDEZCAFZGCDHZDIZJZCKALZJZCDJUFJZCUDUHUJ
    MUEUDCUGUIABDNOPUDABDQUEUHUKMABDSABCUFDRTUEUJCMUDACUAUBUC $.

  $( The value of the converse value of a one-to-one onto function.
     (Contributed by NM, 20-May-2004.) $)
  f1ocnvfv2 $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
                 ( F ` ( `' F ` C ) ) = C ) $=
    ( wf1o wcel ccnv ccom cfv cid cres wceq f1ococnv2 fveq1d adantr f1ocnv f1of
    wa wf syl fvco3 sylan fvresi adantl 3eqtr3d ) ABDEZCBFZRCDDGZHZIZCJBKZIZCUH
    IDIZCUFUJULLUGUFCUIUKABDMNOUFBAUHSZUGUJUMLUFBAUHEUNABDPBAUHQTBACDUHUAUBUGUL
    CLUFBCUCUDUE $.

  $( Relationship between the value of a one-to-one onto function and the value
     of its converse.  (Contributed by Raph Levien, 10-Apr-2004.) $)
  f1ocnvfv $p |- ( ( F : A -1-1-onto-> B /\ C e. A ) ->
                  ( ( F ` C ) = D -> ( `' F ` D ) = C ) ) $=
    ( cfv wceq ccnv wf1o wcel wa fveq2 eqcoms f1ocnvfv1 eqeq2d imbitrid ) CEFZD
    GDEHZFZQRFZGZABEICAJKZSCGUADQDQRLMUBTCSABCENOP $.

  $( Relationship between the value of a one-to-one onto function and the value
     of its converse.  (Contributed by NM, 20-May-2004.) $)
  f1ocnvfvb $p |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) ->
                  ( ( F ` C ) = D <-> ( `' F ` D ) = C ) ) $=
    ( wf1o wcel w3a cfv wceq ccnv wi f1ocnvfv 3adant3 wa fveq2 eqcoms f1ocnvfv2
    eqeq2d imbitrid 3adant2 impbid ) ABEFZCAGZDBGZHCEIZDJZDEKIZCJZUCUDUGUILUEAB
    CDEMNUCUEUIUGLUDUIUFUHEIZJZUCUEOZUGUKCUHCUHEPQULUJDUFABDERSTUAUB $.

  $( The value of the converse of a one-to-one onto function belongs to its
     domain.  (Contributed by NM, 26-May-2006.) $)
  f1ocnvdm $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
                    ( `' F ` C ) e. A ) $=
    ( wf1o ccnv wf f1ocnv f1of syl ffvelcdmda ) ABDEZBACDFZLBAMEBAMGABDHBAMIJK
    $.

  $( If the values of a one-to-one function for two arguments from the range of
     the function are equal, the arguments themselves must be equal.
     (Contributed by Alexander van der Vekens, 12-Nov-2017.) $)
  f1ocnvfvrneq $p |- ( ( F : A -1-1-> B /\ ( C e. ran F /\ D e. ran F ) ) ->
                  ( ( `' F ` C ) = ( `' F ` D ) -> C = D ) ) $=
    ( wf1 crn wcel wa ccnv cfv wceq wi wf1o f1f1orn f1ocnv f1of1 f1veqaeq 4syl
    ex imp ) ABEFZCEGZHDUCHIZCEJZKDUEKLCDLMZUBAUCENUCAUENUCAUEFZUDUFMABEOAUCEPU
    CAUEQUGUDUFUCACDUERTSUA $.

  ${
    $d A x y $.  $d B x y $.  $d F x y $.  $d R x y $.
    $( An application is injective if a retraction exists.  Proposition 8 of
       [BourbakiEns] p.  E.II.18.  (Contributed by FL, 11-Nov-2011.)  (Revised
       by Mario Carneiro, 27-Dec-2014.) $)
    fcof1 $p |- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) )
         -> F : A -1-1-> B ) $=
      ( vx vy wf ccom wceq wa cfv wral wcel fvco3 syl2anc fveq1d 3eqtr3d fvresi
      cv syl cid cres wi wf1 simpl simprr fveq2d simpll simprll simprlr 3eqtr4d
      simplr expr ralrimivva dff13 sylanbrc ) ABDGZCDHZUAAUBZIZJZUQESZDKZFSZDKZ
      IZVBVDIZUCZFALEALABDUDUQUTUEVAVHEFAAVAVBAMZVDAMZJZVFVGVAVKVFJZJZVBUSKZVDU
      SKZVBVDVMVBURKZVDURKZVNVOVMVCCKZVECKZVPVQVMVCVECVAVKVFUFUGVMUQVIVPVRIUQUT
      VLUHZVAVIVJVFUIZABVBCDNOVMUQVJVQVSIVTVAVIVJVFUJZABVDCDNOUKVMVBURUSUQUTVLU
      LZPVMVDURUSWCPQVMVIVNVBIWAAVBRTVMVJVOVDIWBAVDRTQUMUNEFABDUOUP $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d F x y $.  $d S x y $.
    $( An application is surjective if a section exists.  Proposition 8 of
       [BourbakiEns] p.  E.II.18.  (Contributed by FL, 17-Nov-2011.)  (Proof
       shortened by Mario Carneiro, 27-Dec-2014.) $)
    fcofo $p |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) )
         -> F : A -onto-> B ) $=
      ( vy vx wf ccom cid cres wceq w3a cfv wrex wral wfo simp1 wcel 3ad2antl2
      cv wa ffvelcdm simpl3 fveq1d fvresi adantl 3eqtr3rd eqeq2d rspcev syl2anc
      fvco3 fveq2 ralrimiva dffo3 sylanbrc ) ABDGZBACGZDCHZIBJZKZLZUPETZFTZDMZK
      ZFANZEBOABDPUPUQUTQVAVFEBVAVBBRZUAZVBCMZARZVBVIDMZKZVFUQUPVGVJUTBAVBCUBSV
      HVBURMZVBUSMZVKVBVHVBURUSUPUQUTVGUCUDUQUPVGVMVKKUTBAVBDCUKSVGVNVBKVABVBUE
      UFUGVEVLFVIAVCVIKVDVKVBVCVIDULUHUIUJUMFEABDUNUO $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y F $.  $d y ph $.  $d x ps $.
    cbvfo.1 $e |- ( ( F ` x ) = y -> ( ph <-> ps ) ) $.
    $( Change bound variable between domain and range of function.
       (Contributed by NM, 23-Feb-1997.)  (Proof shortened by Mario Carneiro,
       21-Mar-2015.) $)
    cbvfo $p |- ( F : A -onto-> B -> ( A. x e. A ph <-> A. y e. B ps ) ) $=
      ( wfo crn wral wfn wb fofn cv cfv wceq bicomd eqcoms ralrn raleqdv bitr3d
      syl forn ) EFGIZBDGJZKZACEKZBDFKUEGELUGUHMEFGNBADCEGBAMCOGPZDOZUIUJQABHRS
      TUCUEBDUFFEFGUDUAUB $.

    $( Change bound variable between domain and range of function.
       (Contributed by NM, 23-Feb-1997.) $)
    cbvexfo $p |- ( F : A -onto-> B -> ( E. x e. A ph <-> E. y e. B ps ) ) $=
      ( wfo crn wrex wfn wb fofn cv cfv wceq bicomd eqcoms rexrn rexeqdv bitr3d
      syl forn ) EFGIZBDGJZKZACEKZBDFKUEGELUGUHMEFGNBADCEGBAMCOGPZDOZUIUJQABHRS
      TUCUEBDUFFEFGUDUAUB $.
  $}

  ${
    $d A x $.  $d B x $.  $d C x $.  $d F x $.  $d H x $.  $d K x $.
    $( An injection is left-cancelable.  (Contributed by FL, 2-Aug-2009.)
       (Revised by Mario Carneiro, 21-Mar-2015.) $)
    cocan1 $p |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) ->
      ( ( F o. H ) = ( F o. K ) <-> H = K ) ) $=
      ( vx wf ccom cfv wceq wral wcel fvco3 3ad2antl2 wb wfn ffn syl syl2anc cv
      wf1 w3a wa 3ad2antl3 eqeq12d simpl1 ffvelcdm f1fveq syl12anc ralbidva f1f
      bitrd 3ad2ant1 simp2 fnfco simp3 eqfnfv 3bitr4d ) BCDUBZABEHZABFHZUCZGUAZ
      DEIZJZVDDFIZJZKZGALZVDEJZVDFJZKZGALZVEVGKZEFKZVCVIVMGAVCVDAMZUDZVIVKDJZVL
      DJZKZVMVRVFVSVHVTVAUTVQVFVSKVBABVDDENOVBUTVQVHVTKVAABVDDFNUEUFVRUTVKBMZVL
      BMZWAVMPUTVAVBVQUGVAUTVQWBVBABVDEUHOVBUTVQWCVAABVDFUHUEBCVKVLDUIUJUMUKVCV
      EAQZVGAQZVOVJPVCDBQZVAWDVCBCDHZWFUTVAWGVBBCDULUNBCDRSZUTVAVBUOZBADEUPTVCW
      FVBWEWHUTVAVBUQZBADFUPTGAVEVGURTVCEAQZFAQZVPVNPVCVAWKWIABERSVCVBWLWJABFRS
      GAEFURTUS $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.  $d x y H $.  $d x y K $.
    $( A surjection is right-cancelable.  (Contributed by FL, 21-Nov-2011.)
       (Proof shortened by Mario Carneiro, 21-Mar-2015.) $)
    cocan2 $p |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) ->
      ( ( H o. F ) = ( K o. F ) <-> H = K ) ) $=
      ( vy vx wfn cv ccom cfv wceq wral 3ad2ant1 fvco3 sylan eqeq12d wb syl2anc
      fveq2 wfo w3a wcel wa fof ralbidva cbvfo bitrd simp2 fnfco eqfnfv 3bitr4d
      wf simp3 ) ABCUAZDBHZEBHZUBZFIZDCJZKZUSECJZKZLZFAMZGIZDKZVFEKZLZGBMZUTVBL
      ZDELZURVEUSCKZDKZVMEKZLZFAMZVJURVDVPFAURUSAUCZUDVAVNVCVOURABCUMZVRVAVNLUO
      UPVSUQABCUENZABUSDCOPURVSVRVCVOLVTABUSECOPQUFUOUPVQVJRUQVPVIFGABCVMVFLVNV
      GVOVHVMVFDTVMVFETQUGNUHURUTAHZVBAHZVKVERURUPVSWAUOUPUQUIZVTBADCUJSURUQVSW
      BUOUPUQUNZVTBAECUJSFAUTVBUKSURUPUQVLVJRWCWDGBDEUKSUL $.
  $}

  $( Show that two functions are inverse to each other by computing their
     compositions.  (Contributed by Mario Carneiro, 21-Mar-2015.) $)
  fcof1o $p |- ( ( ( F : A --> B /\ G : B --> A ) /\
    ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) ->
    ( F : A -1-1-onto-> B /\ `' F = G ) ) $=
    ( wf wa ccom cid cres wceq wf1o wf1 wfo fcof1 ad2ant2rl fcofo 3expa adantrr
    ccnv df-f1o sylanbrc simprl coeq2d coass f1ococnv1 syl fcoi2 ad2antlr eqtrd
    coeq1d eqtr3id f1ocnv f1of fcoi1 4syl 3eqtr3rd jca ) ABCEZBADEZFZCDGZHBIZJZ
    DCGHAIZJZFZFZABCKZCSZDJVGABCLZABCMZVHURVEVJUSVCABDCNOUTVCVKVEURUSVCVKABDCPQ
    RABCTUAZVGVIVAGZVIVBGZDVIVGVAVBVIUTVCVEUBUCVGVMVICGZDGZDVICDUDVGVPVDDGZDVGV
    OVDDVGVHVOVDJVLABCUEUFUJUSVQDJURVFBADUGUHUIUKVGVHBAVIKBAVIEVNVIJVLABCULBAVI
    UMBAVIUNUOUPUQ $.

  ${
    $d F x y $.  $d G x y $.  $d A x y $.  $d B x y $.
    $( Condition for function equality in terms of vanishing of the composition
       with the converse. _EDITORIAL_:  Is there a relation-algebraic proof of
       this?  (Contributed by Stefan O'Rear, 12-Feb-2015.) $)
    foeqcnvco $p |- ( ( F : A -onto-> B /\ G : A -onto-> B ) ->
        ( F = G <-> ( F o. `' G ) = ( _I |` B ) ) ) $=
      ( vx vy wfo wa wceq ccnv ccom adantr wfn wcel wbr cvv syl syl2anc adantlr
      wb cid cres fococnv2 cnveq coeq2d eqeq1d syl5ibcom fofn ad2antrr ad2antlr
      wi cv cfv wex cop adantl fnopfv anim1i adantll funfvex funfni vex sylancl
      sylan brcnvg df-br bitrdi mpbird sylibr breq2 breq1 anbi12d spcev breq wf
      brcog mpbid fof ffvelcdmda resieq eqcomd eqfnfvd ex impbid ) ABCGZABDGZHZ
      CDIZCDJZKZUABUBZIZWEWHWLUKWFWECCJZKZWKIWHWLABCUCWHWNWJWKWHWMWICCDUDUEUFUG
      LWGWLWHWGWLHZEACDWECAMZWFWLABCUHZUIWFDAMZWEWLABDUHZUJWOEULZANZHZWTDUMZWTC
      UMZXBXCXDWKOZXCXDIZXBXCXDWJOZXEWGXAXGWLWGXAHZXGXCFULZWIOZXIXDCOZHZFUNZXHX
      CWTWIOZWTXDCOZXMXHXNWTXCUODNZWGWRXAXPWFWRWEWSUPAWTDUQVDXHWRXAHZXNXPTWFXAX
      QWEWFWRXAWSURUSZXQXNWTXCDOZXPXQXCPNZWTPNXNXSTXTAWTDWTDUTVAZEVBZXCWTPPDVEV
      CWTXCDVFVGQVHXHWTXDUOCNZXOWGWPXAYCWEWPWFWQLAWTCUQVDWTXDCVFVIXLXNXOHFWTYBX
      IWTIXJXNXKXOXIWTXCWIVJXIWTXDCVKVLVMRXHXTXDPNZXGXMTXHXQXTXRYAQXHWPXAHZYDWE
      XAYEWFWEWPXAWQURSYDAWTCWTCUTVAQFXCXDCWIPPVPRVHSWLXGXETWGXAXCXDWJWKVNUJVQW
      GXAXEXFTZWLXHXCBNXDBNYFWGABWTDWFABDVOWEABDVRUPVSWGABWTCWEABCVOWFABCVRLVSB
      XCXDVTRSVQWAWBWCWD $.

    $( Condition for function equality in terms of vanishing of the composition
       with the inverse.  (Contributed by Stefan O'Rear, 12-Feb-2015.) $)
    f1eqcocnv $p |- ( ( F : A -1-1-> B /\ G : A -1-1-> B ) ->
        ( F = G <-> ( `' F o. G ) = ( _I |` A ) ) ) $=
      ( vx vy wf1 wa wceq ccom wi adantr wfn f1fn adantl cv wcel wbr wb syl cid
      ccnv cres f1cocnv1 coeq2 eqeq1d syl5ibcom equid resieq mpbiri anidms breq
      cfv ad2antlr mpbird wex vex brco cop wfun cdm fnfun fndm biimpar funopfvb
      eleq2d syl2anc bicomd df-br eqcom 3bitr4g biimpd bitr4id anim12d biimtrid
      eximdv cvv anim1i adantll funfvex funfni eqvincg 3syl sylibrd adantlr mpd
      brcnv eqfnfvd eqcomd ex impbid ) ABCGZABDGZHZCDIZCUBZDJZUAAUCZIZWLWOWSKWM
      WLWPCJZWRIWOWSABCUDWOWTWQWRCDWPUEUFUGLWNWSWOWNWSHZDCXAEADCWNDAMZWSWMXBWLA
      BDNZOZLWNCAMZWSWLXEWMABCNLZLXAEPZAQZHZXGXGWQRZXGDUMZXGCUMZIZXIXJXGXGWRRZX
      HXNXAXHXNXHXHHXNXGXGIEUHAXGXGUIUJUKOWSXJXNSWNXHXGXGWQWRULUNUOWNXHXJXMKWSW
      NXHHZXJFPZXKIZXPXLIZHZFUPZXMXJXGXPDRZXPXGWPRZHZFUPXOXTFXGXGWPDEUQZYDURXOY
      CXSFXOYAXQYBXRXOYAXQXOXGXPUSZDQZXKXPIZYAXQXOYGYFXODUTZXGDVAZQZYGYFSWNYHXH
      WNXBYHXDADVBTLWNYJXHWNYIAXGWNXBYIAIXDADVCTVFVDXGXPDVEVGVHXGXPDVIXPXKVJVKV
      LXOYBXRXOXGXPCRZXLXPIZYBXRXOYKYECQZYLXGXPCVIXOCUTZXGCVAZQZYLYMSWNYNXHWNXE
      YNXFACVBTLWNYPXHWNYOAXGWNXEYOAIXFACVCTVFVDXGXPCVEVGVMXPXGCFUQYDWGXPXLVJVK
      VLVNVPVOXOXBXHHZXKVQQZXMXTSWMXHYQWLWMXBXHXCVRVSYRAXGDXGDVTWAFXKXLVQWBWCWD
      WEWFWHWIWJWK $.
  $}

  ${
    $d u v y z A $.  $d u v y z B $.  $d u v x z C $.  $d x y z R $.  $d x Y $.
    $d u v x z D $.  $d u v y z F $.  $d u v x y z ph $.  $d u v x y z X $.
    $d x y z S $.
    flift.1 $e |- F = ran ( x e. X |-> <. A , B >. ) $.
    flift.2 $e |- ( ( ph /\ x e. X ) -> A e. R ) $.
    flift.3 $e |- ( ( ph /\ x e. X ) -> B e. S ) $.
    $( ` F ` , a function lift, is a subset of ` R X. S ` .  (Contributed by
       Mario Carneiro, 23-Dec-2016.) $)
    fliftrel $p |- ( ph -> F C_ ( R X. S ) ) $=
      ( cop cmpt crn cxp wf wss cv wcel wa opelxpi syl2anc eqid fmptd eqsstrid
      frn syl ) AGBHCDLZMZNZEFOZIAHUKUIPUJUKQABHUHUKUIABRHSTCESDFSUHUKSJKCDEFUA
      UBUIUCUDHUKUIUFUGUE $.

    $( Elementhood in the relation ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.) $)
    fliftel $p |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) ) $=
      ( cop wceq wrex wa wcel cvv wb wbr cmpt crn df-br eleq2i bitri wral opexg
      cv syl2anc ralrimiva eqid elrnmptg syl bitrid opthg2 rexbidva bitrd ) AEF
      IUAZEFNZCDNZOZBJPZECOFDOQZBJPUSUTBJVAUBZUCZRZAVCUSUTIRVGEFIUDIVFUTKUEUFAV
      ASRZBJUGVGVCTAVHBJABUIJRQZCGRZDHRZVHLMCDGHUHUJUKBJVAUTVESVEULUMUNUOAVBVDB
      JVIVJVKVBVDTLMEFCDGHUPUJUQUR $.

    $( Elementhood in the relation ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.) $)
    fliftel1 $p |- ( ( ph /\ x e. X ) -> A F B ) $=
      ( cv wcel wa cop wbr cmpt crn cvv opexg syl2anc eqid elrnmpt1 mpdan df-br
      adantll eleqtrrdi sylibr ) ABLHMZNZCDOZGMCDGPUJUKBHUKQZRZGUJUKSMZUKUMMZUJ
      CEMDFMUNJKCDEFTUAUIUNUOABHUKULSULUBUCUFUDIUGCDGUEUH $.

    $( Converse of the relation ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.) $)
    fliftcnv $p |- ( ph -> `' F = ran ( x e. X |-> <. B , A >. ) ) $=
      ( vy vz wrel cop wa wceq cv wbr wcel ccnv cmpt crn cxp wss fliftrel relxp
      eqid relss mpisyl relcnv jctil wrex fliftel vex brcnv ancom rexbii bitr4d
      3bitr4g df-br 3bitr3g eqrelrdv2 mpancom ) GUAZNZBHDCOUBUCZNZPAVEVGQAVHVFA
      VGFEUDZUEVINVHABDCFEVGHVGUHZKJUFFEUGVGVIUIUJGUKULALMVEVGALRZMRZVESZVKVLVG
      SZVKVLOZVETVOVGTAVMVKDQZVLCQZPZBHUMZVNAVLVKGSVQVPPZBHUMVMVSABCDVLVKEFGHIJ
      KUNVKVLGLUOMUOUPVRVTBHVPVQUQURUTABDCVKVLFEVGHVJKJUNUSVKVLVEVAVKVLVGVAVBVC
      VD $.

    ${
      fliftfun.4 $e |- ( x = y -> A = C ) $.
      fliftfun.5 $e |- ( x = y -> B = D ) $.
      $( The function ` F ` is the unique function defined by ` F `` A = B ` ,
         provided that the well-definedness condition holds.  (Contributed by
         Mario Carneiro, 23-Dec-2016.) $)
      fliftfun $p |- ( ph -> ( Fun F <->
        A. x e. X A. y e. X ( A = C -> B = D ) ) ) $=
        ( wceq cv wa wrex vz vu vv wfun wi wral nfv cop cmpt nfmpt1 nfrn nfcxfr
        crn nffun wcel cfv fveq2 wbr simplr fliftel1 ad2ant2r sylc simprr eqidd
        funbrfv eqeq2d anbi12d rspcev syl12anc fliftel ad2antrr mpbird imbitrid
        wb eqeq12d anassrs ralrimiva exp31 ralrimd cbvrexv bitrdi biimpd reeanv
        wal r19.29 eqtr2 imim1i imp simprlr simprrr 3eqtr4d rexlimivw biimtrrid
        syl ex syl9 alrimdv wrel cxp fliftrel relxp relss mpisyl dffun2 sylibrd
        wss baib impbid ) AJUDZDFQZEGQZUEZCKUFZBKUFZAXIXMBKABUGBJBJBKDEUHZUIZUM
        LBXPBKXOUJUKULUNAXIBRZKUOZXMAXISZXRSXLCKXSXRCRZKUOZXLXJDJUPZFJUPZQXSXRY
        ASZSZXKDFJUQYEYBEYCGYEXIDEJURZYBEQAXIYDUSZAXRYFXIYAABDEHIJKLMNUTVADEJVE
        VBYEXIFGJURZYCGQYGYEYHFDQZGEQZSZBKTZYEYAFFQZGGQZYLXSXRYAVCYEFVDYEGVDYKY
        MYNSBXTKXQXTQZYIYMYJYNYODFFOVFYOEGGPVFVGVHVIAYHYLVNXIYDABDEFGHIJKLMNVJV
        KVLFGJVEVBVOVMVPVQVRVSAXNUARZUBRZJURZYPUCRZJURZSZYQYSQZUEZUCWDZUBWDZUAW
        DZXIAXNUUEUAAXNUUDUBAXNUUCUCAUUAYPDQZYQEQZSZBKTZYPFQZYSGQZSZCKTZSZXNUUB
        AUUAUUOAYRUUJYTUUNABDEYPYQHIJKLMNVJAYTUUGYSEQZSZBKTUUNABDEYPYSHIJKLMNVJ
        UUQUUMBCKYOUUGUUKUUPUULYODFYPOVFYOEGYSPVFVGVTWAVGWBUUOUUIUUMSZCKTZBKTZX
        NUUBUUIUUMBCKKWCXNUUTUUBXNUUTSXMUUSSZBKTUUBXMUUSBKWEUVAUUBBKUVAXLUURSZC
        KTUUBXLUURCKWEUVBUUBCKUVBEGYQYSXLUURXKUURXJXKUUGUUKXJUUHUULYPDFWFVAWGWH
        XLUUGUUHUUMWIXLUUIUUKUULWJWKWLWNWLWNWOWMWPWQWQWQAJWRZXIUUFVNAJHIWSZXFUV
        DWRUVCABDEHIJKLMNWTHIXAJUVDXBXCXIUVCUUFUAUBUCJXDXGWNXEXH $.

      fliftfund.6 $e |- ( ( ph /\ ( x e. X /\ y e. X /\ A = C ) ) -> B = D ) $.
      $( The function ` F ` is the unique function defined by ` F `` A = B ` ,
         provided that the well-definedness condition holds.  (Contributed by
         Mario Carneiro, 23-Dec-2016.) $)
      fliftfund $p |- ( ph -> Fun F ) $=
        ( wceq wral cv wfun wi wcel 3exp2 imp32 ralrimivva fliftfun mpbird ) AJ
        UADFRZEGRZUBZCKSBKSAUKBCKKABTKUCZCTKUCZUKAULUMUIUJQUDUEUFABCDEFGHIJKLMN
        OPUGUH $.
    $}

    $( The function ` F ` is the unique function defined by ` F `` A = B ` ,
       provided that the well-definedness condition holds.  (Contributed by
       Mario Carneiro, 23-Dec-2016.) $)
    fliftfuns $p |- ( ph -> ( Fun F <-> A. y e. X A. z e. X
      ( [_ y / x ]_ A = [_ z / x ]_ A -> [_ y / x ]_ B = [_ z / x ]_ B ) ) ) $=
      ( cv csb cop cmpt crn nfcsb1v wcel nfcv nfop csbeq1a opeq12d cbvmpt rneqi
      weq eqtri wral ralrimiva nfel1 eleq1d rspc mpan9 csbeq1 fliftfun ) ACDBCN
      ZEOZBUQFOZBDNZEOBUTFOGHIJIBJEFPZQZRCJURUSPZQZRKVBVDBCJVAVCCVAUABURUSBUQES
      ZBUQFSZUBBCUGZEURFUSBUQEUCZBUQFUCZUDUEUFUHAEGTZBJUIUQJTZURGTZAVJBJLUJVJVL
      BUQJBURGVEUKVGEURGVHULUMUNAFHTZBJUIVKUSHTZAVMBJMUJVMVNBUQJBUSHVFUKVGFUSHV
      IULUMUNBUQUTEUOBUQUTFUOUP $.

    $( The domain and range of the function ` F ` .  (Contributed by Mario
       Carneiro, 23-Dec-2016.) $)
    fliftf $p |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) ) $=
      ( vy vz crn wa wss wceq cv wex wrex wfun cmpt wf wfn cdm simpr wbr cab wb
      fliftel exbidv adantr rexcom4 19.42v elisset syl biantrud bitr4id bitr3id
      wcel rexbidva bitrd abbidv df-dm eqid rnmpt 3eqtr4g sylanbrc cxp fliftrel
      df-fn rnss rnxpss sstrdi df-f ex ffun impbid1 ) AGUAZBHCUBZNZFGUCZAVSWBAV
      SOZGWAUDZGNZFPWBWCVSGUEZWAQWDAVSUFWCLRZMRZGUGZMSZLUHWGCQZBHTZLUHWFWAWCWJW
      LLWCWJWKWHDQZOZBHTZMSZWLAWJWPUIVSAWIWOMABCDWGWHEFGHIJKUJUKULWPWNMSZBHTZWC
      WLWNBMHUMAWRWLUIVSAWQWKBHABRHUTOZWQWKWMMSZOWKWKWMMUNWSWTWKWSDFUTWTKMDFUOU
      PUQURVAULUSVBVCLMGVDBLHCVTVTVEVFVGGWAVKVHWCWEEFVIZNZFWCGXAPZWEXBPAXCVSABC
      DEFGHIJKVJULGXAVLUPEFVMVNWAFGVOVHVPWAFGVQVR $.

    fliftval.4 $e |- ( x = Y -> A = C ) $.
    fliftval.5 $e |- ( x = Y -> B = D ) $.
    fliftval.6 $e |- ( ph -> Fun F ) $.
    $( The value of the function ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.) $)
    fliftval $p |- ( ( ph /\ Y e. X ) -> ( F ` C ) = D ) $=
      ( wa wceq adantr wcel wfun wbr cfv simpr eqidd anim12ci cv eqeq2d anbi12d
      wrex rspcev syl2anc wb fliftel mpbird funbrfv sylc ) AKJUAZRZIUBZEFIUCZEI
      UDFSAVAUSQTUTVBECSZFDSZRZBJUKZUTUSEESZFFSZRZVFAUSUEAVHUSVGAFUFUSEUFUGVEVI
      BKJBUHKSZVCVGVDVHVJCEEOUIVJDFFPUIUJULUMAVBVFUNUSABCDEFGHIJLMNUOTUPEFIUQUR
      $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y H $.  $d x y G $.
    $d x y R $.  $d x y S $.  $d x y T $.
    $( Equality theorem for isomorphisms.  (Contributed by NM, 17-May-2004.) $)
    isoeq1 $p |- ( H = G ->
          ( H Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) ) $=
      ( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq1 fveq1 df-isom breq12d
      bibi2d 2ralbidv anbi12d 3bitr4g ) FEIZABFJZGKZHKZCLZUHFMZUIFMZDLZNZHAOGAO
      ZPABEJZUJUHEMZUIEMZDLZNZHAOGAOZPABCDFQABCDEQUFUGUPUOVAABFERUFUNUTGHAAUFUM
      USUJUFUKUQULURDUHFESUIFESUAUBUCUDGHABCDFTGHABCDETUE $.

    $( Equality theorem for isomorphisms.  (Contributed by NM, 17-May-2004.) $)
    isoeq2 $p |- ( R = T ->
          ( H Isom R , S ( A , B ) <-> H Isom T , S ( A , B ) ) ) $=
      ( vx vy wceq wf1o cv wbr cfv wb wral wa wiso breq bibi1d df-isom 2ralbidv
      anbi2d 3bitr4g ) CEIZABFJZGKZHKZCLZUFFMUGFMDLZNZHAOGAOZPUEUFUGELZUINZHAOG
      AOZPABCDFQABEDFQUDUKUNUEUDUJUMGHAAUDUHULUIUFUGCERSUAUBGHABCDFTGHABEDFTUC
      $.

    $( Equality theorem for isomorphisms.  (Contributed by NM, 17-May-2004.) $)
    isoeq3 $p |- ( S = T ->
          ( H Isom R , S ( A , B ) <-> H Isom R , T ( A , B ) ) ) $=
      ( vx vy wceq wf1o cv wbr cfv wb wral wa wiso breq bibi2d df-isom 2ralbidv
      anbi2d 3bitr4g ) DEIZABFJZGKZHKZCLZUFFMZUGFMZDLZNZHAOGAOZPUEUHUIUJELZNZHA
      OGAOZPABCDFQABCEFQUDUMUPUEUDULUOGHAAUDUKUNUHUIUJDERSUAUBGHABCDFTGHABCEFTU
      C $.

    $( Equality theorem for isomorphisms.  (Contributed by NM, 17-May-2004.) $)
    isoeq4 $p |- ( A = C ->
          ( H Isom R , S ( A , B ) <-> H Isom R , S ( C , B ) ) ) $=
      ( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq2 raleq df-isom anbi12d
      raleqbi1dv 3bitr4g ) ACIZABFJZGKZHKZDLUFFMUGFMELNZHAOZGAOZPCBFJZUHHCOZGCO
      ZPABDEFQCBDEFQUDUEUKUJUMACBFRUIULGACUHHACSUBUAGHABDEFTGHCBDEFTUC $.

    $( Equality theorem for isomorphisms.  (Contributed by NM, 17-May-2004.) $)
    isoeq5 $p |- ( B = C ->
          ( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) ) $=
      ( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq3 anbi1d df-isom
      3bitr4g ) BCIZABFJZGKZHKZDLUDFMUEFMELNHAOGAOZPACFJZUFPABDEFQACDEFQUBUCUGU
      FBCAFRSGHABDEFTGHACDEFTUA $.
  $}

  ${
    $d y z H $.  $d y z R $.  $d y z S $.  $d y z A $.  $d y z B $.
    $d x y z $.
    nfiso.1 $e |- F/_ x H $.
    nfiso.2 $e |- F/_ x R $.
    nfiso.3 $e |- F/_ x S $.
    nfiso.4 $e |- F/_ x A $.
    nfiso.5 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for an isomorphism.  (Contributed by
       NM, 17-May-2004.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
    nfiso $p |- F/ x H Isom R , S ( A , B ) $=
      ( vy vz cv wbr cfv wral nfcv nfbr nffv wiso wf1o wb wa df-isom nff1o nfbi
      nfralxy nfan nfxfr ) BCDEFUABCFUBZLNZMNZDOZULFPZUMFPZEOZUCZMBQZLBQZUDALMB
      CDEFUEUKUTAABCFGJKUFUSALBJURAMBJUNUQAAULUMDAULRZHAUMRZSAUOUPEAULFGVATIAUM
      FGVBTSUGUHUHUIUJ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.  $d x y S $.  $d x y H $.
    $( An isomorphism is a one-to-one onto function.  (Contributed by NM,
       27-Apr-2004.) $)
    isof1o $p |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B ) $=
      ( vx vy wiso wf1o cv wbr cfv wb wral df-isom simplbi ) ABCDEHABEIFJZGJZCK
      QELRELDKMGANFANFGABCDEOP $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.  $d x y S $.  $d x y H $.
    $d x y C $.  $d x y D $.
    $( An isomorphism connects binary relations via its function values.
       (Contributed by NM, 27-Apr-2004.) $)
    isorel $p |- ( ( H Isom R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) ->
                 ( C R D <-> ( H ` C ) S ( H ` D ) ) ) $=
      ( vx vy wiso cv wbr cfv wb wral wcel wa wceq fveq2 bibi12d df-isom breq1d
      wf1o simprbi breq1 breq2 breq2d rspc2v mpan9 ) ABEFGJZHKZIKZELZUKGMZULGMZ
      FLZNZIAOHAOZCAPDAPQCDELZCGMZDGMZFLZNZUJABGUCURHIABEFGUAUDUQVCCULELZUTUOFL
      ZNHICDAAUKCRZUMVDUPVEUKCULEUEVFUNUTUOFUKCGSUBTULDRZVDUSVEVBULDCEUFVGUOVAU
      TFULDGSUGTUHUI $.
  $}

  ${
    $d x y A $.  $d x y F $.  $d x y R $.  $d x y S $.
    $( A consequence of isomorphism on two relations for a function's
       restriction.  (Contributed by Jim Kingdon, 11-Jan-2019.) $)
    isoresbr $p |- ( ( F |` A ) Isom R , S ( A , ( F " A ) ) ->
        A. x e. A A. y e. A ( x R y -> ( F ` x ) S ( F ` y ) ) ) $=
      ( cima cres wiso cv wbr cfv wi wcel wa isorel wb fvres breqan12d adantl
      bitrd biimpd ralrimivva ) CFCGZDEFCHZIZAJZBJZDKZUGFLZUHFLZEKZMABCCUFUGCNZ
      UHCNZOZOZUIULUPUIUGUELZUHUELZEKZULCUDUGUHDEUEPUOUSULQUFUMUNUQUJURUKEUGCFR
      UHCFRSTUAUBUC $.
  $}

  ${
    $d x y A $.  $d x y R $.
    $( Identity law for isomorphism.  Proposition 6.30(1) of [TakeutiZaring]
       p. 33.  (Contributed by NM, 27-Apr-2004.) $)
    isoid $p |- ( _I |` A ) Isom R , R ( A , A ) $=
      ( vx vy cid cres wiso wf1o wbr cfv wral f1oi wcel fvresi breqan12d bicomd
      cv wb wa rgen2a df-isom mpbir2an ) AABBEAFZGAAUCHCQZDQZBIZUDUCJZUEUCJZBIZ
      RZDAKCAKALUJCDAUDAMZUEAMZSUIUFUKULUGUDUHUEBAUDNAUENOPTCDAABBUCUAUB $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z H $.  $d w x y z R $.
    $d w x y z S $.
    $( Converse law for isomorphism.  Proposition 6.30(2) of [TakeutiZaring]
       p. 33.  (Contributed by NM, 27-Apr-2004.) $)
    isocnv $p |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) ) $=
      ( vx vy vz vw wf1o cv wbr cfv wb wral wa wiso wcel wceq f1ocnvfv2 adantrr
      ccnv f1ocnv adantr adantrl breq12d adantlr wf f1of syl ffvelcdm anim12dan
      breq1 fveq2 breq1d bibi12d bicom bitrdi breq2 rspc2va sylan an32s sylanl1
      breq2d bitr3d ralrimivva jca df-isom 3imtr4i ) ABEJZFKZGKZCLZVKEMZVLEMZDL
      ZNZGAOFAOZPZBAEUBZJZHKZIKZDLZWBVTMZWCVTMZCLZNZIBOHBOZPABCDEQBADCVTQVSWAWI
      VJWAVRABEUCZUDVSWHHIBBVSWBBRZWCBRZPZPWEEMZWFEMZDLZWDWGVJWMWPWDNVRVJWMPWNW
      BWOWCDVJWKWNWBSWLABWBETUAVJWLWOWCSWKABWCETUEUFUGVJBAVTUHZVRWMWPWGNZVJWAWQ
      WJBAVTUIUJWQWMVRWRWQWMPWEARZWFARZPVRWRWQWKWSWLWTBAWBVTUKBAWCVTUKULVQWRWNV
      ODLZWEVLCLZNZFGWEWFAAVKWESZVQXBXANXCXDVMXBVPXAVKWEVLCUMXDVNWNVODVKWEEUNUO
      UPXBXAUQURVLWFSZXAWPXBWGXEVOWOWNDVLWFEUNVDVLWFWECUSUPUTVAVBVCVEVFVGFGABCD
      EVHHIBADCVTVHVI $.

    $( Converse law for isomorphism.  (Contributed by Mario Carneiro,
       30-Jan-2014.) $)
    isocnv2 $p |- ( H Isom R , S ( A , B ) <->
                    H Isom `' R , `' S ( A , B ) ) $=
      ( vy vx wiso ccnv isof1o syl cv wbr cfv wb wral wa wcel vex cvv wfn f1ofn
      wf1o ralcom funfvex funfni adantr adantlr brcnvg syl2anc bibi12d ralbidva
      brcnv a1i bitr4id anbi2d df-isom 3bitr4g pm5.21nii ) ABCDEHZEAUAZABCIZDIZ
      EHZUTABEUCZVAABCDEJABEUBZKVDVEVAABVBVCEJVFKVAVEFLZGLZCMZVGENZVHENZDMZOZGA
      PFAPZQVEVHVGVBMZVKVJVCMZOZFAPZGAPZQUTVDVAVNVSVEVAVNVMFAPZGAPVSVMFGAAUDVAV
      RVTGAVAVHARZQZVQVMFAWBVGARZQZVOVIVPVLVOVIOWDVHVGCGSFSUMUNWDVKTRZVJTRZVPVL
      OWBWEWCWEAVHEVHEUEUFUGVAWCWFWAWFAVGEVGEUEUFUHVKVJTTDUIUJUKULULUOUPFGABCDE
      UQGFABVBVCEUQURUS $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d H x y $.  $d R x y $.  $d S x y $.
    $( An isomorphism from one well-order to another can be restricted on
       either well-order.  (Contributed by Mario Carneiro, 15-Jan-2013.) $)
    isores2 $p |- ( H Isom R , S ( A , B ) <->
                      H Isom R , ( S i^i ( B X. B ) ) ( A , B ) ) $=
      ( vx vy wf1o cv wbr cfv wb wral cxp wiso wcel ffvelcdm ralbidva df-isom
      wa cin f1of adantrr adantrl brinxp syl2anc anassrs bibi2d pm5.32i 3bitr4i
      wf sylan ) ABEHZFIZGIZCJZUNEKZUOEKZDJZLZGAMZFAMZTUMUPUQURDBBNUAZJZLZGAMZF
      AMZTABCDEOABCVCEOUMVBVGUMVAVFFAUMUNAPZTZUTVEGAVIUOAPZTUSVDUPUMVHVJUSVDLZU
      MABEUKZVHVJTZVKABEUBVLVMTUQBPZURBPZVKVLVHVNVJABUNEQUCVLVJVOVHABUOEQUDUQUR
      BBDUEUFULUGUHRRUIFGABCDESFGABCVCESUJ $.
  $}

  $( An isomorphism from one well-order to another can be restricted on either
     well-order.  (Contributed by Mario Carneiro, 15-Jan-2013.) $)
  isores1 $p |- ( H Isom R , S ( A , B ) <->
                    H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) $=
    ( wiso cxp cin ccnv isocnv isores2 sylib syl wf1o isof1o isoeq1 sylbi mpbid
    wb 3syl wrel f1orel wceq dfrel2 sylibr impbii ) ABCDEFZABCAAGHZDEFZUGABUHDE
    IZIZFZUIUGBADUHUJFZULUGBADCUJFZUMABCDEJBADCUJKZLBADUHUJJMUGABENZEUAZULUISZA
    BCDEOABEUBZUQUKEUCZUREUDZABUHDEUKPQTRUIABCDUKFZUGUIUNVBUIUMUNABUHDEJUOUEBAD
    CUJJMUIUPUQVBUGSZABUHDEOUSUQUTVCVAABCDEUKPQTRUF $.

  ${
    $d H a b $.  $d R a b $.  $d S a b $.  $d K a b $.  $d A a b $.
    $d B a b $.  $d X a b $.
    $( Induced isomorphism on a subset.  (Contributed by Stefan O'Rear,
       5-Nov-2014.) $)
    isores3 $p |- ( ( H Isom R , S ( A , B ) /\ K C_ A /\ X = ( H " K ) ) ->
        ( H |` K ) Isom R , S ( K , X ) ) $=
      ( va vb wiso wa wf1o cv wbr cfv wb wral ssralv wcel fvres cima wceq f1of1
      wss cres wf1 f1ores expcom syl5 adantr breqan12d adantll biimprd ralimdva
      wi bibi2d syld anim12d df-isom 3imtr4g impcom isoeq5 syl5ibrcom 3impia )
      ABCDEJZFAUDZGEFUAZUBZFGCDEFUEZJZVEVFKVJVHFVGCDVIJZVFVEVKVFABELZHMZIMZCNZV
      MEOZVNEOZDNZPZIAQZHAQZKFVGVILZVOVMVIOZVNVIOZDNZPZIFQZHFQZKVEVKVFVLWBWAWHV
      LABEUFZVFWBABEUCWIVFWBABFEUGUHUIVFWAVTHFQWHVTHFARVFVTWGHFVFVMFSZKZVTVSIFQ
      ZWGVFVTWLUOWJVSIFARUJWKVSWFIFWKVNFSZKZWFVSWNWEVRVOWJWMWEVRPVFWJWMWCVPWDVQ
      DVMFETVNFETUKULUPUMUNUQUNUQURHIABCDEUSHIFVGCDVIUSUTVAFGVGCDVIVBVCVD $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.  $d x y z w C $.  $d x y z w R $.
    $d x y z w S $.  $d x y z w T $.  $d x y z w G $.  $d x y z w H $.
    $( Composition (transitive) law for isomorphism.  Proposition 6.30(3) of
       [TakeutiZaring] p. 33.  (Contributed by NM, 27-Apr-2004.)  (Proof
       shortened by Mario Carneiro, 5-Dec-2016.) $)
    isotr $p |- ( ( H Isom R , S ( A , B ) /\ G Isom S , T ( B , C ) ) ->
               ( G o. H ) Isom R , T ( A , C ) ) $=
      ( vx vy vz vw cv wbr cfv wb wral wa wcel wceq wf1o simpl f1oco syl2anr wf
      ccom wiso f1of simprl ffvelcdmd simprr simplrr breq1 fveq2 breq1d bibi12d
      ad2antrr breq2 breq2d rspc2va syl21anc fvco3 syl2anc bitr4d bibi2d biimpd
      breq12d 2ralbidva impancom imp jca df-isom anbi12i 3imtr4i ) ABHUAZIMZJMZ
      DNZVPHOZVQHOZENZPZJAQIAQZRZBCGUAZKMZLMZENZWFGOZWGGOZFNZPZLBQKBQZRZRZACGHU
      FZUAZVRVPWPOZVQWPOZFNZPZJAQIAQZRABDEHUGZBCEFGUGZRACDFWPUGWOWQXBWNWEVOWQWD
      WEWMUBVOWCUBABCGHUCUDWDWNXBVOWNWCXBVOWNRZWCXBXEWBXAIJAAXEVPASZVQASZRZRZWA
      WTVRXIWAVSGOZVTGOZFNZWTXIVSBSVTBSWMWAXLPZXIABVPHVOABHUEZWNXHABHUHUQZXEXFX
      GUIZUJXIABVQHXOXEXFXGUKZUJVOWEWMXHULWLXMVSWGENZXJWJFNZPKLVSVTBBWFVSTZWHXR
      WKXSWFVSWGEUMXTWIXJWJFWFVSGUNUOUPWGVTTZXRWAXSXLWGVTVSEURYAWJXKXJFWGVTGUNU
      SUPUTVAXIWRXJWSXKFXIXNXFWRXJTXOXPABVPGHVBVCXIXNXGWSXKTXOXQABVQGHVBVCVGVDV
      EVHVFVIVJVKXCWDXDWNIJABDEHVLKLBCEFGVLVMIJACDFWPVLVN $.
  $}

  ${
    $d x y R $.  $d x y S $.
    $( The empty set is an ` R , S ` isomorphism from the empty set to the
       empty set.  (Contributed by Steve Rodriguez, 24-Oct-2015.) $)
    iso0 $p |- (/) Isom R , S ( (/) , (/) ) $=
      ( vx vy c0 wiso wf1o cv wbr cfv wb wral f1o0 ral0 df-isom mpbir2an ) EEAB
      EFEEEGCHZDHZAIQEJREJBIKDELZCELMSCNCDEEABEOP $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.  $d x y S $.  $d x y H $.
    $d x y D $.
    $( Isomorphisms preserve initial segments.  Proposition 6.31(2) of
       [TakeutiZaring] p. 33.  (Contributed by NM, 20-Apr-2004.) $)
    isoini $p |- ( ( H Isom R , S ( A , B ) /\ D e. A ) ->
               ( H " ( A i^i ( `' R " { D } ) ) ) =
               ( B i^i ( `' S " { ( H ` D ) } ) ) ) $=
      ( vx vy wcel wa ccnv csn cima cin cv wbr wrex wb 3syl bitrdi wiso cab cfv
      dfima2 elin wceq crn wf1o wfo isof1o f1ofo forn eleq2d wfn fvelrnb bitr3d
      f1ofn adantr cvv syl anim1i funfvex funfni eliniseg anbi12d anbi2d bitrid
      vex anbi1d anass adantl isorel fnbrfvb bicomd sylan adantrr ancom pm5.32i
      breq1 bitri exp32 com23 imp pm5.32d bitrd rexbidv2 r19.41v bitr4d abbi2dv
      wi eqtr4id ) ABDEFUAZCAIZJZFADKCLMZNZMGOZHOZFPZGWPQZHUBBEKCFUCZLMZNZGHFWP
      UDWNWTHXCWRXCIWRBIZWRXBIZJZWNWTWRBXBUEWNXFWQFUCZWRUFZGAQZWRXAEPZJZWTWNXDX
      IXEXJWLXDXIRWMWLWRFUGZIZXDXIWLABFUHZABFUIZXMXDRABDEFUJZABFUKXOXLBWRABFULU
      MSWLXNFAUNZXMXIRXPABFUQZGAWRFUOSUPURWNXQWMJXAUSIZXEXJRWLXQWMWLXNXQXPXRUTZ
      VAXSACFCFVBVCEXAWRUSHVHVDSVEWNWTXHXJJZGAQXKWNWSYAGWPAWNWQWPIZWSJZWQAIZWQC
      DPZWSJZJZYDYAJWMYCYGRWLWMYCYDYEJZWSJYGWMYBYHWSYBYDWQWOIZJWMYHWQAWOUEWMYIY
      EYDDCWQAGVHVDVFVGVIYDYEWSVJTVKWNYDYFYAWLWMYDYFYARZWJWLYDWMYJWLYDWMYJWLYDW
      MJJZYFXGXAEPZXHJZYAYKYEYLWSXHABWQCDEFVLWLYDWSXHRZWMWLXQYDYNXTXQYDJXHWSAWQ
      WRFVMVNVOVPVEYMXHYLJYAYLXHVQXHYLXJXGWRXAEVSVRVTTWAWBWCWDWEWFXHXJGAWGTWHVG
      WIWK $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y H $.
    $d x y R $.  $d x y S $.
    isoini2.1 $e |- C = ( A i^i ( `' R " { X } ) ) $.
    isoini2.2 $e |- D = ( B i^i ( `' S " { ( H ` X ) } ) ) $.
    $( Isomorphisms are isomorphisms on their initial segments.  (Contributed
       by Mario Carneiro, 29-Mar-2014.) $)
    isoini2 $p |- ( ( H Isom R , S ( A , B ) /\ X e. A ) ->
                    ( H |` C ) Isom R , S ( C , D ) ) $=
      ( vx vy wiso wcel wf1o wbr cfv wb wral cima wa cres cv wf1 wss isof1o syl
      f1of1 adantr ccnv csn inss1 eqsstri f1ores sylancl isoini imaeq2i 3eqtr4g
      cin wceq f1oeq3 mpbid df-isom simprbi ssralv mpsyl fvres breqan12d bibi2d
      ralimdv ralbidva ralbiia sylibr sylanbrc ) ABEFGMZHANZUAZCDGCUBZOZKUCZLUC
      ZEPZVTVRQZWAVRQZFPZRZLCSZKCSZCDEFVRMVQCGCTZVROZVSVQABGUDZCAUEZWJVOWKVPVOA
      BGOZWKABEFGUFABGUHUGUICAEUJHUKTZUSZAIAWNULUMZABCGUNUOVQWIDUTWJVSRVQGWOTBF
      UJHGQUKTUSWIDABHEFGUPCWOGIUQJURWIDCVRVAUGVBVQWBVTGQZWAGQZFPZRZLCSZKCSZWHW
      LVQXAKASZXBWPWLVQWTLASZKASZXCWPVOXEVPVOWMXEKLABEFGVCVDUIWLXDXAKAWTLCAVEVJ
      VFXAKCAVEVFWGXAKCVTCNZWFWTLCXFWACNZUAWEWSWBXFXGWCWQWDWRFVTCGVGWACGVGVHVIV
      KVLVMKLCDEFVRVCVN $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z H $.  $d x y z ph $.  $d x y z R $.
    $d x y z S $.
    isofrlem.1 $e |- ( ph -> H Isom R , S ( A , B ) ) $.
    isofrlem.2 $e |- ( ph -> ( H " x ) e. _V ) $.
    $( Lemma for ~ isose .  (Contributed by Mario Carneiro, 23-Jun-2015.) $)
    isoselem $p |- ( ph -> ( R Se A -> S Se B ) ) $=
      ( vy vz cv csn cima cin cvv wcel wral wi wceq wse ccnv cfv dfse2 r19.21bi
      biimpi expcom adantl imaeq2 eleq1d imbi2d vtoclg com12 adantr wiso isoini
      wa sylan sylibd syld ralrimdva crn wf1o wfn wb isof1o sneq imaeq2d ineq2d
      f1ofn ralrn 4syl wfo f1ofo forn raleqdv bitr3d imbitrrdi ) ACEUAZDFUBZJLZ
      MZNZOZPQZJDRZDFUAAVSDVTKLZGUCZMZNZOZPQZKCRZWFAVSWLKCAWGCQZUQZVSCEUBWGMNOZ
      PQZWLWNVSWQSAVSWNWQVSWQKCVSWQKCRKCEUDUFUEUGUHWOWQGWPNZPQZWLAWQWSSWNWQAWSA
      GBLZNZPQZSAWSSBWPPWTWPTZXBWSAXCXAWRPWTWPGUIUJUKIULUMUNWOWRWKPACDEFGUOZWNW
      RWKTHCDWGEFGUPURUJUSUTVAAWEJGVBZRZWMWFAXDCDGVCZGCVDXFWMVEHCDEFGVFZCDGVJWE
      WLJKCGWAWHTZWDWKPXIWCWJDXIWBWIVTWAWHVGVHVIUJVKVLAWEJXEDAXDXGCDGVMXEDTHXHC
      DGVNCDGVOVLVPVQUSJDFUDVR $.
  $}

  ${
    $d x A $.  $d x B $.  $d x H $.  $d x R $.  $d x S $.
    $( An isomorphism preserves set-like relations.  (Contributed by Mario
       Carneiro, 23-Jun-2015.) $)
    isose $p |- ( H Isom R , S ( A , B ) -> ( R Se A <-> S Se B ) ) $=
      ( vx wiso wse wf1o wfun cima cvv wcel isof1o f1ofun vex funimaex isoselem
      id cv 3syl ccnv isocnv 4syl impbid ) ABCDEGZACHBDHUFFABCDEUFSUFABEIEJEFTZ
      KLMABCDENABEOEUGFPZQUARUFFBADCEUBZABCDEUCZUFBADCUIGBAUIIUIJUIUGKLMUJBADCU
      INBAUIOUIUGUHQUDRUE $.
  $}

  ${
    $d H a b c d e f $.  $d R a b c d e f $.  $d S a b c d e f $.
    $d A a b c d e f $.  $d B a b c d e f $.
    $( Lemma for ~ isopo .  (Contributed by Stefan O'Rear, 16-Nov-2014.) $)
    isopolem $p |- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) ) $=
      ( va vb vc vd ve vf cv wbr wa wi wral wcel ex wb anbi12d wiso wpo w3a cfv
      wn wf1o isof1o f1of ffvelcdm 3anim123d 3syl imp wceq breq12 anidms notbid
      breq1 anbi1d imbi12d breq2 imbi1d anbi2d rspc3v syl simpl simpr1 syl12anc
      wf isorel simpr2 simpr3 sylibrd com23 imp31 ralrimivvva df-po 3imtr4g ) A
      BCDEUAZFLZVSDMZUEZVSGLZDMZWBHLZDMZNZVSWDDMZOZNZHBPGBPFBPZILZWKCMZUEZWKJLZ
      CMZWNKLZCMZNZWKWPCMZOZNZKAPJAPIAPZBDUBACUBVRWJXBVRWJNXAIJKAAAVRWJWKAQZWNA
      QZWPAQZUCZXAVRXFWJXAVRXFWJXAOVRXFNZWJWKEUDZXHDMZUEZXHWNEUDZDMZXKWPEUDZDMZ
      NZXHXMDMZOZNZXAXGXHBQZXKBQZXMBQZUCZWJXROVRXFYBVRABEUFABEVHZXFYBOABCDEUGAB
      EUHYCXCXSXDXTXEYAYCXCXSABWKEUIRYCXDXTABWNEUIRYCXEYAABWPEUIRUJUKULWIXRXJXH
      WBDMZWENZXHWDDMZOZNXJXLXKWDDMZNZYFOZNFGHXHXKXMBBBVSXHUMZWAXJWHYGYKVTXIYKV
      TXISVSXHVSXHDUNUOUPYKWFYEWGYFYKWCYDWEVSXHWBDUQURVSXHWDDUQUSTWBXKUMZYGYJXJ
      YLYEYIYFYLYDXLWEYHWBXKXHDUTWBXKWDDUQTVAVBWDXMUMZYJXQXJYMYIXOYFXPYMYHXNXLW
      DXMXKDUTVBWDXMXHDUTUSVBVCVDXGWMXJWTXQXGWLXIXGVRXCXCWLXISVRXFVEZVRXCXDXEVF
      ZYOABWKWKCDEVIVGUPXGWRXOWSXPXGWOXLWQXNXGVRXCXDWOXLSYNYOVRXCXDXEVJZABWKWNC
      DEVIVGXGVRXDXEWQXNSYNYPVRXCXDXEVKZABWNWPCDEVIVGTXGVRXCXEWSXPSYNYOYQABWKWP
      CDEVIVGUSTVLRVMVNVORFGHBDVPIJKACVPVQ $.

    $( An isomorphism preserves partial ordering.  (Contributed by Stefan
       O'Rear, 16-Nov-2014.) $)
    isopo $p |- ( H Isom R , S ( A , B ) -> ( R Po A <-> S Po B ) ) $=
      ( wiso wpo ccnv wi isocnv isopolem syl impbid ) ABCDEFZACGZBDGZNBADCEHZFO
      PIABCDEJBADCQKLABCDEKM $.
  $}

  ${
    $d H a b c x y z $.  $d R a b c x y z $.  $d S a b c x y z $.
    $d A a b c x y z $.  $d B a b c x y z $.
    $( Lemma for ~ isoso .  (Contributed by Stefan O'Rear, 16-Nov-2014.) $)
    isosolem $p |- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) ) $=
      ( vx vy vz va vb vc cv wbr wo wi wral wa wcel cfv ffvelcdm wpo wor df-3an
      wiso isopolem w3a wf1o wf isof1o f1of ex 3anim123d 3syl wceq breq1 orbi1d
      imp imbi12d breq2 orbi2d orbi12d imbi2d rspc3v wb isorel 3adantr3 ancom2s
      syl 3adantr2 3adantr1 sylibrd anassrs ralrimdva ralrimdvva anim12d df-iso
      sylan2br 3imtr4g ) ABCDEUDZBDUAZFLZGLZDMZWAHLZDMZWDWBDMZNZOZHBPGBPFBPZQAC
      UAZILZJLZCMZWKKLZCMZWNWLCMZNZOZKAPZJAPIAPZQBDUBACUBVSVTWJWIWTABCDEUEVSWIW
      SIJAAVSWKARZWLARZQZQWIWRKAVSXCWNARZWIWROZXCXDQVSXAXBXDUFZXEXAXBXDUCVSXFQZ
      WIWKESZWLESZDMZXHWNESZDMZXKXIDMZNZOZWRXGXHBRZXIBRZXKBRZUFZWIXOOVSXFXSVSAB
      EUGABEUHZXFXSOABCDEUIABEUJXTXAXPXBXQXDXRXTXAXPABWKETUKXTXBXQABWLETUKXTXDX
      RABWNETUKULUMUQWHXOXHWBDMZXHWDDMZWFNZOXJYBWDXIDMZNZOFGHXHXIXKBBBWAXHUNZWC
      YAWGYCWAXHWBDUOYFWEYBWFWAXHWDDUOUPURWBXIUNZYAXJYCYEWBXIXHDUSYGWFYDYBWBXIW
      DDUSUTURWDXKUNZYEXNXJYHYBXLYDXMWDXKXHDUSWDXKXIDUOVAVBVCVHXGWMXJWQXNVSXAXB
      WMXJVDXDABWKWLCDEVEVFXGWOXLWPXMVSXAXDWOXLVDXBABWKWNCDEVEVIVSXBXDWPXMVDZXA
      VSXDXBYIABWNWLCDEVEVGVJVAURVKVQVLVMVNVOFGHBDVPIJKACVPVR $.
  $}

  $( An isomorphism preserves strict ordering.  (Contributed by Stefan O'Rear,
     16-Nov-2014.) $)
  isoso $p |- ( H Isom R , S ( A , B ) -> ( R Or A <-> S Or B ) ) $=
    ( wiso wor ccnv wi isocnv isosolem syl impbid ) ABCDEFZACGZBDGZNBADCEHZFOPI
    ABCDEJBADCQKLABCDEKM $.

  ${
    $d x y z w v u A $.  $d x y v u B $.  $d x y z w v u H $.
    $d x y z w v u R $.  $d v u S $.
    $( Any one-to-one onto function determines an isomorphism with an induced
       relation ` S ` .  Proposition 6.33 of [TakeutiZaring] p. 34.
       (Contributed by NM, 30-Apr-2004.) $)
    f1oiso $p |- ( ( H : A -1-1-onto-> B /\ S = { <. z , w >. |
     E. x e. A E. y e. A ( ( z = ( H ` x ) /\ w = ( H ` y ) ) /\ x R y ) } ) ->
                  H Isom R , S ( A , B ) ) $=
      ( vv vu cv cfv wceq wa wbr wrex wb wcel anbi1d wf1o copab wral wiso simpl
      wf1 f1of1 cop df-br eleq2 wfn cvv funfvex funfni anim12dan eqeq1 2rexbidv
      f1fn anbi2d opelopabg syl sylan anass f1fveq equcom bitrdi anassrs bitrid
      rexbidv r19.42v rexbidva breq1 ceqsrexv adantl sylan9bb anandis sylan9bbr
      bitrd breq2 an32s bitr2id ralrimivva df-isom sylanbrc ) EFIUAZHCLZALZIMZN
      ZDLZBLZIMZNZOZWGWKGPZOZBEQAEQZCDUBZNZOWEJLZKLZGPZWTIMZXAIMZHPZRZKEUCJEUCZ
      EFGHIUDWEWSUEWEEFIUFZWSXGEFIUGXHWSOZXFJKEEXEXCXDUHZHSZXIWTESZXAESZOZOXBXC
      XDHUIXHXNWSXKXBRWSXKXJWRSZXHXNOZXBHWRXJUJXPXOXCWHNZXDWLNZOZWOOZBEQZAEQZXB
      XHIEUKZXNXOYBRZEFIURYCXNOXCULSZXDULSZOYDYCXLYEXMYFYEEWTIWTIUMUNYFEXAIXAIU
      MUNUOWQXQWMOZWOOZBEQAEQYBCDXCXDULULWFXCNZWPYHABEEYIWNYGWOYIWIXQWMWFXCWHUP
      TTUQWJXDNZYHXTABEEYJYGXSWOYJWMXRXQWJXDWLUPUSTUQUTVAVBXHXLXMYBXBRXHXLOZYBX
      RWTWKGPZOZBEQZXHXMOZXBYKYBWGWTNZXRWOOZBEQZOZAEQZYNYKYAYSAEYKWGESZOZYAYPYQ
      OZBEQYSUUBXTUUCBEXTXQYQOUUBUUCXQXRWOVCUUBXQYPYQXHXLUUAXQYPRXHXLUUAOOXQWTW
      GNYPEFWTWGIVDJAVEVFVGTVHVIYPYQBEVJVFVKXLYTYNRXHYRYNAWTEYPYQYMBEYPWOYLXRWG
      WTWKGVLUSVIVMVNVRYOYNWKXANZYLOZBEQZXBYOYMUUEBEYOWKESZOXRUUDYLXHXMUUGXRUUD
      RXHXMUUGOOXRXAWKNUUDEFXAWKIVDKBVEVFVGTVKXMUUFXBRXHYLXBBXAEWKXAWTGVSVMVNVR
      VOVPVRVQVTWAWBVBJKEFGHIWCWD $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.  $d H w x y z $.  $d R w x y z $.
    f1oiso2.1 $e |- S = { <. x , y >. |
      ( ( x e. B /\ y e. B ) /\ ( `' H ` x ) R ( `' H ` y ) ) } $.
    $( Any one-to-one onto function determines an isomorphism with an induced
       relation ` S ` .  (Contributed by Mario Carneiro, 9-Mar-2013.) $)
    f1oiso2 $p |- ( H : A -1-1-onto-> B -> H Isom R , S ( A , B ) ) $=
      ( vz vw cv cfv wceq wa wbr wrex wcel 3adant3 eqcomd syl2anc wf1o wiso w3a
      copab ccnv f1ocnvdm adantrr f1ocnvfv2 anim12dan simp3 fveq2 eqeq2d anbi2d
      adantrl breq2 anbi12d rspcev syl12anc anbi1d breq1 rexbidv 3expib simp3ll
      simp1 simp2l ffvelcdmda eqeltrd simp3lr simp2r simp3r wi f1ocnvfv 3brtr4d
      f1of mpd jca31 3exp rexlimdvv impbid opabbidv eqtrid f1oiso mpdan ) CDGUA
      ZFAKZIKZGLZMZBKZJKZGLZMZNZWFWJEOZNZJCPZICPZABUDZMCDEFGUBWDFWEDQZWIDQZNZWE
      GUEZLZWIXBLZEOZNZABUDWRHWDXFWQABWDXFWQWDXAXEWQWDXAXEUCZXCCQZWEXCGLZMZWLNZ
      XCWJEOZNZJCPZWQWDXAXHXEWDWSXHWTCDWEGUFUGRXGXDCQZXJWIXDGLZMZNZXEXNWDXAXOXE
      WDWTXOWSCDWIGUFUNRWDXAXRXEWDWSXJWTXQWDWSNXIWECDWEGUHSWDWTNXPWICDWIGUHSUIR
      WDXAXEUJXMXRXENJXDCWJXDMZXKXRXLXEXSWLXQXJXSWKXPWIWJXDGUKULUMWJXDXCEUOUPUQ
      URWPXNIXCCWFXCMZWOXMJCXTWMXKWNXLXTWHXJWLXTWGXIWEWFXCGUKULUSWFXCWJEUTUPVAU
      QTVBWDWOXFIJCCWDWFCQZWJCQZNZWOXFWDYCWOUCZWSWTXEYDWEWGDWHWLWNWDYCVCZYDWDYA
      WGDQWDYCWOVDZWDYAYBWOVEZWDCDWFGCDGVNZVFTVGYDWIWKDWHWLWNWDYCVHZYDWDYBWKDQY
      FWDYAYBWOVIZWDCDWJGYHVFTVGYDWFWJXCXDEWDYCWMWNVJYDWGWEMZXCWFMZYDWEWGYESYDW
      DYAYKYLVKYFYGCDWFWEGVLTVOYDWKWIMZXDWJMZYDWIWKYISYDWDYBYMYNVKYFYJCDWJWIGVL
      TVOVMVPVQVRVSVTWAIJABCDEFGWBWC $.
  $}

  $( A different way to write ` F ` is a function.  (Contributed by Thierry
     Arnoux, 7-Dec-2016.) $)
  fdmrn $p |- ( Fun F <-> F : dom F --> ran F ) $=
    ( cdm crn wf wfn wfun wss ssid df-f mpbiran2 wceq eqid df-fn bitr2i ) ABZAC
    ZADZAOEZAFZQRPPGPHOPAIJRSOOKOLAOMJN $.

  ${
    rinvbij.1 $e |- Fun F $.
    rinvbij.2 $e |- `' F = F $.
    rinvbij.3a $e |- ( F " A ) C_ B $.
    rinvbij.3b $e |- ( F " B ) C_ A $.
    rinvbij.4a $e |- A C_ dom F $.
    rinvbij.4b $e |- B C_ dom F $.
    $( Sufficient conditions for the restriction of an involution to be a
       bijection.  (Contributed by Thierry Arnoux, 7-Dec-2016.) $)
    rinvf1o $p |- ( F |` A ) : A -1-1-onto-> B $=
      ( cima cres wf1o cdm crn wf1 wss wfun mpbi mp2an wb wf fdmrn funeqi mpbir
      ccnv df-f1 mpbir2an f1ores funimass3 imaeq1i sseqtri eqssi f1oeq3 ax-mp
      wceq ) ACAJZCAKZLZABUQLZCMZCNZCOZAUTPURVBUTVACUAZCUEZQZCQZVCDCUBRVEVFDVDC
      EUCUDUTVACUFUGHUTVAACUHSUPBUOURUSTUPBFBVDAJZUPCBJAPZBVGPZGVFBUTPVHVITDIBA
      CUISRVDCAEUJUKULUPBAUQUMUNR $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Cantor's Theorem
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y A $.  $d x y F $.
    canth.1 $e |- A e. _V $.
    $( No set ` A ` is equinumerous to its power set (Cantor's theorem), i.e.,
       no function can map ` A ` onto its power set.  Compare Theorem 6B(b) of
       [Enderton] p. 132.  (Use ~ nex if you want the form
       ` -. E. f f : A -onto-> ~P A ` .)  (Contributed by NM, 7-Aug-1994.)
       (Revised by Noah R Kingdon, 23-Jul-2024.) $)
    canth $p |- -. F : A -onto-> ~P A $=
      ( vx vy cpw wfo cv cfv wcel wn crab crn ssrab2 elpwi2 forn eleqtrrid wceq
      cvv wb wrex wa pm5.19 eleq2 weq fveq2 eleq12d notbid elrab3 sylan9bbr mto
      id imnani nrex wfn fofn fvelrnb syl mtbiri pm2.65i ) AAFZBGZDHZVCBIZJZKZD
      ALZBMZJZVBVGVAVHVGASCVFDANOAVABPQVBVIEHZBIZVGRZEAUAZVLEAVJAJZVLVNVLUBVJVK
      JZVOKZTVOUCVLVOVJVGJVNVPVKVGVJUDVFVPDVJADEUEZVEVOVQVCVJVDVKVQULVCVJBUFUGU
      HUIUJUKUMUNVBBAUOVIVMTAVABUPEAVGBUQURUSUT $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Restricted iota (description binder)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c iota_ $.

  $( Extend class notation with restricted description binder. $)
  crio $a class ( iota_ x e. A ph ) $.

  $( Define restricted description binder.  In case there is no unique ` x `
     such that ` ( x e. A /\ ph ) ` holds, it evaluates to the empty set.  See
     also comments for ~ df-iota .  (Contributed by NM, 15-Sep-2011.)  (Revised
     by Mario Carneiro, 15-Oct-2016.)  (Revised by NM, 2-Sep-2018.) $)
  df-riota $a |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) ) $.

  ${
    $d x ph $.
    riotaeqdv.1 $e |- ( ph -> A = B ) $.
    $( Formula-building deduction for iota.  (Contributed by NM,
       15-Sep-2011.) $)
    riotaeqdv $p |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) ) $=
      ( cv wcel wa cio crio eleq2d anbi1d iotabidv df-riota 3eqtr4g ) ACGZDHZBI
      ZCJQEHZBIZCJBCDKBCEKASUACARTBADEQFLMNBCDOBCEOP $.
  $}

  ${
    $d x ph $.
    riotabidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building deduction for restricted iota.  (Contributed by NM,
       15-Sep-2011.) $)
    riotabidv $p |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) ) $=
      ( cv wcel wa cio crio biidd anbi12d iotabidv df-riota 3eqtr4g ) ADGEHZBIZ
      DJQCIZDJBDEKCDEKARSDAQQBCAQLFMNBDEOCDEOP $.
  $}

  ${
    $d x ph $.
    riotaeqbidv.1 $e |- ( ph -> A = B ) $.
    riotaeqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equality deduction for restricted universal quantifier.  (Contributed by
       NM, 15-Sep-2011.) $)
    riotaeqbidv $p |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) ) $=
      ( crio riotabidv riotaeqdv eqtrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.
  $}

  ${
    $d A x $.
    $( Restricted iota is a set.  (Contributed by Jim Kingdon, 15-Jun-2020.) $)
    riotaexg $p |- ( A e. V -> ( iota_ x e. A ps ) e. _V ) $=
      ( wcel crio cv cio cvv df-riota cuni uniexg wss iotass elssuni adantr mpg
      wa wi a1i ssexd eqeltrid ) CDEZABCFBGZCEZARZBHZIABCJUCUGCKZICDLUGUHMZUCUF
      UDUHMZSUIBUFBUHNUEUJAUDCOPQTUAUB $.
  $}

  ${
    $d A x $.
    $( Set existence of an iota expression in which all values are contained
       within a set.  (Contributed by Jim Kingdon, 28-Jun-2025.) $)
    iotaexel $p |- ( ( A e. V /\ A. x ( ph -> x e. A ) )
        -> ( iota x ph ) e. _V ) $=
      ( wcel cv wi wal wa crio cio cvv df-riota wceq pm4.71r albii iotabi sylbi
      wb adantl eqtr4id riotaexg adantr eqeltrrd ) CDEZABFCEZGZBHZIZABCJZABKZLU
      IUJUFAIZBKZUKABCMUHUKUMNZUEUHAULSZBHUNUGUOBAUFOPAULBQRTUAUEUJLEUHABCDUBUC
      UD $.
  $}

  $( An iota restricted to the universe is unrestricted.  (Contributed by NM,
     18-Sep-2011.) $)
  riotav $p |- ( iota_ x e. _V ph ) = ( iota x ph ) $=
    ( cvv crio cv wcel wa cio df-riota vex biantrur iotabii eqtr4i ) ABCDBECFZA
    GZBHABHABCIAOBNABJKLM $.

  $( Restricted iota in terms of class union.  (Contributed by NM,
     11-Oct-2011.) $)
  riotauni $p |- ( E! x e. A ph
                    -> ( iota_ x e. A ph ) = U. { x e. A | ph } ) $=
    ( wreu cv wcel wa cio cab cuni crio crab wceq df-reu iotauni sylbi df-riota
    weu df-rab unieqi 3eqtr4g ) ABCDZBECFAGZBHZUCBIZJZABCKABCLZJUBUCBRUDUFMABCN
    UCBOPABCQUGUEABCSTUA $.

  ${
    $d x A $.
    $( The abstraction variable in a restricted iota descriptor isn't free.
       (Contributed by NM, 12-Oct-2011.)  (Revised by Mario Carneiro,
       15-Oct-2016.) $)
    nfriota1 $p |- F/_ x ( iota_ x e. A ph ) $=
      ( crio cv wcel wa cio df-riota nfiota1 nfcxfr ) BABCDBECFAGZBHABCILBJK $.
  $}

  ${
    $d x y $.
    nfriotadxy.1 $e |- F/ y ph $.
    nfriotadxy.2 $e |- ( ph -> F/ x ps ) $.
    nfriotadxy.3 $e |- ( ph -> F/_ x A ) $.
    $( Deduction version of ~ nfriota .  (Contributed by Jim Kingdon,
       12-Jan-2019.) $)
    nfriotadxy $p |- ( ph -> F/_ x ( iota_ y e. A ps ) ) $=
      ( crio cv wcel wa cio df-riota wnfc nfcv a1i nfeld nfand nfiotadw nfcxfrd
      ) ACBDEIDJZEKZBLZDMBDENAUDCDFAUCBCACUBECUBOACUBPQHRGSTUA $.
  $}

  ${
    $d x y $.
    nfriota.1 $e |- F/ x ph $.
    nfriota.2 $e |- F/_ x A $.
    $( A variable not free in a wff remains so in a restricted iota descriptor.
       (Contributed by NM, 12-Oct-2011.) $)
    nfriota $p |- F/_ x ( iota_ y e. A ph ) $=
      ( crio wnfc wtru nftru wnf a1i nfriotadxy mptru ) BACDGHIABCDCJABKIELBDHI
      FLMN $.
  $}

  ${
    $d x y A $.  $d y ph $.  $d x ps $.
    cbvriotavw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable in a restricted description binder.  Version of
       ~ cbvriotav with a disjoint variable condition.  (Contributed by NM,
       18-Mar-2013.)  (Revised by GG, 30-Sep-2024.) $)
    cbvriotavw $p |- ( iota_ x e. A ph ) = ( iota_ y e. A ps ) $=
      ( cv wcel wa cio crio weq eleq1w anbi12d cbviotavw df-riota 3eqtr4i ) CGE
      HZAIZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $.
  $}

  ${
    $d x z A $.  $d y z A $.  $d z ph $.  $d z ps $.
    cbvriota.1 $e |- F/ y ph $.
    cbvriota.2 $e |- F/ x ps $.
    cbvriota.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable in a restricted description binder.  (Contributed
       by NM, 18-Mar-2013.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    cbvriota $p |- ( iota_ x e. A ph ) = ( iota_ y e. A ps ) $=
      ( vz cv wcel wa cio crio wsb weq eleq1 anbi12d nfv nfan nfs1v sbie bitrdi
      sbequ12 cbviota sbequ nfsb eqtri df-riota 3eqtr4i ) CJZEKZALZCMZDJZEKZBLZ
      DMZACENBDENUNIJZEKZACIOZLZIMURUMVBCICIPULUTAVAUKUSEQACIUDRUMISUTVACUTCSAC
      IUATUEVBUQIDIDPZUTUPVABUSUOEQVCVAACDOBAIDCUFABCDGHUBUCRUTVADUTDSACIDFUGTU
      QISUEUHACEUIBDEUIUJ $.
  $}

  ${
    $d x A $.  $d y A $.  $d y ph $.  $d x ps $.
    cbvriotav.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable in a restricted description binder.  (Contributed
       by NM, 18-Mar-2013.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    cbvriotav $p |- ( iota_ x e. A ph ) = ( iota_ y e. A ps ) $=
      ( nfv cbvriota ) ABCDEADGBCGFH $.
  $}

  ${
    $d y z A $.  $d x z B $.  $d z ph $.  $d x y $.
    $( Interchange class substitution and restricted description binder.
       (Contributed by NM, 24-Feb-2013.) $)
    csbriotag $p |- ( A e. V
      -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) ) $=
      ( vz cv crio csb wsb wceq wsbc csbeq1 dfsbcq2 riotabidv eqeq12d vex nfs1v
      nfcv nfriota weq sbequ12 csbief vtoclg ) BGHZACEIZJZABGKZCEIZLBDUGJZABDMZ
      CEIZLGDFUFDLZUHUKUJUMBUFDUGNUNUIULCEABGDOPQBUFUGUJGRUIBCEABGSBETUABGUBAUI
      CEABGUCPUDUE $.
  $}

  $( Membership law for "the unique element in ` A ` such that ` ph ` ."

     (Contributed by NM, 21-Aug-2011.)  (Revised by Mario Carneiro,
     23-Dec-2016.) $)
  riotacl2 $p |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } ) $=
    ( wreu cv wcel wa cio cab crio crab weu df-reu iotacl sylbi df-riota df-rab
    3eltr4g ) ABCDZBECFAGZBHZTBIZABCJABCKSTBLUAUBFABCMTBNOABCPABCQR $.

  ${
    $d x A $.
    $( Closure of restricted iota.  (Contributed by NM, 21-Aug-2011.) $)
    riotacl $p |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A ) $=
      ( wreu crab crio ssrab2 riotacl2 sselid ) ABCDABCECABCFABCGABCHI $.
  $}

  $( Substitution law for descriptions.  (Contributed by NM, 23-Aug-2011.)
     (Proof shortened by Mario Carneiro, 24-Dec-2016.) $)
  riotasbc $p |- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph ) $=
    ( wreu crio cab wcel wsbc crab rabssab riotacl2 sselid df-sbc sylibr ) ABCD
    ZABCEZABFZGABPHOABCIQPABCJABCKLABPMN $.

  ${
    $d x ph $.
    riotabidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal restricted class abstractions (deduction
       form).  ( ~ rabbidva analog.)  (Contributed by NM, 17-Jan-2012.) $)
    riotabidva $p |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) ) $=
      ( cv wcel wa cio crio pm5.32da iotabidv df-riota 3eqtr4g ) ADGEHZBIZDJPCI
      ZDJBDEKCDEKAQRDAPBCFLMBDENCDENO $.
  $}

  ${
    riotabiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
    $( Equivalent wff's yield equal restricted iotas (inference form).
       ( ~ rabbiia analog.)  (Contributed by NM, 16-Jan-2012.) $)
    riotabiia $p |- ( iota_ x e. A ph ) = ( iota_ x e. A ps ) $=
      ( cvv wceq crio eqid cv wcel wb adantl riotabidva ax-mp ) FFGZACDHBCDHGFI
      PABCDCJDKABLPEMNO $.
  $}

  ${
    $d x A $.
    $( Property of restricted iota.  Compare ~ iota1 .  (Contributed by Mario
       Carneiro, 15-Oct-2016.) $)
    riota1 $p |- ( E! x e. A ph ->
      ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) ) $=
      ( wreu cv wcel wa cio wceq crio weu wb df-reu iota1 sylbi df-riota eqeq1i
      bitr4di ) ABCDZBEZCFAGZUABHZTIZABCJZTISUABKUAUCLABCMUABNOUDUBTABCPQR $.
  $}

  $( Property of iota.  (Contributed by NM, 23-Aug-2011.) $)
  riota1a $p |- ( ( x e. A /\ E! x e. A ph ) ->
          ( ph <-> ( iota x ( x e. A /\ ph ) ) = x ) ) $=
    ( cv wcel wa wreu cio wceq ibar weu wb df-reu iota1 sylbi sylan9bb ) BDZCEZ
    ARAFZABCGZSBHQIZRAJTSBKSUALABCMSBNOP $.

  ${
    $d x A $.
    riota2df.1 $e |- F/ x ph $.
    riota2df.2 $e |- ( ph -> F/_ x B ) $.
    riota2df.3 $e |- ( ph -> F/ x ch ) $.
    riota2df.4 $e |- ( ph -> B e. A ) $.
    riota2df.5 $e |- ( ( ph /\ x = B ) -> ( ps <-> ch ) ) $.
    $( A deduction version of ~ riota2f .  (Contributed by NM, 17-Feb-2013.)
       (Revised by Mario Carneiro, 15-Oct-2016.) $)
    riota2df $p |- ( ( ph /\ E! x e. A ps ) ->
            ( ch <-> ( iota_ x e. A ps ) = B ) ) $=
      ( wreu wa cv wcel cio wceq crio adantr simpr weu df-reu eqeltrd biantrurd
      sylib adantlr bitr3d nfreu1 nfan wnf wnfc iota2df df-riota eqeq1i bitr4di
      wb ) ABDELZMZCDNZEOZBMZDPZFQBDERZFQURVACDFEAFEOZUQJSZURUQVADUAAUQTBDEUBUE
      URUSFQZMZBVACVGUTBVGUSFEURVFTURVDVFVESUCUDAVFBCUPUQKUFUGAUQDGBDEUHUIACDUJ
      UQISADFUKUQHSULVCVBFBDEUMUNUO $.
  $}

  ${
    $d x A $.
    riota2f.1 $e |- F/_ x B $.
    riota2f.2 $e |- F/ x ps $.
    riota2f.3 $e |- ( x = B -> ( ph <-> ps ) ) $.
    $( This theorem shows a condition that allows us to represent a descriptor
       with a class expression ` B ` .  (Contributed by NM, 23-Aug-2011.)
       (Revised by Mario Carneiro, 15-Oct-2016.) $)
    riota2f $p |- ( ( B e. A /\ E! x e. A ph ) ->
            ( ps <-> ( iota_ x e. A ph ) = B ) ) $=
      ( wcel nfel1 wnfc a1i wnf id cv wceq wb adantl riota2df ) EDIZABCDECEDFJC
      EKTFLBCMTGLTNCOEPABQTHRS $.
  $}

  ${
    $d x ps $.  $d x A $.  $d x B $.
    riota2.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
    $( This theorem shows a condition that allows us to represent a descriptor
       with a class expression ` B ` .  (Contributed by NM, 23-Aug-2011.)
       (Revised by Mario Carneiro, 10-Dec-2016.) $)
    riota2 $p |- ( ( B e. A /\ E! x e. A ph ) ->
            ( ps <-> ( iota_ x e. A ph ) = B ) ) $=
      ( nfcv nfv riota2f ) ABCDECEGBCHFI $.
  $}

  ${
    $d I a $.  $d J a $.  $d V a $.  $d X a $.  $d Y a $.
    riotaeqimp.i $e |- I = ( iota_ a e. V X = A ) $.
    riotaeqimp.j $e |- J = ( iota_ a e. V Y = A ) $.
    riotaeqimp.x $e |- ( ph -> E! a e. V X = A ) $.
    riotaeqimp.y $e |- ( ph -> E! a e. V Y = A ) $.
    $( If two restricted iota descriptors for an equality are equal, then the
       terms of the equality are equal.  (Contributed by AV, 6-Dec-2020.) $)
    riotaeqimp $p |- ( ( ph /\ I = J ) -> X = Y ) $=
      ( wceq wa crio csb wb wcel nfcvd adantl eqcomi eqeq2i bicomd wreu riotacl
      a1i biimpa eqeq1i syl eqeltrid nfcsb1d nfeqd id cv csbeq1a eqeq2d syl2anc
      nfv riota2df bitrid wi bitrdi adantr csbeq1 eqcoms eqeq12 syl5ibrcom expd
      eqcom ancoms sylbird mp2d ) ACDMZNZCGBMZHEOZMZFHDBPZMZFGMZAVMVQAVQVMVQVMQ
      AVPDCDVPJUAUBUFUCUGAVMVSVMFBMZHEOZDMZAVSCWBDIUHADERZWAHEUDZWCVSQADVPEJAVO
      HEUDZVPERLVOHEUEUIUJKWDWENVSWCWDWAVSHEDWDHURWDHDSZWDHFVRWDHFSWDHDBWGUKULW
      DUMHUNZDMZWAVSQWDWIBVRFHDBUOUPTUSUCUQUTUGVNVQGHCBPZMZVSVTVAZAWKVQQVMAWKVP
      CMZVQACERZWFWKWMQACWBEIAWEWBERKWAHEUEUIUJLWNVOWKHECWNHURWNHCSZWNHGWJWNHGS
      WNHCBWOUKULWNUMWHCMZVOWKQWNWPBWJGHCBUOUPTUSUQVPCVIVBVCVMWKWLVAAVMWKVSVTVM
      VTWKVSNVRWJMZWQDCHDCBVDVEVSWKVTWQQFVRGWJVFVJVGVHTVKVL $.
  $}

  ${
    $d x A $.
    riotaprop.0 $e |- F/ x ps $.
    riotaprop.1 $e |- B = ( iota_ x e. A ph ) $.
    riotaprop.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
    $( Properties of a restricted definite description operator.  Todo
       ( ~ df-riota update): can some uses of ~ riota2f be shortened with this?
       (Contributed by NM, 23-Nov-2013.) $)
    riotaprop $p |- ( E! x e. A ph -> ( B e. A /\ ps ) ) $=
      ( wreu wcel crio riotacl eqeltrid wa wceq eqcomi nfriota1 nfcxfr riota2f
      mpbiri mpancom jca ) ACDIZEDJZBUCEACDKZDGACDLMZUDUCBUFUDUCNBUEEOEUEGPABCD
      ECEUEGACDQRFHSTUAUB $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y ph $.  $d y ps $.
    riota5f.1 $e |- ( ph -> F/_ x B ) $.
    riota5f.2 $e |- ( ph -> B e. A ) $.
    riota5f.3 $e |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) ) $.
    $( A method for computing restricted iota.  (Contributed by NM,
       16-Apr-2013.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    riota5f $p |- ( ph -> ( iota_ x e. A ps ) = B ) $=
      ( vy cv wceq wb wral ralrimiva wi wcel wa wtru trud nfv crio reu6i adantl
      wsbc wreu nfra1 nfan nfcvd nfvd simprl simpr simplrr simplrl eqeltrd sylc
      mpbird 2thd riota2df mpdan mpbid rspsbc nfeqd nfan1 eqeq2d bibi2d imbi12d
      rsp expr ralbid sbcied mpd ) ABCJZEKZLZCDMZBCDUAZEKZAVNCDHNABVLIJZKZLZCDM
      ZVPVRKZOZIEUDZVOVQOZAEDPWCIDMWDGAWCIDAVRDPZWAWBAWFWAQZQZRWBWHSWHBCDUEZRWB
      LWGWIABCDVRUBUCWHBRCDVRAWGCACTZWFWACWFCTVTCDUFUGUGWHCVRUHWHRCUIAWFWAUJWHV
      SQZBRWKBVSWHVSUKZWKWAVLDPVTAWFWAVSULWKVLVRDWLAWFWAVSUMUNVTCDVGUOUPWKSUQUR
      USUTVHNWCIEDVAUOAWCWEIEDGAVREKZQZWAVOWBVQWNVTVNCDAWMCWJACVREACVRUHFVBVCWN
      VSVMBWNVREVLAWMUKZVDVEVIWNVREVPWOVDVFVJUTVK $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    riota5.1 $e |- ( ph -> B e. A ) $.
    riota5.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) ) $.
    $( A method for computing restricted iota.  (Contributed by NM,
       20-Oct-2011.)  (Revised by Mario Carneiro, 6-Dec-2016.) $)
    riota5 $p |- ( ph -> ( iota_ x e. A ps ) = B ) $=
      ( nfcvd riota5f ) ABCDEACEHFGI $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Restriction of a unique element to a smaller class.  (Contributed by NM,
       21-Aug-2011.)  (Revised by NM, 22-Mar-2013.) $)
    riotass2 $p |- ( ( ( A C_ B /\ A. x e. A ( ph -> ps ) )
           /\ ( E. x e. A ph /\ E! x e. B ps ) )
         -> ( iota_ x e. A ph ) = ( iota_ x e. B ps ) ) $=
      ( wi wral wa wrex wreu crio wsbc wceq reuss2 simplr riotasbc wcel riotacl
      wss syl rspsbc sbcimg sylibd mpid sylc wb ssel ad2antrr mpd simprr nfsbc1
      nfriota1 sbceq1a riota2f syl2anc mpbid eqcomd ) DESZABFZCDGZHZACDIZBCEJZH
      ZHZBCEKZACDKZVEBCVGLZVFVGMZVEACDJZUTVHABCDENZURUTVDOVJUTACVGLZVHACDPVJVGD
      QZUTVLVHFZFACDRZVMUTUSCVGLVNUSCVGDUAABCVGDUBUCTUDUEVEVGEQZVCVHVIUFVEVMVPV
      EVJVMVKVOTURVMVPFUTVDDEVGUGUHUIVAVBVCUJBVHCEVGACDULZBCVGVQUKBCVGUMUNUOUPU
      Q $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Restriction of a unique element to a smaller class.  (Contributed by NM,
       19-Oct-2005.)  (Revised by Mario Carneiro, 24-Dec-2016.) $)
    riotass $p |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) ->
                ( iota_ x e. A ph ) = ( iota_ x e. B ph ) ) $=
      ( wss wrex wreu w3a crio wsbc wceq reuss riotasbc syl wcel riotacl sseldd
      wb simp1 simp3 nfriota1 nfsbc1 sbceq1a riota2f syl2anc mpbid eqcomd ) CDE
      ZABCFZABDGZHZABDIZABCIZUKABUMJZULUMKZUKABCGZUNABCDLZABCMNUKUMDOUJUNUORUKC
      DUMUHUIUJSUKUPUMCOUQABCPNQUHUIUJTAUNBDUMABCUAZABUMURUBABUMUCUDUEUFUG $.

    $( Restriction of a unique element to a smaller class.  (Contributed by NM,
       19-Feb-2006.)  (Revised by NM, 16-Jun-2017.) $)
    moriotass $p |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) ->
                ( iota_ x e. A ph ) = ( iota_ x e. B ph ) ) $=
      ( wss wrex wrmo wreu crio wceq w3a ssrexv imp 3adant3 simp3 reu5 sylanbrc
      riotass syld3an3 ) CDEZABCFZABDGZABDHZABCIABDIJTUAUBKABDFZUBUCTUAUDUBTUAU
      DABCDLMNTUAUBOABDPQABCDRS $.
  $}

  $( A restricted class abstraction with a unique member can be expressed as a
     singleton.  (Contributed by NM, 30-May-2006.) $)
  snriota $p |- ( E! x e. A ph ->
                 { x e. A | ph } = { ( iota_ x e. A ph ) } ) $=
    ( wreu cv wcel wa cab cio csn crab crio weu wceq df-reu sniota sylbi df-rab
    df-riota sneqi 3eqtr4g ) ABCDZBECFAGZBHZUCBIZJZABCKABCLZJUBUCBMUDUFNABCOUCB
    PQABCRUGUEABCSTUA $.

  ${
    $d x y z A $.  $d x z B $.
    eusvobj1.1 $e |- B e. _V $.
    $( Specify the same property in two ways when class ` B ( y ) ` is
       single-valued.  (Contributed by NM, 1-Nov-2010.)  (Proof shortened by
       Mario Carneiro, 24-Dec-2016.) $)
    eusvobj2 $p |- ( E! x E. y e. A x = B
       -> ( E. y e. A x = B <-> A. y e. A x = B ) ) $=
      ( vz cv wceq wrex weu wral cab csn wex wi euabsn2 wcel eleq2 abid exlimiv
      velsn 3bitr3g nfre1 nfab nfeq1 elabrex elsn eqcom bitrdi imbitrid ralrimi
      bitri eqeq1 ralbidv syl5ibrcom sylbid sylbi euex rexm r19.2m 3syl impbid
      ex ) AGZDHZBCIZAJZVFVEBCKZVGVFALZFGZMZHZFNVFVHOZVFAFPVLVMFVLVFVDVJHZVHVLV
      DVIQVDVKQVFVNVIVKVDRVFASAVJUAUBVLVHVNVJDHZBCKVLVOBCBVIVKVFBAVEBCUCUDUEBGC
      QZDVIQZVLVOBACDEUFVLVQDVKQZVOVIVKDRVRDVJHVODVJEUGDVJUHULUIUJUKVNVEVOBCVDV
      JDUMUNUOUPTUQVGVFANVPBNZVHVFOVFAURVFVSAVEBCUSTVSVHVFVEBBCUTVCVAVB $.

    $( Specify the same object in two ways when class ` B ( y ) ` is
       single-valued.  (Contributed by NM, 1-Nov-2010.)  (Proof shortened by
       Mario Carneiro, 19-Nov-2016.) $)
    eusvobj1 $p |- ( E! x E. y e. A x = B
       -> ( iota x E. y e. A x = B ) = ( iota x A. y e. A x = B ) ) $=
      ( cv wceq wrex weu wral wb wal cio nfeu1 eusvobj2 alrimi iotabi syl ) AFD
      GZBCHZAIZTSBCJZKZALTAMUBAMGUAUCATANABCDEOPTUBAQR $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x F $.
    $( There is one domain element for each value of a one-to-one onto
       function.  (Contributed by NM, 26-May-2006.) $)
    f1ofveu $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
                  E! x e. A ( F ` x ) = C ) $=
      ( wf1o wcel wa cv cfv wceq wreu cop ccnv wf f1ocnv f1of syl wb wfn 3com23
      feu sylan f1ocnvfvb dff1o4 simprbi fnopfvb 3adant3 syl3an1 bitrd reubidva
      w3a 3expa mpbird ) BCEFZDCGZHZAIZEJDKZABLDURMENZGZABLZUOCBUTOZUPVBUOCBUTF
      VCBCEPCBUTQRACBDUTUBUCUQUSVAABUOUPURBGZUSVASUOUPVDULUSDUTJURKZVAUOVDUPUSV
      ESBCURDEUDUAUOUTCTZUPVDVEVASZUOEBTVFBCEUEUFVFUPVGVDCDURUTUGUHUIUJUMUKUN
      $.

    $( Value of the converse of a one-to-one onto function.  (Contributed by
       NM, 26-May-2006.)  (Proof shortened by Mario Carneiro, 24-Dec-2016.) $)
    f1ocnvfv3 $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
                  ( `' F ` C ) = ( iota_ x e. A ( F ` x ) = C ) ) $=
      ( wf1o wcel wa cv cfv wceq crio ccnv f1ocnvdm f1ocnvfvb 3expa an32s eqcom
      wb bitr4di riota5 eqcomd ) BCEFZDCGZHZAIZEJDKZABLDEMJZUEUGABUHBCDENUEUFBG
      ZHUGUHUFKZUFUHKUCUIUDUGUJSZUCUIUDUKBCUFDEOPQUFUHRTUAUB $.
  $}

  ${
    $d x A $.
    $( Restricted iota equals the empty set when not meaningful.  (Contributed
       by NM, 16-Jan-2012.)  (Revised by Mario Carneiro, 15-Oct-2016.)
       (Revised by NM, 13-Sep-2018.) $)
    riotaund $p |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) ) $=
      ( wreu wn crio cv wcel cio df-riota weu wceq df-reu iotanul sylnbi eqtrid
      wa c0 ) ABCDZEABCFBGCHAQZBIZRABCJSTBKUARLABCMTBNOP $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u ph $.
    acexmidlem.a $e |- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } $.
    acexmidlem.b $e |- B = { x e. { (/) , { (/) } }
                                                | ( x = { (/) } \/ ph ) } $.
    acexmidlem.c $e |- C = { A , B } $.
    $( Lemma for ~ acexmid .  (Contributed by Jim Kingdon, 6-Aug-2019.) $)
    acexmidlema $p |- ( { (/) } e. A -> ph ) $=
      ( c0 csn wcel wceq wo cv cpr crab eleq2i wb p0ex ax-mp prid2 eqeq1 orbi1d
      elrab3 bitri wn wi noel 0ex snid eleq2 mpbii mto orel1 sylbi ) IJZCKZUPIL
      ZAMZAUQUPBNZILZAMZBIUPOZPZKZUSCVDUPFQUPVCKVEUSRIUPSUAVBUSBUPVCUTUPLVAURAU
      TUPIUBUCUDTUEURUFUSAUGURIIKZIUHURIUPKVFIUIUJUPIIUKULUMURAUNTUO $.

    $( Lemma for ~ acexmid .  (Contributed by Jim Kingdon, 6-Aug-2019.) $)
    acexmidlemb $p |- ( (/) e. B -> ph ) $=
      ( c0 wcel csn wceq wo cv cpr crab eleq2i wb 0ex ax-mp prid1 orbi1d elrab3
      eqeq1 bitri wn wi noel snid eleq2 mpbiri mto orel1 sylbi ) IDJZIIKZLZAMZA
      UOIBNZUPLZAMZBIUPOZPZJZURDVCIGQIVBJVDURRIUPSUAVAURBIVBUSILUTUQAUSIUPUDUBU
      CTUEUQUFURAUGUQIIJZIUHUQVEIUPJISUIIUPIUJUKULUQAUMTUN $.

    $( Lemma for ~ acexmid .  (Contributed by Jim Kingdon, 6-Aug-2019.) $)
    acexmidlemph $p |- ( ph -> A = B ) $=
      ( c0 csn wceq wo wral olc ralrimivw crab eqeq2i rabid2 bitri sylibr cpr
      cv eqtr3d ) AIIJZUAZCDABUBZIKZALZBUEMZUECKZAUHBUEAUGNOUJUEUHBUEPZKUICUKUE
      FQUHBUERSTAUFUDKZALZBUEMZUEDKZAUMBUEAULNOUOUEUMBUEPZKUNDUPUEGQUMBUERSTUC
      $.

    $( Lemma for ~ acexmid .  (Contributed by Jim Kingdon, 6-Aug-2019.) $)
    acexmidlemab $p |- (
        ( ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) = (/) /\
        ( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ) = { (/) } ) ->
        -. ph ) $=
      ( cv wcel wa wrex crio c0 wceq csn noel 0ex eleq2 mpbiri mto acexmidlemph
      snid id eleq1 anbi1d rexbidv riotaeqbidv syl eqeq1d biimpa adantrr simprr
      eqtr3d ex mtoi con2i ) AFELZMZDLVAMZNZECLZOZDFPZQRZGVAMZVCNZEVEOZDGPZQSZR
      ZNZAVOQVMRZVPQQMZQTVPVQQVMMQUAUFQVMQUBUCUDAVOVPAVONVLQVMAVHVLQRZVNAVHVRAV
      GVLQAFGRZVGVLRABFGHIJKUEVSVFVKDFGVSUGVSVDVJEVEVSVBVIVCFGVAUHUIUJUKULUMUNU
      OAVHVNUPUQURUSUT $.

    $( Lemma for ~ acexmid .  Here we divide the proof into cases (based on the
       disjunction implicit in an unordered pair, not the sort of case
       elimination which relies on excluded middle).

       The cases are (1) the choice function evaluated at ` A ` equals
       ` { (/) } ` , (2) the choice function evaluated at ` B ` equals
       ` (/) ` , and (3) the choice function evaluated at ` A ` equals ` (/) `
       and the choice function evaluated at ` B ` equals ` { (/) } ` .

       Because of the way we represent the choice function ` y ` , the choice
       function evaluated at ` A ` is
       ` ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) ` and the choice
       function evaluated at ` B ` is
       ` ( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ) ` .  Other than the
       difference in notation these work just as ` ( y `` A ) ` and
       ` ( y `` B ) ` would if ` y ` were a function as defined by ~ df-fun .

       Although it isn't exactly about the division into cases, it is also
       convenient for this lemma to also include the step that if the choice
       function evaluated at ` A ` equals ` { (/) } ` , then ` { (/) } e. A `
       and likewise for ` B ` .

       (Contributed by Jim Kingdon, 7-Aug-2019.) $)
    acexmidlemcase $p |- ( A. z e. C E! v e. z E. u e. y
      ( z e. u /\ v e. u ) ->
      ( { (/) } e. A \/
        (/) e. B \/
        ( ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) = (/) /\
        ( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ) = { (/) } ) ) ) $=
      ( cv wcel wa c0 wceq wo eleq1 syl wrex wreu wral csn crio w3o cpr wi con0
      crab onsucelsucexmidlem eqeltri prid1g ax-mp eleqtrri anbi1d reueqd rspcv
      rexbidv riotacl elrabi eleq2s elpri 3syl syl5ibcom orim2d mpd pp0ex rabex
      cvv prid2 orim1d anddi sylib simpl jaoi orim2i orcomd simpr orim1i 3orass
      jca sylibr ) DMZFMZNZEMWENZOZFCMZUAZEWDUBZDIUCZPUDZGNZPHNZGWENZWGOZFWIUAZ
      EGUEZPQZHWENZWGOZFWIUAZEHUEZWMQZOZRZRZWNWOXFUFWLWNWTWOOZXFRZRXHWLXJWNWLXJ
      WNWOOZWNXEOZRZRZXJWNRWLWTWNRZWOXERZOXNWLXOXPWLWTWSWMQZRZXOWLWSGNZWSPWMUGZ
      NZXRWLWREGUBZXSGINWLYBUHGGHUGZIGUINGYCNGBMZPQARZBXTUJZUIJABUKULGHUIUMUNLU
      OWKYBDGIWJWREWDGWDGQZWHWQFWIYGWFWPWGWDGWESUPUSUQURUNWREGUTTZYAWSYFGYEBWSX
      TVAJVBWSPWMVCVDWLXQWNWTWLXSXQWNYHWSWMGSVEVFVGWLXDPQZXERZXPWLXDHNZXDXTNZYJ
      WLXCEHUBZYKHINWLYMUHHYCIGHHYDWMQARZBXTUJZVJKYNBXTVHVIULVKLUOWKYMDHIWJXCEW
      DHWDHQZWHXBFWIYPWFXAWGWDHWESUPUSUQURUNXCEHUTTZYLXDYOHYNBXDXTVAKVBXDPWMVCV
      DWLYIWOXEWLYKYIWOYQXDPHSVEVLVGWBWTWNWOXEVMVNXMWNXJXKWNXLWNWOVOWNXEVOVPVQT
      VRXJXGWNXIWOXFWTWOVSVTVQTWNWOXFWAWC $.

    $( Lemma for ~ acexmid .  List the cases identified in ~ acexmidlemcase and
       hook them up to the lemmas which handle each case.  (Contributed by Jim
       Kingdon, 7-Aug-2019.) $)
    acexmidlem1 $p |- ( A. z e. C E! v e. z E. u e. y
        ( z e. u /\ v e. u ) -> ( ph \/ -. ph ) ) $=
      ( cv wcel wa wrex c0 crio wceq orcd csn w3o wn acexmidlemcase acexmidlema
      wreu wral wo acexmidlemb acexmidlemab olcd 3jaoi syl ) DMZFMZNEMUONZOFCMZ
      PEUNUFDIUGQUAZGNZQHNZGUONUPOFUQPEGRQSHUONUPOFUQPEHRURSOZUBAAUCZUHZABCDEFG
      HIJKLUDUSVCUTVAUSAVBABGHIJKLUETUTAVBABGHIJKLUITVAVBAABCEFGHIJKLUJUKULUM
      $.

    $( Lemma for ~ acexmid .  This builds on ~ acexmidlem1 by noting that every
       element of ` C ` is inhabited.

       (Note that ` y ` is not quite a function in the ~ df-fun sense because
       it uses ordered pairs as described in ~ opthreg rather than ~ df-op ).

       The set ` A ` is also found in ~ onsucelsucexmidlem .

       (Contributed by Jim Kingdon, 5-Aug-2019.) $)
    acexmidlem2 $p |- ( A. z e. C A. w e. z E! v e. z E. u e. y
      ( z e. u /\ v e. u ) -> ( ph \/ -. ph ) ) $=
      ( wel cv wral wo wcel wceq c0 wa wrex wreu wn wex wi df-ral 19.23v bitr2i
      wal cpr eleq2i vex elpr bitri csn crab onsucelsucexmidlem1 eleqtrri elex2
      ax-mp eleq2 exbidv mpbiri p0ex prid2 eqid orci eqeq1 orbi1d mpbir2an jaoi
      elrab sylbi pm2.27 syl imp sylan2br ralimiaa acexmidlem1 ) DGNFGNUAGCOUBF
      DOZUCZEWAPZDJPWBDJPAAUDQWCWBDJWCWAJRZEDNZEUEZWBUFZWBWCWEWBUFEUJWGWBEWAUGW
      EWBEUHUIWDWGWBWDWFWGWBUFWDWAHSZWAISZQZWFWDWAHIUKZRWJJWKWAMULWAHIDUMUNUOWH
      WFWIWHWFEOZHRZEUEZTHRWNTBOZTSAQBTTUPZUKZUQHABURKUSETHUTVAWHWEWMEWAHWLVBVC
      VDWIWFWLIRZEUEZWPIRWSWPWOWPSZAQZBWQUQZIWPXBRWPWQRWPWPSZAQZTWPVEVFXCAWPVGV
      HXAXDBWPWQWTWTXCAWOWPWPVIVJVMVKLUSEWPIUTVAWIWEWREWAIWLVBVCVDVLVNWFWBVOVPV
      QVRVSABCDFGHIJKLMVTVP $.
  $}

  ${
    $d ph x y z w v u s t $.
    acexmidlemv.choice $e |- E. y A. z e. x A. w e. z E! v e. z E. u e. y
      ( z e. u /\ v e. u ) $.
    $( Lemma for ~ acexmid .

       This is ~ acexmid with additional disjoint variable conditions, most
       notably between ` ph ` and ` x ` .

       (Contributed by Jim Kingdon, 6-Aug-2019.) $)
    acexmidlemv $p |- ( ph \/ -. ph ) $=
      ( vs vt cv wcel wral c0 wceq wo cpr crab wex cvv wa wrex wreu csn wn con0
      onsucelsucexmidlem pp0ex rabex prexg mp2an raleq exbidv vtocl orbi1d eqid
      eqeq1 cbvrabv acexmidlem2 exlimiv ax-mp ) DKZGKZLFKVCLUAGCKUBFVBUCEVBMZDI
      KZNOZAPZINNUDZQZRZVEVHOZAPZIVIRZQZMZCSZAAUEPZVDDBKZMZCSVPBVNVJUFLVMTLVNTL
      AIUGVLIVIUHUIVJVMUFTUJUKVRVNOVSVOCVDDVRVNULUMHUNVOVQCAJCDEFGVJVMVNVGJKZNO
      ZAPIJVIVEVTOZVFWAAVEVTNUQUOURVLVTVHOZAPIJVIWBVKWCAVEVTVHUQUOURVNUPUSUTVA
      $.
  $}

  ${
    $d x y z w v u $.  $d a b c d e f ph $.  $d a b c d e f u v w x y z $.
    acexmid.choice $e |- E. y A. z e. x A. w e. z E! v e. z E. u e. y
      ( z e. u /\ v e. u ) $.
    $( The axiom of choice implies excluded middle.  Theorem 1.3 in [Bauer]
       p. 483.

       The statement of the axiom of choice given here is ac2 in the Metamath
       Proof Explorer (version of 3-Aug-2019).  In particular, note that the
       choice function ` y ` provides a value when ` z ` is inhabited (as
       opposed to nonempty as in some statements of the axiom of choice).

       Essentially the same proof can also be found at "The axiom of choice
       implies instances of EM", [Crosilla], p.  "Set-theoretic principles
       incompatible with intuitionistic logic".

       Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill
       theorem, after Radu Diaconescu who discovered it in 1975 in the
       framework of topos theory and N. D. Goodman and John Myhill in 1978 in
       the framework of set theory (although it already appeared as an exercise
       in Errett Bishop's book _Foundations of Constructive Analysis_ from
       1967).

       For this theorem stated using the ~ df-ac and ~ df-exmid syntaxes, see
       ~ exmidac .  (Contributed by Jim Kingdon, 4-Aug-2019.) $)
    acexmid $p |- ( ph \/ -. ph ) $=
      ( va vb vc vd vf ve wel wa cv wral weq wb wrex wreu wex wal weu wsb sb8eu
      nfv w3a eleq12 ancoms 3adant3 3ad2antl1 3ad2antl2 simpl3 cbvrexdva2 3expa
      anbi12d 3com23 sbiedv eubidv bitrid df-reu 3bitr4g adantr simpll cbvexdva
      cbvraldva2 adantll cbvalv mpgbir spi acexmidlemv ) AIJKLMNKNOZMNOZPZNJQZU
      AZMKQZUBZLVSRZKIQZRZJUCZIWDIUDDGOZFGOZPZGCQZUAZFDQZUBZEWJRZDBQZRZCUCZBWDW
      OIBIBSZWCWNJCWPJCSZPWAWLKDWBWMWQKDSZWAWLTZWPWRWQWSWRWQPZVTWKLEVSWJWTVTWKT
      LESZWTMKOZVRPZMUEZFDOZWIPZFUEZVTWKXDXCMFUFZFUEWTXGXCMFXCFUHUGWTXHXFFWTXCX
      FMFWRWQMFSZXCXFTZWRXIWQXJWRXIWQUIZXBXEVRWIWRXIXBXETZWQXIWRXLMQZFQZVSWJUJU
      KULXKVPWGNGVQWHXKNGSZPVNWEVOWFWRXIXOVNWETWQVSWJNQZGQZUJUMXIWRXOVOWFTWQXMX
      NXPXQUJUNURWRXIWQXOUOUPURUSUQUTVAVBVRMVSVCWIFWJVCVDVEWRWQXAVFVHUKVIWPWQWR
      VFVHVGVJHVKVLVM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Operations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Extend class notation to include the value of an operation ` F ` (such as
     + ) for two arguments ` A ` and ` B ` .  Note that the syntax is simply
     three class symbols in a row surrounded by parentheses.  Since operation
     values are the only possible class expressions consisting of three class
     expressions in a row surrounded by parentheses, the syntax is
     unambiguous. $)
  co $a class ( A F B ) $.

  $( Extend class notation to include class abstraction (class builder) of
     nested ordered pairs. $)
  coprab $a class { <. <. x , y >. , z >. | ph } $.

  $( Extend the definition of a class to include maps-to notation for defining
     an operation via a rule. $)
  cmpo $a class ( x e. A , y e. B |-> C ) $.

  $( Define the value of an operation.  Definition of operation value in
     [Enderton] p. 79.  Note that the syntax is simply three class expressions
     in a row bracketed by parentheses.  There are no restrictions of any kind
     on what those class expressions may be, although only certain kinds of
     class expressions - a binary operation ` F ` and its arguments ` A ` and
     ` B ` - will be useful for proving meaningful theorems.  For example, if
     class ` F ` is the operation + and arguments ` A ` and ` B ` are 3 and 2 ,
     the expression ( 3 + 2 ) can be proved to equal 5 .  This definition is
     well-defined, although not very meaningful, when classes ` A ` and/or
     ` B ` are proper classes (i.e. are not sets); see ~ ovprc1 and ~ ovprc2 .
     On the other hand, we often find uses for this definition when ` F ` is a
     proper class. ` F ` is normally equal to a class of nested ordered pairs
     of the form defined by ~ df-oprab .  (Contributed by NM, 28-Feb-1995.) $)
  df-ov $a |- ( A F B ) = ( F ` <. A , B >. ) $.

  ${
    $d x w $.  $d y w $.  $d z w $.  $d w ph $.
    $( Define the class abstraction (class builder) of a collection of nested
       ordered pairs (for use in defining operations).  This is a special case
       of Definition 4.16 of [TakeutiZaring] p. 14.  Normally ` x ` , ` y ` ,
       and ` z ` are distinct, although the definition doesn't strictly require
       it.  See ~ df-ov for the value of an operation.  The brace notation is
       called "class abstraction" by Quine; it is also called a "class builder"
       in the literature.  The value of the most common operation class builder
       is given by ~ ovmpo .  (Contributed by NM, 12-Mar-1995.) $)
    df-oprab $a |- { <. <. x , y >. , z >. | ph } =
                  { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } $.
  $}

  ${
    $d x z $.  $d y z $.  $d z A $.  $d z B $.  $d z C $.
    $( Define maps-to notation for defining an operation via a rule.  Read as
       "the operation defined by the map from ` x , y ` (in ` A X. B ` ) to
       ` B ( x , y ) ` ".  An extension of ~ df-mpt for two arguments.
       (Contributed by NM, 17-Feb-2008.) $)
    df-mpo $a |- ( x e. A , y e. B |-> C ) =
             { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } $.
  $}

  $( Equality theorem for operation value.  (Contributed by NM,
     28-Feb-1995.) $)
  oveq $p |- ( F = G -> ( A F B ) = ( A G B ) ) $=
    ( wceq cop cfv co fveq1 df-ov 3eqtr4g ) CDEABFZCGLDGABCHABDHLCDIABCJABDJK
    $.

  $( Equality theorem for operation value.  (Contributed by NM,
     28-Feb-1995.) $)
  oveq1 $p |- ( A = B -> ( A F C ) = ( B F C ) ) $=
    ( wceq cop cfv co opeq1 fveq2d df-ov 3eqtr4g ) ABEZACFZDGBCFZDGACDHBCDHMNOD
    ABCIJACDKBCDKL $.

  $( Equality theorem for operation value.  (Contributed by NM,
     28-Feb-1995.) $)
  oveq2 $p |- ( A = B -> ( C F A ) = ( C F B ) ) $=
    ( wceq cop cfv co opeq2 fveq2d df-ov 3eqtr4g ) ABEZCAFZDGCBFZDGCADHCBDHMNOD
    ABCIJCADKCBDKL $.

  $( Equality theorem for operation value.  (Contributed by NM,
     16-Jul-1995.) $)
  oveq12 $p |- ( ( A = B /\ C = D ) -> ( A F C ) = ( B F D ) ) $=
    ( wceq co oveq1 oveq2 sylan9eq ) ABFCDFACEGBCEGBDEGABCEHCDBEIJ $.

  ${
    oveq1i.1 $e |- A = B $.
    $( Equality inference for operation value.  (Contributed by NM,
       28-Feb-1995.) $)
    oveq1i $p |- ( A F C ) = ( B F C ) $=
      ( wceq co oveq1 ax-mp ) ABFACDGBCDGFEABCDHI $.

    $( Equality inference for operation value.  (Contributed by NM,
       28-Feb-1995.) $)
    oveq2i $p |- ( C F A ) = ( C F B ) $=
      ( wceq co oveq2 ax-mp ) ABFCADGCBDGFEABCDHI $.

    ${
      oveq12i.2 $e |- C = D $.
      $( Equality inference for operation value.  (Contributed by NM,
         28-Feb-1995.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
      oveq12i $p |- ( A F C ) = ( B F D ) $=
        ( wceq co oveq12 mp2an ) ABHCDHACEIBDEIHFGABCDEJK $.
    $}

    $( Equality inference for operation value.  (Contributed by NM,
       24-Nov-2007.) $)
    oveqi $p |- ( C A D ) = ( C B D ) $=
      ( wceq co oveq ax-mp ) ABFCDAGCDBGFECDABHI $.
  $}

  ${
    oveq123i.1 $e |- A = C $.
    oveq123i.2 $e |- B = D $.
    oveq123i.3 $e |- F = G $.
    $( Equality inference for operation value.  (Contributed by FL,
       11-Jul-2010.) $)
    oveq123i $p |- ( A F B ) = ( C G D ) $=
      ( co oveq12i oveqi eqtri ) ABEJCDEJCDFJACBDEGHKEFCDILM $.
  $}

  ${
    oveq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for operation value.  (Contributed by NM,
       13-Mar-1995.) $)
    oveq1d $p |- ( ph -> ( A F C ) = ( B F C ) ) $=
      ( wceq co oveq1 syl ) ABCGBDEHCDEHGFBCDEIJ $.

    $( Equality deduction for operation value.  (Contributed by NM,
       13-Mar-1995.) $)
    oveq2d $p |- ( ph -> ( C F A ) = ( C F B ) ) $=
      ( wceq co oveq2 syl ) ABCGDBEHDCEHGFBCDEIJ $.

    $( Equality deduction for operation value.  (Contributed by NM,
       9-Sep-2006.) $)
    oveqd $p |- ( ph -> ( C A D ) = ( C B D ) ) $=
      ( wceq co oveq syl ) ABCGDEBHDECHGFDEBCIJ $.

    ${
      oveq12d.2 $e |- ( ph -> C = D ) $.
      $( Equality deduction for operation value.  (Contributed by NM,
         13-Mar-1995.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
      oveq12d $p |- ( ph -> ( A F C ) = ( B F D ) ) $=
        ( wceq co oveq12 syl2anc ) ABCIDEIBDFJCEFJIGHBCDEFKL $.
    $}

    ${
      opreqan12i.2 $e |- ( ps -> C = D ) $.
      $( Equality deduction for operation value.  (Contributed by NM,
         10-Aug-1995.) $)
      oveqan12d $p |- ( ( ph /\ ps ) -> ( A F C ) = ( B F D ) ) $=
        ( wceq co oveq12 syl2an ) ACDJEFJCEGKDFGKJBHICDEFGLM $.

      $( Equality deduction for operation value.  (Contributed by NM,
         10-Aug-1995.) $)
      oveqan12rd $p |- ( ( ps /\ ph ) -> ( A F C ) = ( B F D ) ) $=
        ( co wceq oveqan12d ancoms ) ABCEGJDFGJKABCDEFGHILM $.
    $}
  $}

  ${
    oveq123d.1 $e |- ( ph -> F = G ) $.
    oveq123d.2 $e |- ( ph -> A = B ) $.
    oveq123d.3 $e |- ( ph -> C = D ) $.
    $( Equality deduction for operation value.  (Contributed by FL,
       22-Dec-2008.) $)
    oveq123d $p |- ( ph -> ( A F C ) = ( B G D ) ) $=
      ( co oveqd oveq12d eqtrd ) ABDFKBDGKCEGKAFGBDHLABCDEGIJMN $.
  $}

  ${
    fvoveq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for nested function and operation value.
       (Contributed by AV, 23-Jul-2022.) $)
    fvoveq1d $p |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) ) $=
      ( co oveq1d fveq2d ) ABDFHCDFHEABCDFGIJ $.
  $}

  $( Equality theorem for nested function and operation value.  Closed form of
     ~ fvoveq1d .  (Contributed by AV, 23-Jul-2022.) $)
  fvoveq1 $p |- ( A = B -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) ) $=
    ( wceq id fvoveq1d ) ABFZABCDEIGH $.

  ${
    $d B x $.  $d X x $.
    ovanraleqv.1 $e |- ( B = X -> ( ph <-> ps ) ) $.
    $( Equality theorem for a conjunction with an operation values within a
       restricted universal quantification.  Technical theorem to be used to
       reduce the size of a significant number of proofs.  (Contributed by AV,
       13-Aug-2022.) $)
    ovanraleqv $p |- ( B = X -> ( A. x e. V ( ph /\ ( A .x. B ) = C )
                               <-> A. x e. V ( ps /\ ( A .x. X ) = C ) ) ) $=
      ( wceq co wa oveq2 eqeq1d anbi12d ralbidv ) EIKZADEGLZFKZMBDIGLZFKZMCHRAB
      TUBJRSUAFEIDGNOPQ $.
  $}

  ${
    imbrov2fvoveq.1 $e |- ( X = Y -> ( ph <-> ps ) ) $.
    $( Equality theorem for nested function and operation value in an
       implication for a binary relation.  Technical theorem to be used to
       reduce the size of a significant number of proofs.  (Contributed by AV,
       17-Aug-2022.) $)
    imbrov2fvoveq $p |- ( X = Y -> ( ( ph -> ( F ` ( ( G ` X ) .x. O ) ) R A )
                           <-> ( ps -> ( F ` ( ( G ` Y ) .x. O ) ) R A ) ) ) $=
      ( wceq cfv co wbr fveq2 fvoveq1d breq1d imbi12d ) IJLZABIGMZHENFMZCDOJGMZ
      HENFMZCDOKTUBUDCDTUAUCHFEIJGPQRS $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y F $.  $d x y ph $.  $d y Y $.
    $d x y G $.  $d x y X $.
    $( If an operation value is element of a class for all operands of two
       classes, then the operation value is an element of the class for
       specific operands of the two classes.  (Contributed by Mario Carneiro,
       6-Dec-2014.) $)
    ovrspc2v $p |- ( ( ( X e. A /\ Y e. B ) /\
      A. x e. A A. y e. B ( x F y ) e. C ) -> ( X F Y ) e. C ) $=
      ( cv co wcel wceq oveq1 eleq1d oveq2 rspc2va ) AIZBIZFJZEKGHFJZEKGRFJZEKA
      BGHCDQGLSUAEQGRFMNRHLUATERHGFONP $.

    oveqrspc2v.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) ->
                         ( x F y ) = ( x G y ) ) $.
    $( Restricted specialization of operands, using implicit substitution.
       (Contributed by Mario Carneiro, 6-Dec-2014.) $)
    oveqrspc2v $p |- ( ( ph /\ ( X e. A /\ Y e. B ) ) ->
                         ( X F Y ) = ( X G Y ) ) $=
      ( cv co wceq wral wcel wa ralrimivva oveq1 eqeq12d oveq2 rspc2v mpan9 ) A
      BKZCKZFLZUCUDGLZMZCENBDNHDOIEOPHIFLZHIGLZMZAUGBCDEJQUGUJHUDFLZHUDGLZMBCHI
      DEUCHMUEUKUFULUCHUDFRUCHUDGRSUDIMUKUHULUIUDIHFTUDIHGTSUAUB $.
  $}

  ${
    oveqdr.1 $e |- ( ph -> F = G ) $.
    $( Equality of two operations for any two operands.  Useful in proofs using
       *propd theorems.  (Contributed by Mario Carneiro, 29-Jun-2015.) $)
    oveqdr $p |- ( ( ph /\ ps ) -> ( x F y ) = ( x G y ) ) $=
      ( cv co wceq oveqd adantr ) ACHZDHZEIMNFIJBAEFMNGKL $.
  $}

  ${
    nfovd.2 $e |- ( ph -> F/_ x A ) $.
    nfovd.3 $e |- ( ph -> F/_ x F ) $.
    nfovd.4 $e |- ( ph -> F/_ x B ) $.
    $( Deduction version of bound-variable hypothesis builder ~ nfov .
       (Contributed by NM, 13-Dec-2005.)  (Proof shortened by Andrew Salmon,
       22-Oct-2011.) $)
    nfovd $p |- ( ph -> F/_ x ( A F B ) ) $=
      ( co cop cfv df-ov nfopd nffvd nfcxfrd ) ABCDEICDJZEKCDELABPEGABCDFHMNO
      $.
  $}

  ${
    nfov.1 $e |- F/_ x A $.
    nfov.2 $e |- F/_ x F $.
    nfov.3 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for operation value.  (Contributed by
       NM, 4-May-2004.) $)
    nfov $p |- F/_ x ( A F B ) $=
      ( co wnfc wtru a1i nfovd mptru ) ABCDHIJABCDABIJEKADIJFKACIJGKLM $.
  $}

  ${
    $d x z $.  $d y z $.
    $( Slight elaboration of ~ exdistrfor .  A lemma for ~ oprabid .
       (Contributed by Jim Kingdon, 15-Jan-2019.) $)
    oprabidlem $p |-
        ( E. x E. y ( x = z /\ ps ) -> E. x ( x = z /\ E. y ps ) ) $=
      ( weq wal wi wo wnf ax-bndl ax-10 wn dtru pm2.53 mpi df-nf sylibr orim12i
      albii ax-mp exdistrfor ) BDEZABCCBECFZCDECFZUBUBCFGCFZBFZHZHBCEBFZUBCIZBF
      ZHBDCJUCUHUGUJCBKUGUFUJUGUDLUFCDMUDUFNOUIUEBUBCPSQRTUA $.
  $}

  ${
    $d a ph r s t w $.  $d a r s t w x $.  $d a r s t w y $.  $d a r s t w z $.
    $( The law of concretion.  Special case of Theorem 9.5 of [Quine] p. 61.
       Although this theorem would be useful with a distinct variable condition
       between ` x ` , ` y ` , and ` z ` , we use ~ ax-bndl to eliminate that
       constraint.  (Contributed by Mario Carneiro, 20-Mar-2013.) $)
    oprabid $p |- ( <. <. x , y >. , z >. e.
        { <. <. x , y >. , z >. | ph } <-> ph ) $=
      ( vw va vt vr vs cv cop wceq wa wex cvv wcel vex wi oprabidlem weu coprab
      opexg mp2an eqvinop biimpi eqeq1 opth1 biimtrdi opeq1 eqeq2d otth2 df-3an
      opex w3a bitri anbi1i anass 3bitri 3exbii eximi excom 3imtr4i anim2i 3syl
      sylbi euequ1 eupick mpan 3impd biimtrid com12 eqcom bitrdi anbi1d 3exbidv
      syl6 syl5 imbi1d imbi12d mpbiri adantr exlimivv com3l mpdd mpcom 19.8a ex
      impbid df-oprab elab2 ) EJZBJZCJZKZDJZKZLZAMZDNZCNZBNZAEWPABCDUAWNOPWOOPW
      POPWLWMBQZCQZUMZDQZWNWOOOUBUCWQXAAWKFJZGJZKZLZXHWPLZMZGNFNZWQXAARZWQXLFGW
      KWNWOXDXEUDUEXKWQXMRZFGXIXNXJXIWQXFWNLZXMXIWQXJXOWKXHWPUFXFXGWNWOFQGQUGUH
      XOXIWQXMXOXFHJZIJZKZLZXRWNLZMZINHNXIXNRZHIXFWLWMXBXCUDYAYBHIXSYBXTXSXIWKX
      RXGKZLZXNXSXHYCWKXFXRXGUIUJYDXNWPYCLZYEAMZDNCNBNZARZRYGWLXPLZWMXQLZWOXGLZ
      AMZDNZMCNZMZBNZYEAYGYIYJYLMZMZDNZCNBNZYPYFYRBCDYFYIYJMZYKMZAMUUAYLMYRYEUU
      BAYEYIYJYKUNZUUBWLWMXPXQWOXGXBXCXEUKZYIYJYKULUOUPUUAYKAUQYIYJYLUQURUSYTYI
      YQDNZMZCNBNZYIUUECNZMZBNYPYSBNZCNUUFBNZCNYTUUGUUJUUKCYQBDHSUTYSBCVAUUFBCV
      AVBUUEBCHSUUIYOBUUHYNYIYLCDISVCUTVDVEYPYEAYEUUCYPAUUDYPYIYJYKAYPYIYNYJYKA
      RZRYIBTYPYIYNRBHVFYIYNBVGVHYNYJYMUULYJCTYNYJYMRCIVFYJYMCVGVHYKDTYMUULDGVF
      YKADVGVHVPVPVIVJVKVQYDWQYEXMYHYDWQYCWPLYEWKYCWPUFYCWPVLVMZYDXAYGAYDWRYFBC
      DYDWQYEAUUMVNVOVRVSVTUHWAWBVEWCWDWAWBWEWQAXAWRWSWTXAWRDWFWSCWFWTBWFVDWGWH
      ABCDEWIWJ $.
  $}

  $( The result of an operation is a set.  (Contributed by Jim Kingdon,
     15-Jan-2019.) $)
  fnovex $p |- ( ( F Fn ( C X. D ) /\ A e. C /\ B e. D ) ->
      ( A F B ) e. _V ) $=
    ( cxp wfn wcel w3a co cop cfv df-ov wa opelxp funfvex funfni sylan2br 3impb
    cvv eqeltrid ) ECDFZGZACHZBDHZIABEJABKZELZTABEMUCUDUEUGTHZUDUENUCUFUBHUHABC
    DOUHUBUFEUFEPQRSUA $.

  $( Evaluating a set operation at two sets gives a set.  (Contributed by Jim
     Kingdon, 19-Aug-2021.) $)
  ovexg $p |- ( ( A e. V /\ F e. W /\ B e. X ) -> ( A F B ) e. _V ) $=
    ( wcel w3a co cop cfv cvv df-ov simp2 opexg 3adant2 fvexg syl2anc eqeltrid
    ) ADGZCEGZBFGZHZABCIABJZCKZLABCMUCUAUDLGZUELGTUAUBNTUBUFUAABDFOPUDCELQRS $.

  $( The result of an operation value is always a subset of the union of the
     range.  (Contributed by Mario Carneiro, 12-Jan-2017.) $)
  ovssunirng $p |- ( ( X e. V /\ Y e. W ) -> ( X F Y ) C_ U. ran F ) $=
    ( wcel wa co cop cfv crn cuni df-ov cvv wss opexg fvssunirng syl eqsstrid )
    DBFECFGZDEAHDEIZAJZAKLZDEAMTUANFUBUCODEBCPUAAQRS $.

  ${
    ovprc1.1 $e |- Rel dom F $.
    $( The value of an operation when the one of the arguments is a proper
       class.  Note: this theorem is dependent on our particular definitions of
       operation value, function value, and ordered pair.  (Contributed by
       Mario Carneiro, 26-Apr-2015.) $)
    ovprc $p |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) ) $=
      ( cvv wcel wa wn co cop cfv df-ov cdm wceq opprc 0ex eqeltrdi wbr df-br
      c0 wrel brrelex12 mpan sylbir con3i ndmfvg syl2anc eqtrid ) AEFBEFGZHZABC
      IABJZCKZTABCLUJUKEFUKCMZFZHULTNUJUKTEABOPQUNUIUNABUMRZUIABUMSUMUAUOUIDABU
      MUBUCUDUEUKCUFUGUH $.

    $( The value of an operation when the first argument is a proper class.
       (Contributed by NM, 16-Jun-2004.) $)
    ovprc1 $p |- ( -. A e. _V -> ( A F B ) = (/) ) $=
      ( cvv wcel wn wa co c0 wceq simpl con3i ovprc syl ) AEFZGPBEFZHZGABCIJKRP
      PQLMABCDNO $.

    $( The value of an operation when the second argument is a proper class.
       (Contributed by Mario Carneiro, 26-Apr-2015.) $)
    ovprc2 $p |- ( -. B e. _V -> ( A F B ) = (/) ) $=
      ( cvv wcel wn wa co c0 wceq simpr con3i ovprc syl ) BEFZGAEFZPHZGABCIJKRP
      QPLMABCDNO $.
  $}

  ${
    $d y A $.  $d y B $.  $d y C $.  $d y D $.  $d y F $.  $d x y $.
    $( Move class substitution in and out of an operation.  (Contributed by NM,
       12-Nov-2005.)  (Proof shortened by Mario Carneiro, 5-Dec-2016.) $)
    csbov123g $p |- ( A e. D -> [_ A / x ]_ ( B F C ) =
           ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) ) $=
      ( vy cv co csb wceq csbeq1 oveq123d eqeq12d nfcsb1v csbeq1a csbief vtoclg
      vex nfov ) AGHZCDFIZJZAUACJZAUADJZAUAFJZIZKABUBJZABCJZABDJZABFJZIZKGBEUAB
      KZUCUHUGULAUABUBLUMUDUIUEUJUFUKAUABFLAUABCLAUABDLMNAUAUBUGGSAUDUEUFAUACOA
      UAFOAUADOTAHUAKCUDDUEFUFAUAFPAUACPAUADPMQR $.
  $}

  ${
    $d x F $.
    $( Move class substitution in and out of an operation.  (Contributed by NM,
       12-Nov-2005.) $)
    csbov12g $p |- ( A e. V ->
                 [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) ) $=
      ( wcel co csb csbov123g csbconstg oveqd eqtrd ) BFGZABCDEHIABCIZABDIZABEI
      ZHOPEHABCDFEJNQEOPABEFKLM $.
  $}

  ${
    $d x C $.  $d x F $.
    $( Move class substitution in and out of an operation.  (Contributed by NM,
       12-Nov-2005.) $)
    csbov1g $p |- ( A e. V ->
                  [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) ) $=
      ( wcel co csb csbov12g csbconstg oveq2d eqtrd ) BFGZABCDEHIABCIZABDIZEHOD
      EHABCDEFJNPDOEABDFKLM $.
  $}

  ${
    $d x B $.  $d x F $.
    $( Move class substitution in and out of an operation.  (Contributed by NM,
       12-Nov-2005.) $)
    csbov2g $p |- ( A e. V ->
                  [_ A / x ]_ ( B F C ) = ( B F [_ A / x ]_ C ) ) $=
      ( wcel co csb csbov12g csbconstg oveq1d eqtrd ) BFGZABCDEHIABCIZABDIZEHCP
      EHABCDEFJNOCPEABCFKLM $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x y C $.  $d y D $.  $d x y F $.  $d x y S $.
    $( A frequently used special case of ~ rspc2ev for operation values.
       (Contributed by NM, 21-Mar-2007.) $)
    rspceov $p |- ( ( C e. A /\ D e. B /\ S = ( C F D ) ) ->
                 E. x e. A E. y e. B S = ( x F y ) ) $=
      ( cv co wceq oveq1 eqeq2d oveq2 rspc2ev ) GAIZBIZHJZKGEFHJZKGEQHJZKABEFCD
      PEKRTGPEQHLMQFKTSGQFEHNMO $.
  $}

  ${
    elovimad.1 $e |- ( ph -> A e. C ) $.
    elovimad.2 $e |- ( ph -> B e. D ) $.
    elovimad.3 $e |- ( ph -> Fun F ) $.
    elovimad.4 $e |- ( ph -> ( C X. D ) C_ dom F ) $.
    $( Elementhood of the image set of an operation value.  (Contributed by
       Thierry Arnoux, 13-Mar-2017.) $)
    elovimad $p |- ( ph -> ( A F B ) e. ( F " ( C X. D ) ) ) $=
      ( co cop cfv cxp cima df-ov wcel opelxpd wfun cdm sseldd funfvima syl2anc
      wi mpd eqeltrid ) ABCFKBCLZFMZFDENZOZBCFPAUGUIQZUHUJQZABCDEGHRZAFSUGFTZQU
      KULUDIAUIUNUGJUMUAUIUGFUBUCUEUF $.
  $}

  $( Value of a binary operation expressed as a binary relation.  See also
     ~ fnbrfvb for functions on Cartesian products.  (Contributed by BJ,
     15-Feb-2022.) $)
  fnbrovb $p |- ( ( F Fn ( V X. W ) /\ ( A e. V /\ B e. W ) ) ->
                                     ( ( A F B ) = C <-> <. A , B >. F C ) ) $=
    ( co wceq cop cfv cxp wfn wcel wa wbr df-ov eqeq1i fnbrfvb2 bitrid ) ABDGZC
    HABIZDJZCHDEFKLAEMBFMNNUACDOTUBCABDPQABCDEFRS $.

  $( Equivalence of operation value and ordered triple membership, analogous to
     ~ fnopfvb .  (Contributed by NM, 17-Dec-2008.)  (Revised by Mario
     Carneiro, 28-Apr-2015.) $)
  fnotovb $p |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) ->
                 ( ( C F D ) = R <-> <. C , D , R >. e. F ) ) $=
    ( cxp wfn wcel w3a cop cfv wceq co cotp wb wa opelxpi fnopfvb sylan2 eqeq1i
    3impb df-ov df-ot eleq1i 3bitr4g ) FABGZHZCAIZDBIZJCDKZFLZEMZUKEKZFIZCDFNZE
    MCDEOZFIUHUIUJUMUOPZUIUJQUHUKUGIURCDABRUGUKEFSTUBUPULECDFUCUAUQUNFCDEUDUEUF
    $.

  ${
    $d E f p z $.  $d V f p z $.  $d W z $.  $d th z $.  $d ps z $.
    opabbrex.1 $e |- ( ( V e. _V /\ E e. _V ) -> ( f ( V W E ) p -> th ) ) $.
    opabbrex.2 $e |- ( ( V e. _V /\ E e. _V )
                       -> { <. f , p >. | th } e. _V ) $.
    $( A collection of ordered pairs with an extension of a binary relation is
       a set.  (Contributed by Alexander van der Vekens, 1-Nov-2017.) $)
    opabbrex $p |- ( ( V e. _V /\ E e. _V )
                    -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V ) $=
      ( vz cvv wcel wa cv co wbr copab wex cab df-opab eqeltrrid adantrd anim2d
      cop wceq 2eximdv ss2abdv eqsstrid ssexd ) EKLDKLMZCNZGNZEDFOPZAMZCGQZJNUK
      ULUDUEZBMZGRCRZJSZKUJUSBCGQKBCGJTIUAUJUOUPUNMZGRCRZJSUSUNCGJTUJVAURJUJUTU
      QCGUJUNBUPUJUMBAHUBUCUFUGUHUI $.
  $}

  $( The empty set is never an element in an ordered-pair class abstraction.
     (Contributed by Alexander van der Vekens, 5-Nov-2017.) $)
  0neqopab $p |- -. (/) e. { <. x , y >. | ph } $=
    ( c0 copab wcel id cv cop wceq wa wex wn elopab nfopab1 nfel2 nfn vex sylbi
    exlimi nfopab2 wne wi opnzi nesym pm2.21 ax-mp adantr pm2.65i ) DABCEZFZUKU
    KGUKDBHZCHZIZJZAKZCLZBLUKMZABCDNUQURBUKBBDUJABCOPQUPURCUKCCDUJABCUAPQUOURAU
    NDUBZUOURUCZULUMBRCRUDUSUOMUTUNDUEUOURUFSUGUHTTSUI $.

  ${
    $d x y X $.  $d x y Y $.
    $( If two classes are in a relationship given by an ordered-pair class
       abstraction, the classes are sets.  (Contributed by Jim Kingdon,
       16-Jan-2019.) $)
    brabvv $p |- ( X { <. x , y >. | ph } Y -> ( X e. _V /\ Y e. _V ) ) $=
      ( copab wbr cv wceq wex cvv wcel cop df-br elopab bitri exsimpl vex isset
      wa eximi sylbi opth biimpi eqcoms 2eximi syl eeanv sylib anbi12i sylibr )
      DEABCFZGZBHZDIZBJZCHZEIZCJZTZDKLZEKLZTUMUOURTZCJBJZUTUMDEMZUNUQMZIZCJZBJZ
      VDUMVGATCJZBJZVIUMVEULLVKDEULNABCVEOPVJVHBVGACQUAUBVGVCBCVCVFVEVFVEIVCUNU
      QDEBRCRUCUDUEUFUGUOURBCUHUIVAUPVBUSBDSCESUJUK $.
  $}

  ${
    $d x z w v $.  $d y z w v $.  $d w ph v $.
    $( Class abstraction for operations in terms of class abstraction of
       ordered pairs.  (Contributed by NM, 12-Mar-1995.) $)
    dfoprab2 $p |- { <. <. x , y >. , z >. | ph } =
                   { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } $=
      ( vv cv cop wceq wa wex cab coprab copab excom exrot4 opeq1 3bitr3i bitri
      vex eqeq2d pm5.32ri anbi1i anass an32 exbii isseti 19.42v mpbiran2 3exbii
      opex 19.42vv 2exbii abbii df-oprab df-opab 3eqtr4i ) FGZBGZCGZHZDGZHZIZAJ
      ZDKCKBKZFLUREGZVBHZIZVGVAIZAJZCKBKZJZDKEKZFLABCDMVLEDNVFVNFVIVKJZCKBKZEKD
      KZVPDKEKVFVNVPDEOVQVOEKZDKCKBKVFVODEBCPVRVEBCDVRVEVJJZEKZVEVOVSEVIVJJZAJV
      DVJJZAJVOVSWAWBAVJVIVDVJVHVCURVGVAVBQUAUBUCVIVJAUDVDVJAUERUFVTVEVJEKEVAUS
      UTBTCTUKUGVEVJEUHUISUJSVPVMEDVIVKBCULUMRUNABCDFUOVLEDFUPUQ $.

    $( An operation class abstraction is a relation.  (Contributed by NM,
       16-Jun-2004.) $)
    reloprab $p |- Rel { <. <. x , y >. , z >. | ph } $=
      ( vw cv cop wceq wa wex coprab dfoprab2 relopabi ) EFBFCFGHAICJBJEDABCDKA
      BCDELM $.
  $}

$(
  @{
    @d x y z w v @.  @d ph v @.
    dfoprab2f.1 @e |- ( ph -> A. w ph ) @.
    @( Class abstraction for operations in terms of class abstraction of
       ordered pairs.  This is a version of ~ dfoprab2 with bound-variable
       hypothesis instead of distinct variable requirement. @)
    dfoprab2f @p |- { <. <. x , y >. , z >. | ph } =
                   { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } @=
      ( vv coprab cv cop wceq wa wex copab dfoprab2 ax-5 hban hbex weq
      eqeq1 anbi1d 2exbidv cbvopab1
      eqtri ) ABCDHGIZBICIJZKZALZCMZBMZGDNEIZUFKZALZCMBM
      ZEDNABCDGOUJUNGDEUIEBUHECUGAEUGEPFQRRUNGPGESZUHUMBCUOUGULAUEUKUFTUAUBUCUD
      @.
  @}
$)

  ${
    $d w x $.  $d w y $.  $d w z $.  $d w ph $.
    $( The abstraction variables in an operation class abstraction are not
       free.  (Contributed by NM, 25-Apr-1995.)  (Revised by David Abernethy,
       19-Jun-2012.) $)
    nfoprab1 $p |- F/_ x { <. <. x , y >. , z >. | ph } $=
      ( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfab nfcxfr ) BABCDFEGBG
      CGHDGHIAJDKCKZBKZELABCDEMRBEQBNOP $.

    $( The abstraction variables in an operation class abstraction are not
       free.  (Contributed by NM, 25-Apr-1995.)  (Revised by David Abernethy,
       30-Jul-2012.) $)
    nfoprab2 $p |- F/_ y { <. <. x , y >. , z >. | ph } $=
      ( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfex nfab nfcxfr ) CABCD
      FEGBGCGHDGHIAJDKZCKZBKZELABCDEMTCESCBRCNOPQ $.

    $( The abstraction variables in an operation class abstraction are not
       free.  (Contributed by NM, 22-Aug-2013.) $)
    nfoprab3 $p |- F/_ z { <. <. x , y >. , z >. | ph } $=
      ( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfex nfab nfcxfr ) DABCD
      FEGBGCGHDGHIAJZDKZCKZBKZELABCDEMUADETDBSDCRDNOOPQ $.
  $}

  ${
    $d v w x $.  $d v w y $.  $d v w z $.  $d v ph $.
    nfoprab.1 $e |- F/ w ph $.
    $( Bound-variable hypothesis builder for an operation class abstraction.
       (Contributed by NM, 22-Aug-2013.) $)
    nfoprab $p |- F/_ w { <. <. x , y >. , z >. | ph } $=
      ( vv coprab cv cop wceq wa wex cab df-oprab nfv nfan nfex nfab nfcxfr ) E
      ABCDHGIBICIJDIJKZALZDMZCMZBMZGNABCDGOUEEGUDEBUCECUBEDUAAEUAEPFQRRRST $.
  $}

  ${
    $d x z w $.  $d y z w $.  $d w ph $.  $d w ps $.  $d w ch $.
    oprabbid.1 $e |- F/ x ph $.
    oprabbid.2 $e |- F/ y ph $.
    oprabbid.3 $e |- F/ z ph $.
    oprabbid.4 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal operation class abstractions (deduction
       form).  (Contributed by NM, 21-Feb-2004.)  (Revised by Mario Carneiro,
       24-Jun-2014.) $)
    oprabbid $p |- ( ph ->
           { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } ) $=
      ( vw cv cop wceq wa wex cab coprab exbid df-oprab anbi2d abbidv 3eqtr4g )
      AKLDLELMFLMNZBOZFPZEPZDPZKQUDCOZFPZEPZDPZKQBDEFRCDEFRAUHULKAUGUKDGAUFUJEH
      AUEUIFIABCUDJUASSSUBBDEFKTCDEFKTUC $.
  $}

  ${
    $d x z ph $.  $d y z ph $.
    oprabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal operation class abstractions (deduction
       form).  (Contributed by NM, 21-Feb-2004.) $)
    oprabbidv $p |- ( ph ->
           { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } ) $=
      ( nfv oprabbid ) ABCDEFADHAEHAFHGI $.
  $}

  ${
    $d x z w $.  $d y z w $.  $d w ph $.  $d w ps $.
    oprabbii.1 $e |- ( ph <-> ps ) $.
    $( Equivalent wff's yield equal operation class abstractions.  (Contributed
       by NM, 28-May-1995.)  (Revised by David Abernethy, 19-Jun-2012.) $)
    oprabbii $p |- { <. <. x , y >. , z >. | ph }
                 = { <. <. x , y >. , z >. | ps } $=
      ( vw cv wceq coprab eqid wb a1i oprabbidv ax-mp ) GHZPIZACDEJBCDEJIPKQABC
      DEABLQFMNO $.
  $}

  ${
    $d ph w $.  $d ps w $.  $d x w $.  $d y w $.  $d z w $.
    $( Equivalence of ordered pair abstraction subclass and implication.
       Compare ~ ssopab2 .  (Contributed by FL, 6-Nov-2013.)  (Proof shortened
       by Mario Carneiro, 11-Dec-2016.) $)
    ssoprab2 $p |- ( A. x A. y A. z ( ph -> ps ) ->
        { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps } ) $=
      ( vw wi wal cv cop wceq wa wex cab coprab id alimi exim syl df-oprab
      anim2d ss2abdv 3sstr4g ) ABGZEHZDHZCHZFICIDIJEIJKZALZEMZDMZCMZFNUHBLZEMZD
      MZCMZFNACDEOBCDEOUGULUPFUGUKUOGZCHULUPGUFUQCUFUJUNGZDHUQUEURDUEUIUMGZEHUR
      UDUSEUDABUHUDPUAQUIUMERSQUJUNDRSQUKUOCRSUBACDEFTBCDEFTUC $.
  $}

  $( Equivalence of ordered pair abstraction subclass and implication.  Compare
     ~ ssopab2b .  (Contributed by FL, 6-Nov-2013.)  (Proof shortened by Mario
     Carneiro, 11-Dec-2016.) $)
  ssoprab2b $p |- ( { <. <. x , y >. , z >. | ph } C_
    { <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph -> ps ) ) $=
    ( coprab wss wi wal nfoprab1 nfss nfoprab2 nfoprab3 cv wcel oprabid 3imtr3g
    cop ssel alrimi ssoprab2 impbii ) ACDEFZBCDEFZGZABHZEIZDIZCIUEUHCCUCUDACDEJ
    BCDEJKUEUGDDUCUDACDELBCDELKUEUFEEUCUDACDEMBCDEMKUECNDNRENRZUCOUIUDOABUCUDUI
    SACDEPBCDEPQTTTABCDEUAUB $.

  $( Equivalence of ordered pair abstraction subclass and biconditional.
     Compare ~ eqopab2b .  (Contributed by Mario Carneiro, 4-Jan-2017.) $)
  eqoprab2b $p |- ( { <. <. x , y >. , z >. | ph } =
    { <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph <-> ps ) ) $=
    ( coprab wss wa wi wceq wb ssoprab2b anbi12i eqss 2albiim albii 19.26 bitri
    wal 3bitr4i ) ACDEFZBCDEFZGZUBUAGZHABIESDSZCSZBAIESDSZCSZHZUAUBJABKESDSZCSZ
    UCUFUDUHABCDELBACDELMUAUBNUKUEUGHZCSUIUJULCABDEOPUEUGCQRT $.

  ${
    $d x y z A $.  $d y z B $.  $d x y z D $.  $d y z E $.  $d z C $.
    $d z F $.
    $( An equality theorem for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.)  (Revised by Mario Carneiro, 19-Mar-2015.) $)
    mpoeq123 $p |- ( ( A = D /\ A. x e. A ( B = E /\ A. y e. B C = F ) ) ->
                 ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) ) $=
      ( vz wceq wral wa cv wcel coprab cmpo nfv nfra1 nfan wb nfralxy rsp eqeq2
      nfcv pm5.32d eleq2 anbi1d sylan9bbr anass 3bitr4g oprabbid df-mpo 3eqtr4g
      syl6 ) CFJZDGJZEHJZBDKZLZACKZLZAMZCNZBMZDNZLIMZEJZLZABIOVBFNZVDGNZLVFHJZL
      ZABIOABCDEPABFGHPVAVHVLABIUOUTAUOAQUSACRSUOUTBUOBQUSBACBCUDUPURBUPBQUQBDR
      SUASVAIQVAVCVEVGLZLZVIVJVKLZLZVHVLUTVNVCVOLUOVPUTVCVMVOUTVCUSVMVOTUSACUBU
      RVMVEVKLUPVOURVEVGVKURVEUQVGVKTUQBDUBEHVFUCUNUEUPVEVJVKDGVDUFUGUHUNUEUOVC
      VIVOCFVBUFUGUHVCVEVGUIVIVJVKUIUJUKABICDEULABIFGHULUM $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.
    $( An equality theorem for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.) $)
    mpoeq12 $p |- ( ( A = C /\ B = D ) ->
                 ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) ) $=
      ( wceq wral wa cmpo eqid rgenw jctr ralrimivw mpoeq123 sylan2 ) DFHZCEHRG
      GHZBDIZJZACIABCDGKABEFGKHRUAACRTSBDGLMNOABCDGEFGPQ $.
  $}

  ${
    $d z A $.  $d z B $.  $d z C $.  $d z D $.  $d z E $.  $d x z ph $.
    $d z F $.  $d y z ph $.
    mpoeq123dv.1 $e |- ( ph -> A = D ) $.
    ${
      mpoeq123dva.2 $e |- ( ( ph /\ x e. A ) -> B = E ) $.
      mpoeq123dva.3 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F ) $.
      $( An equality deduction for the maps-to notation.  (Contributed by Mario
         Carneiro, 26-Jan-2017.) $)
      mpoeq123dva $p |- ( ph
              -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) ) $=
        ( vz cv wcel wa wceq coprab cmpo pm5.32da eqeq2d eleq2d bitrd oprabbidv
        anbi1d df-mpo 3eqtr4g ) ABNZDOZCNZEOZPZMNZFQZPZBCMRUHGOZUJHOZPZUMIQZPZB
        CMRBCDEFSBCGHISAUOUTBCMAUOULUSPUTAULUNUSAULPFIUMLUATAULURUSAULUIUQPURAU
        IUKUQAUIPEHUJKUBTAUIUPUQADGUHJUBUEUCUEUCUDBCMDEFUFBCMGHIUFUG $.
    $}

    mpoeq123dv.2 $e |- ( ph -> B = E ) $.
    mpoeq123dv.3 $e |- ( ph -> C = F ) $.
    $( An equality deduction for the maps-to notation.  (Contributed by NM,
       12-Sep-2011.) $)
    mpoeq123dv $p |- ( ph
            -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) ) $=
      ( wceq cv wcel adantr wa mpoeq123dva ) ABCDEFGHIJAEHMBNDOZKPAFIMSCNEOQLPR
      $.
  $}

  ${
    mpoeq123i.1 $e |- A = D $.
    mpoeq123i.2 $e |- B = E $.
    mpoeq123i.3 $e |- C = F $.
    $( An equality inference for the maps-to notation.  (Contributed by NM,
       15-Jul-2013.) $)
    mpoeq123i $p |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) $=
      ( cmpo wceq wtru a1i mpoeq123dv mptru ) ABCDELABFGHLMNABCDEFGHCFMNIODGMNJ
      OEHMNKOPQ $.
  $}

  ${
    $d x z ph $.  $d y z ph $.  $d z A $.  $d z B $.  $d z C $.  $d z D $.
    mpoeq3dva.1 $e |- ( ( ph /\ x e. A /\ y e. B ) -> C = D ) $.
    $( Slightly more general equality inference for the maps-to notation.
       (Contributed by NM, 17-Oct-2013.) $)
    mpoeq3dva $p |- ( ph -> ( x e. A , y e. B |-> C )
              = ( x e. A , y e. B |-> D ) ) $=
      ( vz cv wcel wa wceq coprab cmpo 3expb eqeq2d pm5.32da oprabbidv df-mpo
      3eqtr4g ) ABJDKZCJEKZLZIJZFMZLZBCINUDUEGMZLZBCINBCDEFOBCDEGOAUGUIBCIAUDUF
      UHAUDLFGUEAUBUCFGMHPQRSBCIDEFTBCIDEGTUA $.
  $}

  ${
    mpoeq3ia.1 $e |- ( ( x e. A /\ y e. B ) -> C = D ) $.
    $( An equality inference for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.) $)
    mpoeq3ia $p |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) $=
      ( cmpo wceq wtru cv wcel 3adant1 mpoeq3dva mptru ) ABCDEHABCDFHIJABCDEFAK
      CLBKDLEFIJGMNO $.
  $}

  ${
    $d ph x $.  $d ph y $.
    mpoeq3dv.1 $e |- ( ph -> C = D ) $.
    $( An equality deduction for the maps-to notation restricted to the value
       of the operation.  (Contributed by SO, 16-Jul-2018.) $)
    mpoeq3dv $p |- ( ph -> ( x e. A , y e. B |-> C ) =
        ( x e. A , y e. B |-> D ) ) $=
      ( cv wcel wceq 3ad2ant1 mpoeq3dva ) ABCDEFGABIDJFGKCIEJHLM $.
  $}

  ${
    $d z A $.  $d z B $.  $d z C $.  $d z x $.  $d z y $.
    $( Bound-variable hypothesis builder for an operation in maps-to notation.
       (Contributed by NM, 27-Aug-2013.) $)
    nfmpo1 $p |- F/_ x ( x e. A , y e. B |-> C ) $=
      ( vz cmpo cv wcel wa wceq coprab df-mpo nfoprab1 nfcxfr ) AABCDEGAHCIBHDI
      JFHEKJZABFLABFCDEMPABFNO $.

    $( Bound-variable hypothesis builder for an operation in maps-to notation.
       (Contributed by NM, 27-Aug-2013.) $)
    nfmpo2 $p |- F/_ y ( x e. A , y e. B |-> C ) $=
      ( vz cmpo cv wcel wa wceq coprab df-mpo nfoprab2 nfcxfr ) BABCDEGAHCIBHDI
      JFHEKJZABFLABFCDEMPABFNO $.
  $}

  ${
    $d w x z $.  $d w y z $.  $d w A $.  $d w B $.  $d w C $.
    nfmpo.1 $e |- F/_ z A $.
    nfmpo.2 $e |- F/_ z B $.
    nfmpo.3 $e |- F/_ z C $.
    $( Bound-variable hypothesis builder for the maps-to notation.
       (Contributed by NM, 20-Feb-2013.) $)
    nfmpo $p |- F/_ z ( x e. A , y e. B |-> C ) $=
      ( vw cmpo cv wcel wa wceq coprab df-mpo nfcri nfan nfeq2 nfoprab nfcxfr )
      CABDEFKALDMZBLEMZNZJLZFOZNZABJPABJDEFQUHABJCUEUGCUCUDCCADGRCBEHRSCUFFITSU
      AUB $.
  $}

  ${
    $d w x z $.  $d w y z $.  $d w z B $.  $d w z C $.
    $( A mapping operation with empty domain.  (Contributed by Stefan O'Rear,
       29-Jan-2015.)  (Revised by Mario Carneiro, 15-May-2015.) $)
    mpo0 $p |- ( x e. (/) , y e. B |-> C ) = (/) $=
      ( vz vw c0 cmpo cv wcel wceq coprab cop wex cab df-mpo df-oprab noel nex
      wa simprll mto abf 3eqtri ) ABGCDHAIZGJZBIZCJZTEIZDKZTZABELFIUEUGMUIMKZUK
      TZENZBNZANZFOGABEGCDPUKABEFQUPFUOAUNBUMEUMUFUERULUFUHUJUAUBSSSUCUD $.
  $}

  ${
    $d x y z $.
    $( Two ways to state the domain of an operation.  (Contributed by FL,
       24-Jan-2010.) $)
    oprab4 $p |-
      { <. <. x , y >. , z >. | ( <. x , y >. e. ( A X. B ) /\ ph ) } =
        { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } $=
      ( cv cop cxp wcel wa opelxp anbi1i oprabbii ) BGZCGZHEFIJZAKOEJPFJKZAKBCD
      QRAOPEFLMN $.
  $}

  ${
    $d x y z w v $.  $d v ph $.  $d v ps $.
    cbvoprab1.1 $e |- F/ w ph $.
    cbvoprab1.2 $e |- F/ x ps $.
    cbvoprab1.3 $e |- ( x = w -> ( ph <-> ps ) ) $.
    $( Rule used to change first bound variable in an operation abstraction,
       using implicit substitution.  (Contributed by NM, 20-Dec-2008.)
       (Revised by Mario Carneiro, 5-Dec-2016.) $)
    cbvoprab1 $p |- { <. <. x , y >. , z >. | ph }
                  = { <. <. w , y >. , z >. | ps } $=
      ( vv cv cop wceq wa wex copab coprab nfv nfan nfex eqeq2d anbi12d opabbii
      opeq1 exbidv cbvex dfoprab2 3eqtr4i ) JKZCKZDKZLZMZANZDOZCOZJEPUIFKZUKLZM
      ZBNZDOZFOZJEPACDEQBFDEQUPVBJEUOVACFUNFDUMAFUMFRGSTUTCDUSBCUSCRHSTUJUQMZUN
      UTDVCUMUSABVCULURUIUJUQUKUDUAIUBUEUFUCACDEJUGBFDEJUGUH $.
  $}

  ${
    $d v w x y z $.  $d ph v $.  $d ps v $.
    cbvoprab2.1 $e |- F/ w ph $.
    cbvoprab2.2 $e |- F/ y ps $.
    cbvoprab2.3 $e |- ( y = w -> ( ph <-> ps ) ) $.
    $( Change the second bound variable in an operation abstraction.
       (Contributed by Jeff Madsen, 11-Jun-2010.)  (Revised by Mario Carneiro,
       11-Dec-2016.) $)
    cbvoprab2 $p |- { <. <. x , y >. , z >. | ph } =
                    { <. <. x , w >. , z >. | ps } $=
      ( vv cv cop wceq wa wex cab coprab nfv nfan nfex opeq2 opeq1d cbvex exbii
      eqeq2d anbi12d exbidv abbii df-oprab 3eqtr4i ) JKZCKZDKZLZEKZLZMZANZEOZDO
      ZCOZJPUKULFKZLZUOLZMZBNZEOZFOZCOZJPACDEQBCFEQVAVIJUTVHCUSVGDFURFEUQAFUQFR
      GSTVFDEVEBDVEDRHSTUMVBMZURVFEVJUQVEABVJUPVDUKVJUNVCUOUMVBULUAUBUEIUFUGUCU
      DUHACDEJUIBCFEJUIUJ $.
  $}

  ${
    $d x y z w v u $.  $d u ph $.  $d u ps $.
    cbvoprab12.1 $e |- F/ w ph $.
    cbvoprab12.2 $e |- F/ v ph $.
    cbvoprab12.3 $e |- F/ x ps $.
    cbvoprab12.4 $e |- F/ y ps $.
    cbvoprab12.5 $e |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) ) $.
    $( Rule used to change first two bound variables in an operation
       abstraction, using implicit substitution.  (Contributed by NM,
       21-Feb-2004.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
    cbvoprab12 $p |- { <. <. x , y >. , z >. | ph }
                   = { <. <. w , v >. , z >. | ps } $=
      ( vu cv cop wceq wa wex nfv nfan coprab anbi12d opabbii dfoprab2 3eqtr4i
      copab opeq12 eqeq2d cbvex2 ) MNZCNZDNZOZPZAQZDRCRZMEUFUJFNZGNZOZPZBQZGRFR
      ZMEUFACDEUABFGEUAUPVBMEUOVACDFGUNAFUNFSHTUNAGUNGSITUTBCUTCSJTUTBDUTDSKTUK
      UQPULURPQZUNUTABVCUMUSUJUKULUQURUGUHLUBUIUCACDEMUDBFGEMUDUE $.
  $}

  ${
    $d x y z w v $.  $d w v ph $.  $d x y ps $.
    cbvoprab12v.1 $e |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) ) $.
    $( Rule used to change first two bound variables in an operation
       abstraction, using implicit substitution.  (Contributed by NM,
       8-Oct-2004.) $)
    cbvoprab12v $p |- { <. <. x , y >. , z >. | ph }
                    = { <. <. w , v >. , z >. | ps } $=
      ( nfv cbvoprab12 ) ABCDEFGAFIAGIBCIBDIHJ $.
  $}

  ${
    $d x z w v $.  $d y z w v $.  $d v ph $.  $d v ps $.
    cbvoprab3.1 $e |- F/ w ph $.
    cbvoprab3.2 $e |- F/ z ps $.
    cbvoprab3.3 $e |- ( z = w -> ( ph <-> ps ) ) $.
    $( Rule used to change the third bound variable in an operation
       abstraction, using implicit substitution.  (Contributed by NM,
       22-Aug-2013.) $)
    cbvoprab3 $p |- { <. <. x , y >. , z >. | ph } =
                     { <. <. x , y >. , w >. | ps } $=
      ( vv cv wceq wa wex copab coprab nfv nfan nfex dfoprab2 2exbidv cbvopab2
      cop anbi2d 3eqtr4i ) JKCKDKUCLZAMZDNZCNZJEOUFBMZDNZCNZJFOACDEPBCDFPUIULJE
      FUHFCUGFDUFAFUFFQGRSSUKECUJEDUFBEUFEQHRSSEKFKLZUGUJCDUMABUFIUDUAUBACDEJTB
      CDFJTUE $.
  $}

  ${
    $d x z w $.  $d y z w $.  $d w ph $.  $d z ps $.
    cbvoprab3v.1 $e |- ( z = w -> ( ph <-> ps ) ) $.
    $( Rule used to change the third bound variable in an operation
       abstraction, using implicit substitution.  (Contributed by NM,
       8-Oct-2004.)  (Revised by David Abernethy, 19-Jun-2012.) $)
    cbvoprab3v $p |- { <. <. x , y >. , z >. | ph } =
                     { <. <. x , y >. , w >. | ps } $=
      ( nfv cbvoprab3 ) ABCDEFAFHBEHGI $.
  $}

  ${
    $d u w x y z $.  $d u w x y z A $.  $d u w B $.  $d u C $.  $d u y D $.
    $d u E $.
    cbvmpox.1 $e |- F/_ z B $.
    cbvmpox.2 $e |- F/_ x D $.
    cbvmpox.3 $e |- F/_ z C $.
    cbvmpox.4 $e |- F/_ w C $.
    cbvmpox.5 $e |- F/_ x E $.
    cbvmpox.6 $e |- F/_ y E $.
    cbvmpox.7 $e |- ( x = z -> B = D ) $.
    cbvmpox.8 $e |- ( ( x = z /\ y = w ) -> C = E ) $.
    $( Rule to change the bound variable in a maps-to function, using implicit
       substitution.  This version of ~ cbvmpo allows ` B ` to be a function of
       ` x ` .  (Contributed by NM, 29-Dec-2014.) $)
    cbvmpox $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. D |-> E ) $=
      ( vu cv nfan wcel wa wceq coprab cmpo nfv nfcri nfeq2 weq wb eleq1 adantr
      nfcv eleq2d sylan9bb anbi12d eqeq2d cbvoprab12 df-mpo 3eqtr4i ) ASZEUAZBS
      ZFUAZUBZRSZGUCZUBZABRUDCSZEUAZDSZHUAZUBZVFIUCZUBZCDRUDABEFGUECDEHIUEVHVOA
      BRCDVEVGCVBVDCVBCUFCBFJUGTCVFGLUHTVEVGDVBVDDVBDUFDBFDFUMUGTDVFGMUHTVMVNAV
      JVLAVJAUFADHKUGTAVFINUHTVMVNBVMBUFBVFIOUHTACUIZBDUIZUBZVEVMVGVNVRVBVJVDVL
      VPVBVJUJVQVAVIEUKULVPVDVCHUAVQVLVPFHVCPUNVCVKHUKUOUPVRGIVFQUQUPURABREFGUS
      CDREHIUSUT $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.
    cbvmpo.1 $e |- F/_ z C $.
    cbvmpo.2 $e |- F/_ w C $.
    cbvmpo.3 $e |- F/_ x D $.
    cbvmpo.4 $e |- F/_ y D $.
    cbvmpo.5 $e |- ( ( x = z /\ y = w ) -> C = D ) $.
    $( Rule to change the bound variable in a maps-to function, using implicit
       substitution.  (Contributed by NM, 17-Dec-2013.) $)
    cbvmpo $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D ) $=
      ( nfcv weq eqidd cbvmpox ) ABCDEFGFHCFNAFNIJKLACOFPMQ $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w z C $.  $d x y D $.
    cbvmpov.1 $e |- ( x = z -> C = E ) $.
    cbvmpov.2 $e |- ( y = w -> E = D ) $.
    $( Rule to change the bound variable in a maps-to function, using implicit
       substitution.  With a longer proof analogous to ~ cbvmpt , some distinct
       variable requirements could be eliminated.  (Contributed by NM,
       11-Jun-2013.) $)
    cbvmpov $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D ) $=
      ( nfcv weq sylan9eq cbvmpo ) ABCDEFGHCGLDGLAHLBHLACMBDMGIHJKNO $.
  $}

  ${
    $d x z w $.  $d y z w $.  $d w ph $.
    $( The domain of an operation class abstraction.  (Contributed by NM,
       17-Mar-1995.)  (Revised by David Abernethy, 19-Jun-2012.) $)
    dmoprab $p |- dom { <. <. x , y >. , z >. | ph } =
                  { <. x , y >. | E. z ph } $=
      ( vw coprab cdm cv cop wceq wa wex copab cab dfoprab2 dmeqi dmopab exrot3
      19.42v 2exbii bitri abbii df-opab eqtr4i 3eqtri ) ABCDFZGEHBHCHIJZAKZCLBL
      ZEDMZGUIDLZENZADLZBCMZUFUJABCDEOPUIEDQULUGUMKZCLBLZENUNUKUPEUKUHDLZCLBLUP
      UHDBCRUQUOBCUGADSTUAUBUMBCEUCUDUE $.
  $}

  ${
    $d x y z A $.  $d x y z B $.
    $( The domain of an operation class abstraction.  (Contributed by NM,
       24-Aug-1995.) $)
    dmoprabss $p |- dom { <. <. x , y >. , z >. |
           ( ( x e. A /\ y e. B ) /\ ph ) } C_ ( A X. B ) $=
      ( cv wcel wa coprab cdm wex copab dmoprab 19.42v opabbii opabssxp eqsstri
      cxp ) BGEHCGFHIZAIZBCDJKUADLZBCMZEFSZUABCDNUCTADLZIZBCMUDUBUFBCTADOPUEBCE
      FQRR $.
  $}

  ${
    $d x z w $.  $d y z w $.  $d w ph $.
    $( The range of an operation class abstraction.  (Contributed by NM,
       30-Aug-2004.)  (Revised by David Abernethy, 19-Apr-2013.) $)
    rnoprab $p |- ran { <. <. x , y >. , z >. | ph } =
                  { z | E. x E. y ph } $=
      ( vw coprab crn cv cop wceq wa wex copab cab dfoprab2 rneqi rnopab exrot3
      vex opex isseti 19.41v mpbiran 2exbii bitri abbii 3eqtri ) ABCDFZGEHBHZCH
      ZIZJZAKZCLBLZEDMZGUNELZDNACLBLZDNUHUOABCDEOPUNEDQUPUQDUPUMELZCLBLUQUMEBCR
      URABCURULELAEUKUIUJBSCSTUAULAEUBUCUDUEUFUG $.
  $}

  ${
    $d A y $.  $d x y z $.
    $( The range of a restricted operation class abstraction.  (Contributed by
       Scott Fenton, 21-Mar-2012.) $)
    rnoprab2 $p |- ran { <. <. x , y >. , z >. |
                          ( ( x e. A /\ y e. B ) /\ ph ) } =
                    { z | E. x e. A E. y e. B ph } $=
      ( cv wcel wa coprab crn wex cab wrex rnoprab r2ex abbii eqtr4i ) BGEHCGFH
      IAIZBCDJKSCLBLZDMACFNBENZDMSBCDOUATDABCEFPQR $.
  $}

  ${
    $d x y z $.
    $( The domain of an operation class abstraction is a relation.
       (Contributed by NM, 17-Mar-1995.) $)
    reldmoprab $p |- Rel dom { <. <. x , y >. , z >. | ph } $=
      ( wex coprab cdm dmoprab relopabi ) ADEBCABCDFGABCDHI $.

    $( Structure of an operation class abstraction.  (Contributed by NM,
       28-Nov-2006.) $)
    oprabss $p |- { <. <. x , y >. , z >. | ph } C_ ( ( _V X. _V ) X. _V ) $=
      ( coprab cdm crn cxp wrel reloprab relssdmrn ax-mp reldmoprab df-rel mpbi
      cvv wss ssv xpss12 mp2an sstri ) ABCDEZUBFZUBGZHZPPHZPHZUBIUBUEQABCDJUBKL
      UCUFQZUDPQUEUGQUCIUHABCDMUCNOUDRUCUFUDPSTUA $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.  $d x y z w C $.  $d w ph $.
    $d x y z w ps $.
    eloprabga.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
    $( The law of concretion for operation class abstraction.  Compare
       ~ elopab .  (Contributed by NM, 14-Sep-1999.)  (Unnecessary distinct
       variable restrictions were removed by David Abernethy, 19-Jun-2012.)
       (Revised by Mario Carneiro, 19-Dec-2013.) $)
    eloprabga $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
       ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) ) $=
      ( vw wcel cvv cop wa cv wceq wex coprab wb w3a opexg sylan 3impa wi simpr
      elex eqeq1d eqcom vex otth2 bitrdi anbi1d pm5.32i 3exbidv df-oprab eleq2i
      bitri cab abid bitr2i eleq1 bitrid adantl elisset 3anim123i eeeanv sylibr
      19.41vvv biantrurd bitr4id adantr 3bitr3d expcom vtocleg mpcom syl3an ) F
      INFONZGJNGONZHKNHONZFGPZHPZACDEUAZNZBUBZFIUIGJUIHKUIWDONZVTWAWBUCZWGVTWAW
      BWHVTWAQWCONWBWHFGOOUDWCHOOUDUEUFWIWGUGMWDOWIMRZWDSZWGWIWKQZWJCRZDRZPERZP
      ZSZAQZETDTCTZWMFSZWNGSZWOHSZUCZBQZETDTCTZWFBWLWRXDCDEWLWRXCAQXDWLWQXCAWLW
      QWDWPSZXCWLWJWDWPWIWKUHUJXFWPWDSXCWDWPUKWMWNFGWOHCULDULEULUMUTUNUOXCABLUP
      UNUQWKWSWFUBWIWSWJWENZWKWFXGWJWSMVAZNWSWEXHWJACDEMURUSWSMVBVCWJWDWEVDVEVF
      WIXEBUBWKWIXEXCETDTCTZBQBXCBCDEVKWIXIBWIWTCTZXADTZXBETZUCXIVTXJWAXKWBXLCF
      OVGDGOVGEHOVGVHWTXAXBCDEVIVJVLVMVNVOVPVQVRVS $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z th $.
    eloprabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    eloprabg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    eloprabg.3 $e |- ( z = C -> ( ch <-> th ) ) $.
    $( The law of concretion for operation class abstraction.  Compare
       ~ elopab .  (Contributed by NM, 14-Sep-1999.)  (Revised by David
       Abernethy, 19-Jun-2012.) $)
    eloprabg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
       ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) ) $=
      ( cv wceq syl3an9b eloprabga ) ADEFGHIJKLMEQHRABFQIRCGQJRDNOPST $.
  $}

  ${
    $d ph w $.  $d ps w $.  $d x z w $.  $d y z w $.
    ssoprab2i.1 $e |- ( ph -> ps ) $.
    $( Inference of operation class abstraction subclass from implication.
       (Contributed by NM, 11-Nov-1995.)  (Revised by David Abernethy,
       19-Jun-2012.) $)
    ssoprab2i $p |- { <. <. x , y >. , z >. | ph } C_
                    { <. <. x , y >. , z >. | ps } $=
      ( vw cv cop wceq wex copab coprab anim2i 2eximi ssopab2i dfoprab2 3sstr4i
      wa ) GHCHDHIJZASZDKCKZGELTBSZDKCKZGELACDEMBCDEMUBUDGEUAUCCDABTFNOPACDEGQB
      CDEGQR $.
  $}

  ${
    $d x z $.  $d y z $.  $d z C $.
    $( Operation with universal domain in maps-to notation.  (Contributed by
       NM, 16-Aug-2013.) $)
    mpov $p |- ( x e. _V , y e. _V |-> C )
                     = { <. <. x , y >. , z >. | z = C } $=
      ( cvv cmpo cv wcel wceq coprab df-mpo vex pm3.2i biantrur oprabbii eqtr4i
      wa ) ABEEDFAGEHZBGEHZQZCGDIZQZABCJUAABCJABCEEDKUAUBABCTUARSALBLMNOP $.
  $}

  ${
    $d w x y z A $.  $d w y z B $.  $d w x y C $.  $d w z D $.
    mpompt.1 $e |- ( z = <. x , y >. -> C = D ) $.
    $( Express a two-argument function as a one-argument function, or
       vice-versa.  In this version ` B ( x ) ` is not assumed to be constant
       w.r.t ` x ` .  (Contributed by Mario Carneiro, 29-Dec-2014.) $)
    mpomptx $p |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) =
      ( x e. A , y e. B |-> D ) $=
      ( vw cv csn cxp ciun cmpt wcel wceq wa copab wex eqtr4i df-mpt coprab cop
      df-mpo eliunxp anbi1i 19.41vv anass eqeq2d anbi2d pm5.32i 3bitr2i opabbii
      cmpo bitri 2exbii dfoprab2 ) CADAJZKELMZFNCJZUSOZIJZFPZQZCIRZABDEGUNZCIUS
      FUAVFURDOBJZEOQZVBGPZQZABIUBZVEABIDEGUDVEUTURVGUCPZVJQZBSASZCIRVKVDVNCIVD
      VLVHQZBSASZVCQVOVCQZBSASVNVAVPVCABDEUTUEUFVOVCABUGVQVMABVQVLVHVCQZQVMVLVH
      VCUHVLVRVJVLVCVIVHVLFGVBHUIUJUKUOUPULUMVJABICUQTTT $.

    $d x B $.
    $( Express a two-argument function as a one-argument function, or
       vice-versa.  (Contributed by Mario Carneiro, 17-Dec-2013.)  (Revised by
       Mario Carneiro, 29-Dec-2014.) $)
    mpompt $p |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D ) $=
      ( cv csn cxp ciun cmpt cmpo wceq iunxpconst mpteq1 ax-mp mpomptx eqtr3i )
      CADAIJEKLZFMZCDEKZFMZABDEGNUAUCOUBUDOADEPCUAUCFQRABCDEFGHST $.
  $}

  $( A mapping with two arguments with the first argument from a difference set
     with a singleton and a conditional as result.  (Contributed by AV,
     13-Feb-2019.) $)
  mpodifsnif $p |- ( i e. ( A \ { X } ) , j e. B |-> if ( i = X , C , D ) )
                  = ( i e. ( A \ { X } ) , j e. B |-> D ) $=
    ( csn cdif cv wceq cif wcel wa wn wne eldifsn neneq simplbiim adantr
    iffalsed mpoeq3ia ) EFAGHIZBEJZGKZCDLDUDUCMZFJBMZNUECDUFUEOZUGUFUDAMUDGPUHU
    DAGQUDGRSTUAUB $.

  $( A mapping with two arguments with the first argument from a singleton and
     a conditional as result.  (Contributed by AV, 14-Feb-2019.) $)
  mposnif $p |- ( i e. { X } , j e. B |-> if ( i = X , C , D ) )
               = ( i e. { X } , j e. B |-> C ) $=
    ( csn cv wceq cif wcel wa elsni adantr iftrued mpoeq3ia ) DEFGZADHZFIZBCJBR
    QKZEHAKZLSBCTSUARFMNOP $.

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.
    $( Representation of a constant operation using the mapping operation.
       (Contributed by SO, 11-Jul-2018.) $)
    fconstmpo $p |- ( ( A X. B ) X. { C } ) = ( x e. A , y e. B |-> C ) $=
      ( vz cxp csn cmpt cmpo fconstmpt cv cop wceq eqidd mpompt eqtri ) CDGZEHG
      FREIABCDEJFREKABFCDEEFLALBLMNEOPQ $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w ph $.
    $( Restriction of an operation class abstraction.  (Contributed by NM,
       10-Feb-2007.) $)
    resoprab $p |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) =
                  { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } $=
      ( vw cv cop wceq wex copab cxp cres wcel coprab resopab 19.42vv dfoprab2
      wa eleq1 opelxp bitrdi anbi1d pm5.32i bitri 2exbii bitr3i opabbii reseq1i
      an12 eqtri 3eqtr4i ) GHZBHZCHZIZJZATZCKBKZGDLZEFMZNZURUOEOUPFOTZATZTZCKBK
      ZGDLZABCDPZVBNVEBCDPVCUNVBOZUTTZGDLVHUTGDVBQVKVGGDVKVJUSTZCKBKVGVJUSBCRVL
      VFBCVLURVJATZTVFVJURAUKURVMVEURVJVDAURVJUQVBOVDUNUQVBUAUOUPEFUBUCUDUEUFUG
      UHUIULVIVAVBABCDGSUJVEBCDGSUM $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d D x y z $.  $d E z $.
    $( Restriction of an operator abstraction.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    resoprab2 $p |- ( ( C C_ A /\ D C_ B ) -> ( { <. <. x , y >. , z >. |
                  ( ( x e. A /\ y e. B ) /\ ph ) } |` ( C X. D ) ) =
              { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ph ) } ) $=
      ( wss wa cv wcel coprab cxp cres resoprab anass ssel pm4.71d bicomd an4
      bi2anan9 bitrid anbi1d bitr3id oprabbidv eqtrid ) GEIZHFIZJZBKZELZCKZFLZJ
      ZAJZBCDMGHNOUKGLZUMHLZJZUPJZBCDMUSAJZBCDMUPBCDGHPUJUTVABCDUTUSUOJZAJUJVAU
      SUOAQUJVBUSAVBUQULJZURUNJZJUJUSUQURULUNUAUHVCUQUIVDURUHUQVCUHUQULGEUKRSTU
      IURVDUIURUNHFUMRSTUBUCUDUEUFUG $.

    $( Restriction of the mapping operation.  (Contributed by Mario Carneiro,
       17-Dec-2013.) $)
    resmpo $p |- ( ( C C_ A /\ D C_ B ) ->
                    ( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) =
                      ( x e. C , y e. D |-> E ) ) $=
      ( vz wss wa cv wcel wceq coprab cxp cres cmpo resoprab2 df-mpo reseq1i
      3eqtr4g ) ECIFDIJAKZCLBKZDLJHKGMZJABHNZEFOZPUBELUCFLJUDJABHNABCDGQZUFPABE
      FGQUDABHCDEFRUGUEUFABHCDGSTABHEFGSUA $.
  $}

  ${
    $d x y z w $.  $d w ph $.
    $( "At most one" is a sufficient condition for an operation class
       abstraction to be a function.  (Contributed by NM, 28-Aug-2007.) $)
    funoprabg $p |- ( A. x A. y E* z ph ->
                    Fun { <. <. x , y >. , z >. | ph } ) $=
      ( vw wmo wal cv cop wceq wa wex coprab wfun mosubopt alrimiv copab funeqi
      dfoprab2 funopab bitr2i sylib ) ADFCGBGZEHZBHCHIJAKCLBLZDFZEGZABCDMZNZUCU
      FEADBCUDOPUIUEEDQZNUGUHUJABCDESRUEEDTUAUB $.
  $}

  ${
    $d x y z $.
    funoprab.1 $e |- E* z ph $.
    $( "At most one" is a sufficient condition for an operation class
       abstraction to be a function.  (Contributed by NM, 17-Mar-1995.) $)
    funoprab $p |- Fun { <. <. x , y >. , z >. | ph } $=
      ( wmo wal coprab wfun gen2 funoprabg ax-mp ) ADFZCGBGABCDHIMBCEJABCDKL $.
  $}

  ${
    $d x y z $.  $d z ph $.
    $( Functionality and domain of an operation class abstraction.
       (Contributed by NM, 28-Aug-2007.) $)
    fnoprabg $p |- ( A. x A. y ( ph -> E! z ps ) ->
  { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn { <. x , y >. | ph } ) $=
      ( weu wi wal wa coprab wfun cdm copab wceq wfn wmo eumo imim2i wex sps wb
      moanimv sylibr 2alimi funoprabg syl dmoprab nfa1 nfa2 simpl exlimiv ancld
      euex 19.42v imbitrrdi impbid2 opabbid eqtrid df-fn sylanbrc ) ABEFZGZDHZC
      HZABIZCDEJZKZVFLZACDMZNVFVIOVDVEEPZDHCHVGVBVJCDVBABEPZGVJVAVKABEQRABEUBUC
      UDVECDEUEUFVDVHVEESZCDMVIVECDEUGVDVLACDVCCUHVBDCUIVCVLAUAZCVBVMDVBVLAVEAE
      ABUJUKVBAABESZIVLVBAVNVAVNABEUMRULABEUNUOUPTTUQURVFVIUSUT $.
  $}

  ${
    $d A w z $.  $d B w z $.  $d C w z $.  $d x y w z $.
    mpofun.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( The maps-to notation for an operation is always a function.
       (Contributed by Scott Fenton, 21-Mar-2012.) $)
    mpofun $p |- Fun F $=
      ( vz vw wfun cv wcel wa wceq coprab wmo weq wi wal mpbir eqtr3 gen2 eqeq1
      ad2ant2l anbi2d mo4 funoprab cmpo df-mpo eqtri funeqi ) FJAKCLBKDLMZHKZEN
      ZMZABHOZJUOABHUOHPUOULIKZENZMZMHIQZRZISHSVAHIUNURUTULULUMUQEUAUDUBUOUSHIU
      TUNURULUMUQEUCUEUFTUGFUPFABCDEUHUPGABHCDEUIUJUKT $.
  $}

  ${
    $d x y z $.  $d z ph $.
    fnoprab.1 $e |- ( ph -> E! z ps ) $.
    $( Functionality and domain of an operation class abstraction.
       (Contributed by NM, 15-May-1995.) $)
    fnoprab $p |- { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn
                    { <. x , y >. | ph } $=
      ( weu wi wal wa coprab copab wfn gen2 fnoprabg ax-mp ) ABEGHZDICIABJCDEKA
      CDLMQCDFNABCDEOP $.
  $}

  ${
    $d x y w A $.  $d x y w B $.  $d x y w C $.  $d x y w F $.
    $( An operation maps to a class to which all values belong.  (Contributed
       by NM, 7-Feb-2004.) $)
    ffnov $p |- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\
         A. x e. A A. y e. B ( x F y ) e. C ) ) $=
      ( vw cxp wf wfn cv cfv wcel wral wa co ffnfv cop wceq fveq2 df-ov eqtr4di
      eleq1d ralxp anbi2i bitri ) CDHZEFIFUGJZGKZFLZEMZGUGNZOUHAKZBKZFPZEMZBDNA
      CNZOGUGEFQULUQUHUKUPGABCDUIUMUNRZSZUJUOEUSUJURFLUOUIURFTUMUNFUAUBUCUDUEUF
      $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.  $d x y F $.  $d x y R $.  $d x y S $.
    fovcld.1 $e |- ( ph -> F : ( R X. S ) --> C ) $.
    $( Closure law for an operation.  (Contributed by NM, 19-Apr-2007.)
       (Revised by Thierry Arnoux, 17-Feb-2017.) $)
    fovcld $p |- ( ( ph /\ A e. R /\ B e. S ) -> ( A F B ) e. C ) $=
      ( vx vy wcel w3a wa cv co wral 3simpc cxp wceq eleq1d wf simprbi 3ad2ant1
      wfn ffnov syl oveq1 oveq2 rspc2v sylc ) ABEKZCFKZLUKULMINZJNZGOZDKZJFPIEP
      ZBCGOZDKZAUKULQAUKUQULAEFRZDGUAZUQHVAGUTUDUQIJEFDGUEUBUFUCUPUSBUNGOZDKIJB
      CEFUMBSUOVBDUMBUNGUGTUNCSVBURDUNCBGUHTUIUJ $.
  $}

  ${
    fovcl.1 $e |- F : ( R X. S ) --> C $.
    $( Closure law for an operation.  (Contributed by NM, 19-Apr-2007.)  (Proof
       shortened by AV, 9-Mar-2025.) $)
    fovcl $p |- ( ( A e. R /\ B e. S ) -> ( A F B ) e. C ) $=
      ( wcel co cxp wf a1i fovcld 3anidm12 ) ADHZBEHABFICHOABCDEFDEJCFKOGLMN $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d z C $.  $d z D $.  $d x y z F $.
    $d x y z G $.
    $( Equality of two operations is determined by their values.  (Contributed
       by NM, 1-Sep-2005.) $)
    eqfnov $p |- ( ( F Fn ( A X. B ) /\ G Fn ( C X. D ) ) -> ( F = G <->
( ( A X. B ) = ( C X. D ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) ) $=
      ( vz cxp wfn wa wceq cv cfv wral co eqfnfv2 fveq2 df-ov cop eqeq12d ralxp
      eqeq12i bitr4di anbi2i bitrdi ) GCDJZKHEFJZKLGHMUHUIMZINZGOZUKHOZMZIUHPZL
      UJANZBNZGQZUPUQHQZMZBDPACPZLIUHUIGHRUOVAUJUNUTIABCDUKUPUQUAZMZUNVBGOZVBHO
      ZMUTVCULVDUMVEUKVBGSUKVBHSUBURVDUSVEUPUQGTUPUQHTUDUEUCUFUG $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d F x y $.  $d G x y $.
    $( Two operators with the same domain are equal iff their values at each
       point in the domain are equal.  (Contributed by Jeff Madsen,
       7-Jun-2010.) $)
    eqfnov2 $p |- ( ( F Fn ( A X. B ) /\ G Fn ( A X. B ) ) ->
                  ( F = G <-> A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) $=
      ( cxp wfn wa wceq cv co wral eqfnov simpr eqidd ancri impbii bitrdi ) ECD
      GZHFTHIEFJTTJZAKZBKZELUBUCFLJBDMACMZIZUDABCDCDEFNUEUDUAUDOUDUAUDTPQRS $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z F $.
    $( Representation of a function in terms of its values.  (Contributed by
       Jim Kingdon, 16-Jan-2019.) $)
    fnovim $p |- ( F Fn ( A X. B ) ->
                    F = ( x e. A , y e. B |-> ( x F y ) ) ) $=
      ( vz cxp wfn cv cfv cmpt wceq cmpo dffn5im cop fveq2 df-ov eqtr4di mpompt
      co eqeq2i sylib ) ECDGZHEFUCFIZEJZKZLEABCDAIZBIZETZMZLFUCENUFUJEABFCDUEUI
      UDUGUHOZLUEUKEJUIUDUKEPUGUHEQRSUAUB $.
  $}

  ${
    $d x y z A $.  $d y z B $.  $d z C $.  $d z D $.
    $( Bidirectional equality theorem for a mapping abstraction.  Equivalent to
       ~ eqfnov2 .  (Contributed by Mario Carneiro, 4-Jan-2017.) $)
    mpo2eqb $p |- ( A. x e. A A. y e. B C e. V ->
      ( ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) <->
        A. x e. A A. y e. B C = D ) ) $=
      ( vz wcel wral cmpo wceq cv wb wal wa coprab df-mpo wi ralimi pm5.32 r2al
      eqeq12i eqoprab2b albii 19.21v bitr3i 2albii bitr4i 3bitri pm13.183 ralbi
      syl bitr4id ) EGIZBDJZACJZABCDEKZABCDFKZLZHMZELZVAFLZNZHOZBDJZACJZEFLZBDJ
      ZACJZUTAMCIBMDIPZVBPZABHQZVKVCPZABHQZLVLVNNZHOZBOAOZVGURVMUSVOABHCDERABHC
      DFRUCVLVNABHUDVRVKVESZBOAOVGVQVSABVQVKVDSZHOVSVTVPHVKVBVCUAUEVKVDHUFUGUHV
      EABCDUBUIUJUQVIVFNZACJVJVGNUPWAACUPVHVENZBDJWAUOWBBDHEFGUKTVHVEBDULUMTVIV
      FACULUMUN $.
  $}

  ${
    $d w x $.  $d w y z A $.  $d w z B $.  $d w z C $.  $d w z F $.  $d z ps $.
    $d x y z D $.  $d x y ph $.
    rngop.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( The range of an operation given by the maps-to notation.  (Contributed
       by FL, 20-Jun-2011.) $)
    rnmpo $p |- ran F = { z | E. x e. A E. y e. B z = C } $=
      ( crn cv wcel wa wceq coprab wrex cab cmpo df-mpo eqtri rneqi rnoprab2 )
      GIAJDKBJEKLCJFMZLABCNZIUBBEOADOCPGUCGABDEFQUCHABCDEFRSTUBABCDEUAS $.

    $( The domain of an operation defined by maps-to notation is a relation.
       (Contributed by Stefan O'Rear, 27-Nov-2014.) $)
    reldmmpo $p |- Rel dom F $=
      ( vz cdm wrel cv wcel wa wceq coprab reldmoprab cmpo df-mpo eqtri dmeqi
      releqi mpbir ) FIZJAKCLBKDLMHKENMZABHOZIZJUDABHPUCUFFUEFABCDEQUEGABHCDERS
      TUAUB $.

    $( Membership in the range of an operation class abstraction.  (Contributed
       by NM, 27-Aug-2007.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    elrnmpog $p |- ( D e. V ->
                     ( D e. ran F <-> E. x e. A E. y e. B D = C ) ) $=
      ( vz cv wceq wrex crn eqeq1 2rexbidv rnmpo elab2g ) JKZELZBDMACMFELZBDMAC
      MJFGNHSFLTUAABCDSFEOPABJCDEGIQR $.

    ${
      elrnmpo.1 $e |- C e. _V $.
      $( Membership in the range of an operation class abstraction.
         (Contributed by NM, 1-Aug-2004.)  (Revised by Mario Carneiro,
         31-Aug-2015.) $)
      elrnmpo $p |- ( D e. ran F <-> E. x e. A E. y e. B D = C ) $=
        ( vz crn wcel cv wceq wrex cab rnmpo eleq2i cvv rexlimivw eleq1 mpbiri
        eqeq1 2rexbidv elab3 bitri ) FGKZLFJMZENZBDOACOZJPZLFENZBDOZACOZUGUKFAB
        JCDEGHQRUJUNJFUMFSLZACULUOBDULUOESLIFESUAUBTTUHFNUIULABCDUHFEUCUDUEUF
        $.
    $}

    ralrnmpo.2 $e |- ( z = C -> ( ph <-> ps ) ) $.
    $( A restricted quantifier over an image set.  (Contributed by Mario
       Carneiro, 1-Sep-2015.) $)
    ralrnmpo $p |- ( A. x e. A A. y e. B C e. V ->
      ( A. z e. ran F ph <-> A. x e. A A. y e. B ps ) ) $=
      ( vw wral cv wceq wrex wi wal wb crn wcel cab rnmpo raleqi eqeq1 2rexbidv
      ralab ralcom4 r19.23v albii bitr2i 3bitri bitri nfv ceqsalg ralbi bitr3id
      ralimi syl bitrid ) AEIUAZNZEOZHPZDGQZARZESZCFNZHJUBZDGNZCFNZBDGNZCFNZVCA
      EMOZHPZDGQCFQZMUCZNVFCFQZARZESZVIAEVBVRCDMFGHIKUDUEVQVSAEMVOVDPVPVECDFGVO
      VDHUFUGUHVIVGCFNZESWAVGCEFUIWBVTEVFACFUJUKULUMVLVHVMTZCFNVIVNTVKWCCFVHVEA
      RZESZDGNZVKVMWFWDDGNZESVHWDDEGUIWGVGEVEADGUJUKUNVKWEBTZDGNWFVMTVJWHDGABEH
      JBEUOLUPUSWEBDGUQUTURUSVHVMCFUQUTVA $.

    $( A restricted quantifier over an image set.  (Contributed by Mario
       Carneiro, 1-Sep-2015.) $)
    rexrnmpo $p |- ( A. x e. A A. y e. B C e. V ->
      ( E. z e. ran F ph <-> E. x e. A E. y e. B ps ) ) $=
      ( vw wrex cv wceq wa wex wral wb crn wcel cab rnmpo rexeqi eqeq1 2rexbidv
      rexab rexcom4 r19.41v exbii bitr2i 3bitri bitri ceqsexgv ralimi rexbi syl
      bitr3id bitrid ) AEIUAZNZEOZHPZDGNZAQZERZCFNZHJUBZDGSZCFSZBDGNZCFNZVBAEMO
      ZHPZDGNCFNZMUCZNVECFNZAQZERZVHAEVAVQCDMFGHIKUDUEVPVRAEMVNVCPVOVDCDFGVNVCH
      UFUGUHVHVFCFNZERVTVFCEFUIWAVSEVEACFUJUKULUMVKVGVLTZCFSVHVMTVJWBCFVGVDAQZE
      RZDGNZVJVLWEWCDGNZERVGWCDEGUIWFVFEVDADGUJUKUNVJWDBTZDGSWEVLTVIWGDGABEHJLU
      OUPWDBDGUQURUSUPVGVLCFUQURUT $.
  $}

  ${
    $d x y z $.  $d z R $.  $d z S $.
    ovid.1 $e |- ( ( x e. R /\ y e. S ) -> E! z ph ) $.
    ovid.2 $e |- F =
                  { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
    $( The value of an operation class abstraction.  (Contributed by NM,
       16-May-1995.)  (Revised by David Abernethy, 19-Jun-2012.) $)
    ovid $p |- ( ( x e. R /\ y e. S ) -> ( ( x F y ) = z <-> ph ) ) $=
      ( cv co wceq cop cfv wcel wa df-ov eqeq1i copab wfn coprab fnoprab fneq1i
      wb mpbir opabid biimpri fnopfvb sylancr eleq2i oprabid bitri bitrd bitrid
      baib ) BJZCJZGKZDJZLUPUQMZGNZUSLZUPEOUQFOPZAURVAUSUPUQGQRVCVBUTUSMZGOZAVC
      GVCBCSZTZUTVFOZVBVEUDVGVCAPZBCDUAZVFTVCABCDHUBVFGVJIUCUEVHVCVCBCUFUGVFUTU
      SGUHUIVEVCAVEVDVJOVIGVJVDIUJVIBCDUKULUOUMUN $.
  $}

  ${
    $d x y z $.
    ovidig.1 $e |- E* z ph $.
    ovidig.2 $e |- F = { <. <. x , y >. , z >. | ph } $.
    $( The value of an operation class abstraction.  Compare ~ ovidi .  The
       condition ` ( x e. R /\ y e. S ) ` is been removed.  (Contributed by
       Mario Carneiro, 29-Dec-2014.) $)
    ovidig $p |- ( ph -> ( x F y ) = z ) $=
      ( cv co cop cfv df-ov wfun wcel wceq coprab funoprab funeqi mpbir oprabid
      biimpri eleqtrrdi funopfv mpsyl eqtrid ) ABHZCHZEIUFUGJZEKZDHZUFUGELEMZAU
      HUJJZENUIUJOUKABCDPZMABCDFQEUMGRSAULUMEULUMNAABCDTUAGUBUHUJEUCUDUE $.
  $}

  ${
    $d x y z $.  $d z R $.  $d z S $.
    ovidi.2 $e |- ( ( x e. R /\ y e. S ) -> E* z ph ) $.
    ovidi.3 $e |- F =
                  { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
    $( The value of an operation class abstraction (weak version).
       (Contributed by Mario Carneiro, 29-Dec-2014.) $)
    ovidi $p |- ( ( x e. R /\ y e. S ) -> ( ph -> ( x F y ) = z ) ) $=
      ( cv wcel wa co wceq wmo wi moanimv mpbir ovidig ex ) BJZEKCJZFKLZAUAUBGM
      DJNUCALZBCDGUDDOUCADOPHUCADQRIST $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z R $.  $d x y z S $.
    $d x y z th $.
    ov.1 $e |- C e. _V $.
    ov.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ov.3 $e |- ( y = B -> ( ps <-> ch ) ) $.
    ov.4 $e |- ( z = C -> ( ch <-> th ) ) $.
    ov.5 $e |- ( ( x e. R /\ y e. S ) -> E! z ph ) $.
    ov.6 $e |- F =
                  { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
    $( The value of an operation class abstraction.  (Contributed by NM,
       16-May-1995.)  (Revised by David Abernethy, 19-Jun-2012.) $)
    ov $p |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) ) $=
      ( wcel wa co wceq cop cv coprab cfv df-ov fveq1i eqtri eqeq1i wfn fnoprab
      copab eleq1 anbi1d anbi2d opelopabg ibir fnopfvb sylancr anbi12d eloprabg
      wb cvv mp3an3 bitrd bitrid bianabs ) HKTZILTZUAZHIMUBZJUCZDVNHIUDZEUEZKTZ
      FUEZLTZUAZAUAZEFGUFZUGZJUCZVLVLDUAZVMWCJVMVOMUGWCHIMUHVOMWBSUIUJUKVLWDVOJ
      UDWBTZWEVLWBVTEFUNZULVOWGTZWDWFVDVTAEFGRUMVLWHVTVJVSUAZVLEFHIKLVPHUCZVQVJ
      VSVPHKUOUPZVRIUCZVSVKVJVRILUOUQZURUSWGVOJWBUTVAVJVKJVETWFWEVDNWAWIBUAVLCU
      AWEEFGHIJKLVEWJVTWIABWKOVBWLWIVLBCWMPVBGUEJUCCDVLQUQVCVFVGVHVI $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z ps $.
    ovigg.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
    ovigg.4 $e |- E* z ph $.
    ovigg.5 $e |- F = { <. <. x , y >. , z >. | ph } $.
    $( The value of an operation class abstraction.  Compare ~ ovig .  The
       condition ` ( x e. R /\ y e. S ) ` is been removed.  (Contributed by FL,
       24-Mar-2007.)  (Revised by Mario Carneiro, 19-Dec-2013.) $)
    ovigg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
                         ( ps -> ( A F B ) = C ) ) $=
      ( wcel w3a cop wceq cfv coprab eloprabga df-ov fveq1i eqtri wfun funoprab
      co wi funopfv ax-mp eqtrid biimtrrdi ) FJPGKPHLPQBFGRZHRACDEUAZPZFGIUHZHS
      ABCDEFGHJKLMUBUPUQUNUOTZHUQUNITURFGIUCUNIUOOUDUEUOUFUPURHSUIACDENUGUNHUOU
      JUKULUM $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z R $.  $d x y z S $.
    $d x y z ps $.
    ovig.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
    ovig.2 $e |- ( ( x e. R /\ y e. S ) -> E* z ph ) $.
    ovig.3 $e |- F =
                  { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
    $( The value of an operation class abstraction (weak version).
       (Unnecessary distinct variable restrictions were removed by David
       Abernethy, 19-Jun-2012.)  (Contributed by NM, 14-Sep-1999.)  (Revised by
       Mario Carneiro, 19-Dec-2013.) $)
    ovig $p |- ( ( A e. R /\ B e. S /\ C e. D ) ->
                         ( ps -> ( A F B ) = C ) ) $=
      ( wcel w3a wa wceq cv co 3simpa wb eleq1 bi2anan9 3adant3 anbi12d moanimv
      wmo wi mpbir ovigg mpand ) FJPZGKPZHIPZQUNUORZBFGLUAHSUNUOUPUBCTZJPZDTZKP
      ZRZARZUQBRCDEFGHLJKIURFSZUTGSZETHSZQVBUQABVDVEVBUQUCVFVDUSUNVEVAUOURFJUDU
      TGKUDUEUFMUGVCEUIVBAEUIUJNVBAEUHUKOULUM $.
  $}

  ${
    $d x y z $.  $d z A $.  $d z B $.  $d z C $.  $d z F $.
    ovmpt4g.3 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( Value of a function given by the maps-to notation.  (This is the
       operation analog of ~ fvmpt2 .)  (Contributed by NM, 21-Feb-2004.)
       (Revised by Mario Carneiro, 1-Sep-2015.) $)
    ovmpt4g $p |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C ) $=
      ( vz cv wcel co wceq wex wa elisset wmo moeq a1i cmpo coprab df-mpo eqtri
      ovidi eqeq2 mpbidi exlimdv syl5 3impia ) AJZCKZBJZDKZEGKZUJULFLZEMZUNIJZE
      MZINUKUMOZUPIEGPUSURUPIURUOUQMUPUSURABICDFURIQUSIERSFABCDETUSUROABIUAHABI
      CDEUBUCUDUQEUOUEUFUGUHUI $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.
    ovmpos.3 $e |- F = ( x e. C , y e. D |-> R ) $.
    $( Value of a function given by the maps-to notation, expressed using
       explicit substitution.  (Contributed by Mario Carneiro, 30-Apr-2015.) $)
    ovmpos $p |- ( ( A e. C /\ B e. D /\ [_ A / x ]_ [_ B / y ]_ R e. V ) ->
      ( A F B ) = [_ A / x ]_ [_ B / y ]_ R ) $=
      ( wcel csb co wceq cvv cv wi nfcv nfcsb1v eleq1d elex nfel1 nfmpo1 nfcxfr
      wa cmpo nfov nfeq nfim nfmpo2 csbeq1a oveq1 eqeq12d imbi12d oveq2 ovmpt4g
      3expia vtocl2gaf csbcomg eqeq2d 3imtr4d syl5 3impia ) CEKZDFKZACBDGLLZIKZ
      CDHMZVFNZVGVFOKZVDVEUEZVIVFIUAVKBDACGLZLZOKZVHVMNZVJVIGOKZAPZBPZHMZGNZQVL
      OKZCVRHMZVLNZQVNVOQABCDEFACRZBCRZBDRZWAWCAAVLOACGSZUBAWBVLACVRHWDAHABEFGU
      FZJABEFGUCUDAVRRUGWGUHUIVNVOBBVMOBDVLSZUBBVHVMBCDHWEBHWHJABEFGUJUDWFUGWIU
      HUIVQCNZVPWAVTWCWJGVLOACGUKZTWJVSWBGVLVQCVRHULWKUMUNVRDNZWAVNWCVOWLVLVMOB
      DVLUKZTWLWBVHVLVMVRDCHUOWMUMUNVQEKVRFKVPVTABEFGHOJUPUQURVKVFVMOABCDGEFUSZ
      TVKVFVMVHWNUTVAVBVC $.
  $}

  ${
    $d x y C $.  $d x y D $.
    ov2gf.a $e |- F/_ x A $.
    ov2gf.c $e |- F/_ y A $.
    ov2gf.d $e |- F/_ y B $.
    ov2gf.1 $e |- F/_ x G $.
    ov2gf.2 $e |- F/_ y S $.
    ov2gf.3 $e |- ( x = A -> R = G ) $.
    ov2gf.4 $e |- ( y = B -> G = S ) $.
    ov2gf.5 $e |- F = ( x e. C , y e. D |-> R ) $.
    $( The value of an operation class abstraction.  A version of ~ ovmpog
       using bound-variable hypotheses.  (Contributed by NM, 17-Aug-2006.)
       (Revised by Mario Carneiro, 19-Dec-2013.) $)
    ov2gf $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
      ( wcel co wceq wa elex cv wi nfel1 cmpo nfmpo1 nfcxfr nfcv nfov nfeq nfim
      nfmpo2 eleq1d oveq1 eqeq12d imbi12d oveq2 ovmpt4g 3expia vtocl2gaf 3impia
      cvv syl5 ) CETZDFTZHKTZCDIUAZHUBZVIHVETZVGVHUCVKHKUDGVETZAUEZBUEZIUAZGUBZ
      UFJVETZCVOIUAZJUBZUFVLVKUFABCDEFLMNVRVTAAJVEOUGAVSJACVOILAIABEFGUHZSABEFG
      UIUJAVOUKULOUMUNVLVKBBHVEPUGBVJHBCDIMBIWASABEFGUOUJNULPUMUNVNCUBZVMVRVQVT
      WBGJVEQUPWBVPVSGJVNCVOIUQQURUSVODUBZVRVLVTVKWCJHVERUPWCVSVJJHVODCIUTRURUS
      VNETVOFTVMVQABEFGIVESVAVBVCVFVD $.
  $}

  ${
    $d x y $.  $d x A $.  $d y B $.
    ovmpodx.1 $e |- ( ph -> F = ( x e. C , y e. D |-> R ) ) $.
    ovmpodx.2 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S ) $.
    ovmpodx.3 $e |- ( ( ph /\ x = A ) -> D = L ) $.
    ovmpodx.4 $e |- ( ph -> A e. C ) $.
    ovmpodx.5 $e |- ( ph -> B e. L ) $.
    ovmpodx.6 $e |- ( ph -> S e. X ) $.
    ${
      ovmpodxf.px $e |- F/ x ph $.
      ovmpodxf.py $e |- F/ y ph $.
      ovmpodxf.ay $e |- F/_ y A $.
      ovmpodxf.bx $e |- F/_ x B $.
      ovmpodxf.sx $e |- F/_ x S $.
      ovmpodxf.sy $e |- F/_ y S $.
      $( Value of an operation given by a maps-to rule, deduction form.
         (Contributed by Mario Carneiro, 29-Dec-2014.) $)
      ovmpodxf $p |- ( ph -> ( A F B ) = S ) $=
        ( co cmpo oveqd cv wcel cvv w3a wceq wi wsbc eqid ovmpt4g alrimi spsbcd
        a1i wa adantr wb simplr ad2antrr eqeltrd simpr 3eltr4d anassrs elex syl
        biimt syl3anc oveq12d eqeq12d bitr3d nfeq2 nfan wnf nfmpo2 nfcv sbciedf
        nfov nfeq nfmpo1 mpbid eqtrd ) ADEJUEDEBCFGHUFZUEZIAJWGDEMUGABUHZFUIZCU
        HZGUIZHUJUIZUKZWIWKWGUEZHULZUMZCEUNZBDUNWHIULZAWRBDFPAWRBSAWQCEKQAWQCTW
        QABCFGHWGUJWGUOUPUSUQURUQURAWRWSBDFPAWIDULZUTZWQWSCEKAEKUIZWTQVAXAWKEUL
        ZUTZWPWQWSXDWJWLWMWPWQVBXDWIDFAWTXCVCZADFUIWTXCPVDVEXDEKWKGAXBWTXCQVDXA
        XCVFZXAGKULXCOVAVGXDHIUJAWTXCHIULNVHZAIUJUIZWTXCAILUIXHRILVIVJVDVEWNWPV
        KVLXDWOWHHIXDWIDWKEWGXEXFVMXGVNVOAWTCTCWIDUAVPVQWSCVRXACWHICDEWGUABCFGH
        VSCEVTWBUDWCUSWASWSBVRABWHIBDEWGBDVTBCFGHWDUBWBUCWCUSWAWEWF $.
    $}

    $d y A $.  $d x B $.  $d x y S $.  $d x y ph $.
    $( Value of an operation given by a maps-to rule, deduction form.
       (Contributed by Mario Carneiro, 29-Dec-2014.) $)
    ovmpodx $p |- ( ph -> ( A F B ) = S ) $=
      ( nfv nfcv ovmpodxf ) ABCDEFGHIJKLMNOPQRABSACSCDTBETBITCITUA $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y S $.  $d x y ph $.
    ovmpod.1 $e |- ( ph -> F = ( x e. C , y e. D |-> R ) ) $.
    ovmpod.2 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S ) $.
    ovmpod.3 $e |- ( ph -> A e. C ) $.
    ovmpod.4 $e |- ( ph -> B e. D ) $.
    ovmpod.5 $e |- ( ph -> S e. X ) $.
    $( Value of an operation given by a maps-to rule, deduction form.
       (Contributed by Mario Carneiro, 7-Dec-2014.) $)
    ovmpod $p |- ( ph -> ( A F B ) = S ) $=
      ( cv wceq wa eqidd ovmpodx ) ABCDEFGHIJGKLMABQDRSGTNOPUA $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y L $.  $d x y S $.
    ovmpox.1 $e |- ( ( x = A /\ y = B ) -> R = S ) $.
    ovmpox.2 $e |- ( x = A -> D = L ) $.
    ovmpox.3 $e |- F = ( x e. C , y e. D |-> R ) $.
    $( The value of an operation class abstraction.  Variant of ~ ovmpoga which
       does not require ` D ` and ` x ` to be distinct.  (Contributed by Jeff
       Madsen, 10-Jun-2010.)  (Revised by Mario Carneiro, 20-Dec-2013.) $)
    ovmpox $p |- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S ) $=
      ( wcel cvv co wceq cv adantl elex w3a cmpo wa simp1 simp2 ovmpodx syl3an3
      a1i simp3 ) HJOCEOZDKOZHPOZCDIQHRHJUAUKULUMUBZABCDEFGHIKPIABEFGUCRUNNUIAS
      CRZBSDRUDGHRUNLTUOFKRUNMTUKULUMUEUKULUMUFUKULUMUJUGUH $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y S $.
    ovmpoga.1 $e |- ( ( x = A /\ y = B ) -> R = S ) $.
    ovmpoga.2 $e |- F = ( x e. C , y e. D |-> R ) $.
    $( Value of an operation given by a maps-to rule.  (Contributed by Mario
       Carneiro, 19-Dec-2013.) $)
    ovmpoga $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
      ( wcel cvv co wceq elex w3a cmpo cv a1i adantl simp1 simp2 ovmpod syl3an3
      wa simp3 ) HJMCEMZDFMZHNMZCDIOHPHJQUIUJUKRZABCDEFGHINIABEFGSPULLUAATCPBTD
      PUGGHPULKUBUIUJUKUCUIUJUKUDUIUJUKUHUEUF $.

    ovmpoa.4 $e |- S e. _V $.
    $( Value of an operation given by a maps-to rule.  (Contributed by NM,
       19-Dec-2013.) $)
    ovmpoa $p |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S ) $=
      ( wcel cvv co wceq ovmpoga mp3an3 ) CEMDFMHNMCDIOHPLABCDEFGHINJKQR $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y ph $.
    ovmpodf.1 $e |- ( ph -> A e. C ) $.
    ovmpodf.2 $e |- ( ( ph /\ x = A ) -> B e. D ) $.
    ovmpodf.3 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V ) $.
    ovmpodf.4 $e |- ( ( ph /\ ( x = A /\ y = B ) ) ->
      ( ( A F B ) = R -> ps ) ) $.
    ${
      ovmpodf.5 $e |- F/_ x F $.
      ovmpodf.6 $e |- F/ x ps $.
      ovmpodf.7 $e |- F/_ y F $.
      ovmpodf.8 $e |- F/ y ps $.
      $( Alternate deduction version of ~ ovmpo , suitable for iteration.
         (Contributed by Mario Carneiro, 7-Jan-2017.) $)
      ovmpodf $p |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) $=
        ( wcel cv wceq cmpo wi nfv nfmpo1 nfeq nfim cvv wex elex isset sylib wa
        syl nfmpo2 co oveq simprl simprr oveq12d adantr eqeltrd adantrr ovmpt4g
        eqid syl3anc eqtr3d eqeq2d sylbid syl5 expr exlimd mpd exlimdd ) ACUAZE
        UBZJCDGHIUCZUBZBUDZCACUEVSBCCJVRPCDGHIUFUGQUHAEUITZVQCUJAEGTZWALEGUKUOC
        EULUMAVQUNZDUAZFUBZDUJZVTWCFUITZWFWCFHTZWGMFHUKUODFULUMWCWEVTDWCDUEVSBD
        DJVRRCDGHIUPUGSUHAVQWEVTVSEFJUQZEFVRUQZUBZAVQWEUNZUNZBEFJVRURWMWKWIIUBB
        WMWJIWIWMVPWDVRUQZWJIWMVPEWDFVRAVQWEUSZAVQWEUTZVAWMVPGTWDHTIKTWNIUBWMVP
        EGWOAWBWLLVBVCWMWDFHWPAVQWHWEMVDVCNCDGHIVRKVRVFVEVGVHVIOVJVKVLVMVNVO $.
    $}

    $d x y F $.  $d x y ps $.
    $( Alternate deduction version of ~ ovmpo , suitable for iteration.
       (Contributed by Mario Carneiro, 7-Jan-2017.) $)
    ovmpodv $p |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) $=
      ( nfcv nfv ovmpodf ) ABCDEFGHIJKLMNOCJPBCQDJPBDQR $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ph $.  $d x y S $.
    ovmpodv2.1 $e |- ( ph -> A e. C ) $.
    ovmpodv2.2 $e |- ( ( ph /\ x = A ) -> B e. D ) $.
    ovmpodv2.3 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V ) $.
    ovmpodv2.4 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S ) $.
    $( Alternate deduction version of ~ ovmpo , suitable for iteration.
       (Contributed by Mario Carneiro, 7-Jan-2017.) $)
    ovmpodv2 $p |- ( ph ->
      ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = S ) ) $=
      ( co wceq cv wa nfcv cmpo eqidd eqeq2d biimpd nfmpo1 nfeq1 nfmpo2 ovmpodf
      nfov mpd oveq eqeq1d syl5ibrcom ) ADEJPZIQJBCFGHUAZQZDEUOPZIQZAUOUOQURAUO
      UBAURBCDEFGHUOKLMNABRDQCREQSSZUQHQURUSHIUQOUCUDBCFGHUEZBUQIBDEUOBDTUTBETU
      IUFBCFGHUGZCUQICDEUOCDTVACETUIUFUHUJUPUNUQIDEJUOUKULUM $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y S $.
    ovmpog.1 $e |- ( x = A -> R = G ) $.
    ovmpog.2 $e |- ( y = B -> G = S ) $.
    ovmpog.3 $e |- F = ( x e. C , y e. D |-> R ) $.
    $( Value of an operation given by a maps-to rule.  Special case.
       (Contributed by NM, 14-Sep-1999.)  (Revised by David Abernethy,
       19-Jun-2012.) $)
    ovmpog $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
      ( cv wceq sylan9eq ovmpoga ) ABCDEFGHIKAOCPBODPGJHLMQNR $.

    ovmpo.4 $e |- S e. _V $.
    $( Value of an operation given by a maps-to rule.  Special case.
       (Contributed by NM, 16-May-1995.)  (Revised by David Abernethy,
       19-Jun-2012.) $)
    ovmpo $p |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S ) $=
      ( wcel cvv co wceq ovmpog mp3an3 ) CEODFOHPOCDIQHRNABCDEFGHIJPKLMST $.
  $}

  ${
    $d ph c d $.  $d C c d $.  $d A a b c d $.  $d B a b c d $.
    fvmpopr2d.1 $e |- ( ph -> F = ( a e. A , b e. B |-> C ) ) $.
    fvmpopr2d.2 $e |- ( ph -> P = <. a , b >. ) $.
    fvmpopr2d.3 $e |- ( ( ph /\ a e. A /\ b e. B ) -> C e. V ) $.
    $( Value of an operation given by maps-to notation.  (Contributed by Rohan
       Ridenour, 14-May-2024.) $)
    fvmpopr2d $p |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = C ) $=
      ( vc vd cv wcel cmpo co wceq nfcv w3a cfv cop fveq12d eqtr4id csb nfcsb1v
      df-ov 3ad2ant1 nfcsbw csbeq1a sylan9eq cbvmpo oveqi equcom anbi12i sylbir
      eqidd wa eqcomd adantl simp2 simp3 ovmpod eqtrid eqtr3d ) AHOZBPZIOZCPZUA
      ZVGVIHIBCDQZRZEFUBZDVKVMVGVIUCZVLUBVNVGVIVLUHVKEVOFVLAVHFVLSVJJUIAVHEVOSV
      JKUIUDUEVKVMVGVIMNBCINOZHMOZDUFZUFZQZRDVLVTVGVIHIMNBCDVSMDTNDTHIVPVRHVPTH
      VQDUGUJIVPVRUGVGVQSZVIVPSZDVRVSHVQDUKIVPVRUKULZUMUNVKMNVGVIBCVSDVTGVKVTUR
      VQVGSZVPVISZUSZVSDSVKWFDVSWFWAWBUSDVSSWAWDWBWEHMUOINUOUPWCUQUTVAAVHVJVBAV
      HVJVCLVDVEVF $.
  $}

  ${
    $d f u v w x y z A $.  $d f u v w x y z B $.  $d x y z R $.
    $d f u v w y z C $.  $d f u v w y z D $.  $d f u v w x y z H $.
    $d f u v w z S $.
    ovi3.1 $e |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) ->
        S e. ( H X. H ) ) $.
    ovi3.2 $e |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) ->
         R = S ) $.
    ovi3.3 $e |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\
                      y e. ( H X. H ) ) /\
                      E. w E. v E. u E. f ( ( x = <. w , v >. /\
                      y = <. u , f >. ) /\ z = R ) ) } $.
    $( The value of an operation class abstraction.  Special case.
       (Contributed by NM, 28-May-1995.)  (Revised by Mario Carneiro,
       29-Dec-2014.) $)
    ovi3 $p |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) ->
        ( <. A , B >. F <. C , D >. ) = S ) $=
      ( wa wex wcel cv wceq cop co cvv cxp elex syl isset sylib nfv nfcv coprab
      nfoprab3 nfcxfr nfov nfeq1 eqeq2d copsex4g opelxpi nfoprab1 nfim nfoprab2
      eqeq1 anbi1d 4exbidv oveq1 eqeq1d imbi12d anbi2d oveq2 moeq mosubop anass
      wi wmo 2exbii 19.42vv bitri mobii mpbir a1i ovidi vtocl2gaf sylbird eqeq2
      syl2an mpbidi exlimd mpd ) GOUAHOUASZIOUAJOUASZSZCUBZLUCZCTZGHUDZIJUDZNUE
      ZLUCZWNLUFUAZWQWNLOOUGZUAXBPLXCUHUICLUJUKWNWPXACWNCULCWTLCWRWSNCWRUMCNAUB
      ZXCUABUBZXCUASZXDDUBZEUBZUDZUCZXEFUBZMUBZUDZUCZSZWOKUCZSZMTFTZETDTZSZABCU
      NZRXTABCUOUPCWSUMUQURWPWTWOUCZXAWNWNWPWRXIUCZWSXMUCZSZXPSZMTFTETDTZYBXPWP
      DEFMGHIJOOXGGUCXHHUCSXKIUCXLJUCSSKLWOQUSUTWLWRXCUAWSXCUAYGYBVPZWMGHOOVAIJ
      OOVAXSXDXENUEZWOUCZVPYCXNSZXPSZMTFTETDTZWRXENUEZWOUCZVPYHABWRWSXCXCAWRUMZ
      BWRUMZBWSUMZYMYOAYMAULAYNWOAWRXENYPANYARXTABCVBUPAXEUMUQURVCYGYBBYGBULBWT
      WOBWRWSNYQBNYARXTABCVDUPYRUQURVCXDWRUCZXSYMYJYOYSXQYLDEFMYSXOYKXPYSXJYCXN
      XDWRXIVEVFVFVGYSYIYNWOXDWRXENVHVIVJXEWSUCZYMYGYOYBYTYLYFDEFMYTYKYEXPYTXNY
      DYCXEWSXMVEVKVFVGYTYNWTWOXEWSWRNVLVIVJXSABCXCXCNXSCVQZXFUUAXJXNXPSZMTFTZS
      ZETDTZCVQUUCCDEXDXPCFMXECKVMVNVNXSUUECXRUUDDEXRXJUUBSZMTFTUUDXQUUFFMXJXNX
      PVOVRXJUUBFMVSVTVRWAWBWCRWDWEWHWFWOLWTWGWIWJWK $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z C $.  $d w z R $.
    $d w x y z S $.
    ov6g.1 $e |- ( <. x , y >. = <. A , B >. -> R = S ) $.
    ov6g.2 $e |- F = { <. <. x , y >. , z >. | ( <. x , y >. e. C
                      /\ z = R ) } $.
    $( The value of an operation class abstraction.  Special case.
       (Contributed by NM, 13-Nov-2006.) $)
    ov6g $p |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J )
                     -> ( A F B ) = S ) $=
      ( vw wcel wa cv wceq wex cop w3a co cfv df-ov eqid biidd copsex2g 3adant3
      mpbiri adantr wi eqeq1 anbi1d eqeq2d eqcoms pm5.32i bitrdi 2exbidv anbi2d
      wb wmo moeq mosubop a1i coprab copab dfoprab2 eleq1 bitr3i 2exbii 19.42vv
      an12 bitri opabbii 3eqtri fvopab3ig 3ad2antl3 mpd eqtrid ) DJPZEKPZDEUAZF
      PZUBZHLPZQZDEIUCWCIUDZHDEIUEWGWCARZBRZUAZSZHHSZQZBTATZWHHSZWEWOWFWAWBWOWD
      WAWBQWOWMHUFWMWMABDEJKWIDSWJESQWMUGUHUJUIUKWDWAWFWOWPULWBORZWKSZCRZGSZQZB
      TATZWLWSHSZQZBTATWOOCWCHFLIWQWCSZXAXDABXEXAWLWTQXDXEWRWLWTWQWCWKUMUNWLWTX
      CWTXCVAWKWCWKWCSGHWSMUOUPUQURUSXCXDWNABXCXCWMWLWSHHUMUTUSXBCVBWQFPZWTCABW
      QCGVCVDVEIWKFPZWTQZABCVFWRXHQZBTATZOCVGXFXBQZOCVGNXHABCOVHXJXKOCXJXFXAQZB
      TATXKXIXLABXIWRXFWTQZQXLWRXMXHWRXFXGWTWQWKFVIUNUQWRXFWTVMVJVKXFXAABVLVNVO
      VPVQVRVSVT $.
  $}

  ${
    $d ph c $.  $d ps x $.  $d ch x y $.  $d th x y z $.  $d ta x y c $.
    $d R x y z c $.  $d S x y z c $.  $d A x y z c $.  $d B x y z c $.
    $d C x y z c $.
    ovg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ovg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    ovg.3 $e |- ( z = C -> ( ch <-> th ) ) $.
    ovg.4 $e |- ( ( ta /\ ( x e. R /\ y e. S ) ) -> E! z ph ) $.
    ovg.5 $e |- F = { <. <. x , y >. , z >. |
                                            ( ( x e. R /\ y e. S ) /\ ph ) } $.
    $( The value of an operation class abstraction.  (Contributed by Jeff
       Madsen, 10-Jun-2010.) $)
    ovg $p |- ( ( ta /\ ( A e. R /\ B e. S /\ C e. D ) )
                                            -> ( ( A F B ) = C <-> th ) ) $=
      ( vc wcel w3a wa co wceq cop cv coprab df-ov fveq1i eqtri eqeq1i wb eqeq2
      cfv wi opeq2 eleq1d bibi12d imbi2d copab wfn weu wal ex alrimivv fnoprabg
      syl eleq1 anbi1d anbi2d opelopabg fnopfvb syl2an vtoclg com12 exp32 3imp2
      ibir anbi12d eloprabg adantl bitrd bitrid biidd bianabs 3adant3 ) EIMUBZJ
      NUBZKLUBZUCZUDZIJOUEZKUFZWIWJUDZDUDZDWOIJUGZFUHZMUBZGUHZNUBZUDZAUDZFGHUIZ
      UPZKUFZWMWQWNXFKWNWROUPXFIJOUJWROXETUKULUMWMXGWRKUGZXEUBZWQEWIWJWKXGXIUNZ
      EWIWJWKXJUQWKEWPUDZXJXKXFUAUHZUFZWRXLUGZXEUBZUNZUQXKXJUQUAKLXLKUFZXPXJXKX
      QXMXGXOXIXLKXFUOXQXNXHXEXLKWRURUSUTVAEXEXCFGVBZVCZWRXRUBZXPWPEXCAHVDZUQZG
      VEFVEXSEYBFGEXCYASVFVGXCAFGHVHVIWPXTXCWIXBUDZWPFGIJMNWSIUFZWTWIXBWSIMVJVK
      ZXAJUFZXBWJWIXAJNVJVLZVMVTXRWRXLXEVNVOVPVQVRVSWLXIWQUNEXDYCBUDWPCUDWQFGHI
      JKMNLYDXCYCABYEPWAYFYCWPBCYGQWAHUHKUFCDWPRVLWBWCWDWEWLWQDUNZEWIWJYHWKWPWQ
      DWPWQWFWGWHWCWD $.
  $}

  $( The value of a restricted operation.  (Contributed by FL, 10-Nov-2006.) $)
  ovres $p |- ( ( A e. C /\ B e. D ) -> ( A ( F |` ( C X. D ) ) B )
    = ( A F B ) ) $=
    ( wcel wa cop cxp cres cfv co wceq opelxpi fvres syl df-ov 3eqtr4g ) ACFBDF
    GZABHZECDIZJZKZTEKZABUBLABELSTUAFUCUDMABCDNTUAEOPABUBQABEQR $.

  ${
    ovresd.1 $e |- ( ph -> A e. X ) $.
    ovresd.2 $e |- ( ph -> B e. X ) $.
    $( Lemma for converting metric theorems to metric space theorems.
       (Contributed by Mario Carneiro, 2-Oct-2015.) $)
    ovresd $p |- ( ph -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) ) $=
      ( wcel cxp cres co wceq ovres syl2anc ) ABEHCEHBCDEEIJKBCDKLFGBCEEDMN $.
  $}

  $( The value of a member of the domain of a subclass of an operation.
     (Contributed by NM, 23-Aug-2007.) $)
  oprssov $p |- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\
        ( A e. C /\ B e. D ) ) -> ( A F B ) = ( A G B ) ) $=
    ( wfun cxp wfn wss w3a wcel wa cres co wceq ovres adantl cdm eqtr3d reseq2d
    fndm 3ad2ant2 funssres 3adant2 oveqd adantr ) EGZFCDHZIZFEJZKZACLBDLMZMABEU
    INZOZABEOZABFOZUMUOUPPULABCDEQRULUOUQPUMULUNFABULEFSZNZUNFUJUHUSUNPUKUJURUI
    EUIFUBUAUCUHUKUSFPUJEFUDUETUFUGT $.

  $( An operation's value belongs to its codomain.  (Contributed by NM,
     27-Aug-2006.) $)
  fovcdm $p |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) ->
               ( A F B ) e. C ) $=
    ( cxp wf wcel co wa cop opelxpi cfv df-ov ffvelcdm eqeltrid sylan2 3impb )
    DEGZCFHZADIZBEIZABFJZCIZUBUCKUAABLZTIZUEABDEMUAUGKUDUFFNCABFOTCUFFPQRS $.

  ${
    fovcdmd.1 $e |- ( ph -> F : ( R X. S ) --> C ) $.
    $( An operation's value belongs to its codomain.  (Contributed by Mario
       Carneiro, 29-Dec-2016.) $)
    fovcdmda $p |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C ) $=
      ( wcel co cxp wf fovcdm syl3an1 3expb ) ABEIZCFIZBCGJDIZAEFKDGLPQRHBCDEFG
      MNO $.

    fovcdmd.2 $e |- ( ph -> A e. R ) $.
    fovcdmd.3 $e |- ( ph -> B e. S ) $.
    $( An operation's value belongs to its codomain.  (Contributed by Mario
       Carneiro, 29-Dec-2016.) $)
    fovcdmd $p |- ( ph -> ( A F B ) e. C ) $=
      ( cxp wf wcel co fovcdm syl3anc ) AEFKDGLBEMCFMBCGNDMHIJBCDEFGOP $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w z C $.  $d w x y z F $.
    $( The range of an operation expressed as a collection of the operation's
       values.  (Contributed by NM, 29-Oct-2006.) $)
    fnrnov $p |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B
                     z = ( x F y ) } ) $=
      ( vw cxp wfn crn cv cfv wceq wrex cab co fnrnfv cop fveq2 df-ov eqtr4di
      eqeq2d rexxp abbii eqtrdi ) FDEHZIFJCKZGKZFLZMZGUFNZCOUGAKZBKZFPZMZBENADN
      ZCOGCUFFQUKUPCUJUOGABDEUHULUMRZMZUIUNUGURUIUQFLUNUHUQFSULUMFTUAUBUCUDUE
      $.

    $( An onto mapping of an operation expressed in terms of operation values.
       (Contributed by NM, 29-Oct-2006.) $)
    foov $p |- ( F : ( A X. B ) -onto-> C <-> ( F : ( A X. B ) --> C /\
                  A. z e. C E. x e. A E. y e. B z = ( x F y ) ) ) $=
      ( vw cxp wfo wf cv cfv wceq wrex wral wa co dffo3 cop fveq2 df-ov eqtr4di
      eqeq2d rexxp ralbii anbi2i bitri ) DEIZFGJUIFGKZCLZHLZGMZNZHUIOZCFPZQUJUK
      ALZBLZGRZNZBEOADOZCFPZQHCUIFGSUPVBUJUOVACFUNUTHABDEULUQURTZNZUMUSUKVDUMVC
      GMUSULVCGUAUQURGUBUCUDUEUFUGUH $.
  $}

  $( An operation's value belongs to its range.  (Contributed by NM,
     10-Feb-2007.) $)
  fnovrn $p |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) ->
                   ( C F D ) e. ran F ) $=
    ( cxp wfn wcel co crn cop opelxpi cfv df-ov fnfvelrn eqeltrid sylan2 3impb
    wa ) EABFZGZCAHZDBHZCDEIZEJZHZUBUCSUACDKZTHZUFCDABLUAUHSUDUGEMUECDENTUGEOPQ
    R $.

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.  $d x y z F $.
    $( A member of an operation's range is a value of the operation.
       (Contributed by NM, 7-Feb-2007.)  (Revised by Mario Carneiro,
       30-Jan-2014.) $)
    ovelrn $p |- ( F Fn ( A X. B ) -> ( C e. ran F <->
                  E. x e. A E. y e. B C = ( x F y ) ) ) $=
      ( vz cxp wfn crn wcel cv co wceq wrex cab fnrnov cvv wi a1i eleq2d fnovex
      elex w3a eleq1 syl5ibrcom 3expb rexlimdvva eqeq1 2rexbidv elabg pm5.21ndd
      wb bitrd ) FCDHIZEFJZKEGLZALZBLZFMZNZBDOACOZGPZKZEUTNZBDOACOZUOUPVCEABGCD
      FQUAUOERKZVDVFVDVGSUOEVCUCTUOVEVGABCDUOURCKZUSDKZVEVGSUOVHVIUDVGVEUTRKURU
      SCDFUBEUTRUEUFUGUHVGVDVFUMSUOVBVFGERUQENVAVEABCDUQEUTUIUJUKTULUN $.

    $( Membership relation for the values of a function whose image is a
       subclass.  (Contributed by Mario Carneiro, 23-Dec-2013.) $)
    funimassov $p |- ( ( Fun F /\ ( A X. B ) C_ dom F ) ->
        ( ( F " ( A X. B ) ) C_ C <-> A. x e. A A. y e. B ( x F y ) e. C ) ) $=
      ( vz wfun cxp cdm wss wa cima cv cfv wcel wral co funimass4 cop eqtr4di
      wceq fveq2 df-ov eleq1d ralxp bitrdi ) FHCDIZFJKLFUHMEKGNZFOZEPZGUHQANZBN
      ZFRZEPZBDQACQGUHEFSUKUOGABCDUIULUMTZUBZUJUNEUQUJUPFOUNUIUPFUCULUMFUDUAUEU
      FUG $.

    $( Operation value in an image.  (Contributed by Mario Carneiro,
       23-Dec-2013.)  (Revised by Mario Carneiro, 29-Jan-2014.) $)
    ovelimab $p |- ( ( F Fn A /\ ( B X. C ) C_ A ) ->
        ( D e. ( F " ( B X. C ) ) <-> E. x e. B E. y e. C D = ( x F y ) ) ) $=
      ( vz wfn cxp wss wa cima wcel cv cfv wceq wrex co bitrdi fvelimab eqtr4di
      cop fveq2 df-ov eqeq1d eqcom rexxp ) GCIDEJZCKLFGUIMNHOZGPZFQZHUIRFAOZBOZ
      GSZQZBERADRHCUIFGUAULUPHABDEUJUMUNUCZQZULUOFQUPURUKUOFURUKUQGPUOUJUQGUDUM
      UNGUEUBUFUOFUGTUHT $.
  $}

  ${
    oprvalconst2.1 $e |- C e. _V $.
    $( The value of a constant operation.  (Contributed by NM, 5-Nov-2006.) $)
    ovconst2 $p |- ( ( R e. A /\ S e. B ) ->
                       ( R ( ( A X. B ) X. { C } ) S ) = C ) $=
      ( wcel wa cxp csn co cop cfv df-ov wceq opelxpi fvconst2 syl eqtrid ) DAG
      EBGHZDEABIZCJIZKDELZUBMZCDEUBNTUCUAGUDCODEABPUACUCFQRS $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.  $d x y D $.  $d x y E $.  $d x y ph $.
    $d x y F $.
    caovclg.1 $e |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E ) $.
    $( Convert an operation closure law to class notation.  (Contributed by
       Mario Carneiro, 26-May-2014.) $)
    caovclg $p |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E ) $=
      ( cv co wcel wral wa ralrimivva wceq oveq1 eleq1d oveq2 rspc2v mpan9 ) AB
      KZCKZILZHMZCGNBFNDFMEGMODEILZHMZAUFBCFGJPUFUHDUDILZHMBCDEFGUCDQUEUIHUCDUD
      IRSUDEQUIUGHUDEDITSUAUB $.

    caovcld.2 $e |- ( ph -> A e. C ) $.
    caovcld.3 $e |- ( ph -> B e. D ) $.
    $( Convert an operation closure law to class notation.  (Contributed by
       Mario Carneiro, 30-Dec-2014.) $)
    caovcld $p |- ( ph -> ( A F B ) e. E ) $=
      ( wcel co id caovclg syl12anc ) AADFMEGMDEINHMAOKLABCDEFGHIJPQ $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y F $.  $d x y S $.
    caovcl.1 $e |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S ) $.
    $( Convert an operation closure law to class notation.  (Contributed by NM,
       4-Aug-1995.)  (Revised by Mario Carneiro, 26-May-2014.) $)
    caovcl $p |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S ) $=
      ( wtru wcel wa co tru cv adantl caovclg mpan ) HCEIDEIJCDFKEILHABCDEEEFAM
      ZEIBMZEIJQRFKEIHGNOP $.
  $}

  ${
    $( General laws for commutative, associative, distributive operations. $)
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.  $d x y z ph $.
    $d x y z F $.  $d x y z G $.  $d x y z H $.  $d x y z K $.  $d x y z R $.
    $d x y z S $.  $d x y z T $.
    ${
      caovcomg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
                          ( x F y ) = ( y F x ) ) $.
      $( Convert an operation commutative law to class notation.  (Contributed
         by Mario Carneiro, 1-Jun-2013.) $)
      caovcomg $p |- ( ( ph /\ ( A e. S /\ B e. S ) ) ->
                        ( A F B ) = ( B F A ) ) $=
        ( cv co wceq wral wcel wa ralrimivva oveq1 oveq2 eqeq12d rspc2v mpan9 )
        ABIZCIZGJZUBUAGJZKZCFLBFLDFMEFMNDEGJZEDGJZKZAUEBCFFHOUEUHDUBGJZUBDGJZKB
        CDEFFUADKUCUIUDUJUADUBGPUADUBGQRUBEKUIUFUJUGUBEDGQUBEDGPRST $.

      caovcomd.2 $e |- ( ph -> A e. S ) $.
      caovcomd.3 $e |- ( ph -> B e. S ) $.
      $( Convert an operation commutative law to class notation.  (Contributed
         by Mario Carneiro, 30-Dec-2014.) $)
      caovcomd $p |- ( ph -> ( A F B ) = ( B F A ) ) $=
        ( wcel co wceq id caovcomg syl12anc ) AADFKEFKDEGLEDGLMANIJABCDEFGHOP
        $.
    $}

    ${
      caovcom.1 $e |- A e. _V $.
      caovcom.2 $e |- B e. _V $.
      caovcom.3 $e |- ( x F y ) = ( y F x ) $.
      $( Convert an operation commutative law to class notation.  (Contributed
         by NM, 26-Aug-1995.)  (Revised by Mario Carneiro, 1-Jun-2013.) $)
      caovcom $p |- ( A F B ) = ( B F A ) $=
        ( cvv wcel wa co wceq pm3.2i cv a1i caovcomg mp2an ) CIJZSDIJZKCDELDCEL
        MFSTFGNSABCDIEAOZBOZELUBUAELMSUAIJUBIJKKHPQR $.
    $}

    ${
      caovassg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
        ( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
      $( Convert an operation associative law to class notation.  (Contributed
         by Mario Carneiro, 1-Jun-2013.)  (Revised by Mario Carneiro,
         26-May-2014.) $)
      caovassg $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
          ( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
        ( cv co wceq wral wcel oveq1 oveq1d eqeq12d oveq2 oveq2d rspc3v mpan9
        w3a ralrimivvva ) ABKZCKZILZDKZILZUEUFUHILZILZMZDHNCHNBHNEHOFHOGHOUCEFI
        LZGILZEFGILZILZMZAULBCDHHHJUDULUQEUFILZUHILZEUJILZMUMUHILZEFUHILZILZMBC
        DEFGHHHUEEMZUIUSUKUTVDUGURUHIUEEUFIPQUEEUJIPRUFFMZUSVAUTVCVEURUMUHIUFFE
        ISQVEUJVBEIUFFUHIPTRUHGMZVAUNVCUPUHGUMISVFVBUOEIUHGFISTRUAUB $.

      caovassd.2 $e |- ( ph -> A e. S ) $.
      caovassd.3 $e |- ( ph -> B e. S ) $.
      caovassd.4 $e |- ( ph -> C e. S ) $.
      $( Convert an operation associative law to class notation.  (Contributed
         by Mario Carneiro, 30-Dec-2014.) $)
      caovassd $p |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
        ( wcel co wceq id caovassg syl13anc ) AAEHNFHNGHNEFIOGIOEFGIOIOPAQKLMAB
        CDEFGHIJRS $.
    $}

    ${
      caovass.1 $e |- A e. _V $.
      caovass.2 $e |- B e. _V $.
      caovass.3 $e |- C e. _V $.
      caovass.4 $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
      $( Convert an operation associative law to class notation.  (Contributed
         by NM, 26-Aug-1995.)  (Revised by Mario Carneiro, 26-May-2014.) $)
      caovass $p |- ( ( A F B ) F C ) = ( A F ( B F C ) ) $=
        ( cvv wcel co wceq wtru w3a tru cv wa a1i caovassg mpan mp3an ) DLMZELM
        ZFLMZDEGNFGNDEFGNGNOZHIJPUEUFUGQUHRPABCDEFLGASZBSZGNCSZGNUIUJUKGNGNOPUI
        LMUJLMUKLMQTKUAUBUCUD $.
    $}

    ${
      caovcang.1 $e |- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) ->
                         ( ( x F y ) = ( x F z ) <-> y = z ) ) $.
      $( Convert an operation cancellation law to class notation.  (Contributed
         by NM, 20-Aug-1995.)  (Revised by Mario Carneiro, 30-Dec-2014.) $)
      caovcang $p |- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) ->
                       ( ( A F B ) = ( A F C ) <-> B = C ) ) $=
        ( cv co wceq wb wral wcel oveq1 oveq2 bibi12d ralrimivvva bibi1d eqeq1d
        w3a eqeq12d eqeq1 eqeq2d eqeq2 rspc3v mpan9 ) ABLZCLZJMZUKDLZJMZNZULUNN
        ZOZDHPCHPBIPEIQFHQGHQUDEFJMZEGJMZNZFGNZOZAURBCDIHHKUAURVCEULJMZEUNJMZNZ
        UQOUSVENZFUNNZOBCDEFGIHHUKENZUPVFUQVIUMVDUOVEUKEULJRUKEUNJRUEUBULFNZVFV
        GUQVHVJVDUSVEULFEJSUCULFUNUFTUNGNZVGVAVHVBVKVEUTUSUNGEJSUGUNGFUHTUIUJ
        $.

      caovcand.2 $e |- ( ph -> A e. T ) $.
      caovcand.3 $e |- ( ph -> B e. S ) $.
      caovcand.4 $e |- ( ph -> C e. S ) $.
      $( Convert an operation cancellation law to class notation.  (Contributed
         by Mario Carneiro, 30-Dec-2014.) $)
      caovcand $p |- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) ) $=
        ( wcel co wceq wb id caovcang syl13anc ) AAEIOFHOGHOEFJPEGJPQFGQRASLMNA
        BCDEFGHIJKTUA $.

      caovcanrd.5 $e |- ( ph -> A e. S ) $.
      caovcanrd.6 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
                          ( x F y ) = ( y F x ) ) $.
      $( Commute the arguments of an operation cancellation law.  (Contributed
         by Mario Carneiro, 30-Dec-2014.) $)
      caovcanrd $p |- ( ph -> ( ( B F A ) = ( C F A ) <-> B = C ) ) $=
        ( co wceq caovcomd eqeq12d caovcand bitr3d ) AEFJQZEGJQZRFEJQZGEJQZRFGR
        AUCUEUDUFABCEFHJPOMSABCEGHJPONSTABCDEFGHIJKLMNUAUB $.
    $}

    ${
      caovcan.1 $e |- C e. _V $.
      caovcan.2 $e |- ( ( x e. S /\ y e. S ) ->
                   ( ( x F y ) = ( x F z ) -> y = z ) ) $.
      $( Convert an operation cancellation law to class notation.  (Contributed
         by NM, 20-Aug-1995.) $)
      caovcan $p |- ( ( A e. S /\ B e. S ) ->
                   ( ( A F B ) = ( A F C ) -> B = C ) ) $=
        ( cv co wceq wi oveq1 eqeq12d imbi1d oveq2 imbi12d wcel eqeq1d eqeq1 wa
        eqeq2d eqeq2 imbi2d vtocl vtocl2ga ) AKZBKZHLZUIFHLZMZUJFMZNZDUJHLZDFHL
        ZMZUNNDEHLZUQMZEFMZNABDEGGUIDMZUMURUNVBUKUPULUQUIDUJHOUIDFHOPQUJEMZURUT
        UNVAVCUPUSUQUJEDHRUAUJEFUBSUIGTUJGTUCZUKUICKZHLZMZUJVEMZNZNVDUONCFIVEFM
        ZVIUOVDVJVGUMVHUNVJVFULUKVEFUIHRUDVEFUJUESUFJUGUH $.
    $}

    ${
      caovordig.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
                         ( x R y -> ( z F x ) R ( z F y ) ) ) $.
      $( Convert an operation ordering law to class notation.  (Contributed by
         Mario Carneiro, 31-Dec-2014.) $)
      caovordig $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
                      ( A R B -> ( C F A ) R ( C F B ) ) ) $=
        ( cv wbr co wi wral wcel wceq oveq2 imbi12d w3a ralrimivvva breq1 breq2
        breq1d breq2d oveq1 breq12d imbi2d rspc3v mpan9 ) ABLZCLZHMZDLZULJNZUOU
        MJNZHMZOZDIPCIPBIPEIQFIQGIQUAEFHMZGEJNZGFJNZHMZOZAUSBCDIIIKUBUSVDEUMHMZ
        UOEJNZUQHMZOUTVFUOFJNZHMZOBCDEFGIIIULERZUNVEURVGULEUMHUCVJUPVFUQHULEUOJ
        SUETUMFRZVEUTVGVIUMFEHUDVKUQVHVFHUMFUOJSUFTUOGRZVIVCUTVLVFVAVHVBHUOGEJU
        GUOGFJUGUHUIUJUK $.

      caovordid.2 $e |- ( ph -> A e. S ) $.
      caovordid.3 $e |- ( ph -> B e. S ) $.
      caovordid.4 $e |- ( ph -> C e. S ) $.
      $( Convert an operation ordering law to class notation.  (Contributed by
         Mario Carneiro, 31-Dec-2014.) $)
      caovordid $p |- ( ph -> ( A R B -> ( C F A ) R ( C F B ) ) ) $=
        ( wcel wbr co wi id caovordig syl13anc ) AAEIOFIOGIOEFHPGEJQGFJQHPRASLM
        NABCDEFGHIJKTUA $.
    $}

    ${
      caovordg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
                         ( x R y <-> ( z F x ) R ( z F y ) ) ) $.
      $( Convert an operation ordering law to class notation.  (Contributed by
         NM, 19-Feb-1996.)  (Revised by Mario Carneiro, 30-Dec-2014.) $)
      caovordg $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
                      ( A R B <-> ( C F A ) R ( C F B ) ) ) $=
        ( cv wbr co wb wral wcel wceq oveq2 bibi12d w3a ralrimivvva breq1 breq2
        breq1d breq2d oveq1 breq12d bibi2d rspc3v mpan9 ) ABLZCLZHMZDLZULJNZUOU
        MJNZHMZOZDIPCIPBIPEIQFIQGIQUAEFHMZGEJNZGFJNZHMZOZAUSBCDIIIKUBUSVDEUMHMZ
        UOEJNZUQHMZOUTVFUOFJNZHMZOBCDEFGIIIULERZUNVEURVGULEUMHUCVJUPVFUQHULEUOJ
        SUETUMFRZVEUTVGVIUMFEHUDVKUQVHVFHUMFUOJSUFTUOGRZVIVCUTVLVFVAVHVBHUOGEJU
        GUOGFJUGUHUIUJUK $.

      caovordd.2 $e |- ( ph -> A e. S ) $.
      caovordd.3 $e |- ( ph -> B e. S ) $.
      caovordd.4 $e |- ( ph -> C e. S ) $.
      $( Convert an operation ordering law to class notation.  (Contributed by
         Mario Carneiro, 30-Dec-2014.) $)
      caovordd $p |- ( ph -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $=
        ( wcel wbr co wb id caovordg syl13anc ) AAEIOFIOGIOEFHPGEJQGFJQHPRASLMN
        ABCDEFGHIJKTUA $.

      caovord2d.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
                            ( x F y ) = ( y F x ) ) $.
      $( Operation ordering law with commuted arguments.  (Contributed by Mario
         Carneiro, 30-Dec-2014.) $)
      caovord2d $p |- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) ) $=
        ( wbr co caovordd caovcomd breq12d bitrd ) AEFHPGEJQZGFJQZHPEGJQZFGJQZH
        PABCDEFGHIJKLMNRAUBUDUCUEHABCGEIJONLSABCGFIJONMSTUA $.

      caovord3d.5 $e |- ( ph -> D e. S ) $.
      $( Ordering law.  (Contributed by Mario Carneiro, 30-Dec-2014.) $)
      caovord3d $p |- ( ph ->
        ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) ) $=
        ( co wbr wb wceq breq1 caovord2d caovordd bibi12d imbitrrid ) EFKRZGHKR
        ZUAEGISZHFISZTAUGGFKRZISZUHUKISZTUGUHUKIUBAUIULUJUMABCDEGFIJKLMONPUCABC
        DHFGIJKLQNOUDUEUF $.
    $}

    ${
      caovord.1 $e |- A e. _V $.
      caovord.2 $e |- B e. _V $.
      caovord.3 $e |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) ) $.
      $( Convert an operation ordering law to class notation.  (Contributed by
         NM, 19-Feb-1996.) $)
      caovord $p |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $=
        ( wbr cv co wb wceq oveq1 wi oveq2 breq12d bibi2d wcel wa breq1 bibi12d
        breq1d breq2 breq2d sylan9bb imbi2d vtocl2 vtoclga ) DEGMZCNZDIOZUOEIOZ
        GMZPZUNFDIOZFEIOZGMZPCFHUOFQZURVBUNVCUPUTUQVAGUOFDIRUOFEIRUAUBUOHUCZANZ
        BNZGMZUOVEIOZUOVFIOZGMZPZSVDUSSABDEJKVEDQZVFEQZUDVKUSVDVLVKDVFGMZUPVIGM
        ZPVMUSVLVGVNVJVOVEDVFGUEVLVHUPVIGVEDUOITUGUFVMVNUNVOURVFEDGUHVMVIUQUPGV
        FEUOITUIUFUJUKLULUM $.

      $( (We don't bother to eliminate this redundant hypothesis.) $)
      caovord2.3 $e |- C e. _V $.
      caovord2.com $e |- ( x F y ) = ( y F x ) $.
      $( Operation ordering law with commuted arguments.  (Contributed by NM,
         27-Feb-1996.) $)
      caovord2 $p |- ( C e. S -> ( A R B <-> ( A F C ) R ( B F C ) ) ) $=
        ( wcel wbr co caovord caovcom breq12i bitrdi ) FHODEGPFDIQZFEIQZGPDFIQZ
        EFIQZGPABCDEFGHIJKLRUBUDUCUEGABFDIMJNSABFEIMKNSTUA $.

      $( (We don't bother to eliminate redundant hypotheses.) $)
      caovord3.4 $e |- D e. _V $.
      $( Ordering law.  (Contributed by NM, 29-Feb-1996.) $)
      caovord3 $p |- ( ( ( B e. S /\ C e. S ) /\
                ( A F B ) = ( C F D ) ) -> ( A R C <-> D R B ) ) $=
        ( wcel wa co wbr wceq wb caovord2 adantr breq1 sylan9bb ad2antlr bitr4d
        caovord ) EIQZFIQZRZDEJSZFGJSZUAZRDFHTZUNFEJSZHTZGEHTZULUPUMUQHTZUOURUJ
        UPUTUBUKABCDFEHIJKNMLOUCUDUMUNUQHUEUFUKUSURUBUJUOABCGEFHIJPLMUIUGUH $.
    $}

    ${
      caovdig.1 $e |- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) ->
        ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) ) $.
      $( Convert an operation distributive law to class notation.  (Contributed
         by NM, 25-Aug-1995.)  (Revised by Mario Carneiro, 26-Jul-2014.) $)
      caovdig $p |- ( ( ph /\ ( A e. K /\ B e. S /\ C e. S ) ) ->
        ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) ) $=
        ( cv co wceq wral wcel oveq1 eqeq12d oveq12d oveq2d oveq2 oveq1d rspc3v
        w3a ralrimivvva mpan9 ) ABNZCNZDNZIOZJOZUIUJJOZUIUKJOZKOZPZDHQCHQBLQELR
        FHRGHRUFEFGIOZJOZEFJOZEGJOZKOZPZAUQBCDLHHMUGUQVCEULJOZEUJJOZEUKJOZKOZPE
        FUKIOZJOZUTVFKOZPBCDEFGLHHUIEPZUMVDUPVGUIEULJSVKUNVEUOVFKUIEUJJSUIEUKJS
        UATUJFPZVDVIVGVJVLULVHEJUJFUKISUBVLVEUTVFKUJFEJUCUDTUKGPZVIUSVJVBVMVHUR
        EJUKGFIUCUBVMVFVAUTKUKGEJUCUBTUEUH $.

      caovdid.2 $e |- ( ph -> A e. K ) $.
      caovdid.3 $e |- ( ph -> B e. S ) $.
      caovdid.4 $e |- ( ph -> C e. S ) $.
      $( Convert an operation distributive law to class notation.  (Contributed
         by Mario Carneiro, 30-Dec-2014.) $)
      caovdid $p |- ( ph -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) ) $=
        ( wcel co wceq id caovdig syl13anc ) AAELQFHQGHQEFGIRJREFJREGJRKRSATNOP
        ABCDEFGHIJKLMUAUB $.
    $}

    ${
      caovdir2d.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
        ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) ) $.
      caovdir2d.2 $e |- ( ph -> A e. S ) $.
      caovdir2d.3 $e |- ( ph -> B e. S ) $.
      caovdir2d.4 $e |- ( ph -> C e. S ) $.
      caovdir2d.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
                           ( x F y ) e. S ) $.
      caovdir2d.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
                            ( x G y ) = ( y G x ) ) $.
      $( Convert an operation distributive law to class notation.  (Contributed
         by Mario Carneiro, 30-Dec-2014.) $)
      caovdir2d $p |- ( ph
          -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) ) $=
        ( co caovdid caovcld caovcomd oveq12d 3eqtr4d ) AGEFIQZJQGEJQZGFJQZIQUC
        GJQEGJQZFGJQZIQABCDGEFHIJIHKNLMRABCUCGHJPABCEFHHHIOLMSNTAUFUDUGUEIABCEG
        HJPLNTABCFGHJPMNTUAUB $.
    $}

    ${
      caovdirg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) ->
        ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) ) $.
      $( Convert an operation reverse distributive law to class notation.
         (Contributed by Mario Carneiro, 19-Oct-2014.) $)
      caovdirg $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. K ) ) ->
        ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) ) $=
        ( cv co wceq wral wcel oveq1 oveq2 w3a ralrimivvva oveq1d oveq2d rspc3v
        eqeq12d oveq12d mpan9 ) ABNZCNZIOZDNZJOZUIULJOZUJULJOZKOZPZDLQCHQBHQEHR
        FHRGLRUAEFIOZGJOZEGJOZFGJOZKOZPZAUQBCDHHLMUBUQVCEUJIOZULJOZEULJOZUOKOZP
        URULJOZVFFULJOZKOZPBCDEFGHHLUIEPZUMVEUPVGVKUKVDULJUIEUJISUCVKUNVFUOKUIE
        ULJSUCUFUJFPZVEVHVGVJVLVDURULJUJFEITUCVLUOVIVFKUJFULJSUDUFULGPZVHUSVJVB
        ULGURJTVMVFUTVIVAKULGEJTULGFJTUGUFUEUH $.

      caovdird.2 $e |- ( ph -> A e. S ) $.
      caovdird.3 $e |- ( ph -> B e. S ) $.
      caovdird.4 $e |- ( ph -> C e. K ) $.
      $( Convert an operation distributive law to class notation.  (Contributed
         by Mario Carneiro, 30-Dec-2014.) $)
      caovdird $p |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) ) $=
        ( wcel co wceq id caovdirg syl13anc ) AAEHQFHQGLQEFIRGJREGJRFGJRKRSATNO
        PABCDEFGHIJKLMUAUB $.
    $}

    ${
      caovdi.1 $e |- A e. _V $.
      caovdi.2 $e |- B e. _V $.
      caovdi.3 $e |- C e. _V $.
      caovdi.4 $e |- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) $.
      $( Convert an operation distributive law to class notation.  (Contributed
         by NM, 25-Aug-1995.)  (Revised by Mario Carneiro, 28-Jun-2013.) $)
      caovdi $p |- ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) $=
        ( cvv wcel co wceq wtru w3a tru cv wa a1i caovdig mpan mp3an ) DMNZEMNZ
        FMNZDEFGOHODEHODFHOGOPZIJKQUFUGUHRUISQABCDEFMGHGMATZBTZCTZGOHOUJUKHOUJU
        LHOGOPQUJMNUKMNULMNRUALUBUCUDUE $.
    $}

    ${
      caovd.1 $e |- ( ph -> A e. S ) $.
      caovd.2 $e |- ( ph -> B e. S ) $.
      caovd.3 $e |- ( ph -> C e. S ) $.
      caovd.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
        ( x F y ) = ( y F x ) ) $.
      caovd.ass $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
        ( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
      $( Rearrange arguments in a commutative, associative operation.
         (Contributed by NM, 26-Aug-1995.)  (Revised by Mario Carneiro,
         30-Dec-2014.) $)
      caov32d $p |- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) ) $=
        ( co caovcomd oveq2d caovassd 3eqtr4d ) AEFGIOZIOEGFIOZIOEFIOGIOEGIOFIO
        ATUAEIABCFGHIMKLPQABCDEFGHINJKLRABCDEGFHINJLKRS $.

      $( Rearrange arguments in a commutative, associative operation.
         (Contributed by NM, 26-Aug-1995.)  (Revised by Mario Carneiro,
         30-Dec-2014.) $)
      caov12d $p |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) ) $=
        ( co caovcomd oveq1d caovassd 3eqtr3d ) AEFIOZGIOFEIOZGIOEFGIOIOFEGIOIO
        ATUAGIABCEFHIMJKPQABCDEFGHINJKLRABCDFEGHINKJLRS $.

      $( Rearrange arguments in a commutative, associative operation.
         (Contributed by NM, 26-Aug-1995.)  (Revised by Mario Carneiro,
         30-Dec-2014.) $)
      caov31d $p |- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) ) $=
        ( co caovcomd oveq1d caov32d 3eqtr4d ) AEGIOZFIOGEIOZFIOEFIOGIOGFIOEIOA
        TUAFIABCEGHIMJLPQABCDEFGHIJKLMNRABCDGFEHILKJMNRS $.

      $( Rearrange arguments in a commutative, associative operation.
         (Contributed by NM, 26-Aug-1995.)  (Revised by Mario Carneiro,
         30-Dec-2014.) $)
      caov13d $p |- ( ph -> ( A F ( B F C ) ) = ( C F ( B F A ) ) ) $=
        ( co caov31d caovassd 3eqtr3d ) AEFIOGIOGFIOEIOEFGIOIOGFEIOIOABCDEFGHIJ
        KLMNPABCDEFGHINJKLQABCDGFEHINLKJQR $.

      ${
        caovd.4 $e |- ( ph -> D e. S ) $.
        caovd.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S ) $.
        $( Rearrange arguments in a commutative, associative operation.
           (Contributed by NM, 26-Aug-1995.)  (Revised by Mario Carneiro,
           30-Dec-2014.) $)
        caov4d $p |- ( ph -> ( ( A F B ) F ( C F D ) ) =
                             ( ( A F C ) F ( B F D ) ) ) $=
          ( co caovcld caovassd caov12d oveq2d 3eqtr4d ) AEFGHJRZJRZJREGFHJRZJR
          ZJREFJRUDJREGJRUFJRAUEUGEJABCDFGHIJLMPNOUAUBABCDEFUDIJOKLABCGHIIIJQMP
          STABCDEGUFIJOKMABCFHIIIJQLPSTUC $.

        $( Rearrange arguments in a commutative, associative operation.
           (Contributed by NM, 26-Aug-1995.)  (Revised by Mario Carneiro,
           30-Dec-2014.) $)
        caov411d $p |- ( ph -> ( ( A F B ) F ( C F D ) ) =
                               ( ( C F B ) F ( A F D ) ) ) $=
          ( co caovcomd oveq1d caov4d 3eqtr3d ) AFEJRZGHJRZJRFGJRZEHJRZJREFJRZU
          DJRGFJRZUFJRABCDFEGHIJLKMNOPQUAAUCUGUDJABCFEIJNLKSTAUEUHUFJABCFGIJNLM
          STUB $.

        $( Rearrange arguments in a commutative, associative operation.
           (Contributed by NM, 26-Aug-1995.)  (Revised by Mario Carneiro,
           30-Dec-2014.) $)
        caov42d $p |- ( ph -> ( ( A F B ) F ( C F D ) ) =
                              ( ( A F C ) F ( D F B ) ) ) $=
          ( co caov4d caovcomd oveq2d eqtrd ) AEFJRGHJRJREGJRZFHJRZJRUCHFJRZJRA
          BCDEFGHIJKLMNOPQSAUDUEUCJABCFHIJNLPTUAUB $.
      $}
    $}

    ${
      caov.1 $e |- A e. _V $.
      caov.2 $e |- B e. _V $.
      caov.3 $e |- C e. _V $.
      caov.com $e |- ( x F y ) = ( y F x ) $.
      caov.ass $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
      $( Rearrange arguments in a commutative, associative operation.
         (Contributed by NM, 26-Aug-1995.) $)
      caov32 $p |- ( ( A F B ) F C ) = ( ( A F C ) F B ) $=
        ( co caovcom oveq2i caovass 3eqtr4i ) DEFGMZGMDFEGMZGMDEGMFGMDFGMEGMRSD
        GABEFGIJKNOABCDEFGHIJLPABCDFEGHJILPQ $.

      $( Rearrange arguments in a commutative, associative operation.
         (Contributed by NM, 26-Aug-1995.) $)
      caov12 $p |- ( A F ( B F C ) ) = ( B F ( A F C ) ) $=
        ( co caovcom oveq1i caovass 3eqtr3i ) DEGMZFGMEDGMZFGMDEFGMGMEDFGMGMRSF
        GABDEGHIKNOABCDEFGHIJLPABCEDFGIHJLPQ $.

      $( Rearrange arguments in a commutative, associative operation.
         (Contributed by NM, 26-Aug-1995.) $)
      caov31 $p |- ( ( A F B ) F C ) = ( ( C F B ) F A ) $=
        ( co caovass caov12 eqtri caov32 eqtr3i 3eqtr4i ) DFGMEGMZFDEGMZGMZUAFG
        MFEGMZDGMZTDUCGMUBABCDFEGHJILNABCDFEGHJIKLOPABCDEFGHIJKLQFDGMEGMUDUBABC
        FDEGJHIKLQABCFDEGJHILNRS $.

      $( Rearrange arguments in a commutative, associative operation.
         (Contributed by NM, 26-Aug-1995.) $)
      caov13 $p |- ( A F ( B F C ) ) = ( C F ( B F A ) ) $=
        ( co caov31 caovass 3eqtr3i ) DEGMFGMFEGMDGMDEFGMGMFEDGMGMABCDEFGHIJKLN
        ABCDEFGHIJLOABCFEDGJIHLOP $.
    $}
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.
    $d r s t x y z ph $.  $d r s t x y z F $.  $d r s t x y z G $.
    $d x y z H $.  $d x y z R $.  $d r s t x y z S $.
    caovdilemd.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
      ( x G y ) = ( y G x ) ) $.
    caovdilemd.distr $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
      ( ( x F y ) G z ) = ( ( x G z ) F ( y G z ) ) ) $.
    caovdilemd.ass $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
      ( ( x G y ) G z ) = ( x G ( y G z ) ) ) $.
    caovdilemd.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
      ( x G y ) e. S ) $.
    caovdilemd.a $e |- ( ph -> A e. S ) $.
    caovdilemd.b $e |- ( ph -> B e. S ) $.
    caovdilemd.c $e |- ( ph -> C e. S ) $.
    caovdilemd.d $e |- ( ph -> D e. S ) $.
    caovdilemd.h $e |- ( ph -> H e. S ) $.
    $( Lemma used by real number construction.  (Contributed by Jim Kingdon,
       16-Sep-2019.) $)
    caovdilemd $p |- ( ph -> ( ( ( A G C ) F ( B G D ) ) G H ) =
                     ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) ) $=
      ( co caovcld caovdird caovassd oveq12d eqtrd ) AEGKUBZFHKUBZJUBLKUBUHLKUB
      ZUILKUBZJUBEGLKUBKUBZFHLKUBKUBZJUBABCDUHUILIJKJINABCEGIIIKPQSUCABCFHIIIKP
      RTUCUAUDAUJULUKUMJABCDEGLIKOQSUAUEABCDFHLIKORTUAUEUFUG $.

    caovdl2.6 $e |- ( ph -> R e. S ) $.
    caovdl2.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
      ( x F y ) = ( y F x ) ) $.
    caovdl2.ass $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
      ( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
    caovdl2.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
      ( x F y ) e. S ) $.
    $( Rearrangement of expression involving multiplication ( ` G ` ) and
       addition ( ` F ` ).  (Contributed by Jim Kingdon, 3-Jan-2020.) $)
    caovlem2d $p |- ( ph -> ( ( ( ( A G C ) F ( B G D ) ) G H ) F
          ( ( ( A G D ) F ( B G C ) ) G R ) ) =
        ( ( A G ( ( C G H ) F ( D G R ) ) ) F
          ( B G ( ( C G R ) F ( D G H ) ) ) ) ) $=
      ( vr vs vt co caovcld caov42d caovdilemd oveq12d cv wcel w3a wceq caovclg
      wa simpr1 3adantr1 caovcomg syldan 3anrot caovdirg sylan2b eqcomd ancom2s
      jca 3adantr3 3adantr2 3eqtrd caovdid 3eqtr4d ) AEGMLUJZLUJZFHMLUJZLUJZKUJ
      ZEHILUJZLUJZFGILUJZLUJZKUJZKUJVQWBKUJZWDVSKUJZKUJEGLUJFHLUJKUJMLUJZEHLUJF
      GLUJKUJILUJZKUJEVPWAKUJLUJZFWCVRKUJLUJZKUJABCDVQVSWBWDJKABCEVPJJJLQRABCGM
      JJJLQTUBUKZUKABCFVRJJJLQSABCHMJJJLQUAUBUKZUKABCEWAJJJLQRABCHIJJJLQUAUCUKZ
      UKUDUEABCFWCJJJLQSABCGIJJJLQTUCUKZUKUFULAWHVTWIWEKABCDEFGHJKLMNOPQRSTUAUB
      UMABCDEFHGJKLINOPQRSUATUCUMUNAWJWFWKWGKABCDEVPWAJKLKJABUOZJUPZCUOZJUPZDUO
      ZJUPZUQZUTZWPWRWTKUJZLUJZXDWPLUJZWRWPLUJZWTWPLUJZKUJZWPWRLUJZWPWTLUJZKUJA
      XBWQXDJUPZUTXEXFURXCWQXLAWQWSXAVAAWSXAXLWQAUGUHWRWTJJJKABCUGUOZUHUOZJJJKU
      FUSUSVBVJAUGUHWPXDJLABCXMXNJLNVCZVCVDXBAWSXAWQUQXFXIURWQWSXAVEAUGUHUIWRWT
      WPJKLKJABCDXMXNUIUOJKLKJOVFVFVGXCXGXJXHXKKAWQWSXGXJURXAAWQWSUTUTXJXGNVHVK
      AWQXAXHXKURZWSAXAWQXPAUGUHWTWPJLXOVCVIVLUNVMZRWLWNVNABCDFWCVRJKLKJXQSWOWM
      VNUNVO $.
  $}

  ${
    $d A v w x y z $.  $d B v w x y $.  $d F v w x y z $.  $d S v w x y z $.
    caovimo.idel $e |- B e. S $.
    caovimo.com $e |- ( ( x e. S /\ y e. S ) -> ( x F y ) = ( y F x ) ) $.
    caovimo.ass $e |- ( ( x e. S /\ y e. S /\ z e. S ) ->
      ( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
    caovimo.id $e |- ( x e. S -> ( x F B ) = x ) $.
    $( Uniqueness of inverse element in commutative, associative operation with
       identity.  The identity element is ` B ` .  (Contributed by Jim Kingdon,
       18-Sep-2019.) $)
    caovimo $p |- ( A e. S -> E* w ( w e. S /\ ( A F w ) = B ) ) $=
      ( vv wcel cv co wceq wa w3a adantl wi wal wmo oveq1 3ad2ant2 df-3an simp1
      simp2 simp3 caovassd caov12d eqtrd adantr oveq2 eqeq12d vtoclga sylan9eqr
      id 3ad2antl2 sylanbr anasss 3adant2r a1i caovcomd 3ad2ant3 3eqtr3d 3expib
      3impa alrimivv eleq1 eqeq1d anbi12d mo4 sylibr ) EGNZDOZGNZEVPHPZFQZRZMOZ
      GNZEWAHPZFQZRZRVPWAQZUAZMUBDUBVTDUCVOWGDMVOVTWEWFVOVTWESVRWAHPZFWAHPZVPWA
      VTVOWHWIQZWEVSWJVQVRFWAHUDTUEVOVQWEWHVPQZVSVOVQWEWKVOVQRZWBWDWKWLWBRVOVQW
      BSZWDWKVOVQWBUFWMWDRWHVPWCHPZVPWMWHWNQWDWMWHEVPWAHPHPWNWMABCEVPWAGHAOZGNZ
      BOZGNZCOZGNSWOWQHPZWSHPWOWQWSHPHPQWMKTZVOVQWBUGZVOVQWBUHZVOVQWBUIZUJWMABC
      EVPWAGHXBXCXDWPWRRZWTWQWOHPQZWMJTXAUKULUMVQVOWDWNVPQWBWDVQWNVPFHPZVPWCFVP
      HUNWOFHPZWOQZXGVPQAVPGWOVPQZXHXGWOVPWOVPFHUDXJURUOLUPUQUSULUTVAVHVBWEVOWI
      WAQZVTWBXKWDWBWIWAFHPZWAWBABFWAGHXEXFWBJTFGNWBIVCWBURVDXIXLWAQAWAGWOWAQZX
      HXLWOWAWOWAFHUDXMURUOLUPULUMVEVFVGVIVTWEDMWFVQWBVSWDVPWAGVJWFVRWCFVPWAEHU
      NVKVLVMVN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Maps-to notation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A x y z $.  $d B x y z $.  $d C z $.
    elmpocl.f $e |- F = ( x e. A , y e. B |-> C ) $.
    $( If a two-parameter class is inhabited, constrain the implicit pair.
       (Contributed by Stefan O'Rear, 7-Mar-2015.) $)
    elmpocl $p |- ( X e. ( S F T ) -> ( S e. A /\ T e. B ) ) $=
      ( vz co wcel cop cxp wa cdm cv wceq coprab df-mpo eqtri dmoprabss eqsstri
      cmpo dmeqi cfv wrel wfun mpofun funrel ax-mp relelfvdm mpan eleq2s sselid
      df-ov opelxp sylib ) IFGHLZMZFGNZCDOZMFCMGDMPVAHQZVCVBVDARCMBRDMPKRESZPAB
      KTZQVCHVFHABCDEUEVFJABKCDEUAUBUFVEABKCDUCUDVBVDMZIVBHUGZUTHUHZIVHMVGHUIVI
      ABCDEHJUJHUKULIVBHUMUNFGHUQUOUPFGCDURUS $.

    $( If a two-parameter class is inhabited, the first argument is in its
       nominal domain.  (Contributed by FL, 15-Oct-2012.)  (Revised by Stefan
       O'Rear, 7-Mar-2015.) $)
    elmpocl1 $p |- ( X e. ( S F T ) -> S e. A ) $=
      ( co wcel elmpocl simpld ) IFGHKLFCLGDLABCDEFGHIJMN $.

    $( If a two-parameter class is inhabited, the second argument is in its
       nominal domain.  (Contributed by FL, 15-Oct-2012.)  (Revised by Stefan
       O'Rear, 7-Mar-2015.) $)
    elmpocl2 $p |- ( X e. ( S F T ) -> T e. B ) $=
      ( co wcel elmpocl simprd ) IFGHKLFCLGDLABCDEFGHIJMN $.
  $}

  ${
    $d D a b $.  $d X a b $.  $d Y a b $.  $d ph a b $.
    elovmpod.o $e |- O = ( a e. A , b e. B |-> C ) $.
    elovmpod.x $e |- ( ph -> X e. A ) $.
    elovmpod.y $e |- ( ph -> Y e. B ) $.
    elovmpod.d $e |- ( ph -> D e. V ) $.
    elovmpod.c $e |- ( ( a = X /\ b = Y ) -> C = D ) $.
    $( Utility lemma for two-parameter classes.  (Contributed by Stefan O'Rear,
       21-Jan-2015.)  Variant of ~ elovmpo in deduction form.  (Revised by AV,
       20-Apr-2025.) $)
    elovmpod $p |- ( ph -> ( E e. ( X O Y ) <-> E e. D ) ) $=
      ( co wceq cv cmpo a1i wa adantl ovmpod eleq2d ) AIJGREFAKLIJBCDEGHGKLBCDU
      ASAMUBKTISLTJSUCDESAQUDNOPUEUF $.
  $}

  ${
    $d A a b $.  $d B a b $.  $d E a b $.  $d F a b $.  $d X a b $.
    $d Y a b $.
    elovmpo.d $e |- D = ( a e. A , b e. B |-> C ) $.
    elovmpo.c $e |- C e. _V $.
    elovmpo.e $e |- ( ( a = X /\ b = Y ) -> C = E ) $.
    $( Utility lemma for two-parameter classes.  (Contributed by Stefan O'Rear,
       21-Jan-2015.) $)
    elovmpo $p |- ( F e. ( X D Y ) <-> ( X e. A /\ Y e. B /\ F e. E ) ) $=
      ( co wcel wa cvv wceq wal cv w3a elmpocl eleq1d spc2gv mpi ovmpoga eleq2d
      gen2 mpd3an3 biadan2 df-3an bitr4i ) FGHDNZOZGAOZHBOZPZFEOZPUOUPURUAUNUQU
      RIJABCGHDFKUBUQUMEFUOUPEQOZUMERUQCQOZJSISUSUTIJLUHUTUSIJGHABITGRJTHRPCEQM
      UCUDUEIJGHABCEDQMKUFUIUGUJUOUPURUKUL $.
  $}

  ${
    $d M x y z $.  $d X x y z $.  $d Y x y z $.  $d Z z $.
    elovmporab.o $e |- O = ( x e. _V , y e. _V |-> { z e. M | ph } ) $.
    elovmporab.v $e |- ( ( X e. _V /\ Y e. _V ) -> M e. _V ) $.
    $( Implications for the value of an operation, defined by the maps-to
       notation with a class abstraction as a result, having an element.
       (Contributed by Alexander van der Vekens, 15-Jul-2018.) $)
    elovmporab $p |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) ) $=
      ( cvv wcel wa crab wsbc wceq cv nfcv nfel1 co w3a elmpocl cmpo wb sbceq1a
      a1i sylan9bbr adantl rabbidv eqidd simpl simpr rabexg nfan nfsbc1v nfrabw
      syl nfsbcw ovmpodxf eleq2d df-3an simplbi2com elrabi syl11 sylbid mpcom )
      GLMZHLMZNZIGHFUAZMZVHVIIEMZUBZBCLLADEOZGHFIJUCVJVLIACHPZBGPZDEOZMZVNVJVKV
      RIVJBCGHLLVOVRFLLFBCLLVOUDQVJJUGVJBRGQZCRHQZNZNAVQDEWBAVQUEVJWAAVPVTVQACH
      UFVPBGUFUHUIUJVJVTNLUKVHVIULVHVIUMVJELMVRLMKVQDELUNURVHVIBBGLBGSTBHLBHSZT
      UOVHVICCGLCGSZTCHLCHSTUOWDWCVQBDEVPBGUPBESUQVQCDEVPCBGWDACHUPUSCESUQUTVAV
      MVJVNVSVNVJVMVHVIVMVBVCVQDIEVDVEVFVG $.
  $}

  ${
    $d M x y z $.  $d X x y z $.  $d Y x y z $.  $d Z z $.  $d m x y z $.
    elovmporab1w.o $e |- O = ( x e. _V , y e. _V
                              |-> { z e. [_ x / m ]_ M | ph } ) $.
    elovmporab1w.v $e |- ( ( X e. _V /\ Y e. _V ) -> [_ X / m ]_ M e. _V ) $.
    $( Implications for the value of an operation, defined by the maps-to
       notation with a class abstraction as a result, having an element.  Here,
       the base set of the class abstraction depends on the first operand.
       (Contributed by Alexander van der Vekens, 15-Jul-2018.)  (Revised by GG,
       26-Jan-2024.) $)
    elovmporab1w $p |- ( Z e. ( X O Y )
                         -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) ) $=
      ( cvv wcel wa csb cv wceq nfcv nfel1 co w3a crab elmpocl wsbc cmpo csbeq1
      a1i ad2antrl wb sbceq1a sylan9bbr adantl rabeqbidv eqidd simpl rabexg syl
      simpr nfan nfcsbw nfrabw nfsbcw ovmpodxf eleq2d df-3an simplbi2com elrabi
      nfsbc1v syl11 sylbid mpcom ) HMNZIMNZOZJHIGUAZNZVMVNJEHFPZNZUBZBCMMADEBQZ
      FPZUCZHIGJKUDVOVQJACIUEZBHUEZDVRUCZNZVTVOVPWFJVOBCHIMMWCWFGMMGBCMMWCUFRVO
      KUHVOWAHRZCQIRZOZOAWEDWBVRWHWBVRRVOWIEWAHFUGUIWJAWEUJVOWIAWDWHWEACIUKWDBH
      UKULUMUNVOWHOMUOVMVNUPVMVNUSVOVRMNWFMNLWEDVRMUQURVMVNBBHMBHSZTBIMBISZTUTV
      MVNCCHMCHSZTCIMCISTUTWMWLWEBDVRWDBHVIBEHFWKBFSVAVBWECDVRWDCBHWMACIVIVCCEH
      FWMCFSVAVBVDVEVSVOVTWGVTVOVSVMVNVSVFVGWEDJVRVHVJVKVL $.
  $}

  ${
    $d x A $.  $d B r $.  $d F r $.
    relmptopab.1 $e |- F = ( x e. A |-> { <. y , z >. | ph } ) $.
    $( Any function to sets of ordered pairs produces a relation on function
       value unconditionally.  (Contributed by Mario Carneiro, 7-Aug-2014.)
       (Proof shortened by Mario Carneiro, 24-Dec-2016.) $)
    relmptopab $p |- Rel ( F ` B ) $=
      ( vr cfv wrel cvv cxp wss cv wcel cdm copab wral df-rel wfun funrel ax-mp
      funmpt2 relelfvdm mpan relopab rgenw fvmptssdm sylancl mpcom ssriv mpbir
      mpbi ssel ) FGJZKUPLLMZNZIUPUQURIOZUPPZUSUQPUTFGQPZACDRZUQNZBESURGKZUTVAG
      UAVDBEVBGHUDGUBUCUSFGUEUFVCBEVBKVCACDUGVBTUNUHBEVBUQFGHUIUJUPUQUSUOUKULUP
      TUM $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.  $d x y ph $.
    f1od.1 $e |- F = ( x e. A |-> C ) $.
    ${
      f1od.2 $e |- ( ( ph /\ x e. A ) -> C e. W ) $.
      f1od.3 $e |- ( ( ph /\ y e. B ) -> D e. X ) $.
      f1od.4 $e |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) ) $.
      $( Describe an implicit one-to-one onto function.  (Contributed by Mario
         Carneiro, 30-Apr-2015.) $)
      f1ocnvd $p |- ( ph ->
        ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) ) $=
        ( ccnv cmpt wceq wfn wcel copab wf1o wral ralrimiva fnmpt eqid opabbidv
        cv wa df-mpt eqtri cnveqi cnvopab 3eqtr4g fneq1d mpbird dff1o4 sylanbrc
        syl jca ) ADEHUAZHOZCEGPZQAHDRZVAERZUTAFISZBDUBVCAVEBDLUCBDFHIKUDURAVDV
        BERZAGJSZCEUBVFAVGCEMUCCEGVBJVBUEUDURAEVAVBABUGZDSCUGZFQUHZCBTZVIESVHGQ
        UHZCBTVAVBAVJVLCBNUFVAVJBCTZOVKHVMHBDFPVMKBCDFUIUJUKVJBCULUJCBEGUIUMZUN
        UODEHUPUQVNUS $.

      $( Describe an implicit one-to-one onto function.  (Contributed by Mario
         Carneiro, 12-May-2014.) $)
      f1od $p |- ( ph -> F : A -1-1-onto-> B ) $=
        ( wf1o ccnv cmpt wceq f1ocnvd simpld ) ADEHOHPCEGQRABCDEFGHIJKLMNST $.
    $}

    f1o2d.2 $e |- ( ( ph /\ x e. A ) -> C e. B ) $.
    f1o2d.3 $e |- ( ( ph /\ y e. B ) -> D e. A ) $.
    f1o2d.4 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) ->
                    ( x = D <-> y = C ) ) $.
    $( Describe an implicit one-to-one onto function.  (Contributed by Mario
       Carneiro, 30-Apr-2015.) $)
    f1ocnv2d $p |- ( ph ->
      ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) ) $=
      ( cv wcel wceq wa wi eleq1a syl impr biimpar exp42 com34 imp32 jcai com23
      biimpa impbida f1ocnvd ) ABCDEFGHEDIJKABMZDNZCMZFOZPZULENZUJGOZPZAUNPUOUP
      AUKUMUOAUKPFENUMUOQJFEULRSTAUKUMUOUPQAUKUOUMUPAUKUOUMUPAUKUOPPZUPUMLUAUBU
      CUDUEAUQPUKUMAUOUPUKAUOPGDNUPUKQKGDUJRSTAUOUPUKUMQAUOUKUPUMAUKUOUPUMQAUKU
      OUPUMURUPUMLUGUBUFUCUDUEUHUI $.

    $( Describe an implicit one-to-one onto function.  (Contributed by Mario
       Carneiro, 12-May-2014.) $)
    f1o2d $p |- ( ph -> F : A -1-1-onto-> B ) $=
      ( wf1o ccnv cmpt wceq f1ocnv2d simpld ) ADEHMHNCEGOPABCDEFGHIJKLQR $.
  $}

  ${
    $d a b A $.  $d a b B $.  $d a b F $.  $d a b ph $.
    f1opw2.1 $e |- ( ph -> F : A -1-1-onto-> B ) $.
    f1opw2.2 $e |- ( ph -> ( `' F " a ) e. _V ) $.
    f1opw2.3 $e |- ( ph -> ( F " b ) e. _V ) $.
    $( A one-to-one mapping induces a one-to-one mapping on power sets.  This
       version of ~ f1opw avoids the Axiom of Replacement.  (Contributed by
       Mario Carneiro, 26-Jun-2015.) $)
    f1opw2 $p |- ( ph ->
        ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B ) $=
      ( cpw cv cima wcel wss crn imassrn wceq syl cvv adantr ccnv cmpt eqid wfo
      wf1o f1ofo forn sseqtrid wb elpwg mpbird cdm dfdm4 f1odm eqtr3id wa elpwi
      adantl foimacnv syl2an eqcomd imaeq2 eqeq2d syl5ibrcom wf1 f1of1 f1imacnv
      impbid f1o2d ) AFEBJZCJZDFKZLZDUAZEKZLZFVJVMUBZVQUCAVMVKMZVLVJMZAVRVMCNZA
      DOZVMCDVLPABCDUDZWACQABCDUEZWBGBCDUFRZBCDUGRUHAVMSMVRVTUIIVMCSUJRUKTAVPVJ
      MZVOVKMZAWEVPBNZAVNOZVPBVNVOPAWHDULZBDUMAWCWIBQGBCDUNRUOUHAVPSMWEWGUIHVPB
      SUJRUKTAVSWFUPZUPZVLVPQZVOVMQZWKWMWLVODVPLZQWKWNVOAWBVOCNZWNVOQWJWDWFWOVS
      VOCUQURBCVODUSUTVAWLVMWNVOVLVPDVBVCVDWKWLWMVLVNVMLZQWKWPVLABCDVEZVLBNZWPV
      LQWJAWCWQGBCDVFRVSWRWFVLBUQTBCVLDVGUTVAWMVPWPVLVOVMVNVBVCVDVHVI $.
  $}

  ${
    $d a b A $.  $d a b B $.  $d a b F $.
    $( A one-to-one mapping induces a one-to-one mapping on power sets.
       (Contributed by Stefan O'Rear, 18-Nov-2014.)  (Revised by Mario
       Carneiro, 26-Jun-2015.) $)
    f1opw $p |- ( F : A -1-1-onto-> B -> ( b e. ~P A |-> ( F " b ) ) :
          ~P A -1-1-onto-> ~P B ) $=
      ( va wf1o id ccnv wfun cv cima cvv wfo dff1o3 simprbi vex funimaex f1ofun
      wcel syl f1opw2 ) ABCFZABCEDUBGUBCHZIZUCEJZKLSUBABCMUDABCNOUCUEEPQTUBCICD
      JZKLSABCRCUFDPQTUA $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.  $d x y ph $.
    f1o3d.1 $e |- ( ph -> F = ( x e. A |-> C ) ) $.
    f1o3d.2 $e |- ( ( ph /\ x e. A ) -> C e. B ) $.
    f1o3d.3 $e |- ( ( ph /\ y e. B ) -> D e. A ) $.
    f1o3d.4 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) ->
                    ( x = D <-> y = C ) ) $.
    $( Describe an implicit one-to-one onto function.  (Contributed by Thierry
       Arnoux, 23-Apr-2017.) $)
    f1o3d $p |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) ) $=
      ( ccnv wceq wfn wcel syl wa copab wi wf1o cmpt wral ralrimiva eqid fneq1d
      fnmpt mpbird cv eleq1a impr biimpar exp42 com34 imp32 jcai biimpa impbida
      com23 opabbidv df-mpt eqtrdi cnveqd cnvopab a1i 3eqtr4d dff1o4 sylanbrc
      jca ) ADEHUAZHMZCEGUBZNAHDOZVKEOZVJAVMBDFUBZDOZAFEPZBDUCVPAVQBDJUDBDFVOEV
      OUEUGQADHVOIUFUHAVNVLEOZAGDPZCEUCVRAVSCEKUDCEGVLDVLUEUGQAEVKVLABUIZDPZCUI
      ZFNZRZCBSZWBEPZVTGNZRZCBSZVKVLAWDWHCBAWDWHAWDRWFWGAWAWCWFAWARVQWCWFTJFEWB
      UJQUKAWAWCWFWGTAWAWFWCWGAWAWFWCWGAWAWFRRZWGWCLULUMUNUOUPAWHRWAWCAWFWGWAAW
      FRVSWGWATKGDVTUJQUKAWFWGWAWCTAWFWAWGWCAWAWFWGWCTAWAWFWGWCWJWGWCLUQUMUSUNU
      OUPURUTAVKWDBCSZMWEAHWKAHVOWKIBCDFVAVBVCWDBCVDVBVLWINACBEGVAVEVFZUFUHDEHV
      GVHWLVI $.
  $}

  ${
    $d ph v $.  $d ph x $.  $d B v $.  $d O v $.  $d R v $.  $d Y v $.
    $d Y x $.  $d Z v $.  $d Z x $.
    suppssov1.s $e |- ( ph ->
        ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L ) $.
    suppssov1.o $e |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z ) $.
    suppssov1.a $e |- ( ( ph /\ x e. D ) -> A e. V ) $.
    suppssov1.b $e |- ( ( ph /\ x e. D ) -> B e. R ) $.
    $( Formula building theorem for support restrictions: operator with left
       annihilator.  (Contributed by Stefan O'Rear, 9-Mar-2015.) $)
    suppssov1 $p |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) "
            ( _V \ { Z } ) ) C_ L ) $=
      ( co cvv wcel wceq cmpt ccnv csn cdif cima crab cv wa wne elex syl adantr
      eldifsni oveq2 eqeq1d wral ralrimiva rspcdva oveq1 syl5ibrcom necon3d imp
      syl5 eldifsn sylanbrc ex ss2rabdv eqid mptpreima 3sstr4g sstrd ) ABFDEIQZ
      UAZUBRLUCUDZUEZBFDUAZUBRKUCUDZUEZHAVLVNSZBFUFDVQSZBFUFVOVRAVSVTBFABUGFSZU
      HZVSVTWBVSUHDRSZDKUIZVTWBWCVSWBDJSWCODJUJUKULWBVSWDVSVLLUIWBWDVLRLUMWBDKV
      LLWBVLLTDKTZKEIQZLTZWBKCUGZIQZLTZWGCGEWHETWIWFLWHEKIUNUOAWJCGUPWAAWJCGNUQ
      ULPURWEVLWFLDKEIUSUOUTVAVCVBDRKVDVEVFVGBFVLVNVMVMVHVIBFDVQVPVPVHVIVJMVK
      $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Function operation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c oF $.
  $c oR $.

  $( Extend class notation to include mapping of an operation to a function
     operation. $)
  cof $a class oF R $.

  $( Extend class notation to include mapping of a binary relation to a
     function relation. $)
  cofr $a class oR R $.

  ${
    $d f g x R $.
    $( Define the function operation map.  The definition is designed so that
       if ` R ` is a binary operation, then ` oF R ` is the analogous operation
       on functions which corresponds to applying ` R ` pointwise to the values
       of the functions.  (Contributed by Mario Carneiro, 20-Jul-2014.) $)
    df-of $a |- oF R = ( f e. _V , g e. _V |->
     ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) ) $.

    $( Define the function relation map.  The definition is designed so that if
       ` R ` is a binary relation, then ` oF R ` is the analogous relation on
       functions which is true when each element of the left function relates
       to the corresponding element of the right function.  (Contributed by
       Mario Carneiro, 28-Jul-2014.) $)
    df-ofr $a |- oR R = { <. f , g >. |
      A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) } $.
  $}

  ${
    $d ph f g x $.  $d R f g x $.  $d S f g x $.
    ofeqd.1 $e |- ( ph -> R = S ) $.
    $( Equality theorem for function operation, deduction form.  (Contributed
       by SN, 11-Nov-2024.) $)
    ofeqd $p |- ( ph -> oF R = oF S ) $=
      ( vf vg vx cvv cv cdm cin cfv cmpt cmpo cof oveqd mpteq2dv mpoeq3dv df-of
      co 3eqtr4g ) AEFHHGEIZJFIZJKZGIZUBLZUEUCLZBTZMZNEFHHGUDUFUGCTZMZNBOCOAEFH
      HUIUKAGUDUHUJABCUFUGDPQRGBEFSGCEFSUA $.
  $}

  ${
    $d f g x R $.  $d f g x S $.
    $( Equality theorem for function operation.  (Contributed by Mario
       Carneiro, 20-Jul-2014.) $)
    ofeq $p |- ( R = S -> oF R = oF S ) $=
      ( vf vg vx wceq cvv cv cdm cin cfv co cmpt cmpo cof w3a simp1 oveqd df-of
      wcel mpteq2dv mpoeq3dva 3eqtr4g ) ABFZCDGGECHZIDHZIJZEHZUEKZUHUFKZALZMZNC
      DGGEUGUIUJBLZMZNAOBOUDCDGGULUNUDUEGTZUFGTZPZEUGUKUMUQABUIUJUDUOUPQRUAUBEA
      CDSEBCDSUC $.

    $( Equality theorem for function relation.  (Contributed by Mario Carneiro,
       28-Jul-2014.) $)
    ofreq $p |- ( R = S -> oR R = oR S ) $=
      ( vx vf vg wceq cv cfv wbr cdm cin wral cofr breq ralbidv opabbidv df-ofr
      copab 3eqtr4g ) ABFZCGZDGZHZUAEGZHZAIZCUBJUDJKZLZDERUCUEBIZCUGLZDERAMBMTU
      HUJDETUFUICUGUCUEABNOPCADEQCBDEQS $.

    $( A function operation restricted to a set is a set.  (Contributed by NM,
       28-Jul-2014.) $)
    ofexg $p |- ( A e. V -> ( oF R |` A ) e. _V ) $=
      ( vf vg vx cof wfun wcel cres cvv cv cdm cin cfv co cmpt mpofun resfunexg
      df-of mpan ) BGZHACIUBAJKIDEKKFDLZMELZMNFLZUCOUEUDOBPQUBFBDETRUBACSUA $.
  $}

  ${
    $d R u v w $.  $d u v w x $.
    nfof.1 $e |- F/_ x R $.
    $( Hypothesis builder for function operation.  (Contributed by Mario
       Carneiro, 20-Jul-2014.) $)
    nfof $p |- F/_ x oF R $=
      ( vu vv vw cof cvv cv cdm cin cfv cmpt cmpo df-of nfcv nfov nfmpt nfmpo
      co nfcxfr ) ABGDEHHFDIZJEIZJKZFIZUBLZUEUCLZBTZMZNFBDEODEAHHUIAHPZUJAFUDUH
      AUDPAUFUGBAUFPCAUGPQRSUA $.

    $( Hypothesis builder for function relation.  (Contributed by Mario
       Carneiro, 28-Jul-2014.) $)
    nfofr $p |- F/_ x oR R $=
      ( vw vu vv cofr cv cfv wbr cdm wral copab df-ofr nfcv nfbr nfralxy nfopab
      cin nfcxfr ) ABGDHZEHZIZUAFHZIZBJZDUBKUDKSZLZEFMDBEFNUHEFAUFADUGAUGOAUCUE
      BAUCOCAUEOPQRT $.
  $}

  ${
    $d x A $.  $d f g x F $.  $d f g x G $.  $d x ph $.  $d x S $.  $d x X $.
    $d f g x R $.
    offval.1 $e |- ( ph -> F Fn A ) $.
    offval.2 $e |- ( ph -> G Fn B ) $.
    offval.3 $e |- ( ph -> A e. V ) $.
    offval.4 $e |- ( ph -> B e. W ) $.
    offval.5 $e |- ( A i^i B ) = S $.
    ${
      offval.6 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = C ) $.
      offval.7 $e |- ( ( ph /\ x e. B ) -> ( G ` x ) = D ) $.
      $( Value of an operation applied to two functions.  (Contributed by Mario
         Carneiro, 20-Jul-2014.) $)
      offval $p |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) ) $=
        ( cvv vf vg cof co cdm cin cfv cmpt wcel wceq wfn fnex syl2anc fndm syl
        cv ineq12d eqtrdi mpteq1d inex1g eqeltrrid mptexg 3syl eqeltrd wa fveq1
        dmeq ineqan12d oveqan12d mpteq12dv df-of ovmpoga syl3anc eleq2i adantrr
        elin bitr3i adantrl oveq12d sylan2b mpteq2dva 3eqtrd ) AIJGUCZUDZBIUEZJ
        UEZUFZBUPZIUGZWHJUGZGUDZUHZBHWKUHZBHEFGUDZUHAITUIZJTUIZWLTUIWDWLUJAICUK
        ZCKUIZWOMOCKIULUMAJDUKZDLUIWPNPDLJULUMAWLWMTABWGHWKAWGCDUFZHAWECWFDAWQW
        ECUJMCIUNUOAWSWFDUJNDJUNUOUQQURUSZAWRHTUIWMTUIOWRHWTTQCDKUTVABHWKTVBVCV
        DUAUBIJTTBUAUPZUEZUBUPZUEZUFZWHXBUGZWHXDUGZGUDZUHWLWCTXBIUJZXDJUJZVEBXF
        XIWGWKXJXKXCWEXEWFXBIVGXDJVGVHXJXKXGWIXHWJGWHXBIVFWHXDJVFVIVJBGUAUBVKVL
        VMXAABHWKWNWHHUIZAWHCUIZWHDUIZVEZWKWNUJXLWHWTUIXOWTHWHQVNWHCDVPVQAXOVEW
        IEWJFGAXMWIEUJXNRVOAXNWJFUJXMSVRVSVTWAWB $.

      $( Value of a relation applied to two functions.  (Contributed by Mario
         Carneiro, 28-Jul-2014.) $)
      ofrfval $p |- ( ph -> ( F oR R G <-> A. x e. S C R D ) ) $=
        ( wcel vf vg cofr wbr cv cfv cdm cin wral cvv wb fnex syl2anc wceq dmeq
        wfn wa ineqan12d fveq1 breqan12d raleqbidv df-ofr brabga ineq12d eqtrdi
        fndm raleqdv inss1 eqsstrri sseli sylan2 inss2 breq12d ralbidva 3bitrd
        syl ) AIJGUCZUDZBUEZIUFZVSJUFZGUDZBIUGZJUGZUHZUIZWBBHUIEFGUDZBHUIAIUJTZ
        JUJTZVRWFUKAICUPZCKTWHMOCKIULUMAJDUPZDLTWINPDLJULUMVSUAUEZUFZVSUBUEZUFZ
        GUDZBWLUGZWNUGZUHZUIWFUAUBIJVQUJUJWLIUNZWNJUNZUQWPWBBWSWEWTXAWQWCWRWDWL
        IUOWNJUOURWTXAWMVTWOWAGVSWLIUSVSWNJUSUTVABGUAUBVBVCUMAWBBWEHAWECDUHZHAW
        CCWDDAWJWCCUNMCIVFVPAWKWDDUNNDJVFVPVDQVEVGAWBWGBHAVSHTZUQVTEWAFGXCAVSCT
        VTEUNHCVSHXBCQCDVHVIVJRVKXCAVSDTWAFUNHDVSHXBDQCDVLVIVJSVKVMVNVO $.
    $}

    ${
      ofval.6 $e |- ( ( ph /\ X e. A ) -> ( F ` X ) = C ) $.
      ofval.7 $e |- ( ( ph /\ X e. B ) -> ( G ` X ) = D ) $.
      ofval.8 $e |- ( ( ph /\ X e. S ) -> ( C R D ) e. U ) $.
      $( Evaluate a function operation at a point.  (Contributed by Mario
         Carneiro, 20-Jul-2014.)  (Revised by Jim Kingdon, 22-Nov-2023.) $)
      ofvalg $p |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) ) $=
        ( vx wcel wa cof co cfv cmpt wceq eqidd offval fveq1d adantr eqid fveq2
        cv oveq12d simpr cin inss1 eqsstrri sseli sylan2 eqeltrd fvmptd3 3eqtrd
        inss2 ) AMGUCZUDZMIJFUEUFZUGZMUBGUBUPZIUGZVLJUGZFUFZUHZUGZMIUGZMJUGZFUF
        ZDEFUFZAVKVQUIVHAMVJVPAUBBCVMVNFGIJKLNOPQRAVLBUCUDVMUJAVLCUCUDVNUJUKULU
        MVIUBMVOVTGVPHVPUNVLMUIVMVRVNVSFVLMIUOVLMJUOUQAVHURVIVTWAHVIVRDVSEFVHAM
        BUCVRDUIGBMGBCUSZBRBCUTVAVBSVCVHAMCUCVSEUIGCMGWBCRBCVGVAVBTVCUQZUAVDVEW
        CVF $.
    $}

    ${
      ofrval.6 $e |- ( ( ph /\ X e. A ) -> ( F ` X ) = C ) $.
      ofrval.7 $e |- ( ( ph /\ X e. B ) -> ( G ` X ) = D ) $.
      $( Exhibit a function relation at a point.  (Contributed by Mario
         Carneiro, 28-Jul-2014.) $)
      ofrval $p |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D ) $=
        ( wcel vx cofr wbr w3a cfv wa cv wral eqidd ofrfval biimpa wceq breq12d
        wi fveq2 rspccv syl 3impia simp1 cin inss1 eqsstrri simp3 syl2anc inss2
        sselid 3brtr3d ) AHIFUBUCZLGTZUDZLHUEZLIUEZDEFAVHVIVKVLFUCZAVHUFUAUGZHU
        EZVNIUEZFUCZUAGUHZVIVMUNAVHVRAUABCVOVPFGHIJKMNOPQAVNBTUFVOUIAVNCTUFVPUI
        UJUKVQVMUALGVNLULVOVKVPVLFVNLHUOVNLIUOUMUPUQURVJALBTVKDULAVHVIUSZVJGBLG
        BCUTZBQBCVAVBAVHVIVCZVFRVDVJALCTVLEULVSVJGCLGVTCQBCVEVBWAVFSVDVG $.
    $}
  $}

  ${
    ofmresval.f $e |- ( ph -> F e. A ) $.
    ofmresval.g $e |- ( ph -> G e. B ) $.
    $( Value of a restriction of the function operation map.  (Contributed by
       NM, 20-Oct-2014.) $)
    ofmresval $p |- ( ph -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) ) $=
      ( wcel cof cxp cres co wceq ovres syl2anc ) AEBIFCIEFDJZBCKLMEFQMNGHEFBCQ
      OP $.
  $}

  ${
    $d z A $.  $d z C $.  $d y z G $.  $d x y z ph $.  $d x y S $.  $d x y T $.
    $d x y z F $.  $d x y z R $.  $d x y z U $.
    off.1 $e |- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U ) $.
    off.2 $e |- ( ph -> F : A --> S ) $.
    off.3 $e |- ( ph -> G : B --> T ) $.
    off.4 $e |- ( ph -> A e. V ) $.
    off.5 $e |- ( ph -> B e. W ) $.
    off.6 $e |- ( A i^i B ) = C $.
    $( The function operation produces a function.  (Contributed by Mario
       Carneiro, 20-Jul-2014.) $)
    off $p |- ( ph -> ( F oF R G ) : C --> U ) $=
      ( vz cof co wf cv cfv cmpt wcel wa wral cin inss1 eqsstrri sseli ffvelcdm
      syl2an inss2 ralrimivva adantr wceq oveq1 eleq1d oveq2 rspc2va eqid fmptd
      syl21anc wfn ffn syl eqidd offval feq1d mpbird ) AFJKLGUBUCZUDFJUAFUAUEZK
      UFZVPLUFZGUCZUGZUDAUAFVSJVTAVPFUHZUIVQHUHZVRIUHZBUEZCUEZGUCZJUHZCIUJBHUJZ
      VSJUHZADHKUDZVPDUHZWBWAPFDVPFDEUKZDTDEULUMUNDHVPKUOUPAEILUDZVPEUHZWCWAQFE
      VPFWLETDEUQUMUNEIVPLUOUPAWHWAAWGBCHIOURUSWGWIVQWEGUCZJUHBCVQVRHIWDVQUTWFW
      OJWDVQWEGVAVBWEVRUTWOVSJWEVRVQGVCVBVDVGVTVEVFAFJVOVTAUADEVQVRGFKLMNAWJKDV
      HPDHKVIVJAWMLEVHQEILVIVJRSTAWKUIVQVKAWNUIVRVKVLVMVN $.

    ${
      $d H x $.  $d G x $.  $d C x $.
      offeq.4 $e |- ( ph -> H : C --> U ) $.
      offeq.5 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = D ) $.
      offeq.6 $e |- ( ( ph /\ x e. B ) -> ( G ` x ) = E ) $.
      ${
        offeq.7 $e |- ( ( ph /\ x e. C ) -> ( D R E ) = ( H ` x ) ) $.
        $( Convert an identity of the operation to the analogous identity on
           the function operation.  (Contributed by Jim Kingdon,
           26-Nov-2023.) $)
        offeq $p |- ( ph -> ( F oF R G ) = H ) $=
          ( cof co off ffnd cv wcel cfv ffvelcdmda eqeltrd ofvalg eqtrd eqfnfvd
          wa ) ABFMNHUHUIZOAFKVAABCDEFHIJKMNPQRSTUAUBUCUJUKAFKOUDUKABULZFUMUTZV
          BVAUNGLHUIZVBOUNZADEGLHFKMNPQVBADIMSUKAEJNTUKUAUBUCUEUFVCVDVEKUGAFKVB
          OUDUOUPUQUGURUS $.
      $}
    $}
  $}

  ${
    $d x A $.  $d x C $.  $d x F $.  $d x G $.  $d x ph $.  $d x R $.
    ofres.1 $e |- ( ph -> F Fn A ) $.
    ofres.2 $e |- ( ph -> G Fn B ) $.
    ofres.3 $e |- ( ph -> A e. V ) $.
    ofres.4 $e |- ( ph -> B e. W ) $.
    ofres.5 $e |- ( A i^i B ) = C $.
    $( Restrict the operands of a function operation to the same domain as that
       of the operation itself.  (Contributed by Mario Carneiro,
       15-Sep-2014.) $)
    ofres $p |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) ) $=
      ( vx co cfv wcel cvv wfn cof cv cmpt cres eqidd offval wss inss1 eqsstrri
      wa cin fnssres sylancl inss2 ssexg sylancr inidm wceq fvres adantl eqtr4d
      ) AFGEUAZPODOUBZFQZVCGQZEPUCFDUDZGDUDZVBPAOBCVDVEEDFGHIJKLMNAVCBRUJVDUEAV
      CCRUJVEUEUFAODDVDVEEDVFVGSSAFBTDBUGZVFDTJDBCUKZBNBCUHUIZBDFULUMAGCTDCUGVG
      DTKDVICNBCUNUICDGULUMAVHBHRDSRVJLDBHUOUPZVKDUQVCDRZVCVFQVDURAVCDFUSUTVLVC
      VGQVEURAVCDGUSUTUFVA $.
  $}

  ${
    $d x y A $.  $d y B $.  $d y C $.  $d y F $.  $d y G $.  $d x y ph $.
    $d x y R $.
    offval2.1 $e |- ( ph -> A e. V ) $.
    offval2.2 $e |- ( ( ph /\ x e. A ) -> B e. W ) $.
    offval2.3 $e |- ( ( ph /\ x e. A ) -> C e. X ) $.
    offval2.4 $e |- ( ph -> F = ( x e. A |-> B ) ) $.
    offval2.5 $e |- ( ph -> G = ( x e. A |-> C ) ) $.
    $( The function operation expressed as a mapping.  (Contributed by Mario
       Carneiro, 20-Jul-2014.) $)
    offval2 $p |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) ) $=
      ( vy co cmpt wceq cof cv cfv wcel wral ralrimiva eqid fnmpt fneq1d mpbird
      wfn inidm wa adantr fveq1d offval nffvmpt1 nfcv nfov fveq2 oveq12d cbvmpt
      syl simpr fvmpt2 syl2anc mpteq2dva eqtrid eqtrd ) AGHFUARQCQUBZBCDSZUCZVJ
      BCESZUCZFRZSZBCDEFRZSZAQCCVLVNFCGHIIAGCUKVKCUKZADJUDZBCUEVSAVTBCMUFBCDVKJ
      VKUGZUHVCACGVKOUIUJAHCUKVMCUKZAEKUDZBCUEWBAWCBCNUFBCEVMKVMUGZUHVCACHVMPUI
      UJLLCULAVJCUDZUMZVJGVKAGVKTWEOUNUOWFVJHVMAHVMTWEPUNUOUPAVPBCBUBZVKUCZWGVM
      UCZFRZSVRQBCVOWJBVLVNFBCDVJUQBFURBCEVJUQUSQWJURVJWGTVLWHVNWIFVJWGVKUTVJWG
      VMUTVAVBABCWJVQAWGCUDZUMZWHDWIEFWLWKVTWHDTAWKVDZMBCDJVKWAVEVFWLWKWCWIETWM
      NBCEKVMWDVEVFVAVGVHVI $.

    $( The function relation acting on maps.  (Contributed by Mario Carneiro,
       20-Jul-2014.) $)
    ofrfval2 $p |- ( ph -> ( F oR R G <-> A. x e. A B R C ) ) $=
      ( vy wbr wral wceq cofr cmpt cfv wfn wcel ralrimiva eqid fnmpt syl fneq1d
      cv mpbird inidm wa adantr fveq1d ofrfval nffvmpt1 nfcv nfbr fveq2 breq12d
      nfv cbvral simpr fvmpt2 syl2anc ralbidva bitrid bitrd ) AGHFUARQUKZBCDUBZ
      UCZVKBCEUBZUCZFRZQCSZDEFRZBCSZAQCCVMVOFCGHIIAGCUDVLCUDZADJUEZBCSVTAWABCMU
      FBCDVLJVLUGZUHUIACGVLOUJULAHCUDVNCUDZAEKUEZBCSWCAWDBCNUFBCEVNKVNUGZUHUIAC
      HVNPUJULLLCUMAVKCUEZUNZVKGVLAGVLTWFOUOUPWGVKHVNAHVNTWFPUOUPUQVQBUKZVLUCZW
      HVNUCZFRZBCSAVSVPWKQBCBVMVOFBCDVKURBFUSBCEVKURUTWKQVCVKWHTVMWIVOWJFVKWHVL
      VAVKWHVNVAVBVDAWKVRBCAWHCUEZUNZWIDWJEFWMWLWAWIDTAWLVEZMBCDJVLWBVFVGWMWLWD
      WJETWNNBCEKVNWEVFVGVBVHVIVJ $.
  $}

  ${
    $d ph v x $.  $d A x $.  $d B v x $.  $d D x $.  $d O v x $.  $d R v $.
    $d Y v x $.  $d Z v x $.
    suppssof1.s $e |- ( ph -> ( `' A " ( _V \ { Y } ) ) C_ L ) $.
    suppssof1.o $e |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z ) $.
    suppssof1.a $e |- ( ph -> A : D --> V ) $.
    suppssof1.b $e |- ( ph -> B : D --> R ) $.
    suppssof1.d $e |- ( ph -> D e. W ) $.
    $( Formula building theorem for support restrictions: vector operation with
       left annihilator.  (Contributed by Stefan O'Rear, 9-Mar-2015.) $)
    suppssof1 $p |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L ) $=
      ( vx ccnv cvv cof co csn cdif cima cv cfv cmpt wf wfn ffn syl inidm eqidd
      wa offval cnveqd imaeq1d feqmptd eqsstrrd funfvex funfni sylan ffvelcdmda
      wcel suppssov1 eqsstrd ) ACDHUAUBZSZTLUCUDZUERERUFZCUGZVKDUGZHUBUHZSZVJUE
      GAVIVOVJAVHVNAREEVLVMHECDJJAEICUICEUJZOEICUKULZAEFDUIDEUJPEFDUKULQQEUMAVK
      EVEZUOZVLUNVSVMUNUPUQURARBVLVMEFGHTKLAREVLUHZSZTKUCUDZUECSZWBUEGAWCWAWBAC
      VTAREICOUSUQURMUTNAVPVRVLTVEZVQWDEVKCVKCVAVBVCAEFVKDPVDVFVG $.
  $}

  ${
    $d y A $.  $d x y C $.  $d x y F $.  $d x y G $.  $d x y H $.  $d x y ph $.
    $d x D $.  $d x y R $.
    ofco.1 $e |- ( ph -> F Fn A ) $.
    ofco.2 $e |- ( ph -> G Fn B ) $.
    ofco.3 $e |- ( ph -> H : D --> C ) $.
    ofco.4 $e |- ( ph -> A e. V ) $.
    ofco.5 $e |- ( ph -> B e. W ) $.
    ofco.6 $e |- ( ph -> D e. X ) $.
    ofco.7 $e |- ( A i^i B ) = C $.
    $( The composition of a function operation with another function.
       (Contributed by Mario Carneiro, 19-Dec-2014.) $)
    ofco $p |- ( ph ->
      ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) ) $=
      ( cfv vx vy cof co ccom cv cmpt ffvelcdmda feqmptd wcel eqidd offval wceq
      wa fveq2 oveq12d fmptco wfn wf wss cin inss1 eqsstrri fss sylancl syl2anc
      fnfco inss2 inidm ffn syl fvco2 sylan eqtr4d ) AGHFUCZUDZIUEUAEUAUFZITZGT
      ZVRHTZFUDZUGGIUEZHIUEZVOUDAUAUBEDVRUBUFZGTZWDHTZFUDWAIVPAEDVQIOUHAUAEDIOU
      IAUBBCWEWFFDGHJKMNPQSAWDBUJUNWEUKAWDCUJUNWFUKULWDVRUMWEVSWFVTFWDVRGUOWDVR
      HUOUPUQAUAEEVSVTFEWBWCLLAGBUREBIUSZWBEURMAEDIUSZDBUTWGODBCVAZBSBCVBVCEDBI
      VDVEBEGIVGVFAHCURECIUSZWCEURNAWHDCUTWJODWICSBCVHVCEDCIVDVECEHIVGVFRREVIAI
      EURZVQEUJZVQWBTVSUMAWHWKOEDIVJVKZEGIVQVLVMAWKWLVQWCTVTUMWMEHIVQVLVMULVN
      $.
  $}

  ${
    $d x A $.  $d x F $.  $d x G $.  $d x H $.  $d x ph $.  $d x R $.
    offveq.1 $e |- ( ph -> A e. V ) $.
    offveq.2 $e |- ( ph -> F Fn A ) $.
    offveq.3 $e |- ( ph -> G Fn A ) $.
    offveq.4 $e |- ( ph -> H Fn A ) $.
    offveq.5 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = B ) $.
    offveq.6 $e |- ( ( ph /\ x e. A ) -> ( G ` x ) = C ) $.

    $d x F $.
    $( Equivalent expressions for equality with a function operation.
       (Contributed by NM, 9-Oct-2014.)  (Proof shortened by Mario Carneiro,
       5-Dec-2016.) $)
    offveqb $p |- ( ph
          -> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) ) $=
      ( co wceq cmpt wral cof cfv wfn dffn5im syl inidm offval eqeq12d cvv wcel
      cv wb funfvex funfni sylan ralrimiva mpteqb bitrd ) AIGHFUAQZRBCBUKZIUBZS
      ZBCDEFQZSZRZVAVCRBCTZAIVBUSVDAICUCZIVBRNBCIUDUEABCCDEFCGHJJLMKKCUFOPUGUHA
      VAUIUJZBCTVEVFULAVHBCAVGUTCUJVHNVHCUTIUTIUMUNUOUPBCVAVCUIUQUEUR $.

    ${
      offveq.7 $e |- ( ( ph /\ x e. A ) -> ( B R C ) = ( H ` x ) ) $.
      $( Convert an identity of the operation to the analogous identity on the
         function operation.  (Contributed by Mario Carneiro, 24-Jul-2014.) $)
      offveq $p |- ( ph -> ( F oF R G ) = H ) $=
        ( co wceq eqcomd cof cv cfv wral wcel wa ralrimiva offveqb mpbird ) AIG
        HFUARZAIUJSBUBZIUCZDEFRZSZBCUDAUNBCAUKCUEUFUMULQTUGABCDEFGHIJKLMNOPUHUI
        T $.
    $}
  $}

  ${
    ofc1.1 $e |- ( ph -> A e. V ) $.
    ofc1.2 $e |- ( ph -> B e. W ) $.
    ofc1.3 $e |- ( ph -> F Fn A ) $.
    ofc1.4 $e |- ( ( ph /\ X e. A ) -> ( F ` X ) = C ) $.
    ofc1g.ex $e |- ( ( ph /\ X e. A ) -> ( B R C ) e. U ) $.
    $( Left operation by a constant.  (Contributed by Mario Carneiro,
       24-Jul-2014.) $)
    ofc1g $p |- ( ( ph /\ X e. A ) ->
      ( ( ( A X. { B } ) oF R F ) ` X ) = ( B R C ) ) $=
      ( csn cxp wcel wfn fnconstg syl inidm cfv wceq fvconst2g sylan ofvalg ) A
      BBCDEBFBCPQZGHHJACIRZUHBSLBCITUAMKKBUBAUIJBRJUHUCCUDLBCJIUEUFNOUG $.
  $}

  ${
    ofc2.1 $e |- ( ph -> A e. V ) $.
    ofc2.2 $e |- ( ph -> B e. W ) $.
    ofc2.3 $e |- ( ph -> F Fn A ) $.
    ofc2.4 $e |- ( ( ph /\ X e. A ) -> ( F ` X ) = C ) $.
    ofc2g.ex $e |- ( ( ph /\ X e. A ) -> ( C R B ) e. U ) $.
    $( Right operation by a constant.  (Contributed by NM, 7-Oct-2014.) $)
    ofc2g $p |- ( ( ph /\ X e. A ) ->
      ( ( F oF R ( A X. { B } ) ) ` X ) = ( C R B ) ) $=
      ( csn cxp wcel wfn fnconstg syl inidm cfv wceq fvconst2g sylan ofvalg ) A
      BBDCEBFGBCPQZHHJMACIRZUHBSLBCITUAKKBUBNAUIJBRJUHUCCUDLBCJIUEUFOUG $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ph $.  $d x R $.  $d x W $.
    $d x X $.
    ofc12.1 $e |- ( ph -> A e. V ) $.
    ofc12.2 $e |- ( ph -> B e. W ) $.
    ofc12.3 $e |- ( ph -> C e. X ) $.
    $( Function operation on two constant functions.  (Contributed by Mario
       Carneiro, 28-Jul-2014.) $)
    ofc12 $p |- ( ph ->
      ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) ) $=
      ( vx csn cxp co cmpt wcel adantr wceq fconstmpt cof a1i offval2 eqtr4di
      cv ) ABCMNZBDMNZEUAOLBCDEOZPBUHMNALBCDEUFUGFGHIACGQLUEBQZJRADHQUIKRUFLBCP
      SALBCTUBUGLBDPSALBDTUBUCLBUHTUD $.
  $}

  ${
    $d w x B $.  $d w x y z F $.  $d w x y z G $.  $d w x y z H $.
    $d w x y z ph $.  $d w x y z R $.  $d w A $.  $d w x y z S $.
    $d w x y z T $.  $d w x y z U $.
    caofref.1 $e |- ( ph -> A e. V ) $.
    caofref.2 $e |- ( ph -> F : A --> S ) $.
    ${
      caofref.3 $e |- ( ( ph /\ x e. S ) -> x R x ) $.
      $( Transfer a reflexive law to the function relation.  (Contributed by
         Mario Carneiro, 28-Jul-2014.) $)
      caofref $p |- ( ph -> F oR R F ) $=
        ( vw cofr wbr cv cfv wral wcel wa wceq ralrimiva breq12d adantr rspcdva
        id ffvelcdmda wf wfn ffn syl inidm eqidd ofrfval mpbird ) AFFDLMKNZFOZU
        ODMZKCPAUPKCAUNCQZRZBNZUSDMZUPBEUOUSUOSZUSUOUSUODVAUDZVBUAAUTBEPUQAUTBE
        JTUBACEUNFIUEUCTAKCCUOUODCFFGGACEFUFFCUGICEFUHUIZVCHHCUJURUOUKZVDULUM
        $.
    $}

    ${
      $d v A $.  $d v F $.  $d x v N $.  $d v S $.  $d v ph $.  $d v w $.
      caofinv.3 $e |- ( ph -> B e. W ) $.
      caofinv.4 $e |- ( ph -> N : S --> S ) $.
      caofinv.5 $e |- ( ph -> G = ( v e. A |-> ( N ` ( F ` v ) ) ) ) $.
      ${
        caofinvl.6 $e |- ( ( ph /\ x e. S ) -> ( ( N ` x ) R x ) = B ) $.
        $( Transfer a left inverse law to the function operation.  (Contributed
           by NM, 22-Oct-2014.) $)
        caofinvl $p |- ( ph -> ( G oF R F ) = ( A X. { B } ) ) $=
          ( vw cfv cof co cmpt csn cxp cv wcel adantr ffvelcdmda ffvelcdmd eqid
          wf wa fmptd feq1d mpbird wfn wceq wral ralrimiva fnmpt fneq1d dffn5im
          feqmptd offval2 fveq1d simpr fveq2 fveq2d fvmptg syl2anc eqtrd oveq1d
          syl id oveq12d eqeq1d rspcdva mpteq2dva fconstmpt eqtr4di ) AIHFUAUBZ
          SDEUCZDEUDUEAWBSDSUFZITZWDHTZFUBZUCWCASDWEWFFIHKGGMADGWDIADGIULDGCDCU
          FZHTZJTZUCZULACDWJGWKAWHDUGZUMGGWIJAGGJULZWLPUHADGWHHNUIUJZWKUKZUNADG
          IWKQUOUPUIADGWDHNUIZAIDUQZISDWEUCURAWQWKDUQZAWJGUGZCDUSWRAWSCDWNUTCDW
          JWKGWOVAVNADIWKQVBUPSDIVCVNASDGHNVDVEASDWGEAWDDUGZUMZWGWFJTZWFFUBZEXA
          WEXBWFFXAWEWDWKTZXBAWEXDURWTAWDIWKQVFUHXAWTXBGUGXDXBURAWTVGXAGGWFJAWM
          WTPUHWPUJCWDWJXBDGWKWHWDURWIWFJWHWDHVHVIWOVJVKVLVMXABUFZJTZXEFUBZEURZ
          XCEURBGWFXEWFURZXGXCEXIXFXBXEWFFXEWFJVHXIVOVPVQAXHBGUSWTAXHBGRUTUHWPV
          RVLVSVLSDEVTWA $.
      $}
    $}

    ${
      caofid0.3 $e |- ( ph -> B e. W ) $.
      ${
        caofid0l.5 $e |- ( ( ph /\ x e. S ) -> ( B R x ) = x ) $.
        $( Transfer a left identity law to the function operation.
           (Contributed by NM, 21-Oct-2014.) $)
        caofid0l $p |- ( ph -> ( ( A X. { B } ) oF R F ) = F ) $=
          ( vw cv cfv csn wcel wceq co cxp wfn fnconstg ffnd fvconst2g sylan wa
          syl eqidd wral ralrimiva ffvelcdmda oveq2 id eqeq12d rspccva syl2an2r
          offveq ) ANCDNOZGPZECDQUAZGGHJADIRZVACUBLCDIUCUHACFGKUDZVCAVBUSCRZUSV
          APDSLCDUSIUEUFAVDUGUTUIADBOZETZVESZBFUJVDUTFRDUTETZUTSZAVGBFMUKACFUSG
          KULVGVIBUTFVEUTSZVFVHVEUTVEUTDEUMVJUNUOUPUQUR $.
      $}

      ${
        caofid0r.5 $e |- ( ( ph /\ x e. S ) -> ( x R B ) = x ) $.
        $( Transfer a right identity law to the function operation.
           (Contributed by NM, 21-Oct-2014.) $)
        caofid0r $p |- ( ph -> ( F oF R ( A X. { B } ) ) = F ) $=
          ( vw cv cfv csn wcel wceq co cxp ffnd wfn fnconstg wa eqidd fvconst2g
          syl sylan wral ralrimiva ffvelcdmda oveq1 id eqeq12d rspccva syl2an2r
          offveq ) ANCNOZGPZDEGCDQUAZGHJACFGKUBZADIRZVACUCLCDIUDUHVBAUSCRZUEUTU
          FAVCVDUSVAPDSLCDUSIUGUIABOZDETZVESZBFUJVDUTFRUTDETZUTSZAVGBFMUKACFUSG
          KULVGVIBUTFVEUTSZVFVHVEUTVEUTDEUMVJUNUOUPUQUR $.
      $}

      caofid1.4 $e |- ( ph -> C e. X ) $.
      ${
        $d C w x $.
        caofid1.5 $e |- ( ( ph /\ x e. S ) -> ( x R B ) = C ) $.
        $( Transfer a right absorption law to the function operation.
           (Contributed by Mario Carneiro, 28-Jul-2014.) $)
        caofid1 $p |- ( ph -> ( F oF R ( A X. { B } ) ) = ( A X. { C } ) ) $=
          ( vw cfv wcel wceq cv csn cxp ffnd wfn fnconstg eqidd fvconst2g sylan
          syl wa wral ralrimiva ffvelcdmda oveq1 eqeq1d rspccva syl2an2r eqtr4d
          co offveq ) AQCQUAZHRZDFHCDUBUCZCEUBUCZILACGHMUDADJSZVDCUENCDJUFUJAEK
          SZVECUEOCEKUFUJAVBCSZUKZVCUGAVFVHVBVDRDTNCDVBJUHUIVIVCDFUTZEVBVERZABU
          AZDFUTZETZBGULVHVCGSVJETZAVNBGPUMACGVBHMUNVNVOBVCGVLVCTVMVJEVLVCDFUOU
          PUQURAVGVHVKETOCEVBKUHUIUSVA $.
      $}

      $d C w x $.
      caofid2.5 $e |- ( ( ph /\ x e. S ) -> ( B R x ) = C ) $.
      $( Transfer a right absorption law to the function operation.
         (Contributed by Mario Carneiro, 28-Jul-2014.) $)
      caofid2 $p |- ( ph -> ( ( A X. { B } ) oF R F ) = ( A X. { C } ) ) $=
        ( vw cfv wcel wceq cv csn cxp wfn fnconstg syl fvconst2g sylan wa eqidd
        ffnd co wral ralrimiva ffvelcdmda eqeq1d rspccva syl2an2r eqtr4d offveq
        oveq2 ) AQCDQUAZHRZFCDUBUCZHCEUBUCZILADJSZVDCUDNCDJUEUFACGHMUKAEKSZVECU
        DOCEKUEUFAVFVBCSZVBVDRDTNCDVBJUGUHAVHUIZVCUJVIDVCFULZEVBVERZADBUAZFULZE
        TZBGUMVHVCGSVJETZAVNBGPUNACGVBHMUOVNVOBVCGVLVCTVMVJEVLVCDFVAUPUQURAVGVH
        VKETOCEVBKUGUHUSUT $.
    $}

    caofcom.3 $e |- ( ph -> G : A --> S ) $.
    ${
      caofcom.4 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
                        ( x R y ) = ( y R x ) ) $.
      $( Transfer a commutative law to the function operation.  (Contributed by
         Mario Carneiro, 26-Jul-2014.) $)
      caofcom $p |- ( ph -> ( F oF R G ) = ( G oF R F ) ) $=
        ( vw cfv co cmpt wcel wa ffvelcdmda cv cof wceq caovcomg syldan feqmptd
        jca mpteq2dva offval2 3eqtr4d ) ANDNUAZGOZUKHOZEPZQNDUMULEPZQGHEUBZPHGU
        PPANDUNUOAUKDRZULFRZUMFRZSUNUOUCAUQSURUSADFUKGKTZADFUKHLTZUGABCULUMFEMU
        DUEUHANDULUMEGHIFFJUTVAANDFGKUFZANDFHLUFZUIANDUMULEHGIFFJVAUTVCVBUIUJ
        $.
    $}

    ${
      caofrss.4 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
                        ( x R y -> x T y ) ) $.
      $( Transfer a relation subset law to the function relation.  (Contributed
         by Mario Carneiro, 28-Jul-2014.) $)
      caofrss $p |- ( ph -> ( F oR R G -> F oR T G ) ) $=
        ( vw cv wbr wral wcel wi cfv cofr wa ffvelcdmda ralrimivva adantr breq1
        wceq imbi12d breq2 rspc2va syl21anc ralimdva wf wfn ffn syl inidm eqidd
        ofrfval 3imtr4d ) AOPZHUAZVBIUAZEQZODRVCVDGQZODRHIEUBQHIGUBQAVEVFODAVBD
        SZUCZVCFSVDFSBPZCPZEQZVIVJGQZTZCFRBFRZVEVFTZADFVBHLUDADFVBIMUDAVNVGAVMB
        CFFNUEUFVMVOVCVJEQZVCVJGQZTBCVCVDFFVIVCUHVKVPVLVQVIVCVJEUGVIVCVJGUGUIVJ
        VDUHVPVEVQVFVJVDVCEUJVJVDVCGUJUIUKULUMAODDVCVDEDHIJJADFHUNHDUOLDFHUPUQZ
        ADFIUNIDUOMDFIUPUQZKKDURZVHVCUSZVHVDUSZUTAODDVCVDGDHIJJVRVSKKVTWAWBUTVA
        $.
    $}

    caofass.4 $e |- ( ph -> H : A --> S ) $.

    ${
      caoftrn.5 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
                        ( ( x R y /\ y T z ) -> x U z ) ) $.
      $( Transfer a transitivity law to the function relation.  (Contributed by
         Mario Carneiro, 28-Jul-2014.) $)
      caoftrn $p |- ( ph -> ( ( F oR R G /\ G oR T H ) -> F oR U H ) ) $=
        ( vw wbr cv cfv wa wral cofr wcel wi ralrimivvva adantr ffvelcdmda wceq
        breq1 anbi1d imbi12d breq2 anbi12d imbi1d anbi2d rspc3v mpd ralimdva wf
        syl3anc wfn ffn syl inidm eqidd ofrfval r19.26 bitr4di 3imtr4d ) ASUAZJ
        UBZVMKUBZFTZVOVMLUBZHTZUCZSEUDZVNVQITZSEUDJKFUETZKLHUETZUCZJLIUETAVSWAS
        EAVMEUFZUCZBUAZCUAZFTZWHDUAZHTZUCZWGWJITZUGZDGUDCGUDBGUDZVSWAUGZAWOWEAW
        NBCDGGGRUHUIWFVNGUFVOGUFVQGUFWOWPUGAEGVMJOUJAEGVMKPUJAEGVMLQUJWNWPVNWHF
        TZWKUCZVNWJITZUGVPVOWJHTZUCZWSUGBCDVNVOVQGGGWGVNUKZWLWRWMWSXBWIWQWKWGVN
        WHFULUMWGVNWJIULUNWHVOUKZWRXAWSXCWQVPWKWTWHVOVNFUOWHVOWJHULUPUQWJVQUKZX
        AVSWSWAXDWTVRVPWJVQVOHUOURWJVQVNIUOUNUSVCUTVAAWDVPSEUDZVRSEUDZUCVTAWBXE
        WCXFASEEVNVOFEJKMMAEGJVBJEVDOEGJVEVFZAEGKVBKEVDPEGKVEVFZNNEVGZWFVNVHZWF
        VOVHZVIASEEVOVQHEKLMMXHAEGLVBLEVDQEGLVEVFZNNXIXKWFVQVHZVIUPVPVRSEVJVKAS
        EEVNVQIEJLMMXGXLNNXIXJXMVIVL $.
    $}
  $}

  ${
    $d w x y z A $.  $d w x y z F $.  $d w x y z G $.  $d w x y z ph $.
    $d w x y z H $.  $d w x y z K $.  $d w x y z O $.  $d w x y z R $.
    $d w x y z S $.  $d w x y z T $.  $d V x y $.  $d W x y $.
    caofdi.1 $e |- ( ph -> A e. V ) $.
    caofdi.2 $e |- ( ph -> F : A --> K ) $.
    caofdi.3 $e |- ( ph -> G : A --> S ) $.
    caofdi.4 $e |- ( ph -> H : A --> S ) $.
    caofdig.r $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y ) e. V ) $.
    ${
      caofdig.t $e |- ( ( ph /\ ( x e. K /\ y e. S ) ) -> ( x T y ) e. W ) $.
      caofdi.5 $e |- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) ->
                          ( x T ( y R z ) ) = ( ( x T y ) O ( x T z ) ) ) $.
      $( Transfer a distributive law to the function operation.  (Contributed
         by Mario Carneiro, 26-Jul-2014.) $)
      caofdig $p |- ( ph ->
        ( F oF T ( G oF R H ) ) = ( ( F oF T G ) oF O ( F oF T H ) ) ) $=
        ( vw cv cfv cmpt cof wcel w3a wceq adantlr ffvelcdmda caovdid mpteq2dva
        co wa oveq2 eleq1d wral oveq1 ralbidv ralrimivva adantr rspcdva feqmptd
        offval2 3eqtr4d ) AUCEUCUDZIUEZVHJUEZVHKUEZFUOZHUOZUFUCEVIVJHUOZVIVKHUO
        ZMUOZUFIJKFUGUOZHUGZUOIJVRUOZIKVRUOZMUGUOAUCEVMVPAVHEUHZUPZBCDVIVJVKGFH
        MLABUDZLUHCUDZGUHDUDZGUHUIWCWDWEFUOHUOWCWDHUOZWCWEHUOMUOUJWAUBUKAELVHIQ
        ULZAEGVHJRULZAEGVHKSULZUMUNAUCEVIVLHIVQNLNPWGWBVJWDFUOZNUHZVLNUHCGVKWDV
        KUJZWJVLNWDVKVJFUQURWBWCWDFUOZNUHZCGUSZWKCGUSBGVJWCVJUJZWNWKCGWPWMWJNWC
        VJWDFUTURVAAWOBGUSWAAWNBCGGTVBVCWHVDWIVDAUCELIQVEZAUCEVJVKFJKNGGPWHWIAU
        CEGJRVEZAUCEGKSVEZVFVFAUCEVNVOMVSVTNOOPWBVIWDHUOZOUHZVNOUHCGVJWDVJUJWTV
        NOWDVJVIHUQURWBWFOUHZCGUSZXACGUSBLVIWCVIUJZXBXACGXDWFWTOWCVIWDHUTURVAAX
        CBLUSWAAXBBCLGUAVBVCWGVDZWHVDWBXAVOOUHCGVKWLWTVOOWDVKVIHUQURXEWIVDAUCEV
        IVJHIJNLGPWGWHWQWRVFAUCEVIVKHIKNLGPWGWIWQWSVFVFVG $.
    $}
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Functions (continued)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( The restriction of a function to a set exists.  Compare Proposition 6.17
     of [TakeutiZaring] p. 28.  This version has a shorter proof than
     ~ resfunexg but requires ~ ax-pow and ~ ax-un .  (Contributed by NM,
     7-Apr-1995.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  resfunexgALT $p |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V ) $=
    ( wfun wcel cres cdm cvv crn cxp dmresexg adantl df-ima funimaexg eqeltrrid
    wa cima jca xpexg wss wrel relres relssdmrn ax-mp ssexg mpan 3syl ) ADZBCEZ
    PZABFZGZHEZUKIZHEZPULUNJZHEZUKHEZUJUMUOUIUMUHABCKLUJUNABQHABMABCNORULUNHHSU
    KUPTZUQURUKUAUSABUBUKUCUDUKUPHUEUFUG $.

  $( Existence of a composition when the first member is a function.
     (Contributed by NM, 8-Oct-2007.) $)
  cofunexg $p |- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V ) $=
    ( wfun wcel wa ccom cdm crn cxp wss wrel relco relssdmrn ax-mp dmcoss ssexg
    cvv sylancr rnexg dmexg adantl cres rnco resfunexg sylan2 syl xpexg syl2anc
    eqeltrid ) ADZBCEZFZABGZUNHZUNIZJZKZUQREZUNREUNLURABMUNNOUMUOREZUPREUSULUTU
    KULUOBHZKVAREUTABPBCUAUOVARQSUBUMUPABIZUCZIZRABUDUMVCREZVDREULUKVBREVEBCTAV
    BRUEUFVCRTUGUJUOUPRRUHUIUNUQRQS $.

  $( Existence of a composition when the second member is one-to-one.
     (Contributed by NM, 8-Oct-2007.) $)
  cofunex2g $p |- ( ( A e. V /\ Fun `' B ) -> ( A o. B ) e. _V ) $=
    ( ccnv wfun wcel ccom cvv wa cnvexg cofunexg sylan2 cnvco cocnvcnv2 3eqtrri
    cocnvcnv1 eqeltrid syl ancoms ) BDZEZACFZABGZHFZUAUBITADZGZHFZUDUBUAUEHFUGA
    CJTUEHKLUGUCUFDZHUHUEDZTDGUIBGUCTUEMUIBNABPOUFHJQRS $.

  $( If the domain of a function is a set, the function is a set.  Theorem
     6.16(1) of [TakeutiZaring] p. 28.  This theorem is derived using the Axiom
     of Replacement in the form of ~ funimaexg .  This version of ~ fnex uses
     ~ ax-pow and ~ ax-un , whereas ~ fnex does not.  (Contributed by NM,
     14-Aug-1994.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  fnexALT $p |- ( ( F Fn A /\ A e. B ) -> F e. _V ) $=
    ( wfn wcel wa cdm crn cxp wss cvv fnrel relssdmrn syl adantr eleq1d biimpar
    wrel cima syl2anc fndm fnfun funimaexg sylan imadmrn imaeq2d eqtr3id syldan
    wfun xpexg ssexg ) CADZABEZFZCCGZCHZIZJZUQKEZCKEULURUMULCRURACLCMNOUNUOBEZU
    PKEZUSULUTUMULUOABACUAZPQULUMCASZKEZVAULCUIUMVDACUBCABUCUDULVAVDULUPVCKULUP
    CUOSVCCUEULUOACVBUFUGPQUHUOUPBKUJTCUQKUKT $.

  $( Weak version of ~ funex that holds without ~ ax-coll .  If the domain and
     codomain of a function exist, so does the function.  (Contributed by Rohan
     Ridenour, 13-Aug-2023.) $)
  funexw $p |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V ) $=
    ( wfun cdm wcel crn w3a cxp cvv xpexg 3adant1 wss wrel funrel relssdmrn syl
    3ad2ant1 ssexd ) CDZCEZAFZCGZBFZHCUAUCIZJUBUDUEJFTUAUCABKLTUBCUEMZUDTCNUFCO
    CPQRS $.
  $( $j usage 'funexw' avoids 'ax-coll' ; $)

  ${
    $d x A $.  $d x C $.
    mptexw.1 $e |- A e. _V $.
    mptexw.2 $e |- C e. _V $.
    mptexw.3 $e |- A. x e. A B e. C $.
    $( Weak version of ~ mptex that holds without ~ ax-coll .  If the domain
       and codomain of a function given by maps-to notation are sets, the
       function is a set.  (Contributed by Rohan Ridenour, 13-Aug-2023.) $)
    mptexw $p |- ( x e. A |-> B ) e. _V $=
      ( cmpt wfun cdm cvv wcel crn funmpt eqid dmmptss ssexi wral wss rnmptss
      ax-mp funexw mp3an ) ABCHZIUDJZKLUDMZKLUDKLABCNUEBEABCUDUDOZPQUFDFCDLABRU
      FDSGABCDUDUGTUAQKKUDUBUC $.
    $( $j usage 'mptexw' avoids 'ax-coll' ; $)
  $}

  $( If the domain of a function exists, so does its range.  Part of Theorem
     4.15(v) of [Monk1] p. 46.  This theorem is derived using the Axiom of
     Replacement in the form of ~ funex .  (Contributed by NM, 11-Nov-1995.) $)
  funrnex $p |- ( dom F e. B -> ( Fun F -> ran F e. _V ) ) $=
    ( wfun cdm wcel cvv crn funex ex rnexg syl6com ) BCZBDAEZBFEZBGFELMNABHIBFJ
    K $.

  $( If the domain of an onto function exists, so does its codomain.
     (Contributed by NM, 23-Jul-2004.) $)
  focdmex $p |- ( A e. C -> ( F : A -onto-> B -> B e. _V ) ) $=
    ( wfo wcel cvv cdm crn wfun funrnex syl5com wf wceq fof fdm syl eleq1d forn
    fofun 3imtr3d com12 ) ABDEZACFZBGFZUCDHZCFZDIZGFZUDUEUCDJUGUIABDTCDKLUCUFAC
    UCABDMUFANABDOABDPQRUCUHBGABDSRUAUB $.

  $( If the codomain of a one-to-one function exists, so does its domain.  This
     can be thought of as a form of the Axiom of Replacement.  (Contributed by
     NM, 4-Sep-2004.) $)
  f1dmex $p |- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V ) $=
    ( wf1 wcel cvv crn wss f1rn ssexg sylan ex ccnv wfo f1cnv f1ofo syl focdmex
    wf1o syl5com syld imp ) ABDEZBCFZAGFZUDUEDHZGFZUFUDUEUHUDUGBIUEUHABDJUGBCKL
    MUDUGADNZOZUHUFUDUGAUITUJABDPUGAUIQRUGAGUISUAUBUC $.

  ${
    $d x y A $.  $d y B $.
    abrexex.1 $e |- A e. _V $.
    $( Existence of a class abstraction of existentially restricted sets. ` x `
       is normally a free-variable parameter in the class expression
       substituted for ` B ` , which can be thought of as ` B ( x ) ` .  This
       simple-looking theorem is actually quite powerful and appears to involve
       the Axiom of Replacement in an intrinsic way, as can be seen by tracing
       back through the path ~ mptexg , ~ funex , ~ fnex , ~ resfunexg , and
       ~ funimaexg .  See also ~ abrexex2 .  (Contributed by NM, 16-Oct-2003.)
       (Proof shortened by Mario Carneiro, 31-Aug-2015.) $)
    abrexex $p |- { y | E. x e. A y = B } e. _V $=
      ( cmpt crn cv wceq wrex cab cvv eqid rnmpt mptex rnex eqeltrri ) ACDFZGBH
      DIACJBKLABCDRRMNRACDEOPQ $.
  $}

  ${
    $d x y A $.  $d y B $.
    $( Existence of a class abstraction of existentially restricted sets. ` x `
       is normally a free-variable parameter in ` B ` .  The antecedent assures
       us that ` A ` is a set.  (Contributed by NM, 3-Nov-2003.) $)
    abrexexg $p |- ( A e. V -> { y | E. x e. A y = B } e. _V ) $=
      ( wcel wceq wrex cab cmpt crn cvv eqid rnmpt mptexg rnexg syl eqeltrrid
      cv ) CEFZBSDGACHBIACDJZKZLABCDUAUAMNTUALFUBLFACDEOUALPQR $.
  $}

  ${
    $d x y A $.  $d y B $.
    $( The existence of an indexed union. ` x ` is normally a free-variable
       parameter in ` B ` .  (Contributed by NM, 23-Mar-2006.) $)
    iunexg $p |- ( ( A e. V /\ A. x e. A B e. W ) -> U_ x e. A B e. _V ) $=
      ( vy wcel wral ciun wceq wrex cab cuni cvv dfiun2g adantl abrexexg uniexg
      wa cv syl adantr eqeltrd ) BDGZCEGABHZSABCIZFTCJABKFLZMZNUEUFUHJUDAFBCEOP
      UDUHNGZUEUDUGNGUIAFBCDQUGNRUAUBUC $.
  $}

  ${
    $d A x y z $.  $d V x y z $.  $d W x y z $.  $d ph z $.
    $( Existence of an existentially restricted class abstraction.
       (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    abrexex2g $p |- ( ( A e. V /\ A. x e. A { y | ph } e. W )
                                  -> { y | E. x e. A ph } e. _V ) $=
      ( vz wcel cab wral wa wrex cv cvv wsb nfv nfcv nfs1v nfrexw wceq eqeltrid
      sbequ12 rexbidv cbvab df-clab rexbii abbii eqtr4i df-iun iunexg eqeltrrid
      ciun ) DEHACIZFHBDJKZABDLZCIZGMZUMHZBDLZGIZNUPACGOZBDLZGIUTUOVBCGUOGPVACB
      DCDQACGRSCMUQTAVABDACGUBUCUDUSVBGURVABDAGCUEUFUGUHUNUTBDUMULNBGDUMUIBDUME
      FUJUKUA $.
  $}

  ${
    $d A x y v w z $.  $d ps v w z $.  $d ph x $.
    opabex3d.1 $e |- ( ph -> A e. _V ) $.
    opabex3d.2 $e |- ( ( ph /\ x e. A ) -> { y | ps } e. _V ) $.
    $( Existence of an ordered pair abstraction, deduction version.
       (Contributed by Alexander van der Vekens, 19-Oct-2017.) $)
    opabex3d $p |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V ) $=
      ( vz vv vw cv wcel wa cvv wex cop wceq an12 exbii bitri copab csn cab cxp
      ciun 19.42v excom velsn anbi1i vex opeq1 eqeq2d anbi1d ceqsexv nfv nfsab1
      elxp nfan opeq2 wsb df-clab sbequ12 equcoms bitr4id anbi12d 3bitri anbi2i
      wb cbvex 3bitr4ri wrex eliun df-rex elopab 3bitr4i eqriv wral snexg ax-mp
      xpexg sylancr ralrimiva iunexg syl2anc eqeltrrid ) ACKZELZBMZCDUAZCEWFUBZ
      BDUCZUDZUEZNHWMWIWGHKZWLLZMZCOZWNWFDKZPZQZWHMZDOZCOWNWMLZWNWILWPXBCWGWTBM
      ZMZDOWGXDDOZMXBWPWGXDDUFXAXEDWTWGBRSWOXFWGWOWNIKZJKZPZQZXGWJLZXHWKLZMMZJO
      IOZWNWFXHPZQZXLMZJOZXFIJWNWJWKUQXNXMIOZJOXRXMIJUGXSXQJXSXGWFQZXJXLMZMZIOX
      QXMYBIXMXKYAMYBXJXKXLRXKXTYAIWFUHUITSYAXQIWFCUJZXTXJXPXLXTXIXOWNXGWFXHUKU
      LUMUNTSTXQXDJDXPXLDXPDUOBDJUPURXDJUOXHWRQZXPWTXLBYDXOWSWNXHWRWFUSULYDXLBD
      JUTZBBJDVABYEVHDJBDJVBVCVDVEVIVFVGVJSXCWOCEVKWQCWNEWLVLWOCEVMTWHCDWNVNVOV
      PAENLWLNLZCEVQWMNLFAYFCEAWGMWJNLZWKNLYFWFNLYGYCWFNVRVSGWJWKNNVTWAWBCEWLNN
      WCWDWE $.
  $}

  ${
    $d A x y v w z $.  $d ph v w z $.
    opabex3.1 $e |- A e. _V $.
    opabex3.2 $e |- ( x e. A -> { y | ph } e. _V ) $.
    $( Existence of an ordered pair abstraction.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    opabex3 $p |- { <. x , y >. | ( x e. A /\ ph ) } e. _V $=
      ( vz vv vw cv wcel wa cvv wex cop wceq an12 exbii bitri eqeq2d ciun copab
      csn cab cxp 19.42v excom velsn anbi1i vex opeq1 anbi1d ceqsexv nfv nfsab1
      elxp nfan opeq2 wsb df-clab sbequ12 equcoms bitr4id anbi12d 3bitri anbi2i
      wb cbvex 3bitr4ri wrex eliun df-rex elopab 3bitr4i eqriv wral snexg ax-mp
      xpexg sylancr rgen iunexg mp2an eqeltrri ) BDBJZUCZACUDZUEZUAZWEDKZALZBCU
      BZMGWIWLWJGJZWHKZLZBNZWMWECJZOZPZWKLZCNZBNWMWIKZWMWLKWOXABWJWSALZLZCNWJXC
      CNZLXAWOWJXCCUFWTXDCWSWJAQRWNXEWJWNWMHJZIJZOZPZXFWFKZXGWGKZLLZINHNZWMWEXG
      OZPZXKLZINZXEHIWMWFWGUPXMXLHNZINXQXLHIUGXRXPIXRXFWEPZXIXKLZLZHNXPXLYAHXLX
      JXTLYAXIXJXKQXJXSXTHWEUHUISRXTXPHWEBUJZXSXIXOXKXSXHXNWMXFWEXGUKTULUMSRSXP
      XCICXOXKCXOCUNACIUOUQXCIUNXGWQPZXOWSXKAYCXNWRWMXGWQWEURTYCXKACIUSZAAICUTA
      YDVGCIACIVAVBVCVDVHVEVFVIRXBWNBDVJWPBWMDWHVKWNBDVLSWKBCWMVMVNVODMKWHMKZBD
      VPWIMKEYEBDWJWFMKZWGMKYEWEMKYFYBWEMVQVRFWFWGMMVSVTWABDWHMMWBWCWD $.
  $}

  ${
    $d x A $.
    iunex.1 $e |- A e. _V $.
    iunex.2 $e |- B e. _V $.
    $( The existence of an indexed union. ` x ` is normally a free-variable
       parameter in the class expression substituted for ` B ` , which can be
       read informally as ` B ( x ) ` .  (Contributed by NM, 13-Oct-2003.) $)
    iunex $p |- U_ x e. A B e. _V $=
      ( cvv wcel wral ciun rgenw iunexg mp2an ) BFGCFGZABHABCIFGDMABEJABCFFKL
      $.
  $}

  ${
    $d x y z A $.  $d z ph $.
    abrexex2.1 $e |- A e. _V $.
    abrexex2.2 $e |- { y | ph } e. _V $.
    $( Existence of an existentially restricted class abstraction. ` ph ` is
       normally has free-variable parameters ` x ` and ` y ` .  See also
       ~ abrexex .  (Contributed by NM, 12-Sep-2004.) $)
    abrexex2 $p |- { y | E. x e. A ph } e. _V $=
      ( vz wrex cab cv wcel cvv wsb nfv nfcv nfs1v nfrexw weq sbequ12 rexbidv
      cbvab df-clab rexbii abbii eqtr4i ciun df-iun iunex eqeltrri eqeltri ) AB
      DHZCIZGJACIZKZBDHZGIZLULACGMZBDHZGIUPUKURCGUKGNUQCBDCDOACGPQCGRAUQBDACGST
      UAUOURGUNUQBDAGCUBUCUDUEBDUMUFUPLBGDUMUGBDUMEFUHUIUJ $.

    $( Existence of a class abstraction with an existentially quantified
       expression.  Both ` x ` and ` y ` can be free in ` ph ` .  (Contributed
       by NM, 29-Jul-2006.) $)
    abexssex $p |- { y | E. x ( x C_ A /\ ph ) } e. _V $=
      ( cpw wrex cab cv wss wa wex cvv wcel df-rex velpw anbi1i exbii bitri
      abbii pwex abrexex2 eqeltrri ) ABDGZHZCIBJZDKZALZBMZCINUFUJCUFUGUEOZALZBM
      UJABUEPULUIBUKUHABDQRSTUAABCUEDEUBFUCUD $.
  $}

  ${
    $d x y A $.
    abexex.1 $e |- A e. _V $.
    abexex.2 $e |- ( ph -> x e. A ) $.
    abexex.3 $e |- { y | ph } e. _V $.
    $( A condition where a class builder continues to exist after its wff is
       existentially quantified.  (Contributed by NM, 4-Mar-2007.) $)
    abexex $p |- { y | E. x ph } e. _V $=
      ( wrex cab wex cvv cv wcel wa df-rex pm4.71ri exbii bitr4i abbii abrexex2
      eqeltrri ) ABDHZCIABJZCIKUBUCCUBBLDMZANZBJUCABDOAUEBAUDFPQRSABCDEGTUA $.
  $}

  ${
    $d x y A $.  $d y B $.  $d y C $.
    elabreximd.1 $e |- F/ x ph $.
    elabreximd.2 $e |- F/ x ch $.
    elabreximd.3 $e |- ( A = B -> ( ch <-> ps ) ) $.
    elabreximd.4 $e |- ( ph -> A e. V ) $.
    elabreximd.5 $e |- ( ( ph /\ x e. C ) -> ps ) $.
    $( Class substitution in an image set.  (Contributed by Thierry Arnoux,
       30-Dec-2016.) $)
    elabreximd $p |- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> ch ) $=
      ( cv wceq wrex cab wcel wa wb eqeq1 rexbidv elabg syl biimpa simpr adantr
      biimpar syl2anc exp31 rexlimd imp syldan ) AFEOZGPZDHQZERSZFGPZDHQZCAURUT
      AFISURUTUAMUQUTEFIUOFPUPUSDHUOFGUBUCUDUEUFAUTCAUSCDHJKADOHSZUSCAVATZUSTUS
      BCVBUSUGVBBUSNUHUSCBLUIUJUKULUMUN $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.  $d x ch $.  $d x ph $.
    elabreximdv.1 $e |- ( A = B -> ( ch <-> ps ) ) $.
    elabreximdv.2 $e |- ( ph -> A e. V ) $.
    elabreximdv.3 $e |- ( ( ph /\ x e. C ) -> ps ) $.
    $( Class substitution in an image set.  (Contributed by Thierry Arnoux,
       30-Dec-2016.) $)
    elabreximdv $p |- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> ch ) $=
      ( nfv elabreximd ) ABCDEFGHIADMCDMJKLN $.
  $}

  ${
    abrexss.1 $e |- F/_ x C $.
    $d x y z $.  $d y z A $.  $d z C $.  $d y z B $.
    $( A necessary condition for an image set to be a subset.  (Contributed by
       Thierry Arnoux, 6-Feb-2017.) $)
    abrexss $p |- ( A. x e. A B e. C -> { y | E. x e. A y = B } C_ C ) $=
      ( vz wcel wral cv wceq wrex cab cvv nfra1 nfcri eleq1 vex a1i rspa ssrdv
      elabreximd ex ) DEHZACIZGBJDKACLBMZEUEGJZUFHUGEHZUEUDUHABUGDCNUDACOAGEFPU
      GDEQUGNHUEGRSUDACTUBUCUA $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y F $.
    funimass4f.1 $e |- F/_ x A $.
    funimass4f.2 $e |- F/_ x B $.
    funimass4f.3 $e |- F/_ x F $.
    $( use ~ ffnfvf $)
    $( Membership relation for the values of a function whose image is a
       subclass.  (Contributed by Thierry Arnoux, 24-Apr-2017.) $)
    funimass4f $p |- ( ( Fun F /\ A C_ dom F ) ->
                    ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) ) $=
      ( vy wfun cdm wss wa cima cv cfv wcel wral nfss nfan wceq nffun funfvima2
      nfdm nfima ssel sylan9 ralrimi cab dfimafnf adantr abrexss adantl eqsstrd
      wrex impbida ) DIZBDJZKZLZDBMZCKZANZDOZCPZABQZUSVALVDABUSVAAUPURAADGUAABU
      QEADGUCRSAUTCADBGEUDFRSUSVBBPVCUTPVAVDBVBDUBUTCVCUEUFUGUSVELUTHNVCTABUNHU
      HZCUSUTVFTVEAHBDEGUIUJVEVFCKUSAHBVCCFUKULUMUO $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d ph x y z $.
    oprabexd.1 $e |- ( ph -> A e. _V ) $.
    oprabexd.2 $e |- ( ph -> B e. _V ) $.
    oprabexd.3 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> E* z ps ) $.
    oprabexd.4 $e |- ( ph -> F = { <. <. x , y >. , z >. |
                                        ( ( x e. A /\ y e. B ) /\ ps ) } ) $.
    $( Existence of an operator abstraction.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    oprabexd $p |- ( ph -> F e. _V ) $=
      ( cv wcel wa coprab cvv wmo wal syl2anc cdm wi ex moanimv sylibr alrimivv
      wfun funoprabg syl cxp wss dmoprabss xpexg ssexg sylancr funex eqeltrd )
      AHCMFNDMGNOZBOZCDEPZQLAUTUGZUTUAZQNZUTQNAUSERZDSCSVAAVDCDAURBERZUBVDAURVE
      KUCURBEUDUEUFUSCDEUHUIAVBFGUJZUKVFQNZVCBCDEFGULAFQNGQNVGIJFGQQUMTVBVFQUNU
      OQUTUPTUQ $.
  $}

  ${
    $d x y z A $.  $d x y z B $.
    oprabex.1 $e |- A e. _V $.
    oprabex.2 $e |- B e. _V $.
    oprabex.3 $e |- ( ( x e. A /\ y e. B ) -> E* z ph ) $.
    oprabex.4 $e |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B )
             /\ ph ) } $.
    $( Existence of an operation class abstraction.  (Contributed by NM,
       19-Oct-2004.) $)
    oprabex $p |- F e. _V $=
      ( cv wcel wa coprab cvv wfun cdm wmo wi mpbir funoprab cxp xpex dmoprabss
      moanimv ssexi funex mp2an eqeltri ) GBLEMCLFMNZANZBCDOZPKUMQUMRZPMUMPMULB
      CDULDSUKADSTJUKADUFUAUBUNEFUCEFHIUDABCDEFUEUGPUMUHUIUJ $.
  $}

  ${
    $d x y z w v u f H $.  $d x y z R $.
    oprabex3.1 $e |- H e. _V $.
    oprabex3.2 $e |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\
                      y e. ( H X. H ) ) /\
                      E. w E. v E. u E. f ( ( x = <. w , v >. /\
                      y = <. u , f >. ) /\ z = R ) ) } $.
    $( Existence of an operation class abstraction (special case).
       (Contributed by NM, 19-Oct-2004.) $)
    oprabex3 $p |- F e. _V $=
      ( cv cop wceq wa wex wmo wcel mosubop cxp xpex anass 2exbii 19.42vv bitri
      moeq mobii mpbir a1i oprabex ) AMZDMEMNOZBMZFMHMNOZPCMGOZPZHQFQZEQDQZABCJ
      JUAZUTIJJKKUBZVAUSCRZULUTSUNUTSPVBUMUOUPPZHQFQZPZEQDQZCRVDCDEULUPCFHUNCGU
      GTTUSVFCURVEDEURUMVCPZHQFQVEUQVGFHUMUOUPUCUDUMVCFHUEUFUDUHUIUJLUK $.
  $}

  ${
    $d A v x y z w $.  $d ph v $.
    oprabrexex2.1 $e |- A e. _V $.
    oprabrexex2.2 $e |- { <. <. x , y >. , z >. | ph } e. _V $.
    $( Existence of an existentially restricted operation abstraction.
       (Contributed by Jeff Madsen, 11-Jun-2010.) $)
    oprabrexex2 $p |- { <. <. x , y >. , z >. | E. w e. A ph } e. _V $=
      ( vv wrex coprab cv cop wa wex cab cvv df-oprab rexcom4 exbii wceq bitr2i
      r19.42v bitri abbii eqtri eqeltrri abrexex2 eqeltri ) AEFJZBCDKZILBLCLMDL
      MUAZANZDOZCOZBOZEFJZIPZQUKULUJNZDOZCOZBOZIPURUJBCDIRVBUQIUQUOEFJZBOVBUOEB
      FSVCVABVCUNEFJZCOVAUNECFSVDUTCVDUMEFJZDOUTUMEDFSVEUSDULAEFUCTUDTUDTUBUEUF
      UPEIFGABCDKUPIPQABCDIRHUGUHUI $.
  $}

  ${
    $d x z A $.  $d y z B $.  $d z C $.
    ab2rexex.1 $e |- A e. _V $.
    ab2rexex.2 $e |- B e. _V $.
    $( Existence of a class abstraction of existentially restricted sets.
       Variables ` x ` and ` y ` are normally free-variable parameters in the
       class expression substituted for ` C ` , which can be thought of as
       ` C ( x , y ) ` .  See comments for ~ abrexex .  (Contributed by NM,
       20-Sep-2011.) $)
    ab2rexex $p |- { z | E. x e. A E. y e. B z = C } e. _V $=
      ( cv wceq wrex abrexex abrexex2 ) CIFJBEKACDGBCEFHLM $.
  $}

  ${
    $d x z A $.  $d y z B $.
    ab2rexex2.1 $e |- A e. _V $.
    ab2rexex2.2 $e |- B e. _V $.
    ab2rexex2.3 $e |- { z | ph } e. _V $.
    $( Existence of an existentially restricted class abstraction. ` ph `
       normally has free-variable parameters ` x ` , ` y ` , and ` z ` .
       Compare ~ abrexex2 .  (Contributed by NM, 20-Sep-2011.) $)
    ab2rexex2 $p |- { z | E. x e. A E. y e. B ph } e. _V $=
      ( wrex abrexex2 ) ACFJBDEGACDFHIKK $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d V y $.
    $( The cross product of two sets is a set.  Proposition 6.2 of
       [TakeutiZaring] p. 23.  This version is proven using Replacement; see
       ~ xpexg for a version that uses the Power Set axiom instead.
       (Contributed by Mario Carneiro, 20-May-2013.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    xpexgALT $p |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V ) $=
      ( vy vx wcel wa cxp cv csn ciun iunid xpeq2i xpiundi eqtr3i wral eqeltrid
      cvv id cmpt fconstmpt mptexg ralrimivw iunexg syl2anr ) ACGZBDGZHABIZEBAE
      JZKZIZLZSAEBUKLZIUIUMUNBAEBMNEBUKAOPUHUHULSGZEBQUMSGUGUHTUGUOEBUGULFAUJUA
      SFAUJUBFAUJCUCRUDEBULDSUEUFR $.
  $}

  ${
    $d F x a b $.  $d G x a b $.  $d V x $.  $d W x $.  $d R x a b $.
    $d D x $.
    $( General value of ` ( F oF R G ) ` with no assumptions on functionality
       of ` F ` and ` G ` .  (Contributed by Stefan O'Rear, 24-Jan-2015.) $)
    offval3 $p |- ( ( F e. V /\ G e. W ) -> ( F oF R G ) =
        ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) ) $=
      ( va vb wcel wa cvv cdm cin cv cfv co cmpt wceq elex adantr adantl inex1g
      cof dmexg mptexg 3syl ineqan12d fveq1 oveqan12d mpteq12dv ovmpoga syl3anc
      dmeq df-of ) CEIZDFIZJCKIZDKIZACLZDLZMZANZCOZVBDOZBPZQZKIZCDBUCZPVFRUOUQU
      PCESTUPURUODFSUAUOVGUPUOUSKIVAKIVGCEUDUSUTKUBAVAVEKUEUFTGHCDKKAGNZLZHNZLZ
      MZVBVIOZVBVKOZBPZQVFVHKVICRZVKDRZJAVMVPVAVEVQVRVJUSVLUTVICUMVKDUMUGVQVRVN
      VCVOVDBVBVICUHVBVKDUHUIUJABGHUNUKUL $.

    $( Pointwise combination commutes with restriction.  (Contributed by Stefan
       O'Rear, 24-Jan-2015.) $)
    offres $p |- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) =
        ( ( F |` D ) oF R ( G |` D ) ) ) $=
      ( vx wcel cdm cin cfv co cmpt cres wceq fvres dmres 3eqtr4ri offval3 cvv
      wa cv cof inss2 sseli oveq12d mpteq2ia inindi incom ineq12i eqid mpteq12i
      syl resmpt3 reseq1d resexg syl2an 3eqtr4a ) CEHZDFHZUAZGCIZDIZJZGUBZCKZVE
      DKZBLZMZANZGCANZIZDANZIZJZVEVKKZVEVMKZBLZMZCDBUCZLZANVKVMVTLZGVDAJZVRMGWC
      VHMVSVJGWCVRVHVEWCHVEAHZVRVHOWCAVEVDAUDUEWDVPVFVQVGBVEACPVEADPUFUMUGGVOVR
      WCVRAVDJAVBJZAVCJZJWCVOAVBVCUHVDAUIVLWEVNWFCAQDAQUJRVRUKULGVDAVHUNRVAWAVI
      AGBCDEFSUOUSVKTHVMTHWBVSOUTCAEUPDAFUPGBVKVMTTSUQUR $.
  $}

  ${
    $d f g A $.  $d f g B $.  $d f g x R $.
    $( Equivalent expressions for a restriction of the function operation map.
       Unlike ` oF R ` which is a proper class, ` ( oF R |`` ( A X. B ) ) ` can
       be a set by ~ ofmresex , allowing it to be used as a function or
       structure argument.  By ~ ofmresval , the restricted operation map
       values are the same as the original values, allowing theorems for
       ` oF R ` to be reused.  (Contributed by NM, 20-Oct-2014.) $)
    ofmres $p |- ( oF R |` ( A X. B ) ) =
         ( f e. A , g e. B |-> ( f oF R g ) ) $=
      ( vx cvv cv cdm cin cfv co cmpo cres wss wceq ssv eqid wcel vex cxp mp2an
      cmpt cof resmpo df-of reseq1i inex1 mptex ovmpt4g mp3an mpoeq123i 3eqtr4i
      dmex ) DEGGFDHZIZEHZIZJZFHZUOKUTUQKCLZUCZMZABUAZNZDEABVBMZCUDZVDNDEABUOUQ
      VGLZMAGOBGOVEVFPAQBQDEGGABVBUEUBVGVCVDFCDEUFZUGDEABVHABVBARBRUOGSUQGSVBGS
      VHVBPDTZETFUSVAUPURUOVJUNUHUIDEGGVBVGGVIUJUKULUM $.

    ofmresex.a $e |- ( ph -> A e. V ) $.
    ofmresex.b $e |- ( ph -> B e. W ) $.
    $( Existence of a restriction of the function operation map.  (Contributed
       by NM, 20-Oct-2014.) $)
    ofmresex $p |- ( ph -> ( oF R |` ( A X. B ) ) e. _V ) $=
      ( cxp cvv wcel cof cres xpexg syl2anc ofexg syl ) ABCIZJKZDLRMJKABEKCFKSG
      HBCEFNORDJPQ $.
  $}

  ${
    $d A f u v x y $.  $d V u v $.  $d f ph u v $.
    $( Principle of unique choice.  This is also called non-choice.  The name
       choice results in its similarity to something like ~ acfun (with the key
       difference being the change of ` E. ` to ` E! ` ) but unique choice in
       fact follows from the axiom of collection and our other axioms.  This is
       somewhat similar to Corollary 3.9.2 of [HoTT], p.  (varies) but is
       better described by the paragraph at the end of Section 3.9 which starts
       "A similar issue arises in set-theoretic mathematics".  (Contributed by
       Jim Kingdon, 13-Sep-2025.) $)
    uchoice $p |- ( ( A e. V /\ A. x e. A E! y ph ) -> E. f
        ( f Fn A /\ A. x e. A [. ( f ` x ) / y ]. ph ) ) $=
      ( vv vu wcel wral wa cv cfv wceq wsbc wb wal wex nfv vex weu wfn cvv eqid
      copab fnopabg biimpi adantl simpl fnex syl2anc cop fnopfvb nfsbc1v eleq1w
      sbceq1a anbi12d anbi2d opelopabf bitrdi ralrimiva alrimiv ancli syl fneq1
      nfan fveq1 eqeq1d bibi1d ralbidv albidv elabd ralcom4 anbi2i exbii sylibr
      nfcv nfsbc nfbi fveqeq2 sbceq2a sbcbidv bibi12d cbvral sylib eqidd dfsbcq
      nfal fvex eqeq2 spcv mpbid simprd ralimi anim2i eximi ) DFIZACUABDJZKZELZ
      DUBZBLZWTMZGLZNZXBDIZACXDOZKZPZGQZBDJZKZERZXAACXCOZBDJZKZERWSXAHLZWTMZXDN
      ZXQDIZABXQOZCXDOZKZPZGQZHDJZKZERZXMWSXAYDHDJZGQZKZERYHWSYKXFAKZBCUEZDUBZX
      QYMMZXDNZYCPZHDJZGQZKZEYMWSYNWQYMUCIWRYNWQWRYNABCDYMYMUDUFUGUHZWQWRUIDFYM
      UJUKWSYNYTUUAYNYSYNYRGYNYQHDYNXTKYPXQXDULYMIYCDXQXDYMUMYLXTYAKYCBCXQXDXTY
      ABXTBSZABXQUNZVFXTYBCXTCSYACXDUNVFHTGTXBXQNXFXTAYABHDUOABXQUPUQCLXDNYAYBX
      TYACXDUPURUSUTVAVBVCVDWTYMNZXAYNYJYSDWTYMVEUUDYIYRGUUDYDYQHDUUDXSYPYCUUDX
      RYOXDXQWTYMVGVHVIVJVKUQVLYGYKEYFYJXAYDHGDVMVNVOVPYGXLEYFXKXAYEXJHBDYDBGXS
      YCBXSBSXTYBBUUBYABCXDBXDVQUUCVRVFVSWHXJHSXQXBNZYDXIGUUEXSXEYCXHXQXBXDWTVT
      UUEXTXFYBXGHBDUOUUEYAACXDABXQWAWBUQWCVKWDVNVOWEXLXPEXKXOXAXJXNBDXJXFXNXJX
      CXCNZXFXNKZXJXCWFXIUUFUUGPGXCXBWTUCUCETBTWIXDXCNZXEUUFXHUUGXDXCXCWJUUHXGX
      NXFACXDXCWGURWCWKWLWMWNWOWPVD $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  First and second members of an ordered pair
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c 1st $. $( First member of an ordered pair $)
  $c 2nd $. $( Second member of an ordered pair $)

  $( Extend the definition of a class to include the first member an ordered
     pair function. $)
  c1st $a class 1st $.

  $( Extend the definition of a class to include the second member an ordered
     pair function. $)
  c2nd $a class 2nd $.

  $( Define a function that extracts the first member, or abscissa, of an
     ordered pair.  Theorem ~ op1st proves that it does this.  For example,
     ( ` 1st `` <. ` 3 , 4 ` >. ` ) = 3 .  Equivalent to Definition 5.13 (i) of
     [Monk1] p. 52 (compare ~ op1sta and ~ op1stb ).  The notation is the same
     as Monk's.  (Contributed by NM, 9-Oct-2004.) $)
  df-1st $a |- 1st = ( x e. _V |-> U. dom { x } ) $.

  $( Define a function that extracts the second member, or ordinate, of an
     ordered pair.  Theorem ~ op2nd proves that it does this.  For example,
     ` ( 2nd `` <. ` 3 , 4 ` >. ` ) = 4 .  Equivalent to Definition 5.13 (ii)
     of [Monk1] p. 52 (compare ~ op2nda and ~ op2ndb ).  The notation is the
     same as Monk's.  (Contributed by NM, 9-Oct-2004.) $)
  df-2nd $a |- 2nd = ( x e. _V |-> U. ran { x } ) $.

  ${
    $d x A $.
    $( The value of the function that extracts the first member of an ordered
       pair.  (Contributed by NM, 9-Oct-2004.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    1stvalg $p |- ( A e. _V -> ( 1st ` A ) = U. dom { A } ) $=
      ( vx cvv wcel csn cdm cuni c1st cfv wceq snexg dmexg uniexg 3syl cv dmeqd
      sneq unieqd df-1st fvmptg mpdan ) ACDZAEZFZGZCDZAHIUEJUBUCCDUDCDUFACKUCCL
      UDCMNBABOZEZFZGUECCHUGAJZUIUDUJUHUCUGAQPRBSTUA $.

    $( The value of the function that extracts the second member of an ordered
       pair.  (Contributed by NM, 9-Oct-2004.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    2ndvalg $p |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } ) $=
      ( vx cvv wcel csn crn cuni c2nd cfv wceq snexg rnexg uniexg 3syl cv rneqd
      sneq unieqd df-2nd fvmptg mpdan ) ACDZAEZFZGZCDZAHIUEJUBUCCDUDCDUFACKUCCL
      UDCMNBABOZEZFZGUECCHUGAJZUIUDUJUHUCUGAQPRBSTUA $.
  $}

  $( The value of the first-member function at the empty set.  (Contributed by
     NM, 23-Apr-2007.) $)
  1st0 $p |- ( 1st ` (/) ) = (/) $=
    ( c1st cfv csn cdm cuni cvv wcel wceq 0ex 1stvalg ax-mp dmsn0 unieqi 3eqtri
    c0 uni0 ) OABZOCDZEZOEOOFGQSHIOJKROLMPN $.

  $( The value of the second-member function at the empty set.  (Contributed by
     NM, 23-Apr-2007.) $)
  2nd0 $p |- ( 2nd ` (/) ) = (/) $=
    ( c0 c2nd cfv csn crn cuni cvv wcel wceq 0ex 2ndvalg ax-mp cdm dmsn0 dm0rn0
    mpbi unieqi uni0 3eqtri ) ABCZADZEZFZAFAAGHTUCIJAKLUBAUAMAIUBAINUAOPQRS $.

  ${
    op1st.1 $e |- A e. _V $.
    op1st.2 $e |- B e. _V $.
    $( Extract the first member of an ordered pair.  (Contributed by NM,
       5-Oct-2004.) $)
    op1st $p |- ( 1st ` <. A , B >. ) = A $=
      ( cop c1st cfv csn cdm cuni cvv wcel wceq opexg mp2an 1stvalg ax-mp eqtri
      op1sta ) ABEZFGZTHIJZATKLZUAUBMAKLBKLUCCDABKKNOTPQABCDSR $.

    $( Extract the second member of an ordered pair.  (Contributed by NM,
       5-Oct-2004.) $)
    op2nd $p |- ( 2nd ` <. A , B >. ) = B $=
      ( cop c2nd cfv csn crn cuni cvv wcel wceq opexg mp2an 2ndvalg ax-mp eqtri
      op2nda ) ABEZFGZTHIJZBTKLZUAUBMAKLBKLUCCDABKKNOTPQABCDSR $.

    $( Extract the first member of an ordered pair.  (Contributed by Mario
       Carneiro, 31-Aug-2015.) $)
    op1std $p |- ( C = <. A , B >. -> ( 1st ` C ) = A ) $=
      ( cop wceq c1st cfv fveq2 op1st eqtrdi ) CABFZGCHIMHIACMHJABDEKL $.

    $( Extract the second member of an ordered pair.  (Contributed by Mario
       Carneiro, 31-Aug-2015.) $)
    op2ndd $p |- ( C = <. A , B >. -> ( 2nd ` C ) = B ) $=
      ( cop wceq c2nd cfv fveq2 op2nd eqtrdi ) CABFZGCHIMHIBCMHJABDEKL $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Extract the first member of an ordered pair.  (Contributed by NM,
       19-Jul-2005.) $)
    op1stg $p |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A ) $=
      ( vx vy cv cop c1st cfv wceq opeq1 fveq2d id eqeq12d opeq2 eqeq1d vtocl2g
      vex op1st ) EGZFGZHZIJZUAKAUBHZIJZAKABHZIJZAKEFABCDUAAKZUDUFUAAUIUCUEIUAA
      UBLMUINOUBBKZUFUHAUJUEUGIUBBAPMQUAUBESFSTR $.

    $( Extract the second member of an ordered pair.  (Contributed by NM,
       19-Jul-2005.) $)
    op2ndg $p |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B ) $=
      ( vx vy cv cop c2nd cfv wceq opeq1 fveq2d eqeq1d opeq2 id eqeq12d vtocl2g
      vex op2nd ) EGZFGZHZIJZUBKAUBHZIJZUBKABHZIJZBKEFABCDUAAKZUDUFUBUIUCUEIUAA
      UBLMNUBBKZUFUHUBBUJUEUGIUBBAOMUJPQUAUBESFSTR $.

    $( Extract the first member of an ordered triple.  (Due to infrequent
       usage, it isn't worthwhile at this point to define special extractors
       for triples, so we reuse the ordered pair extractors for ~ ot1stg ,
       ~ ot2ndg , ~ ot3rdgg .)  (Contributed by NM, 3-Apr-2015.)  (Revised by
       Mario Carneiro, 2-May-2015.) $)
    ot1stg $p |- ( ( A e. V /\ B e. W /\ C e. X )
         -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = A ) $=
      ( wcel w3a cotp c1st cfv cop df-ot fveq2i wceq wa cvv opexg op1stg sylan
      3impa eqtrid fveq2d 3adant3 eqtrd ) ADGZBEGZCFGZHZABCIZJKZJKABLZJKZAUIUKU
      LJUIUKULCLZJKZULUJUNJABCMNUFUGUHUOULOZUFUGPULQGUHUPABDERULCQFSTUAUBUCUFUG
      UMAOUHABDESUDUE $.

    $( Extract the second member of an ordered triple.  (See ~ ot1stg comment.)
       (Contributed by NM, 3-Apr-2015.)  (Revised by Mario Carneiro,
       2-May-2015.) $)
    ot2ndg $p |- ( ( A e. V /\ B e. W /\ C e. X )
         -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B ) $=
      ( wcel w3a cotp c1st cfv c2nd cop df-ot fveq2i wceq wa cvv opexg op1stg
      sylan 3impa eqtrid fveq2d op2ndg 3adant3 eqtrd ) ADGZBEGZCFGZHZABCIZJKZLK
      ABMZLKZBUKUMUNLUKUMUNCMZJKZUNULUPJABCNOUHUIUJUQUNPZUHUIQUNRGUJURABDESUNCR
      FTUAUBUCUDUHUIUOBPUJABDEUEUFUG $.

    $( Extract the third member of an ordered triple.  (See ~ ot1stg comment.)
       (Contributed by NM, 3-Apr-2015.) $)
    ot3rdgg $p |- ( ( A e. V /\ B e. W /\ C e. X )
        -> ( 2nd ` <. A , B , C >. ) = C ) $=
      ( wcel w3a cotp c2nd cfv cop df-ot fveq2i wceq wa cvv opexg op2ndg sylan
      3impa eqtrid ) ADGZBEGZCFGZHABCIZJKABLZCLZJKZCUFUHJABCMNUCUDUEUICOZUCUDPU
      GQGUEUJABDERUGCQFSTUAUB $.

    $( Alternate value of the function that extracts the first member of an
       ordered pair.  Definition 5.13 (i) of [Monk1] p. 52.  (Contributed by
       NM, 18-Aug-2006.) $)
    1stval2 $p |- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A ) $=
      ( vx vy cvv cxp wcel cv cop wceq wex c1st cfv cint elvv vex op1stb eqtr4i
      op1st fveq2 inteq inteqd 3eqtr4a exlimivv sylbi ) ADDEFABGZCGZHZIZCJBJAKL
      ZAMZMZIZBCANUHULBCUHUGKLZUGMZMZUIUKUMUEUOUEUFBOZCOZRUEUFUPUQPQAUGKSUHUJUN
      AUGTUAUBUCUD $.

    $( Alternate value of the function that extracts the second member of an
       ordered pair.  Definition 5.13 (ii) of [Monk1] p. 52.  (Contributed by
       NM, 18-Aug-2006.) $)
    2ndval2 $p |- ( A e. ( _V X. _V )
               -> ( 2nd ` A ) = |^| |^| |^| `' { A } ) $=
      ( vx vy cvv cxp wcel cv cop wceq wex c2nd cfv ccnv cint elvv op2nd op2ndb
      csn vex inteqd eqtr4i fveq2 sneq cnveqd 3eqtr4a exlimivv sylbi ) ADDEFABG
      ZCGZHZIZCJBJAKLZARZMZNZNZNZIZBCAOUKURBCUKUJKLZUJRZMZNZNZNZULUQUSUIVDUHUIB
      SZCSZPUHUIVEVFQUAAUJKUBUKUPVCUKUOVBUKUNVAUKUMUTAUJUCUDTTTUEUFUG $.
  $}

  ${
    $d x y z A $.  $d x y z B $.
    $( The ` 1st ` function maps the universe onto the universe.  (Contributed
       by NM, 14-Oct-2004.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    fo1st $p |- 1st : _V -onto-> _V $=
      ( vx vy cvv c1st wfo wfn crn wceq csn cdm cuni vex snex dmex uniex df-1st
      cv fnmpti wrex wcel cab rnmpt opex op1sta eqcomi sneq dmeqd unieqd eqeq2d
      cop rspcev mp2an 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAQZIZJZ
      KZDVAUTUSALMNOAPZRURBQZVBHZACSZBUACABCVBDVCUBVFBCVDCTVFBLZVDVDUJZCTVDVHIZ
      JZKZHZVFVDVDVGVGUCVKVDVDVDVGVGUDUEVEVLAVHCUSVHHZVBVKVDVMVAVJVMUTVIUSVHUFU
      GUHUIUKULUMUNUOCCDUPUQ $.

    $( The ` 2nd ` function maps the universe onto the universe.  (Contributed
       by NM, 14-Oct-2004.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    fo2nd $p |- 2nd : _V -onto-> _V $=
      ( vx vy cvv c2nd wfo wfn crn wceq cv csn cuni vex snex rnex df-2nd fnmpti
      uniex wrex cab wcel rnmpt cop opex op2nda eqcomi sneq rneqd unieqd eqeq2d
      rspcev mp2an 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAIZJZGZKZDU
      TUSURALMNQAOZPUQBIZVAHZACRZBSCABCVADVBUAVEBCVCCTVEBLZVCVCUBZCTVCVGJZGZKZH
      ZVEVCVCVFVFUCVJVCVCVCVFVFUDUEVDVKAVGCURVGHZVAVJVCVLUTVIVLUSVHURVGUFUGUHUI
      UJUKULUMUNCCDUOUP $.

    $( Mapping of a restriction of the ` 1st ` (first member of an ordered
       pair) function.  (Contributed by NM, 11-Oct-2004.)  (Revised by Mario
       Carneiro, 8-Sep-2013.) $)
    f1stres $p |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A $=
      ( vx vy vz cv csn cdm cuni wcel cxp wral c1st cres cop vex wceq cvv cmpt
      wf op1sta eleq1i adantr rgen2 sneq dmeqd unieqd eleq1d ralxp mpbir df-1st
      biimpri reseq1i wss ssv resmpt ax-mp eqtri fmpt mpbi ) CFZGZHZIZAJZCABKZL
      ZVFAMVFNZTVGDFZEFZOZGZHZIZAJZEBLDALVODEABVIAJZVOVJBJVOVPVNVIAVIVJDPEPUAUB
      ULUCUDVEVOCDEABVAVKQZVDVNAVQVCVMVQVBVLVAVKUEUFUGUHUIUJCVFAVDVHVHCRVDSZVFN
      ZCVFVDSZMVRVFCUKUMVFRUNVSVTQVFUOCRVFVDUPUQURUSUT $.

    $( Mapping of a restriction of the ` 2nd ` (second member of an ordered
       pair) function.  (Contributed by NM, 7-Aug-2006.)  (Revised by Mario
       Carneiro, 8-Sep-2013.) $)
    f2ndres $p |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B $=
      ( vx vy vz cv csn crn cuni wcel cxp wral c2nd cres cop vex wceq cvv cmpt
      wf op2nda eleq1i adantl rgen2 sneq rneqd unieqd eleq1d ralxp mpbir df-2nd
      biimpri reseq1i wss ssv resmpt ax-mp eqtri fmpt mpbi ) CFZGZHZIZBJZCABKZL
      ZVFBMVFNZTVGDFZEFZOZGZHZIZBJZEBLDALVODEABVJBJZVOVIAJVOVPVNVJBVIVJDPEPUAUB
      ULUCUDVEVOCDEABVAVKQZVDVNBVQVCVMVQVBVLVAVKUEUFUGUHUIUJCVFBVDVHVHCRVDSZVFN
      ZCVFVDSZMVRVFCUKUMVFRUNVSVTQVFUOCRVFVDUPUQURUSUT $.
  $}

  ${
    $d y B $.  $d u v y $.  $d u v A $.  $d u v B $.
    $( Onto mapping of a restriction of the ` 1st ` (first member of an ordered
       pair) function.  (Contributed by Jim Kingdon, 24-Jan-2019.) $)
    fo1stresm $p |- ( E. y y e. B ->
                 ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A ) $=
      ( vv vu cv wcel wex cxp c1st wfo wa wss cfv vex ax-mp sylbir jctil sylibr
      cres eleq1 cbvexv wf crn wceq wi cop opelxp fvres eqtr2di wfn f1stres ffn
      op1st fnfvelrn mpan eqeltrd expcom exlimiv ssrdv frn eqss dffo2 ) AFZCGZA
      HDFZCGZDHZBCIZBJVITZKZVGVEDAVFVDCUAUBVHVIBVJUCZVJUDZBUEZLVKVHVNVLVHVMBMZB
      VMMZLVNVHVPVOVHEBVMVGEFZBGZVQVMGZUFDVRVGVSVRVGLVQVFUGZVIGZVSVQVFBCUHWAVQV
      TVJNZVMWAWBVTJNVQVTVIJUIVQVFEODOUNUJVJVIUKZWAWBVMGVLWCBCULZVIBVJUMPVIVTVJ
      UOUPUQQURUSUTVLVOWDVIBVJVAPRVMBVBSWDRVIBVJVCSQ $.
  $}

  ${
    $d x A $.  $d u v x $.  $d u v A $.  $d u v B $.
    $( Onto mapping of a restriction of the ` 2nd ` (second member of an
       ordered pair) function.  (Contributed by Jim Kingdon, 24-Jan-2019.) $)
    fo2ndresm $p |- ( E. x x e. A ->
                 ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B ) $=
      ( vu vv cv wcel wex cxp c2nd wfo wa wss cfv vex ax-mp sylbir jctil sylibr
      cres eleq1 cbvexv wf crn wceq wi cop opelxp fvres eqtr2di wfn f2ndres ffn
      op2nd fnfvelrn mpan eqeltrd ex exlimiv ssrdv frn eqss dffo2 ) AFZBGZAHDFZ
      BGZDHZBCIZCJVITZKZVGVEDAVFVDBUAUBVHVICVJUCZVJUDZCUEZLVKVHVNVLVHVMCMZCVMMZ
      LVNVHVPVOVHECVMVGEFZCGZVQVMGZUFDVGVRVSVGVRLVFVQUGZVIGZVSVFVQBCUHWAVQVTVJN
      ZVMWAWBVTJNVQVTVIJUIVFVQDOEOUNUJVJVIUKZWAWBVMGVLWCBCULZVICVJUMPVIVTVJUOUP
      UQQURUSUTVLVOWDVICVJVAPRVMCVBSWDRVICVJVCSQ $.
  $}

  $( Composition of the first member function with another function.
     (Contributed by NM, 12-Oct-2007.) $)
  1stcof $p |- ( F : A --> ( B X. C ) -> ( 1st o. F ) : A --> B ) $=
    ( cxp wf c1st ccom wfn crn wss cvv wfo fo1st fofn ax-mp ffn dffn2 cres frn
    sylib fnfco sylancr rnco ssres2 rnss 3syl f1stres sstrdi eqsstrid sylanbrc
    df-f ) ABCEZDFZGDHZAIZUOJZBKABUOFUNGLIZALDFZUPLLGMURNLLGOPUNDAIUSAUMDQADRUA
    LAGDUBUCUNUQGDJZSZJZBGDUDUNVBGUMSZJZBUNUTUMKVAVCKVBVDKAUMDTUTUMGUEVAVCUFUGU
    MBVCFVDBKBCUHUMBVCTPUIUJABUOULUK $.

  $( Composition of the second member function with another function.
     (Contributed by FL, 15-Oct-2012.) $)
  2ndcof $p |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) : A --> C ) $=
    ( cxp wf c2nd ccom wfn crn wss cvv wfo fo2nd fofn ax-mp ffn dffn2 cres frn
    sylib fnfco sylancr rnco ssres2 rnss 3syl f2ndres sstrdi eqsstrid sylanbrc
    df-f ) ABCEZDFZGDHZAIZUOJZCKACUOFUNGLIZALDFZUPLLGMURNLLGOPUNDAIUSAUMDQADRUA
    LAGDUBUCUNUQGDJZSZJZCGDUDUNVBGUMSZJZCUNUTUMKVAVCKVBVDKAUMDTUTUMGUEVAVCUFUGU
    MCVCFVDCKBCUHUMCVCTPUIUJACUOULUK $.

  ${
    $d A b c $.  $d B b c $.  $d C b c $.
    $( Location of the first element of a Cartesian product.  (Contributed by
       Jeff Madsen, 2-Sep-2009.) $)
    xp1st $p |- ( A e. ( B X. C ) -> ( 1st ` A ) e. B ) $=
      ( vb vc cxp wcel cv cop wceq wex c1st cfv elxp vex op1std biimpar adantrr
      wa eleq1d exlimivv sylbi ) ABCFGADHZEHZIJZUCBGZUDCGZSSZEKDKALMZBGZDEABCNU
      HUJDEUEUFUJUGUEUJUFUEUIUCBUCUDADOEOPTQRUAUB $.
  $}

  ${
    $d A b c $.  $d B b c $.  $d C b c $.
    $( Location of the second element of a Cartesian product.  (Contributed by
       Jeff Madsen, 2-Sep-2009.) $)
    xp2nd $p |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C ) $=
      ( vb vc cxp wcel cv cop wceq wex c2nd cfv elxp vex op2ndd biimpar adantrl
      wa eleq1d exlimivv sylbi ) ABCFGADHZEHZIJZUCBGZUDCGZSSZEKDKALMZCGZDEABCNU
      HUJDEUEUGUJUFUEUJUGUEUIUDCUCUDADOEOPTQRUAUB $.
  $}

  $( Existence of the first member of a set.  (Contributed by Jim Kingdon,
     26-Jan-2019.) $)
  1stexg $p |- ( A e. V -> ( 1st ` A ) e. _V ) $=
    ( wcel cvv c1st cfv elex wfn wfo fo1st fofn ax-mp funfvex funfni mpan syl )
    ABCADCZAEFDCZABGEDHZQRDDEISJDDEKLRDAEAEMNOP $.

  $( Existence of the first member of a set.  (Contributed by Jim Kingdon,
     26-Jan-2019.) $)
  2ndexg $p |- ( A e. V -> ( 2nd ` A ) e. _V ) $=
    ( wcel cvv c2nd cfv elex wfn wfo fo2nd fofn ax-mp funfvex funfni mpan syl )
    ABCADCZAEFDCZABGEDHZQRDDEISJDDEKLRDAEAEMNOP $.

  $( Membership in a cross product.  This version requires no quantifiers or
     dummy variables.  See also ~ elxp4 .  (Contributed by NM, 9-Oct-2004.) $)
  elxp6 $p |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >.
               /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) $=
    ( cxp wcel cvv c1st cfv c2nd wceq wa elex opexg adantl wb eleq1 cuni eleq1d
    cop anbi12d adantr mpbird csn cdm crn elxp4 1stvalg 2ndvalg opeq12d bitr4id
    eqeq2d pm5.21nii ) ABCDZEZAFEZAAGHZAIHZSZJZUPBEZUQCEZKZKZAUMLVCUOURFEZVBVDU
    SUPUQBCMNUSUOVDOVBAURFPUAUBUOUNAAUCZUDQZVEUEQZSZJZVFBEZVGCEZKZKVCABCUFUOUSV
    IVBVLUOURVHAUOUPVFUQVGAUGZAUHZUIUKUOUTVJVAVKUOUPVFBVMRUOUQVGCVNRTTUJUL $.

  $( Membership in a cross product.  This version requires no quantifiers or
     dummy variables.  See also ~ elxp4 .  (Contributed by NM, 19-Aug-2006.) $)
  elxp7 $p |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V )
               /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) $=
    ( cxp wcel cvv c1st cfv c2nd wa elex adantr cop wceq 1stexg 2ndexg biantrud
    elxp6 jca bitr4id anbi1d pm5.21nii ) ABCDZEZAFEZAFFDZEZAGHZBEAIHZCEJZJZAUCK
    UGUEUJAUFKLUEUDAUHUIMNZUJJUKABCRUEUGULUJUEUGULUHFEZUIFEZJZJULAFFRUEUOULUEUM
    UNAFOAFPSQTUATUB $.

  ${
    $d F u v x y z $.  $d S u x y z $.  $d ph u x y z $.
    oprssdmm.m $e |- ( ( ph /\ u e. S ) -> E. v v e. u ) $.
    oprssdmm.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S ) $.
    oprssdmm.f $e |- ( ph -> Rel F ) $.
    $( Domain of closure of an operation.  (Contributed by Jim Kingdon,
       23-Oct-2023.) $)
    oprssdmm $p |- ( ph -> ( S X. S ) C_ dom F ) $=
      ( vz cv wcel wa cfv cop wceq wral wex adantr cxp c1st elxp6 biimpi adantl
      c2nd simpld simprd wrel eleq2 exbidv ralrimiva co df-ov eqeltrrid rspcdva
      relelfvdm ex exlimdv sylc ralrimivva opeq1 eleq1d rspc2va syl2anc eqeltrd
      cdm opeq2 ssrdv ) AKFFUAZGVGZAKLZVJMZVLVKMAVMNZVLVLUBOZVLUFOZPZVKVNVLVQQZ
      VOFMVPFMNZVMVRVSNZAVMVTVLFFUCUDUEZUGVNVSBLZCLZPZVKMZCFRBFRZVQVKMZVNVRVSWA
      UHAWFVMAWEBCFFAWBFMWCFMNZNZGUIZDLZWDGOZMZDSZWEAWJWHJTWIWKELZMZDSZWNEFWLWO
      WLQWPWMDWOWLWKUJUKAWQEFRWHAWQEFHULTWIWLWBWCGUMFWBWCGUNIUOUPWJWMWEDWJWMWEW
      KWDGUQURUSUTVATWEWGVOWCPZVKMBCVOVPFFWBVOQWDWRVKWBVOWCVBVCWCVPQWRVQVKWCVPV
      OVHVCVDVEVFURVI $.
  $}

  $( Equality with an ordered pair.  (Contributed by NM, 15-Dec-2008.)
     (Revised by Mario Carneiro, 23-Feb-2014.) $)
  eqopi $p |- ( ( A e. ( V X. W ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) )
              -> A = <. B , C >. ) $=
    ( cxp wcel cvv c1st cfv wceq c2nd wa cop xpss sseli simplbi opeq12 sylan9eq
    elxp6 sylan ) ADEFZGAHHFZGZAIJZBKALJZCKMZABCNZKUBUCADEOPUDUGAUEUFNZUHUDAUIK
    UEHGUFHGMAHHTQUEUFBCRSUA $.

  ${
    $d x A $.  $d x B $.
    $( Representation of cross product based on ordered pair component
       functions.  (Contributed by NM, 16-Sep-2006.) $)
    xp2 $p |- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\
              ( 2nd ` x ) e. B ) } $=
      ( cxp cv cvv wcel c1st cfv c2nd wa cab crab elxp7 abbi2i df-rab eqtr4i )
      BCDZAEZFFDZGSHIBGSJICGKZKZALUAATMUBARSBCNOUAATPQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( The membership relation for a cross product is inherited by union.
       (Contributed by NM, 16-Sep-2006.) $)
    unielxp $p |- ( A e. ( B X. C ) -> U. A e. U. ( B X. C ) ) $=
      ( vx cxp wcel cvv c1st cfv c2nd wa cuni elxp7 elvvuni adantr cv cab fveq2
      eleq1d anbi12d wex simprl wceq eleq2 eleq1 spcegv eluniab sylibr crab xp2
      mpcom df-rab eqtri unieqi eleqtrrdi mpancom sylbi ) ABCEZFAGGEZFZAHIZBFZA
      JIZCFZKZKZALZURLZFZABCMVGAFZVFVIUTVJVEANOVJVFKZVGDPZUSFZVLHIZBFZVLJIZCFZK
      ZKZDQZLZVHVKVGVLFZVSKZDUAZVGWAFUTVKWDVJUTVEUBWCVKDAUSVLAUCZWBVJVSVFVLAVGU
      DWEVMUTVRVEVLAUSUEWEVOVBVQVDWEVNVABVLAHRSWEVPVCCVLAJRSTTTUFUKVSDVGUGUHURV
      TURVRDUSUIVTDBCUJVRDUSULUMUNUOUPUQ $.
  $}

  $( Reconstruction of a member of a cross product in terms of its ordered pair
     components.  (Contributed by NM, 20-Oct-2013.) $)
  1st2nd2 $p |- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) $=
    ( cxp wcel c1st cfv c2nd cop wceq wa elxp6 simplbi ) ABCDEAAFGZAHGZIJNBEOCE
    KABCLM $.

  $( An ordered pair theorem for members of cross products.  (Contributed by
     NM, 20-Jun-2007.) $)
  xpopth $p |- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) ->
       ( ( ( 1st ` A ) = ( 1st ` B ) /\
     ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) ) $=
    ( cxp wcel wa wceq c1st cfv c2nd cop 1st2nd2 eqeqan12d wb cvv 1stexg 2ndexg
    opthg syl2anc adantr bitr2d ) ACDGZHZBEFGHZIABJAKLZAMLZNZBKLZBMLZNZJZUHUKJU
    IULJIZUFUGAUJBUMACDOBEFOPUFUNUOQZUGUFUHRHUIRHUPAUESAUETUHUIUKULRRUAUBUCUD
    $.

  $( Two ways to express equality with an ordered pair.  (Contributed by NM,
     3-Sep-2007.)  (Proof shortened by Mario Carneiro, 26-Apr-2015.) $)
  eqop $p |- ( A e. ( V X. W ) -> ( A = <. B , C >.
            <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) ) $=
    ( cxp wcel cop wceq c1st cfv c2nd wa 1st2nd2 eqeq1d cvv 1stexg 2ndexg opthg
    wb syl2anc bitrd ) ADEFZGZABCHZIAJKZALKZHZUEIZUFBIUGCIMZUDAUHUEADENOUDUFPGU
    GPGUIUJTAUCQAUCRUFUGBCPPSUAUB $.

  ${
    eqop2.1 $e |- B e. _V $.
    eqop2.2 $e |- C e. _V $.
    $( Two ways to express equality with an ordered pair.  (Contributed by NM,
       25-Feb-2014.) $)
    eqop2 $p |- ( A = <. B , C >.
       <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) ) $=
      ( cop wceq cvv cxp wcel c1st cfv c2nd wa opelvv eleq1 mpbiri eqop biadan2
      ) ABCFZGZAHHIZJZAKLBGAMLCGNUAUCTUBJBCDEOATUBPQABCHHRS $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Two ways of expressing that an element is the first member of an ordered
       pair.  (Contributed by NM, 22-Sep-2013.)  (Revised by Mario Carneiro,
       23-Feb-2014.) $)
    op1steq $p |- ( A e. ( V X. W )
         -> ( ( 1st ` A ) = B <-> E. x A = <. B , x >. ) ) $=
      ( cxp wcel cvv c1st cfv wceq cv cop wex wb xpss sseli wa c2nd syl eqid wi
      eqopi mpanr2 2ndexg opeq2 eqeq2d spcegv adantr mpd ex eqop simpl biimtrdi
      exlimdv impbid ) BDEFZGBHHFZGZBIJCKZBCALZMZKZANZOUQURBDEPQUSUTVDUSUTVDUSU
      TRBCBSJZMZKZVDUSUTVEVEKVGVEUABCVEHHUCUDUSVGVDUBZUTUSVEHGVHBURUEVCVGAVEHVA
      VEKVBVFBVAVECUFUGUHTUIUJUKUSVCUTAUSVCUTVEVAKZRUTBCVAHHULUTVIUMUNUOUPT $.
  $}

  $( Swap the members of an ordered pair.  (Contributed by NM, 31-Dec-2014.) $)
  2nd1st $p |- ( A e. ( B X. C ) ->
    U. `' { A } = <. ( 2nd ` A ) , ( 1st ` A ) >. ) $=
    ( cxp wcel csn ccnv cuni c1st cfv c2nd cop 1st2nd2 sneqd cnveqd unieqd wceq
    cvv 1stexg 2ndexg opswapg syl2anc eqtrd ) ABCDZEZAFZGZHAIJZAKJZLZFZGZHZUIUH
    LZUEUGULUEUFUKUEAUJABCMNOPUEUHREUIREUMUNQAUDSAUDTUHUIRRUAUBUC $.

  $( Reconstruction of a member of a relation in terms of its ordered pair
     components.  (Contributed by NM, 29-Aug-2006.) $)
  1st2nd $p |- ( ( Rel B /\ A e. B ) ->
               A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) $=
    ( wrel wcel wa cvv cxp c1st cfv c2nd cop wceq wss df-rel sylanb 1st2nd2 syl
    ssel2 ) BCZABDZEAFFGZDZAAHIAJIKLSBUAMTUBBNBUAAROAFFPQ $.

  $( The first ordered pair component of a member of a relation belongs to the
     domain of the relation.  (Contributed by NM, 17-Sep-2006.) $)
  1stdm $p |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R ) $=
    ( wrel wcel c1st cfv cint cdm cvv cxp wceq wss df-rel biimpi sselda 1stval2
    wa syl elreldm eqeltrd ) BCZABDQZAEFZAGGZBHUBAIIJZDUCUDKUABUEAUABUELBMNOAPR
    BAST $.

  $( The second ordered pair component of a member of a relation belongs to the
     range of the relation.  (Contributed by NM, 17-Sep-2006.) $)
  2ndrn $p |- ( ( Rel R /\ A e. R ) -> ( 2nd ` A ) e. ran R ) $=
    ( wrel wcel wa c1st cfv cop crn simpr 1st2nd eqeltrrd cvv 1stexg 2ndexg jca
    c2nd opelrng 3expa sylan syl2anc ) BCZABDZEZUCAFGZAQGZHZBDZUFBIDZUBUCJZUDAU
    GBABKUJLUCUEMDZUFMDZEUHUIUCUKULABNABOPUKULUHUIUEUFBMMRSTUA $.

  $( Express an element of a relation as a relationship between first and
     second components.  (Contributed by Mario Carneiro, 22-Jun-2016.) $)
  1st2ndbr $p |- ( ( Rel B /\ A e. B ) -> ( 1st ` A ) B ( 2nd ` A ) ) $=
    ( wrel wcel wa c1st cfv c2nd cop wbr 1st2nd simpr eqeltrrd df-br sylibr ) B
    CZABDZEZAFGZAHGZIZBDSTBJRAUABABKPQLMSTBNO $.

  ${
    $d x y A $.  $d x y B $.
    $( Two ways of expressing membership in the domain of a relation.
       (Contributed by NM, 22-Sep-2013.) $)
    releldm2 $p |- ( Rel A
         -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) ) $=
      ( vy wrel cdm wcel cv c1st cfv wceq wrex cvv wa elex anim2i id vex wex wb
      1stexg ax-mp eqeltrrdi rexlimivw cop eldm2g adantl cxp wss wi df-rel ssel
      sylbi imp op1steq syl rexbidva adantr rexcom4 risset bitr4i bitrdi bitr4d
      exbii pm5.21nd ) BEZCBFZGZAHZIJZCKZABLZVFCMGZNZVHVMVFCVGOPVLVMVFVKVMABVKC
      VJMVKQVIMGVJMGARVIMUAUBUCUDPVNVHCDHUEZBGZDSZVLVMVHVQTVFDCBMUFUGVNVLVIVOKZ
      DSZABLZVQVFVLVTTVMVFVKVSABVFVIBGZNVIMMUHZGZVKVSTVFWAWCVFBWBUIWAWCUJBUKBWB
      VIULUMUNDVICMMUOUPUQURVTVRABLZDSVQVRADBUSVPWDDAVOBUTVDVAVBVCVE $.
  $}

  ${
    $d x y z A $.
    $( An expression for the domain of a relation.  (Contributed by NM,
       22-Sep-2013.) $)
    reldm $p |- ( Rel A -> dom A = ran ( x e. A |-> ( 1st ` x ) ) ) $=
      ( vy vz wrel cdm c1st cfv cmpt crn wcel wceq wrex releldm2 cvv vex 1stexg
      cv wb ax-mp wfn eqid fnmpti fvelrnb fveq2 fvmpt rexbiia a1i bitr2id bitrd
      eqeq1d eqrdv ) BEZCBFZABARZGHZIZJZUMCRZUNKDRZGHZUSLZDBMZUSURKZDBUSNVDUTUQ
      HZUSLZDBMZUMVCUQBUAVDVGSABUPUQUOOKUPOKAPUOOQTUQUBZUCDBUSUQUDTVGVCSUMVFVBD
      BUTBKVEVAUSAUTUPVABUQUOUTGUEVHUTOKVAOKDPUTOQTUFUKUGUHUIUJUL $.
  $}

  $( Equality theorem for substitution of a class for an ordered pair (analog
     of ~ sbceq1a that avoids the existential quantifiers of ~ copsexg ).
     (Contributed by NM, 19-Aug-2006.)  (Revised by Mario Carneiro,
     31-Aug-2015.) $)
  sbcopeq1a $p |- ( A = <. x , y >. ->
      ( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) ) $=
    ( cv cop wceq c2nd cfv wsbc c1st wb vex op2ndd eqcomd sbceq1a op1std bitr2d
    syl ) DBEZCEZFGZAACDHIZJZUDBDKIZJZUBUAUCGAUDLUBUCUATUADBMZCMZNOACUCPSUBTUEG
    UDUFLUBUETTUADUGUHQOUDBUEPSR $.

  $( Equality theorem for substitution of a class ` A ` for an ordered pair
     ` <. x , y >. ` in ` B ` (analog of ~ csbeq1a ).  (Contributed by NM,
     19-Aug-2006.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
  csbopeq1a $p |- ( A = <. x , y >. ->
      [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B = B ) $=
    ( cv cop wceq c2nd cfv csb c1st vex op2ndd eqcomd csbeq1a syl op1std eqtr2d
    ) CAEZBEZFGZDBCHIZDJZACKIZUCJZUATUBGDUCGUAUBTSTCALZBLZMNBUBDOPUASUDGUCUEGUA
    UDSSTCUFUGQNAUDUCOPR $.

  ${
    $d z ph $.  $d x y z $.
    $( A way to define an ordered-pair class abstraction without using
       existential quantifiers.  (Contributed by NM, 18-Aug-2006.)  (Revised by
       Mario Carneiro, 31-Aug-2015.) $)
    dfopab2 $p |- { <. x , y >. | ph } = { z e. ( _V X. _V ) |
                  [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph } $=
      ( cv cop wceq wex cab cvv cxp wcel c2nd cfv wsbc c1st nfsbc1v 19.41 exbii
      wa copab crab sbcopeq1a pm5.32i nfsbc bitr3i anbi1i 3bitr4i abbii df-opab
      nfcv elvv df-rab 3eqtr4i ) DEZBECEFGZATZCHZBHZDIUOJJKZLZACUOMNZOZBUOPNZOZ
      TZDIABCUAVEDUTUBUSVFDUPCHZVETZBHVGBHZVETUSVFVGVEBVCBVDQRURVHBURUPVETZCHVH
      VJUQCUPVEAABCUOUCUDSUPVECVCCBVDCVDUKACVBQUERUFSVAVIVEBCUOULUGUHUIABCDUJVE
      DUTUMUN $.
  $}

  ${
    $d w ph $.  $d x y z w $.
    $( A way to define an operation class abstraction without using existential
       quantifiers.  (Contributed by NM, 18-Aug-2006.)  (Revised by Mario
       Carneiro, 31-Aug-2015.) $)
    dfoprab3s $p |- { <. <. x , y >. , z >. | ph } = { <. w , z >. |
       ( w e. ( _V X. _V )
           /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) } $=
      ( coprab cv cop wceq wex copab cvv cxp wcel cfv wsbc nfsbc1v 19.41 exbii
      wa c2nd c1st dfoprab2 sbcopeq1a pm5.32i nfcv nfsbc bitr3i 3bitr4i opabbii
      elvv anbi1i eqtri ) ABCDFEGZBGCGHIZATZCJZBJZEDKUNLLMNZACUNUAOZPZBUNUBOZPZ
      TZEDKABCDEUCURVDEDUOCJZVCTZBJVEBJZVCTURVDVEVCBVABVBQRUQVFBUQUOVCTZCJVFVHU
      PCUOVCAABCUNUDUESUOVCCVACBVBCVBUFACUTQUGRUHSUSVGVCBCUNUKULUIUJUM $.
  $}

  ${
    $d x y ph $.  $d w ps $.  $d x y z w $.
    dfoprab3.1 $e |- ( w = <. x , y >. -> ( ph <-> ps ) ) $.
    $( Operation class abstraction expressed without existential quantifiers.
       (Contributed by NM, 16-Dec-2008.) $)
    dfoprab3 $p |- { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) } =
       { <. <. x , y >. , z >. | ps } $=
      ( coprab cv cvv cxp wcel cfv wsbc wa copab ax-mp wceq wb eqcom vex 1stexg
      c2nd c1st dfoprab3s 2ndexg cop anbi12i eqopi sylan2b syl sbc2iedv pm5.32i
      bicomd ex opabbii eqtr2i ) BCDEHFIZJJKLZBDURUCMZNCURUDMZNZOZFEPUSAOZFEPBC
      DEFUEVCVDFEUSVBAUSBACDVAUTURJLZVAJLFUAZURJUBQVEUTJLVFURJUFQUSCIZVARZDIZUT
      RZOZBASUSVKOZABVLURVGVIUGRZABSVKUSVAVGRZUTVIRZOVMVHVNVJVOVGVATVIUTTUHURVG
      VIJJUIUJGUKUNUOULUMUPUQ $.
  $}

  ${
    $d w x y A $.  $d w x y B $.  $d x y ph $.  $d w ps $.  $d w x y z $.
    dfoprab4.1 $e |- ( w = <. x , y >. -> ( ph <-> ps ) ) $.
    $( Operation class abstraction expressed without existential quantifiers.
       (Contributed by NM, 3-Sep-2007.)  (Revised by Mario Carneiro,
       31-Aug-2015.) $)
    dfoprab4 $p |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } =
      { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } $=
      ( cv cxp wcel wa copab cvv coprab xpss sseli adantr pm4.71ri opabbii wceq
      cop eleq1 opelxp bitrdi anbi12d dfoprab3 eqtri ) FJZGHKZLZAMZFENUJOOKZLZU
      MMZFENCJZGLDJZHLMZBMZCDEPUMUPFEUMUOULUOAUKUNUJGHQRSTUAUMUTCDEFUJUQURUCZUB
      ZULUSABVBULVAUKLUSUJVAUKUDUQURGHUEUFIUGUHUI $.
  $}

  ${
    $d t u w x y z $.  $d t u w x y A $.  $d t u w x y B $.  $d t u w ps $.
    $d t u ph $.
    dfoprab4f.x $e |- F/ x ph $.
    dfoprab4f.y $e |- F/ y ph $.
    dfoprab4f.1 $e |- ( w = <. x , y >. -> ( ph <-> ps ) ) $.
    $( Operation class abstraction expressed without existential quantifiers.
       (Unnecessary distinct variable restrictions were removed by David
       Abernethy, 19-Jun-2012.)  (Contributed by NM, 20-Dec-2008.)  (Revised by
       Mario Carneiro, 31-Aug-2015.) $)
    dfoprab4f $p |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } =
      { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } $=
      ( vt vu cv wcel wa cop wceq wb nfv cxp copab coprab nfs1v nfbi nfim opeq1
      wsb wi eqeq2d sbequ12 bibi2d imbi12d opeq2 chvar dfoprab4 nfan nfsb eleq1
      bi2anan9 sylan9bbr anbi12d cbvoprab12 eqtr4i ) FNZGHUAOAPFEUBLNZGOZMNZHOZ
      PZBDMUHZCLUHZPZLMEUCCNZGOZDNZHOZPZBPZCDEUCAVLLMEFGHVEVNVHQZRZAVKSZUIZVEVF
      VHQZRZAVLSZUICLWEWFCWECTAVLCIVKCLUDZUEUFVNVFRZWAWEWBWFWHVTWDVEVNVFVHUGUJW
      HVKVLAVKCLUKZULUMVEVNVPQZRZABSZUIWCDMWAWBDWADTAVKDJBDMUDZUEUFVPVHRZWKWAWL
      WBWNWJVTVEVPVHVNUNUJWNBVKABDMUKZULUMKUOUOUPVSVMCDELMVSLTVSMTVJVLCVJCTWGUQ
      VJVLDVJDTVKCLDWMURUQWHWNPVRVJBVLWHVOVGWNVQVIVNVFGUSVPVHHUSUTWNBVKWHVLWOWI
      VAVBVCVD $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ph $.
    opabex2.1 $e |- ( ph -> A e. V ) $.
    opabex2.2 $e |- ( ph -> B e. W ) $.
    opabex2.3 $e |- ( ( ph /\ ps ) -> x e. A ) $.
    opabex2.4 $e |- ( ( ph /\ ps ) -> y e. B ) $.
    $( Condition for an operation to be a set.  (Contributed by Thierry Arnoux,
       25-Jun-2019.) $)
    opabex2 $p |- ( ph -> { <. x , y >. | ps } e. _V ) $=
      ( copab cxp cvv xpexd opabssxpd ssexd ) ABCDMEFNOAEFGHIJPABCDEFKLQR $.
  $}

  ${
    $d x y $.  $d ph x $.
    $( An ordered-pair class abstraction which does not depend on the first
       abstraction variable is a proper class.  There must be, however, at
       least one set which satisfies the restricting wff.  (Contributed by AV,
       27-Dec-2020.) $)
    opabn1stprc $p |- ( E. y ph -> { <. x , y >. | ph } e/ _V ) $=
      ( wex copab cvv wcel wn wnel cdm cv wa vex biantrur opabbii dmeqi wral id
      wceq ralrimivw dmopab3 sylib eqtrid vprc a1i eqneltrd dmexg df-nel sylibr
      nsyl ) ACDZABCEZFGZHULFIUKULJZFGUMUKUNFFUKUNBKFGZALZBCEZJZFULUQAUPBCUOABM
      NOPUKUKBFQURFSUKUKBFUKRTABCFUAUBUCFFGHUKUDUEUFULFUGUJULFUHUI $.
  $}

  ${
    $d x y z u A $.  $d x y z u B $.  $d x y z u C $.
    $( Define the cross product of three classes.  Compare ~ df-xp .
       (Contributed by FL, 6-Nov-2013.)  (Proof shortened by Mario Carneiro,
       3-Nov-2015.) $)
    dfxp3 $p |- ( ( A X. B ) X. C ) =
      { <. <. x , y >. , z >. | ( x e. A /\ y e. B /\ z e. C ) } $=
      ( vu cv cxp wcel wa copab coprab w3a cop wceq biidd dfoprab4 df-xp df-3an
      oprabbii 3eqtr4i ) GHZDEIZJCHFJZKGCLAHZDJZBHZEJZKUEKZABCMUDFIUGUIUENZABCM
      UEUEABCGDEUCUFUHOPUEQRGCUDFSUKUJABCUGUIUETUAUB $.
  $}

  ${
    $d x y A $.  $d x y ch $.
    elopabi.1 $e |- ( x = ( 1st ` A ) -> ( ph <-> ps ) ) $.
    elopabi.2 $e |- ( y = ( 2nd ` A ) -> ( ps <-> ch ) ) $.
    $( A consequence of membership in an ordered-pair class abstraction, using
       ordered pair extractors.  (Contributed by NM, 29-Aug-2006.) $)
    elopabi $p |- ( A e. { <. x , y >. | ph } -> ch ) $=
      ( copab wcel c1st cfv c2nd cop wrel wceq relopab 1st2nd mpan cvv eqeltrrd
      id wb 1stexg 2ndexg opelopabg syl2anc mpbid ) FADEIZJZFKLZFMLZNZUIJZCUJFU
      MUIUIOUJFUMPADEQFUIRSUJUBUAUJUKTJULTJUNCUCFUIUDFUIUEABCDEUKULTTGHUFUGUH
      $.
  $}

  ${
    $d w x y z A $.  $d w ph $.  $d x y z th $.
    eloprabi.1 $e |- ( x = ( 1st ` ( 1st ` A ) ) -> ( ph <-> ps ) ) $.
    eloprabi.2 $e |- ( y = ( 2nd ` ( 1st ` A ) ) -> ( ps <-> ch ) ) $.
    eloprabi.3 $e |- ( z = ( 2nd ` A ) -> ( ch <-> th ) ) $.
    $( A consequence of membership in an operation class abstraction, using
       ordered pair extractors.  (Contributed by NM, 6-Nov-2006.)  (Revised by
       David Abernethy, 19-Jun-2012.) $)
    eloprabi $p |- ( A e. { <. <. x , y >. , z >. | ph } -> th ) $=
      ( vw cv wceq wex c1st cfv wb syl c2nd coprab wcel wa eqeq1 anbi1d 3exbidv
      cop df-oprab elab2g ibi vex opex op1std fveq2d op1st eqtr2di op2nd op2ndd
      eqcomd 3bitrd biimpa exlimiv ) HAEFGUAZUBZHEMZFMZUGZGMZUGZNZAUCZGOZFOZEOZ
      DVDVNLMZVINZAUCZGOFOEOVNLHVCVCVOHNZVQVKEFGVRVPVJAVOHVIUDUEUFAEFGLUHUIUJVM
      DEVLDFVKDGVJADVJABCDVJVEHPQZPQZNABRVJVTVGPQVEVJVSVGPVGVHHVEVFEUKZFUKZULZG
      UKZUMZUNVEVFWAWBUOUPISVJVFVSTQZNBCRVJWFVGTQVFVJVSVGTWEUNVEVFWAWBUQUPJSVJV
      HHTQZNCDRVJWGVHVGVHHWCWDURUSKSUTVAVBVBVBS $.
  $}

  ${
    $d u v x y z A $.  $d u v y z B $.  $d u v z C $.
    $( Express a two-argument function as a one-argument function, or
       vice-versa.  (Contributed by Mario Carneiro, 24-Dec-2016.) $)
    mpomptsx $p |- ( x e. A , y e. B |-> C ) = ( z e. U_ x e. A ( { x } X. B )
        |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) $=
      ( vu vv cv csn csb cxp ciun cfv cmpt cmpo wceq nfcv nfcsb1v csbeq1a eqtrd
      c1st c2nd cop vex op1std csbeq1d op2ndd csbeq2dv mpomptx nfxp sneq cbviun
      xpeq12d mpteq1 ax-mp nfcsb sylan9eqr cbvmpox 3eqtr4ri ) CGDGIZJZAVAEKZLZM
      ZACIZUBNZBVFUCNZFKZKZOZGHDVCAVABHIZFKZKZPCADAIZJZELZMZVJOZABDEFPGHCDVCVJV
      NVFVAVLUDQZVJAVAVIKVNVTAVGVAVIVAVLVFGUEZHUEZUFUGVTAVAVIVMVTBVHVLFVAVLVFWA
      WBUHUGUIUAUJVRVEQVSVKQAGDVQVDGVQRAVBVCAVBRAVAESZUKVOVAQZVPVBEVCVOVAULAVAE
      TZUNUMCVRVEVJUOUPABGHDEFVCVNGERWCGFRHFRAVAVMSBAVAVMBVARBVLFSUQWEBIVLQWDFV
      MVNBVLFTAVAVMTURUSUT $.

    $d x B $.
    $( Express a two-argument function as a one-argument function, or
       vice-versa.  (Contributed by Mario Carneiro, 24-Sep-2015.) $)
    mpompts $p |- ( x e. A , y e. B |-> C ) =
      ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) $=
      ( cmpo cv csn cxp ciun c1st cfv c2nd cmpt mpomptsx wceq iunxpconst mpteq1
      csb ax-mp eqtri ) ABDEFGCADAHIEJKZACHZLMBUDNMFTTZOZCDEJZUEOZABCDEFPUCUGQU
      FUHQADERCUCUGUESUAUB $.
  $}

  ${
    $d t u v w x y z A $.  $d t u v w y z B $.  $d t u v w z C $.
    $d v w x y z D $.
    fmpox.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( The domain of a mapping is a subset of its base class.  (Contributed by
       Mario Carneiro, 9-Feb-2015.) $)
    dmmpossx $p |- dom F C_ U_ x e. A ( { x } X. B ) $=
      ( vu vt vv cv csn csb cxp ciun cfv nfcv nfcsb1v csbeq1a wceq c1st cbvmpox
      cdm c2nd cmpo cmpt nfcsb sylan9eqr cop vex op1std csbeq1d op2ndd csbeq2dv
      eqtrd mpomptx 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtrri ) FUCHCHK
      ZLZAVDDMZNZOZACAKZLZDNZOIVHAIKZUAPZBVLUDPZEMZMZFABCDEUEHJCVFAVDBJKZEMZMZU
      EFIVHVPUFABHJCDEVFVSHDQAVDDRZHEQJEQAVDVRRBAVDVRBVDQBVQERUGAVDDSZBKVQTVIVD
      TZEVRVSBVQESAVDVRSUHUBGHJICVFVPVSVLVDVQUITZVPAVDVOMVSWCAVMVDVOVDVQVLHUJZJ
      UJZUKULWCAVDVOVRWCBVNVQEVDVQVLWDWEUMULUNUOUPUQURAHCVKVGHVKQAVEVFAVEQVTUSW
      BVJVEDVFVIVDUTWAVAVBVC $.

    $( Functionality, domain and codomain of a class given by the maps-to
       notation, where ` B ( x ) ` is not constant but depends on ` x ` .
       (Contributed by NM, 29-Dec-2014.) $)
    fmpox $p |- ( A. x e. A A. y e. B C e. D <->
      F : U_ x e. A ( { x } X. B ) --> D ) $=
      ( vz vw vv cv csb wcel wral wceq eleq1d wa nfv nfcsb1v csn ciun c1st c2nd
      cxp cfv cop vex op1std csbeq1d op2ndd csbeq2dv eqtrd raliunxp cmpo coprab
      wf cmpt nfcri nfan nfeq2 nfcv nfcsb eleq1 adantr csbeq1a eleq2d sylan9bbr
      wb anbi12d sylan9eqr eqeq2d cbvoprab12 df-mpo 3eqtr4i mpomptx fmpt bitr3i
      nfel1 nfralxy cbvral raleqbidv bitrid nfxp xpeq12d cbviun feq2i 3bitr4i
      sneq ) AILZBJLZEMZMZFNZJAWJDMZOZICOZICWJUAZWOUEZUBZFGUQZEFNZBDOZACOACALZU
      AZDUEZUBZFGUQWQAKLZUCUFZBXHUDUFZEMZMZFNZKWTOXAXMWNKIJCWOXHWJWKUGPZXLWMFXN
      XLAWJXKMWMXNAXIWJXKWJWKXHIUHZJUHZUIUJXNAWJXKWLXNBXJWKEWJWKXHXOXPUKUJULUMZ
      QUNKWTFXLGABCDEUOZIJCWOWMUOZGKWTXLURXDCNZBLZDNZRZXHEPZRZABKUPWJCNZWKWONZR
      ZXHWMPZRZIJKUPXRXSYEYJABKIJYEISYEJSYHYIAYFYGAYFASAJWOAWJDTZUSUTAXHWMAWJWL
      TZVAUTYHYIBYHBSBXHWMBAWJWLBWJVBBWKETZVCVAUTXDWJPZYAWKPZRZYCYHYDYIYPXTYFYB
      YGYNXTYFVIYOXDWJCVDVEYOYBWKDNYNYGYAWKDVDYNDWOWKAWJDVFZVGVHVJYPEWMXHYOYNEW
      LWMBWKEVFZAWJWLVFZVKVLVJVMABKCDEVNIJKCWOWMVNVOHIJKCWOXLWMXQVPVOVQVRXCWPAI
      CXCISWNAJWOYKAWMFYLVSVTXCWLFNZJDOYNWPXBYTBJDXBJSBWLFYMVSYOEWLFYRQWAYNYTWN
      JDWOYQYNWLWMFYSQWBWCWAXGWTFGAICXFWSIXFVBAWRWOAWRVBYKWDYNXEWRDWOXDWJWIYQWE
      WFWGWH $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C z $.  $d D x y z $.
    fmpo.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( Functionality, domain and range of a class given by the maps-to
       notation.  (Contributed by FL, 17-May-2010.) $)
    fmpo $p |- ( A. x e. A A. y e. B C e. D <-> F : ( A X. B ) --> D ) $=
      ( wcel wral cv csn cxp ciun wf fmpox iunxpconst feq2i bitri ) EFIBDJACJAC
      AKLDMNZFGOCDMZFGOABCDEFGHPTUAFGACDQRS $.

    $( Functionality and domain of a class given by the maps-to notation.
       (Contributed by FL, 17-May-2010.) $)
    fnmpo $p |- ( A. x e. A A. y e. B C e. V -> F Fn ( A X. B ) ) $=
      ( wcel wral cvv cxp wfn elex ralimi wf fmpo dffn2 bitr4i sylib ) EGIZBDJZ
      ACJEKIZBDJZACJZFCDLZMZUBUDACUAUCBDEGNOOUEUFKFPUGABCDEKFHQUFFRST $.

    fnmpoi.2 $e |- C e. _V $.
    $( Functionality and domain of a class given by the maps-to notation.
       (Contributed by FL, 17-May-2010.) $)
    fnmpoi $p |- F Fn ( A X. B ) $=
      ( cvv wcel wral cxp wfn rgen2w fnmpo ax-mp ) EIJZBDKACKFCDLMQABCDHNABCDEF
      IGOP $.

    $( Domain of a class given by the maps-to notation.  (Contributed by FL,
       17-May-2010.) $)
    dmmpo $p |- dom F = ( A X. B ) $=
      ( cxp wfn cdm wceq fnmpoi fndm ax-mp ) FCDIZJFKPLABCDEFGHMPFNO $.
  $}

  ${
    $d A x y z $.  $d B y z $.  $d C z $.  $d D x y z $.  $d R z $.  $d S z $.
    $d V z $.
    mpofvex.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( Sufficient condition for an operation maps-to notation to be set-like.
       (Contributed by Mario Carneiro, 3-Jul-2019.) $)
    mpofvex $p |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) ->
      ( R F S ) e. _V ) $=
      ( vz wcel wal cfv cvv cv csb alimi w3a co cop df-ov c1st c2nd elex wi vex
      2ndexg nfcv nfcsb1v nfel1 wceq csbeq1a eleq1d spcgf mp2b alrimiv 3ad2ant1
      syl 1stexg opexg 3adant1 csn cxp ciun cmpo mpomptsx eqtri mptfvex syl2anc
      cmpt eqeltrid ) EINZBOZAOZFJNZGKNZUAZFGHUBFGUCZHPZQFGHUDVTAMRZUEPZBWCUFPZ
      ESZSZQNZMOZWAQNZWBQNVQVRWIVSVQWHMVQWFQNZAOZWHVPWKAVPEQNZBOZWKVOWMBEIUGTWC
      QNZWEQNWNWKUHMUIZWCQUJWMWKBWEQBWEUKBWFQBWEEULUMBRWEUNEWFQBWEEUOUPUQURVATW
      OWDQNWLWHUHWPWCQVBWKWHAWDQAWDUKAWGQAWDWFULUMARZWDUNWFWGQAWDWFUOUPUQURVAUS
      UTVRVSWJVQFGJKVCVDMACWQVEDVFVGZWGWAHQQHABCDEVHMWRWGVMLABMCDEVIVJVKVLVN $.

    mpofvexi.c $e |- C e. _V $.
    mpofvexi.3 $e |- R e. _V $.
    mpofvexi.4 $e |- S e. _V $.
    $( Sufficient condition for an operation maps-to notation to be set-like.
       (Contributed by Mario Carneiro, 3-Jul-2019.) $)
    mpofvexi $p |- ( R F S ) e. _V $=
      ( cvv wcel wal co gen2 mpofvex mp3an ) EMNZBOAOFMNGMNFGHPMNTABJQKLABCDEFG
      HMMMIRS $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d M x y $.
    ovmpoelrn.o $e |- O = ( x e. A , y e. B |-> C ) $.
    $( An operation's value belongs to its range.  (Contributed by AV,
       27-Jan-2020.) $)
    ovmpoelrn $p |- ( ( A. x e. A A. y e. B C e. M /\ X e. A /\ Y e. B )
                       -> ( X O Y ) e. M ) $=
      ( wcel wral cxp wf co fmpo fovcdm syl3an1b ) EFKBDLACLCDMFGNHCKIDKHIGOFKA
      BCDEFGJPHIFCDGQR $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d V x y $.
    dmmpog.f $e |- F = ( x e. A , y e. B |-> C ) $.
    $( Domain of an operation given by the maps-to notation, closed form of
       ~ dmmpo .  (Contributed by Alexander van der Vekens, 10-Feb-2019.) $)
    dmmpoga $p |- ( A. x e. A A. y e. B C e. V -> dom F = ( A X. B ) ) $=
      ( wcel wral cxp wfn cdm wceq fnmpo fndm syl ) EGIBDJACJFCDKZLFMRNABCDEFGH
      ORFPQ $.

    $d C x y $.
    $( Domain of an operation given by the maps-to notation, closed form of
       ~ dmmpo .  Caution:  This theorem is only valid in the very special case
       where the value of the mapping is a constant!  (Contributed by Alexander
       van der Vekens, 1-Jun-2017.)  (Proof shortened by AV, 10-Feb-2019.) $)
    dmmpog $p |- ( C e. V -> dom F = ( A X. B ) ) $=
      ( wcel wral cdm cxp wceq cv wa simpl ralrimivva dmmpoga syl ) EGIZTBDJACJ
      FKCDLMTTABCDTANCIBNDIOPQABCDEFGHRS $.
  $}

  ${
    $d A x y $.  $d B y $.
    mpoexg.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( Existence of an operation class abstraction (version for dependent
       domains).  (Contributed by Mario Carneiro, 30-Dec-2016.) $)
    mpoexxg $p |- ( ( A e. R /\ A. x e. A B e. S ) -> F e. _V ) $=
      ( wcel wral wa wfun cdm cvv mpofun cv csn cxp sylancr ciun dmmpossx snexg
      wss vex ax-mp xpexg mpan ralimi iunexg sylan2 ssexg funex ) CFJZDGJZACKZL
      ZHMHNZOJZHOJABCDEHIPUQURACAQZRZDSZUAZUDVCOJZUSABCDEHIUBUPUNVBOJZACKVDUOVE
      ACVAOJZUOVEUTOJVFAUEUTOUCUFVADOGUGUHUIACVBFOUJUKURVCOULTOHUMT $.

    $d x B $.
    $( Existence of an operation class abstraction (special case).
       (Contributed by FL, 17-May-2010.)  (Revised by Mario Carneiro,
       1-Sep-2015.) $)
    mpoexg $p |- ( ( A e. R /\ B e. S ) -> F e. _V ) $=
      ( wcel cvv wral elex ralrimivw syl mpoexxg sylan2 ) DGJZCFJDKJZACLZHKJRST
      DGMSSACDKMNOABCDEFKHIPQ $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( If the domain of an operation given by maps-to notation is a set, the
       operation is a set.  (Contributed by NM, 12-Sep-2011.) $)
    mpoexga $p |- ( ( A e. V /\ B e. W )
                       -> ( x e. A , y e. B |-> C ) e. _V ) $=
      ( cmpo eqid mpoexg ) ABCDEFGABCDEHZKIJ $.
  $}

  ${
    $d z C $.  $d x y z A $.  $d x y z B $.  $d x y z D $.
    mpoexw.1 $e |- A e. _V $.
    mpoexw.2 $e |- B e. _V $.
    mpoexw.3 $e |- D e. _V $.
    mpoexw.4 $e |- A. x e. A A. y e. B C e. D $.
    $( Weak version of ~ mpoex that holds without ~ ax-coll .  If the domain
       and codomain of an operation given by maps-to notation are sets, the
       operation is a set.  (Contributed by Rohan Ridenour, 14-Aug-2023.) $)
    mpoexw $p |- ( x e. A , y e. B |-> C ) e. _V $=
      ( vz cmpo wfun cvv wcel wral wceq eqeltri cv wrex cdm crn eqid mpofun cxp
      dmmpoga ax-mp xpex cab rnmpo wa wi r19.21bi eleq1a syl rexlimdva rexlimiv
      rspec abssi ssexi funexw mp3an ) ABCDELZMVCUAZNOVCUBZNOVCNOABCDEVCVCUCZUD
      VDCDUEZNEFOZBDPZACPVDVGQJABCDEVCFVFUFUGCDGHUHRVEKSZEQZBDTZACTZKUIZNABKCDE
      VCVFUJVNFIVMKFVLVJFOZACASCOZVKVOBDVPBSDOUKVHVKVOULVPVHBDVIACJURUMEFVJUNUO
      UPUQUSUTRNNVCVAVB $.
    $( $j usage 'mpoexw' avoids 'ax-coll' ; $)
  $}

  ${
    $d x y A $.  $d y B $.
    mpoex.1 $e |- A e. _V $.
    mpoex.2 $e |- B e. _V $.
    $( If the domain of an operation given by maps-to notation is a set, the
       operation is a set.  (Contributed by Mario Carneiro, 20-Dec-2013.) $)
    mpoex $p |- ( x e. A , y e. B |-> C ) e. _V $=
      ( cvv wcel wral cmpo rgenw eqid mpoexxg mp2an ) CHIDHIZACJABCDEKZHIFPACGL
      ABCDEHHQQMNO $.
  $}

  ${
    $d A a b i j $.  $d B a b i j $.  $d C i j $.  $d D a b $.  $d M i j $.
    $d ph a b i j $.
    fnmpoovd.m $e |- ( ph -> M Fn ( A X. B ) ) $.
    fnmpoovd.s $e |- ( ( i = a /\ j = b ) -> D = C ) $.
    fnmpoovd.d $e |- ( ( ph /\ i e. A /\ j e. B ) -> D e. U ) $.
    fnmpoovd.c $e |- ( ( ph /\ a e. A /\ b e. B ) -> C e. V ) $.
    $( A function with a Cartesian product as domain is a mapping with two
       arguments defined by its operation values.  (Contributed by AV,
       20-Feb-2019.)  (Revised by AV, 3-Jul-2022.) $)
    fnmpoovd $p |- ( ph -> ( M = ( a e. A , b e. B |-> C )
                              <-> A. i e. A A. j e. B ( i M j ) = D ) ) $=
      ( wceq cv wral wcel cmpo co cxp wfn wb 3expb ralrimivva fnmpo syl eqfnov2
      eqid syl2anc nfcv cbvmpo eqcomi a1i eqeq2d 2ralbidv simprl simprr ovmpt4g
      oveqd wa syl3anc 2ralbidva 3bitrd ) AIKLBCDUAZQZGRZHRZIUBZVIVJVGUBZQZHCSG
      BSZVKVIVJGHBCEUAZUBZQZHCSGBSVKEQZHCSGBSAIBCUCZUDVGVSUDZVHVNUEMADJTZLCSKBS
      VTAWAKLBCAKRBTLRCTWAPUFUGKLBCDVGJVGUKUHUIGHBCIVGUJULAVMVQGHBCAVLVPVKAVGVO
      VIVJVGVOQAVOVGGHKLBCEDKEUMLEUMGDUMHDUMNUNUOUPVBUQURAVQVRGHBCAVIBTZVJCTZVC
      VCZVPEVKWDWBWCEFTZVPEQAWBWCUSAWBWCUTAWBWCWEOUFGHBCEVOFVOUKVAVDUQVEVF $.
  $}

  ${
    $d u v w x y B $.  $d u w x y z C $.  $d x y ph $.  $d u v w x y S $.
    $d u v w x y A $.  $d u v w z R $.  $d z T $.
    fmpoco.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> R e. C ) $.
    fmpoco.2 $e |- ( ph -> F = ( x e. A , y e. B |-> R ) ) $.
    fmpoco.3 $e |- ( ph -> G = ( z e. C |-> S ) ) $.
    fmpoco.4 $e |- ( z = R -> S = T ) $.
    $( Composition of two functions.  Variation of ~ fmptco when the second
       function has two arguments.  (Contributed by Mario Carneiro,
       8-Feb-2015.) $)
    fmpoco $p |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) ) $=
      ( vw vu vv csb ccom cxp c2nd cfv c1st cmpt cmpo wcel wral ralrimivva eqid
      cv fmpo sylib nfcv nfcsb1v nfcsb weq csbeq1a sylan9eq cbvmpo cop wceq vex
      wf op2ndd csbeq1d op1std csbeq2dv eqtrd mpompt eqtr4i fmpt sylibr fmptcos
      eqtrdi wa w3a 3impb nfcvd csbiegf syl mpoeq3dva eqtrid ) ALKUAQEFUBZDCQUL
      ZUCUDZBWFUEUDZHTZTZITZUFZBCEFJUGZAQDWEGWJIKLAWEGBCEFHUGZVEZWJGUHQWEUIAHGU
      HZCFUIBEUIWOAWPBCEFMUJBCEFHGWNWNUKUMUNQWEGWJWNWNRSEFCSULZBRULZHTZTZUGQWEW
      JUFZBCRSEFHWTRHUOSHUOBCWQWSBWQUOBWRHUPUQZCWQWSUPZBRURZCSURZHWSWTBWRHUSCWQ
      WSUSUTZVARSQEFWJWTWFWRWQVBVCZWJCWQWITWTXGCWGWQWIWRWQWFRVDZSVDZVFVGXGCWQWI
      WSXGBWHWRHWRWQWFXHXIVHVGVIVJZVKVLZVMVNAKWNXANXKVPOVOAWLBCEFDHITZUGZWMWLRS
      EFDWTITZUGXMRSQEFWKXNXGDWJWTIXJVGVKBCRSEFXLXNRXLUOSXLUOBDWTIXBBIUOUQCDWTI
      XCCIUOUQXDXEVQDHWTIXFVGVAVLABCEFXLJABULEUHZCULFUHZVRWPXLJVCAXOXPWPMVSDHIJ
      GWPDJVTPWAWBWCWDVJ $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z D $.  $d x y z H $.  $d z C $.
    oprabco.1 $e |- ( ( x e. A /\ y e. B ) -> C e. D ) $.
    oprabco.2 $e |- F = ( x e. A , y e. B |-> C ) $.
    oprabco.3 $e |- G = ( x e. A , y e. B |-> ( H ` C ) ) $.
    $( Composition of a function with an operator abstraction.  (Contributed by
       Jeff Madsen, 2-Sep-2009.)  (Proof shortened by Mario Carneiro,
       26-Sep-2015.) $)
    oprabco $p |- ( H Fn D -> G = ( H o. F ) ) $=
      ( vz wfn cfv cmpo ccom cv wcel wa adantl a1i dffn5im fveq2 fmpoco eqtr4id
      wceq ) IFNZHABCDEIOZPIGQLUHABMCDFEMRZIOUIGIARCSBRDSTEFSUHJUAGABCDEPUGUHKU
      BMFIUCUJEIUDUEUF $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y M $.  $d x y R $.  $d x y S $.
    oprab2co.1 $e |- ( ( x e. A /\ y e. B ) -> C e. R ) $.
    oprab2co.2 $e |- ( ( x e. A /\ y e. B ) -> D e. S ) $.
    oprab2co.3 $e |- F = ( x e. A , y e. B |-> <. C , D >. ) $.
    oprab2co.4 $e |- G = ( x e. A , y e. B |-> ( C M D ) ) $.
    $( Composition of operator abstractions.  (Contributed by Jeff Madsen,
       2-Sep-2009.)  (Revised by David Abernethy, 23-Apr-2013.) $)
    oprab2co $p |- ( M Fn ( R X. S ) -> G = ( M o. F ) ) $=
      ( cop cxp cv wcel cmpo wa opelxpi syl2anc co cfv df-ov a1i mpoeq3ia eqtri
      wceq oprabco ) ABCDEFPZGHQZIJKARCSBRDSUAZEGSFHSULUMSLMEFGHUBUCNJABCDEFKUD
      ZTABCDULKUEZTOABCDUOUPUOUPUJUNEFKUFUGUHUIUK $.
  $}

  ${
    $d w x y z $.
    $( An alternate possible definition of the ` 1st ` function.  (Contributed
       by NM, 14-Oct-2004.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    df1st2 $p |- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) ) $=
      ( vw c1st cvv cxp cres cv cfv wceq copab wcel wa coprab wfo wfn fo1st vex
      cmpt fofn dffn5im mp2b mptv eqtri reseq1i resopab op1std dfoprab3 3eqtrri
      cop eqeq2d ) EFFGZHCIZDIZEJZKZDCLZUMHUOUMMUQNDCLUNAIZKZABCOEURUMEDFUPTZUR
      FFEPEFQEVAKRFFEUADFEUBUCDCUPUDUEUFUQDCUMUGUQUTABCDUOUSBIZUKKUPUSUNUSVBUOA
      SBSUHULUIUJ $.

    $( An alternate possible definition of the ` 2nd ` function.  (Contributed
       by NM, 10-Aug-2006.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    df2nd2 $p |- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) ) $=
      ( vw c2nd cvv cxp cres cv cfv wceq copab wcel wa coprab wfo wfn fo2nd vex
      cmpt fofn dffn5im mp2b mptv eqtri reseq1i resopab op2ndd dfoprab3 3eqtrri
      cop eqeq2d ) EFFGZHCIZDIZEJZKZDCLZUMHUOUMMUQNDCLUNBIZKZABCOEURUMEDFUPTZUR
      FFEPEFQEVAKRFFEUADFEUBUCDCUPUDUEUFUQDCUMUGUQUTABCDUOAIZUSUKKUPUSUNVBUSUOA
      SBSUHULUIUJ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y V $.
    $( The mapping of a restriction of the ` 1st ` function to a constant
       function.  (Contributed by NM, 14-Dec-2008.) $)
    1stconst $p |- ( B e. V ->
             ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A ) $=
      ( vx vy wcel csn cxp c1st wfo cv wbr wmo wceq wa vex cfv cvv jca syl2anc
      cres ccnv wfun wf1o wex snmg fo1stresm syl wal cop moeq moani brres fo1st
      wfn wb fofn ax-mp fnbrfvb mp2an anbi1i elxp7 eleq1 biimpa adantrr adantrl
      c2nd elsni eqopi an12s sylanr2 adantrrl adantl simprr fveq2d simprl simpl
      sylan2b op1stg eqtrd adantr opelxpi eqeltrd impbida bitr3id bitrid mobidv
      snidg mpbiri alrimiv funcnv2 sylibr dff1o3 sylanbrc ) BCFZABGZHZAIWQUAZJZ
      WRUBUCZWQAWRUDWODKZWPFDUEWSDBCUFDAWPUGUHWOXAEKZWRLZDMZEUIWTWOXDEWOXDXBAFZ
      XAXBBUJZNZOZDMXGXEDDXFUKULWOXCXHDXCXAXBILZXAWQFZOZWOXHXAXBIWQEPUMXKXAIQZX
      BNZXJOZWOXHXMXIXJIRUOZXARFXMXIUPRRIJXOUNRRIUQURDPRXAXBIUSUTVAWOXNXHXNXHWO
      XJXMXARRHFZXLAFZXAVGQZWPFZOZOZXHXAAWPVBXMYAOXEXGXMXTXEXPXMXQXEXSXMXQXEXLX
      BAVCVDVEVFXMXPXSXGXQXSXMXPXRBNZXGXRBVHXPXMYBXGXAXBBRRVIVJVKVLSVRVMWOXHOZX
      MXJYCXLXFIQZXBYCXAXFIWOXEXGVNZVOYCXEWOYDXBNWOXEXGVPZWOXHVQXBBACVSTVTYCXAX
      FWQYEYCXEBWPFZXFWQFYFWOYGXHBCWHWAXBBAWPWBTWCSWDWEWFWGWIWJDEWRWKWLWQAWRWMW
      N $.

    $( The mapping of a restriction of the ` 2nd ` function to a converse
       constant function.  (Contributed by NM, 27-Mar-2008.) $)
    2ndconst $p |- ( A e. V ->
             ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -1-1-onto-> B ) $=
      ( vx vy wcel csn cxp c2nd wfo cv wbr wmo wceq wa vex cfv cvv adantrl jca
      cres ccnv wfun wf1o wex snmg fo2ndresm syl wal cop moeq moani brres fo2nd
      wfn fofn ax-mp fnbrfvb mp2an anbi1i c1st elxp7 eleq1 biimpa elsni ancom2s
      eqopi an12s sylanr2 adantrrr sylan2b adantl fveq2 op2ndg sylan9eqr simprr
      mpan2 adantr simprl opelxpi syl2anc eqeltrd impbida bitr3id bitrid mobidv
      wb snidg mpbiri alrimiv funcnv2 sylibr dff1o3 sylanbrc ) ACFZAGZBHZBIWQUA
      ZJZWRUBUCZWQBWRUDWODKZWPFDUEWSDACUFDWPBUGUHWOXAEKZWRLZDMZEUIWTWOXDEWOXDXB
      BFZXAAXBUJZNZOZDMXGXEDDXFUKULWOXCXHDXCXAXBILZXAWQFZOZWOXHXAXBIWQEPZUMXKXA
      IQZXBNZXJOZWOXHXNXIXJIRUOZXARFXNXIWGRRIJXPUNRRIUPUQDPRXAXBIURUSUTWOXOXHXO
      XHWOXJXNXARRHFZXAVAQZWPFZXMBFZOZOZXHXAWPBVBXNYBOXEXGXNYAXEXQXNXTXEXSXNXTX
      EXMXBBVCVDSSXNXQXSXGXTXSXNXQXRANZXGXRAVEXQXNYCXGXQYCXNXGXAAXBRRVGVFVHVIVJ
      TVKVLWOXHOZXNXJWOXGXNXEXGWOXMXFIQZXBXAXFIVMWOXBRFYEXBNXLAXBCRVNVQVOSYDXAX
      FWQWOXEXGVPYDAWPFZXEXFWQFWOYFXHACWHVRWOXEXGVSAXBWPBVTWAWBTWCWDWEWFWIWJDEW
      RWKWLWQBWRWMWN $.
  $}

  ${
    $d w x y A $.  $d w x y B $.  $d w C $.
    dfmpo.1 $e |- C e. _V $.
    $( Alternate definition for the maps-to notation ~ df-mpo (although it
       requires that ` C ` be a set).  (Contributed by NM, 19-Dec-2008.)
       (Revised by Mario Carneiro, 31-Aug-2015.) $)
    dfmpo $p |- ( x e. A , y e. B |-> C )
                = U_ x e. A U_ y e. B { <. <. x , y >. , C >. } $=
      ( vw cv cfv csb cop csn ciun cvv wcel ax-mp csbexa nfcv nfcsb1v nfop cmpo
      cxp c1st c2nd cmpt mpompts vex 1stexg 2ndexg dfmpt nfsn wceq id csbopeq1a
      nfcsb opeq12d sneqd iunxpf 3eqtri ) ABCDEUAGCDUBZAGHZUCIZBVAUDIZEJZJZUEGU
      TVAVEKZLZMACBDAHBHKZEKZLZMMABGCDEUFGUTVEAVBVDVANOZVBNOGUGZVANUHPBVCEVKVCN
      OVLVANUIPFQQUJGABCDVGVJAVFAVAVEAVARAVBVDSTUKBVFBVAVEBVARBAVBVDBVBRBVCESUO
      TUKGVJRVAVHULZVFVIVMVAVHVEEVMUMABVAEUNUPUQURUS $.
  $}

  $( Lemma for ~ cnvf1o .  (Contributed by Mario Carneiro, 27-Apr-2014.) $)
  cnvf1olem $p |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) ->
                    ( C e. `' A /\ B = U. `' { C } ) ) $=
    ( wrel wcel csn ccnv cuni wceq c2nd cfv cop sneqd cnveqd unieqd cvv opswapg
    wa syl2anc ad2antrl c1st simprr 1st2nd 1stexg 2ndexg 3eqtrd simprl eqeltrrd
    adantrr wb opelcnvg mpbird eqeltrd eqcomd 3eqtr4d jca ) ADZBAEZCBFZGZHZIZRR
    ZCAGZEBCFZGZHZIVCCBJKZBUAKZLZVDVCCVAVIVHLZFZGZHZVJUQURVBUBVCUTVMVCUSVLVCBVK
    UQURBVKIVBBAUCUIZMNOURVNVJIZUQVBURVIPEZVHPEZVPBAUDZBAUEZVIVHPPQSTUFZVCVJVDE
    ZVKAEZVCBVKAVOUQURVBUGUHURWBWCUJZUQVBURVRVQWDVTVSVHVIPPAUKSTULUMVCVKVJFZGZH
    ZBVGURVKWGIUQVBURWGVKURVRVQWGVKIVTVSVHVIPPQSUNTVOVCVFWFVCVEWEVCCVJWAMNOUOUP
    $.

  ${
    $d x y A $.
    $( Describe a function that maps the elements of a set to its converse
       bijectively.  (Contributed by Mario Carneiro, 27-Apr-2014.) $)
    cnvf1o $p |- ( Rel A ->
                   ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A ) $=
      ( vy wrel ccnv cv csn cuni cmpt wcel snexg cnvexg uniexg 3syl adantl wceq
      cvv wa cnvf1olem wb relcnv simpr sylancr dfrel2 eleq2 sylbi anbi1d adantr
      eqid mpbid impbida f1od ) BDZACBBEZAFZGZEZHZCFZGZEZHZABURIZQQVCUIUOBJZURQ
      JZUMVDUPQJUQQJVEUOBKUPQLUQQMNOUSUNJZVBQJZUMVFUTQJVAQJVGUSUNKUTQLVAQMNOUMV
      DUSURPZRZVFUOVBPRZBUOUSSUMVJRZUOUNEZJZVHRZVIVKUNDVJVNBUAUMVJUBUNUSUOSUCUM
      VNVITVJUMVMVDVHUMVLBPVMVDTBUDVLBUOUEUFUGUHUJUKUL $.
  $}

  $( The ` 2nd ` (second component of an ordered pair) function restricted to a
     function ` F ` is a function from ` F ` into the codomain of ` F ` .
     (Contributed by Alexander van der Vekens, 4-Feb-2018.) $)
  f2ndf $p |- ( F : A --> B -> ( 2nd |` F ) : F --> B ) $=
    ( wf c2nd cres cxp wss f2ndres fssxp fssres sylancr wceq resabs1 syl eqcomd
    feq1d mpbird ) ABCDZCBECFZDCBEABGZFZCFZDZSUABUBDCUAHZUDABIABCJZUABCUBKLSCBT
    UCSUCTSUEUCTMUFECUANOPQR $.

  ${
    $d A x y $.  $d B x y $.  $d F x y $.
    $( The ` 2nd ` (second component of an ordered pair) function restricted to
       a function ` F ` is a function from ` F ` onto the range of ` F ` .
       (Contributed by Alexander van der Vekens, 4-Feb-2018.) $)
    fo2ndf $p |- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F ) $=
      ( vy vx wf crn c2nd cres wceq wfo wfn ffn dffn3 f2ndf syl cv wcel cfv vex
      sylib wss sylbi frn cop wex elrn2g wa fvres adantl op2nd eqtr2di fnfvelrn
      ibi sylan eqeltrd ex exlimdv syl5 ssrdv eqssd dffo2 sylanbrc ) ABCFZCCGZH
      CIZFZVFGZVEJCVEVFKVDAVECFZVGVDCALZVIABCMZACNZUAAVECOZPVDVHVEVDVGVHVEUBVDV
      JVGVKVJVIVGVLVMUCPCVEVFUDPVDDVEVHDQZVERZEQZVNUEZCRZEUFZVDVNVHRZVOVSEVNCVE
      UGUNVDVRVTEVDVRVTVDVRUHZVNVQVFSZVHWAWBVQHSZVNVRWBWCJVDVQCHUIUJVPVNETDTUKU
      LVDVFCLZVRWBVHRVDCBVFFWDABCOCBVFMPCVQVFUMUOUPUQURUSUTVACVEVFVBVC $.
  $}

  ${
    $d A a b v w x y $.  $d B a b v w x y $.  $d F a b v w x y $.
    $( The ` 2nd ` (second component of an ordered pair) function restricted to
       a one-to-one function ` F ` is a one-to-one function from ` F ` onto the
       range of ` F ` .  (Contributed by Alexander van der Vekens,
       4-Feb-2018.) $)
    f1o2ndf1 $p |- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F ) $=
      ( vx vy va vv vb vw c2nd syl cv cfv wceq wi wcel wa wrex ex com23 wf1 crn
      cres wfo ccnv wfun wf1o f1f fo2ndf weq wral f2ndf cxp wss fssxp cop ssel2
      elxp2 sylib anim12dan fvres adantr ad2antlr eqeq12d op2nd eqeq12i funopfv
      wf vex f1fun anim12d eqcom biimpi eqeqan12d simpl anim12i f1veqaeq sylan2
      opeq12 syl6 com14 biimtrdi syld com13 impcom biimtrid sylbid adantl com12
      pm2.43i adantlr eleq1 bi2anan9 anbi2d fveq2 simpllr simpr imbi12d 3imtr4d
      wb imbi2d rexlimdvva rexlimivv imp mpcom ralrimivv dff13 sylanbrc simprbi
      df-f1 dff1o3 ) ABCUAZCCUBZJCUCZUDZXNUEUFZCXMXNUGXLABCVHZXOABCUHZABCUIKXLC
      BXNUAZXPXLCBXNVHZDLZXNMZELZXNMZNZDEUJZOZECUKDCUKXSXLXQXTXRABCULKXLYGDECCC
      ABUMZUNZXLYACPZYCCPZQZYGOXLXQYIXRABCUOKYIYLXLYGYIYLXLYGOZYAFLZGLZUPZNZGBR
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      YIYKQYCYHPUUCCYHYCUQHIYCABURUSUTYRUUCUUDYMOZYQUUCUUEOZFGABYNAPZYOBPZQZYQU
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      MZUUAXNMZNZYPUUANZOZOZUUDYMUUNUUSUVEOZUUBUUIUUMUVFYQUUSUUIUUMQZUVEUURUVGU
      VEOYIUURUVGUVEUURUVGQZUVBXLUVCUVHUVBYPJMZUUAJMZNZXLUVCOZUVHUUTUVIUVAUVJUU
      RUUTUVINZUVGUUPUVMUUQYPCJVAVBVBUUQUVAUVJNUUPUVGUUACJVAVCVDUVKGIUJZUVHUVLU
      VIYOUVJYTYNYOFVIGVIVEYSYTHVIIVIVEVFUVHXLUVNUVCUVGUURXLUVNUVCOZOXLUURUVGUV
      OXLUURYNCMZYONZYSCMZYTNZQZUVGUVOOXLCUFZUURUVTOABCVJUWAUUPUVQUUQUVSYNYOCVG
      YSYTCVGVKKXLUVGUVTUVOUVNUVGUVTXLUVCUVNUVGUVTUVLOOUVTUVNUVGUVNUVLUVTUVNUVP
      UVRNZUVGUVNUVLOOUVQUVSYOUVPYTUVRUVQYOUVPNUVPYOVLVMUVSYTUVRNUVRYTVLVMVNXLU
      VGUVNUWBUVCXLUVGUVNUWBUVCOOXLUVGQZUWBUVNUVCUWCUWBFHUJZUVOUVGXLUUGUUKQUWBU
      WDOUUIUUGUUMUUKUUGUUHVOUUKUULVOVPABYNYSCVQVRUWDUVNUVCYNYOYSYTVSSVTTSWAWBW
      AWJWATWCWDWETWFWGTSWHWIWKVBUUOYLUURYIUUNYJUUPUUBYKUUQYQYJUUPWTUUIUUMYAYPC
      WLVCYCUUACWLWMWNUUOYGUVDXLUUOYEUVBYFUVCUUNUUBYBUUTYDUVAYQYBUUTNUUIUUMYAYP
      XNWOVCYCUUAXNWOVNUUOYAYPYCUUAUUIYQUUMUUBWPUUNUUBWQVDWRXAWSSXBSXCXDXESTXEX
      FDECBXNXGXHXSXTXPCBXNXJXIKCXMXNXKXH $.
  $}

  ${
    algrflem.1 $e |- B e. _V $.
    algrflem.2 $e |- C e. _V $.
    $( Lemma for algrf and related theorems.  (Contributed by Mario Carneiro,
       28-May-2014.)  (Revised by Mario Carneiro, 30-Apr-2015.) $)
    algrflem $p |- ( B ( F o. 1st ) C ) = ( F ` B ) $=
      ( c1st ccom co cop cfv df-ov cvv wf wcel wceq wfo fo1st fof ax-mp mp2an
      opexg fvco3 op1st fveq2i 3eqtri ) ABCFGZHABIZUFJZUGFJZCJZACJABUFKLLFMZUGL
      NZUHUJOLLFPUKQLLFRSALNBLNULDEABLLUATLLUGCFUBTUIACABDEUCUDUE $.
  $}

  $( Lemma for ~ algrf and related theorems.  (Contributed by Mario Carneiro,
     28-May-2014.)  (Revised by Jim Kingdon, 22-Jul-2021.) $)
  algrflemg $p |- ( ( B e. V /\ C e. W ) ->
      ( B ( F o. 1st ) C ) = ( F ` B ) ) $=
    ( wcel wa c1st ccom co cop cfv df-ov cvv wf wceq wfo fo1st fof ax-mp op1stg
    opexg fvco3 sylancr fveq2d eqtrd eqtrid ) ADFBEFGZABCHIZJABKZUILZACLZABUIMU
    HUKUJHLZCLZULUHNNHOZUJNFUKUNPNNHQUORNNHSTABDEUBNNUJCHUCUDUHUMACABDEUAUEUFUG
    $.

  ${
    $d A x y $.  $d B x y $.  $d R x y $.  $d S x y $.  $d a x y $.
    $d b x y $.  $d c x y $.  $d d x y $.
    xporderlem.1 $e |- T = { <. x , y >. | ( ( x e. ( A X. B )
     /\ y e. ( A X. B ) )
     /\ ( ( 1st ` x ) R ( 1st ` y )
       \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } $.
    $( Lemma for lexicographical ordering theorems.  (Contributed by Scott
       Fenton, 16-Mar-2011.) $)
    xporderlem $p |- ( <. a , b >. T <. c , d >. <->
                        ( ( ( a e. A /\ c e. A ) /\ ( b e. B /\ d e. B ) ) /\
                          ( a R c \/ ( a = c /\ b S d ) ) ) ) $=
      ( cv wbr wcel wa cfv wceq vex anbi12d cop c1st c2nd wo copab df-br eleq2i
      bitri opex eleq1 opelxp bitrdi anbi1d op1std breq1d eqeq1d op2ndd orbi12d
      cxp anbi2d breq2d eqeq2d opelopab an4 anbi1i 3bitri ) HMZIMZUAZJMZKMZUAZG
      NZVIVLUAZAMZCDUSZOZBMZVPOZPZVOUBQZVRUBQZENZWAWBRZVOUCQZVRUCQZFNZPZUDZPZAB
      UEZOZVGCOZVHDOZPZVJCOZVKDOZPZPZVGVJENZVGVJRZVHVKFNZPZUDZPZWMWPPWNWQPPZXDP
      VMVNGOWLVIVLGUFGWKVNLUGUHWJWOVSPZVGWBENZVGWBRZVHWFFNZPZUDZPXEABVIVLVGVHHS
      ZISZUIVJVKJSZKSZUIVOVIRZVTXGWIXLXQVQWOVSXQVQVIVPOWOVOVIVPUJVGVHCDUKULUMXQ
      WCXHWHXKXQWAVGWBEVGVHVOXMXNUNZUOXQWDXIWGXJXQWAVGWBXRUPXQWEVHWFFVGVHVOXMXN
      UQUOTURTVRVLRZXGWSXLXDXSVSWRWOXSVSVLVPOWRVRVLVPUJVJVKCDUKULUTXSXHWTXKXCXS
      WBVJVGEVJVKVRXOXPUNZVAXSXIXAXJXBXSWBVJVGXTVBXSWFVKVHFVJVKVRXOXPUQVATURTVC
      WSXFXDWMWNWPWQVDVEVF $.
  $}

  ${
    $d A a b c d e f t u v x y $.  $d B a b c d e f t u v x y $.
    $d R a b c d e f t u v x y $.  $d S a b c d e f t u v x y $.
    $d T a b c d e f t u v $.
    poxp.1 $e |- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) )
      /\ ( ( 1st ` x ) R ( 1st ` y ) \/
           ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } $.
    $( A lexicographical ordering of two posets.  (Contributed by Scott Fenton,
       16-Mar-2011.)  (Revised by Mario Carneiro, 7-Mar-2013.) $)
    poxp $p |- ( ( R Po A /\ S Po B ) -> T Po ( A X. B ) ) $=
      ( va vb vc vd ve vf wa cv wbr wn wi wcel vt vu wpo cxp wral cop wceq elxp
      vv wex w3a weq wo poirr ex intnand im2anan9 ioran imbitrrdi imp 3ad2antr1
      3an6 an4 potr 3impia orcd 3expia expdimp breq2 biimpa expcom adantrd jaod
      adantl anim2d orim2d breq1 equequ1 anbi1d imbi2d imbitrrid expd impd jaao
      orbi12d an4s sylan2b biimpi 3adant2 jctild adantld biimtrid jca wb breq12
      anidms notbid 3ad2ant1 3adant3 3adant1 anbi12d imbi12d xporderlem anbi12i
      com12 notbii imbi12i expcomd sylbi 3exp com3l exlimivv 3imp syl3anb com3r
      bitrdi ralrimiv ralrimivva df-po sylibr ) CEUCZDFUCZOZUAPZYDGQZRZYDUBPZGQ
      ZYGUIPZGQZOZYDYIGQZSZOZUICDUDZUEZUBYOUEUAYOUEYOGUCYCYPUAUBYOYOYCYDYOTZYGY
      OTZOZOYNUIYOYCYSYIYOTZYNSYSYTYCYNYQYRYTYCYNSZYQYDIPZJPZUFZUGZUUBCTZUUCDTZ
      OZOZJUJIUJZYRYGKPZLPZUFZUGZUUKCTZUULDTZOZOZLUJKUJZYTYIMPZNPZUFZUGZUUTCTZU
      VADTZOZOZNUJMUJZUUAIJYDCDUHKLYGCDUHMNYICDUHUUJUUSUVHUUAUUIUUSUVHUUASSIJUV
      HUUIUUSUUAUVGUUIUUSUUASSMNUUSUVGUUIUUAUURUVGUUIUUASSKLUUIUURUVGUUAUUIUURU
      VGUUAUUIUURUVGUKUUEUUNUVCUKZUUHUUQUVFUKZOUUAUUEUUHUUNUUQUVCUVFVBUVIUVJUUA
      UVIYCUVJYNYCUVJOZYNUVIUUFUUFOUUGUUGOOZUUBUUBEQZIIULZUUCUUCFQZOZUMZOZRZUUF
      UUOOUUGUUPOOZUUBUUKEQZIKULZUUCUULFQZOZUMZOZUUOUVDOUUPUVEOOZUUKUUTEQZKMULZ
      UULUVAFQZOZUMZOZOZUUFUVDOUUGUVEOOZUUBUUTEQZIMULZUUCUVAFQZOZUMZOZSZOZUVKUV
      SUXBYCUUQUUHUVSUVFYCUUHOUVQUVLYCUUHUVQRZYCUUHUVMRZUVPRZOUXDYAUUFUXEYBUUGU
      XFYAUUFUXECUUBEUNUOYBUUGUXFYBUUGOUVOUVNDUUCFUNUPUOUQUVMUVPURUSUTUPVAUWNUV
      TUWGOZUWEUWLOZOUVKUXAUVTUWEUWGUWLVCUVKUXHUXAUXGUVKUXHUWTUWOUVJYCUUFUUOUVD
      UKZUUGUUPUVEUKZOUXHUWTSZUUFUUGUUOUUPUVDUVEVBYAUXIYBUXJUXKYAUXIOZYBUXJOZOU
      WEUWLUWTUXLUWAUWLUWTSZUXMUWDUXLUWAUXNUXLUWAOUWHUWTUWKUXLUWAUWHUWTYAUXIUWA
      UWHOZUWTYAUXIUXOUKUWPUWSYAUXIUXOUWPCUUBUUKUUTEVDVEVFVGVHUWAUWKUWTSUXLUWAU
      WIUWTUWJUWIUWAUWTUWIUWAOUWPUWSUWIUWAUWPUUKUUTUUBEVIVJVFVKVLVNVMUOUXMUWBUW
      CUXNUWBUXMUWCUXNSUWBUXMUWCUXNUXMUWCOZUXNUWBUWLUWHUWIUWROZUMZSUXPUWKUXQUWH
      UXPUWJUWRUWIUXMUWCUWJUWRDUUCUULUVAFVDVHVOVPUWBUWTUXRUWLUWBUWPUWHUWSUXQUUB
      UUKUUTEVQUWBUWQUWIUWRIKMVRVSWEVTWAWBXEWCWDWCWFWGUVJUWOYCUUHUVFUWOUUQUUHUV
      FOUWOUUFUUGUVDUVEVCWHWIVNWJWKWLWMUVIYNUUDUUDGQZRZUUDUUMGQZUUMUVBGQZOZUUDU
      VBGQZSZOUXCUVIYFUXTYMUYEUUEUUNYFUXTWNUVCUUEYEUXSUUEYEUXSWNYDUUDYDUUDGWOWP
      WQWRUVIYKUYCYLUYDUVIYHUYAYJUYBUUEUUNYHUYAWNUVCYDUUDYGUUMGWOWSUUNUVCYJUYBW
      NUUEYGUUMYIUVBGWOWTXAUUEUVCYLUYDWNUUNYDUUDYIUVBGWOWIXBXAUXTUVSUYEUXBUXSUV
      RABCDEFGIJIJHXCXFUYCUWNUYDUXAUYAUWFUYBUWMABCDEFGIJKLHXCABCDEFGKLMNHXCXDAB
      CDEFGIJMNHXCXGXDXPWAXHUTXIXJXKXLXKXLXKXLXMXNVGXOUTXQXRUAUBUIYOGXSXT $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ph $.
    spc2ed.x $e |- F/ x ch $.
    spc2ed.y $e |- F/ y ch $.
    spc2ed.1 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) ) $.
    $( Existential specialization with 2 quantifiers, using implicit
       substitution.  (Contributed by Thierry Arnoux, 23-Aug-2017.) $)
    spc2ed $p |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( ch -> E. x E. y ps ) )
      $=
      ( wcel wa cv wceq wex elisset nfv nfan wi eeanv sylibr anass ancom anbi1i
      anim12i bitr3i biimparc sylbir ex eximd impancom sylan2 ) FHMZGIMZNZADOFP
      ZEOGPZNZEQZDQZCBEQZDQZUAUQURDQZUSEQZNVBUOVEUPVFDFHREGIRUGURUSDEUBUCACVBVD
      ACNZVAVCDACDADSJTVGUTBEACEAESKTVGUTBVGUTNZCAUTNZNZBVJCANZUTNVHCAUTUDVKVGU
      TCAUEUFUHVIBCLUIUJUKULULUMUN $.
  $}

  ${
    $d a w x y z $.  $d a w ph $.  $d w ps $.
    cnvoprab.x $e |- F/ x ps $.
    cnvoprab.y $e |- F/ y ps $.
    cnvoprab.1 $e |- ( a = <. x , y >. -> ( ps <-> ph ) ) $.
    cnvoprab.2 $e |- ( ps -> a e. ( _V X. _V ) ) $.
    $( TODO - simplify with ~ dfoprab3 $)
    $( The converse of a class abstraction of nested ordered pairs.
       (Contributed by Thierry Arnoux, 17-Aug-2017.) $)
    cnvoprab $p |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps } $=
      ( vw copab cv cop wceq wa wex vex cvv wcel ccnv coprab cab excom nfv nfan
      nfex opex opeq1 eqeq2d anbi12d spcev exlimi cxp adantl c1st cfv wi 1stexg
      c2nd ax-mp 2ndexg wb eqcom anbi12i eqopi sylan2br bicomd syl spc2ed mpcom
      mpanr12 exlimiv impbii exbii exrot3 3bitr2ri abbii df-opab cnveqi cnvopab
      df-oprab 3eqtr4ri eqtr3i ) BFELZUAACDEUBZUABEFLWEWFKMZCMZDMZNZEMZNZOZAPZE
      QDQCQZKUCWGFMZWKNZOZBPZEQFQZKUCWFWEWOWTKWTWSFQZEQWNDQZCQZEQWOWSFEUDXCXAEX
      CXAXBXACWSCFWRBCWRCUEGUFZUGWNXADWSDFWRBDWRDUEHUFZUGWSWNFWJWHWICRDRUHWPWJO
      ZWRWMBAXFWQWLWGWPWJWKUIUJIUKZULUMUMWSXCFWPSSUNTZWSXCBXHWRJUOXHWPUPUQZSTZW
      PUTUQZSTZWSXCURWPSTZXJFRZWPSUSVAXMXLXNWPSVBVAXHWNWSCDXIXKSSXDXEXHWHXIOZWI
      XKOZPZPXFWNWSVCXQXHXIWHOZXKWIOZPXFXRXOXSXPXIWHVDXKWIVDVEWPWHWISSVFVGXFWSW
      NXGVHVIVJVLVKVMVNVOWNECDVPVQVRACDEKWBBFEKVSWCVTBFEWAWD $.
  $}

  ${
    $d a x y z A $.  $d a x y z B $.  $d a z C $.  $d a x y z D $.
    $d a x y I $.  $d a x y J $.  $d a x y z ph $.
    f1od2.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    f1od2.2 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C e. W ) $.
    f1od2.3 $e |- ( ( ph /\ z e. D ) -> ( I e. X /\ J e. Y ) ) $.
    f1od2.4 $e |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <->
                            ( z e. D /\ ( x = I /\ y = J ) ) ) ) $.
    $( Describe an implicit one-to-one onto function of two variables.
       (Contributed by Thierry Arnoux, 17-Aug-2017.) $)
    f1od2 $p |- ( ph -> F : ( A X. B ) -1-1-onto-> D ) $=
      ( wa wsbc va cxp wfn ccnv wf1o wcel wral ralrimivva fnmpo syl cop cmpt cv
      opelxpi ralrimiva eqid fnmpt c1st cfv c2nd csb wceq copab cvv elxp7 anass
      anbi1i sbcbidv sbcan vex 2ndexg ax-mp sbcg sbcel1v anbi12i sbceq2g sbcbii
      wb 1stexg 3bitri sbceq1g csbvarg eqeq1i 3bitr3g anbi2d bitrid xpss simprr
      bitri adantrr eqeltrd sselid pm4.71rd eqop pm5.32i bitr2di bitrd opabbidv
      coprab cmpo df-mpo eqtri cnveqi nfv nfcsb1v nfeq2 nfan nfcsb eleq1 opelxp
      nfcv bitrdi csbopeq1a eqeq2d anbi12d sseli adantr cnvoprab df-mpt 3eqtr4g
      ex fneq1d mpbird dff1o4 sylanbrc ) AIEFUBZUCZIUDZHUCZYFHIUEAGLUFZCFUGBEUG
      YGAYJBCEFPUHBCEFGILOUIUJAYIDHJKUKZULZHUCZAYKMNUBZUFZDHUGYMAYODHADUMZHUFZS
      JMUFKNUFSYOQJKMNUNUJZUODHYKYLYNYLUPUQUJAHYHYLAUAUMZYFUFZYPBYSURUSZCYSUTUS
      ZGVAZVAZVBZSZDUAVCZYQYSYKVBZSZDUAVCYHYLAUUFUUIDUAUUFYSVDVDUBZUFZUUAEUFZUU
      BFUFZSZSZUUESZAUUIYTUUOUUEYSEFVEVGAUUPUUKYQUUAJVBZUUBKVBZSZSZSZUUIUUPUUKU
      UNUUESZSAUVAUUKUUNUUEVFAUVBUUTUUKABUMZEUFZCUMZFUFZSZYPGVBZSZCUUBTZBUUATZY
      QUVCJVBZUVEKVBZSZSZCUUBTZBUUATZUVBUUTAUVJUVPBUUAAUVIUVOCUUBRVHVHUVKUVDUUM
      SZYPUUCVBZSZBUUATUVRBUUATZUVSBUUATZSUVBUVJUVTBUUAUVJUVGCUUBTZUVHCUUBTZSUV
      TUVGUVHCUUBVIUWCUVRUWDUVSUWCUVDCUUBTZUVFCUUBTZSUVRUVDUVFCUUBVIUWEUVDUWFUU
      MUUBVDUFZUWEUVDVRYSVDUFZUWGUAVJZYSVDVKVLZUVDCUUBVDVMVLCUUBFVNVOWIUWGUWDUV
      SVRUWJCUUBYPGVDVPVLVOWIVQUVRUVSBUUAVIUWAUUNUWBUUEUWAUVDBUUATZUUMBUUATZSUU
      NUVDUUMBUUAVIUWKUULUWLUUMBUUAEVNUUAVDUFZUWLUUMVRUWHUWMUWIYSVDVSVLZUUMBUUA
      VDVMVLVOWIUWMUWBUUEVRUWNBUUAYPUUCVDVPVLVOVTUVQYQUVLUURSZSZBUUATYQBUUATZUW
      OBUUATZSUUTUVPUWPBUUAUVPYQCUUBTZUVNCUUBTZSUWPYQUVNCUUBVIUWSYQUWTUWOUWGUWS
      YQVRUWJYQCUUBVDVMVLUWTUVLCUUBTZUVMCUUBTZSUWOUVLUVMCUUBVIUXAUVLUXBUURUWGUX
      AUVLVRUWJUVLCUUBVDVMVLUXBCUUBUVEVAZKVBZUURUWGUXBUXDVRUWJCUUBUVEKVDWAVLUXC
      UUBKUWGUXCUUBVBUWJCUUBVDWBVLWCWIVOWIVOWIVQYQUWOBUUAVIUWQYQUWRUUSUWMUWQYQV
      RUWNYQBUUAVDVMVLUWRUVLBUUATZUURBUUATZSUUSUVLUURBUUAVIUXEUUQUXFUURUXEBUUAU
      VCVAZJVBZUUQUWMUXEUXHVRUWNBUUAUVCJVDWAVLUXGUUAJUWMUXGUUAVBUWNBUUAVDWBVLWC
      WIUWMUXFUURVRUWNUURBUUAVDVMVLVOWIVOVTWDWEWFAUUIUUKUUISUVAAUUIUUKAUUIUUKAU
      UISZYNUUJYSMNWGUXIYSYKYNAYQUUHWHAYQYOUUHYRWJWKWLYAWMUUKUUIUUTUUKUUHUUSYQY
      SJKVDVDWNWEWOWPWQWFWRYHUVIBCDWSZUDUUGIUXJIBCEFGWTUXJOBCDEFGXAXBXCUVIUUFBC
      DUAYTUUEBYTBXDBYPUUDBUUAUUCXEXFXGYTUUECYTCXDCYPUUDCBUUAUUCCUUAXKCUUBGXEXH
      XFXGYSUVCUVEUKZVBZYTUVGUUEUVHUXLYTUXKYFUFUVGYSUXKYFXIUVCUVEEFXJXLUXLUUDGY
      PBCYSGXMXNXOYTUUKUUEYFUUJYSEFWGXPXQXRXBDUAHYKXSXTYBYCYFHIYDYE $.
  $}

  ${
    $d A x y z w $.  $d B y z w $.  $d C y z w $.  $d ph x y z w $.
    disjxp1.1 $e |- ( ph -> Disj_ x e. A B ) $.
    $( The sets of a cartesian product are disjoint if the sets in the first
       argument are disjoint.  (Contributed by Glauco Siliprandi,
       11-Oct-2020.) $)
    disjxp1 $p |- ( ph -> Disj_ x e. A ( B X. C ) ) $=
      ( vy vw vz cv cxp wcel wrmo wal wdisj csb wa wceq wi wral cfv xp1st sylib
      c1st df-disj cvv 1stexg elv eleq1 rmobidv spcv nfcv nfcsb1v nfel2 csbeq1a
      syl eleq2d rmo4f r19.21bi syl2ani ralrimiva nfxp xpeq12d sylibr alrimiv )
      AGJZDEKZLZBCMZGNBCVGOAVIGAVHVFBHJZDPZBVJEPZKZLZQBJZVJRZSZHCTZBCTVIAVRBCAV
      OCLQZVQHCVHVSVJCLQVFUDUAZDLZVTVKLZVPVNVFDEUBVFVKVLUBVSWAWBQVPSZHCAWCHCTZB
      CAWABCMZWDBCTAIJZDLZBCMZINZWEABCDOWIFBICDUEUCWHWEIVTVTUFLGVFUFUGUHWFVTRWG
      WABCWFVTDUIUJUKUPWAWBBHCBCULZHCULZBVTVKBVJDUMZUNVPDVKVTBVJDUOZUQURUCUSUSU
      TVAVAVHVNBHCWJWKBVFVMBVKVLWLBVJEUMVBUNVPVGVMVFVPDVKEVLWMBVJEUOVCUQURVDVEB
      GCVGUEVD $.
  $}

  ${
    $d A j $.
    $( The sets in the cartesian product of singletons with other sets, are
       disjoint.  (Contributed by Glauco Siliprandi, 11-Oct-2020.) $)
    disjsnxp $p |- Disj_ j e. A ( { j } X. B ) $=
      ( cv csn cxp wdisj wtru sndisj a1i disjxp1 mptru ) CACDEZBFGHCAMBCAMGHCAI
      JKL $.
  $}

  ${
    $d A r x y $.  $d A z $.  $d B r x y $.  $d B w $.  $d C r $.  $d D r $.
    $d D s $.  $d F s $.
    elmpoex.f $e |- F = ( x e. A , y e. B |-> C ) $.
    $( If a maps-to operation is inhabited, the first class it is defined with
       is inhabited.  (Contributed by Jim Kingdon, 4-Mar-2026.) $)
    elmpom $p |- ( D e. F -> E. z z e. A ) $=
      ( vw vr vs wcel cv wex wa cfv cdm wceq cop c1st coprab df-mpo eqtri dmeqi
      cxp cmpo dmoprabss eqsstri c2nd 2ndexg wrel wfun mpofun funrel ax-mp mpan
      1st2nd eleq1d ibi opeq2 elabd wb 1stexg eldm2g mpbird sselid elex2 sylibr
      cvv syl xpm simpld ) GHMZCNDMCOZJNEMJOZVNKNZDEUFZMKOZVOVPPVNGUAQZVRMVSVNH
      RZVRVTWAANDMBNEMPVQFSZPABKUBZRVRHWCHABDEFUGWCIABKDEFUCUDUEWBABKDEUHUIVNVT
      WAMZVTLNZTZHMZLOZVNWGVTGUJQZTZHMZLWIGHUKVNWKVNGWJHHULZVNGWJSHUMWLABDEFHIU
      NHUOUPGHURUQUSUTWEWISWFWJHWEWIVTVAUSVBVNVTVJMWDWHVCGHVDLVTHVJVEVKVFVGKVTV
      RVHVKCJKDEVLVIVM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The support of functions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  In this section, the support of functions is defined and corresponding
  theorems are provided.  Since basic properties (see ~ suppval ) are based on
  the Axiom of Union (usage of ~ dmexg ), these definition and theorems cannot
  be provided earlier.  Until April 2019, the support of a function was
  represented by the expression ` ( ``' R " ( _V \ { Z } ) ) ` (see
  ~ suppimacnvfn ).  The theorems which are based on this representation and
  which are provided in previous sections could be moved into this section to
  have all related theorems in one section, although they do not depend on the
  Axiom of Union.  This was possible because they are not used before.  The
  current theorems differ from the original ones by requiring that the classes
  representing the "function" (or its "domain") and the "zero element" are
  sets.  Actually, this does not cause any problem (until now).

$)

  $c supp $.  $( Class for the support of a function. $)

  $( Extend class definition to include the support of functions. $)
  csupp $a class supp $.

  ${
    $d i x z $.
    $( Define the support of a function against a "zero" value.  The support of
       a function is the subset of its domain which is mapped to a value which
       is not equal to a designed value called the zero value.  Note that this
       definition uses not equal rather than being in terms of an apartness
       relation ( ~ df-ap or any other apartness relation), and thus is
       sometimes called "support" rather than "strong support".  It is
       therefore probably most useful when the function has a codomain which
       has decidable equality and contains the zero value.  (Contributed by AV,
       31-Mar-2019.)  (Revised by AV, 6-Apr-2019.) $)
    df-supp $a |- supp = ( x e. _V , z e. _V
                           |-> { i e. dom x | ( x " { i } ) =/= { z } } ) $.
  $}

  ${
    $d V x z $.  $d W x z $.  $d X i x z $.  $d Z i x z $.
    $( The value of the operation constructing the support of a function.
       (Contributed by AV, 31-Mar-2019.)  (Revised by AV, 6-Apr-2019.) $)
    suppval $p |- ( ( X e. V /\ Z e. W )
                -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } ) $=
      ( vx vz wcel wa cvv cv csn cima wne cdm crab csupp wceq adantr adantl a1i
      cmpo df-supp dmeq imaeq1 sneq neeq12d rabeqbidv elex dmexg rabexg ovmpod
      syl ) DBHZECHZIZFGDEJJFKZAKLZMZGKZLZNZAUQOZPZDURMZELZNZADOZPZQJQFGJJVDUBR
      UPFGAUCUAUQDRZUTERZIZVDVIRUPVLVBVGAVCVHVJVCVHRVKUQDUDSVLUSVEVAVFVJUSVERVK
      UQDURUESVKVAVFRVJUTEUFTUGUHTUNDJHUODBUISUOEJHUNECUITUPVHJHZVIJHUNVMUODBUJ
      SVGAVHJUKUMUL $.
  $}

  ${
    $d Z i $.
    $( The support of the empty set is the empty set.  (Contributed by AV,
       12-Apr-2019.) $)
    supp0 $p |- ( Z e. W -> ( (/) supp Z ) = (/) ) $=
      ( vi wcel c0 csupp co csn cima wne cdm crab cvv wceq 0ex suppval mpan dm0
      cv rabeq mp1i rab0 a1i 3eqtrd ) BADZEBFGZECSHIBHJZCEKZLZUGCELZEEMDUEUFUIN
      OCMAEBPQUHENUIUJNUERUGCUHETUAUJENUEUGCUBUCUD $.
  $}

  ${
    $d V i $.  $d W i $.  $d X i $.  $d Z i $.
    $( The value of the operation constructing the support of a function.
       (Contributed by AV, 6-Apr-2019.) $)
    suppval1 $p |- ( ( Fun X /\ X e. V /\ Z e. W )
                    -> ( X supp Z ) = { i e. dom X | ( X ` i ) =/= Z } ) $=
      ( wfun wcel w3a csupp co cv csn cima wne cdm crab cfv wceq suppval cvv wa
      3adant1 wfn funfn biimpi 3ad2ant1 fnsnfv sylan eqcomd neeq1d wb simp2 vex
      fvexg sylancl sneqbg syl adantr necon3bid bitrd rabbidva eqtrd ) DFZDBGZE
      CGZHZDEIJZDAKZLMZELZNZADOZPZVHDQZENZAVLPVDVEVGVMRVCABCDESUBVFVKVOAVLVFVHV
      LGZUAZVKVNLZVJNVOVQVIVRVJVQVRVIVFDVLUCZVPVRVIRVCVDVSVEVCVSDUDUEUFVLVHDUGU
      HUIUJVQVRVJVNEVFVRVJRVNERUKZVPVFVNTGZVTVFVDVHTGWAVCVDVEULAUMVHDBTUNUOVNET
      UPUQURUSUTVAVB $.

    $d F i $.
    $( The value of the operation constructing the support of a function with a
       given domain.  This version of ~ suppvalfn assumes ` F ` is a set rather
       than its domain ` X ` , avoiding ~ ax-coll .  (Contributed by SN,
       5-Aug-2024.) $)
    suppvalfng $p |- ( ( F Fn X /\ F e. V /\ Z e. W )
                    -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } ) $=
      ( wfn wcel w3a csupp co cv cfv wne cdm crab wfun wceq fnfun suppval1 fndm
      syl3an1 3ad2ant1 rabeqdv eqtrd ) BEGZBCHZFDHZIZBFJKZALBMFNZABOZPZUKAEPUFB
      QUGUHUJUMREBSACDBFTUBUIUKAULEUFUGULERUHEBUAUCUDUE $.
    $( $j usage 'suppvalfng' avoids 'ax-coll'; $)

    $( The value of the operation constructing the support of a function with a
       given domain.  (Contributed by Stefan O'Rear, 1-Feb-2015.)  (Revised by
       AV, 22-Apr-2019.) $)
    suppvalfn $p |- ( ( F Fn X /\ X e. V /\ Z e. W )
                    -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } ) $=
      ( wfn wcel w3a csupp co cv cfv wne cdm crab wfun cvv wceq 3ad2ant1 fnfun
      fnex 3adant3 simp3 suppval1 syl3anc fndm rabeqdv eqtrd ) BEGZECHZFDHZIZBF
      JKZALBMFNZABOZPZUOAEPUMBQZBRHZULUNUQSUJUKURULEBUATUJUKUSULECBUBUCUJUKULUD
      ARDBFUEUFUMUOAUPEUJUKUPESULEBUGTUHUI $.

    $d S i $.
    $( An element of the support of a function with a given domain.  This
       version of ~ elsuppfn assumes ` F ` is a set rather than its domain
       ` X ` , avoiding ~ ax-coll .  (Contributed by SN, 5-Aug-2024.) $)
    elsuppfng $p |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( S e. ( F supp Z )
                               <-> ( S e. X /\ ( F ` S ) =/= Z ) ) ) $=
      ( vi wfn wcel w3a csupp co cv cfv wne crab wa suppvalfng eleq2d wceq
      fveq2 neeq1d elrab bitrdi ) BEHBCIFDIJZABFKLZIAGMZBNZFOZGEPZIAEIABNZFOZQU
      EUFUJAGBCDEFRSUIULGAEUGATUHUKFUGABUAUBUCUD $.
    $( $j usage 'elsuppfng' avoids 'ax-coll'; $)

    $( An element of the support of a function with a given domain.
       (Contributed by AV, 27-May-2019.) $)
    elsuppfn $p |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( S e. ( F supp Z )
                               <-> ( S e. X /\ ( F ` S ) =/= Z ) ) ) $=
      ( vi wfn wcel w3a csupp co cv cfv wne crab wa suppvalfn eleq2d wceq fveq2
      neeq1d elrab bitrdi ) BEHECIFDIJZABFKLZIAGMZBNZFOZGEPZIAEIABNZFOZQUEUFUJA
      GBCDEFRSUIULGAEUGATUHUKFUGABUAUBUCUD $.
  $}

  ${
    $d B x y $.  $d F x y $.  $d X x y $.  $d Z x y $.
    fvdifsuppst.1 $e |- ( ph -> F : A --> B ) $.
    fvdifsupp.2 $e |- ( ph -> A e. V ) $.
    fvdifsuppst.st $e |- ( ph -> A. x e. B A. y e. B STAB x = y ) $.
    fvdifsuppst.3 $e |- ( ph -> Z e. B ) $.
    fvdifsupp.4 $e |- ( ph -> X e. ( A \ ( F supp Z ) ) ) $.
    $( Function value is zero outside of its support.  (Contributed by Thierry
       Arnoux, 21-Jan-2024.) $)
    fvdifsuppst $p |- ( ph -> ( F ` X ) = Z ) $=
      ( wceq wn wcel wa wstab wi cfv csupp co eldifbd wne df-ne eldifad wb ffnd
      elsuppfn syl3anc mpbirand biimprd biimtrrid mtod cv wral ffvelcdmd eqeq12
      wfn stbid rspc2gv syl2anc mpd df-stab sylib ) AHFUAZIOZPZPZVHAVIHFIUBUCZQ
      ZAHDVKNUDVIVGIUEZAVLVGIUFAVLVMAVLHDQZVMAHDVKNUGZAFDUTDGQIEQZVLVNVMRUHADEF
      JUIKMHFGEDIUJUKULUMUNUOAVHSZVJVHTABUPZCUPZOZSZCEUQBEUQZVQLAVGEQVPWBVQTADE
      HFJVOURMWAVQBCVGIEEVRVGOVSIORVTVHVRVGVSIUSVAVBVCVDVHVEVFVD $.
  $}

  ${
    $d R x y $.  $d Z x y $.
    $( The support of functions "defined" by inverse images expressed by binary
       relations.  (Contributed by AV, 7-Apr-2019.) $)
    cnvimadfsn $p |- ( `' R " ( _V \ { Z } ) )
                     = { x | E. y ( x R y /\ y =/= Z ) } $=
      ( ccnv cvv csn cdif cima cv wcel cop wa wex cab wbr wne dfima3 wb vex elv
      eldifvsn opelcnv df-br bitr4i anbi12ci exbii abbii eqtri ) CEZFDGHZIBJZUK
      KZULAJZLUJKZMZBNZAOUNULCPZULDQZMZBNZAOBAUJUKRUQVAAUPUTBUMUSUOURUMUSSBULDF
      UBUAUOUNULLCKURULUNCBTATUCUNULCUDUEUFUGUHUI $.
  $}

  ${
    $d F x $.  $d V x $.  $d W x $.  $d X x $.  $d Z x $.
    $( Support sets of functions expressed by inverse images.  (Contributed by
       AV, 31-Mar-2019.)  (Revised by AV, 7-Apr-2019.) $)
    suppimacnvfn $p |- ( ( F Fn X /\ F e. V /\ Z e. W )
                       -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) ) $=
      ( vx wfn wcel w3a csupp co ccnv cvv csn cdif cima cv cfv wa wb wceq simp2
      wne vex fvexg sylancl elsng syl necon3bbid biantrurd bitr3d eldif bitr4di
      wn anbi2d elsuppfng elpreima 3ad2ant1 3bitr4d eqrdv ) ADGZABHZECHZIZFAEJK
      ZALMENZOZPZVDFQZDHZVIARZEUCZSVJVKVGHZSZVIVEHVIVHHZVDVLVMVJVDVLVKMHZVKVFHZ
      UNZSZVMVDVRVLVSVDVQVKEVDVPVQVKEUATVDVBVIMHVPVAVBVCUBFUDVIABMUEUFZVKEMUGUH
      UIVDVPVRVTUJUKVKMVFULUMUOVIABCDEUPVAVBVOVNTVCDVIVGAUQURUSUT $.
  $}

  $( Two ways of writing the support of a function with known codomain.
     (Contributed by Stefan O'Rear, 9-Jul-2015.)  (Revised by AV,
     7-Jul-2019.) $)
  fsuppeq $p |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S
                  -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) ) $=
    ( wcel wa wf csupp co ccnv csn cdif cima wceq cvv adantl cin eqtrd wfn wfun
    ffn fex expcom adantr imp simplr suppimacnvfn syl3anc ffun inpreima syl wss
    wi cdm cnvimass fdm fimacnv eqtr4d sseqtrid sseqin2 sylib invdif imaeq2i ex
    eqtr3di ) CDGZFEGZHZCABIZBFJKZBLZAFMZNZOZPVJVKHZVLVMQVNNZOZVPVQBCUAZBQGZVIV
    LVSPVKVTVJCABUCRVJVKWAVHVKWAUOVIVKVHWACADBUDUEUFUGVHVIVKUHBQECFUIUJVKVSVPPV
    JVKVMAVRSZOZVSVPVKWCVMAOZVSSZVSVKBUBWCWEPCABUKAVRBULUMVKVSWDUNWEVSPVKBUPZVS
    WDBVRUQVKWFCWDCABURCABUSUTVAVSWDVBVCTWBVOVMAVNVDVEVGRTVF $.

  $( Version of ~ fsuppeq avoiding ~ ax-coll by assuming ` F ` is a set rather
     than its domain ` I ` .  (Contributed by SN, 30-Jul-2024.) $)
  fsuppeqg $p |- ( ( F e. V /\ Z e. W ) -> ( F : I --> S
                   -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) ) $=
    ( wcel wa wf csupp co ccnv csn cdif cima wceq cvv adantl cin eqtrd wfn wfun
    ffn simpll simplr suppimacnvfn syl3anc inpreima syl wss cdm cnvimass eqtr4d
    ffun fdm fimacnv sseqtrid sseqin2 sylib invdif imaeq2i eqtr3di ex ) BDGZFEG
    ZHZCABIZBFJKZBLZAFMZNZOZPVFVGHZVHVIQVJNZOZVLVMBCUAZVDVEVHVOPVGVPVFCABUCRVDV
    EVGUDVDVEVGUEBDECFUFUGVGVOVLPVFVGVIAVNSZOZVOVLVGVRVIAOZVOSZVOVGBUBVRVTPCABU
    NAVNBUHUIVGVOVSUJVTVOPVGBUKZVOVSBVNULVGWACVSCABUOCABUPUMUQVOVSURUSTVQVKVIAV
    JUTVAVBRTVC $.
  $( $j usage 'fsuppeqg' avoids 'ax-coll'; $)

  ${
    $d F i $.  $d Z i $.
    $( The support of a function is a subset of the function's domain.
       (Contributed by AV, 30-May-2019.) $)
    suppssdmg $p |- ( ( F e. V /\ Z e. W ) -> ( F supp Z ) C_ dom F ) $=
      ( vi wcel wa csupp co cv csn cima wne cdm crab suppval ssrab2 eqsstrdi )
      ABFDCFGADHIAEJKLDKMZEANZOTEBCADPSETQR $.
  $}

  ${
    $d F x $.  $d X x $.  $d Z x $.
    suppsnop.f $e |- F = { <. X , Y >. } $.
    suppsnopdc.x $e |- ( ph -> X e. V ) $.
    suppsnopdc.y $e |- ( ph -> Y e. W ) $.
    suppsnopdc.z $e |- ( ph -> Z e. U ) $.
    suppsnopdc.dc $e |- ( ph -> DECID Y = Z ) $.
    $( The support of a singleton of an ordered pair.  (Contributed by AV,
       12-Apr-2019.) $)
    suppsnopdc $p |- ( ph
                   -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) ) $=
      ( vx csn wne c0 wceq cvv wcel csupp co cv cima cdm crab cif wf w3a cop wa
      wf1o f1osng f1of syl 3adant3 feq1i sylibr syl3anc snexg fexd suppval fdmd
      syl2anc rabeqdv sneq imaeq2d neeq1d rabsnif eqtrdi cfv wfn snidg 3ad2ant1
      wn ffnd fnsnfv eqcomd fveq1i fvsng eqtrid sneqd wb sneqbg 3ad2ant2 3bitrd
      necon3abid ifbid wdc ifnotdc eqtrd 3eqtrd ) ACHUAUBZCNUCZOZUDZHOZPZNCUEZU
      FZCFOZUDZWQPZXAQUGZGHRZQXAUGZACSTHBTZWMWTRAXAGOZSCAFDTZGETZXGXAXHCUHZJKLX
      IXJXGUIZXAXHFGUJOZUHZXKXIXJXNXGXIXJUKXAXHXMULXNFGDEUMXAXHXMUNUOUPXAXHCXMI
      UQURZUSAXIXASTJFDUTUOVALNSBCHVBVDAXIXJXGWTXDRJKLXLWTWRNXAUFXDXLWRNWSXAXLX
      AXHCXOVCVEWRXCNFWNFRZWPXBWQXPWOXACWNFVFVGVHVIVJUSAXDXEVOZXAQUGZXFAXIXJXGX
      DXRRJKLXLXCXQXAQXLXCFCVKZOZWQPXHWQPXQXLXBXTWQXLCXAVLZFXATZXBXTRXLXAXHCXOV
      PXIXJYBXGFDVMVNYAYBUKXTXBXAFCVQVRVDVHXLXTXHWQXLXSGXLXSFXMVKZGFCXMIVSXIXJY
      CGRXGFGDEVTUPWAWBVHXLXEXHWQXJXIXHWQRXEWCXGGHEWDWEWGWFWHUSAXEWIXRXFRMXEXAQ
      WJUOWKWL $.
  $}

  $( If the function value for a given argument is not empty, the argument
     belongs to the support of the function with the empty set as zero.
     (Contributed by AV, 2-Jul-2019.)  (Revised by AV, 4-Apr-2020.) $)
  fvn0elsupp $p |- ( ( ( B e. V /\ X e. B )
                       /\ ( G Fn B /\ ( G ` X ) =/= (/) ) )
                       -> X e. ( G supp (/) ) ) $=
    ( wcel wa wfn cfv c0 wne csupp co simpr anim12i cvv simprl simpll 0ex a1i
    wb elsuppfn syl3anc mpbird ) ACEZDAEZFZBAGZDBHIJZFZFZDBIKLEZUEUHFZUFUEUIUHU
    DUEMUGUHMNUJUGUDIOEZUKULTUFUGUHPUDUEUIQUMUJRSDBCOAIUAUBUC $.

  $( The function value for a given argument is not empty iff the argument
     belongs to the support of the function with the empty set as zero.
     (Contributed by AV, 4-Apr-2020.) $)
  fvn0elsuppb $p |- ( ( B e. V /\ X e. B /\ G Fn B )
                      -> ( ( G ` X ) =/= (/) <-> X e. ( G supp (/) ) ) ) $=
    ( wcel wfn w3a cfv c0 wne csupp co wi fvn0elsupp exp43 3imp wa cvv wb simp3
    simp1 0ex a1i elsuppfn syl3anc simpr biimtrdi impbid ) ACEZDAEZBAFZGZDBHIJZ
    DBIKLEZUIUJUKUMUNMUIUJUKUMUNABCDNOPULUNUJUMQZUMULUKUIIREZUNUOSUIUJUKTUIUJUK
    UAUPULUBUCDBCRAIUDUEUJUMUFUGUH $.

  ${
    $d F x $.  $d V x $.  $d W x $.  $d X x $.  $d Z x $.
    $( Existential quantification restricted to a support.  (Contributed by
       Stefan O'Rear, 23-Mar-2015.)  (Revised by AV, 27-May-2019.) $)
    rexsupp $p |- ( ( F Fn X /\ X e. V /\ Z e. W )
                    -> ( E. x e. ( F supp Z ) ph
                         <-> E. x e. X ( ( F ` x ) =/= Z /\ ph ) ) ) $=
      ( wfn wcel w3a cv cfv wne wa csupp co elsuppfn anbi1d anass bitrdi
      rexbidv2 ) CFHFDIGEIJZABKZCLGMZANZBCGOPZFUBUCUFIZANUCFIZUDNZANUHUENUBUGUI
      AUCCDEFGQRUHUDASTUA $.
  $}

  ${
    $d B b $.  $d F b $.  $d V b $.  $d W b $.  $d Z b $.
    $( The support of the restriction of a function is a subset of the support
       of the function itself.  (Contributed by AV, 22-Apr-2019.) $)
    ressuppss $p |- ( ( F e. V /\ Z e. W )
                      -> ( ( F |` B ) supp Z ) C_ ( F supp Z ) ) $=
      ( vb wcel wa csn cima wne cdm crab csupp cab eleq2s ad2antrl wceq adantld
      co cv cin elinel2 dmres elinel1 wss snssi resima2 syl neeq1d biimpd mpcom
      cres jca ex ss2abdv df-rab 3sstr4g cvv resexg suppval sylan 3sstr4d ) BCG
      ZEDGZHZBAUMZFUAZIZJZEIZKZFVGLZMZBVIJZVKKZFBLZMZVGENTZBENTVFVHVMGZVLHZFOVH
      VQGZVPHZFOVNVRVFWAWCFVFWAWCVFWAHZWBVPVTWBVFVLWBVHAVQUBZVMVHAVQUCBAUDZPQVH
      AGZWDVPVTWGVFVLWGVHWEVMVHAVQUEWFPQWGWAVPVFWGVLVPVTWGVLVPWGVJVOVKWGVIAUFVJ
      VORVHAUGBVIAUHUIUJUKSSULUNUOUPVLFVMUQVPFVQUQURVDVGUSGVEVSVNRBACUTFUSDVGEV
      AVBFCDBEVAVC $.
  $}

  ${
    $d A x $.  $d Z x $.
    mptsuppdifd.f $e |- F = ( x e. A |-> B ) $.
    mptsuppdifd.a $e |- ( ph -> A e. V ) $.
    mptsuppdifd.z $e |- ( ph -> Z e. W ) $.
    $( The support of a function in maps-to notation with a class difference.
       (Contributed by AV, 28-May-2019.) $)
    mptsuppdifd $p |- ( ph
                        -> ( F supp Z ) = { x e. A | B e. ( _V \ { Z } ) } ) $=
      ( csupp co ccnv cvv csn cdif cima wcel crab cdm wfn wceq wfun funmpt2 a1i
      funfnd cmpt mptexd eqeltrid suppimacnvfn syl3anc mptpreima eqtrdi ) AEHLM
      ZENOHPQZRZDUPSBCTAEEUAZUBEOSHGSUOUQUCAEEUDABCDEIUEUFUGAEBCDUHOIABCDFJUIUJ
      KEOGURHUKULBCDUPEIUMUN $.

    $d ph x $.
    mptsuppd.b $e |- ( ( ph /\ x e. A ) -> B e. U ) $.
    $( The support of a function in maps-to notation.  (Contributed by AV,
       10-Apr-2019.)  (Revised by AV, 28-May-2019.) $)
    mptsuppd $p |- ( ph -> ( F supp Z ) = { x e. A | B =/= Z } ) $=
      ( csupp co cvv csn wcel crab wa cdif wne mptsuppdifd cv eldifsn biantrurd
      elexd bitr4id rabbidva eqtrd ) AFINODPIQUARZBCSDIUBZBCSABCDFGHIJKLUCAUKUL
      BCABUDCRTZUKDPRZULTULDPIUEUMUNULUMDEMUGUFUHUIUJ $.
  $}

  ${
    $d A x y $.  $d B y $.  $d F x y $.  $d G x y $.  $d V y $.  $d W y $.
    $d Z x y $.
    $( The support of a function which has the same zero values (in its domain)
       as another function is a subset of the support of this other function.
       (Contributed by AV, 30-Apr-2019.)  (Proof shortened by AV,
       6-Jun-2022.) $)
    suppfnss $p |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) )
                     -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z )
                          -> ( F supp Z ) C_ ( G supp Z ) ) ) $=
      ( vy wfn wa wss wcel cv cfv wceq wi csupp ad2antrr cvv w3a wral co simpr1
      wne cdm crab fndm ad2antlr 3sstr4d adantr wb eleq2d fveqeq2 imbi12d rspcv
      weq biimtrdi com23 imp31 necon3d ex 3imp rabssrabd wfun fnfun simpl ssexg
      3adant3 fnex syl2an simpr3 suppval1 syl3anc simpr simp2 sseq12d mpbird )
      DBJZECJZKZBCLZCFMZHGMZUAZKZANZEOHPZWGDOHPZQZABUBZDHRUCZEHRUCZLZWFWKKZWNIN
      ZDOZHUEZIDUFZUGZWPEOZHUEZIEUFZUGZLZWOWRXBIWSXCWFWSXCLWKWFBCWSXCWAWBWCWDUD
      VSWSBPVTWEBDUHZSVTXCCPVSWECEUHUIUJUKWOWRWPWSMZXBWOXGWRXBWOXGWRXBQWOXGKXAH
      WQHWFWKXGXAHPZWQHPZQZWFXGWKXJWFXGWPBMZWKXJQVSXGXKULVTWEVSWSBWPXFUMSWJXJAW
      PBAIUQWHXHWIXIWGWPHEUNWGWPHDUNUOUPURUSUTVAVBUSVCVDWFWNXEULWKWFWLWTWMXDWFD
      VEZDTMZWDWLWTPVSXLVTWEBDVFSWAVSBTMZXMWEVSVTVGWBWCXNWDBCFVHVIBTDVJVKWAWBWC
      WDVLZITGDHVMVNWFEVEZETMZWDWMXDPVTXPVSWECEVFUIWAVTWCXQWEVSVTVOWBWCWDVPCFEV
      JVKXOITGEHVMVNVQUKVRVB $.
  $}

  ${
    $d F x $.  $d G x $.  $d Z x $.  $d V x $.  $d f i z $.
    $( The support of a function which is a subset of another function is a
       subset of the support of this other function.  (Contributed by AV,
       27-Jul-2019.) $)
    funsssuppss $p |- ( ( Fun G /\ F C_ G /\ G e. V )
                        -> ( F supp Z ) C_ ( G supp Z ) ) $=
      ( vx vf vz vi wfun wss wcel w3a csupp co cv wa cvv cdm wceq adantr csn wi
      cima wne crab df-supp elmpocl2 wfn cfv wral funss impcom funfnd funfn jca
      biimpi 3adant3 dmss 3ad2ant2 dmexg 3ad2ant3 simpr 3jca 3expa eqeq1 biimpd
      funssfv syl ralrimiva suppfnss sylc sylan2 sseldd ex ssrdv ) BIZABJZBCKZL
      ZEADMNZBDMNZVSEOZVTKZWBWAKVSWCPVTWAWBWCVSDQKZVTWAJZFGQQFOZHOUAUCGOUAUDHWF
      RUEADMWBFGHUFUGVSWDPZAARZUHZBBRZUHZPZWHWJJZWJQKZWDLZPWBBUIZDSZWBAUIZDSZUB
      ZEWHUJZWEWGWLWOVSWLWDVPVQWLVRVPVQPZWIWKXBAVQVPAIABUKULUMVPWKVQVPWKBUNUPTU
      OUQTWGWMWNWDVSWMWDVQVPWMVRABURUSTVSWNWDVRVPWNVQBCUTVATVSWDVBVCUOVSXAWDVPV
      QXAVRXBWTEWHXBWBWHKZPWPWRSZWTVPVQXCXDWBBAVGVDXDWQWSWPWRDVEVFVHVIUQTEWHWJA
      BQQDVJVKVLVSWCVBVMVNVO $.
  $}

  ${
    $d B x $.  $d Z x $.  $d Z y $.  $d f q z $.
    $( The support of a constant function with value zero is empty.
       (Contributed by AV, 30-Jun-2019.) $)
    fczsupp0 $p |- ( ( B X. { Z } ) supp Z ) = (/) $=
      ( vx vf vz vq csn cxp csupp co cv wcel cfv wceq cvv wne cdm simprd simpld
      cima crab df-supp elmpocl wa wfn fnconstg syl elsuppfng syl3anc fvconst2g
      wb ibi syl2anc neneqd pm2.65i nel0 ) CABGHZBIJZCKZURLZUSUQMZBNZUTBOLZUSAL
      ZVBUTUQOLZVCDEOODKZFKGTEKGPFVFQUAUQBIUSDEFUBUCZRZUTVDVABPZUTVDVIUDZUTUQAU
      EZVEVCUTVJUKUTVCVKVHABOUFUGUTVEVCVGSVHUSUQOOABUHUIULZSABUSOUJUMUTVABUTVDV
      IVLRUNUOUP $.
  $}

  ${
    $d k F $.  $d k ph $.  $d k x W $.  $d k Z $.  $d f i z $.  $d A x $.
    suppss.f $e |- ( ph -> F : A --> B ) $.
    suppss.n $e |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z ) $.
    suppssdc.dc $e |- ( ph -> A. x e. A DECID x e. W ) $.
    $( Show that the support of a function is contained in a set.  (Contributed
       by Mario Carneiro, 19-Dec-2014.)  (Revised by AV, 28-May-2019.)  (Proof
       shortened by SN, 5-Aug-2024.) $)
    suppssdc $p |- ( ph -> ( F supp Z ) C_ W ) $=
      ( vf vz vi csupp cv wcel cvv wa wi co csn wne cdm df-supp elmpocl cfv wfn
      cima crab wb ffnd adantl simpll simplr elsuppfng syl3anc wdc eleq1w dcbid
      wceq wral ad2antlr simpr rspcdva wn cdif adantll sylan2br expr necon1addc
      eldif a1d mpd expimpd sylbid expcom com23 mpdi ssrdv ) AEFHOUAZGAEPZWAQZF
      RQZHRQZSZWBGQZLMRRLPZNPUBUIMPUBUCNWHUDUJFHOWBLMNUEUFAWFWCWGWFAWCWGTWFASZW
      CWBCQZWBFUGZHUCZSZWGWIFCUHZWDWEWCWMUKAWNWFACDFIULUMWDWEAUNWDWEAUOWBFRRCHU
      PUQWIWJWLWGWIWJSZWGURZWLWGTWOBPZGQZURZWPBCWBWQWBVAWRWGBEGUSUTAWSBCVBWFWJK
      VCWIWJVDVEWOWGWKHWOWGVFZWKHVAZTWPWIWJWTXAWJWTSWIWBCGVGQZXAWBCGVLAXBXAWFJV
      HVIVJVMVKVNVOVPVQVRVSVT $.
    $( $j usage 'suppssdc' avoids 'ax-coll'; $)
  $}

  ${
    $d B u v $.  $d F u v $.  $d X u v $.  $d Z v $.
    suppssr.f $e |- ( ph -> F : A --> B ) $.
    suppssr.n $e |- ( ph -> ( F supp Z ) C_ W ) $.
    suppssr.a $e |- ( ph -> A e. V ) $.
    suppssrst.z $e |- ( ph -> Z e. B ) $.
    suppssrst.st $e |- ( ph -> A. u e. B A. v e. B STAB u = v ) $.
    $( A function is zero outside its support.  (Contributed by Mario Carneiro,
       19-Dec-2014.)  (Revised by AV, 28-May-2019.) $)
    suppssrst $p |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z ) $=
      ( wcel wn wa wceq cvv cdif cfv eldif wne df-ne fexd fvexg sylan biantrurd
      csn eldifsn bitr4di csupp co wfn wb elsuppfn syl3anc pm5.32da bitrd sseld
      ffnd sylbird expdimp sylbid biimtrrid con3d wstab wi cv eqeq2 stbid eqeq1
      wral ralbidv adantr ffvelcdmda rspcdva df-stab sylib syld impr sylan2b )
      IDHUAPAIDPZIHPZQZRIFUBZJSZIDHUCAWDWFWHAWDRZWFWHQZQZWHWIWJWEWJWGJUDZWIWEWG
      JUEWIWLWGTJUJUAPZWEWIWLWGTPZWLRWMWIWNWLAFTPWDWNADEGFKMUFIFTDUGUHUIWGTJUKU
      LZAWDWMWEAWDWMRZIFJUMUNZPZWEAWRWDWLRZWPAFDUODGPJEPZWRWSUPADEFKVBMNIFGEDJU
      QURAWDWLWMWOUSUTAWQHILVAVCVDVEVFVGWIWHVHZWKWHVIWIWGBVJZSZVHZXABEJXBJSXCWH
      XBJWGVKVLWICVJZXBSZVHZBEVNZXDBEVNCEWGXEWGSZXGXDBEXIXFXCXEWGXBVMVLVOAXHCEV
      NWDOVPADEIFKVQVRAWTWDNVPVRWHVSVTWAWBWC $.
  $}

  ${
    $d B u v $.  $d F u v $.  $d X u v $.  $d Z v $.
    suppssrg.f $e |- ( ph -> F : A --> B ) $.
    suppssrg.n $e |- ( ph -> ( F supp Z ) C_ W ) $.
    suppssrg.a $e |- ( ph -> F e. V ) $.
    suppssrgst.z $e |- ( ph -> Z e. B ) $.
    suppssrgst.st $e |- ( ph -> A. u e. B A. v e. B STAB u = v ) $.
    $( A function is zero outside its support.  Version of ~ suppssrst avoiding
       ~ ax-coll by assuming ` F ` is a set rather than its domain ` A ` .
       (Contributed by SN, 5-May-2024.) $)
    suppssrgst $p |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z ) $=
      ( wcel wn wa wceq cvv cfv eldif wne df-ne csn fvexg sylan eldifsn bitr4di
      cdif biantrurd csupp co wfn ffnd elsuppfng syl3anc pm5.32da bitrd sylbird
      wb sseld expdimp sylbid biimtrrid con3d wstab wi eqeq2 stbid wral ralbidv
      cv eqeq1 adantr ffvelcdmda rspcdva df-stab sylib syld impr sylan2b ) IDHU
      JPAIDPZIHPZQZRIFUAZJSZIDHUBAWCWEWGAWCRZWEWGQZQZWGWHWIWDWIWFJUCZWHWDWFJUDW
      HWKWFTJUEUJPZWDWHWKWFTPZWKRWLWHWMWKAFGPZWCWMMIFGDUFUGUKWFTJUHUIZAWCWLWDAW
      CWLRZIFJULUMZPZWDAWRWCWKRZWPAFDUNWNJEPZWRWSVAADEFKUOMNIFGEDJUPUQAWCWKWLWO
      URUSAWQHILVBUTVCVDVEVFWHWGVGZWJWGVHWHWFBVMZSZVGZXABEJXBJSXCWGXBJWFVIVJWHC
      VMZXBSZVGZBEVKZXDBEVKCEWFXEWFSZXGXDBEXIXFXCXEWFXBVNVJVLAXHCEVKWCOVOADEIFK
      VPVQAWTWCNVOVQWGVRVSVTWAWB $.
    $( $j usage 'suppssrgst' avoids 'ax-coll'; $)
  $}

  ${
    $d ph x y $.  $d D x y $.  $d Y x $.  $d Z x y $.  $d A y $.  $d F y $.
    $d L y $.  $d f i z $.
    suppssfv.a $e |- ( ph -> ( ( x e. D |-> A ) supp Y ) C_ L ) $.
    suppssfv.f $e |- ( ph -> ( F ` Y ) = Z ) $.
    suppssfv.v $e |- ( ( ph /\ x e. D ) -> A e. V ) $.
    suppssfv.y $e |- ( ph -> Y e. U ) $.
    suppssfvg.d $e |- ( ph -> D e. W ) $.
    $( Formula building theorem for support restriction, on a function which
       preserves zero.  (Contributed by Stefan O'Rear, 9-Mar-2015.)  (Revised
       by AV, 28-May-2019.) $)
    suppssfvg $p |- ( ph -> ( ( x e. D |-> ( F ` A ) ) supp Z ) C_ L ) $=
      ( cv wcel wa cvv vy vf vz vi cfv cmpt csupp wss adantr elexd csn cima wne
      co crab df-supp elmpocl2 adantl simpl cdif eldifsni ad4ant23 wceq fveqeq2
      cdm wi syl5ibrcom necon3d ad2antlr eldifsn sylanbrc ex syl5 ss2rabdv eqid
      imp simpll simplr mptsuppdifd 3sstr4d sstrd syl21anc simpr sseldd ssrdv )
      AUABDCFUEZUFZKUGUNZGAUAQZWHRZWIGRAWJSZWHGWIWKDTRZKTRZAWHGUHWKDIADIRWJPUIU
      JWJWMAUBUCTTUBQZUDQUKULUCQUKUMUDWNVEUOWGKUGWIUBUCUDUPUQURAWJUSWLWMSZASZWH
      BDCUFZJUGUNZGWPWFTKUKUTRZBDUOCTJUKUTRZBDUOWHWRWPWSWTBDWSWFKUMZWPBQDRZSZWT
      WFTKVAXCXAWTXCXASCTRZCJUMZWTAXBXDWOXAAXBSCHNUJVBXCXAXEAXAXEVFWOXBACJWFKAW
      FKVCCJVCJFUEKVCMCJKFVDVGVHVIVPCTJVJVKVLVMVNWPBDWFWGTTKWGVOWLWMAVQZWLWMAVR
      VSWPBDCWQTEJWQVOXFAJERWOOURVSVTAWRGUHWOLURWAWBAWJWCWDVLWE $.
  $}

  ${
    $d A x y $.  $d B x $.  $d F x y $.  $d G x y $.  $d X x y $.  $d Z x y $.
    $d ph x y $.  $d B u v $.  $d ph u v $.
    suppofssd.1 $e |- ( ph -> A e. V ) $.
    suppofssd.2 $e |- ( ph -> Z e. B ) $.
    suppofssd.3 $e |- ( ph -> F : A --> B ) $.
    suppofssd.4 $e |- ( ph -> G : A --> B ) $.
    suppofss1dcl.cl $e |- ( ( ph /\ ( u e. B /\ v e. B ) )
      -> ( u X v ) e. B ) $.

    ${
      $d F u v y $.  $d G v $.  $d X u v $.
      suppofss1d.5 $e |- ( ( ph /\ x e. B ) -> ( Z X x ) = Z ) $.
      $( Condition for the support of a function operation to be a subset of
         the support of the left function term.  (Contributed by Thierry
         Arnoux, 21-Jun-2019.) $)
      suppofss1dcl $p |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) ) $=
        ( wceq co wcel vy cv cfv cof wi wral csupp wss inidm eqidd oveq2 eleq1d
        wa ffnd oveq1 ralbidv ralrimivva adantr ffvelcdmda rspcdva ofvalg simpr
        oveq1d ralrimiva oveq2d eqeq1d rspcdv mpd 3eqtrd wfn off ssidd suppfnss
        ex syl23anc ) AUAUBZGUCZKRZVPGHJUDSZUCZKRZUEZUAEUFZVSKUGSGKUGSUHZAWBUAE
        AVPETZUMZVRWAWFVRUMZVTVQVPHUCZJSZKWHJSZKWFVTWIRVRAEEVQWHJEFGHIIVPAEFGNU
        NZAEFHOUNLLEUIZWFVQUJWFWHUJWFVQCUBZJSZFTZWIFTCFWHWMWHRWNWIFWMWHVQJUKULW
        FDUBZWMJSZFTZCFUFZWOCFUFDFVQWPVQRZWRWOCFWTWQWNFWPVQWMJUOULUPAWSDFUFWEAW
        RDCFFPUQURAEFVPGNUSUTAEFVPHOUSZUTVAURWGVQKWHJWFVRVBVCWFWJKRZVRWFKBUBZJS
        ZKRZBFUFZXBAXFWEAXEBFQVDURWFXEXBBWHFXAWFXCWHRZUMZXDWJKXHXCWHKJWFXGVBVEV
        FVGVHURVIVNVDAVSEVJGEVJEEUHEITKFTWCWDUEAEFVSADCEEEJFFFGHIIPNOLLWLVKUNWK
        AEVLLMUAEEVSGIFKVMVOVH $.
    $}

    ${
      $d F u v y $.  $d G v $.  $d X u v $.
      suppofss2d.5 $e |- ( ( ph /\ x e. B ) -> ( x X Z ) = Z ) $.
      $( Condition for the support of a function operation to be a subset of
         the support of the right function term.  (Contributed by Thierry
         Arnoux, 21-Jun-2019.) $)
      suppofss2dcl $p |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) ) $=
        ( wceq co wcel vy cv cfv cof wi wral csupp wss inidm eqidd oveq2 eleq1d
        wa ffnd oveq1 ralbidv ralrimivva adantr ffvelcdmda rspcdva ofvalg simpr
        oveq2d ralrimiva oveq1d eqeq1d rspcdv mpd 3eqtrd wfn off ssidd suppfnss
        ex syl23anc ) AUAUBZHUCZKRZVPGHJUDSZUCZKRZUEZUAEUFZVSKUGSHKUGSUHZAWBUAE
        AVPETZUMZVRWAWFVRUMZVTVPGUCZVQJSZWHKJSZKWFVTWIRVRAEEWHVQJEFGHIIVPAEFGNU
        NAEFHOUNZLLEUIZWFWHUJWFVQUJWFWHCUBZJSZFTZWIFTCFVQWMVQRWNWIFWMVQWHJUKULW
        FDUBZWMJSZFTZCFUFZWOCFUFDFWHWPWHRZWRWOCFWTWQWNFWPWHWMJUOULUPAWSDFUFWEAW
        RDCFFPUQURAEFVPGNUSZUTAEFVPHOUSUTVAURWGVQKWHJWFVRVBVCWFWJKRZVRWFBUBZKJS
        ZKRZBFUFZXBAXFWEAXEBFQVDURWFXEXBBWHFXAWFXCWHRZUMZXDWJKXHXCWHKJWFXGVBVEV
        FVGVHURVIVNVDAVSEVJHEVJEEUHEITKFTWCWDUEAEFVSADCEEEJFFFGHIIPNOLLWLVKUNWK
        AEVLLMUAEEVSHIFKVMVOVH $.
    $}
  $}

  ${
    $d F x $.  $d G x $.  $d V x $.  $d W x $.  $d Z x $.  $d f i z $.
    $( The support of the composition of two functions is the inverse image by
       the inner function of the support of the outer function.  (Contributed
       by AV, 30-May-2019.)  (Revised by SN, 15-Sep-2023.) $)
    suppcofn $p |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) ->
                         ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) ) $=
      ( vf vz vi wcel wa wfun csupp ccnv cima cvv cv csn cdm wfn funfnd vx ccom
      co wi wne crab df-supp elmpocl2 a1i cfv simprr elpreima syl simplbda cdif
      wb ex funco adantl adantr coexg ad2antrr simpr suppimacnvfn syl3anc cnvco
      wceq imaeq1i imaco simprl simplll imaeq2d eqtr4id 3eqtrd eleq2d pm5.21ndd
      eqrdv ) ACIZBDIZJZAKZBKZJZJZUAABUBZELUCZBMZAELUCZNZWDEOIZUAPZWFIZWKWIIZWL
      WJUDWDFGOOFPZHPQNGPQUEHWNRUFZWEELWKFGHUGZUHUIWDWMWJWDWMJWKBUJZWHIZWJWDWMW
      KBRZIZWRWDBWSSWMWTWRJUPWDBVTWAWBUKTWSWKWHBULUMUNFGOOWOAELWQWPUHUMUQWDWJWL
      WMUPWDWJJZWFWIWKXAWFWEMZOEQUOZNZWGAMZUBZXCNZWIXAWEWERZSZWEOIZWJWFXDVGWDXI
      WJWDWEWCWEKVTABURUSTUTVTXJWCWJABCDVAVBWDWJVCZWEOOXHEVDVEXDXGVGXAXBXFXCABV
      FVHUIXAXGWGXEXCNZNWIWGXEXCVIXAWHXLWGXAAARZSZVRWJWHXLVGWDXNWJWDAVTWAWBVJTU
      TVRVSWCWJVKXKACOXMEVDVEVLVMVNVOUQVPVQ $.
  $}

  $( The support of the composition of two functions is empty if the support of
     the outer function is empty.  (Contributed by AV, 30-May-2019.) $)
  supp0cosupp0fn $p |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) )
                  -> ( ( F supp Z ) = (/) -> ( ( F o. G ) supp Z ) = (/) ) ) $=
    ( wcel wa wfun csupp co wceq ccom ccnv cima suppcofn imaeq2 eqtrdi sylan9eq
    c0 ima0 ex ) ACFBDFGAHBHGGZAEIJZSKZABLEIJZSKUBUDUEBMZUCNZSABCDEOUDUGUFSNSUC
    SUFPUFTQRUA $.

  $( The image of the support of the composition of two functions is the
     support of the outer function.  (Contributed by AV, 30-May-2019.) $)
  imacosuppfn $p |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) )
                    -> ( ( Fun G /\ ( F supp Z ) C_ ran G )
                         -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) ) $=
    ( wcel wa wfun csupp crn wss ccom cima wceq ccnv suppcofn imaeq2d cdm wfo
    co funforn foimacnv sylanb sylan9eq ex ) ACFBDFGAHBHZGGZUFAEITZBJZKZGZBABLE
    ITZMZUHNUGUKUMBBOUHMZMZUHUGULUNBABCDEPQUFBRZUIBSUJUOUHNBUAUPUIUHBUBUCUDUE
    $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Special maps-to operations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The following theorems are about maps-to operations (see ~ df-mpo ) where
  the domain of the second argument depends on the domain of the first
  argument, especially when the first argument is a pair and the base set of
  the second argument is the first component of the first argument, in short
  "x-maps-to operations".  For labels, the abbreviations "mpox" are used
  (since "x" usually denotes the first argument).  This is in line with the
  currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
  ~ fmpox ).  If the first argument is an ordered pair, as in the following,
  the abbreviation is extended to "mpoxop", and the maps-to operations are
  called "x-op maps-to operations" for short.

$)

  ${
    $d x A $.  $d x C $.  $d x D $.
    opeliunxp2f.f $e |- F/_ x E $.
    opeliunxp2f.e $e |- ( x = C -> B = E ) $.
    $( Membership in a union of Cartesian products, using bound-variable
       hypothesis for ` E ` instead of distinct variable conditions as in
       ~ opeliunxp2 .  (Contributed by AV, 25-Oct-2020.) $)
    opeliunxp2f $p |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <->
      ( C e. A /\ D e. E ) ) $=
      ( cop cv csn cxp ciun wcel cvv wa wbr wrel wb nfel2 df-br relxp brrelex1i
      wral rgenw reliun mpbir sylbir elex adantr nfiu1 nfv nfan nfbi wceq opeq1
      eleq1d eleq1 eleq2d anbi12d bibi12d opeliunxp vtoclg1f pm5.21nii ) DEIZAB
      AJZKZCLZMZNZDONZDBNZEFNZPZVJDEVIQVKDEVIUADEVIVIRVHRZABUDVOABVGCUBUEABVHUF
      UGUCUHVLVKVMDBUIUJVFEIZVINZVFBNZECNZPZSVJVNSADOVJVNAAVEVIABVHUKTVLVMAVLAU
      LAEFGTUMUNVFDUOZVQVJVTVNWAVPVEVIVFDEUPUQWAVRVLVSVMVFDBURWACFEHUSUTVAABCEV
      BVCVD $.
  $}

  ${
    $d x y $.  $d K x $.  $d V x $.  $d W x $.
    mpoxopn0yelv.f $e |- F = ( x e. _V , y e. ( 1st ` x ) |-> C ) $.
    $( If there is an element of the value of an operation given by a maps-to
       rule, where the first argument is a pair and the base set of the second
       argument is the first component of the first argument, then the second
       argument is an element of the first component of the first argument.
       (Contributed by Alexander van der Vekens, 10-Oct-2017.) $)
    mpoxopn0yelv $p |- ( ( V e. X /\ W e. Y )
                          -> ( N e. ( <. V , W >. F K ) -> K e. V ) ) $=
      ( cop co wcel c1st cfv wa cvv cv csn cxp ciun dmmpossx wrel mpofun funrel
      cdm wfun ax-mp relelfvdm df-ov eleq2s sselid fveq2 opeliunxp2 simprbi syl
      mpan op1stg eleq2d imbitrid ) FGHLZEDMZNZEVBOPZNZGINHJNQZEGNVDVBELZARASZT
      VIOPZUAUBZNZVFVDDUGZVKVHABRVJCDKUCVHVMNZFVHDPZVCDUDZFVONVNDUHVPABRVJCDKUE
      DUFUIFVHDUJURVBEDUKULUMVLVBRNVFARVJVBEVEVIVBOUNUOUPUQVGVEGEGHIJUSUTVA $.
  $}

  ${
    $d K n x y $.  $d V n x y $.  $d W n x y $.  $d X n x y $.  $d Y n x y $.
    mpoxopoveq.f $e |- F = ( x e. _V , y e. ( 1st ` x )
                            |-> { n e. ( 1st ` x ) | ph } ) $.
    $( Value of an operation given by a maps-to rule, where the first argument
       is a pair and the base set of the second argument is the first component
       of the first argument.  (Contributed by Alexander van der Vekens,
       11-Oct-2017.) $)
    mpoxopoveq $p |- ( ( ( V e. X /\ W e. Y ) /\ K e. V )
                 -> ( <. V , W >. F K )
                      = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } ) $=
      ( wcel wa cvv cv c1st wceq adantr adantl nfcv vz cop crab wsbc cmpo fveq2
      cfv op1stg sylan9eqr adantrr sbceq1a bitrd rabeqbidv opexg simpr ad2antrr
      a1i wb rabexg wnf equid nfvd ax-mp nfsbc1v nfrabw nfsbc ovmpodxf ) GILZHJ
      LZMZFGLZMZBCGHUBZFNBOZPUGZADVOUCZACFUDZBVMUDZDGUCZEGNEBCNVOVPUEQVLKUQVLVN
      VMQZCOFQZMZMZAVRDVOGVLVTVOGQWAVTVLVOVMPUGZGVNVMPUFVJWDGQVKGHIJUHRUIZUJWCA
      VQVRWBAVQURZVLWAWFVTACFUKSSWBVQVRURZVLVTWGWAVQBVMUKRSULUMWEVJVMNLVKGHIJUN
      RVJVKUOVHVSNLVIVKVRDGIUSUPUAOZWHQZVLBUTUAVAZWIVLBVBVCWIVLCUTWJWIVLCVBVCCV
      MTZBFTVRBDGVQBVMVDBGTVEVRCDGVQCBVMWKACFVDVFCGTVEVG $.

    $d N x y $.
    $( Element of the value of an operation given by a maps-to rule, where the
       first argument is a pair and the base set of the second argument is the
       first component of the first argument.  (Contributed by Alexander van
       der Vekens and Mario Carneiro, 10-Oct-2017.) $)
    mpoxopovel $p |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K )
                <-> ( K e. V /\ N e. V /\ [. <. V , W >. / x ].
                      [. K / y ]. [. N / n ]. ph ) ) ) $=
      ( wcel wa cop co wsbc crab sbccom bitri w3a cv c1st mpoxopn0yelv pm4.71rd
      mpoxopoveq eleq2d nfcv elrabsf sbcbii anbi2i bitrdi pm5.32da 3anass bitrd
      cfv bitr4di ) HJMIKMNZGHIOZFEPZMZFHMZVANZVBGHMZADGQCFQZBUSQZUAZURVAVBBCAD
      BUBUCUPREFGHIJKLUDUEURVCVBVDVFNZNVGURVBVAVHURVBNZVAGACFQZBUSQZDHRZMZVHVIU
      TVLGABCDEFHIJKLUFUGVMVDVKDGQZNVHVKDGHDHUHUIVNVFVDVNVJDGQZBUSQVFVJDBGUSSVO
      VEBUSADCGFSUJTUKTULUMVBVDVFUNUQUO $.
  $}

  ${
    $d F f p $.  $d P f p $.  $d W f p $.  $d ch f p $.
    rbropapd.1 $e |- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } ) $.
    rbropapd.2 $e |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) ) $.
    $( Properties of a pair in an extended binary relation.  (Contributed by
       Alexander van der Vekens, 30-Oct-2017.) $)
    rbropapd $p |- ( ph -> ( ( F e. X /\ P e. Y )
                     -> ( F M P <-> ( F W P /\ ch ) ) ) ) $=
      ( wcel wa wbr wb cop cv wceq copab df-br eleq2d bitrid anbi12d opelopabga
      breq12 sylan9bb ex ) AFINDJNOZFDGPZFDHPZCOZQAUKFDRZESZKSZHPZBOZEKUAZNZUJU
      MUKUNGNAUTFDGUBAGUSUNLUCUDURUMEKFDIJUOFTUPDTOUQULBCUOFUPDHUGMUEUFUHUI $.

    $( Properties of a pair in a restricted binary relation ` M ` expressed as
       an ordered-pair class abstraction: ` M ` is the binary relation ` W `
       restricted by the condition ` ps ` .  (Contributed by AV,
       31-Jan-2021.) $)
    rbropap $p |- ( ( ph /\ F e. X /\ P e. Y )
                    -> ( F M P <-> ( F W P /\ ch ) ) ) $=
      ( wcel wbr wa wb rbropapd 3impib ) AFINDJNFDGOFDHOCPQABCDEFGHIJKLMRS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Function transposition
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c tpos $. $( Function transposition $)

  $( The transposition of a function. $)
  ctpos $a class tpos F $.

  ${
    $d F x $.
    $( Define the transposition of a function, which is a function
       ` G = tpos F ` satisfying ` G ( x , y ) = F ( y , x ) ` .  (Contributed
       by Mario Carneiro, 10-Sep-2015.) $)
    df-tpos $a |- tpos F = ( F o.
        ( x e. ( `' dom F u. { (/) } ) |-> U. `' { x } ) ) $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d w x y z F $.  $d x G $.
    $( Subset theorem for transposition.  (Contributed by Mario Carneiro,
       10-Sep-2015.) $)
    tposss $p |- ( F C_ G -> tpos F C_ tpos G ) $=
      ( vx wss cdm ccnv c0 csn cun cv cuni cmpt ccom ctpos coss1 cres wceq dmss
      cnvss df-tpos unss1 resmpt 4syl resss eqsstrrdi coss2 syl sstrd 3sstr4g )
      ABDZACAEZFZGHZIZCJHFKZLZMZBCBEZFZUMIZUOLZMZANBNUJUQBUPMZVBABUPOUJUPVADVCV
      BDUJUPVAUNPZVAUJUKURDULUSDUNUTDVDUPQABRUKURSULUSUMUACUTUNUOUBUCVAUNUDUEUP
      VABUFUGUHCATCBTUI $.

    $( Equality theorem for transposition.  (Contributed by Mario Carneiro,
       10-Sep-2015.) $)
    tposeq $p |- ( F = G -> tpos F = tpos G ) $=
      ( wceq ctpos wss eqimss tposss syl eqimss2 eqssd ) ABCZADZBDZKABELMEABFAB
      GHKBAEMLEBAIBAGHJ $.

    ${
      tposeqd.1 $e |- ( ph -> F = G ) $.
      $( Equality theorem for transposition.  (Contributed by Mario Carneiro,
         7-Jan-2017.) $)
      tposeqd $p |- ( ph -> tpos F = tpos G ) $=
        ( wceq ctpos tposeq syl ) ABCEBFCFEDBCGH $.
    $}

    $( The transposition is a subset of a cross product.  (Contributed by Mario
       Carneiro, 12-Jan-2017.) $)
    tposssxp $p |- tpos F C_ ( ( `' dom F u. { (/) } ) X. ran F ) $=
      ( vx ctpos cdm ccnv csn cun cuni cmpt crn cxp ccom df-tpos cossxp eqsstri
      c0 cv wss eqid dmmptss xpss1 ax-mp sstri ) ACZBADEPFGZBQFEHZIZDZAJZKZUEUI
      KZUDAUGLUJBAMAUGNOUHUERUJUKRBUEUFUGUGSTUHUEUIUAUBUC $.

    $( The transposition is a relation.  (Contributed by Mario Carneiro,
       10-Sep-2015.) $)
    reltpos $p |- Rel tpos F $=
      ( ctpos cdm ccnv c0 csn cun crn cxp wss wrel tposssxp relxp relss mp2 ) A
      BZACDEFGZAHZIZJSKPKALQRMPSNO $.

    $( Value of the transposition at a pair ` <. A , B >. ` .  (Contributed by
       Mario Carneiro, 10-Sep-2015.) $)
    brtpos2 $p |- ( B e. V -> ( A tpos F B <->
      ( A e. ( `' dom F u. { (/) } ) /\ U. `' { A } F B ) ) ) $=
      ( vy vx wcel cvv wbr cdm ccnv csn cuni wa wi wb wex wceq syl bitri c0 cun
      ctpos reltpos brrelex1i a1i elex adantr cv cmpt ccom df-tpos breqi bitrid
      brcog cfv wfun funmpt funbrfv2b ax-mp vex snexg cnvex uniex dmmpti eleq2i
      eqid eqcom anbi12i cnvexg uniexg sneq cnveqd unieqd fvmptg eqeq2d pm5.32i
      mpdan ancom anbi1i anass exbii exsimpr exsimpl 19.9v sylib simpl ceqsexgv
      breq1 anbi2d pm5.21nii bitrdi expcom pm5.21ndd ) BDGZAHGZABCUCZIZACJKUALU
      BZGZALZKZMZBCIZNZWRWPOWOABWQCUDUEUFXEWPOWOWTWPXDAWSUGUHUFWPWOWRXEPWPWONZW
      RAEUIZFWSFUIZLZKZMZUJZIZXGBCIZNZEQZXEWRABCXLUKZIXFXPABWQXQFCULUMEABCXLHDU
      OUNXPXGXCRZWTXNNZNZEQZXEXOXTEXOXRWTNZXNNXTXMYBXNXMWTXRNZYBXMAXLJZGZAXLUPZ
      XGRZNZYCXLUQXMYHPFWSXKURAXGXLUSUTYHWTXGYFRZNYCYEWTYGYIYDWSAFWSXKXLXJXIXHH
      GXIHGFVAXHHVBUTVCVDXLVGZVEVFYFXGVHVIWTYIXRWTYFXCXGWTXCHGZYFXCRWTXBHGZYKWT
      XAHGYLAWSVBXAHVJSXBHVKSZFAXKXCWSHXLXHARZXJXBYNXIXAXHAVLVMVNYJVOVRVPVQTTWT
      XRVSTVTXRWTXNWATWBYAWTXEYAXSEQZWTXRXSEWCYOWTEQWTWTXNEWDWTEWEWFSWTXDWGWTYK
      YAXEPYMXSXEEXCHXRXNXDWTXGXCBCWIWJWHSWKTWLWMWN $.

    $( The behavior of ` tpos ` when the left argument is the empty set (which
       is not an ordered pair but is the "default" value of an ordered pair
       when the arguments are proper classes).  (Contributed by Mario Carneiro,
       10-Sep-2015.) $)
    brtpos0 $p |- ( A e. V -> ( (/) tpos F A <-> (/) F A ) ) $=
      ( wcel c0 ctpos wbr cdm ccnv csn cun cuni brtpos2 ssun2 0ex snid biantrur
      wa sselii cnvsn0 unieqi uni0 eqtri breq1i bitr3i bitrdi ) ACDEABFGEBHIZEJ
      ZKZDZUHIZLZABGZRZEABGZEABCMUNUMUOUJUMUHUIEUHUGNEOPSQULEABULELEUKETUAUBUCU
      DUEUF $.

    $( Necessary and sufficient condition for ` dom tpos F ` to be a relation.
       (Contributed by Mario Carneiro, 10-Sep-2015.) $)
    reldmtpos $p |- ( Rel dom tpos F <-> -. (/) e. dom F ) $=
      ( vy vx cdm wrel c0 wcel wn cv wbr wex 0ex eldm cvv wb vex ax-mp wi sylbi
      wss ctpos brtpos0 cxp 0nelxp df-rel ssel mtoi breldm nsyl3 sylbir exlimiv
      con2i ccnv csn relcnv mpbi sseli a1i elsni breq1d pm2.24d biimtrdi impcom
      wa com3l wo cun cuni brtpos2 simplbi elun sylib adantl mpjaod ex biimtrid
      exlimdv ssrdv sylibr impbii ) AUAZDZEZFADZGZHZWEWCWEFBIZAJZBKWCHZBFALMWHW
      IBWHFWGWAJZWIWGNGZWJWHOBPZWGANUBQZWCFWBGZWJWCWNFNNUCZGZNNUDWCWBWOTZWNWPRW
      BUEZWBWOFUFSUGFWGWALWLUHUIUJUKSULWFWQWCWFCWBWOCIZWBGWSWGWAJZBKWFWSWOGZBWS
      WACPMWFWTXABWFWTXAWFWTVDZWSWDUMZGZXAWSFUNZGZXDXARXBXCWOWSXCEXCWOTWDUOXCUE
      UPUQURWTWFXFXARXFWTWFXAXFWTWJWFXARZXFWSFWGWAWSFUSUTWJWHXGWMWHWEXAFWGALWLU
      HVASVBVEVCWTXDXFVFZWFWTWSXCXEVGGZXHWTXIWSUNUMVHWGAJZWKWTXIXJVDOWLWSWGANVI
      QVJWSXCXEVKVLVMVNVOVQVPVRWRVSVT $.

    $( The transposition swaps arguments of a three-parameter relation.
       (Contributed by Jim Kingdon, 31-Jan-2019.) $)
    brtposg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
        ( <. A , B >. tpos F C <-> <. B , A >. F C ) ) $=
      ( wcel w3a cop cdm ccnv c0 csn cun wbr wa wb 3adant3 cvv opswapg 3ad2ant3
      cuni ctpos breq1d anbi2d brtpos2 opexg ancoms anim1i 3impa breldmg 3expia
      wi syl opelcnvg sylibrd elun1 syl6 pm4.71rd 3bitr4d ) AEHZBFHZCGHZIZABJZD
      KZLZMNZOHZVFNLUCZCDPZQZVJBAJZCDPZQVFCDUDPZVOVEVLVOVJVBVCVLVORVDVBVCQZVKVN
      CDABEFUAUESUFVDVBVPVMRVCVFCDGUGUBVEVOVJVEVOVFVHHZVJVEVOVNVGHZVRVEVNTHZVDQ
      ZVOVSUNVBVCVDWAVQVTVDVCVBVTBAFEUHUIUJUKVTVDVOVSVNCTGDULUMUOVBVCVRVSRVDABE
      FVGUPSUQVFVHVIURUSUTVA $.

    $( The transposition swaps the first two elements in a collection of
       ordered triples.  (Contributed by Mario Carneiro, 1-Dec-2014.) $)
    ottposg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
        ( <. A , B , C >. e. tpos F <-> <. B , A , C >. e. F ) ) $=
      ( wcel w3a cop ctpos cotp wbr brtposg df-br 3bitr3g df-ot eleq1i 3bitr4g
      ) AEHBFHCGHIZABJZCJZDKZHZBAJZCJZDHZABCLZUCHBACLZDHTUACUCMUECDMUDUGABCDEFG
      NUACUCOUECDOPUHUBUCABCQRUIUFDBACQRS $.

    $( The domain of ` tpos F ` when ` dom F ` is a relation.  (Contributed by
       Mario Carneiro, 10-Sep-2015.) $)
    dmtpos $p |- ( Rel dom F -> dom tpos F = `' dom F ) $=
      ( vx vy vz ctpos cdm wrel ccnv wa wceq cvv wcel cop wbr wex vex opex eldm
      c0 cv cxp wss wn 0nelxp ssel mtoi df-rel reldmtpos 3imtr4i relcnv brtposg
      jctir wb mp3an a1i exbidv opelcnv bitri 3bitr4g eqrelrdv2 mpancom ) AEZFZ
      GZAFZHZGZIVEGZVCVFJVHVDVGVEKKUAZUBZSVELZUCVHVDVJVKSVILKKUDVEVISUEUFVEUGAU
      HUIVEUJULVHBCVCVFVHBTZCTZMZDTZVBNZDOVMVLMZVOANZDOZVNVCLVNVFLZVHVPVRDVPVRU
      MZVHVLKLVMKLVOKLWABPZCPZDPVLVMVOAKKKUKUNUOUPDVNVBVLVMWBWCQRVTVQVELVSVLVMV
      EWBWCUQDVQAVMVLWCWBQRURUSUTVA $.

    $( The range of ` tpos F ` when ` dom F ` is a relation.  (Contributed by
       Mario Carneiro, 10-Sep-2015.) $)
    rntpos $p |- ( Rel dom F -> ran tpos F = ran F ) $=
      ( vx vy vw vz cdm wrel crn cv wcel wbr wex vex elrn cop wceq breldm elrel
      wi cvv ctpos ccnv dmtpos eleq2d imbitrid mpan syl6 breq1 wb brtposg mp3an
      relcnv bitrdi opex brelrn biimtrdi exlimivv syli exlimdv biimtrid ex syl5
      bitr4di impbid eqrdv ) AFZGZBAUAZHZAHZVGBIZVIJZVKVJJZVLCIZVKVHKZCLVGVMCVK
      VHBMZNVGVOVMCVOVGVNDIZEIZOZPZELDLZVMVGVOVNVFUBZJZWAVOVNVHFZJVGWCVNVKVHCMZ
      VPQVGWDWBVNAUCUDUEWBGWCWAVFULDEVNWBRUFUGVTVOVMSDEVTVOVRVQOZVKAKZVMVTVOVSV
      KVHKZWGVNVSVKVHUHVQTJVRTJVKTJWHWGUIDMZEMZVPVQVRVKATTTUJUKZUMWFVKAVRVQWJWI
      UNVPUOUPUQURUSUTVMVNVKAKZCLVGVLCVKAVPNVGWLVLCWLVGVNWFPZDLELZVLWLVNVFJZVGW
      NVNVKAWEVPQVGWOWNEDVNVFRVAVBWMWLVLSEDWMWLWHVLWMWLWGWHVNWFVKAUHWKVCVSVKVHV
      QVRWIWJUNVPUOUPUQURUSUTVDVE $.

    $( The transposition of a set is a set.  (Contributed by Mario Carneiro,
       10-Sep-2015.) $)
    tposexg $p |- ( F e. V -> tpos F e. _V ) $=
      ( wcel ctpos cdm ccnv c0 csn cun crn cxp wss cvv tposssxp cnvexg syl p0ex
      dmexg unexg sylancl rnexg xpexg syl2anc ssexg sylancr ) ABCZADZAEZFZGHZIZ
      AJZKZLUMMCZUGMCANUFUKMCZULMCUNUFUIMCZUJMCUOUFUHMCUPABRUHMOPQUIUJMMSTABUAU
      KULMMUBUCUGUMMUDUE $.
  $}

  ${
    $d y A $.  $d y B $.  $d y F $.  $d y V $.  $d y W $.
    $( The transposition swaps the arguments in a two-argument function.  When
       ` F ` is a matrix, which is to say a function from ( 1 ... m ) ` X. `
       ( 1 ... n ) to the reals or some ring, ` tpos F ` is the transposition
       of ` F ` , which is where the name comes from.  (Contributed by Mario
       Carneiro, 10-Sep-2015.) $)
    ovtposg $p |- ( ( A e. V /\ B e. W ) -> ( A tpos F B ) = ( B F A ) ) $=
      ( vy wcel wa cop ctpos cfv co cv wbr cio cvv wb df-fv 3eqtr4g df-ov vex
      brtposg mp3an3 iotabidv ) ADGZBEGZHZABIZCJZKZBAIZCKZABUILBACLUGUHFMZUINZF
      OUKUMCNZFOUJULUGUNUOFUEUFUMPGUNUOQFUAABUMCDEPUBUCUDFUHUIRFUKCRSABUITBACTS
      $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d w x y z F $.  $d x G $.
    $( The transposition of a function is a function.  (Contributed by Mario
       Carneiro, 10-Sep-2015.) $)
    tposfun $p |- ( Fun F -> Fun tpos F ) $=
      ( vx wfun cdm ccnv c0 csn cun cv cuni cmpt ccom ctpos funco mpan2 df-tpos
      funmpt funeqi sylibr ) ACZABADEFGHZBIGEJZKZLZCZAMZCTUCCUEBUAUBQAUCNOUFUDB
      APRS $.

    $( Alternate definition of ` tpos ` when ` F ` has relational domain.
       (Contributed by Mario Carneiro, 10-Sep-2015.) $)
    dftpos2 $p |- ( Rel dom F -> tpos F =
        ( F o. ( x e. `' dom F |-> U. `' { x } ) ) ) $=
      ( cdm wrel ctpos cres ccnv csn cuni cmpt ccom dmtpos reseq2d wceq reltpos
      cv resdm ax-mp c0 cun df-tpos reseq1i resco resmpt coeq2i 3eqtri 3eqtr3g
      wss ssun1 ) BCZDZBEZULCZFZULUJGZFZULBAUOAPHGIZJZKZUKUMUOULBLMULDUNULNBOUL
      QRUPBAUOSHZTZUQJZKZUOFBVBUOFZKUSULVCUOABUAUBBVBUOUCVDURBUOVAUHVDURNUOUTUI
      AVAUOUQUDRUEUFUG $.

    $( Alternate definition of ` tpos ` when ` F ` has relational domain.
       Compare ~ df-cnv .  (Contributed by Mario Carneiro, 10-Sep-2015.) $)
    dftpos3 $p |- ( Rel dom F -> tpos F =
      { <. <. x , y >. , z >. | <. y , x >. F z } ) $=
      ( vw cdm wrel ctpos cv cop wceq wbr wa wex wcel cvv cxp vex bitr3i bitrdi
      cab coprab wss ccnv relcnv dmtpos releqd mpbiri reltpos jctil sylib sseld
      relrelss elvvv imbitrdi pm4.71rd 19.41vvv eleq1 wb brtposg pm5.32i 3exbii
      df-br mp3an abbi2dv df-oprab eqtr4di ) DFZGZDHZEIZAIZBIZJZCIZJZKZVMVLJVOD
      LZMZCNBNANZEUAVRABCUBVIVTEVJVIVKVJOZVQCNBNANZWAMZVTVIWAWBVIWAVKPPQPQZOWBV
      IVJWDVKVIVJGZVJFZGZMVJWDUCVIWGWEVIWGVHUDZGVHUEVIWFWHDUFUGUHDUIUJVJUMUKULA
      BCVKUNUOUPWCVQWAMZCNBNANVTVQWAABCUQWIVSABCVQWAVRVQWAVPVJOZVRVKVPVJURWJVNV
      OVJLZVRVNVOVJVCVLPOVMPOVOPOWKVRUSARBRCRVLVMVODPPPUTVDSTVAVBSTVEVRABCEVFVG
      $.

    $( Alternate definition of ` tpos ` .  (Contributed by Mario Carneiro,
       4-Oct-2015.) $)
    dftpos4 $p |- tpos F =
        ( F o. ( x e. ( ( _V X. _V ) u. { (/) } ) |-> U. `' { x } ) ) $=
      ( vy vz vw cvv csn cun cv ccnv cuni wss wceq cop wcel wbr wa wex vex wb
      ctpos cxp cmpt ccom cdm df-tpos cres wrel relcnv df-rel mpbi unss1 resmpt
      c0 mp2b resss eqsstrri coss2 ax-mp eqsstri relco opelco eleq1 sneq cnveqd
      unieqd eqeq2d anbi12d anbi2d df-mpt brab wi simplr breldm adantl eqeltrrd
      eqeq1 opswapg mp2an eleq1i opelcnv bitr4i eleq1d bibi12d exlimivv biimpcd
      elvv mpbiri sylbi elun1 syl6 syl elun2 a1i simpll elun sylib mpjaod simpr
      wo eqbrtrrd jca sylanb brtpos2 sylibr df-br exlimiv relssi eqssi ) BUAZBA
      FFUBZUNGZHZAIZGZJZKZUCZUDZXJBABUEZJZXLHZXQUCZUDZXSABUFYCXRLYDXSLYCXRYBUGZ
      XRYAXKLZYBXMLYEYCMYAUHYFXTUIYAUJUKYAXKXLULAXMYBXQUMUOXRYBUPUQYCXRBURUSUTC
      DXSXJBXRVACIZDIZNZXSOYGEIZXRPZYJYHBPZQZERYIXJOZEYGYHBXRCSZDSZVBYMYNEYMYGY
      HXJPZYNYMYGYBOZYGGZJZKZYHBPZQZYQYKYGXMOZYJUUAMZQZYLUUCXNXMOZYHXQMZQUUDYHU
      UAMZQUUFADYGYJXRYOESZXNYGMZUUGUUDUUHUUIXNYGXMVCUUKXQUUAYHUUKXPYTUUKXOYSXN
      YGVDVEVFVGVHYHYJMUUIUUEUUDYHYJUUAVQVIADXMXQVJVKUUFYLQZYRUUBUULYGXKOZYRYGX
      LOZUULUUAXTOZUUMYRVLUULYJUUAXTUUDUUEYLVMZYLYJXTOUUFYJYHBUUJYPVNVOVPUUOUUM
      YGYAOZYRUUMUUOUUQUUMYGYHYJNZMZERDRUUOUUQTZDEYGWGUUSUUTDEUUSUUTUURGZJZKZXT
      OZUURYAOZTUVDYJYHNZXTOUVEUVCUVFXTYHFOZYJFOUVCUVFMYPUUJYHYJFFVRVSVTYHYJXTY
      PUUJWAWBUUSUUOUVDUUQUVEUUSUUAUVCXTUUSYTUVBUUSYSUVAYGUURVDVEVFWCYGUURYAVCW
      DWHWEWIWFYGYAXLWJWKWLUUNYRVLUULYGXLYAWMWNUULUUDUUMUUNWTUUDUUEYLWOYGXKXLWP
      WQWRUULYJUUAYHBUUPUUFYLWSXAXBXCUVGYQUUCTYPYGYHBFXDUSXEYGYHXJXFWQXGWIXHXI
      $.

    $( Value of the double transposition for a general class ` F ` .
       (Contributed by Mario Carneiro, 16-Sep-2015.) $)
    tpostpos $p |- tpos tpos F =
        ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) $=
      ( vz vx vy cvv c0 csn wrel cv ccnv wcel wbr wa wo wb bitrdi eqtrdi breq1d
      cuni vex vw ctpos cxp cun cin reltpos wss inss2 relxp relss relcnv df-rel
      mp2 cdm mpbi simpl sselid simpr cop wceq elvv eleq1 opelcnv cnveqd unieqd
      wex sneq opswapg mp2an anbi12d breldm pm4.71ri brtposg mp3an bitr3i breq1
      opex bitr4d exlimivv sylbi iba bitrd pm5.21nii elsni sneqd cnvsn0 brtpos0
      uni0 ax-mp pm5.32i ancom bitri orbi12i andir andi 3bitr4i anbi1i mpbiran2
      elun brxp anbi2i brtpos2 brin eqbrriv ) UABAUBZUBZAEEUCZFGZUDZEUCZUEZXEUF
      XKXJUGXJHXKHAXJUHXIEUIXKXJUJUMUAIZXEUNZJZXHUDKZXLGZJZSZBIZXELZMZXLXSALZXL
      XSXJLZMZXLXSXFLZXLXSXKLXLXNKZXLXHKZNZXTMZYBXLXGKZYGNZMZYAYDYFXTMZYGXTMZNY
      BYJMZYBYGMZNYIYLYMYOYNYPYMYJYOYMXNXGXLXNHXNXGUGXMUKXNULUOYFXTUPUQYBYJURYJ
      YMYBYOYJXLCIZDIZUSZUTZDVFCVFYMYBOZCDXLVAYTUUACDYTYMYSXSALZYBYTYMYRYQUSZXM
      KZUUCXSXELZMZUUBYTYFUUDXTUUEYTYFYSXNKUUDXLYSXNVBYQYRXMCTZDTZVCPYTXRUUCXSX
      EYTXRYSGZJZSZUUCYTXQUUJYTXPUUIXLYSVGVDVEYQEKZYREKZUUKUUCUTUUGUUHYQYREEVHV
      IQRVJUUFUUEUUBUUEUUDUUCXSXEYRYQUUHUUGVQBTZVKVLUUMUULXSEKZUUEUUBOUUHUUGUUN
      YRYQXSAEEEVMVNVOPXLYSXSAVPVRVSVTYJYBWAWBWCYNYGYBMYPYGXTYBYGXTFXSALZYBYGXT
      FXSXELZUUPYGXRFXSXEYGXRFSFYGXQFYGXQXHJFYGXPXHYGXLFXLFWDZWEVDWFQVEWHQRUUOU
      UQUUPOUUNXSAEWGWIPYGXLFXSAUURRVRWJYGYBWKWLWMYFYGXTWNYBYJYGWOWPXOYHXTXLXNX
      HWSWQYCYKYBYCXLXIKZYKYCUUSUUOUUNXLXSXIEWTWRXLXGXHWSWLXAWPUUOYEYAOUUNXLXSX
      EEXBWIXLXSAXJXCWPXD $.

    $( Value of the double transposition for a relation on triples.
       (Contributed by Mario Carneiro, 16-Sep-2015.) $)
    tpostpos2 $p |- ( ( Rel F /\ Rel dom F ) -> tpos tpos F = F ) $=
      ( wrel cdm wa ctpos cvv cxp csn cun cin tpostpos wss relrelss ssun1 xpss1
      c0 wceq ax-mp sstr mpan2 sylbi df-ss sylib eqtrid ) ABACBDZAEEAFFGZPHZIZF
      GZJZAAKUEAUILZUJAQUEAUFFGZLZUKAMUMULUILZUKUFUHLUNUFUGNUFUHFORAULUISTUAAUI
      UBUCUD $.

    $( The domain of a transposition.  (Contributed by NM, 10-Sep-2015.) $)
    tposfn2 $p |- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) ) $=
      ( wrel wfun cdm wceq wa ctpos ccnv wfn wi tposfun a1i dmtpos releq eqeq2d
      cnveq 3imtr3d com12 df-fn anim12d 3imtr4g ) ACZBDZBEZAFZGBHZDZUGEZAIZFZGB
      AJUGUJJUCUDUHUFUKUDUHKUCBLMUFUCUKUFUECZUIUEIZFZUCUKULUNKUFBNMUEAOUFUMUJUI
      UEAQPRSUABATUGUJTUB $.

    $( Condition for a surjective transposition.  (Contributed by NM,
       10-Sep-2015.) $)
    tposfo2 $p |- ( Rel A ->
        ( F : A -onto-> B -> tpos F : `' A -onto-> B ) ) $=
      ( wrel wfn crn wceq wa ctpos ccnv wfo tposfn2 adantrd cdm releqd biimparc
      fndm rntpos syl df-fo eqeq1d biimprd expimpd jcad 3imtr4g ) ADZCAEZCFZBGZ
      HZCIZAJZEZUKFZBGZHABCKULBUKKUFUJUMUOUFUGUMUIACLMUFUGUIUOUFUGHZUOUIUPUNUHB
      UPCNZDZUNUHGUGURUFUGUQAACQOPCRSUAUBUCUDABCTULBUKTUE $.

    $( The domain and codomain of a transposition.  (Contributed by NM,
       10-Sep-2015.) $)
    tposf2 $p |- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) ) $=
      ( wrel wf ccnv ctpos crn wss wfo wfn ffn dffn4 sylib tposfo2 syl5 imp fof
      wa syl frn adantl fss syl2anc ex ) ADZABCEZAFZBCGZEZUFUGSZUHCHZUIEZULBIZU
      JUKUHULUIJZUMUFUGUOUGAULCJZUFUOUGCAKUPABCLACMNAULCOPQUHULUIRTUGUNUFABCUAU
      BUHULBUIUCUDUE $.

    $( Condition for an injective transposition.  (Contributed by NM,
       10-Sep-2015.) $)
    tposf12 $p |- ( Rel A -> ( F : A -1-1-> B -> tpos F : `' A -1-1-> B ) ) $=
      ( vx wrel wf1 ccnv ctpos wa cdm cv csn cuni cmpt wceq wb syl f1eq1 mpbird
      3syl ccom simpr relcnv cnvf1o f1of1 simpl dfrel2 sylib f1eq3 mpbii cnveqd
      wf1o mp2b f1dm mpteq1 f1co syl2anc releqd biimparc dftpos2 ex ) AEZABCFZA
      GZBCHZFZVBVCIZVFVDBCDCJZGZDKLGMZNZUAZFZVGVCVDAVKFZVMVBVCUBZVGVNVDADVDVJNZ
      FZVGVDVDGZVPFZVQVDEVDVRVPULVSAUCDVDUDVDVRVPUEUMVGVRAOZVSVQPVGVBVTVBVCUFAU
      GUHVRAVDVPUIQUJVGVIVDOVKVPOVNVQPVGVHAVGVCVHAOVOABCUNZQUKDVIVDVJUOVDAVKVPR
      TSVDABCVKUPUQVGVHEZVEVLOVFVMPVCWBVBVCVHAWAURUSDCUTVDBVEVLRTSVA $.

    $( Condition of a bijective transposition.  (Contributed by NM,
       10-Sep-2015.) $)
    tposf1o2 $p |- ( Rel A ->
      ( F : A -1-1-onto-> B -> tpos F : `' A -1-1-onto-> B ) ) $=
      ( wrel wf1 wfo wa ccnv ctpos wf1o tposf12 tposfo2 anim12d df-f1o 3imtr4g
      ) ADZABCEZABCFZGAHZBCIZEZSBTFZGABCJSBTJPQUARUBABCKABCLMABCNSBTNO $.

    $( The domain and codomain/range of a transposition.  (Contributed by NM,
       10-Sep-2015.) $)
    tposfo $p |- ( F : ( A X. B ) -onto-> C ->
              tpos F : ( B X. A ) -onto-> C ) $=
      ( cxp wfo ccnv ctpos wrel wi relxp tposfo2 ax-mp wceq cnvxp foeq2 sylib
      wb ) ABEZCDFZSGZCDHZFZBAEZCUBFZSITUCJABKSCDLMUAUDNUCUERABOUAUDCUBPMQ $.

    $( The domain and codomain of a transposition.  (Contributed by NM,
       10-Sep-2015.) $)
    tposf $p |- ( F : ( A X. B ) --> C -> tpos F : ( B X. A ) --> C ) $=
      ( cxp wf ccnv ctpos wrel wi relxp tposf2 ax-mp cnvxp feq2i sylib ) ABEZCD
      FZQGZCDHZFZBAEZCTFQIRUAJABKQCDLMSUBCTABNOP $.

    $( Functionality of a transposition.  (Contributed by Mario Carneiro,
       4-Oct-2015.) $)
    tposfn $p |- ( F Fn ( A X. B ) -> tpos F Fn ( B X. A ) ) $=
      ( cxp cvv wf ctpos wfn tposf dffn2 3imtr4i ) ABDZECFBADZECGZFCLHNMHABECIL
      CJMNJK $.

    $( Transposition of the empty set.  (Contributed by NM, 10-Sep-2015.) $)
    tpos0 $p |- tpos (/) = (/) $=
      ( c0 ctpos wfn wceq ccnv wrel rel0 eqid fn0 mpbir tposfn2 mp2 cnv0 fneq2i
      mpbi ) ABZACZPADPAEZCZQAFAACZSGTAADAHAIJAAKLRAPMNOPIO $.

    $( Transposition of a composition.  (Contributed by Mario Carneiro,
       4-Oct-2015.) $)
    tposco $p |- tpos ( F o. G ) = ( F o. tpos G ) $=
      ( vx ccom cvv cxp c0 csn cun cv ccnv cuni cmpt ctpos coass dftpos4 coeq2i
      3eqtr4i ) ABDZCEEFGHICJHKLMZDABTDZDSNABNZDABTOCSPUBUAACBPQR $.

    $( Two ways to say a function is symmetric.  (Contributed by Mario
       Carneiro, 4-Oct-2015.) $)
    tpossym $p |- ( F Fn ( A X. A ) ->
      ( tpos F = F <-> A. x e. A A. y e. A ( x F y ) = ( y F x ) ) ) $=
      ( cxp wfn ctpos wceq cv co wral tposfn eqfnov2 mpancom eqcom cvv wcel vex
      wb ovtposg mp2an eqeq2i bitri 2ralbii bitrdi ) DCCEZFZDGZDHZAIZBIZUHJZUJU
      KDJZHZBCKACKZUMUKUJDJZHZBCKACKUHUFFUGUIUOSCCDLABCCUHDMNUNUQABCCUNUMULHUQU
      LUMOULUPUMUJPQUKPQULUPHARBRUJUKDPPTUAUBUCUDUE $.
  $}

  ${
    tposeqi.1 $e |- F = G $.
    $( Equality theorem for transposition.  (Contributed by Mario Carneiro,
       10-Sep-2015.) $)
    tposeqi $p |- tpos F = tpos G $=
      ( wceq ctpos tposeq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    tposex.1 $e |- F e. _V $.
    $( A transposition is a set.  (Contributed by Mario Carneiro,
       10-Sep-2015.) $)
    tposex $p |- tpos F e. _V $=
      ( cvv wcel ctpos tposexg ax-mp ) ACDAECDBACFG $.
  $}

  ${
    $d x y $.  $d y F $.
    nftpos.1 $e |- F/_ x F $.
    $( Hypothesis builder for transposition.  (Contributed by Mario Carneiro,
       10-Sep-2015.) $)
    nftpos $p |- F/_ x tpos F $=
      ( vy ctpos cvv cxp c0 csn cv ccnv cuni cmpt ccom dftpos4 nfcv nfco nfcxfr
      cun ) ABEBDFFGHISDJIKLMZNDBOABTCATPQR $.
  $}

  ${
    $d a b c x y z $.  $d a b c ph $.
    tposoprab.1 $e |- F = { <. <. x , y >. , z >. | ph } $.
    $( Transposition of a class of ordered triples.  (Contributed by Mario
       Carneiro, 10-Sep-2015.) $)
    tposoprab $p |- tpos F = { <. <. y , x >. , z >. | ph } $=
      ( vb va vc ctpos coprab cv cop wbr tposeqi wceq nfcv nfbr nfv weq dftpos3
      cdm reldmoprab ax-mp nfoprab2 nfoprab1 wa opeq12 ancoms breq1d cbvoprab12
      wrel nfoprab3 breq2 df-br oprabid bitri bitrdi cbvoprab3 eqtri 3eqtri
      wcel ) EJABCDKZJZGLZHLZMZILZVCNZHGIKZACBDKZEVCFOVCUBULVDVJPABCDUCHGIVCUAU
      DVJBLZCLZMZVHVCNZCBIKVKVIVOHGICBCVGVHVCCVGQABCDUECVHQRBVGVHVCBVGQABCDUFBV
      HQRVOHSVOGSHCTZGBTZUGVGVNVHVCVQVPVGVNPVEVFVLVMUHUIUJUKVOACBIDDVNVHVCDVNQA
      BCDUMDVHQRAISIDTVOVNDLZVCNZAVHVRVNVCUNVSVNVRMVCVBAVNVRVCUOABCDUPUQURUSUTV
      A $.
  $}

  ${
    $d x y z $.  $d z A $.  $d z B $.  $d z C $.
    tposmpo.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( Transposition of a two-argument mapping.  (Contributed by Mario
       Carneiro, 10-Sep-2015.) $)
    tposmpo $p |- tpos F = ( y e. B , x e. A |-> C ) $=
      ( vz ctpos cv wcel wceq coprab cmpo df-mpo ancom anbi1i oprabbii 3eqtri
      wa tposoprab eqtr4i ) FIBJDKZAJCKZTZHJELZTZBAHMBADCENUGABHFFABCDENUDUCTZU
      FTZABHMUGABHMGABHCDEOUIUGABHUHUEUFUDUCPQRSUABAHDCEOUB $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Undefined values
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( The power set of the union of a set does not belong to the set.  This
     theorem provides a way of constructing a new set that doesn't belong to a
     given set.  (Contributed by Stefan O'Rear, 22-Feb-2015.) $)
  pwuninel2 $p |- ( U. A e. V -> -. ~P U. A e. A ) $=
    ( cuni wcel cpw wss pwnss elssuni nsyl ) ACZBDJEZJFKADJBGKAHI $.

  $( The power set of the power set of the union of a set does not belong to
     the set.  This theorem provides a way of constructing a new set that
     doesn't belong to a given set.  (Contributed by Jim Kingdon,
     14-Jan-2020.) $)
  2pwuninelg $p |- ( A e. V -> -. ~P ~P U. A e. A ) $=
    ( wcel cuni cpw wa en2lp wi wss pwuni elpwg mpbiri ax-ia3 syl mtoi ) ABCZAD
    EZEZACZARCZSFZARGPTSUAHPTAQIAJAQBKLTSMNO $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Functions on ordinals; strictly monotone ordinal functions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x A $.
    $( The indexed union of a set of ordinal numbers ` B ( x ) ` is an ordinal
       number.  (Contributed by NM, 13-Oct-2003.)  (Revised by Mario Carneiro,
       5-Dec-2016.) $)
    iunon $p |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On ) $=
      ( wcel con0 wral wa ciun cmpt crn cuni wceq dfiun3g adantl cvv wss mptexg
      rnexg syl wf eqid fmpt frn sylbi ssonuni imp syl2an eqeltrd ) BDEZCFEABGZ
      HABCIZABCJZKZLZFUKULUOMUJABCFNOUJUNPEZUNFQZUOFEZUKUJUMPEUPABCDRUMPSTUKBFU
      MUAUQABFCUMUMUBUCBFUMUDUEUPUQURUNPUFUGUHUI $.
  $}

  $c Smo $.

  $( Introduce the strictly monotone ordinal function.  A strictly monotone
     function is one that is constantly increasing across the ordinals. $)
  wsmo $a wff Smo A $.

  ${
    $d x y A $.
    $( Definition of a strictly monotone ordinal function.  Definition 7.46 in
       [TakeutiZaring] p. 50.  (Contributed by Andrew Salmon, 15-Nov-2011.) $)
    df-smo $a |- ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A
       A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) ) $.
  $}

  ${
    $d F x y $.
    $( Alternate definition of a strictly monotone ordinal function.
       (Contributed by Mario Carneiro, 4-Mar-2013.) $)
    dfsmo2 $p |- ( Smo F <-> ( F : dom F --> On /\ Ord dom F /\ A. x e. dom F
       A. y e. x ( F ` y ) e. ( F ` x ) ) ) $=
      ( wsmo cdm con0 wf word wel cfv wcel wral w3a df-smo ralcom impexp 3anass
      cv wi wa simpr ordtr1 3impib 3com23 simp3 3expia impbid2 bitr3id ralbidv2
      jca imbi1d ralbidva bitrid pm5.32i anbi2i 3bitr4i bitri ) CDCEZFCGZURHZBA
      IZBRZCJARZCJKZSZAURLBURLZMZUSUTVDBVCLZAURLZMZBACNUSUTVFTZTUSUTVITZTVGVJVK
      VLUSUTVFVIVFVEBURLZAURLUTVIVEBAURUROUTVMVHAURUTVCURKZTZVEVDBURVCVBURKZVES
      VPVATZVDSVOVEVPVAVDPVOVQVAVDVOVQVAVPVAUAUTVNVAVQUTVNVAMVPVAUTVAVNVPUTVAVN
      VPVBVCURUBUCUDUTVNVAUEUJUFUGUKUHUIULUMUNUOUSUTVFQUSUTVIQUPUQ $.
  $}

  ${
    $d x y A $.
    issmo.1 $e |- A : B --> On $.
    issmo.2 $e |- Ord B $.
    issmo.3 $e |- ( ( x e. B /\ y e. B ) -> ( x e. y -> ( A ` x ) e. ( A ` y
       ) ) ) $.
    issmo.4 $e |- dom A = B $.
    $( Conditions for which ` A ` is a strictly monotone ordinal function.
       (Contributed by Andrew Salmon, 15-Nov-2011.) $)
    issmo $p |- Smo A $=
      ( wsmo cdm con0 wf word cv wcel cfv wi wral mpbir eleq2i feq2i wceq ordeq
      wb ax-mp syl2anb rgen2a df-smo mpbir3an ) CICJZKCLZUJMZANZBNZOUMCPUNCPOQZ
      BUJRAUJRUKDKCLEUJDKCHUASULDMZFUJDUBULUPUDHUJDUCUESUOABUJUMUJOUMDOUNDOUOUN
      UJOUJDUMHTUJDUNHTGUFUGABCUHUI $.
  $}

  ${
    $d A x $.  $d F x y $.
    $( Alternate definition of a strictly monotone ordinal function.
       (Contributed by Mario Carneiro, 12-Mar-2013.) $)
    issmo2 $p |- ( F : A --> B -> ( ( B C_ On /\ Ord A /\
       A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) ) -> Smo F ) ) $=
      ( wf con0 wss word cv cfv wcel wral w3a cdm wsmo fss ex fdm biimprd feq2d
      sylibrd wceq wb ordeq syl raleqdv 3anim123d dfsmo2 imbitrrdi ) CDEFZDGHZC
      IZBJEKAJZEKLBUNMZACMZNEOZGEFZUQIZUOAUQMZNEPUKULURUMUSUPUTUKULCGEFZURUKULV
      ACDGEQRUKUQCGECDESZUAUBUKUSUMUKUQCUCUSUMUDVBUQCUEUFTUKUTUPUKUOAUQCVBUGTUH
      ABEUIUJ $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Equality theorem for strictly monotone functions.  (Contributed by
       Andrew Salmon, 16-Nov-2011.) $)
    smoeq $p |- ( A = B -> ( Smo A <-> Smo B ) ) $=
      ( vx vy wceq cdm con0 wf word cv wcel cfv wi wral w3a wsmo raleqdv df-smo
      id fveq1 dmeq feq12d wb ordeq syl eleq12d imbi2d ralbidv 3bitrd 3anbi123d
      2ralbidv 3bitr4g ) ABEZAFZGAHZUNIZCJZDJZKZUQALZURALZKZMZDUNNCUNNZOBFZGBHZ
      VEIZUSUQBLZURBLZKZMZDVENZCVENZOAPBPUMUOVFUPVGVDVMUMUNVEGABUMSABUAZUBUMUNV
      EEUPVGUCVNUNVEUDUEUMVDVKDUNNZCUNNVLCUNNVMUMVCVKCDUNUNUMVBVJUSUMUTVHVAVIUQ
      ABTURABTUFUGUKUMVOVLCUNUMVKDUNVEVNQUHUMVLCUNVEVNQUIUJCDARCDBRUL $.

    $( The domain of a strictly monotone function is an ordinal.  (Contributed
       by Andrew Salmon, 16-Nov-2011.) $)
    smodm $p |- ( Smo A -> Ord dom A ) $=
      ( vx vy wsmo cdm con0 wf word cv wcel cfv wi wral df-smo simp2bi ) ADAEZF
      AGPHBIZCIZJQAKRAKJLCPMBPMBCANO $.

    $( A strictly monotone function restricted to an ordinal remains strictly
       monotone.  (Contributed by Andrew Salmon, 16-Nov-2011.)  (Proof
       shortened by Mario Carneiro, 5-Dec-2016.) $)
    smores $p |- ( ( Smo A /\ B e. dom A ) -> Smo ( A |` B ) ) $=
      ( vx vy cdm wcel wsmo con0 wf word cv cfv wi wral w3a wss ax-mp wceq syl
      wa cres wfn crn wfun funres funfn 3imtr3i resss rnss sstr anim12i 3imtr4i
      mpan df-f a1i ordelord expcom ordin ex syli wb dmres ordeq imbitrrdi dmss
      cin ssralv ralimi inss1 eqsstri simpl sselid fvres simpr eleq12d ralbidva
      imbi2d ralbiia sylibr 3anim123d df-smo 3imtr4g impcom ) BAEZFZAGZABUAZGZW
      EWDHAIZWDJZCKZDKZFZWKALZWLALZFZMZDWDNZCWDNZOWGEZHWGIZWTJZWMWKWGLZWLWGLZFZ
      MZDWTNZCWTNZOWFWHWEWIXAWJXBWSXHWIXAMWEAWDUBZAUCZHPZTWGWTUBZWGUCZHPZTWIXAX
      IXLXKXNAUDWGUDXIXLBAUEAUFWGUFUGXMXJPZXKXNWGAPZXOABUHZWGAUIQXMXJHUJUMUKWDH
      AUNWTHWGUNULUOWEWJBWDVFZJZXBWJWEBJZXSWJWEXTWDBUPUQXTWJXSBWDURUSUTWTXRRXBX
      SVAABVBZWTXRVCQVDWSXHMWEWSWQDWTNZCWTNZXHWSWRCWTNZYCWTWDPZWSYDMXPYEXQWGAVE
      QZWRCWTWDVGQWRYBCWTYEWRYBMYFWQDWTWDVGQVHSXGYBCWTWKWTFZXFWQDWTYGWLWTFZTZXE
      WPWMYIXCWNXDWOYIWKBFXCWNRYIWTBWKWTXRBYABWDVIVJZYGYHVKVLWKBAVMSYIWLBFXDWOR
      YIWTBWLYJYGYHVNVLWLBAVMSVOVQVPVRVSUOVTCDAWACDWGWAWBWC $.

    $( A strictly monotone function restricted to an ordinal remains strictly
       monotone.  (Contributed by Andrew Salmon, 19-Nov-2011.) $)
    smores3 $p |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo
       ( A |` C ) ) $=
      ( cres wsmo cdm cin wcel word w3a dmres incom eqtri eleq2i smores 3adant3
      sylan2br wss wceq wb inss2 sseli ordelss sylan 3adant1 resabs1 smoeq 3syl
      ancoms mpbid ) ABDZEZCAFZBGZHZBIZJZUKCDZEZACDZEZULUOUSUPUOULCUKFZHUSVBUNC
      VBBUMGUNABKBUMLMNUKCOQPUQCBRZURUTSUSVATUOUPVCULUOCBHZUPVCUNBCUMBUAUBUPVDV
      CBCUCUIUDUEACBUFURUTUGUHUJ $.
  $}

  ${
    $d A x y $.  $d F x y $.
    $( A strictly monotone ordinal function restricted to an ordinal is still
       monotone.  (Contributed by Mario Carneiro, 15-Mar-2013.) $)
    smores2 $p |- ( ( Smo F /\ Ord A ) -> Smo ( F |` A ) ) $=
      ( vy vx wsmo word wa cdm con0 wf cv cfv wcel wral crn wss wfun dfsmo2 syl
      wceq cres wfn simp1bi ffun funres funfn sylib df-ima imassrn eqsstrri frn
      cima sstrid df-f sylanbrc adantr smodm cin ordin dmres ordeq ax-mp sylibr
      wb ancoms sylan resss dmss simp3bi ssralv mpsyl wel wi ordtr1 inss1 sseli
      eqsstri expcomd imp31 fvres ad2antlr eleq12d ralbidva mpbird syl3anbrc
      syl6 ) BEZAFZGZBAUAZHZIWJJZWKFZCKZWJLZDKZWJLZMZCWPNZDWKNZWJEWGWLWHWGWJWKU
      BZWJOZIPWLWGBQZXAWGBHZIBJZXCWGXEXDFZWNBLZWPBLZMZCWPNZDXDNZDCBRZUCZXDIBUDS
      XCWJQXAABUEWJUFUGSWGXBBOZIXBBAULXNBAUHBAUIUJWGXEXNIPXMXDIBUKSUMWKIWJUNUOU
      PWGXFWHWMBUQWHXFWMWHXFGAXDURZFZWMAXDUSWKXOTWMXPVDBAUTZWKXOVAVBVCVEVFZWIWT
      XJDWKNZWGXSWHWKXDPZWGXKXSWJBPXTBAVGWJBVHVBWGXEXFXKXLVIXJDWKXDVJVKUPWIWSXJ
      DWKWIWPWKMZGZWRXICWPYBCDVLZGZWOXGWQXHYDWNAMZWOXGTWIYAYCYEWIYCYAYEWIYCYAGZ
      WNWKMZYEWIWMYFYGVMXRWNWPWKVNSWKAWNWKXOAXQAXDVOVQZVPWFVRVSWNABVTSYAWQXHTZW
      IYCYAWPAMYIWKAWPYHVPWPABVTSWAWBWCWCWDDCWJRWE $.
  $}

  $( The domain of a strictly monotone ordinal function is an ordinal.
     (Contributed by Mario Carneiro, 12-Mar-2013.) $)
  smodm2 $p |- ( ( F Fn A /\ Smo F ) -> Ord A ) $=
    ( wsmo wfn cdm word smodm wceq wb fndm ordeq syl biimpa sylan2 ) BCBADZBEZF
    ZAFZBGOQROPAHQRIABJPAKLMN $.

  ${
    $d F x y $.
    $( The function values of a strictly monotone ordinal function are
       ordinals.  (Contributed by Mario Carneiro, 12-Mar-2013.) $)
    smofvon2dm $p |- ( ( Smo F /\ B e. dom F ) -> ( F ` B ) e. On ) $=
      ( vy vx wsmo cdm con0 wf word cv cfv wcel wral dfsmo2 simp1bi ffvelcdmda
      ) BEZBFZGABQRGBHRICJBKDJZBKLCSMDRMDCBNOP $.
  $}

  ${
    $d x y A $.
    iordsmo.1 $e |- Ord A $.
    $( The identity relation restricted to the ordinals is a strictly monotone
       function.  (Contributed by Andrew Salmon, 16-Nov-2011.) $)
    iordsmo $p |- Smo ( _I |` A ) $=
      ( vx vy cid cres con0 wf wfn crn wss fnresi rnresi word ordsson wcel wceq
      cv cfv fvresi ax-mp eqsstri df-f mpbir2an wa adantr adantl eleq12d dmresi
      biimprd issmo ) CDEAFZAAGULHULAIULJZGKALUMAGAMANAGKBAOUAUBAGULUCUDBCRZAPZ
      DRZAPZUEZUNULSZUPULSZPUNUPPURUSUNUTUPUOUSUNQUQAUNTUFUQUTUPQUOAUPTUGUHUJAU
      IUK $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y F $.
    $( The empty set is a strictly monotone ordinal function.  (Contributed by
       Andrew Salmon, 20-Nov-2011.) $)
    smo0 $p |- Smo (/) $=
      ( cid c0 cres wsmo ord0 iordsmo wceq wb res0 smoeq ax-mp mpbi ) ABCZDZBDZ
      BEFMBGNOHAIMBJKL $.

    $( If ` B ` is a strictly monotone ordinal function, and ` A ` is in the
       domain of ` B ` , then the value of the function at ` A ` is an ordinal.
       (Contributed by Andrew Salmon, 20-Nov-2011.) $)
    smofvon $p |- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On ) $=
      ( vx vy wsmo cdm con0 wf word cv wcel cfv wral df-smo simp1bi ffvelcdmda
      wi ) BEZBFZGABRSGBHSICJZDJZKTBLUABLKQDSMCSMCDBNOP $.

    $( If ` x ` is less than ` y ` then a strictly monotone function's value
       will be strictly less at ` x ` than at ` y ` .  (Contributed by Andrew
       Salmon, 22-Nov-2011.) $)
    smoel $p |- ( ( Smo B /\ A e. dom B /\ C e. A ) -> ( B ` C ) e. ( B ` A )
       ) $=
      ( vx vy wsmo cdm wcel cfv wa wi word smodm ordtr1 expdimp wral wceq fveq2
      cv imbi12d ancomsd sylan con0 w3a df-smo eleq1 eleq1d eleq2 eleq2d rspc2v
      wf ancoms com12 3ad2ant3 sylbi syld pm2.43d 3impia ) BFZABGZHZCAHZCBIZABI
      ZHZUSVAJZVBVEVFVBCUTHZVBVEKZUSUTLZVAVBVGKBMVIVAVBVGVIVBVAVGCAUTNUAOUBUSVA
      VGVHUSUTUCBUKZVIDSZESZHZVKBIZVLBIZHZKZEUTPDUTPZUDVAVGJZVHKZDEBUEVRVJVTVIV
      SVRVHVGVAVRVHKVQVHCVLHZVCVOHZKDECAUTUTVKCQZVMWAVPWBVKCVLUFWCVNVCVOVKCBRUG
      TVLAQZWAVBWBVEVLACUHWDVOVDVCVLABRUITUJULUMUNUOOUPUQUR $.

    $( The value of a strictly monotone ordinal function contains its indexed
       union.  (Contributed by Andrew Salmon, 22-Nov-2011.) $)
    smoiun $p |- ( ( Smo B /\ A e. dom B ) -> U_ x e. A ( B ` x )
       C_ ( B ` A ) ) $=
      ( vy wsmo cdm wcel wa cv cfv ciun wrex eliun con0 wi smofvon smoel 3expia
      ontr1 expcomd sylsyld rexlimdv biimtrid ssrdv ) CEZBCFGZHZDABAIZCJZKZBCJZ
      DIZUJGULUIGZABLUGULUKGZAULBUIMUGUMUNABUGUKNGZUHBGZUIUKGZUMUNOBCPUEUFUPUQB
      CUHQRUOUMUQUNULUIUKSTUAUBUCUD $.

    $( If ` F ` is an isomorphism from an ordinal ` A ` onto ` B ` , which is a
       subset of the ordinals, then ` F ` is a strictly monotonic function.
       Exercise 3 in [TakeutiZaring] p. 50.  (Contributed by Andrew Salmon,
       24-Nov-2011.) $)
    smoiso $p |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On )
        -> Smo F ) $=
      ( vx vy cep word con0 wss wf cv wcel cfv wral syl wb wa eleq2d wbr cvv wi
      wiso w3a cdm wsmo wf1o isof1o f1of ffdm simpld sylan 3adant2 syl3an1 wceq
      fss fdm eqcomd ordeq 4syl biimpa 3adant3 anbi12d 3syl epel isorel bitr3id
      wfn ffn adantr simprr funfvex funfni epelg syl2anc bitrd biimpd ex sylbid
      ralrimivv 3ad2ant1 df-smo syl3anbrc ) ABFFCUBZAGZBHIZUCCUDZHCJZWFGZDKZEKZ
      LZWICMZWJCMZLZUAZEWFNDWFNZCUEWCABCJZWDWEWGWCABCUFZWQABFFCUGZABCUHZOZWQWEW
      GWDWQWFBCJZWEWGWQXBWFAIABCUIUJWFBHCUOUKULUMWCWDWHWEWCWDWHWCWRWQAWFUNWDWHP
      WSWTWQWFAABCUPZUQAWFURUSUTVAWCWDWPWEWCWODEWFWFWCWIWFLZWJWFLZQZWIALZWJALZQ
      ZWOWCWRWQXFXIPWSWTWQXDXGXEXHWQWFAWIXCRWQWFAWJXCRVBVCWCXIWOWCXIQZWKWNXJWKW
      LWMFSZWNWKWIWJFSXJXKDEVDABWIWJFFCVEVFXJCAVGZXHXKWNPZWCXLXIWCWQXLXAABCVHOV
      IWCXGXHVJXLXHQWMTLZXMXNAWJCWJCVKVLWLWMTVMOVNVOVPVQVRVSVTDECWAWB $.
  $}

  $( A strictly monotone ordinal function preserves the epsilon relation.
     (Contributed by Mario Carneiro, 12-Mar-2013.) $)
  smoel2 $p |- ( ( ( F Fn A /\ Smo F ) /\ ( B e. A /\ C e. B ) )
    -> ( F ` C ) e. ( F ` B ) ) $=
    ( wfn wsmo wcel cfv cdm fndm eleq2d anbi1d biimprd smoel 3expib sylan9 imp
    wa ) DAEZDFZRBAGZCBGZRZCDHBDHGZSUCBDIZGZUBRZTUDSUGUCSUFUAUBSUEABADJKLMTUFUB
    UDBDCNOPQ $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  "Strong" transfinite recursion
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c recs $.

  $( Notation for a function defined by strong transfinite recursion. $)
  crecs $a class recs ( F ) $.

  ${
    $d F f x y $.
    $( Define a function ` recs ( F ) ` on ` On ` , the class of ordinal
       numbers, by transfinite recursion given a rule ` F ` which sets the next
       value given all values so far.  See ~ df-irdg for more details on why
       this definition is desirable.  Unlike ~ df-irdg which restricts the
       update rule to use only the previous value, this version allows the
       update rule to use _all_ previous values, which is why it is described
       as "strong", although it is actually more primitive.  See ~ tfri1d and
       ~ tfri2d for the primary contract of this definition.

       (Contributed by Stefan O'Rear, 18-Jan-2015.) $)
    df-recs $a |- recs ( F ) = U. { f | E. x e. On ( f Fn x /\
        A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
  $}

  ${
    $d F a b c $.  $d G a b c $.  $d x a b c $.
    $( Equality theorem for ` recs ` .  (Contributed by Stefan O'Rear,
       18-Jan-2015.) $)
    recseq $p |- ( F = G -> recs ( F ) = recs ( G ) ) $=
      ( va vb vc wceq cv wfn cfv cres wral wa con0 wrex cuni crecs fveq1 eqeq2d
      cab df-recs ralbidv anbi2d rexbidv abbidv unieqd 3eqtr4g ) ABFZCGZDGZHZEG
      ZUHIZUHUKJZAIZFZEUIKZLZDMNZCSZOUJULUMBIZFZEUIKZLZDMNZCSZOAPBPUGUSVEUGURVD
      CUGUQVCDMUGUPVBUJUGUOVAEUIUGUNUTULUMABQRUAUBUCUDUEDECATDECBTUF $.

    nfrecs.f $e |- F/_ x F $.
    $( Bound-variable hypothesis builder for ` recs ` .  (Contributed by Stefan
       O'Rear, 18-Jan-2015.) $)
    nfrecs $p |- F/_ x recs ( F ) $=
      ( va vb vc crecs cv wfn cfv cres wceq wral wa con0 wrex cuni df-recs nfcv
      cab nfv nffv nfeq2 nfralxy nfan nfrexw nfab nfuni nfcxfr ) ABGDHZEHZIZFHZ
      UJJZUJUMKZBJZLZFUKMZNZEOPZDTZQEFDBRAVAUTADUSAEOAOSULURAULAUAUQAFUKAUKSAUN
      UPAUOBCAUOSUBUCUDUEUFUGUHUI $.
  $}

  ${
    $d A u w x y z $.  $d B x $.  $d F u w x y z $.  $d G u w x y z $.
    $d ph u w y z $.
    tfrlem1.1 $e |- ( ph -> A e. On ) $.
    tfrlem1.2 $e |- ( ph -> ( Fun F /\ A C_ dom F ) ) $.
    tfrlem1.3 $e |- ( ph -> ( Fun G /\ A C_ dom G ) ) $.
    tfrlem1.4 $e |- ( ph -> A. x e. A ( F ` x ) = ( B ` ( F |` x ) ) ) $.
    tfrlem1.5 $e |- ( ph -> A. x e. A ( G ` x ) = ( B ` ( G |` x ) ) ) $.
    $( A technical lemma for transfinite recursion.  Compare Lemma 1 of
       [TakeutiZaring] p. 47.  (Contributed by NM, 23-Mar-1995.)  (Revised by
       Mario Carneiro, 24-May-2019.) $)
    tfrlem1 $p |- ( ph -> A. x e. A ( F ` x ) = ( G ` x ) ) $=
      ( vz wss cv cfv wceq wral wi wa syl vy vw vu ssid con0 wcel sseq1 imbi12d
      raleq imbi2d r19.21v cres cdm wfun simplll adantr simpld funfn sylib word
      wfn simpllr eloni ordelss sylan simplr sstrd simprd fnssres syl2anc fveq2
      eqeq12d ad2antrr rspcv syl3c rspcdva adantl 3eqtr4d eqfnfvd fveq2d reseq2
      simpr fvres sselda ralrimiva cbvralv sylibr exp31 expcom biimtrid vtoclga
      a2d tfis2 mpcom mpi ) ACCMZBNZEOZWQFOZPZBCQZCUDCUEUFAWPXARZGAUANZCMZWTBXC
      QZRZRZAXBRUACUEXCCPZXFXBAXHXDWPXEXAXCCCUGWTBXCCUIUHUJXGALNZCMZWTBXIQZRZRZ
      UALXCXIPZXFXLAXNXDXJXEXKXCXICUGWTBXCXIUIUHUJXMLXCQAXLLXCQZRXCUEUFZXGAXLLX
      CUKXPAXOXFAXPXOXFRAXPSZXOXDXEXQXOSZXDSZUBNZEOZXTFOZPZUBXCQXEXSYCUBXCXSXTX
      CUFZSZEXTULZDOZFXTULZDOZYAYBYEYFYHDYEUCXTYFYHYEEEUMZVAZXTYJMYFXTVAYEEUNZY
      KYEYLCYJMZYEAYLYMSXSAYDAXPXOXDUOUPZHTZUQEURUSYEXTCYJYEXTXCCXSXCUTZYDXTXCM
      XSXPYPAXPXOXDVBXCVCTXCXTVDVEXRXDYDVFVGZYEYLYMYOVHVGYJXTEVIVJYEFFUMZVAZXTY
      RMYHXTVAYEFUNZYSYEYTCYRMZYEAYTUUASYNITZUQFURUSYEXTCYRYQYEYTUUAUUBVHVGYRXT
      FVIVJYEUCNZXTUFZSZUUCEOZUUCFOZUUCYFOZUUCYHOZUUEWTUUFUUGPBXTUUCWQUUCPWRUUF
      WSUUGWQUUCEVKWQUUCFVKVLUUEYDXOXTCMZWTBXTQZXSYDUUDVFXSXOYDUUDXQXOXDVFVMYEU
      UJUUDYQUPXLUUJUUKRLXTXCXIXTPXJUUJXKUUKXIXTCUGWTBXIXTUIUHVNVOYEUUDWBVPUUDU
      UHUUFPYEUUCXTEWCVQUUDUUIUUGPYEUUCXTFWCVQVRVSVTYEWREWQULZDOZPZYAYGPBCXTWQX
      TPZWRYAUUMYGWQXTEVKZUUOUULYFDWQXTEWAVTVLYEAUUNBCQYNJTXSXCCXTXRXDWBWDZVPYE
      WSFWQULZDOZPZYBYIPBCXTUUOWSYBUUSYIWQXTFVKZUUOUURYHDWQXTFWAVTVLYEAUUTBCQYN
      KTUUQVPVRWEWTYCBUBXCUUOWRYAWSYBUUPUVAVLWFWGWHWIWLWJWMWKWNWO $.
  $}

  ${
    $d g A $.  $d f g w x y z F $.  $d f w x y z G $.
    tfrlem3.1 $e |- A = { f | E. x e. On ( f Fn x /\
                A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
    $( Lemma for transfinite recursion.  This lemma just changes some bound
       variables in ` A ` for later use.  (Contributed by Jim Kingdon,
       5-Jul-2019.) $)
    tfrlem3ag $p |- ( G e. _V -> ( G e. A <-> E. z e. On ( G Fn z /\
                A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) ) $=
      ( cv wfn cfv cres wceq wral wa con0 wrex cvv fneq12 simpll simpr reseq12d
      fveq12d fveq2d eqeq12d simplr cbvraldva2 anbi12d cbvrexdva elab2g ) FJZAJ
      ZKZBJZULLZULUOMZGLZNZBUMOZPZAQRHCJZKZDJZHLZHVDMZGLZNZDVBOZPZCQRFHESULHNZV
      AVJACQVKUMVBNZPZUNVCUTVIUMVBULHTVMUSVHBDUMVBVMUOVDNZPZUPVEURVGVOUOVDULHVK
      VLVNUAZVMVNUBZUDVOUQVFGVOULHUOVDVPVQUCUEUFVKVLVNUGUHUIUJIUK $.

    ${
      tfrlem3.2 $e |- G e. _V $.
      $( Lemma for transfinite recursion.  Let ` A ` be the class of
         "acceptable" functions.  The final thing we're interested in is the
         union of all these acceptable functions.  This lemma just changes some
         bound variables in ` A ` for later use.  (Contributed by NM,
         9-Apr-1995.) $)
      tfrlem3a $p |- ( G e. A <-> E. z e. On ( G Fn z /\
                  A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) $=
        ( cv wfn cfv cres wceq wral wa con0 wrex simpr fveq12d reseq12d eqeq12d
        fneq12 simpll fveq2d adantr cbvraldva2 anbi12d cbvrexdva elab2 ) FKZAKZ
        LZBKZULMZULUONZGMZOZBUMPZQZARSHCKZLZDKZHMZHVDNZGMZOZDVBPZQZCRSFHEJULHOZ
        VAVJACRVKUMVBOZQZUNVCUTVIUMVBULHUDVMUSVHBDUMVBVMUOVDOZQZUPVEURVGVOUOVDU
        LHVKVLVNUEZVMVNTZUAVOUQVFGVOULHUOVDVPVQUBUFUCVMVLVNVKVLTUGUHUIUJIUK $.
    $}

    $( Lemma for transfinite recursion.  Let ` A ` be the class of "acceptable"
       functions.  The final thing we're interested in is the union of all
       these acceptable functions.  This lemma just changes some bound
       variables in ` A ` for later use.  (Contributed by NM, 9-Apr-1995.) $)
    tfrlem3 $p |- A = { g | E. z e. On ( g Fn z /\
                A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) } $=
      ( cv wfn cfv cres wceq wral wa con0 wrex vex tfrlem3a abbi2i ) GJZCJZKDJZ
      UBLUBUDMHLNDUCOPCQRGEABCDEFHUBIGSTUA $.
  $}

  ${
    $d g x F $.
    tfrlem3-2d.1 $e |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) ) $.
    $( Lemma for transfinite recursion which changes a bound variable
       (Contributed by Jim Kingdon, 2-Jul-2019.) $)
    tfrlem3-2d $p |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) ) $=
      ( wfun cv cfv cvv wcel wal weq fveq2 eleq1d anbi2d cbvalv sylib 19.21bi
      wa ) ADFZCGZDHZIJZSZCATBGZDHZIJZSZBKUDCKEUHUDBCBCLZUGUCTUIUFUBIUEUADMNOPQ
      R $.
  $}

  ${
    $d f g x y z B $.  $d a f g h u v w x y z F $.  $d g h z A $.
    tfrlem.1 $e |- A = { f | E. x e. On ( f Fn x /\
                A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
    $( Lemma for transfinite recursion. ` A ` is the class of all "acceptable"
       functions, and ` F ` is their union.  First we show that an acceptable
       function is in fact a function.  (Contributed by NM, 9-Apr-1995.) $)
    tfrlem4 $p |- ( g e. A -> Fun g ) $=
      ( vz vw cv wcel wfn cfv cres wceq wral wa con0 wrex wfun abeq2i rexlimivw
      tfrlem3 fnfun adantr sylbi ) EJZCKUGHJZLZIJZUGMUGUJNFMOIUHPZQZHRSZUGTZUME
      CABHICDEFGUCUAULUNHRUIUNUKUHUGUDUEUBUF $.

    $( Lemma for transfinite recursion.  The values of two acceptable functions
       are the same within their domains.  (Contributed by NM, 9-Apr-1995.)
       (Revised by Mario Carneiro, 24-May-2019.) $)
    tfrlem5 $p |- ( ( g e. A /\ h e. A ) ->
                    ( ( x g u /\ x h v ) -> u = v ) ) $=
      ( vz va vw cv wcel cfv wceq wral wa con0 wfn cres wrex wi tfrlem3a reeanv
      wbr vex w3a cin fveq2 eqeq12d onin 3ad2ant1 wfun cdm wss simp2ll syl fndm
      fnfun inss1 sseqtrrid simp2rl inss2 simp2lr ssralv simp2rr tfrlem1 simp3l
      mpsyl fnbr syl2anc simp3r elin sylanbrc rspcdva funbrfv sylc 3eqtr3d 3exp
      jca rexlimdva rexlimiv sylbir syl2anb ) GNZEOWGKNZUAZLNZWGPZWGWJUBIPQZLWH
      RZSZKTUCZHNZMNZUAZWJWPPZWPWJUBIPQZLWQRZSZMTUCZANZDNZWGUGZXDCNZWPUGZSZXEXG
      QZUDZWPEOABKLEFIWGJGUHUEABMLEFIWPJHUHUEWOXCSWNXBSZMTUCZKTUCXKWNXBKMTTUFXM
      XKKTWHTOZXLXKMTXNWQTOSZXLXIXJXOXLXIUIZXDWGPZXDWPPZXEXGXPWKWSQXQXRQLWHWQUJ
      ZXDWJXDQWKXQWSXRWJXDWGUKWJXDWPUKULXPLXSIWGWPXOXLXSTOXIWHWQUMUNXPWGUOZXSWG
      UPZUQXPWIXTWIWMXBXOXIURZWHWGVAUSZXPWHXSYAWHWQVBZXPWIYAWHQYBWHWGUTUSVCWBXP
      WPUOZXSWPUPZUQXPWRYEWRXAWNXOXIVDZWQWPVAUSZXPWQXSYFWHWQVEZXPWRYFWQQYGWQWPU
      TUSVCWBXSWHUQXPWMWLLXSRYDWIWMXBXOXIVFWLLXSWHVGVKXSWQUQXPXAWTLXSRYIWRXAWNX
      OXIVHWTLXSWQVGVKVIXPXDWHOZXDWQOZXDXSOXPWIXFYJYBXOXLXFXHVJZWHXDXEWGVLVMXPW
      RXHYKYGXOXLXFXHVNZWQXDXGWPVLVMXDWHWQVOVPVQXPXTXFXQXEQYCYLXDXEWGVRVSXPYEXH
      XRXGQYHYMXDXGWPVRVSVTWAWCWDWEWF $.

    $( Lemma for transfinite recursion.  The definition ` recs ` is the union
       of all acceptable functions.  (Contributed by Mario Carneiro,
       9-May-2015.) $)
    recsfval $p |- recs ( F ) = U. A $=
      ( crecs cv wfn cfv cres wceq wral wa con0 wrex cab cuni df-recs unieqi
      eqtr4i ) EGDHZAHZIBHZUBJUBUDKEJLBUCMNAOPDQZRCRABDESCUEFTUA $.

    $( Lemma for transfinite recursion.  The union of all acceptable functions
       is a relation.  (Contributed by NM, 8-Aug-1994.)  (Revised by Mario
       Carneiro, 9-May-2015.) $)
    tfrlem6 $p |- Rel recs ( F ) $=
      ( vg crecs wrel cuni reluni wcel wfun tfrlem4 funrel syl mprgbir recsfval
      cv releqi mpbir ) EHZICJZIZUDGSZIZGCGCKUECLUEMUFABCDGEFNUEOPQUBUCABCDEFRT
      UA $.

    $( Lemma for transfinite recursion.  The union of all acceptable functions
       is a function.  (Contributed by NM, 9-Aug-1994.)  (Revised by Mario
       Carneiro, 24-May-2019.) $)
    tfrlem7 $p |- Fun recs ( F ) $=
      ( vu vv vg vh cv cop wcel wa wal wex eleq2i eluni bitri anbi12i wfun wrel
      crecs wceq wi tfrlem6 cuni recsfval eeanv bitr4i wbr df-br tfrlem5 impcom
      sylanbr an4s exlimivv sylbi ax-gen gen2 dffun4 mpbir2an ) EUCZUAVCUBAKZGK
      ZLZVCMZVDHKZLZVCMZNZVEVHUDZUEZHOZGOAOABCDEFUFVNAGVMHVKVFIKZMZVOCMZNZVIJKZ
      MZVSCMZNZNZJPIPZVLVKVRIPZWBJPZNWDVGWEVJWFVGVFCUGZMWEVCWGVFABCDEFUHZQIVFCR
      SVJVIWGMWFVCWGVIWHQJVICRSTVRWBIJUIUJWCVLIJVPVTVQWAVLVPVTNVDVEVOUKZVDVHVSU
      KZNZVQWANZVLWIVPWJVTVDVEVOULVDVHVSULTWLWKVLABHGCDIJEFUMUNUOUPUQURUSUTAGHV
      CVAVB $.

    $( Lemma for transfinite recursion.  The domain of ` recs ` is ordinal.
       (Contributed by NM, 14-Aug-1994.)  (Proof shortened by Alan Sare,
       11-Mar-2008.) $)
    tfrlem8 $p |- Ord dom recs ( F ) $=
      ( vz vg vw cdm word cv wceq wrex cuni con0 wcel cfv rexlimiv ax-mp cab wi
      crecs wss wfn cres wral tfrlem3 abeq2i fndm adantr eleq1d biimprcd eleq1a
      wa sylbi syl abssi ssorduni wb ciun recsfval dmeqi vex dmex dfiun2 3eqtri
      dmuni ordeq mpbir ) EUCZJZKZGLZHLZJZMZHCNZGUAZOZKZVSPUDWAVRGPVQVNPQZHCVOC
      QZVPPQZVQWBUBWCVOVNUEZILZVORVOWFUFERMIVNUGZUOZGPNZWDWIHCABGICDHEFUHUIWHWD
      GPWHWDWBWHVPVNPWEVPVNMWGVNVOUJUKULUMSUPVPPVNUNUQSURVSUSTVLVTMVMWAUTVLCOZJ
      HCVPVAVTVKWJABCDEFVBVCHCVHHGCVPVOHVDVEVFVGVLVTVITVJ $.

    $( Lemma for transfinite recursion.  Here we compute the value of ` recs `
       (the union of all acceptable functions).  (Contributed by NM,
       17-Aug-1994.) $)
    tfrlem9 $p |- ( B e. dom recs ( F ) ->
      ( recs ( F ) ` B ) = ( F ` ( recs ( F ) |` B ) ) ) $=
      ( vz cdm wcel cv wex cfv cres wceq wa con0 wi syl com3l crecs cop ibi wfn
      eldm2g wral wrex cab cuni df-recs eleq2i eluniab bitri wss abeq2i elssuni
      fnop rspe recsfval sseqtrrdi sylbir fveq2 reseq2 fveq2d rspcv fndm eleq2d
      eqeq12d wfun tfrlem7 funssfv mp3an1 adantrl eleq1d onelss biimtrrdi imp31
      w3a fun2ssres sylan2 exbiri exp31 com34 com24 sylbird syld imp4d exp4d ex
      mpdi com4r pm2.43i imp4a rexlimdv imp exlimiv sylbi ) DFUAZIZJZDHKZUBZWRJ
      ZHLZDWRMZWRDNZFMZOZWTXDHDWRWSUEUCXCXHHXCXBEKZJZXIAKZUDZBKZXIMZXIXMNZFMZOZ
      BXKUFZPZAQUGZPZELZXHXCXBXTEUHUIZJYBWRYCXBABEFUJUKXTEXBULUMYAXHEXJXTXHXJXS
      XHAQXJXKQJZXLXRXHXLXJYDXRXHRZXLXJYDYERRXLXJYDXLYEXLXJYDXLYERRXLXJPZYDXLXR
      XHYFDXKJZYDXSPZXHRXKDXAXIUQYGYHXIWRUNZXHYHXTYIXSAQURXTXICJZYIXTECGUOYJXIC
      UIWRXICUPABCEFGUSUTVASYGYDXLXRYIXHRZYGXRXLYDYKYGXRDXIMZXIDNZFMZOZXLYDYKRZ
      RXQYOBDXKXMDOZXNYLXPYNXMDXIVBYQXOYMFXMDXIVCVDVHVEXLYGYOYPXLYGDXIIZJZYOYPR
      XLYRXKDXKXIVFZVGXLYDYOYSYKXLYDYSYOYKXLYDYSYOYKRYIXLYDPZYSPZYOXHYIUUBXHYOY
      IUUBPXEYLXGYNYIYSXEYLOZUUAWRVIZYIYSUUCABCEFGVJZDWRXIVKVLVMUUBYIDYRUNZXGYN
      OZXLYDYSUUFXLYDYRQJYSUUFRXLYRXKQYTVNYRDVOVPVQUUDYIUUFUUGUUEUUDYIUUFVRXFYM
      FDWRXIVSVDVLVTVHWATWBWCWDWETWFWDWGWJSWHWIWKWLTWMWNWOWPWQWPS $.
  $}

  ${
    $d F f x y $.
    $( Transfinite recursion produces a function.  (Contributed by Jim Kingdon,
       20-Aug-2021.) $)
    tfrfun $p |- Fun recs ( F ) $=
      ( vx vy vf cv wfn cfv cres wceq wral wa con0 wrex cab eqid tfrlem7 ) BCDE
      ZBEZFCEZQGQSHAGICRJKBLMDNZDATOP $.
  $}

  ${
    $d f x y A $.  $d x y F $.  $d f x y G $.
    tfr.1 $e |- F = recs ( G ) $.
    $( A weak version of transfinite recursion.  (Contributed by Mario
       Carneiro, 24-Jun-2015.) $)
    tfr2a $p |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) ) $=
      ( vx vy vf cdm wcel crecs cfv cres wceq cv wfn wral wa con0 wrex cab eqid
      tfrlem9 dmeqi eleq2s fveq1i reseq1i fveq2i 3eqtr4g ) ABHZIACJZKZUJALZCKZA
      BKBALZCKUKUMMAUJHUIEFGNZENZOFNZUOKUOUQLCKMFUPPQERSGTZAGCURUAUBBUJDUCUDABU
      JDUEUNULCBUJADUFUGUH $.

    $( Transfinite recursion is defined at the empty set.  (Contributed by Jim
       Kingdon, 8-Mar-2022.) $)
    tfr0dm $p |- ( ( G ` (/) ) e. V -> (/) e. dom F ) $=
      ( vx vy vf c0 cfv wcel cv wfn cres wceq wral wa con0 cvv 0ex mpan cop cdm
      csn wrex opexg snidg fnsng fvsng res0 fveq2i eqtr4di fveq2 reseq2 eqeq12d
      syl fveq2d ralsn sylibr csuc suc0 0elon onsuci fneq2 raleq anbi12d rspcev
      eqeltrri syl2anc wi snexg eleq2 fneq1 fveq1 reseq1 ralbidv rexbidv spcegv
      wex 3syl crecs cab eleq2i df-recs eluniab 3bitri imbitrrdi mp2and opeldmg
      cuni mpd ) HBIZCJZHWKUAZAJZHAUBJZWLWMWMUCZJZWPEKZLZFKZWPIZWPWTMZBIZNZFWRO
      ZPZEQUDZWNWLWMRJZWQHRJZWLXHSHWKRCUETZWMRUFUOWLWPHUCZLZXDFXKOZXGXIWLXLSHWK
      RCUGTWLHWPIZWPHMZBIZNZXMWLXNWKXPXIWLXNWKNSHWKRCUHTXOHBWPUIUJUKXDXQFHSWTHN
      ZXAXNXCXPWTHWPULXRXBXOBWTHWPUMUPUNUQURXKQJXLXMPZXGHUSXKQUTHVAVBVGXFXSEXKQ
      WRXKNWSXLXEXMWRXKWPVCXDFWRXKVDVEVFTVHWLWQXGPZWMGKZJZYAWRLZWTYAIZYAWTMZBIZ
      NZFWROZPZEQUDZPZGVRZWNWLXHWPRJXTYLVIXJWMRVJYKXTGWPRYAWPNZYBWQYJXGYAWPWMVK
      YMYIXFEQYMYCWSYHXEWRYAWPVLYMYGXDFWRYMYDXAYFXCWTYAWPVMYMYEXBBYAWPWTVNUPUNV
      OVEVPVEVQVSWNWMBVTZJWMYJGWAWIZJYLAYNWMDWBYNYOWMEFGBWCWBYJGWMWDWEWFWGXIWLW
      NWOVISHWKARCWHTWJ $.

    $( Transfinite recursion at the empty set.  (Contributed by Jim Kingdon,
       8-May-2020.) $)
    tfr0 $p |- ( ( G ` (/) ) e. V -> ( F ` (/) ) = ( G ` (/) ) ) $=
      ( c0 cfv wcel cres cdm wceq tfr0dm tfr2a syl res0 fveq2i eqtrdi ) EBFZCGZ
      EAFZAEHZBFZQREAIGSUAJABCDKEABDLMTEBANOP $.
  $}

  ${
    $d e f g h k t u v w x y z A $.  $d e f g h k t u v w x y z F $.
    $d ph v w y $.
    tfrlemisucfn.1 $e |- A = { f | E. x e. On ( f Fn x /\
                A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
    tfrlemisucfn.2 $e |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) ) $.
    ${
      $d u w y ph $.
      tfrlemisucfn.3 $e |- ( ph -> z e. On ) $.
      tfrlemisucfn.4 $e |- ( ph -> g Fn z ) $.
      tfrlemisucfn.5 $e |- ( ph -> g e. A ) $.
      $( We can extend an acceptable function by one element to produce a
         function.  Lemma for ~ tfrlemi1 .  (Contributed by Jim Kingdon,
         2-Jul-2019.) $)
      tfrlemisucfn $p |- ( ph -> ( g u. { <. z , ( F ` g ) >. } ) Fn suc z ) $=
        ( cv csuc cfv cop cvv wcel a1i csn cun wfun tfrlem3-2d simprd df-suc wn
        vex eqid elirrv fnunsn ) ADNZULOGNZUMULUMHPZQUAUBZULUNULRSADUHTAHUCUNRS
        ABGHJUDUELUOUIULUFULULSUGADUJTUK $.

      $( We can extend an acceptable function by one element to produce an
         acceptable function.  Lemma for ~ tfrlemi1 .  (Contributed by Jim
         Kingdon, 4-Mar-2019.)  (Proof shortened by Mario Carneiro,
         24-May-2019.) $)
      tfrlemisucaccv $p |- ( ph -> ( g u. { <. z , ( F ` g ) >. } ) e. A ) $=
        ( vu cfv wcel wceq wa con0 cvv vw vv cv cop csn cun cres wral wrex csuc
        wfn onsuc syl tfrlemisucfn wo vex elsuc tfrlem3a simprrr simprrl adantr
        sylib fndmu syl2anc raleqdv rexlimddv r19.21bi wne elirrv elequ2 mtbiri
        mpbid necon2ai adantl fvunsng sylancr wss word eloni ordelss resabs1 wn
        sylan fsnunres sylancl reseq1d eqtr3d fveq2d cdm wfun tfrlem3-2d simprd
        3eqtr4d eleq2d fsnunfv syl3anc simpr reseq2 sylan9eqr sylan2b ralrimiva
        fndm jaodan fneq2 raleq anbi12d syl12anc wb opexg snexg unexg tfrlem3ag
        rspcev mpbird ) AGUCZDUCZXOHOZUDZUEZUFZEPZXTUAUCZUKZNUCZXTOZXTYDUGZHOZQ
        ZNYBUHZRZUASUIZAXPUJZSPZXTYLUKZYHNYLUHZYKAXPSPZYMKXPULUMABCDEFGHIJKLMUN
        AYHNYLYDYLPAYDXPPZYDXPQZUOYHYDXPNUPZUQAYQYHYRAYQRZYDXOOZXOYDUGZHOZYEYGA
        UUAUUCQZNXPAXOUBUCZUKZUUDNUUEUHZRZUUDNXPUHZUBSAXOEPUUHUBSUIMBCUBNEFHXOI
        GUPZURVBAUUESPZUUHRZRZUUGUUIAUUKUUFUUGUSUUMUUDNUUEXPUUMUUFXOXPUKZUUEXPQ
        AUUKUUFUUGUTAUUNUULLVAUUEXPXOVCVDVEVLVFVGYTYDTPXPYDVHZYEUUAQYSYQUUOAYQX
        PYDXPYDQYQYDYDPNVIDNNVJVKVMVNXOXPXQYDTVOVPYTYFUUBHYTXTXPUGZYDUGZYFUUBYT
        YDXPVQZUUQYFQAXPVRZYQUURAYPUUSKXPVSUMXPYDVTWCXTYDXPWAUMAUUQUUBQYQAUUPXO
        YDAUUNXPXPPZWBUUPXOQLDVIZXPXOXPXQWDWEZWFVAWGWHWMAYRRZXPXTOZXQYEYGAUVDXQ
        QZYRAYPXQTPZXPXOWIZPZWBUVEKAHWJUVFABGHJWKWLZAUVHUUTUVAAUVGXPXPAUUNUVGXP
        QLXPXOXBUMWNVKXOSTXPXQWOWPVAUVCYDXPXTAYRWQWHUVCYFXOHYRAYFUUPXOYDXPXTWRU
        VBWSWHWMXCWTXAYJYNYORUAYLSYBYLQYCYNYIYOYBYLXTXDYHNYBYLXEXFXMXGAXTTPZYAY
        KXHAXOTPXSTPZUVJUUJAXRTPZUVKAXPTPUVFUVLDUPUVIXPXQTTXIVPXRTXJUMXOXSTTXKV
        PBCUANEFHXTIXLUMXN $.
    $}

    ${
      $d u w f g h z B $.  $d g h y z w ph $.
      tfrlemi1.3 $e |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\
         h = ( g u. { <. z , ( F ` g ) >. } ) ) } $.
      tfrlemi1.4 $e |- ( ph -> x e. On ) $.
      tfrlemi1.5 $e |- ( ph -> A. z e. x E. g ( g Fn z /\
        A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) ) $.
      $( Each element of ` B ` is an acceptable function.  Lemma for
         ~ tfrlemi1 .  (Contributed by Jim Kingdon, 14-Mar-2019.)  (Proof
         shortened by Mario Carneiro, 24-May-2019.) $)
      tfrlemibacc $p |- ( ph -> B C_ A ) $=
        ( cv wcel cfv wa wfn cop csn cun wceq w3a wex wrex simpr3 wfun ad2antrr
        cab cvv con0 simplr onelon syl2anc simpr1 simpr2 tfrlemisucaccv eqeltrd
        wal ex exlimdv rexlimdva abssdv eqsstrid ) AGIQZDQZUAZVHFRZJQZVHVIVHKSU
        BUCUDZUEZUFZIUGZDBQZUHZJULFNAVRJFAVPVLFRZDVQAVIVQRZTZVOVSIWAVOVSWAVOTZV
        LVMFWAVJVKVNUIWBBCDFHIKLAKUJVQKSUMRTBVBVTVOMUKWBVQUNRZVTVIUNRAWCVTVOOUK
        AVTVOUOVQVIUPUQWAVJVKVNURWAVJVKVNUSUTVAVCVDVEVFVG $.

      $( The union of ` B ` is defined on all ordinals.  Lemma for ~ tfrlemi1 .
         (Contributed by Jim Kingdon, 18-Mar-2019.)  (Proof shortened by Mario
         Carneiro, 24-May-2019.) $)
      tfrlemibxssdm $p |- ( ph -> x C_ dom U. B ) $=
        ( cv wcel wa cvv cuni cdm wral wss wfn cfv cres wceq con0 wi tfrlem3-2d
        wex w3a cop wfun simprd 3ad2ant1 csn cun opexg sylancr snidg elun2 3syl
        simp2r simp3l wrex onelon rspe sylan tfrlem3a sylibr 3adant1 3jca snexg
        vex unexg isset sylib simpr3 19.8a abeq2i sylan2 eqeltrrd 3exp2 exlimdv
        mpan 3imp sylc elunii syl2anc opeq2 eleq1d spcegv eldm2 3expia ralimdva
        imbitrrdi anassrs mpdan mpd dfss3 ) ADQZGUAZUBZRZDBQZUCZXGXEUDAIQZXCUEZ
        EQZXIUFXIXKUGKUFUHEXCUCZSZIULZDXGUCZXHPAXGUIRZXOXHUJOAXPSXNXFDXGAXPXCXG
        RZXNXFUJAXPXQSZSXMXFIAXRXMXFAXRXMUMZXIKUFZTRZXCXTUNZXDRZXFAXRYAXMAKUOYA
        ABIKMUKUPZUQXSYBXIYBURZUSZRZYFGRZYCAXRYGXMAYBTRZYBYERYGAXCTRYAYIDVPZYDX
        CXTTTUTVAZYBTVBYBYEXIVCVDUQXSXQXJXIFRZUMZJQZYFUHZJULZYHXSXQXJYLAXPXQXMV
        EAXRXJXLVFXRXMYLAXRXMSXMDUIVGZYLXRXCUIRXMYQXGXCVHXMDUIVIVJBCDEFHKXILIVP
        ZVKVLVMVNAXRYPXMAYFTRZYPAYIYETRZYSYKYBTVOXITRYTYSYRXIYETTVQWGVDJYFVRVSU
        QYMYOYHJXQXJYLYOYHUJXQXJYLYOYHXQXJYLYOUMZSYNYFGXQXJYLYOVTUUAXQUUAIULZYN
        GRZUUAIWAXQUUBSUUBDXGVGZUUCUUBDXGVIUUDJGNWBVLWCWDWEWHWFWIYBYFGWJWKYAYCX
        CXKUNZXDRZEULXFUUFYCEXTTXKXTUHUUEYBXDXKXTXCWLWMWNEXCXDYJWOWRWIWPWFWSWQW
        TXADXGXEXBVL $.

      $( The union of ` B ` is a function defined on ` x ` .  Lemma for
         ~ tfrlemi1 .  (Contributed by Jim Kingdon, 18-Mar-2019.)  (Proof
         shortened by Mario Carneiro, 24-May-2019.) $)
      tfrlemibfn $p |- ( ph -> U. B Fn x ) $=
        ( wss cvv wcel syl cuni wfun cdm cv wceq wfn crecs tfrlemibacc recsfval
        unissd sseqtrrdi tfrlem7 funss cxp cpw cfv cop csn cun w3a wex wrex cab
        mpisyl wa simpr3 csuc wf wal ad2antrr con0 simplr onelon syl2anc simpr1
        simpr2 tfrlemisucfn dffn2 sylib fssxp word eloni ordsucss sylc xpss1 wb
        sstrd vex tfrlem3-2d simprd opexg sylancr snexg unexg mpbird eqeltrd ex
        elpwg exlimdv rexlimdva abssdv sspwuni dmss dmxpss sstrdi tfrlemibxssdm
        eqsstrid eqssd df-fn sylanbrc ) AGUAZUBZXKUCZBUDZUEXKXNUFAXKKUGZQXOUBXL
        AXKFUAXOAGFABCDEFGHIJKLMNOPUHUJBCFHKLUIUKBCFHKLULXKXOUMVDAXMXNAXMXNRUNZ
        UCZXNAXKXPQZXMXQQAGXPUOZQXRAGIUDZDUDZUFZXTFSZJUDZXTYAXTKUPZUQZURZUSZUEZ
        UTZIVAZDXNVBZJVCXSNAYLJXSAYKYDXSSZDXNAYAXNSZVEZYJYMIYOYJYMYOYJVEZYDYHXS
        YOYBYCYIVFYPYHXSSZYHXPQZYPYHYAVGZRUNZXPYPYSRYHVHZYHYTQYPYHYSUFUUAYPBCDF
        HIKLAKUBZXNKUPRSVEBVIYNYJMVJYPXNVKSZYNYAVKSAUUCYNYJOVJZAYNYJVLZXNYAVMVN
        YOYBYCYIVOYOYBYCYIVPVQYSYHVRVSYSRYHVTTYPYSXNQZYTXPQYPXNWAZYNUUFYPUUCUUG
        UUDXNWBTUUEYAXNWCWDYSXNRWETWGAYQYRWFZYNYJAYHRSZUUHAXTRSYGRSZUUIIWHAYFRS
        ZUUJAYARSYERSZUUKDWHAUUBUULABIKMWIWJYAYERRWKWLYFRWMTXTYGRRWNWLYHXPRWRTV
        JWOWPWQWSWTXAXGGXPXBVSXKXPXCTXNRXDXEABCDEFGHIJKLMNOPXFXHXKXNXIXJ $.

      $( The set ` B ` exists.  Lemma for ~ tfrlemi1 .  (Contributed by Jim
         Kingdon, 17-Mar-2019.)  (Proof shortened by Mario Carneiro,
         24-May-2019.) $)
      tfrlemibex $p |- ( ph -> B e. _V ) $=
        ( cuni cvv wcel cv wfn tfrlemibfn vex fnex sylancl uniexb sylibr ) AGQZ
        RSZGRSAUHBTZUAUJRSUIABCDEFGHIJKLMNOPUBBUCUJRUHUDUEGUFUG $.

      $( The union of ` B ` satisfies the recursion rule (lemma for
         ~ tfrlemi1 ).  (Contributed by Jim Kingdon, 22-Apr-2019.)  (Proof
         shortened by Mario Carneiro, 24-May-2019.) $)
      tfrlemiubacc $p |- ( ph ->
          A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) $=
        ( cfv wceq wcel cv cuni cres wral crecs cdm wfn tfrlemibfn fndm syl wss
        tfrlemibacc unissd recsfval sseqtrrdi dmss eqsstrrd sselda tfrlem9 wfun
        wa tfrlem7 a1i adantr eleq2d biimpar funssfv syl3anc word eloni ordelss
        sylan sseqtrrd fun2ssres fveq2d 3eqtr3d ralrimiva fveq2 eqeq12d cbvralv
        con0 reseq2 sylibr ) AEUAZHUBZRZWEWDUCZLRZSZEBUAZUDFUAZWERZWEWKUCZLRZSZ
        FWJUDAWIEWJAWDWJTZVAZWDLUEZRZWRWDUCZLRZWFWHWQWDWRUFZTWSXASAWJXBWDAWJWEU
        FZXBAWEWJUGXCWJSZABCDEGHIJKLMNOPQUHWJWEUIUJZAWEWRUKZXCXBUKAWEGUBWRAHGAB
        CDEGHIJKLMNOPQULUMBCGILMUNUOZWEWRUPUJUQURBCGWDILMUSUJWQWRUTZXFWDXCTZWSW
        FSXHWQBCGILMVBVCZAXFWPXGVDZAXIWPAXCWJWDXEVEVFWDWRWEVGVHWQWTWGLWQXHXFWDX
        CUKWTWGSXJXKWQWDWJXCAWJVIZWPWDWJUKAWJWATXLPWJVJUJWJWDVKVLAXDWPXEVDVMWDW
        RWEVNVHVOVPVQWOWIFEWJWKWDSZWLWFWNWHWKWDWEVRXMWMWGLWKWDWEWBVOVSVTWC $.

      $( Lemma for ~ tfrlemi1 .  (Contributed by Jim Kingdon, 18-Mar-2019.)
         (Proof shortened by Mario Carneiro, 24-May-2019.) $)
      tfrlemiex $p |- ( ph ->
          E. f ( f Fn x /\ A. u e. x ( f ` u ) = ( F ` ( f |` u ) ) ) ) $=
        ( cvv cv cfv cuni wcel wfn cres wceq wral wex tfrlemibex syl tfrlemibfn
        wa uniexg tfrlemiubacc fneq1 fveq1 reseq1 fveq2d eqeq12d ralbidv spcegv
        jca anbi12d sylc ) AHUAZRUBZVDBSZUCZFSZVDTZVDVHUDZLTZUEZFVFUFZUKZISZVFU
        CZVHVOTZVOVHUDZLTZUEZFVFUFZUKZIUGAHRUBVEABCDEGHIJKLMNOPQUHHRULUIAVGVMAB
        CDEGHIJKLMNOPQUJABCDEFGHIJKLMNOPQUMVAWBVNIVDRVOVDUEZVPVGWAVMVFVOVDUNWCV
        TVLFVFWCVQVIVSVKVHVOVDUOWCVRVJLVOVDVHUPUQURUSVBUTVC $.
    $}

    $d g u z C $.  $d ph e f h z $.
    $( We can define an acceptable function on any ordinal.

       As with many of the transfinite recursion theorems, we have a hypothesis
       that states that ` F ` is a function and that it is defined for all
       ordinals.  (Contributed by Jim Kingdon, 4-Mar-2019.)  (Proof shortened
       by Mario Carneiro, 24-May-2019.) $)
    tfrlemi1 $p |- ( ( ph /\ C e. On ) -> E. g ( g Fn C /\
        A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) ) $=
      ( vz vk vw vv cv cfv wral wa weq ve vt vh con0 wcel wfn cres wex wi simpr
      wceq simpl fneq12d fveq1d reseq1d fveq2d eqeq12d raleqbidv anbi12d imbi2d
      cbvexdva fneq2 raleq exbidv r19.21v cop csn cun w3a wrex cab tfrlem3 wfun
      cvv fveq2 eleq1d anbi2d cbvalv sylib adantr simplr simpll opeq12d uneq12d
      wal 3anbi123d cbvrexdva cbvabv adantl fveq12d reseq12d cbvraldva2 cbvralv
      sneqd tfrlemiex expr expcom a2d biimtrid tfis3 impcom ) FUDUEAHPZFUFZDPZX
      BQZXBXDUGZIQZUKZDFRZSZHUHZAXBLPZUFZXHDXLRZSZHUHZUIZAMPZNPZUFZXDXRQZXRXDUG
      ZIQZUKZDXSRZSZMUHZUIZAXKUILNFLNTZXPYGAYIXOYFHMYIHMTZSZXMXTXNYEYKXLXSXBXRY
      IYJUJZYIYJULZUMYKXHYDDXLXSYMYKXEYAXGYCYKXDXBXRYLUNYKXFYBIYKXBXRXDYLUOUPUQ
      URUSVAUTXLFUKZXPXKAYNXOXJHYNXMXCXNXIXLFXBVBXHDXLFVCUSVDUTYHNXLRAYGNXLRZUI
      XLUDUEZXQAYGNXLVEYPAYOXPAYPYOXPUIAYPYOXPAYPYOSZSLUAOCDEXTXREUEZUBPZXRXSXR
      IQZVFZVGZVHZUKZVIZMUHZNXLVJZUBVKHGUCIBCLUAEGHIJVLAIVMZXLIQZVNUEZSZLWEZYQA
      UUHBPZIQZVNUEZSZBWEUULKUUPUUKBLBLTZUUOUUJUUHUUQUUNUUIVNUUMXLIVOVPVQVRVSVT
      UUGGPZOPZUFZUUREUEZUCPZUURUUSUURIQZVFZVGZVHZUKZVIZGUHZOXLVJUBUCUBUCTZUUFU
      VINOXLUVJNOTZSZUUEUVHMGUVLMGTZSZXTUUTYRUVAUUDUVGUVNXSUUSXRUURUVLUVMUJZUVJ
      UVKUVMWAZUMUVNXRUUREUVOVPUVNYSUVBUUCUVFUVJUVKUVMWBUVNXRUURUUBUVEUVOUVNUUA
      UVDUVNXSUUSYTUVCUVPUVNXRUURIUVOUPWCWNWDUQWFVAWGWHYQYPAYPYOULWIYQUUTCPZUUR
      QZUURUVQUGZIQZUKZCUUSRZSZGUHZOXLRZAYQYOUWEYPYOUJYGUWDNOXLUVKYFUWCMGUVKUVM
      SZXTUUTYEUWBUWFXSUUSXRUURUVKUVMUJUVKUVMULUMUWFYDUWADCXSUUSUWFDCTZSZYAUVRY
      CUVTUWHXDUVQXRUURUVKUVMUWGWAZUWFUWGUJZWJUWHYBUVSIUWHXRUURXDUVQUWIUWJWKUPU
      QUVKUVMUWGWBWLUSVAWMVSWIWOWPWQWRWSWTXA $.
  $}

  ${
    $d f g h u w x y z A $.  $d f g h u w x y z F $.  $d f g ph w y z $.
    tfrlemi14d.1 $e |- A = { f | E. x e. On ( f Fn x /\
                A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
    tfrlemi14d.2 $e |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) ) $.
    $( The domain of ` recs ` is all ordinals (lemma for transfinite
       recursion).  (Contributed by Jim Kingdon, 9-Jul-2019.) $)
    tfrlemi14d $p |- ( ph -> dom recs ( F ) = On ) $=
      ( vh vz vg vu vw cdm con0 cv wcel wa cfv cvv crecs word tfrlem8 mp1i ciun
      wss ordsson wrex wfn cres wceq wral tfrlemi1 cop csn wfun ad2antrr simplr
      cun wal simprl fneq2 raleq anbi12d rspcev adantll tfrlem3a tfrlemisucaccv
      vex sylibr tfrlem3-2d simprd opexg sylancr snidg 3syl wi opeldmg mpd dmeq
      elun2 eleq2d syl2anc exlimddv eliun ssrdv cuni recsfval dmeqi dmuni eqtri
      ex sseqtrrdi eqssd ) AFUAZNZOWPUBWPOUFABCDEFGUCWPUGUDAOIDIPZNZUEZWPAJOWSA
      JPZOQZWTWSQZAXARZWTWRQZIDUHZXBXCKPZWTUIZLPZXFSXFXHUJFSUKZLWTULZRZXEKABCLD
      WTEKFGHUMXCXKRZXFWTXFFSZUNZUOZUSZDQWTXPNZQZXEXLBCJDEKFGAFUPZBPFSTQRBUTXAX
      KHUQAXAXKURXCXGXJVAXLXFMPZUIZXILXTULZRZMOUHZXFDQXAXKYDAYCXKMWTOXTWTUKYAXG
      YBXJXTWTXFVBXILXTWTVCVDVEVFBCMLDEFXFGKVIVGVJVHXLXNXPQZXRAYEXAXKAXNTQZXNXO
      QYEAWTTQZXMTQZYFJVIZAXSYHABKFHVKVLZWTXMTTVMVNXNTVOXNXOXFWAVPUQAYEXRVQZXAX
      KAYGYHYKYIYJWTXMXPTTVRVNUQVSXDXRIXPDWQXPUKWRXQWTWQXPVTWBVEWCWDIWTDWRWEVJW
      LWFWPDWGZNWSWOYLBCDEFGWHWIIDWJWKWMWN $.
  $}

  ${
    $d e f g h u v t w x y z A $.  $d e f g h u v t w x y z F $.  $d z C $.
    $d u v z ph $.
    tfrexlem.1 $e |- A = { f | E. x e. On ( f Fn x /\
                A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
    tfrexlem.2 $e |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) ) $.
    $( The transfinite recursion function is set-like if the input is.
       (Contributed by Mario Carneiro, 3-Jul-2019.) $)
    tfrexlem $p |- ( ( ph /\ C e. V ) -> ( recs ( F ) ` C ) e. _V ) $=
      ( vz vv wcel cfv cvv cv wceq con0 wa syl vg vw vu vh vt ve crecs wi fveq2
      eleq1d imbi2d csuc cin cuni wfn cres wral wex word wss inss2 ssorduni vex
      ax-mp sucex inex1 uniex elon2 mpbir2an tfrlem3 wal anbi2d cbvalv tfrlemi1
      wfun sylib mpan2 crn wbr cop recsfval breqi df-br eluni 3bitri sucid wrex
      simpr tfrlem3a simprl simprrl simpll fnop syl2anc onelon rexlimddv adantl
      cdm onsuc onss df-ss unieqd wtr eloni ordtr 3syl eqtrd eleqtrrid ad2antrr
      unisuc fndm eleqtrrd eldm fneq2 raleq anbi12d rspcev mpan simplrr simplrl
      sylibr tfrlem5 syl22anc breqtrd exlimddv elssuni exlimdv biimtrid alrimiv
      imp brelrn ex fvss rnex ssex exlimiv vtoclg impcom ) EHMAEGUGZNZOMZAKPZYS
      NZOMZUHAUUAUHKEHUUBEQZUUDUUAAUUEUUCYTOUUBEYSUIUJUKAUAPZUUBULZULZRUMZUNZUO
      ZUBPZUUFNUUFUULUPGNQZUBUUJUQZSZUAURZUUDAUUJRMZUUPUUQUUJUSZUUJOMUUIRUTUURU
      UHRVAUUIVBVDUUIUUHRUUGUUBKVCZVEZVEVFVGUUJVHVIZAKUCUBDUUJLUAGBCKUCDFLGIVJZ
      AGVOZBPZGNZOMZSZBVKUVCUUBGNZOMZSZKVKJUVGUVJBKUVDUUBQZUVFUVIUVCUVKUVEUVHOU
      VDUUBGUIUJVLVMVPVNVQUUOUUDUAUUOUUCUUFVRZUNZUTZUUDUUOUUBCPZYSVSZUVOUVMUTZU
      HZCVKUVNUUOUVRCUVPUUBUVOVTZUDPZMZUVTDMZSZUDURZUUOUVQUVPUUBUVODUNZVSUVSUWE
      MUWDUUBUVOYSUWEKUCDLGUVBWAWBUUBUVOUWEWCUDUVSDWDWEUUOUWCUVQUDUUOUWCUVQUUOU
      WCSZUVOUVLMZUVQUWFUUBUVOUUFVSZUWGUWFUUBUVDUUFVSZUWHBUWFUUBUUFWRZMUWIBURUW
      FUUBUUJUWJUWFUUBUUGUUJUUBUUSWFUWFUUJUUHUNZUUGUWFUUIUUHUWFUUHRUTZUUIUUHQUW
      FUUHRMZUWLUWFUUGRMZUWMUWFUUBRMZUWNUWCUWOUUOUWCUVTUEPZUOZUFPZUVTNUVTUWRUPG
      NQUFUWPUQZSZUWOUERUWCUWBUWTUERWGUWAUWBWHBCUEUFDFGUVTIUDVCWIVPUWCUWPRMZUWT
      SZSZUXAUUBUWPMZUWOUWCUXAUWTWJUXCUWQUWAUXDUWCUXAUWQUWSWKUWAUWBUXBWLUWPUUBU
      VOUVTWMWNUWPUUBWOWNWPWQUUBWSTZUUGWSTUUHWTTUUHRXAVPXBUWFUUGXCZUWKUUGQUWFUW
      NUUGUSUXFUXEUUGXDUUGXEXFUUGUUTXJVPXGXHUUKUWJUUJQUUNUWCUUJUUFXKXIXLBUUBUUF
      UUSXMVPUWFUWISZUUBUVDUVOUUFUWFUWIWHZUXGUUFDMZUWBUWIUUBUVOUVTVSZUVDUVOQZUU
      OUXIUWCUWIUUOUUFLPZUOZUUMUBUXLUQZSZLRWGZUXIUUQUUOUXPUVAUXOUUOLUUJRUXLUUJQ
      UXMUUKUXNUUNUXLUUJUUFXNUUMUBUXLUUJXOXPXQXRBCLUBDFGUUFIUAVCZWIYAXIUUOUWAUW
      BUWIXSUXHUXGUWAUXJUUOUWAUWBUWIXTUUBUVOUVTWCYAUXIUWBSUWIUXJSUXKKUCCBDLUAUD
      GUVBYBYJYCYDYEUUBUVOUUFUUSCVCYKTUVOUVLYFTYLYGYHYICUUBUVMYSYMTUUCUVMUVLUUF
      UXQYNVGYOTYPTYQYR $.
  $}

  ${
    $d G f g u w x y z $.  $d ph f y $.
    tfri1d.1 $e |- F = recs ( G ) $.
    tfri1d.2 $e |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) ) $.
    $( Principle of Transfinite Recursion, part 1 of 3.  Theorem 7.41(1) of
       [TakeutiZaring] p. 47, with an additional condition.

       The condition is that ` G ` is defined "everywhere", which is stated
       here as ` ( G `` x ) e. _V ` .  Alternately,
       ` A. x e. On A. f ( f Fn x -> f e. dom G ) ` would suffice.

       Given a function ` G ` satisfying that condition, we define a class
       ` A ` of all "acceptable" functions.  The final function we're
       interested in is the union ` F = recs ( G ) ` of them. ` F ` is then
       said to be defined by transfinite recursion.  The purpose of the 3 parts
       of this theorem is to demonstrate properties of ` F ` .  In this first
       part we show that ` F ` is a function whose domain is all ordinal
       numbers.  (Contributed by Jim Kingdon, 4-May-2019.)  (Revised by Mario
       Carneiro, 24-May-2019.) $)
    tfri1d $p |- ( ph -> F Fn On ) $=
      ( vy vg vz vu vf vw con0 wfn wceq wa cv cfv cres wral crecs wfun cdm wrex
      cab eqid tfrlem3 tfrlemi14d tfrlem7 jctil df-fn sylibr fneq1i ) ADUAZMNZC
      MNAUNUBZUNUCMOZPUOAUQUPABGHQZIQZNJQZURRURUTSDROJUSTPIMUDHUEZKDIJBGVAHKDVA
      UFUGFUHGILQZGQZNUSVBRVBUSSDROIVCTPGMUDLUEZLDVDUFUIUJUNMUKULMCUNEUMUL $.

    $( Principle of Transfinite Recursion, part 2 of 3.  Theorem 7.41(2) of
       [TakeutiZaring] p. 47, with an additional condition on the recursion
       rule ` G ` ( as described at ~ tfri1 ).  Here we show that the function
       ` F ` has the property that for any function ` G ` satisfying that
       condition, the "next" value of ` F ` is ` G ` recursively applied to all
       "previous" values of ` F ` .  (Contributed by Jim Kingdon,
       4-May-2019.) $)
    tfri2d $p |- ( ( ph /\ A e. On ) -> ( F ` A ) = ( G ` ( F |` A ) ) ) $=
      ( con0 wcel wa cdm cfv cres wceq wfn tfri1d fndm syl eleq2d biimpar tfr2a
      ) ACHIZJCDKZIZCDLDCMELNAUDUBAUCHCADHOUCHNABDEFGPHDQRSTCDEFUAR $.
  $}

  ${
    $d G f w x y z $.  $d H f w x y z $.  $d X f x z $.
    tfr1onlem3ag.1 $e |- A = { f | E. x e. X ( f Fn x
      /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.
    $( Lemma for transfinite recursion.  This lemma changes some bound
       variables in ` A ` (version of ~ tfrlem3ag but for ~ tfr1on related
       lemmas).  (Contributed by Jim Kingdon, 13-Mar-2022.) $)
    tfr1onlem3ag $p |- ( H e. V -> ( H e. A <-> E. z e. X ( H Fn z
        /\ A. w e. z ( H ` w ) = ( G ` ( H |` w ) ) ) ) ) $=
      ( cv wfn cfv cres wceq wral wa wrex fneq12 simpll fveq12d reseq12d fveq2d
      simpr eqeq12d simplr cbvraldva2 anbi12d cbvrexdva elab2g ) FLZALZMZBLZULN
      ZULUOOZGNZPZBUMQZRZAJSHCLZMZDLZHNZHVDOZGNZPZDVBQZRZCJSFHEIULHPZVAVJACJVKU
      MVBPZRZUNVCUTVIUMVBULHTVMUSVHBDUMVBVMUOVDPZRZUPVEURVGVOUOVDULHVKVLVNUAZVM
      VNUEZUBVOUQVFGVOULHUOVDVPVQUCUDUFVKVLVNUGUHUIUJKUK $.

    $d A g $.  $d f g w x y z $.
    $( Lemma for transfinite recursion.  This lemma changes some bound
       variables in ` A ` (version of ~ tfrlem3 but for ~ tfr1on related
       lemmas).  (Contributed by Jim Kingdon, 14-Mar-2022.) $)
    tfr1onlem3 $p |- A = { g | E. z e. X ( g Fn z
        /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) } $=
      ( cv wfn cfv cres wceq wral wa wrex cvv wcel wb tfr1onlem3ag ax-mp abbi2i
      vex ) GKZCKZLDKZUFMUFUHNHMODUGPQCIRZGEUFSTUFETUIUAGUEABCDEFHUFSIJUBUCUD
      $.
  $}

  ${
    $d G f x y $.  $d X x $.  $d f ph $.
    tfr1onlemssrecs.1 $e |- A = { f | E. x e. X ( f Fn x
        /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.
    tfr1onlemssrecs.x $e |- ( ph -> Ord X ) $.
    $( Lemma for ~ tfr1on .  The union of functions acceptable for ~ tfr1on is
       a subset of ` recs ` .  (Contributed by Jim Kingdon, 15-Mar-2022.) $)
    tfr1onlemssrecs $p |- ( ph -> U. A C_ recs ( G ) ) $=
      ( cuni cv wfn cfv cres wceq wral wa con0 wrex cab word wss ordsson ssrexv
      crecs wi 3syl ss2abdv eqsstrid unissd df-recs sseqtrrdi ) ADJEKZBKZLCKZUM
      MUMUONFMOCUNPQZBRSZETZJFUEADURADUPBGSZETURHAUSUQEAGUAGRUBUSUQUFIGUCUPBGRU
      DUGUHUIUJBCEFUKUL $.
  $}

  ${
    tfr1on.f $e |- F = recs ( G ) $.
    tfr1on.g $e |- ( ph -> Fun G ) $.
    tfr1on.x $e |- ( ph -> Ord X ) $.
    tfr1on.ex $e |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V ) $.
    ${
      tfr1onlemsucfn.1 $e |- A = { f | E. x e. X ( f Fn x
          /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.

      ${
        $d G f x $.  $d X f x $.  $d f g $.  $d f ph x $.  $d f x z $.
        tfr1onlemsucfn.3 $e |- ( ph -> z e. X ) $.
        tfr1onlemsucfn.4 $e |- ( ph -> g Fn z ) $.
        tfr1onlemsucfn.5 $e |- ( ph -> g e. A ) $.
        $( We can extend an acceptable function by one element to produce a
           function.  Lemma for ~ tfr1on .  (Contributed by Jim Kingdon,
           12-Mar-2022.) $)
        tfr1onlemsucfn $p |- ( ph
            -> ( g u. { <. z , ( G ` g ) >. } ) Fn suc z ) $=
          ( cv wcel csuc cfv cop csn cun elexd wfn cvv wi wal weq imbi1d albidv
          fneq2 3expia alrimiv ralrimiva rspcdva fneq1 fveq2 eleq1d imbi12d spv
          wa sylc eqid df-suc con0 word wn ordelon syl2anc eloni ordirr fnunsn
          3syl ) ADSZVQUAGSZVRVQVRIUBZUCUDUEZVQVSAVQJPUFAFSZVQUGZWAIUBZUHTZUIZF
          UJZVRVQUGZVSUHTZAWABSZUGZWDUIZFUJZWFBJVQBDUKZWKWEFWMWJWBWDWIVQWAUNULU
          MAWLBJAWIJTZVDWKFAWNWJWDNUOUPUQPURQWEWGWHUIFGFGUKZWBWGWDWHVQWAVRUSWOW
          CVSUHWAVRIUTVAVBVCVEQVTVFVQVGAVQVHTZVQVIVQVQTVJAJVIVQJTWPMPJVQVKVLVQV
          MVQVNVPVO $.
      $}

      ${
        $d G f u v x y $.  $d G f u w x y $.  $d X f v x $.  $d X f w x $.
        $d f g u v x y $.  $d f ph u v x $.  $d f u v x y z $.  $d g u w x y $.
        $d w x y z $.
        tfr1onlemsucaccv.yx $e |- ( ph -> Y e. X ) $.
        tfr1onlemsucaccv.zy $e |- ( ph -> z e. Y ) $.
        tfr1onlemsucaccv.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
        tfr1onlemsucaccv.gfn $e |- ( ph -> g Fn z ) $.
        tfr1onlemsucaccv.gacc $e |- ( ph -> g e. A ) $.
        $( Lemma for ~ tfr1on .  We can extend an acceptable function by one
           element to produce an acceptable function.  (Contributed by Jim
           Kingdon, 12-Mar-2022.) $)
        tfr1onlemsucaccv $p |- ( ph
            -> ( g u. { <. z , ( G ` g ) >. } ) e. A ) $=
          ( vw vu vv cv cfv cop csn cun wcel wfn cres wceq wral wrex csuc suceq
          wa cuni eleq1d ralrimiva elunii syl2anc rspcdva ordtr1 tfr1onlemsucfn
          word jca wo vex elsuc cvv wb tfr1onlem3ag ax-mp sylib simprrr simprrl
          sylc adantr fndmu raleqdv mpbid rexlimddv r19.21bi wne con0 wn onelon
          ordelon sylan eloni ordirr 3syl wi elequ2 biimpcd adantl mtod fvunsng
          neqned sylancr wss syl ordelss resabs1 fsnunres reseq1d eqtr3d fveq2d
          3eqtr4d cdm wal fneq2 imbi1d albidv 3expia alrimiv fneq1 imbi12d fndm
          fveq2 neleqtrrd fsnunfv syl3anc simpr reseq2 sylan9eqr jaodan sylan2b
          spv raleq anbi12d rspcev syl12anc opexg snexg unexg mpbird ) AGUEZDUE
          ZYTIUFZUGZUHZUIZEUJZUUEUBUEZUKZUCUEZUUEUFZUUEUUIULZIUFZUMZUCUUGUNZURZ
          UBJUOZAUUAUPZJUJZUUEUUQUKZUUMUCUUQUNZUUPABUEZUPZJUJZUURBJUSZUUAUVAUUA
          UMZUVBUUQJUVAUUAUQUTAUVCBUVDSVAAUUAKUJZKJUJZUUAUVDUJRQUUAKJVBVCVDABCD
          EFGHIJLMNOPAJVGZUVFUVGURUUAJUJZNAUVFUVGRQVHUUAKJVEVSZTUAVFAUUMUCUUQUU
          IUUQUJAUUIUUAUJZUUIUUAUMZVIUUMUUIUUAUCVJZVKAUVKUUMUVLAUVKURZUUIYTUFZY
          TUUIULZIUFZUUJUULAUVOUVQUMZUCUUAAYTUDUEZUKZUVRUCUVSUNZURZUVRUCUUAUNZU
          DJAYTEUJZUWBUDJUOZUAYTVLUJZUWDUWEVMGVJZBCUDUCEFIYTVLJPVNVOVPAUVSJUJZU
          WBURZURZUWAUWCAUWHUVTUWAVQUWJUVRUCUVSUUAUWJUVTYTUUAUKZUVSUUAUMAUWHUVT
          UWAVRAUWKUWITVTUVSUUAYTWAVCWBWCWDWEUVNUUIVLUJUUAUUIWFUUJUVOUMUVMUVNUU
          AUUIUVNUUAUUIUMZUUIUUIUJZUVNUUIWGUJZUUIVGUWMWHAUUAWGUJZUVKUWNAUVHUVIU
          WONUVJJUUAWJVCZUUAUUIWIWKUUIWLUUIWMWNUVKUWLUWMWOAUWLUVKUWMDUCUCWPWQWR
          WSXAYTUUAUUBUUIVLWTXBUVNUUKUVPIUVNUUEUUAULZUUIULZUUKUVPUVNUUIUUAXCZUW
          RUUKUMAUUAVGZUVKUWSAUWOUWTUWPUUAWLXDZUUAUUIXEWKUUEUUIUUAXFXDAUWRUVPUM
          UVKAUWQYTUUIAUWKUUAUUAUJWHZUWQYTUMTAUWTUXBUXAUUAWMXDZUUAYTUUAUUBXGVCZ
          XHVTXIXJXKAUVLURZUUAUUEUFZUUBUUJUULAUXFUUBUMZUVLAUVFUUBVLUJZUUAYTXLZU
          JWHUXGRAFUEZUUAUKZUXJIUFZVLUJZWOZFXMZUWKUXHAUXJUVAUKZUXMWOZFXMZUXOBJU
          UAUVEUXQUXNFUVEUXPUXKUXMUVAUUAUXJXNXOXPAUXRBJAUVAJUJZURUXQFAUXSUXPUXM
          OXQXRVAUVJVDTUXNUWKUXHWOFGUXJYTUMZUXKUWKUXMUXHUUAUXJYTXSUXTUXLUUBVLUX
          JYTIYBUTXTYKVSZAUXIUUAUUAUXCAUWKUXIUUAUMTUUAYTYAXDYCYTKVLUUAUUBYDYEVT
          UXEUUIUUAUUEAUVLYFXJUXEUUKYTIUVLAUUKUWQYTUUIUUAUUEYGUXDYHXJXKYIYJVAUU
          OUUSUUTURUBUUQJUUGUUQUMUUHUUSUUNUUTUUGUUQUUEXNUUMUCUUGUUQYLYMYNYOAUUE
          VLUJZUUFUUPVMAUWFUUDVLUJZUYBUWGAUUCVLUJZUYCAUUAVLUJUXHUYDDVJUYAUUAUUB
          VLVLYPXBUUCVLYQXDYTUUDVLVLYRXBBCUBUCEFIUUEVLJPVNXDYS $.
      $}

      ${
        $d A f g h x z $.  $d D f g x $.  $d G f x y $.  $d X f x $.
        $d f g h ph x z $.  $d f g x y z $.
        tfr1onlembacc.3 $e |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\
           h = ( g u. { <. z , ( G ` g ) >. } ) ) } $.
        tfr1onlembacc.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
        tfr1onlembacc.4 $e |- ( ph -> D e. X ) $.
        tfr1onlembacc.5 $e |- ( ph -> A. z e. D E. g ( g Fn z /\
          A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) ) $.
        $( Lemma for ~ tfr1on .  Each element of ` B ` is an acceptable
           function.  (Contributed by Jim Kingdon, 14-Mar-2022.) $)
        tfr1onlembacc $p |- ( ph -> B C_ A ) $=
          ( wfn wcel cfv cop csn cun wceq w3a wex wrex cab simpr3 wfun ad2antrr
          cv word cvv 3adant1r simplr cuni csuc adantlr simpr1 tfr1onlemsucaccv
          wa simpr2 eqeltrd ex exlimdv rexlimdva abssdv eqsstrid ) AGJURZDURZUD
          ZVPFUEZKURZVPVQVPMUFUGUHUIZUJZUKZJULZDHUMZKUNFTAWEKFAWDVTFUEZDHAVQHUE
          ZVHZWCWFJWHWCWFWHWCVHZVTWAFWHVRVSWBUOWIBCDFIJLMNHOAMUPWGWCPUQANUSWGWC
          QUQWHBURZNUEZIURZWJUDZWLMUFUTUEZWCAWKWMWNWGRVAVASAHNUEWGWCUBUQAWGWCVB
          WHWJNVCUEZWJVDNUEZWCAWOWPWGUAVEVEWHVRVSWBVFWHVRVSWBVIVGVJVKVLVMVNVO
          $.

        ${
          $d A f g h x z $.  $d B g h z $.  $d B g w z $.  $d D f g h x z $.
          $d G f h x z $.  $d G f w x y z $.  $d X f g x z $.
          $d f g h ph x z $.  $d f g x y z $.
          $( Lemma for ~ tfr1on .  The union of ` B ` is defined on all
             elements of ` X ` .  (Contributed by Jim Kingdon, 14-Mar-2022.) $)
          tfr1onlembxssdm $p |- ( ph -> D C_ dom U. B ) $=
            ( cv cuni cdm wcel wral wss wfn cfv cres wceq wex w3a cvv cop simp1
            wa simp2 syl word wi ordtr1 imp syl12anc simp3l fneq2 imbi1d albidv
            wal 3expia alrimiv ralrimiva adantr simpr rspcdva fneq1 imbi12d spv
            fveq2 eleq1d syl21anc csn cun opexg sylancr snidg elun2 3syl simp3r
            vex wrex rspe tfr1onlem3ag ax-mp sylibr 3jca snexg unexg mpan isset
            sylib simpr3 19.8a abeq2i sylan2 eqeltrrd 3exp2 3imp exlimdv elunii
            wb sylc syl2anc opeq2 spcegv eldm2 imbitrrdi ralimdva mpd dfss3 ) A
            DUDZGUEZUFZUGZDHUHZHYEUIAJUDZYCUJZEUDZYHUKYHYJULMUKUMEYCUHZUSZJUNZD
            HUHYGUCAYMYFDHAYCHUGZUSYLYFJAYNYLYFAYNYLUOZYHMUKZUPUGZYCYPUQZYDUGZY
            FYOAYCNUGZYIYQAYNYLURZYOAYNHNUGZYTUUAAYNYLUTZYOAUUBUUAUBVAAYNUUBUSZ
            YTANVBUUDYTVCQYCHNVDVAVEVFZAYNYIYKVGZAYTUSZYIYQUUGIUDZYCUJZUUHMUKZU
            PUGZVCZIVKZYIYQVCZUUGUUHBUDZUJZUUKVCZIVKZUUMBNYCUUOYCUMZUUQUULIUUSU
            UPUUIUUKUUOYCUUHVHVIVJAUURBNUHYTAUURBNAUUONUGZUSUUQIAUUTUUPUUKRVLVM
            VNVOAYTVPVQUULUUNIJUUHYHUMZUUIYIUUKYQYCUUHYHVRUVAUUJYPUPUUHYHMWAWBV
            SVTVAVEWCZYOYRYHYRWDZWEZUGZUVDGUGZYSYOYRUPUGZYRUVCUGUVEYOYCUPUGYQUV
            GDWLZUVBYCYPUPUPWFWGZYRUPWHYRUVCYHWIWJYOYNYIYHFUGZUOZKUDZUVDUMZKUNZ
            UVFYOYNYIUVJUUCUUFYOYLDNWMZUVJYOYTYIYKUVOUUEUUFAYNYIYKWKYLDNWNVFYHU
            PUGZUVJUVOXMJWLZBCDEFIMYHUPNSWOWPWQWRYOUVDUPUGZUVNYOUVGUVCUPUGZUVRU
            VIYRUPWSUVPUVSUVRUVQYHUVCUPUPWTXAWJKUVDXBXCUVKUVMUVFKYNYIUVJUVMUVFV
            CYNYIUVJUVMUVFYNYIUVJUVMUOZUSUVLUVDGYNYIUVJUVMXDUVTYNUVTJUNZUVLGUGZ
            UVTJXEYNUWAUSUWADHWMZUWBUWADHWNUWCKGTXFWQXGXHXIXJXKXNYRUVDGXLXOYQYS
            YCYJUQZYDUGZEUNYFUWEYSEYPUPYJYPUMUWDYRYDYJYPYCXPWBXQEYCYDUVHXRXSXNV
            LXKXTYADHYEYBWQ $.
        $}

        ${
          $d A f g h x z $.  $d B g h z w $.  $d D f g h x z $.
          $d G f h x y z w $.  $d X f g x z $.  $d f g h ph x z $.
          $d f g x y z w $.
          $( Lemma for ~ tfr1on .  The union of ` B ` is a function defined on
             ` x ` .  (Contributed by Jim Kingdon, 15-Mar-2022.) $)
          tfr1onlembfn $p |- ( ph -> U. B Fn D ) $=
            ( cuni wfun cdm wceq wfn crecs tfr1onlembacc unissd tfr1onlemssrecs
            wss sstrd tfrfun mpisyl cvv cxp cpw cv wcel cfv cop csn cun w3a wex
            funss wrex cab wa simpr3 csuc wf simpl word adantr simpr jca ordtr1
            sylc ad2antrr simplr simpr1 simpr2 tfr1onlemsucfn sylan dffn2 sylib
            3adant1r fssxp syl con0 ordelon syl2anc eloni ordsucss xpss1 wb vex
            wi fneq2 imbi1d albidv wral 3expia alrimiv ralrimiva 3ad2ant1 simp2
            wal rspcdva simp3 fneq1 fveq2 imbi12d spv syl3anc opexg snexg unexg
            eleq1d sylancr mpbird eqeltrd ex exlimdv rexlimdva eqsstrid sspwuni
            elpwg abssdv dmxpss sstrdi tfr1onlembxssdm eqssd df-fn sylanbrc
            dmss ) AGUDZUEZYTUFZHUGYTHUHAYTMUIZUMUUCUEUUAAYTFUDUUCAGFABCDEFGHIJ
            KLMNOPQRSTUAUBUCUJUKABCFIMNSQULUNMUOYTUUCVHUPAUUBHAUUBHUQURZUFZHAYT
            UUDUMZUUBUUEUMAGUUDUSZUMUUFAGJUTZDUTZUHZUUHFVAZKUTZUUHUUIUUHMVBZVCZ
            VDZVEZUGZVFZJVGZDHVIZKVJUUGTAUUTKUUGAUUSUULUUGVAZDHAUUIHVAZVKZUURUV
            AJUVCUURUVAUVCUURVKZUULUUPUUGUVCUUJUUKUUQVLUVDUUPUUGVAZUUPUUDUMZUVD
            UUPUUIVMZUQURZUUDUVDUVGUQUUPVNZUUPUVHUMUVDUUPUVGUHZUVIUVCAUUINVAZVK
            ZUURUVJUVCAUVKAUVBVOZUVCNVPZUVBHNVAZVKUVKAUVNUVBQVQUVCUVBUVOAUVBVRA
            UVOUVBUBVQVSUUIHNVTWAZVSUVLUURVKBCDFIJLMNOAMUEUVKUURPWBAUVNUVKUURQW
            BUVLBUTZNVAZIUTZUVQUHZUVSMVBZUQVAZUURAUVRUVTUWBUVKRWJWJSAUVKUURWCUV
            LUUJUUKUUQWDUVLUUJUUKUUQWEWFWGUVGUUPWHWIUVGUQUUPWKWLUVDUVGHUMZUVHUU
            DUMUVDHVPZUVBUWCAUWDUVBUURAHWMVAZUWDAUVNUVOUWEQUBNHWNWOHWPWLWBAUVBU
            URWCUUIHWQWAUVGHUQWRWLUNUVDUUPUQVAZUVEUVFWSUVDUUHUQVAUUOUQVAZUWFJWT
            UVDUUNUQVAZUWGUVDUUIUQVAUUMUQVAZUWHDWTUVDAUVKUUJUWIUVCAUURUVMVQUVCU
            VKUURUVPVQUVCUUJUUKUUQWDAUVKUUJVFZUVSUUIUHZUWBXAZIXKZUUJUWIUWJUVTUW
            BXAZIXKZUWMBNUUIUVQUUIUGZUWNUWLIUWPUVTUWKUWBUVQUUIUVSXBXCXDAUVKUWOB
            NXEUUJAUWOBNAUVRVKUWNIAUVRUVTUWBRXFXGXHXIAUVKUUJXJXLAUVKUUJXMUWLUUJ
            UWIXAIJUVSUUHUGZUWKUUJUWBUWIUUIUVSUUHXNUWQUWAUUMUQUVSUUHMXOYBXPXQWA
            XRUUIUUMUQUQXSYCUUNUQXTWLUUHUUOUQUQYAYCUUPUUDUQYKWLYDYEYFYGYHYLYIGU
            UDYJWIYTUUDYSWLHUQYMYNABCDEFGHIJKLMNOPQRSTUAUBUCYOYPYTHYQYR $.
        $}

        ${
          $d A f g h x z $.  $d B g h w z $.  $d D f g h x z $.
          $d G f h w x y z $.  $d X f g x z $.  $d f g h ph x z $.
          $d f g x y z $.
          $( Lemma for ~ tfr1on .  The set ` B ` exists.  (Contributed by Jim
             Kingdon, 14-Mar-2022.) $)
          tfr1onlembex $p |- ( ph -> B e. _V ) $=
            ( cuni cvv wcel wfn tfr1onlembfn fnex syl2anc uniexb sylibr ) AGUDZ
            UEUFZGUEUFAUMHUGHNUFUNABCDEFGHIJKLMNOPQRSTUAUBUCUHUBHNUMUIUJGUKUL
            $.
        $}

        ${
          $d A f g h x z $.  $d B g h w z $.  $d B u w $.  $d D f g h w x z $.
          $d D u w $.  $d G f h x y z $.  $d G u w $.  $d X f g x z $.
          $d f g h ph x z $.  $d f g x y z $.  $d ph w x z $.  $d w x y z $.
          $( Lemma for ~ tfr1on .  The union of ` B ` satisfies the recursion
             rule.  (Contributed by Jim Kingdon, 15-Mar-2022.) $)
          tfr1onlemubacc $p |- ( ph ->
              A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) $=
            ( cv cuni cfv cres wceq wral wcel wa crecs cdm wfn tfr1onlembfn syl
            fndm wss tfr1onlembacc unissd tfr1onlemssrecs sstrd eqsstrrd sselda
            dmss con0 wrex eqid tfrlem9 tfrfun eleq2d biimpar funssfv mp3an2ani
            cab wfun word ordelon syl2anc eloni sylan adantr sseqtrrd fun2ssres
            ordelss fveq2d 3eqtr3d ralrimiva reseq2 eqeq12d cbvralv sylibr
            fveq2 ) AEUEZHUFZUGZWPWOUHZNUGZUIZEIUJFUEZWPUGZWPXAUHZNUGZUIZFIUJAW
            TEIAWOIUKZULZWONUMZUGZXHWOUHZNUGZWQWSXGWOXHUNZUKXIXKUIAIXLWOAIWPUNZ
            XLAWPIUOXMIUIZABCDEGHIJKLMNOPQRSTUAUBUCUDUPIWPURUQZAWPXHUSZXMXLUSAW
            PGUFXHAHGABCDEGHIJKLMNOPQRSTUAUBUCUDUTVAABCGJNOTRVBVCZWPXHVFUQVDVEB
            CJUEZBUEZUOCUEZXRUGXRXTUHNUGUICXSUJULBVGVHJVPZWOJNYAVIVJUQXHVQZAXPX
            FWOXMUKZXIWQUINVKZXQAYCXFAXMIWOXOVLVMWOXHWPVNVOXGXJWRNYBAXPXFWOXMUS
            XJWRUIYDXQXGWOIXMAIVRZXFWOIUSAIVGUKZYEAOVRIOUKYFRUCOIVSVTIWAUQIWOWF
            WBAXNXFXOWCWDWOXHWPWEVOWGWHWIXEWTFEIXAWOUIZXBWQXDWSXAWOWPWNYGXCWRNX
            AWOWPWJWGWKWLWM $.
        $}

        ${
          $d A f g h x z $.  $d B f g h w z $.  $d B f u w $.
          $d D f g h w x z $.  $d D f u w $.  $d G f h x y z $.  $d G f u w $.
          $d X f g x z $.  $d f g h ph x z $.  $d f g x y z $.  $d ph w x z $.
          $d w x y z $.
          $( Lemma for ~ tfr1on .  (Contributed by Jim Kingdon,
             16-Mar-2022.) $)
          tfr1onlemex $p |- ( ph ->
              E. f ( f Fn D /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) ) $=
            ( cuni cvv wcel wfn cv cfv cres wceq wa wex tfr1onlembex uniexg syl
            wral tfr1onlembfn tfr1onlemubacc fneq1 fveq1 reseq1 eqeq12d ralbidv
            jca fveq2d anbi12d spcegv sylc ) AHUEZUFUGZVKIUHZFUIZVKUJZVKVNUKZNU
            JZULZFIURZUMZJUIZIUHZVNWAUJZWAVNUKZNUJZULZFIURZUMZJUNAHUFUGVLABCDEG
            HIJKLMNOPQRSTUAUBUCUDUOHUFUPUQAVMVSABCDEGHIJKLMNOPQRSTUAUBUCUDUSABC
            DEFGHIJKLMNOPQRSTUAUBUCUDUTVFWHVTJVKUFWAVKULZWBVMWGVSIWAVKVAWIWFVRF
            IWIWCVOWEVQVNWAVKVBWIWDVPNWAVKVNVCVGVDVEVHVIVJ $.
        $}
      $}

      ${
        $d A a b c d h u x $.  $d A a b c d h r s t u $.  $d C g u z $.
        $d G a b c d g h u w x z $.  $d G f x y z $.  $d G a b c d h r s t u $.
        $d X a b c d h w x z $.  $d X f h x z $.  $d a b c d h ph w x z $.
        $d a b c d g h x y z $.  $d f h ph x z $.  $d s t u z $.
        tfr1onlemaccex.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
        $( We can define an acceptable function on any element of ` X ` .

           As with many of the transfinite recursion theorems, we have
           hypotheses that state that ` F ` is a function and that it is
           defined up to ` X ` .  (Contributed by Jim Kingdon, 16-Mar-2022.) $)
        tfr1onlemaccex $p |- ( ( ph /\ C e. X ) -> E. g ( g Fn C /\
            A. u e. C ( g ` u ) = ( G ` ( g |` u ) ) ) ) $=
          ( wa cv cfv vz vw vh vb vc vr vt vs va vd con0 wcel wfn cres wceq wex
          wral word ordelon sylan weq eleq1 anbi2d fneq2 anbi12d exbidv imbi12d
          wi raleq cop csn cun w3a wrex cab wfun ad3antrrr cvv wal 3expia fneq1
          alrimiv fveq2 eleq1d cbvalv sylib 3impia 3adant1r fveq1 reseq1 fveq2d
          19.21bi eqeq12d ralbidv rexbidv cbvabv eqtri id opeq2d uneq12d eqeq2d
          sneqd 3anbi123d cbvexv rexbii opeq1 uneq2d 3anbi13d bitri abbii eqeq1
          cbvrexv 3anbi3d cuni csuc adantlr simpr simplr ordtr1 ad4antr simpllr
          syl mp2and rspcdva reseq2 cbvralv anbi2i exbii imbitrdi mpd ralrimiva
          tfr1onlemex exp31 expcom a2d impexp ralbii r19.21v 3imtr4g tfis3
          mpcom ) FUKULZAFKULZRZHSZFUMZDSZUUETZUUEUUGUNZJTZUOZDFUQZRZHUPZAKURZU
          UCUUBNKFUSUTAUASZKULZRZUUEUUPUMZUUKDUUPUQZRZHUPZVHZAUBSZKULZRZUUEUVDU
          MZUUKDUVDUQZRZHUPZVHZUUDUUNVHUAUBFUAUBVAZUURUVFUVBUVJUVLUUQUVEAUUPUVD
          KVBVCUVLUVAUVIHUVLUUSUVGUUTUVHUUPUVDUUEVDUUKDUUPUVDVIVEVFVGUUPFUOZUUR
          UUDUVBUUNUVMUUQUUCAUUPFKVBVCUVMUVAUUMHUVMUUSUUFUUTUULUUPFUUEVDUUKDUUP
          FVIVEVFVGUUPUKULZAUVEUVJVHZUBUUPUQZVHZAUUQUVBVHZVHUVKUBUUPUQZUVCUVNAU
          VPUVRAUVNUVPUVRVHAUVNRZUVPUUQUVBUVTUVPRZUUQRZUCSZUUPUMZUUGUWCTZUWCUUG
          UNZJTZUOZDUUPUQZRZUCUPUVBUWBBCUDUEDEUFSZUGSZUMZUWKEULZUHSZUWKUWLUWKJT
          ZVJZVKZVLZUOZVMZUFUPZUGUUPVNZUHVOZUUPUCUIUJIJKLAJVPUVNUVPUUQMVQAUUOUV
          NUVPUUQNVQUWABSZKULZUWCUXEUMZUWCJTZVRULZUUQUVTUXFUXGUXIUVPAUXFUXGUXIU
          VNAUXFUXGUXIAUXFRZUXGUXIVHZUCUXJGSZUXEUMZUXLJTZVRULZVHZGVSUXKUCVSUXJU
          XPGAUXFUXMUXOOVTWBUXPUXKGUCGUCVAZUXMUXGUXOUXIUXEUXLUWCWAZUXQUXNUXHVRU
          XLUWCJWCWDVGWEWFWLWGWHWHWHEUXMCSZUXLTZUXLUXSUNZJTZUOZCUXEUQZRZBKVNZGV
          OUXGUXSUWCTZUWCUXSUNZJTZUOZCUXEUQZRZBKVNZUCVOPUYFUYMGUCUXQUYEUYLBKUXQ
          UXMUXGUYDUYKUXRUXQUYCUYJCUXEUXQUXTUYGUYBUYIUXSUXLUWCWIUXQUYAUYHJUXLUW
          CUXSWJWKWMWNVEWOWPWQUXDUISZUDSZUMZUYNEULZUWOUYNUYOUYNJTZVJZVKZVLZUOZV
          MZUIUPZUDUUPVNZUHVOUYPUYQUJSZVUAUOZVMZUIUPZUDUUPVNZUJVOUXCVUEUHUXCUYN
          UWLUMZUYQUWOUYNUWLUYRVJZVKZVLZUOZVMZUIUPZUGUUPVNVUEUXBVUQUGUUPUXAVUPU
          FUIUFUIVAZUWMVUKUWNUYQUWTVUOUWLUWKUYNWAUWKUYNEVBVURUWSVUNUWOVURUWKUYN
          UWRVUMVURWRVURUWQVULVURUWPUYRUWLUWKUYNJWCWSXBWTXAXCXDXEVUQVUDUGUDUUPU
          GUDVAZVUPVUCUIVUSVUKUYPVUOVUBUYQUWLUYOUYNVDVUSVUNVUAUWOVUSVUMUYTUYNVU
          SVULUYSUWLUYOUYRXFXBXGXAXHVFXLXIXJVUEVUJUHUJUHUJVAZVUDVUIUDUUPVUTVUCV
          UHUIVUTVUBVUGUYPUYQUWOVUFVUAXKXMVFWOWPWQUWAUXEKXNULZUXEXOKULZUUQUVTVV
          AVVBUVPAVVAVVBUVNQXPXPXPUWAUUQXQUWBUYPUESZUYNTZUYNVVCUNZJTZUOZUEUYOUQ
          ZRZUIUPZUDUUPUWBUYOUUPULZRZUYOKULZVVJVVLVVKUUQVVMUWBVVKXQZUWAUUQVVKXR
          AVVKUUQRVVMVHZUVNUVPUUQVVKAUUOVVONUYOUUPKXSYBXTYCVVLVVMUUEUYOUMZUUKDU
          YOUQZRZHUPZVVJVVLUVOVVMVVSVHUBUUPUYOUBUDVAZUVEVVMUVJVVSUVDUYOKVBVVTUV
          IVVRHVVTUVGVVPUVHVVQUVDUYOUUEVDUUKDUVDUYOVIVEVFVGUVTUVPUUQVVKYAVVNYDV
          VSUYPUUGUYNTZUYNUUGUNZJTZUOZDUYOUQZRZUIUPVVJVVRVWFHUIHUIVAZVVPUYPVVQV
          WEUYOUUEUYNWAVWGUUKVWDDUYOVWGUUHVWAUUJVWCUUGUUEUYNWIVWGUUIVWBJUUEUYNU
          UGWJWKWMWNVEXDVWFVVIUIVWEVVHUYPVWDVVGDUEUYODUEVAZVWAVVDVWCVVFUUGVVCUY
          NWCVWHVWBVVEJUUGVVCUYNYEWKWMYFYGYHXIYIYJYKYLUWJUVAUCHUCHVAZUWDUUSUWIU
          UTUUPUWCUUEWAVWIUWHUUKDUUPVWIUWEUUHUWGUUJUUGUWCUUEWIVWIUWFUUIJUWCUUEU
          UGWJWKWMWNVEXDWFYMYNYOUVSAUVOVHZUBUUPUQUVQUVKVWJUBUUPAUVEUVJYPYQAUVOU
          BUUPYRXIAUUQUVBYPYSYTUUA $.
      $}

      ${
        $d A g h z $.  $d A g u x z $.  $d G f g u w x y $.  $d G g h $.
        $d X f w x $.  $d Y f g x z $.  $d f g ph x z $.  $d w x y z $.
        tfr1onlemres.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
        tfr1onlemres.yx $e |- ( ph -> Y e. X ) $.
        $( Lemma for ~ tfr1on .  Recursion is defined on an ordinal if the
           characteristic function is defined up to a suitable point.
           (Contributed by Jim Kingdon, 18-Mar-2022.) $)
        tfr1onlemres $p |- ( ph -> Y C_ dom F ) $=
          ( vg cv wcel cvv vh vz vu vw crecs cdm ciun wa wrex wfn cfv cres wceq
          wral wex word adantr simpr jca ordtr1 sylc tfr1onlemaccex cop csn cun
          syldan wfun ad2antrr 3adant1r cuni adantlr simprl fneq2 raleq anbi12d
          rspcev sylan wb vex tfr1onlem3ag ax-mp sylibr tfr1onlemsucaccv imbi1d
          csuc wi wal 3expia alrimiv fneq1 fveq2 eleq1d imbi12d spv syl rspcdva
          ralrimiva imp opexg sylancr snidg elun2 3syl opeldmg mpd dmeq syl2anc
          eleq2d exlimddv eliun ssrdv dmuni wss tfr1onlemssrecs eqsstrrid sstrd
          ex dmss dmeqi sseqtrrdi ) AIGUEZUFZFUFAIUADUARZUFZUGZYBAUBIYEAUBRZISZ
          YFYESZAYGUHZYFYDSZUADUIZYHYIQRZYFUJZUCRZYLUKYLYNULGUKUMZUCYFUNZUHZYKQ
          AYGYFHSZYQQUOYIHUPZYGIHSZUHYRAYSYGLUQYIYGYTAYGURZAYTYGPUQUSYFIHUTVAZA
          BCUCDYFEQFGHJKLMNOVBVFYIYQUHZYLYFYLGUKZVCZVDZVEZDSYFUUGUFZSZYKUUCBCUB
          DEQFGHIJAGVGYGYQKVHAYSYGYQLVHYIBRZHSZERZUUJUJZUULGUKZTSZYQAUUKUUMUUOY
          GMVIVINAYTYGYQPVHYIYGYQUUAUQYIUUJHVJSZUUJWEHSZYQAUUPUUQYGOVKVKYIYMYPV
          LZUUCYLUDRZUJZYOUCUUSUNZUHZUDHUIZYLDSZYIYRYQUVCUUBUVBYQUDYFHUUSYFUMUU
          TYMUVAYPUUSYFYLVMYOUCUUSYFVNVOVPVQYLTSUVDUVCVRQVSBCUDUCDEGYLTHNVTWAWB
          WCUUCUUEUUGSZUUIUUCUUETSZUUEUUFSUVEUUCYFTSZUUDTSZUVFUBVSZYIYQYMUVHUUR
          YIYMUVHYIYLUUJUJZUVHWFZYMUVHWFBHYFUUJYFUMUVJYMUVHUUJYFYLVMWDAUVKBHUNY
          GAUVKBHAUUKUHZUUMUUOWFZEWGUVKUVLUVMEAUUKUUMUUOMWHWIUVMUVKEQUULYLUMZUU
          MUVJUUOUVHUUJUULYLWJUVNUUNUUDTUULYLGWKWLWMWNWOWQUQUUBWPWRVFZYFUUDTTWS
          WTUUETXAUUEUUFYLXBXCUUCUVGUVHUVEUUIWFUVIUVOYFUUDUUGTTXDWTXEYJUUIUAUUG
          DYCUUGUMYDUUHYFYCUUGXFXHVPXGXIUAYFDYDXJWBXQXKAYEDVJZUFZYBUADXLAUVPYAX
          MUVQYBXMABCDEGHNLXNUVPYAXRWOXOXPFYAJXSXT $.
      $}
    $}

    $d G a b c f x y $.  $d X a b f x $.  $d Y f x $.  $d f ph x $.
    tfr1on.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
    tfr1on.yx $e |- ( ph -> Y e. X ) $.
    $( Recursion is defined on an ordinal if the characteristic function is
       defined up to a suitable point.  (Contributed by Jim Kingdon,
       12-Mar-2022.) $)
    tfr1on $p |- ( ph -> Y C_ dom F ) $=
      ( vy va vb vc cv wfn cfv cres wceq wral wrex eqid tfr1onlem3 tfr1onlemres
      wa cab ) ABNORZPRZSQRZUJTUJULUAETUBQUKUCUHPFUDOUIZCDEFGHIJKPQBNUMOCEFUMUE
      UFLMUG $.
  $}

  ${
    $d F z $.  $d G a b c $.  $d G f x y z $.  $d f ph y z $.  $d G f x y z $.
    $d ph f y $.
    tfri1dALT.1 $e |- F = recs ( G ) $.
    tfri1dALT.2 $e |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) ) $.
    $( Alternate proof of ~ tfri1d in terms of ~ tfr1on .

       Although this does show that the ~ tfr1on proof is general enough to
       also prove ~ tfri1d , the ~ tfri1d proof is simpler in places because it
       does not need to deal with ` X ` being any ordinal.  For that reason, we
       have both proofs.  (Proof modification is discouraged.)
       (New usage is discouraged.)  (Contributed by Jim Kingdon,
       20-Mar-2022.) $)
    tfri1dALT $p |- ( ph -> F Fn On ) $=
      ( vb vc va vz vf wfun con0 wceq wfn word cv cfv wa wcel cdm tfrfun funeqi
      vy crecs mpbir a1i wss cres wral wrex eqid tfrlem8 wb dmeqi ordeq ordsson
      cab ax-mp mp1i csuc cvv wal simpl alimi 19.21bi adantr ordon simpr eleq1d
      syl fveq2 spv 3syl 3ad2ant1 cuni onsuc unon eleq2s adantl vex sucid ssel2
      tfr1on sylancl ex ssrdv eqssd df-fn sylanbrc ) ACLZCUAZMNCMOWKAWKDUEZLDUB
      CWMEUCUFUGAWLMWLPZWLMUHAWNWMUAZPZGHIQZGQZOHQZWQRWQWSUIDRNHWRUJSGMUKIURZID
      WTULUMWLWONWNWPUNCWMEUOWLWOUPUSUFWLUQUTAJMWLAJQZMTZXAWLTZAXBSZXAVAZWLUHXA
      XETXCXDUDKCDMXEEADLZXBAXFBAXFBQZDRZVBTZSZBVCZXFBVCFXJXFBXFXIVDVEVKVFVGMPX
      DVHUGXDUDQZMTKQZDRZVBTZXMXLOAXOXBAXKXIBVCXOFXJXIBXFXIVIVEXIXOBKXGXMNXHXNV
      BXGXMDVLVJVMVNVGVOXLMVPZTXLVAMTZXDXQXLMXPXLVQVRVSVTXBXEMTAXAVQVTWDXAJWAWB
      XEWLXAWCWEWFWGWHCMWIWJ $.
  $}

  ${
    $d G f x y $.  $d X x $.  $d f ph $.
    tfrcllemssrecs.1 $e |- A = { f | E. x e. X ( f : x --> S
          /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.
    tfrcllemssrecs.x $e |- ( ph -> Ord X ) $.
    $( Lemma for ~ tfrcl .  The union of functions acceptable for ~ tfrcl is a
       subset of ` recs ` .  (Contributed by Jim Kingdon, 25-Mar-2022.) $)
    tfrcllemssrecs $p |- ( ph -> U. A C_ recs ( G ) ) $=
      ( cuni cv wf cfv cres wceq wa con0 wrex cab wral crecs word wss wi ssrexv
      ordsson 3syl ss2abdv eqsstrid unissd wfn ffn anim1i reximi ss2abi df-recs
      unissi sseqtrri sstrdi ) ADKBLZEFLZMZCLZVBNVBVDOGNPCVAUAZQZBRSZFTZKZGUBZA
      DVHADVFBHSZFTVHIAVKVGFAHUCHRUDVKVGUEJHUGVFBHRUFUHUIUJUKVIVBVAULZVEQZBRSZF
      TZKVJVHVOVGVNFVFVMBRVCVLVEVAEVBUMUNUOUPURBCFGUQUSUT $.
  $}

  ${
    tfrcl.f $e |- F = recs ( G ) $.
    tfrcl.g $e |- ( ph -> Fun G ) $.
    tfrcl.x $e |- ( ph -> Ord X ) $.
    tfrcl.ex $e |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S ) $.

    ${
      tfrcllemsucfn.1 $e |- A = { f | E. x e. X ( f : x --> S
          /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.

      ${
        $d G f x $.  $d S f x $.  $d X f x $.  $d f g $.  $d f ph x $.
        $d f x z $.
        tfrcllemsucfn.3 $e |- ( ph -> z e. X ) $.
        tfrcllemsucfn.4 $e |- ( ph -> g : z --> S ) $.
        tfrcllemsucfn.5 $e |- ( ph -> g e. A ) $.
        $( We can extend an acceptable function by one element to produce a
           function.  Lemma for ~ tfrcl .  (Contributed by Jim Kingdon,
           24-Mar-2022.) $)
        tfrcllemsucfn $p |- ( ph
            -> ( g u. { <. z , ( G ` g ) >. } ) : suc z --> S ) $=
          ( wcel cv csn cun cfv cop wf csuc cvv elexd con0 word ordelon syl2anc
          eloni ordirr 3syl wal weq feq2 imbi1d albidv 3expia alrimiv ralrimiva
          wn wi wa rspcdva feq1 eleq1d imbi12d spv sylc fsnunf syl121anc df-suc
          fveq2 feq2i sylibr ) ADUAZVTUBUCZFHUAZVTWBJUDZUEUBUCZUFZVTUGZFWDUFAVT
          FWBUFZVTUHTVTVTTVEZWCFTZWERAVTKQUIAVTUJTZVTUKWHAKUKVTKTWJNQKVTULUMVTU
          NVTUOUPAVTFGUAZUFZWKJUDZFTZVFZGUQZWGWIABUAZFWKUFZWNVFZGUQZWPBKVTBDURZ
          WSWOGXAWRWLWNWQVTFWKUSUTVAAWTBKAWQKTZVGWSGAXBWRWNOVBVCVDQVHRWOWGWIVFG
          HGHURZWLWGWNWIVTFWKWBVIXCWMWCFWKWBJVQVJVKVLVMVTFWBUHVTWCVNVOWFWAFWDVT
          VPVRVS $.
      $}

      ${
        $d G f x y $.  $d G w x y $.  $d S f x $.  $d S w x $.  $d X f x $.
        $d X w x $.  $d f g x y $.  $d f ph x y $.  $d f x y z $.
        $d g w x y $.  $d w x y z $.
        tfrcllemsucaccv.yx $e |- ( ph -> Y e. X ) $.
        tfrcllemsucaccv.zy $e |- ( ph -> z e. Y ) $.
        tfrcllemsucaccv.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
        tfrcllemsucaccv.gfn $e |- ( ph -> g : z --> S ) $.
        tfrcllemsucaccv.gacc $e |- ( ph -> g e. A ) $.
        $( Lemma for ~ tfrcl .  We can extend an acceptable function by one
           element to produce an acceptable function.  (Contributed by Jim
           Kingdon, 24-Mar-2022.) $)
        tfrcllemsucaccv $p |- ( ph
            -> ( g u. { <. z , ( G ` g ) >. } ) e. A ) $=
          ( vw cv cfv cop csn cun wcel wf cres wceq wral wa wrex csuc weq suceq
          cuni eleq1d ralrimiva elunii syl2anc rspcdva jca ordtr1 tfrcllemsucfn
          word sylc wel wo vex elsuc feq1 reseq1 fveq2d eqeq12d ralbidv anbi12d
          fveq1 rexbidv elab2 sylib simprrr wfn simprrl ffn fndmu raleqdv mpbid
          syl adantr rexlimddv r19.21bi cvv wne con0 ordelon onelon sylan eloni
          ordirr 3syl elequ2 biimpcd adantl mtod neqned fvunsng sylancr ordelss
          wn wi wss resabs1 fsnunres reseq1d eqtr3d 3eqtr4d cdm wal feq2 imbi1d
          albidv alrimiv fveq2 imbi12d spv fndm neleqtrrd fsnunfv syl3anc simpr
          3expia reseq2 sylan9eqr jaodan sylan2b raleq rspcev syl12anc wb opexg
          cbvrexv snexg unexg elab2g mpbird ) AHUDZDUDZUUIJUEZUFZUGZUHZEUIZBUDZ
          FUUNUJZCUDZUUNUEZUUNUURUKZJUEZULZCUUPUMZUNZBKUOZAUUJUPZKUIZUVFFUUNUJZ
          UVBCUVFUMZUVEAUUPUPZKUIZUVGBKUSZUUJBDUQZUVJUVFKUUPUUJURUTAUVKBUVLTVAA
          UUJLUIZLKUIZUUJUVLUISRUUJLKVBVCVDABCDEFGHIJKMNOPQAKVHZUVNUVOUNUUJKUIZ
          OAUVNUVOSRVEUUJLKVFVIZUAUBVGAUVBCUVFUURUVFUIACDVJZCDUQZVKUVBUURUUJCVL
          ZVMAUVSUVBUVTAUVSUNZUURUUIUEZUUIUURUKZJUEZUUSUVAAUWCUWEULZCUUJAUUPFUU
          IUJZUWFCUUPUMZUNZUWFCUUJUMZBKAUUIEUIUWIBKUOZUBUUPFGUDZUJZUURUWLUEZUWL
          UURUKZJUEZULZCUUPUMZUNZBKUOZUWKGUUIEHVLZGHUQZUWSUWIBKUXBUWMUWGUWRUWHU
          UPFUWLUUIVNUXBUWQUWFCUUPUXBUWNUWCUWPUWEUURUWLUUIVTUXBUWOUWDJUWLUUIUUR
          VOVPVQVRVSWAQWBWCAUUPKUIZUWIUNZUNZUWHUWJAUXCUWGUWHWDUXEUWFCUUPUUJUXEU
          UIUUPWEZUUIUUJWEZUVMUXEUWGUXFAUXCUWGUWHWFUUPFUUIWGWKUXEUUJFUUIUJZUXGA
          UXHUXDUAWLUUJFUUIWGZWKUUPUUJUUIWHVCWIWJWMWNUWBUURWOUIUUJUURWPUUSUWCUL
          UWAUWBUUJUURUWBDCUQZCCVJZUWBUURWQUIZUURVHUXKXLAUUJWQUIZUVSUXLAUVPUVQU
          XMOUVRKUUJWRVCZUUJUURWSWTUURXAUURXBXCUVSUXJUXKXMAUXJUVSUXKDCCXDXEXFXG
          XHUUIUUJUUKUURWOXIXJUWBUUTUWDJUWBUUNUUJUKZUURUKZUUTUWDUWBUURUUJXNZUXP
          UUTULAUUJVHZUVSUXQAUXMUXRUXNUUJXAWKZUUJUURXKWTUUNUURUUJXOWKAUXPUWDULU
          VSAUXOUUIUURAUXGDDVJXLZUXOUUIULAUXHUXGUAUXIWKZAUXRUXTUXSUUJXBWKZUUJUU
          IUUJUUKXPVCZXQWLXRVPXSAUVTUNZUUJUUNUEZUUKUUSUVAAUYEUUKULZUVTAUVNUUKFU
          IZUUJUUIXTZUIXLUYFSAUUJFUWLUJZUWLJUEZFUIZXMZGYAZUXHUYGAUWMUYKXMZGYAZU
          YMBKUUJUVMUYNUYLGUVMUWMUYIUYKUUPUUJFUWLYBYCYDAUYOBKAUXCUNUYNGAUXCUWMU
          YKPYNYEVAUVRVDUAUYLUXHUYGXMGHUXBUYIUXHUYKUYGUUJFUWLUUIVNUXBUYJUUKFUWL
          UUIJYFUTYGYHVIZAUYHUUJUUJUYBAUXGUYHUUJULUYAUUJUUIYIWKYJUUILFUUJUUKYKY
          LWLUYDUURUUJUUNAUVTYMVPUYDUUTUUIJUVTAUUTUXOUUIUURUUJUUNYOUYCYPVPXSYQY
          RVAUVGUVHUVIUNZUNUCUDZFUUNUJZUVBCUYRUMZUNZUCKUOUVEVUAUYQUCUVFKUYRUVFU
          LUYSUVHUYTUVIUYRUVFFUUNYBUVBCUYRUVFYSVSYTVUAUVDUCBKUCBUQUYSUUQUYTUVCU
          YRUUPFUUNYBUVBCUYRUUPYSVSUUDWCUUAAUUNWOUIZUUOUVEUUBAUUIWOUIUUMWOUIZVU
          BUXAAUULWOUIZVUCAUUJWOUIUYGVUDDVLUYPUUJUUKWOFUUCXJUULWOUUEWKUUIUUMWOW
          OUUFXJUWTUVEGUUNEWOUWLUUNULZUWSUVDBKVUEUWMUUQUWRUVCUUPFUWLUUNVNVUEUWQ
          UVBCUUPVUEUWNUUSUWPUVAUURUWLUUNVTVUEUWOUUTJUWLUUNUURVOVPVQVRVSWAQUUGW
          KUUH $.
      $}

      ${
        $d A f g h x y z $.  $d D f g x y $.  $d G f x y $.  $d S f x y $.
        $d X f x $.  $d f g h ph x y z $.
        tfrcllembacc.3 $e |- B = { h | E. z e. D E. g ( g : z --> S
           /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) } $.
        tfrcllembacc.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
        tfrcllembacc.4 $e |- ( ph -> D e. X ) $.
        tfrcllembacc.5 $e |- ( ph -> A. z e. D E. g ( g : z --> S
          /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) ) $.
        ${
          $( Lemma for ~ tfrcl .  Each element of ` B ` is an acceptable
             function.  (Contributed by Jim Kingdon, 25-Mar-2022.) $)
          tfrcllembacc $p |- ( ph -> B C_ A ) $=
            ( cv wf wcel cfv cop csn cun wceq w3a wex wrex cab wa wfun ad2antrr
            simpr3 word simp1ll syld3an1 simplr cuni csuc adantlr simpr1 simpr2
            tfrcllemsucaccv eqeltrd ex exlimdv rexlimdva abssdv eqsstrid ) AGDU
            EZIKUEZUFZVRFUGZLUEZVRVQVRNUHUIUJUKZULZUMZKUNZDHUOZLUPFUAAWFLFAWEWA
            FUGZDHAVQHUGZUQZWDWGKWIWDWGWIWDUQZWAWBFWIVSVTWCUTWJBCDFIJKMNOHPANUR
            WHWDQUSAOVAWHWDRUSABUEZOUGZWJWKIJUEZUFZWMNUHIUGAWHWDWLWNVBSVCTAHOUG
            WHWDUCUSAWHWDVDWIWKOVEUGZWKVFOUGZWDAWOWPWHUBVGVGWIVSVTWCVHWIVSVTWCV
            IVJVKVLVMVNVOVP $.
        $}

        ${
          $d A f g h x y z $.  $d B g h z $.  $d B g w z $.
          $d D f g h x y z $.  $d G f h x y z $.  $d G w y z $.
          $d S f h x y z $.  $d X f x z $.  $d f g h ph x y z $.
          $( Lemma for ~ tfrcl .  The union of ` B ` is defined on all elements
             of ` X ` .  (Contributed by Jim Kingdon, 25-Mar-2022.) $)
          tfrcllembxssdm $p |- ( ph -> D C_ dom U. B ) $=
            ( cv cuni cdm wcel wral wss wf cfv cres wceq wa fveq2 reseq2 fveq2d
            wex weq eqeq12d cbvralv anbi2i exbii ralbii sylib w3a cop simp1 syl
            simp2 word ordtr1 imp syl12anc simp3l wal feq2 imbi1d albidv 3expia
            alrimiv ralrimiva adantr simpr rspcdva feq1 eleq1d imbi12d syl21anc
            spv csn cun cvv vex opexg sylancr snidg elun2 3syl wrex simp3r rspe
            raleq anbi12d cbvrexv fveq1 reseq1 ralbidv rexbidv elab2 3jca snexg
            wi sylibr unexg mpan isset simpr3 19.8a abeq2i sylan2 eqeltrrd 3imp
            3exp2 exlimdv sylc elunii opeq2 spcegv eldm2 imbitrrdi ralimdva mpd
            syl2anc dfss3 ) ADUEZGUFZUGZUHZDHUIZHYSUJAYQIKUEZUKZCUEZUUBULZUUBUU
            DUMZNULZUNZCYQUIZUOZKUSZDHUIZUUAAUUCEUEZUUBULZUUBUUMUMZNULZUNZEYQUI
            ZUOZKUSZDHUIUULUDUUTUUKDHUUSUUJKUURUUIUUCUUQUUHECYQECUTZUUNUUEUUPUU
            GUUMUUDUUBUPUVAUUOUUFNUUMUUDUUBUQURVAVBVCVDVEVFAUUKYTDHAYQHUHZUOUUJ
            YTKAUVBUUJYTAUVBUUJVGZUUBNULZIUHZYQUVDVHZYRUHZYTUVCAYQOUHZUUCUVEAUV
            BUUJVIZUVCAUVBHOUHZUVHUVIAUVBUUJVKZUVCAUVJUVIUCVJAUVBUVJUOZUVHAOVLU
            VLUVHXNRYQHOVMVJVNVOZAUVBUUCUUIVPZAUVHUOZUUCUVEUVOYQIJUEZUKZUVPNULZ
            IUHZXNZJVQZUUCUVEXNZUVOBUEZIUVPUKZUVSXNZJVQZUWABOYQBDUTZUWEUVTJUWGU
            WDUVQUVSUWCYQIUVPVRVSVTAUWFBOUIUVHAUWFBOAUWCOUHZUOUWEJAUWHUWDUVSSWA
            WBWCWDAUVHWEWFUVTUWBJKJKUTZUVQUUCUVSUVEYQIUVPUUBWGUWIUVRUVDIUVPUUBN
            UPWHWIWKVJVNWJZUVCUVFUUBUVFWLZWMZUHZUWLGUHZUVGUVCUVFWNUHZUVFUWKUHUW
            MUVCYQWNUHUVEUWODWOZUWJYQUVDWNIWPWQZUVFWNWRUVFUWKUUBWSWTUVCUVBUUCUU
            BFUHZVGZLUEZUWLUNZLUSZUWNUVCUVBUUCUWRUVKUVNUVCUWCIUUBUKZUUHCUWCUIZU
            OZBOXAZUWRUVCUUJDOXAZUXFUVCUVHUUCUUIUXGUVMUVNAUVBUUCUUIXBUUJDOXCVOU
            UJUXEDBODBUTUUCUXCUUIUXDYQUWCIUUBVRUUHCYQUWCXDXEXFVFUWDUUDUVPULZUVP
            UUDUMZNULZUNZCUWCUIZUOZBOXAUXFJUUBFKWOZUWIUXMUXEBOUWIUWDUXCUXLUXDUW
            CIUVPUUBWGUWIUXKUUHCUWCUWIUXHUUEUXJUUGUUDUVPUUBXGUWIUXIUUFNUVPUUBUU
            DXHURVAXIXEXJTXKXOXLUVCUWLWNUHZUXBUVCUWOUWKWNUHZUXOUWQUVFWNXMUUBWNU
            HUXPUXOUXNUUBUWKWNWNXPXQWTLUWLXRVFUWSUXAUWNLUVBUUCUWRUXAUWNXNUVBUUC
            UWRUXAUWNUVBUUCUWRUXAVGZUOUWTUWLGUVBUUCUWRUXAXSUXQUVBUXQKUSZUWTGUHZ
            UXQKXTUVBUXRUOUXRDHXAZUXSUXRDHXCUXTLGUAYAXOYBYCYEYDYFYGUVFUWLGYHYOU
            VEUVGYQUUMVHZYRUHZEUSYTUYBUVGEUVDIUUMUVDUNUYAUVFYRUUMUVDYQYIWHYJEYQ
            YRUWPYKYLYGWAYFYMYNDHYSYPXO $.
        $}

        ${
          $d A f g h x y z $.  $d B g h z $.  $d B g w z $.
          $d D f g h x y z $.  $d G f h x y z $.  $d G w y z $.
          $d S f g h x y z $.  $d X f x z $.  $d f g h ph x y z $.
          $( Lemma for ~ tfrcl .  The union of ` B ` is a function defined on
             ` x ` .  (Contributed by Jim Kingdon, 25-Mar-2022.) $)
          tfrcllembfn $p |- ( ph -> U. B : D --> S ) $=
            ( cuni wfn crn wss wfun cdm wceq tfrcllembacc unissd tfrcllemssrecs
            wf crecs sstrd tfrfun funss mpisyl cxp cpw wcel cfv cop csn cun w3a
            cv wex wrex cab simpr3 csuc simpl word adantr simpr jca ordtr1 sylc
            wa ad2antrr 3adant1r simplr simpr1 simpr2 tfrcllemsucfn sylan fssxp
            syl con0 ordelon syl2anc eloni ordsucss xpss1 cvv wb vex wi wal weq
            feq2 imbi1d albidv wral 3expia alrimiv 3ad2ant1 simp2 rspcdva simp3
            ralrimiva feq1 fveq2 eleq1d imbi12d spv syl3anc opexg sylancr snexg
            unexg elpwg mpbird eqeltrd exlimdv rexlimdva eqsstrid sspwuni sylib
            ex abssdv dmss dmxpss sstrdi tfrcllembxssdm eqssd sylanbrc rnxpss
            df-fn rnss df-f ) AGUEZHUFZUUEUGZIUHHIUUEUOAUUEUIZUUEUJZHUKUUFAUUEN
            UPZUHUUJUIUUHAUUEFUEUUJAGFABCDEFGHIJKLMNOPQRSTUAUBUCUDULUMABCFIJNOT
            RUNUQNURUUEUUJUSUTAUUIHAUUIHIVAZUJZHAUUEUUKUHZUUIUULUHAGUUKVBZUHUUM
            AGDVIZIKVIZUOZUUPFVCZLVIZUUPUUOUUPNVDZVEZVFZVGZUKZVHZKVJZDHVKZLVLUU
            NUAAUVGLUUNAUVFUUSUUNVCZDHAUUOHVCZWBZUVEUVHKUVJUVEUVHUVJUVEWBZUUSUV
            CUUNUVJUUQUURUVDVMUVKUVCUUNVCZUVCUUKUHZUVKUVCUUOVNZIVAZUUKUVKUVNIUV
            CUOZUVCUVOUHUVJAUUOOVCZWBZUVEUVPUVJAUVQAUVIVOZUVJOVPZUVIHOVCZWBUVQA
            UVTUVIRVQUVJUVIUWAAUVIVRAUWAUVIUCVQVSUUOHOVTWAZVSUVRUVEWBBCDFIJKMNO
            PANUIUVQUVEQWCAUVTUVQUVERWCUVRBVIZOVCZUWCIJVIZUOZUWENVDZIVCZUVEAUWD
            UWFUWHUVQSWDWDTAUVQUVEWEUVRUUQUURUVDWFUVRUUQUURUVDWGWHWIUVNIUVCWJWK
            UVKUVNHUHZUVOUUKUHUVKHVPZUVIUWIAUWJUVIUVEAHWLVCZUWJAUVTUWAUWKRUCOHW
            MWNHWOWKWCAUVIUVEWEUUOHWPWAUVNHIWQWKUQUVKUVCWRVCZUVLUVMWSUVKUUPWRVC
            UVBWRVCZUWLKWTUVKUVAWRVCZUWMUVKUUOWRVCUUTIVCZUWNDWTUVKAUVQUUQUWOUVJ
            AUVEUVSVQUVJUVQUVEUWBVQUVJUUQUURUVDWFAUVQUUQVHZUUOIUWEUOZUWHXAZJXBZ
            UUQUWOUWPUWFUWHXAZJXBZUWSBOUUOBDXCZUWTUWRJUXBUWFUWQUWHUWCUUOIUWEXDX
            EXFAUVQUXABOXGUUQAUXABOAUWDWBUWTJAUWDUWFUWHSXHXIXNXJAUVQUUQXKXLAUVQ
            UUQXMUWRUUQUWOXAJKJKXCZUWQUUQUWHUWOUUOIUWEUUPXOUXCUWGUUTIUWEUUPNXPX
            QXRXSWAXTUUOUUTWRIYAYBUVAWRYCWKUUPUVBWRWRYDYBUVCUUKWRYEWKYFYGYMYHYI
            YNYJGUUKYKYLZUUEUUKYOWKHIYPYQABCDEFGHIJKLMNOPQRSTUAUBUCUDYRYSUUEHUU
            BYTAUUGUUKUGZIAUUMUUGUXEUHUXDUUEUUKUUCWKHIUUAYQHIUUEUUDYT $.
        $}

        ${
          $d A f g h x y z $.  $d B g h z $.  $d B g w z $.
          $d D f g h x y z $.  $d G f h x y z $.  $d G w y z $.
          $d S f g h x y z $.  $d X f x z $.  $d f g h ph x y z $.
          $( Lemma for ~ tfrcl .  The set ` B ` exists.  (Contributed by Jim
             Kingdon, 25-Mar-2022.) $)
          tfrcllembex $p |- ( ph -> B e. _V ) $=
            ( cuni cvv wcel wf tfrcllembfn fex syl2anc uniexb sylibr ) AGUEZUFU
            GZGUFUGAHIUNUHHOUGUOABCDEFGHIJKLMNOPQRSTUAUBUCUDUIUCHIOUNUJUKGULUM
            $.
        $}

        ${
          $d A f g h x y z $.  $d B g h z $.  $d B u w $.  $d D f g h x y z $.
          $d D u w $.  $d G e t v w $.  $d G f h x y z $.  $d G u w $.
          $d S f g h x y z $.  $d X f x z $.  $d f g h ph x y z $.
          $d g ph w y z $.
          $( Lemma for ~ tfrcl .  The union of ` B ` satisfies the recursion
             rule.  (Contributed by Jim Kingdon, 25-Mar-2022.) $)
          tfrcllemubacc $p |- ( ph ->
              A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) $=
            ( vv vt ve cv cuni cfv cres wceq wral wcel wa crecs cdm tfrcllembfn
            wf fdm syl wss tfrcllembacc unissd tfrcllemssrecs sstrd dmss sselda
            eqsstrrd wfn con0 wrex cab eqid tfrlem9 wfun tfrfun biimpar funssfv
            eleq2d mp3an2ani word ordelon syl2anc eloni ordelss adantr sseqtrrd
            sylan fun2ssres fveq2d 3eqtr3d fveq2 reseq2 eqeq12d cbvralv sylibr
            ralrimiva ) AEUIZHUJZUKZXAWTULZOUKZUMZEIUNFUIZXAUKZXAXFULZOUKZUMZFI
            UNAXEEIAWTIUOZUPZWTOUQZUKZXMWTULZOUKZXBXDXLWTXMURZUOXNXPUMAIXQWTAIX
            AURZXQAIJXAUTXRIUMZABCDEGHIJKLMNOPQRSTUAUBUCUDUEUSIJXAVAVBZAXAXMVCZ
            XRXQVCAXAGUJXMAHGABCDEGHIJKLMNOPQRSTUAUBUCUDUEVDVEABCGJKOPUASVFVGZX
            AXMVHVBVJVIUFUGUHUIZUFUIZVKUGUIZYCUKYCYEULOUKUMUGYDUNUPUFVLVMUHVNZW
            TUHOYFVOVPVBXMVQZAYAXKWTXRUOZXNXBUMOVRZYBAYHXKAXRIWTXTWAVSWTXMXAVTW
            BXLXOXCOYGAYAXKWTXRVCXOXCUMYIYBXLWTIXRAIWCZXKWTIVCAIVLUOZYJAPWCIPUO
            YKSUDPIWDWEIWFVBIWTWGWJAXSXKXTWHWIWTXMXAWKWBWLWMWSXJXEFEIXFWTUMZXGX
            BXIXDXFWTXAWNYLXHXCOXFWTXAWOWLWPWQWR $.
        $}

        ${
          $d A f g h x y z $.  $d B f g h z $.  $d B f u $.  $d B g w z $.
          $d D f g h x y z $.  $d D f u $.  $d D g w y z $.  $d G f h x y z $.
          $d G f u $.  $d G u w $.  $d S f g h x y z $.  $d X f x z $.
          $d f g h ph x y z $.  $d ph w y z $.
          $( Lemma for ~ tfrcl .  (Contributed by Jim Kingdon, 26-Mar-2022.) $)
          tfrcllemex $p |- ( ph ->
              E. f ( f : D --> S
                /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) ) $=
            ( cuni cvv wcel wf cv cfv cres wceq wral wex tfrcllembex uniexg syl
            wa tfrcllembfn tfrcllemubacc jca feq1 reseq1 fveq2d eqeq12d ralbidv
            fveq1 anbi12d spcegv sylc ) AHUFZUGUHZIJVLUIZFUJZVLUKZVLVOULZOUKZUM
            ZFIUNZUSZIJKUJZUIZVOWBUKZWBVOULZOUKZUMZFIUNZUSZKUOAHUGUHVMABCDEGHIJ
            KLMNOPQRSTUAUBUCUDUEUPHUGUQURAVNVTABCDEGHIJKLMNOPQRSTUAUBUCUDUEUTAB
            CDEFGHIJKLMNOPQRSTUAUBUCUDUEVAVBWIWAKVLUGWBVLUMZWCVNWHVTIJWBVLVCWJW
            GVSFIWJWDVPWFVRVOWBVLVHWJWEVQOWBVLVOVDVEVFVGVIVJVK $.
        $}
      $}

      ${
        $d A a b c r s t u $.  $d A a b d h r s t u $.  $d A a b d h u x y $.
        $d C g u z $.  $d G a b c g u w y z $.  $d G a b d g h u w x y z $.
        $d G f h x y $.  $d G a b c r s t u $.  $d S a b c g u w y z $.
        $d S a b d g h u w x y z $.  $d S f h x y $.  $d S a b c r s t u $.
        $d X a b c w y z $.  $d X a b d h w x y z $.  $d X f h x y $.
        $d a b c ph w y z $.  $d d h ph w x y z $.  $d f h ph x y $.
        $d s t u z $.
        tfrcllemaccex.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
        $( We can define an acceptable function on any element of ` X ` .

           As with many of the transfinite recursion theorems, we have
           hypotheses that state that ` F ` is a function and that it is
           defined up to ` X ` .  (Contributed by Jim Kingdon, 26-Mar-2022.) $)
        tfrcllemaccex $p |- ( ( ph /\ C e. X ) -> E. g ( g : C --> S
            /\ A. u e. C ( g ` u ) = ( G ` ( g |` u ) ) ) ) $=
          ( wa cfv vz vw vh vb vc vt vr vs va vd con0 wcel cv wf cres wceq wral
          wex word ordelon sylan wi weq eleq1 anbi2d feq2 raleq anbi12d imbi12d
          exbidv cop csn cun w3a wrex cab wfun ad3antrrr wal alrimiv feq1 fveq2
          3expia eleq1d cbvalv sylib 19.21bi 3impia fveq1 reseq1 fveq2d eqeq12d
          3adant1r ralbidv rexbidv cbvabv eqtri opeq2d uneq12d eqeq2d 3anbi123d
          sneqd cbvexv rexbii opeq1 uneq2d 3anbi13d cbvrexv bitri abbii 3anbi3d
          id eqeq1 cuni csuc adantlr simpr simplr ordtr1 ad4antr mp2and simpllr
          syl rspcdva reseq2 cbvralv anbi2i exbii imbitrdi ralrimiva tfrcllemex
          mpd exp31 expcom a2d impexp ralbii r19.21v 3imtr4g tfis3 mpcom ) FUKU
          LZAFLULZSZFGIUMZUNZDUMZUUETZUUEUUGUOZKTZUPZDFUQZSZIURZALUSZUUCUUBOLFU
          TVAAUAUMZLULZSZUUPGUUEUNZUUKDUUPUQZSZIURZVBZAUBUMZLULZSZUVDGUUEUNZUUK
          DUVDUQZSZIURZVBZUUDUUNVBUAUBFUAUBVCZUURUVFUVBUVJUVLUUQUVEAUUPUVDLVDVE
          UVLUVAUVIIUVLUUSUVGUUTUVHUUPUVDGUUEVFUUKDUUPUVDVGVHVJVIUUPFUPZUURUUDU
          VBUUNUVMUUQUUCAUUPFLVDVEUVMUVAUUMIUVMUUSUUFUUTUULUUPFGUUEVFUUKDUUPFVG
          VHVJVIUUPUKULZAUVEUVJVBZUBUUPUQZVBZAUUQUVBVBZVBUVKUBUUPUQZUVCUVNAUVPU
          VRAUVNUVPUVRVBAUVNSZUVPUUQUVBUVTUVPSZUUQSZUUPGUCUMZUNZUUGUWCTZUWCUUGU
          OZKTZUPZDUUPUQZSZUCURUVBUWBBCUDUEDEUFUMZGUGUMZUNZUWLEULZUHUMZUWLUWKUW
          LKTZVKZVLZVMZUPZVNZUGURZUFUUPVOZUHVPZUUPGUCUIUJJKLMAKVQUVNUVPUUQNVRAU
          UOUVNUVPUUQOVRUWABUMZLULZUXEGUWCUNZUWCKTZGULZUUQUVTUXFUXGUXIUVPAUXFUX
          GUXIUVNAUXFUXGUXIAUXFSZUXGUXIVBZUCUXJUXEGHUMZUNZUXLKTZGULZVBZHVSUXKUC
          VSUXJUXPHAUXFUXMUXOPWCVTUXPUXKHUCHUCVCZUXMUXGUXOUXIUXEGUXLUWCWAZUXQUX
          NUXHGUXLUWCKWBWDVIWEWFWGWHWMWMWMEUXMCUMZUXLTZUXLUXSUOZKTZUPZCUXEUQZSZ
          BLVOZHVPUXGUXSUWCTZUWCUXSUOZKTZUPZCUXEUQZSZBLVOZUCVPQUYFUYMHUCUXQUYEU
          YLBLUXQUXMUXGUYDUYKUXRUXQUYCUYJCUXEUXQUXTUYGUYBUYIUXSUXLUWCWIUXQUYAUY
          HKUXLUWCUXSWJWKWLWNVHWOWPWQUXDUDUMZGUIUMZUNZUYOEULZUWOUYOUYNUYOKTZVKZ
          VLZVMZUPZVNZUIURZUDUUPVOZUHVPUYPUYQUJUMZVUAUPZVNZUIURZUDUUPVOZUJVPUXC
          VUEUHUXCUWKGUYOUNZUYQUWOUYOUWKUYRVKZVLZVMZUPZVNZUIURZUFUUPVOVUEUXBVUQ
          UFUUPUXAVUPUGUIUGUIVCZUWMVUKUWNUYQUWTVUOUWKGUWLUYOWAUWLUYOEVDVURUWSVU
          NUWOVURUWLUYOUWRVUMVURXLVURUWQVULVURUWPUYRUWKUWLUYOKWBWRXBWSWTXAXCXDV
          UQVUDUFUDUUPUFUDVCZVUPVUCUIVUSVUKUYPVUOVUBUYQUWKUYNGUYOVFVUSVUNVUAUWO
          VUSVUMUYTUYOVUSVULUYSUWKUYNUYRXEXBXFWTXGVJXHXIXJVUEVUJUHUJUHUJVCZVUDV
          UIUDUUPVUTVUCVUHUIVUTVUBVUGUYPUYQUWOVUFVUAXMXKVJWOWPWQUWAUXELXNULZUXE
          XOLULZUUQUVTVVAVVBUVPAVVAVVBUVNRXPXPXPUWAUUQXQUWBUYPUEUMZUYOTZUYOVVCU
          OZKTZUPZUEUYNUQZSZUIURZUDUUPUWBUYNUUPULZSZUYNLULZVVJVVLVVKUUQVVMUWBVV
          KXQZUWAUUQVVKXRAVVKUUQSVVMVBZUVNUVPUUQVVKAUUOVVOOUYNUUPLXSYCXTYAVVLVV
          MUYNGUUEUNZUUKDUYNUQZSZIURZVVJVVLUVOVVMVVSVBUBUUPUYNUBUDVCZUVEVVMUVJV
          VSUVDUYNLVDVVTUVIVVRIVVTUVGVVPUVHVVQUVDUYNGUUEVFUUKDUVDUYNVGVHVJVIUVT
          UVPUUQVVKYBVVNYDVVSUYPUUGUYOTZUYOUUGUOZKTZUPZDUYNUQZSZUIURVVJVVRVWFIU
          IIUIVCZVVPUYPVVQVWEUYNGUUEUYOWAVWGUUKVWDDUYNVWGUUHVWAUUJVWCUUGUUEUYOW
          IVWGUUIVWBKUUEUYOUUGWJWKWLWNVHXCVWFVVIUIVWEVVHUYPVWDVVGDUEUYNDUEVCZVW
          AVVDVWCVVFUUGVVCUYOWBVWHVWBVVEKUUGVVCUYOYEWKWLYFYGYHXIYIYLYJYKUWJUVAU
          CIUCIVCZUWDUUSUWIUUTUUPGUWCUUEWAVWIUWHUUKDUUPVWIUWEUUHUWGUUJUUGUWCUUE
          WIVWIUWFUUIKUWCUUEUUGWJWKWLWNVHXCWFYMYNYOUVSAUVOVBZUBUUPUQUVQUVKVWJUB
          UUPAUVEUVJYPYQAUVOUBUUPYRXIAUUQUVBYPYSYTUUA $.
      $}

      ${
        $d A g h z $.  $d A g u x y z $.  $d G f g u x y $.  $d G g h $.
        $d S f g u x y $.  $d X f x y $.  $d Y f g x y z $.  $d f g ph x y z $.
        $d x y z $.
        tfrcllemres.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
        tfrcllemres.yx $e |- ( ph -> Y e. X ) $.
        $( Lemma for ~ tfr1on .  Recursion is defined on an ordinal if the
           characteristic function is defined up to a suitable point.
           (Contributed by Jim Kingdon, 18-Mar-2022.) $)
        tfrcllemres $p |- ( ph -> Y C_ dom F ) $=
          ( wcel wa cfv vh vz vg vu crecs cdm ciun wrex cres wceq wral wex word
          cv adantr simpr jca ordtr1 sylc tfrcllemaccex syldan cop csn cun wfun
          wf ad2antrr 3adant1r cuni csuc adantlr simprl feq2 raleq fveq2 reseq2
          anbi12d fveq2d eqeq12d cbvralv anbi2i bitrdi rspcev sylan feq1 reseq1
          vex fveq1 ralbidv rexbidv elab2 sylibr tfrcllemsucaccv cvv imbi1d wal
          3expia alrimiv eleq1d imbi12d spv syl ralrimiva rspcdva opexg sylancr
          wi imp snidg elun2 3syl opeldmg dmeq eleq2d syl2anc exlimddv eliun ex
          mpd ssrdv dmuni tfrcllemssrecs dmss eqsstrrid sstrd dmeqi sseqtrrdi
          wss ) AJHUEZUFZGUFAJUADUAUNZUFZUGZYJAUBJYMAUBUNZJRZYNYMRZAYOSZYNYLRZU
          ADUHZYPYQYNEUCUNZVFZUDUNZYTTZYTUUBUIZHTZUJZUDYNUKZSZYSUCAYOYNIRZUUHUC
          ULYQIUMZYOJIRZSUUIAUUJYOMUOYQYOUUKAYOUPZAUUKYOQUOUQYNJIURUSZABCUDDYNE
          FUCGHIKLMNOPUTVAYQUUHSZYTYNYTHTZVBZVCZVDZDRYNUURUFZRZYSUUNBCUBDEFUCGH
          IJKAHVEYOUUHLVGAUUJYOUUHMVGYQBUNZIRZUVAEFUNZVFZUVCHTZERZUUHAUVBUVDUVF
          YONVHVHOAUUKYOUUHQVGYQYOUUHUULUOYQUVAIVIRZUVAVJIRZUUHAUVGUVHYOPVKVKYQ
          UUAUUGVLZUUNUVAEYTVFZCUNZYTTZYTUVKUIZHTZUJZCUVAUKZSZBIUHZYTDRYQUUIUUH
          UVRUUMUVQUUHBYNIUVAYNUJZUVQUUAUVOCYNUKZSUUHUVSUVJUUAUVPUVTUVAYNEYTVMZ
          UVOCUVAYNVNVQUVTUUGUUAUVOUUFCUDYNUVKUUBUJZUVLUUCUVNUUEUVKUUBYTVOUWBUV
          MUUDHUVKUUBYTVPVRVSVTWAWBWCWDUVDUVKUVCTZUVCUVKUIZHTZUJZCUVAUKZSZBIUHU
          VRFYTDUCWGUVCYTUJZUWHUVQBIUWIUVDUVJUWGUVPUVAEUVCYTWEZUWIUWFUVOCUVAUWI
          UWCUVLUWEUVNUVKUVCYTWHUWIUWDUVMHUVCYTUVKWFVRVSWIVQWJOWKWLWMUUNUUPUURR
          ZUUTUUNUUPWNRZUUPUUQRUWKUUNYNWNRZUUOERZUWLUBWGZYQUUHUUAUWNUVIYQUUAUWN
          YQUVJUWNXGZUUAUWNXGBIYNUVSUVJUUAUWNUWAWOAUWPBIUKYOAUWPBIAUVBSZUVDUVFX
          GZFWPUWPUWQUWRFAUVBUVDUVFNWQWRUWRUWPFUCUWIUVDUVJUVFUWNUWJUWIUVEUUOEUV
          CYTHVOWSWTXAXBXCUOUUMXDXHVAZYNUUOWNEXEXFUUPWNXIUUPUUQYTXJXKUUNUWMUWNU
          WKUUTXGUWOUWSYNUUOUURWNEXLXFXSYRUUTUAUURDYKUURUJYLUUSYNYKUURXMXNWCXOX
          PUAYNDYLXQWLXRXTAYMDVIZUFZYJUADYAAUWTYIYHUXAYJYHABCDEFHIOMYBUWTYIYCXB
          YDYEGYIKYFYG $.
      $}
    $}

    tfrcl.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
    tfrcl.yx $e |- ( ph -> Y e. U. X ) $.

    ${
      $d F z $.  $d G a b f x y $.  $d S a f x y $.  $d X a f x y $.
      $d X f x y z $.  $d Y f x y z $.  $d f ph x y z $.
      $( Recursion is defined on an ordinal if the characteristic function
         satisfies a closure hypothesis up to a suitable point.  (Contributed
         by Jim Kingdon, 26-Mar-2022.) $)
      tfrcldm $p |- ( ph -> Y e. dom F ) $=
        ( vz vb cv wcel wa cfv vy va cdm cuni wex eluni sylib wf cres wceq wral
        wrex wfun adantr word 3adant1r feq2 raleq anbi12d cbvrexv reseq2 fveq2d
        cab fveq2 eqeq12d cbvralv anbi2i rexbii bitri abbii csuc adantlr simprr
        tfrcllemres simprl sseldd exlimddv ) AHOQZRZVRGRZSZHEUCZROAHGUDZRWAOUEN
        OHGUFUGAWASZVRWBHWDBUAUBQZCDQZUHZPQZWFTZWFWHUIZFTZUJZPWEUKZSZUBGULZDVCC
        DEFGVRIAFUMWAJUNAGUOWAKUNABQZGRWPCWFUHZWFFTCRWALUPWOWQUAQZWFTZWFWRUIZFT
        ZUJZUAWPUKZSZBGULZDWOWQWLPWPUKZSZBGULXEWNXGUBBGWEWPUJWGWQWMXFWEWPCWFUQW
        LPWEWPURUSUTXGXDBGXFXCWQWLXBPUAWPWHWRUJZWIWSWKXAWHWRWFVDXHWJWTFWHWRWFVA
        VBVEVFVGVHVIVJAWPWCRWPVKGRWAMVLAVSVTVMVNAVSVTVOVPVQ $.
    $}

    ${
      $d F f k w x $.  $d G f x $.  $d S f k w x $.  $d X f k w x $.  $d Y w $.
      $d f k ph w x $.
      $( Closure for transfinite recursion.  As with ~ tfr1on , the
         characteristic function must be defined up to a suitable point, not
         necessarily on all ordinals.  (Contributed by Jim Kingdon,
         25-Mar-2022.) $)
      tfrcl $p |- ( ph -> ( F ` Y ) e. S ) $=
        ( vk wcel wa cfv wi ad2antrl vw con0 cuni word orduni syl ordelon ancli
        syl2anc cv wceq eleq1 anbi2d fveq2 eleq1d imbi12d wral cres cdm wfun wf
        3adant1r 3adant1l csuc adantlr adantll simprr tfrcldm tfr2a cvv simplrl
        wal adantr syl3an1 sylan simpr simplrr jca ordtr1 sylc imim1i impcom ex
        ralimdva imp an32s crecs tfrfun funeqi mpbir a1i ffvresb mpbird vex fex
        sylancl feq2 imbi1d albidv 3expia alrimiv ralrimiva sucid suceq rspcdva
        wb feq1 spcgv syl3c eqeltrd exp31 tfis3 ) AHUBPZAHGUCZPZQZHERZCPZAXNUDZ
        XOXMAGUDZXSKGUEZUFNXNHUGUIAXONUHAUAUJZXNPZQZYBERZCPZSAOUJZXNPZQZYGERZCP
        ZSZXPXRSUAOHYBYGUKZYDYIYFYKYMYCYHAYBYGXNULUMYMYEYJCYBYGEUNUOUPYBHUKZYDX
        PYFXRYNYCXOAYBHXNULUMYNYEXQCYBHEUNUOUPYBUBPZYLOYBUQZYDYFYOYPQZYDQZYEEYB
        URZFRZCYRYBEUSZPYEYTUKYRBCDEFGYBIAFUTZYQYCJTAXTYQYCKTZYDBUJZGPZUUDCDUJZ
        VAZUUFFRZCPZYQAUUEUUGUUIYCLVBVCYDUUDXNPZUUDVDZGPZYQAUUJUULYCMVEVFYQAYCV
        GZVHYBEFIVIUFYRYSVJPZYBCUUFVAZUUISZDVLZYBCYSVAZYTCPZYRUURYBVJPUUNYRUURY
        GUUAPZYKQZOYBUQZYOYDYPUVBYOYDQZYPUVBUVCYLUVAOYBUVCYGYBPZQZYLUVAUVEYLQZU
        UTYKUVFBCDEFGYGIUVEUUBYLUVCUUBUVDAUUBYOYCJTVMVMUVEXTYLUVCXTUVDAXTYOYCKT
        VMZVMUVEUUEUUGUUIYLUVEAUUEUUGUUIYOAYCUVDVKZLVNVBUVEUUJUULYLUVEAUUJUULUV
        HMVOVEUVEYHYLUVEXSUVDYCQYHUVEXTXSUVGYAUFUVEUVDYCUVCUVDVPYOAYCUVDVQVRYGY
        BXNVSVTZVMVHYLUVEYKUVEYIYKUVEAYHUVHUVIVRWAWBVRWCWDWEWFYREUTZUURUVBXFUVJ
        YRUVJFWGZUTFWHEUVKIWIWJWKOYBCEWLUFWMZUAWNZYBCVJYSWOWPYRUUGUUISZDVLZUUQB
        GYBUUDYBUKZUVNUUPDUVPUUGUUOUUIUUDYBCUUFWQWRWSAUVOBGUQYQYCAUVOBGAUUEQUVN
        DAUUEUUGUUILWTXAXBTYRXTYBYBVDZPZUVQGPZQYBGPUUCYRUVRUVSUVRYRYBUVMXCWKYRU
        ULUVSBXNYBUVPUUKUVQGUUDYBXDUOAUULBXNUQYQYCAUULBXNMXBTUUMXEVRYBUVQGVSVTX
        EUVLUUPUURUUSSDYSVJUUFYSUKZUUOUURUUIUUSYBCUUFYSXGUVTUUHYTCUUFYSFUNUOUPX
        HXIXJXKXLVT $.
    $}
  $}

  ${
    $d G x $.
    tfri1.1 $e |- F = recs ( G ) $.
    tfri1.2 $e |- ( Fun G /\ ( G ` x ) e. _V ) $.
    $( Principle of Transfinite Recursion, part 1 of 3.  Theorem 7.41(1) of
       [TakeutiZaring] p. 47, with an additional condition.

       The condition is that ` G ` is defined "everywhere", which is stated
       here as ` ( G `` x ) e. _V ` .  Alternately,
       ` A. x e. On A. f ( f Fn x -> f e. dom G ) ` would suffice.

       Given a function ` G ` satisfying that condition, we define a class
       ` A ` of all "acceptable" functions.  The final function we're
       interested in is the union ` F = recs ( G ) ` of them. ` F ` is then
       said to be defined by transfinite recursion.  The purpose of the 3 parts
       of this theorem is to demonstrate properties of ` F ` .  In this first
       part we show that ` F ` is a function whose domain is all ordinal
       numbers.  (Contributed by Jim Kingdon, 4-May-2019.)  (Revised by Mario
       Carneiro, 24-May-2019.) $)
    tfri1 $p |- F Fn On $=
      ( con0 wfn wtru wfun cv cfv cvv wcel wa wal ax-gen a1i tfri1d mptru ) BFG
      HABCDCIAJCKLMNZAOHTAEPQRS $.

    $( Principle of Transfinite Recursion, part 2 of 3.  Theorem 7.41(2) of
       [TakeutiZaring] p. 47, with an additional condition on the recursion
       rule ` G ` ( as described at ~ tfri1 ).  Here we show that the function
       ` F ` has the property that for any function ` G ` satisfying that
       condition, the "next" value of ` F ` is ` G ` recursively applied to all
       "previous" values of ` F ` .  (Contributed by Jim Kingdon,
       4-May-2019.) $)
    tfri2 $p |- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) ) $=
      ( con0 wcel cdm cfv cres wceq wfn tfri1 fndm ax-mp eleq2i tfr2a sylbir )
      BGHBCIZHBCJCBKDJLTGBCGMTGLACDEFNGCOPQBCDERS $.
  $}

  ${
    $d B x y $.  $d F x y $.  $d G x y $.
    tfri3.1 $e |- F = recs ( G ) $.
    tfri3.2 $e |- ( Fun G /\ ( G ` x ) e. _V ) $.
    $( Principle of Transfinite Recursion, part 3 of 3.  Theorem 7.41(3) of
       [TakeutiZaring] p. 47, with an additional condition on the recursion
       rule ` G ` ( as described at ~ tfri1 ).  Finally, we show that ` F ` is
       unique.  We do this by showing that any class ` B ` with the same
       properties of ` F ` that we showed in parts 1 and 2 is identical to
       ` F ` .  (Contributed by Jim Kingdon, 4-May-2019.) $)
    tfri3 $p |- ( ( B Fn On /\
             A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F ) $=
      ( vy con0 wfn cv cfv cres wceq wral wa nfv wi fveq2 wb imp nfan wcel nfim
      nfra1 eqeq12d imbi2d r19.21v rsp wss onss fvreseq mpanl2 biimtrrdi sylan2
      tfri1 ancoms adantr tfri2 jctr jcab sylibr eqeq12 syl6 adantl exp43 com4t
      mpbird exp4a pm2.43d syl com3l impd a2d tfis2f com12 ralrimi eqfnfv mpan2
      biimtrid biimpar syldan ) BHIZAJZBKZBWCLZDKZMZAHNZWDWCCKZMZAHNZBCMZWBWHOZ
      WJAHWBWHAWBAPWGAHUDUAZWCHUBZWMWJWMWJQZWMGJZBKZWQCKZMZQZAGWMWTAWNWTAPUCWCW
      QMZWJWTWMXBWDWRWIWSWCWQBRWCWQCRUEUFXAGWCNWMWTGWCNZQWOWPWMWTGWCUGWOWMXCWJW
      OWBWHXCWJQZWHWOWBXDWHWOWGQZWOWBXDQZQWGAHUHXEWOXFXEWOWOWBXDWOWBOZXCXEWOWJX
      GXCXEWOWJXGXCOZXEWOOZOWJWFCWCLZDKZMZXHXLXIXGXCXLWBWOXCXLQZWOWBWCHUIZXMWCU
      JWBXNOXCWEXJMZXLWBCHIZXNXOXCSACDEFUOZGHWCBCUKULWEXJDRUMUNUPTUQXIWJXLSZXHX
      EWOXRXEWOWGWIXKMZOZXRXEXEWOXSQZOWOXTQXEYAAWCCDEFURUSWOWGXSUTVAWDWFWIXKVBV
      CTVDVGVEVFVHVIVJVKVLVMVSVNVOVPWBWLWKWBXPWLWKSXQAHBCVQVRVTWA $.
  $}

  ${
    $d f g u x y z G $.
    tfrex.1 $e |- F = recs ( G ) $.
    tfrex.2 $e |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) ) $.
    $( The transfinite recursion function is set-like if the input is.
       (Contributed by Mario Carneiro, 3-Jul-2019.) $)
    tfrex $p |- ( ( ph /\ A e. V ) -> ( F ` A ) e. _V ) $=
      ( vy vg vz vu vf wcel wa cfv crecs cvv fveq1i cv cres wceq wral con0 wrex
      wfn cab eqid tfrlem3 tfrexlem eqeltrid ) ACFNOCDPCEQZPRCDULGSABIJTZKTZUFL
      TZUMPUMUOUAEPUBLUNUCOKUDUEJUGZCMEFKLBIUPJMEUPUHUIHUJUK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Recursive definition generator
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Define a constant for the recursive definition generator. $)
  $c rec $.

  $( Extend class notation with the recursive definition generator, with
     characteristic function ` F ` and initial value ` I ` . $)
  crdg $a class rec ( F , I ) $.

  ${
    $d x g F $.  $d x g I $.
    $( Define a recursive definition generator on ` On ` (the class of ordinal
       numbers) with characteristic function ` F ` and initial value ` I ` .
       This rather amazing operation allows us to define, with compact direct
       definitions, functions that are usually defined in textbooks only with
       indirect self-referencing recursive definitions.  A recursive definition
       requires advanced metalogic to justify - in particular, eliminating a
       recursive definition is very difficult and often not even shown in
       textbooks.  On the other hand, the elimination of a direct definition is
       a matter of simple mechanical substitution.  The price paid is the
       daunting complexity of our ` rec ` operation (especially when ~ df-recs
       that it is built on is also eliminated).  But once we get past this
       hurdle, definitions that would otherwise be recursive become relatively
       simple.  In classical logic it would be easier to divide this definition
       into cases based on whether the domain of ` g ` is zero, a successor, or
       a limit ordinal.  Cases do not (in general) work that way in
       intuitionistic logic, so instead we choose a definition which takes the
       union of all the results of the characteristic function for ordinals in
       the domain of ` g ` .  This means that this definition has the expected
       properties for increasing and continuous ordinal functions, which
       include ordinal addition and multiplication.

       For finite recursion we also define ~ df-frec and for suitable
       characteristic functions ~ df-frec yields the same result as ` rec `
       restricted to ` _om ` , as seen at ~ frecrdg .

       _Note:  We introduce_ ` rec ` _with the philosophical goal of being_
       _able to eliminate all definitions with direct mechanical substitution_
       _and to verify easily the soundness of definitions.  Metamath itself_
       _has no built-in technical limitation that prevents multiple-part_
       _recursive definitions in the traditional textbook style_.  (Contributed
       by Jim Kingdon, 19-May-2019.) $)
    df-irdg $a |- rec ( F , I ) = recs ( ( g e. _V |-> ( I u.
      U_ x e. dom g ( F ` ( g ` x ) ) ) ) ) $.
  $}

  ${
    $d x g F $.  $d x g G $.  $d x g A $.  $d x g B $.
    $( Equality theorem for the recursive definition generator.  (Contributed
       by NM, 9-Apr-1995.)  (Revised by Mario Carneiro, 9-May-2015.) $)
    rdgeq1 $p |- ( F = G -> rec ( F , A ) = rec ( G , A ) ) $=
      ( vg vx wceq cvv cv cdm cfv ciun cun cmpt crecs crdg fveq1 iuneq2d uneq2d
      mpteq2dv df-irdg recseq syl 3eqtr4g ) BCFZDGAEDHZIZEHUEJZBJZKZLZMZNZDGAEU
      FUGCJZKZLZMZNZBAOCAOUDUKUPFULUQFUDDGUJUOUDUIUNAUDEUFUHUMUGBCPQRSUKUPUAUBE
      DBATEDCATUC $.

    $( Equality theorem for the recursive definition generator.  (Contributed
       by NM, 9-Apr-1995.)  (Revised by Mario Carneiro, 9-May-2015.) $)
    rdgeq2 $p |- ( A = B -> rec ( F , A ) = rec ( F , B ) ) $=
      ( vg vx wceq cvv cv cdm cfv ciun cun cmpt crecs uneq1 mpteq2dv recseq syl
      crdg df-irdg 3eqtr4g ) ABFZDGAEDHZIEHUCJCJKZLZMZNZDGBUDLZMZNZCASCBSUBUFUI
      FUGUJFUBDGUEUHABUDOPUFUIQREDCATEDCBTUA $.
  $}

  ${
    $d x y z f g F $.  $d x y z f g A $.
    $( The recursive definition generator is a function.  (Contributed by Mario
       Carneiro, 16-Nov-2014.) $)
    rdgfun $p |- Fun rec ( F , A ) $=
      ( vg vx vy vz vf crdg wfun cvv cdm cfv ciun cun cmpt crecs wfn cres wceq
      cv wral wa con0 wrex cab eqid tfrlem7 df-irdg funeqi mpbir ) BAHZICJADCTZ
      KDTULLBLMNOZPZIEFGTZETZQFTZUOLUOUQRUMLSFUPUAUBEUCUDGUEZGUMURUFUGUKUNDCBAU
      HUIUJ $.
  $}

  ${
    $d A g $.  $d g x z F $.
    $( The recursion rule for the recursive definition generator is defined
       everywhere.  (Contributed by Jim Kingdon, 14-May-2020.) $)
    rdgtfr $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> (
        Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) )
        /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f )
          e. _V ) ) $=
      ( wcel cv cfv cvv wal cdm wa vex wral wceq ralrimivw sylancr sylan2 fveq2
      ciun cun cmpt wfun elex funmpt dmex fvex eleq1d spcv iunexg ancoms dmmptg
      unexg syl eleqtrrid funfvex jctil ) CGHBIZFJZKHZBLZCKHZEKCAEIZMZAIZVEJZFJ
      ZUBZUCZUDZUEZDIZVLJKHZNCGUFVCVDNZVOVMVPVMVNVLMZHVOEKVKUGZVPVNKVQDOVPVKKHZ
      EKPVQKQVPVSEKVDVCVSVCVDVJKHZVSVCVFKHVIKHZAVFPVTVEEOZUHVCWAAVFVBWABVHVGVEK
      KWBAOUIUTVHQVAVIKUTVHFUAUJUKRAVFVIKKULSCVJKKUOTUMREKVKKUNUPUQVNVLURSVRUST
      $.
  $}

  ${
    $d A g $.  $d g x F $.
    $( The recursion rule for the recursive definition generator is defined
       everywhere.  (Contributed by Jim Kingdon, 4-Jul-2019.) $)
    rdgruledefgg $p |- ( ( F Fn _V /\ A e. V ) -> (
        Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) )
        /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f )
          e. _V ) ) $=
      ( wcel cvv wfn cv cdm cfv ciun wa vex wral funfvex ralrimivw syl sylan2
      cun cmpt wfun elex funmpt wceq fvex funfni mpan2 dmex iunexg unexg ancoms
      mpan dmmptg eleqtrrid sylancr jctil ) BFGEHIZBHGZDHBADJZKZAJZVALZELZMZUAZ
      UBZUCZCJZVHLHGZNBFUDUSUTNZVKVIVLVIVJVHKZGVKDHVGUEZVLVJHVMCOVLVGHGZDHPVMHU
      FVLVODHUTUSVOUSUTVFHGZVOUSVEHGZAVBPZVPUSVQAVBUSVDHGVQVCVAHHDOZAOUGVQHVDEV
      DEQUHUIRVBHGVRVPVAVSUJAVBVEHHUKUNSBVFHHULTUMRDHVGHUOSUPVJVHQUQVNURT $.
  $}

  ${
    $d A g $.  $d V g $.  $d g x F $.
    rdgruledefg.1 $e |- F Fn _V $.
    $( The recursion rule for the recursive definition generator is defined
       everywhere.  (Contributed by Jim Kingdon, 4-Jul-2019.) $)
    rdgruledefg $p |- ( A e. V -> (
        Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) )
        /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f )
          e. _V ) ) $=
      ( cvv wfn wcel cv cdm cfv ciun cun cmpt wfun wa rdgruledefgg mpan ) EHIBF
      JDHBADKZLAKUAMEMNOPZQCKUBMHJRGABCDEFST $.
  $}

  ${
    $d A g x y $.  $d F g x y $.  $d V g y $.
    $( The recursive definition generator produces a set on a set input.
       (Contributed by Jim Kingdon, 4-Jul-2019.) $)
    rdgexggg $p |- ( ( F Fn _V /\ A e. V /\ B e. W ) ->
        ( rec ( F , A ) ` B ) e. _V ) $=
      ( vy vg vx cvv wfn wcel crdg cfv wa cv cdm ciun cun cmpt df-irdg alrimiv
      wfun rdgruledefgg tfrex 3impa ) CIJZADKZBEKBCALZMIKUFUGNZFBUHGIAHGOZPHOUJ
      MCMQRSZEHGCATUIUKUBFOUKMIKNFHAFGCDUCUAUDUE $.
  $}

  ${
    rdgexgg.1 $e |- F Fn _V $.
    $( The recursive definition generator produces a set on a set input.
       (Contributed by Jim Kingdon, 4-Jul-2019.) $)
    rdgexgg $p |- ( ( A e. V /\ B e. W ) -> ( rec ( F , A ) ` B ) e. _V ) $=
      ( cvv wfn wcel crdg cfv rdgexggg mp3an1 ) CGHADIBEIBCAJKGIFABCDELM $.
  $}

  ${
    $d A f g x $.  $d F f g x $.  $d V f $.
    $( The recursive definition generator is a function on ordinal numbers.
       The ` F Fn _V ` condition states that the characteristic function is
       defined for all sets (being defined for all ordinals might be enough if
       being used in a manner similar to ~ rdgon ; in cases like ~ df-oadd
       either presumably could work).  (Contributed by Jim Kingdon,
       13-Jul-2019.) $)
    rdgifnon $p |- ( ( F Fn _V /\ A e. V ) ->
      rec ( F , A ) Fn On ) $=
      ( vf vg vx cvv wfn wcel wa crdg cv cdm cfv ciun cmpt df-irdg rdgruledefgg
      cun wfun alrimiv tfri1d ) BGHACIJZDBAKEGAFELZMFLUDNBNOSPZFEBAQUCUETDLUENG
      IJDFADEBCRUAUB $.
  $}

  ${
    $d A f g x $.  $d F f g x z $.  $d V f $.
    $( The recursive definition generator is a function on ordinal numbers.
       (Contributed by Jim Kingdon, 14-May-2020.) $)
    rdgifnon2 $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) ->
        rec ( F , A ) Fn On ) $=
      ( vf vg vx cv cfv cvv wcel wal wa crdg cdm ciun cun cmpt df-irdg wfun
      rdgtfr alrimiv tfri1d ) AHCIJKALBDKMZECBNFJBGFHZOGHUEICIPQRZGFCBSUDUFTEHU
      FIJKMEGABEFCDUAUBUC $.
  $}

  ${
    $d A g x y $.  $d B g x $.  $d F g x y $.  $d V g x y $.
    $( Value of the recursive definition generator.  Lemma for ~ rdgival which
       simplifies the value further.  (Contributed by Jim Kingdon,
       13-Jul-2019.)  (New usage is discouraged.) $)
    rdgivallem $p |- ( ( F Fn _V /\ A e. V /\ B e. On ) ->
        ( rec ( F , A ) ` B ) =
        ( A u. U_ x e. B ( F ` ( ( rec ( F , A ) |` B ) ` x ) ) ) ) $=
      ( vg vy cvv wfn wcel con0 cfv cv cdm ciun cun wceq wa wfun wss w3a tfri2d
      crdg cres cmpt df-irdg rdgruledefgg alrimiv 3impa eqidd 3ad2ant3 rdgifnon
      dmeq onss fndm syl 3adant3 sseqtrrd ssdmres sylib sylan9eqr fveq2d adantl
      fveq1 iuneq12d uneq2d rdgfun resfunexg mpan simpr fvexg sylancl ralrimivw
      wral vex wi funfvex funfni ex ralimdv adantr iunexg syl2anc 3adant2 unexg
      mpd 3ad2ant2 fvmptd eqtrd ) DHIZBEJZCKJZUAZCDBUCZLZWNCUDZFHBAFMZNZAMZWQLZ
      DLZOZPZUEZLZBACWSWPLZDLZOZPZWJWKWLWOXEQWJWKRZGCWNXDAFDBUFXJXDSGMXDLHJRGAB
      GFDEUGUHUBUIWMFWPXCXIHXDHWMXDUJWMWQWPQZRZXBXHBXLAWRCXAXGXKWMWRWPNZCWQWPUM
      WMCWNNZTXMCQWMCKXNWLWJCKTWKCUNUKWJWKXNKQZWLXJWNKIXOBDEULKWNUOUPUQURCWNUSU
      TVAXKXAXGQWMXKWTXFDWSWQWPVDVBVCVEVFWLWJWPHJZWKWNSWLXPBDVGWNCKVHVIZUKWMXHH
      JZXIHJZWJWLXRWKWJWLRZWLXGHJZACVNZXRWJWLVJXTXFHJZACVNZYBWLYDWJWLYCACWLXPWS
      HJYCXQAVOWSWPHHVKVLVMVCWJYDYBVPWLWJYCYAACWJYCYAYAHXFDXFDVQVRVSVTWAWFACXGK
      HWBWCWDWKWJXRXSVPWLWKXRXSBXHEHWEVSWGWFWHWI $.

    $( Value of the recursive definition generator.  (Contributed by Jim
       Kingdon, 26-Jul-2019.) $)
    rdgival $p |- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) =
        ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) ) $=
      ( cvv wfn wcel con0 w3a crdg cfv cv cres ciun cun rdgivallem fvres fveq2d
      iuneq2i uneq2i eqtrdi ) DFGBEHCIHJCDBKZLBACAMZUCCNLZDLZOZPBACUDUCLZDLZOZP
      ABCDEQUGUJBACUFUIUDCHUEUHDUDCUCRSTUAUB $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d F x y $.  $d I x y $.  $d V x y $.  $d ph x $.
    rdgss.1 $e |- ( ph -> F Fn _V ) $.
    rdgss.2 $e |- ( ph -> I e. V ) $.
    rdgss.3 $e |- ( ph -> A e. On ) $.
    rdgss.4 $e |- ( ph -> B e. On ) $.
    rdgss.5 $e |- ( ph -> A C_ B ) $.
    $( Subset and recursive definition generator.  (Contributed by Jim Kingdon,
       15-Jul-2019.) $)
    rdgss $p |- ( ph -> ( rec ( F , I ) ` A ) C_ ( rec ( F , I ) ` B ) ) $=
      ( vx vy cv cfv ciun cun wss wcel wceq crdg wrex wral fveq2d sseq2d rspcev
      ssel ssid fveq2 mpan2 syl6 ralrimiv syl iunss2 unss2 3syl cvv wfn rdgival
      con0 syl3anc 3sstr4d ) AELBLNZDEUAZOZDOZPZQZEMCMNZVDOZDOZPZQZBVDOZCVDOZAV
      FVKRZMCUBZLBUCZVGVLRVHVMRABCRZVRKVSVQLBVSVCBSVCCSZVQBCVCUGVTVFVFRZVQVFUHV
      PWAMVCCVIVCTZVKVFVFWBVJVEDVIVCVDUIUDUEUFUJUKULUMLMBCVFVKUNVGVLEUOUPADUQUR
      ZEFSZBUTSVNVHTGHILEBDFUSVAAWCWDCUTSVOVMTGHJMECDFUSVAVB $.
  $}

  ${
    $d F x $.  $d A x $.  $d B x $.  $d V x $.
    rdgisuc1.1 $e |- ( ph -> F Fn _V ) $.
    rdgisuc1.2 $e |- ( ph -> A e. V ) $.
    rdgisuc1.3 $e |- ( ph -> B e. On ) $.
    $( One way of describing the value of the recursive definition generator at
       a successor.  There is no condition on the characteristic function ` F `
       other than ` F Fn _V ` .  Given that, the resulting expression
       encompasses both the expected successor term
       ` ( F `` ( rec ( F , A ) `` B ) ) ` but also terms that correspond to
       the initial value ` A ` and to limit ordinals
       ` U_ x e. B ( F `` ( rec ( F , A ) `` x ) ) ` .

       If we add conditions on the characteristic function, we can show tighter
       results such as ~ rdgisucinc .  (Contributed by Jim Kingdon,
       9-Jun-2019.) $)
    rdgisuc1 $p |- ( ph -> ( rec ( F , A ) ` suc B ) =
        ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) )
          u. ( F ` ( rec ( F , A ) ` B ) ) ) ) ) $=
      ( csuc crdg cfv cv ciun cun wcel con0 wceq syl uneq2d cvv rdgival syl3anc
      wfn onsuc csn df-suc iuneq1 ax-mp iunxun eqtri fveq2 fveq2d iunxsng eqtrd
      eqtrid ) ADJZECKZLZCBUQBMZURLZELZNZOZCBDVBNZDURLZELZOZOZAEUAUDCFPUQQPZUSV
      DRGHADQPZVJIDUESBCUQEFUBUCAVKVDVIRIVKVCVHCVKVCVEBDUFZVBNZOZVHVCBDVLOZVBNZ
      VNUQVORVCVPRDUGBUQVOVBUHUIBDVLVBUJUKVKVMVGVEBDVBVGQUTDRVAVFEUTDURULUMUNTU
      PTSUO $.

    ${
      $d A x $.  $d B x $.  $d F x $.
      rdgisucinc.inc $e |- ( ph -> A. x x C_ ( F ` x ) ) $.
      $( Value of the recursive definition generator at a successor.

         This can be thought of as a generalization of ~ oasuc and ~ omsuc .
         (Contributed by Jim Kingdon, 29-Aug-2019.) $)
      rdgisucinc $p |- ( ph -> ( rec ( F , A ) ` suc B ) =
          ( F ` ( rec ( F , A ) ` B ) ) ) $=
        ( csuc crdg cfv cun cvv wcel con0 wceq syl3anc wss cv ciun rdgisuc1 wfn
        unass eqtr4di rdgival uneq1d wal rdgexggg id fveq2 sseq12d sylc ssequn1
        spcgv sylib 3eqtr2d ) ADKECLZMZCBDBUAZUSMEMUBZNZDUSMZEMZNZVDVENZVEAUTCV
        BVENNVFABCDEFGHIUCCVBVEUEUFAVDVCVEAEOUDZCFPZDQPZVDVCRGHIBCDEFUGSUHAVDVE
        TZVGVERAVDOPZVAVAEMZTZBUIVKAVHVIVJVLGHICDEFQUJSJVNVKBVDOVAVDRZVAVDVMVEV
        OUKVAVDEULUMUPUNVDVEUOUQUR $.
    $}
  $}

  ${
    $d A f g x y $.  $d B f x y z $.  $d F f g x y $.  $d F f x y z $.
    $d f ph x y z $.
    rdgon.2 $e |- ( ph -> A e. On ) $.
    rdgon.3 $e |- ( ph -> A. x e. On ( F ` x ) e. On ) $.
    $( Evaluating the recursive definition generator produces an ordinal.
       There is a hypothesis that the characteristic function produces ordinals
       on ordinal arguments.  (Contributed by Jim Kingdon, 26-Jul-2019.)
       (Revised by Jim Kingdon, 13-Apr-2022.) $)
    rdgon $p |- ( ( ph /\ B e. On ) -> ( rec ( F , A ) ` B ) e. On ) $=
      ( vy vf vg vz con0 wcel wa cvv cv cfv wceq adantr wral crdg cdm ciun cmpt
      cun df-irdg wfun funmpt a1i word ordon w3a vex 3ad2ant1 dmex fveq2 eleq1d
      wf simpl3 simpr fdm eleq2d syl ffvelcdmd rspcdva ralrimiva fveq2d cbvralv
      wb mpbid sylibr iunon sylancr onun2 syl2anc dmeq fveq1 uneq2d eqid fvmptg
      iuneq12d eqeltrd cuni csuc unon eleq2i biimpi adantl onsuc biimpri tfrcl
      ) ADLMZNZHLIECUAJOCBJPZUBZBPZWNQZEQZUCZUEZUDZLDBJECUFXAUGWMJOWTUHUILUJWMU
      KUIWMHPZLMZXBLIPZURZULZXDXAQZCBXDUBZWPXDQZEQZUCZUEZLXFXDOMXLLMZXGXLRIUMZX
      FCLMZXKLMZXMWMXCXOXEAXOWLFSUNXFXHOMXJLMZBXHTZXPXDXNUOXFKPZXDQZEQZLMZKXHTX
      RXFYBKXHXFXSXHMZNZWPEQZLMZYBBLXTWPXTRYEYALWPXTEUPUQXFYFBLTZYCWMXCYGXEAYGW
      LGSUNSYDXBLXSXDWMXCXEYCUSZYDYCXSXBMZXFYCUTYDXEYCYIVIYHXEXHXBXSXBLXDVAVBVC
      VJVDVEVFXQYBBKXHWPXSRZXJYALYJXIXTEWPXSXDUPVGUQVHVKBXHXJOVLVMCXKVNVOZJXDWT
      XLOLXAWNXDRZWSXKCYLBWOXHWRXJWNXDVPYLWQXIEWPWNXDVQVGWAVRXAVSVTVMYKWBWMXBLW
      CZMZNXCXBWDLMYNXCWMYNXCYMLXBWEWFWGWHXBWIVCWLDYMMZAYOWLYMLDWEWFWJWHWK $.
  $}

  ${
    $d x g F $.  $d x g A $.
    ${
      rdg.1 $e |- A e. _V $.
      $( The initial value of the recursive definition generator.  (Contributed
         by NM, 23-Apr-1995.)  (Revised by Mario Carneiro, 14-Nov-2014.) $)
      rdg0 $p |- ( rec ( F , A ) ` (/) ) = A $=
        ( vg vx c0 crdg cfv cvv cdm ciun cun cmpt wcel wceq ax-mp eqtri eqeltri
        cv 0ex dmeq fveq1 fveq2d iuneq12d uneq2d eqid iuneq1 0iun fvmpt df-irdg
        dm0 unex tfr0 uneq2i un0 ) FBAGZHZFDIAEDSZJZESZURHZBHZKZLZMZHZAVFINUQVF
        OVFAEFJZUTFHZBHZKZLZIFINVFVKOTDFVDVKIVEURFOZVCVJAVLEUSVGVBVIURFUAVLVAVH
        BUTURFUBUCUDUEVEUFAVJCVJFIVJEFVIKZFVGFOVJVMOUKEVGFVIUGPEVIUHQZTRULZUIPZ
        VORUPVEIEDBAUJUMPVFAFLZAVFVKVQVPVJFAVNUNQAUOQQ $.
    $}
  $}

  ${
    $d x A $.  $d x F $.
    $( The initial value of the recursive definition generator.  (Contributed
       by NM, 25-Apr-1995.) $)
    rdg0g $p |- ( A e. C -> ( rec ( F , A ) ` (/) ) = A ) $=
      ( vx c0 cv crdg cfv wceq rdgeq2 fveq1d id eqeq12d vex rdg0 vtoclg ) ECDFZ
      GZHZQIECAGZHZAIDABQAIZSUAQAUBERTQACJKUBLMQCDNOP $.
  $}

  ${
    rdg0.1 $e |- A e. _V $.
    rdg0.2 $e |- F Fn _V $.
    $( The recursive definition generator produces a set on a set input.
       (Contributed by Mario Carneiro, 3-Jul-2019.) $)
    rdgexg $p |- ( B e. V -> ( rec ( F , A ) ` B ) e. _V ) $=
      ( cvv wcel crdg cfv rdgexgg mpan ) AGHBDHBCAIJGHEABCGDFKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Finite recursion
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Define a constant for the finite recursive definition generator. $)
  $c frec $.

  $( Extend class notation with the finite recursive definition generator, with
     characteristic function ` F ` and initial value ` I ` . $)
  cfrec $a class frec ( F , I ) $.

  ${
    $d x g m F $.  $d x g m I $.
    $( Define a recursive definition generator on ` _om ` (the class of finite
       ordinals) with characteristic function ` F ` and initial value ` I ` .
       This rather amazing operation allows us to define, with compact direct
       definitions, functions that are usually defined in textbooks only with
       indirect self-referencing recursive definitions.  A recursive definition
       requires advanced metalogic to justify - in particular, eliminating a
       recursive definition is very difficult and often not even shown in
       textbooks.  On the other hand, the elimination of a direct definition is
       a matter of simple mechanical substitution.  The price paid is the
       daunting complexity of our ` frec ` operation (especially when ~ df-recs
       that it is built on is also eliminated).  But once we get past this
       hurdle, definitions that would otherwise be recursive become relatively
       simple; see ~ frec0g and ~ frecsuc .

       Unlike with transfinite recursion, finite recurson can readily divide
       definitions and proofs into zero and successor cases, because even
       without excluded middle we have theorems such as ~ nn0suc .  The
       analogous situation with transfinite recursion - being able to say that
       an ordinal is zero, successor, or limit - is enabled by excluded middle
       and thus is not available to us.  For the characteristic functions which
       satisfy the conditions given at ~ frecrdg , this definition and
       ~ df-irdg restricted to ` _om ` produce the same result.

       _Note:  We introduce_ ` frec ` _with the philosophical goal of being_
       _able to eliminate all definitions with direct mechanical substitution_
       _and to verify easily the soundness of definitions.  Metamath itself_
       _has no built-in technical limitation that prevents multiple-part_
       _recursive definitions in the traditional textbook style_.  (Contributed
       by Mario Carneiro and Jim Kingdon, 10-Aug-2019.) $)
    df-frec $a |- frec ( F , I ) = ( recs ( ( g e. _V |-> { x |
      ( E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/
      ( dom g = (/) /\ x e. I ) ) } ) ) |` _om ) $.
  $}

  ${
    $d x g m F $.  $d x g m G $.  $d x g m A $.  $d x g m B $.
    $( Equality theorem for the finite recursive definition generator.
       (Contributed by Jim Kingdon, 30-May-2020.) $)
    freceq1 $p |- ( F = G -> frec ( F , A ) = frec ( G , A ) ) $=
      ( vg vm vx wceq cvv cv cfv wcel wa com wrex wo cab cmpt crecs cres cfrec
      csuc c0 simpl fveq1d eleq2d anbi2d rexbidv orbi1d abbidv mpteq2dva recseq
      cdm syl reseq1d df-frec 3eqtr4g ) BCGZDHDIZULZEIZUAGZFIZUTURJZBJZKZLZEMNZ
      USUBGVBAKLZOZFPZQZRZMSDHVAVBVCCJZKZLZEMNZVHOZFPZQZRZMSBATCATUQVLVTMUQVKVS
      GVLVTGUQDHVJVRUQURHKZLZVIVQFWBVGVPVHWBVFVOEMWBVEVNVAWBVDVMVBWBVCBCUQWAUCU
      DUEUFUGUHUIUJVKVSUKUMUNFDEBAUOFDECAUOUP $.

    $( Equality theorem for the finite recursive definition generator.
       (Contributed by Jim Kingdon, 30-May-2020.) $)
    freceq2 $p |- ( A = B -> frec ( F , A ) = frec ( F , B ) ) $=
      ( vg vm vx wceq cvv cv cfv wcel wa com cab cmpt crecs cres cfrec df-frec
      wo cdm csuc c0 simpl eleq2d anbi2d orbi2d abbidv mpteq2dva recseq reseq1d
      wrex syl 3eqtr4g ) ABGZDHDIZUAZEIZUBGFIZURUPJCJKLEMULZUQUCGZUSAKZLZTZFNZO
      ZPZMQDHUTVAUSBKZLZTZFNZOZPZMQCARCBRUOVGVMMUOVFVLGVGVMGUODHVEVKUOUPHKZLZVD
      VJFVOVCVIUTVOVBVHVAVOABUSUOVNUDUEUFUGUHUIVFVLUJUMUKFDECASFDECBSUN $.
  $}

  ${
    $d A g m x $.  $d F g m x $.
    $( Finite recursion produces a set.  (Contributed by Jim Kingdon,
       20-Aug-2021.) $)
    frecex $p |- frec ( F , A ) e. _V $=
      ( vg vm vx cfrec cvv cv cdm csuc wceq cfv wcel wa com wrex c0 wo cab cmpt
      crecs cres df-frec wfun tfrfun omex resfunexg mp2an eqeltri ) BAFCGCHZIZD
      HZJKEHZULUJLBLMNDOPUKQKUMAMNRESTZUAZOUBZGECDBAUCUOUDOGMUPGMUNUEUFUOOGUGUH
      UI $.
  $}

  ${
    $d A g m x $.  $d F g m x $.
    $( Finite recursion produces a function.  See also ~ frecfnom which also
       states that the domain of that function is ` _om ` but which puts
       conditions on ` A ` and ` F ` .  (Contributed by Jim Kingdon,
       13-Feb-2022.) $)
    frecfun $p |- Fun frec ( F , A ) $=
      ( vg vm vx cfrec wfun cvv cv cdm csuc wceq cfv wcel wa com wrex c0 wo cab
      cmpt crecs cres tfrfun funres ax-mp df-frec funeqi mpbir ) BAFZGCHCIZJZDI
      ZKLEIZUMUKMBMNODPQULRLUNANOSETUAZUBZPUCZGZUPGURUOUDPUPUEUFUJUQECDBAUGUHUI
      $.
  $}

  ${
    $d g m y F $.  $d g m y A $.  $d x g m y $.
    nffrec.1 $e |- F/_ x F $.
    nffrec.2 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for the finite recursive definition
       generator.  (Contributed by Jim Kingdon, 30-May-2020.) $)
    nffrec $p |- F/_ x frec ( F , A ) $=
      ( vg vm vy cfrec cvv cv wceq cfv wcel wa com nfcv nfv nfcri nfan cdm csuc
      wrex c0 wo cab cmpt crecs cres df-frec nffv nfrexya nfor nfab nfmpt nfres
      nfrecs nfcxfr ) ACBIFJFKZUAZGKZUBLZHKZVAUSMZCMZNZOZGPUCZUTUDLZVCBNZOZUEZH
      UFZUGZUHZPUIHFGCBUJAVOPAVNAFJVMAJQVLAHVHVKAVGAGPAPQZVBVFAVBARAHVEAVDCDAVD
      QUKSTULVIVJAVIARAHBESTUMUNUOUQVPUPUR $.
  $}

  ${
    $d A g m x $.  $d F g m x $.
    $( The initial value resulting from finite recursive definition generation.
       (Contributed by Jim Kingdon, 7-May-2020.) $)
    frec0g $p |- ( A e. V -> ( frec ( F , A ) ` (/) ) = A ) $=
      ( vg vm vx wcel c0 cfv cvv cv cdm wceq wa com wrex wo cab dm0 ax-mp cfrec
      csuc cmpt biantrur wn wne vex nsuceq0g nesymi eqeq1i mtbir intnanr biorfi
      a1i nrex orcom 3bitri abbii abid2 eqtr3i elex eqeltrid dmeq eqeq1d fveq2d
      0ex fveq1 eleq2d anbi12d rexbidv anbi1d orbi12d abbidv eqid fvmptg eqtrdi
      mpan syl eqeltrd crecs cres df-frec fveq1i peano1 fvres eqtri tfr0 eqtrid
      eqtrd ) ACGZHBAUAZIZHDJDKZLZEKZUBZMZFKZWOWMIZBIZGZNZEOPZWNHMZWRAGZNZQZFRZ
      UCZIZAWJXJJGZWLXJMWJXJAJWJHLZWPMZWRWOHIZBIZGZNZEOPZXLHMZXENZQZFRZJGZXJAMW
      JYBAJXEFRYBAXEYAFXEXTXTXRQYAXSXESUDXRXTXQEOXQUEWOOGXMXPXMHWPMWPHWOJGWPHUF
      EUGWOJUHTUIXLHWPSUJUKULUNUOUMXTXRUPUQURFAUSUTZACVAZVBYCXJYBAHJGYCXJYBMVFD
      HXHYBJJXIWMHMZXGYAFYFXCXRXFXTYFXBXQEOYFWQXMXAXPYFWNXLWPWMHVCZVDYFWTXOWRYF
      WSXNBWOWMHVGVEVHVIVJYFXDXSXEYFWNXLHYGVDVKVLVMXIVNVOVQYDVPVRZYEVSXKWLHXIVT
      ZIZXJWLHYIOWAZIZYJHWKYKFDEBAWBWCHOGYLYJMWDHOYIWETWFYIXIJYIVNWGWHVRYHWI $.
  $}

  ${
    $d A x $.  $d F x $.  $d S x y $.  $d m ph $.  $d m x y $.  $d F y $.
    frecabex.sex $e |- ( ph -> S e. V ) $.
    frecabex.fvex $e |- ( ph -> A. y ( F ` y ) e. _V ) $.
    frecabex.aex $e |- ( ph -> A e. W ) $.
    $( The class abstraction from ~ df-frec exists.  This is a lemma for other
       finite recursion proofs.  (Contributed by Jim Kingdon, 13-May-2020.) $)
    frecabex $p |- ( ph -> { x |
        ( E. m e. _om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) \/
        ( dom S = (/) /\ x e. A ) ) } e. _V ) $=
      ( cv wceq cfv wcel wa com cab cvv cdm csuc wrex c0 wo wral omex wss simpr
      abssi wal fvexg sylancl fveq2 eleq1d spcgv sylc ssexg ralrimivw abrexex2g
      vex sylancr jca cun unexb unab eleq1i bitri sylib ) AEUAZFMZUBNZBMZVKEOZG
      OZPZQZFRUCZBSZTPZVJUDNZVMDPZQZBSZTPZQZVRWCUEBSZTPZAVTWEARTPVQBSZTPZFRUFVT
      UGAWJFRAWIVOUHVOTPZWJVQBVOVLVPUIUJAVNTPZCMZGOZTPZCUKWKAEHPVKTPWLJFVAVKEHT
      ULUMKWOWKCVNTWMVNNWNVOTWMVNGUNUOUPUQWIVOTURVBUSVQFBRTTUTVBAWDDUHDIPWEWCBD
      WAWBUIUJLWDDIURVBVCWFVSWDVDZTPWHVSWDVEWPWGTVRWCBVFVGVHVI $.
  $}

  ${
    $d A k m x z $.  $d F k m x y $.  $d F k m x z $.  $d G k m x y $.
    $d G k m x z $.  $d N k m x y $.  $d S k x y $.  $d S k x z $.
    $d k m ph x y $.
    frecabcl.n $e |- ( ph -> N e. _om ) $.
    frecabcl.g $e |- ( ph -> G : N --> S ) $.
    frecabcl.fs $e |- ( ph -> A. y e. S ( F ` y ) e. S ) $.
    frecabcl.as $e |- ( ph -> A e. S ) $.
    $( The class abstraction from ~ df-frec exists.  Unlike ~ frecabex the
       function ` F ` only needs to be defined on ` S ` , not all sets.  This
       is a lemma for other finite recursion proofs.  (Contributed by Jim
       Kingdon, 21-Mar-2022.) $)
    frecabcl $p |- ( ph -> { x |
        ( E. m e. _om ( dom G = suc m /\ x e. ( F ` ( G ` m ) ) ) \/
        ( dom G = (/) /\ x e. A ) ) } e. S ) $=
      ( cv wcel wceq wa com c0 wb vz vk va cdm csuc cfv wrex wal wex cab adantr
      wo peano1 elex2 r19.9rmv mp2b wf fdm syl eqeq2 adantl mpbid biantrurd wne
      peano3 neeqtrrd necomd neneqd intnanrd biorf bitrd rexbidva bitrid bitrdi
      r19.44mv alrimiv jca eleq1 eleq2 bibi1d albidv anbi12d spcegv sylc eleq1d
      wn fveq2 ad2antrr vex sucid mpbiri ffvelcdmd rspcdva simpllr simplr eqtrd
      simpr suceq eqeq2d fveq2d eleq2d rspcev syl2anc ex simprl 3eqtr3rd peano4
      wral mpbird rexlimdva impbid neeq1 eqnetrd orcom bibi2d imp nn0suc clabel
      mpjaodan sylibr ) AUANZEOZBNZYAOZHUDZFNZUEZPZYCYFHUFZGUFZOZQZFRUGZYESPZYC
      DOZQZULZTZBUHZQZUAUIZYQBUJEOAISPZUUAIUBNZUEZPZUBRUGZAUUBQZDEOZUUHYOYQTZBU
      HZQZUUAAUUHUUBMUKZUUGUUHUUJUULUUGUUIBUUGYOYLYPULZFRUGZYQYOYOFRUGZUUGUUNSR
      OZUCNROUCUIZYOUUOTUMUCSRUNZYOFUCRUOUPUUGYOUUMFRUUGYFROZQZYOYPUUMUUTYNYOUU
      GYNUUSUUGYEIPZYNAUVAUUBAIEHUQZUVAKIEHURZUSUKUUBUVAYNTAISYEUTVAVBUKZVCUUTY
      LWFYPUUMTUUTYHYKUUTYEYGUUTYGYEUUTYGSYEUUSYGSVDUUGYFVEVAUVDVFVGVHVIYLYPVJU
      SVKVLVMUUPUUQUUNYQTUMUURYLYPFUCRVOUPVNVPVQYTUUKUADEYADPZYBUUHYSUUJYADEVRU
      VEYRUUIBUVEYDYOYQYADYCVSVTWAWBWCWDAUUFUUAAUUEUUAUBRAUUCROZQZUUEUUAUVGUUEQ
      ZUUCHUFZGUFZEOZUVKYCUVJOZYQTZBUHZQZUUAUVHCNZGUFZEOZUVKCEUVIUVPUVIPUVQUVJE
      UVPUVIGWGWEAUVRCEXHUVFUUELWHUVHIEUUCHAUVBUVFUUEKWHZUUEUUCIOZUVGUUEUVTUUCU
      UDOUUCUBWIWJIUUDUUCVSWKVAWLWMZUVHUVKUVNUWAUVHUVLYMTZBUHUVNUVHUWBBUVHUVLYM
      UVHUVLYMUVHUVLQZUVFYEUUDPZUVLQZYMAUVFUUEUVLWNUWCUWDUVLUWCYEIUUDUVHUVAUVLU
      VHUVBUVAUVSUVCUSZUKUVGUUEUVLWOWPUVHUVLWQVQYLUWEFUUCRYFUUCPZYHUWDYKUVLUWGY
      GUUDYEYFUUCWRWSUWGYJUVJYCUWGYIUVIGYFUUCHWGWTXAWBXBXCXDUVHYLUVLFRUVHUUSQZY
      LUVLUWHYLQZUVLYKYLYKUWHYHYKWQVAUWIUVJYJYCUWIUVIYIGUWIUUCYFHUWIUUDYGPZUUCY
      FPZUWIYEIYGUUDUVHUVAUUSYLUWFWHUWHYHYKXEUVGUUEUUSYLWNXFUWIUVFUUSUWJUWKTUVH
      UVFUUSYLAUVFUUEWOZWHUVHUUSYLWOUUCYFXGXCVBWTWTXAXIXDXJXKVPUVHUWBUVMBUVHYMY
      QUVLUVHYMYPYMULZYQUVHYPWFYMUWMTUVHYNYOUVHYESUVHYEISUWFUVHISVDZUUDSVDZUVHU
      VFUWOUWLUUCVEUSUUEUWNUWOTUVGIUUDSXLVAXIXMVHVIYPYMVJUSYPYMXNVNXOWAVBVQYTUV
      OUAUVJEYAUVJPZYBUVKYSUVNYAUVJEVRUWPYRUVMBUWPYDUVLYQYAUVJYCVSVTWAWBWCWDXDX
      JXPAIROUUBUUFULJUBIXQUSXSYQBUAEXRXT $.
  $}

  ${
    $d g m x y A $.  $d g m x y z F $.  $d g m y V $.
    frectfr.1 $e |- G = ( g e. _V |-> { x | ( E. m e. _om ( dom g = suc m /\
          x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) $.
    $( Lemma to connect transfinite recursion theorems with finite recursion.
       That is, given the conditions ` F Fn _V ` and ` A e. V ` on
       ` frec ( F , A ) ` , we want to be able to apply ~ tfri1d or ~ tfri2d ,
       and this lemma lets us satisfy hypotheses of those theorems.

       (Contributed by Jim Kingdon, 15-Aug-2019.) $)
    frectfr $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) ->
        A. y ( Fun G /\ ( G ` y ) e. _V ) ) $=
      ( cv cfv cvv wcel wal wa wfun wfn wceq vex cdm csuc com wrex cab wral a1i
      c0 wo simpl simpr frecabex ralrimivw fnmpt funfvex funfni sylancl funmpt2
      syl jctil alrimiv ) CKGLMNCOZDINZPZHQZBKZHLMNZPBVDVGVEVDHMRZVFMNVGVDEKZUA
      ZFKZUBSAKZVKVILGLNPFUCUDVJUHSVLDNPUIAUEZMNZEMUFVHVDVNEMVDACDVIFGMIVIMNVDE
      TUGVBVCUJVBVCUKULUMEMVMHMJUNUSBTVGMVFHVFHUOUPUQEMVMHJURUTVA $.
  $}

  ${
    $d A g m x y z $.  $d F g m x y z $.  $d V g m x y $.

    $( The function generated by finite recursive definition generation is a
       function on omega.  (Contributed by Jim Kingdon, 13-May-2020.) $)
    frecfnom $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) ->
        frec ( F , A ) Fn _om ) $=
      ( vg vm vx vy cv cfv cvv wcel wal wa wceq com con0 cres wfn eqid cdm csuc
      wrex c0 wo cab cmpt crecs cin cfrec frectfr tfri1d fnresin1 syl wb omsson
      wss sseqin2 mpbi reseq2i df-frec eqtr4i fneq12 mp2an sylib ) AICJKLAMBDLN
      ZEKEIZUAZFIZUBOGIZVIVGJCJLNFPUCVHUDOVJBLNUEGUFUGZUHZQPUIZRZVMSZCBUJZPSZVF
      VLQSVOVFHVLVKVLTGHABEFCVKDVKTUKULQPVLUMUNVNVPOVMPOZVOVQUOVNVLPRVPVMPVLPQU
      QVRUPPQURUSZUTGEFCBVAVBVSVMPVNVPVCVDVE $.
  $}

  ${
    freccl.a $e |- ( ph -> A e. S ) $.
    freccl.cl $e |- ( ( ph /\ z e. S ) -> ( F ` z ) e. S ) $.
    freccl.b $e |- ( ph -> B e. _om ) $.

    ${
      $d A f g m x y $.  $d A f m x y z $.  $d B x $.  $d F f g m x y $.
      $d F f m x y z $.  $d G f y $.  $d S f m x y z $.  $d f m ph x y z $.
      freccllem.g $e |- G = recs ( ( g e. _V |-> { x |
        ( E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
        \/ ( dom g = (/) /\ x e. A ) ) } ) ) $.
      $( Lemma for ~ freccl .  Just giving a name to a common expression to
         simplify the proof.  (Contributed by Jim Kingdon, 27-Mar-2022.) $)
      freccllem $p |- ( ph -> ( frec ( F , A ) ` B ) e. S ) $=
        ( cfv com cvv cv wceq wcel vy vf cfrec cres cdm csuc wa wrex c0 wo cmpt
        cab crecs df-frec reseq1i eqtr4i fveq1i fvres syl wfun funmpt a1i ordom
        word wf w3a vex simp2 simp3 wral ralrimiva 3ad2ant1 frecabcl dmeq fveq1
        eqeq1d fveq2d eleq2d anbi12d rexbidv anbi1d orbi12d abbidv eqid sylancr
        fvmptg eqeltrd cuni limom limuni eleq2i peano2 adantl sylan2br eleqtrdi
        wlim ax-mp tfrcl eqeltrid ) AEIDUCZOEJPUDZOZFEWTXAWTGQGRZUEZHRZUFZSZBRZ
        XEXCOZIOZTZUGZHPUHZXDUISZXHDTZUGZUJZBULZUKZUMZPUDXABGHIDUNJXTPNUOUPUQAX
        BEJOZFAEPTXBYASMEPJURUSAUAFUBJXSPENXSUTAGQXRVAVBPVDAVCVBAUARZPTZYBFUBRZ
        VEZVFZYDXSOZYDUEZXFSZXHXEYDOZIOZTZUGZHPUHZYHUISZXOUGZUJZBULZFYFYDQTYRFT
        YGYRSUBVGYFBCDFHIYDYBAYCYEVHAYCYEVIAYCCRIOFTZCFVJYEAYSCFLVKVLAYCDFTYEKV
        LVMZGYDXRYRQFXSXCYDSZXQYQBUUAXMYNXPYPUUAXLYMHPUUAXGYIXKYLUUAXDYHXFXCYDV
        NZVPUUAXJYKXHUUAXIYJIXEXCYDVOVQVRVSVTUUAXNYOXOUUAXDYHUIUUBVPWAWBWCXSWDW
        FWEYTWGYBPWHZTAYCYBUFPTZPUUCYBPWPPUUCSWIPWJWQZWKYCUUDAYBWLWMWNAEPUUCMUU
        EWOWRWGWS $.
    $}

    $d ph x z m $.  $d S x z m $.  $d F x z g m $.  $d A x z g m $.  $d B x $.
    $( Closure for finite recursion.  (Contributed by Jim Kingdon,
       27-Mar-2022.) $)
    freccl $p |- ( ph -> ( frec ( F , A ) ` B ) e. S ) $=
      ( vx vg vm cvv cv cdm csuc wceq cfv wcel wa com wrex c0 wo cab cmpt crecs
      eqid freccllem ) AJBCDEKLFKMKNZOZLNZPQJNZULUJRFRSTLUAUBUKUCQUMCSTUDJUEUFU
      GZGHIUNUHUI $.
  $}

  ${
    ${
      $d A f g m x y $.  $d A f k m w x y $.  $d F f g m x y $.
      $d F f k m w x y z $.  $d G f k y $.  $d S f k m w x y z $.  $d m x z $.
      frecfcllem.g $e |- G = recs ( ( g e. _V |-> { x |
        ( E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
        \/ ( dom g = (/) /\ x e. A ) ) } ) ) $.
      $( Lemma for ~ frecfcl .  Just giving a name to a common expression to
         simplify the proof.  (Contributed by Jim Kingdon, 30-Mar-2022.) $)
      frecfcllem $p |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S )
          -> frec ( F , A ) : _om --> S ) $=
        ( vk vy vf vw cv cfv wcel wa com cvv wceq wral cres cfrec cdm csuc wrex
        wf c0 cab cmpt wfun funmpt a1i word ordom w3a simp2 simp3 simp1ll fveq2
        wo vex eleq1d cbvralv sylib simp1lr frecabcl eqeq1d fveq1 fveq2d eleq2d
        dmeq anbi12d rexbidv anbi1d orbi12d abbidv fvmptg sylancr eqeltrd limom
        eqid cuni limuni ax-mp eleq2i peano2 sylbir adantl biimpi tfrcldm tfrcl
        wlim jca ralrimiva wb crecs tfrfun funeqi mpbir ffvresb df-frec reseq1i
        sylibr eqtr4i feq1i ) BNZGOZDPZBDUAZCDPZQZRDHRUBZUGZRDGCUCZUGXLJNZHUDPZ
        XPHODPZQZJRUAZXNXLXSJRXLXPRPZQZXQXRYBKDLHESENZUDZFNZUEZTZANZYEYCOZGOZPZ
        QZFRUFZYDUHTZYHCPZQZVAZAUIZUJZRXPIYSUKYBESYRULUMZRUNYBUOUMZYBKNZRPZUUBD
        LNZUGZUPZUUDYSOZUUDUDZYFTZYHYEUUDOZGOZPZQZFRUFZUUHUHTZYOQZVAZAUIZDUUFUU
        DSPUURDPUUGUURTLVBUUFAMCDFGUUDUUBYBUUCUUEUQYBUUCUUEURUUFXJMNZGOZDPZMDUA
        XJXKYAUUCUUEUSXIUVABMDXGUUSTXHUUTDXGUUSGUTVCVDVEXJXKYAUUCUUEVFVGZEUUDYR
        UURSDYSYCUUDTZYQUUQAUVCYMUUNYPUUPUVCYLUUMFRUVCYGUUIYKUULUVCYDUUHYFYCUUD
        VLZVHUVCYJUUKYHUVCYIUUJGYEYCUUDVIVJVKVMVNUVCYNUUOYOUVCYDUUHUHUVDVHVOVPV
        QYSWBVRVSUVBVTZUUBRWCZPZUUBUERPZYBUVGUUCUVHRUVFUUBRWMRUVFTWARWDWEZWFUUB
        WGWHWIZYAXPUVFPZXLYAUVKRUVFXPUVIWFWJWIZWKYBKDLHYSRXPIYTUUAUVEUVJUVLWLWN
        WOHUKZXNXTWPUVMYSWQZUKYSWRHUVNIWSWTJRDHXAWEXDRDXOXMXOUVNRUBXMAEFGCXBHUV
        NRIXCXEXFXD $.
    $}

    $d A g m x $.  $d F g m x $.  $d F m x z $.  $d S m x z $.
    $( Finite recursion yields a function on the natural numbers.  (Contributed
       by Jim Kingdon, 30-Mar-2022.) $)
    frecfcl $p |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S )
        -> frec ( F , A ) : _om --> S ) $=
      ( vx vg vm cvv cv cdm csuc wceq cfv wcel wa com wrex c0 wo cab cmpt crecs
      eqid frecfcllem ) EABCFGDFHFIZJZGIZKLEIZUGUEMDMNOGPQUFRLUHBNOSETUAUBZUIUC
      UD $.
  $}

  ${
    $d A f g m x $.  $d A f m w x y $.  $d B f g m x $.  $d B f m w x y $.
    $d F f g m x $.  $d F f m w x y z $.  $d G f g m x $.  $d G f m w x y $.
    $d S f m w x y z $.
    frecsuclem.g $e |- G = ( g e. _V |-> { x | (
      E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
      \/ ( dom g = (/) /\ x e. A ) ) } ) $.
    $( Lemma for ~ frecsuc .  Just giving a name to a common expression to
       simplify the proof.  (Contributed by Jim Kingdon, 29-Mar-2022.) $)
    frecsuclem $p |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S /\ B e. _om ) ->
        ( frec ( F , A ) ` suc B ) = ( F ` ( frec ( F , A ) ` B ) ) ) $=
      ( vw cv cfv wcel com wceq wa c0 cvv 3ad2ant3 vy wral w3a csuc cfrec crecs
      vf cres wrex cdm wo cab df-frec recseq ax-mp reseq1i eqtr4i fveq1i peano2
      cmpt fvres syl eqtrid eqid wfun funmpt2 a1i word ordom wf vex simp2 simp3
      simp11 fveq2 eleq1d sylib simp12 frecabcl dmeq eqeq1d fveq1 fveq2d eleq2d
      cbvralv anbi12d rexbidv anbi1d orbi12d abbidv fvmptg syl2anc eqeltrd cuni
      wlim limom limuni eleq2i sylbir adantl tfrcldm tfr2a tfrfun resfunexg wss
      eqtrd frecfcl feq1i 3adant3 ordelsuc mpan2 mpbid fssres2 simp1 abeq2d fdm
      wb wn wne peano3 eqnetrd neneqd intnanrd biorf orcom bitrdi bitrd 3bitr2d
      suc11g eqcom ceqsrexbv 3anibar eqrdv sucidg eqtr4d ) BLZHMZENZBEUBZCENZDO
      NZUCZDUDZHCUEZMZDIUFZUUCUHZMZHMZDUUDMZHMUUBAUUEUUIYSYTUUAALZUUENZUUKUUINZ
      UUBUULGLZDPZUUKUUNUUGMZHMZNZQZGOUIZUUAUUMQUUBUULUUGUJZUUNUDZPZUURQZGOUIZU
      VARPZUUKCNZQZUKZUVEUUTUUBUVIAUUEUUBUUEUUGIMZUVIAULZUUBUUEUUCUUFMZUVJUUAYS
      UUEUVLPYTUUAUUEUUCUUFOUHZMZUVLUUCUUDUVMUUDFSFLZUJZUVBPZUUKUUNUVOMZHMZNZQZ
      GOUIZUVPRPZUVGQZUKZAULZUTZUFZOUHUVMAFGHCUMUUFUWHOIUWGPUUFUWHPJIUWGUNUOUPU
      QZURUUAUUCONZUVNUVLPDUSZUUCOUUFVAVBVCTUUBUUCUUFUJNUVLUVJPUUBUAEUGUUFIOUUC
      UUFVDZIVEUUBFSUWFIJVFVGOVHZUUBVIVGUUBUALZONZUWNEUGLZVJZUCZUWPIMZUWPUJZUVB
      PZUUKUUNUWPMZHMZNZQZGOUIZUWTRPZUVGQZUKZAULZEUWRUWPSNZUXJENUWSUXJPUXKUWRUG
      VKVGUWRAKCEGHUWPUWNUUBUWOUWQVLUUBUWOUWQVMUWRYSKLZHMZENZKEUBZYSYTUUAUWOUWQ
      VNYRUXNBKEYPUXLPYQUXMEYPUXLHVOVPWEZVQYSYTUUAUWOUWQVRVSZFUWPUWFUXJSEIUVOUW
      PPZUWEUXIAUXRUWBUXFUWDUXHUXRUWAUXEGOUXRUVQUXAUVTUXDUXRUVPUWTUVBUVOUWPVTZW
      AUXRUVSUXCUUKUXRUVRUXBHUUNUVOUWPWBWCWDWFWGUXRUWCUXGUVGUXRUVPUWTRUXSWAWHWI
      WJJWKWLUXQWMUWNOWNZNZUWNUDONZUUBUYAUWOUYBOUXTUWNOWOOUXTPWPOWQUOZWRUWNUSWS
      WTUUAYSUUCUXTNZYTUUAUWJUYDUWKOUXTUUCUYCWRVQTXAUUCUUFIUWLXBVBXFUUBUUGSNZUV
      KENUVJUVKPUUBUUFVEZUWJUYEUYFUUBIXCVGUUAYSUWJYTUWKTZUUFUUCOXDWLUUBAKCEGHUU
      GUUCUYGUUBOEUVMVJZUUCOXEZUUCEUUGVJZYSYTUYHUUAYSYTQOEUUDVJUYHBCEHXGOEUUDUV
      MUWIXHVQXIUUBUUAUYIYSYTUUAVMUUAYSUUAUYIXQZYTUUAUWMUYKVIDOOXJXKTXLOEUUCUUF
      XMWLZUUBYSUXOYSYTUUAXNUXPVQYSYTUUAVLVSFUUGUWFUVKSEIUVOUUGPZUWEUVIAUYMUWBU
      VEUWDUVHUYMUWAUVDGOUYMUVQUVCUVTUURUYMUVPUVAUVBUVOUUGVTZWAUYMUVSUUQUUKUYMU
      VRUUPHUUNUVOUUGWBWCWDWFWGUYMUWCUVFUVGUYMUVPUVARUYNWAWHWIWJJWKWLXFXOUUBUVE
      UVHUVEUKZUVIUUBUVHXRUVEUYOXQUUBUVFUVGUUBUVARUUBUVAUUCRUUBUYJUVAUUCPUYLUUC
      EUUGXPVBZUUAYSUUCRXSYTDXTTYAYBYCUVHUVEYDVBUVHUVEYEYFUUBUVDUUSGOUUBUVCUUOU
      URUUBUVCDUUNPZUUOUUBUVCUUCUVBPZUYQUUBUVAUUCUVBUYPWAUUAYSUYRUYQXQZYTUUAUUN
      SNUYSGVKDUUNOSYIXKTYGDUUNYJYFWHWGYHUURUUMGDOUUOUUQUUIUUKUUOUUPUUHHUUNDUUG
      VOWCWDYKYFYLYMUUBUUHUUJHUUAYSUUHUUJPYTUUAUUHDUUFMZUUJUUADUUCNUUHUYTPDOYND
      UUCUUFVAVBUUAUUJDUVMMUYTDUUDUVMUWIURDOUUFVAVCYOTWCXF $.
  $}

  ${
    $d A f g m x y $.  $d B g m x $.  $d F f g m n x y $.  $d F m x z $.
    $d S m x z $.
    $( The successor value resulting from finite recursive definition
       generation.  (Contributed by Jim Kingdon, 31-Mar-2022.) $)
    frecsuc $p |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S /\ B e. _om ) ->
        ( frec ( F , A ) ` suc B ) = ( F ` ( frec ( F , A ) ` B ) ) ) $=
      ( vx vg vm vn vy cvv cv wceq cfv wcel wa com wrex wo cab vf cdm csuc cmpt
      c0 dmeq eqeq1d fveq1 fveq2d eleq2d anbi12d rexbidv anbi1d orbi12d cbvmptv
      abbidv eleq1 anbi2d cbvabv mpteq2i suceq eqeq2d fveq2 orbi1i abbii 3eqtri
      cbvrexv frecsuclem ) FABCDGHEUAKUALZUBZILZUCZMZJLZVKVINZENZOZPZIQRZVJUEMZ
      VNBOZPZSZJTZUDZWEGKGLZUBZVLMZVNVKWFNZENZOZPZIQRZWGUEMZWAPZSZJTZUDGKWHFLZW
      JOZPZIQRZWNWRBOZPZSZFTZUDGKWGHLZUCZMZWRXFWFNZENZOZPZHQRZXCSZFTZUDUAGKWDWQ
      VIWFMZWCWPJXPVSWMWBWOXPVRWLIQXPVMWHVQWKXPVJWGVLVIWFUFZUGXPVPWJVNXPVOWIEVK
      VIWFUHUIUJUKULXPVTWNWAXPVJWGUEXQUGUMUNUPUOGKWQXEWPXDJFVNWRMZWMXAWOXCXRWLW
      TIQXRWKWSWHVNWRWJUQURULXRWAXBWNVNWRBUQURUNUSUTGKXEXOXDXNFXAXMXCWTXLIHQVKX
      FMZWHXHWSXKXSVLXGWGVKXFVAVBXSWJXJWRXSWIXIEVKXFWFVCUIUJUKVGVDVEUTVFVH $.
  $}

  ${
    frecrdg.1 $e |- ( ph -> F Fn _V ) $.
    frecrdg.2 $e |- ( ph -> A e. V ) $.
    frecrdg.inc $e |- ( ph -> A. x x C_ ( F ` x ) ) $.
    $d A x y z $.  $d F x y z $.  $d V x $.  $d ph x y $.
    $( Transfinite recursion restricted to omega.

       Given a suitable characteristic function, ~ df-frec produces the same
       results as ~ df-irdg restricted to ` _om ` .

       Presumably the theorem would also hold if ` F Fn _V ` were changed to
       ` A. z ( F `` z ) e. _V ` .  (Contributed by Jim Kingdon,
       29-Aug-2019.) $)
    frecrdg $p |- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` _om ) ) $=
      ( vz com cfv cvv wcel wfn syl con0 wceq c0 fveq2 adantr vy cfrec crdg wal
      cv vex funfvex funfni mpan2 alrimiv frecfnom syl2anc wss rdgifnon2 omsson
      cres fnssres sylancl csuc eqeq12d frec0g peano1 fvres ax-mp eqtrid eqtr4d
      rdg0g wi simpr ad2antlr eqtrd fveq2d jca w3a wral simp1 ralv sylibr simp2
      wa elexd simp3 frecsuc syl3anc 3expa sylan nnon rdgisucinc 3eqtr4d peano2
      ex expcom finds2 impcom eqfnfvd ) ABJDCUBZDCUCZJUPZAIUEZDKLMZIUDZCEMZWPJN
      ADLNZXAFXCWTIXCWSLMWTIUFWTLWSDWSDUGUHUIUJOZGICDEUKULAWQPNZJPUMWRJNAXAXBXE
      XDGICDEUNULUOPJWQUQURBUEZJMAXFWPKZXFWRKZQZXIRWPKZRWRKZQUAUEZWPKZXLWRKZQZX
      LUSZWPKZXPWRKZQZABUAXFRQXGXJXHXKXFRWPSXFRWRSUTXFXLQXGXMXHXNXFXLWPSXFXLWRS
      UTXFXPQXGXQXHXRXFXPWPSXFXPWRSUTAXJCXKAXBXJCQGCDEVAOAXKRWQKZCRJMXKXTQVBRJW
      QVCVDAXBXTCQGCEDVGOVEVFAXLJMZXOXSVHAYAVTZXOXSYBXOVTZXQXPWQKZXRYCXMDKZXLWQ
      KZDKZXQYDYCXMYFDYCXMXNYFYBXOVIYAXNYFQAXOXLJWQVCVJVKVLYBXQYEQZXOAXAXBVTYAY
      HAXAXBXDGVMXAXBYAYHXAXBYAVNZWTILVOZCLMYAYHYIXAYJXAXBYAVPWTIVQVRYICEXAXBYA
      VSWAXAXBYAWBICXLLDWCWDWEWFTYBYDYGQXOYBBCXLDEAXCYAFTAXBYAGTYBYAXLPMAYAVIXL
      WGOAXFXFDKUMBUDYAHTWHTWIYAXRYDQZAXOYAXPJMYKXLWJXPJWQVCOVJVFWKWLWMWNWO $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordinal arithmetic
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c 1o $. $( The ordinal number 1 $)
  $c 2o $. $( The ordinal number 2 $)
  $c 3o $. $( The ordinal number 3 $)
  $c 4o $. $( The ordinal number 4 $)
  $c +o $.  $( Ordinal addition operation (plus subscript o) $)
  $c .o $.  $( Ordinal multiplication operation (center dot subscript o) $)
  $c ^oi $.  $( Ordinal exponentiation operation (caret subscript oi) $)

  $( Extend the definition of a class to include the ordinal number 1. $)
  c1o $a class 1o $.

  $( Extend the definition of a class to include the ordinal number 2. $)
  c2o $a class 2o $.

  $( Extend the definition of a class to include the ordinal number 3. $)
  c3o $a class 3o $.

  $( Extend the definition of a class to include the ordinal number 4. $)
  c4o $a class 4o $.

  $( Extend the definition of a class to include the ordinal addition
     operation. $)
  coa $a class +o $.

  $( Extend the definition of a class to include the ordinal multiplication
     operation. $)
  comu $a class .o $.

  $( Extend the definition of a class to include the ordinal exponentiation
     operation. $)
  coei $a class ^oi $.

  $( Define the ordinal number 1.  (Contributed by NM, 29-Oct-1995.) $)
  df-1o $a |- 1o = suc (/) $.

  $( Define the ordinal number 2.  (Contributed by NM, 18-Feb-2004.) $)
  df-2o $a |- 2o = suc 1o $.

  $( Define the ordinal number 3.  (Contributed by Mario Carneiro,
     14-Jul-2013.) $)
  df-3o $a |- 3o = suc 2o $.

  $( Define the ordinal number 4.  (Contributed by Mario Carneiro,
     14-Jul-2013.) $)
  df-4o $a |- 4o = suc 3o $.

  ${
    $d x y z $.
    $( Define the ordinal addition operation.  (Contributed by NM,
       3-May-1995.) $)
    df-oadd $a |- +o = ( x e. On , y e. On |->
                ( rec ( ( z e. _V |-> suc z ) , x ) ` y ) ) $.

    $( Define the ordinal multiplication operation.  (Contributed by NM,
       26-Aug-1995.) $)
    df-omul $a |- .o = ( x e. On , y e. On |->
                ( rec ( ( z e. _V |-> ( z +o x ) ) , (/) ) ` y ) ) $.

    $( Define the ordinal exponentiation operation.

       This definition is similar to a conventional definition of
       exponentiation except that it defines ` (/) ^oi A ` to be ` 1o ` for all
       ` A e. On ` , in order to avoid having different cases for whether the
       base is ` (/) ` or not.

       We do not yet have an extensive development of ordinal exponentiation.
       For background on ordinal exponentiation without excluded middle, see
       Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu
       (2025), "Ordinal Exponentiation in Homotopy Type Theory",
       arXiv:2501.14542 , ~ https://arxiv.org/abs/2501.14542 which is
       formalized in the TypeTopology proof library at
       ~ https://ordinal-exponentiation-hott.github.io/ .

       (Contributed by Mario Carneiro, 4-Jul-2019.) $)
    df-oexpi $a |- ^oi = ( x e. On , y e. On |->
                 ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) ) $.
  $}

  $( Ordinal 1 is an ordinal number.  (Contributed by NM, 29-Oct-1995.) $)
  1on $p |- 1o e. On $=
    ( c1o c0 csuc con0 df-1o 0elon onsuci eqeltri ) ABCDEBFGH $.

  $( Ordinal 1 is a set.  (Contributed by BJ, 4-Jul-2022.) $)
  1oex $p |- 1o e. _V $=
    ( c1o con0 1on elexi ) ABCD $.

  $( Ordinal 2 is an ordinal number.  (Contributed by NM, 18-Feb-2004.)  (Proof
     shortened by Andrew Salmon, 12-Aug-2011.) $)
  2on $p |- 2o e. On $=
    ( c2o c1o csuc con0 df-2o 1on onsuci eqeltri ) ABCDEBFGH $.

  $( Ordinal two is not zero.  (Contributed by Scott Fenton, 17-Jun-2011.) $)
  2on0 $p |- 2o =/= (/) $=
    ( c2o c1o csuc c0 df-2o con0 wcel wne 1on nsuceq0g ax-mp eqnetri ) ABCZDEBF
    GMDHIBFJKL $.

  $( Ordinal 3 is an ordinal number.  (Contributed by Mario Carneiro,
     5-Jan-2016.) $)
  3on $p |- 3o e. On $=
    ( c3o c2o csuc con0 df-3o 2on onsuci eqeltri ) ABCDEBFGH $.

  $( Ordinal 3 is an ordinal class.  (Contributed by BTernaryTau,
     6-Jan-2025.) $)
  ord3 $p |- Ord 3o $=
    ( c3o 3on onordi ) ABC $.

  $( Ordinal 4 is an ordinal number.  (Contributed by Mario Carneiro,
     5-Jan-2016.) $)
  4on $p |- 4o e. On $=
    ( c4o c3o csuc con0 df-4o 3on onsuci eqeltri ) ABCDEBFGH $.

  $( Expanded value of the ordinal number 1.  (Contributed by NM,
     4-Nov-2002.) $)
  df1o2 $p |- 1o = { (/) } $=
    ( c1o c0 csuc csn df-1o suc0 eqtri ) ABCBDEFG $.

  $( Expanded value of the ordinal number 2.  (Contributed by Mario Carneiro,
     14-Aug-2015.) $)
  df2o3 $p |- 2o = { (/) , 1o } $=
    ( c2o c1o csuc csn cun c0 cpr df-2o df-suc df1o2 uneq1i df-pr eqtr4i 3eqtri
    ) ABCBBDZEZFBGZHBIPFDZOEQBROJKFBLMN $.

  $( Expanded value of the ordinal number 2.  (Contributed by NM,
     29-Jan-2004.) $)
  df2o2 $p |- 2o = { (/) , { (/) } } $=
    ( c2o c0 c1o cpr csn df2o3 df1o2 preq2i eqtri ) ABCDBBEZDFCJBGHI $.

  $( ` 2o ` is a set.  (Contributed by BJ, 6-Apr-2019.)  (Proof shortened by
     Zhi Wang, 19-Sep-2024.) $)
  2oex $p |- 2o e. _V $=
    ( c2o c0 c1o cpr cvv df2o3 wcel 0ex 1oex prexg mp2an eqeltri ) ABCDZEFBEGCE
    GMEGHIBCEEJKL $.

  $( Ordinal one is not equal to ordinal zero.  (Contributed by NM,
     26-Dec-2004.) $)
  1n0 $p |- 1o =/= (/) $=
    ( c1o c0 csn df1o2 0ex snnz eqnetri ) ABCBDBEFG $.

  $( Cartesian products with the singletons of ordinals 0 and 1 are disjoint.
     (Contributed by NM, 2-Jun-2007.) $)
  xp01disj $p |- ( ( A X. { (/) } ) i^i ( C X. { 1o } ) ) = (/) $=
    ( c0 c1o wne csn cxp cin wceq 1n0 necomi xpsndisj ax-mp ) CDEACFGBDFGHCIDCJ
    KACBDLM $.

  $( Cartesian products with the singletons of ordinals 0 and 1 are disjoint.
     (Contributed by Jim Kingdon, 11-Jul-2023.) $)
  xp01disjl $p |- ( ( { (/) } X. A ) i^i ( { 1o } X. C ) ) = (/) $=
    ( c0 c1o wne csn cin wceq cxp 1n0 necomi disjsn2 xpdisj1 mp2b ) CDECFZDFZGC
    HOAIPBIGCHDCJKCDLOPABMN $.

  $( Two ways to express that an ordinal class is positive.  (Contributed by
     NM, 21-Dec-2004.) $)
  ordgt0ge1 $p |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) ) $=
    ( word c0 wcel csuc wss c1o con0 0elon ordelsuc mpan df-1o sseq1i bitr4di
    wb ) ABZCADZCEZAFZGAFCHDPQSOICAHJKGRALMN $.

  $( An ordinal greater than or equal to 1 is nonzero.  (Contributed by Jim
     Kingdon, 26-Jun-2019.) $)
  ordge1n0im $p |- ( Ord A -> ( 1o C_ A -> A =/= (/) ) ) $=
    ( word c1o wss c0 wcel wne ordgt0ge1 ne0i biimtrrdi ) ABCADEAFAEGAHAEIJ $.

  $( Membership in ordinal one.  (Contributed by NM, 5-Jan-2005.) $)
  el1o $p |- ( A e. 1o <-> A = (/) ) $=
    ( c1o wcel c0 csn wceq df1o2 eleq2i 0ex elsn2 bitri ) ABCADEZCADFBLAGHADIJK
    $.

  $( Two ways to say that ` A ` is a nonzero number of the set ` B ` .
     (Contributed by Mario Carneiro, 21-May-2015.) $)
  dif1o $p |- ( A e. ( B \ 1o ) <-> ( A e. B /\ A =/= (/) ) ) $=
    ( c1o cdif wcel c0 csn wne wa df1o2 difeq2i eleq2i eldifsn bitri ) ABCDZEAB
    FGZDZEABEAFHIOQACPBJKLABFMN $.

  $( Closure of the pair swapping function on ` 2o ` .  (Contributed by Mario
     Carneiro, 27-Sep-2015.) $)
  2oconcl $p |- ( A e. 2o -> ( 1o \ A ) e. 2o ) $=
    ( c1o cdif c2o wcel c0 cpr wceq wo elpri difeq2 eqtrdi difid orim12i orcomd
    dif0 syl con0 cvv df2o3 1on difexg ax-mp elpr sylibr eleqtrrdi eleq2s ) BAC
    ZDEAFBGZDAUIEZUHUIDUJUHFHZUHBHZIZUHUIEUJAFHZABHZIZUMAFBJUPULUKUNULUOUKUNUHB
    FCBAFBKBPLUOUHBBCFABBKBMLNOQUHFBBREUHSEUABARUBUCUDUETUFTUG $.

  $( Ordinal zero is less than ordinal one.  (Contributed by NM,
     5-Jan-2005.) $)
  0lt1o $p |- (/) e. 1o $=
    ( c0 c1o wcel wceq eqid el1o mpbir ) ABCAADAEAFG $.

  $( Ordinal zero is less than ordinal two.  (Contributed by Jim Kingdon,
     31-Jul-2022.) $)
  0lt2o $p |- (/) e. 2o $=
    ( c0 c1o cpr c2o 0ex prid1 df2o3 eleqtrri ) AABCDABEFGH $.

  $( Ordinal one is less than ordinal two.  (Contributed by Jim Kingdon,
     31-Jul-2022.) $)
  1lt2o $p |- 1o e. 2o $=
    ( c1o c0 cpr c2o 1oex prid2 df2o3 eleqtrri ) ABACDBAEFGH $.

  $( Being an element of ordinal two implies being a subset of ordinal one.
     The converse is equivalent to excluded middle by ~ ss1oel2o .
     (Contributed by Jim Kingdon, 8-Aug-2022.) $)
  el2oss1o $p |- ( A e. 2o -> A C_ 1o ) $=
    ( c2o wcel c0 wceq c1o wo wss cpr elpri df2o3 eleq2s 0ss mpbiri eqimss jaoi
    sseq1 syl ) ABCADEZAFEZGZAFHZUAADFIBADFJKLSUBTSUBDFHFMADFQNAFOPR $.
  $( $j usage 'el2oss1o' avoids 'ax-coll' 'ax-sep' 'ax-bdsep' 'ax-nul' 'ax-pow'
     'ax-pr' 'ax-un' 'ax-setind' 'ax-iinf'; $)

  $( The characteristic function for ordinal addition is defined everywhere.
     (Contributed by Jim Kingdon, 27-Jul-2019.) $)
  oafnex $p |- ( x e. _V |-> suc x ) Fn _V $=
    ( cvv cv csuc cmpt vex sucex eqid fnmpti ) ABACZDZABKEZJAFGLHI $.

  ${
    $d x z $.
    sucinc.1 $e |- F = ( z e. _V |-> suc z ) $.
    $( Successor is increasing.  (Contributed by Jim Kingdon, 25-Jun-2019.) $)
    sucinc $p |- A. x x C_ ( F ` x ) $=
      ( cv cfv wss csuc sssucid cvv wcel wceq sucex suceq fvmptg mp2an sseqtrri
      vex ax-gen ) AEZTCFZGATTHZUATITJKUBJKUAUBLARZTUCMBTBEZHUBJJCUDTNDOPQS $.

    $d A z $.  $d B z $.
    $( Successor is increasing.  (Contributed by Jim Kingdon, 14-Jul-2019.) $)
    sucinc2 $p |- ( ( B e. On /\ A e. B ) ->
        ( F ` A ) C_ ( F ` B ) ) $=
      ( con0 wcel wa csuc cfv wss word syl wceq cvv sucexg suceq fvmptg syl2anc
      elex wi eloni ordsucss imp sssucid sstrdi onelon cv adantr 3sstr4d ) CFGZ
      BCGZHZBIZCIZBDJZCDJZUMUNCUOUKULUNCKZUKCLULURUACUBBCUCMUDCUEUFUMBFGZUPUNNZ
      CBUGUSBOGUNOGUTBFTBFPABAUHZIZUNOODVABQERSMUKUQUONZULUKCOGUOOGVCCFTCFPACVB
      UOOODVACQERSUIUJ $.
  $}

  ${
    $d x y z $.
    $( Functionality and domain of ordinal addition.  (Contributed by NM,
       26-Aug-1995.)  (Proof shortened by Mario Carneiro, 3-Jul-2019.) $)
    fnoa $p |- +o Fn ( On X. On ) $=
      ( vx vy vz con0 cvv csuc cmpt crdg cfv coa df-oadd wcel vex oafnex rdgexg
      cv ax-mp fnmpoi ) ABDDBPZCECPFGZAPZHIZJABCKSELUBELBMUASTEAMCNOQR $.

    $( Ordinal addition is a set.  (Contributed by Mario Carneiro,
       3-Jul-2019.) $)
    oaexg $p |- ( ( A e. V /\ B e. W ) -> ( A +o B ) e. _V ) $=
      ( vy vz vx cv cvv csuc cmpt crdg cfv wcel wal coa co vex oafnex con0 gen2
      rdgexg ax-mp df-oadd mpofvex mp3an1 ) EHZFIFHJKZGHZLMZINZEOGOACNBDNABPQIN
      UKGEUGINUKERUIUGUHIGRFSUBUCUAGETTUJABPICDGEFUDUEUF $.
  $}

  ${
    $d x A $.  $d x V $.
    $( The characteristic function for ordinal multiplication is defined
       everywhere.  (Contributed by Jim Kingdon, 23-Aug-2019.) $)
    omfnex $p |- ( A e. V -> ( x e. _V |-> ( x +o A ) ) Fn _V ) $=
      ( wcel cv coa cvv wral cmpt wfn vex oaexg mpan ralrimivw eqid fnmpt syl
      co ) BCDZAEZBFRZGDZAGHAGUAIZGJSUBAGTGDSUBAKTBGCLMNAGUAUCGUCOPQ $.
  $}

  ${
    $d x y z $.
    $( Functionality and domain of ordinal multiplication.  (Contributed by NM,
       26-Aug-1995.)  (Revised by Mario Carneiro, 3-Jul-2019.) $)
    fnom $p |- .o Fn ( On X. On ) $=
      ( vx vy vz con0 cv cvv coa co cmpt crdg cfv comu df-omul wcel vex 0ex wfn
      c0 omfnex ax-mp rdgexg fnmpoi ) ABDDBEZCFCEAEZGHIZRJKZLABCMUCFNUFFNBORUCU
      EFPUDFNUEFQAOCUDFSTUATUB $.

    $( Ordinal multiplication is a set.  (Contributed by Mario Carneiro,
       3-Jul-2019.) $)
    omexg $p |- ( ( A e. V /\ B e. W ) -> ( A .o B ) e. _V ) $=
      ( vy vz vx cv cvv coa co cmpt c0 crdg wcel wal comu vex ax-mp con0 omfnex
      cfv 0ex wfn rdgexg gen2 df-omul mpofvex mp3an1 ) EHZFIFHGHZJKLZMNUBZIOZEP
      GPACOBDOABQKIOUNGEUJIOUNERMUJULIUCUKIOULIUDGRFUKIUASUESUFGETTUMABQICDGEFU
      GUHUI $.

    $( Functionality and domain of ordinal exponentiation.  (Contributed by
       Mario Carneiro, 29-May-2015.)  (Revised by Mario Carneiro,
       3-Jul-2019.) $)
    fnoei $p |- ^oi Fn ( On X. On ) $=
      ( vx vy vz con0 cv cvv comu co cmpt c1o crdg cfv coei df-oexpi wcel elexi
      vex 1on omexg mp2an eqid fnmpti rdgexg ax-mp fnmpoi ) ABDDBEZCFCEZAEZGHZI
      ZJKLZMABCNUFFOUKFOBQJUFUJFJDRPCFUIUJUGFOUHFOUIFOCQAQUGUHFFSTUJUAUBUCUDUE
      $.

    $( Ordinal exponentiation is a set.  (Contributed by Mario Carneiro,
       3-Jul-2019.) $)
    oeiexg $p |- ( ( A e. V /\ B e. W ) -> ( A ^oi B ) e. _V ) $=
      ( vy vz vx cv cvv comu co cmpt c1o crdg cfv wcel wal coei vex con0 fnmpti
      1on elexi omexg mp2an eqid rdgexg ax-mp gen2 df-oexpi mpofvex mp3an1 ) EH
      ZFIFHZGHZJKZLZMNOZIPZEQGQACPBDPABRKIPUSGEUMIPUSESMUMUQIMTUBUCFIUPUQUNIPUO
      IPUPIPFSGSUNUOIIUDUEUQUFUAUGUHUIGETTURABRICDGEFUJUKUL $.
  $}

  ${
    $d x y z A $.  $d y z B $.
    $( Value of ordinal addition.  (Contributed by NM, 3-May-1995.)  (Revised
       by Mario Carneiro, 8-Sep-2013.) $)
    oav $p |- ( ( A e. On /\ B e. On ) ->
             ( A +o B ) = ( rec ( ( x e. _V |-> suc x ) , A ) ` B ) ) $=
      ( vy vz con0 wcel cvv cv csuc cmpt crdg cfv co wceq oafnex rdgexgg rdgeq2
      coa fveq1d fveq2 df-oadd ovmpog mpd3an3 ) BFGCFGCAHAIJKZBLZMZHGBCSNUGOBCU
      EFFAPQDEBCFFEIZUEDIZLZMUGSUHUFMHUIBOUHUJUFUIBUERTUHCUFUADEAUBUCUD $.

    $( Value of ordinal multiplication.  (Contributed by NM, 17-Sep-1995.)
       (Revised by Mario Carneiro, 23-Aug-2014.) $)
    omv $p |- ( ( A e. On /\ B e. On ) ->
     ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) ) $=
      ( vy vz con0 wcel cvv cv coa co cmpt crdg cfv comu wceq 0elon wfn omfnex
      c0 rdgexggg syl3an1 mp3an2 oveq2 mpteq2dv rdgeq1 syl fveq1d fveq2 df-omul
      ovmpog mpd3an3 ) BFGZCFGZCAHAIZBJKZLZTMZNZHGZBCOKUSPUMTFGZUNUTQUMUQHRVAUN
      UTABFSTCUQFFUAUBUCDEBCFFEIZAHUODIZJKZLZTMZNUSOVBURNHVCBPZVBVFURVGVEUQPVFU
      RPVGAHVDUPVCBUOJUDUETVEUQUFUGUHVBCURUIDEAUJUKUL $.
  $}

  ${
    $d x y z A $.  $d y z B $.
    $( Value of ordinal exponentiation.  (Contributed by Jim Kingdon,
       9-Jul-2019.) $)
    oeiv $p |- ( ( A e. On /\ B e. On ) -> ( A ^oi B ) =
        ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) $=
      ( vy vz con0 wcel cvv cv comu co cmpt c1o crdg cfv coei wceq 1on wfn syl
      wral vex omexg mpan ralrimivw eqid fnmpt rdgexggg syl3an1 mp3an2 mpteq2dv
      oveq2 rdgeq1 fveq1d fveq2 df-oexpi ovmpog mpd3an3 ) BFGZCFGZCAHAIZBJKZLZM
      NZOZHGZBCPKVEQUSMFGZUTVFRUSVCHSZVGUTVFUSVBHGZAHUAVHUSVIAHVAHGUSVIAUBVABHF
      UCUDUEAHVBVCHVCUFUGTMCVCFFUHUIUJDEBCFFEIZAHVADIZJKZLZMNZOVEPVJVDOHVKBQZVJ
      VNVDVOVMVCQVNVDQVOAHVLVBVKBVAJULUKMVMVCUMTUNVJCVDUODEAUPUQUR $.
  $}

  ${
    $d x A $.
    $( Addition with zero.  Proposition 8.3 of [TakeutiZaring] p. 57.
       (Contributed by NM, 3-May-1995.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    oa0 $p |- ( A e. On -> ( A +o (/) ) = A ) $=
      ( vx con0 wcel c0 coa co cvv cv csuc cmpt crdg cfv wceq 0elon mpan2 rdg0g
      oav eqtrd ) ACDZAEFGZEBHBIJKZALMZATECDUAUCNOBAERPACUBQS $.

    $( Ordinal multiplication with zero.  Definition 8.15 of [TakeutiZaring]
       p. 62.  (Contributed by NM, 17-Sep-1995.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    om0 $p |- ( A e. On -> ( A .o (/) ) = (/) ) $=
      ( vx con0 wcel c0 comu co cvv coa cmpt crdg cfv wceq 0elon omv mpan2 rdg0
      cv 0ex eqtrdi ) ACDZAEFGZEBHBRAIGJZEKLZEUAECDUBUDMNBAEOPEUCSQT $.

    $( Ordinal exponentiation with zero exponent.  Definition 8.30 of
       [TakeutiZaring] p. 67.  (Contributed by NM, 31-Dec-2004.)  (Revised by
       Mario Carneiro, 8-Sep-2013.) $)
    oei0 $p |- ( A e. On -> ( A ^oi (/) ) = 1o ) $=
      ( vx con0 wcel c0 coei co cvv cv comu cmpt c1o crdg wceq 0elon oeiv mpan2
      cfv 1on rdg0g ax-mp eqtrdi ) ACDZAEFGZEBHBIAJGKZLMRZLUCECDUDUFNOBAEPQLCDU
      FLNSLCUETUAUB $.
  $}

  ${
    $d x y z w A $.  $d x y w B $.
    $( Closure law for ordinal addition.  Proposition 8.2 of [TakeutiZaring]
       p. 57.  (Contributed by NM, 5-May-1995.)  (Constructive proof by Jim
       Kingdon, 26-Jul-2019.) $)
    oacl $p |- ( ( A e. On /\ B e. On ) -> ( A +o B ) e. On ) $=
      ( vz vw con0 wcel wa coa co cvv cv csuc cmpt crdg cfv oav wral wceq vex
      id suceq eqid sucex fvmpt ax-mp eleq1i ralbii onsuc mprgbir rdgon eqeltrd
      a1i ) AEFZBEFGABHIBCJCKZLZMZANOECABPUMDABUPUMTDKZUPOZEFZDEQZUMUTUQLZEFZDE
      USVBDEURVAEUQJFURVARDSZCUQUOVAJUPUNUQUAUPUBUQVCUCUDUEUFUGUQUHUIULUJUK $.

    $( Closure law for ordinal multiplication.  Proposition 8.16 of
       [TakeutiZaring] p. 57.  (Contributed by NM, 3-Aug-2004.)  (Constructive
       proof by Jim Kingdon, 26-Jul-2019.) $)
    omcl $p |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On ) $=
      ( vx vy con0 wcel wa comu co cvv cv coa cmpt c0 cfv omv 0elon a1i eqeltrd
      crdg wceq vex oacl oveq1 eqid fvmptg sylancr ancoms ralrimiva rdgon ) AEF
      ZBEFGABHIBCJCKZALIZMZNTOECABPUKDNBUNNEFUKQRUKDKZUNOZEFZDEUOEFZUKUQURUKGZU
      PUOALIZEUSUOJFUTEFUPUTUADUBUOAUCZCUOUMUTJEUNULUOALUDUNUEUFUGVASUHUIUJS $.

    $( Closure law for ordinal exponentiation.  (Contributed by Jim Kingdon,
       26-Jul-2019.) $)
    oeicl $p |- ( ( A e. On /\ B e. On ) -> ( A ^oi B ) e. On ) $=
      ( vx vy con0 wcel wa coei cvv comu cmpt c1o crdg cfv oeiv 1on a1i eqeltrd
      co cv wceq vex omcl oveq1 eqid fvmptg sylancr ancoms ralrimiva rdgon ) AE
      FZBEFGABHSBCICTZAJSZKZLMNECABOUKDLBUNLEFUKPQUKDTZUNNZEFZDEUOEFZUKUQURUKGZ
      UPUOAJSZEUSUOIFUTEFUPUTUADUBUOAUCZCUOUMUTIEUNULUOAJUDUNUEUFUGVARUHUIUJR
      $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Value of ordinal addition.  (Contributed by Mario Carneiro and Jim
       Kingdon, 12-Aug-2019.) $)
    oav2 $p |- ( ( A e. On /\ B e. On ) ->
        ( A +o B ) = ( A u. U_ x e. B suc ( A +o x ) ) ) $=
      ( vy con0 wcel wa cvv cv csuc cmpt crdg cfv ciun cun coa co wfn wceq oav
      oafnex rdgival mp3an1 onelon vex oaexg mpan2 sucexg syl suceq eqid fvmptg
      syl2anc adantr fveq2d eqtr3d sylan2 anassrs iuneq2dv uneq2d 3eqtr4d ) BEF
      ZCEFZGZCDHDIZJZKZBLZMZBACAIZVHMZVGMZNZOZBCPQBACBVJPQZJZNZOVGHRVBVCVIVNSDU
      AABCVGEUBUCDBCTVDVQVMBVDACVPVLVBVCVJCFZVPVLSZVCVRGVBVJEFZVSCVJUDVBVTGZVOV
      GMZVPVLVBWBVPSZVTVBVOHFZVPHFZWCVBVJHFWDAUEBVJEHUFUGZVBWDWEWFVOHUHUIDVOVFV
      PHHVGVEVOUJVGUKULUMUNWAVOVKVGDBVJTUOUPUQURUSUTVA $.

    $( Addition with successor.  Definition 8.1 of [TakeutiZaring] p. 56.
       (Contributed by NM, 3-May-1995.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    oasuc $p |- ( ( A e. On /\ B e. On ) ->
                ( A +o suc B ) = suc ( A +o B ) ) $=
      ( vx con0 wcel wa csuc coa co cun cv ciun wceq onsuc sylan2 df-suc iuneq1
      oav2 csn uneq2d ax-mp iunxun eqtri oveq2 suceq iunxsng eqtrid eqtrd unass
      syl adantl eqtr4di uneq1d eqtr4d wss sssucid ssequn1 mpbi eqtrdi ) ADEZBD
      EZFZABGZHIZABHIZVEGZJZVFVBVDACBACKZHIZGZLZJZVFJZVGVBVDAVKVFJZJZVMVBVDACVC
      VJLZJZVOVAUTVCDEVDVQMBNCAVCROVAVQVOMUTVAVPVNAVAVPVKCBSZVJLZJZVNVPCBVRJZVJ
      LZVTVCWAMVPWBMBPCVCWAVJQUACBVRVJUBUCVAVSVFVKCBVJVFDVHBMVIVEMVJVFMVHBAHUDV
      IVEUEUJUFTUGTUKUHAVKVFUIULVBVEVLVFCABRUMUNVEVFUOVGVFMVEUPVEVFUQURUS $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Value of ordinal multiplication.  (Contributed by Jim Kingdon,
       23-Aug-2019.) $)
    omv2 $p |- ( ( A e. On /\ B e. On ) ->
        ( A .o B ) = U_ x e. B ( ( A .o x ) +o A ) ) $=
      ( vy con0 wcel wa comu co c0 coa ciun cun cvv cmpt cfv wceq omv syl2anc
      cv crdg wfn omfnex 0elon rdgival mp3an2 sylan omexg omcl simpl oacl oveq1
      onelon eqid fvmptg fveq2d eqtr3d sylan2 anassrs iuneq2dv uneq2d uncom un0
      3eqtr4d eqtri eqtrdi ) BEFZCEFZGZBCHIZJACBATZHIZBKIZLZMZVNVICDNDTZBKIZOZJ
      UAZPZJACVKVSPZVRPZLZMZVJVOVGVRNUBZVHVTWDQZDBEUCWEJEFVHWFUDAJCVREUEUFUGDBC
      RVIVNWCJVIACVMWBVGVHVKCFZVMWBQZVHWGGVGVKEFZWHCVKUMVGWIGZVLVRPZVMWBWJVLNFV
      MEFZWKVMQBVKEEUHWJVLEFVGWLBVKUIVGWIUJVLBUKSDVLVQVMNEVRVPVLBKULVRUNUOSWJVL
      WAVRDBVKRUPUQURUSUTVAVDVOVNJMVNJVNVBVNVCVEVF $.
  $}

  $( Addition with successor.  Theorem 4I(A2) of [Enderton] p. 79.
     (Contributed by Mario Carneiro, 16-Nov-2014.) $)
  onasuc $p |- ( ( A e. On /\ B e. _om ) ->
                ( A +o suc B ) = suc ( A +o B ) ) $=
    ( com wcel con0 csuc coa co wceq nnon oasuc sylan2 ) BCDAEDBEDABFGHABGHFIBJ
    ABKL $.

  $( Addition with 1 is same as successor.  Proposition 4.34(a) of [Mendelson]
     p. 266.  (Contributed by NM, 29-Oct-1995.)  (Revised by Mario Carneiro,
     16-Nov-2014.) $)
  oa1suc $p |- ( A e. On -> ( A +o 1o ) = suc A ) $=
    ( con0 wcel c1o coa co c0 csuc df-1o oveq2i wceq peano1 onasuc mpan2 eqtrid
    com oa0 suceq syl eqtrd ) ABCZADEFZAGEFZHZAHZUAUBAGHZEFZUDDUFAEIJUAGPCUGUDK
    LAGMNOUAUCAKUDUEKAQUCARST $.

  $( 1 + 1 = 2 for ordinal numbers.  (Contributed by NM, 18-Feb-2004.) $)
  o1p1e2 $p |- ( 1o +o 1o ) = 2o $=
    ( c1o coa co csuc c2o con0 wcel wceq 1on oa1suc ax-mp df-2o eqtr4i ) AABCZA
    DZEAFGNOHIAJKLM $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Weak ordering property of ordinal addition.  (Contributed by Jim
       Kingdon, 27-Jul-2019.) $)
    oawordi $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                 ( A C_ B -> ( C +o A ) C_ ( C +o B ) ) ) $=
      ( vx con0 wcel w3a wss coa co wa cvv cv csuc cmpt cfv wceq jca oav syl ex
      crdg wfn oafnex a1i simpl3 simpl1 simpl2 simpr rdgss 3sstr4d ) AEFZBEFZCE
      FZGZABHZCAIJZCBIJZHUOUPKZADLDMNOZCUBZPZBVAPZUQURUSABUTCEUTLUCUSDUDUEULUMU
      NUPUFZULUMUNUPUGZULUMUNUPUHZUOUPUIUJUSUNULKUQVBQUSUNULVDVERDCASTUSUNUMKUR
      VCQUSUNUMVDVFRDCBSTUKUA $.
  $}

  ${
    $d a b c $.  $d a ph $.
    oawordriexmid.1 $e |- ( ( a e. On /\ b e. On /\ c e. On )
      -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) ) $.
    $( A weak ordering property of ordinal addition which implies excluded
       middle.  The property is proposition 8.7 of [TakeutiZaring] p. 59.
       Compare with ~ oawordi .  (Contributed by Jim Kingdon, 15-May-2022.) $)
    oawordriexmid $p |- ( ph \/ -. ph ) $=
      ( cv wss csuc wi con0 wcel wa c1o coa co wceq oveq2 sseq12d imbi2d oa1suc
      wral 1on 3expa expcom rgen rspcv mp2 adantr adantl sylibd onsucsssucexmid
      rgen2a ) ABCBFZCFZGZUMHZUNHZGZIBCJUMJKZUNJKZLZUOUMMNOZUNMNOZGZURMJKVAUOUM
      DFZNOZUNVENOZGZIZIZDJUAVAUOVDIZIZUBVJDJVAVEJKZVIUSUTVMVIEUCUDUEVJVLDMJVEM
      PZVIVKVAVNVHVDUOVNVFVBVGVCVEMUMNQVEMUNNQRSSUFUGVAVBUPVCUQUSVBUPPUTUMTUHUT
      VCUQPUSUNTUIRUJULUK $.
  $}

  $( An ordinal is less than or equal to its sum with another.  Part of
     Exercise 5 of [TakeutiZaring] p. 62.  (Contributed by NM, 6-Dec-2004.) $)
  oaword1 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( A +o B ) ) $=
    ( con0 wcel wa c0 coa co wceq oa0 adantr wss wi 0elon oawordi 3com13 mp3an3
    0ss mpi eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBRUAUBFCDZUF
    UGMZNUHUBUAUIFBAOPQST $.

  ${
    $d x A $.  $d x B $.
    $( Multiplication with successor.  Definition 8.15 of [TakeutiZaring]
       p. 62.  (Contributed by NM, 17-Sep-1995.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    omsuc $p |- ( ( A e. On /\ B e. On ) ->
                ( A .o suc B ) = ( ( A .o B ) +o A ) ) $=
      ( vx con0 wcel wa csuc comu co coa cun ciun wceq csn df-suc iuneq1 iunxun
      cv ax-mp omv2 eqtri oveq2 oveq1d uneq2d eqtrid adantl onsuc sylan2 uneq1d
      iunxsng 3eqtr4d omcl simpl wss oaword1 ssequn1 sylib syl2anc eqtrd ) ADEZ
      BDEZFZABGZHIZABHIZVEAJIZKZVFVBCVCACRZHIZAJIZLZCBVJLZVFKZVDVGVAVKVMMUTVAVK
      VLCBNZVJLZKZVMVKCBVNKZVJLZVPVCVQMVKVRMBOCVCVQVJPSCBVNVJQUAVAVOVFVLCBVJVFD
      VHBMVIVEAJVHBAHUBUCUJUDUEUFVAUTVCDEVDVKMBUGCAVCTUHVBVEVLVFCABTUIUKVBVEDEZ
      UTVGVFMZABULUTVAUMVSUTFVEVFUNVTVEAUOVEVFUPUQURUS $.
  $}

  $( Multiplication with successor.  Theorem 4J(A2) of [Enderton] p. 80.
     (Contributed by NM, 20-Sep-1995.)  (Revised by Mario Carneiro,
     14-Nov-2014.) $)
  onmsuc $p |- ( ( A e. On /\ B e. _om ) ->
                ( A .o suc B ) = ( ( A .o B ) +o A ) ) $=
    ( com wcel con0 csuc comu co coa wceq nnon omsuc sylan2 ) BCDAEDBEDABFGHABG
    HAIHJBKABLM $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Natural number arithmetic
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Addition with zero.  Theorem 4I(A1) of [Enderton] p. 79.  (Contributed by
     NM, 20-Sep-1995.) $)
  nna0 $p |- ( A e. _om -> ( A +o (/) ) = A ) $=
    ( com wcel con0 c0 coa co wceq nnon oa0 syl ) ABCADCAEFGAHAIAJK $.

  $( Multiplication with zero.  Theorem 4J(A1) of [Enderton] p. 80.
     (Contributed by NM, 20-Sep-1995.) $)
  nnm0 $p |- ( A e. _om -> ( A .o (/) ) = (/) ) $=
    ( com wcel con0 c0 comu co wceq nnon om0 syl ) ABCADCAEFGEHAIAJK $.

  $( Addition with successor.  Theorem 4I(A2) of [Enderton] p. 79.
     (Contributed by NM, 20-Sep-1995.)  (Revised by Mario Carneiro,
     14-Nov-2014.) $)
  nnasuc $p |- ( ( A e. _om /\ B e. _om ) ->
                ( A +o suc B ) = suc ( A +o B ) ) $=
    ( com wcel con0 csuc coa co wceq nnon onasuc sylan ) ACDAEDBCDABFGHABGHFIAJ
    ABKL $.

  $( Multiplication with successor.  Theorem 4J(A2) of [Enderton] p. 80.
     (Contributed by NM, 20-Sep-1995.)  (Revised by Mario Carneiro,
     14-Nov-2014.) $)
  nnmsuc $p |- ( ( A e. _om /\ B e. _om ) ->
                ( A .o suc B ) = ( ( A .o B ) +o A ) ) $=
    ( com wcel con0 csuc comu co coa wceq nnon onmsuc sylan ) ACDAEDBCDABFGHABG
    HAIHJAKABLM $.

  ${
    $d x y A $.  $d x B $.
    $( Addition to zero.  Remark in proof of Theorem 4K(2) of [Enderton] p. 81.
       (Contributed by NM, 20-Sep-1995.)  (Revised by Mario Carneiro,
       14-Nov-2014.) $)
    nna0r $p |- ( A e. _om -> ( (/) +o A ) = A ) $=
      ( vx vy c0 cv coa co wceq csuc oveq2 id eqeq12d con0 wcel 0elon oa0 ax-mp
      com peano1 nnasuc mpan suceq eqeq2d syl5ibcom finds ) DBEZFGZUFHDDFGZDHZD
      CEZFGZUJHZDUJIZFGZUMHZDAFGZAHBCAUFDHZUGUHUFDUFDDFJUQKLUFUJHZUGUKUFUJUFUJD
      FJURKLUFUMHZUGUNUFUMUFUMDFJUSKLUFAHZUGUPUFAUFADFJUTKLDMNUIODPQUJRNZUNUKIZ
      HZULUODRNVAVCSDUJTUAULVBUMUNUKUJUBUCUDUE $.

    $( Multiplication with zero.  Exercise 16 of [Enderton] p. 82.
       (Contributed by NM, 20-Sep-1995.)  (Revised by Mario Carneiro,
       15-Nov-2014.) $)
    nnm0r $p |- ( A e. _om -> ( (/) .o A ) = (/) ) $=
      ( vx vy c0 cv comu co wceq csuc oveq2 eqeq1d con0 0elon om0 ax-mp com coa
      wcel oveq1 oa0 eqtrdi peano1 nnmsuc mpan imbitrrid finds ) DBEZFGZDHDDFGZ
      DHZDCEZFGZDHZDUKIZFGZDHZDAFGZDHBCAUGDHUHUIDUGDDFJKUGUKHUHULDUGUKDFJKUGUNH
      UHUODUGUNDFJKUGAHUHUQDUGADFJKDLRZUJMDNOUMUPUKPRZULDQGZDHUMUTDDQGZDULDDQSU
      RVADHMDTOUAUSUOUTDDPRUSUOUTHUBDUKUCUDKUEUF $.

    $( Closure of addition of natural numbers.  Proposition 8.9 of
       [TakeutiZaring] p. 59.  (Contributed by NM, 20-Sep-1995.)  (Proof
       shortened by Andrew Salmon, 22-Oct-2011.) $)
    nnacl $p |- ( ( A e. _om /\ B e. _om ) -> ( A +o B ) e. _om ) $=
      ( vx vy com wcel coa co cv wceq oveq2 eleq1d imbi2d csuc nna0 ibir peano2
      wi c0 wa nnasuc imbitrrid expcom finds2 vtoclga impcom ) BEFAEFZABGHZEFZU
      GACIZGHZEFZRUGUIRCBEUJBJZULUIUGUMUKUHEUJBAGKLMULASGHZEFZADIZGHZEFZAUPNZGH
      ZEFZUGCDUJSJUKUNEUJSAGKLUJUPJUKUQEUJUPAGKLUJUSJUKUTEUJUSAGKLUGUOUGUNAEAOL
      PUGUPEFZURVARURVAUGVBTZUQNZEFUQQVCUTVDEAUPUALUBUCUDUEUF $.

    $( Closure of multiplication of natural numbers.  Proposition 8.17 of
       [TakeutiZaring] p. 63.  (Contributed by NM, 20-Sep-1995.)  (Proof
       shortened by Andrew Salmon, 22-Oct-2011.) $)
    nnmcl $p |- ( ( A e. _om /\ B e. _om ) -> ( A .o B ) e. _om ) $=
      ( vx vy wcel comu co cv wi wceq oveq2 eleq1d imbi2d c0 csuc nnm0 eqeltrdi
      com peano1 expcom coa nnacl adantr nnmsuc sylibrd finds2 vtoclga impcom
      wa ) BREAREZABFGZREZUJACHZFGZREZIUJULICBRUMBJZUOULUJUPUNUKRUMBAFKLMUOANFG
      ZREADHZFGZREZAUROZFGZREZUJCDUMNJUNUQRUMNAFKLUMURJUNUSRUMURAFKLUMVAJUNVBRU
      MVAAFKLUJUQNRAPSQUJURREZUTVCIUJVDUIZUTUSAUAGZREZVCUJUTVGIVDUTUJVGUSAUBTUC
      VEVBVFRAURUDLUETUFUGUH $.
  $}

  ${
    nncli.1 $e |- A e. _om $.
    nncli.2 $e |- B e. _om $.
    $( ` _om ` is closed under addition.  Inference form of ~ nnacl .
       (Contributed by Scott Fenton, 20-Apr-2012.) $)
    nnacli $p |- ( A +o B ) e. _om $=
      ( com wcel coa co nnacl mp2an ) AEFBEFABGHEFCDABIJ $.

    $( ` _om ` is closed under multiplication.  Inference form of ~ nnmcl .
       (Contributed by Scott Fenton, 20-Apr-2012.) $)
    nnmcli $p |- ( A .o B ) e. _om $=
      ( com wcel comu co nnmcl mp2an ) AEFBEFABGHEFCDABIJ $.
  $}

  ${
    $d x y A $.  $d x y z B $.
    $( Addition of natural numbers is commutative.  Theorem 4K(2) of [Enderton]
       p. 81.  (Contributed by NM, 6-May-1995.)  (Revised by Mario Carneiro,
       15-Nov-2014.) $)
    nnacom $p |- ( ( A e. _om /\ B e. _om ) -> ( A +o B ) = ( B +o A ) ) $=
      ( vx vy com wcel coa co wceq cv wi oveq1 oveq2 eqeq12d c0 csuc nna0 suceq
      syl nnasuc vz imbi2d nna0r eqtr4d wa peano2 sylan imbitrrid expcom finds2
      vtoclga imp ) AEFBEFZABGHZBAGHZIZUMCJZBGHZBUQGHZIZKUMUPKCAEUQAIZUTUPUMVAU
      RUNUSUOUQABGLUQABGMNUBUTOBGHZBOGHZIDJZBGHZBVDGHZIZVDPZBGHZBVHGHZIZUMCDUQO
      IZURVBUSVCUQOBGLUQOBGMNUQVDIURVEUSVFUQVDBGLUQVDBGMNUQVHIURVIUSVJUQVHBGLUQ
      VHBGMNUMVBBVCBUCBQUDUMVDEFZVGVKKVGVKUMVMUEZVEPZVFPZIVEVFRVNVIVOVJVPUMVMVI
      VOIZVMVHUQGHZVDUQGHZPZIZKVMVQKCBEUQBIZWAVQVMWBVRVIVTVOUQBVHGMWBVSVEIVTVOI
      UQBVDGMVSVERSNUBWAVHOGHZVDOGHZPZIVHUAJZGHZVDWFGHZPZIZVHWFPZGHZVDWKGHZPZIZ
      VMCUAVLVRWCVTWEUQOVHGMVLVSWDIVTWEIUQOVDGMVSWDRSNUQWFIZVRWGVTWIUQWFVHGMWPV
      SWHIVTWIIUQWFVDGMVSWHRSNUQWKIZVRWLVTWNUQWKVHGMWQVSWMIVTWNIUQWKVDGMVSWMRSN
      VMWCVHWEVMVHEFZWCVHIVDUFZVHQSVMWDVDIWEVHIVDQWDVDRSUDVMWFEFZWJWOKWJWOVMWTU
      EZWGPZWIPZIWGWIRXAWLXBWNXCVMWRWTWLXBIWSVHWFTUGXAWMWIIWNXCIVDWFTWMWIRSNUHU
      IUJUKULBVDTNUHUIUJUKUL $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x C $.
    $( Addition of natural numbers is associative.  Theorem 4K(1) of [Enderton]
       p. 81.  (Contributed by NM, 20-Sep-1995.)  (Revised by Mario Carneiro,
       15-Nov-2014.) $)
    nnaass $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                 ( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) ) $=
      ( vx vy com wcel coa co wceq wa cv wi oveq2 oveq2d eqeq12d c0 csuc nnasuc
      nnacl imbi2d syl adantl eqtr4d suceq sylan sylan2 eqtrd anassrs imbitrrid
      nna0 expcom finds2 vtoclga com12 3impia ) AFGZBFGZCFGZABHIZCHIZABCHIZHIZJ
      ZUSUQURKZVDVEUTDLZHIZABVFHIZHIZJZMVEVDMDCFVFCJZVJVDVEVKVGVAVIVCVFCUTHNVKV
      HVBAHVFCBHNOPUAVJUTQHIZABQHIZHIZJUTELZHIZABVOHIZHIZJZUTVORZHIZABVTHIZHIZJ
      ZVEDEVFQJZVGVLVIVNVFQUTHNWEVHVMAHVFQBHNOPVFVOJZVGVPVIVRVFVOUTHNWFVHVQAHVF
      VOBHNOPVFVTJZVGWAVIWCVFVTUTHNWGVHWBAHVFVTBHNOPVEVLUTVNVEUTFGZVLUTJABTZUTU
      KUBURVNUTJUQURVMBAHBUKOUCUDVEVOFGZVSWDMVSWDVEWJKZVPRZVRRZJVPVRUEWKWAWLWCW
      MVEWHWJWAWLJWIUTVOSUFUQURWJWCWMJUQURWJKZKWCAVQRZHIZWMWNWCWPJUQWNWBWOAHBVO
      SOUCWNUQVQFGWPWMJBVOTAVQSUGUHUIPUJULUMUNUOUP $.

    $( Distributive law for natural numbers (left-distributivity).  Theorem
       4K(3) of [Enderton] p. 81.  (Contributed by NM, 20-Sep-1995.)  (Revised
       by Mario Carneiro, 15-Nov-2014.) $)
    nndi $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                 ( A .o ( B +o C ) ) = ( ( A .o B ) +o ( A .o C ) ) ) $=
      ( vx vy com wcel coa co comu wceq wa cv wi oveq2 oveq2d eqeq12d c0 eqtr4d
      csuc imbi2d nna0 adantl nnmcl syl nnm0 adantr oveq1 nnasuc 3adant1 nnmsuc
      w3a nnacl sylan2 3impb eqtrd 3adant2 nnaass syl3an1 syl3an2 exp4b pm2.43a
      3exp com4r pm2.43i 3imp imbitrrid com3r impd finds2 vtoclga expdcom ) AFG
      ZBFGZCFGZABCHIZJIZABJIZACJIZHIZKZVOVMVNWAVMVNLZABDMZHIZJIZVRAWCJIZHIZKZNW
      BWANDCFWCCKZWHWAWBWIWEVQWGVTWIWDVPAJWCCBHOPWIWFVSVRHWCCAJOPQUAWHABRHIZJIZ
      VRARJIZHIZKABEMZHIZJIZVRAWNJIZHIZKZABWNTZHIZJIZVRAWTJIZHIZKZWBDEWCRKZWEWK
      WGWMXFWDWJAJWCRBHOPXFWFWLVRHWCRAJOPQWCWNKZWEWPWGWRXGWDWOAJWCWNBHOPXGWFWQV
      RHWCWNAJOPQWCWTKZWEXBWGXDXHWDXAAJWCWTBHOPXHWFXCVRHWCWTAJOPQWBWKVRRHIZWMWB
      WKVRXIWBWJBAJVNWJBKVMBUBUCPWBVRFGZXIVRKABUDZVRUBUESWBWLRVRHVMWLRKVNAUFUGP
      SWNFGZVMVNWSXENZVMVNXLXMVMVNXLXMWSXEVMVNXLULZWPAHIZWRAHIZKWPWRAHUHXNXBXOX
      DXPXNXBAWOTZJIZXOXNXAXQAJVNXLXAXQKVMBWNUIUJPVMVNXLXRXOKZVNXLLVMWOFGXSBWNU
      MAWOUKUNUOUPXNXDVRWQAHIZHIZXPXNXCXTVRHVMXLXCXTKVNAWNUKUQPVMVNXLXPYAKZVMVN
      XLYBNNVMVNXLVMYBVNVMXLVMYBNZNVMVNVMXLYCWBVMXLLZVMYBYDWBWQFGZVMYBAWNUDWBXJ
      YEVMYBXKVRWQAURUSUTVCVAVBVDVEVFSQVGVCVHVIVJVKVLVF $.

    $( Multiplication of natural numbers is associative.  Theorem 4K(4) of
       [Enderton] p. 81.  (Contributed by NM, 20-Sep-1995.)  (Revised by Mario
       Carneiro, 15-Nov-2014.) $)
    nnmass $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                 ( ( A .o B ) .o C ) = ( A .o ( B .o C ) ) ) $=
      ( vx vy com wcel comu co wceq wa cv wi oveq2 oveq2d eqeq12d c0 nnmcl nnm0
      coa imbi2d csuc syl sylan9eqr eqtr4d w3a oveq1 nnmsuc sylan 3impa 3adant1
      nndi syl3an2 3exp expd com34 pm2.43d eqtrd imbitrrid com3r finds2 vtoclga
      3imp impd expdcom ) AFGZBFGZCFGZABHIZCHIZABCHIZHIZJZVHVFVGVMVFVGKZVIDLZHI
      ZABVOHIZHIZJZMVNVMMDCFVOCJZVSVMVNVTVPVJVRVLVOCVIHNVTVQVKAHVOCBHNOPUAVSVIQ
      HIZABQHIZHIZJVIELZHIZABWDHIZHIZJZVIWDUBZHIZABWIHIZHIZJZVNDEVOQJZVPWAVRWCV
      OQVIHNWNVQWBAHVOQBHNOPVOWDJZVPWEVRWGVOWDVIHNWOVQWFAHVOWDBHNOPVOWIJZVPWJVR
      WLVOWIVIHNWPVQWKAHVOWIBHNOPVNWAQWCVNVIFGZWAQJABRZVISUCVGVFWCAQHIQVGWBQAHB
      SOASUDUEWDFGZVFVGWHWMMZVFVGWSWTVFVGWSWTWHWMVFVGWSUFZWEVITIZWGVITIZJWEWGVI
      TUGXAWJXBWLXCVFVGWSWJXBJZVNWQWSXDWRVIWDUHUIUJXAWLAWFBTIZHIZXCXAWKXEAHVGWS
      WKXEJVFBWDUHUKOVFVGWSXFXCJZVFVGWSXGMVFVGWSVGXGVFVGWSVGXGMVFVGWSKZVGXGXHVF
      WFFGVGXGBWDRAWFBULUMUNUOUPUQVCURPUSUNUTVDVAVBVEVC $.

    $( Multiplication with successor.  Exercise 16 of [Enderton] p. 82.
       (Contributed by NM, 21-Sep-1995.)  (Proof shortened by Andrew Salmon,
       22-Oct-2011.) $)
    nnmsucr $p |- ( ( A e. _om /\ B e. _om ) ->
                  ( suc A .o B ) = ( ( A .o B ) +o B ) ) $=
      ( vx vy com wcel csuc comu co coa wceq cv wi oveq2 id oveq12d eqeq12d syl
      c0 eqtr4d imbi2d peano2 nnm0 peano1 nnmcl mpan2 nna0 oveq1 peano2b nnmsuc
      wa sylanb nnaass syl3an3b syl3an1 3expb anidms oveq1d an42s nnacom nnasuc
      suceq ancoms 3eqtr4d oveq2d imbitrrid expcom finds2 vtoclga impcom ) BEFA
      EFZAGZBHIZABHIZBJIZKZVKVLCLZHIZAVQHIZVQJIZKZMVKVPMCBEVQBKZWAVPVKWBVRVMVTV
      OVQBVLHNWBVSVNVQBJVQBAHNWBOPQUAWAVLSHIZASHIZSJIZKVLDLZHIZAWFHIZWFJIZKZVLW
      FGZHIZAWKHIZWKJIZKZVKCDVQSKZVRWCVTWEVQSVLHNWPVSWDVQSJVQSAHNWPOPQVQWFKZVRW
      GVTWIVQWFVLHNWQVSWHVQWFJVQWFAHNWQOPQVQWKKZVRWLVTWNVQWKVLHNWRVSWMVQWKJVQWK
      AHNWROPQVKWCWDWEVKWCSWDVKVLEFZWCSKAUBVLUCRAUCTVKWDEFZWEWDKVKSEFWTUDASUEUF
      WDUGRTVKWFEFZWJWOMWJWOVKXAUKZWGVLJIZWIVLJIZKWGWIVLJUHXBWLXCWNXDVKWSXAWLXC
      KAUIZVLWFUJULXBWHAJIZWKJIZWHAWKJIZJIZWNXDXBXGXIKZXBVKXAXJXBWHEFZVKXAXJAWF
      UEZXAXKVKWKEFXJWFUIWHAWKUMUNUOUPUQXBWMXFWKJAWFUJURXBXDWHWFVLJIZJIZXIXBXDX
      NKZVKXAXAVKXOXBXAVKXOXBXKXAVKXOXLVKXKXAWSXOXEWHWFVLUMUNUOUPUSUQXBXHXMWHJX
      BAWFJIZGZWFAJIZGZXHXMXBXPXRKXQXSKAWFUTXPXRVBRAWFVAXAVKXMXSKWFAVAVCVDVETVD
      QVFVGVHVIVJ $.

    $( Multiplication of natural numbers is commutative.  Theorem 4K(5) of
       [Enderton] p. 81.  (Contributed by NM, 21-Sep-1995.)  (Proof shortened
       by Andrew Salmon, 22-Oct-2011.) $)
    nnmcom $p |- ( ( A e. _om /\ B e. _om ) -> ( A .o B ) = ( B .o A ) ) $=
      ( vx vy com wcel comu co wceq cv wi oveq1 oveq2 eqeq12d imbi2d csuc nnm0r
      c0 nnm0 coa eqtr4d wa nnmsucr nnmsuc ancoms imbitrrid finds2 vtoclga imp
      ex ) AEFBEFZABGHZBAGHZIZUKCJZBGHZBUOGHZIZKUKUNKCAEUOAIZURUNUKUSUPULUQUMUO
      ABGLUOABGMNOURRBGHZBRGHZIDJZBGHZBVBGHZIZVBPZBGHZBVFGHZIZUKCDUORIUPUTUQVAU
      ORBGLUORBGMNUOVBIUPVCUQVDUOVBBGLUOVBBGMNUOVFIUPVGUQVHUOVFBGLUOVFBGMNUKUTR
      VABQBSUAVBEFZUKVEVIKVEVIVJUKUBZVCBTHZVDBTHZIVCVDBTLVKVGVLVHVMVBBUCUKVJVHV
      MIBVBUDUENUFUJUGUHUI $.
  $}

  $( Distributive law for natural numbers (right-distributivity).  (Contributed
     by Jim Kingdon, 3-Dec-2019.) $)
  nndir $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
               ( ( A +o B ) .o C ) = ( ( A .o C ) +o ( B .o C ) ) ) $=
    ( com wcel w3a coa co comu wceq nndi 3coml nnacl nnmcom sylan2 ancoms 3impa
    wa 3adant2 3adant1 oveq12d 3eqtr3d ) ADEZBDEZCDEZFZCABGHZIHZCAIHZCBIHZGHZUG
    CIHZACIHZBCIHZGHUEUCUDUHUKJCABKLUCUDUEUHULJZUEUCUDRZUOUPUEUGDEUOABMCUGNOPQU
    FUIUMUJUNGUCUEUIUMJZUDUEUCUQCANPSUDUEUJUNJZUCUEUDURCBNPTUAUB $.

  ${
    $d x y A $.  $d x y B $.
    $( Membership is inherited by successors.  The reverse direction holds for
       all ordinals, as seen at ~ onsucelsucr , but the forward direction, for
       all ordinals, implies excluded middle as seen as ~ onsucelsucexmid .
       (Contributed by Jim Kingdon, 25-Aug-2019.) $)
    nnsucelsuc $p |- ( B e. _om -> ( A e. B <-> suc A e. suc B ) ) $=
      ( vx vy com wcel csuc cv wi c0 wceq eleq2 suceq eleq2d imbi12d pm2.21i wa
      noel wo a1i elsuci adantl simpl orim12d mpd vex sucex elsuc2 sylibr finds
      ex con0 nnon onsucelsucr syl impbid ) BEFZABFZAGZBGZFZACHZFZUSVBGZFZIAJFZ
      USJGZFZIADHZFZUSVIGZFZIZAVKFZUSVKGZFZIZURVAICDBVBJKZVCVFVEVHVBJALVRVDVGUS
      VBJMNOVBVIKZVCVJVEVLVBVIALVSVDVKUSVBVIMNOVBVKKZVCVNVEVPVBVKALVTVDVOUSVBVK
      MNOVBBKZVCURVEVAVBBALWAVDUTUSVBBMNOVFVHARPVMVQIVIEFVMVNVPVMVNQZVLUSVKKZSZ
      VPWBVJAVIKZSZWDVNWFVMAVIUAUBWBVJVLWEWCVMVNUCWEWCIWBAVIMTUDUEVKUSVIDUFUGUH
      UIUKTUJUQBULFVAURIBUMABUNUOUP $.

    $( Membership is inherited by successors.  The reverse direction holds for
       all ordinals, as seen at ~ onsucsssucr , but the forward direction, for
       all ordinals, implies excluded middle as seen as ~ onsucsssucexmid .
       (Contributed by Jim Kingdon, 25-Aug-2019.) $)
    nnsucsssuc $p |- ( ( A e. _om /\ B e. _om ) ->
        ( A C_ B <-> suc A C_ suc B ) ) $=
      ( vx vy com wcel wa wss csuc wi wceq sseq1 suceq sseq1d imbi12d c0 syl wb
      ordelsuc mpbid cv imbi2d wn peano3 neneqd peano2 0elnn ord mpd word nnord
      ordsucim cvv 0ex mpan 3syl a1d w3a simp3 simp1l simp1r syl2anc nnsucelsuc
      wo mpbird 3expia exp31 finds2 vtoclga con0 nnon onsucsssucr syl2an impbid
      imp ) AEFZBEFZGABHZAIZBIZHZVPVQVRWAJZVQCUAZBHZWCIZVTHZJZJVQWBJCAEWCAKZWGW
      BVQWHWDVRWFWAWCABLWHWEVSVTWCAMNOUBWGPBHZPIZVTHZJDUAZBHZWLIZVTHZJZWNBHZWNI
      ZVTHZJZVQCDWCPKZWDWIWFWKWCPBLXAWEWJVTWCPMNOWCWLKZWDWMWFWOWCWLBLXBWEWNVTWC
      WLMNOWCWNKZWDWQWFWSWCWNBLXCWEWRVTWCWNMNOVQWKWIVQPVTFZWKVQVTPKZUCXDVQVTPBU
      DUEVQXEXDVQVTEFXEXDVDBUFVTUGQUHUIVQBUJZVTUJZXDWKRZBUKZBULZPUMFXGXHUNPVTUM
      SUOUPTUQWLEFZVQWPWTXKVQGZWPWQWSXLWPWQURZWNVTFZWSXMWLBFZXNXMXOWQXLWPWQUSXM
      XKXFXOWQRXKVQWPWQUTZXMVQXFXKVQWPWQVAZXIQZWLBESVBVEXMVQXOXNRXQWLBVCQTXMWNE
      FZXGXNWSRXMXKXSXPWLUFQXMXFXGXRXJQWNVTESVBTVFVGVHVIVOVPAVJFXFWAVRJVQAVKXIA
      BVLVMVN $.

    $( Trichotomy for natural numbers.  (Contributed by Jim Kingdon,
       25-Aug-2019.) $)
    nntri3or $p |- ( ( A e. _om /\ B e. _om ) ->
        ( A e. B \/ A = B \/ B e. A ) ) $=
      ( vx vy com wcel wceq cv wi eleq2 eqeq2 eleq1 3orbi123d c0 csuc wo 3orass
      w3o sylibr syl imbi2d olc wa df-3or cvv elex elsuc2g biimtrrdi nnsucelsuc
      0elnn 3mix1 elsuci biimtrdi eqcom orbi2i biimpi orcomd olcd syl6 biimtrid
      jaao ex finds2 vtoclga impcom ) BEFAEFZABFZABGZBAFZRZVFACHZFZAVKGZVKAFZRZ
      IVFVJICBEVKBGZVOVJVFVPVLVGVMVHVNVIVKBAJVKBAKVKBALMUAVOANFZANGZNAFZRZADHZF
      ZAWAGZWAAFZRZAWAOZFZAWFGZWFAFZRZVFCDVKNGVLVQVMVRVNVSVKNAJVKNAKVKNALMVKWAG
      VLWBVMWCVNWDVKWAAJVKWAAKVKWAALMVKWFGVLWGVMWHVNWIVKWFAJVKWFAKVKWFALMVFVRVS
      PZVTAUJWKVQWKPVTWKVQUBVQVRVSQSTWAEFZVFWEWJIWEWBWCPZWDPWLVFUCWJWBWCWDUDWLW
      MWJVFWDWLWAUEFZWMWJIWAEUFWNWMWGWJAWAUEUGWGWHWIUKUHTVFWDWIWFAGZPZWJVFWDWFA
      OFWPWAAUIWFAULUMWPWGWHWIPZPWJWPWQWGWPWIWHWPWIWHPWOWHWIWFAUNUOUPUQURWGWHWI
      QSUSVAUTVBVCVDVE $.

    $( A trichotomy law for natural numbers.  (Contributed by Jim Kingdon,
       28-Aug-2019.) $)
    nntri2 $p |- ( ( A e. _om /\ B e. _om ) ->
        ( A e. B <-> -. ( A = B \/ B e. A ) ) ) $=
      ( com wcel wa wceq wo elirr eleq2 mtbii con2i en2lp imnani ioran sylanbrc
      wn w3o nntri3or 3orass sylib orcomd ord impbid2 ) ACDBCDEZABDZABFZBADZGZP
      ZUEUFPUGPUIUFUEUFAADUEAHABAIJKUEUGABLMUFUGNOUDUHUEUDUEUHUDUEUFUGQUEUHGABR
      UEUFUGSTUAUBUC $.
  $}

  ${
    $d A n $.  $d B n $.
    $( Given an element ` A ` of the union of a natural number ` B ` ,
       ` suc A ` is an element of ` B ` itself.  The reverse direction holds
       for all ordinals ( ~ sucunielr ).  The forward direction for all
       ordinals implies excluded middle ( ~ ordsucunielexmid ).  (Contributed
       by Jim Kingdon, 13-Mar-2022.) $)
    nnsucuniel $p |- ( B e. _om -> ( A e. U. B <-> suc A e. B ) ) $=
      ( vn wcel cuni csuc c0 wceq wi cv wrex noel uni0 unieq eleq2d wa wb mpbid
      com syl eleq2i mtbir mtbiri pm2.21d adantl ad2antll biimpa wtr word nnord
      simplrl ordtr vex unisuc sylib nnsucelsuc simplrr eleqtrrd rexlimdvaa imp
      ex nn0suc mpjaodan sucunielr impbid1 ) BSDZABEZDZAFZBDZVFBGHZVHVJIZBCJZFZ
      HZCSKZVKVLVFVKVHVJVKVHAGEZDZVRAGDALVQGAMUAUBVKVGVQABGNOUCUDUEVFVPVLVFVOVL
      CSVFVMSDZVOPPZVHVJVTVHPZVIVNBWAAVMDZVIVNDZWAAVNEZDZWBVTVHWEVOVHWEQVFVSVOV
      GWDABVNNOUFUGWAWDVMAWAVSWDVMHZVFVSVOVHUKZVSVMUHZWFVSVMUIWHVMUJVMULTVMCUMU
      NUOTORWAVSWBWCQWGAVMUPTRVFVSVOVHUQURVAUSUTCBVBVCABVDVE $.
  $}

  $( A trichotomy law for natural numbers.  (Contributed by Jim Kingdon,
     28-Aug-2019.) $)
  nntri1 $p |- ( ( A e. _om /\ B e. _om ) -> ( A C_ B <-> -. B e. A ) ) $=
    ( com wcel wa wss wn ssnel wceq wo w3o wi nntri3or df-3or biimpi orcomd ord
    syl word nnord ordelss sylan ex adantl eqimss a1i jaod syld impbid2 ) ACDZB
    CDZEZABFZBADZGZABHULUOABDZABIZJZUMULUPUQUNKZUOURLABMUSUNURUSURUNUSURUNJUPUQ
    UNNOPQRULUPUMUQUKUPUMLUJUKUPUMUKBSUPUMBTBAUAUBUCUDUQUMLULABUEUFUGUHUI $.

  $( A trichotomy law for natural numbers.  (Contributed by Jim Kingdon,
     15-May-2020.) $)
  nntri3 $p |- ( ( A e. _om /\ B e. _om ) ->
      ( A = B <-> ( -. A e. B /\ -. B e. A ) ) ) $=
    ( com wcel wa wceq wn wb elirr eleq2 mtbii con2i adantl simpl 2falsed simpr
    eleq1 jca 2thd nntri3or mpjao3dan ) ACDBCDEZABDZABFZUCGZBADZGZEZHUDUFUBUCEU
    DUHUCUDGZUBUDUCUDAADZUCAIZABAJKZLMUCUHGZUBUHUCUEUGNLMOUBUDEUDUHUBUDPUDUHUBU
    DUEUGULUDUJUFUKABAQKZRMSUBUFEUDUHUFUIUBUDUFUNLMUFUMUBUHUFUEUGPLMOABTUA $.

  $( A trichotomy law for natural numbers.  (Contributed by Jim Kingdon,
     15-Sep-2021.) $)
  nntri2or2 $p |- ( ( A e. _om /\ B e. _om ) -> ( A C_ B \/ B C_ A ) ) $=
    ( com wcel wa wss wo wceq con0 wi nnon adantl onelss syl orcd eqimss adantr
    imp olcd nntri3or mpjao3dan ) ACDZBCDZEZABDZABFZBAFZGABHZBADZUDUEEUFUGUDUEU
    FUDBIDZUEUFJUCUJUBBKLBAMNROUDUHEUFUGUHUFUDABPLOUDUIEUGUFUDUIUGUDAIDZUIUGJUB
    UKUCAKQABMNRSABTUA $.

  $( Equality of natural numbers is decidable.  Theorem 7.2.6 of [HoTT], p.
     (varies).  For the specific case where ` B ` is zero, see ~ nndceq0 .
     (Contributed by Jim Kingdon, 31-Aug-2019.) $)
  nndceq $p |- ( ( A e. _om /\ B e. _om ) -> DECID A = B ) $=
    ( com wcel wa wceq wn wo wdc w3o nntri3or elirr eleq2 mtbii olcd orc mtbiri
    con2i 3jaoi syl df-dc sylibr ) ACDBCDEZABFZUDGZHZUDIUCABDZUDBADZJUFABKUGUFU
    DUHUGUEUDUDUGUDAADUGALABAMNROUDUEPUHUEUDUDUHUDUHBBDBLABBMQROSTUDUAUB $.

  $( Set membership between two natural numbers is decidable.  (Contributed by
     Jim Kingdon, 6-Sep-2019.) $)
  nndcel $p |- ( ( A e. _om /\ B e. _om ) -> DECID A e. B ) $=
    ( com wcel wa wn wo wdc wceq w3o nntri3or orc elirr eleq1 mtbiri olcd en2lp
    imnani 3jaoi syl df-dc sylibr ) ACDBCDEZABDZUDFZGZUDHUCUDABIZBADZJUFABKUDUF
    UGUHUDUELUGUEUDUGUDBBDBMABBNOPUHUEUDUHUDBAQRPSTUDUAUB $.

  $( For natural numbers, inclusion is equivalent to membership or equality.
     (Contributed by Jim Kingdon, 16-Sep-2021.) $)
  nnsseleq $p |- ( ( A e. _om /\ B e. _om ) ->
      ( A C_ B <-> ( A e. B \/ A = B ) ) ) $=
    ( com wcel wa wss wceq wo wn nntri1 w3o nntri3or df-3or sylib orcomd sylbid
    ord word wi nnord adantl ordelss ex syl eqimss a1i jaod impbid ) ACDZBCDZEZ
    ABFZABDZABGZHZUKULBADZIUOABJUKUPUOUKUOUPUKUMUNUPKUOUPHABLUMUNUPMNOQPUKUMULU
    NUKBRZUMULSUJUQUIBTUAUQUMULBAUBUCUDUNULSUKABUEUFUGUH $.

  $( A natural number is a subset of another natural number if and only if it
     belongs to its successor.  (Contributed by Jim Kingdon, 22-Jul-2023.) $)
  nnsssuc $p |- ( ( A e. _om /\ B e. _om ) -> ( A C_ B <-> A e. suc B ) ) $=
    ( com wcel wa wss wceq wo csuc nnsseleq wb elsucg adantr bitr4d ) ACDZBCDZE
    ABFABDABGHZABIDZABJORQKPABCLMN $.

  $( Transitive law for natural numbers.  (Contributed by Jim Kingdon,
     22-Jul-2023.) $)
  nntr2 $p |- ( ( A e. _om /\ C e. _om ) ->
      ( ( A C_ B /\ B e. C ) -> A e. C ) ) $=
    ( com wcel wa wss wceq con0 nnon ad3antlr simpr simprr adantr ontr1 eqeltrd
    jca sylc simplrl syl2anc sseldd word simplr elnn nnord ordirr 3syl pm2.21dd
    wn w3o simpll nntri3or mpjao3dan ex ) ADEZCDEZFZABGZBCEZFZACEZUQUTFZABEZVAA
    BHZBAEZVBVCFZCIEZVCUSFVAUPVGUOUTVCCJKVFVCUSVBVCLVBUSVCUQURUSMZNQABCORVBVDFA
    BCVBVDLVBUSVDVHNPVBVEFZBBEZVAVIABBUQURUSVESVBVELUAVIBDEZBUBVJUIVBVKVEVBUSUP
    VKVHUOUPUTUCBCUDTZNBUEBUFUGUHVBUOVKVCVDVEUJUOUPUTUKVLABULTUMUN $.

  ${
    $d A x y z $.  $d B x y z $.
    $( If we remove a single element from a set with decidable equality then
       put it back in, we end up with the original set.  This strengthens
       ~ difsnss from subset to equality but the proof relies on equality being
       decidable.  (Contributed by Jim Kingdon, 17-Jun-2022.) $)
    dcdifsnid $p |- ( ( A. x e. A A. y e. A DECID x = y /\ B e. A ) ->
        ( ( A \ { B } ) u. { B } ) = A ) $=
      ( vz weq wdc wral wcel wa csn cdif cun wss cv wceq simpr syl simplr dcbid
      difsnss adantl wn velsn sylibr elun2 sylnibr eldifd elun1 wo simpll eqeq2
      wi equequ1 rspc2v syl2anc mpd exmiddc mpjaodan ex ssrdv eqssd ) ABFZGZBCH
      ACHZDCIZJZCDKZLZVHMZCVFVJCNVECDUAUBVGECVJVGEOZCIZVKVJIZVGVLJZVKDPZVMVOUCZ
      VNVOJZVKVHIZVMVQVOVRVNVOQEDUDZUEVKVHVIUFRVNVPJZVKVIIVMVTVKCVHVGVLVPSVTVOV
      RVNVPQVSUGUHVKVIVHUIRVNVOGZVOVPUJVNVEWAVEVFVLUKVNVLVFVEWAUMVGVLQVEVFVLSVD
      WAEBFZGABVKDCCAEFVCWBAEBUNTBOZDPWBVOWCDVKULTUOUPUQVOURRUSUTVAVB $.
  $}

  ${
    $d A x y $.  $d X x y $.
    $( Split a function into a single point and all the rest.  (Contributed by
       Stefan O'Rear, 27-Feb-2015.)  (Revised by Jim Kingdon, 29-Jan-2023.) $)
    fnsnsplitdc $p |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A )
      -> F = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) ) $=
      ( weq wdc wral wfn wcel w3a cres csn cdif cfv cop wceq fnresdm 3ad2ant2
      cun resundi dcdifsnid 3adant2 reseq2d wa fnressn uneq2d 3adant1 3eqtr3a
      eqtr3d ) ABFGBCHACHZDCIZECJZKZDCLZDDCEMZNZLZEEDOPMZTZULUKUODQUMCDRSUNDUQU
      PTZLURDUPLZTZUOUTDUQUPUAUNVACDUKUMVACQULABCEUBUCUDULUMVCUTQUKULUMUEVBUSUR
      CEDUFUGUHUIUJ $.
  $}

  ${
    $d F x y $.  $d X x y $.
    $( Restricting a function to a domain without one element of the domain of
       the function, and adding a pair of this element and the function value
       of the element results in the function itself, where equality is
       decidable.  (Contributed by AV, 2-Dec-2018.)  (Revised by Jim Kingdon,
       30-Jan-2023.) $)
    funresdfunsndc $p |- (
        ( A. x e. dom F A. y e. dom F DECID x = y /\ Fun F /\ X e. dom F )
        -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = F ) $=
      ( weq wdc cdm wral wfun wcel w3a cvv csn cdif cres cfv cop cun wceq wrel
      funrel resdmdfsn syl 3ad2ant2 uneq1d funfn fnsnsplitdc syl3an2b eqtr4d
      wfn ) ABEFBCGZHAUKHZCIZDUKJZKZCLDMZNOZDDCPQMZRCUKUPNOZURRZCUOUQUSURUMULUQ
      USSZUNUMCTVACUACDUBUCUDUEUMULCUKUJUNCUTSCUFABUKCDUGUHUI $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( If we remove a single element from a natural number then put it back in,
       we end up with the original natural number.  This strengthens ~ difsnss
       from subset to equality but the proof relies on equality being
       decidable.  (Contributed by Jim Kingdon, 31-Aug-2021.) $)
    nndifsnid $p |- ( ( A e. _om /\ B e. A ) ->
        ( ( A \ { B } ) u. { B } ) = A ) $=
      ( vx vy com wcel cv wceq wdc wral csn cdif cun elnn expcom anim12d nndceq
      wa syl6 ralrimivv dcdifsnid sylan ) AEFZCGZDGZHIZDAJCAJBAFABKZLUGMAHUCUFC
      DAAUCUDAFZUEAFZRUDEFZUEEFZRUFUCUHUJUIUKUHUCUJUDANOUIUCUKUEANOPUDUEQSTCDAB
      UAUB $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C x $.

    $( Ordering property of addition.  Proposition 8.4 of [TakeutiZaring]
       p. 58, limited to natural numbers.  (Contributed by NM, 3-Feb-1996.)
       (Revised by Mario Carneiro, 15-Nov-2014.) $)
    nnaordi $p |- ( ( B e. _om /\ C e. _om ) ->
                 ( A e. B -> ( C +o A ) e. ( C +o B ) ) ) $=
      ( vx vy com wcel coa co wa wi wceq oveq2 eleq12d c0 csuc syl syl2anc con0
      nnon cv imbi2d simpr elnn ancoms nna0 adantr 3eltr4d wb simprl nnsucelsuc
      simpl nnacl adantl oasuc ad2antrl bitr4d biimpd finds2 vtoclga imp nnacom
      ex adantrr 3eltr3d 3impb 3com12 3expia ) BFGZCFGZABGZCAHIZCBHIZGZVJVIVKVN
      VJVIVKVNVJVIVKJZJZACHIZBCHIZVLVMVJVOVQVRGZVOADUAZHIZBVTHIZGZKVOVSKDCFVTCL
      ZWCVSVOWDWAVQWBVRVTCAHMVTCBHMNUBWCAOHIZBOHIZGAEUAZHIZBWGHIZGZAWGPZHIZBWKH
      IZGZVODEVTOLWAWEWBWFVTOAHMVTOBHMNVTWGLWAWHWBWIVTWGAHMVTWGBHMNVTWKLWAWLWBW
      MVTWKAHMVTWKBHMNVOABWEWFVIVKUCVOAFGZWEALVKVIWOABUDUEZAUFQVIWFBLVKBUFUGUHW
      GFGZVOWJWNKWQVOJZWJWNWRWJWHPZWIPZGZWNWRWIFGZWJXAUIWRVIWQXBWQVIVKUJWQVOULB
      WGUMRWHWIUKQWRWLWSWMWTWRASGZWGSGZWLWSLWRWOXCVOWOWQWPUNATQWQXDVOWGTUGZAWGU
      ORWRBSGZXDWMWTLVIXFWQVKBTUPXEBWGUORNUQURVCUSUTVAVPWOVJVQVLLVOWOVJWPUNVJVO
      ULACVBRVJVIVRVMLZVKVIVJXGBCVBUEVDVEVFVGVH $.
  $}

  $( Ordering property of addition.  Proposition 8.4 of [TakeutiZaring] p. 58,
     limited to natural numbers, and its converse.  (Contributed by NM,
     7-Mar-1996.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
  nnaord $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                 ( A e. B <-> ( C +o A ) e. ( C +o B ) ) ) $=
    ( com wcel w3a coa co wi nnaordi 3adant1 wo wn oveq2 a1i wa wb nnacl nntri2
    wceq 3adant2 orim12d con3d df-3an ancom anandi 3bitri anim12i sylbi 3adant3
    syl 3imtr4d impbid ) ADEZBDEZCDEZFZABEZCAGHZCBGHZEZUOUPURVAIUNABCJKUQUSUTTZ
    UTUSEZLZMZABTZBAEZLZMZVAURUQVHVDUQVFVBVGVCVFVBIUQABCGNOUNUPVGVCIUOBACJUAUBU
    CUQUSDEZUTDEZPZVAVEQUQUPUNPZUPUOPZPZVLUQUNUOPZUPPUPVPPVOUNUOUPUDVPUPUEUPUNU
    OUFUGVMVJVNVKCARCBRUHUIUSUTSUKUNUOURVIQUPABSUJULUM $.

  $( Ordering property of addition of natural numbers.  (Contributed by NM,
     9-Nov-2002.) $)
  nnaordr $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                 ( A e. B <-> ( A +o C ) e. ( B +o C ) ) ) $=
    ( com wcel w3a coa nnaord wceq nnacom ancoms 3adant2 3adant1 eleq12d bitrd
    co ) ADEZBDEZCDEZFZABECAGPZCBGPZEACGPZBCGPZEABCHTUAUCUBUDQSUAUCIZRSQUECAJKL
    RSUBUDIZQSRUFCBJKMNO $.

  ${
    $d A x y $.  $d B x y $.  $d C x $.

    $( Weak ordering property of addition.  (Contributed by NM, 17-Sep-1995.)
       (Revised by Mario Carneiro, 15-Nov-2014.) $)
    nnaword $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                  ( A C_ B <-> ( C +o A ) C_ ( C +o B ) ) ) $=
      ( vx vy com wcel wss co wb wa wi wceq oveq1 sseq12d bibi2d c0 csuc nnacom
      coa cv imbi2d nna0r eqcomd adantr adantl nnacl 3adant3 3adant2 nnsucsssuc
      w3a syl2anc nnasuc peano2 sylan2 syl 3eqtr3rd ancoms biimpd 3expib finds2
      suceq bitrd vtoclga impcom 3impa ) AFGZBFGZCFGZABHZCATIZCBTIZHZJZVIVGVHKZ
      VNVOVJDUAZATIZVPBTIZHZJZLVOVNLDCFVPCMZVTVNVOWAVSVMVJWAVQVKVRVLVPCATNVPCBT
      NOPUBVTVJQATIZQBTIZHZJVJEUAZATIZWEBTIZHZJZVJWERZATIZWJBTIZHZJZVODEVPQMZVS
      WDVJWOVQWBVRWCVPQATNVPQBTNOPVPWEMZVSWHVJWPVQWFVRWGVPWEATNVPWEBTNOPVPWJMZV
      SWMVJWQVQWKVRWLVPWJATNVPWJBTNOPVOAWBBWCVGAWBMVHVGWBAAUCUDUEVHBWCMVGVHWCBB
      UCUDUFOWEFGZVGVHWIWNLWRVGVHUKZWIWNWSWHWMVJWSWHWFRZWGRZHZWMWSWFFGZWGFGZWHX
      BJWRVGXCVHWEAUGUHWRVHXDVGWEBUGUIWFWGUJULWSWTWKXAWLWRVGWTWKMZVHVGWRXEVGWRK
      ZAWJTIZAWETIZRZWKWTAWEUMWRVGWJFGZXGWKMWEUNZAWJSUOXFXHWFMXIWTMAWESXHWFVBUP
      UQURUHWRVHXAWLMZVGVHWRXLVHWRKZBWJTIZBWETIZRZWLXABWEUMWRVHXJXNWLMXKBWJSUOX
      MXOWGMXPXAMBWESXOWGVBUPUQURUIOVCPUSUTVAVDVEVF $.
  $}

  $( Cancellation law for addition of natural numbers.  (Contributed by NM,
     27-Oct-1995.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
  nnacan $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                   ( ( A +o B ) = ( A +o C ) <-> B = C ) ) $=
    ( com wcel w3a coa co wss wa wb nnaword 3comr 3com13 anbi12d bicomd 3bitr4g
    wceq eqss ) ADEZBDEZCDEZFZABGHZACGHZIZUEUDIZJZBCIZCBIZJZUDUERBCRUCUKUHUCUIU
    FUJUGUAUBTUIUFKBCALMUBUATUJUGKCBALNOPUDUESBCSQ $.

  $( Weak ordering property of addition.  (Contributed by NM, 9-Nov-2002.)
     (Revised by Mario Carneiro, 15-Nov-2014.) $)
  nnaword1 $p |- ( ( A e. _om /\ B e. _om ) -> A C_ ( A +o B ) ) $=
    ( com wcel con0 coa co wss nnon oaword1 syl2an ) ACDAEDBEDAABFGHBCDAIBIABJK
    $.

  $( Weak ordering property of addition.  (Contributed by NM, 9-Nov-2002.) $)
  nnaword2 $p |- ( ( A e. _om /\ B e. _om ) -> A C_ ( B +o A ) ) $=
    ( com wcel wa coa co nnaword1 nnacom sseqtrd ) ACDBCDEAABFGBAFGABHABIJ $.

  $( Adding to both sides of an inequality in ` _om ` .  (Contributed by Scott
     Fenton, 16-Apr-2012.)  (Revised by Mario Carneiro, 12-May-2012.) $)
  nnawordi $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
           ( A C_ B -> ( A +o C ) C_ ( B +o C ) ) ) $=
    ( com wcel w3a wss co nnaword biimpd nnacom 3adant2 3adant1 sseq12d sylibrd
    coa wceq ) ADEZBDEZCDEZFZABGZCAPHZCBPHZGZACPHZBCPHZGUAUBUEABCIJUAUFUCUGUDRT
    UFUCQSACKLSTUGUDQRBCKMNO $.

  ${
    $d x y A $.  $d x B $.  $d x y C $.
    $( Ordering property of multiplication.  Half of Proposition 8.19 of
       [TakeutiZaring] p. 63, limited to natural numbers.  (Contributed by NM,
       18-Sep-1995.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
    nnmordi $p |- ( ( ( B e. _om /\ C e. _om ) /\ (/) e. C ) ->
                    ( A e. B -> ( C .o A ) e. ( C .o B ) ) ) $=
      ( vx vy com wcel c0 comu co wi wa wceq eleq2 oveq2 eleq2d imbi12d coa imp
      cv elnn expcom imbi2d csuc pm2.21i a1i wo elsuci nnmcl simpl jca nnaword1
      noel sseld imim2d adantrl nna0 ad2antrr ancoms eqeltrrd eleq1d syl5ibrcom
      nnaordi adantrr jaod sylan syl5 nnmsuc adantr sylibrd exp43 com12 adantld
      wb impd finds2 vtoclga com23 exp4a mpdd com34 com24 imp31 ) BFGZCFGZHCGZA
      BGZCAIJZCBIJZGZKZWDWGWFWEWJWDWGWEWFWJWDWGAFGZWEWFWJKZKWGWDWLABUAUBWDWGWLW
      EWMWDWGWLWELZWFWJWDWNWFLZWGWJWOADTZGZWHCWPIJZGZKZKWOWKKDBFWPBMZWTWKWOXAWQ
      WGWSWJWPBANXAWRWIWHWPBCIOPQUCWTAHGZWHCHIJZGZKZAETZGZWHCXFIJZGZKZAXFUDZGZW
      HCXKIJZGZKZWODEWPHMZWQXBWSXDWPHANXPWRXCWHWPHCIOPQWPXFMZWQXGWSXIWPXFANXQWR
      XHWHWPXFCIOPQWPXKMZWQXLWSXNWPXKANXRWRXMWHWPXKCIOPQXEWOXBXDAUMUEUFXFFGZWNW
      FXJXOKZXSWEWFXTKZWLWEXSYAWEXSWFXJXOWEXSLZWFXJLZLZXLWHXHCRJZGZXNXLXGAXFMZU
      GZYDYFAXFUHYBXHFGZWELZYCYHYFKYBYIWECXFUIWEXSUJUKYJYCLXGYFYGYJXJXGYFKZWFYJ
      XJYKYJXIYFXGYJXHYEWHXHCULUNUOSUPYJWFYGYFKXJYJWFLZYFYGXHYEGYLXHHRJZXHYEYIY
      MXHMWEWFXHUQURYJWFYMYEGZWEYIWFYNKHCXHVCUSSUTYGWHXHYEAXFCIOVAVBVDVEVFVGYBX
      NYFVNYCYBXMYEWHCXFVHPVIVJVKVLVMVOVPVQVRVSVSVTWAWBWC $.
  $}

  $( Ordering property of multiplication.  Proposition 8.19 of [TakeutiZaring]
     p. 63, limited to natural numbers.  (Contributed by NM, 22-Jan-1996.)
     (Revised by Mario Carneiro, 15-Nov-2014.) $)
  nnmord $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
               ( ( A e. B /\ (/) e. C ) <-> ( C .o A ) e. ( C .o B ) ) ) $=
    ( com wcel c0 wa comu co wi nnmordi ex com23 3adant1 wne wceq wb wo syl2anc
    wn w3a impd ne0i nnm0r eqeq1d syl5ibrcom necon3d syl5 adantr nn0eln0 adantl
    oveq1 sylibrd oveq2 a1i 3adantl2 orim12d con3d simpl3 simpl1 simpl2 3imtr4d
    nnmcl nntri2 mpdd jcad impbid ) ADEZBDEZCDEZUAZABEZFCEZGZCAHIZCBHIZEZVIVJVN
    VQJVHVIVJGZVLVMVQVRVMVLVQVRVMVLVQJABCKLMUBNVKVQVLVMVKVQVMVLVIVJVQVMJVHVRVQC
    FOZVMVIVQVSJVJVQVPFOVIVSVPVOUCVICFVPFVIVPFPCFPZFBHIZFPBUDVTVPWAFCFBHULUEUFU
    GUHUIVJVMVSQVICUJUKUMNZVKVMVQVLVKVMVQVLJVKVMGZVOVPPZVPVOEZRZTZABPZBAEZRZTZV
    QVLWCWJWFWCWHWDWIWEWHWDJWCABCHUNUOVHVJVMWIWEJVIBACKUPUQURWCVODEZVPDEZVQWGQW
    CVJVHWLVHVIVJVMUSZVHVIVJVMUTZCAVCSWCVJVIWMWNVHVIVJVMVAZCBVCSVOVPVDSWCVHVIVL
    WKQWOWPABVDSVBLMVEWBVFVG $.

  $( Weak ordering property of ordinal multiplication.  (Contributed by Mario
     Carneiro, 17-Nov-2014.) $)
  nnmword $p |- ( ( ( A e. _om /\ B e. _om /\ C e. _om ) /\ (/) e. C ) ->
                  ( A C_ B <-> ( C .o A ) C_ ( C .o B ) ) ) $=
    ( com wcel w3a c0 wa wn comu wss iba nnmord 3com12 sylan9bbr nntri1 syl2anc
    co wb nnmcl notbid simpl1 simpl2 simpl3 3bitr4d ) ADEZBDEZCDEZFZGCEZHZBAEZI
    ZCBJRZCAJRZEZIZABKZUOUNKZUKULUPUJULULUJHZUIUPUJULLUGUFUHUTUPSBACMNOUAUKUFUG
    URUMSUFUGUHUJUBZUFUGUHUJUCZABPQUKUODEZUNDEZUSUQSUKUHUFVCUFUGUHUJUDZVACATQUK
    UHUGVDVEVBCBTQUOUNPQUE $.

  $( Cancellation law for multiplication of natural numbers.  (Contributed by
     NM, 26-Oct-1995.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
  nnmcan $p |- ( ( ( A e. _om /\ B e. _om /\ C e. _om ) /\ (/) e. A ) ->
                   ( ( A .o B ) = ( A .o C ) <-> B = C ) ) $=
    ( com wcel w3a c0 comu wss wceq 3anrot nnmword sylanb 3anrev anbi12d bicomd
    wa co wb eqss 3bitr4g ) ADEZBDEZCDEZFZGAEZQZABHRZACHRZIZUIUHIZQZBCIZCBIZQZU
    HUIJBCJUGUOULUGUMUJUNUKUEUCUDUBFUFUMUJSUBUCUDKBCALMUEUDUCUBFUFUNUKSUBUCUDNC
    BALMOPUHUITBCTUA $.

  $( One is a natural number.  (Contributed by NM, 29-Oct-1995.) $)
  1onn $p |- 1o e. _om $=
    ( c1o c0 csuc com df-1o wcel peano1 peano2 ax-mp eqeltri ) ABCZDEBDFKDFGBHI
    J $.

  $( The ordinal 2 is a natural number.  (Contributed by NM, 28-Sep-2004.) $)
  2onn $p |- 2o e. _om $=
    ( c2o c1o csuc com df-2o wcel 1onn peano2 ax-mp eqeltri ) ABCZDEBDFKDFGBHIJ
    $.

  $( The ordinal 3 is a natural number.  (Contributed by Mario Carneiro,
     5-Jan-2016.) $)
  3onn $p |- 3o e. _om $=
    ( c3o c2o csuc com df-3o wcel 2onn peano2 ax-mp eqeltri ) ABCZDEBDFKDFGBHIJ
    $.

  $( The ordinal 4 is a natural number.  (Contributed by Mario Carneiro,
     5-Jan-2016.) $)
  4onn $p |- 4o e. _om $=
    ( c4o c3o csuc com df-4o wcel 3onn peano2 ax-mp eqeltri ) ABCZDEBDFKDFGBHIJ
    $.

  $( The ordinal 2 is included in the set of natural number ordinals.
     (Contributed by BJ, 5-Aug-2024.) $)
  2ssom $p |- 2o C_ _om $=
    ( c2o com wcel wss 2onn elomssom ax-mp ) ABCABDEAFG $.

  $( Multiply an element of ` _om ` by ` 1o ` .  (Contributed by Mario
     Carneiro, 17-Nov-2014.) $)
  nnm1 $p |- ( A e. _om -> ( A .o 1o ) = A ) $=
    ( com wcel c1o comu co c0 csuc df-1o oveq2i wceq peano1 nnmsuc mpan2 oveq1d
    coa nnm0 nna0r 3eqtrd eqtrid ) ABCZADEFAGHZEFZADUBAEIJUAUCAGEFZAPFZGAPFAUAG
    BCUCUEKLAGMNUAUDGAPAQOARST $.

  $( Multiply an element of ` _om ` by ` 2o ` .  (Contributed by Scott Fenton,
     18-Apr-2012.)  (Revised by Mario Carneiro, 17-Nov-2014.) $)
  nnm2 $p |- ( A e. _om -> ( A .o 2o ) = ( A +o A ) ) $=
    ( com wcel c2o comu c1o csuc coa df-2o oveq2i wceq 1onn nnmsuc mpan2 oveq1d
    co nnm1 eqtrd eqtrid ) ABCZADEPAFGZEPZAAHPZDUAAEIJTUBAFEPZAHPZUCTFBCUBUEKLA
    FMNTUDAAHAQORS $.

  $( Multiply an element of ` _om ` by ` 2o ` .  (Contributed by Scott Fenton,
     16-Apr-2012.)  (Revised by Mario Carneiro, 17-Nov-2014.) $)
  nn2m $p |- ( A e. _om -> ( 2o .o A ) = ( A +o A ) ) $=
    ( com wcel c2o comu co coa wceq 2onn nnmcom mpan nnm2 eqtrd ) ABCZDAEFZADEF
    ZAAGFDBCNOPHIDAJKALM $.

  ${
    $d A b x y $.  $d A x y z $.  $d B b x y $.
    $( Equivalence for ordering.  Compare Exercise 23 of [Enderton] p. 88.
       (Contributed by NM, 5-Dec-1995.)  (Revised by Mario Carneiro,
       15-Nov-2014.) $)
    nnaordex $p |- ( ( A e. _om /\ B e. _om ) -> ( A e. B <->
                  E. x e. _om ( (/) e. x /\ ( A +o x ) = B ) ) ) $=
      ( vz com wcel wa c0 cv co wceq wrex wi eleq2 eqeq2 anbi2d rexbidv imbi12d
      coa c1o vb vy imbi2d csuc noel pm2.21i a1i wo elsuci simpr peano2 elelsuc
      ad2antlr nnasuc suceq sylan9eq ex anim12d imp oveq2 eqeq1d anbi12d rspcev
      syl2anc rexlimdva imbitrdi syld 0lt1o con0 nnon oa1suc syl 1onn mpan jaod
      cbvrexv syl5 exp31 finds2 vtoclga impcom peano1 nnaord mp3an1 ancoms nna0
      wb adantr eleq1d bitrd anbi1d biimpac biimtrdi impbid ) BEFZCEFZGBCFZHAIZ
      FZBWRSJZCKZGZAELZWPWOWQXCMZWOBUAIZFZWSWTXEKZGZAELZMZMWOXDMUACEXECKZXJXDWO
      XKXFWQXIXCXECBNXKXHXBAEXKXGXAWSXECWTOPQRUCXJBHFZWSWTHKZGZAELZMZBUBIZFZWSW
      TXQKZGZAELZMZBXQUDZFZWSWTYCKZGZAELZMZWOUAUBXEHKZXFXLXIXOXEHBNYIXHXNAEYIXG
      XMWSXEHWTOPQRXEXQKZXFXRXIYAXEXQBNYJXHXTAEYJXGXSWSXEXQWTOPQRXEYCKZXFYDXIYG
      XEYCBNYKXHYFAEYKXGYEWSXEYCWTOPQRXPWOXLXOBUEUFUGXQEFZWOYBYHYDXRBXQKZUHYLWO
      GZYBGZYGBXQUIYOXRYGYMYOXRYAYGYNYBUJWOYAYGMYLYBWOYAHDIZFZBYPSJZYCKZGZDELZY
      GWOXTUUAAEWOWREFZGZXTUUAUUCXTGWRUDZEFZHUUDFZBUUDSJZYCKZGZUUAUUBUUEWOXTWRU
      KUMUUCXTUUIUUCWSUUFXSUUHWSUUFMUUCHWRULUGUUCXSUUHUUCXSUUGWTUDYCBWRUNWTXQUO
      UPUQURUSYTUUIDUUDEYPUUDKZYQUUFYSUUHYPUUDHNUUJYRUUGYCYPUUDBSUTVAVBVCVDUQVE
      YTYFDAEYPWRKZYQWSYSYEYPWRHNUUKYRWTYCYPWRBSUTVAVBVPVFUMVGWOYMYGMYLYBWOYMYG
      WOYMGZHTFZBTSJZYCKZYGUUMUULVHUGWOYMUUNBUDZYCWOBVIFUUNUUPKBVJBVKVLBXQUOUPT
      EFUUMUUOGZYGVMYFUUQATEWRTKZWSUUMYEUUOWRTHNUURWTUUNYCWRTBSUTVAVBVCVNVDUQUM
      VOVQVRVSVTWAWOXCWQMWPWOXBWQAEUUCXBBWTFZXAGWQUUCWSUUSXAUUCWSBHSJZWTFZUUSUU
      BWOWSUVAWGZHEFUUBWOUVBWBHWRBWCWDWEUUCUUTBWTWOUUTBKUUBBWFWHWIWJWKXAUUSWQWT
      CBNWLWMVEWHWN $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Equivalence for weak ordering of natural numbers.  (Contributed by NM,
       8-Nov-2002.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
    nnawordex $p |- ( ( A e. _om /\ B e. _om ) -> ( A C_ B <->
                  E. x e. _om ( A +o x ) = B ) ) $=
      ( com wcel wa wss cv coa co wceq wrex w3a nntri3or 3adant3 wi c0 nnaordex
      w3o syl5ibcom simpr reximi biimtrdi nna0 eqeq2 peano1 oveq2 eqeq1d rspcev
      3ad2ant1 mpan syl6 nntri1 biimp3a pm2.21d 3jaod mpd 3expia nnaword1 sseq2
      wn rexlimdva adantr impbid ) BDEZCDEZFZBCGZBAHZIJZCKZADLZVEVFVHVLVEVFVHMZ
      BCEZBCKZCBEZSZVLVEVFVQVHBCNOVMVNVLVOVPVEVFVNVLPVHVGVNQVIEZVKFZADLVLABCRVS
      VKADVRVKUAUBUCOVMVOBQIJZCKZVLVMVTBKZVOWAVEVFWBVHBUDUJBCVTUETQDEWAVLUFVKWA
      AQDVIQKVJVTCVIQBIUGUHUIUKULVMVPVLVEVFVHVPVABCUMUNUOUPUQURVEVLVHPVFVEVKVHA
      DVEVIDEFBVJGVKVHBVIUSVJCBUTTVBVCVD $.
  $}

  $( The product of two natural numbers is zero iff at least one of them is
     zero.  (Contributed by Jim Kingdon, 11-Nov-2004.) $)
  nnm00 $p |- ( ( A e. _om /\ B e. _om ) -> ( ( A .o B ) = (/) <->
              ( A = (/) \/ B = (/) ) ) ) $=
    ( com wcel wa comu co c0 wceq wo wi simpl jaoi orcd a1i adantr ex sylan9eqr
    jaod 0elnn olcd simplr wn nnmordi expimpd ancoms nnm0 eleq1d sylibd imp n0i
    simpr pm2.21dd anim12i anddi sylib mpjaod oveq1 nnm0r adantl oveq2 impbid
    syl ) ACDZBCDZEZABFGZHIZAHIZBHIZJZVFVHVKVFVHEZVIVJEZVIHBDZEZJZVKHADZVJEZVQV
    NEZJZVPVKKVLVPVIVJVMVIVOVIVJLVIVNLMNOVLVRVKVSVRVKKVLVRVJVIVQVJULUAOVLVSVKVL
    VSEZVHVKVFVHVSUBWAHVGDZVHUCVLVSWBVFVSWBKVHVFVSAHFGZVGDZWBVEVDVSWDKVEVDEVQVN
    WDHBAUDUEUFVFWCHVGVDWCHIVEAUGZPUHUIPUJVGHUKVCUMQSVFVPVTJZVHVFVIVQJZVJVNJZEW
    FVDWGVEWHATBTUNVIVQVJVNUOUPPUQQVFVIVHVJVEVIVHKVDVEVIVHVIVEVGHBFGHAHBFURBUSR
    QUTVDVJVHKVEVDVJVHVJVDVGWCHBHAFVAWERQPSVB $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Equivalence relations and classes
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c Er $.  $( Equivalence predicate $)
  $c /. $.  $( Long slash for quotient set $)

  $( Extend the definition of a wff to include the equivalence predicate. $)
  wer $a wff R Er A $.

  $( Extend the definition of a class to include equivalence class. $)
  cec $a class [ A ] R $.

  $( Extend the definition of a class to include quotient set. $)
  cqs $a class ( A /. R ) $.

  $( Define the equivalence relation predicate.  Our notation is not standard.
     A formal notation doesn't seem to exist in the literature; instead only
     informal English tends to be used.  The present definition, although
     somewhat cryptic, nicely avoids dummy variables.  In ~ dfer2 we derive a
     more typical definition.  We show that an equivalence relation is
     reflexive, symmetric, and transitive in ~ erref , ~ ersymb , and ~ ertr .
     (Contributed by NM, 4-Jun-1995.)  (Revised by Mario Carneiro,
     2-Nov-2015.) $)
  df-er $a |- ( R Er A <->
    ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) ) $.

  ${
    $d x y z R $.
    $( Alternate definition of equivalence predicate.  (Contributed by NM,
       3-Jan-1997.)  (Revised by Mario Carneiro, 12-Aug-2015.) $)
    dfer2 $p |- ( R Er A <->
    ( Rel R /\ dom R = A /\ A. x A. y A. z
          ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) ) ) $=
      ( wer wrel cdm wceq ccnv wss w3a cv wbr wi wa wal albii 19.26 bitri df-er
      ccom cun cnvsym cotr anbi12i unss 19.28v bitr2i 3bitr3i 3anbi3i ) DEFEGZE
      HDIZEJZEEUBZUCEKZLULUMAMZBMZENZURUQENOZUSURCMZENPUQVAENOZPCQZBQZAQZLDEUAU
      PVEULUMUNEKZUOEKZPUTBQZAQZVBCQZBQZAQZPZUPVEVFVIVGVLABEUDABCEUEUFUNUOEUGVE
      VHVKPZAQVMVDVNAVDUTVJPZBQVNVCVOBUTVBCUHRUTVJBSTRVHVKASUIUJUKT $.
  $}

  $( Define the ` R ` -coset of ` A ` .  Exercise 35 of [Enderton] p. 61.  This
     is called the equivalence class of ` A ` modulo ` R ` when ` R ` is an
     equivalence relation (i.e. when ` Er R ` ; see ~ dfer2 ).  In this case,
     ` A ` is a representative (member) of the equivalence class ` [ A ] R ` ,
     which contains all sets that are equivalent to ` A ` .  Definition of
     [Enderton] p. 57 uses the notation ` [ A ] ` (subscript) ` R ` , although
     we simply follow the brackets by ` R ` since we don't have subscripted
     expressions.  For an alternate definition, see ~ dfec2 .  (Contributed by
     NM, 23-Jul-1995.) $)
  df-ec $a |- [ A ] R = ( R " { A } ) $.

  ${
    $d y A $.  $d y R $.
    $( Alternate definition of ` R ` -coset of ` A ` .  Definition 34 of
       [Suppes] p. 81.  (Contributed by NM, 3-Jan-1997.)  (Proof shortened by
       Mario Carneiro, 9-Jul-2014.) $)
    dfec2 $p |- ( A e. V -> [ A ] R = { y | A R y } ) $=
      ( wcel cec csn cima cv wbr cab df-ec imasng eqtrid ) BDEBCFCBGHBAICJAKBCL
      ABDCMN $.
  $}

  $( An equivalence class modulo a set is a set.  (Contributed by NM,
     24-Jul-1995.) $)
  ecexg $p |- ( R e. B -> [ A ] R e. _V ) $=
    ( wcel cec csn cima cvv df-ec imaexg eqeltrid ) CBDACECAFZGHACICLBJK $.

  ${
    $d A x $.  $d B x $.  $d R x $.
    $( An inhabited equivalence class implies the representative is a set.
       (Contributed by Mario Carneiro, 9-Jul-2014.) $)
    ecexr $p |- ( A e. [ B ] R -> B e. _V ) $=
      ( vx cec wcel cv wceq wex cvv wbr csn wrex elimag ibi df-ec eleq2s df-rex
      cima wa simpl velsn sylib eximi sylbi syl isset sylibr ) ABCEZFZDGZBHZDIZ
      BJFUJUKACKZDBLZMZUMUPACUOSZUIAUQFUPDACUOUQNOBCPQUPUKUOFZUNTZDIUMUNDUORUSU
      LDUSURULURUNUADBUBUCUDUEUFDBUGUH $.
  $}

  ${
    $d x y A $.  $d x y R $.
    $( Define quotient set. ` R ` is usually an equivalence relation.
       Definition of [Enderton] p. 58.  (Contributed by NM, 23-Jul-1995.) $)
    df-qs $a |- ( A /. R ) = { y | E. x e. A y = [ x ] R } $.
  $}

  $( Equality theorem for equivalence predicate.  (Contributed by NM,
     4-Jun-1995.)  (Revised by Mario Carneiro, 12-Aug-2015.) $)
  ereq1 $p |- ( R = S -> ( R Er A <-> S Er A ) ) $=
    ( wceq wrel cdm ccnv ccom cun wss releq dmeq eqeq1d cnveq coeq1 coeq2 eqtrd
    w3a wer df-er uneq12d sseq1d sseq2 bitrd 3anbi123d 3bitr4g ) BCDZBEZBFZADZB
    GZBBHZIZBJZRCEZCFZADZCGZCCHZIZCJZRABSACSUGUHUOUJUQUNVABCKUGUIUPABCLMUGUNUTB
    JVAUGUMUTBUGUKURULUSBCNUGULCBHUSBCBOBCCPQUAUBBCUTUCUDUEABTACTUF $.

  $( Equality theorem for equivalence predicate.  (Contributed by Mario
     Carneiro, 12-Aug-2015.) $)
  ereq2 $p |- ( A = B -> ( R Er A <-> R Er B ) ) $=
    ( wceq wrel cdm ccnv ccom cun wss w3a wer eqeq2 3anbi2d df-er 3bitr4g ) ABD
    ZCEZCFZADZCGCCHICJZKRSBDZUAKACLBCLQTUBRUAABSMNACOBCOP $.

  $( An equivalence relation is a relation.  (Contributed by Mario Carneiro,
     12-Aug-2015.) $)
  errel $p |- ( R Er A -> Rel R ) $=
    ( wer wrel cdm wceq ccnv ccom cun wss df-er simp1bi ) ABCBDBEAFBGBBHIBJABKL
    $.

  $( The domain of an equivalence relation.  (Contributed by Mario Carneiro,
     12-Aug-2015.) $)
  erdm $p |- ( R Er A -> dom R = A ) $=
    ( wer wrel cdm wceq ccnv ccom cun wss df-er simp2bi ) ABCBDBEAFBGBBHIBJABKL
    $.

  ${
    ersym.1 $e |- ( ph -> R Er X ) $.
    ersym.2 $e |- ( ph -> A R B ) $.
    $( Elementhood in the field of an equivalence relation.  (Contributed by
       Mario Carneiro, 12-Aug-2015.) $)
    ercl $p |- ( ph -> A e. X ) $=
      ( cdm wrel wbr wcel wer errel syl releldm syl2anc wceq erdm eleqtrd ) ABD
      HZEADIZBCDJBTKAEDLZUAFEDMNGBCDOPAUBTEQFEDRNS $.

    $( An equivalence relation is symmetric.  (Contributed by NM, 4-Jun-1995.)
       (Revised by Mario Carneiro, 12-Aug-2015.) $)
    ersym $p |- ( ph -> B R A ) $=
      ( ccnv wbr cvv wcel wa wb wrel wer errel syl brrelex12 syl2anc brcnvg cun
      ancoms mpbird ccom wss cdm wceq df-er simp3bi unssad ssbrd mpd ) ACBDHZIZ
      CBDIAUNBCDIZGABJKZCJKZLZUNUOMZADNZUOURAEDOZUTFEDPQGBCDRSUQUPUSCBJJDTUBQUC
      AUMDCBAUMDDUDZDAVAUMVBUADUEZFVAUTDUFEUGVCEDUHUIQUJUKUL $.

    $( Elementhood in the field of an equivalence relation.  (Contributed by
       Mario Carneiro, 12-Aug-2015.) $)
    ercl2 $p |- ( ph -> B e. X ) $=
      ( ersym ercl ) ACBDEFABCDEFGHI $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ph $.  $d x R $.
    ersymb.1 $e |- ( ph -> R Er X ) $.
    $( An equivalence relation is symmetric.  (Contributed by NM, 30-Jul-1995.)
       (Revised by Mario Carneiro, 12-Aug-2015.) $)
    ersymb $p |- ( ph -> ( A R B <-> B R A ) ) $=
      ( wbr wa wer adantr simpr ersym impbida ) ABCDGZCBDGZANHBCDEAEDIZNFJANKLA
      OHCBDEAPOFJAOKLM $.

    $( An equivalence relation is transitive.  (Contributed by NM, 4-Jun-1995.)
       (Revised by Mario Carneiro, 12-Aug-2015.) $)
    ertr $p |- ( ph -> ( ( A R B /\ B R C ) -> A R C ) ) $=
      ( vx wbr wa ccom cv wex cvv wcel syl simpr brrelex syl2an wceq wrel errel
      breq2 breq1 anbi12d spcegv sylc wb simpl brrelex2 brcog syl2anc mpbird ex
      wer ccnv cun wss cdm df-er simp3bi unssbd ssbrd syld ) ABCEIZCDEIZJZBDEEK
      ZIZBDEIAVGVIAVGJZVIBHLZEIZVKDEIZJZHMZVJCNOZVGVOAEUAZVFVPVGAFEUOZVQGFEUBPZ
      VEVFQZCDERSAVGQVNVGHCNVKCTVLVEVMVFVKCBEUCVKCDEUDUEUFUGVJBNOZDNOZVIVOUHAVQ
      VEWAVGVSVEVFUIBCERSAVQVFWBVGVSVTCDEUJSHBDEENNUKULUMUNAVHEBDAEUPZVHEAVRWCV
      HUQEURZGVRVQEUSFTWDFEUTVAPVBVCVD $.

    ${
      ertrd.5 $e |- ( ph -> A R B ) $.
      ertrd.6 $e |- ( ph -> B R C ) $.
      $( A transitivity relation for equivalences.  (Contributed by Mario
         Carneiro, 9-Jul-2014.) $)
      ertrd $p |- ( ph -> A R C ) $=
        ( wbr ertr mp2and ) ABCEJCDEJBDEJHIABCDEFGKL $.

      $( A transitivity relation for equivalences.  (Contributed by Mario
         Carneiro, 9-Jul-2014.) $)
      ertr2d $p |- ( ph -> C R A ) $=
        ( ertrd ersym ) ABDEFGABCDEFGHIJK $.
    $}

    ${
      ertr3d.5 $e |- ( ph -> B R A ) $.
      ertr3d.6 $e |- ( ph -> B R C ) $.
      $( A transitivity relation for equivalences.  (Contributed by Mario
         Carneiro, 9-Jul-2014.) $)
      ertr3d $p |- ( ph -> A R C ) $=
        ( ersym ertrd ) ABCDEFGACBEFGHJIK $.
    $}

    ${
      ertr4d.5 $e |- ( ph -> A R B ) $.
      ertr4d.6 $e |- ( ph -> C R B ) $.
      $( A transitivity relation for equivalences.  (Contributed by Mario
         Carneiro, 9-Jul-2014.) $)
      ertr4d $p |- ( ph -> A R C ) $=
        ( ersym ertrd ) ABCDEFGHADCEFGIJK $.
    $}

    erref.2 $e |- ( ph -> A e. X ) $.
    $( An equivalence relation is reflexive on its field.  Compare Theorem 3M
       of [Enderton] p. 56.  (Contributed by Mario Carneiro, 6-May-2013.)
       (Revised by Mario Carneiro, 12-Aug-2015.) $)
    erref $p |- ( ph -> A R A ) $=
      ( vx cv wbr cdm wcel wex wer wceq erdm syl eleqtrrd wb eldmg mpbid adantr
      wa simpr ertr4d exlimddv ) ABGHZCIZBBCIGABCJZKZUGGLZABDUHFADCMZUHDNEDCOPQ
      ABDKUIUJRFGBCDSPTAUGUBBUFBCDAUKUGEUAAUGUCZULUDUE $.
  $}

  ${
    $d x y A $.  $d x y R $.
    $( The converse of an equivalence relation is itself.  (Contributed by
       Mario Carneiro, 12-Aug-2015.) $)
    ercnv $p |- ( R Er A -> `' R = R ) $=
      ( vx vy wrel wer ccnv wceq errel relcnv cv wbr wcel id ersymb brcnv df-br
      cop vex bitr3i 3bitr3g eqrelrdv2 mpanl1 mpancom ) BEZABFZBGZBHZABIUGEUEUF
      UHBJUFCDUGBUFDKZCKZBLZUJUIBLUJUIRZUGMZULBMUFUIUJBAUFNOUKUJUIUGLUMUJUIBCSD
      SPUJUIUGQTUJUIBQUAUBUCUD $.
  $}

  $( The range and domain of an equivalence relation are equal.  (Contributed
     by Rodolfo Medina, 11-Oct-2010.)  (Revised by Mario Carneiro,
     12-Aug-2015.) $)
  errn $p |- ( R Er A -> ran R = A ) $=
    ( wer crn ccnv cdm df-rn ercnv dmeqd erdm eqtrd eqtrid ) ABCZBDBEZFZABGMOBF
    AMNBABHIABJKL $.

  $( An equivalence relation is a subset of the cartesian product of the field.
     (Contributed by Mario Carneiro, 12-Aug-2015.) $)
  erssxp $p |- ( R Er A -> R C_ ( A X. A ) ) $=
    ( wer cdm crn cxp wrel wss errel relssdmrn syl erdm errn xpeq12d sseqtrd )
    ABCZBBDZBEZFZAAFPBGBSHABIBJKPQARAABLABMNO $.

  $( An equivalence relation is a set if its domain is a set.  (Contributed by
     Rodolfo Medina, 15-Oct-2010.)  (Proof shortened by Mario Carneiro,
     12-Aug-2015.) $)
  erex $p |- ( R Er A -> ( A e. V -> R e. _V ) ) $=
    ( wer wcel cvv cxp wss erssxp xpexg anidms ssexg syl2an ex ) ABDZACEZBFEZOB
    AAGZHRFEZQPABIPSAACCJKBRFLMN $.

  $( An equivalence relation is a set if and only if its domain is a set.
     (Contributed by Rodolfo Medina, 15-Oct-2010.)  (Revised by Mario Carneiro,
     12-Aug-2015.) $)
  erexb $p |- ( R Er A -> ( R e. _V <-> A e. _V ) ) $=
    ( wer cvv wcel cdm dmexg erdm eleq1d imbitrid erex impbid ) ABCZBDEZADEZNBF
    ZDEMOBDGMPADABHIJABDKL $.

  ${
    $d x y z R $.  $d x A $.  $d x y z ph $.
    iserd.1 $e |- ( ph -> Rel R ) $.
    iserd.2 $e |- ( ( ph /\ x R y ) -> y R x ) $.
    iserd.3 $e |- ( ( ph /\ ( x R y /\ y R z ) ) -> x R z ) $.
    iserd.4 $e |- ( ph -> ( x e. A <-> x R x ) ) $.
    $( A reflexive, symmetric, transitive relation is an equivalence relation
       on its domain.  (Contributed by Mario Carneiro, 9-Jul-2014.)  (Revised
       by Mario Carneiro, 12-Aug-2015.) $)
    iserd $p |- ( ph -> R Er A ) $=
      ( wer wceq cv wbr wi wa wal ex alrimiv wcel cdm eqidd jca dfer2 syl3anbrc
      wrel adantr simpr erref vex breldm impbid1 bitr4d eqrdv ereq2 syl mpbid
      wb ) AFUAZFKZEFKZAFUFUSUSLBMZCMZFNZVCVBFNZOZVDVCDMZFNPZVBVGFNZOZPZDQZCQZB
      QUTGAUSUBAVMBAVLCAVKDAVFVJAVDVEHRAVHVIIRUCSSSBCDUSFUDUEZAUSELUTVAURABUSEA
      VBUSTZVBVBFNZVBETAVOVPAVOVPAVOPVBFUSAUTVOVNUGAVOUHUIRVBVBFBUJZVQUKULJUMUN
      USEFUOUPUQ $.
  $}

  ${
    swoer.1 $e |- R = ( ( X X. X ) \ ( .< u. `' .< ) ) $.
    $( Evaluate the incomparability relation.  (Contributed by Mario Carneiro,
       9-Jul-2014.) $)
    brdifun $p |- ( ( A e. X /\ B e. X ) ->
      ( A R B <-> -. ( A .< B \/ B .< A ) ) ) $=
      ( wcel wa wbr ccnv cun wn wo cxp wb cop opelxpi df-br sylibr cdif brcnvg
      breqi brdif bitri baib syl brun orbi2d bitrid notbid bitrd ) AEGBEGHZABCI
      ZABDDJZKZIZLZABDIZBADIZMZLULABEENZIZUMUQOULABPVAGVBABEEQABVARSUMVBUQUMABV
      AUOTZIVBUQHABCVCFUBABVAUOUCUDUEUFULUPUTUPURABUNIZMULUTABDUNUGULVDUSURABEE
      DUAUHUIUJUK $.

    $d u v w R $.  $d x y z .< $.  $d x y z A $.  $d x y z B $.  $d x y z C $.
    $d u v w x y z ph $.  $d u x y z X $.
    swoer.2 $e |- ( ( ph /\ ( y e. X /\ z e. X ) ) ->
                    ( y .< z -> -. z .< y ) ) $.
    swoer.3 $e |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) ->
                    ( x .< y -> ( x .< z \/ z .< y ) ) ) $.
    $( Incomparability under a strict weak partial order is an equivalence
       relation.  (Contributed by Mario Carneiro, 9-Jul-2014.)  (Revised by
       Mario Carneiro, 12-Aug-2015.) $)
    swoer $p |- ( ph -> R Er X ) $=
      ( cv wbr wa wo wn wb wcel brdifun syl syl2anc vu vv wrel cxp wss ccnv cun
      vw cdif difss eqsstri relxp relss mp2 a1i simpr orcom notbid ssbri adantl
      brxp sylib simprd simpld 3bitr4d mpbid simprl simplbi simprbi simprr brel
      ad2antrl wi simpl swopolem syl13anc imbitrrdi orim12d or4 imbitrdi mpbird
      mtord wpo swopo poirr sylan pm1.2 nsyl impbida iserd ) AUAUBUHGEEUCZAEGGU
      DZUEWLUCWKEWLFFUFUGZUIWLHWLWMUJUKZGGULEWLUMUNUOAUAKZUBKZELZMZWQWPWOELZAWQ
      UPWRWOWPFLZWPWOFLZNZOZXAWTNZOZWQWSWRXBXDXBXDPWRWTXAUQUOURWRWOGQZWPGQZMZWQ
      XCPZWRWOWPWLLZXHWQXJAEWLWOWPWNUSZUTWOWPGGVAZVBZWOWPEFGHRZSWRXGXFWSXEPWRXF
      XGXMVCWRXFXGXMVDWPWOEFGHRTVEVFAWQWPUHKZELZMZMZWOXOELZWOXOFLZXOWOFLZNZOZXR
      YBXBWPXOFLZXOWPFLZNZXRWQXCAWQXPVGXRXFXGXIXRXJXFWQXJAXPXKVLZXJXFXGXLVHSZXR
      XJXGYGXJXFXGXLVISZXNTVFXRXPYFOZAWQXPVJZXRXGXOGQZXPYJPYIXRXPYLYKXPXGYLWPXO
      GGEWNVKVCSZWPXOEFGHRTVFXRYBWTYDNZXAYENZNXBYFNXRXTYNYAYOXRAXFYLXGXTYNVMAXQ
      VNZYHYMYIABCDGFWOXOWPJVOVPXRYAYEXANZYOXRAYLXFXGYAYQVMYPYMYHYIABCDGFXOWOWP
      JVOVPXAYEUQVQVRWTYDXAYEVSVTWBXRXFYLXSYCPYHYMWOXOEFGHRTWAAXFWOWOELZAXFMZYR
      WOWOFLZYTNZOZYSYTUUAAGFWCXFYTOABCDGFIJWDGWOFWEWFYTWGWHYSXFXFYRUUBPAXFUPZU
      UCWOWOEFGHRTWAYRXFAYRWOWOWLLZXFEWLWOWOWNUSUUDXFXFWOWOGGVAVHSUTWIWJ $.

    ${
      swoord.4 $e |- ( ph -> B e. X ) $.
      swoord.5 $e |- ( ph -> C e. X ) $.
      swoord.6 $e |- ( ph -> A R B ) $.
      $( The incomparability equivalence relation is compatible with the
         original order.  (Contributed by Mario Carneiro, 31-Dec-2014.) $)
      swoord1 $p |- ( ph -> ( A .< C <-> B .< C ) ) $=
        ( wbr wo wcel wn wi cxp ccnv cun cdif difss eqsstri ssbri df-br opelxp1
        id cop sylbi 3syl swopolem syl13anc wb brdifun syl2anc mpbid nsyl biorf
        orc syl sylibrd olc impbid ) AEGIQZFGIQZAVHEFIQZVIRZVIAAEJSZGJSZFJSZVHV
        KUAAUKZAEFHQZEFJJUBZQZVLPHVQEFHVQIIUCUDZUEVQKVQVSUFUGUHVREFULVQSVLEFVQU
        IEFJJUJUMUNZONABCDJIEGFMUOUPAVJTVIVKUQAVJFEIQZRZVJAVPWBTZPAVLVNVPWCUQVT
        NEFHIJKURUSUTZVJWAVCVAVJVIVBVDVEAVIWAVHRZVHAAVNVMVLVIWEUAVONOVTABCDJIFG
        EMUOUPAWATVHWEUQAWBWAWDWAVJVFVAWAVHVBVDVEVG $.

      $( The incomparability equivalence relation is compatible with the
         original order.  (Contributed by Mario Carneiro, 31-Dec-2014.) $)
      swoord2 $p |- ( ph -> ( C .< A <-> C .< B ) ) $=
        ( wbr wo wcel wi id cxp ccnv cun cdif difss eqsstri ssbri df-br opelxp1
        cop sylbi swopolem syl13anc idd wn wb brdifun syl2anc mpbid olc pm2.21d
        3syl nsyl jaod syld orc impbid ) AGEIQZGFIQZAVIVJFEIQZRZVJAAGJSZEJSZFJS
        ZVIVLTAUAZOAEFHQZEFJJUBZQZVNPHVREFHVRIIUCUDZUEVRKVRVTUFUGUHVSEFUKVRSVNE
        FVRUIEFJJUJULVCZNABCDJIGEFMUMUNAVJVJVKAVJUOAVKVJAEFIQZVKRZVKAVQWCUPZPAV
        NVOVQWDUQWANEFHIJKURUSUTZVKWBVAVDVBVEVFAVJVIWBRZVIAAVMVOVNVJWFTVPONWAAB
        CDJIGFEMUMUNAVIVIWBAVIUOAWBVIAWCWBWEWBVKVGVDVBVEVFVH $.
    $}
  $}

  ${
    $d w x y $.  $d x y z $.  $d y A $.  $d v x B $.
    eqer.1 $e |- ( x = y -> A = B ) $.
    eqer.2 $e |- R = { <. x , y >. | A = B } $.
    $( Lemma for ~ eqer .  (Contributed by NM, 17-Mar-2008.)  (Proof shortened
       by Mario Carneiro, 6-Dec-2016.) $)
    eqerlem $p |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A ) $=
      ( cv wceq wsbc csb cvv wcel wb vex nfcsb1v weq sbciegf brabsb nfeq csbief
      wbr nfv nfcv csbeq1 eqtr3id eqeq2d ax-mp csbeq1a eqeq1d bitrid bitri ) CJ
      ZDJZGUDEFKZBUPLZAUOLZAUOEMZAUPEMZKZUQABUOUPGIUAUONOUSVBPCQURVBAUONAUTVAAU
      OERAUPERUBUREVAKZACSZVBUPNOURVCPDQUQVCBUPNVCBUEBDSZFVAEVEFABJZEMVAAVFEFBQ
      AFUFHUCAVFUPEUGUHUITUJVDEUTVAAUOEUKULUMTUJUN $.

    $d v w x y z $.  $d v w z R $.
    $( Equivalence relation involving equality of dependent classes ` A ( x ) `
       and ` B ( y ) ` .  (Contributed by NM, 17-Mar-2008.)  (Revised by Mario
       Carneiro, 12-Aug-2015.) $)
    eqer $p |- R Er _V $=
      ( vz vw vv cvv wtru wceq a1i cv wbr csb eqerlem 3imtr4i adantl wer eqcomd
      wrel relopabi id wa eqtr anbi12i wcel wb vex eqid mpbir 2th iserd mptru )
      KEUALHIJKEEUCLCDMABEGUDNHOZIOZEPZURUQEPZLAUQCQZAURCQZMZVBVAMUSUTVCVAVBVCU
      EUBABHICDEFGRZABIHCDEFGRSTUSURJOZEPZUFZUQVEEPZLVCVBAVECQZMZUFVAVIMVGVHVAV
      BVIUGUSVCVFVJVDABIJCDEFGRUHABHJCDEFGRSTUQKUIZUQUQEPZUJLVKVLHUKVLVAVAMVAUL
      ABHHCDEFGRUMUNNUOUP $.
  $}

  ${
    $d x y z $.
    $( The identity relation is an equivalence relation.  (Contributed by NM,
       10-May-1998.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.)  (Proof
       shortened by Mario Carneiro, 9-Jul-2014.) $)
    ider $p |- _I Er _V $=
      ( vx vy cv cid wceq id df-id eqer ) ABACZBCZDIJEFABGH $.

    $( The empty set is an equivalence relation on the empty set.  (Contributed
       by Mario Carneiro, 5-Sep-2015.) $)
    0er $p |- (/) Er (/) $=
      ( vx vy vz c0 wer wtru wrel rel0 a1i cv wbr wcel df-br noel pm2.21i sylbi
      cop adantl ad2antrl wb 2false bitr4i iserd mptru ) DDEFABCDDDGFHIAJZBJZDK
      ZUFUEDKZFUGUEUFQZDLZUHUEUFDMZUJUHUINZOPRUGUECJZDKZFUFUMDKUGUJUNUKUJUNULOP
      SUEDLZUEUEDKZTFUOUEUEQZDLZUPUOURUENUQNUAUEUEDMUBIUCUD $.
  $}

  $( Equality theorem for equivalence class.  (Contributed by NM,
     23-Jul-1995.) $)
  eceq1 $p |- ( A = B -> [ A ] C = [ B ] C ) $=
    ( wceq csn cima cec sneq imaeq2d df-ec 3eqtr4g ) ABDZCAEZFCBEZFACGBCGLMNCAB
    HIACJBCJK $.

  ${
    eceq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for equivalence class (deduction form).  (Contributed
       by Jim Kingdon, 31-Dec-2019.) $)
    eceq1d $p |- ( ph -> [ A ] C = [ B ] C ) $=
      ( wceq cec eceq1 syl ) ABCFBDGCDGFEBCDHI $.
  $}

  $( Equality theorem for equivalence class.  (Contributed by NM,
     23-Jul-1995.) $)
  eceq2 $p |- ( A = B -> [ C ] A = [ C ] B ) $=
    ( wceq csn cima cec imaeq1 df-ec 3eqtr4g ) ABDACEZFBKFCAGCBGABKHCAICBIJ $.

  ${
    eceq2i.1 $e |- A = B $.
    $( Equality theorem for the ` A ` -coset and ` B ` -coset of ` C ` ,
       inference version.  (Contributed by Peter Mazsa, 11-May-2021.) $)
    eceq2i $p |- [ C ] A = [ C ] B $=
      ( wceq cec eceq2 ax-mp ) ABECAFCBFEDABCGH $.
  $}

  ${
    eceq2d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for the ` A ` -coset and ` B ` -coset of ` C ` ,
       deduction version.  (Contributed by Peter Mazsa, 23-Apr-2021.) $)
    eceq2d $p |- ( ph -> [ C ] A = [ C ] B ) $=
      ( wceq cec eceq2 syl ) ABCFDBGDCGFEBCDHI $.
  $}

  $( Membership in an equivalence class.  Theorem 72 of [Suppes] p. 82.
     (Contributed by Mario Carneiro, 9-Jul-2014.) $)
  elecg $p |- ( ( A e. V /\ B e. W ) -> ( A e. [ B ] R <-> B R A ) ) $=
    ( wcel wa csn cima cop cec wbr elimasng ancoms df-ec eleq2i df-br 3bitr4g
    wb ) ADFZBEFZGACBHIZFZBAJCFZABCKZFBACLUATUCUDSCBAEDMNUEUBABCOPBACQR $.

  ${
    elec.1 $e |- A e. _V $.
    elec.2 $e |- B e. _V $.
    $( Membership in an equivalence class.  Theorem 72 of [Suppes] p. 82.
       (Contributed by NM, 23-Jul-1995.) $)
    elec $p |- ( A e. [ B ] R <-> B R A ) $=
      ( cvv wcel cec wbr wb elecg mp2an ) AFGBFGABCHGBACIJDEABCFFKL $.
  $}

  $( Membership in an equivalence class when ` R ` is a relation.  (Contributed
     by Mario Carneiro, 11-Sep-2015.) $)
  relelec $p |- ( Rel R -> ( A e. [ B ] R <-> B R A ) ) $=
    ( wrel cec wcel wbr cvv wa ecexr jca adantl brrelex12 ancomd elecg pm5.21nd
    elex ) CDZABCEZFZBACGZAHFZBHFZIZTUDRTUBUCASQABCJKLRUAIUCUBBACMNABCHHOP $.

  ${
    ecss.1 $e |- ( ph -> R Er X ) $.
    $( An equivalence class is a subset of the domain.  (Contributed by NM,
       6-Aug-1995.)  (Revised by Mario Carneiro, 12-Aug-2015.) $)
    ecss $p |- ( ph -> [ A ] R C_ X ) $=
      ( crn cec csn cima df-ec imassrn eqsstri wer wceq errn syl sseqtrid ) ACF
      ZBCGZDSCBHZIRBCJCTKLADCMRDNEDCOPQ $.
  $}

  ${
    $d x R $.  $d x A $.
    $( A representative of an inhabited equivalence class belongs to the domain
       of the equivalence relation.  (Contributed by Jim Kingdon,
       21-Aug-2019.) $)
    ecdmn0m $p |- ( A e. dom R <-> E. x x e. [ A ] R ) $=
      ( cdm wcel cvv cv cec wex elex ecexr exlimiv wbr eldmg wb vex mpan exbidv
      elecg bitr4d pm5.21nii ) BCDZEZBFEZAGZBCHEZAIZBUBJUFUDAUEBCKLUDUCBUECMZAI
      UGABCFNUDUFUHAUEFEUDUFUHOAPUEBCFFSQRTUA $.
  $}

  ${
    $d A x $.  $d B x $.  $d R x $.  $d ph x $.
    ereldm.1 $e |- ( ph -> R Er X ) $.
    ereldm.2 $e |- ( ph -> [ A ] R = [ B ] R ) $.
    $( Equality of equivalence classes implies equivalence of domain
       membership.  (Contributed by NM, 28-Jan-1996.)  (Revised by Mario
       Carneiro, 12-Aug-2015.) $)
    ereldm $p |- ( ph -> ( A e. X <-> B e. X ) ) $=
      ( vx cdm wcel cv cec wex eleq2d exbidv ecdmn0m 3bitr4g wer wceq erdm syl
      3bitr3d ) ABDIZJZCUCJZBEJCEJAHKZBDLZJZHMUFCDLZJZHMUDUEAUHUJHAUGUIUFGNOHBD
      PHCDPQAUCEBAEDRUCESFEDTUAZNAUCECUKNUB $.
  $}

  ${
    $d x A $.  $d x B $.  $d x R $.  $d x ph $.
    erth.1 $e |- ( ph -> R Er X ) $.
    erth.2 $e |- ( ph -> A e. X ) $.
    $( Basic property of equivalence relations.  Theorem 73 of [Suppes] p. 82.
       (Contributed by NM, 23-Jul-1995.)  (Revised by Mario Carneiro,
       6-Jul-2015.) $)
    erth $p |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) ) $=
      ( vx wbr cec wa wcel ertr impl sylan impbida cvv wb adantr elecg cv simpl
      wceq ersymb biimpa jca vex sylancr wrel wer errel syl 3bitr4d eqrdv erref
      brrelex2 syl2anc mpbird simpr eleqtrd ereldm mpbid ersym ) ABCDIZBDJZCDJZ
      UCZAVDKZHVEVFVHBHUAZDIZCVIDIZVIVELZVIVFLZVHVJVKVHACBDIZKVJVKVHAVNAVDUBAVD
      VNABCDEFUDUEUFAVNVJVKACBVIDEFMNOAVDVKVJABCVIDEFMNPVHVIQLZBELZVLVJRHUGZAVP
      VDGSVIBDQETUHVHVOCQLZVMVKRVQADUIZVDVRAEDUJZVSFEDUKULBCDUPOVICDQQTUHUMUNAV
      GKZCBDEAVTVGFSZWABVFLZVNWABVEVFWABVELZBBDIZAWEVGABDEFGUOSWAVPVPWDWERAVPVG
      GSZWFBBDEETUQURAVGUSZUTWAVPCELZWCVNRWFWAVPWHWFWABCDEWBWGVAVBBCDEETUQVBVCP
      $.
  $}

  ${
    erth2.1 $e |- ( ph -> R Er X ) $.
    erth2.2 $e |- ( ph -> B e. X ) $.
    $( Basic property of equivalence relations.  Compare Theorem 73 of [Suppes]
       p. 82.  Assumes membership of the second argument in the domain.
       (Contributed by NM, 30-Jul-1995.)  (Revised by Mario Carneiro,
       6-Jul-2015.) $)
    erth2 $p |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) ) $=
      ( wbr cec wceq ersymb erth eqcom bitrdi bitrd ) ABCDHCBDHZBDIZCDIZJZABCDE
      FKAPRQJSACBDEFGLRQMNO $.
  $}

  ${
    erthi.1 $e |- ( ph -> R Er X ) $.
    erthi.2 $e |- ( ph -> A R B ) $.
    $( Basic property of equivalence relations.  Part of Lemma 3N of [Enderton]
       p. 57.  (Contributed by NM, 30-Jul-1995.)  (Revised by Mario Carneiro,
       9-Jul-2014.) $)
    erthi $p |- ( ph -> [ A ] R = [ B ] R ) $=
      ( wbr cec wceq ercl erth mpbid ) ABCDHBDICDIJGABCDEFABCDEFGKLM $.
  $}

  $( An equivalence class modulo the identity relation is a singleton.
     (Contributed by NM, 24-Oct-2004.) $)
  ecidsn $p |- [ A ] _I = { A } $=
    ( cid cec csn cima df-ec imai eqtri ) ABCBADZEIABFIGH $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Equality theorem for quotient set.  (Contributed by NM, 23-Jul-1995.) $)
    qseq1 $p |- ( A = B -> ( A /. C ) = ( B /. C ) ) $=
      ( vy vx wceq cv cec wrex cab cqs rexeq abbidv df-qs 3eqtr4g ) ABFZDGEGCHF
      ZEAIZDJQEBIZDJACKBCKPRSDQEABLMEDACNEDBCNO $.

    $( Equality theorem for quotient set.  (Contributed by NM, 23-Jul-1995.) $)
    qseq2 $p |- ( A = B -> ( C /. A ) = ( C /. B ) ) $=
      ( vy vx wceq cec wrex cab cqs eceq2 eqeq2d rexbidv abbidv df-qs 3eqtr4g
      cv ) ABFZDQZEQZAGZFZECHZDISTBGZFZECHZDICAJCBJRUCUFDRUBUEECRUAUDSABTKLMNED
      CAOEDCBOP $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.
    $( Closed form of ~ elqs .  (Contributed by Rodolfo Medina,
       12-Oct-2010.) $)
    elqsg $p |- ( B e. V ->
      ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) ) $=
      ( vy cv cec wceq wrex cqs eqeq1 rexbidv df-qs elab2g ) FGZAGDHZIZABJCQIZA
      BJFCBDKEPCIRSABPCQLMAFBDNO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x R $.
    elqs.1 $e |- B e. _V $.
    $( Membership in a quotient set.  (Contributed by NM, 23-Jul-1995.) $)
    elqs $p |- ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) $=
      ( cvv wcel cqs cv cec wceq wrex wb elqsg ax-mp ) CFGCBDHGCAIDJKABLMEABCDF
      NO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x R $.
    $( Membership in a quotient set.  (Contributed by NM, 23-Jul-1995.) $)
    elqsi $p |- ( B e. ( A /. R ) -> E. x e. A B = [ x ] R ) $=
      ( cqs wcel cv cec wceq wrex elqsg ibi ) CBDEZFCAGDHIABJABCDMKL $.
  $}

  ${
    $d R x $.  $d B x $.  $d A x $.
    $( Membership of an equivalence class in a quotient set.  (Contributed by
       Jeff Madsen, 10-Jun-2010.)  (Revised by Mario Carneiro, 9-Jul-2014.) $)
    ecelqsg $p |- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) ) $=
      ( vx wcel cec cv wceq wrex cqs eqid eceq1 eqeq2d rspcev mpan2 ecexg elqsg
      cvv wb syl biimpar sylan2 ) BAFZCDFZBCGZEHZCGZIZEAJZUFACKFZUDUFUFIZUJUFLU
      IULEBAUGBIUHUFUFUGBCMNOPUEUKUJUEUFSFUKUJTBDCQEAUFCSRUAUBUC $.
  $}

  ${
    ecelqsi.1 $e |- R e. _V $.
    $( Membership of an equivalence class in a quotient set.  (Contributed by
       NM, 25-Jul-1995.)  (Revised by Mario Carneiro, 9-Jul-2014.) $)
    ecelqsi $p |- ( B e. A -> [ B ] R e. ( A /. R ) ) $=
      ( cvv wcel cec cqs ecelqsg mpan ) CEFBAFBCGACHFDABCEIJ $.
  $}

  ${
    ecopqsi.1 $e |- R e. _V $.
    ecopqsi.2 $e |- S = ( ( A X. A ) /. R ) $.
    $( "Closure" law for equivalence class of ordered pairs.  (Contributed by
       NM, 25-Mar-1996.) $)
    ecopqsi $p |- ( ( B e. A /\ C e. A ) -> [ <. B , C >. ] R e. S ) $=
      ( wcel wa cop cxp cec opelxpi cqs ecelqsi eleqtrrdi syl ) BAHCAHIBCJZAAKZ
      HZRDLZEHBCAAMTUASDNESRDFOGPQ $.
  $}

  ${
    $d x y A $.  $d x y R $.
    $( A quotient set exists.  (Contributed by FL, 19-May-2007.)  (Revised by
       Mario Carneiro, 9-Jul-2014.) $)
    qsexg $p |- ( A e. V -> ( A /. R ) e. _V ) $=
      ( vy vx wcel cqs cv cec wceq wrex cab cvv df-qs abrexexg eqeltrid ) ACFAB
      GDHEHBIZJEAKDLMEDABNEDAQCOP $.
  $}

  ${
    qsex.1 $e |- A e. _V $.
    $( A quotient set exists.  (Contributed by NM, 14-Aug-1995.) $)
    qsex $p |- ( A /. R ) e. _V $=
      ( cvv wcel cqs qsexg ax-mp ) ADEABFDECABDGH $.
  $}

  ${
    $d x y A $.  $d x y R $.  $d x V $.
    $( The union of a quotient set.  (Contributed by NM, 9-Dec-2008.) $)
    uniqs $p |- ( R e. V -> U. ( A /. R ) = ( R " A ) ) $=
      ( vy vx wcel cv cec wceq wrex cab cuni ciun cqs cima wral ecexg ralrimivw
      cvv dfiun2g syl eqcomd df-qs unieqi csn df-ec a1i iuneq2i imaiun 3eqtr2ri
      iunid imaeq2i 3eqtr4g ) BCFZDGEGZBHZIEAJDKZLZEAUPMZABNZLBAOZUNUSURUNUPSFZ
      EAPUSURIUNVBEAUOCBQREDAUPSTUAUBUTUQEDABUCUDUSEABUOUEZOZMBEAVCMZOVAEAUPVDU
      PVDIUOAFUOBUFUGUHEBAVCUIVEABEAUKULUJUM $.
  $}

  ${
    $d x y A $.  $d x y ph $.  $d x y R $.
    qsss.1 $e |- ( ph -> R Er A ) $.
    $( A quotient set is a set of subsets of the base set.  (Contributed by
       Mario Carneiro, 9-Jul-2014.)  (Revised by Mario Carneiro,
       12-Aug-2015.) $)
    qsss $p |- ( ph -> ( A /. R ) C_ ~P A ) $=
      ( vx vy cqs cpw cv wcel cec wceq wrex vex elqs wss sseq1 syl5ibrcom velpw
      ecss imbitrrdi rexlimdvw biimtrid ssrdv ) AEBCGZBHZEIZUEJUGFIZCKZLZFBMAUG
      UFJZFBUGCENOAUJUKFBAUJUGBPZUKAULUJUIBPAUHCBDTUGUIBQREBSUAUBUCUD $.

    qsss.2 $e |- ( ph -> R e. V ) $.
    $( The union of a quotient set.  (Contributed by Mario Carneiro,
       11-Jul-2014.) $)
    uniqs2 $p |- ( ph -> U. ( A /. R ) = A ) $=
      ( cqs cuni crn cdm cima wcel wceq uniqs syl erdm imaeq2d eqtr4d imadmrn
      wer eqtrdi errn eqtrd ) ABCGHZCIZBAUDCCJZKZUEAUDCBKZUGACDLUDUHMFBCDNOAUFB
      CABCTZUFBMEBCPOQRCSUAAUIUEBMEBCUBOUC $.
  $}

  ${
    $d x y A $.  $d x y R $.
    snec.1 $e |- A e. _V $.
    $( The singleton of an equivalence class.  (Contributed by NM,
       29-Jan-1999.)  (Revised by Mario Carneiro, 9-Jul-2014.) $)
    snec $p |- { [ A ] R } = ( { A } /. R ) $=
      ( vy vx cv cec wceq csn wrex cab cqs eceq1 eqeq2d rexsn abbii df-qs df-sn
      3eqtr4ri ) DFZEFZBGZHZEAIZJZDKTABGZHZDKUDBLUFIUEUGDUCUGEACUAAHUBUFTUAABMN
      OPEDUDBQDUFRS $.
  $}

  ${
    ecqs.1 $e |- R e. _V $.
    $( Equivalence class in terms of quotient set.  (Contributed by NM,
       29-Jan-1999.) $)
    ecqs $p |- [ A ] R = U. ( { A } /. R ) $=
      ( cec csn cima cqs cuni df-ec cvv wcel wceq uniqs ax-mp eqtr4i ) ABDBAEZF
      ZPBGHZABIBJKRQLCPBJMNO $.
  $}

  ${
    $d y A $.
    ecid.1 $e |- A e. _V $.
    $( A set is equal to its converse epsilon coset.  (Note: converse epsilon
       is not an equivalence relation.)  (Contributed by NM, 13-Aug-1995.)
       (Revised by Mario Carneiro, 9-Jul-2014.) $)
    ecid $p |- [ A ] `' _E = A $=
      ( vy cep ccnv cec cv wcel wbr vex elec brcnv epelc 3bitri eqriv ) CADEZFZ
      ACGZQHARPIRADIRAHRAPCJZBKARDBSLRABMNO $.
  $}

  ${
    $d y A $.  $d y V $.
    $( A set is equal to its converse epsilon coset.  (Note: converse epsilon
       is not an equivalence relation.)  (Contributed by Jim Kingdon,
       8-Jan-2020.) $)
    ecidg $p |- ( A e. V -> [ A ] `' _E = A ) $=
      ( vy wcel cep ccnv cec cv wbr cvv wb elecg mpan brcnvg mpan2 epelg 3bitrd
      vex eqrdv ) ABDZCAEFZGZATCHZUBDZAUCUAIZUCAEIZUCADUCJDZTUDUEKCRZUCAUAJBLMT
      UGUEUFKUHAUCBJENOUCABPQS $.
  $}

  ${
    $d x y A $.
    $( A set is equal to its quotient set mod converse epsilon.  (Note:
       converse epsilon is not an equivalence relation.)  (Contributed by NM,
       13-Aug-1995.)  (Revised by Mario Carneiro, 9-Jul-2014.) $)
    qsid $p |- ( A /. `' _E ) = A $=
      ( vy vx cep ccnv cqs cv cec wceq wrex wcel vex eqeq2i equcom bitri rexbii
      ecid elqs risset 3bitr4i eqriv ) BADEZFZABGZCGZUBHZIZCAJUEUDIZCAJUDUCKUDA
      KUGUHCAUGUDUEIUHUFUEUDUECLQMBCNOPCAUDUBBLRCUDASTUA $.
  $}

  ${
    $d x A $.  $d x B $.  $d x R $.  $d x ps $.  $d x ch $.
    ectocl.1 $e |- S = ( B /. R ) $.
    ectocl.2 $e |- ( [ x ] R = A -> ( ph <-> ps ) ) $.
    ${
      ectocld.3 $e |- ( ( ch /\ x e. B ) -> ph ) $.
      $( Implicit substitution of class for equivalence class.  (Contributed by
         Mario Carneiro, 9-Jul-2014.) $)
      ectocld $p |- ( ( ch /\ A e. S ) -> ps ) $=
        ( wcel cv cec wceq wrex cqs elqsi eleq2s wa wb syl5ibcom rexlimdva syl5
        eqcoms imp ) CEHLZBUGEDMZGNZOZDFPZCBUKEFGQHDFEGRISCUJBDFCUHFLTAUJBKABUA
        UIEJUEUBUCUDUF $.
    $}

    ectocl.3 $e |- ( x e. B -> ph ) $.
    $( Implicit substitution of class for equivalence class.  (Contributed by
       NM, 23-Jul-1995.)  (Revised by Mario Carneiro, 9-Jul-2014.) $)
    ectocl $p |- ( A e. S -> ps ) $=
      ( wtru wcel tru cv adantl ectocld mpan ) KDGLBMABKCDEFGHICNELAKJOPQ $.
  $}

  ${
    $d x y R $.  $d x y A $.  $d x y B $.
    $( An element of a quotient set is inhabited.  (Contributed by Jim Kingdon,
       21-Aug-2019.) $)
    elqsn0m $p |- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B ) $=
      ( vy cv cec wcel wex cdm wceq cqs eqid eleq2 exbidv biimpar ecdmn0m sylib
      wa ectocld ) AFZEFZDGZHZAIZUACHZAIDJZBKZECBDBDLZUIMUCCKUDUFAUCCUANOUHUBBH
      ZSUBUGHZUEUHUKUJUGBUBNPAUBDQRT $.

    $( A quotient set doesn't contain the empty set.  (Contributed by NM,
       24-Aug-1995.) $)
    elqsn0 $p |- ( ( dom R = A /\ B e. ( A /. R ) ) -> B =/= (/) ) $=
      ( vx cdm wceq cqs wcel wa cv wex c0 wne elqsn0m n0r syl ) CEAFBACGHIDJBHD
      KBLMDABCNDBOP $.
  $}

  ${
    $d A x $.  $d B x $.  $d R x $.
    $( Membership of an equivalence class in a quotient set.  (Contributed by
       NM, 30-Jul-1995.) $)
    ecelqsdm $p |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A ) $=
      ( vx cdm wceq cec cqs wcel wa cv wex elqsn0m ecdmn0m sylibr simpl eleqtrd
      ) CEZAFZBCGZACHIZJZBRAUBDKTIDLBRIDATCMDBCNOSUAPQ $.
  $}

  $( A square Cartesian product is an equivalence relation (in general it's not
     a poset).  (Contributed by FL, 31-Jul-2009.)  (Revised by Mario Carneiro,
     12-Aug-2015.) $)
  xpider $p |- ( A X. A ) Er A $=
    ( cxp wer wrel cdm wceq ccnv ccom cun relxp dmxpid cnvxp xpidtr uneq1 unss2
    wss wi unidm wa eqtr sseq2 biimpd syl mpan2 syl2im mp2 df-er mpbir3an ) AAA
    BZCUIDUIEAFUIGZUIUIHZIZUIPZAAJAKUJUIFZUKUIPZUMAALAMUNUJUIIZUIUIIZFZUOULUPPZ
    UMUJUIUINUKUIUJOURUQUIFZUSUMQZUIRURUTSUPUIFZVAUPUQUITVBUSUMUPUIULUAUBUCUDUE
    UFAUIUGUH $.

  ${
    $d a u v w x A $.  $d u v w x B $.  $d u v w R $.  $d a y A $.
    $( The intersection of a nonempty family of equivalence relations is an
       equivalence relation.  (Contributed by Mario Carneiro, 27-Sep-2015.) $)
    iinerm $p |- ( ( E. y y e. A /\ A. x e. A R Er B ) ->
                  |^|_ x e. A R Er B ) $=
      ( va vu cv wcel wral bitri wa cvv wbr cop df-br wb opex eliin ax-mp vv vw
      wex wer ciin eleq1w cbvexv cxp wrel wrex r19.2m errel df-rel sylib reximi
      wss iinss 3syl sylibr wi ersymb biimpd 3imtr3g ral2imi adantl vex 3imtr4g
      id imp r19.26 ertr anbi12i biimtrrid simpl simpr erref expcom ralimdv cdm
      com12 opeldm eleq2d biimpa sylan2 rexlimivw syl ex expdimp impbid bitr4di
      erdm iserd sylanbr ) BHCIZBUCZAHCIZAUCZDEUDZACJZDACEUEZUDWQFHCIZFUCWOWPXA
      AFAFCUFUGXAWNFBFBCUFUGKWQWSLZGUAUBDWTXBWTMMUHZUPZWTUIXBWRACUJEXCUPZACUJXD
      WRAACUKWRXEACWREUIXEDEULEUMUNUOACEXCUQURWTUMUSXBGHZUAHZWTNZXGXFWTNZXBXFXG
      OZEIZACJZXGXFOZEIZACJZXHXIWSXLXOUTWQWRXKXNACWRXFXGENZXGXFENZXKXNWRXPXQWRX
      FXGEDWRVHZVAVBXFXGEPZXGXFEPVCVDVEXHXJWTIZXLXFXGWTPXJMIXTXLQXFXGGVFZUAVFZR
      AXJCEMSTKZXIXMWTIZXOXGXFWTPXMMIYDXOQXGXFYBYARAXMCEMSTKVGVIXBXHXGUBHZWTNZL
      ZXFYEWTNZXBXLXGYEOZEIZACJZLZXFYEOZEIZACJZYGYHYLXKYJLZACJZXBYOXKYJACVJWSYQ
      YOUTWQWRYPYNACWRXPXGYEENZLXFYEENYPYNWRXFXGYEEDXRVKXPXKYRYJXSXGYEEPVLXFYEE
      PVCVDVEVMXHXLYFYKYCYFYIWTIZYKXGYEWTPYIMIYSYKQXGYEYBUBVFZRAYICEMSTKVLYHYMW
      TIZYOXFYEWTPYMMIUUAYOQXFYEYAYTRAYMCEMSTKVGVIXBXFDIZXFXFOZEIZACJZXFXFWTNZX
      BUUBUUEWSUUBUUEUTWQUUBWSUUEUUBWRUUDACWRUUBUUDWRUUBLZXFXFENUUDUUGXFEDWRUUB
      VNWRUUBVOVPXFXFEPUNVQVRVTVEWQWSUUEUUBWSUUELWRUUDLZACJZWQUUBWRUUDACVJWQUUI
      UUBWQUUILUUHACUJUUBUUHAACUKUUHUUBACUUDWRXFEVSZIZUUBXFXFEYAYAWAWRUUKUUBWRU
      UJDXFDEWKWBWCWDWEWFWGVMWHWIUUFUUCWTIZUUEXFXFWTPUUCMIUULUUEQXFXFYAYARAUUCC
      EMSTKWJWLWM $.

    $( The relative intersection of a family of equivalence relations is an
       equivalence relation.  (Contributed by Mario Carneiro, 27-Sep-2015.) $)
    riinerm $p |- ( ( E. y y e. A /\ A. x e. A R Er B ) ->
      ( ( B X. B ) i^i |^|_ x e. A R ) Er B ) $=
      ( va cv wcel wex wer wral wa cxp ciin cin iinerm wb eleq1 cbvexv bitri
      wceq wss erssxp ralimi riinm sylan ereq1 syl sylan2br ancoms mpbird ) BGZ
      CHZBIZDEJZACKZLDDDMZACENZOZJZDURJZABCDEPUPUNUTVAQZUNUPAGZCHZAIZVBVEFGZCHZ
      FIUNVDVGAFVCVFCRSVGUMFBVFULCRSTUPVELUSURUAZVBUPEUQUBZACKVEVHUOVIACDEUCUDA
      UQECUEUFDUSURUGUHUIUJUK $.
  $}

  ${
    $d x y z B $.  $d x y z ph $.  $d x y z R $.
    erinxp.r $e |- ( ph -> R Er A ) $.
    erinxp.a $e |- ( ph -> B C_ A ) $.
    $( A restricted equivalence relation is an equivalence relation.
       (Contributed by Mario Carneiro, 10-Jul-2015.)  (Revised by Mario
       Carneiro, 12-Aug-2015.) $)
    erinxp $p |- ( ph -> ( R i^i ( B X. B ) ) Er B ) $=
      ( vx vy vz wrel cv wbr wa wcel w3a brinxp2 sylib simp2d adantr simp3d cxp
      cin wss inss2 relxp relss mp2 a1i simpr simp1d wer ersym syl3anbrc simprr
      adantrr ertrd sselda erref ex pm4.71rd brin brxp anidm bitri anbi2i iserd
      bitr4di ) AGHICDCCUAZUBZVIJZAVIVHUCVHJVJDVHUDCCUEVIVHUFUGUHAGKZHKZVILZMZV
      LCNZVKCNZVLVKDLVLVKVILVNVPVOVKVLDLZVNVMVPVOVQOAVMUIVKVLCCDPQZRVNVPVOVQVRU
      JZVNVKVLDBABDUKZVMESVNVPVOVQVRTZULVLVKCCDPUMAVMVLIKZVILZMZMZVPWBCNZVKWBDL
      VKWBVILAVMVPWCVSUOWEVOWFVLWBDLZWEWCVOWFWGOAVMWCUNVLWBCCDPQZRWEVKVLWBDBAVT
      WDESAVMVQWCWAUOWEVOWFWGWHTUPVKWBCCDPUMAVPVKVKDLZVPMZVKVKVILZAVPWIAVPWIAVP
      MVKDBAVTVPESACBVKFUQURUSUTWKWIVKVKVHLZMWJVKVKDVHVAWLVPWIWLVPVPMVPVKVKCCVB
      VPVCVDVEVDVGVF $.
  $}

  $( Restrict the relation in an equivalence class to a base set.  (Contributed
     by Mario Carneiro, 10-Jul-2015.) $)
  ecinxp $p |- ( ( ( R " A ) C_ A /\ B e. A ) ->
      [ B ] R = [ B ] ( R i^i ( A X. A ) ) ) $=
    ( cima wss wcel csn cxp cin cec wceq simpr snssd df-ss sylib imaeq2d ineq1d
    wa imass2 df-ec syl simpl sstrd eqtr2d imainrect eqtr4di 3eqtr4g ) CADZAEZB
    AFZRZCBGZDZCAAHIZULDZBCJBUNJUKUMCULAIZDZAIZUOUKURUMAIZUMUKUQUMAUKUPULCUKULA
    EZUPULKUKBAUIUJLMZULANOPQUKUMAEUSUMKUKUMUHAUKUTUMUHEVAULACSUAUIUJUBUCUMANOU
    DAACULUEUFBCTBUNTUG $.

  ${
    $d x y A $.  $d x y R $.
    $( Restrict the equivalence relation in a quotient set to the base set.
       (Contributed by Mario Carneiro, 23-Feb-2015.) $)
    qsinxp $p |- ( ( R " A ) C_ A ->
        ( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) ) $=
      ( vy vx cima wss cv cec wceq wrex cab cxp cin cqs wcel wa ecinxp rexbidva
      eqeq2d df-qs abbidv 3eqtr4g ) BAEAFZCGZDGZBHZIZDAJZCKUDUEBAALMZHZIZDAJZCK
      ABNAUINUCUHULCUCUGUKDAUCUEAOPUFUJUDAUEBQSRUADCABTDCAUITUB $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x R $.  $d x X $.
    $( If an element of a quotient set contains a given element, it is equal to
       the equivalence class of the element.  (Contributed by Mario Carneiro,
       12-Aug-2015.) $)
    qsel $p |- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R ) $=
      ( vx wer cqs wcel cec wceq cv wi eqid eleq2 eqeq1 imbi12d wbr wa cvv syl5
      wb vex elecg mpan2 ibi simpll simpr erthi ex ectocld 3impia ) EDGZBADHZIC
      BIZBCDJZKZCFLZDJZIZUSUPKZMUOUQMUMFBADUNUNNUSBKUTUOVAUQUSBCOUSBUPPQUTURCDR
      ZUMURAIZSZVAUTVBUTURTIUTVBUBFUCCURDUSTUDUEUFVDVBVAVDVBSURCDEUMVCVBUGVDVBU
      HUIUJUAUKUL $.
  $}

  ${
    $d y z A $.  $d x B $.  $d x C $.  $d x D $.  $d x y z ph $.  $d x y z R $.
    $d y z F $.  $d x y z X $.  $d x y z Y $.
    qlift.1 $e |- F = ran ( x e. X |-> <. [ x ] R , A >. ) $.
    qlift.2 $e |- ( ( ph /\ x e. X ) -> A e. Y ) $.
    qlift.3 $e |- ( ph -> R Er X ) $.
    qlift.4 $e |- ( ph -> X e. _V ) $.
    $( ` F ` , a function lift, is a subset of ` R X. S ` .  (Contributed by
       Mario Carneiro, 23-Dec-2016.) $)
    qliftlem $p |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) ) $=
      ( cvv wcel cv cec cqs wer erex sylc ecelqsg sylan ) ADLMZBNZFMUCDOFDPMAFD
      QFLMUBJKFDLRSFUCDLTUA $.

    $( ` F ` , a function lift, is a subset of ` R X. S ` .  (Contributed by
       Mario Carneiro, 23-Dec-2016.) $)
    qliftrel $p |- ( ph -> F C_ ( ( X /. R ) X. Y ) ) $=
      ( cv cec cqs qliftlem fliftrel ) ABBLDMCFDNGEFHABCDEFGHIJKOIP $.

    $( Elementhood in the relation ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.) $)
    qliftel $p |- ( ph -> ( [ C ] R F D <-> E. x e. X ( C R x /\ D = A ) ) ) $=
      ( cec wbr cv wceq wa wrex cqs qliftlem wcel wer adantr simpr erth2 anbi1d
      fliftel rexbidva bitr4d ) ADFNZEGOUKBPZFNZQZECQZRZBHSDULFOZUORZBHSABUMCUK
      EHFTIGHJABCFGHIJKLMUAKUHAURUPBHAULHUBZRZUQUNUOUTDULFHAHFUCUSLUDAUSUEUFUGU
      IUJ $.

    $( Elementhood in the relation ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.) $)
    qliftel1 $p |- ( ( ph /\ x e. X ) -> [ x ] R F A ) $=
      ( cv cec cqs qliftlem fliftel1 ) ABBLDMCFDNGEFHABCDEFGHIJKOIP $.

    ${
      qliftfun.4 $e |- ( x = y -> A = B ) $.
      $( The function ` F ` is the unique function defined by
         ` F `` [ x ] = A ` , provided that the well-definedness condition
         holds.  (Contributed by Mario Carneiro, 23-Dec-2016.) $)
      qliftfun $p |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) ) $=
        ( cv cec wceq wi wal wa wfun wral wbr cqs qliftlem fliftfun wcel adantr
        eceq1 wer simpr ercl ercl2 jca ex pm4.71rd simprl pm5.32da bitrd imbi1d
        erth impexp bitrdi 2albidv r2al bitr4di bitr4d ) AGUABOZFPZCOZFPZQZDEQZ
        RZCHUBBHUBZVHVJFUCZVMRZCSBSZABCVIDVKEHFUDIGHJABDFGHIJKLMUEKVHVJFUINUFAV
        RVHHUGZVJHUGZTZVNRZCSBSVOAVQWBBCAVQWAVLTZVMRWBAVPWCVMAVPWAVPTWCAVPWAAVP
        WAAVPTZVSVTWDVHVJFHAHFUJZVPLUHZAVPUKZULWDVHVJFHWFWGUMUNUOUPAWAVPVLAWATV
        HVJFHAWEWALUHAVSVTUQVAURUSUTWAVLVMVBVCVDVNBCHHVEVFVG $.

      qliftfund.6 $e |- ( ( ph /\ x R y ) -> A = B ) $.
      $( The function ` F ` is the unique function defined by
         ` F `` [ x ] = A ` , provided that the well-definedness condition
         holds.  (Contributed by Mario Carneiro, 23-Dec-2016.) $)
      qliftfund $p |- ( ph -> Fun F ) $=
        ( wfun cv wbr wceq wal wi ex alrimivv qliftfun mpbird ) AGPBQCQFRZDESZU
        AZCTBTAUHBCAUFUGOUBUCABCDEFGHIJKLMNUDUE $.
    $}

    $( The function ` F ` is the unique function defined by
       ` F `` [ x ] = A ` , provided that the well-definedness condition holds.
       (Contributed by Mario Carneiro, 23-Dec-2016.) $)
    qliftfuns $p |- ( ph -> ( Fun F <->
      A. y A. z ( y R z -> [_ y / x ]_ A = [_ z / x ]_ A ) ) ) $=
      ( cv csb cec cop cmpt crn wcel nfcv nfcsb1v nfop weq eceq1 csbeq1a cbvmpt
      opeq12d rneqi eqtri wral ralrimiva nfel1 eleq1d mpan9 csbeq1 qliftfun
      rspc ) ACDBCNZEOZBDNZEOFGHIGBHBNZFPZEQZRZSCHUSFPZUTQZRZSJVEVHBCHVDVGCVDUA
      BVFUTBVFUABUSEUBZUCBCUDZVCVFEUTVBUSFUEBUSEUFZUHUGUIUJAEITZBHUKUSHTUTITZAV
      LBHKULVLVMBUSHBUTIVIUMVJEUTIVKUNURUOLMBUSVAEUPUQ $.

    $( The domain and codomain of the function ` F ` .  (Contributed by Mario
       Carneiro, 23-Dec-2016.) $)
    qliftf $p |- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) ) $=
      ( vy wfun cv cec cmpt crn wf cqs wceq qliftlem fliftf wrex cab df-qs eqid
      rnmpt eqtr4i a1i feq2d bitr4d ) AEMBFBNDOZPZQZGERFDSZGERABULCUOGEFHABCDEF
      GHIJKUAIUBAUOUNGEUOUNTAUOLNULTBFUCLUDUNBLFDUEBLFULUMUMUFUGUHUIUJUK $.

    qliftval.4 $e |- ( x = C -> A = B ) $.
    qliftval.6 $e |- ( ph -> Fun F ) $.
    $( The value of the function ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.) $)
    qliftval $p |- ( ( ph /\ C e. X ) -> ( F ` [ C ] R ) = B ) $=
      ( cv cec cqs qliftlem eceq1 fliftval ) ABBPZFQCEFQDHFRIGHEJABCFGHIJKLMSKU
      BEFTNOUA $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z R $.  $d x y z ps $.
    ecoptocl.1 $e |- S = ( ( B X. C ) /. R ) $.
    ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
    ecoptocl.3 $e |- ( ( x e. B /\ y e. C ) -> ph ) $.
    $( Implicit substitution of class for equivalence class of ordered pair.
       (Contributed by NM, 23-Jul-1995.) $)
    ecoptocl $p |- ( A e. S -> ps ) $=
      ( vz cxp cqs wcel cv cec wceq wi wrex elqsi cop eceq1 eqeq2d imbi1d wa wb
      eqid eqcoms syl5ibcom optocl rexlimiv syl eleq2s ) BEFGNZHOZIEUQPEMQZHRZS
      ZMUPUABMUPEHUBUTBMUPECQZDQZUCZHRZSZBTUTBTCDURFGUPUPUIVCURSZVEUTBVFVDUSEVC
      URHUDUEUFVAFPVBGPUGAVEBLABUHVDEKUJUKULUMUNJUO $.
  $}

  ${
    $d x y z w A $.  $d z w B $.  $d x y z w C $.  $d x y z w D $.  $d z w S $.
    $d x y z w R $.  $d x y ps $.  $d z w ch $.
    2ecoptocl.1 $e |- S = ( ( C X. D ) /. R ) $.
    2ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
    2ecoptocl.3 $e |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) ) $.
    2ecoptocl.4 $e |- ( ( ( x e. C /\ y e. D ) /\
                      ( z e. C /\ w e. D ) ) -> ph ) $.
    $( Implicit substitution of classes for equivalence classes of ordered
       pairs.  (Contributed by NM, 23-Jul-1995.) $)
    2ecoptocl $p |- ( ( A e. S /\ B e. S ) -> ch ) $=
      ( wcel wi cv cop cec wceq imbi2d wa ex ecoptocl com12 impcom ) IMRHMRZCUJ
      BSUJCSFGIJKLMNFTZGTZUALUBIUCBCUJPUDUJUKJRULKRUEZBUMASUMBSDEHJKLMNDTZETZUA
      LUBHUCABUMOUDUNJRUOKRUEUMAQUFUGUHUGUI $.
  $}

  ${
    $d x y z w v u A $.  $d z w v u B $.  $d v u C $.  $d x y z w v u D $.
    $d z w v u S $.  $d x y z w v u R $.  $d x y ps $.  $d z w ch $.
    $d v u th $.
    3ecoptocl.1 $e |- S = ( ( D X. D ) /. R ) $.
    3ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
    3ecoptocl.3 $e |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) ) $.
    3ecoptocl.4 $e |- ( [ <. v , u >. ] R = C -> ( ch <-> th ) ) $.
    3ecoptocl.5 $e |- ( ( ( x e. D /\ y e. D ) /\
                      ( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ph ) $.
    $( Implicit substitution of classes for equivalence classes of ordered
       pairs.  (Contributed by NM, 9-Aug-1995.) $)
    3ecoptocl $p |- ( ( A e. S /\ B e. S /\ C e. S ) -> th ) $=
      ( wcel wa wi cop cec wceq imbi2d 3expib ecoptocl com12 2ecoptocl 3impib
      cv ) KPUBZLPUBZMPUBZDUPUQUCUODUOBUDUOCUDUODUDGHIJLMNNOPQGUNZHUNZUEOUFLUGB
      CUOSUHIUNZJUNZUEOUFMUGCDUOTUHUOURNUBUSNUBUCZUTNUBVANUBUCZUCZBVDAUDVDBUDEF
      KNNOPQEUNZFUNZUEOUFKUGABVDRUHVENUBVFNUBUCVBVCAUAUIUJUKULUKUM $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u D $.  $d x y z w v u .~ $.  $d x y H $.  $d z w v u G $.
    $d x y ph $.  $d z w v u ps $.
    brecop.1 $e |- .~ e. _V $.
    brecop.2 $e |- .~ Er ( G X. G ) $.
    brecop.4 $e |- H = ( ( G X. G ) /. .~ ) $.
    brecop.5 $e |- .<_ = { <. x , y >. | ( ( x e. H /\ y e. H ) /\
                      E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\
                      y = [ <. v , u >. ] .~ ) /\ ph ) ) } $.
    brecop.6 $e |- ( ( ( ( z e. G /\ w e. G ) /\ ( A e. G /\ B e. G ) ) /\
                       ( ( v e. G /\ u e. G ) /\ ( C e. G /\ D e. G ) ) ) ->
                     ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\
                         [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) ->
                     ( ph <-> ps ) ) ) $.
    $( Binary relation on a quotient set.  Lemma for real number construction.
       (Contributed by NM, 29-Jan-1996.) $)
    brecop $p |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) ->
                ( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> ps ) ) $=
      ( wcel wa cop cec wbr cv wceq wex ecopqsi copab df-br eleq2i bitri anbi1d
      wb eqeq1 4exbidv anbi2d opelopab2 bitrid syl2an opeq12 eceq1d anim12i cxp
      wi opelxpi opelxp wer a1i id ereldm bitr3id imbitrrid im2anan9 an4s com13
      ex mpdd pm5.74d cgsex4g eqcom anbi12i biimt anbi12d 3bitr4d bitrd ) INUBJ
      NUBUCZKNUBLNUBUCZUCZIJUDZMUEZKLUDZMUEZPUFZWMEUGZFUGZUDZMUEZUHZWOGUGZHUGZU
      DZMUEZUHZUCZAUCZHUIGUIFUIEUIZBWIWMOUBZWOOUBZWPXIUPWJNIJMOQSUJNKLMOQSUJWPW
      MWOUDZCUGZOUBDUGZOUBUCXMWTUHZXNXEUHZUCZAUCZHUIGUIFUIEUIZUCCDUKZUBZXJXKUCX
      IWPXLPUBYAWMWOPULPXTXLTUMUNXSXAXPUCZAUCZHUIGUIFUIEUIXICDWMWOOOXMWMUHZXRYC
      EFGHYDXQYBAYDXOXAXPXMWMWTUQUOUOURXNWOUHZYCXHEFGHYEYBXGAYEXPXFXAXNWOXEUQUS
      UOURUTVAVBWKWTWMUHZXEWOUHZUCZWKAVGZUCZHUIGUIFUIEUIWKBVGZXIBYIYKYHEFGHIJKL
      NNWQIUHWRJUHUCZYFXBKUHXCLUHUCZYGYLWSWLMWQWRIJVCVDYMXDWNMXBXCKLVCVDVEYHWKA
      BYHWKWQNUBWRNUBUCZXBNUBXCNUBUCZUCZABUPZYFWIYNYGWJYOWIYNYFWLNNVFZUBZIJNNVH
      YNWSYRUBYFYSWQWRNNVIYFWSWLMYRYRMVJZYFRVKYFVLVMVNVOWJYOYGWNYRUBZKLNNVHYOXD
      YRUBYGUUAXBXCNNVIYGXDWNMYRYTYGRVKYGVLVMVNVOVPYPWKYHYQYPWKYHYQVGZYNWIYOWJU
      UBUAVQVSVRVTWAWBWKXHYJEFGHWKXGYHAYIXGYHUPWKXAYFXFYGWMWTWCWOXEWCWDVKWKAWEW
      FURWKBWEWGWH $.
  $}

  ${
    $d p q r s t u w x y z A $.  $d p q r s t u w x y z B $.  $d p q x y z L $.
    $d p q w x y z J $.  $d p q r s t u w x y z R $.  $d p q w x y z K $.
    $d p q r s t u w x y z S $.  $d p q r s t u w x y z .+ $.
    $d p q r s t u w x y z ph $.  $d p q r s t u w x y z T $.
    $d p q r s t u w z X $.  $d p q r s t u w z Y $.
    eropr.1 $e |- J = ( A /. R ) $.
    eropr.2 $e |- K = ( B /. S ) $.
    eropr.3 $e |- ( ph -> T e. Z ) $.
    eropr.4 $e |- ( ph -> R Er U ) $.
    eropr.5 $e |- ( ph -> S Er V ) $.
    eropr.6 $e |- ( ph -> T Er W ) $.
    eropr.7 $e |- ( ph -> A C_ U ) $.
    eropr.8 $e |- ( ph -> B C_ V ) $.
    eropr.9 $e |- ( ph -> C C_ W ) $.
    eropr.10 $e |- ( ph -> .+ : ( A X. B ) --> C ) $.
    eropr.11 $e |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) )
                      -> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) ) $.
    $( Lemma for ~ eroprf .  (Contributed by Jeff Madsen, 10-Jun-2010.)
       (Revised by Mario Carneiro, 9-Jul-2014.) $)
    eroveu $p |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> E! z E. p e. A E. q e. B
                ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) $=
      ( vw wcel wa cv cec wceq co wrex wex weq wal weu cqs elqsi eleq2s anim12i
      wi adantl reeanv sylibr adantr ecexg elisset 3syl biantrud 2rexbidv mpbid
      19.42v bicomi rexbii rexcom4 bitri sylib eceq1 eqeq2d anbi1d oveq1 eceq1d
      cvv anbi12d anbi2d oveq2 cbvrex2v anbi12i bitr4i wbr wer wss simprll erth
      sseldd simprrl wf fovcdmd 3imtr3d eqeq2 biimprcd syl6 impd eqeq1 bi2anan9
      cxp imbi12d syl5ibrcom anassrs rexlimdvva biimtrrid alrimivv eu4 sylanbrc
      wb ) AQMUPZRNUPZUQZUQZQUCURZIUSZUTZRUBURZJUSZUTZUQZBURZYJYMHVAZKUSZUTZUQZ
      UBFVBZUCEVBZBVCZUUCYPUOURZYSUTZUQZUBFVBUCEVBZUQZBUOVDZVKZUOVEBVEUUCBVFYIY
      PYTBVCZUQZUBFVBZUCEVBZUUDYIYPUBFVBUCEVBZUUOYIYLUCEVBZYOUBFVBZUQZUUPYHUUSA
      YFUUQYGUURUUQQEIVGMUCEQIVHUDVIUURRFJVGNUBFRJVHUEVIVJVLYLYOUCUBEFVMVNYIYPU
      UMUCUBEFYIUULYPYIKSUPZYSWMUPUULAUUTYHUFVOYRSKVPBYSWMVQVRVSVTWAUUOUUBBVCZU
      CEVBUUDUUNUVAUCEUUNUUABVCZUBFVBUVAUUMUVBUBFUVBUUMYPYTBWBWCWDUUAUBBFWEWFWD
      UUBUCBEWEWFWGYIUUKBUOAUUKYHUUIQUAURZIUSZUTZRDURZJUSZUTZUQZYQUVCUVFHVAZKUS
      ZUTZUQZDFVBZQTURZIUSZUTZRCURZJUSZUTZUQZUUEUVOUVRHVAZKUSZUTZUQZCFVBZUQZTEV
      BUAEVBZAUUJUWHUVNUAEVBZUWFTEVBZUQUUIUVNUWFUATEEVMUUCUWIUUHUWJUUAUVMUVEYOU
      QZYQUVCYMHVAZKUSZUTZUQUCUBUADEFUCUAVDZYPUWKYTUWNUWOYLUVEYOUWOYKUVDQYJUVCI
      WHWIWJUWOYSUWMYQUWOYRUWLKYJUVCYMHWKWLWIWNUBDVDZUWKUVIUWNUVLUWPYOUVHUVEUWP
      YNUVGRYMUVFJWHWIWOUWPUWMUVKYQUWPUWLUVJKYMUVFUVCHWPWLWIWNWQUUGUWEUVQYOUQZU
      UEUVOYMHVAZKUSZUTZUQUCUBTCEFUCTVDZYPUWQUUFUWTUXAYLUVQYOUXAYKUVPQYJUVOIWHW
      IWJUXAYSUWSUUEUXAYRUWRKYJUVOYMHWKWLWIWNUBCVDZUWQUWAUWTUWDUXBYOUVTUVQUXBYN
      UVSRYMUVRJWHWIWOUXBUWSUWCUUEUXBUWRUWBKYMUVRUVOHWPWLWIWNWQWRWSAUWGUUJUATEE
      UWGUVMUWEUQZCFVBDFVBAUVCEUPZUVOEUPZUQZUQZUUJUVMUWEDCFFVMUXGUXCUUJDCFFAUXF
      UVFFUPZUVRFUPZUQZUXCUUJVKAUXFUXJUQZUQZUVMUWEUUJUXLUWEUUJVKUVMUVDUVPUTZUVG
      UVSUTZUQZUWDUQZUVKUUEUTZVKUXLUXOUWDUXQUXLUXOUVKUWCUTZUWDUXQVKUXLUVCUVOIWT
      ZUVFUVRJWTZUQUVJUWBKWTUXOUXRUNUXLUXSUXMUXTUXNUXLUVCUVOILALIXAUXKUGVOUXLEL
      UVCAELXBUXKUJVOAUXDUXEUXJXCZXEXDUXLUVFUVRJOAOJXAUXKUHVOUXLFOUVFAFOXBUXKUK
      VOAUXFUXHUXIXFZXEXDWNUXLUVJUWBKPAPKXAUXKUIVOUXLGPUVJAGPXBUXKULVOUXLUVCUVF
      GEFHAEFXPGHXGUXKUMVOUYAUYBXHXEXDXIUWDUXQUXRUUEUWCUVKXJXKXLXMUVMUWEUXPUUJU
      XQUVIUWEUXPYEUVLUVIUWAUXOUWDUVEUVQUXMUVHUVTUXNQUVDUVPXNRUVGUVSXNXOWJVOUVL
      UUJUXQYEUVIYQUVKUUEXNVLXQXRXMXSXTYAXTYAVOYBUUCUUHBUOUUJUUAUUGUCUBEFUUJYTU
      UFYPYQUUEYSXNWOVTYCYD $.

    eropr.12 $e |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B
                ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) } $.
    $( Lemma for ~ eroprf .  (Contributed by Jeff Madsen, 10-Jun-2010.)
       (Revised by Mario Carneiro, 30-Dec-2014.) $)
    erovlem $p |- ( ph -> .+^ = ( x e. J , y e. K |->
      ( iota z E. p e. A E. q e. B
        ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) ) $=
      ( vw cv cec wceq wa wrex coprab wcel cio cmpo simpl reximi cqs eleq2i vex
      co elqs bitri anbi12i reeanv bitr4i sylibr pm4.71ri wb eroveu iota1 eqcom
      weu syl bitrdi pm5.32da bitrid oprabbidv df-mpo nfiota1 nfeq2 nfan anbi2d
      nfv eqeq1 cbvoprab3 eqtr4i 3eqtr4g ) ABURZUDURZLUSUTZCURZUCURZMUSUTZVAZDU
      RZXAXDJVLNUSUTZVAZUCHVBZUDGVBZBCDVCWTPVDZXCQVDZVAZXGXKDVEZUTZVAZBCDVCZKBC
      PQXOVFZAXKXQBCDXKXNXKVAAXQXKXNXKXFUCHVBZUDGVBZXNXJXTUDGXIXFUCHXFXHVGVHVHX
      NXBUDGVBZXEUCHVBZVAYAXLYBXMYCXLWTGLVIZVDYBPYDWTUEVJUDGWTLBVKVMVNXMXCHMVIZ
      VDYCQYEXCUFVJUCHXCMCVKVMVNVOXBXEUDUCGHVPVQVRVSAXNXKXPAXNVAZXKXOXGUTZXPYFX
      KDWDXKYGVTADEFGHIJLMNOPQRSWTXCTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOWAXKDWBWEXOX
      GWCWFWGWHWIUPXSXNUQURZXOUTZVAZBCUQVCXRBCUQPQXOWJXQYJBCDUQXQUQWOXNYIDXNDWO
      DYHXOXKDWKWLWMXGYHUTXPYIXNXGYHXOWPWNWQWRWS $.

    eropr.13 $e |- ( ph -> R e. X ) $.
    eropr.14 $e |- ( ph -> S e. Y ) $.
    eropr.15 $e |- L = ( C /. T ) $.
    $( Functionality of an operation defined on equivalence classes.
       (Contributed by Jeff Madsen, 10-Jun-2010.)  (Revised by Mario Carneiro,
       30-Dec-2014.) $)
    eroprf $p |- ( ph -> .+^ : ( J X. K ) --> L ) $=
      ( cxp wf cv cec wceq wa co wrex cio cmpo wcel wral cab wi ad2antrr adantr
      cqs ecelqsg syl2anc eleqtrrdi eleq1a syl adantld rexlimdvva abssdv eroveu
      fovcdmda iotacl sseldd ralrimivva eqid fmpo sylib erovlem feq1d mpbird
      weu ) APQVCZRKVDWTRBCPQBVEZUGVEZLVFVGCVEZUFVEZMVFVGVHZDVEZXBXDJVIZNVFZVGZ
      VHZUFHVJUGGVJZDVKZVLZVDZAXLRVMZCQVNBPVNXNAXOBCPQAXAPVMXCQVMVHZVHZXKDVOZRX
      LXQXKDRXQXJXFRVMZUGUFGHXQXBGVMXDHVMVHZVHZXIXSXEYAXHRVMXIXSVPYAXHINVSZRYAN
      UCVMZXGIVMXHYBVMAYCXPXTUJVQXQXBXDIGHJAGHVCIJVDXPUQVRWIIXGNUCVTWAVBWBXHRXF
      WCWDWEWFWGXQXKDWSXLXRVMADEFGHIJLMNOPQSTXAXCUCUDUEUFUGUHUIUJUKULUMUNUOUPUQ
      URWHXKDWJWDWKWLBCPQXLRXMXMWMWNWOAWTRKXMABCDEFGHIJKLMNOPQSTUCUDUEUFUGUHUIU
      JUKULUMUNUOUPUQURUSWPWQWR $.
  $}

  ${
    $d p q r s t u x y z A $.  $d p q r s t u z X $.
    $d p q r s t u x y z .+ $.  $d p q r s t u x y z .~ $.  $d p q x y z J $.
    $d p q r s t u x y z ph $.
    eropr2.1 $e |- J = ( A /. .~ ) $.
    eropr2.2 $e |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. A
             ( ( x = [ p ] .~ /\ y = [ q ] .~ ) /\ z = [ ( p .+ q ) ] .~ ) } $.
    eropr2.3 $e |- ( ph -> .~ e. X ) $.
    eropr2.4 $e |- ( ph -> .~ Er U ) $.
    eropr2.5 $e |- ( ph -> A C_ U ) $.
    eropr2.6 $e |- ( ph -> .+ : ( A X. A ) --> A ) $.
    eropr2.7 $e |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. A /\ u e. A ) ) )
      -> ( ( r .~ s /\ t .~ u ) -> ( r .+ t ) .~ ( s .+ u ) ) ) $.
    $( Functionality of an operation defined on equivalence classes.
       (Contributed by Jeff Madsen, 10-Jun-2010.) $)
    eroprf2 $p |- ( ph -> .+^ : ( J X. J ) --> J ) $=
      ( eroprf ) ABCDEFGGGHIJJJKLLLKKMMMNOPQRRTUAUAUAUBUBUBUCUDSTTRUE $.
  $}

  ${
    $d f g h t s r A $.  $d f g h t s r B $.  $d f g h t s r C $.
    $d f g h t s r D $.  $d x y z w v u f g h t s r .+ $.  $d f g h t s r .~ $.
    $d x y z w v u f g h t s r S $.
    ecopopr.1 $e |- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S )
          ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\
          ( z .+ u ) = ( w .+ v ) ) ) } $.
    ${
      $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
      $d x y z w v u D $.
      $( This is the first of several theorems about equivalence relations of
         the kind used in construction of fractions and signed reals, involving
         operations on equivalent classes of ordered pairs.  This theorem
         expresses the relation ` .~ ` (specified by the hypothesis) in terms
         of its operation ` F ` .  (Contributed by NM, 16-Aug-1995.) $)
      ecopoveq $p |- ( ( ( A e. S /\ B e. S ) /\
                     ( C e. S /\ D e. S ) ) ->
          ( <. A , B >. .~ <. C , D >. <-> ( A .+ D ) = ( B .+ C ) ) ) $=
        ( cv co wceq wb wa oveq12 eqeqan12d an42s opbrop ) COZFOZKPZDOZEOZKPZQZ
        GJKPZHIKPZQZABCDEFGHIJLMUDGQZUEJQZUGHQZUHIQZUJUMRUNUOSUPUQSUFUKUIULUDGU
        EJKTUGHUHIKTUAUBNUC $.
    $}

    ${
      ecopopr.com $e |- ( x .+ y ) = ( y .+ x ) $.
      $( Assuming the operation ` F ` is commutative, show that the relation
         ` .~ ` , specified by the first hypothesis, is symmetric.
         (Contributed by NM, 27-Aug-1995.)  (Revised by Mario Carneiro,
         26-Apr-2015.) $)
      ecopovsym $p |- ( A .~ B -> B .~ A ) $=
        ( wbr wcel wa wb cv wceq co vf vg vh cxp cop wex copab opabssxp eqsstri
        brel eqid breq1 breq2 bibi12d ecopoveq vex caovcom eqeq12i eqcom bitrdi
        vt bitri ancoms bitr4d 2optocl syl ibi ) GHJNZHGJNZVHGKKUDZOHVJOPVHVIQZ
        GHVJVJJJARZVJOBRZVJOPVLCRZDRZUESVMERZFRZUESPVNVQITVOVPITSPFUFEUFDUFCUFZ
        PABUGVJVJUDLVRABVJVJUHUIUJUARZUBRZUEZUCRZVARZUEZJNZWDWAJNZQGWDJNZWDGJNZ
        QVKUAUBUCVAGHKKVJVJUKWAGSWEWGWFWHWAGWDJULWAGWDJUMUNWDHSWGVHWHVIWDHGJUMW
        DHGJULUNVSKOVTKOPZWBKOWCKOPZPZWEWBVTITZWCVSITZSZWFWKWEVSWCITZVTWBITZSZW
        NABCDEFVSVTWBWCIJKLUOWQWMWLSWNWOWMWPWLABVSWCIUAUPVAUPMUQABVTWBIUBUPUCUP
        MUQURWMWLUSVBUTWJWIWFWNQABCDEFWBWCVSVTIJKLUOVCVDVEVFVG $.

      ${
        $( Assume the operation is closed. $)
        ecopopr.cl $e |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S ) $.
        $( Assume the operation is associative. $)
        ecopopr.ass $e |- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) $.
        $( Assume the operation has the cancellation property. $)
        ecopopr.can $e |- ( ( x e. S /\ y e. S ) ->
                   ( ( x .+ y ) = ( x .+ z ) -> y = z ) ) $.
        $( Assuming that operation ` F ` is commutative (second hypothesis),
           closed (third hypothesis), associative (fourth hypothesis), and has
           the cancellation property (fifth hypothesis), show that the relation
           ` .~ ` , specified by the first hypothesis, is transitive.
           (Contributed by NM, 11-Feb-1996.)  (Revised by Mario Carneiro,
           26-Apr-2015.) $)
        ecopovtrn $p |- ( ( A .~ B /\ B .~ C ) -> A .~ C ) $=
          ( wcel wa co vf vg vh vt vs vr cxp w3a wbr cv cop wceq copab opabssxp
          wex eqsstri brel simpld anim12i 3anass sylibr wi breq1 anbi1d imbi12d
          breq2 anbi12d imbi1d anbi2d wb ecopoveq 3adant3 3adant1 oveq12 simp2l
          eqid simp2r simp1l simp3r adantl caov411d simp1r simp3l caov4d eqtr3d
          eqeq12d imbitrrid sylbid oveq2 impbid1 caovcl syl2anc caovcand sylibd
          a1i caovcld 3adant2 sylibrd 3optocl mpcom ) GLLUGZRZHXARZIXARZUHZGHKU
          IZHIKUIZSZGIKUIZXHXBXCXDSZSXEXFXBXGXJXFXBXCGHXAXAKKAUJZXARBUJZXARSXKC
          UJZDUJZUKULXLEUJZFUJZUKULSXMXPJTXNXOJTULSFUOEUODUOCUOZSABUMXAXAUGMXQA
          BXAXAUNUPZUQURHIXAXAKXRUQUSXBXCXDUTVAUAUJZUBUJZUKZUCUJZUDUJZUKZKUIZYD
          UEUJZUFUJZUKZKUIZSZYAYHKUIZVBGYDKUIZYISZGYHKUIZVBXFHYHKUIZSZYNVBXHXIV
          BUAUBUCUDUEUFGHILXALXAVPYAGULZYJYMYKYNYQYEYLYIYAGYDKVCVDYAGYHKVCVEYDH
          ULZYMYPYNYRYLXFYIYOYDHGKVFYDHYHKVCVGVHYHIULZYPXHYNXIYSYOXGXFYHIHKVFVI
          YHIGKVFVEXSLRZXTLRZSZYBLRZYCLRZSZYFLRZYGLRZSZUHZYJXSYGJTZXTYFJTZULZYK
          UUIYJYBYCJTZUUJJTZUUMUUKJTZULZUULUUIYJXSYCJTZXTYBJTZULZYBYGJTZYCYFJTZ
          ULZSZUUPUUIYEUUSYIUVBUUBUUEYEUUSVJUUHABCDEFXSXTYBYCJKLMVKVLUUEUUHYIUV
          BVJUUBABCDEFYBYCYFYGJKLMVKVMVGUVCUUPUUIUUQUUTJTZUURUVAJTZULUUQUURUUTU
          VAJVNUUIUUNUVDUUOUVEUUIABCYBYCXSYGLJUUBUUCUUDUUHVOZUUBUUCUUDUUHVQZYTU
          UAUUEUUHVRZXKXLJTZXLXKJTULUUIXKLRZXLLRZSZSNWOZUVIXMJTXKXLXMJTJTULUUIU
          VJUVKXMLRZUHZSPWOZUUBUUEUUFUUGVSZUVLUVILRUUIOVTZWAUUIXTYCJTYBYFJTJTUU
          OUVEUUIABCXTYCYBYFLJYTUUAUUEUUHWBZUVGUVFUVMUVPUUBUUEUUFUUGWCZUVRWAUUI
          ABCXTYCYBYFLJUVSUVGUVFUVMUVPUVTUVRWDWEWFWGWHUUIABCUUMUUJUUKLLJUVOUVIX
          KXMJTULZXLXMULZVJUUIUVOUWAUWBUVJUVKUWAUWBVBUVNQVLXLXMXKJWIWJVTUUIUUCU
          UDUUMLRUVFUVGABYBYCLJOWKWLUUIYTUUGUUJLRUVHUVQABXSYGLJOWKWLUUIABXTYFLL
          LJUVRUVSUVTWPWMWNUUBUUHYKUULVJUUEABCDEFXSXTYFYGJKLMVKWQWRWSWT $.

        $( Assuming that operation ` F ` is commutative (second hypothesis),
           closed (third hypothesis), associative (fourth hypothesis), and has
           the cancellation property (fifth hypothesis), show that the relation
           ` .~ ` , specified by the first hypothesis, is an equivalence
           relation.  (Contributed by NM, 16-Feb-1996.)  (Revised by Mario
           Carneiro, 12-Aug-2015.) $)
        ecopover $p |- .~ Er ( S X. S ) $=
          ( vg vh wtru cv wa wbr vf cxp wer wrel wcel cop wceq wex relopabi a1i
          co ecopovsym adantl ecopovtrn wral vex caovcom ecopoveq mpbiri anidms
          wi rgen2a ralxp mpbir rspec copab opabssxp eqsstri ssbri brxp simplbi
          wb breq12 syl impbid1 iserd mptru ) IIUBZHUCQUAOPVRHHUDQARZVRUEBRZVRU
          ESVSCRZDRZUFUGVTERZFRZUFUGSWAWDGUKWBWCGUKUGSFUHEUHDUHCUHZSZABHJUIUJUA
          RZORZHTZWHWGHTQABCDEFWGWHGHIJKULUMWIWHPRZHTSWGWJHTQABCDEFWGWHWJGHIJKL
          MNUNUMQWGVRUEZWGWGHTZWKWLVAQWLUAVRWLUAVRUOWHWJUFZWMHTZPIUOOIUOWNOPIWH
          IUEWJIUESZWNWOWOSWNWHWJGUKWJWHGUKUGABWHWJGOUPPUPKUQABCDEFWHWJWHWJGHIJ
          URUSUTVBWLWNUAOPIIWGWMUGWLWNVLWGWMWGWMHVMUTVCVDVEUJWLWGWGVRVRUBZTZWKH
          WPWGWGHWFABVFWPJWEABVRVRVGVHVIWQWKWKWGWGVRVRVJVKVNVOVPVQ $.
      $}
    $}

    ${
      ecopoprg.com $e |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) ) $.
      $( Assuming the operation ` F ` is commutative, show that the relation
         ` .~ ` , specified by the first hypothesis, is symmetric.
         (Contributed by Jim Kingdon, 1-Sep-2019.) $)
      ecopovsymg $p |- ( A .~ B -> B .~ A ) $=
        ( wbr wcel wa wb cv wceq co vf vg vh cxp cop wex copab opabssxp eqsstri
        vt brel breq1 breq2 bibi12d adantl simpll simprr caovcomd simplr simprl
        eqid eqeq12d eqcom bitrdi ecopoveq ancoms 3bitr4d 2optocl syl ibi ) GHJ
        NZHGJNZVKGKKUDZOHVMOPVKVLQZGHVMVMJJARZVMOBRZVMOPVOCRZDRZUESVPERZFRZUESP
        VQVTITVRVSITSPFUFEUFDUFCUFZPABUGVMVMUDLWAABVMVMUHUIUKUARZUBRZUEZUCRZUJR
        ZUEZJNZWGWDJNZQGWGJNZWGGJNZQVNUAUBUCUJGHKKVMVMVAWDGSWHWJWIWKWDGWGJULWDG
        WGJUMUNWGHSWJVKWKVLWGHGJUMWGHGJULUNWBKOZWCKOZPZWEKOZWFKOZPZPZWBWFITZWCW
        EITZSZWEWCITZWFWBITZSZWHWIWRXAXCXBSXDWRWSXCWTXBWRABWBWFKIVOKOVPKOPVOVPI
        TVPVOITSWRMUOZWLWMWQUPWNWOWPUQURWRABWCWEKIXEWLWMWQUSWNWOWPUTURVBXCXBVCV
        DABCDEFWBWCWEWFIJKLVEWQWNWIXDQABCDEFWEWFWBWCIJKLVEVFVGVHVIVJ $.

      ${
        $( Assume the operation is closed. $)
        ecopoprg.cl $e |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S ) $.
        $( Assume the operation is associative. $)
        ecopoprg.ass $e |- ( ( x e. S /\ y e. S /\ z e. S ) ->
          ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) $.
        $( Assume the operation has the cancellation property. $)
        ecopoprg.can $e |- ( ( x e. S /\ y e. S /\ z e. S ) ->
                   ( ( x .+ y ) = ( x .+ z ) -> y = z ) ) $.
        $( Assuming that operation ` F ` is commutative (second hypothesis),
           closed (third hypothesis), associative (fourth hypothesis), and has
           the cancellation property (fifth hypothesis), show that the relation
           ` .~ ` , specified by the first hypothesis, is transitive.
           (Contributed by Jim Kingdon, 1-Sep-2019.) $)
        ecopovtrng $p |- ( ( A .~ B /\ B .~ C ) -> A .~ C ) $=
          ( wcel wa co vf vg vh vt vs vr cxp w3a wbr cv cop wceq copab opabssxp
          wex eqsstri brel simpld anim12i 3anass sylibr wi breq1 anbi1d imbi12d
          breq2 anbi12d imbi1d anbi2d wb ecopoveq 3adant3 3adant1 oveq12 simp2l
          eqid simp2r simp1l adantl simp3r caov411d simp1r simp3l caov4d eqtr3d
          eqeq12d imbitrrid sylbid oveq2 impbid1 caovcl syl2anc caovcand sylibd
          caovcld 3adant2 sylibrd 3optocl mpcom ) GLLUGZRZHWTRZIWTRZUHZGHKUIZHI
          KUIZSZGIKUIZXGXAXBXCSZSXDXEXAXFXIXEXAXBGHWTWTKKAUJZWTRBUJZWTRSXJCUJZD
          UJZUKULXKEUJZFUJZUKULSXLXOJTXMXNJTULSFUOEUODUOCUOZSABUMWTWTUGMXPABWTW
          TUNUPZUQURHIWTWTKXQUQUSXAXBXCUTVAUAUJZUBUJZUKZUCUJZUDUJZUKZKUIZYCUEUJ
          ZUFUJZUKZKUIZSZXTYGKUIZVBGYCKUIZYHSZGYGKUIZVBXEHYGKUIZSZYMVBXGXHVBUAU
          BUCUDUEUFGHILWTLWTVPXTGULZYIYLYJYMYPYDYKYHXTGYCKVCVDXTGYGKVCVEYCHULZY
          LYOYMYQYKXEYHYNYCHGKVFYCHYGKVCVGVHYGIULZYOXGYMXHYRYNXFXEYGIHKVFVIYGIG
          KVFVEXRLRZXSLRZSZYALRZYBLRZSZYELRZYFLRZSZUHZYIXRYFJTZXSYEJTZULZYJUUHY
          IYAYBJTZUUIJTZUULUUJJTZULZUUKUUHYIXRYBJTZXSYAJTZULZYAYFJTZYBYEJTZULZS
          ZUUOUUHYDUURYHUVAUUAUUDYDUURVJUUGABCDEFXRXSYAYBJKLMVKVLUUDUUGYHUVAVJU
          UAABCDEFYAYBYEYFJKLMVKVMVGUVBUUOUUHUUPUUSJTZUUQUUTJTZULUUPUUQUUSUUTJV
          NUUHUUMUVCUUNUVDUUHABCYAYBXRYFLJUUAUUBUUCUUGVOZUUAUUBUUCUUGVQZYSYTUUD
          UUGVRZXJLRZXKLRZSZXJXKJTZXKXJJTULUUHNVSZUVHUVIXLLRUHZUVKXLJTXJXKXLJTJ
          TULUUHPVSZUUAUUDUUEUUFVTZUVJUVKLRUUHOVSZWAUUHXSYBJTYAYEJTJTUUNUVDUUHA
          BCXSYBYAYELJYSYTUUDUUGWBZUVFUVEUVLUVNUUAUUDUUEUUFWCZUVPWAUUHABCXSYBYA
          YELJUVQUVFUVEUVLUVNUVRUVPWDWEWFWGWHUUHABCUULUUIUUJLLJUVMUVKXJXLJTULZX
          KXLULZVJUUHUVMUVSUVTQXKXLXJJWIWJVSUUHUUBUUCUULLRUVEUVFABYAYBLJOWKWLUU
          HYSUUFUUILRUVGUVOABXRYFLJOWKWLUUHABXSYELLLJUVPUVQUVRWOWMWNUUAUUGYJUUK
          VJUUDABCDEFXRXSYEYFJKLMVKWPWQWRWS $.

        $( Assuming that operation ` F ` is commutative (second hypothesis),
           closed (third hypothesis), associative (fourth hypothesis), and has
           the cancellation property (fifth hypothesis), show that the relation
           ` .~ ` , specified by the first hypothesis, is an equivalence
           relation.  (Contributed by Jim Kingdon, 1-Sep-2019.) $)
        ecopoverg $p |- .~ Er ( S X. S ) $=
          ( wtru cv wcel wa wceq wbr vf vg vh cxp wer wrel cop wex relopabi a1i
          co ecopovsymg adantl ecopovtrng wi wral simpll simplr caovcomd mpbird
          ecopoveq anidms rgen2a ralxp mpbir rspec copab opabssxp eqsstri ssbri
          wb breq12 brxp simplbi syl impbid1 iserd mptru ) IIUDZHUEOUAUBUCVSHHU
          FOAPZVSQBPZVSQRVTCPZDPZUGSWAEPZFPZUGSRWBWEGUKWCWDGUKSRFUHEUHDUHCUHZRZ
          ABHJUIUJUAPZUBPZHTZWIWHHTOABCDEFWHWIGHIJKULUMWJWIUCPZHTRWHWKHTOABCDEF
          WHWIWKGHIJKLMNUNUMOWHVSQZWHWHHTZWLWMUOOWMUAVSWMUAVSUPWIWKUGZWNHTZUCIU
          PUBIUPWOUBUCIWIIQZWKIQZRZWOWRWRRZWOWIWKGUKWKWIGUKSWSABWIWKIGVTIQWAIQR
          VTWAGUKWAVTGUKSWSKUMWPWQWRUQWPWQWRURUSABCDEFWIWKWIWKGHIJVAUTVBVCWMWOU
          AUBUCIIWHWNSWMWOVKWHWNWHWNHVLVBVDVEVFUJWMWHWHVSVSUDZTZWLHWTWHWHHWGABV
          GWTJWFABVSVSVHVIVJXAWLWLWHWHVSVSVMVNVOVPVQVR $.
      $}
    $}
  $}

  ${
    $d x y z w v u .+ $.  $d x y z w v u .~ $.  $d x y z w v u S $.
    $d x y z w v u A $.  $d x y z w v u B $.
    th3qlem1.1 $e |- .~ Er S $.
    $( Compatibility assumption. $)
    th3qlem1.3 $e |- ( ( ( y e. S /\ w e. S ) /\ ( z e. S /\ v e. S ) ) ->
                 ( ( y .~ w /\ z .~ v ) -> ( y .+ z ) .~ ( w .+ v ) ) ) $.
    $( Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60.  The
       third hypothesis is the compatibility assumption.  (Contributed by NM,
       3-Aug-1995.)  (Revised by Mario Carneiro, 9-Jul-2014.) $)
    th3qlem1 $p |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> E* x
       E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\
       x = [ ( y .+ z ) ] .~ ) ) $=
      ( vu wcel wa cv cec wceq wex ecelqsdm cqs co wi wal ee4anv eleq1 bi2anan9
      wmo an4 wb adantr biimpac anim12i an4s adantl wer a1i wbr simprl cdm erdm
      eqtr2 ax-mp simpll sylancr mpbird simprr simplr eqeltrrd syl22anc syl2anc
      erth mp2and erthi syl5ibrcom expimpd biimtrid exlimdvv biimtrrid alrimivv
      eqeq12 eqeq1 anbi2d 2exbidv eceq1 eqeq2d oveq12 eceq1d anbi12d bitrdi mo4
      cbvex2v sylibr ) FJIUAZNZGWNNZOZFBPZIQZRZGCPZIQZRZOZAPZWRXAHUBZIQZRZOZCSB
      SZFDPZIQZRZGEPZIQZRZOZMPZXKXNHUBZIQZRZOZESDSZOZXEXRRZUCZMUDAUDXJAUHWQYFAM
      YDXIYBOZESDSZCSBSWQYEXIYBBCDEUEWQYHYEBCWQYGYEDEYGXDXQOZXHYAOZOWQYEXDXHXQY
      AUIWQYIYJYEWQYIOZYEYJXGXTRZYKWSWNNZXBWNNZOZWSXLRZXBXORZOZYLYIWQYOXDWQYOUJ
      XQWTWOYMXCWPYNFWSWNUFGXBWNUFUGUKULYIYRWQWTXMXCXPYRWTXMOYPXCXPOYQFWSXLVBGX
      BXOVBUMUNUOYOYROZXFXSIJJIUPZYSKUQZYSWRXKIURZXAXNIURZXFXSIURZYSUUBYPYOYPYQ
      USZYSWRXKIJUUAYSIUTJRZYMWRJNZYTUUFKJIVAVCZYMYNYRVDZJWRITVEZVLVFYSUUCYQYOY
      PYQVGZYSXAXNIJUUAYSUUFYNXAJNZUUHYMYNYRVHZJXAITVEZVLVFYSUUGXKJNZUULXNJNZUU
      BUUCOUUDUCUUJYSUUFXLWNNUUOUUHYSWSXLWNUUEUUIVIJXKITVEUUNYSUUFXOWNNUUPUUHYS
      XBXOWNUUKUUMVIJXNITVELVJVMVNVKXEXGXRXTWAVOVPVQVRVRVSVTXJYCAMYEXJXDXRXGRZO
      ZCSBSYCYEXIUURBCYEXHUUQXDXEXRXGWBWCWDUURYBBCDEWRXKRZXAXNRZOZXDXQUUQYAUUSW
      TXMUUTXCXPUUSWSXLFWRXKIWEWFUUTXBXOGXAXNIWEWFUGUVAXGXTXRUVAXFXSIWRXKXAXNHW
      GWHWFWIWLWJWKWM $.
  $}

  ${
    $d x y z w v u t s f g h .~ $.  $d x y z w v u t s f g h S $.
    $d x y z w v u t s f A $.  $d x y z w v u t s f B $.  $d x y z w v u t C $.
    $d x y z w v u t D $.  $d x y z w v u t s f g h .+ $.
    th3q.1 $e |- .~ e. _V $.
    th3q.2 $e |- .~ Er ( S X. S ) $.
    $( Compatibility assumption. $)
    th3q.4 $e |- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) )
       /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) ->
       ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) ->
       ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) ) $.
    $( Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60,
       extended to operations on ordered pairs.  The fourth hypothesis is the
       compatibility assumption.  (Contributed by NM, 4-Aug-1995.)  (Revised by
       Mario Carneiro, 12-Aug-2015.) $)
    th3qlem2 $p |- ( ( A e. ( ( S X. S ) /. .~ ) /\ B e. ( ( S X. S ) /. .~ ) )
           -> E* z E. w E. v E. u E. t (
           ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\
                  z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) $=
      ( wcel wa wceq vx vy cxp cqs cv cec co wex wmo cop wbr breq1 anbi1d oveq1
      eqid breq1d imbi12d imbi2d breq2 breq2d anbi2d oveq2 expcom 2optocl com12
      wi imp th3qlem1 opex eceq1 eqeq2d bi2anan9 oveq12 eceq1d anbi12d exlimivv
      vex spc2ev moimi syl ) FJJUCZIUDZRGWBRSFNUEZIUFZTZGUAUEZIUFZTZSZAUEZWCWFH
      UGZIUFZTZSZUAUHNUHZAUIFBUEZCUEZUJZIUFZTZGDUEZEUEZUJZIUFZTZSZWJWRXCHUGZIUF
      ZTZSZEUHDUHZCUHBUHZAUIANUAKUBFGHIWAPWCWARKUEZWARSWFWARUBUEZWARSZWCXMIUKZW
      FXNIUKZSZWKXMXNHUGZIUKZVFZXOWRXCIUKZXQSZWRWFHUGZXCXNHUGZIUKZVFZVFXOWCXCIU
      KZXQSZWKYEIUKZVFZVFXOYAVFBCDEWCXMJJWAWAUOZWRWCTZYGYKXOYMYCYIYFYJYMYBYHXQW
      RWCXCIULUMYMYDWKYEIWRWCWFHUNUPUQURXCXMTZYKYAXOYNYIXRYJXTYNYHXPXQXCXMWCIUS
      UMYNYEXSWKIXCXMXNHUNUTUQURXOWPJRWQJRSXAJRXBJRSSZYGYOYBWCXMUJZLUEZMUEZUJZI
      UKZSZWRYPHUGZXCYSHUGZIUKZVFZVFYOYBWFYSIUKZSZYDUUCIUKZVFZVFYOYGVFNKLMWFXNJ
      JWAYLYPWFTZUUEUUIYOUUJUUAUUGUUDUUHUUJYTUUFYBYPWFYSIULVAUUJUUBYDUUCIYPWFWR
      HVBUPUQURYSXNTZUUIYGYOUUKUUGYCUUHYFUUKUUFXQYBYSXNWFIUSVAUUKUUCYEYDIYSXNXC
      HVBUTUQURYOWCJRXMJRSYQJRYRJRSSUUEQVCVDVEVDVGVHXLWOAXKWOBCXJWODEWNXJNUAWRX
      CWPWQBVQCVQVIXAXBDVQEVQVIWCWRTZWFXCTZSZWIXFWMXIUULWEWTUUMWHXEUULWDWSFWCWR
      IVJVKUUMWGXDGWFXCIVJVKVLUUNWLXHWJUUNWKXGIWCWRWFXCHVMVNVKVOVRVPVPVSVT $.

    ${
      th3q.5 $e |- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ )
           /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( (
           x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\
           z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) } $.
      $( Corollary of Theorem 3Q of [Enderton] p. 60.  (Contributed by NM,
         12-Nov-1995.)  (Revised by David Abernethy, 4-Jun-2013.) $)
      th3qcor $p |- Fun G $=
        ( cv wfun cxp cqs wcel wa cop cec co wex coprab wmo wi th3qlem2 moanimv
        wceq mpbir funoprab funeqi ) NUAATZJJUBIUCZUDBTZUTUDUEZUSDTETUFZIUGUOVA
        FTGTUFZIUGUOUECTVCVDHUHIUGUOUEGUIFUIEUIDUIZUEZABCUJZUAVFABCVFCUKVBVECUK
        ULCDEFGUSVAHIJKLMOPQRUMVBVECUNUPUQNVGSURUP $.

      $( Theorem 3Q of [Enderton] p. 60, extended to operations on ordered
         pairs.  (Contributed by NM, 4-Aug-1995.)  (Revised by Mario Carneiro,
         19-Dec-2013.) $)
      th3q $p |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
                 ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) =
                 [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) $=
        ( wcel wa cop cec cxp cqs cv wceq co wex opelxpi ecelqsi anim12i pm3.2i
        syl eqid opeq12 eceq1 eqeq2d anbi1d oveq1 eceq1d anbi12d spc2egv anbi2d
        wb oveq2 2eximdv sylan9 mp2ani cvv ecexg ax-mp w3a eqeq1 bi2anan9 3impa
        wi 4exbidv th3qlem2 ovig mp3an3 sylc ) HNUDINUDUEZJNUDKNUDUEZUEZHIUFZMU
        GZNNUHZMUIZUDZJKUFZMUGZWMUDZUEWKDUJZEUJZUFZMUGZUKZWPFUJZGUJZUFZMUGZUKZU
        EZWJWOLULZMUGZWTXELULZMUGZUKZUEZGUMFUMZEUMDUMZWKWPRULXJUKZWGWNWHWQWGWJW
        LUDWNHINNUNWLWJMTUOURWHWOWLUDWQJKNNUNWLWOMTUOURUPWIWKWKUKZWPWPUKZUEZXJX
        JUKZXPXRXSWKUSWPUSUQXJUSWGXTYAUEZXBXSUEZXJWTWOLULZMUGZUKZUEZEUMDUMWHXPY
        GYBDEHINNWRHUKWSIUKUEWTWJUKZYGYBVIWRWSHIUTYHYCXTYFYAYHXBXRXSYHXAWKWKWTW
        JMVAVBVCYHYEXJXJYHYDXIMWTWJWOLVDVEVBVFURVGWHYGXODEXNYGFGJKNNXCJUKXDKUKU
        EXEWOUKZXNYGVIXCXDJKUTYIXHYCXMYFYIXGXSXBYIXFWPWPXEWOMVAVBVHYIXLYEXJYIXK
        YDMXEWOWTLVJVEVBVFURVGVKVLVMWNWQXJVNUDZXPXQWAMVNUDYJTXIVNMVOVPAUJZXAUKZ
        BUJZXFUKZUEZCUJZXLUKZUEZGUMFUMEUMDUMXPABCWKWPXJVNWMWMRYKWKUKZYMWPUKZYPX
        JUKZVQYRXNDEFGYSYTUUAYRXNVIYSYTUEYOXHUUAYQXMYSYLXBYTYNXGYKWKXAVRYMWPXFV
        RVSYPXJXLVRVSVTWBCDEFGYKYMLMNOPQSTUAUBWCUCWDWEWF $.
    $}
  $}

  ${
    $d a b c d f u v w x y z C $.  $d a b c d f u v w x y z D $.  $d x y z J $.
    $d a b c d f g h u v w x y z A $.  $d u v w z ch $.  $d f u v w x y z H $.
    $d a b c d f g h u v w x y z B $.  $d f u v w x y z K $.  $d u v w z ps $.
    $d f u v w x y z L $.  $d x y ph $.  $d a b c d f g h s t u v w x y z S $.
    $d a b c d g h s t x y z .+ $.  $d a b c d g h s t x y z .~ $.
    oviec.1 $e |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
        H e. ( S X. S ) ) $.
    oviec.2 $e |- ( ( ( a e. S /\ b e. S ) /\ ( g e. S /\ h e. S ) ) ->
        K e. ( S X. S ) ) $.
    oviec.3 $e |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) ->
        L e. ( S X. S ) ) $.
    oviec.4 $e |- .~ e. _V $.
    oviec.5 $e |- .~ Er ( S X. S ) $.
    $( Equivalence relation. $)
    oviec.7 $e |- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\
                      y e. ( S X. S ) ) /\
                      E. z E. w E. v E. u ( ( x = <. z , w >. /\
                      y = <. v , u >. ) /\ ph ) ) } $.
    $( Substitutions for equivalence relation. $)
    oviec.8 $e |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) ->
                     ( ph <-> ps ) ) $.
    oviec.9 $e |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) ->
                     ( ph <-> ch ) ) $.
    $( Pre-operation. $)
    oviec.10 $e |- .+ = { <. <. x , y >. , z >. | ( ( x e. ( S X. S ) /\
                      y e. ( S X. S ) ) /\
                      E. w E. v E. u E. f ( ( x = <. w , v >. /\
                      y = <. u , f >. ) /\ z = J ) ) } $.
    $( Substitutions for pre-operation. $)
    oviec.11 $e |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> J = K ) $.
    oviec.12 $e |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> J = L ) $.
    oviec.13 $e |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> J = H ) $.
    $( Main operation. $)
    oviec.14 $e |- .+^ = { <. <. x , y >. , z >. | ( ( x e. Q /\ y e. Q ) /\
           E. a E. b E. c E. d ( (
           x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\
           z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) } $.
    $( Quotient class for main operation. $)
    oviec.15 $e |- Q = ( ( S X. S ) /. .~ ) $.
    $( Lemma to establish compatibility of pre-operation. $)
    oviec.16 $e |- ( ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) /\
                       ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) ) ->
                   ( ( ps /\ ch ) -> K .~ L ) ) $.
    $( Express an operation on equivalence classes of ordered pairs in terms of
       equivalence class of operations on ordered pairs.  See iset.mm for
       additional comments describing the hypotheses.  (Unnecessary distinct
       variable restrictions were removed by David Abernethy, 4-Jun-2013.)
       (Contributed by NM, 6-Aug-1995.)  (Revised by Mario Carneiro,
       4-Jun-2013.) $)
    oviec $p |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
                ( [ <. A , B >. ] .~ .+^ [ <. C , D >. ] .~ ) =
                   [ H ] .~ ) $=
      ( wcel wa cop cec co wbr opbrop bi2anan9 ovi3 breqan12d an4s 3imtr4d wceq
      cv wb wex coprab cxp cqs eleq2i anbi12i anbi1i oprabbii eqtri th3q eceq1d
      eqtrd ) KSVGLSVGVHMSVGNSVGVHVHZKLVIZRVJMNVIZRVJPVKWOWPOVKZRVJUCRVJDEFUHUI
      UJUKKLMNORSUBJUGPUAUOUPUHVTZSVGUIVTZSVGVHZUJVTZSVGUKVTZSVGVHZVHZUAVTZSVGU
      BVTZSVGVHZJVTZSVGUGVTZSVGVHZVHZVHBCVHUEUFRVLZWRWSVIZXAXBVIZRVLZXEXFVIZXHX
      IVIZRVLZVHXMXPOVKZXNXQOVKZRVLZVFXDXOBXKXRCABDEFGHIWRWSXAXBRSURUQVMACDEFGH
      IXEXFXHXIRSUSUQVMVNWTXGXCXJYAXLWAWTXGVHXCXJVHXSUEXTUFRDEFGHIWRWSXEXFUDUET
      OSUMVAUTVODEFGHIXAXBXHXIUDUFTOSUNVBUTVOVPVQVRPDVTZQVGZEVTZQVGZVHZYBXMRVJV
      SYDXNRVJVSVHFVTXMXNOVKRVJVSVHUKWBUJWBUIWBUHWBZVHZDEFWCYBSSWDRWEZVGZYDYIVG
      ZVHZYGVHZDEFWCVDYHYMDEFYFYLYGYCYJYEYKQYIYBVEWFQYIYDVEWFWGWHWIWJWKWNWQUCRD
      EFGHIKLMNUDUCTOSULVCUTVOWLWM $.
  $}

  ${
    $d x y z w A $.  $d z w B $.  $d x y z w .+ $.  $d x y z w .~ $.
    $d x y z w S $.  $d z w C $.
    ecovcom.1 $e |- C = ( ( S X. S ) /. .~ ) $.
    $( D,G for AFB $)
    ecovcom.2 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
 ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ ) $.
    $( H,J for BFA $)
    ecovcom.3 $e |- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) ->
 ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ ) $.
    ecovcom.4 $e |- D = H $.
    ecovcom.5 $e |- G = J $.
    $( Lemma used to transfer a commutative law via an equivalence relation.
       Most uses will want ~ ecovicom instead.  (Contributed by NM,
       29-Aug-1995.)  (Revised by David Abernethy, 4-Jun-2013.) $)
    ecovcom $p |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) ) $=
      ( wceq cv cop cec co oveq1 oveq2 eqeq12d wcel opeq12 eceq1d mp2an 3eqtr4a
      wa ancoms 2ecoptocl ) AUAZBUAZUBJUCZCUAZDUAZUBJUCZIUDZVAURIUDZTEVAIUDZVAE
      IUDZTEFIUDZFEIUDZTABCDEFKKJGOURETVBVDVCVEUREVAIUEUREVAIUFUGVAFTVDVFVEVGVA
      FEIUFVAFEIUEUGUPKUHUQKUHUMZUSKUHUTKUHUMZUMHLUBZJUCZMNUBZJUCZVBVCHMTZLNTZV
      KVMTRSVNVOUMVJVLJHLMNUIUJUKPVIVHVCVMTQUNULUO $.
  $}

  ${
    $d x y z w A $.  $d z w B $.  $d x y z w .+ $.  $d x y z w .~ $.
    $d x y z w S $.  $d z w C $.
    ecovicom.1 $e |- C = ( ( S X. S ) /. .~ ) $.
    $( D,G for AFB $)
    ecovicom.2 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
 ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ ) $.
    $( H,J for BFA $)
    ecovicom.3 $e |- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) ->
 ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ ) $.
    ecovicom.4 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
      D = H ) $.
    ecovicom.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
      G = J ) $.
    $( Lemma used to transfer a commutative law via an equivalence relation.
       (Contributed by Jim Kingdon, 15-Sep-2019.) $)
    ecovicom $p |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) ) $=
      ( wceq cv cop co oveq1 oveq2 eqeq12d wcel wa opeq12 eceq1d syl2anc ancoms
      cec 3eqtr4d 2ecoptocl ) AUAZBUAZUBJUMZCUAZDUAZUBJUMZIUCZVAURIUCZTEVAIUCZV
      AEIUCZTEFIUCZFEIUCZTABCDEFKKJGOURETVBVDVCVEUREVAIUDUREVAIUEUFVAFTVDVFVEVG
      VAFEIUEVAFEIUDUFUPKUGUQKUGUHZUSKUGUTKUGUHZUHZHLUBZJUMZMNUBZJUMZVBVCVJHMTZ
      LNTZVLVNTRSVOVPUHVKVMJHLMNUIUJUKPVIVHVCVNTQULUNUO $.
  $}

  ${
    $d x y z w v u A $.  $d z w v u B $.  $d x y z w v u C $.
    $d x y z w v u .+ $.  $d x y z w v u .~ $.  $d x y z w v u S $.
    $d z w v u D $.
    ecovass.1 $e |- D = ( ( S X. S ) /. .~ ) $.
    ecovass.2 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
 ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ ) $.
    ecovass.3 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
 ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ ) $.
    ecovass.4 $e |- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) ->
 ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ ) $.
    ecovass.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) ->
 ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ ) $.
    ecovass.6 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
                     ( G e. S /\ H e. S ) ) $.
    ecovass.7 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
                     ( N e. S /\ Q e. S ) ) $.
    ecovass.8 $e |- J = L $.
    ecovass.9 $e |- K = M $.
    $( Lemma used to transfer an associative law via an equivalence relation.
       In most cases ~ ecoviass will be more useful.  (Contributed by NM,
       31-Aug-1995.)  (Revised by David Abernethy, 4-Jun-2013.) $)
    ecovass $p |- ( ( A e. D /\ B e. D /\ C e. D ) ->
                 ( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) ) $=
      ( cv cop cec co wceq oveq1 oveq1d eqeq12d oveq2 oveq2d wcel wa w3a opeq12
      eceq1d mp2an adantr sylan eqtrd 3impa adantl sylan2 3eqtr4a 3ecoptocl
      3impb ) AUKZBUKZULMUMZCUKZDUKZULMUMZKUNZEUKZFUKZULMUMZKUNZVRWAWEKUNZKUNZU
      OGWAKUNZWEKUNZGWGKUNZUOGHKUNZWEKUNZGHWEKUNZKUNZUOWLIKUNZGHIKUNZKUNZUOABCD
      EFGHINMJUBVRGUOZWFWJWHWKWSWBWIWEKVRGWAKUPUQVRGWGKUPURWAHUOZWJWMWKWOWTWIWL
      WEKWAHGKUSUQWTWGWNGKWAHWEKUPUTURWEIUOZWMWPWOWRWEIWLKUSXAWNWQGKWEIHKUSUTUR
      VPNVAVQNVAVBZVSNVAVTNVAVBZWCNVAWDNVAVBZVCQRULZMUMZSTULZMUMZWFWHQSUOZRTUOZ
      XFXHUOUIUJXIXJVBXEXGMQRSTVDVEVFXBXCXDWFXFUOXBXCVBZXDVBWFOPULMUMZWEKUNZXFX
      KWFXMUOXDXKWBXLWEKUCUQVGXKONVAPNVAVBXDXMXFUOUGUEVHVIVJXBXCXDWHXHUOXBXCXDV
      BZVBWHVRUALULMUMZKUNZXHXNWHXPUOXBXNWGXOVRKUDUTVKXNXBUANVALNVAVBXPXHUOUHUF
      VLVIVOVMVN $.
  $}

  ${
    $d x y z w v u A $.  $d z w v u B $.  $d x y z w v u C $.
    $d x y z w v u .+ $.  $d x y z w v u .~ $.  $d x y z w v u S $.
    $d z w v u D $.
    ecoviass.1 $e |- D = ( ( S X. S ) /. .~ ) $.
    ecoviass.2 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
 ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ ) $.
    ecoviass.3 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
 ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ ) $.
    ecoviass.4 $e |- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) ->
 ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ ) $.
    ecoviass.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) ->
 ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ ) $.
    ecoviass.6 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
                     ( G e. S /\ H e. S ) ) $.
    ecoviass.7 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
                     ( N e. S /\ Q e. S ) ) $.
    ecoviass.8 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S )
        /\ ( v e. S /\ u e. S ) ) -> J = L ) $.
    ecoviass.9 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S )
        /\ ( v e. S /\ u e. S ) ) -> K = M ) $.
    $( Lemma used to transfer an associative law via an equivalence relation.
       (Contributed by Jim Kingdon, 16-Sep-2019.) $)
    ecoviass $p |- ( ( A e. D /\ B e. D /\ C e. D ) ->
                 ( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) ) $=
      ( cv cop cec co wceq oveq1 oveq1d eqeq12d oveq2 oveq2d wcel wa w3a opeq12
      eceq1d syl2anc adantr sylan eqtrd 3impa adantl sylan2 3eqtr4d 3ecoptocl
      3impb ) AUKZBUKZULMUMZCUKZDUKZULMUMZKUNZEUKZFUKZULMUMZKUNZVRWAWEKUNZKUNZU
      OGWAKUNZWEKUNZGWGKUNZUOGHKUNZWEKUNZGHWEKUNZKUNZUOWLIKUNZGHIKUNZKUNZUOABCD
      EFGHINMJUBVRGUOZWFWJWHWKWSWBWIWEKVRGWAKUPUQVRGWGKUPURWAHUOZWJWMWKWOWTWIWL
      WEKWAHGKUSUQWTWGWNGKWAHWEKUPUTURWEIUOZWMWPWOWRWEIWLKUSXAWNWQGKWEIHKUSUTUR
      VPNVAVQNVAVBZVSNVAVTNVAVBZWCNVAWDNVAVBZVCZQRULZMUMZSTULZMUMZWFWHXEQSUOZRT
      UOZXGXIUOUIUJXJXKVBXFXHMQRSTVDVEVFXBXCXDWFXGUOXBXCVBZXDVBWFOPULMUMZWEKUNZ
      XGXLWFXNUOXDXLWBXMWEKUCUQVGXLONVAPNVAVBXDXNXGUOUGUEVHVIVJXBXCXDWHXIUOXBXC
      XDVBZVBWHVRUALULMUMZKUNZXIXOWHXQUOXBXOWGXPVRKUDUTVKXOXBUANVALNVAVBXQXIUOU
      HUFVLVIVOVMVN $.
  $}

  ${
    $d x y z w v u A $.  $d z w v u B $.  $d w v u C $.  $d x y z w v u .+ $.
    $d x y z w v u .~ $.  $d x y z w v u S $.  $d x y z w v u .x. $.
    $d z w v u D $.
    ecovdi.1 $e |- D = ( ( S X. S ) /. .~ ) $.
    $( M,N for BFC $)
    ecovdi.2 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
 ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ ) $.
    $( H,J for AG(BFC) $)
    ecovdi.3 $e |- ( ( ( x e. S /\ y e. S ) /\ ( M e. S /\ N e. S ) ) ->
 ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ ) $.
    $( W,X for AGB $)
    ecovdi.4 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
 ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ ) $.
    $( Y,Z for AGC $)
    ecovdi.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ->
 ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = [ <. Y , Z >. ] .~ ) $.
    $( K,L for (AGB)F(AGC) $)
    ecovdi.6 $e |- ( ( ( W e. S /\ X e. S ) /\ ( Y e. S /\ Z e. S ) ) ->
 ( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ ) $.
    ecovdi.7 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
                     ( M e. S /\ N e. S ) ) $.
    ecovdi.8 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
                     ( W e. S /\ X e. S ) ) $.
    ecovdi.9 $e |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ->
                     ( Y e. S /\ Z e. S ) ) $.
    ecovdi.10 $e |- H = K $.
    ecovdi.11 $e |- J = L $.
    $( Lemma used to transfer a distributive law via an equivalence relation.
       Most likely ~ ecovidi will be more helpful.  (Contributed by NM,
       2-Sep-1995.)  (Revised by David Abernethy, 4-Jun-2013.) $)
    ecovdi $p |- ( ( A e. D /\ B e. D /\ C e. D ) ->
                 ( A .x. ( B .+ C ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) ) $=
      ( cv cop co wceq oveq1 oveq12d eqeq12d oveq2d oveq2 oveq1d wcel wa opeq12
      cec w3a eceq1d mp2an adantl sylan2 eqtrd 3impb oveqan12d syl2an 3ecoptocl
      3impdi 3eqtr4a ) AUPZBUPZUQLVIZCUPZDUPZUQLVIZEUPZFUPZUQLVIZKURZNURZWDWGNU
      RZWDWJNURZKURZUSGWKNURZGWGNURZGWJNURZKURZUSGHWJKURZNURZGHNURZWRKURZUSGHIK
      URZNURZXBGINURZKURZUSABCDEFGHIMLJUEWDGUSZWLWPWOWSWDGWKNUTXHWMWQWNWRKWDGWG
      NUTWDGWJNUTVAVBWGHUSZWPXAWSXCXIWKWTGNWGHWJKUTVCXIWQXBWRKWGHGNVDVEVBWJIUSZ
      XAXEXCXGXJWTXDGNWJIHKVDVCXJWRXFXBKWJIGNVDVCVBWBMVFWCMVFVGZWEMVFWFMVFVGZWH
      MVFWIMVFVGZVJOPUQZLVIZQRUQZLVIZWLWOOQUSZPRUSZXOXQUSUNUOXRXSVGXNXPLOPQRVHV
      KVLXKXLXMWLXOUSXKXLXMVGZVGWLWDSTUQLVIZNURZXOXTWLYBUSXKXTWKYAWDNUFVCVMXTXK
      SMVFTMVFVGYBXOUSUKUGVNVOVPXKXLXMWOXQUSXKXLVGZXKXMVGZVGWOUAUBUQLVIZUCUDUQL
      VIZKURZXQYCYDWMYEWNYFKUHUIVQYCUAMVFUBMVFVGUCMVFUDMVFVGYGXQUSYDULUMUJVRVOV
      TWAVS $.
  $}

  ${
    $d x y z w v u A $.  $d z w v u B $.  $d w v u C $.  $d x y z w v u .+ $.
    $d x y z w v u .~ $.  $d x y z w v u S $.  $d x y z w v u .x. $.
    $d z w v u D $.
    ecovidi.1 $e |- D = ( ( S X. S ) /. .~ ) $.
    $( M,N for BFC $)
    ecovidi.2 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
 ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ ) $.
    $( H,J for AG(BFC) $)
    ecovidi.3 $e |- ( ( ( x e. S /\ y e. S ) /\ ( M e. S /\ N e. S ) ) ->
 ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ ) $.
    $( W,X for AGB $)
    ecovidi.4 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
 ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ ) $.
    $( Y,Z for AGC $)
    ecovidi.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ->
 ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = [ <. Y , Z >. ] .~ ) $.
    $( K,L for (AGB)F(AGC) $)
    ecovidi.6 $e |- ( ( ( W e. S /\ X e. S ) /\ ( Y e. S /\ Z e. S ) ) ->
 ( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ ) $.
    ecovidi.7 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
                     ( M e. S /\ N e. S ) ) $.
    ecovidi.8 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
                     ( W e. S /\ X e. S ) ) $.
    ecovidi.9 $e |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ->
                     ( Y e. S /\ Z e. S ) ) $.
    ecovidi.10 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S )
        /\ ( v e. S /\ u e. S ) ) -> H = K ) $.
    ecovidi.11 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S )
        /\ ( v e. S /\ u e. S ) ) -> J = L ) $.
    $( Lemma used to transfer a distributive law via an equivalence relation.
       (Contributed by Jim Kingdon, 17-Sep-2019.) $)
    ecovidi $p |- ( ( A e. D /\ B e. D /\ C e. D ) ->
                 ( A .x. ( B .+ C ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) ) $=
      ( cv cop co wceq oveq1 oveq12d eqeq12d oveq2d oveq2 oveq1d wcel wa opeq12
      cec w3a eceq1d syl2anc adantl sylan2 eqtrd 3impb oveqan12d syl2an 3eqtr4d
      3impdi 3ecoptocl ) AUPZBUPZUQLVIZCUPZDUPZUQLVIZEUPZFUPZUQLVIZKURZNURZWDWG
      NURZWDWJNURZKURZUSGWKNURZGWGNURZGWJNURZKURZUSGHWJKURZNURZGHNURZWRKURZUSGH
      IKURZNURZXBGINURZKURZUSABCDEFGHIMLJUEWDGUSZWLWPWOWSWDGWKNUTXHWMWQWNWRKWDG
      WGNUTWDGWJNUTVAVBWGHUSZWPXAWSXCXIWKWTGNWGHWJKUTVCXIWQXBWRKWGHGNVDVEVBWJIU
      SZXAXEXCXGXJWTXDGNWJIHKVDVCXJWRXFXBKWJIGNVDVCVBWBMVFWCMVFVGZWEMVFWFMVFVGZ
      WHMVFWIMVFVGZVJZOPUQZLVIZQRUQZLVIZWLWOXNOQUSZPRUSZXPXRUSUNUOXSXTVGXOXQLOP
      QRVHVKVLXKXLXMWLXPUSXKXLXMVGZVGWLWDSTUQLVIZNURZXPYAWLYCUSXKYAWKYBWDNUFVCV
      MYAXKSMVFTMVFVGYCXPUSUKUGVNVOVPXKXLXMWOXRUSXKXLVGZXKXMVGZVGWOUAUBUQLVIZUC
      UDUQLVIZKURZXRYDYEWMYFWNYGKUHUIVQYDUAMVFUBMVFVGUCMVFUDMVFVGYHXRUSYEULUMUJ
      VRVOVTVSWA $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The mapping operation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c ^m $.  $( The mapping operation. $)
  $c ^pm $.  $( The mapping operation. $)

  $( Extend the definition of a class to include the mapping operation.  (Read
     for ` A ^m B ` , "the set of all functions that map from ` B ` to
     ` A ` .) $)
  cmap $a class ^m $.

  $( Extend the definition of a class to include the partial mapping operation.
     (Read for ` A ^pm B ` , "the set of all partial functions that map from
     ` B ` to ` A ` .) $)
  cpm $a class ^pm $.

  ${
    $d x y f $.
    $( Define the mapping operation or set exponentiation.  The set of all
       functions that map from ` B ` to ` A ` is written ` ( A ^m B ) ` (see
       ~ mapval ).  Many authors write ` A ` followed by ` B ` as a superscript
       for this operation and rely on context to avoid confusion other
       exponentiation operations (e.g., Definition 10.42 of [TakeutiZaring]
       p. 95).  Other authors show ` B ` as a prefixed superscript, which is
       read " ` A ` pre ` B ` " (e.g., definition of [Enderton] p. 52).
       Definition 8.21 of [Eisenberg] p. 125 uses the notation Map( ` B ` ,
       ` A ` ) for our ` ( A ^m B ) ` .  The up-arrow is used by Donald Knuth
       for iterated exponentiation (_Science_ 194, 1235-1242, 1976).  We adopt
       the first case of his notation (simple exponentiation) and subscript it
       with _m_ to distinguish it from other kinds of exponentiation.
       (Contributed by NM, 8-Dec-2003.) $)
    df-map $a |- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } ) $.

    $( Define the partial mapping operation.  A partial function from ` B ` to
       ` A ` is a function from a subset of ` B ` to ` A ` .  The set of all
       partial functions from ` B ` to ` A ` is written ` ( A ^pm B ) ` (see
       ~ pmvalg ).  A notation for this operation apparently does not appear in
       the literature.  We use ` ^pm ` to distinguish it from the less general
       set exponentiation operation ` ^m ` ( ~ df-map ) .  See ~ mapsspm for
       its relationship to set exponentiation.  (Contributed by NM,
       15-Nov-2007.) $)
    df-pm $a |- ^pm = ( x e. _V , y e. _V |->
                    { f e. ~P ( y X. x ) | Fun f } ) $.
  $}

  ${
    $d f g A $.  $d f g B $.
    $( When ` A ` is a proper class, the class of all functions mapping ` A `
       to ` B ` is empty.  Exercise 4.41 of [Mendelson] p. 255.  (Contributed
       by NM, 8-Dec-2003.) $)
    mapprc $p |- ( -. A e. _V -> { f | f : A --> B } = (/) ) $=
      ( vg cvv wcel wn cv wf cab wex c0 wceq cdm fdm vex dmex eqeltrrdi exlimiv
      abn0m sylbi con3i notm0 sylib ) AEFZGDHABCHZIZCJZFDKZGUHLMUIUEUIUGCKUEUGC
      DTUGUECUGAUFNEABUFOUFCPQRSUAUBDUHUCUD $.

    $( The class of all partial functions from one set to another is a set.
       (Contributed by NM, 15-Nov-2007.) $)
    pmex $p |- ( ( A e. C /\ B e. D ) ->
                   { f | ( Fun f /\ f C_ ( A X. B ) ) } e. _V ) $=
      ( wcel wa cv wfun cxp wss cab cvv ancom abbii xpexg abssexg syl eqeltrid
      ) ACFBDFGZEHZIZUAABJZKZGZELUDUBGZELZMUEUFEUBUDNOTUCMFUGMFABCDPUBEUCMQRS
      $.

    $( The class of all functions mapping one set to another is a set.  Remark
       after Definition 10.24 of [Kunen] p. 31.  (Contributed by Raph Levien,
       4-Dec-2003.) $)
    mapex $p |- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V ) $=
      ( wcel wa cv cab cxp cpw wss cvv fssxp ss2abi df-pw sseqtrri xpexg pwexg
      wf syl ssexg sylancr ) ACFBDFGZABEHZTZEIZABJZKZLUIMFZUGMFUGUEUHLZEIUIUFUK
      EABUENOEUHPQUDUHMFUJABCDRUHMSUAUGUIMUBUC $.
  $}

  ${
    $d f x y $.
    $( Set exponentiation has a universal domain.  (Contributed by NM,
       8-Dec-2003.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    fnmap $p |- ^m Fn ( _V X. _V ) $=
      ( vx vy vf cvv cv wf cab cmap df-map wcel vex mapex mp2an fnmpoi ) ABDDBE
      ZAEZCEFCGZHABCIODJPDJQDJBKAKOPDDCLMN $.

    $( Partial function exponentiation has a universal domain.  (Contributed by
       Mario Carneiro, 14-Nov-2013.) $)
    fnpm $p |- ^pm Fn ( _V X. _V ) $=
      ( vx vy vf cvv cv wfun cxp cpw crab cpm df-pm vex xpex pwex rabex fnmpoi
      ) ABDDCEFZCBEZAEZGZHZIJABCKQCUATRSBLALMNOP $.

    $( Set exponentiation is a well-behaved binary operator.  (Contributed by
       Stefan O'Rear, 27-Feb-2015.) $)
    reldmmap $p |- Rel dom ^m $=
      ( vx vy vf cvv cv wf cab cmap df-map reldmmpo ) ABDDBEAECEFCGHABCIJ $.
  $}

  ${
    $d x y f A $.  $d x y f B $.
    $( The value of set exponentiation. ` ( A ^m B ) ` is the set of all
       functions that map from ` B ` to ` A ` .  Definition 10.24 of [Kunen]
       p. 24.  (Contributed by NM, 8-Dec-2003.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    mapvalg $p |- ( ( A e. C /\ B e. D ) ->
                  ( A ^m B ) = { f | f : B --> A } ) $=
      ( vx vy wcel wa cv wf cab cvv cmap co wceq mapex ancoms elex abbidv feq3
      wi feq2 df-map ovmpog 3expia syl2an mpd ) ACHZBDHZIBAEJZKZELZMHZABNOUMPZU
      JUIUNBADCEQRUIAMHZBMHZUNUOUBUJACSBDSUPUQUNUOFGABMMGJZFJZUKKZELUMNURAUKKZE
      LMUSAPUTVAEUSAURUKUATURBPVAULEURBAUKUCTFGEUDUEUFUGUH $.

    $( The value of the partial mapping operation. ` ( A ^pm B ) ` is the set
       of all partial functions that map from ` B ` to ` A ` .  (Contributed by
       NM, 15-Nov-2007.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    pmvalg $p |- ( ( A e. C /\ B e. D ) ->
                  ( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } ) $=
      ( vx vy wcel wa cv cxp cpw crab cvv cpm wceq syl elex pweqd rabeq wfun co
      wss ssrab2 xpexg ancoms pwexg ssexg sylancr wi xpeq2 ovmpog 3expia syl2an
      xpeq1 df-pm mpd ) ACHZBDHZIZEJUAZEBAKZLZMZNHZABOUBVDPZUTVDVCUCVCNHZVEVAEV
      CUDUTVBNHZVGUSURVHBADCUEUFVBNUGQVDVCNUHUIURANHZBNHZVEVFUJUSACRBDRVIVJVEVF
      FGABNNVAEGJZFJZKZLZMZVDOVAEVKAKZLZMZNVLAPZVNVQPVOVRPVSVMVPVLAVKUKSVAEVNVQ
      TQVKBPZVQVCPVRVDPVTVPVBVKBAUOSVAEVQVCTQFGEUPULUMUNUQ $.
  $}

  ${
    $d f A $.  $d f B $.
    mapval.1 $e |- A e. _V $.
    mapval.2 $e |- B e. _V $.
    $( The value of set exponentiation (inference version). ` ( A ^m B ) ` is
       the set of all functions that map from ` B ` to ` A ` .  Definition
       10.24 of [Kunen] p. 24.  (Contributed by NM, 8-Dec-2003.) $)
    mapval $p |- ( A ^m B ) = { f | f : B --> A } $=
      ( cvv wcel cmap co cv wf cab wceq mapvalg mp2an ) AFGBFGABHIBACJKCLMDEABF
      FCNO $.
  $}

  ${
    $d g A $.  $d g B $.  $d g C $.
    $( Membership relation for set exponentiation.  (Contributed by NM,
       17-Oct-2006.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
    elmapg $p |- ( ( A e. V /\ B e. W ) ->
                  ( C e. ( A ^m B ) <-> C : B --> A ) ) $=
      ( vg wcel wa cmap co cv wf cab mapvalg eleq2d cvv wi wb fex2 3com13 bitrd
      3expia feq1 elab3g syl ) ADGZBEGZHZCABIJZGCBAFKZLZFMZGZBACLZUHUIULCABDEFN
      OUHUNCPGZQUMUNRUFUGUNUOUNUGUFUOBACEDSTUBUKUNFCPBAUJCUCUDUEUA $.
  $}

  ${
    elmapd.a $e |- ( ph -> A e. V ) $.
    elmapd.b $e |- ( ph -> B e. W ) $.
    $( Deduction form of ~ elmapg .  (Contributed by BJ, 11-Apr-2020.) $)
    elmapd $p |- ( ph -> ( C e. ( A ^m B ) <-> C : B --> A ) ) $=
      ( wcel cmap co wf wb elmapg syl2anc ) ABEICFIDBCJKICBDLMGHBCDEFNO $.
  $}

  ${
    $d B f $.  $d V f $.
    $( The empty set is the only map with empty domain.  (Contributed by Glauco
       Siliprandi, 11-Oct-2020.)  (Proof shortened by Thierry Arnoux,
       3-Dec-2021.) $)
    mapdm0 $p |- ( B e. V -> ( B ^m (/) ) = { (/) } ) $=
      ( vf wcel c0 cmap co csn cv wceq wf cvv 0ex elmapg mpan2 f0bi bitrdi elsn
      wb vex bitr4di eqrdv ) ABDZCAEFGZEHZUCCIZUDDZUFEJZUFUEDUCUGEAUFKZUHUCELDU
      GUISMAEUFBLNOUFAPQUFECTRUAUB $.
  $}

  ${
    $d g A $.  $d g B $.  $d g C $.
    $( The predicate "is a partial function".  (Contributed by Mario Carneiro,
       14-Nov-2013.) $)
    elpmg $p |- ( ( A e. V /\ B e. W ) ->
                  ( C e. ( A ^pm B ) <-> ( Fun C /\ C C_ ( B X. A ) ) ) ) $=
      ( vg wcel wa cpm co wfun cxp cpw wss cv crab bitrdi cvv wi a1i funeq elex
      pmvalg eleq2d elrab ancom xpexg ancoms ssexg expcom elpwg pm5.21ndd bitrd
      syl wb anbi2d ) ADGZBEGZHZCABIJZGZCKZCBALZMZGZHZVBCVCNZHUSVAVEVBHZVFUSVAC
      FOZKZFVDPZGVHUSUTVKCABDEFUCUDVJVBFCVDVICUAUEQVEVBUFQUSVEVGVBUSCRGZVEVGVEV
      LSUSCVDUBTUSVCRGZVGVLSURUQVMBAEDUGUHVGVMVLCVCRUIUJUNVLVEVGUOSUSCVCRUKTULU
      PUM $.
  $}

  $( The predicate "is a partial function".  (Contributed by NM,
     31-Dec-2013.) $)
  elpm2g $p |- ( ( A e. V /\ B e. W ) ->
                ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) ) $=
    ( wcel wa cpm co wfun cxp wss cdm wf elpmg funssxp bitrdi ) ADFBEFGCABHIFCJ
    CBAKLGCMZACNRBLGABCDEOBACPQ $.

  $( Sufficient condition for being a partial function.  (Contributed by NM,
     31-Dec-2013.) $)
  elpm2r $p |- ( ( ( A e. V /\ B e. W ) /\ ( F : C --> A /\ C C_ B ) ) ->
                   F e. ( A ^pm B ) ) $=
    ( wcel wa wf wss cpm co cdm wb fdm feq2d sseq1d anbi12d adantr ibir elpm2g
    imbitrrid imp ) AEGBFGHZCADIZCBJZHZDABKLGZUGUHUDDMZADIZUIBJZHZUGULUEULUGNUF
    UEUJUEUKUFUEUICADCADOZPUEUICBUMQRSTABDEFUAUBUC $.

  ${
    $d f x y $.
    $( A partial function is a function.  (Contributed by Mario Carneiro,
       15-Sep-2015.) $)
    elpmi $p |- ( F e. ( A ^pm B ) -> ( F : dom F --> A /\ dom F C_ B ) ) $=
      ( vx vy vf cpm co wcel cdm wf wss wa cvv wb cv wfun cxp cpw crab elmpocl
      df-pm elpm2g syl ibi ) CABGHIZCJZACKUGBLMZUFANIBNIMUFUHODENNFPQFEPDPRSTAB
      GCDEFUBUAABCNNUCUDUE $.
  $}

  $( A partial function is a function.  (Contributed by Mario Carneiro,
     30-Jan-2014.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  pmfun $p |- ( F e. ( A ^pm B ) -> Fun F ) $=
    ( cpm co wcel cdm wf wss wa wfun elpmi ffun adantr syl ) CABDEFCGZACHZPBIZJ
    CKZABCLQSRPACMNO $.

  ${
    $d f x y $.
    $( Eliminate antecedent for mapping theorems: domain can be taken to be a
       set.  (Contributed by Stefan O'Rear, 8-Oct-2014.) $)
    elmapex $p |- ( A e. ( B ^m C ) -> ( B e. _V /\ C e. _V ) ) $=
      ( vx vy vf cvv cv wf cab cmap df-map elmpocl ) DEGGEHDHFHIFJBCKADEFLM $.
  $}

  $( A mapping is a function, forward direction only with superfluous
     antecedent removed.  (Contributed by Stefan O'Rear, 10-Oct-2014.) $)
  elmapi $p |- ( A e. ( B ^m C ) -> A : C --> B ) $=
    ( cmap co wcel wf cvv wa wb elmapex elmapg syl ibi ) ABCDEFZCBAGZOBHFCHFIOP
    JABCKBCAHHLMN $.

  ${
    $d A f m $.  $d B f m $.  $d V m $.
    $( If ` B ` is a set, the value of the set exponentiation ` ( B ^m A ) ` is
       the class of all functions from ` A ` to ` B ` .  Generalisation of
       ~ mapvalg to arbitrary domains.  Note that the class
       ` { f | f : A --> B } ` can only contain set-functions, as opposed to
       arbitrary class-functions.  When ` A ` is a proper class, there can be
       no set-functions on it, so the above class is empty (see also
       ~ fsetdmprc0 ), hence a set.  In this case, both sides of the equality
       in this theorem are the empty set.  (Contributed by AV, 8-Aug-2024.) $)
    mapfset $p |- ( B e. V -> { f | f : A --> B } = ( B ^m A ) ) $=
      ( vm wcel cv wf cab cmap co vex feq1 elab wa cvv simpr dmfex mpan adantr
      elmapd exbiri pm2.43b elmapi impbid1 bitrid eqrdv ) BDFZEABCGZHZCIZBAJKZE
      GZUKFABUMHZUHUMULFZUJUNCUMELZABUIUMMNUHUNUOUHUNUOUNUHUOUNUNUHOBAUMDPUNUHQ
      UNAPFZUHUMPFUNUQUPABPUMRSTUAUBUCUMBAUDUEUFUG $.
  $}

  ${
    $d A f $.  $d B f $.
    $( The value of the set exponentiation ` ( B ^m A ) ` is a subset of the
       class of functions from ` A ` to ` B ` .  (Contributed by AV,
       10-Aug-2024.) $)
    mapssfsetg $p |- ( B e. V -> ( B ^m A ) C_ { f | f : A --> B } ) $=
      ( wcel cv wf cab cmap co wceq wss mapfset eqimss2 syl ) BDEABCFGCHZBAIJZK
      QPLABCDMQPNO $.

    $d A f m $.  $d B m $.
    $( The value of the set exponentiation ` ( B ^m A ) ` is a superset of the
       set of all functions from ` A ` onto ` B ` .  (Contributed by AV,
       7-Aug-2024.) $)
    mapfoss $p |- { f | f : A -onto-> B } C_ ( B ^m A ) $=
      ( vm cv wfo cab cmap co wcel vex foeq1 elab wf fof cvv crn forn eqeltrrdi
      rnex dmfex sylancr elmapd mpbird sylbi ssriv ) DABCEZFZCGZBAHIZDEZUIJABUK
      FZUKUJJZUHULCUKDKZABUGUKLMULUMABUKNZABUKOZULBAUKPPULBUKQPABUKRUKUNTSULUKP
      JUOAPJUNUPABPUKUAUBUCUDUEUF $.
  $}

  ${
    $d A f g $.  $d B f g $.
    $( The class of all functions from ` A ` into ` B ` is a subclass of the
       power class of the cartesion product of ` A ` and ` B ` .  (Contributed
       by AV, 13-Sep-2024.) $)
    fsetsspwxp $p |- { f | f : A --> B } C_ ~P ( A X. B ) $=
      ( vg cv wf cab cxp cpw wss wcel fssxp vex feq1 elab velpw 3imtr4i ssriv )
      DABCEZFZCGZABHZIZABDEZFZUDUBJUDUAKUDUCKABUDLTUECUDDMABSUDNODUBPQR $.
  $}

  $( The set of functions from the empty set is the singleton containing the
     empty set.  (Contributed by AV, 13-Sep-2024.) $)
  fset0 $p |- { f | f : (/) --> B } = { (/) } $=
    ( c0 cv wf cab wceq csn f0bi abbii df-sn eqtr4i ) CABDZEZBFMCGZBFCHNOBMAIJB
    CKL $.

  ${
    $d A f g $.
    $( The set of functions with a proper class as domain is empty.
       (Contributed by AV, 22-Aug-2024.) $)
    fsetdmprc0 $p |- ( A e/ _V -> { f | f Fn A } = (/) ) $=
      ( vg cvv wnel cv wfn wn wal cab c0 wceq wcel df-nel vex a1i fndmexd con3i
      id sylbi alrimiv fneq1 ab0w sylibr ) ADEZCFZAGZHZCIBFZAGZBJKLUEUHCUEADMZH
      UHADNUGUKUGAUFDUFDMUGCOPUGSQRTUAUJUGBCAUIUFUBUCUD $.

    $d B f $.
    $( The set of injections between two sets exists.  (Contributed by AV,
       14-Aug-2024.) $)
    f1setexg $p |- ( ( A e. V /\ B e. W ) -> { f | f : A -1-1-> B } e. _V ) $=
      ( cv ccnv wfun wf1 cab wf wa df-f1 abbii fabexg ) CFZGHZCABDEABPIZCJRABPK
      QLCABPMNO $.
  $}

  $( A mapping is a function with the appropriate domain.  (Contributed by AV,
     6-Apr-2019.) $)
  elmapfn $p |- ( A e. ( B ^m C ) -> A Fn C ) $=
    ( cmap co wcel wf wfn elmapi ffn syl ) ABCDEFCBAGACHABCICBAJK $.

  $( A mapping is always a function.  (Contributed by Stefan O'Rear,
     9-Oct-2014.)  (Revised by Stefan O'Rear, 5-May-2015.) $)
  elmapfun $p |- ( A e. ( B ^m C ) -> Fun A ) $=
    ( cmap co wcel wf wfun elmapi ffun syl ) ABCDEFCBAGAHABCICBAJK $.

  $( A restricted mapping is a mapping.  (Contributed by Stefan O'Rear,
     9-Oct-2014.)  (Revised by Mario Carneiro, 5-May-2015.) $)
  elmapssres $p |- ( ( A e. ( B ^m C ) /\ D C_ C ) ->
      ( A |` D ) e. ( B ^m D ) ) $=
    ( cmap co wcel wss wa cres wf elmapi fssres sylan cvv elmapex simpld adantr
    simprd ssexg ancoms elmapd mpbird ) ABCEFGZDCHZIZADJZBDEFGDBUGKZUDCBAKUEUHA
    BCLCBDAMNUFBDUGOOUDBOGZUEUDUICOGZABCPZQRUDUJUEDOGZUDUIUJUKSUEUJULDCOTUANUBU
    C $.

  $( A total function is a partial function.  (Contributed by Mario Carneiro,
     31-Dec-2013.) $)
  fpmg $p |- ( ( A e. V /\ B e. W /\ F : A --> B ) -> F e. ( B ^pm A ) ) $=
    ( wcel wf cpm co wa wss ssid elpm2r mpanr2 3impa 3com12 ) BEFZADFZABCGZCBAH
    IFZQRSTQRJSAAKTALBAACEDMNOP $.

  ${
    $d f A $.  $d f B $.  $d f C $.  $d f D $.  $d f V $.  $d f W $.
    $( Subset relation for the set of partial functions.  (Contributed by Mario
       Carneiro, 31-Dec-2013.) $)
    pmss12g $p |- ( ( ( A C_ C /\ B C_ D ) /\ ( C e. V /\ D e. W ) ) ->
                   ( A ^pm B ) C_ ( C ^pm D ) ) $=
      ( vf wss wa wcel cpm co cv wfun cxp wi wb cvv ssexg elpmg xpss12 sstr syl
      ancoms expcom anim2d adantr syl2an an4s adantl 3imtr4d ssrdv ) ACHZBDHZIZ
      CEJZDFJZIZIZGABKLZCDKLZUSGMZNZVBBAOZHZIZVCVBDCOZHZIZVBUTJZVBVAJZUOVFVIPUR
      UOVEVHVCUOVDVGHZVEVHPUNUMVLBDACUAUDVEVLVHVBVDVGUBUEUCUFUGUMUPUNUQVJVFQZUM
      UPIARJBRJVMUNUQIACESBDFSABVBRRTUHUIURVKVIQUOCDVBEFTUJUKUL $.
  $}

  ${
    $d f x y $.
    $( Elementhood of a restricted function in the set of partial functions.
       (Contributed by Mario Carneiro, 31-Dec-2013.) $)
    pmresg $p |- ( ( B e. V /\ F e. ( A ^pm C ) ) ->
                 ( F |` B ) e. ( A ^pm B ) ) $=
      ( vx vy vf wcel cpm co wa cvv cres wf cv wfun adantl wss wceq cdm cin cxp
      cpw crab df-pm elmpocl1 simpl elpmi simpld inss1 fssres sylancl ffun wrel
      resres funrel resdm reseq1 3syl eqtr3id feq1d mpbid inss2 elpm2r syl21anc
      mpanr2 ) BEIZDACJKIZLZAMIZVHDUAZBUBZADBNZOZVNABJKIZVIVKVHFGMMHPQHGPFPUCUD
      UEACJDFGHUFUGRVHVIUHVJVMADVMNZOZVOVJVLADOZVMVLSVRVIVSVHVIVSVLCSACDUIUJRZV
      LBUKVLAVMDULUMVJVMAVQVNVJVSDQZVQVNTVTVLADUNWAVQDVLNZBNZVNDVLBUPWADUOWBDTW
      CVNTDUQDURWBDBUSUTVAUTVBVCVKVHLVOVMBSVPVLBVDABVMVNMEVEVGVF $.
  $}

  ${
    $d g A $.  $d f g B $.  $d g F $.
    elmap.1 $e |- A e. _V $.
    elmap.2 $e |- B e. _V $.
    $( Membership relation for set exponentiation.  (Contributed by NM,
       8-Dec-2003.) $)
    elmap $p |- ( F e. ( A ^m B ) <-> F : B --> A ) $=
      ( cvv wcel cmap co wf wb elmapg mp2an ) AFGBFGCABHIGBACJKDEABCFFLM $.

    $( Alternate expression for the value of set exponentiation.  (Contributed
       by NM, 3-Nov-2007.) $)
    mapval2 $p |- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } ) $=
      ( vg cmap co cxp cpw cv wfn cab cin wf wss wa wcel dff2 bitri ancom elmap
      elin velpw vex fneq1 elab anbi12i 3bitr4i eqriv ) FABGHZBAIZJZCKZBLZCMZNZ
      BAFKZOZURULPZURBLZQZURUKRURUQRZUSVAUTQVBBAURSVAUTUATABURDEUBVCURUMRZURUPR
      ZQVBURUMUPUCVDUTVEVAFULUDUOVACURFUEBUNURUFUGUHTUIUJ $.

    $( The predicate "is a partial function".  (Contributed by NM,
       15-Nov-2007.)  (Revised by Mario Carneiro, 14-Nov-2013.) $)
    elpm $p |- ( F e. ( A ^pm B ) <-> ( Fun F /\ F C_ ( B X. A ) ) ) $=
      ( cvv wcel cpm co wfun cxp wss wa wb elpmg mp2an ) AFGBFGCABHIGCJCBAKLMND
      EABCFFOP $.

    $( The predicate "is a partial function".  (Contributed by NM,
       15-Nov-2007.)  (Revised by Mario Carneiro, 31-Dec-2013.) $)
    elpm2 $p |- ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) $=
      ( cvv wcel cpm co cdm wf wss wa wb elpm2g mp2an ) AFGBFGCABHIGCJZACKQBLMN
      DEABCFFOP $.

    $( A total function is a partial function.  (Contributed by NM,
       15-Nov-2007.)  (Revised by Mario Carneiro, 31-Dec-2013.) $)
    fpm $p |- ( F : A --> B -> F e. ( B ^pm A ) ) $=
      ( cvv wcel wf cpm co fpmg mp3an12 ) AFGBFGABCHCBAIJGDEABCFFKL $.
  $}

  ${
    $d f A $.  $d f B $.  $d f x y $.
    $( Set exponentiation is a subset of partial maps.  (Contributed by NM,
       15-Nov-2007.)  (Revised by Mario Carneiro, 27-Feb-2016.) $)
    mapsspm $p |- ( A ^m B ) C_ ( A ^pm B ) $=
      ( vf cmap co cpm cv wcel cvv wf elmapex simprd simpld elmapi fpmg syl3anc
      ssriv ) CABDEZABFEZCGZRHZBIHZAIHZBATJTSHUAUCUBTABKZLUAUCUBUDMTABNBATIIOPQ
      $.

    $( Partial maps are a subset of the power set of the Cartesian product of
       its arguments.  (Contributed by Mario Carneiro, 2-Jan-2017.) $)
    pmsspw $p |- ( A ^pm B ) C_ ~P ( B X. A ) $=
      ( vf vx vy cpm co cxp cpw cv wcel wss wfun wa wb crab df-pm elmpocl elpmg
      cvv syl ibi simprd velpw sylibr ssriv ) CABFGZBAHZIZCJZUGKZUJUHLZUJUIKUKU
      JMZULUKUMULNZUKATKBTKNUKUNODETTUMCEJDJHIPABFUJDECQRABUJTTSUAUBUCCUHUDUEUF
      $.

    $( Set exponentiation is a subset of the power set of the Cartesian product
       of its arguments.  (Contributed by NM, 8-Dec-2006.)  (Revised by Mario
       Carneiro, 26-Apr-2015.) $)
    mapsspw $p |- ( A ^m B ) C_ ~P ( B X. A ) $=
      ( cmap co cpm cxp cpw mapsspm pmsspw sstri ) ABCDABEDBAFGABHABIJ $.
  $}

  ${
    $d x A $.  $d x C $.  $d x D $.  $d x R $.
    fvmptmap.1 $e |- C e. _V $.
    fvmptmap.2 $e |- D e. _V $.
    fvmptmap.3 $e |- R e. _V $.
    fvmptmap.4 $e |- ( x = A -> B = C ) $.
    fvmptmap.5 $e |- F = ( x e. ( R ^m D ) |-> B ) $.
    $( Special case of ~ fvmpt for operator theorems.  (Contributed by NM,
       27-Nov-2007.) $)
    fvmptmap $p |- ( A : D --> R -> ( F ` A ) = C ) $=
      ( wf cmap co wcel cfv wceq elmap fvmpt sylbir ) EFBMBFENOZPBGQDRFEBJISABC
      DUBGKLHTUA $.
  $}

  ${
    $d f A $.  $d f B $.  $d f V $.  $d f W $.
    $( Set exponentiation with an empty exponent (ordinal number 0) is ordinal
       number 1.  Exercise 4.42(a) of [Mendelson] p. 255.  (Contributed by NM,
       10-Dec-2003.)  (Revised by Mario Carneiro, 30-Apr-2015.) $)
    map0e $p |- ( A e. V -> ( A ^m (/) ) = 1o ) $=
      ( vf wcel c0 cmap co c1o cv wf cvv 0ex elmapg mpan2 wceq f0bi el1o bitr4i
      wb bitrdi eqrdv ) ABDZCAEFGZHUBCIZUCDZEAUDJZUDHDZUBEKDUEUFSLAEUDBKMNUFUDE
      OUGUDAPUDQRTUA $.

    $( Set exponentiation with an empty base is the empty set, provided the
       exponent is nonempty.  Theorem 96 of [Suppes] p. 89.  (Contributed by
       NM, 10-Dec-2003.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
    map0b $p |- ( A =/= (/) -> ( (/) ^m A ) = (/) ) $=
      ( vf c0 wne cmap co cv wcel wf wceq elmapi cdm fdm crn wss frn ss0 dm0rn0
      syl sylibr eqtr3d necon3ai eq0rdv ) ACDBCAEFZBGZUDHZACUFACUEIZACJUECAKUGU
      ELZACACUEMUGUENZCJZUHCJUGUICOUJACUEPUIQSUERTUASUBUC $.

    $( Set exponentiation is empty iff the base is empty and the exponent is
       not empty.  Theorem 97 of [Suppes] p. 89.  (Contributed by Mario
       Carneiro, 30-Apr-2015.) $)
    map0g $p |- ( ( A e. V /\ B e. W ) ->
      ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) ) $=
      ( vf wcel wa cmap co c0 wceq wne cv wex wn wf elmapg imbitrrid ne0i syl6
      csn cxp fconst6g exlimdv necon2bd notm0 imbitrdi feq2 mpbiri necon2d jcad
      f0 oveq1 map0b sylan9eq impbid1 ) ACFBDFGZABHIZJKZAJKZBJLZGUQUSUTVAUQUSEM
      ZAFZENZOUTUQVDURJUQVCURJLZEUQVCBVBUAUBZURFZVEVCVGUQBAVFPBVBAUCABVFCDQRURV
      FSTUDUEEAUFUGUQBJURJUQBJKZJURFZVEVHVIUQBAJPZVHVJJAJPAULBJAJUHUIABJCDQRURJ
      STUJUKUTVAURJBHIJAJBHUMBUNUOUP $.
  $}

  ${
    $d A f y $.  $d B f y $.  $d f ph y $.
    mapsnd.1 $e |- ( ph -> A e. V ) $.
    mapsnd.2 $e |- ( ph -> B e. W ) $.
    $( The value of set exponentiation with a singleton exponent.  Theorem 98
       of [Suppes] p. 89.  (Contributed by NM, 10-Dec-2003.)  (Revised by
       Glauco Siliprandi, 24-Dec-2020.) $)
    mapsnd $p |- ( ph -> ( A ^m { B } ) = { f | E. y e. A f = { <. B , y >. } }
       ) $=
      ( cv csn wceq wcel wf cvv syl wa wex cima mpbid wrex cmap co snexg elmapd
      cop crn wbr weu wfn ffn snidg fneu syl2anr cab euabsn cdm imaeq2d imadmrn
      imasng fdm eqtr3di sylan9req eqeq1d exbidv bitrid sseld vsnid eleq2 impel
      frn mpbiri adantll ffrn feq3 syl5ibcom imp wb ad2antrr vex sylancl jca ex
      fsng eximdv mpd df-rex sylibr w3a wf1o f1osng adantr f1oeq1 bicomd adantl
      f1of 3adant2 wss snssi 3ad2ant2 fssd rexlimdv3a impbid bitrd eqabdv ) AEJ
      ZDBJZUFKZLZBCUAZECDKZUBUCZAXFXLMXKCXFNZXJACXKXFFOHADGMZXKOMIDGUDPUEAXMXJA
      XMXJAXMQZXGCMZXIQZBRZXJXOXFUGZXGKZLZBRZXRXODXGXFUHZBUIZYBXMXFXKUJDXKMZYDA
      XKCXFUKAXNYEIDGULPBXKDXFUMUNYDYCBUOZXTLZBRXOYBYCBUPXOYGYABXOYFXSXTAXMYFXF
      XKSZXSAXNYHYFLIBDGXFUTPXMXFXFUQZSYHXSXMYIXKXFXKCXFVAURXFUSVBVCVDVEVFTXOYA
      XQBXOYAXQXOYAQZXPXIXMYAXPAXMXGXSMZXPYAXMXSCXGXKCXFVKVGYAYKXGXTMBVHXSXTXGV
      IVLVJVMYJXKXTXFNZXIXMYAYLAXMYAYLXMXKXSXFNYAYLXKCXFVNXSXTXKXFVOVPVQVMYJXNX
      GOMZYLXIVRAXNXMYAIVSBVTZDXGGOXFWDWATWBWCWEWFXIBCWGWHWCAXIXMBCAXPXIWIXKXTC
      XFAXIYLXPAXIQZXKXTXFWJZYLYOXKXTXHWJZYPAYQXIAXNYMYQIYNDXGGOWKWAWLXIYQYPVRA
      XIYPYQXKXTXFXHWMWNWOTXKXTXFWPPWQXPAXTCWRXIXGCWSWTXAXBXCXDXE $.
  $}

  ${
    $d f y A $.  $d f y B $.
    map0.1 $e |- A e. _V $.
    map0.2 $e |- B e. _V $.
    $( Set exponentiation is empty iff the base is empty and the exponent is
       not empty.  Theorem 97 of [Suppes] p. 89.  (Contributed by NM,
       10-Dec-2003.) $)
    map0 $p |- ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) $=
      ( cvv wcel cmap co c0 wceq wne wa wb map0g mp2an ) AEFBEFABGHIJAIJBIKLMCD
      ABEENO $.

    $( The value of set exponentiation with a singleton exponent.  Theorem 98
       of [Suppes] p. 89.  (Contributed by NM, 10-Dec-2003.) $)
    mapsn $p |- ( A ^m { B } ) = { f | E. y e. A f = { <. B , y >. } } $=
      ( cv cop csn wceq wrex wcel wf wex snid cima cvv mpbiri syl wf1o co elmap
      cmap snex wa crn wbr weu wfn ffn fneu sylancl cab euabsn imasng ax-mp cdm
      fdm imaeq2d imadmrn eqtr3di eqtr3id eqeq1d exbidv bitrid mpbid eleq2 syl5
      vex frn sseld wfo dffn4 sylib fof feq3 syl5ibcom fsn imbitrdi jcad eximdv
      mpd df-rex sylibr wss f1osn f1oeq1 f1of fss syl2an expcom rexlimiv impbii
      snssi bitri abbi2i ) DGZCAGZHIZJZABKZDBCIZUCUAZWQXCLXBBWQMZXABXBWQECFUDUB
      XDXAXDWRBLZWTUEZANZXAXDWQUFZWRIZJZANZXGXDCWRWQUGZAUHZXKXDWQXBUIZCXBLXMXBB
      WQUJZCFOAXBCWQUKULXMXLAUMZXIJZANXDXKXLAUNXDXQXJAXDXPXHXIXDXPWQXBPZXHCQLXR
      XPJFACQWQUOUPXDWQWQUQZPXRXHXDXSXBWQXBBWQURUSWQUTVAVBVCVDVEVFXDXJXFAXDXJXE
      WTXJWRXHLZXDXEXJXTWRXILWRAVIZOXHXIWRVGRXDXHBWRXBBWQVJVKVHXDXJXBXIWQMZWTXD
      XBXHWQMZXJYBXDXBXHWQVLZYCXDXNYDXOXBWQVMVNXBXHWQVOSXHXIXBWQVPVQCWRWQFYAVRV
      SVTWAWBWTABWCWDWTXDABWTXEXDWTYBXIBWEXDXEWTXBXIWQTZYBWTYEXBXIWSTCWRFYAWFXB
      XIWQWSWGRXBXIWQWHSWRBWNXBXIBWQWIWJWKWLWMWOWP $.
  $}

  ${
    $d f A $.  $d f B $.  $d f C $.  $d f V $.
    $( Subset inheritance for set exponentiation.  Theorem 99 of [Suppes]
       p. 89.  (Contributed by NM, 10-Dec-2003.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    mapss $p |- ( ( B e. V /\ A C_ B ) -> ( A ^m C ) C_ ( B ^m C ) ) $=
      ( vf wcel wss wa cmap co cv wf elmapi adantl simplr simpll elmapex simprd
      fssd cvv elmapd mpbird ex ssrdv ) BDFZABGZHZEACIJZBCIJZUGEKZUHFZUJUIFZUGU
      KHZULCBUJLUMCABUJUKCAUJLUGUJACMNUEUFUKOSUMBCUJDTUEUFUKPUKCTFZUGUKATFUNUJA
      CQRNUAUBUCUD $.
  $}

  ${
    $d B x $.  $d I x $.  $d V x $.  $d W x $.  $d X x $.
    fdiagfn.f $e |- F = ( x e. B |-> ( I X. { x } ) ) $.
    $( Functionality of the diagonal map.  (Contributed by Stefan O'Rear,
       24-Jan-2015.) $)
    fdiagfn $p |- ( ( B e. V /\ I e. W ) -> F : B --> ( B ^m I ) ) $=
      ( wcel wa cv csn cxp cmap co wf fconst6g adantl wb elmapg adantr mpbird
      fmptd ) BEHDFHIZABDAJZKLZBDMNZCUCUDBHZIUEUFHZDBUEOZUGUIUCDUDBPQUCUHUIRUGB
      DUEEFSTUAGUB $.

    $( Functionality of the diagonal map.  (Contributed by Stefan O'Rear,
       24-Jan-2015.) $)
    fvdiagfn $p |- ( ( I e. W /\ X e. B ) -> ( F ` X ) = ( I X. { X } ) ) $=
      ( wcel wa csn cxp cvv cfv wceq simpr snexg xpexg sylan2 cv sneq syl2anc
      xpeq2d fvmptg ) DEHZFBHZIUEDFJZKZLHZFCMUGNUDUEOUEUDUFLHUHFBPDUFELQRAFDASZ
      JZKUGBLCUIFNUJUFDUIFTUBGUCUA $.
  $}

  ${
    $d B x y $.  $d S x y $.  $d X y $.
    mapsncnv.s $e |- S = { X } $.
    mapsncnv.b $e |- B e. _V $.
    mapsncnv.x $e |- X e. _V $.
    $( Every singleton map is a constant function.  (Contributed by Stefan
       O'Rear, 25-Mar-2015.) $)
    mapsnconst $p |- ( F e. ( B ^m S ) -> F = ( S X. { ( F ` X ) } ) ) $=
      ( cmap co wcel cfv cop csn cxp wceq wf snex elmap fsn2 cvv simprbi oveq2i
      sylbi eleq2s xpeq1i fvexg mpan2 xpsng sylancr eqtr2id eqtrd ) CABHIZJZCDD
      CKZLMZBUNMZNZCUOOZCADMZHIZULCUTJUSACPZURAUSCFDGQRVAUNAJURDACGSUAUCBUSAHEU
      BUDUMUQUSUPNZUOBUSUPEUEUMDTJZUNTJZVBUOOGUMVCVDGDCULTUFUGDUNTTUHUIUJUK $.

    ${
      mapsncnv.f $e |- F = ( x e. ( B ^m S ) |-> ( x ` X ) ) $.
      $( Expression for the inverse of the canonical map between a set and its
         set of singleton functions.  (Contributed by Stefan O'Rear,
         21-Mar-2015.) $)
      mapsncnv $p |- `' F = ( y e. B |-> ( S X. { y } ) ) $=
        ( cv cmap co wcel cfv wceq wa copab csn cxp ccnv cmpt wf elmapi sylancl
        snid ffvelcdm eqid mapsnconst jca sneq xpeq2d eqeq2d anbi12d syl5ibrcom
        eleq1 imp fconst6g snex elmap sylibr fvconst2 mp1i eqcomd impbii oveq2i
        fveq1 eleq2i anbi1i xpeq1i eqeq2i anbi2i 3bitr4i opabbii df-mpt cnvopab
        vex eqtri cnveqi 3eqtr4i ) AKZCDLMZNZBKZFWAOZPZQZBARZWDCNZWADWDSZTZPZQZ
        BAREUAZBCWKUBWGWMBAWACFSZLMZNZWFQZWIWAWOWJTZPZQZWGWMWRXAWQWFXAWQXAWFWEC
        NZWAWOWESZTZPZQWQXBXEWQWOCWAUCFWONZXBWACWOUDFIUFZWOCFWAUGUECWOWAFWOUHHI
        UIUJWFWIXBWTXEWDWECUPWFWSXDWAWFWJXCWOWDWEUKULUMUNUOUQWIWTWRWIWRWTWSWPNZ
        WDFWSOZPZQWIXHXJWIWOCWSUCXHWOWDCURCWOWSHFIUSUTVAWIXIWDXFXIWDPWIXGWOWDFB
        VQVBVCVDUJWTWQXHWFXJWAWSWPUPWTWEXIWDFWAWSVGUMUNUOUQVEWCWQWFWBWPWADWOCLG
        VFVHVIWLWTWIWKWSWADWOWJGVJVKVLVMVNWNWGABRZUAWHEXKEAWBWEUBXKJABWBWEVOVRV
        SWGABVPVRBACWKVOVT $.

      $( Explicit bijection between a set and its singleton functions.
         (Contributed by Stefan O'Rear, 21-Mar-2015.) $)
      mapsnf1o2 $p |- F : ( B ^m S ) -1-1-onto-> B $=
        ( vy cmap co wf1o wfn cv cvv vex fnmpti csn snex ccnv cfv fvex cxp xpex
        eqeltri mapsncnv dff1o4 mpbir2an ) BCKLZBDMDUJNDUAZBNAUJEAOZUBDEULPPAQH
        UCIRJBCJOZSZUDUKCUNCESPFEHTUFUMJQTUEAJBCDEFGHIUGRUJBDUHUI $.
    $}

    mapsnf1o3.f $e |- F = ( y e. B |-> ( S X. { y } ) ) $.
    $( Explicit bijection in the reverse of ~ mapsnf1o2 .  (Contributed by
       Stefan O'Rear, 24-Mar-2015.) $)
    mapsnf1o3 $p |- F : B -1-1-onto-> ( B ^m S ) $=
      ( vx cmap co wf1o cv cfv cmpt ccnv eqid mapsnf1o2 ax-mp f1ocnv wb csn cxp
      wceq mapsncnv eqtr4i f1oeq1 mpbir ) BBCKLZDMZBUJJUJEJNOPZQZMZUJBULMUNJBCU
      LEFGHULRZSUJBULUATDUMUEUKUNUBDABCANUCUDPUMIJABCULEFGHUOUFUGBUJDUMUHTUI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Infinite Cartesian products
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c X_ $.

  $( Extend class notation to include infinite Cartesian products. $)
  cixp $a class X_ x e. A B $.

  ${
    $d f x $.  $d f A $.  $d f B $.
    $( Definition of infinite Cartesian product of [Enderton] p. 54.  Enderton
       uses a bold "X" with ` x e. A ` written underneath or as a subscript, as
       does Stoll p. 47.  Some books use a capital pi, but we will reserve that
       notation for products of numbers.  Usually ` B ` represents a class
       expression containing ` x ` free and thus can be thought of as
       ` B ( x ) ` .  Normally, ` x ` is not free in ` A ` , although this is
       not a requirement of the definition.  (Contributed by NM,
       28-Sep-2006.) $)
    df-ixp $a |- X_ x e. A B =
               { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) } $.

    $d x A $.
    $( Eliminate the expression ` { x | x e. A } ` in ~ df-ixp , under the
       assumption that ` A ` and ` x ` are disjoint.  This way, we can say that
       ` x ` is bound in ` X_ x e. A B ` even if it appears free in ` A ` .
       (Contributed by Mario Carneiro, 12-Aug-2016.) $)
    dfixp $p |- X_ x e. A B =
                 { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) } $=
      ( cixp cv wcel cab wfn cfv wral wa df-ixp abid2 fneq2i anbi1i abbii eqtri
      ) ABCEDFZAFZBGAHZIZTSJCGABKZLZDHSBIZUCLZDHABCDMUDUFDUBUEUCUABSABNOPQR $.
  $}

  ${
    $d B f $.  $d V f $.  $d X f x $.
    $( The value of an infinite Cartesian product with a singleton.
       (Contributed by AV, 3-Dec-2018.) $)
    ixpsnval $p |- ( X e. V -> X_ x e. { X } B
                = { f | ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ B ) } ) $=
      ( wcel csn cixp cv wfn cfv wral wa cab dfixp wsbc ralsnsg sbcel12g csbfvg
      csb eleq1d 3bitrd anbi2d abbidv eqtrid ) EDFZAEGZBHCIZUGJZAIUHKZBFZAUGLZM
      ZCNUIEUHKZAEBTZFZMZCNAUGBCOUFUMUQCUFULUPUIUFULUKAEPAEUJTZUOFUPUKAEDQAEUJB
      DRUFURUNUOAEDUHSUAUBUCUDUE $.
  $}

  ${
    $d f x A $.  $d f B $.  $d x C $.  $d x D $.  $d f x F $.
    $( Membership in an infinite Cartesian product.  See ~ df-ixp for
       discussion of the notation.  (Contributed by NM, 28-Sep-2006.) $)
    elixp2 $p |- ( F e. X_ x e. A B <->
                ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) ) $=
      ( vf cvv wcel cixp wa wfn cv cfv wral w3a wceq fneq1 fveq1 eleq1d ralbidv
      anbi12d dfixp elab2g pm5.32i elex pm4.71ri 3anass 3bitr4i ) DFGZDABCHZGZI
      UHDBJZAKZDLZCGZABMZIZIUJUHUKUONUHUJUPEKZBJZULUQLZCGZABMZIUPEDUIFUQDOZURUK
      VAUOBUQDPVBUTUNABVBUSUMCULUQDQRSTABCEUAUBUCUJUHDUIUDUEUHUKUOUFUG $.

    fvixp.1 $e |- ( x = C -> B = D ) $.
    $( Projection of a factor of an indexed Cartesian product.  (Contributed by
       Mario Carneiro, 11-Jun-2016.) $)
    fvixp $p |- ( ( F e. X_ x e. A B /\ C e. A ) -> ( F ` C ) e. D ) $=
      ( cixp wcel cv cfv wral cvv wfn elixp2 simp3bi wceq fveq2 eleq12d rspccva
      sylan ) FABCHIZAJZFKZCIZABLZDBIDFKZEIZUBFMIFBNUFABCFOPUEUHADBUCDQUDUGCEUC
      DFRGSTUA $.
  $}

  ${
    $d A f x $.  $d B f $.  $d F f $.  $d f x $.
    $( A nuple is a function.  (Contributed by FL, 6-Jun-2011.)  (Revised by
       Mario Carneiro, 31-May-2014.) $)
    ixpfn $p |- ( F e. X_ x e. A B -> F Fn A ) $=
      ( vf cv wfn cixp fneq1 wcel cvv cfv wral elixp2 simp2bi vtoclga ) EFZBGZD
      BGEDABCHZBQDIQSJQKJRAFQLCJABMABCQNOP $.
  $}

  ${
    $d x F $.  $d x A $.
    elixp.1 $e |- F e. _V $.
    $( Membership in an infinite Cartesian product.  (Contributed by NM,
       28-Sep-2006.) $)
    elixp $p |- ( F e. X_ x e. A B <->
                ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) $=
      ( cixp wcel cvv wfn cv cfv wral w3a wa elixp2 3anass mpbiran bitri ) DABC
      FGDHGZDBIZAJDKCGABLZMZTUANZABCDOUBSUCESTUAPQR $.

    $d x B $.
    $( Membership in an infinite Cartesian product of a constant ` B ` .
       (Contributed by NM, 12-Apr-2008.) $)
    elixpconst $p |- ( F e. X_ x e. A B <-> F : A --> B ) $=
      ( cixp wcel wfn cv cfv wral wa wf elixp ffnfv bitr4i ) DABCFGDBHAIDJCGABK
      LBCDMABCDENABCDOP $.
  $}

  ${
    $d f x A $.  $d f x B $.
    $( Infinite Cartesian product of a constant ` B ` .  (Contributed by Mario
       Carneiro, 11-Jan-2015.) $)
    ixpconstg $p |- ( ( A e. V /\ B e. W ) -> X_ x e. A B = ( B ^m A ) ) $=
      ( vf wcel cixp cmap co wa cv wf cab vex elixpconst abbi2i mapvalg eqtr4id
      wceq ancoms ) CEGZBDGZABCHZCBIJZTUBUCKUDBCFLZMZFNUEUGFUDABCUFFOPQCBEDFRSU
      A $.

    ixpconst.1 $e |- A e. _V $.
    ixpconst.2 $e |- B e. _V $.
    $( Infinite Cartesian product of a constant ` B ` .  (Contributed by NM,
       28-Sep-2006.) $)
    ixpconst $p |- X_ x e. A B = ( B ^m A ) $=
      ( cvv wcel cixp cmap co wceq ixpconstg mp2an ) BFGCFGABCHCBIJKDEABCFFLM
      $.
  $}

  ${
    $d f x A $.  $d f x B $.  $d f C $.
    $( Equality theorem for infinite Cartesian product.  (Contributed by NM,
       29-Sep-2006.) $)
    ixpeq1 $p |- ( A = B -> X_ x e. A C = X_ x e. B C ) $=
      ( vf wceq cv wfn cfv wcel wral wa cixp fneq2 raleq anbi12d abbidv 3eqtr4g
      cab dfixp ) BCFZEGZBHZAGUBIDJZABKZLZESUBCHZUDACKZLZESABDMACDMUAUFUIEUAUCU
      GUEUHBCUBNUDABCOPQABDETACDETR $.

    ixpeq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for infinite Cartesian product.  (Contributed by Mario
       Carneiro, 11-Jun-2016.) $)
    ixpeq1d $p |- ( ph -> X_ x e. A C = X_ x e. B C ) $=
      ( wceq cixp ixpeq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
  $}

  ${
    $d f A $.  $d f B $.  $d f C $.  $d f x $.
    $( Subclass theorem for infinite Cartesian product.  (Contributed by NM,
       29-Sep-2006.)  (Revised by Mario Carneiro, 12-Aug-2016.) $)
    ss2ixp $p |- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C ) $=
      ( vf wss wral cv wcel cab wfn cfv cixp ssel ral2imi anim2d ss2abdv df-ixp
      wa 3sstr4g ) CDFZABGZEHZAHZBIAJKZUDUCLZCIZABGZSZEJUEUFDIZABGZSZEJABCMABDM
      UBUIULEUBUHUKUEUAUGUJABCDUFNOPQABCERABDERT $.
  $}

  $( Equality theorem for infinite Cartesian product.  (Contributed by NM,
     29-Sep-2006.) $)
  ixpeq2 $p |- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C ) $=
    ( wss wral wa cixp wceq ss2ixp anim12i eqss ralbii r19.26 bitri 3imtr4i ) C
    DEZABFZDCEZABFZGZABCHZABDHZEZUCUBEZGCDIZABFZUBUCIRUDTUEABCDJABDCJKUGQSGZABF
    UAUFUHABCDLMQSABNOUBUCLP $.

  ${
    $d x ph $.
    ixpeq2dva.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
    $( Equality theorem for infinite Cartesian product.  (Contributed by Mario
       Carneiro, 11-Jun-2016.) $)
    ixpeq2dva $p |- ( ph -> X_ x e. A B = X_ x e. A C ) $=
      ( wceq wral cixp ralrimiva ixpeq2 syl ) ADEGZBCHBCDIBCEIGAMBCFJBCDEKL $.
  $}

  ${
    $d x ph $.
    ixpeq2dv.1 $e |- ( ph -> B = C ) $.
    $( Equality theorem for infinite Cartesian product.  (Contributed by Mario
       Carneiro, 11-Jun-2016.) $)
    ixpeq2dv $p |- ( ph -> X_ x e. A B = X_ x e. A C ) $=
      ( wceq cv wcel adantr ixpeq2dva ) ABCDEADEGBHCIFJK $.
  $}

  ${
    $d A f x y $.  $d B f $.  $d C f $.
    cbvixp.1 $e |- F/_ y B $.
    cbvixp.2 $e |- F/_ x C $.
    cbvixp.3 $e |- ( x = y -> B = C ) $.
    $( Change bound variable in an indexed Cartesian product.  (Contributed by
       Jeff Madsen, 20-Jun-2011.) $)
    cbvixp $p |- X_ x e. A B = X_ y e. A C $=
      ( vf cv wfn cfv wcel wral wa cab cixp nfel2 weq dfixp fveq2 cbvral anbi2i
      eleq12d abbii 3eqtr4i ) IJZCKZAJZUGLZDMZACNZOZIPUHBJZUGLZEMZBCNZOZIPACDQB
      CEQUMURIULUQUHUKUPABCBUJDFRAUOEGRABSUJUODEUIUNUGUAHUDUBUCUEACDITBCEITUF
      $.
  $}

  ${
    $d A x y $.  $d B y $.  $d C x $.
    cbvixpv.1 $e |- ( x = y -> B = C ) $.
    $( Change bound variable in an indexed Cartesian product.  (Contributed by
       Jeff Madsen, 2-Sep-2009.) $)
    cbvixpv $p |- X_ x e. A B = X_ y e. A C $=
      ( nfcv cbvixp ) ABCDEBDGAEGFH $.
  $}

  ${
    $d x y z $.  $d A z $.  $d B z $.
    nfixp.1 $e |- F/_ y A $.
    nfixp.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for indexed Cartesian product.
       (Contributed by Mario Carneiro, 15-Oct-2016.)  (Revised by Jim Kingdon,
       15-Feb-2023.) $)
    nfixpxy $p |- F/_ y X_ x e. A B $=
      ( vz cixp cv wcel cab wfn cfv wral wa wnfc wtru a1i nfeld mptru nfcv nffn
      df-ixp nftru nfcvd nfabd wal df-ral wnf nffvd nfimd nfald nfxfr nfan nfab
      wi nfcxfr ) BACDHGIZAIZCJZAKZLZUSURMZDJZACNZOZGKACDGUCVFBGVBVEBBVAURBURUA
      ZBVAPQUTBAAUDZQBUSCQBUSUEZBCPQERSZUFTUBVEUTVDUPZAUGZBVDACUHVLBUIQVKBAVHQU
      TVDBVJQBVCDQBUSURBURPQVGRVIUJBDPQFRSUKULTUMUNUOUQ $.
  $}

  ${
    $d x y $.  $d A y $.  $d B y $.
    $( The index variable in an indexed Cartesian product is not free.
       (Contributed by Jeff Madsen, 19-Jun-2011.)  (Revised by Mario Carneiro,
       15-Oct-2016.) $)
    nfixp1 $p |- F/_ x X_ x e. A B $=
      ( vy cixp cv wcel cab wfn cfv wral df-ixp nfcv nfab1 nffn nfra1 nfan nfab
      wa nfcxfr ) AABCEDFZAFZBGZAHZIZUBUAJCGZABKZSZDHABCDLUHADUEUGAAUDUAAUAMUCA
      NOUFABPQRT $.
  $}

  ${
    $d f x A $.  $d f B $.
    $( A cartesian product of proper-class many sets is empty, because any
       function in the cartesian product has to be a set with domain ` A ` ,
       which is not possible for a proper class domain.  (Contributed by Mario
       Carneiro, 25-Jan-2015.) $)
    ixpprc $p |- ( -. A e. _V -> X_ x e. A B = (/) ) $=
      ( vf cvv wcel wn cv cixp wex c0 wceq wfn ixpfn cdm fndm vex eqeltrrdi syl
      dmex exlimiv con3i notm0 sylib ) BEFZGDHZABCIZFZDJZGUGKLUIUEUHUEDUHUFBMZU
      EABCUFNUJBUFOEBUFPUFDQTRSUAUBDUGUCUD $.
  $}

  ${
    $d f x A $.  $d f B $.  $d x F $.  $d f C $.
    $( A member of an infinite Cartesian product maps to the indexed union of
       the product argument.  Remark in [Enderton] p. 54.  (Contributed by NM,
       28-Sep-2006.) $)
    ixpf $p |- ( F e. X_ x e. A B -> F : A --> U_ x e. A B ) $=
      ( cixp wcel cvv wfn cv cfv wral w3a ciun wf elixp2 wa ssiun2 ralimia nfcv
      sseld anim2i nfiu1 ffnfvf sylibr 3adant1 sylbi ) DABCEFDGFZDBHZAIZDJZCFZA
      BKZLBABCMZDNZABCDOUHULUNUGUHULPUHUJUMFZABKZPUNULUPUHUKUOABUIBFCUMUJABCQTR
      UAABUMDABSABCUBADSUCUDUEUF $.

    $( The union of an infinite Cartesian product is included in a Cartesian
       product.  (Contributed by NM, 28-Sep-2006.)  (Revised by Mario Carneiro,
       24-Jun-2015.) $)
    uniixp $p |- U. X_ x e. A B C_ ( A X. U_ x e. A B ) $=
      ( vf cixp ciun cxp cpw wss cuni cv wcel ixpf fssxp syl velpw sylibr ssriv
      wf sspwuni mpbi ) ABCEZBABCFZGZHZIUBJUDIDUBUEDKZUBLZUFUDIZUFUELUGBUCUFSUH
      ABCUFMBUCUFNODUDPQRUBUDTUA $.

    $( The existence of an infinite Cartesian product. ` x ` is normally a
       free-variable parameter in ` B ` .  Remark in Enderton p. 54.
       (Contributed by NM, 28-Sep-2006.)  (Revised by Jim Kingdon,
       15-Feb-2023.) $)
    ixpexgg $p |- ( ( A e. W /\ A. x e. A B e. V ) -> X_ x e. A B e. _V ) $=
      ( wcel wral cixp cuni cvv ciun cxp wss uniixp iunexg xpexg syldan sylancr
      wa ssexg uniexb sylibr ) BEFZCDFABGZSZABCHZIZJFZUFJFUEUGBABCKZLZMUJJFZUHA
      BCNUCUDUIJFUKABCEDOBUIEJPQUGUJJTRUFUAUB $.

    $( The intersection of two infinite Cartesian products.  (Contributed by
       Mario Carneiro, 3-Feb-2015.) $)
    ixpin $p |- X_ x e. A ( B i^i C ) = ( X_ x e. A B i^i X_ x e. A C ) $=
      ( vf cin cixp cv wfn wcel wral wa anandi elin ralbii r19.26 bitri 3bitr4i
      cfv elixp anbi2i vex anbi12i eqriv ) EABCDFZGZABCGZABDGZFZEHZBIZAHUJSZUEJ
      ZABKZLZUJUGJZUJUHJZLZUJUFJUJUIJUKULCJZABKZULDJZABKZLZLUKUTLZUKVBLZLUOURUK
      UTVBMUNVCUKUNUSVALZABKVCUMVFABULCDNOUSVAABPQUAUPVDUQVEABCUJEUBZTABDUJVGTU
      CRABUEUJVGTUJUGUHNRUD $.
  $}

  ${
    $d f x y A $.  $d f x y B $.  $d y z B $.  $d f C $.
    $( The indexed intersection of a collection of infinite Cartesian products.
       (Contributed by Mario Carneiro, 6-Feb-2015.)  (Revised by Jim Kingdon,
       15-Feb-2023.) $)
    ixpiinm $p |- ( E. z z e. B ->
      X_ x e. A |^|_ y e. B C = |^|_ y e. B X_ x e. A C ) $=
      ( vf cv wcel wex cixp ciin wral wa wb cvv eliin vex elixp bitri wceq fvex
      eleq1w cbvexv wfn cfv elv ralbii ax-mp ralcom anbi2i 3bitr4g eqrdv sylbir
      r19.28mv eqcomd ) CHEIZCJZBEADFKZLZADBEFLZKZURBHEIZBJZUTVBUAVCUQBCBCEUCUD
      VDGUTVBVDGHZDUEZAHZVEUFZFIZADMZNZBEMZVFVJBEMZNZVEUTIZVEVBIZVFVJBEUOVOVEUS
      IZBEMZVLVOVROGBVEEUSPQUGVQVKBEADFVEGRZSUHTVPVFVHVAIZADMZNVNADVAVEVSSWAVMV
      FWAVIBEMZADMVMVTWBADVHPIVTWBOVGVEPPVSARUBBVHEFPQUIUHVIABDEUJTUKTULUMUNUP
      $.

    $( The intersection of a collection of infinite Cartesian products.
       (Contributed by Mario Carneiro, 3-Feb-2015.)  (Revised by Jim Kingdon,
       15-Feb-2023.) $)
    ixpintm $p |- ( E. z z e. B
        -> X_ x e. A |^| B = |^|_ y e. B X_ x e. A y ) $=
      ( cv wcel wex cint cixp ciin wceq ixpeq2 intiin a1i mprg ixpiinm eqtrid )
      CFEGCHADEIZJZADBEBFZKZJZBEADUAJKSUBLZTUCLADADSUBMUDAFDGBENOPABCDEUAQR $.
  $}

  ${
    $d f x $.  $d f A $.
    $( An infinite Cartesian product with an empty index set.  (Contributed by
       NM, 21-Sep-2007.) $)
    ixp0x $p |- X_ x e. (/) A = { (/) } $=
      ( vf c0 cixp cv wfn cfv wcel wral wa cab csn dfixp wceq velsn fn0 biantru
      ral0 3bitr2i abbi2i eqtr4i ) ADBECFZDGZAFUCHBIZADJZKZCLDMZADBCNUGCUHUCUHI
      UCDOUDUGCDPUCQUFUDUEASRTUAUB $.
  $}

  ${
    $d A f x $.  $d B f $.  $d V f $.
    $( An infinite Cartesian product is a subset of set exponentiation.  This
       version of ~ ixpssmapg avoids ~ ax-coll .  (Contributed by Mario
       Carneiro, 16-Nov-2014.) $)
    ixpssmap2g $p |- ( U_ x e. A B e. V ->
        X_ x e. A B C_ ( U_ x e. A B ^m A ) ) $=
      ( vf ciun wcel cixp cmap co cv wa wf ixpf adantl cvv wb wfn ixpfn cdm vex
      fndm dmex eqeltrrdi syl elmapg sylan2 mpbird ex ssrdv ) ABCFZDGZEABCHZUKB
      IJZULEKZUMGZUOUNGZULUPLUQBUKUOMZUPURULABCUONOUPULBPGZUQURQUPUOBRZUSABCUOS
      UTBUOTPBUOUBUOEUAUCUDUEUKBUODPUFUGUHUIUJ $.

    $( An infinite Cartesian product is a subset of set exponentiation.
       (Contributed by Jeff Madsen, 19-Jun-2011.) $)
    ixpssmapg $p |- ( A. x e. A B e. V ->
        X_ x e. A B C_ ( U_ x e. A B ^m A ) ) $=
      ( vf wcel wral cixp ciun cmap co cv wa cvv wss wfn ixpfn cdm fndm syl vex
      dmex eqeltrrdi id iunexg syl2anr ixpssmap2g simpr sseldd ex ssrdv ) CDFAB
      GZEABCHZABCIZBJKZULELZUMFZUPUOFULUQMZUMUOUPURUNNFZUMUOOUQBNFZULUSULUQUPBP
      ZUTABCUPQVABUPRNBUPSUPEUAUBUCTULUDABCNDUEUFABCNUGTULUQUHUIUJUK $.
  $}

  $( Membership of the empty set in an infinite Cartesian product.
     (Contributed by Steve Rodriguez, 29-Sep-2006.) $)
  0elixp $p |- (/) e. X_ x e. (/) A $=
    ( c0 csn cixp 0ex snid ixp0x eleqtrri ) CCDACBECFGABHI $.

  ${
    $d f A $.  $d f z B $.  $d f x z $.
    $( If an infinite Cartesian product of a family ` B ( x ) ` is inhabited,
       every ` B ( x ) ` is inhabited.  (Contributed by Mario Carneiro,
       22-Jun-2016.)  (Revised by Jim Kingdon, 16-Feb-2023.) $)
    ixpm $p |- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B ) $=
      ( cv cixp wcel wex wral cab wfn cfv wa df-ixp abeq2i elex2 ralimi exlimiv
      simplbiim ) EFZACDGZHZBFDHBIZACJZEUCUAAFZCHAKLZUFUAMZDHZACJZUEUGUJNEUBACD
      EOPUIUDACBUHDQRTS $.

    $( The infinite Cartesian product of a family ` B ( x ) ` with an empty
       member is empty.  (Contributed by NM, 1-Oct-2006.)  (Revised by Jim
       Kingdon, 16-Feb-2023.) $)
    ixp0 $p |- ( E. x e. A B = (/) -> X_ x e. A B = (/) ) $=
      ( vz vf c0 wceq wrex cv wcel wex wral wn cixp rexbii rexnalim sylbir ixpm
      notm0 con3i sylib syl ) CFGZABHZDICJDKZABLZMZABCNZFGZUDUEMZABHUGUJUCABDCS
      OUEABPQUGEIUHJEKZMUIUKUFADBCERTEUHSUAUB $.
  $}

  ${
    $d x A $.
    ixpssmap.2 $e |- B e. _V $.
    $( An infinite Cartesian product is a subset of set exponentiation.  Remark
       in [Enderton] p. 54.  (Contributed by NM, 28-Sep-2006.) $)
    ixpssmap $p |- X_ x e. A B C_ ( U_ x e. A B ^m A ) $=
      ( cvv wcel wral cixp ciun cmap co wss rgenw ixpssmapg ax-mp ) CEFZABGABCH
      ABCIBJKLPABDMABCENO $.
  $}

  ${
    $d A x $.  $d B x $.  $d F x $.
    $( Restriction of an element of an infinite Cartesian product.
       (Contributed by FL, 7-Nov-2011.)  (Proof shortened by Mario Carneiro,
       31-May-2014.) $)
    resixp $p |- ( ( B C_ A /\ F e. X_ x e. A C )
       -> ( F |` B ) e. X_ x e. B C ) $=
      ( wss cixp wcel wa cres cvv wfn cfv wral resexg adantl w3a simpr elixp2
      cv sylib simp2d simpl fnssres syl2anc simp3d ssralv eleq1d ralbiia sylibr
      sylc fvres syl3anbrc ) CBFZEABDGZHZIZECJZKHZURCLZATZURMZDHZACNZURACDGHUPU
      SUNECUOOPUQEBLZUNUTUQEKHZVEVAEMZDHZABNZUQUPVFVEVIQUNUPRABDESUAZUBUNUPUCZB
      CEUDUEUQVHACNZVDUQUNVIVLVKUQVFVEVIVJUFVHACBUGUKVCVHACVACHVBVGDVACEULUHUIU
      JACDURSUM $.
  $}

  ${
    $d I x y $.  $d J y $.  $d K y $.
    $( Condition for an explicit member of an indexed product.  (Contributed by
       Stefan O'Rear, 4-Jan-2015.) $)
    mptelixpg $p |- ( I e. V -> ( ( x e. I |-> J ) e. X_ x e. I K <->
          A. x e. I J e. K ) ) $=
      ( vy wcel cvv cmpt cixp wral wb elex cv cfv wa fvmpt2 ralimiaa wi syl wfn
      csb w3a nfcv nfcsb1v csbeq1a cbvixp eleq2i elixp2 3anass eqid fnmpt simpr
      3bitri eqeltrd jca dffn2 fmpt eleq1d biimpd ralim sylbir sylbi imp impbii
      wf nfv nffvmpt1 nfel weq fveq2 eleq12d cbvral anbi2i bitri mptexg bitr2id
      biantrurd bitrid ) BEGBHGZABCIZABDJZGZCDGZABKZLBEMWCWAHGZWABUAZFNZWAOZAWH
      DUBZGZFBKZPZPZVTWEWCWAFBWJJZGWFWGWLUCWNWBWOWAAFBDWJFDUDAWHDUEZAWHDUFZUGUH
      FBWJWAUIWFWGWLUJUNWEWMVTWNWEWGANZWAOZDGZABKZPZWMWEXBWEWGXAABCWADWAUKZULWD
      WTABWRBGZWDPWSCDABCDWAXCQXDWDUMUORUPWGXAWEWGBHWAVFZXAWESZBWAUQXECHGZABKZX
      FABHCWAXCURXHWTWDSZABKXFXGXIABXDXGPZWTWDXJWSCDABCHWAXCQUSUTRWTWDABVATVBVC
      VDVEXAWLWGWTWKAFBWTFVGAWIWJABCWHVHWPVIAFVJWSWIDWJWRWHWAVKWQVLVMVNVOVTWFWM
      ABCHVPVRVQVST $.
  $}

  ${
    $d B x y z w $.  $d F x y z w $.  $d A x y z w $.  $d V x y z w $.
    $( Membership in a class of singleton functions.  (Contributed by Stefan
       O'Rear, 24-Jan-2015.) $)
    elixpsn $p |- ( A e. V -> ( F e. X_ x e. { A } B <->
        E. y e. B F = { <. A , y >. } ) ) $=
      ( vz vw cv csn cixp wcel cop wceq wrex cvv vex cfv wa eleq1d sneq ixpeq1d
      eleq2d opeq1 sneqd eqeq2d elex opex snex eleq1 mpbiri rexlimivw eqeq1 wfn
      rexbidv wral elixp fveq2 ralsn anbi2i simpl biimpri adantl ffnfv sylanbrc
      wf fsn2 sylib opeq2 rspceeqv fvsn eqeltrid fnsn jctil fneq1 fveq1 anbi12d
      syl id syl5ibrcom rexlimiv impbii 3bitri vtoclbg pm5.21nii ) EAGIZJZDKZLZ
      EWFBIZMZJZNZBDOZEACJZDKZLECWJMZJZNZBDOGCFWFCNZWHWPEWTAWGWODWFCUAUBUCWTWMW
      SBDWTWLWREWTWKWQWFCWJUDUEUFUOWIEPLZWNEWHUGWMXABDWMXAWLPLWKWFWJGQZBQZUHUIE
      WLPUJUKULHIZWHLZXDWLNZBDOZWIWNHEPXDEWHUJXDENXFWMBDXDEWLUMUOXEXDWGUNZAIZXD
      RZDLZAWGUPZSXHWFXDRZDLZSZXGAWGDXDHQUQXLXNXHXKXNAWFXBXIWFNXJXMDXIWFXDURTUS
      UTXOXGXOXNXDWFXMMZJZNSZXGXOWGDXDVFZXRXOXHWJXDRZDLZBWGUPZXSXHXNVAXNYBXHYBX
      NYAXNBWFXBWJWFNXTXMDWJWFXDURTUSVBVCBWGDXDVDVEWFDXDXBVGVHBXMDWLXQXDWJXMNWK
      XPWJXMWFVIUEVJVRXFXOBDWJDLZXOXFWLWGUNZWFWLRZDLZSYCYFYDYCYEWJDWFWJXBXCVKYC
      VSVLWFWJXBXCVMVNXFXHYDXNYFWGXDWLVOXFXMYEDWFXDWLVPTVQVTWAWBWCWDWEWD $.
  $}

  ${
    $d I a b c x y $.  $d A a b c x y $.  $d V a b c x y $.  $d F a b c y $.
    $d W a b c x y $.
    ixpsnf1o.f $e |- F = ( x e. A |-> ( { I } X. { x } ) ) $.
    $( A bijection between a class and single-point functions to it.
       (Contributed by Stefan O'Rear, 24-Jan-2015.) $)
    ixpsnf1o $p |- ( I e. V -> F : A -1-1-onto-> X_ y e. { I } A ) $=
      ( va vb vc wcel csn cv crn cuni cvv vex wa wceq eqeq2d cixp snexg sylancl
      cxp snex xpexg adantr rnex a1i cop wrex sneq xpeq1d anbi2d wb elixpsn elv
      uniex ixpeq1d eleq2d bitr3id anbi1d xpsn eqeq2i eqid opeq2 sneqd rspceeqv
      anbi2i mpan2 op2nda eqcomi jctir eqeq1 rexbidv rneq unieqd syl5ibrcom imp
      anbi12d eqidd ancli eleq1w biimtrid imbi12d rexlimiv impbii bitri vtoclbg
      wi f1od ) EFKZAHCBELZCUAZWMAMZLZUDZHMZNZOZDPPGWLWQPKZWOCKZWLWMPKWPPKXAEFU
      BWOAQZUEWMWPPPUFUCUGWTPKWLWRWNKZRWSWRHQUHURUIXBWRIMZLZWPUDZSZRZWRXEJMZUJZ
      LZSZJCUKZWOWTSZRZXBWRWQSZRXDXORIEFXEESZXHXQXBXRXGWQWRXRXFWMWPXEEULZUMTUNX
      RXNXDXOXNWRBXFCUAZKZXRXDYAXNUOIBJXECWRPUPUQXRXTWNWRXRBXFWMCXSUSUTVAVBXIXB
      WRXEWOUJZLZSZRZXPXHYDXBXGYCWRXEWOIQZXCVCVDVIYEXPXBYDXPXBXPYDYCXLSZJCUKZWO
      YCNZOZSZRXBYHYKXBYCYCSYHYCVEJWOCXLYCYCXJWOSXKYBXJWOXEVFVGVHVJYJWOXEWOYFXC
      VKVLVMYDXNYHXOYKYDXMYGJCWRYCXLVNVOYDWTYJWOYDWSYIWRYCVPVQTVTVRVSXNXOYEXMXO
      YEWJZJCXJCKZYLXMWOXLNZOZSZXBXLYCSZRZWJYPWOXJSZYMYRYOXJWOXEXJYFJQVKVDYMYRY
      SYMXLXLSZRYMYTYMXLWAWBYSXBYMYQYTAJCWCYSYCXLXLYSYBXKWOXJXEVFVGTVTVRWDXMXOY
      PYEYRXMWTYOWOXMWSYNWRXLVPVQTXMYDYQXBWRXLYCVNUNWEVRWFVSWGWHWIWK $.

    $( A bijection between a set and single-point functions to it.
       (Contributed by Stefan O'Rear, 24-Jan-2015.) $)
    mapsnf1o $p |- ( ( A e. V /\ I e. W ) ->
        F : A -1-1-onto-> ( A ^m { I } ) ) $=
      ( vy wcel wa csn cmap co wf1o cixp ixpsnf1o adantl wceq wb cvv eqcomd syl
      snexg simpl ixpconstg syl2an2 f1oeq3 mpbird ) BEIZDFIZJZBBDKZLMZCNZBHULBO
      ZCNZUJUPUIAHBCDFGPQUKUMUORZUNUPSUJULTIZUIUIUQDFUCUIUJUDURUIJUOUMHULBTEUEU
      AUFUMUOBCUGUBUH $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Equinumerosity
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c ~~ $.  $( Equinumerosity relation ("approximately equals" symbol) $)
  $c ~<_ $.  $( Dominance relation (curly less-than-or-equal) $)
  $c Fin $.  $( Class of all finite sets. $)

  $( Extend class definition to include the equinumerosity relation
     ("approximately equals" symbol) $)
  cen $a class ~~ $.

  $( Extend class definition to include the dominance relation (curly
     less-than-or-equal) $)
  cdom $a class ~<_ $.

  $( Extend class definition to include the class of all finite sets. $)
  cfn $a class Fin $.

  ${
    $d x y f $.
    $( Define the equinumerosity relation.  Definition of [Enderton] p. 129.
       We define ` ~~ ` to be a binary relation rather than a connective, so
       its arguments must be sets to be meaningful.  This is acceptable because
       we do not consider equinumerosity for proper classes.  We derive the
       usual definition as ~ bren .  (Contributed by NM, 28-Mar-1998.) $)
    df-en $a |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y } $.

    $( Define the dominance relation.  Compare Definition of [Enderton] p. 145.
       Typical textbook definitions are derived as ~ brdom and ~ domen .
       (Contributed by NM, 28-Mar-1998.) $)
    df-dom $a |- ~<_ = { <. x , y >. | E. f f : x -1-1-> y } $.

    $( Define the (proper) class of all finite sets.  Similar to Definition
       10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to
       our " ` a e. Fin ` ".  This definition is meaningful whether or not we
       accept the Axiom of Infinity ~ ax-inf2 .  (Contributed by NM,
       22-Aug-2008.) $)
    df-fin $a |- Fin = { x | E. y e. _om x ~~ y } $.
  $}

  ${
    $d x y f $.  $d x y f $.
    $( Equinumerosity is a relation.  (Contributed by NM, 28-Mar-1998.) $)
    relen $p |- Rel ~~ $=
      ( vx vy vf cv wf1o wex cen df-en relopabi ) ADBDCDECFABGABCHI $.

    $( Dominance is a relation.  (Contributed by NM, 28-Mar-1998.) $)
    reldom $p |- Rel ~<_ $=
      ( vx vy vf cv wf1 wex cdom df-dom relopabi ) ADBDCDECFABGABCHI $.
  $}

  $( If two classes are equinumerous, both classes are sets.  (Contributed by
     AV, 21-Mar-2019.) $)
  encv $p |- ( A ~~ B -> ( A e. _V /\ B e. _V ) ) $=
    ( cen wrel wbr cvv wcel wa relen brrelex12 mpan ) CDABCEAFGBFGHIABCJK $.

  ${
    $d f x y A $.  $d f x y B $.  $d y C $.
    $( Equinumerosity relation.  This variation of ~ bren does not require the
       Axiom of Union.  (Contributed by NM, 15-Jun-1998.)  Extract from a
       subproof of ~ bren .  (Revised by BTernaryTau, 23-Sep-2024.) $)
    breng $p |- ( ( A e. V /\ B e. W ) ->
        ( A ~~ B <-> E. f f : A -1-1-onto-> B ) ) $=
      ( vx vy cv wf1o wex cen wceq f1oeq2 exbidv f1oeq3 df-en brabg ) FHZGHZCHZ
      IZCJASTIZCJABTIZCJFGABDEKRALUAUBCRASTMNSBLUBUCCSBATONFGCPQ $.
    $( $j usage 'breng' avoids 'ax-un'; $)
  $}

  ${
    $d f x y A $.  $d f x y B $.  $d y C $.
    $( Equinumerosity relation.  (Contributed by NM, 15-Jun-1998.) $)
    bren $p |- ( A ~~ B <-> E. f f : A -1-1-onto-> B ) $=
      ( vx vy cen wbr cvv wcel wa cv wf1o wex encv wfn f1ofn eqeltrrdi syl wceq
      exbidv cdm fndm vex dmex crn wfo f1ofo forn rnex jca exlimiv f1oeq2 df-en
      f1oeq3 brabg pm5.21nii ) ABFGAHIZBHIZJZABCKZLZCMZABNVAUSCVAUQURVAUTAOZUQA
      BUTPVCAUTUAHAUTUBUTCUCZUDQRVABUTUEZHVAABUTUFVEBSABUTUGABUTUHRUTVDUIQUJUKD
      KZEKZUTLZCMAVGUTLZCMVBDEABHHFVFASVHVICVFAVGUTULTVGBSVIVACVGBAUTUNTDECUMUO
      UP $.

    $( Dominance relation.  This variation of ~ brdomg does not require the
       Axiom of Union.  (Contributed by NM, 15-Jun-1998.)  Extract from a
       subproof of ~ brdomg .  (Revised by BTernaryTau, 29-Nov-2024.) $)
    brdom2g $p |- ( ( A e. V /\ B e. W ) ->
        ( A ~<_ B <-> E. f f : A -1-1-> B ) ) $=
      ( vx vy cv wf1 wex cdom wceq f1eq2 exbidv f1eq3 df-dom brabg ) FHZGHZCHZI
      ZCJASTIZCJABTIZCJFGABDEKRALUAUBCRASTMNSBLUBUCCSBATONFGCPQ $.
    $( $j usage 'brdom2g' avoids 'ax-un'; $)

    $( Dominance relation.  (Contributed by NM, 15-Jun-1998.) $)
    brdomg $p |- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) $=
      ( vx vy wcel cvv cdom wbr cv wf1 wex wi reldom brrelex1i a1i wceq exbidv
      wf f1f cdm fdm vex dmex eqeltrrdi syl exlimiv wb f1eq2 f1eq3 df-dom brabg
      expcom pm5.21ndd ) BCGZAHGZABIJZABDKZLZDMZURUQNUPABIOPQVAUQNUPUTUQDUTABUS
      TZUQABUSUAVBAUSUBHABUSUCUSDUDUEUFUGUHQUQUPURVAUIEKZFKZUSLZDMAVDUSLZDMVAEF
      ABHCIVCARVEVFDVCAVDUSUJSVDBRVFUTDVDBAUSUKSEFDULUMUNUO $.

    $( Dominance relation.  (Contributed by Mario Carneiro, 26-Apr-2015.) $)
    brdomi $p |- ( A ~<_ B -> E. f f : A -1-1-> B ) $=
      ( cdom wbr cv wf1 wex cvv wcel wb reldom brrelex2i brdomg syl ibi ) ABDEZ
      ABCFGCHZQBIJQRKABDLMABICNOP $.
  $}

  ${
    $d f x A $.  $d f x B $.
    bren.1 $e |- B e. _V $.
    $( Dominance relation.  (Contributed by NM, 15-Jun-1998.) $)
    brdom $p |- ( A ~<_ B <-> E. f f : A -1-1-> B ) $=
      ( cvv wcel cdom wbr cv wf1 wex wb brdomg ax-mp ) BEFABGHABCIJCKLDABECMN
      $.

    $( Dominance in terms of equinumerosity.  Example 1 of [Enderton] p. 146.
       (Contributed by NM, 15-Jun-1998.) $)
    domen $p |- ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) $=
      ( vf cdom wbr cv wf1 wex cen wss brdom wf1o vex f11o exbii bitri bitr4i
      wa excom bren anbi1i 19.41v ) BCFGBCEHZIZEJZBAHZKGZUHCLZTZAJZBCEDMUGBUHUE
      NZUJTZEJZAJZULUGUNAJZEJUPUFUQEABCUEEOPQUNEAUARUKUOAUKUMEJZUJTUOUIURUJBUHE
      UBUCUMUJEUDSQSR $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Dominance in terms of equinumerosity, with the sethood requirement
       expressed as an antecedent.  Example 1 of [Enderton] p. 146.
       (Contributed by NM, 24-Apr-2004.) $)
    domeng $p |- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) ) $=
      ( vy cv cdom wbr cen wss wex breq2 wceq sseq2 anbi2d exbidv domen vtoclbg
      wa vex ) BEFZGHBAFZIHZUBUAJZSZAKBCGHUCUBCJZSZAKECDUACBGLUACMZUEUGAUHUDUFU
      CUACUBNOPABUAETQR $.
  $}

  $( A class dominated by ` _om ` is a set.  See also ~ ctfoex which says that
     a countable class is a set.  (Contributed by Thierry Arnoux, 29-Dec-2016.)
     (Proof shortened by Jim Kingdon, 13-Mar-2023.) $)
  ctex $p |- ( A ~<_ _om -> A e. _V ) $=
    ( com cdom reldom brrelex1i ) ABCDE $.

  ${
    $d f A $.  $d f B $.  $d f F $.
    $( The domain and range of a one-to-one, onto set function are
       equinumerous.  This variation of ~ f1oeng does not require the Axiom of
       Collection nor the Axiom of Union.  (Contributed by BTernaryTau,
       7-Dec-2024.) $)
    f1oen4g $p |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B )
        -> A ~~ B ) $=
      ( vf wcel w3a wf1o wa cen wbr cv wex f1oeq1 spcegv imp 3ad2antl1 wb breng
      3adant1 adantr mpbird ) CDHZAEHZBFHZIZABCJZKABLMZABGNZJZGOZUEUFUIUMUGUEUI
      UMULUIGCDABUKCPQRSUHUJUMTZUIUFUGUNUEABGEFUAUBUCUD $.
    $( $j usage 'f1oen4g' avoids 'ax-coll' 'ax-un'; $)

    $( The domain of a one-to-one set function is dominated by its codomain
       when the latter is a set.  This variation of ~ f1domg does not require
       the Axiom of Collection nor the Axiom of Union.  (Contributed by
       BTernaryTau, 7-Dec-2024.) $)
    f1dom4g $p |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B )
        -> A ~<_ B ) $=
      ( vf wcel w3a wf1 wa cdom wbr cv wex f1eq1 spcegv imp 3ad2antl1 wb adantr
      brdom2g 3adant1 mpbird ) CDHZAEHZBFHZIZABCJZKABLMZABGNZJZGOZUEUFUIUMUGUEU
      IUMULUIGCDABUKCPQRSUHUJUMTZUIUFUGUNUEABGEFUBUCUAUD $.
    $( $j usage 'f1dom4g' avoids 'ax-coll' 'ax-un'; $)
  $}

  ${
    $d f A $.  $d f B $.  $d f F $.
    $( The domain and range of a one-to-one, onto function are equinumerous.
       This variation of ~ f1oeng does not require the Axiom of Replacement.
       (Contributed by NM, 13-Jan-2007.)  (Revised by Mario Carneiro,
       10-Sep-2015.) $)
    f1oen3g $p |- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
      ( vf wcel wf1o wa cv wex cen wbr f1oeq1 spcegv imp bren sylibr ) CDFZABCG
      ZHABEIZGZEJZABKLRSUBUASECDABTCMNOABEPQ $.
    $( $j usage 'f1oen3g' avoids 'ax-coll'; $)

    $( The domain and range of a one-to-one, onto function are equinumerous.
       This variation of ~ f1oeng does not require the Axiom of Replacement.
       (Contributed by Mario Carneiro, 10-Sep-2015.) $)
    f1oen2g $p |- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
      ( wcel wf1o w3a cvv cen wbr f1of fex2 syl3an1 3coml simp3 f1oen3g syl2anc
      wf ) ADFZBEFZABCGZHCIFZUBABJKUBTUAUCUBABCSTUAUCABCLABCDEMNOTUAUBPABCIQR
      $.
    $( $j usage 'f1oen2g' avoids 'ax-coll'; $)

    $( The domain of a one-to-one function is dominated by its codomain.  This
       variation of ~ f1domg does not require the Axiom of Replacement.
       (Contributed by Mario Carneiro, 24-Jun-2015.) $)
    f1dom2g $p |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B ) $=
      ( vf wcel wf1 w3a cdom wbr cv wex cvv wf f1f fex2 syl3an1 3coml simp3 wb
      f1eq1 spcegv sylc brdomg 3ad2ant2 mpbird ) ADGZBEGZABCHZIZABJKZABFLZHZFMZ
      UKCNGZUJUOUJUHUIUPUJABCOUHUIUPABCPABCDEQRSUHUIUJTUNUJFCNABUMCUBUCUDUIUHUL
      UOUAUJABEFUEUFUG $.
    $( $j usage 'f1dom2g' avoids 'ax-coll'; $)

    $( The domain and range of a one-to-one, onto function are equinumerous.
       (Contributed by NM, 19-Jun-1998.) $)
    f1oeng $p |- ( ( A e. C /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
      ( wcel wf1o cvv cen wbr wfo f1ofo focdmex syl5 imp f1oen2g 3com23 mpd3an3
      ) ACEZABDFZBGEZABHIZRSTSABDJRTABDKABCDLMNRTSUAABDCGOPQ $.

    $( The domain of a one-to-one function is dominated by its codomain.
       (Contributed by NM, 4-Sep-2004.) $)
    f1domg $p |- ( B e. C -> ( F : A -1-1-> B -> A ~<_ B ) ) $=
      ( vf wcel wf1 cv wex cdom wbr cvv f1dmex wf f1f sylan syldan expcom f1eq1
      fex spcegv syli brdomg sylibrd ) BCFZABDGZABEHZGZEIZABJKUFUEDLFZUIUFUEUJU
      FUEALFZUJABCDMUFABDNUKUJABDOABLDTPQRUHUFEDLABUGDSUAUBABCEUCUD $.
  $}

  ${
    f1oen.1 $e |- A e. _V $.
    $( The domain and range of a one-to-one, onto function are equinumerous.
       (Contributed by NM, 19-Jun-1998.) $)
    f1oen $p |- ( F : A -1-1-onto-> B -> A ~~ B ) $=
      ( cvv wcel wf1o cen wbr f1oeng mpan ) AEFABCGABHIDABECJK $.
  $}

  ${
    f1dom.1 $e |- B e. _V $.
    $( The domain of a one-to-one function is dominated by its codomain.
       (Contributed by NM, 19-Jun-1998.) $)
    f1dom $p |- ( F : A -1-1-> B -> A ~<_ B ) $=
      ( cvv wcel wf1 cdom wbr wi f1domg ax-mp ) BEFABCGABHIJDABECKL $.
  $}

  ${
    $d x y f A $.
    $( Express " ` A ` is finite".  Definition 10.29 of [TakeutiZaring] p. 91
       (whose " ` Fin ` " is a predicate instead of a class).  (Contributed by
       NM, 22-Aug-2008.) $)
    isfi $p |- ( A e. Fin <-> E. x e. _om A ~~ x ) $=
      ( vy cfn wcel cv cen wbr com wrex df-fin eleq2i relen brrelex1i rexlimivw
      cab cvv wceq breq1 rexbidv elab3 bitri ) BDEBCFZAFZGHZAIJZCPZEBUDGHZAIJZD
      UGBCAKLUFUICBUHBQEAIBUDGMNOUCBRUEUHAIUCBUDGSTUAUB $.

    $( Equinumerosity implies dominance.  (Contributed by NM, 31-Mar-1998.) $)
    enssdom $p |- ~~ C_ ~<_ $=
      ( vx vy vf cen cdom relen cop wf1o wex copab wcel wf1 f1of1 eximi 3imtr4i
      cv opabid df-en eleq2i df-dom relssi ) ABDEFAPZBPZGZUBUCCPZHZCIZABJZKZUDU
      BUCUELZCIZABJZKZUDDKUDEKUGUKUIUMUFUJCUBUCUEMNUGABQUKABQODUHUDABCRSEULUDAB
      CTSOUA $.
  $}

  $( Equinumerosity implies dominance.  Theorem 15 of [Suppes] p. 94.
     (Contributed by NM, 28-May-1998.) $)
  endom $p |- ( A ~~ B -> A ~<_ B ) $=
    ( cen cdom enssdom ssbri ) CDABEF $.

  $( Equinumerosity is reflexive.  Theorem 1 of [Suppes] p. 92.  (Contributed
     by NM, 18-Jun-1998.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  enrefg $p |- ( A e. V -> A ~~ A ) $=
    ( wcel cen wbr cid cres wf1o f1oi f1oen2g mp3an3 anidms ) ABCZAADEZMMAAFAGZ
    HNAIAAOBBJKL $.

  ${
    enref.1 $e |- A e. _V $.
    $( Equinumerosity is reflexive.  Theorem 1 of [Suppes] p. 92.  (Contributed
       by NM, 25-Sep-2004.) $)
    enref $p |- A ~~ A $=
      ( cvv wcel cen wbr enrefg ax-mp ) ACDAAEFBACGH $.
  $}

  $( Equality implies equinumerosity.  (Contributed by NM, 26-Oct-2003.) $)
  eqeng $p |- ( A e. V -> ( A = B -> A ~~ B ) ) $=
    ( wcel cen wbr wceq enrefg breq2 syl5ibcom ) ACDAAEFABGABEFACHABAEIJ $.

  $( Dominance is reflexive.  (Contributed by NM, 18-Jun-1998.) $)
  domrefg $p |- ( A e. V -> A ~<_ A ) $=
    ( wcel cen wbr cdom enrefg endom syl ) ABCAADEAAFEABGAAHI $.

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.  $d x y ph $.
    en2d.1 $e |- ( ph -> A e. V ) $.
    en2d.2 $e |- ( ph -> B e. W ) $.
    en2d.3 $e |- ( ph -> ( x e. A -> C e. X ) ) $.
    en2d.4 $e |- ( ph -> ( y e. B -> D e. Y ) ) $.
    en2d.5 $e |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) ) $.
    $( Equinumerosity inference from an implicit one-to-one onto function.
       (Contributed by NM, 27-Jul-2004.)  (Revised by Mario Carneiro,
       12-May-2014.)  (Revised by AV, 4-Aug-2024.) $)
    en2d $p |- ( ph -> A ~~ B ) $=
      ( wcel cmpt cv imp wf1o cen wbr eqid f1od f1oen2g syl3anc ) ADHQEIQDEBDFR
      ZUADEUBUCLMABCDEFGUHJKUHUDABSDQFJQNTACSEQGKQOTPUEDEUHHIUFUG $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.  $d x y ph $.
    en3d.1 $e |- ( ph -> A e. V ) $.
    en3d.2 $e |- ( ph -> B e. W ) $.
    en3d.3 $e |- ( ph -> ( x e. A -> C e. B ) ) $.
    en3d.4 $e |- ( ph -> ( y e. B -> D e. A ) ) $.
    en3d.5 $e |- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) ) $.
    $( Equinumerosity inference from an implicit one-to-one onto function.
       (Contributed by NM, 27-Jul-2004.)  (Revised by Mario Carneiro,
       12-May-2014.)  (Revised by AV, 4-Aug-2024.) $)
    en3d $p |- ( ph -> A ~~ B ) $=
      ( wcel cmpt wf1o cv imp wceq cen wbr eqid wa wb f1o2d f1oen2g syl3anc ) A
      DHOEIODEBDFPZQDEUAUBJKABCDEFGUIUIUCABRZDOZFEOLSACRZEOZGDOMSAUKUMUDUJGTULF
      TUENSUFDEUIHIUGUH $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.
    en2i.1 $e |- A e. _V $.
    en2i.2 $e |- B e. _V $.
    en2i.3 $e |- ( x e. A -> C e. _V ) $.
    en2i.4 $e |- ( y e. B -> D e. _V ) $.
    en2i.5 $e |- ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) $.
    $( Equinumerosity inference from an implicit one-to-one onto function.
       (Contributed by NM, 4-Jan-2004.) $)
    en2i $p |- A ~~ B $=
      ( cen wtru cvv wcel a1i cv wi wceq wa wbr wb en2d mptru ) CDLUAMABCDEFNNN
      NCNOMGPDNOMHPAQZCOZENORMIPBQZDOZFNORMJPUFUGESTUHUEFSTUBMKPUCUD $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.
    en3i.1 $e |- A e. _V $.
    en3i.2 $e |- B e. _V $.
    en3i.3 $e |- ( x e. A -> C e. B ) $.
    en3i.4 $e |- ( y e. B -> D e. A ) $.
    en3i.5 $e |- ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) $.
    $( Equinumerosity inference from an implicit one-to-one onto function.
       (Contributed by NM, 19-Jul-2004.) $)
    en3i $p |- A ~~ B $=
      ( cen wbr wtru cvv wcel a1i cv wi wceq wa wb en3d mptru ) CDLMNABCDEFOOCO
      PNGQDOPNHQARZCPZEDPSNIQBRZDPZFCPSNJQUFUHUAUEFTUGETUBSNKQUCUD $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d y z C $.  $d x z D $.  $d x y ph $.
    dom2d.1 $e |- ( ph -> ( x e. A -> C e. B ) ) $.
    dom2d.2 $e |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) ) $.
    $( A mapping (first hypothesis) that is one-to-one (second hypothesis)
       implies its domain is dominated by its codomain.  (Contributed by NM,
       24-Jul-2004.) $)
    dom2lem $p |- ( ph -> ( x e. A |-> C ) : A -1-1-> B ) $=
      ( cv cfv wceq wi wral wcel wa imp anbi2d wb eqeq12d cmpt wf ralrimiv eqid
      wf1 fmpt sylib fvmpt2 adantll mpdan adantrr nfv nffvmpt1 nfeq1 nfim eleq1
      imbi1d anbi1d anidm bitrdi fveq2 adantr biimparc ex sylbird pm5.74d bitrd
      chvar adantrl biimpd sylbid ralrimivva nfmpt1 nfcv dff13f sylanbrc ) ADEB
      DFUAZUBZBJZVQKZCJZVQKZLZVSWALZMZCDNBDNDEVQUEAFEOZBDNVRAWFBDHUCBDEFVQVQUDZ
      UFUGAWEBCDDAVSDOZWADOZPZPZWCFGLZWDWKVTFWBGAWHVTFLZWIAWHPZWFWMAWHWFHQWHWFW
      MABDFEVQWGUHUIUJZUKAWIWBGLZWHWNWMMZAWIPZWPMZBCWRWPBWRBULBWBGBDFWAUMUNUOWD
      WQWRWMMWSWDWNWRWMWDWHWIAVSWADUPZRUQWDWRWMWPWDWRWKWMWPSZWDWJWIAWDWJWIWIPWI
      WDWHWIWIWTURWIUSUTRWDWKXAWDWKPVTWBFGWDWCWKVSWAVQVAVBWKWLWDAWJWLWDSIQZVCTV
      DVEVFVGWOVHVITWKWLWDXBVJVKVLBCDEVQBDFVMCVQVNVOVP $.

    $( A mapping (first hypothesis) that is one-to-one (second hypothesis)
       implies its domain is dominated by its codomain.  (Contributed by NM,
       24-Jul-2004.)  (Revised by Mario Carneiro, 20-May-2013.) $)
    dom2d $p |- ( ph -> ( B e. R -> A ~<_ B ) ) $=
      ( cmpt wf1 wcel cdom wbr dom2lem f1domg syl5com ) ADEBDFKZLEHMDENOABCDEFG
      IJPDEHSQR $.

    dom3d.3 $e |- ( ph -> A e. V ) $.
    dom3d.4 $e |- ( ph -> B e. W ) $.
    $( A mapping (first hypothesis) that is one-to-one (second hypothesis)
       implies its domain is dominated by its codomain.  (Contributed by Mario
       Carneiro, 20-May-2013.) $)
    dom3d $p |- ( ph -> A ~<_ B ) $=
      ( vz cdom wbr wf1 cvv wcel syl cv wex cmpt dom2lem f1f fex2 syl3anc f1eq1
      wf spcegv sylc wb brdomg mpbird ) ADEOPZDENUAZQZNUBZABDFUCZRSZDEUSQZURADE
      USUIZDHSEISZUTAVAVBABCDEFGJKUDZDEUSUETLMDEUSHIUFUGVDUQVANUSRDEUPUSUHUJUKA
      VCUOURULMDEINUMTUN $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.  $d x D $.
    dom2.1 $e |- ( x e. A -> C e. B ) $.
    dom2.2 $e |- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) $.
    $( A mapping (first hypothesis) that is one-to-one (second hypothesis)
       implies its domain is dominated by its codomain. ` C ` and ` D ` can be
       read ` C ( x ) ` and ` D ( y ) ` , as can be inferred from their
       distinct variable conditions.  (Contributed by NM, 26-Oct-2003.) $)
    dom2 $p |- ( B e. V -> A ~<_ B ) $=
      ( wceq wcel cdom wbr wi eqid cv a1i wa wb dom2d ax-mp ) CCJZDGKCDLMNCOUBA
      BCDEFGAPZCKZEDKNUBHQUDBPZCKREFJUCUEJSNUBIQTUA $.

    $d x y V $.  $d x y W $.
    $( A mapping (first hypothesis) that is one-to-one (second hypothesis)
       implies its domain is dominated by its codomain. ` C ` and ` D ` can be
       read ` C ( x ) ` and ` D ( y ) ` , as can be inferred from their
       distinct variable conditions.  (Contributed by Mario Carneiro,
       20-May-2013.) $)
    dom3 $p |- ( ( A e. V /\ B e. W ) -> A ~<_ B ) $=
      ( wcel wa cv wi a1i wceq wb simpl simpr dom3d ) CGKZDHKZLZABCDEFGHAMZCKZE
      DKNUCIOUEBMZCKLEFPUDUFPQNUCJOUAUBRUAUBST $.
  $}

  ${
    $d x y $.
    $( Equality implies equinumerosity.  (Contributed by NM, 30-Apr-1998.)
       (Revised by Mario Carneiro, 15-Nov-2014.) $)
    idssen $p |- _I C_ ~~ $=
      ( vx vy cid cen reli cv wbr cop wcel weq vex ideq eqeng ax-mp sylbi df-br
      cvv wi 3imtr3i relssi ) ABCDEAFZBFZCGZUAUBDGZUAUBHZCIUEDIUCABJZUDUAUBBKLU
      AQIUFUDRAKUAUBQMNOUAUBCPUAUBDPST $.
  $}

  ${
    $d A f $.  $d B f $.  $d C f $.  $d V f $.
    $( If ` C ` is a superset of ` B ` and ` B ` dominates ` A ` , then ` C `
       also dominates ` A ` .  (Contributed by BTernaryTau, 7-Dec-2024.) $)
    domssr $p |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A ~<_ C ) $=
      ( vf wcel wss cdom wbr w3a cv wf1 wex cvv wa brdomi 3ad2ant3 simp2 reldom
      brrelex1i simp1 jca32 wi f1ss vex f1dom4g mp3anl1 sylan expl exlimiv sylc
      ancoms ) CDFZBCGZABHIZJZABEKZLZEMZUNANFZUMOZOZACHIZUOUMUSUNABEPQUPUNUTUMU
      MUNUORUOUMUTUNABHSTQUMUNUOUAUBURVBVCUCEURUNVAVCURUNOACUQLZVAVCABCUQUDVAVD
      VCUQNFUTUMVDVCEUEACUQNNDUFUGULUHUIUJUK $.
  $}

  $( A set dominates its subsets.  Theorem 16 of [Suppes] p. 94.  (Contributed
     by NM, 19-Jun-1998.)  (Revised by Mario Carneiro, 24-Jun-2015.) $)
  ssdomg $p |- ( B e. V -> ( A C_ B -> A ~<_ B ) ) $=
    ( wss wcel cdom wbr wa cvv cid cres wf1 ssexg simpr wf ccnv wfun wf1o ax-mp
    wfo f1oi dff1o3 mpbi simpli fof fss mpan funi cnvi funeqi mpbir jctir df-f1
    funres11 sylibr adantr f1dom2g syl3anc expcom ) ABDZBCEZABFGZUTVAHAIEVAABJA
    KZLZVBABCMUTVANUTVDVAUTABVCOZVCPQZHVDUTVEVFAAVCOZUTVEAAVCTZVGVHVFAAVCRVHVFH
    AUAAAVCUBUCUDAAVCUESAABVCUFUGJPZQZVFVJJQUHVIJUIUJUKAJUNSULABVCUMUOUPABVCICU
    QURUS $.

  ${
    $d f g x y z $.
    $( Equinumerosity is an equivalence relation.  (Contributed by NM,
       19-Mar-1998.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
    ener $p |- ~~ Er _V $=
      ( vx vy vz vf cvv cen wtru a1i wbr wf1o wex bren wcel vex f1oen2g mp3an12
      vg cv syl wa wrel relen ccnv f1ocnv exlimiv sylbi adantl eeanv ccom f1oco
      wer ancoms exlimivv sylbir syl2anb wb enref 2th iserd mptru ) EFUKGABCEFF
      UAGUBHARZBRZFIZVBVAFIZGVCVAVBDRZJZDKVDVAVBDLVFVDDVFVBVAVEUCZJZVDVAVBVEUDV
      BEMVAEMZVHVDBNANZVBVAVGEEOPSUEUFUGVCVBCRZFIZTVAVKFIZGVCVAVBQRZJZQKZVBVKVE
      JZDKZVMVLVAVBQLVBVKDLVPVRTVOVQTZDKQKVMVOVQQDUHVSVMQDVSVAVKVEVNUIZJZVMVQVO
      WAVAVBVKVEVNUJULVIVKEMWAVMVJCNVAVKVTEEOPSUMUNUOUGVIVAVAFIZUPGVIWBVJVAVJUQ
      URHUSUT $.
  $}

  $( Symmetry of equinumerosity.  Theorem 2 of [Suppes] p. 92.  (Contributed by
     Mario Carneiro, 26-Apr-2015.) $)
  ensymb $p |- ( A ~~ B <-> B ~~ A ) $=
    ( cen wbr wb wtru cvv wer ener a1i ersymb mptru ) ABCDBACDEFABCGGCHFIJKL $.

  $( Symmetry of equinumerosity.  Theorem 2 of [Suppes] p. 92.  (Contributed by
     NM, 26-Oct-2003.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  ensym $p |- ( A ~~ B -> B ~~ A ) $=
    ( cen wbr ensymb biimpi ) ABCDBACDABEF $.

  ${
    ensymi.2 $e |- A ~~ B $.
    $( Symmetry of equinumerosity.  Theorem 2 of [Suppes] p. 92.  (Contributed
       by NM, 25-Sep-2004.) $)
    ensymi $p |- B ~~ A $=
      ( cen wbr ensym ax-mp ) ABDEBADECABFG $.
  $}

  ${
    ensymd.1 $e |- ( ph -> A ~~ B ) $.
    $( Symmetry of equinumerosity.  Deduction form of ~ ensym .  (Contributed
       by David Moews, 1-May-2017.) $)
    ensymd $p |- ( ph -> B ~~ A ) $=
      ( cen wbr ensym syl ) ABCEFCBEFDBCGH $.
  $}

  ${
    $d x y z f g h A $.  $d x y z f g B $.  $d x y z f g h C $.
    $( Transitivity of equinumerosity.  Theorem 3 of [Suppes] p. 92.
       (Contributed by NM, 9-Jun-1998.) $)
    entr $p |- ( ( A ~~ B /\ B ~~ C ) -> A ~~ C ) $=
      ( cen wbr wa wi wtru cvv wer ener a1i ertr mptru ) ABDEBCDEFACDEGHABCDIID
      JHKLMN $.

    $( Transitivity of dominance relation.  Theorem 17 of [Suppes] p. 94.
       (Contributed by NM, 4-Jun-1998.)  (Revised by Mario Carneiro,
       15-Nov-2014.) $)
    domtr $p |- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C ) $=
      ( vx vy vz vg vf vh cdom reldom cv wbr wf1 wex vex brdom wa eeanv ccom
      f1co ancoms coex f1eq1 spcev syl sylibr exlimivv sylbir syl2anb vtoclr )
      DEFABCJKDLZELZJMULUMGLZNZGOZUMFLZHLZNZHOZULUQJMZUMUQJMULUMGEPQUMUQHFPZQUP
      UTRUOUSRZHOGOVAUOUSGHSVCVAGHVCULUQILZNZIOZVAVCULUQURUNTZNZVFUSUOVHULUMUQU
      RUNUAUBVEVHIVGURUNHPGPUCULUQVDVGUDUEUFULUQIVBQUGUHUIUJUK $.
  $}

  ${
    entri.1 $e |- A ~~ B $.
    entri.2 $e |- B ~~ C $.
    $( A chained equinumerosity inference.  (Contributed by NM,
       25-Sep-2004.) $)
    entri $p |- A ~~ C $=
      ( cen wbr entr mp2an ) ABFGBCFGACFGDEABCHI $.
  $}

  ${
    entr2i.1 $e |- A ~~ B $.
    entr2i.2 $e |- B ~~ C $.
    $( A chained equinumerosity inference.  (Contributed by NM,
       25-Sep-2004.) $)
    entr2i $p |- C ~~ A $=
      ( entri ensymi ) ACABCDEFG $.
  $}

  ${
    entr3i.1 $e |- A ~~ B $.
    entr3i.2 $e |- A ~~ C $.
    $( A chained equinumerosity inference.  (Contributed by NM,
       25-Sep-2004.) $)
    entr3i $p |- B ~~ C $=
      ( ensymi entri ) BACABDFEG $.
  $}

  ${
    entr4i.1 $e |- A ~~ B $.
    entr4i.2 $e |- C ~~ B $.
    $( A chained equinumerosity inference.  (Contributed by NM,
       25-Sep-2004.) $)
    entr4i $p |- A ~~ C $=
      ( ensymi entri ) ABCDCBEFG $.
  $}

  $( Transitivity of equinumerosity and dominance.  (Contributed by NM,
     7-Jun-1998.) $)
  endomtr $p |- ( ( A ~~ B /\ B ~<_ C ) -> A ~<_ C ) $=
    ( cen wbr cdom endom domtr sylan ) ABDEABFEBCFEACFEABGABCHI $.

  $( Transitivity of dominance and equinumerosity.  (Contributed by NM,
     7-Jun-1998.) $)
  domentr $p |- ( ( A ~<_ B /\ B ~~ C ) -> A ~<_ C ) $=
    ( cen wbr cdom endom domtr sylan2 ) BCDEABFEBCFEACFEBCGABCHI $.

  $( A one-to-one function's image under a subset of its domain is equinumerous
     to the subset.  (Contributed by Mario Carneiro, 15-May-2015.) $)
  f1imaeng $p |- ( ( F : A -1-1-> B /\ C C_ A /\ C e. V ) ->
    ( F " C ) ~~ C ) $=
    ( wf1 wss wcel w3a cima cres wf1o cen f1ores f1oeng ancoms stoic3 ensymd
    wbr ) ABDFZCAGZCEHZICDCJZTUACUCDCKZLZUBCUCMSZABCDNUBUEUFCUCEUDOPQR $.

  $( A one-to-one function's image under a subset of its domain is equinumerous
     to the subset.  (This version of ~ f1imaen does not need ~ ax-setind .)
     (Contributed by Mario Carneiro, 16-Nov-2014.)  (Revised by Mario Carneiro,
     25-Jun-2015.) $)
  f1imaen2g $p |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) ->
    ( F " C ) ~~ C ) $=
    ( wf1 wcel wa wss cima cvv cres wf1o cen wbr simprr simplr wf f1f crn ssexd
    imassrn frn sstrid syl ad2antrr f1ores ad2ant2r f1oen2g syl3anc ensymd ) AB
    DFZBEGZHZCAIZCEGZHZHZCDCJZURUPUSKGCUSDCLZMZCUSNOUNUOUPPURUSBEULUMUQQULUSBIZ
    UMUQULABDRZVBABDSVCUSDTBDCUBABDUCUDUEUFUAULUOVAUMUPABCDUGUHCUSUTEKUIUJUK $.

  ${
    f1imaen.1 $e |- C e. _V $.
    $( A one-to-one function's image under a subset of its domain is
       equinumerous to the subset.  (Contributed by NM, 30-Sep-2004.) $)
    f1imaen $p |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F " C ) ~~ C ) $=
      ( wf1 wss cvv wcel cima cen wbr f1imaeng mp3an3 ) ABDFCAGCHIDCJCKLEABCDHM
      N $.
  $}

  ${
    $d f A $.
    $( The empty set is equinumerous only to itself.  Exercise 1 of
       [TakeutiZaring] p. 88.  (Contributed by NM, 27-May-1998.) $)
    en0 $p |- ( A ~~ (/) <-> A = (/) ) $=
      ( vf c0 cen wbr wceq cv wf1o wex bren ccnv f1ocnv f1o00 simprbi syl sylbi
      exlimiv 0ex enref breq1 mpbiri impbii ) ACDEZACFZUCACBGZHZBIUDACBJUFUDBUF
      CAUEKZHZUDACUELUHUGCFUDAUGMNOQPUDUCCCDECRSACCDTUAUB $.
  $}

  ${
    $d f A $.
    ensn1.1 $e |- A e. _V $.
    $( A singleton is equinumerous to ordinal one.  (Contributed by NM,
       4-Nov-2002.) $)
    ensn1 $p |- { A } ~~ 1o $=
      ( vf csn c0 c1o cen wbr cv wf1o wex cop 0ex f1osn opex f1oeq1 spcev ax-mp
      snex bren mpbir df1o2 breqtrri ) ADZEDZFGUDUEGHUDUECIZJZCKZUDUEAELZDZJZUH
      AEBMNUGUKCUJUIAEBMOSUDUEUFUJPQRUDUECTUAUBUC $.
  $}

  ${
    $d x A $.
    $( A singleton is equinumerous to ordinal one.  (Contributed by NM,
       23-Apr-2004.) $)
    ensn1g $p |- ( A e. V -> { A } ~~ 1o ) $=
      ( vx cv csn c1o cen wbr wceq sneq breq1d vex ensn1 vtoclg ) CDZEZFGHAEZFG
      HCABOAIPQFGOAJKOCLMN $.
  $}

  $( ` { A , A } ` has only one element.  (Contributed by FL, 15-Feb-2010.) $)
  enpr1g $p |- ( A e. V -> { A , A } ~~ 1o ) $=
    ( wcel cpr csn c1o cen dfsn2 ensn1g eqbrtrrid ) ABCAADAEFGAHABIJ $.

  ${
    $d x f y A $.  $d y ph $.
    $( A set is equinumerous to ordinal one iff it is a singleton.
       (Contributed by NM, 25-Jul-2004.) $)
    en1 $p |- ( A ~~ 1o <-> E. x A = { x } ) $=
      ( vf c1o cen wbr cv csn wceq wex c0 wf1o df1o2 breq2i crn syl 0ex exlimiv
      wcel cvv bren bitri ccnv cfv f1ocnv wfo f1ofo forn cop wf f1of fsn2 rneqd
      simprbi rnsnop eqtrdi eqtr3d wfn f1ofn snid funfvex funfni sylancl eqeq2d
      wi sneq spcegv sylc sylbi vex ensn1 breq1 mpbiri impbii ) BDEFZBAGZHZIZAJ
      ZVOBKHZCGZLZCJZVSVOBVTEFWCDVTBEMNBVTCUAUBWBVSCWBVTBWAUCZLZBKWDUDZHZIZVSBV
      TWAUEZWBWEWHWIWEWDOZBWGWEVTBWDUFWJBIVTBWDUGVTBWDUHPWEWJKWFUIHZOWGWEWDWKWE
      VTBWDUJZWDWKIZVTBWDUKWLWFBSWMKBWDQULUNPUMKWFQUOUPUQPWEWFTSZWHVSVEWEWDVTUR
      KVTSWNVTBWDUSKQUTWNVTKWDKWDVAVBVCVRWHAWFTVPWFIVQWGBVPWFVFVDVGPVHRVIVRVOAV
      RVOVQDEFVPAVJVKBVQDEVLVMRVN $.

    $( A set is equinumerous to ordinal one iff it is a singleton.
       (Contributed by Jim Kingdon, 13-Apr-2020.) $)
    en1bg $p |- ( A e. V -> ( A ~~ 1o <-> A = { U. A } ) ) $=
      ( vx wcel c1o cen wbr cuni csn cv wex en1 id unieq vex unisn eqtrdi sneqd
      wceq cvv eqtr4d exlimiv sylbi uniexg ensn1g syl breq1 syl5ibrcom impbid2
      ) ABDZAEFGZAAHZIZSZUKACJZIZSZCKUNCALUQUNCUQAUPUMUQMUQULUOUQULUPHUOAUPNUOC
      OPQRUAUBUCUJUKUNUMEFGZUJULTDURABUDULTUEUFAUMEFUGUHUI $.

    $( Two ways to express "exactly one".  (Contributed by Stefan O'Rear,
       28-Oct-2014.) $)
    reuen1 $p |- ( E! x e. A ph <-> { x e. A | ph } ~~ 1o ) $=
      ( vy wreu crab cv csn wceq wex c1o cen wbr reusn en1 bitr4i ) ABCEABCFZDG
      HIDJQKLMABDCNDQOP $.

    $( Two ways to express "exactly one".  (Contributed by Stefan O'Rear,
       28-Oct-2014.) $)
    euen1 $p |- ( E! x ph <-> { x | ph } ~~ 1o ) $=
      ( cvv wreu crab c1o cen wbr weu cab reuen1 reuv rabab breq1i 3bitr3i ) AB
      CDABCEZFGHABIABJZFGHABCKABLPQFGABMNO $.

    $( Two ways to express " ` A ` has a unique element".  (Contributed by
       Mario Carneiro, 9-Apr-2015.) $)
    euen1b $p |- ( A ~~ 1o <-> E! x x e. A ) $=
      ( cv wcel weu cab c1o cen wbr euen1 abid2 breq1i bitr2i ) ACBDZAENAFZGHIB
      GHINAJOBGHABKLM $.
  $}

  $( A singleton contains its sole element.  (Contributed by Stefan O'Rear,
     16-Aug-2015.) $)
  en1uniel $p |- ( S ~~ 1o -> U. S e. S ) $=
    ( c1o cen wbr cuni csn cvv wcel relen brrelex1i uniexg snidg 3syl wceq encv
    wb simpld en1bg syl ibi eleqtrrd ) ABCDZAEZUCFZAUBAGHZUCGHUCUDHABCIJAGKUCGL
    MUBAUDNZUBUEUBUFPUBUEBGHABOQAGRSTUA $.

  ${
    $d A x $.
    $( A set with one element is inhabited.  (Contributed by Jim Kingdon,
       3-Jan-2026.) $)
    en1m $p |- ( A ~~ 1o -> E. x x e. A ) $=
      ( c1o cen wbr cuni wcel cv wex en1uniel elex2 syl ) BCDEBFZBGAHBGAIBJAMBK
      L $.
  $}

  ${
    $d x y f A $.
    $( A set that dominates ordinal 2 has at least 2 different members.
       (Contributed by NM, 25-Jul-2004.) $)
    2dom $p |- ( 2o ~<_ A -> E. x e. A E. y e. A -. x = y ) $=
      ( vf c2o cdom wbr c0 csn cpr cv wf1 wceq wn wrex cfv wcel ffvelcdm notbid
      sylancl wex df2o2 breq1i brdomi sylbi f1f 0ex prid1 p0ex prid2 0nep0 neii
      wf wb f1fveq mpanr12 mtbiri eqeq1 eqeq2 rspc2ev syl3anc exlimiv syl ) ECF
      GZHHIZJZCDKZLZDUAZAKZBKZMZNZBCOACOZVDVFCFGVIEVFCFUBUCVFCDUDUEVHVNDVHHVGPZ
      CQZVEVGPZCQZVOVQMZNZVNVHVFCVGUMZHVFQZVPVFCVGUFZHVEUGUHZVFCHVGRTVHWAVEVFQZ
      VRWCHVEUIUJZVFCVEVGRTVHVSHVEMZHVEUKULVHWBWEVSWGUNWDWFVFCHVEVGUOUPUQVMVTVO
      VKMZNABVOVQCCVJVOMVLWHVJVOVKURSVKVQMWHVSVKVQVOUSSUTVAVBVC $.
  $}

  ${
    $d x y z w F $.
    fundmen.1 $e |- F e. _V $.
    $( A function is equinumerous to its domain.  Exercise 4 of [Suppes] p. 98.
       (Contributed by NM, 28-Jul-2004.)  (Revised by Mario Carneiro,
       15-Nov-2014.) $)
    fundmen $p |- ( Fun F -> dom F ~~ F ) $=
      ( vx vy vz vw cv cop cint cvv wcel a1i ex wi wa wceq wb wex vex adantl
      wfun cdm cfv dmex funfvop wrel funrel elreldm syl cxp df-rel sylib sselda
      elvv inteq inteqd op1stb eqtrdi eqeq1 imbitrrid opeq1 syl6 eqeq2 biimprcd
      wss imp mpd ancoms eleq1d funopfv adantr sylbid exp32 com24 opeq2d eqtr4d
      imp43 exlimdvv adantrl funfvex op1stbg sylancr eqtr2d adantrr impbid en3d
      ) AUAZCDAUBZACGZWIAUCZHZDGZIZIZJJWHJKWGABUDLAJKWGBLWGWIWHKZWKAKWIAUEMWGAU
      FZWLAKZWNWHKZNAUGZWPWQWRAWLUHMUIWGWOWQOZWIWNPZWLWKPZQWGWTOXAXBWGWQXAXBNZW
      OWGWQOZWLEGZFGZHZPZFRERZXCXDWLJJUJZKXIWGAXJWLWGWPAXJVEWSAUKULUMEFWLUNULXD
      XHXCEFXDXHXAXBXDXHXAOZOZWLWIXFHZWKXKWLXMPZXDXAXHXNXAXHOZXMXGPZXNXAXHXPXAX
      HWIXEPZXPXHXQXAWNXEPXHWNXGIZIXEXHWMXRWLXGUOUPXEXFESFSUQURWIWNXEUSUTWIXEXF
      VAVBVFXHXPXNNXAXPXNXHXMXGWLVCVDTVGZVHTXLWJXFWIWGWQXHXAWJXFPZWGXAXHWQXTWGX
      AXHWQXTNWGXOOWQXMAKZXTXOWQYAQWGXOWLXMAXSVITWGYAXTNXOWIXFAVJVKVLVMVNVQVOVP
      VMVRVGVSWGWOXBXANWQWGWOOZXBXAYBXBOWNWKIZIZWIXBWNYDPYBXBWMYCWLWKUOUPTYBYDW
      IPZXBYBWIJKWJJKYECSWIAVTWIWJJJWAWBVKWCMWDWEMWF $.
  $}

  ${
    $d x A $.  $d x F $.
    $( A function is equinumerous to its domain.  Exercise 4 of [Suppes] p. 98.
       (Contributed by NM, 17-Sep-2013.) $)
    fundmeng $p |- ( ( F e. V /\ Fun F ) -> dom F ~~ F ) $=
      ( vx wcel wfun cdm cen wbr cv wceq funeq dmeq breq12d imbi12d vex fundmen
      wi id vtoclg imp ) ABDAEZAFZAGHZCIZEZUDFZUDGHZQUAUCQCABUDAJZUEUAUGUCUDAKU
      HUFUBUDAGUDALUHRMNUDCOPST $.

    $( A relational set is equinumerous to its converse.  (Contributed by Mario
       Carneiro, 28-Dec-2014.) $)
    cnven $p |- ( ( Rel A /\ A e. V ) -> A ~~ `' A ) $=
      ( vx wrel wcel wa ccnv cvv csn cuni cmpt wf1o cen wbr simpr cnvexg adantl
      cv cnvf1o adantr f1oen2g syl3anc ) ADZABEZFUDAGZHEZAUECACRIGJKZLZAUEMNUCU
      DOUDUFUCABPQUCUHUDCASTAUEUGBHUAUB $.
  $}

  $( If a set is dominated by ` _om ` , so is its converse.  (Contributed by
     Thierry Arnoux, 29-Dec-2016.) $)
  cnvct $p |- ( A ~<_ _om -> `' A ~<_ _om ) $=
    ( ccnv cdom wbr com cen wrel cvv wcel relcnv ctex cnvexg syl cnven cnvcnvss
    sylancr wss ssdomg mpisyl endomtr syl2anc domtr mpancom ) ABZACDZAECDZUDECD
    UFUDUDBZFDZUGACDZUEUFUDGUDHIZUHAJUFAHIZUJAKZAHLMUDHNPUFUKUGAQUIULAOUGAHRSUD
    UGATUAUDAEUBUC $.

  $( A function is equinumerate to its domain.  (Contributed by Paul Chapman,
     22-Jun-2011.) $)
  fndmeng $p |- ( ( F Fn A /\ A e. C ) -> A ~~ F ) $=
    ( wfn wcel wa cdm cen wbr wfun fnex fnfun adantr fundmeng syl2anc wb breq1d
    cvv fndm mpbid ) CADZABEZFZCGZCHIZACHIZUCCRECJZUEABCKUAUGUBACLMCRNOUAUEUFPU
    BUAUDACHACSQMT $.

  ${
    $d A w y z $.  $d B w y z $.  $d ph w y z $.
    mapsnend.a $e |- ( ph -> A e. V ) $.
    mapsnend.b $e |- ( ph -> B e. W ) $.
    $( Set exponentiation to a singleton exponent is equinumerous to its base.
       Exercise 4.43 of [Mendelson] p. 255.  (Contributed by NM, 17-Dec-2003.)
       (Revised by Mario Carneiro, 15-Nov-2014.)  (Revised by Glauco
       Siliprandi, 24-Dec-2020.) $)
    mapsnend $p |- ( ph -> ( A ^m { B } ) ~~ A ) $=
      ( vz vw vy csn cmap cv cvv wcel vex wceq wa wb a1i co cfv cop fnmap elexd
      cxp wfn snexg syl fnovex mp3an2i fvexg sylancr a1d opexg sylancl wex wrex
      mapsnd eqabrd anbi1d r19.41v bicomi df-rex 3bitrd sylan9eqr eqeq2d equcom
      fveq1 fvsng bitrdi pm5.32da anbi2d anass ancom 3bitr2d exbidv opeq2 sneqd
      eleq1w anbi12d equsexvw en2d ) AHIBCKZLUAZBCHMZUBZCIMZUCZKZNDNNLNNUFUGABN
      OWDNOZWENOUDABDFUEACEOZWKGCEUHUIBWDNNLUJUKFAWGNOZWFWEOZAWFNOWLWMHPGCWFNEU
      LUMUNAWJNOZWHBOZAWINOZWOAWLWHNOWQGIPCWHENUOUPWINUHUIUNAWNWHWGQZRZJMZBOZWF
      CWTUCZKZQZWRRZRZJUQZWTWHQZXAXDRZRZJUQZWPWFWJQZRZAWSXDJBURZWRRZXEJBURZXGAW
      NXNWRAXNHWEAJBCHDEFGUSUTVAXOXPSAXPXOXDWRJBVBVCTXPXGSAXEJBVDTVEAXFXJJAXFXA
      XDXHRZRZXIXHRZXJAXEXQXAAXDWRXHAXDRZWRWHWTQXHXTWGWTWHXDAWGCXCUBZWTCWFXCVIA
      WLWTNOYAWTQGJPCWTENVJUPVFVGIJVHVKVLVMXSXRSAXAXDXHVNTXSXJSAXIXHVOTVPVQXKXM
      SAXIXMJIXHXAWPXDXLJIBVTXHXCWJWFXHXBWIWTWHCVRVSVGWAWBTVEWC $.
  $}

  ${
    $d w y z A $.  $d w y z B $.
    mapsnen.1 $e |- A e. _V $.
    mapsnen.2 $e |- B e. _V $.
    $( Set exponentiation to a singleton exponent is equinumerous to its base.
       Exercise 4.43 of [Mendelson] p. 255.  (Contributed by NM, 17-Dec-2003.)
       (Revised by Mario Carneiro, 15-Nov-2014.) $)
    mapsnen $p |- ( A ^m { B } ) ~~ A $=
      ( vz vw vy csn cmap cv cfv cop cvv wcel snex vex a1i wceq wa wex co fnmap
      cxp wfn fnovex mp3an fvex opex mapsn abeq2i anbi1i r19.41v df-rex 3bitr2i
      fveq1 eqtrdi eqeq2d equcom bitrdi pm5.32i anbi2i anass ancom exbii eleq1w
      wrex fvsn opeq2 sneqd anbi12d ceqsexv 3bitri en2i ) EFABHZIUAZABEJZKZBFJZ
      LZHZIMMUCUDAMNVNMNVOMNUBCBDOAVNMMIUEUFCVQMNVPVONZBVPMMEPDUGQVTMNVRANZVSBV
      RDFPZUHOQWAVRVQRZSZGJZANZVPBWFLZHZRZWDSZSZGTZWFVRRZWGWJSZSZGTWBVPVTRZSZWE
      WJGAVFZWDSWKGAVFWMWAWSWDWSEVOGABECDUIUJUKWJWDGAULWKGAUMUNWLWPGWLWGWJWNSZS
      WOWNSWPWKWTWGWJWDWNWJWDVRWFRWNWJVQWFVRWJVQBWIKWFBVPWIUOBWFDGPVGUPUQFGURUS
      UTVAWGWJWNVBWOWNVCUNVDWOWRGVRWCWNWGWBWJWQGFAVEWNWIVTVPWNWHVSWFVRBVHVIUQVJ
      VKVLVM $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y V $.
    $( Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1.
       Exercise 4.42(b) of [Mendelson] p. 255.  (Contributed by NM,
       17-Dec-2003.) $)
    map1 $p |- ( A e. V -> ( 1o ^m A ) ~~ 1o ) $=
      ( vx vy wcel c1o cmap co c0 cxp cvv 1oex a1i cv 0ex p0ex wceq wa bitr2di
      wb csn wfn fnmap elex fnovex mp3an12i 2a1i xpexg mpan2 a1d el1o wf oveq1i
      df1o2 eleq2i elmapg mpan bitrid fconst2 anbi12d ancom en2d ) ABEZCDFAGHZF
      IAIUAZJZKKKKGKKJUBFKEZVCAKEVDKEUCLABUDFAKKGUEUFVGVCLMVCCNZVDEZIKEOUGVCVFK
      EZDNZFEZVCVEKEZVJPAVEBKUHUIUJVCVLVHVFQZRVKIQZVIRVIVORVCVLVOVNVIVLVOTVCVKU
      KMVCVIAVEVHULZVNVIVHVEAGHZEZVCVPVDVQVHFVEAGUNUMUOVMVCVRVPTPVEAVHKBUPUQURA
      IVHOUSSUTVOVIVASVB $.

    $( Two singletons are equinumerous.  (Contributed by NM, 9-Nov-2003.) $)
    en2sn $p |- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } ) $=
      ( wcel csn c1o cen wbr ensn1g ensymd entr syl2an ) ACEAFZGHIGBFZHINOHIBDE
      ZACJPOGBDJKNGOLM $.

    $( A singleton is finite.  For the proper class case, see ~ snprc .
       (Contributed by Jim Kingdon, 13-Apr-2020.) $)
    snfig $p |- ( A e. V -> { A } e. Fin ) $=
      ( vx wcel csn cv cen wbr com wrex cfn c1o 1onn ensn1g rspcev sylancr isfi
      breq2 sylibr ) ABDZAEZCFZGHZCIJZUAKDTLIDUALGHZUDMABNUCUECLIUBLUAGROPCUAQS
      $.

    $( The class of finite sets is a proper class.  (Contributed by Jeff
       Hankins, 3-Oct-2008.) $)
    fiprc $p |- Fin e/ _V $=
      ( vx vy cv csn wceq wex cab cvv wnel cfn snnex wcel wss snfig ax-mp eleq1
      wn vex mpbiri df-nel exlimiv abssi ssexg mpan con3i 3imtr4i ) ACZBCZDZEZB
      FZAGZHIZJHIZABKULHLZQJHLZQUMUNUPUOULJMUPUOUKAJUJUGJLZBUJUQUIJLZUHHLURBRUH
      HNOUGUIJPSUAUBULJHUCUDUEULHTJHTUFO $.

    $( Equinumerosity of union of disjoint sets.  Theorem 4 of [Suppes] p. 92.
       (Contributed by NM, 11-Jun-1998.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    unen $p |- ( ( ( A ~~ B /\ C ~~ D ) /\
  ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( A u. C ) ~~ ( B u. D ) ) $=
      ( vx vy cen wbr wa cin c0 wceq cun cv wf1o wex wi bren cvv vex eeanv wcel
      unex f1oun f1oen3g sylancr ex exlimivv sylbir syl2anb imp ) ABGHZCDGHZIAC
      JKLBDJKLIZACMZBDMZGHZULABENZOZEPZCDFNZOZFPZUNUQQZUMABERCDFRUTVCIUSVBIZFPE
      PVDUSVBEFUAVEVDEFVEUNUQVEUNIURVAMZSUBUOUPVFOUQURVAETFTUCABCDURVAUDUOUPVFS
      UEUFUGUHUIUJUK $.
  $}

  ${
    $d A f $.  $d B f $.  $d C f $.  $d D f $.
    en2prd.1 $e |- ( ph -> A e. V ) $.
    en2prd.2 $e |- ( ph -> B e. W ) $.
    en2prd.3 $e |- ( ph -> C e. X ) $.
    en2prd.4 $e |- ( ph -> D e. Y ) $.
    en2prd.5 $e |- ( ph -> A =/= B ) $.
    en2prd.6 $e |- ( ph -> C =/= D ) $.
    $( Two proper unordered pairs are equinumerous.  (Contributed by
       BTernaryTau, 23-Dec-2024.) $)
    en2prd $p |- ( ph -> { A , B } ~~ { C , D } ) $=
      ( vf cpr cvv wcel syl2anc cen wbr cv wex cop opexg prexg wne wa wi f1oprg
      wf1o syl22anc mp2and f1oeq1 elabd wb breng mpbird ) ABCQZDEQZUAUBZUTVAPUC
      ZULZPUDZAVDUTVABDUEZCEUEZQZULZPVHAVFRSZVGRSZVHRSABFSZDHSZVJJLBDFHUFTACGSZ
      EISZVKKMCEGIUFTVFVGRRUGTABCUHZDEUHZVINOAVLVMVNVOVPVQUIVIUJJLKMBDCEFHGIUKU
      MUNUTVAVCVHUOUPAUTRSZVARSZVBVEUQAVLVNVRJKBCFGUGTAVMVOVSLMDEHIUGTUTVAPRRUR
      TUS $.
    $( $j usage 'en2prd' avoids 'ax-un'; $)
  $}

  ${
    $d A f $.  $d B f $.  $d C f $.
    $( If a set is dominated by one, then any two of its elements are equal.
       (Contributed by Jim Kingdon, 23-Apr-2025.) $)
    1dom1el $p |- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C ) $=
      ( vf c1o cdom wbr wcel w3a cv wf1 wceq wex brdomi 3ad2ant1 ffvelcdmd el1o
      cfv c0 sylib wa wf adantl simpl2 simpl3 eqtr4d wi simpr f1veqaeq syl12anc
      f1f mpd exlimddv ) AEFGZBAHZCAHZIZAEDJZKZBCLZDUNUOUSDMUPAEDNOUQUSUAZBURRZ
      CURRZLZUTVAVBSVCVAVBEHVBSLVAAEBURUSAEURUBUQAEURUKUCZUNUOUPUSUDZPVBQTVAVCE
      HVCSLVAAECURVEUNUOUPUSUEZPVCQTUFVAUSUOUPVDUTUGUQUSUHVFVGAEBCURUIUJULUM $.
  $}

  ${
    $d ph u v y $.  $d u v x y $.
    $( Two ways to express "at most one".  (Contributed by Stefan O'Rear,
       28-Oct-2014.) $)
    modom $p |- ( E* x ph <-> { x | ph } ~<_ 1o ) $=
      ( vu vv vy c1o cdom cvv wcel c0 cv wa weq wsb df-clab wal nfv nfcv nfs1v
      wi wmo cab wbr 1oex 0lt1o 2a1i wceq wb anbi12i mo3 nfan nfim nfal sbequ12
      eqidd anbi1d equequ1 imbi12d albidv spcgf elv sylbi sbequ anbi2d biimtrid
      equequ2 syl imp 2thd ex dom2d mpi nfab1 nfbr simpl abid ad2antrl ad2antll
      biimpri 1dom1el syl3anc alrimiv alrimi sylibr impbii ) ABUAZABUBZFGUCZWFF
      HIWHUDWFCDWGFJJHWFCKZWGIZJFIUEUFWFWJDKZWGIZLZJJUGZCDMZUHWFWMLZWNWOWPJUOWF
      WMWOWMABCNZABDNZLZWFWOWJWQWLWRACBOADBOUIWFWQABENZLZCEMZTZEPZWSWOTZWFAWTLZ
      BEMZTZEPZBPZXDABEAEQUJZXJXDTCXIXDBWIHBWIRXCBEXAXBBWQWTBABCSABESUKXBBQULUM
      BCMZXHXCEXLXFXAXGXBXLAWQWTABCUNUPBCEUQURUSUTVAVBXDXETDXCXEEWKHEWKRXEEQEDM
      ZXAWSXBWOXMWTWRWQAEDBVCVDEDCVFURUTVAVGVEVHVIVJVKVLWHXJWFWHXIBBWGFGABVMBGR
      BFRVNWHXHEWHXFXGWHXFLWHBKZWGIZEKZWGIZXGWHXFVOAXOWHWTXOAABVPVSVQWTXQWHAXQW
      TAEBOVSVRWGXNXPVTWAVJWBWCXKWDWE $.
  $}

  ${
    $d x A $.
    $( Two ways to express "at most one".  (Contributed by Mario Carneiro,
       24-Dec-2016.) $)
    modom2 $p |- ( E* x x e. A <-> A ~<_ 1o ) $=
      ( cv wcel wmo cab c1o cdom wbr modom abid2 breq1i bitri ) ACBDZAENAFZGHIB
      GHINAJOBGHABKLM $.
  $}

  ${
    $d A x y $.
    $( A set that has at least 2 different members dominates ordinal 2.
       (Contributed by BTernaryTau, 30-Dec-2024.) $)
    rex2dom $p |- ( ( A e. V /\ E. x e. A E. y e. A x =/= y ) -> 2o ~<_ A ) $=
      ( wcel cv wne wrex c2o cdom wbr cvv wi cpr cen c0 c1o a1i vex syl elex wa
      wss prssi df2o3 0ex 1oex 1n0 necomi en2prd eqbrtrid domssr 3expib syl2ani
      id endom expd rexlimdvv imp ) CDEZAFZBFZGZBCHACHZICJKZUTCLEZVDVEMCDUAVFVC
      VEABCCVFVACEVBCEUBZVCVEVGVFVAVBNZCUCZIVHJKZVEVCVAVBCUDVCIVHOKVJVCIPQNVHOU
      EVCPQVAVBLLLLPLEVCUFRQLEVCUGRVALEVCASRVBLEVCBSRPQGVCQPUHUIRVCUOUJUKIVHUPT
      VFVIVJVEIVHCLULUMUNUQURTUS $.
  $}

  ${
    enpr2d.1 $e |- ( ph -> A e. C ) $.
    enpr2d.2 $e |- ( ph -> B e. D ) $.
    enpr2d.3 $e |- ( ph -> -. A = B ) $.
    $( A pair with distinct elements is equinumerous to ordinal two.
       (Contributed by Rohan Ridenour, 3-Aug-2023.) $)
    enpr2d $p |- ( ph -> { A , B } ~~ 2o ) $=
      ( c1o cen csn cun wbr cin c0 wceq wcel syl con0 1on cpr csuc ensn1g en2sn
      c2o sylancl wne neqned disjsn2 wn onirri a1i disjsn sylibr syl22anc df-pr
      unen df-suc 3brtr4g df-2o breqtrrdi ) ABCUAZIUBZUEJABKZCKZLZIIKZLZVBVCJAV
      DIJMZVEVGJMZVDVENOPZIVGNOPZVFVHJMABDQVIFBDUCRACEQISQVJGTCIESUDUFABCUGVKAB
      CHUHBCUIRAIIQUJZVLVMAITUKULIIUMUNVDIVEVGUQUOBCUPIURUSUTVA $.
  $}

  ${
    $d A f x y $.
    $( A set equinumerous to ordinal 2 is an unordered pair.  (Contributed by
       Mario Carneiro, 5-Jan-2016.) $)
    en2 $p |- ( A ~~ 2o -> E. x E. y A = { x , y } ) $=
      ( vf c2o cv wf1o cpr wceq wex cfv c1o cima wfn adantl wcel syl 0lt2o a1i
      c0 cen wbr bren biimpi ccnv cdm crn cnvimarndm wfun dff1o2 simp3bi eqtrdi
      wa df2o3 imaeq2d eqtr3id f1odm f1ocnv f1ofn 1lt2o fnimapr syl3anc 3eqtr3d
      simpr f1ocnvdm sylancl wi preq2 eqeq2d spcegv exbidv sylsyld mpd exlimddv
      preq1 ) CEUAUBZCEDFZGZCAFZBFZHZIZBJZAJZDVPVRDJCEDUCUDVPVRUMZCTVQUEZKZLWFK
      ZHZIZWDWEVQUFZWFTLHZMZCWIWEWKWFVQUGZMZWMVQUHVRWOWMIVPVRWNWLWFVRWNEWLVRVQC
      NWFUIWNEICEVQUJUKUNULUOOUPVRWKCIVPCEVQUQOWEWFENZTEPZLEPZWMWIIWEECWFGZWPVR
      WSVPCEVQUROECWFUSQWQWERSWRWEUTSETLWFVAVBVCWEWGCPZWJCWGVTHZIZBJZWDWEVRWQWT
      VPVRVDZRCETVQVEVFWEWHCPZWJXCVGWEVRWRXEXDUTCELVQVEVFXBWJBWHCVTWHIXAWICVTWH
      WGVHVIVJQWCXCAWGCVSWGIZWBXBBXFWAXACVSWGVTVOVIVKVJVLVMVN $.
  $}

  ${
    $d A x y $.
    $( A set with two elements is inhabited.  (Contributed by Jim Kingdon,
       3-Jan-2026.) $)
    en2m $p |- ( A ~~ 2o -> E. x x e. A ) $=
      ( vy c2o cen wbr cv cpr wceq wex en2 wi vex prid2 eleq2 mpbiri a1i eximdv
      wcel imp exlimddv ) BDEFZBCGZAGZHZIZAJZUDBSZAJZCCABKUBUGUIUBUFUHAUFUHLUBU
      FUHUDUESUCUDAMNBUEUDOPQRTUA $.
  $}

  $( A subset of a set dominated by ` _om ` is dominated by ` _om ` .
     (Contributed by Thierry Arnoux, 31-Jan-2017.) $)
  ssct $p |- ( ( A C_ B /\ B ~<_ _om ) -> A ~<_ _om ) $=
    ( wss com cdom wbr cvv wcel wi ctex ssdomg syl impcom domtr sylancom ) ABCZ
    BDEFZABEFZADEFQPRQBGHPRIBJABGKLMABDNO $.

  ${
    $d A x y $.
    $( A singleton (whether of a set or a proper class) is dominated by one.
       (Contributed by Jim Kingdon, 1-Mar-2022.) $)
    1domsn $p |- { A } ~<_ 1o $=
      ( vx vy csn c1o c0 cmpt wf1 cdom wbr wcel wral cv wi 0lt1o rgenw wa elsni
      wceq adantr adantl eqtr4d a1d rgen2a eqid eqidd f1mpt mpbir2an 1oex f1dom
      ax-mp ) ADZEBULFGZHZULEIJUNFEKZBULLFFSZBMZCMZSZNZCULLBULLUOBULOPUTBCULUQU
      LKZURULKZQZUSUPVCUQAURVAUQASVBUQARTVBURASVAURARUAUBUCUDBCULEFFUMUMUEUSFUF
      UGUHULEUMUIUJUK $.
  $}

  ${
    $d A f j $.  $d V f $.
    $( Two ways of saying that a set is inhabited.  (Contributed by Jim
       Kingdon, 3-Jan-2026.) $)
    dom1o $p |- ( A e. V -> ( 1o ~<_ A <-> E. j j e. A ) ) $=
      ( vf wcel c1o cdom wbr cv wex wf1 brdomg wi wf c0 cfv f1f 0lt1o a1i csn
      ffvelcdm mpan2 elex2 3syl exlimdv sylbid cop cvv 0ex opex snex f1sng mpan
      vex wceq wb df1o2 f1eq2 ax-mp sylibr f1eq1 elabd exlimiv imbitrrid impbid
      ) ACEZFAGHZBIZAEZBJZVFVGFADIZKZDJZVJFACDLZVFVLVJDVLVJMVFVLFAVKNZOVKPZAEZV
      JFAVKQVOOFEZVQRFAOVKUAUBBVPAUCUDSUEUFVJVGVFVMVIVMBVIVLFAOVHUGZTZKZDVTVTUH
      EVIVSOVHUIBUNUJUKSVIOTZAVTKZWAVRVIWCROVHFAULUMFWBUOWAWCUPUQFWBAVTURUSUTFA
      VKVTVAVBVCVNVDVE $.
  $}

  ${
    $d A j $.  $d B j $.
    $( A set with an element dominates one.  (Contributed by Jim Kingdon,
       3-Feb-2026.) $)
    dom1oi $p |- ( ( A e. V /\ B e. A ) -> 1o ~<_ A ) $=
      ( vj wcel wa c1o cdom wbr cv wex elex2 adantl wb dom1o adantr mpbird ) AC
      EZBAEZFGAHIZDJAEDKZSUARDBALMRTUANSADCOPQ $.
  $}

  ${
    $d f x y A $.  $d f x y B $.
    $( A set equinumerous to an inhabited set is inhabited.  (Contributed by
       Jim Kingdon, 19-May-2020.) $)
    enm $p |- ( ( A ~~ B /\ E. x x e. A ) -> E. y y e. B ) $=
      ( vf cv wcel wex cen wbr wi wf1o bren wf f1of wa cfv ffvelcdm syl exlimiv
      elex2 ex sylbi com12 impcom ) AFZCGZAHCDIJZBFDGBHZUGUHUIKAUHUGUIUHCDEFZLZ
      EHUGUIKZCDEMUKULEUKCDUJNZULCDUJOUMUGUIUMUGPUFUJQZDGUICDUFUJRBUNDUASUBSTUC
      UDTUE $.
  $}

  ${
    $d x y z A $.  $d x y z B $.
    xpsnen.1 $e |- A e. _V $.
    xpsnen.2 $e |- B e. _V $.
    $( A set is equinumerous to its Cartesian product with a singleton.
       Proposition 4.22(c) of [Mendelson] p. 254.  (Contributed by NM,
       4-Jan-2004.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
    xpsnen $p |- ( A X. { B } ) ~~ A $=
      ( vy vx vz cv cint cop wcel wceq wex cvv inteq inteqd vex op1stb adantr
      wa csn cxp snex xpex elxp eqtrdi eqeltrdi exlimivv sylbi opex wb eqvisset
      a1i ancom anass velsn anbi1i 3bitr3i exbii eqeq2d anbi1d ceqsexv pm4.71ri
      opeq2 eqtr2di bitri 3bitri opeq1 anbi12d ceqsexgv bitrid pm5.32ri pm4.71i
      eleq1 syl bitr2i en2i ) EFABUAZUBZAEHZIZIZFHZBJZAVRCBDUCUDCVTVSKZVTWCGHZJ
      ZLZWCAKZWFVRKZTZTZGMZFMZWBNKZFGVTAVRUEZWLWOFGWHWOWKWHWBWCNWHWBWGIZIWCWHWA
      WQVTWGOPWCWFFQZGQRUFWRUGSUHUIWDNKWIWCBWRDUJUMWEWCWBLZTVTWBBJZLZWBAKZTZWST
      ZVTWDLZWITZWIXETWSWEXCWSWOWEXCUKFWBULWEWSXFTZFMZWOXCWEWNXHWPWMXGFWMWFBLZW
      HWITZTZGMXFXGWLXKGXJWJTWJXJTWLXKXJWJUNWHWIWJUOWJXIXJGBUPUQURUSXJXFGBDXIWH
      XEWIXIWGWDVTWFBWCVDUTVAVBXFWSXETZWITXGXEXLWIXEWSXEWBWDIZIWCXEWAXMVTWDOPWC
      BWRDRVEZVCUQWSXEWIUOVFVGUSVFXFXCFWBNWSXEXAWIXBWSWDWTVTWCWBBVHUTWCWBAVNVIZ
      VJVKVOVLXFXFWSTXDXFWSXEWSWIXNSVMWSXFXCXOVLVPXEWIUNVGVQ $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( A set is equinumerous to its Cartesian product with a singleton.
       Proposition 4.22(c) of [Mendelson] p. 254.  (Contributed by NM,
       22-Oct-2004.) $)
    xpsneng $p |- ( ( A e. V /\ B e. W ) -> ( A X. { B } ) ~~ A ) $=
      ( vx vy cv csn cxp cen wbr wceq xpeq1 id breq12d xpeq2d breq1d vex xpsnen
      sneq vtocl2g ) EGZFGZHZIZUBJKAUDIZAJKABHZIZAJKEFABCDUBALZUEUFUBAJUBAUDMUI
      NOUCBLZUFUHAJUJUDUGAUCBTPQUBUCERFRSUA $.
  $}

  $( One times a cardinal number.  (Contributed by NM, 27-Sep-2004.)  (Revised
     by Mario Carneiro, 29-Apr-2015.) $)
  xp1en $p |- ( A e. V -> ( A X. 1o ) ~~ A ) $=
    ( wcel c1o cxp c0 csn cen df1o2 xpeq2i cvv wbr 0ex xpsneng mpan2 eqbrtrid )
    ABCZADEAFGZEZAHDRAIJQFKCSAHLMAFBKNOP $.

  ${
    $d x y A $.  $d x y B $.
    endisj.1 $e |- A e. _V $.
    endisj.2 $e |- B e. _V $.
    $( Any two sets are equinumerous to disjoint sets.  Exercise 4.39 of
       [Mendelson] p. 255.  (Contributed by NM, 16-Apr-2004.) $)
    endisj $p |- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) ) $=
      ( c0 csn cxp cen wbr c1o wa cin wceq cv wex xpsnen xpex breq1 con0 pm3.2i
      0ex elexi xp01disj p0ex snex bi2anan9 ineq12 eqeq1d anbi12d spc2ev mp2an
      1on ) CGHZIZCJKZDLHZIZDJKZMZUPUSNZGOZAPZCJKZBPZDJKZMZVDVFNZGOZMZBQAQUQUTC
      GEUCRDLFLUAUNUDZRUBCDUEVKVAVCMABUPUSCUOEUFSDURFLVLUGSVDUPOZVFUSOZMZVHVAVJ
      VCVMVEUQVNVGUTVDUPCJTVFUSDJTUHVOVIVBGVDUPVFUSUIUJUKULUM $.
  $}

  ${
    $d u v w x y z A $.  $d u v w x y z B $.  $d u v C $.  $d u v w y z F $.
    $d u v w G $.
    xpcomf1o.1 $e |- F = ( x e. ( A X. B ) |-> U. `' { x } ) $.
    $( The canonical bijection from ` ( A X. B ) ` to ` ( B X. A ) ` .
       (Contributed by Mario Carneiro, 23-Apr-2014.) $)
    xpcomf1o $p |- F : ( A X. B ) -1-1-onto-> ( B X. A ) $=
      ( cxp ccnv wf1o cv cuni cmpt wrel relxp cnvf1o ax-mp wceq wb f1oeq1 mpbir
      csn cnvxp f1oeq3 mpbi ) BCFZUDGZDHZUDCBFZDHZUFUDUEAUDAITGJKZHZUDLUJBCMAUD
      NODUIPUFUJQEUDUEDUIROSUEUGPUFUHQBCUAUEUGUDDUBOUC $.

    xpcomco.1 $e |- G = ( y e. B , z e. A |-> C ) $.
    $( Composition with the bijection of ~ xpcomf1o swaps the arguments to a
       mapping.  (Contributed by Mario Carneiro, 30-May-2015.) $)
    xpcomco $p |- ( G o. F ) = ( z e. A , y e. B |-> C ) $=
      ( vu vw vv cv wbr wa wex wceq wcel cvv copab cop ccom cmpo cfv cxp cdm wb
      wf1o wfun xpcomf1o f1ofun funbrfv2b mp2b ancom eqcom f1odm eleq2i anbi12i
      ax-mp 3bitri anbi1i anass bitri exbii csn ccnv cuni wal vex mptfvex mpan2
      snex cnvex uniex breq1 anbi2d ceqsexv elxp nfcv nfmpo2 nfcxfr nfbr nfmpo1
      mpg 19.41 fveq2 opelxpi sneq cnveqd unieqd opswapg mp2an eqtrdi fvmpt syl
      opex sylan9eq breq1d coprab df-br df-mpo eqtri oprabid baib ancoms adantl
      bitrd pm5.32da pm5.32i bitr3i 3bitr2i opabbii df-co dfoprab2 3eqtr4i ) KN
      ZLNZGOZXRMNZHOZPZLQZKMUAXQCNZBNZUBZRZYDDSZYEESZPZXTFRZPZPZBQZCQZKMUAZHGUC
      CBDEFUDZYCYOKMYCXRXQGUEZRZXQDEUFZSZYAPZPZLQUUAYRXTHOZPZYOYBUUCLYBYSUUAPZY
      APUUCXSUUFYAXSXQGUGZSZYRXRRZPZUUIUUHPUUFYTEDUFZGUIZGUJXSUUJUHADEGIUKZYTUU
      KGULXQXRGUMUNUUHUUIUOUUIYSUUHUUAYRXRUPUUGYTXQUULUUGYTRUUMYTUUKGUQUTURUSVA
      VBYSUUAYAVCVDVEUUBUUELYRANZVFZVGZVHZTSZYRTSZAUURAVIXQTSUUSKVJAYTUUQXQGTTI
      VKVLUUPUUOUUNAVJVMVNVOWEYSYAUUDUUAXRYRXTHVPVQVRUUEYGYJPZBQZCQZUUDPUVAUUDP
      ZCQYOUUAUVBUUDCBXQDEVSVBUVAUUDCCYRXTHCYRVTCHBCEDFUDZJBCEDFWAWBCXTVTWCWFUV
      CYNCUVCUUTUUDPZBQYNUUTUUDBBYRXTHBYRVTBHUVDJBCEDFWDWBBXTVTWCWFUVEYMBUVEYGY
      JUUDPZPYMYGYJUUDVCYGUVFYLYGYJUUDYKUUTUUDYEYDUBZXTHOZYKUUTYRUVGXTHYGYJYRYF
      GUEZUVGXQYFGWGYJYFYTSUVIUVGRYDYEDEWHAYFUUQUVGYTGUUNYFRZUUQYFVFZVGZVHZUVGU
      VJUUPUVLUVJUUOUVKUUNYFWIWJWKYDTSYETSUVMUVGRCVJZBVJZYDYETTWLWMWNIYEYDUVOUV
      NWQWOWPWRWSYJUVHYKUHZYGYIYHUVPUVHYIYHPZYKUVHUVGXTUBZHSUVRUVQYKPZBCMWTZSUV
      SUVGXTHXAHUVTUVRHUVDUVTJBCMEDFXBXCURUVSBCMXDVAXEXFXGXHXIXJVDVEXKVEXLVAXMK
      MLHGXNYQYLCBMWTYPCBMDEFXBYLCBMKXOXCXP $.
  $}

  ${
    $d x A $.  $d x B $.
    xpcomen.1 $e |- A e. _V $.
    xpcomen.2 $e |- B e. _V $.
    $( Commutative law for equinumerosity of Cartesian product.  Proposition
       4.22(d) of [Mendelson] p. 254.  (Contributed by NM, 5-Jan-2004.)
       (Revised by Mario Carneiro, 15-Nov-2014.) $)
    xpcomen $p |- ( A X. B ) ~~ ( B X. A ) $=
      ( cxp cvv wcel csn ccnv cuni cmpt wf1o cen wbr xpex eqid xpcomf1o f1oen2g
      vx cv mp3an ) ABEZFGBAEZFGUBUCSUBSTHIJKZLUBUCMNABCDOBADCOSABUDUDPQUBUCUDF
      FRUA $.
  $}

  ${
    $d x y A $.  $d y B $.
    $( Commutative law for equinumerosity of Cartesian product.  Proposition
       4.22(d) of [Mendelson] p. 254.  (Contributed by NM, 27-Mar-2006.) $)
    xpcomeng $p |- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) ) $=
      ( vx vy cv cxp cen wbr wceq xpeq1 xpeq2 breq12d vex xpcomen vtocl2g ) EGZ
      FGZHZSRHZIJASHZSAHZIJABHZBAHZIJEFABCDRAKTUBUAUCIRASLRASMNSBKUBUDUCUEISBAM
      SBALNRSEOFOPQ $.
  $}

  $( A set is equinumerous to its Cartesian product with a singleton on the
     left.  (Contributed by Stefan O'Rear, 21-Nov-2014.) $)
  xpsnen2g $p |- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ B ) $=
    ( wcel csn cxp cen wbr cvv snexg xpcomeng sylan xpsneng ancoms entr syl2anc
    wa ) ACEZBDEZRAFZBGZBUAGZHIZUCBHIZUBBHISUAJETUDACKUABJDLMTSUEBADCNOUBUCBPQ
    $.

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    xpassen.1 $e |- A e. _V $.
    xpassen.2 $e |- B e. _V $.
    xpassen.3 $e |- C e. _V $.
    $( Associative law for equinumerosity of Cartesian product.  Proposition
       4.22(e) of [Mendelson] p. 254.  (Contributed by NM, 22-Jan-2004.)
       (Revised by Mario Carneiro, 15-Nov-2014.) $)
    xpassen $p |- ( ( A X. B ) X. C ) ~~ ( A X. ( B X. C ) ) $=
      ( vz vw vv vu csn cdm cuni crn cop wcel uniex wceq wa wex vx cxp xpex cvv
      vy vex snex dmex rnex opex a1i sneq dmeqd unieqd sneqd op1sta sneqi dmeqi
      unieqi eqtri eqtr2di rneqd rneqi op2nda opeq12d eq2tri anass anbi12i an32
      3bitr4i exbii 19.41v 3bitr3i 2exbii 19.41vv elxp excom anbi1i an12 exrot4
      cv opeq1 eqeq2d anbi1d anbi2d ceqsexv 3bitri 3exbii anbi2i 19.42vv bitr3i
      bitri exrot3 opeq2 en2i ) UAUEABUBZCUBZABCUBZUBZUAWAZKZLZMZKZLZMZXDNZMZXA
      NZMZOZOZUEWAZKZLZMZXNNZMZKZLZMZOZXSNZMZOZWPCABDEUCFUCAWRDBCEFUCUCXLUDPWTW
      QPZXFXKXEXDXCXBXAWTUAUFUGZUHQUGZUHQXHXJXGXDYHUIQXIXAYGUIQUJUJUKYEUDPXMWSP
      ZYBYDXPYAXOXNXMUEUFUGZUHQXTXSXRXQXNYJUIQUGZUHQUJYCXSYKUIQUJUKWTGWAZHWAZOZ
      IWAZOZRZYLAPZYMBPZSZYOCPZSZSZITZHTGTZXMXLRZSZXMYLYMYOOZOZRZYRYSUUASZSZSZI
      TZHTZGTZWTYERZSZYFUUFSYIUUQSUUDUUFSZHTGTUUNUUQSZHTGTUUGUURUUSUUTGHUUCUUFS
      ZITUUMUUQSZITUUSUUTUVAUVBIYQUUFSZUUBSUUJUUQSZUULSUVAUVBUVCUVDUUBUULWTXMYP
      UUIXLYEYQYLXFUUHXKYQXFYPKZLZMZKZLZMZYLYQXEUVIYQXDUVHYQXCUVGYQXBUVFYQXAUVE
      WTYPULZUMUNUOZUMUNUVJYNKZLZMYLUVIUVNUVHUVMUVGYNYNYOYLYMGUFZHUFZUJZIUFZUPU
      QZURUSYLYMUVOUVPUPUTVAYQYMXHYOXJYQXHUVHNZMZYMYQXGUVTYQXDUVHUVLVBUNUWAUVMN
      ZMYMUVTUWBUVHUVMUVSVCUSYLYMUVOUVPVDUTVAYQXJUVENZMYOYQXIUWCYQXAUVEUVKVBUNY
      NYOUVQUVRVDVAVEVEUUJYNYBYOYDUUJYLXPYMYAUUJXPUUIKZLZMYLUUJXOUWEUUJXNUWDXMU
      UIULZUMUNYLUUHUVOYMYOUVPUVRUJZUPVAUUJYAUWDNZMZKZLZMZYMUUJXTUWKUUJXSUWJUUJ
      XRUWIUUJXQUWHUUJXNUWDUWFVBUNUOZUMUNUWLUUHKZLZMYMUWKUWOUWJUWNUWIUUHYLUUHUV
      OUWGVDUQZURUSYMYOUVPUVRUPUTVAVEUUJYDUWJNZMZYOUUJYCUWQUUJXSUWJUWMVBUNUWRUW
      NNZMYOUWQUWSUWJUWNUWPVCUSYMYOUVPUVRVDUTVAVEVFYRYSUUAVGVHYQUUBUUFVIUUJUULU
      UQVIVJVKUUCUUFIVLUUMUUQIVLVMVNUUDUUFGHVOUUNUUQGHVOVMYFUUEUUFYFWTJWAZYOOZR
      ZUWTWPPZUUASSZITJTUXDJTITZUUEJIWTWPCVPUXDJIVQUXEUWTYNRZYTSZUXBUUASZSZHTGT
      ZJTITUXIJTZITHTGTUUEUXDUXJIJUXCUXHSUXGHTGTZUXHSUXDUXJUXCUXLUXHGHUWTABVPVR
      UXBUXCUUAVSUXGUXHGHVOVJVNUXIIJGHVTUXKUUCGHIUXKUXFYTUXHSZSZJTYTYQUUASZSZUU
      CUXIUXNJUXFYTUXHVGVKUXMUXPJYNUVQUXFUXHUXOYTUXFUXBYQUUAUXFUXAYPWTUWTYNYOWB
      WCWDWEWFYTYQUUAVSWGWHWGWGVRYIUUPUUQYIXMYLUWTOZRZYRUWTWRPZSSZJTZGTUUPGJXMA
      WRVPUYAUUOGUYAUWTUUHRZUXRUULSZSZITHTZJTUYDJTZITHTUUOUXTUYEJUXRYRSZUXSSUYG
      UYBUUKSZITHTZSZUXTUYEUXSUYIUYGHIUWTBCVPWIUXRYRUXSVGUYJUYGUYHSZITHTUYEUYGU
      YHHIWJUYKUYDHIUYKUYBUYGUUKSZSUYDUYGUYBUUKVSUYLUYCUYBUXRYRUUKVGWIWLVNWKVMV
      KUYDJHIWMUYFUUMHIUYCUUMJUUHUWGUYBUXRUUJUULUYBUXQUUIXMUWTUUHYLWNWCWDWFVNWG
      VKWLVRVJWO $.
  $}

  ${
    $d f u v w x y z A $.  $d f u v w x y z B $.  $d f u v w x y z C $.
    xpdom.2 $e |- C e. _V $.
    $( Dominance law for Cartesian product.  Proposition 10.33(2) of
       [TakeutiZaring] p. 92.  (Contributed by NM, 24-Jul-2004.)  (Revised by
       Mario Carneiro, 15-Nov-2014.) $)
    xpdom2 $p |- ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) $=
      ( vf vz vw vv vu cdom cv wa cuni cop cvv wcel wi wceq wb vex vx vy brdomi
      wbr wf1 cxp csn cdm crn cfv wf f1f ffvelcdm ex syl anim2d adantld 3imtr4g
      elxp4 opelxp adantl wrex elxp2 fvex f1fveq ancoms anbi2d ad2ant2l adantlr
      opth bitrid sneq dmeqd unieqd op1sta eqtrdi rneqd op2nda fveq2d eqeqan12d
      imp opeq12d eqeq12 bitrdi 3bitr4d exp53 com23 rexlimivv rexlimdvv syl2anb
      ad2antlr com12 reldom brrelex1i xpexg sylancr adantr brrelex2i exlimddv
      dom3d ) ABJUDZABEKZUEZCAUFZCBUFZJUDEABEUCXAXCLUAUBXDXEUAKZUGZUHZMZXGUIZMZ
      XBUJZNZUBKZUGZUHZMZXOUIZMZXBUJZNZOOXCXFXDPZXMXEPZQXAXCXFXIXKNRZXICPZXKAPZ
      LZLYEXLBPZLZYBYCXCYGYIYDXCYFYHYEXCABXBUKZYFYHQABXBULYJYFYHABXKXBUMUNUOUPU
      QXFCAUSXIXLCBUTURVAXCYBXNXDPZLZXMYARZXFXNRZSZQXAYLXCYOYBXFFKZGKZNZRZGAVBF
      CVBZXNHKZIKZNZRZIAVBHCVBZXCYOQZYKFGXFCAVCHIXNCAVCYTUUEUUFYTUUDUUFHICAYSUU
      ACPZUUBAPZLZUUDUUFQZQFGCAYPCPZYQAPZLZUUIYSUUJUUMUUIYSUUDXCYOUUMUUILZYSUUD
      LZLXCLYPYQXBUJZNZUUAUUBXBUJZNZRZYPUUARZYQUUBRZLZYMYNUUNXCUUTUVCSZUUOUUNXC
      UVDUULUUHXCUVDQUUKUUGUULUUHLZXCUVDUUTUVAUUPUURRZLUVEXCLZUVCYPUUPUUAUURFTZ
      YQXBOOETGTZVDVJUVGUVFUVBUVAXCUVEUVFUVBSABYQUUBXBVEVFVGVKUNVHWAVIUUOYMUUTS
      UUNXCYSUUDXMUUQYAUUSYSXIYPXLUUPYSXIYRUGZUHZMYPYSXHUVKYSXGUVJXFYRVLZVMVNYP
      YQUVHUVIVOVPYSXKYQXBYSXKUVJUIZMYQYSXJUVMYSXGUVJUVLVQVNYPYQUVHUVIVRVPVSWBU
      UDXQUUAXTUURUUDXQUUCUGZUHZMUUAUUDXPUVOUUDXOUVNXNUUCVLZVMVNUUAUUBHTZITZVOV
      PUUDXSUUBXBUUDXSUVNUIZMUUBUUDXRUVSUUDXOUVNUVPVQVNUUAUUBUVQUVRVRVPVSWBVTWK
      UUOYNUVCSUUNXCUUOYNYRUUCRUVCXFYRXNUUCWCYPYQUUAUUBUVHUVIVJWDWKWEWFWGWHWIWA
      WJWLVAXAXDOPZXCXACOPZAOPUVTDABJWMWNCAOOWOWPWQXAXEOPZXCXAUWABOPUWBDABJWMWR
      CBOOWOWPWQWTWS $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x V $.  $d x W $.
    $( Dominance law for Cartesian product.  Theorem 6L(c) of [Enderton]
       p. 149.  (Contributed by Mario Carneiro, 26-Apr-2015.) $)
    xpdom2g $p |- ( ( C e. V /\ A ~<_ B ) ->
                  ( C X. A ) ~<_ ( C X. B ) ) $=
      ( vx wcel cdom wbr cxp cv wceq xpeq1 breq12d imbi2d vex xpdom2 vtoclg imp
      wi ) CDFABGHZCAIZCBIZGHZTEJZAIZUDBIZGHZSTUCSECDUDCKZUGUCTUHUEUAUFUBGUDCAL
      UDCBLMNABUDEOPQR $.

    $( Dominance law for Cartesian product.  Theorem 6L(c) of [Enderton]
       p. 149.  (Contributed by NM, 25-Mar-2006.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    xpdom1g $p |- ( ( C e. V /\ A ~<_ B ) ->
                  ( A X. C ) ~<_ ( B X. C ) ) $=
      ( wcel wbr wa cxp cen cvv reldom brrelex1i xpcomeng ancoms sylan2 xpdom2g
      cdom brrelex2i domentr syl2anc endomtr ) CDEZABQFZGZACHZCAHZIFZUFBCHZQFZU
      EUHQFUCUBAJEZUGABQKLUJUBUGACJDMNOUDUFCBHZQFUKUHIFZUIABCDPUCUBBJEULABQKRCB
      DJMOUFUKUHSTUEUFUHUAT $.

    $( A set is dominated by its Cartesian product with an inhabited set.
       Exercise 6 of [Suppes] p. 98.  (Contributed by Jim Kingdon,
       15-Apr-2020.) $)
    xpdom3m $p |- ( ( A e. V /\ B e. W /\ E. x x e. B ) ->
        A ~<_ ( A X. B ) ) $=
      ( wcel cv wex cxp cdom wbr wa w3a csn cen xpsneng 3adant2 ensymd cvv wss
      xpexg 3adant3 simp3 snssd xpss2 syl ssdomg endomtr syl2anc 3expia exlimdv
      sylc 3impia ) BDFZCEFZAGZCFZAHBBCIZJKZUNUOLUQUSAUNUOUQUSUNUOUQMZBBUPNZIZO
      KVBURJKZUSUTVBBUNUQVBBOKUOBUPDCPQRUTURSFZVBURTZVCUNUOVDUQBCDEUAUBUTVACTVE
      UTUPCUNUOUQUCUDVACBUEUFVBURSUGULBVBURUHUIUJUKUM $.
  $}

  ${
    xpdom1.2 $e |- C e. _V $.
    $( Dominance law for Cartesian product.  Theorem 6L(c) of [Enderton]
       p. 149.  (Contributed by NM, 28-Sep-2004.)  (Revised by NM,
       29-Mar-2006.)  (Revised by Mario Carneiro, 7-May-2015.) $)
    xpdom1 $p |- ( A ~<_ B -> ( A X. C ) ~<_ ( B X. C ) ) $=
      ( cvv wcel cdom wbr cxp xpdom1g mpan ) CEFABGHACIBCIGHDABCEJK $.
  $}

  ${
    $d A p q x y $.  $d A x y z $.  $d B x y z $.  $d C x y z $.  $d G x y $.
    $d S q x y $.  $d S x y z $.  $d ph x y $.
    pw2f1o.1 $e |- ( ph -> A e. V ) $.
    pw2f1o.2 $e |- ( ph -> B e. W ) $.
    pw2f1o.3 $e |- ( ph -> C e. W ) $.
    pw2f1o.4 $e |- ( ph -> B =/= C ) $.
    pw2f1odc.4 $e |- ( ph -> A. p e. A A. q e. ~P A DECID p e. q ) $.
    $( Lemma for ~ pw2f1odc .  (Contributed by Mario Carneiro, 6-Oct-2014.) $)
    pw2f1odclem $p |- ( ph ->
      ( ( S e. ~P A /\ G = ( z e. A |-> if ( z e. S , C , B ) ) ) <->
        ( G e. ( { B , C } ^m A ) /\ S = ( `' G " { C } ) ) ) ) $=
      ( vy wcel wceq wa vx wss cv cif cmpt cpr wf ccnv csn cima cpw cmap prid2g
      co syl ad2antrr prid1g wel wdc eleq2 dcbid wral weq ralbidv simpr rspcdva
      elequ1 simplr cvv wb ssexd elpwg mpbird ifcldcd fmpttd adantrr simprr cfv
      feq1d iftrue simplrl wne iffalse neeq1d syl5ibrcom a1d necon4bddc impbid2
      wi wn mpd simplrr fveq1d eqid eleq1w ifbid fvmptd3 eqeq1d bitr4d pm5.32da
      eqtrd sseld pm4.71rd wfn ffn fniniseg 3syl 3bitr4d eqrdv jca cdm cnvimass
      simprl ad2antrl sseqtrid eqsstrd dffn5im eleq2d baibd bitrd biimpa adantl
      fdm eqtr4d notbid ffvelcdmda elpri adantr ecased syldanl exmiddc mpjaodan
      wo mpteq2dva impbida elpw2g cbvmptv a1i eqeq2d anbi12d syl2anc elmapd
      prexg anbi1d ) AFCUBZGQCQUCZFRZEDUDZUEZSZTZCDEUFZGUGZFGUHEUIZUJZSZTZFCUKZ
      RZGBCBUCFRZEDUDZUEZSZTGUULCULUNRZUUPTAUUKUUQAUUKTZUUMUUPUVEUUMCUULUUIUGZA
      UUEUVFUUJAUUETZQCUUHUULUVGUUFCRZTZUUGEDUULAEUULRZUUEUVHAEIRZUVJNDEIUMUOZU
      PADUULRZUUEUVHADIRZUVMMDEIUQUOZUPUVIQJURZUSZUUGUSZJUURFJUCZFSZUVPUUGUVSFU
      UFUTVAUVIKJURZUSZJUURVBZUVQJUURVBKCUUFKQVCZUWBUVQJUURUWDUWAUVPKQJVGVAVDAU
      WCKCVBZUUEUVHPUPUVGUVHVEVFUVIUUSUUEAUUEUVHVHZUVIFVIRZUUSUUEVJZUVIFCHACHRZ
      UUEUVHLUPUWFVKFCVIVLZUOVMVFZVNVOVPUVECUULGUUIAUUEUUJVQVSVMZUVEUAFUUOUVEUA
      UCZCRZUWMFRZTUWNUWMGVRZESZTZUWOUWMUUORZUVEUWNUWOUWQUVEUWNTZUWOUWOEDUDZESZ
      UWQUWTUWOUXBUWOEDVTUWTUWOUSZUXBUWOWIUWTUAJURZUSZUXCJUURFUVTUXDUWOUVSFUWMU
      TVAUWTUWCUXEJUURVBKCUWMKUAVCZUWBUXEJUURUXFUWAUXDKUAJVGVAVDAUWEUUKUWNPUPUV
      EUWNVEZVFUWTUUSUUEAUUEUUJUWNWAZUWTUWGUWHUWTFCHAUWIUUKUWNLUPUXHVKUWJUOVMVF
      ZUWTUWOUXAEUWTUWOWJZUXAEWBZWIUXCUWTUXKUXJDEWBZAUXLUUKUWNOUPUXJUXADEUWOEDW
      CWDWEWFWGWKWHUWTUWPUXAEUWTUWPUWMUUIVRUXAUWTUWMGUUIAUUEUUJUWNWLWMUWTQUWMUU
      HUXACUUIUULUUIWNQUAVCUUGUWOEDQUAFWOWPUXGUWTUWOEDUULAUVJUUKUWNUVLUPAUVMUUK
      UWNUVOUPUXIVNWQXAWRWSWTUVEUWOUWNUVEFCUWMAUUEUUJXMXBXCUVEUUMGCXDZUWSUWRVJU
      WLCUULGXEZCEUWMGXFXGXHXIXJAUUQTZUUEUUJUXOFUUOCAUUMUUPVQUXOGXKZUUOCGUUNXLU
      UMUXPCSAUUPCUULGYCXNXOXPZUXOGQCUUFGVRZUEZUUIUXOUXMGUXSSUUMUXMAUUPUXNXNZQC
      GXQUOUXOQCUXRUUHUXOUVHTZUUGUXRUUHSUUGWJZUYAUUGTUXREUUHUYAUUGUXRESZUYAUUGU
      UFUUORZUYCUYAFUUOUUFAUUMUUPUVHWLXRUXOUYDUVHUYCUXOUXMUYDUVHUYCTVJUXTCEUUFG
      XFUOXSXTZYAUUGUUHESUYAUUGEDVTYBYDUYAUYBTZUXRDUUHUYFUXRDSZUYCUYAUYBUYCWJUY
      AUUGUYCUYEYEYAUYAUYGUYCYMZUYBUYAUXRUULRUYHUXOCUULUUFGAUUMUUPXMYFUXRDEYGUO
      YHYIUYBUUHDSUYAUUGEDWCYBYDUYAUVRUUGUYBYMAUUQUUEUVHUVRUXQUWKYJUUGYKUOYLYNX
      AXJYOAUUSUUEUVCUUJAUWIUWHLFCHYPUOAUVBUUIGUVBUUISABQCUVAUUHBQVCUUTUUGEDBQF
      WOWPYQYRYSYTAUVDUUMUUPAUULCGVIHAUVNUVKUULVIRMNDEIIUUCUUALUUBUUDXH $.

    pw2f1o.5 $e |- F = ( x e. ~P A |->
      ( z e. A |-> if ( z e. x , C , B ) ) ) $.
    $( The power set of a set is equinumerous to set exponentiation with an
       unordered pair base of ordinal 2.  Generalized from Proposition 10.44 of
       [TakeutiZaring] p. 96.  (Contributed by Mario Carneiro, 6-Oct-2014.) $)
    pw2f1odc $p |- ( ph -> F : ~P A -1-1-onto-> ( { B , C } ^m A ) ) $=
      ( vy wcel wa cpw cpr cmap co cif cmpt ccnv csn cima wceq eqid pw2f1odclem
      cv cvv biimpa mpanr2 simpld vex cnvex imaex a1i f1od ) ABRDUAZEFUBDUCUDZC
      DCUMBUMZSFEUEUFZRUMZUGZFUHZUIZGVDUNQAVEVCSZTVFVDSZVEVFUGVIUIUJZAVKVFVFUJZ
      VLVMTZVFUKAVKVNTVOACDEFVEVFHIJKLMNOPULUOUPUQVJUNSAVGVDSTVHVIVGRURUSUTVAAC
      DEFVEVGHIJKLMNOPULVB $.
  $}

  ${
    $d A a b $.  $d B a b $.  $d F a b $.
    $( Covering implies injection on power sets.  (Contributed by Stefan
       O'Rear, 6-Nov-2014.)  (Revised by Mario Carneiro, 24-Jun-2015.) $)
    fopwdom $p |- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A ) $=
      ( va vb cvv wcel wa cpw cv cima wss crn syl adantl imaeq2 elpwid foimacnv
      wceq wb wfo ccnv imassrn cdm dfdm4 fof fdm eqtr3id sseqtrid cnvexg adantr
      wf imaexg 3syl mpbird a1d simpllr simplrl syl2anc simplrr 3eqtr3d impbid1
      elpwg ex rnexg forn eleq1d syl5ibcom imp pwexg dmfex sylan2 dom3d ) CFGZA
      BCUAZHZDEBIZAIZCUBZDJZKZVSEJZKZFFVPWAVRGZVTVQGZVPWDWAALZVOWFVNVOVSMZWAAVS
      VTUCVOWGCUDZACUEVOABCULZWHASABCUFZABCUGNUHUIOVPVSFGZWAFGWDWFTVNWKVOCFUJUK
      VSVTFUMWAAFVCUNUOUPVPWEWBVQGZHZWAWCSZVTWBSZTVPWMHZWNWOWPWNWOWPWNHZCWAKZCW
      CKZVTWBWNWRWSSWPWAWCCPOWQVOVTBLWRVTSVNVOWMWNUQZWQVTBVPWEWLWNURQABVTCRUSWQ
      VOWBBLWSWBSWTWQWBBVPWEWLWNUTQABWBCRUSVAVDVTWBVSPVBVDVPBFGZVQFGVNVOXAVNCMZ
      FGVOXACFVEVOXBBFABCVFVGVHVIBFVJNVPAFGZVRFGVOVNWIXCWJABFCVKVLAFVJNVM $.
  $}

  $( Any set dominates the empty set.  (Contributed by NM, 26-Oct-2003.)
     (Revised by Mario Carneiro, 26-Apr-2015.) $)
  0domg $p |- ( A e. V -> (/) ~<_ A ) $=
    ( wcel c0 wss cdom wbr 0ss ssdomg mpi ) ABCDAEDAFGAHDABIJ $.

  ${
    $d A x $.
    $( A set dominated by the empty set is empty.  (Contributed by NM,
       22-Nov-2004.) $)
    dom0 $p |- ( A ~<_ (/) <-> A = (/) ) $=
      ( vx c0 cdom wbr cv wf1 brdomi wf wa f1f f00 sylib simprd adantl exlimddv
      wceq cvv wcel 0ex domrefg ax-mp breq1 mpbiri impbii ) ACDEZACQZUFACBFZGZU
      GBACBHUIUGUFUIUHCQZUGUIACUHIUJUGJACUHKAUHLMNOPUGUFCCDEZCRSUKTCRUAUBACCDUC
      UDUE $.
  $}

  ${
    0sdom.1 $e |- A e. _V $.
    $( Any set dominates the empty set.  (Contributed by NM, 26-Oct-2003.)
       (Revised by Mario Carneiro, 26-Apr-2015.) $)
    0dom $p |- (/) ~<_ A $=
      ( cvv wcel c0 cdom wbr 0domg ax-mp ) ACDEAFGBACHI $.
  $}

  $( Equality-like theorem for equinumerosity.  (Contributed by NM,
     18-Dec-2003.) $)
  enen1 $p |- ( A ~~ B -> ( A ~~ C <-> B ~~ C ) ) $=
    ( cen wbr ensym entr sylan impbida ) ABDEZACDEZBCDEZJBADEKLABFBACGHABCGI $.

  $( Equality-like theorem for equinumerosity.  (Contributed by NM,
     18-Dec-2003.) $)
  enen2 $p |- ( A ~~ B -> ( C ~~ A <-> C ~~ B ) ) $=
    ( cen wbr entr ancoms ensym sylan impbida ) ABDEZCADEZCBDEZLKMCABFGKBADEZML
    ABHMNLCBAFGIJ $.

  $( Equality-like theorem for equinumerosity and dominance.  (Contributed by
     NM, 8-Nov-2003.) $)
  domen1 $p |- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) ) $=
    ( cen wbr cdom ensym endomtr sylan impbida ) ABDEZACFEZBCFEZKBADELMABGBACHI
    ABCHJ $.

  $( Equality-like theorem for equinumerosity and dominance.  (Contributed by
     NM, 8-Nov-2003.) $)
  domen2 $p |- ( A ~~ B -> ( C ~<_ A <-> C ~<_ B ) ) $=
    ( cen wbr cdom domentr ancoms ensym sylan impbida ) ABDEZCAFEZCBFEZMLNCABGH
    LBADEZNMABINOMCBAGHJK $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Equinumerosity (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d s t u v w x y z A $.  $d s t u v w x y z C $.  $d s t u v w y z X $.
    $d u v w x z B $.  $d u v w y z D $.  $d s t u v w x z Y $.  $d u w z ph $.
    xpf1o.1 $e |- ( ph -> ( x e. A |-> X ) : A -1-1-onto-> B ) $.
    xpf1o.2 $e |- ( ph -> ( y e. C |-> Y ) : C -1-1-onto-> D ) $.
    $( Construct a bijection on a Cartesian product given bijections on the
       factors.  (Contributed by Mario Carneiro, 30-May-2015.) $)
    xpf1o $p |- ( ph -> ( x e. A , y e. C |-> <. X , Y >. ) :
        ( A X. C ) -1-1-onto-> ( B X. D ) ) $=
      ( vu vz vw vs vt wcel wral wceq wa vv c1st cfv csb c2nd cop cxp wreu cmpo
      cv wf1o xp1st adantl cmpt eqid f1ompt sylib simpld adantr nfcsb1v csbeq1a
      nfel1 eleq1d rspc sylc xp2nd opelxpi syl2anc ralrimiva weq wb wrex simprd
      r19.21bi reu6 anim12dan reeanv pm4.38 ex ralimdv impcom reximi sylbir syl
      com12 op1std csbeq1d eqeq2d op2ndd anbi12d eqeq1 bibi12d ralxp nfcv nfeq2
      vex nfv nfan nfbi nfralxy anbi2d opeq2 eqeq1d cbvral anbi1d opeq1 ralbidv
      bitrid bitr4i eqeq2 opth bitrdi bibi2d 2ralbidv sylibr ralrimivva reubidv
      rexxp nfop opeq12 syl2an cbvmpo opeq12d mpompt eqtr4i sylanbrc ) ABLUJZUB
      UCZHUDZCYGUEUCZIUDZUFZEGUGZQZLDFUGZRUAUJZYLSZLYOUHZUAYMRZYOYMBCDFHIUFZUIZ
      UKAYNLYOAYGYOQZTZYIEQZYKGQZYNUUCYHDQZHEQZBDRZUUDUUBUUFAYGDFULUMAUUHUUBAUU
      HMUJZHSZBDUHZMERZADEBDHUNZUKUUHUULTJBMDEHUUMUUMUOUPUQZURUSUUGUUDBYHDBYIEB
      YHHUTVBBUJZYHSHYIEBYHHVAVCVDVEUUCYJFQZIGQZCFRZUUEUUBUUPAYGDFVFUMAUURUUBAU
      URNUJZISZCFUHZNGRZAFGCFIUNZUKUURUVBTKCNFGIUVCUVCUOUPUQZURUSUUQUUECYJFCYKG
      CYJIUTVBCUJZYJSIYKGCYJIVAVCVDVEYIYKEGVGVHVIAUUIYISZUUSYKSZTZLYOUHZNGRMERY
      SAUVIMNEGAUUIEQZUUSGQZTTZUVHLUAVJZVKZLYORZUAYOVLZUVIUVLUUJUUTTZBOVJZCPVJZ
      TZVKZCFRZBDRZPFVLZODVLZUVPUVLUUJUVRVKZBDRZODVLZUUTUVSVKZCFRZPFVLZTZUWEAUV
      JUWHUVKUWKAUVJTUUKUWHAUUKMEAUUHUULUUNVMVNUUJBODVOUQAUVKTUVAUWKAUVANGAUURU
      VBUVDVMVNUUTCPFVOUQVPUWLUWGUWJTZPFVLZODVLUWEUWGUWJOPDFVQUWNUWDODUWMUWCPFU
      WJUWGUWCUWJUWFUWBBDUWFUWJUWBUWFUWIUWACFUWFUWIUWAUUJUUTUVRUVSVRVSVTWEVTWAW
      BWBWCWDUVOUWCUAOPDFUVOUVQUUOUVEUFZYPSZVKZCFRZBDRZYPOUJZPUJZUFZSZUWCUVOUUI
      BUWTHUDZSZUUSCUXAIUDZSZTZUXBYPSZVKZPFRZODRUWSUVNUXJLOPDFYGUXBSZUVHUXHUVMU
      XIUXLUVFUXEUVGUXGUXLYIUXDUUIUXLBYHUWTHUWTUXAYGOWPZPWPZWFWGWHUXLYKUXFUUSUX
      LCYJUXAIUWTUXAYGUXMUXNWIWGWHWJYGUXBYPWKWLWMUWRUXKBODUWROWQUXJBPFBFWNUXHUX
      IBUXEUXGBBUUIUXDBUWTHUTWOUXGBWQWRUXIBWQWSWTUWRUUJUXGTZUUOUXAUFZYPSZVKZPFR
      UVRUXKUWQUXRCPFUWQPWQUXOUXQCUUJUXGCUUJCWQCUUSUXFCUXAIUTWOWRUXQCWQWSUVSUVQ
      UXOUWPUXQUVSUUTUXGUUJUVSIUXFUUSCUXAIVAWHXAUVSUWOUXPYPUVEUXAUUOXBXCWLXDUVR
      UXRUXJPFUVRUXOUXHUXQUXIUVRUUJUXEUXGUVRHUXDUUIBUWTHVAWHXEUVRUXPUXBYPUUOUWT
      UXAXFXCWLXGXHXDXIUXCUWQUWABCDFUXCUWPUVTUVQUXCUWPUWOUXBSUVTYPUXBUWOXJUUOUV
      EUWTUXABWPCWPXKXLXMXNXHXRXOUVHLUAYOVOXOXPYRUVIUAMNEGYPUUIUUSUFZSZYQUVHLYO
      UXTYQUXSYLSUVHYPUXSYLWKUUIUUSYIYKMWPZNWPZXKXLXQWMXOLUAYOYMYLUUAUUAMNDFBUU
      IHUDZCUUSIUDZUFZUILYOYLUNBCMNDFYTUYEMYTWNNYTWNBUYCUYDBUUIHUTBUYDWNXSCUYCU
      YDCUYCWNCUUSIUTXSBMVJHUYCSIUYDSYTUYESCNVJBUUIHVACUUSIVAHIUYCUYDXTYAYBMNLD
      FYLUYEYGUXSSZYIUYCYKUYDUYFBYHUUIHUUIUUSYGUYAUYBWFWGUYFCYJUUSIUUIUUSYGUYAU
      YBWIWGYCYDYEUPYF $.
  $}

  ${
    $d A f g x y $.  $d B f g x $.  $d C f g x y $.  $d D f g y $.
    $( Equinumerosity law for Cartesian product.  Proposition 4.22(b) of
       [Mendelson] p. 254.  (Contributed by NM, 24-Jul-2004.) $)
    xpen $p |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) ~~ ( B X. D ) ) $=
      ( vf vg vx vy cen wbr wa cv wf1o cxp wex bren biimpi cvv wcel cfv cmpo wb
      adantr ad2antlr cop relen brrelex1i xpexg syl2an ad2antrr cmpt simplr wfn
      wceq f1ofn dffn5im f1oeq1 3syl mpbid simpr xpf1o f1oeng syl2anc exlimddv
      syl ) ABIJZCDIJZKZABELZMZACNZBDNZIJZEVFVJEOZVGVFVNABEPQUCVHVJKZCDFLZMZVMF
      VGVQFOZVFVJVGVRCDFPQUDVOVQKZVKRSZVKVLGHACGLVITZHLVPTZUEUAZMVMVHVTVJVQVFAR
      SCRSVTVGABIUFUGCDIUFUGACRRUHUIUJVSGHABCDWAWBVSVJABGAWAUKZMZVHVJVQULZVSVJV
      IWDUNZVJWEUBWFVJVIAUMWGABVIUOGAVIUPVEABVIWDUQURUSVSVQCDHCWBUKZMZVOVQUTZVS
      VQVPWHUNZVQWIUBWJVQVPCUMWKCDVPUOHCVPUPVECDVPWHUQURUSVAVKVLRWCVBVCVDVD $.
  $}

  ${
    $d f g x y A $.  $d f g x y B $.  $d f g x y C $.  $d f g x y D $.
    $( Two set exponentiations are equinumerous when their bases and exponents
       are equinumerous.  Theorem 6H(c) of [Enderton] p. 139.  (Contributed by
       NM, 16-Dec-2003.)  (Proof shortened by Mario Carneiro, 26-Apr-2015.) $)
    mapen $p |- ( ( A ~~ B /\ C ~~ D ) -> ( A ^m C ) ~~ ( B ^m D ) ) $=
      ( vf vg cv wf1o wex cmap wa ccom cvv wcel wceq adantr eqeltrrdi adantl wf
      syl vx vy cen wbr bren eeanv ccnv cxp wfn fnmap cdm f1odm vex dmex fnovex
      co mp3an2i crn wfo f1ofo forn rnex elmapi f1of fco sylan f1ocnv fcod syl5
      ex elmapd sylibrd id syl2anr syl2an2r coass cid f1ococnv2 ad2antrr coeq1d
      cres adantrl fcoi2 eqtrd eqtr3id eqeq2d f1ococnv1 ad2antlr coeq2d adantrr
      fcoi1 eqtrid eqcom bitrdi bitr4d wf1 f1of1 simprl cocan1 syl3anc ad2antll
      wb ffn ffnd cocan2 3bitr3d syl2ani en3d exlimivv sylbir syl2anb ) ABUCUDA
      BEGZHZEIZCDFGZHZFIZACJUPZBDJUPZUCUDZCDUCUDABEUECDFUEXNXQKXMXPKZFIEIXTXMXP
      EFUFYAXTEFYAUAUBXRXSXLUAGZLZXOUGZLZXLUGZUBGZXOLZLZMMJMMUHUIZYAAMNCMNXRMNU
      JYAAXLUKZMXMYKAOXPABXLULPXLEUMZUNQZYACXOUKZMXPYNCOXMCDXOULRXOFUMZUNQZACMM
      JUOUQYJYABMNDMNXSMNUJYABXLURZMYAABXLUSZYQBOXMYRXPABXLUTPABXLVATXLYLVBQZYA
      DXOURZMYACDXOUSZYTDOXPUUAXMCDXOUTRZCDXOVATXOYOVBQZBDMMJUOUQYAYBXRNZDBYESZ
      YEXSNUUDCAYBSZYAUUEYBACVCZYAUUFUUEYAUUFKDCBYCYDYAABXLSZUUFCBYCSZXMUUHXPAB
      XLVDPCABXLYBVEVFZYADCYDSZUUFYADCYDHZUUKXPUULXMCDXOVGRDCYDVDTPVHZVJVIYABDY
      EMMYSUUCVKVLYAYGXSNZCAYISZYIXRNUUNDBYGSZYAUUOYGBDVCZYAUUPUUOYABAYFSZUUPCB
      YHSZUUOYABAYFHZUURXMUUTXPABXLVGPBAYFVDTUUPUUPCDXOSZUUSYAUUPVMXPUVAXMCDXOV
      DRCDBYGXOVEVNZCBAYFYHVEVOZVJVIYAACYIMMYMYPVKVLUUDYAUUFUUPYBYIOZYGYEOZXBZU
      UNUUGUUQYAUUFUUPKZUVFYAUVGKZYCXLYILZOZYHYEXOLZOZUVDUVEUVHUVJYCYHOZUVLUVHU
      VIYHYCUVHUVIXLYFLZYHLZYHXLYFYHVPUVHUVOVQBWAZYHLZYHUVHUVNUVPYHXMUVNUVPOXPU
      VGABXLVRVSVTUVHUUSUVQYHOYAUUPUUSUUFUVBWBCBYHWCTWDWEWFUVHUVLYHYCOUVMUVHUVK
      YCYHUVHUVKYCYDXOLZLZYCYCYDXOVPUVHUVSYCVQCWAZLZYCUVHUVRUVTYCXPUVRUVTOXMUVG
      CDXOWGWHWIUVHUUIUWAYCOYAUUFUUIUUPUUJWJCBYCWKTWDWLWFYHYCWMWNWOUVHABXLWPZUU
      FUUOUVJUVDXBXMUWBXPUVGABXLWQVSYAUUFUUPWRYAUUPUUOUUFUVCWBCABXLYBYIWSWTUVHU
      UAYGDUIZYEDUIUVLUVEXBYAUUAUVGUUBPUUPUWCYAUUFDBYGXCXAUVHDBYEYAUUFUUEUUPUUM
      WJXDCDXOYGYEXEWTXFVJXGXHXIXJXK $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x V $.
    $( Order-preserving property of set exponentiation.  (Contributed by Jim
       Kingdon, 15-Jul-2022.) $)
    mapdom1g $p |- ( ( A ~<_ B /\ C e. V ) -> ( A ^m C ) ~<_ ( B ^m C ) ) $=
      ( vx cdom wbr wcel wa cv cen wss cmap co wex cvv wb syl adantr syl2anc wi
      reldom brrelex2i domeng simpl enrefg adantl mapen syl2anr ad2antrr simprr
      ibi mapss cxp wfn fnmap elex fnovex mp3an3an ssdomg mpd endomtr exlimddv
      ) ABFGZCDHZIZAEJZKGZVGBLZIZACMNZBCMNZFGZEVDVJEOZVEVDVNVDBPHZVDVNQABFUBUCZ
      EABPUDRULSVFVJIZVKVGCMNZKGZVRVLFGZVMVJVHCCKGZVSVFVHVIUEVEWAVDCDUFUGAVGCCU
      HUIVQVRVLLZVTVQVOVIWBVDVOVEVJVPUJVFVHVIUKVGBCPUMTVFWBVTUAZVJVFVLPHZWCMPPU
      NUOVDVOVECPHWDUPVPCDUQBCPPMURUSVRVLPUTRSVAVKVRVLVBTVC $.
  $}

  ${
    $d f g x y A $.  $d f g x y B $.  $d f g x y C $.  $d f g x y V $.
    $d f g x y W $.  $d f g x y X $.
    $( Equinumerosity law for double set exponentiation.  Proposition 10.45 of
       [TakeutiZaring] p. 96.  (Contributed by NM, 21-Feb-2004.)  (Revised by
       Mario Carneiro, 24-Jun-2015.) $)
    mapxpen $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
      ( ( A ^m B ) ^m C ) ~~ ( A ^m ( B X. C ) ) ) $=
      ( vx vy wcel cmap co cv cfv cmpt cvv wf wral wa eqid wceq vf w3a cxp cmpo
      vg wfn fnmap elex 3ad2ant1 3ad2ant2 fnovex mp3an2i 3ad2ant3 xpexg 3adant1
      elmapi ffvelcdmda syl an32s ralrimiva sylib simp1 elmapd imbitrrid adantl
      fmpo fovcdm 3expa sylanl1 fmpttd wb elmapg 3adant3 ad2antrr simp3 sylibrd
      mpbird ex elmapfn ad2antll fnovim adantlrl 3adant2 simp1l2 simp1l1 fvmpt2
      fex2 syl3anc syl2anc fveq1d simp2 vex ovexg mp3an sylancl eqtrd mpoeq3dva
      eqtr4d nfmpt1 nfmpt nfeq2 fveq1 a1d ralrimi jctil mpoeq123 sylancr eqeq2d
      nfcv syl5ibrcom ad2antrl feqmptd simprl sylan mpteq2dva nfmpo2 nfmpo1 nfv
      eqidd fvex ovmpt4g mp3an3 oveq eqeq1d expcomd ralrimd mpteq12 syl6an en3d
      impbid ) ADIZBEIZCFIZUBZUAUEABJKZCJKZABCUCZJKZGHBCGLZHLZUALZMZMZUDZHCGBYS
      YTUELZKZNZNZOOJOOUCUFZYNYOOIZCOIZYPOIUGUUIYNAOIZBOIZUUJUGYKYLUULYMADUHUIZ
      YLYKUUMYMBEUHUJABOOJUKULZYMYKUUKYLCFUHUMYOCOOJUKULUUIYNUULYQOIZYROIUGUUNY
      LYMUUPYKBCEFUNUOZAYQOOJUKULUUAYPIZUUDYRIYNYQAUUDPZUURUUCAIZHCQZGBQUUSUURU
      VAGBUURYSBIZRUUTHCUURYTCIZUVBUUTUURUVCRZBAYSUUBUVDUUBYOIBAUUBPZUURCYOYTUU
      AUUAYOCUPZUQUUBABUPURZUQUSUTUTGHBCUUCAUUDUUDSZVFVAYNAYQUUDDOYKYLYMVBUUQVC
      VDYNUUEYRIZCYOUUHPZUUHYPIYNUVIUVJYNUVIRZHCUUGYOUVKUVCRZUUGYOIZBAUUGPZUVLG
      BUUFAUVKYQAUUEPZUVCUVBUUFAIZUVIUVOYNUUEAYQUPVEUVOUVBUVCUVPUVOUVBUVCUVPYSY
      TABCUUEVGVHUSVIVJZYNUVMUVNVKZUVIUVCYKYLUVRYMABUUGDEVLVMVNVQVJVRYNYOCUUHOF
      UUOYKYLYMVOVCVPYNUURUVIRZUUAUUHTZUUEUUDTZVKYNUVSRZUVTUWAUWBUWAUVTUUEGHBCY
      SYTUUHMZMZUDZTUWBUUEGHBCUUFUDZUWEUWBUUEYQUFZUUEUWFTUVIUWGYNUURUUEAYQVSVTG
      HBCUUEWAURUWBGHBCUWDUUFUWBUVBUVCUBZUWDYSUUGMZUUFUWHYSUWCUUGUWHUVCUUGOIZUW
      CUUGTUWBUVBUVCVOUWHUVNYLYKUWJUWBUVCUVNUVBYNUVIUVCUVNUURUVQWBWCYKYLYMUVSUV
      BUVCWDYKYLYMUVSUVBUVCWEBAUUGEDWGWHHCUUGOUUHUUHSWFWIWJUWHUVBUUFOIZUWIUUFTU
      WBUVBUVCWKYSOIUUEOIYTOIUWKGWLZUEWLHWLZYSYTUUEOOOWMWNGBUUFOUUGUUGSWFWOWPWQ
      WRUVTUUDUWEUUEUVTBBTZCCTZUUCUWDTZHCQZRZGBQUUDUWETBSUVTUWRGBGUUAUUHGHCUUGG
      CXIGBUUFWSWTXAUVTUWRUVBUVTUWQUWOUVTUWPHCHUUAUUHHCUUGWSXAUVTUWPUVCUVTYSUUB
      UWCYTUUAUUHXBWJXCXDCSZXEXCXDGHBCUUCBCUWDXFXGXHXJUWBUVTUWAUUAHCGBUUCNZNZTU
      WBUUAHCUUBNUXAUWBHCYOUUAUURCYOUUAPYNUVIUVFXKXLUWBHCUUBUWTUWBUVCRGBAUUBUWB
      UURUVCUVEYNUURUVIXMUVGXNXLXOWPUWAUUHUXAUUAUWAUWOUUGUWTTZHCQUUHUXATUWSUWAU
      XBHCHUUEUUDGHBCUUCXPXAUWAUWNUVCUUFUUCTZGBQUXBUWABXSUWAUVCUXCGBGUUEUUDGHBC
      UUCXQXAUVCGXRUWAUVBUVCUXCUVBUVCRUXCUWAYSYTUUDKZUUCTZUVBUVCUUCOIUXEYSUUBOO
      YTUUAOOUAWLUWMXTUWLXTGHBCUUCUUDOUVHYAYBUWAUUFUXDUUCYSYTUUEUUDYCYDVDYEYFGB
      UUFBUUCYGYHXDHCUUGCUWTYGXGXHXJYJVRYI $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z w C $.
    xpmapen.1 $e |- A e. _V $.
    xpmapen.2 $e |- B e. _V $.
    xpmapen.3 $e |- C e. _V $.
    ${
      $d y z D $.  $d y z R $.  $d x z S $.
      xpmapenlem.4 $e |- D = ( z e. C |-> ( 1st ` ( x ` z ) ) ) $.
      xpmapenlem.5 $e |- R = ( z e. C |-> ( 2nd ` ( x ` z ) ) ) $.
      xpmapenlem.6 $e |- S = ( z e. C |->
               <. ( ( 1st ` y ) ` z ) , ( ( 2nd ` y ) ` z ) >. ) $.
      $( Lemma for ~ xpmapen .  (Contributed by NM, 1-May-2004.)  (Revised by
         Mario Carneiro, 16-Nov-2014.) $)
      xpmapenlem $p |- ( ( A X. B ) ^m C ) ~~ ( ( A ^m C ) X. ( B ^m C ) ) $=
        ( cmap cvv wcel cfv wceq cxp co cop wfn fnmap xpex fnovex mp3an cv c1st
        wf wa elmap ffvelcdm sylanb xp1st syl fmptd sylibr c2nd opelxpi syl2anc
        xp2nd ffvelcdmda 1st2nd2 ad2antlr cmpt feqmptd simplr fveq1d vex 1stexg
        sylib ax-mp fvex 2ndexg opex fvmpt2 mpan2 adantl eqtrd fveq2d mpteq2dva
        op1st eqtrdi eqtrid eqtr4d op2nd opeq12d simpll simpr ad2antrr sylan9eq
        op1stg op2ndg impbida en3i ) ABDEUAZFPUBZDFPUBZEFPUBZUAZGHUCZIPQQUAUDZW
        RQRFQRZWSQRUEDEJKUFZLWRFQQPUGUHWTXAXDDQRXEWTQRUEJLDFQQPUGUHXDEQRXEXAQRU
        EKLEFQQPUGUHUFAUIZWSRZGWTRZHXARZXCXBRXHFDGUKXIXHCFCUIZXGSZUJSZDGXHXKFRZ
        ULZXLWRRZXMDRXHFWRXGUKZXNXPWRFXGXFLUMZFWRXKXGUNUOZXLDEUPUQMURDFGJLUMUSZ
        XHFEHUKXJXHCFXLUTSZEHXOXPYAERXSXLDEVCUQNUREFHKLUMUSZGHWTXAVAVBBUIZXBRZF
        WRIUKIWSRYDCFXKYCUJSZSZXKYCUTSZSZUCZWRIYDXNULYFDRYHERYIWRRYDFDXKYEYDYEW
        TRFDYEUKYCWTXAUPDFYEJLUMVMZVDYDFEXKYGYDYGXARFEYGUKYCWTXAVCEFYGKLUMVMZVD
        YFYHDEVAVBOURWRFIXFLUMUSXHYDULZXGITZYCXCTZYLYMULZYCYEYGUCZXCYDYCYPTXHYM
        YCWTXAVEVFYOYEGYGHYOYECFYFVGZGYDYEYQTXHYMYDCFDYEYJVHVFYOGCFXMVGYQMYOCFX
        MYFYOXNULZXMYIUJSYFYRXLYIUJYRXLXKISZYIYRXKXGIYLYMXNVIVJXNYSYITZYOXNYIQR
        YTYFYHXKYEQQYCQRZYEQRBVKZYCQVLVNCVKZVOZXKYGQQUUAYGQRUUBYCQVPVNUUCVOZVQC
        FYIQIOVRVSVTWAZWBYFYHUUDUUEWDWEWCWFWGYOYGCFYHVGZHYDYGUUGTXHYMYDCFEYGYKV
        HVFYOHCFYAVGUUGNYOCFYAYHYRYAYIUTSYHYRXLYIUTUUFWBYFYHUUDUUEWHWEWCWFWGWIW
        AYLYNULZXGCFXLVGZIUUHCFWRXGUUHXHXQXHYDYNWJXRVMZVHUUHICFYIVGUUIOUUHCFYIX
        LUUHXNULZYIXMYAUCZXLUUKYFXMYHYAUUHXNYFXKGSZXMUUHXKYEGUUHYEXCUJSZGUUHYCX
        CUJYLYNWKZWBUUHXIXJUUNGTXHXIYDYNXTWLZXHXJYDYNYBWLZGHWTXAWNVBWAVJXNXMQRZ
        UUMXMTXLQRZUURXKXGQQAVKUUCVOZXLQVLVNCFXMQGMVRVSWMUUHXNYHXKHSZYAUUHXKYGH
        UUHYGXCUTSZHUUHYCXCUTUUOWBUUHXIXJUVBHTUUPUUQGHWTXAWOVBWAVJXNYAQRZUVAYAT
        UUSUVCUUTXLQVPVNCFYAQHNVRVSWMWIUUKXPXLUULTUUHFWRXKXGUUJVDXLDEVEUQWGWCWF
        WGWPWQ $.
    $}

    $( Equinumerosity law for set exponentiation of a Cartesian product.
       Exercise 4.47 of [Mendelson] p. 255.  (Contributed by NM, 23-Feb-2004.)
       (Proof shortened by Mario Carneiro, 16-Nov-2014.) $)
    xpmapen $p |- ( ( A X. B ) ^m C ) ~~ ( ( A ^m C ) X. ( B ^m C ) ) $=
      ( vx vy vz vw cv cfv c1st cmpt c2nd cop weq fveq2 fveq2d cbvmptv opeq12d
      xpmapenlem ) GHIABCJCJKZGKZLZMLZNJCUEOLZNJCUCHKZMLZLZUCUHOLZLZPZNDEFJICUF
      IKZUDLZMLJIQZUEUOMUCUNUDRZSTJICUGUOOLUPUEUOOUQSTJICUMUNUILZUNUKLZPUPUJURU
      LUSUCUNUIRUCUNUKRUATUB $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y V $.  $d x y W $.
    $d x y X $.
    $( Equinumerosity law for set exponentiation of a disjoint union.  Exercise
       4.45 of [Mendelson] p. 255.  (Contributed by NM, 23-Sep-2004.)  (Revised
       by Mario Carneiro, 29-Apr-2015.) $)
    mapunen $p |- ( ( ( A e. V /\ B e. W /\ C e. X ) /\ ( A i^i B ) = (/) ) ->
                  ( C ^m ( A u. B ) ) ~~ ( ( C ^m A ) X. ( C ^m B ) ) ) $=
      ( vx wcel wceq wa cun cmap co cres cop cvv fnmap elexd fnovex wf vy c0 cv
      w3a cin cxp c1st cfv c2nd simpl3 simpl1 simpl2 unexd mp3an2i xpexd elmapi
      wfn wss ssun1 fssres sylancl ssun2 opelxp elmapd anbi12d bitrid imbitrrid
      jca xp1st adantl xp2nd simplr fun2d ex sylibrd wb 1st2nd2 fresaunres1disj
      syl ad2antll adantrl syl3anc fresaunres2disj opeq12d eqtr4d reseq1 eqeq2d
      syl5ibrcom ffn fnresdm 3syl ad2antrl eqcomd op1std op2ndd uneq12d resundi
      vex resex eqtr4di impbid en3d ) ADHZBEHZCFHZUDZABUEUBIZJZGUACABKZLMZCALMZ
      CBLMZUFZGUCZANZXNBNZOZUAUCZUGUHZXRUIUHZKZPPLPPUFUQZXHCPHZXIPHXJPHQXHCFXCX
      DXEXGUJZRZXHABDEXCXDXEXGUKZXCXDXEXGULZUMZCXIPPLSUNXHXKXLPPYBXHYCAPHXKPHQY
      EXHADYFRCAPPLSUNYBXHYCBPHXLPHQYEXHBEYGRCBPPLSUNUOXNXJHZXQXMHZXHACXOTZBCXP
      TZJZYIYKYLYIXICXNTZAXIURYKXNCXIUPZABUSXICAXNUTVAYIYNBXIURYLYOBAVBXICBXNUT
      VAVHYJXOXKHZXPXLHZJXHYMXOXPXKXLVCXHYPYKYQYLXHCAXOFDYDYFVDXHCBXPFEYDYGVDVE
      VFVGXHXRXMHZXICYATZYAXJHXHYRYSXHYRJZABCXSXTYTXSXKHZACXSTZYRUUAXHXRXKXLVIV
      JXSCAUPVSZYTXTXLHZBCXTTZYRUUDXHXRXKXLVKVJXTCBUPVSZXFXGYRVLVMVNXHCXIYAFPYD
      YHVDVOXHYIYRJZXNYAIZXRXQIZVPXHUUGJZUUHUUIUUJUUIUUHXRYAANZYABNZOZIUUJXRXSX
      TOZUUMYRXRUUNIXHYIXRXKXLVQVTUUJUUKXSUULXTUUJUUBUUEXGUUKXSIXHYRUUBYIUUCWAZ
      XHYRUUEYIUUFWAZXFXGUUGVLZABCXSXTVRWBUUJUUBUUEXGUULXTIUUOUUPUUQABCXSXTWCWB
      WDWEUUHXQUUMXRUUHXOUUKXPUULXNYAAWFXNYABWFWDWGWHUUJUUHUUIXNXNXINZIUUJUURXN
      YIUURXNIZXHYRYIYNXNXIUQUUSYOXICXNWIXIXNWJWKWLWMUUIYAUURXNUUIYAXOXPKUURUUI
      XSXOXTXPXOXPXRXNAGWRZWSZXNBUUTWSZWNXOXPXRUVAUVBWOWPXNABWQWTWGWHXAVNXB $.
  $}

  ${
    $d x y z f A $.  $d x y z f B $.  $d x y z f C $.
    $( Equinumerosity of equinumerous subsets of a set.  (Contributed by NM,
       30-Sep-2004.)  (Revised by Mario Carneiro, 16-Nov-2014.) $)
    ssenen $p |- ( A ~~ B ->
             { x | ( x C_ A /\ x ~~ C ) } ~~ { x | ( x C_ B /\ x ~~ C ) } ) $=
      ( vf cen wbr cv cab cin wss wa cima cvv wcel wceq syl sylan elin elpw cpw
      vy vz wf1o wex bren ccnv cdm f1odm vex dmex eqeltrrdi inex1g 3syl crn wfo
      pwexg f1ofo forn rnex wf1 f1of1 adantr simpr f1imaen2g syl22anc entr expl
      a1i imassrn sseqtrid jctild breq1 elab anbi12i bitri imaex 3imtr4g f1ocnv
      wi f1f1orn f1imaen cnvex wb simpl elpwid sylbi imaeq2 f1orel dfrel2 sylib
      imaeq1d f1imacnv eqtr3d sylan9eqr eqcomd ex sylan2 adantrl adantrr impbid
      wrel en3d exlimiv df-pw ineq1i inab eqtri 3brtr3g ) BCFGZBUAZAHZDFGZAIZJZ
      CUAZXNJZXLBKZXMLAIZXLCKZXMLAIZFXJBCEHZUDZEUEXOXQFGZBCEUFYCYDEYCUBUCXOXQYB
      UBHZMZYBUGZUCHZMZNNYCBNOXKNOXONOYCBYBUHNBCYBUIYBEUJZUKULBNUQXKXNNUMUNYCCN
      OZXPNOXQNOYCCYBUOZNYCBCYBUPZYLCPBCYBURZBCYBUSZQYBYJUTULZCNUQXPXNNUMUNYCYE
      BKZYEDFGZLZYFCKZYFDFGZLZYEXOOZYFXQOZYCYSUUAYTYCYQYRUUAYCYQLZYFYEFGZYRUUAU
      UEBCYBVAZYKYQYENOZUUFYCUUGYQBCYBVBZVCYCYKYQYPVCYCYQVDUUHUUEUBUJZVIBCYEYBN
      VEVFYFYEDVGRVHYCYMYTYNYMYLYFCYBYEVJYOVKQVLUUCYEXKOZYEXNOZLZYSYEXKXNSZUUKY
      QUULYRYEBUUJTXMYRAYEUUJXLYEDFVMVNVOVPUUDYFXPOZYFXNOZLUUBYFXPXNSUUOYTUUPUU
      AYFCYBYEYJVQZTXMUUAAYFUUQXLYFDFVMVNVOVPVRYCYHCKZYHDFGZLZYIBKZYIDFGZLZYHXQ
      OZYIXOOZYCCBYGUDZUUTUVCVTBCYBVSZUVFUUTUVBUVAUVFUURUUSUVBUVFUURLYIYHFGZUUS
      UVBUVFCYGUOZYGVAZUURUVHUVFCBYGVAZCUVIYGUDUVJCBYGVBZCBYGWACUVIYGVBUNCUVIYH
      YGUCUJZWBRYIYHDVGRVHUVFCBYGUPZUVACBYGURUVNUVIYIBYGYHVJCBYGUSVKQVLQUVDYHXP
      OZYHXNOZLZUUTYHXPXNSZUVOUURUVPUUSYHCUVMTXMUUSAYHUVMXLYHDFVMVNVOVPUVEYIXKO
      ZYIXNOZLUVCYIXKXNSUVSUVAUVTUVBYIBYGYHYBYJWCVQZTXMUVBAYIUWAXLYIDFVMVNVOVPV
      RYCUUCUVDLZYEYIPZYHYFPZWDYCUWBLUWCUWDYCUVDUWCUWDVTZUUCUVDYCUURUWEUVDUVQUU
      RUVRUVQYHCUVOUVPWEWFWGYCUURLZUWCUWDUWFUWCLYFYHUWCUWFYFYBYIMZYHYEYIYBWHUWF
      YGUGZYIMZUWGYHYCUWIUWGPUURYCUWHYBYIYCYBXBUWHYBPBCYBWIYBWJWKWLVCYCUVKUURUW
      IYHPYCUVFUVKUVGUVLQCBYHYGWMRWNWOWPWQWRWSYCUUCUWDUWCVTZUVDUUCYCYQUWJUUCUUM
      YQUUNUUMYEBUUKUULWEWFWGUUEUWDUWCUUEUWDLYIYEUWDUUEYIYGYFMZYEYHYFYGWHYCUUGY
      QUWKYEPUUIBCYEYBWMRWOWPWQWRWTXAWQXCXDWGXOXRAIZXNJXSXKUWLXNABXEXFXRXMAXGXH
      XQXTAIZXNJYAXPUWMXNACXEXFXTXMAXGXHXI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Pigeonhole Principle
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Lemma for Pigeonhole Principle.  If we join a natural number to itself
     minus an element, we end up with its successor minus the same element.
     (Contributed by NM, 25-May-1998.) $)
  phplem1 $p |- ( ( A e. _om /\ B e. A ) ->
             ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) ) $=
    ( com wcel wa csn cin c0 wceq cdif cun csuc word nnord nordeq disjsn2 sylan
    wne syl undif4 df-suc equncomi difeq1i eqtr4di ) ACDZBADZEAFZBFZGHIZUGAUHJK
    ZALZUHJZIUEAMZUFUIANUMUFEABRUIABOABPSQUIUJUGAKZUHJULUGAUHTUKUNUHUKAUGAUAUBU
    CUDS $.

  ${
    $d f A $.  $d f B $.
    phplem2.1 $e |- A e. _V $.
    phplem2.2 $e |- B e. _V $.
    $( Lemma for Pigeonhole Principle.  A natural number is equinumerous to its
       successor minus one of its elements.  (Contributed by NM, 11-Jun-1998.)
       (Revised by Mario Carneiro, 16-Nov-2014.) $)
    phplem2 $p |- ( ( A e. _om /\ B e. A ) -> A ~~ ( suc A \ { B } ) ) $=
      ( com wcel wa csn cdif cun csuc cen wbr cin c0 wceq cvv wss syl eqtrid
      cop wf1o opex snex f1osn mp2an difss ssexi enref pm3.2i incom ssrin ax-mp
      f1oen3g word nnord orddisj sseqtrid ss0 disjdif jctil unen sylancr adantr
      uncom nndifsnid phplem1 3brtr3d ) AEFZBAFZGZBHZAVLIZJZAHZVMJZAAKVLILVIVNV
      PLMZVJVIVLVOLMZVMVMLMZGVLVMNOPZVOVMNZOPZGVQVRVSBAUAZHZQFVLVOWDUBVRWCBADCU
      CUDBADCUEVLVOWDQUNUFVMVMACAVLUGZUHUIUJVIWBVTVIWAVMVONZOVOVMUKVIWFORWFOPVI
      AVONZWFOVMARWFWGRWEVMAVOULUMVIAUOWGOPAUPAUQSURWFUSSTVLAUTVAVLVOVMVMVBVCVD
      VKVNVMVLJAVLVMVEABVFTABVGVH $.

    $( Lemma for Pigeonhole Principle.  A natural number is equinumerous to its
       successor minus any element of the successor.  For a version without the
       redundant hypotheses, see ~ phplem3g .  (Contributed by NM,
       26-May-1998.) $)
    phplem3 $p |- ( ( A e. _om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) $=
      ( csuc wcel com wceq wo csn cdif cen wbr elsuci phplem2 enref word orddif
      wa nnord syl sneq difeq2d eqcoms sylan9eq breqtrid jaodan sylan2 ) BAEZFA
      GFZBAFZBAHZIAUIBJZKZLMZBANUJUKUOULABCDOUJULSAAUNLACPUJULAUIAJZKZUNUJAQAUQ
      HATARUAUQUNHABABHUPUMUIABUBUCUDUEUFUGUH $.

    $( Lemma for Pigeonhole Principle.  Equinumerosity of successors implies
       equinumerosity of the original natural numbers.  (Contributed by NM,
       28-May-1998.)  (Revised by Mario Carneiro, 24-Jun-2015.) $)
    phplem4 $p |- ( ( A e. _om /\ B e. _om )
                    -> ( suc A ~~ suc B -> A ~~ B ) ) $=
      ( csuc cen wbr com wcel csn cdif cima cvv sylancl ensymd wceq syl imaeq2d
      vf wa cv wf1o wex bren cfv wf1 f1of1 adantl wss sssucid f1imaen2g mpanr12
      sucex word nnord orddif wfn f1ofn sucid fnsnfv difeq2d crn imadmrn eqcomi
      cdm wfo f1ofo forn f1odm 3eqtr3a difeq1d ccnv wfun dff1o3 imadif 3eqtr4rd
      simprbi sylan9eq breqtrd fnfvelrn wb eleq2d mpbid vex fvex phplem3 sylan2
      entr syl2an anandirs ex exlimdv biimtrid ) AEZBEZFGWNWOSUAZUBZSUCAHIZBHIZ
      TZABFGZWNWOSUDWTWQXASWTWQXAWRWSWQXAWRWQTZAWOAWPUEZJZKZFGXEBFGXAWSWQTZXBAW
      PALZXEFXBXGAXBWNWOWPUFZWOMIZXGAFGZWQXHWRWNWOWPUGUHBDUMXHXITAWNUIAMIXJAUJC
      WNWOAWPMUKULNOWRWQXGWPWNAJZKZLZXEWRAXLWPWRAUNAXLPAUOAUPQRWQWPWNLZXDKXNWPX
      KLZKZXEXMWQXDXOXNWQWPWNUQZAWNIZXDXOPWNWOWPURZACUSZWNAWPUTNVAWQWOXNXDWQWPV
      BZWPWPVEZLZWOXNYCYAWPVCVDWQWNWOWPVFZYAWOPWNWOWPVGZWNWOWPVHZQWQYBWNWPWNWOW
      PVIRVJVKWQWPVLVMZXMXPPWQYDYGWNWOWPVNVQWNXKWPVOQVPVRVSXFBXEWQWSXCWOIZBXEFG
      WQXCYAIZYHWQXQXRYIXSXTWNAWPVTNWQYDYIYHWAYEYDYAWOXCYFWBQWCBXCDAWPMMSWDCWEW
      FWGOAXEBWHWIWJWKWLWM $.
  $}

  ${
    $d A a b $.  $d B b $.
    $( A natural number is equinumerous to its successor minus any element of
       the successor.  Version of ~ phplem3 with unnecessary hypotheses
       removed.  (Contributed by Jim Kingdon, 1-Sep-2021.) $)
    phplem3g $p |- ( ( A e. _om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) $=
      ( vb va com wcel csuc csn cdif cen wbr cv wa wi wceq eleq1 anbi2d imbi12d
      vex vtoclg difeq2d breq2d suceq eleq2d anbi12d id difeq1d breq12d phplem3
      sneq anabsi5 anabsi7 ) AEFZBAGZFZAUNBHZIZJKZUMCLZUNFZMZAUNUSHZIZJKZNZUMUO
      MZURNCBUNUSBOZVAVFVDURVGUTUOUMUSBUNPQVGVCUQAJVGVBUPUNUSBUJUAUBRUMUTVDDLZE
      FZUSVHGZFZMZVHVJVBIZJKZNVEDAEVHAOZVLVAVNVDVOVIUMVKUTVHAEPVOVJUNUSVHAUCZUD
      UEVOVHAVMVCJVOUFVOVJUNVBVPUGUHRVHUSDSCSUITUKTUL $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Two equinumerous natural numbers are equal.  Proposition 10.20 of
       [TakeutiZaring] p. 90 and its converse.  Also compare Corollary 6E of
       [Enderton] p. 136.  (Contributed by NM, 28-May-1998.) $)
    nneneq $p |- ( ( A e. _om /\ B e. _om ) -> ( A ~~ B <-> A = B ) ) $=
      ( vz vy vw com wcel cen wceq cv wi wral breq1 eqeq1 imbi12d ralbidv breq2
      wbr c0 eqeq2 vx wa csuc ensym en0 eqcom bitri sylib rgenw wrex wo bibi12d
      nn0suc wb mpbiri biimpd a1i nfv nfra1 nfan rsp vex phplem4 imim1d ex syl5
      a2d imp suceq syl8 biimprcd syl6 rexlimd jaod syl7 ralrimdv cbvralv finds
      imbitrdi rspcv mpan9 eqeng adantr impbid ) AFGZBFGZUBABHRZABIZWEACJZHRZAW
      IIZKZCFLZWFWGWHKZUAJZWIHRZWOWIIZKZCFLSWIHRZSWIIZKZCFLDJZWIHRZXBWIIZKZCFLZ
      XBUCZWIHRZXGWIIZKZCFLZWMUADAWOSIZWRXACFXLWPWSWQWTWOSWIHMWOSWINOPWOXBIZWRX
      ECFXMWPXCWQXDWOXBWIHMWOXBWINOPWOXGIZWRXJCFXNWPXHWQXIWOXGWIHMWOXGWINOPWOAI
      ZWRWLCFXOWPWJWQWKWOAWIHMWOAWINOPXACFWSWISHRZWTSWIUDXPWISIWTWIUEWISUFUGUHU
      IXBFGZXFXGEJZHRZXGXRIZKZEFLXKXQXFYAEFXRFGXRSIZXRWIUCZIZCFUJZUKZXQXFYACXRU
      MXQXFYFYAKXQXFUBZYBYAYEYBYAKYGYBXSXTYBXSXTUNXGSHRZXGSIZUNXGUEYBXSYHXTYIXR
      SXGHQXRSXGTULUOUPUQYGYDYACFXQXFCXQCURXECFUSUTYACURYGWIFGZXGYCHRZXGYCIZKZY
      DYAKYGYJYKXDYLXQXFYJYKXDKZKZXFYJXEKXQYOXECFVAXQYJXEYNXQYJXEYNKXQYJUBYKXCX
      DXBWIDVBCVBVCVDVEVGVFVHXBWIVIVJYDYAYMYDXSYKXTYLXRYCXGHQXRYCXGTOVKVLVMVNVE
      VOVPYAXJECFXRWIIXSXHXTXIXRWIXGHQXRWIXGTOVQVSVRWLWNCBFWIBIWJWGWKWHWIBAHQWI
      BATOVTWAWEWHWGKWFABFWBWCWD $.
  $}

  ${
    $d A w $.  $d k w $.
    $( A natural number is not equinumerous to its successor.  Corollary
       10.21(1) of [TakeutiZaring] p. 90.  (Contributed by NM, 26-Jul-2004.) $)
    php5 $p |- ( A e. _om -> -. A ~~ suc A ) $=
      ( vw vk cv csuc cen wbr wn c0 wceq id suceq breq12d notbid com wne peano1
      wcel peano3 ax-mp en0 nemtbir ensymb mtbi wi peano2 vex sucex mpdan con3d
      phplem4 finds ) BDZUMEZFGZHIIEZFGZHCDZUREZFGZHUSUSEZFGZHAAEZFGZHBCAUMIJZU
      OUQVEUMIUNUPFVEKUMILMNUMURJZUOUTVFUMURUNUSFVFKUMURLMNUMUSJZUOVBVGUMUSUNVA
      FVGKUMUSLMNUMAJZUOVDVHUMAUNVCFVHKUMALMNUPIFGZUQVIUPIIORUPIPQISTUPUAUBUPIU
      CUDURORZVBUTVJUSORVBUTUEURUFURUSCUGZURVKUHUKUIUJUL $.
  $}

  $( A singleton ` { A } ` is never equinumerous with the ordinal number 2.  If
     ` A ` is a proper class, see ~ snnen2oprc .  (Contributed by Jim Kingdon,
     1-Sep-2021.) $)
  snnen2og $p |- ( A e. V -> -. { A } ~~ 2o ) $=
    ( c1o csuc cen wbr wn wcel csn c2o com 1onn php5 ax-mp ensn1g eqcomi breq2i
    wi df-2o ensymb entr ex sylbi con3rr3 sylnbi mpsyl ) CCDZEFZGZABHAIZCEFZUJJ
    EFZGZCKHUILCMNABOUHCJEFZUKUMRUGJCEJUGSPQUKULUNUKCUJEFZULUNRUJCTUOULUNCUJJUA
    UBUCUDUEUF $.

  $( A singleton ` { A } ` is never equinumerous with the ordinal number 2.  If
     ` A ` is a set, see ~ snnen2og .  (Contributed by Jim Kingdon,
     1-Sep-2021.) $)
  snnen2oprc $p |- ( -. A e. _V -> -. { A } ~~ 2o ) $=
    ( cvv wcel wn csn c2o cen wbr c0 2on0 ensymb en0 bitri nemtbir snprc biimpi
    wceq breq1d mtbiri ) ABCDZAEZFGHIFGHZUBFIJUBFIGHFIQIFKFLMNTUAIFGTUAIQAOPRS
    $.

  $( One and two are not equinumerous.  (Contributed by Jim Kingdon,
     25-Jan-2022.) $)
  1nen2 $p |- -. 1o ~~ 2o $=
    ( c1o c2o cen wbr csuc com wcel wn 1onn php5 ax-mp df-2o breq2i mtbir ) ABC
    DAAEZCDZAFGPHIAJKBOACLMN $.

  ${
    $d A f $.  $d B f $.
    $( Dominance of successors implies dominance of the original natural
       numbers.  (Contributed by Jim Kingdon, 1-Sep-2021.) $)
    phplem4dom $p |- ( ( A e. _om /\ B e. _om ) ->
        ( suc A ~<_ suc B -> A ~<_ B ) ) $=
      ( com wcel csuc cdom wbr adantl syl csn cdif cen cima wss ensymd cvv wceq
      vf wa syl2anc cv wf1 wex wb peano2 brdomg biimpa cfv ad2antrr sssucid a1i
      simpr simplll f1imaen2g syl22anc difexg word nnord orddif wfn f1fn sucidg
      imaeq2d fnsnfv difeq2d ccnv wfun wf simprbi imadif eqtr4d f1f crn imassrn
      df-f1 frn sstrid ssdifd eqsstrrd eqsstrd ssdomg endomtr simpllr ffvelcdmd
      sylc phplem3g domentr exlimddv ex ) ACDZBCDZSZAEZBEZFGZABFGZWLWOSZWMWNRUA
      ZUBZWPRWLWOWSRUCZWLWNCDZWOWTUDWKXAWJBUEHZWMWNCRUFIUGWQWSSZAWNAWRUHZJZKZFG
      ZXFBLGWPXCAWRAMZLGXHXFFGZXGXCXHAXCWSXAAWMNZWJXHALGWQWSULWLXAWOWSXBUIZXJXC
      AUJUKWJWKWOWSUMZWMWNAWRCUNUOOXCXFPDZXHXFNXIXCXAXMXKWNXECUPIXCXHWRWMAJZKZM
      ZXFXCWJXHXPQXLWJAXOWRWJAUQAXOQAURAUSIVCIXCXPWRWMMZXEKZXFXCXRXQWRXNMZKZXPX
      CXEXSXQXCWRWMUTZAWMDZXEXSQWSYAWQWMWNWRVAHXCWJYBXLACVBIZWMAWRVDTVEWSXPXTQZ
      WQWSWRVFVGZYDWSWMWNWRVHZYEWMWNWRVOVIWMXNWRVJIHVKXCXQWNXEXCYFXQWNNWSYFWQWM
      WNWRVLHZYFXQWRVMWNWRWMVNWMWNWRVPVQIVRVSVTXHXFPWAWEAXHXFWBTXCBXFXCWKXDWNDB
      XFLGWJWKWOWSWCXCWMWNAWRYGYCWDBXDWFTOAXFBWGTWHWI $.
  $}

  ${
    $d A w $.  $d k w $.
    $( A natural number does not dominate its successor.  (Contributed by Jim
       Kingdon, 1-Sep-2021.) $)
    php5dom $p |- ( A e. _om -> -. suc A ~<_ A ) $=
      ( vw vk vx cv csuc cdom wbr wn c0 wceq suceq id breq12d notbid cen peano1
      com wcel php5 ax-mp wss wa wex 0ex domen ss0 en0 sylibr entr sylan2 sylbi
      exlimiv ensymd mto wi peano2 phplem4dom mpancom con3d finds ) BEZFZVBGHZI
      JFZJGHZICEZFZVGGHZIVHFZVHGHZIAFZAGHZIBCAVBJKZVDVFVNVCVEVBJGVBJLVNMNOVBVGK
      ZVDVIVOVCVHVBVGGVBVGLVOMNOVBVHKZVDVKVPVCVJVBVHGVBVHLVPMNOVBAKZVDVMVQVCVLV
      BAGVBALVQMNOVFJVEPHZJRSVRIQJTUAVFVEJVFVEDEZPHZVSJUBZUCZDUDVEJPHZDVEJUEUFW
      BWCDWAVTVSJPHZWCWAVSJKWDVSUGVSUHUIVEVSJUJUKUMULUNUOVGRSZVKVIVHRSWEVKVIUPV
      GUQVHVGURUSUTVA $.
  $}

  $( Cardinal ordering agrees with natural number ordering.  Example 3 of
     [Enderton] p. 146.  (Contributed by NM, 17-Jun-1998.) $)
  nndomo $p |- ( ( A e. _om /\ B e. _om ) -> ( A ~<_ B <-> A C_ B ) ) $=
    ( com wcel wa cdom wbr wn csuc php5dom ad2antlr wi domtr expcom adantl mtod
    wss ssdomg ad2antrr word nnord ordsucss syl wb nntri1 adantr mpbird impbid
    ex ) ACDZBCDZEZABFGZABQZULUMUNULUMEZUNBADZHZUOUPBIZAQZUOUSURAFGZUOUTURBFGZU
    KVAHUJUMBJKUMUTVALULUTUMVAURABMNOPUJUSUTLUKUMURACRSPUJUPUSLZUKUMUJATVBAUABA
    UBUCSPULUNUQUDUMABUEUFUGUIUKUNUMLUJABCROUH $.

  $( Two is not dominated by one.  (Contributed by Jim Kingdon,
     10-Jan-2026.) $)
  1ndom2 $p |- -. 2o ~<_ 1o $=
    ( c2o c1o cdom wbr wss wcel com word wn 1onn nnord ordirr mp2b ssel mpi mto
    1lt2o wb 2onn nndomo mp2an mtbir ) ABCDZABEZUDBBFZBGFZBHUEIJBKBLMUDBAFUEQAB
    BNOPAGFUFUCUDRSJABTUAUB $.

  ${
    $d x y A $.  $d x y B $.
    $( Pigeonhole Principle.  A natural number is not equinumerous to a proper
       subset of itself.  By "proper subset" here we mean that there is an
       element which is in the natural number and not in the subset, or in
       symbols ` E. x x e. ( A \ B ) ` (which is stronger than not being equal
       in the absence of excluded middle).  Theorem (Pigeonhole Principle) of
       [Enderton] p. 134.  The theorem is so-called because you can't put _n_ +
       1 pigeons into _n_ holes (if each hole holds only one pigeon).  The
       proof consists of lemmas ~ phplem1 through ~ phplem4 , ~ nneneq , and
       this final piece of the proof.  (Contributed by NM, 29-May-1998.) $)
    phpm $p |- ( ( A e. _om /\ B C_ A /\ E. x x e. ( A \ B ) ) ->
        -. A ~~ B ) $=
      ( vy com wcel wss cv cdif cen wn wa c0 wceq simpr ad2antlr simplr syl2anc
      wbr cdom wex csuc wrex wne eldifi syl neneqd pm2.21dd php5dom csn cvv vex
      sucex difss ssexi eldifn ad3antlr simpllr adantr sseqtrd ssdif cin disjsn
      ne0i disj3 bitr3i sseq1 sylbi imbitrrid sylc ssdomg mpsyl phplem3g ensymd
      wb eleqtrd domentr endomtr eqbrtrrd mtand ex rexlimdva wo nn0suc ad2antrr
      imp mpjaodan exlimdv 3impia ) BEFZCBGZAHZBCIFZAUABCJSZKZWJWKLZWMWOAWPWMWO
      WPWMLZBMNZWOBDHZUBZNZDEUCZWQWRLWRWOWQWROWMWRKWPWRWMBMWMWLBFZBMUDWLBCUEZBW
      LVDUFUGPUHWQXBWOWQXAWODEWQWSEFZLZXAWOXFXALZWNWTWSTSZXEXHKWQXAWSUIPXGWNLZB
      WTWSTXFXAWNQXIWNCWSTSZBWSTSXGWNOXGXJWNXGCWTWLUJZIZTSZXLWSJSZXJXLUKFXGCXLG
      ZXMXLWTWSDULUMWTXKUNUOXGWLCFKZCWTGZXOWMXPWPXEXAWLBCUPUQXGCBWTXFWKXAWJWKWM
      XEURUSXFXAOZUTXQXOXPCXKIZXLGZCWTXKVAXPCXSNZXOXTVOXPCXKVBMNYACWLVCCXKVEVFC
      XSXLVGVHVIVJCXLUKVKVLXGXEWLWTFZXNWQXEXAQXGWLBWTWMXCWPXEXAXDUQXRVPXEYBLWSX
      LWSWLVMVNRCXLWSVQRUSBCWSVRRVSVTWAWBWFWJWRXBWCWKWMDBWDWEWGWAWHWI $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Pigeonhole Principle.  A natural number is not equinumerous to an
       element of itself.  (Contributed by Jim Kingdon, 6-Sep-2021.) $)
    phpelm $p |- ( ( A e. _om /\ B e. A ) -> -. A ~~ B ) $=
      ( vx com wcel wa wss cv cdif wex cen wbr wn simpl con0 wi nnon onelss syl
      imp simpr elirr a1i eldifd eleq1 spcegv sylc phpm syl3anc ) ADEZBAEZFZUJB
      AGZCHZABIZEZCJZABKLMUJUKNUJUKUMUJAOEUKUMPAQABRSTULUKBUOEZUQUJUKUAZULBABUS
      BBEMULBUBUCUDUPURCBAUNBUOUEUFUGCABUHUI $.
  $}

  ${
    $d A f $.  $d B f $.
    $( Equinumerosity of successors of an ordinal and a natural number implies
       equinumerosity of the originals.  (Contributed by Jim Kingdon,
       5-Sep-2021.) $)
    phplem4on $p |- ( ( A e. On /\ B e. _om ) ->
        ( suc A ~~ suc B -> A ~~ B ) ) $=
      ( vf con0 wcel com wa csuc cen wbr cv adantl csn cdif cima ensymd imaeq2d
      syl wceq syl2anc wf1o wex bren biimpi cfv wf1 wss peano2 ad3antlr sssucid
      f1of1 nnon a1i simplll f1imaen2g syl22anc word eloni orddif ad3antrrr wfn
      f1ofn sucidg fnsnfv difeq2d crn cdm imadmrn eqcomi wfo f1ofo forn 3eqtr3a
      difeq1d ccnv wfun dff1o3 simprbi imadif 3eqtr4rd breqtrd simpllr fnfvelrn
      f1odm eqtrd wb eleq2d mpbid phplem3g entr exlimddv ex ) ADEZBFEZGZAHZBHZI
      JZABIJZWOWRGZWPWQCKZUAZWSCWRXBCUBZWOWRXCWPWQCUCUDLWTXBGZAWQAXAUEZMZNZIJXG
      BIJWSXDAXAAOZXGIXDXHAXDWPWQXAUFZWQDEZAWPUGZWMXHAIJXBXIWTWPWQXAUKLWNXJWMWR
      XBWNWQFEXJBUHWQULRUIXKXDAUJUMWMWNWRXBUNZWPWQAXADUOUPPXDXHXAWPAMZNZOZXGWMX
      HXOSWNWRXBWMAXNXAWMAUQAXNSAURAUSRQUTXDXAWPOZXFNZXPXAXMOZNZXGXOXDXFXRXPXDX
      AWPVAZAWPEZXFXRSXBXTWTWPWQXAVBLZXDWMYAXLADVCRZWPAXAVDTVEXBXGXQSWTXBWQXPXF
      XBXAVFZXAXAVGZOZWQXPYFYDXAVHVIXBWPWQXAVJZYDWQSWPWQXAVKZWPWQXAVLZRXBYEWPXA
      WPWQXAWDQVMVNLXBXOXSSZWTXBXAVOVPZYJXBYGYKWPWQXAVQVRWPXMXAVSRLVTWEWAXDBXGX
      DWNXEWQEZBXGIJWMWNWRXBWBXDXEYDEZYLXDXTYAYMYBYCWPAXAWCTXBYMYLWFZWTXBYGYNYH
      YGYDWQXEYIWGRLWHBXEWITPAXGBWJTWKWL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Finite sets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A n $.
    $( A finite set is dominated by ` _om ` .  Also see ~ finct .  (Contributed
       by Thierry Arnoux, 27-Mar-2018.) $)
    fict $p |- ( A e. Fin -> A ~<_ _om ) $=
      ( vn cfn wcel cv cen wbr com cdom wrex isfi biimpi wa simprr cvv wss omex
      word ordom ordelss mpan ssdomg mpsyl ad2antrl endomtr syl2anc rexlimddv )
      ACDZABEZFGZAHIGZBHUHUJBHJBAKLUHUIHDZUJMMUJUIHIGZUKUHULUJNULUMUHUJHODULUIH
      PZUMQHRULUNSHUITUAUIHOUBUCUDAUIHUEUFUG $.
  $}

  ${
    $d A f x $.  $d B f x $.  $d C f x $.
    $( Equality of members of a finite set is decidable.  This may be
       counterintuitive: cannot any two sets be elements of a finite set?
       Well, to show, for example, that ` { B , C } ` is finite would require
       showing it is equinumerous to ` 1o ` or to ` 2o ` but to show that you'd
       need to know ` B = C ` or ` -. B = C ` , respectively.  (Contributed by
       Jim Kingdon, 5-Sep-2021.) $)
    fidceq $p |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C ) $=
      ( vx vf wcel cv wceq wdc com biimpi wa wn wo cfv adantl ffvelcdmd syl2anc
      elnn wi cfn w3a cen wbr wrex isfi 3ad2ant1 wf1o wex bren ad2antll wf f1of
      simpll2 simplrl simpll3 nndceq exmiddc syl wf1 f1of1 f1veqaeq fveq2 con3i
      syl12anc a1i orim12d mpd df-dc sylibr exlimddv rexlimddv ) AUAFZBAFZCAFZU
      BZADGZUCUDZBCHZIZDJVMVNVRDJUEZVOVMWADAUFKUGVPVQJFZVRLZLZAVQEGZUHZVTEVRWFE
      UIZVPWBVRWGAVQEUJKUKWDWFLZVSVSMZNZVTWHBWEOZCWEOZHZWMMZNZWJWHWMIZWOWHWKJFZ
      WLJFZWPWHWKVQFWBWQWHAVQBWEWFAVQWEULWDAVQWEUMPZVMVNVOWCWFUNZQVPWBVRWFUOZWK
      VQSRWHWLVQFWBWRWHAVQCWEWSVMVNVOWCWFUPZQXAWLVQSRWKWLUQRWMURUSWHWMVSWNWIWHA
      VQWEUTZVNVOWMVSTWFXCWDAVQWEVAPWTXBAVQBCWEVBVEWNWITWHVSWMBCWEVCVDVFVGVHVSV
      IVJVKVL $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d X x y $.
    $( All decrements of a finite set are equinumerous.  (Contributed by Jim
       Kingdon, 9-Sep-2021.) $)
    fidifsnen $p |- ( ( X e. Fin /\ A e. X /\ B e. X ) ->
        ( X \ { A } ) ~~ ( X \ { B } ) ) $=
      ( vx vy wcel wceq wn wa adantr syl adantl eqeltrd sylanbrc ad2antlr simpr
      neneqad mpjaodan ad2antrr eqeq2d cfn w3a csn cdif cen wbr difexg 3ad2ant1
      cvv enrefg sneq difeq2d breqtrd cv cif cmpt wf1o eqid wne simpll2 simpllr
      iftrue eqeq1d mtbird eldifsn iffalse eldifi wdc wo simpll1 simpll3 fidceq
      syl3anc exmiddc simpl3 necomd wb wi biimpar a1d mpbid eqtr3d simprr velsn
      eldifbd sylnib pm2.21dd ex ad2antll biimprcd sylbid impbid eqtrd ad3antlr
      simplr eqtr2d simprl impbida adantrr f1o2d f1oeng syl2anc ) CUAFZACFZBCFZ
      UBZABGZCAUCZUDZCBUCZUDZUEUFZXGHZXFXGIZXIXIXKUEXNXIUIFZXIXIUEUFXFXOXGXCXDX
      OXECXHUAUGUHZJXIUIUJKXGXIXKGXFXGXHXJCABUKULLUMXFXMIZXOXIXKDXIDUNZBGZAXRUO
      ZUPZUQXLXFXOXMXPJXQDEXIXKXTEUNZAGZBYBUOZYAYAURXQXRXIFZIZXSXTXKFZXSHZYFXSI
      ZXTCFZXTBUSZYGYIXTACXSXTAGZYFXSAXRVBZLZYFXDXSXCXDXEXMYEUTJMYIXTBYIXTBGZXG
      XFXMYEXSVAYIXTABYNVCVDQXTCBVEZNYFYHIZYJYKYGYQXTXRCYHXTXRGZYFXSAXRVFZLZYEX
      RCFZXQYHXRCXHVGZOMYQXTBYQYOXSYFYHPYQXTXRBYTVCVDQYPNYFXSVHZXSYHVIZYFXCUUAX
      EUUCXCXDXEXMYEVJYEUUAXQUUBLXCXDXEXMYEVKCXRBVLVMXSVNKZRXQYBXKFZIZYCYDXIFYC
      HZUUGYCIYDBXIYCYDBGZUUGYCBYBVBZLXQBXIFZUUFYCXQXEBAUSUUKXCXDXEXMVOXQABXQAB
      XFXMPQVPBCAVENSMUUGUUHIZYDYBXIUUHYDYBGZUUGYCBYBVFZLUULYBCFZYBAUSYBXIFUUFU
      UOXQUUHYBCXJVGZOUULYBAUUGUUHPQYBCAVENMUUGYCVHZYCUUHVIZUUGXCUUOXDUUQXCXDXE
      XMUUFVJUUFUUOXQUUPLXCXDXEXMUUFUTCYBAVLZVMYCVNZKRXQYEUUFIZIZXSXRYDGZYBXTGZ
      VQYHUVBXSIZUVCUVDUVEYCUVCUVDVRUUHUVEYCIUVDUVCUVEUVDYCUVEXTAYBXSYLUVBYMLTZ
      VSVTUVEUUHIZUVCUVDUVGUVCIZYBBGZUVDUVHXRYBBUVHUVCXRYBGZUVGUVCPUUHUVCUVJVQU
      VEUVCUUHYDYBXRUUNTOWAUVBXSUUHUVCVAWBUVHYBXJFZUVIUVGUVKHUVCUVGYBCXJUVBUUFX
      SUUHXQYEUUFWCSWEJEBWDWFWGWHUVBUURXSUVBUUQUURUVBXCUUOXDUUQXCXDXEXMUVAVJUUF
      UUOXQYEUUPWIXCXDXEXMUVAUTUUSVMUUTKZJRUVEUVDYCUVCUVFXSYCUVCVRUVBYCUVCXSYCY
      DBXRUUJTWJLWKWLUVBYHIZUVCUVDUVMUVCIZYCUVDUUHUVNYCIZXSUVDUVOXRYDBUVMUVCYCW
      OYCUUIUVNUUJLWMUVBYHUVCYCVAWGUVNUUHIZXTXRYBYHYRUVBUVCUUHYSWNUVPXRYDYBUVMU
      VCUUHWOUUHUUMUVNUUNLWMWPUVBUURYHUVCUVLSRUVMUVDIZYDYBXRUVQUUHUUMUVQYCXRAGZ
      UVBUVRHYHUVDUVBXRXHFUVRUVBXRCXHXQYEUUFWQWEDAWDWFSUVQYBXRAUVQUVDYBXRGZUVMU
      VDPYHUVDUVSVQUVBUVDYHXTXRYBYSTOWAZVCVDUUNKUVTWPWRXQYEUUDUUFUUEWSRWTXIXKUI
      YAXAXBXFXGVHXGXMVICABVLXGVNKR $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( If we remove a single element from a finite set then put it back in, we
       end up with the original finite set.  This strengthens ~ difsnss from
       subset to equality when the set is finite.  (Contributed by Jim Kingdon,
       9-Sep-2021.) $)
    fidifsnid $p |- ( ( A e. Fin /\ B e. A ) ->
        ( ( A \ { B } ) u. { B } ) = A ) $=
      ( vx vy cfn wcel cv wceq wdc wral csn cdif cun 3expb ralrimivva dcdifsnid
      fidceq sylan ) AEFZCGZDGZHIZDAJCAJBAFABKZLUCMAHSUBCDAASTAFUAAFUBATUAQNOCD
      ABPR $.
  $}

  ${
    $d A x $.
    $( Natural numbers are finite sets.  (Contributed by Stefan O'Rear,
       21-Mar-2015.) $)
    nnfi $p |- ( A e. _om -> A e. Fin ) $=
      ( vx com wcel cv cen wbr wrex cfn enrefg breq2 rspcev mpdan isfi sylibr )
      ACDZABEZFGZBCHZAIDPAAFGZSACJRTBACQAAFKLMBANO $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Equinumerous sets have the same finiteness.  (Contributed by NM,
       22-Aug-2008.) $)
    enfi $p |- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) ) $=
      ( vx cen wbr cv com wrex cfn wcel enen1 rexbidv isfi 3bitr4g ) ABDEZACFZD
      EZCGHBPDEZCGHAIJBIJOQRCGABPKLCAMCBMN $.

    $( A set equinumerous to a finite set is finite.  (Contributed by Mario
       Carneiro, 12-Mar-2015.) $)
    enfii $p |- ( ( B e. Fin /\ A ~~ B ) -> A e. Fin ) $=
      ( cen wbr cfn wcel enfi biimparc ) ABCDAEFBEFABGH $.
  $}

  ${
    $d ph n x y z $.
    ssfilem.1 $e |- { z e. { (/) } | ph } e. Fin $.
    $( Lemma for ~ ssfiexmid .  (Contributed by Jim Kingdon, 3-Feb-2022.) $)
    ssfilem $p |- ( ph \/ -. ph ) $=
      ( vn vw vy vx c0 csn crab cv cen wbr com wo wcel wceq wa wex ax-mp wn cfn
      wrex isfi mpbi 0elnn breq2 en0 bitrdi biimpac wral rabeq0 wb 0ex r19.3rmv
      snm bitr4i sylib olcd ensym elex2 syl2an biidd elrab simprbi orcd exlimiv
      enm syl jaodan sylan2 ancoms rexlimiva ) ABHIZJZDKZLMZDNUCZAAUAZOZVOUBPVR
      CDVOUDUEVQVTDNVQVPNPZVTWAVQVPHQZHVPPZOVTVPUFVQWBVTWCVQWBRZVSAWDVOHQZVSWBV
      QWEWBVQVOHLMWEVPHVOLUGVOUHUIUJWEVSBVNUKZVSABVNULEKVNPESVSWFUMEHUNUPVSBEVN
      UOTUQURUSVQWCRFKZVOPZFSZVTVQVPVOLMGKVPPGSWIWCVOVPUTGHVPVAGFVPVOVHVBWHVTFW
      HAVSWHWGVNPAAABWGVNBKWGQAVCVDVEVFVGVIVJVKVLVMT $.
  $}

  ${
    $d ph x y z $.
    ssfiexmid.1 $e |- A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin ) $.
    $( If any subset of a finite set is finite, excluded middle follows.  One
       direction of Theorem 2.1 of [Bauer], p. 485.  (Contributed by Jim
       Kingdon, 19-May-2020.) $)
    ssfiexmid $p |- ( ph \/ -. ph ) $=
      ( vz c0 csn cfn wcel wss cvv ax-mp cv wa wi wal p0ex wceq eleq1 spcv crab
      snfig ssrab2 sseq2 anbi12d imbi1d albidv rabex sseq1 anbi2d imbi12d mp2an
      0ex ssfilem ) AEFGZHIZAEUOUAZUOJZUQHIZFKIUPUMFKUBLAEUOUCUPCMZUOJZNZUTHIZO
      ZCPZUPURNZUSOZBMZHIZUTVHJZNZVCOZCPZBPVEDVMVEBUOQVHUORZVLVDCVNVKVBVCVNVIUP
      VJVAVHUOHSVHUOUTUDUEUFUGTLVDVGCUQAEUOQUHUTUQRZVBVFVCUSVOVAURUPUTUQUOUIUJU
      TUQHSUKTLULUN $.
  $}

  ${
    $d ps n x y z $.
    ssfilemd.1 $e |- ( ph -> { z e. { (/) } | ps } e. Fin ) $.
    $( Lemma for ~ ssfiexmidt .  (Contributed by Jim Kingdon, 3-Feb-2022.) $)
    ssfilemd $p |- ( ph -> ( ps \/ -. ps ) ) $=
      ( vn vw vy vx c0 cv cen wbr com wo wcel sylib wceq wa wex syl csn crab wn
      wrex cfn isfi 0elnn breq2 en0 bitrdi biimpac wral rabeq0 0ex snm r19.3rmv
      ax-mp bitr4i olcd ensym elex2 enm syl2an biidd elrab simprbi orcd exlimiv
      wb jaodan sylan2 ancoms rexlimiva ) ABCIUAZUBZEJZKLZEMUDZBBUCZNZAVOUEOVRD
      EVOUFPVQVTEMVQVPMOZVTWAVQVPIQZIVPOZNVTVPUGVQWBVTWCVQWBRZVSBWDVOIQZVSWBVQW
      EWBVQVOIKLWEVPIVOKUHVOUIUJUKWEVSCVNULZVSBCVNUMFJVNOFSVSWFVIFIUNUOVSCFVNUP
      UQURPUSVQWCRGJZVOOZGSZVTVQVPVOKLHJVPOHSWIWCVOVPUTHIVPVAHGVPVOVBVCWHVTGWHB
      VSWHWGVNOBBBCWGVNCJWGQBVDVEVFVGVHTVJVKVLVMT $.
  $}

  ${
    $d ph x y z $.
    $( If any subset of a finite set is finite, excluded middle follows.  One
       direction of Theorem 2.1 of [Bauer], p. 485.  (Contributed by Jim
       Kingdon, 19-May-2020.) $)
    ssfiexmidt $p |- ( A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin )
        -> ( ph \/ -. ph ) ) $=
      ( vz cv cfn wcel wss wa wi wal c0 csn crab p0ex wceq eleq1 sseq2 spcv cvv
      anbi12d imbi1d albidv snfig ax-mp ssrab2 pm3.2i rabex sseq1 anbi2d mpisyl
      0ex imbi12d ssfilemd ) BEZFGZCEZUOHZIZUQFGZJZCKZBKZADVCLMZFGZUQVDHZIZUTJZ
      CKZVEADVDNZVDHZIZVJFGZVBVIBVDOUOVDPZVAVHCVNUSVGUTVNUPVEURVFUOVDFQUOVDUQRU
      AUBUCSVEVKLTGVEULLTUDUEADVDUFUGVHVLVMJCVJADVDOUHUQVJPZVGVLUTVMVOVFVKVEUQV
      JVDUIUJUQVJFQUMSUKUN $.
  $}

  ${
    $d ph x y $.
    infiexmid.1 $e |- ( x e. Fin -> ( x i^i y ) e. Fin ) $.
    $( If the intersection of any finite set and any other set is finite,
       excluded middle follows.  (Contributed by Jim Kingdon, 5-Feb-2022.) $)
    infiexmid $p |- ( ph \/ -. ph ) $=
      ( cv cfn wcel wss wa wi cin wceq dfss1 biimpi adantl adantr eqeltrrd gen2
      ssfiexmid ) ABCBEZFGZCEZTHZIZUBFGJBCUDTUBKZUBFUCUEUBLZUAUCUFUBTMNOUAUEFGU
      CDPQRS $.
  $}

  ${
    $d ph y z $.  $d x y $.
    domfiexmid.1 $e |- ( ( x e. Fin /\ y ~<_ x ) -> y e. Fin ) $.
    $( If any set dominated by a finite set is finite, excluded middle follows.
       (Contributed by Jim Kingdon, 3-Feb-2022.) $)
    domfiexmid $p |- ( ph \/ -. ph ) $=
      ( vz c0 cfn wcel cdom wbr cvv ax-mp cv wa wi wal p0ex wceq eleq1 spcv csn
      crab 0ex snfig wss ssrab2 ssdomg mp2 gen2 breq2 imbi1d albidv rabex breq1
      anbi12d anbi2d imbi12d mp2an ssfilem ) AEFUAZGHZAEUTUBZUTIJZVBGHZFKHVAUCF
      KUDLZVAVBUTUEVCVEAEUTUFVBUTGUGUHVACMZUTIJZNZVFGHZOZCPZVAVCNZVDOZBMZGHZVFV
      NIJZNZVIOZCPZBPVKVRBCDUIVSVKBUTQVNUTRZVRVJCVTVQVHVIVTVOVAVPVGVNUTGSVNUTVF
      IUJUOUKULTLVJVMCVBAEUTQUMVFVBRZVHVLVIVDWAVGVCVAVFVBUTIUNUPVFVBGSUQTLURUS
      $.
  $}

  ${
    $d A f $.  $d X f $.  $d M f $.
    $( If a set ` A ` is equinumerous to the successor of a natural number
       ` M ` , then ` A ` with an element removed is equinumerous to ` M ` .
       (Contributed by Jeff Madsen, 2-Sep-2009.)  (Revised by Stefan O'Rear,
       16-Aug-2015.) $)
    dif1en $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) ->
        ( A \ { X } ) ~~ M ) $=
      ( vf com wcel cen wbr wf1o csn cdif ensymd 3ad2ant1 syl2anc adantr adantl
      syl wceq cres wfo csuc w3a cv wex simp2 bren sylib wa cfv cfn peano2 nnfi
      enfii simpl3 wf f1of sucidg ffvelcdmd fidifsnen syl3anc word nnord orddif
      eleq1d ibi ccnv wfun wfn crn dff1o2 simp2bi f1ofo wb cdm wrel resdm f1odm
      f1orel reseq2d eqtr3d foeq1 mpbid cop simpl1 f1osng fnressn mpbird resdif
      f1ofn f1oeng eqbrtrd entr exlimddv ) BEFZABUAZGHZCAFZUBZWOADUCZIZACJKZBGH
      ZDWRWOAGHWTDUDWRAWOWNWPWQUEZLWOADUFUGWRWTUHZXAABWSUIZJZKZGHZXGBGHXBXDAUJF
      ZWQXEAFZXHWRXIWTWRWOUJFZWPXIWNWPXKWQWNWOEFXKBUKWOULQMXCAWOUMNOWNWPWQWTUNX
      DWOABWSWTWOAWSUOWRWOAWSUPPWRBWOFZWTWNWPXLWQBEUQMOZURZCXEAUSUTXDBXGXDBWOBJ
      ZKZXGGWRBXPRZWTWNWPXQWQWNBVAXQBVBBVCQZMOXDXPEFZXPXGWSXPSZIZXPXGGHWRXSWTWN
      WPXSWQWNXSWNBXPEXRVDVEMOXDWSVFVGZWOAWSWOSZTZXOXFWSXOSZTZYAWTYBWRWTWSWOVHZ
      YBWSVIARWOAWSVJVKPXDWOAWSTZYDWTYHWRWOAWSVLPXDWSYCRYHYDVMXDWSWSVNZSZWSYCXD
      WSVOZYJWSRWTYKWRWOAWSVRPWSVPQWTYJYCRWRWTYIWOWSWOAWSVQVSPVTWOAWSYCWAQWBXDY
      FXOXFBXEWCJZTZXDXOXFYLIZYMXDWNXJYNWNWPWQWTWDXNBXEEAWENXOXFYLVLQXDYEYLRZYF
      YMVMXDYGXLYOWTYGWRWOAWSWIPXMWOBWSWFNXOXFYEYLWAQWGWOXOAXFWSWHUTXPXGEXTWJNW
      KLXAXGBWLNWM $.
  $}

  ${
    $d A m n $.  $d B m n $.  $d C m n $.  $d D m n $.  $d m n ph $.
    dif1enen.a $e |- ( ph -> A e. Fin ) $.
    dif1enen.ab $e |- ( ph -> A ~~ B ) $.
    dif1enen.c $e |- ( ph -> C e. A ) $.
    dif1enen.d $e |- ( ph -> D e. B ) $.
    $( Subtracting one element from each of two equinumerous finite sets.
       (Contributed by Jim Kingdon, 5-Jun-2022.) $)
    dif1enen $p |- ( ph -> ( A \ { C } ) ~~ ( B \ { D } ) ) $=
      ( vn vm cv cen wbr com wcel wa c0 wceq ad3antrrr csn cdif wrex isfi sylib
      cfn simplrr wb breq2 adantl mpbid en0 wn n0i syl ad2antrr pm2.21dd simplr
      csuc simprr dif1en syl3anc ensymd entr syl2anc ex rexlimdva imp wo nn0suc
      ad2antrl mpjaodan rexlimddv ) ABJLZMNZBDUAUBZCEUAUBZMNZJOABUFPVOJOUCFJBUD
      UEAVNOPZVOQZQZVNRSZVRVNKLZUSZSZKOUCZWAWBQZBRSZVRWGBRMNZWHWGVOWIAVSVOWBUGW
      BVOWIUHWAVNRBMUIUJUKBULUEAWHUMZVTWBADBPZWJHBDUNUOUPUQWAWFVRWAWEVRKOWAWCOP
      ZQZWEVRWMWEQZVPWCMNZWCVQMNVRWNWLBWDMNZWKWOWAWLWEURZWNVOWPWAVOWLWEAVSVOUTU
      PWEVOWPUHWMVNWDBMUIUJUKZAWKVTWLWEHTBWCDVAVBWNVQWCWNWLCWDMNZECPZVQWCMNWQWN
      CBMNWPWSWNBCABCMNVTWLWEGTVCWRCBWDVDVEAWTVTWLWEITCWCEVAVBVCVPWCVQVDVEVFVGV
      HVSWBWFVIAVOKVNVJVKVLVM $.
  $}

  $( Adding one element to a finite set which is equinumerous to a natural
     number.  (Contributed by Jim Kingdon, 13-Sep-2021.) $)
  fiunsnnn $p |- ( ( ( A e. Fin /\ B e. ( _V \ A ) )
      /\ ( N e. _om /\ A ~~ N ) ) ->
      ( A u. { B } ) ~~ suc N ) $=
    ( cfn wcel cvv cdif wa com cen wbr csn cun csuc cin c0 wceq simprr disjsn
    wn en2sn ad2ant2lr simplr eldifbd sylibr elirr mpbir a1i syl22anc breqtrrdi
    unen df-suc ) ADEZBFAGZEZHZCIEZACJKZHZHZABLZMZCCLZMZCNJUTURVAVCJKZAVAOPQZCV
    COPQZVBVDJKUPUQURRUOUQVEUMURBCUNIUAUBUTBAETVFUTBFAUMUOUSUCUDABSUEVGUTVGCCET
    CUFCCSUGUHACVAVCUKUICULUJ $.

  ${
    $d A n $.  $d B n $.
    $( A finite set is not equinumerous to a set which adds one element.
       (Contributed by Jim Kingdon, 13-Sep-2021.) $)
    php5fin $p |- ( ( A e. Fin /\ B e. ( _V \ A ) ) ->
        -. A ~~ ( A u. { B } ) ) $=
      ( vn cfn wcel cvv cdif wa cv cen wbr csn cun wn com wrex biimpi adantr wb
      isfi csuc php5 ad2antrl enen1 ad2antll fiunsnnn enen2 syl bitrd rexlimddv
      mtbird ) ADEZBFAGEZHZACIZJKZAABLMZJKZNCOULUPCOPZUMULUSCATQRUNUOOEZUPHHZUR
      UOUOUAZJKZUTVCNUNUPUOUBUCVAURUOUQJKZVCUPURVDSUNUTAUOUQUDUEVAUQVBJKVDVCSAB
      UOUFUQVBUOUGUHUIUKUJ $.
  $}

  ${
    $d A m n $.  $d B m n $.
    $( Schroeder-Bernstein Theorem for finite sets.  (Contributed by Jim
       Kingdon, 12-Sep-2021.) $)
    fisbth $p |- ( ( ( A e. Fin /\ B e. Fin ) /\
        ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B ) $=
      ( vn vm cfn wcel wa cdom wbr cen com wrex isfi biimpi ad2antrr wss ensymd
      cv simprl syl2anc ad3antlr simplrr endomtr simprr domentr wb nndomo mpbid
      simplrl eqssd breqtrd entr rexlimddv ) AEFZBEFZGZABHIZBAHIZGZGZACRZJIZABJ
      IZCKUNVBCKLZUOUSUNVDCAMNOUTVAKFZVBGZGZBDRZJIZVCDKUOVIDKLZUNUSVFUOVJDBMNUA
      VGVHKFZVIGZGZAVHJIVHBJIZVCVMAVAVHJUTVEVBVLUBZVMVAVHVMVAVHHIZVAVHPZVMVABHI
      ZVIVPVMVAAJIUQVRVMAVAVOQUTUQVFVLUPUQURSOVAABUCTVGVKVIUDZVABVHUETVMVEVKVPV
      QUFUTVEVBVLUIZVGVKVISZVAVHUGTUHVMVHVAHIZVHVAPZVMVHAHIZVBWBVMVNURWDVMBVHVS
      QZUTURVFVLUPUQURUDOVHBAUCTVOVHAVAUETVMVKVEWBWCUFWAVTVHVAUGTUHUJUKWEAVHBUL
      TUMUM $.
  $}

  $( The empty set is finite.  (Contributed by FL, 14-Jul-2008.) $)
  0fi $p |- (/) e. Fin $=
    ( c0 com wcel cfn peano1 nnfi ax-mp ) ABCADCEAFG $.

  ${
    $d A f m n x $.
    $( A nonempty finite set has at least one element.  (Contributed by Jim
       Kingdon, 10-Sep-2021.) $)
    fin0 $p |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) ) $=
      ( vn vm vf cfn wcel cv cen wbr c0 wex com wrex wa wceq simplrr sylib syl
      wn wne wb isfi biimpi csuc simpr breqtrd en0 nner necon2bi 2falsed adantr
      n0r wf1o ensymd bren cfv adantl sucidg ad3antlr simplr eleqtrrd ffvelcdmd
      wf f1of elex2 2thd exlimddv ex rexlimdva imp wo nn0suc ad2antrl rexlimddv
      mpjaodan ) BFGZBCHZIJZBKUAZAHBGALZUBZCMVQVSCMNCBUCUDVQVRMGZVSOOZVRKPZWBVR
      DHZUEZPZDMNZWDWEOZVTWAWJBKPZVTTWJBKIJWKWJBVRKIVQWCVSWEQWDWEUFUGBUHRZBKUIS
      WJWKWATWLWABKABUMZUJSUKWDWIWBWDWHWBDMWDWFMGZOZWHWBWOWHOZVRBEHZUNZWBEWPVRB
      IJWRELWPBVRWOVSWHVQWCVSWNQULUOVRBEUPRWPWROZVTWAWSWAVTWSWFWQUQZBGWAWSVRBWF
      WQWRVRBWQVDWPVRBWQVEURWSWFWGVRWNWFWGGWDWHWRWFMUSUTWOWHWRVAVBVCAWTBVFSZWMS
      XAVGVHVIVJVKWCWEWIVLVQVSDVRVMVNVPVO $.
  $}

  ${
    $d A f m n x $.
    $( A finite set is either empty or inhabited.  (Contributed by Jim Kingdon,
       30-Sep-2021.) $)
    fin0or $p |- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) ) $=
      ( vn vm vf cfn wcel cv cen wbr c0 wceq wex wo com wrex wa simplrr sylib
      ex isfi biimpi csuc nn0suc ad2antrl simpr breqtrd wf1o adantr ensymd bren
      en0 cfv wf f1of adantl sucidg ad3antlr simplr eleqtrrd ffvelcdmd exlimddv
      elex2 syl rexlimdva orim12d mpd rexlimddv ) BFGZBCHZIJZBKLZAHBGAMZNZCOVIV
      KCOPCBUAUBVIVJOGZVKQQZVJKLZVJDHZUCZLZDOPZNZVNVOWBVIVKDVJUDUEVPVQVLWAVMVPV
      QVLVPVQQZBKIJVLWCBVJKIVIVOVKVQRVPVQUFUGBULSTVPVTVMDOVPVROGZQZVTVMWEVTQZVJ
      BEHZUHZVMEWFVJBIJWHEMWFBVJWEVKVTVIVOVKWDRUIUJVJBEUKSWFWHQZVRWGUMZBGVMWIVJ
      BVRWGWHVJBWGUNWFVJBWGUOUPWIVRVSVJWDVRVSGVPVTWHVROUQURWEVTWHUSUTVAAWJBVCVD
      VBTVEVFVGVH $.
  $}

  ${
    $d a b $.  $d b ph x $.  $d n ph w x y $.
    diffitest.1 $e |- A. a e. Fin A. b ( a \ b ) e. Fin $.
    $( If subtracting any set from a finite set gives a finite set, any
       proposition of the form ` -. ph ` is decidable.  This is not a proof of
       full excluded middle, but it is close enough to show we won't be able to
       prove ` A e. Fin -> ( A \ B ) e. Fin ` .  (Contributed by Jim Kingdon,
       8-Sep-2021.) $)
    diffitest $p |- ( -. ph \/ -. -. ph ) $=
      ( vx vn vw vy c0 cdif cv cen wbr com cfn wcel cvv ax-mp wceq wex csn crab
      wrex wn wo wal wral 0ex snfig difeq1 eleq1d albidv mp2 rabexg difeq2 spcv
      rspcv isfi mpbi 0elnn wa breq2 en0 bitrdi biimpac rabeq0 notrab eqeq1i wb
      snm r19.3rmv 3bitr4i sylib olcd ensym elex2 enm syl2an biidd simprbi orcd
      elrab eleq2s exlimiv syl jaodan sylan2 ancoms rexlimiva ) IUAZAEWJUBZJZFK
      ZLMZFNUCZAUDZWPUDZUEZWLOPZWOWJCKZJZOPZCUFZWSWJOPZBKZWTJZOPZCUFZBOUGXCIQPX
      DUHIQUIRZDXHXCBWJOXEWJSZXGXBCXJXFXAOXEWJWTUJUKULUQUMXBWSCWKXDWKQPXIAEWJOU
      NRWTWKSXAWLOWTWKWJUOUKUPRFWLURUSWNWRFNWNWMNPZWRXKWNWMISZIWMPZUEWRWMUTWNXL
      WRXMWNXLVAZWQWPXNWLISZWQXLWNXOXLWNWLILMXOWMIWLLVBWLVCVDVEWPEWJUBZISWQEWJU
      GZXOWQWPEWJVFWLXPIAEWJVGZVHGKZWJPGTWQXQVIGIUHVJWQEGWJVKRVLVMVNWNXMVAHKZWL
      PZHTZWRWNWMWLLMXSWMPGTYBXMWLWMVOGIWMVPGHWMWLVQVRYAWRHWRXTXPWLXTXPPZWPWQYC
      XTWJPWPWPWPEXTWJEKXTSWPVSWBVTWAXRWCWDWEWFWGWHWIR $.
  $}

  ${
    $d v w x y z $.  $d A x y z $.  $d ps x $.  $d ch x $.  $d th x $.
    $d ta x $.  $d ph v w y z $.
    findcard.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    findcard.2 $e |- ( x = ( y \ { z } ) -> ( ph <-> ch ) ) $.
    findcard.3 $e |- ( x = y -> ( ph <-> th ) ) $.
    findcard.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    findcard.5 $e |- ps $.
    findcard.6 $e |- ( y e. Fin -> ( A. z e. y ch -> th ) ) $.
    $( Schema for induction on the cardinality of a finite set.  The inductive
       hypothesis is that the result is true on the given set with any one
       element removed.  The result is then proven to be true for all finite
       sets.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    findcard $p |- ( A e. Fin -> ta ) $=
      ( vw wcel cen wbr com vv cfn cv wrex isfi wi wal csuc breq2 imbi1d albidv
      c0 wceq en0 mpbiri sylbi ax-gen w3a wa peano2 rspcev sylan sylibr 3adant2
      wral csn cdif dif1en 3expa cvv vex difexg ax-mp imbi12d syl5com ralrimdva
      breq1 spcv an32s 3impa sylc 3exp alrimdv cbvalv imbitrrdi finds1 rexlimiv
      imp 19.21bi vtoclga ) AEFIUBMFUCZUBQWKPUCZRSZPTUDAPWKUEWMAPTWLTQWMAUFZFWN
      FUGWKULRSZAUFZFUGWKUAUCZRSZAUFZFUGZWKWQUHZRSZAUFZFUGZPUAWLULUMZWNWPFXEWMW
      OAWLULWKRUIUJUKWLWQUMZWNWSFXFWMWRAWLWQWKRUIUJUKWLXAUMZWNXCFXGWMXBAWLXAWKR
      UIUJUKWPFWOWKULUMZAWKUNXHABNJUOUPUQWQTQZWTGUCZXARSZDUFZGUGXDXIWTXLGXIWTXK
      DXIWTXKURXJUBQZCHXJVEZDXIXKXMWTXIXKUSZXJWLRSZPTUDZXMXIXATQXKXQWQUTXPXKPXA
      TWLXAXJRUIVAVBPXJUEVCVDXIWTXKXNXIXKWTXNXOWTXNXOWTCHXJXOHUCZXJQZUSXJXRVFZV
      GZWQRSZWTCXIXKXSYBXJWQXRVHVIWSYBCUFFYAXJVJQYAVJQGVKXJXTVJVLVMWKYAUMWRYBAC
      WKYAWQRVQKVNVRVOVPWHVSVTOWAWBWCXCXLFGWKXJUMXBXKADWKXJXARVQLVNWDWEWFWIWGUP
      WJ $.
  $}

  ${
    $d v w x y z $.  $d A w x y z $.  $d ps x $.  $d ch x $.  $d th x $.
    $d ta x $.  $d ph v w y z $.
    findcard2.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    findcard2.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    findcard2.3 $e |- ( x = ( y u. { z } ) -> ( ph <-> th ) ) $.
    findcard2.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    findcard2.5 $e |- ps $.
    findcard2.6 $e |- ( y e. Fin -> ( ch -> th ) ) $.
    $( Schema for induction on the cardinality of a finite set.  The inductive
       step shows that the result is true if one more element is added to the
       set.  The result is then proven to be true for all finite sets.
       (Contributed by Jeff Madsen, 8-Jul-2010.) $)
    findcard2 $p |- ( A e. Fin -> ta ) $=
      ( vw wcel cen wi c0 vv cfn wbr com wrex isfi wal csuc breq2 imbi1d albidv
      cv wceq en0 mpbiri sylbi ax-gen wsbc wne wa peano3 adantr breq1 anbi2d wb
      peano1 peano2 nneneq sylancr biimpa eqcomd biimtrdi com12 necon3d mpd wex
      ex nnfi syl enfi adantl mpbird fin0 csn simpll dif1en 3expa cun fidifsnid
      sylan cvv vex difexg ax-mp uneq1 sbceq1d imbi2d imbi12d spv sylibr pm2.27
      cdif rspe sylsyld syl5 snex sbcie imbitrrdi vtocl dfsbcq imbitrid exlimdv
      unex mp2and sylbid mpdd com23 alrimdv nfsbc1v nfim sbceq1a finds1 19.21bi
      nfv cbval rexlimiv vtoclga ) AEFIUBMFULZUBQYHPULZRUCZPUDUEAPYHUFYJAPUDYIU
      DQYJASZFYKFUGYHTRUCZASZFUGYHUAULZRUCZASZFUGZYHYNUHZRUCZASZFUGZPUAYITUMZYK
      YMFUUBYJYLAYITYHRUIUJUKYIYNUMZYKYPFUUCYJYOAYIYNYHRUIUJUKYIYRUMZYKYTFUUDYJ
      YSAYIYRYHRUIUJUKYMFYLYHTUMZAYHUNUUEABNJUOUPUQYNUDQZYQYIYRRUCZAFYIURZSZPUG
      UUAUUFYQUUIPUUFUUGYQUUHUUFUUGYITUSZYQUUHSZUUFUUGUUJUUFUUGUTZYRTUSZUUJUUFU
      UMUUGYNVAVBUULYITYRTUUBUULYRTUMZUUBUULUUFTYRRUCZUTZUUNUUBUUGUUOUUFYITYRRV
      CVDUUPTYRUUFUUOTYRUMZUUFTUDQYRUDQZUUOUUQVEVFYNVGZTYRVHVIVJVKVLVMVNVOVQUUF
      UUGUUJUUKSUULUUJHULZYIQZHVPZUUKUULYIUBQZUUJUVBVEUULUVCYRUBQZUUFUVDUUGUUFU
      URUVDUUSYRVRVSVBUUGUVCUVDVEUUFYIYRVTWAWBZHYIWCVSUULUVAUUKHUULUVAUUKUULUVA
      UTZUUFYIUUTWDZXBZYNRUCZUUKUUFUUGUVAWEUUFUUGUVAUVIYIYNUUTWFWGUVFUVHUVGWHZY
      IUMZUUFUVIUTZUUKSUULUVCUVAUVKUVEYIUUTWIWJUVLYQAFUVJURZSZUVKUUKUUFGULZYNRU
      CZUTZYQAFUVOUVGWHZURZSZSUVLUVNSGUVHYIWKQUVHWKQPWLYIUVGWKWMWNUVOUVHUMZUVQU
      VLUVTUVNUWAUVPUVIUUFUVOUVHYNRVCVDUWAUVSUVMYQUWAAFUVRUVJUVOUVHUVGWOWPWQWRU
      VQYQDUVSYQUVPCSZUVQDYPUWBFGYHUVOUMYOUVPACYHUVOYNRVCKWRWSUVQUVOUBQZUWBCDUV
      QUVPUAUDUEUWCUVPUAUDXCUAUVOUFWTUVPUWBCSUUFUVPCXAWAOXDXEADFUVRUVOUVGGWLUUT
      HWLXFXMLXGXHXIUVKUVMUUHYQAFUVJYIXJWQXKVSXNVQXLXOVQXPXQXRYTUUIFPYTPYDUUGUU
      HFUUGFYDAFYIXSXTYHYIUMYSUUGAUUHYHYIYRRVCAFYIYAWRYEXHYBYCYFUPYG $.
  $}

  ${
    $d A x y z $.  $d ch x $.  $d ph v w y z $.  $d ps x $.  $d ta x $.
    $d th x $.  $d v w x y z $.
    findcard2s.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    findcard2s.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    findcard2s.3 $e |- ( x = ( y u. { z } ) -> ( ph <-> th ) ) $.
    findcard2s.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    findcard2s.5 $e |- ps $.
    findcard2s.6 $e |- ( ( y e. Fin /\ -. z e. y ) -> ( ch -> th ) ) $.
    $( Variation of ~ findcard2 requiring that the element added in the
       induction step not be a member of the original set.  (Contributed by
       Paul Chapman, 30-Nov-2012.) $)
    findcard2s $p |- ( A e. Fin -> ta ) $=
      ( vw wcel cen wi c0 vv cfn wbr com wrex isfi wal csuc breq2 imbi1d albidv
      cv wceq en0 mpbiri sylbi ax-gen wsbc wne wa peano3 adantr breq1 anbi2d wb
      peano1 peano2 nneneq sylancr biimpa eqcomd biimtrdi com12 necon3d mpd wex
      ex nnfi syl enfi adantl mpbird fin0 csn simpll dif1en 3expa cun fidifsnid
      sylan wn neldifsn cvv vex difexg ax-mp eleq2 notbid anbi12d uneq1 sbceq1d
      cdif imbi2d imbi12d spv pm2.27 rspe sylibr syld syl5 snex sbcie imbitrrdi
      unex vtocl mpan2 dfsbcq imbitrid mp2and exlimdv sylbid mpdd com23 alrimdv
      nfv nfsbc1v nfim sbceq1a cbval finds1 19.21bi rexlimiv vtoclga ) AEFIUBMF
      ULZUBQYNPULZRUCZPUDUEAPYNUFYPAPUDYOUDQYPASZFYQFUGYNTRUCZASZFUGYNUAULZRUCZ
      ASZFUGZYNYTUHZRUCZASZFUGZPUAYOTUMZYQYSFUUHYPYRAYOTYNRUIUJUKYOYTUMZYQUUBFU
      UIYPUUAAYOYTYNRUIUJUKYOUUDUMZYQUUFFUUJYPUUEAYOUUDYNRUIUJUKYSFYRYNTUMZAYNU
      NUUKABNJUOUPUQYTUDQZUUCYOUUDRUCZAFYOURZSZPUGUUGUULUUCUUOPUULUUMUUCUUNUULU
      UMYOTUSZUUCUUNSZUULUUMUUPUULUUMUTZUUDTUSZUUPUULUUSUUMYTVAVBUURYOTUUDTUUHU
      URUUDTUMZUUHUURUULTUUDRUCZUTZUUTUUHUUMUVAUULYOTUUDRVCVDUVBTUUDUULUVATUUDU
      MZUULTUDQUUDUDQZUVAUVCVEVFYTVGZTUUDVHVIVJVKVLVMVNVOVQUULUUMUUPUUQSUURUUPH
      ULZYOQZHVPZUUQUURYOUBQZUUPUVHVEUURUVIUUDUBQZUULUVJUUMUULUVDUVJUVEUUDVRVSV
      BUUMUVIUVJVEUULYOUUDVTWAWBZHYOWCVSUURUVGUUQHUURUVGUUQUURUVGUTZUULYOUVFWDZ
      XBZYTRUCZUUQUULUUMUVGWEUULUUMUVGUVOYOYTUVFWFWGUVLUVNUVMWHZYOUMZUULUVOUTZU
      UQSUURUVIUVGUVQUVKYOUVFWIWJUVRUUCAFUVPURZSZUVQUUQUVRUVFUVNQZWKZUVTUVFYOWL
      UULGULZYTRUCZUTZUVFUWCQZWKZUTZUUCAFUWCUVMWHZURZSZSUVRUWBUTZUVTSGUVNYOWMQU
      VNWMQPWNYOUVMWMWOWPUWCUVNUMZUWHUWLUWKUVTUWMUWEUVRUWGUWBUWMUWDUVOUULUWCUVN
      YTRVCVDUWMUWFUWAUWCUVNUVFWQWRWSUWMUWJUVSUUCUWMAFUWIUVPUWCUVNUVMWTXAXCXDUW
      HUUCDUWJUUCUWDCSZUWHDUUBUWNFGYNUWCUMUUAUWDACYNUWCYTRVCKXDXEUWHUWNCDUWEUWN
      CSZUWGUWDUWOUULUWDCXFWAVBUWEUWCUBQZUWGCDSUWEUWDUAUDUEUWPUWDUAUDXGUAUWCUFX
      HOWJXIXJADFUWIUWCUVMGWNUVFHWNXKXNLXLXMXOXPUVQUVSUUNUUCAFUVPYOXQXCXRVSXSVQ
      XTYAVQYBYCYDUUFUUOFPUUFPYEUUMUUNFUUMFYEAFYOYFYGYNYOUMUUEUUMAUUNYNYOUUDRVC
      AFYOYHXDYIXMYJYKYLUPYM $.
  $}

  ${
    $d A x y z $.  $d ph x y z $.  $d ps y z $.  $d ch x $.  $d th x $.
    $d ta x $.  $d et x $.
    findcard2d.ch $e |- ( x = (/) -> ( ps <-> ch ) ) $.
    findcard2d.th $e |- ( x = y -> ( ps <-> th ) ) $.
    findcard2d.ta $e |- ( x = ( y u. { z } ) -> ( ps <-> ta ) ) $.
    findcard2d.et $e |- ( x = A -> ( ps <-> et ) ) $.
    findcard2d.z $e |- ( ph -> ch ) $.
    findcard2d.i $e |- ( ( ph /\ ( y C_ A /\ z e. ( A \ y ) ) ) ->
        ( th -> ta ) ) $.
    findcard2d.a $e |- ( ph -> A e. Fin ) $.
    $( Deduction version of ~ findcard2 .  If you also need ` y e. Fin ` (which
       doesn't come for free due to ~ ssfiexmid ), use ~ findcard2sd instead.
       (Contributed by SO, 16-Jul-2018.) $)
    findcard2d $p |- ( ph -> et ) $=
      ( wcel wa wi wss ssid cfn adantr cv csn cun wceq sseq1 anbi2d imbi12d weq
      c0 wel wn simprl simprr unssad jca cdif id vsnid elun2 mp1i sseldd simplr
      ad2antll eldifd syl12anc embantd ex com23 findcard2s mpcom mpan2 ) AJJUAZ
      FJUBJUCRZAVPSZFAVQVPQUDAGUEZJUAZSZBTAUMJUAZSZCTAHUEZJUAZSZDTZAWDIUEZUFZUG
      ZJUAZSZETVRFTGHIJVSUMUHZWAWCBCWMVTWBAVSUMJUIUJKUKGHULZWAWFBDWNVTWEAVSWDJU
      IUJLUKVSWJUHZWAWLBEWOVTWKAVSWJJUIUJMUKVSJUHZWAVRBFWPVTVPAVSJJUIUJNUKACWBO
      UDWDUCRZIHUNUOZSZWLWGEWSWLWGETWSWLSZWFDEWTAWEWSAWKUPZWTWDWIJWSAWKUQURZUSW
      TAWEWHJWDUTRDETXAXBWTWHJWDWKWHJRWSAWKWJJWHWKVAWHWIRWHWJRWKIVBWHWIWDVCVDVE
      VGWQWRWLVFVHPVIVJVKVLVMVNVO $.
  $}

  ${
    $d A x y z $.  $d ph x y z $.  $d ps y z $.  $d ch x $.  $d th x $.
    $d ta x $.  $d et x $.
    findcard2sd.ch $e |- ( x = (/) -> ( ps <-> ch ) ) $.
    findcard2sd.th $e |- ( x = y -> ( ps <-> th ) ) $.
    findcard2sd.ta $e |- ( x = ( y u. { z } ) -> ( ps <-> ta ) ) $.
    findcard2sd.et $e |- ( x = A -> ( ps <-> et ) ) $.
    findcard2sd.z $e |- ( ph -> ch ) $.
    findcard2sd.i $e |- ( ( ( ph /\ y e. Fin ) /\
        ( y C_ A /\ z e. ( A \ y ) ) ) -> ( th -> ta ) ) $.
    findcard2sd.a $e |- ( ph -> A e. Fin ) $.
    $( Deduction form of finite set induction .  (Contributed by Jim Kingdon,
       14-Sep-2021.) $)
    findcard2sd $p |- ( ph -> et ) $=
      ( wcel wa wi wss ssid cfn adantr cv c0 csn cun wceq anbi2d imbi12d simprl
      sseq1 wn simprr unssad jca cdif simpll vsnid elun2 sseldd ad2antll simplr
      id mp1i eldifd syl22anc embantd ex com23 findcard2s mpcom mpan2 ) AJJUAZF
      JUBJUCRZAVOSZFAVPVOQUDAGUEZJUAZSZBTAUFJUAZSZCTAHUEZJUAZSZDTZAWCIUEZUGZUHZ
      JUAZSZETVQFTGHIJVRUFUIZVTWBBCWLVSWAAVRUFJUMUJKUKVRWCUIZVTWEBDWMVSWDAVRWCJ
      UMUJLUKVRWIUIZVTWKBEWNVSWJAVRWIJUMUJMUKVRJUIZVTVQBFWOVSVOAVRJJUMUJNUKACWA
      OUDWCUCRZWGWCRUNZSZWKWFEWRWKWFETWRWKSZWEDEWSAWDWRAWJULZWSWCWHJWRAWJUOUPZU
      QWSAWPWDWGJWCURRDETWTWPWQWKUSXAWSWGJWCWJWGJRWRAWJWIJWGWJVEWGWHRWGWIRWJIUT
      WGWHWCVAVFVBVCWPWQWKVDVGPVHVIVJVKVLVMVN $.
  $}

  ${
    $d A m n $.  $d A x $.  $d B m n $.  $d B x $.
    $( Subtracting a singleton from a finite set produces a finite set.
       (Contributed by Jim Kingdon, 11-Sep-2021.) $)
    diffisn $p |- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin ) $=
      ( vn vm vx cfn wcel wa cv cen wbr com wrex adantr wceq adantl simplrr en0
      c0 wb csn cdif isfi biimpi csuc wne elex2 fin0 mpbird neneqd biimpri entr
      syl2anc sylib mtand wo nn0suc orcomd ad2antrl ecased nnfi simprl ad2antll
      wex breq2 mpbid simpllr dif1en syl3anc enfii rexlimddv ) AFGZBAGZHZACIZJK
      ZABUAUBZFGZCLVLVPCLMZVMVLVSCAUCUDNVNVOLGZVPHZHZVODIZUEZOZVRDLWBWEDLMZVOSO
      ZWBWGASOZWBASVNASUFZWAVNWIEIAGEVDZVMWJVLEBAUGPVLWIWJTVMEAUHNUINUJWBWGHZAS
      JKZWHWKVPVOSJKZWLVNVTVPWGQWGWMWBWMWGVORUKPAVOSULUMARUNUOVTWFWGUPVNVPVTWGW
      FDVOUQURUSUTWBWCLGZWEHZHZWCFGZVQWCJKZVRWNWQWBWEWCVAUSWPWNAWDJKZVMWRWBWNWE
      VBWPVPWSVNVTVPWOQWEVPWSTWBWNVOWDAJVEVCVFVLVMWAWOVGAWCBVHVIVQWCVJUMVKVK $.
  $}

  ${
    $d A w y z $.  $d B w y z $.
    $( Subtracting one finite set from another produces a finite set.
       (Contributed by Jim Kingdon, 8-Sep-2021.) $)
    diffifi $p |- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) ->
        ( A \ B ) e. Fin ) $=
      ( vw vy vz cfn wcel wss cdif wa cv wi c0 wceq sseq1 anbi2d difeq2 imbi12d
      eleq1d simprr w3a simp2 simp1 simp3 csn cun dif0 eleq1i biimpri adantr wn
      difun1 simprl unssad simplr mp2and vsnid unssbd sseld mpi adantllr eldifd
      simpllr diffisn syl2anc eqeltrid exp31 findcard2s imp syl12anc ) AFGZBFGZ
      BAHZUAVLVKVMABIZFGZVKVLVMUBVKVLVMUCVKVLVMUDVLVKVMJZVOVKCKZAHZJZAVQIZFGZLV
      KMAHZJZAMIZFGZLVKDKZAHZJZAWFIZFGZLZVKWFEKZUEZUFZAHZJZAWNIZFGZLVPVOLCDEBVQ
      MNZVSWCWAWEWSVRWBVKVQMAOPWSVTWDFVQMAQSRVQWFNZVSWHWAWJWTVRWGVKVQWFAOPWTVTW
      IFVQWFAQSRVQWNNZVSWPWAWRXAVRWOVKVQWNAOPXAVTWQFVQWNAQSRVQBNZVSVPWAVOXBVRVM
      VKVQBAOPXBVTVNFVQBAQSRVKWEWBWEVKWDAFAUGUHUIUJWFFGZWLWFGUKZJZWKWPWRXEWKJZW
      PJZWQWIWMIZFAWFWMULXGWJWLWIGXHFGXGVKWGWJXFVKWOUMXGWFWMAXFVKWOTUNXEWKWPUOU
      PXGWLAWFXCWKWPWLAGZXDXCWKJZWPJZWLWMGXIEUQXKWMAWLXKWFWMAXJVKWOTURUSUTVAXCX
      DWKWPVCVBWIWLVDVEVFVGVHVIVJ $.
  $}

  ${
    $d A n $.
    $( An infinite set is not finite.  (Contributed by Jim Kingdon,
       20-Feb-2022.) $)
    infnfi $p |- ( _om ~<_ A -> -. A e. Fin ) $=
      ( vn com cdom wbr cfn wcel wa cv cen wrex isfi biimpi adantl csuc cvv wss
      wn omex ad2antrl word ordom peano2 ordelss sylancr ssdomg mpsyl ad2ant2rl
      domentr domtr syl2anc php5dom pm2.21dd rexlimddv pm2.01da ) CADEZAFGZUPUQ
      HZABIZJEZUQRZBCUQUTBCKZUPUQVBBALMNURUSCGZUTHHZUSOZUSDEZVAVDVECDEZCUSDEZVF
      CPGVDVECQZVGSVDCUAVECGZVIUBVCVJURUTUSUCTCVEUDUEVECPUFUGUPUTVHUQVCCAUSUIUH
      VECUSUJUKVCVFRURUTUSULTUMUNUO $.
  $}

  $( The set of natural numbers is not finite.  Although we supply this theorem
     because we can, the more natural way to express " ` _om ` is infinite" is
     ` _om ~<_ _om ` which is an instance of ~ domrefg .  (Contributed by NM,
     2-Jun-1998.) $)
  ominf $p |- -. _om e. Fin $=
    ( com cvv wcel cdom wbr cfn wn omex domrefg infnfi mp2b ) ABCAADEAFCGHABIAJ
    K $.

  ${
    $d A f n x $.
    $( An infinite set contains subsets of arbitrarily large finite
       cardinality.  (Contributed by Jim Kingdon, 15-Jun-2022.) $)
    isinfinf $p |- ( _om ~<_ A -> A. n e. _om E. x ( x C_ A /\ x ~~ n ) ) $=
      ( vf com cdom wbr cv wss cen wa wex wcel wf1 brdomi adantr cvv vex imaexg
      cima ax-mp crn imassrn wf simpr f1f frn 3syl sstrid word ordelss ad2antlr
      ordom mpan simplr f1imaeng syl3anc jca wceq sseq1 anbi12d spcegv exlimddv
      breq1 mpsyl ralrimiva ) EBFGZAHZBIZVHCHZJGZKZALZCEVGVJEMZKZEBDHZNZVMDVGVQ
      DLVNEBDOPVPVJTZQMZVOVQKZVRBIZVRVJJGZKZVMVPQMVSDRVPVJQSUAVTWAWBVTVRVPUBZBV
      PVJUCVTVQEBVPUDWDBIVOVQUEZEBVPUFEBVPUGUHUIVTVQVJEIZVNWBWEVNWFVGVQEUJVNWFU
      MEVJUKUNULVGVNVQUOEBVJVPEUPUQURVLWCAVRQVHVRUSVIWAVKWBVHVRBUTVHVRVJJVDVAVB
      VEVCVF $.
  $}

  ${
    $d f u w x z A $.  $d f g u w x y z B $.  $d f g u w z ph $.
    $d g u w y z ps $.
    ac6sfi.1 $e |- ( y = ( f ` x ) -> ( ph <-> ps ) ) $.
    $( Existence of a choice function for finite sets.  (Contributed by Jeff
       Hankins, 26-Jun-2009.)  (Proof shortened by Mario Carneiro,
       29-Jan-2014.) $)
    ac6sfi $p |- ( ( A e. Fin /\ A. x e. A E. y e. B ph ) ->
                E. f ( f : A --> B /\ A. x e. A ps ) ) $=
      ( wcel wral cv wf wa wex wi c0 wceq raleq anbi12d cvv vu vw vz vg cfn csn
      wrex cun cfv wsbc feq2 exbidv imbi12d feq1 vex fvex sbcie sbceq1d bitr3id
      fveq1 ralbidv cbvexv bitrdi f0 0ex ral0 biantru spcev wn wss ssun1 ssralv
      mp1i ax-mp imim1i ssun2 wb ralsnsg sbcrex bitri sylib nfv nfsbc1v nfralxy
      nfcv nfan nfex nfim w3a cop cin simprl wf1o f1osn f1of simpl2 fssd simpl1
      snssd disjsn sylibr fun2 syl21anc simprr wne eleq1a necon3bd fvunsng mpan
      impcom dfsbcq bitr2di 3syl ralbidva syl mpbid simpl3 wfun ffun cdm dmsnop
      vsnid eleqtrri funssfv mp3an23 fvsn eqtr2di elsni fveq2d biimparc sbceq1a
      eqeq2d ralun syl2anc opex snex unex ex exlimdv syl5 a2d adantl findcard2s
      3exp rexlimd imp ) EUEIADFUGZCEJZEFGKZLZBCEJZMZGNZUUGCUAKZJZUUNFUUILZBCUU
      NJZMZGNZOUUGCPJZPFUUILZBCPJZMZGNZOUUGCUBKZJZUVEFUUILZBCUVEJZMZGNZOZUUGCUV
      EUCKZUFZUHZJZUVNFUDKZLZADCKZUVPUIZUJZCUVNJZMZUDNZOZUUHUUMOUAUBUCEUUNPQZUU
      OUUTUUSUVDUUGCUUNPRUWEUURUVCGUWEUUPUVAUUQUVBUUNPFUUIUKBCUUNPRSULUMUUNUVEQ
      ZUUOUVFUUSUVJUUGCUUNUVERUWFUURUVIGUWFUUPUVGUUQUVHUUNUVEFUUIUKBCUUNUVERSUL
      UMUUNUVNQZUUOUVOUUSUWCUUGCUUNUVNRUWGUUSUVNFUUILZBCUVNJZMZGNUWCUWGUURUWJGU
      WGUUPUWHUUQUWIUUNUVNFUUIUKBCUUNUVNRSULUWJUWBGUDUUIUVPQZUWHUVQUWIUWAUVNFUU
      IUVPUNUWKBUVTCUVNBADUVRUUIUIZUJZUWKUVTABDUWLUVRUUITTGUOZCUOZUPHUQZUWKADUW
      LUVSUVRUUIUVPUTURUSVASVBVCUMUUNEQZUUOUUHUUSUUMUUGCUUNERUWQUURUULGUWQUUPUU
      JUUQUUKUUNEFUUIUKBCUUNERSULUMPFPLZUVDUUTFVDUVCUWRGPVEUVCUVAUUIPQUWRUVBUVA
      BCVFVGPFUUIPUNUSVHVMUVLUVEIZVIZUVKUWDOUVEUEIUVKUVOUVJOUWTUWDUVOUVFUVJUVEU
      VNVJUVOUVFOUVEUVMVKUUGCUVEUVNVLVNVOUWTUVOUVJUWCUVOACUVLUJZDFUGZUWTUVJUWCO
      ZUVOUUGCUVMJZUXBUVMUVNVJUVOUXDOUVMUVEVPUUGCUVMUVNVLVNUXDUUGCUVLUJZUXBUVLT
      IZUXDUXEVQUCUOZUUGCUVLTVRVNACDUVLFVSVTWAUWTUXAUXCDFUWTDWBUVJUWCDUVJDWBUWB
      DUDUVQUWADUVQDWBUVTDCUVNDUVNWEADUVSWCWDWFWGWHUWTDKZFIZUXAUXCUWTUXIUXAWIZU
      VIUWCGUXJUVIUWCUXJUVIMZUVNFUUIUVLUXHWJZUFZUHZLZADUVRUXNUIZUJZCUVNJZUWCUXK
      UVGUVMFUXMLUVEUVMWKPQZUXOUXJUVGUVHWLUXKUVMUXHUFZFUXMUVMUXTUXMWMUVMUXTUXML
      UXKUVLUXHUXGDUOZWNUVMUXTUXMWOVMUXKUXHFUWTUXIUXAUVIWPWSWQUXKUWTUXSUWTUXIUX
      AUVIWRZUVEUVLWTXAUVEUVMFUUIUXMXBXCZUXKUXQCUVEJZUXQCUVMJZUXRUXKUVHUYDUXJUV
      GUVHXDUXKUWTUVHUYDVQUYBUWTBUXQCUVEUWTUVRUVEIZMUVLUVRXEZUXPUWLQZBUXQVQUYFU
      WTUYGUYFUWSUVLUVRUVRUVEUVLXFXGXJUVRTIUYGUYHUWOUUIUVLUXHUVRTXHXIUYHUXQUWMB
      ADUXPUWLXKUWPXLXMXNXOXPUXKUXAUYEUWTUXIUXAUVIXQUXKUXHUVLUXNUIZQZUXAUYEVQUX
      KUYIUVLUXMUIZUXHUXKUXOUXNXRZUYIUYKQZUYCUVNFUXNXSUYLUXMUXNVJUVLUXMXTZIUYMU
      XMUUIVPUVLUVMUYNUCYBUVLUXHUYAYAYCUVLUXNUXMYDYEXMUVLUXHUXGUYAYFYGUXAACUVMJ
      ZUYJUYEUXFUYOUXAVQUXGACUVLTVRVNUYJAUXQCUVMUYJUVRUVMIZMUXHUXPQZAUXQVQUYPUY
      QUYJUYPUXPUYIUXHUYPUVRUVLUXNUVRUVLYHYIYLYJADUXPYKXOXNUSXOXPUXQCUVEUVMYMYN
      UWBUXOUXRMUDUXNUUIUXMUWNUXLUVLUXHUXGUYAYOYPYQUVPUXNQZUVQUXOUWAUXRUVNFUVPU
      XNUNUYRUVTUXQCUVNUYRADUVSUXPUVRUVPUXNUTURVASVHYNYRYSUUDUUEYTUUAYTUUBUUCUU
      F $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( Equinumerosity of finite sets is decidable.  (Contributed by Jim
       Kingdon, 10-Feb-2026.) $)
    fidcen $p |- ( ( A e. Fin /\ B e. Fin ) -> DECID A ~~ B ) $=
      ( vx vy cfn wcel wa cv cen wbr wdc wrex isfi biimpi wn wo simplrl simplrr
      com entr adantr ad2antlr weq simprl nndceq syl2anc exmiddc simpr breqtrrd
      syl ensymd syl2an2r ex sylan wb nneneq mpbid orim12d mpd sylibr rexlimddv
      con3d df-dc ) AEFZBEFZGZACHZIJZABIJZKZCSVDVHCSLZVEVDVKCAMNUAVFVGSFZVHGZGZ
      BDHZIJZVJDSVEVPDSLZVDVMVEVQDBMNUBVNVOSFZVPGZGZVIVIOZPZVJVTCDUCZWCOZPZWBVT
      WCKZWEVTVLVRWFVFVLVHVSQZVNVRVPUDVGVOUEUFWCUGUJVTWCVIWDWAVTWCVIVTVHWCVGBIJ
      ZVIVFVLVHVSRZVTWCGZBVGWJBVOVGIVNVRVPWCRVTWCUHUIUKAVGBTULUMVTVIWCVTVIWCVTV
      IGZVGVOIJZWCWKWHVPWLVTVGAIJVIWHVTAVGWIUKVGABTUNVNVRVPVIRVGBVOTUFVTVLVIVRW
      LWCUOWGVNVRVPVIQVGVOUPULUQUMVBURUSVIVCUTVAVA $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C y $.  $d R x y $.
    tridc.po $e |- ( ph -> R Po A ) $.
    tridc.tri $e |- ( ph
      -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) $.
    tridc.b $e |- ( ph -> B e. A ) $.
    tridc.c $e |- ( ph -> C e. A ) $.
    $( A trichotomous order is decidable.  (Contributed by Jim Kingdon,
       5-Sep-2022.) $)
    tridc $p |- ( ph -> DECID B R C ) $=
      ( wbr wceq wa wn simpr sylibr wcel adantr w3o wdc wo orcd df-dc wpo poirr
      syl2anc breq1d mtbird olcd po2nr syl12anc simplr jca mtand cv breq1 eqeq1
      wral breq2 3orbi123d eqeq2 rspc2va syl21anc mpjao3dan ) AEFGLZVFUAZEFMZFE
      GLZAVFNZVFVFOZUBZVGVJVFVKAVFPUCVFUDZQAVHNZVLVGVNVKVFVNVFFFGLZAVOOZVHADGUE
      ZFDRZVPHKDFGUFUGSVNEFFGAVHPUHUIUJVMQAVINZVLVGVSVKVFVSVFVIVFNZAVTOZVIAVQVR
      EDRZWAHKJDFEGUKULSVSVFNVIVFAVIVFUMVSVFPUNUOUJVMQAWBVRBUPZCUPZGLZWCWDMZWDW
      CGLZTZCDUSBDUSVFVHVITZJKIWHWIEWDGLZEWDMZWDEGLZTBCEFDDWCEMWEWJWFWKWGWLWCEW
      DGUQWCEWDURWCEWDGUTVAWDFMWJVFWKVHWLVIWDFEGUTWDFEVBWDFEGUQVAVCVDVE $.
  $}

  ${
    $d R x y z $.  $d A x y z $.
    fimax2gtri.po $e |- ( ph -> R Po A ) $.
    fimax2gtri.tri $e |- ( ph
      -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) $.
    fimax2gtri.fin $e |- ( ph -> A e. Fin ) $.
    fimax2gtri.n0 $e |- ( ph -> A =/= (/) ) $.

    ${
      $d A x y $.  $d R x y w $.  $d U x y w $.  $d V x y w $.  $d Z x y w $.
      $d ph w $.
      fimax2gtri.ufin $e |- ( ph -> U e. Fin ) $.
      fimax2gtri.uss $e |- ( ph -> U C_ A ) $.
      fimax2gtri.za $e |- ( ph -> Z e. A ) $.
      fimax2gtri.va $e |- ( ph -> V e. A ) $.
      fimax2gtri.vu $e |- ( ph -> -. V e. U ) $.
      fimax2gtri.zb $e |- ( ph -> A. y e. U -. Z R y ) $.
      $( Lemma for ~ fimax2gtri .  The induction step.  (Contributed by Jim
         Kingdon, 5-Sep-2022.) $)
      fimax2gtrilemstep $p |- ( ph
          -> E. x e. A A. y e. ( U u. { V } ) -. x R y ) $=
        ( wbr wn vw cif wcel cv csn cun wral wrex tridc ifcldcd wa simplr simpr
        iftrued breq1d biimpa adantllr wi wpo ad2antrr wss sseldd potr syl13anc
        adantr mp2and wceq breq2 notbid cbvralv sylib r19.21bi pm2.65da adantlr
        wb iffalsed mtbird wo wdc exmiddc syl mpjaodan ralrimiva syl2anc ralsng
        poirr mpbird ralun breq1 ralbidv rspcev ) AHGESZGHUBZDUCWMCUDZESZTZCFGU
        EZUFZUGZBUDZWNESZTZCWRUGZBDUHAWLGHDPOABCDHGEIJOPUIZUJAWPCFUGZWPCWQUGZWS
        AWMUAUDZESZTZUAFUGXEAXIUAFAXGFUCZUKZWLXIWLTZXKWLUKZXHHXGESZXMXHUKWLGXGE
        SZXNXKWLXHULAWLXHXOXJAWLUKZXHXOXPWMGXGEXPWLGHAWLUMUNZUOUPUQXMWLXOUKXNUR
        ZXHXMDEUSZHDUCZGDUCZXGDUCXRAXSXJWLIUTAXTXJWLOUTAYAXJWLPUTXMFDXGAFDVAXJW
        LNUTAXJWLULVBDHGXGEVCVDVEVFXKXNTZWLXHAYBUAFAHWNESZTZCFUGYBUAFUGRYDYBCUA
        FWNXGVGYCXNWNXGHEVHVIVJVKVLZUTVMXKXLUKXHXNXKYBXLYEVEAXLXHXNVOXJAXLUKZWM
        HXGEYFWLGHAXLUMZVPZUOVNVQAWLXLVRZXJAWLVSYIXDWLVTWAZVEWBWCXIWPUACFXGWNVG
        XHWOXGWNWMEVHVIVJVKAXFWMGESZTZAWLYLXLXPYKGGESZXPXSYAYMTAXSWLIVEAYAWLPVE
        DGEWFWDXPWMGGEXQUOVQYFYKWLYGYFWMHGEYHUOVQYJWBAYAXFYLVOPWPYLCGDWNGVGWOYK
        WNGWMEVHVIWEWAWGWPCFWQWHWDXCWSBWMDWTWMVGZXBWPCWRYNXAWOWTWMWNEWIVIWJWKWD
        $.
    $}

    ${
      $d A u v w x y $.  $d A u v x y z $.  $d R u v w x y $.
      $d R u v x y z $.  $d ph u v w $.  $d ph u v z $.
      $( A finite set has a maximum under a trichotomous order.  (Contributed
         by Jim Kingdon, 5-Sep-2022.) $)
      fimax2gtri $p |- ( ph -> E. x e. A A. y e. A -. x R y ) $=
        ( vz cv wral wrex c0 wceq raleq rexbidv wcel wa ad4antr vw vu vv wbr wn
        csn cun wex wne cfn wb fin0 syl mpbid biantru exbii sylib df-rex sylibr
        ral0 wss cdif notbid ralbidv cbvrexv wpo simp-4r simprl ad2antrr simplr
        breq1 simprr eldifad eldifbd simpr fimax2gtrilemstep rexlimdva biimtrid
        w3o ex findcard2sd ) ABKZCKZEUDZUEZCUAKZLZBDMWECNLZBDMZWECUBKZLZBDMZWEC
        WJUCKZUFUGZLZBDMZWECDLZBDMUAUBUCDWFNOWGWHBDWECWFNPQWFWJOWGWKBDWECWFWJPQ
        WFWNOWGWOBDWECWFWNPQWFDOWGWQBDWECWFDPQAWBDRZWHSZBUHZWIAWRBUHZWTADNUIZXA
        IADUJRZXBXAUKHBDULUMUNWRWSBWHWRWECUTUOUPUQWHBDURUSWLJKZWCEUDZUEZCWJLZJD
        MAWJUJRZSZWJDVAZWMDWJVBRZSZSZWPWKXGBJDWBXDOZWEXFCWJXNWDXEWBXDWCEVKVCVDV
        EXMXGWPJDXMXDDRZSZXGWPXPXGSZBCDEWJWMXDADEVFXHXLXOXGFTAWDWBWCOWCWBEUDVSC
        DLBDLXHXLXOXGGTAXCXHXLXOXGHTAXBXHXLXOXGITAXHXLXOXGVGXMXJXOXGXIXJXKVHVIX
        MXOXGVJXQWMDWJXMXKXOXGXIXJXKVLVIZVMXQWMDWJXRVNXPXGVOVPVTVQVRHWA $.
    $}
  $}

  ${
    $d A w x y z $.  $d ph w y z $.
    $( Decidability of existence, over a finite set and defined by a decidable
       proposition.  (Contributed by Jim Kingdon, 12-Jul-2022.) $)
    finexdc $p |- ( ( A e. Fin /\ A. x e. A DECID ph )
        -> DECID E. x e. A ph ) $=
      ( vw vy vz wcel wdc wa cv wrex c0 wceq rexeq dcbid wn simpr sylibr orcd
      wo cfn wral csn cun rex0 olci df-dc mpbir a1i wss cdif wsbc rexsns bitr4i
      wsb sbsbc sylib olcd rexun notbii adantr ioran sylanbrc ad2antlr mpjaodan
      exmiddc simplrr eldifad simp-4r nfs1v nfdc sbequ12 rspc sylc syl ex simpl
      findcard2sd ) CUAGZAHZBCUBZIZABDJZKZHABLKZHZABEJZKZHZABWGFJZUCZUDZKZHZABC
      KZHDEFCWCLMWDWEABWCLNOWCWGMWDWHABWCWGNOWCWLMWDWMABWCWLNOWCCMWDWOABWCCNOWF
      WBWFWEWEPZTWPWEABUEUFWEUGUHUIWBWGUAGZIZWGCUJZWJCWGUKGZIZIZWIWNXBWIIZABFUO
      ZWNXDPZXCXDIZWMWMPZTZWNXFWMXGXFWHABWKKZTZWMXFXIWHXFXDXIXCXDQXDABWJULXIABF
      UPABWJUMUNZUQURABWGWKUSZRSWMUGZRXCXEIZWHWNWHPZXNWHIZXHWNXPWMXGXPXJWMXPWHX
      IXNWHQSXLRSXMRXNXOIZXHWNXQXGWMXQXJPZXGXQXOXIPZXRXNXOQXNXSXOXNXEXSXCXEQXDX
      IXKUTUQVAWHXIVBVCWMXJXLUTRURXMRWIWHXOTXBXEWHVFVDVEXCXDHZXDXETXCWJCGWAXTXC
      WJCWGWRWSWTWIVGVHVSWAWQXAWIVIVTXTBWJCXDBABFVJVKBJWJMAXDABFVLOVMVNXDVFVOVE
      VPVSWAVQVR $.
  $}

  ${
    $d A x $.
    $( Relationship between universal and existential quantifiers over a finite
       set.  Remark in Section 2.2.1 of [Pierik], p. 8.  Although Pierik does
       not mention the decidability condition explicitly, it does say "only
       finitely many x to check" which means there must be some way of checking
       each value of x.  (Contributed by Jim Kingdon, 11-Jul-2022.) $)
    dfrex2fin $p |- ( ( A e. Fin /\ A. x e. A DECID ph )
        -> ( E. x e. A ph <-> -. A. x e. A -. ph ) ) $=
      ( cfn wcel wdc wral wa wrex wn wb finexdc dfrex2dc syl ) CDEAFBCGHABCIZFO
      AJBCGJKABCLABCMN $.
  $}

  ${
    $d A u v w $.  $d B x y $.  $d X u v w $.  $d X x y $.  $d ph u v w $.
    $d v y $.
    elssdc.b $e |- ( ph -> A. x e. B A. y e. B DECID x = y ) $.
    elssdc.x $e |- ( ph -> X e. B ) $.
    elssdc.ss $e |- ( ph -> A C_ B ) $.
    elssdc.a $e |- ( ph -> A e. Fin ) $.
    $( Membership in a finite subset of a set with decidable equality is
       decidable.  (Contributed by Jim Kingdon, 11-Feb-2026.) $)
    elssdc $p |- ( ph -> DECID X e. A ) $=
      ( vv cv wcel wdc wceq eleq2 dcbid wn wo wa vw vu csn cun noel df-dc mpbir
      c0 olci a1i cfn cdif vsnid wb eleq1 adantl mpbiri elun2 orcd sylibr elun1
      wss syl simpr elsni con3i ad2antlr sylanbrc elun sylnibr exmiddc mpjaodan
      ioran olcd eqeq2 wral ralbidv rspcdva ad3antrrr simplrr eldifad sseldd ex
      eqeq1 findcard2sd ) AFUALZMZNFUHMZNZFUBLZMZNZFWJKLZUCZUDZMZNZFDMZNUAUBKDW
      FUHOWGWHWFUHFPQWFWJOWGWKWFWJFPQWFWOOWGWPWFWOFPQWFDOWGWRWFDFPQWIAWIWHWHRZS
      WSWHFUEUIWHUFUGUJAWJUKMZTZWJDVBZWMDWJULMZTZTZWLWQXEWLTZFWMOZWQXGRZXFXGTZF
      WNMZWQXIXJWMWNMZKUMXGXJXKUNXFFWMWNUOUPUQXJWPWPRZSZWQXJWPXLFWNWJURUSWPUFZU
      TVCXFXHTZWKWQWKRZWKWQXOWKXMWQWKWPXLFWJWNVAUSXNUTUPXOXPTZXMWQXQXLWPXQWKXJS
      ZWPXQXPXJRZXRRXOXPVDXHXSXFXPXJXGFWMVEVFVGWKXJVMVHFWJWNVIVJVNXNUTWLWKXPSXE
      XHWKVKVGVLXFXGNZXGXHSXFFCLZOZNZXTCEWMYAWMOYBXGYAWMFVOQAYCCEVPZWTXDWLABLZY
      AOZNZCEVPYDBEFYEFOZYGYCCEYHYFYBYEFYAWDQVQGHVRVSXFDEWMADEVBWTXDWLIVSXFWMDW
      JXAXBXCWLVTWAWBVRXGVKVCVLWCJWE $.

    ${
      $d u x $.  $d u y $.
      $( Decidability of equality between a finite subset of a set with
         decidable equality, and a singleton whose element is an element of the
         larger set.  (Contributed by Jim Kingdon, 15-Feb-2026.) $)
      eqsndc $p |- ( ph -> DECID A = { X } ) $=
        ( vu cen wbr wceq wdc wa cv c1o wcel syl csn wn wex simpr ensn1g adantr
        entr syl2anc en1 sylib weq wral wss ad2antrr vsnid mpbiri adantl sseldd
        eleq2 eqeq1 dcbid eqeq2 rspc2va syl21anc cvv sneqbg elv bitrdi exlimddv
        wb mpbird wo cfn eqeng con3dimp olcd df-dc sylibr snfig fidcen mpjaodan
        wi exmiddc ) ADFUAZLMZDWDNZOZWEUBZAWEPZDKQZUAZNZWGKWIDRLMZWLKUCWIWEWDRL
        MZWMAWEUDAWNWEAFESZWNHFEUETUFDWDRUGUHKDUIUJWIWLPZWGWJFNZOZWPWJESWOBCUKZ
        OZCEULBEULZWRWPDEWJADEUMWEWLIUNWLWJDSZWIWLXBWJWKSKUODWKWJUSUPUQURAWOWEW
        LHUNAXAWEWLGUNWTWRKCUKZOBCWJFEEBKUKWSXCBQWJCQZUTVAXDFNXCWQXDFWJVBVAVCVD
        WPWFWQWPWFWKWDNZWQWLWFXEVJWIDWKWDUTUQXEWQVJKWJFVEVFVGVHVAVKVIAWHPZWFWFU
        BZVLWGXFXGWFAWFWEADVMSZWFWEWBJDWDVMVNTVOVPWFVQVRAWEOZWEWHVLAXHWDVMSZXIJ
        AWOXJHFEVSTDWDVTUHWEWCTWA $.
    $}
  $}

  ${
    $d A f x $.
    $( An infinite set is inhabited.  (Contributed by Jim Kingdon,
       18-Feb-2022.) $)
    infm $p |- ( _om ~<_ A -> E. x x e. A ) $=
      ( vf com cdom wbr cv wf1 wcel wex brdomi wa c0 wf adantl peano1 ffvelcdmd
      cfv f1f a1i elex2 syl exlimddv ) DBEFZDBCGZHZAGBIAJZCDBCKUDUFLZMUERZBIUGU
      HDBMUEUFDBUENUDDBUESOMDIUHPTQAUIBUAUBUC $.

    $( An infinite set is not empty.  (Contributed by NM, 23-Oct-2004.) $)
    infn0 $p |- ( _om ~<_ A -> A =/= (/) ) $=
      ( vx com cdom wbr cv wcel wex c0 wne infm n0r syl ) CADEBFAGBHAIJBAKBALM
      $.
  $}

  ${
    $d ph x y z $.
    inffiexmid.1 $e |- ( x e. Fin \/ _om ~<_ x ) $.
    $( If any given set is either finite or infinite, excluded middle follows.
       For another example, ` ~P 1o ` is not infinite, by ~ pw1ninf , but also
       cannot be shown to be finite by ~ pw1fin .  (Contributed by Jim Kingdon,
       15-Jun-2022.) $)
    inffiexmid $p |- ( ph \/ -. ph ) $=
      ( vy vw vz wn wo com crab cfn wcel cdom wbr cv omex wceq wex c0 ax-mp syl
      rabex eleq1 breq2 orbi12d vtocl ominf peano1 elex2 r19.3rmv rabid2 sylbb2
      wral wb eleq1d mtbii con2i infm biidd elrab simprbi exlimiv orim12i orcom
      mpbi ) AGZAHZAVFHADIJZKLZIVHMNZHZVGBOZKLZIVLMNZHVKBVHADIPUBVLVHQVMVIVNVJV
      LVHKUCVLVHIMUDUECUFVIVFVJAAVIAIKLVIUGAIVHKAADIUMZIVHQEOILERZAVOUNSILVPUHE
      SIUITADEIUJTADIUKULUOUPUQVJFOZVHLZFRAFVHURVRAFVRVQILAAADVQIDOVQQAUSUTVAVB
      UAVCTVFAVDVE $.
  $}

  ${
    $d A f x $.  $d B f x $.  $d C f x $.
    $( Building a set with two elements.  (Contributed by FL, 11-Aug-2008.)
       (Revised by Mario Carneiro, 10-Sep-2015.) $)
    en2eqpr $p |- ( ( C ~~ 2o /\ A e. C /\ B e. C )
       -> ( A =/= B -> C = { A , B } ) ) $=
      ( vf c2o wcel wa adantr c0 c1o simplr simpr eqtr4d ad2antrr mpbid eqeltrd
      wceq syl ad3antrrr mpjaodan vx cen wbr w3a wne cpr wf1o wex bren 3ad2ant1
      cv biimpi cfv wb wf1 f1of1 adantl simpll3 f1fveq syl12anc simpllr simpll2
      prid2g prid1g wn neneqd ad4antr pm2.21dd wo wf f1of ffvelcdmd elpri df2o3
      mtbird eleq2s ffvelcdmda ex ssrdv wss prssi syl2anc eqssd exlimddv ) CEUB
      UCZACFZBCFZUDZABUEZCABUFZQZWHWIGZCEDUKZUGZWKDWHWNDUHZWIWEWFWOWGWEWOCEDUIU
      LUJHWLWNGZCWJWPUACWJWPUAUKZCFZWQWJFZWPWRGZWQWMUMZIQZWSXAJQZWTXBGZBWMUMZIQ
      ZWSXEJQZXDXFGZWQBWJXHXAXEQZWQBQZXHXAIXEWTXBXFKXDXFLMWTXIXJUNZXBXFWTCEWMUO
      ZWRWGXKWPXLWRWNXLWLCEWMUPUQZHZWPWRLZWPWGWRWEWFWGWIWNURZHZCEWQBWMUSUTZNOWT
      BWJFZXBXFWTWGXSXQABCVCRZNPXDXGGZAWMUMZIQZWSYBJQZYAYCGZWQAWJYEXAYBQZWQAQZY
      EXAIYBWTXBXGYCVAYAYCLMWTYFYGUNZXBXGYCWTXLWRWFYHXNXOWPWFWRWEWFWGWIWNVBZHZC
      EWQAWMUSUTZSOWTAWJFZXBXGYCWTWFYLYJABCVDRZSPYAYDGZYBXEQZWSYNYBJXEYAYDLXDXG
      YDKMWPYOVEZWRXBXGYDWPYOABQZWPABWHWIWNKVFWPXLWFWGYOYQUNXMYIXPCEABWMUSUTVOZ
      VGVHWPYCYDVIZWRXBXGWPYBEFYSWPCEAWMWNCEWMVJWLCEWMVKUQZYIVLYSYBIJUFZEYBIJVM
      VNVPRZSTWPXFXGVIZWRXBWPXEEFUUCWPCEBWMYTXPVLUUCXEUUAEXEIJVMVNVPRZNTWTXCGZX
      FWSXGUUEXFGZYCWSYDUUFYCGZYOWSUUGYBIXEUUFYCLUUEXFYCKMWPYPWRXCXFYCYRVGVHUUF
      YDGZWQAWJUUHYFYGUUHXAJYBWTXCXFYDVAUUFYDLMWTYHXCXFYDYKSOWTYLXCXFYDYMSPWPYS
      WRXCXFUUBSTUUEXGGZWQBWJUUIXIXJUUIXAJXEWTXCXGKUUEXGLMWTXKXCXGXRNOWTXSXCXGX
      TNPWPUUCWRXCUUDNTWTXAEFXBXCVIZWPCEWQWMYTVQUUJXAUUAEXAIJVMVNVPRTVRVSWPWFWG
      WJCVTYIXPABCWAWBWCWDVR $.
  $}

  $( Excluded middle is equivalent to the power set of ` 1o ` having two
     elements.  Remark of [PradicBrown2022], p. 2.  (Contributed by Jim
     Kingdon, 30-Jun-2022.) $)
  exmidpw $p |- ( EXMID <-> ~P 1o ~~ 2o ) $=
    ( vx wem c1o cpw c2o cen wbr cvv wcel wceq c0 csn df1o2 p0ex eqeltri cpr cv
    pwex wss wi wo wal exmid01 biimpi 19.21bi pweqi eleq2i velpw bitri vex elpr
    3imtr4g ssrdv pwpw0ss sseqtrri a1i eqssd df2o2 eqtr4di eqeng mpsyl 0nep0 wa
    wne 0ex prss mpbir en2eqpr 3expb mpan2 eleq2d 3bitr3g biimpd alrimiv sylibr
    mpi impbii ) BCDZEFGZVRHIBVREJVSCCKLZHMNORBVRKVTPZEBVRWABAVRWABAQZVTSZWBKJW
    BVTJUAZWBVRIZWBWAIZBWCWDTZABWGAUBZAUCZUDUEWEWBVTDZIWCVRWJWBCVTMUFZUGAVTUHUI
    ZWBKVTAUJUKZULUMWAVRSZBWAWJVRUNWKUOZUPUQURUSVREHUTVAVSWHBVSWGAVSWCWDVSWEWFW
    CWDVSVRWAWBVSKVTVDZVRWAJZVBVSKVRIZVTVRIZVCZWPWQTZWTWNWOKVTVRVENVFVGVSWRWSXA
    KVTVRVHVIVJVPVKWLWMVLVMVNWIVOVQ $.

  $( Excluded middle is equivalent to the power set of ` 1o ` being ` 2o ` .
     (Contributed by Jim Kingdon, 28-Jul-2024.) $)
  exmidpweq $p |- ( EXMID <-> ~P 1o = 2o ) $=
    ( vx wem c1o cpw c2o wceq c0 csn cpr cv wss wcel wal exmid01 biimpi 19.21bi
    wo wi df2o2 sylibr df1o2 pweqi eleq2i velpw bitri vex 3imtr4g ssrdv pwpw0ss
    sseqtrri a1i eqssd eqtr4di wa simpr eleqtrrdi simpl eqtrdi eleqtrd sylib ex
    elpr alrimiv impbii ) BCDZEFZBVEGGHZIZEBVEVHBAVEVHBAJZVGKZVIGFVIVGFQZVIVELZ
    VIVHLZBVJVKRZABVNAMZANZOPVLVIVGDZLZVJVEVQVICVGUAUBZUCAVGUDZUEVIGVGAUFVBZUGU
    HVHVEKBVHVQVEUIVSUJUKULSUMVFVOBVFVNAVFVJVKVFVJUNZVMVKWBVIVEVHWBVIVQVEWBVJVR
    VFVJUOVTTVSUPWBVEEVHVFVJUQSURUSWAUTVAVCVPTVD $.

  $( Excluded middle is equivalent to the power set of ` 1o ` being finite.
     (Contributed by SN and Jim Kingdon, 7-Aug-2024.) $)
  pw1fin $p |- ( EXMID <-> ~P 1o e. Fin ) $=
    ( vx wem c1o cpw cfn wcel c2o wceq exmidpweq biimpi com 2onn ax-mp eqeltrdi
    nnfi cv wss wdc df1o2 sylan2br c0 wa sseq2i velpw 1oex fidceq mp3an3 eqeq2i
    csn pwid dcbii sylib exmid1dc impbii ) BCDZEFZBUOGEBUOGHIJGKFGEFLGOMNUPAUPA
    PZUAUIZQZUBUQCHZRZUQURHZRUSUPUQCQZVACURUQSUCVCUPUQUOFZVAACUDUPVDCUOFVACUEUJ
    UOUQCUFUGTTUTVBCURUQSUHUKULUMUN $.

  $( Another equivalent of excluded middle, which is a mere reformulation of
     the definition.  (Contributed by BJ, 9-Aug-2024.) $)
  pw1dc0el $p |- ( EXMID <-> A. x e. ~P 1o DECID (/) e. x ) $=
    ( cv c0 csn wss wcel wdc wi wal c1o cpw wem wral df1o2 eqcomi sseq2i bitr4i
    velpw imbi1i albii df-exmid df-ral 3bitr4i ) ABZCDZEZCUDFGZHZAIUDJKZFZUGHZA
    ILUGAUIMUHUKAUFUJUGUFUDJEUJUEJUDJUENOPAJRQSTAUAUGAUIUBUC $.

  ${
    $d f p q x y z $.
    $( The power set of a set being equinumerous to set exponentiation with a
       base of ordinal ` 2o ` is equivalent to excluded middle.  This is
       Metamath 100 proof #52.  The forward direction uses excluded middle
       expressed as ` EXMID ` to show this equinumerosity.

       The reverse direction is the one which establishes that power set being
       equinumerous to set exponentiation implies excluded middle.  This
       resolves the question of whether we will be able to prove this
       equinumerosity theorem in the negative.  (Contributed by Jim Kingdon,
       13-Aug-2022.) $)
    exmidpw2en $p |- ( EXMID <-> A. x ~P x ~~ ( 2o ^m x ) ) $=
      ( vy vz vf vq vp wem cv cpw c2o cmap co cen wbr c0 cvv wcel pp0ex a1i c1o
      wal csn cpr cif cmpt wf1o vpwex wf cab vex mapval mapex mp2an eqeltri 0ex
      p0ex wne 0nep0 wdc wral exmidexmid ralrimivw eqid pw2f1odc mp3an12i df2o2
      f1oen2g oveq1i breqtrrdi alrimiv 1oex wceq pweq oveq2 breq12d spcv oveq2i
      df1o2 mapsnen eqbrtri entr sylancl exmidpw sylibr impbii ) GAHZIZJWFKLZMN
      ZAUAZGWIAGWGOOUBZUCZWFKLZWHMWGPQWMPQGWGWMBWGCWFCHBHQWKOUDUEUEZUFWGWMMNAUG
      WMWFWLDHUHDUIZPWLWFDRAUJZUKWFPQZWLPQWOPQWPRWFWLPPDULUMUNGBCWFOWKWNPPEFWQG
      WPSOPQGUOSWKPQGUPSOWKUQGURSGFHEHQZUSZEWGUTFWFGWSEWGWRVAVBVBWNVCVDWGWMWNPP
      VGVEJWLWFKVFVHVIVJWJTIZJMNZGWJWTJTKLZMNZXBJMNXAWIXCATVKWFTVLWGWTWHXBMWFTV
      MWFTJKVNVOVPXBJWKKLJMTWKJKVRVQJOJWLPVFRUNUOVSVTWTXBJWAWBWCWDWE $.
  $}

  ${
    $d A x $.
    $( Reformulation of ~ ss1o0el1 using ` 1o ` instead of ` { (/) } ` .
       (Contributed by BJ, 9-Aug-2024.) $)
    ss1o0el1o $p |- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) ) $=
      ( c1o wss c0 csn wcel wceq wb eqcomi sseq2i ss1o0el1 eqeq2i bitrdi sylbir
      df1o2 ) ABCADEZCZDAFZABGZHPBABPOIZJQRAPGSAKPBATLMN $.
  $}

  $( If, in the set of truth values (the powerset of 1o), equality to 1o is
     decidable, then excluded middle holds (and conversely).  (Contributed by
     BJ and Jim Kingdon, 8-Aug-2024.) $)
  pw1dc1 $p |- ( EXMID <-> A. x e. ~P 1o DECID x = 1o ) $=
    ( wem c0 cv wcel wdc c1o cpw wral pw1dc0el wss wb elpwi ss1o0el1o syl dcbid
    wceq ralbiia bitri ) BCADZEZFZAGHZITGQZFZAUCIAJUBUEAUCTUCEZUAUDUFTGKUAUDLTG
    MTNOPRS $.

  ${
    $d A m n $.  $d B m n $.
    $( Trichotomy of dominance for finite sets.  (Contributed by Jim Kingdon,
       15-Sep-2021.) $)
    fientri3 $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) ) $=
      ( vn vm cfn wcel wa cv cen wbr cdom wo wrex biimpi adantr simplrr simplrl
      com isfi syl2anc ad2antlr wss simpr wb nndomo mpbird endomtr domentr orcd
      ensymd olcd simprl nntri2or2 mpjaodan rexlimddv ) AEFZBEFZGZACHZIJZABKJZB
      AKJZLZCRUPUTCRMZUQUPVDCASNOURUSRFZUTGZGZBDHZIJZVCDRUQVIDRMZUPVFUQVJDBSNUA
      VGVHRFZVIGZGZUSVHUBZVCVHUSUBZVMVNGZVAVBVPAVHKJZVHBIJVAVPUTUSVHKJZVQVMUTVN
      URVEUTVLPZOVPVRVNVMVNUCVPVEVKVRVNUDVMVEVNURVEUTVLQZOVGVKVIVNQUSVHUETUFAUS
      VHUGTVPBVHVGVKVIVNPUJAVHBUHTUIVMVOGZVBVAWABUSKJZUSAIJVBWAVIVHUSKJZWBVGVKV
      IVOPWAWCVOVMVOUCWAVKVEWCVOUDVGVKVIVOQVMVEVOVTOVHUSUETUFBVHUSUGTWAAUSVMUTV
      OVSOUJBUSAUHTUKVMVEVKVNVOLVTVGVKVIULUSVHUMTUNUOUO $.
  $}

  ${
    $d A x y $.
    $( A natural number is well-ordered by ` _E ` .  More specifically, this
       order both satisfies ` We ` and is trichotomous.  (Contributed by Jim
       Kingdon, 25-Sep-2021.) $)
    nnwetri $p |- ( A e. _om -> ( _E We A /\
        A. x e. A A. y e. A ( x _E y \/ x = y \/ y _E x ) ) ) $=
      ( com wcel cep wwe cv wbr weq w3o wral word nnord ordwe elnn syl2anc epel
      syl wa simprl simpl simprr nntri3or biid 3orbi123i sylibr ralrimivva jca
      ) CDEZCFGZAHZBHZFIZABJZUMULFIZKZBCLACLUJCMUKCNCOSUJUQABCCUJULCEZUMCEZTZTZ
      ULDEZUMDEZUQVAURUJVBUJURUSUAUJUTUBZULCPQVAUSUJVCUJURUSUCVDUMCPQVBVCTULUME
      ZUOUMULEZKUQULUMUDUNVEUOUOUPVFABRUOUEBARUFUGQUHUI $.
  $}

  ${
    $d A x $.  $d B x $.  $d V x $.
    $( Adding a singleton to create an ordinal.  (Contributed by Jim Kingdon,
       20-Oct-2021.) $)
    onunsnss $p |- ( ( B e. V /\ ( A u. { B } ) e. On ) ->
        B C_ A ) $=
      ( vx wcel csn cun con0 wa cv elirr wceq elsni adantl simplr eqeltrrd mtoi
      ex wo snidg elun2 syl adantr wi ontr1 mpan2d imp elun sylib ecased ssrdv
      ) BCEZABFZGZHEZIZDBAUPDJZBEZUQAEZUPURIZUSUQUMEZUTVABBEZBKUTVAVBUTVAIUQBBV
      AUQBLUTUQBMNUPURVAOPRQUTUQUNEZUSVASUPURVCUPURBUNEZVCULVDUOULBUMEVDBCTBUMA
      UAUBUCUOURVDIVCUDULUQBUNUENUFUGUQAUMUHUIUJRUK $.
  $}

  ${
    $d ph x y z $.
    unfiexmid.1 $e |- ( ( x e. Fin /\ y e. Fin ) -> ( x u. y ) e. Fin ) $.
    $( If the union of any two finite sets is finite, excluded middle follows.
       Remark 8.1.17 of [AczelRathjen], p. 74.  (Contributed by Mario Carneiro
       and Jim Kingdon, 5-Mar-2022.) $)
    unfiexmid $p |- ( ph \/ -. ph ) $=
      ( vz c1o crab wceq wn wo cfn wcel csn cun cv wral con0 c0 ax-mp com df-pr
      wdc cpr rgen2a df1o2 rabeq ordtriexmidlem eqeltri snfig 1onn uneq1 eleq1d
      wi uneq2 rspc2v mp2an elexi prid1 prid2 fidceq mp3an exmiddc eqeq2i biidd
      0ex rabsnt sylbi wa cab df-rab iba abbi2dv eqtr4id con3i orim12i ) AEFGZF
      HZVQIZJZAAIZJVQUBZVSVPFUCZKLVPWBLFWBLWAWBVPMZFMZNZKVPFUABOZCOZNZKLZCKPBKP
      ZWEKLZWIBCKDUDWCKLZWDKLZWJWKUMVPQLWLVPAERMZGZQFWNHVPWOHUEAEFWNUFSAEUGUHZV
      PQUISFTLWMUJFTUISWIWKWCWGNZKLBCWCWDKKWFWCHWHWQKWFWCWGUKULWGWDHWQWEKWGWDWC
      UNULUOUPSUHVPFVPQWPUQURVPFFTUJUQUSWBVPFUTVAVQVBSVQAVRVTVQVPWNHAFWNVPUEVCA
      AEFRVEEOZRHAVDVFVGAVQAVPWRFLZAVHZEVIFAEFVJAWTEFAWSVKVLVMVNVOS $.
  $}

  ${
    $d A m n $.  $d B m n $.  $d V n $.
    $( Adding a singleton to a finite set yields a finite set.  (Contributed by
       Jim Kingdon, 3-Feb-2022.) $)
    unsnfi $p |- ( ( A e. Fin /\ B e. V /\ -. B e. A )
        -> ( A u. { B } ) e. Fin ) $=
      ( vn vm cfn wcel wn cv cen wbr csn cun com wrex isfi wa ad2antrl cin c0
      w3a biimpi 3ad2ant1 csuc peano2 simprr simpl2 simprl en2sn syl2anc disjsn
      wceq biimpri 3ad2ant3 adantr word nnord ordirr syl sylibr syl22anc df-suc
      unen breqtrrdi breq2 rspcev rexlimddv ) AFGZBCGZBAGHZUAZADIZJKZABLZMZFGZD
      NVHVIVMDNOZVJVHVQDAPUBUCVKVLNGZVMQZQZVOEIZJKZENOZVPVTVLUDZNGZVOWDJKZWCVRW
      EVKVMVLUERVTVOVLVLLZMZWDJVTVMVNWGJKZAVNSTULZVLWGSTULZVOWHJKVKVRVMUFVTVIVR
      WIVHVIVJVSUGVKVRVMUHBVLCNUIUJVKWJVSVJVHWJVIWJVJABUKUMUNUOVRWKVKVMVRVLVLGH
      ZWKVRVLUPWLVLUQVLURUSVLVLUKUTRAVLVNWGVCVAVLVBVDWBWFEWDNWAWDVOJVEVFUJEVOPU
      TVG $.
  $}

  ${
    $d A m n $.  $d B m n $.
    $( The ` B e. V ` condition in ~ unsnfi .  This is intended to show that
       ~ unsnfi without that condition would not be provable but it probably
       would need to be strengthened (for example, to imply included middle) to
       fully show that.  (Contributed by Jim Kingdon, 6-Feb-2022.) $)
    unsnfidcex $p |- ( ( A e. Fin /\ -. B e. A /\ ( A u. { B } ) e. Fin )
        -> DECID -. B e. _V ) $=
      ( vn vm cfn wcel wn cun cvv wo wdc cen wbr com biimpi wa ad2antrr syl2anc
      wceq c0 csn cv wrex isfi 3ad2ant1 3ad2ant3 adantr simprr ad3antrrr simplr
      w3a breqtrrd ensymd entr cdif simp1 ad4antr simpr eldifd php5fin pm2.65da
      simp2 orcd snprc uneq2d un0 eqtrdi adantl eqbrtrrd wb simprl nneneq mpbid
      simplrl eqcomd olcd nndceq exmiddc syl mpjaodan rexlimddv df-dc sylibr )
      AEFZBAFGZABUAZHZEFZUKZBIFZGZWKGZJZWKKWIACUBZLMZWMCNWDWEWOCNUCZWHWDWPCAUDO
      UEWIWNNFZWOPZPZWGDUBZLMZWMDNWIXADNUCZWRWHWDXBWEWHXBDWGUDOUFUGWSWTNFZXAPZP
      ZWTWNSZWMXFGZXEXFPZWKWLXHWJAWGLMZXHWJPZAWTLMZWTWGLMXIXJAWNWTLWSWOXDXFWJWI
      WQWOUHZUIXEXFWJUJULXJWGWTXEXAXFWJWSXCXAUHZQUMAWTWGUNRXJWDBIAUOFXIGWIWDWRX
      DXFWJWDWEWHUPUQXJBIAXHWJURWIWEWRXDXFWJWDWEWHVBUQUSABUTRVAVCXEXGPZWLWKXNWK
      XFXNWKPZWNWTXOWNWTLMZWNWTSZXOWNALMXKXPXOAWNWSWOXDXGWKXLUIUMXOWGAWTLWKWGAS
      XNWKWGATHAWKWFTAWKWFTSBVDOVEAVFVGVHXEXAXGWKXMQVIWNAWTUNRXOWQXCXPXQVJXEWQX
      GWKWIWQWOXDVNZQXEXCXGWKWSXCXAVKZQWNWTVLRVMVOXEXGWKUJVAVPXEXFKZXFXGJXEXCWQ
      XTXSXRWTWNVQRXFVRVSVTWAWAWKWBWC $.
  $}

  ${
    $d A m n $.  $d B m n $.  $d V m n $.
    $( The ` -. B e. A ` condition in ~ unsnfi .  This is intended to show that
       ~ unsnfi without that condition would not be provable but it probably
       would need to be strengthened (for example, to imply included middle) to
       fully show that.  (Contributed by Jim Kingdon, 6-Feb-2022.) $)
    unsnfidcel $p |- ( ( A e. Fin /\ B e. V /\ ( A u. { B } ) e. Fin )
        -> DECID -. B e. A ) $=
      ( vn vm cfn wcel wn wo wdc cv cen wbr com wrex ad2antrr ad3antrrr syl2anc
      wa ensymd csn cun isfi biimpi 3ad2ant1 3ad2ant3 adantr wceq simprr simplr
      w3a breqtrrd entr cdif simp1 ad4antr simpl2 elexd eldifd php5fin pm2.65da
      cvv simpr olcd wb wss snssi ssequn2 sylib breq1d adantl mpbid simprl orcd
      nneneq nndceq exmiddc syl mpjaodan rexlimddv df-dc sylibr ) AFGZBCGZABUAZ
      UBZFGZUKZBAGZHZWJHZIZWJJWHADKZLMZWLDNWCWDWNDNOZWGWCWODAUCUDUEWHWMNGZWNSZS
      ZWFEKZLMZWLENWHWTENOZWQWGWCXAWDWGXAEWFUCUDUFUGWRWSNGZWTSZSZWSWMUHZWLXEHZX
      DXESZWKWJXGWJAWFLMZXGWJSZWFAXIWTWSALMZWFALMXDWTXEWJWRXBWTUIZPXIAWSXIAWMWS
      LWRWNXCXEWJWHWPWNUIZQXDXEWJUJULTWFWSAUMRTXIWCBVBAUNGXHHWHWCWQXCXEWJWCWDWG
      UOUPXIBVBAXIBCWRWDXCXEWJWCWDWGWQUQQURXGWJVCUSABUTRVAVDXDXFSZWJWKXMWIXEXMW
      ISZWSWMLMZXEXNXJWNXOXNAWSXNWTAWSLMZXDWTXFWIXKPWIWTXPVEXMWIWFAWSLWIWEAVFWF
      AUHBAVGWEAVHVIVJVKVLTWRWNXCXFWIXLQWSAWMUMRXNXBWPXOXEVEXDXBXFWIWRXBWTVMZPW
      RWPXCXFWIWHWPWNVMZQWSWMVORVLXDXFWIUJVAVNXDXEJZXEXFIXDXBWPXSXQWRWPXCXRUGWS
      WMVPRXEVQVRVSVTVTWJWAWB $.
  $}

  ${
    $d A w y z $.  $d B w y z $.
    $( The union of two disjoint finite sets is finite.  (Contributed by Jim
       Kingdon, 25-Feb-2022.) $)
    unfidisj $p |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) )
        -> ( A u. B ) e. Fin ) $=
      ( vw vy vz cfn wcel cin c0 wceq w3a cv cun csn uneq2 eleq1d un0 wa cvv wn
      simp1 eqeltrid wss unass simpr vex a1i wo simplrr eldifad simp3 ad3antrrr
      cdif minel syl2anc eldifbd sylanbrc elun sylnibr unsnfi syl3anc eqeltrrid
      ioran ex simp2 findcard2sd ) AFGZBFGZABHIJZKZACLZMZFGAIMZFGADLZMZFGZAVNEL
      ZNZMZMZFGZABMZFGCDEBVKIJVLVMFVKIAOPVKVNJVLVOFVKVNAOPVKVSJVLVTFVKVSAOPVKBJ
      VLWBFVKBAOPVJVMAFAQVGVHVIUAUBVJVNFGZRZVNBUCZVQBVNUMGZRZRZVPWAWHVPRZVTVOVR
      MZFAVNVRUDWIVPVQSGZVQVOGZTWJFGWHVPUEWKWIEUFUGWIVQAGZVQVNGZUHZWLWIWMTZWNTW
      OTWIVQBGVIWPWIVQBVNWDWEWFVPUIZUJVJVIWCWGVPVGVHVIUKULVQBAUNUOWIVQBVNWQUPWM
      WNVCUQVQAVNURUSVOVQSUTVAVBVDVGVHVIVEVF $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( Union of complementary parts into whole and decidability.  (Contributed
       by Jim Kingdon, 17-Jun-2022.) $)
    undifdcss $p |- ( A = ( B u. ( A \ B ) ) <->
        ( B C_ A /\ A. x e. A DECID x e. B ) ) $=
      ( vy cdif cun wceq wss cv wcel wdc wral wa eqimss2 sylibr wn wo syl simpr
      simplr undifss eleq2 biimpa sylib eldifn orim2i df-dc ralrimiva jca elun1
      adantl eldifd elun2 eleq1 dcbid rspcdva exmiddc mpjaodan ex biimpi adantr
      elun ssrdv eqssd impbii ) BCBCEZFZGZCBHZAIZCJZKZABLZMZVHVIVMVHVGBHZVIVGBN
      CBUAZOVHVLABVHVJBJZMZVKVKPZQZVLVRVKVJVFJZQZVTVRVJVGJZWBVHVQWCBVGVJUBUCVJC
      VFVBUDWAVSVKVJBCUEUFRVKUGOUHUIVNBVGVNDBVGVNDIZBJZWDVGJZVNWEMZWDCJZWFWHPZW
      HWFWGWDCVFUJUKWGWIMZWDVFJWFWJWDBCVNWEWITWGWISULWDVFCUMRWGWHKZWHWIQWGVLWKA
      BWDVJWDGVKWHVJWDCUNUOVIVMWETVNWESUPWHUQRURUSVCVIVOVMVIVOVPUTVAVDVE $.
  $}

  ${
    $d A v w x y z $.  $d B v w y z $.

    $( Union of complementary parts into whole.  This is a case where we can
       strengthen ~ undifss from subset to equality.  (Contributed by Jim
       Kingdon, 17-Jun-2022.) $)
    undifdc $p |- ( ( A. x e. A A. y e. A DECID x = y /\ B e. Fin /\ B C_ A )
        -> A = ( B u. ( A \ B ) ) ) $=
      ( vw cv wceq wcel wss cun c0 id difeq2 uneq12d eqeq2d uncom wa cin eqtrid
      cdif vv vz wdc wral cfn w3a csn un0 dif0 3eqtr3ri a1i difundi uneq2i undi
      eqtri uneq1i unass simp3 simplrr eldifad sseldd snssd ssequn1 sylib simpr
      ad3antrrr uneq2d eqtr3d eqtr4id simp1 syl2anc simplrl sstrd eqtrd ineq12d
      dcdifsnid inidm eqtr2di ex simp2 findcard2sd ) AFBFGUCBCUDACUDZDUEHZDCIZU
      FZCEFZCWFTZJZGCKCKTZJZGZCUAFZCWLTZJZGZCWLUBFZUGZJZCWRTZJZGZCDCDTZJZGEUAUB
      DWFKGZWHWJCXDWFKWGWIXDLWFKCMNOWFWLGZWHWNCXEWFWLWGWMXELWFWLCMNOWFWRGZWHWTC
      XFWFWRWGWSXFLWFWRCMNOWFDGZWHXCCXGWFDWGXBXGLWFDCMNOWKWEWIKJWIWJCWIUHWIKPCU
      IUJUKWEWLUEHZQZWLDIZWPDWLTHZQZQZWOXAXMWOQZWTCCRZCXNWTWRWMJZWRCWQTZJZRZXOW
      TWRWMXQRZJXSWSXTWRCWLWQULUMWRWMXQUNUOXNXPCXRCXNXPWQWNJZCXPWQWLJZWMJYAWRYB
      WMWLWQPUPWQWLWMUQUOXNWQCJZCYAXNWQCIYCCGXNWPCXNDCWPWEWDXHXLWOWBWCWDURVFZXN
      WPDWLXIXJXKWOUSUTVAZVBWQCVCVDXNCWNWQXMWOVEVGVHVIXNXRWLCJZCXNXRWLWQXQJZJYF
      WLWQXQUQXNYGCWLXNYGXQWQJZCWQXQPXNWBWPCHYHCGWEWBXHXLWOWBWCWDVJVFYEABCWPVPV
      KSVGSXNWLCIYFCGXNWLDCXIXJXKWOVLYDVMWLCVCVDVNVOSCVQVRVSWBWCWDVTWA $.
  $}

  ${
    $d A x y $.  $d B y $.
    $( Union of complementary parts into whole.  This is a case where we can
       strengthen ~ undifss from subset to equality.  (Contributed by Jim
       Kingdon, 2-Mar-2022.) $)
    undiffi $p |- ( ( A e. Fin /\ B e. Fin /\ B C_ A )
        -> A = ( B u. ( A \ B ) ) ) $=
      ( vx vy cfn wcel cv wceq wdc wral wss cun fidceq 3expb ralrimivva undifdc
      cdif syl3an1 ) AEFZCGZDGZHIZDAJCAJBEFBAKABABQLHSUBCDAASTAFUAAFUBATUAMNOCD
      ABPR $.
  $}

  $( The union of two finite sets is finite if their intersection is.
     (Contributed by Jim Kingdon, 2-Mar-2022.) $)
  unfiin $p |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) e. Fin )
      -> ( A u. B ) e. Fin ) $=
    ( cfn wcel cin cun wa cdif wss simpll a1i undiffi syl3anc c0 diffifi eqtr3i
    wceq ineq2i disjdif unfidisj simpr inss1 simplr inss2 uneq12d eqtr4di incom
    unundi difeq2i difin difss ssdisj mp2an eqtri difundir eqeltrd 3impa ) ACDZ
    BCDZABEZCDZABFZCDURUSGZVAGZVBUTAUTHZBUTHZFZFZCVDVBUTVEFZUTVFFZFVHVDAVIBVJVD
    URVAUTAIZAVIQURUSVAJZVCVAUAZVKVDABUBKZAUTLMVDUSVAUTBIZBVJQURUSVAUCZVMVOVDAB
    UDKZBUTLMUEUTVEVFUHUFVDVAVGCDZUTVGEZNQZVHCDVMVDVECDZVFCDZVEVFEZNQZVRVDURVAV
    KWAVLVMVNAUTOMVDUSVAVOWBVPVMVQBUTOMWDVDWCVEBAHZEZNVFWEVEBBAEZHVFWEWGUTBBAUG
    UIBAUJPRVEAIAWEENQWFNQAUTUKABSVEAWEULUMUNKVEVFTMVTVDUTVBUTHZEVSNWHVGUTABUTU
    ORUTVBSPKUTVGTMUPUQ $.

  $( A pair is finite if it consists of two unequal sets.  For the case where
     ` A = B ` , see ~ snfig .  For the cases where one or both is a proper
     class, see ~ prprc1 , ~ prprc2 , or ~ prprc .  (Contributed by Jim
     Kingdon, 31-May-2022.) $)
  prfidisj $p |- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. Fin ) $=
    ( wcel wne w3a cpr csn cun cfn df-pr cin wceq snfig disjsn2 unfidisj syl3an
    c0 eqeltrid ) ACEZBDEZABFZGABHAIZBIZJZKABLUAUDKEUBUEKEUCUDUEMSNUFKEACOBDOAB
    PUDUEQRT $.

  ${
    $d A x y $.  $d B y $.  $d C x y $.
    prfidceq.a $e |- ( ph -> A e. C ) $.
    prfidceq.b $e |- ( ph -> B e. C ) $.
    prfidceq.dc $e |- ( ph -> A. x e. C A. y e. C DECID x = y ) $.
    $( A pair is finite if it consists of elements of a class with decidable
       equality.  (Contributed by Jim Kingdon, 13-Oct-2025.) $)
    prfidceq $p |- ( ph -> { A , B } e. Fin ) $=
      ( wceq cpr cfn wcel wn wa syl wdc cv wral dcbid csn snfig adantr wb dfsn2
      preq2 eqtrid eleq1d adantl mpbid wne neqne prfidisj syl2an3an wo wi eqeq1
      eqeq2 rspc2v syl2anc mpd exmiddc mpjaodan ) ADEJZDEKZLMZVDNZAVDODUAZLMZVF
      AVIVDADFMZVIGDFUBPUCVDVIVFUDAVDVHVELVDVHDDKVEDUEDEDUFUGUHUIUJAVJEFMZVGDEU
      KVFGHDEULDEFFUMUNAVDQZVDVGUOABRZCRZJZQZCFSBFSZVLIAVJVKVQVLUPGHVPVLDVNJZQB
      CDEFFVMDJVOVRVMDVNUQTVNEJVRVDVNEDURTUSUTVAVDVBPVC $.
  $}

  ${
    tpfidisj.a $e |- ( ph -> A e. V ) $.
    tpfidisj.b $e |- ( ph -> B e. W ) $.
    tpfidisj.c $e |- ( ph -> C e. X ) $.
    tpfidisj.ab $e |- ( ph -> A =/= B ) $.
    tpfidisj.ac $e |- ( ph -> A =/= C ) $.
    tpfidisj.bc $e |- ( ph -> B =/= C ) $.
    $( A triple is finite if it consists of three unequal sets.  (Contributed
       by Jim Kingdon, 1-Oct-2022.) $)
    tpfidisj $p |- ( ph -> { A , B , C } e. Fin ) $=
      ( ctp cpr csn cfn wcel syl3anc necomd cun df-tp cin c0 wne prfidisj snfig
      wceq syl wn nelprd disjsn sylibr unfidisj eqeltrid ) ABCDNBCOZDPZUAZQBCDU
      BAUPQRZUQQRZUPUQUCUDUHZURQRABERCFRBCUEUSHIKBCEFUFSADGRUTJDGUGUIADUPRUJVAA
      DBCABDLTACDMTUKUPDULUMUPUQUNSUO $.
  $}

  ${
    $d A x y $.  $d B y $.  $d C x y $.  $d D x y $.
    tpfidceq.a $e |- ( ph -> A e. D ) $.
    tpfidceq.b $e |- ( ph -> B e. D ) $.
    tpfidceq.c $e |- ( ph -> C e. D ) $.
    tpfidceq.dc $e |- ( ph -> A. x e. D A. y e. D DECID x = y ) $.
    $( A triple is finite if it consists of elements of a class with decidable
       equality.  (Contributed by Jim Kingdon, 13-Oct-2025.) $)
    tpfidceq $p |- ( ph -> { A , B , C } e. Fin ) $=
      ( csn cfn wcel wceq wb syl adantr wdc dcbid ctp cpr cun df-tp wn wa snssg
      wss biimpa ssequn2 sylib prfidceq eqeltrd simpr unsnfi syl3anc wo cv wral
      eqeq1 eqeq2 rspc2va syl21anc elsng mpbird dcun df-pr eleq2i dcbii exmiddc
      sylibr mpjaodan eqeltrid ) ADEFUADEUBZFLZUCZMDEFUDAFVNNZVPMNZVQUEZAVQUFZV
      PVNMVTVOVNUHZVPVNOAVQWAAFGNZVQWAPJFVNGUGQUIVOVNUJUKAVNMNZVQABCDEGHIKULZRU
      MAVSUFWCWBVSVRAWCVSWDRAWBVSJRAVSUNVNFGUOUPAVQSZVQVSUQAFDLZELZUCZNZSWEAWFW
      GFAFWFNZSFDOZSZAWBDGNBURZCURZOZSZCGUSBGUSZWLJHKWPWLFWNOZSZBCFDGGWMFOWOWRW
      MFWNUTTZWNDOWRWKWNDFVATVBVCAWJWKAWBWJWKPJFDGVDQTVEAFWGNZSFEOZSZAWBEGNWQXC
      JIKWPXCWSBCFEGGWTWNEOWRXBWNEFVATVBVCAXAXBAWBXAXBPJFEGVDQTVEVFVQWIVNWHFDEV
      GVHVIVKVQVJQVLVM $.
  $}

  ${
    $d x y z w v f A $.
    $( If a class is closed under pairwise intersections, then it is closed
       under nonempty finite intersections.  The converse would appear to
       require an additional condition, such as ` x ` and ` y ` not being
       equal, or ` A ` having decidable equality.

       This theorem is applicable to a topology, which (among other axioms) is
       closed under finite intersections.  Some texts use a pairwise
       intersection and some texts use a finite intersection, but most topology
       texts assume excluded middle (in which case the two intersection
       properties would be equivalent).  (Contributed by NM, 22-Sep-2002.)
       (Revised by Jim Kingdon, 14-Jan-2023.) $)
    fiintim $p |- ( A. x e. A A. y e. A ( x i^i y ) e. A ->
      A. x ( ( x C_ A /\ x =/= (/) /\ x e. Fin ) -> |^| x e. A ) ) $=
      ( vz vw vf cv cin wcel wss c0 wa wi wal cen wbr wceq syl cvv eleq1d ensym
      vv wral wne cfn cint w3a com wrex isfi breq1 anbi2d imbi1d albidv weq en0
      csuc sylib anim1i ancoms adantll wn pm3.24 pm2.21i sylan2b ax-gen a1i nfv
      df-ne nfa1 simpl wf1o wex bren cima cfv ssel wf f1of vex ffvelcdm sylancl
      sucid impel adantr adantlll imaeq2 ima0 eqtrdi inteq int0 ineq1d ssv mpbi
      sseqin2 biimprd adantl cdm f1ofun ad3antlr elelsuc wb f1odm eleq2d mpbird
      wfun jca simpr funfvima sylc ne0i crn imassrn wfn dff1o2 simp3bi sseqtrid
      ccnv sstr2 anim1d wf1 f1of1 sssucid f1imaen2g mpanr12 ensymd jctird imaex
      sseq1 neeq1 anbi12d breq2 ineq1 ineq2 ex syl6 mpd csn cun com24 sylan9 wo
      imbi12d spcv rspc2v com4r exp5c com14 imp43 0elnn ad3antrrr mpjaodan fvex
      intunsn f1ofn fnsnfv uneq2d df-suc imaeq2i imaundi eqtr2i wfo f1ofo foima
      eqtrd inteqd eqtr3id ad2antlr mpbid exp43 exlimdv biimtrid expimpd sylani
      imp alrimd finds2 sp exp4a syl5 rexlimiv sylbi com13 impd cbvral2v df-3an
      alrimiv imbi1i albii 3imtr4i ) DGZEGZHZCIZECUCDCUCZAGZCJZUWPKUDZLZUWPUEIZ
      LZUWPUFZCIZMZANUWPBGZHZCIZBCUCACUCUWQUWRUWTUGZUXCMZANUWOUXDAUWOUWSUWTUXCU
      WTUWSUWOUXCUWTUWPUXEOPZBUHUIUWSUWOUXCMZMZBUWPUJUXJUXLBUHUXJUXEUWPOPZUXEUH
      IZUXLUWPUXEUAUXNUWOUWSUXMUXCUXNUWOUWSUXMUXCUXNUWOUWSUXMLZUXCMZANZUXPUXQUW
      SKUWPOPZLZUXCMZANZUWSUBGZUWPOPZLZUXCMZANZUWSUYBUQZUWPOPZLZUXCMZANZUWOBUBU
      XEKQZUXPUXTAUYLUXOUXSUXCUYLUXMUXRUWSUXEKUWPOUKULUMUNBUBUOZUXPUYEAUYMUXOUY
      DUXCUYMUXMUYCUWSUXEUYBUWPOUKULUMUNUXEUYGQZUXPUYJAUYNUXOUYIUXCUYNUXMUYHUWS
      UXEUYGUWPOUKULUMUNUYAUWOUXTAUXSUWPKQZUWRLZUXCUWRUXRUYPUWQUXRUWRUYPUXRUYOU
      WRUXRUWPKOPUYOKUWPUAUWPUPURUSUTVAUWRUYOUYOVBZUXCUWPKVIUYOUYQLUXCUYOVCVDVE
      RVFVGUYBUHIZUWOUYFUYKMUYRUWOLZUYFUYJAUYSAVHUYEAVJUYRUWOUYFUYJMUYRUYIUYFUW
      OUXCUWSUYRUWQUYHUYFUXKMZUWQUWRVKUYRUWQUYHUYTUYHUYGUWPFGZVLZFVMUYRUWQLZUYT
      UYGUWPFVNVUCVUBUYTFVUCVUBUYFUWOUXCVUCVUBLUYFUWOLZLZVUAUYBVOZUFZUYBVUAVPZH
      ZCIZUXCVUEVUHCIZVUJUWQVUBVUDVUKUYRUWQVUBLVUKVUDUWQVUHUWPIZVUKVUBUWPCVUHVQ
      VUBUYGUWPVUAVRUYBUYGIZVULUYGUWPVUAVSUYBUBVTZWCZUYGUWPUYBVUAWAWBWDWEWFVUEU
      YBKQZVUKVUJMZKUYBIZVUPVUQVUEVUPVUFKQZVUQVUPVUFVUAKVOKUYBKVUAWGVUAWHWIVUSV
      UJVUKVUSVUIVUHCVUSVUISVUHHZVUHVUSVUGSVUHVUSVUGKUFSVUFKWJWKWIWLVUHSJVUTVUH
      QVUHWMVUHSWOWNWITWPRWQVUEVURLZVUFKUDZVUQVVAKVUAVPZVUFIZVVBVVAVUAXFZKVUAWR
      ZIZLVURVVDVVAVVEVVGVUBVVEVUCVUDVURUYGUWPVUAWSWTVVAVVGKUYGIZVURVVHVUEKUYBX
      AWQVUBVVGVVHXBVUCVUDVURVUBVVFUYGKUYGUWPVUAXCXDWTXEXGVUEVURXHUYBKVUAXIXJVU
      FVVCXKRVUEVVBVUQMZVURUWQVUBVUDVVIUYRUWQVUBUYFUWOVVIUWOVUBUYFUWQVVIUWOVUBU
      YFUWQVVBVUQVUBUYFLZUWQVVBLZVUKUWOVUJVVJVVKVUGCIZVUKUWOVUJMZMVUBVVKVUFCJZV
      VBLZUYBVUFOPZLZUYFVVLVUBVVKVVOVVPVUBUWQVVNVVBVUBVUFUWPJUWQVVNMVUBVUAXLZVU
      FUWPVUAUYBXMVUBVUAUYGXNZVUAXRXFVVRUWPQUYGUWPVUAXOXPXQVUFUWPCXSRXTVUBVUFUY
      BVUBUYGUWPVUAYAZUWPSIZVUFUYBOPZUYGUWPVUAYBAVTVVTVWALUYBUYGJUYBSIVWBUYBYCV
      UNUYGUWPUYBVUASYDYEWBYFYGUYEVVQVVLMAVUFVUAUYBFVTZYHUWPVUFQZUYDVVQUXCVVLVW
      DUWSVVOUYCVVPVWDUWQVVNUWRVVBUWPVUFCYIUWPVUFKYJYKUWPVUFUYBOYLYKVWDUXBVUGCU
      WPVUFWJTUUCUUDUUAVVLVUKVVMUWNVUJVUGUWLHZCIDEVUGVUHCCUWKVUGQUWMVWECUWKVUGU
      WLYMTUWLVUHQVWEVUICUWLVUHVUGYNTUUEYOYPUUFUUGUUHUUIWFWEYQUYRVUPVURUUBUWQVU
      BVUDUYBUUJUUKUULYQVUBVUJUXCXBVUCVUDVUBVUIUXBCVUBVUIVUFVUHYRZYSZUFUXBVUFVU
      HUYBVUASSVWCVUNUUMUUNVUBVWGUWPVUBVWGVUAUYGVOZUWPVUBVWGVUFVUAUYBYRZVOZYSZV
      WHVUBVWFVWJVUFVUBVVSVUMVWFVWJQUYGUWPVUAUUOVUOUYGUYBVUAUUPWBUUQVWHVUAUYBVW
      IYSZVOVWKUYGVWLVUAUYBUURUUSVUAUYBVWIUUTUVAWIVUBUYGUWPVUAUVBVWHUWPQUYGUWPV
      UAUVCUYGUWPVUAUVDRUVEUVFUVGTUVHUVIUVJUVKUVLUVMUVNYTUVOUVPYOUVQUXPAUVRYPUV
      SYTUVTUWAUWBUWCUWDUWGUXGUWNUWKUXEHZCIABDECCADUOUXFVWMCUWPUWKUXEYMTBEUOVWM
      UWMCUXEUWLUWKYNTUWEUXIUXDAUXHUXAUXCUWQUWRUWTUWFUWHUWIUWJ $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.
    $( The Cartesian product of two finite sets is finite.  Lemma 8.1.16 of
       [AczelRathjen], p. 74.  (Contributed by Jeff Madsen, 2-Sep-2009.)
       (Revised by Mario Carneiro, 12-Mar-2015.) $)
    xpfi $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin ) $=
      ( vx vy vz vw cfn wcel cxp cv wi c0 csn wceq xpeq1 eleq1d imbi2d cvv cin
      wa cdif 0xp 0fi eqeltri a1i wral eqeltrdi a1i13 sneq difeq2d xpeq1d rspcv
      wex adantl pm2.27 ad2antlr cen wbr c2nd cres wf1o vex snex xpexg 2ndconst
      mpan id mp1i f1oen2g syl3anc enfii mpdan cun incom disjdif eqtr3i xpdisj1
      ax-mp unfidisj mp3an3 xpundir fidifsnid adantlr eqtr3id imbitrid 3syld ex
      mpan2d exlimdv wo fin0or adantr mpjaod com23 findcard imp ) AGHBGHZABIZGH
      ZWQCJZBIZGHZKWQLBIZGHZKWQDJZEJZMZUAZBIZGHZKZWQXEBIZGHZKWQWSKCDEAWTLNZXBXD
      WQXNXAXCGWTLBOPQWTXHNZXBXJWQXOXAXIGWTXHBOPQWTXENZXBXMWQXPXAXLGWTXEBOPQWTA
      NZXBWSWQXQXAWRGWTABOPQXDWQXCLGBUBUCUDZUEXEGHZWQXKEXEUFZXMXSWQXTXMKZXSWQTZ
      XELNZYAFJZXEHZFUMZYBYCXTXMYCXLXCGXELBOXRUGUHYBYEYAFYBYEYAYBYETZXTWQXEYDMZ
      UAZBIZGHZKZYKXMYEXTYLKYBXKYLEYDXEXFYDNZXJYKWQYMXIYJGYMXHYIBYMXGYHXEXFYDUI
      UJUKPQULUNWQYLYKKXSYEWQYKUOUPYGYKYHBIZGHZXMWQYOXSYEWQYNBUQURZYOWQYNRHZWQY
      NBUSYNUTZVAZYPYHRHWQYQYDFVBZVCYHBRGVDVFWQVGYDRHYSWQYTYDBRVEVHYNBYRRGVIVJY
      NBVKVLUPYKYOTYJYNVMZGHZYGXMYKYOYJYNSLNZUUBYIYHSZLNUUCYHYISUUDLYHYIVNYHXEV
      OVPYIYHBBVQVRYJYNVSVTYGUUAXLGYGUUAYIYHVMZBIXLYIYHBWAYGUUEXEBXSYEUUEXENWQX
      EYDWBWCUKWDPWEWHWFWGWIXSYCYFWJWQFXEWKWLWMWGWNWOWP $.
  $}

  $( The image of a finite set under a one-to-one mapping is finite.
     (Contributed by Jim Kingdon, 28-Mar-2026.) $)
  imaf1fi $p |- ( ( F : A -1-1-> B /\ X C_ A /\ X e. Fin )
      -> ( F " X ) e. Fin ) $=
    ( wf1 wss cfn wcel w3a cima cen wbr simp3 f1imaeng enfii syl2anc ) ABCEZDAF
    ZDGHZISCDJZDKLTGHQRSMABDCGNTDOP $.

  $( The Cartesian product of three finite sets is a finite set.  (Contributed
     by Alexander van der Vekens, 11-Mar-2018.) $)
  3xpfi $p |- ( V e. Fin -> ( ( V X. V ) X. V ) e. Fin ) $=
    ( cxp cfn wcel xpfi anidms mpancom ) AABZCDZACDZHABCDJIAAEFHAEG $.

  ${
    $d A w x y z $.  $d B x $.  $d a x y z $.
    $( A finite set is equal to its subset if they are equinumerous.
       (Contributed by FL, 11-Aug-2008.) $)
    fisseneq $p |- ( ( B e. Fin /\ A C_ B /\ A ~~ B ) -> A = B ) $=
      ( vx vz va cfn wcel wss cen wbr cv wa wi wal c0 sseq1 anbi12d imbi12d cvv
      wceq vw vy w3a enfii 3adant2 csn cun breq1 eqeq1 albidv en0 eqcomd adantl
      ensym sylib ax-gen wn sseq2 breq2 eqeq2 cbvalv cdif simplr difsn ad3antlr
      difun2 eqtrid simprl ssdifd simplll vex a1i simpllr unsnfi syl3anc simprr
      eqsstrrd vsnid elun2 sseldd dif1enen eqbrtrrd jca difexg spcv sylc uneq1d
      ax-mp ensymd syl2anc fidifsnid eqtrd ex alrimiv biimtrid findcard2s spcgv
      syl 3simpc 3ad2ant1 mp2d ) BFGZABHZABIJZUCZACKZHZAXFIJZLZAXFTZMZCNZXCXDLZ
      ABTZXEAFGZXLXBXDXOXCABUDUEUAKZXFHZXPXFIJZLZXPXFTZMZCNOXFHZOXFIJZLZOXFTZMZ
      CNUBKZXFHZYGXFIJZLZYGXFTZMZCNZYGDKZUFZUGZXFHZYPXFIJZLZYPXFTZMZCNZXLUAUBDA
      XPOTZYAYFCUUCXSYDXTYEUUCXQYBXRYCXPOXFPXPOXFIUHQXPOXFUIRUJXPYGTZYAYLCUUDXS
      YJXTYKUUDXQYHXRYIXPYGXFPXPYGXFIUHQXPYGXFUIRUJXPYPTZYAUUACUUEXSYSXTYTUUEXQ
      YQXRYRXPYPXFPXPYPXFIUHQXPYPXFUIRUJXPATZYAXKCUUFXSXIXTXJUUFXQXGXRXHXPAXFPX
      PAXFIUHQXPAXFUIRUJYFCYCYEYBYCXFOYCXFOIJXFOTOXFUNXFUKUOULUMUPYMYGEKZHZYGUU
      GIJZLZYGUUGTZMZENZYGFGZYNYGGUQZLZUUBYLUULCEXFUUGTZYJUUJYKUUKUUQYHUUHYIUUI
      XFUUGYGURXFUUGYGIUSQXFUUGYGUTRVAUUPUUMUUBUUPUUMLZUUACUURYSYTUURYSLZYPXFYO
      VBZYOUGZXFUUSYGUUTYOUUSUUMYGUUTHZYGUUTIJZLZYGUUTTZUUPUUMYSVCUUSUVBUVCUUSY
      GYPYOVBZUUTUUSUVFYGYOVBZYGYGYOVFUUOUVGYGTUUNUUMYSYNYGVDVEVGZUUSYPXFYOUURY
      QYRVHZVIVQUUSUVFYGUUTIUVHUUSYPXFYNYNUUSUUNYNSGZUUOYPFGZUUNUUOUUMYSVJUVJUU
      SDVKVLUUNUUOUUMYSVMYGYNSVNVOZUURYQYRVPZYNYPGZUUSYNYOGUVNDVRYNYOYGVSWHVLZU
      USYPXFYNUVIUVOVTZWAWBWCUULUVDUVEMEUUTXFSGUUTSGCVKXFYOSWDWHUUGUUTTZUUJUVDU
      UKUVEUVQUUHUVBUUIUVCUUGUUTYGURUUGUUTYGIUSQUUGUUTYGUTRWEWFWGUUSXFFGZYNXFGU
      VAXFTUUSUVKXFYPIJUVRUVLUUSYPXFUVMWIXFYPUDWJUVPXFYNWKWJWLWMWNWMWOWPWRXBXCX
      DWSXBXCXLXMXNMZMXDXKUVSCBFXFBTZXIXMXJXNUVTXGXCXHXDXFBAURXFBAIUSQXFBAUTRWQ
      WTXA $.
  $}

  ${
    phpeqd.1 $e |- ( ph -> A e. Fin ) $.
    phpeqd.2 $e |- ( ph -> B C_ A ) $.
    phpeqd.3 $e |- ( ph -> A ~~ B ) $.
    $( Corollary of the Pigeonhole Principle using equality.  Strengthening of
       ~ phpm expressed without negation.  (Contributed by Rohan Ridenour,
       3-Aug-2023.) $)
    phpeqd $p |- ( ph -> A = B ) $=
      ( cfn wcel wss cen wbr wceq w3a ensymb fisseneq syl3an3br eqcomd syl3anc
      ) ABGHZCBIZBCJKZBCLDEFSTUAMCBUASTCBJKCBLCBNCBOPQR $.
  $}

  ${
    $d A w x y z $.  $d ph w y z $.  $d ps w y z $.
    ssfirab.a $e |- ( ph -> A e. Fin ) $.
    ssfirab.dc $e |- ( ph -> A. x e. A DECID ps ) $.
    $( A subset of a finite set is finite if it is defined by a decidable
       property.  (Contributed by Jim Kingdon, 27-May-2022.) $)
    ssfirab $p |- ( ph -> { x e. A | ps } e. Fin ) $=
      ( vz cv crab cfn wcel c0 cun wceq rabeq eleq1d wa wn wral cvv vw csn rab0
      vy 0fi eqeltri a1i wss cdif wsb rabun2 wsbc sbsbc vex ralsns ax-mp bitr4i
      wb rabid2 sylbb2 adantl uneq2d simplr simprr ad2antrr eldifbd elrabi nsyl
      unsnfi syl3anc eqeltrrd eqeltrid sbn rabeq0 un0 eqtrdi eqtrid eqeltrd wdc
      3bitr2ri simplrr eldifad ad3antrrr nfs1v nfdc sbequ12 dcbid rspc sylc syl
      wo exmiddc mpjaodan ex findcard2sd ) ABCUAHZIZJKBCLIZJKZBCUDHZIZJKZBCWTGH
      ZUBZMZIZJKZBCDIZJKUAUDGDWPLNWQWRJBCWPLOPWPWTNWQXAJBCWPWTOPWPXENWQXFJBCWPX
      EOPWPDNWQXHJBCWPDOPWSAWRLJBCUCUEUFUGAWTJKZQZWTDUHZXCDWTUIKZQZQZXBXGXNXBQZ
      BCGUJZXGXPRZXOXPQZXFXABCXDIZMZJBCWTXDUKZXRXAXDMZXTJXRXDXSXAXPXDXSNZXOXPBC
      XDSZYCXPBCXCULZYDBCGUMXCTKZYDYEURGUNZBCXCTUOUPUQBCXDUSUTVAVBXRXBYFXCXAKZR
      YBJKXNXBXPVCYFXRYGUGXRXCWTKYHXRXCDWTXNXLXBXPXJXKXLVDVEVFBCXCWTVGVHXAXCTVI
      VJVKVLXOXQQZXFXAJYIXFXTXAYAYIXTXALMXAYIXSLXAXQXSLNZXOXQBRZCXDSZYJYLYKCXCU
      LZYKCGUJXQYFYLYMURYGYKCXCTUOUPYKCGUMBCGVMVTBCXDVNUTVAVBXAVOVPVQXNXBXQVCVR
      XOXPVSZXPXQWKXOXCDKBVSZCDSZYNXOXCDWTXJXKXLXBWAWBAYPXIXMXBFWCYOYNCXCDXPCBC
      GWDWECHXCNBXPBCGWFWGWHWIXPWLWJWMWNEWO $.
  $}

  ${
    $d A x $.  $d B x $.
    $( A subset of a finite set is finite if membership in the subset is
       decidable.  (Contributed by Jim Kingdon, 27-May-2022.) $)
    ssfidc $p |- ( ( A e. Fin /\ B C_ A /\ A. x e. A DECID x e. B )
        -> B e. Fin ) $=
      ( cfn wcel wss cv wdc wral w3a cin wceq dfss1 biimpi 3ad2ant2 dfin5 simp1
      crab simp3 ssfirab eqeltrid eqeltrrd ) BDEZCBFZAGCEZHABIZJZBCKZCDUDUCUHCL
      ZUFUDUICBMNOUGUHUEABRDABCPUGUEABUCUDUFQUCUDUFSTUAUB $.
  $}

  ${
    $d w x y $.  $d x y z $.
    $( Excluded middle is equivalent to any subset of a finite set being
       finite.  Theorem 2.1 of [Bauer], p. 485.  (Contributed by Jim Kingdon,
       20-Mar-2026.) $)
    exmidssfi $p |- ( EXMID
        <-> A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin ) ) $=
      ( vw vz wem cv cfn wcel wss wa wi wal wdc simprl simprr exmidexmid adantr
      wral ralrimivw ssfidc syl3anc ex alrimivv c0 csn wceq wn ssfiexmidt df-dc
      wo sylibr exmid1dc impbii ) EAFZGHZBFZUNIZJZUPGHZKZBLALZEUTABEURUSEURJZUO
      UQCFUPHZMZCUNRUSEUOUQNEUOUQOVBVDCUNEVDURVCPQSCUNUPTUAUBUCVADVADFZUDUEZUFZ
      MZVEVFIVAVGVGUGUJVHVGABUHVGUIUKQULUM $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d S z $.  $d ph z $.
    opabfi.s $e |- S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } $.
    opabfi.a $e |- ( ph -> A e. Fin ) $.
    opabfi.b $e |- ( ph -> B e. Fin ) $.
    opabfi.dc $e |- ( ph -> A. x e. A A. y e. B DECID ps ) $.
    $( Finiteness of an ordered pair abstraction which is a decidable subset of
       finite sets.  (Contributed by Jim Kingdon, 16-Sep-2025.) $)
    opabfi $p |- ( ph -> S e. Fin ) $=
      ( vz cfn wcel cv wdc wral syl2anc wa adantl cxp wss xpfi opabssxp eqsstri
      copab a1i c1st wsbc c2nd xp2nd xp1st adantr nfcv nfsbc1v nfdc nfralw wceq
      cfv sbceq1a dcbid ralbidv rspc sylc wb cop nfv nfan anbi1d anbi12d anbi2d
      eleq1 opelopabgf 1st2nd2 eleq12d 3bitr4d mpbird ralrimiva ssfidc syl3anc
      ibar ) AEFUAZMNZGWBUBZLOZGNZPZLWBQGMNAEMNFMNWCIJEFUCRWDAGCOZENZDOZFNZSZBS
      ZCDUFZWBHBCDEFUDUEUGAWGLWBAWEWBNZSZWGBCWEUHUSZUIZDWEUJUSZUIZPZWPWSFNZWRPZ
      DFQZXAWOXBAWEEFUKZTWPWQENZBPZDFQZCEQZXDWOXFAWEEFULZTAXIWOKUMXHXDCWQEXCCDF
      CFUNWRCBCWQUOZUPUQWHWQURZXGXCDFXLBWRBCWQUTZVAVBVCVDXCXADWSFWTDWRDWSUOZUPW
      JWSURZWRWTWRDWSUTZVAVCVDWOWGXAVEAWOWFWTWOWQWSVFZWNNZXFXBSZWTSZWFWTWOXFXBX
      RXTVEXJXEWMXFWKSZWRSXTCDWQWSEFYAWRCYACVGXKVHXSWTDXSDVGXNVHXLWLYABWRXLWIXF
      WKWHWQEVLVIXMVJXOYAXSWRWTXOWKXBXFWJWSFVLVKXPVJVMRWOWEXQGWNWEEFVNGWNURWOHU
      GVOWOXFXBWTXTVEXJXEXSWTWARVPVATVQVRLWBGVSVT $.
  $}

  ${
    $d A x $.  $d B x $.
    $( The intersection of two sets is finite if one of them is and the other
       is decidable.  (Contributed by Jim Kingdon, 24-May-2025.) $)
    infidc $p |- ( ( A e. Fin /\ A. x e. A DECID x e. B )
        -> ( A i^i B ) e. Fin ) $=
      ( cfn wcel cv wdc wral wa cin wss simpl inss1 a1i elin baibr dcbid biimpd
      wi adantl ralimdva imp ssfidc syl3anc ) BDEZAFZCEZGZABHZIZUEBCJZBKZUFUKEZ
      GZABHZUKDEUEUILULUJBCMNUEUIUOUEUHUNABUFBEZUHUNSUEUPUHUNUPUGUMUMUPUGUFBCOP
      QRTUAUBABUKUCUD $.
  $}

  ${
    $d A x $.  $d V x $.
    $( An ordinal which is a singleton is ` { (/) } ` .  (Contributed by Jim
       Kingdon, 19-Oct-2021.) $)
    snon0 $p |- ( ( A e. V /\ { A } e. On ) -> A = (/) ) $=
      ( vx wcel con0 wa cv elirr wceq snidg adantr wi ontr1 adantl mpan2d elsni
      csn syl6 eleq1 biimpcd sylcom mtoi eq0rdv ) ABDZAQZEDZFZCAUGCGZADZAADZAHU
      GUIUHAIZUJUGUIUHUEDZUKUGUIAUEDZULUDUMUFABJKUFUIUMFULLUDUHAUEMNOUHAPRUKUIU
      JUHAASTUAUBUC $.
  $}

  ${
    $d y z w A $.  $d y z w F $.
    $( A version of ~ fnex for finite sets.  (Contributed by Mario Carneiro,
       16-Nov-2014.)  (Revised by Mario Carneiro, 24-Jun-2015.) $)
    fnfi $p |- ( ( F Fn A /\ A e. Fin ) -> F e. Fin ) $=
      ( vw vy vz cfn wcel wa cres cv c0 csn cun reseq2 eleq1d syl2anc cvv simpr
      wceq cdm wfn fnresdm adantr res0 0fi eqeltri a1i wss cdif resundi cfv cop
      simp-4l simplrr eldifad fnressn uneq2d elexd funfvex funfni opexg eldifbd
      wn wi opeldmg dmres eleq2i imbitrdi elin simplbi syl6 mtod unsnfi syl3anc
      cin eqeltrd eqeltrid ex findcard2sd eqeltrrd ) BAUAZAFGZHZBAIZBFWAWDBSWBA
      BUBUCWCBCJZIZFGBKIZFGZBDJZIZFGZBWIEJZLZMZIZFGZWDFGCDEAWEKSWFWGFWEKBNOWEWI
      SWFWJFWEWIBNOWEWNSWFWOFWEWNBNOWEASWFWDFWEABNOWHWCWGKFBUDUEUFUGWCWIFGZHZWI
      AUHZWLAWIUIZGZHZHZWKWPXCWKHZWOWJBWMIZMZFBWIWMUJXDXFWJWLWLBUKZULZLZMZFXDXE
      XIWJXDWAWLAGZXEXISWAWBWQXBWKUMZXDWLAWIWRWSXAWKUNZUOZAWLBUPPUQXDWKXHQGZXHW
      JGZVCXJFGXCWKRXDWLQGXGQGZXOXDWLWTXMURXDWAXKXQXLXNXQAWLBWLBUSUTPZWLXGQQVAP
      XDXPWLWIGZXDWLAWIXMVBXDXPWLWIBTZVOZGZXSXDXPWLWJTZGZYBXDXKXQXPYDVDXNXRWLXG
      WJAQVEPYCYAWLBWIVFVGVHYBXSWLXTGWLWIXTVIVJVKVLWJXHQVMVNVPVQVRWAWBRVSVT $.
  $}

  $( The domain of a finite function is finite.  (Contributed by Jim Kingdon,
     5-Feb-2022.) $)
  fundmfi $p |- ( ( A e. Fin /\ Fun A ) -> dom A e. Fin ) $=
    ( cfn wcel wfun cdm cen wbr fundmeng enfii syldan ) ABCADAEZAFGKBCABHKAIJ
    $.

  $( A function is finite if and only if its domain is finite.  (Contributed by
     AV, 10-Jan-2020.) $)
  fundmfibi $p |- ( Fun F -> ( F e. Fin <-> dom F e. Fin ) ) $=
    ( wfun cfn wcel cdm fundmfi ancoms wfn funfn fnfi sylanb impbida ) ABZACDZA
    EZCDZNMPAFGMAOHPNAIOAJKL $.

  $( The restriction of a function to a finite subset of its domain is finite.
     (Contributed by Alexander van der Vekens, 3-Feb-2018.) $)
  resfnfinfinss $p |- ( ( F Fn A /\ B e. Fin /\ B C_ A )
      -> ( F |` B ) e. Fin ) $=
    ( wfn cfn wcel wss w3a cres fnssres 3adant2 simp2 fnfi syl2anc ) CADZBEFZBA
    GZHCBIZBDZPREFOQSPABCJKOPQLBRMN $.

  $( A restricted identity function is finite iff the restricting class is
     finite.  (Contributed by AV, 10-Jan-2020.) $)
  residfi $p |- ( ( _I |` A ) e. Fin <-> A e. Fin ) $=
    ( cid cres cfn wcel cdm dmresi wfun funi funres ax-mp fundmfi eqeltrrid cvv
    mpan2 wfn wss wf1o f1ovi f1ofn ssv resfnfinfinss mp3an13 impbii ) BACZDEZAD
    EZUFAUEFZDAGUFUEHZUHDEBHUIIABJKUELOMBNPZUGANQUFNNBRUJSNNBTKAUANABUBUCUD $.

  $( If a relation is finite, its converse is as well.  (Contributed by Jim
     Kingdon, 5-Feb-2022.) $)
  relcnvfi $p |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin ) $=
    ( wrel cfn wcel wa ccnv cen wbr wceq dfrel2 biimpi adantr simpr eqeltrd cvv
    relcnv cnvexg cnven sylancr adantl enfii syl2anc ) ABZACDZEZAFZFZCDUFUGGHZU
    FCDUEUGACUCUGAIZUDUCUIAJKLUCUDMNUDUHUCUDUFBUFODUHAPACQUFORSTUFUGUAUB $.

  $( The range of a finite relation is finite if its converse is a function.
     (Contributed by Jim Kingdon, 5-Feb-2022.) $)
  funrnfi $p |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin ) $=
    ( wrel ccnv wfun cfn wcel w3a crn cdm df-rn relcnvfi 3adant2 simp2 eqeltrid
    fundmfi syl2anc ) ABZACZDZAEFZGZAHRIZEAJUAREFZSUBEFQTUCSAKLQSTMROPN $.

  $( If a 1-1 and onto function has a finite domain, its range is finite.
     (Contributed by Jim Kingdon, 21-Feb-2022.) $)
  f1ofi $p |- ( ( A e. Fin /\ F : A -1-1-onto-> B ) -> B e. Fin ) $=
    ( cfn wcel wf1o cen wbr wa f1oeng ensymd enfii syldan ) ADEZABCFZBAGHBDENOI
    ABABDCJKBALM $.

  $( A one-to-one function whose domain is a set is finite if and only if its
     range is finite.  See also ~ f1vrnfibi .  (Contributed by AV,
     10-Jan-2020.) $)
  f1dmvrnfibi $p |- ( ( A e. V /\ F : A -1-1-> B )
                      -> ( F e. Fin <-> ran F e. Fin ) ) $=
    ( wcel wf1 wa cfn crn wrel ccnv wfun f1rel ad2antlr wf1o f1cnv syl simpr wb
    wceq f1ofun funrnfi syl3anc cdm cen wbr wi f1dm f1f1orn eleq1 f1oeq2 eqcoms
    anbi12d biimpd expcomd sylc impcom adantr f1oeng enfii syl2anc f1fun mpbird
    fundmfibi impbida ) ADEZABCFZGZCHEZCIZHEZVHVIGCJZCKZLZVIVKVGVLVFVIABCMNVGVN
    VFVIVGVJAVMOVNABCPVJAVMUAQNVHVIRCUBUCVHVKGZVICUDZHEZVOVKVPVJUEUFZVQVHVKRVOV
    PDEZVPVJCOZGZVRVHWAVKVGVFWAVGVPATZAVJCOZVFWAUGABCUHABCUIWBVFWCWAWBVFWCGZWAW
    DWASAVPAVPTVFVSWCVTAVPDUJAVPVJCUKUMULUNUOUPUQURVPVJDCUSQVPVJUTVAVOCLZVIVQSV
    GWEVFVKABCVBNCVDQVCVE $.

  $( A one-to-one function which is a set is finite if and only if its range is
     finite.  See also ~ f1dmvrnfibi .  (Contributed by AV, 10-Jan-2020.) $)
  f1vrnfibi $p |- ( ( F e. V /\ F : A -1-1-> B )
                   -> ( F e. Fin <-> ran F e. Fin ) ) $=
    ( wcel wf1 cvv cfn crn wb cdm wceq wi f1dm dmexg eleq1 eqcoms imbitrrid syl
    impcom f1dmvrnfibi sylancom ) CDEZABCFZAGEZCHECIHEJUDUCUEUDCKZALZUCUEMABCNU
    CUEUGUFGEZCDOUEUHJAUFAUFGPQRSTABCGUAUB $.

  ${
    $d A w x y z $.  $d B w y z $.
    $( The finite union of disjoint finite sets is finite.  Note that ` B `
       depends on ` x ` , i.e. can be thought of as ` B ( x ) ` .  (Contributed
       by NM, 23-Mar-2006.)  (Revised by Jim Kingdon, 7-Oct-2022.) $)
    iunfidisj $p |- ( ( A e. Fin /\ A. x e. A B e. Fin /\ Disj_ x e. A B )
        -> U_ x e. A B e. Fin ) $=
      ( vw vy vz cfn wcel wral cv ciun c0 cun iuneq1 eleq1d wss wa cin eqeltrid
      wceq wdisj w3a csn 0iun 0fi eqeltri a1i cdif iunxun simpr csb cvv nfcsb1v
      csbeq1a iunxsngf simplrr eldifad simpll2 nfel1 rspc simpll3 simplrl snssd
      elv sylc wn eldifbd disjsn sylibr disjiun syl13anc unfidisj syl3anc simp1
      ex findcard2d ) BGHZCGHZABIZABCUAZUBZADJZCKZGHALCKZGHZAEJZCKZGHZAWFFJZUCZ
      MZCKZGHZABCKZGHDEFBWBLTWCWDGAWBLCNOWBWFTWCWGGAWBWFCNOWBWKTWCWLGAWBWKCNOWB
      BTWCWNGAWBBCNOWEWAWDLGACUDUEUFUGWAWFBPZWIBWFUHHZQZQZWHWMWRWHQZWLWGAWJCKZM
      ZGAWFWJCUIWSWHWTGHWGWTRLTZXAGHWRWHUJWSWTAWICUKZGWTXCTFAWICXCULAWICUMZAWIC
      UNZUOVDWSWIBHVSXCGHZWSWIBWFWAWOWPWHUPZUQZVQVSVTWQWHURVRXFAWIBAXCGXDUSAJWI
      TCXCGXEOUTVESWSVTWOWJBPWFWJRLTZXBVQVSVTWQWHVAWAWOWPWHVBWSWIBXHVCWSWIWFHVF
      XIWSWIBWFXGVGWFWIVHVIABCWFWJVJVKWGWTVLVMSVOVQVSVTVNVP $.
  $}

  ${
    $d A w y z $.  $d B w y z $.
    $( Set exponentiation of finite sets is finite.  (Contributed by Jeff
       Madsen, 19-Jun-2011.) $)
    mapfi $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A ^m B ) e. Fin ) $=
      ( vw vy vz cfn wcel wa cv cmap co c0 csn oveq2 eleq1d cvv cen a1i syl2anc
      wceq cun mapdm0 0ex snfig ax-mp eqeltrdi adantr wss cxp wbr simpr simp-4l
      cdif vex mapsnend enfii xpfi cin wn simplrr eldifbd disjsn sylibr mapunen
      snex syl31anc ex findcard2sd ) AFGZBFGZHZACIZJKZFGALJKZFGZADIZJKZFGZAVPEI
      ZMZUAZJKZFGZABJKZFGCDEBVLLTVMVNFVLLAJNOVLVPTVMVQFVLVPAJNOVLWATVMWBFVLWAAJ
      NOVLBTVMWDFVLBAJNOVIVOVJVIVNLMZFAFUBLPGWEFGUCLPUDUEUFUGVKVPFGZHZVPBUHZVSB
      VPUMGZHZHZVRWCWKVRHZVQAVTJKZUIZFGZWBWNQUJZWCWLVRWMFGZWOWKVRUKWLVIWMAQUJWQ
      VIVJWFWJVRULZWLAVSFPWRVSPGWLEUNZRUOWMAUPSVQWMUQSWLVPPGZVTPGZVIVPVTURLTZWP
      WTWLDUNRXAWLVSWSVERWRWLVSVPGUSXBWLVSBVPWGWHWIVRUTVAVPVSVBVCVPVTAPPFVDVFWB
      WNUPSVGVIVJUKVH $.
  $}

  $( Membership in a class of finite subsets.  (Contributed by Stefan O'Rear,
     4-Apr-2015.)  (Revised by Mario Carneiro, 22-Aug-2015.) $)
  elfpw $p |- ( A e. ( ~P B i^i Fin ) <-> ( A C_ B /\ A e. Fin ) ) $=
    ( cpw cfn cin wcel wa wss elin elpwg pm5.32ri bitri ) ABCZDEFAMFZADFZGABHZO
    GAMDIONPABDJKL $.

  ${
    $d A u v w x $.  $d S u v w x $.
    $( A finite subset of a finite set is a decidable subset.  (Contributed by
       Jim Kingdon, 18-May-2026.) $)
    fissfi $p |- ( ( S C_ A /\ A e. Fin /\ S e. Fin )
        -> A. x e. A DECID x e. S ) $=
      ( vw vu vv wss cfn wcel w3a cv wdc wa c0 csn cun wceq eleq2 dcbid ad4antr
      wn wo noel olci df-dc mpbir a1i simpr simp2 simp-4r simp1 simplrr eldifad
      cdif sseldd fidceq syl3anc velsn sylibr dcun simpl3 findcard2sd ralrimiva
      dcbii ex ) CBGZBHIZCHIZJZAKZCIZLZABVIVJBIZMZVJDKZIZLVJNIZLZVJEKZIZLZVJVSF
      KZOZPZIZLZVLDEFCVONQVPVQVONVJRSVOVSQVPVTVOVSVJRSVOWDQVPWEVOWDVJRSVOCQVPVK
      VOCVJRSVRVNVRVQVQUAZUBWGVQVJUCUDVQUEUFUGVNVSHIZMZVSCGZWBCVSUNIZMZMZWAWFWM
      WAMZVSWCVJWMWAUHWNVJWBQZLZVJWCIZLWNVGVMWBBIWPVIVGVMWHWLWAVFVGVHUITVIVMWHW
      LWAUJWNCBWBVIVFVMWHWLWAVFVGVHUKTWNWBCVSWIWJWKWAULUMUOBVJWBUPUQWQWOAWBURVD
      USUTVEVFVGVHVMVAVBVC $.
  $}

  $( Any injection from one finite set to another of equal size must be a
     bijection.  (Contributed by Jeff Madsen, 5-Jun-2010.) $)
  f1finf1o $p |- ( ( A ~~ B /\ B e. Fin ) ->
      ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) ) $=
    ( cen wbr cfn wcel wa wf1 wf1o crn wceq wss simplr f1rn adantl cima syl2anc
    simpr syl3anc wfn f1fn fnima syl ssid a1i simpll f1imaeng eqbrtrrd fisseneq
    enfii entr dff1o5 sylanbrc ex f1of1 impbid1 ) ABDEZBFGZHZABCIZABCJZUTVAVBUT
    VAHZVACKZBLZVBUTVASZVCUSVDBMZVDBDEZVEURUSVANZVAVGUTABCOPVCVDADEURVHVCCAQZVD
    ADVAVJVDLZUTVACAUAVKABCUBACUCUDPVCVAAAMZAFGZVJADEVFVLVCAUEUFVCUSURVMVIURUSV
    AUGZABUKRABACFUHTUIVNVDABULRVDBUJTABCUMUNUOABCUPUQ $.

  $( A set with one element is a singleton.  (Contributed by FL,
     18-Aug-2008.) $)
  en1eqsn $p |- ( ( A e. B /\ B ~~ 1o ) -> B = { A } ) $=
    ( wcel c1o cen wbr wa csn cfn wss wceq com 1onn nnfi ax-mp enfii mpan snssi
    adantl adantr ensn1g ensym entr syl2an fisseneq syl3anc eqcomd ) ABCZBDEFZG
    ZAHZBUJBICZUKBJZUKBEFZUKBKUIULUHDICZUIULDLCUOMDNOBDPQSUHUMUIABRTUHUKDEFDBEF
    UNUIABUABDUBUKDBUCUDUKBUEUFUG $.

  $( A set containing an element has exactly one element iff it is a singleton.
     (Contributed by FL, 13-Feb-2010.)  (Revised by AV, 25-Jan-2020.) $)
  en1eqsnbi $p |- ( A e. B -> ( B ~~ 1o <-> B = { A } ) ) $=
    ( wcel c1o cen wbr csn wceq en1eqsn ex ensn1g breq1 syl5ibrcom impbid ) ABC
    ZBDEFZBAGZHZOPRABIJOPRQDEFABKBQDELMN $.

  ${
    $d ph x y $.
    $( A case where the antecedent of ~ snexg is not needed.  The class
       ` { x | ph } ` is from ~ dcextest .  (Contributed by Mario Carneiro and
       Jim Kingdon, 4-Jul-2022.) $)
    snexxph $p |- { { x | ph } } e. _V $=
      ( vy cab csn c1o con0 1on elexi cv wcel c0 elsni wn wceq cvv vprc weq syl
      df-v wb equid pm5.1im ax-mp abbidv eqtr2id eleq1d mtbiri wex 19.8a sylibr
      wi isset nsyl3 vex biidd notbii biimpri eq0rdv eqtrd 0lt1o eqeltrdi ssriv
      elab ssexi ) ABDZEZFFGHICVGFCJZVGKZVHLFVIVHVFLVHVFMZVIANZVFLOAVFPKZVIAVLP
      PKQAVFPPAPBBRZBDVFBTAVMABVMAVMAUAULBUBVMAUCUDUEUFUGUHVIVHVFOZCUIZVLVIVNVO
      VJVNCUJSCVFUMUKUNVKCVFVHVFKZNVKVPAAABVHCUOBCRAUPVDUQURUSSUTVAVBVCVE $.
  $}

  ${
    preimaf1ofi.ss $e |- ( ph -> C C_ B ) $.
    preimaf1ofi.f $e |- ( ph -> F : A -1-1-onto-> B ) $.
    preimaf1ofi.c $e |- ( ph -> C e. Fin ) $.
    $( The preimage of a finite set under a one-to-one, onto function is
       finite.  (Contributed by Jim Kingdon, 24-Sep-2022.) $)
    preimaf1ofi $p |- ( ph -> ( `' F " C ) e. Fin ) $=
      ( cfn wcel ccnv cima cres wf1o wf1 wss f1ocnv f1of1 3syl syl2anc f1ores
      f1ofi ) ADIJDEKZDLZUCDMZNZUDIJHACBUCOZDCPUFABCENCBUCNUGGBCEQCBUCRSFCBDUCU
      ATDUDUEUBT $.
  $}

  ${
    $d A f n $.  $d A x y $.
    $( Lemma for ~ fidcenum .  Forward direction.  (Contributed by Jim Kingdon,
       19-Oct-2022.) $)
    fidcenumlemim $p |- ( A e. Fin -> ( A. x e. A A. y e. A DECID x = y
        /\ E. n e. _om E. f f : n -onto-> A ) ) $=
      ( cfn wcel weq wdc wral cv wfo wex com wrex fidceq 3expb cen wbr sylbi
      ralrimivva isfi ensym wf1o bren f1ofo eximi syl reximi jca ) CFGZABHIZBCJ
      ACJEKZCDKZLZDMZENOZUKULABCCUKAKZCGBKZCGULCURUSPQUAUKCUMRSZENOUQECUBUTUPEN
      UTUMCRSZUPCUMUCVAUMCUNUDZDMUPUMCDUEVBUODUMCUNUFUGTUHUITUJ $.
  $}

  ${
    fidcenumlemr.dc $e |- ( ph -> A. x e. A A. y e. A DECID x = y ) $.
    fidcenumlemr.f $e |- ( ph -> F : N -onto-> A ) $.

    ${
      $d A x y $.  $d F y $.  $d J y $.  $d X x y $.
      fidcenumlemrks.j $e |- ( ph -> J e. _om ) $.
      fidcenumlemrks.jn $e |- ( ph -> suc J C_ N ) $.
      fidcenumlemrks.h $e |- ( ph
        -> ( X e. ( F " J ) \/ -. X e. ( F " J ) ) ) $.
      fidcenumlemrks.x $e |- ( ph -> X e. A ) $.
      $( Lemma for ~ fidcenum .  Induction step for ~ fidcenumlemrk .
         (Contributed by Jim Kingdon, 20-Oct-2022.) $)
      fidcenumlemrks $p |- ( ph
          -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) ) $=
        ( cima wcel wn wa syl wceq csuc wo csn cun simpr df-suc imaeq2i imaundi
        elun1 eqtri eleq2i sylibr orcd cfv elsng wfn wfo fofn com sucidg sseldd
        wb fnsnfv syl2anc eleq2d bitr3d ad2antrr mpbid elun2 simplr mtbid ioran
        sylanbrc elun sylnibr olcd wdc cv wral wf ffvelcdmd eqeq1 dcbid rspc2va
        fof eqeq2 syl21anc exmiddc adantr mpjaodan ) AHEFOZPZHEFUAZOZPZWOQZUBZW
        LQZAWLRZWOWPWSHWKEFUCZOZUDZPZWOWSWLXCAWLUEHWKXAUISWNXBHWNEFWTUDZOXBWMXD
        EFUFUGEFWTUHUJUKZULUMAWRRZHFEUNZTZWQXHQZXFXHRZWOWPXJXCWOXJHXAPZXCXJXHXK
        XFXHUEAXHXKVBZWRXHAHXGUCZPZXHXKAHDPZXNXHVBNHXGDUOSAXMXAHAEGUPZFGPXMXATA
        GDEUQZXPJGDEURSAWMGFLAFUSPFWMPKFUSUTSVAZGFEVCVDVEVFZVGVHHXAWKVISXEULUMX
        FXIRZWPWOXTXCWOXTWLXKUBZXCXTWRXKQYAQAWRXIVJXTXHXKXFXIUEAXLWRXIXSVGVKWLX
        KVLVMHWKXAVNVOXEVOVPAXHXIUBZWRAXHVQZYBAXOXGDPBVRZCVRZTZVQZCDVSBDVSYCNAG
        DFEAXQGDEVTJGDEWESXRWAIYGYCHYETZVQBCHXGDDYDHTYFYHYDHYEWBWCYEXGTYHXHYEXG
        HWFWCWDWGXHWHSWIWJMWJ $.
    $}

    ${
      $d F j w x y $.  $d K w $.  $d N j w $.  $d X j w x y $.  $d j ph w $.
      $d A x y $.
      fidcenumlemrk.k $e |- ( ph -> K e. _om ) $.
      fidcenumlemrk.kn $e |- ( ph -> K C_ N ) $.
      fidcenumlemrk.x $e |- ( ph -> X e. A ) $.
      $( Lemma for ~ fidcenum .  (Contributed by Jim Kingdon, 20-Oct-2022.) $)
      fidcenumlemrk $p |- ( ph -> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) ) $=
        ( wcel wss wa cima wn wo c0 vw vj com ancli cv csuc sseq1 anbi2d imaeq2
        wi wceq eleq2d notbid orbi12d imbi12d weq noel ima0 eleq2i a1i olcd wdc
        wral ad2antrl simpll simprr simprl sssucid sstrid simplr fidcenumlemrks
        mtbir wfo mp2and exp31 finds sylc ) AFUCNAFGOZPZHEFQZNZWARZSZKAVRLUDAUA
        UEZGOZPZHEWDQZNZWHRZSZUJATGOZPZHETQZNZWNRZSZUJAUBUEZGOZPZHEWQQZNZXARZSZ
        UJZAWQUFZGOZPZHEXEQZNZXIRZSZUJVSWCUJUAUBFWDTUKZWFWLWJWPXLWEWKAWDTGUGUHX
        LWHWNWIWOXLWGWMHWDTEUIULZXLWHWNXMUMUNUOUAUBUPZWFWSWJXCXNWEWRAWDWQGUGUHX
        NWHXAWIXBXNWGWTHWDWQEUIULZXNWHXAXOUMUNUOWDXEUKZWFXGWJXKXPWEXFAWDXEGUGUH
        XPWHXIWIXJXPWGXHHWDXEEUIULZXPWHXIXQUMUNUOWDFUKZWFVSWJWCXRWEVRAWDFGUGUHX
        RWHWAWIWBXRWGVTHWDFEUIULZXRWHWAXSUMUNUOWLWOWNWOWLWNHTNHUQWMTHEURUSVLUTV
        AWQUCNZXDXGXKXTXDPZXGPZBCDEWQGHABCUPVBCDVCBDVCYAXFIVDAGDEVMYAXFJVDXTXDX
        GVEYAAXFVFZYBAWRXCYAAXFVGYBWQXEGWQVHYCVIXTXDXGVJVNAHDNYAXFMVDVKVOVPVQ
        $.
    $}

    ${
      $d A x y $.  $d F k w $.  $d F k x y $.  $d N k w $.  $d k ph w $.
      fidcenumlemr.n $e |- ( ph -> N e. _om ) $.
      $( Lemma for ~ fidcenum .  Reverse direction (put into deduction form).
         (Contributed by Jim Kingdon, 19-Oct-2022.) $)
      fidcenumlemr $p |- ( ph -> A e. Fin ) $=
        ( cima cfn wceq syl wss wcel wa adantr cv wi c0 vw vk wfo ssid com csuc
        foima sseq1 anbi2d imaeq2 eleq1d imbi12d 0fi eqeltri a1i cfv wn csn cun
        ima0 wfn simprl fofn simprr sseldd fnsnfv syl2anc uneq2d df-suc imaeq2i
        vex sucid imaundi eqtri eqtr4di simpr snssd ssequn2 eqtr3d sssucid sstr
        sylib mpan ad2antll simplr mp2and eqeltrd fof ffvelcdmd unsnfi eqeltrrd
        syl3anc wdc wral simpll fidcenumlemrk mpjaodan exp31 finds mpcom mpan2
        wf ) AEFJZDKAFDEUCZXCDLHFDEUGMAFFNZXCKOZFUDFUEOZAXEPZXFAXGXEIQAUARZFNZP
        ZEXIJZKOZSATFNZPZETJZKOZSAUBRZFNZPZEXRJZKOZSZAXRUFZFNZPZEYDJZKOZSXHXFSU
        AUBFXITLZXKXOXMXQYIXJXNAXITFUHUIYIXLXPKXITEUJUKULXIXRLZXKXTXMYBYJXJXSAX
        IXRFUHUIYJXLYAKXIXREUJUKULXIYDLZXKYFXMYHYKXJYEAXIYDFUHUIYKXLYGKXIYDEUJU
        KULXIFLZXKXHXMXFYLXJXEAXIFFUHUIYLXLXCKXIFEUJUKULXQXOXPTKEUTUMUNUOXRUEOZ
        YCYFYHYMYCPZYFPZXREUPZYAOZYHYQUQZYOYQPZYGYAKYSYAYPURZUSZYGYAYOUUAYGLZYQ
        YOUUAYAEXRURZJZUSZYGYOYTUUDYAYOEFVAZXRFOYTUUDLYOAUUFYNAYEVBZAXDUUFHFDEV
        CMMYOYDFXRYNAYEVDXRYDOYOXRUBVKVLUOVEZFXREVFVGVHYGEXRUUCUSZJUUEYDUUIEXRV
        IVJEXRUUCVMVNVOZQYSYTYANUUAYALYSYPYAYOYQVPVQYTYAVRWBVSYOYBYQYOAXSYBUUGY
        EXSYNAXRYDNYEXSXRVTXRYDFWAWCWDZYMYCYFWEWFZQWGYOYRPZUUAYGKYOUUBYRUUJQUUM
        YBYPDOZYRUUAKOYOYBYRUULQYOUUNYRYOFDXREYOXDFDEXBYOAXDUUGHMZFDEWHMUUHWIZQ
        YOYRVPYAYPDWJWLWKYOBCDEXRFYPYOABRCRLWMCDWNBDWNUUGGMUUOYMYCYFWOUUKUUPWPW
        QWRWSWTXAWK $.
    $}
  $}

  ${
    $d A f n x y $.
    $( A set is finite if and only if it has decidable equality and is finitely
       enumerable.  Proposition 8.1.11 of [AczelRathjen], p. 72.  The
       definition of "finitely enumerable" as
       ` E. n e. _om E. f f : n -onto-> A ` is Definition 8.1.4 of
       [AczelRathjen], p. 71.  (Contributed by Jim Kingdon, 19-Oct-2022.) $)
    fidcenum $p |- ( A e. Fin <-> ( A. x e. A A. y e. A DECID x = y
        /\ E. n e. _om E. f f : n -onto-> A ) ) $=
      ( cfn wcel weq wdc wral cv wfo wex com wrex wa fidcenumlemim simpll simpr
      simplr fidcenumlemr ex exlimdv rexlimdva imp impbii ) CFGZABHIBCJACJZEKZC
      DKZLZDMZENOZPABCDEQUHUMUGUHULUGENUHUINGZPZUKUGDUOUKUGUOUKPABCUJUIUHUNUKRU
      OUKSUHUNUKTUAUBUCUDUEUF $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Schroeder-Bernstein Theorem
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x A $.  $d x B $.  $d x D $.  $d x f $.  $d x g $.
    sbthlem.1 $e |- A e. _V $.
    sbthlem.2 $e |- D = { x |
                   ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) } $.
    $( Lemma for ~ isbth .  (Contributed by NM, 22-Mar-1998.) $)
    sbthlem1 $p |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) $=
      ( cuni cv cima cdif wss unissb wcel wa abeq2i imass2 sscon 3syl wi difss2
      ssconb exbiri syl5 pm2.43d imp sylbi elssuni sstrd mprgbir ) DIZBFJZCEJZU
      LKZLZKZLZMAJZURMADADURNUSDOZUSBUMCUNUSKZLZKZLZURUTUSBMZVCBUSLMZPZUSVDMZVG
      ADHQVEVFVHVEVFVHVFVCBMZVEVFVHUAVCBUSUBVEVIVHVFUSVCBUCUDUEUFUGUHUTUPVBMZUQ
      VCMVDURMUTUSULMVAUOMVJUSDUIUSULUNRVAUOCSTUPVBUMRUQVCBSTUJUK $.

    $( Lemma for ~ isbth .  (Contributed by NM, 22-Mar-1998.) $)
    sbthlem2 $p |- ( ran g C_ A ->
                     ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D ) $=
      ( cv crn wss cuni cima cdif wcel wa cab imass2 sscon mp2b sbthlem1 ssconb
      wb imassrn sstr2 ax-mp difss sylancl mpbiri jctil ssexi wceq sseq1 imaeq2
      difeq2d imaeq2d difeq2 sseq12d anbi12d elab sylibr eleqtrrdi elssuni syl
      wi ) FIZJZBKZBVFCEIZDLZMZNZMZNZDOVNVJKVHVNAIZBKZVFCVIVOMZNZMZBVONZKZPZAQZ
      DVHVNBKZVFCVIVNMZNZMZBVNNZKZPZVNWCOVHWIWDVHWIVNBWGNKZWFVLKZWGVMKWKVJVNKVK
      WEKWLABCDEFGHUAVJVNVIRVKWECSTWFVLVFRWGVMBSTVHWGBKZWDWIWKUCWGVGKVHWMVEVFWF
      UDWGVGBUEUFBVMUGZWGVNBUBUHUIWNUJWBWJAVNVNBGWNUKVOVNULZVPWDWAWIVOVNBUMWOVS
      WGVTWHWOVRWFVFWOVQWECVOVNVIUNUOUPVOVNBUQURUSUTVAHVBVNDVCVD $.

    ${
      $d D y $.  $d B y $.  $d f y $.  $d g y $.
      $( Lemma for ~ isbth .  (Contributed by NM, 22-Mar-1998.) $)
      sbthlemi3 $p |- ( ( EXMID /\ ran g C_ A ) ->
                    ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) ) $=
        ( vy wem cv crn wss wa cuni cdif cima wceq sbthlem2 syl sbthlem1 sylibr
        jctil eqss difeq2d adantl imassrn sstr2 ax-mp wcel wstab wal exmidexmid
        wi wb wdc dcstab alrimiv dfss4st imbitrid imp eqtr2d ) JFKZLZBMZNBDOZPZ
        BBVCCEKVFQPZQZPZPZVIVEVGVKRJVEVFVJBVEVFVJMZVJVFMZNVFVJRVEVMVLABCDEFGHSA
        BCDEFGHUAUCVFVJUDUBUEUFJVEVKVIRZVEVIBMZJVNVIVDMVEVOUNVCVHUGVIVDBUHUIJIK
        VIUJZUKZIULVOVNUOJVQIJVPUPVQVPUMVPUQTURIVIBUSTUTVAVB $.
    $}

    $( Lemma for ~ isbth .  (Contributed by NM, 27-Mar-1998.) $)
    sbthlemi4 $p |- ( ( EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ->
                  ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) ) $=
      ( wem cv cdm wceq crn wss wa ccnv w3a cdif cima cres df-ima difss ssdmres
      wfun sseq2 mpbiri sylib dfdm4 eqtr3di adantr 3ad2ant2 funcnvres sbthlemi3
      cuni 3ad2ant3 reseq2d 3adant3 eqtrd rneqd 3adant2l eqtr4id ) IFJZKZCLZVBM
      BNZOZVBPZUDZQZVGBDUNZRZSVGVKTZMZCEJVJSZRZVGVKUAVIVOVBVOTZPZMZVMVFIVOVRLZV
      HVDVSVEVDVPKZVOVRVDVOVCNZVTVOLVDWAVOCNCVNUBVCCVOUEUFVOVBUCUGVPUHUIUJUKIVE
      VHVRVMLVDIVEVHQZVQVLWBVQVGVBVOSZTZVLVHIVQWDLVEVOVBULUOIVEWDVLLVHIVEOWCVKV
      GABCDEFGHUMUPUQURUSUTURVA $.

    ${
      $d x H $.  $d A y $.  $d B y $.  $d D y $.  $d f y $.
      sbthlem.3 $e |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) $.
      $( Lemma for ~ isbth .  (Contributed by NM, 22-Mar-1998.) $)
      sbthlemi5 $p |- ( ( EXMID /\ ( dom f = A /\ ran g C_ A ) )
          -> dom H = A ) $=
        ( vy wem cv cdm wceq wss wa cdif cun cin crn cuni cres ccnv dmeqi dmres
        dmun df-rn eqcomi ineq2i eqtri uneq12i 3eqtri cima sbthlem1 difss sstri
        sseq2 mpbiri dfss sylib uneq1d sbthlemi3 imassrn eqsstrrdi uneq2d an12s
        sylan9eq eqtr4id wcel wdc undifdcss exmidexmid ralrimivw bitr4id adantr
        wral biantrud eqtr4d ) LEMZNZBOZFMZUAZBPZQZQZGNZDUBZBWIRZSZBWGWHWIWATZW
        JWDTZSZWKWHVTWIUCZWCUDZWJUCZSZNWONZWQNZSWNGWRJUEWOWQUGWSWLWTWMVTWIUFWTW
        JWPNZTWMWPWJUFXAWDWJWDXAWCUHUIUJUKULUMWBLWEWKWNOWBLWEQZWKWLWJSWNWBWIWLW
        JWBWIWAPZWIWLOWBXCWIBPZWIBWCCVTWIUNRZUNZRBABCDEFHIUOBXFUPUQZWABWIURUSWI
        WAUTVAVBXBWJWMWLXBWJWDPWJWMOXBWJXFWDABCDEFHIVCWCXEVDVEWJWDUTVAVFVHVGVIL
        BWKOZWFLXHXDXGLXHXDKMWIVJZVKZKBVQZQXDKBWIVLLXKXDLXJKBXIVMVNVRVOUSVPVS
        $.

      $( Lemma for ~ isbth .  (Contributed by NM, 27-Mar-1998.) $)
      sbthlemi6 $p |- ( ( ( EXMID /\ ran f C_ B ) /\
               ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B ) $=
        ( vy wem cv crn wss wa wceq cima cdif cun ccnv wfun cuni simpll simprll
        cdm simprlr simprr w3a cres rnun rneqi df-ima 3eqtr4i sbthlemi4 eqtr3di
        uneq1i uneq2d eqtr4id syl121anc wi imassrn sstr2 ax-mp adantl wcel wral
        wb undifdcss exmidexmid ralrimivw biantrud bitr4id adantr mpbird eqcomd
        wdc eqtrd ) LEMZNZCOZPZFMZUFCQZWCNBOZPZWCUAZUBZPZPZGNZVSDUCZRZCWMSZTZCW
        JLWDWEWHWKWOQLWAWIUDWBWDWEWHUEWBWDWEWHUGWBWFWHUHLWFWHUIZWKWMWGBWLSZUJZN
        ZTZWOVSWLUJZWRTZNXANZWSTWKWTXAWRUKGXBJULWMXCWSVSWLUMUQUNWPWNWSWMWPWGWQR
        WNWSABCDEFHIUOWGWQUMUPURUSUTWBWOCQWIWBCWOWBCWOQZWMCOZWAXELWMVTOWAXEVAVS
        WLVBWMVTCVCVDVELXDXEVHWALXDXEKMWMVFZVQZKCVGZPXEKCWMVILXHXELXGKCXFVJVKVL
        VMVNVOVPVNVR $.

      $( Lemma for ~ isbth .  (Contributed by NM, 27-Mar-1998.) $)
      sbthlem7 $p |- ( ( Fun f /\ Fun `' g ) -> Fun H ) $=
        ( cv wfun wa cres funres cdm cin c0 wss ax-mp ccnv cuni cdif wceq dmres
        cun inss1 eqsstri ssrin sslin sstri disjdif sseqtri funun syl2an funeqi
        ss0 mpan2 sylibr ) EKZLZFKUAZLZMUTDUBZNZVBBVDUCZNZUFZLZGLVAVELZVGLZVIVC
        VDUTOVFVBOVJVKMVEPZVGPZQZRUDZVIVNRSVOVNVDVFQZRVNVDVMQZVPVLVDSVNVQSVLVDU
        TPZQVDUTVDUEVDVRUGUHVLVDVMUITVMVFSVQVPSVMVFVBPZQVFVBVFUEVFVSUGUHVMVFVDU
        JTUKVDBULUMVNUQTVEVGUNURUOGVHJUPUS $.

      $( Lemma for ~ isbth .  (Contributed by NM, 27-Mar-1998.) $)
      sbthlemi8 $p |- ( ( ( EXMID /\ Fun `' f )
          /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) )
          -> Fun `' H ) $=
        ( wem cv ccnv wfun wa cdm wceq crn cres cdif wss cuni funres11 ad2antlr
        cun cin c0 funcnvcnv syl ad2antrr ad2antrl simpll simprll simprd simprr
        simprlr cima df-ima df-rn eqtr2i eqtri sbthlemi4 eqtr3id ineq12 sylancr
        w3a disjdif eqtrdi syl121anc funun syl21anc cnveqi cnvun funeqi sylibr
        ) KELZMNZOZFLZNZVSPCQZOZVSRBUAZOZVSMZNZOZOZVPDUBZSZMZWEBWITZSZMZUEZNZGM
        ZNWHWKNZWNNZWKPZWNPZUFZUGQZWPVQWRKWGWIVPUCUDWDWSVRWFVTWSWAWCVTWEMNWSVSU
        HWLWEUCUIUJUKWHKWAWCWFXCKVQWGULWHVTWAVRWBWCWFUMUNVRWBWCWFUPVRWDWFUOKWAW
        COWFVFZXBVPWIUQZCXETZUFZUGXDWTXEQXAXFQXBXGQXEWJRWTVPWIURWJUSUTXDXAWEWLU
        QZXFXHWMRXAWEWLURWMUSVAABCDEFHIVBVCWTXEXAXFVDVEXECVGVHVIWKWNVJVKWQWOWQW
        JWMUEZMWOGXIJVLWJWMVMVAVNVO $.

      $( Lemma for ~ isbth .  (Contributed by NM, 28-Mar-1998.) $)
      sbthlemi9 $p |- ( ( EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) ->
                    H : A -1-1-onto-> B ) $=
        ( wem wfn wfun cdm wceq wa sylib simpld df-fn simprd cv wf1 w3a crn wss
        ccnv wf1o wf simp2 df-f1 df-f simp3 sbthlem7 syl2anc sbthlemi5 syl12anc
        simp1 sylanbrc jca sbthlemi8 syl22anc df-rn sbthlemi6 eqtr3id dff1o4 )
        KBCEUAZUBZCBFUAZUBZUCZGBLZGUFZCLZBCGUGVJGMZGNBOZVKVJVFMZVHUFMZVNVJVPVFN
        BOZVJVFBLZVPVRPVJVSVFUDCUEZVJBCVFUHZVSVTPVJWAVFUFMZVJVGWAWBPKVGVIUIBCVF
        UJQZRBCVFUKQZRVFBSQZRVJCBVHUHZVQVJVIWFVQPKVGVIULCBVHUJQZTZABCDEFGHIJUMU
        NVJKVRVHUDBUEZVOKVGVIUQZVJVPVRWETVJVHCLZWIVJWFWKWIPVJWFVQWGRCBVHUKQZTZA
        BCDEFGHIJUOUPGBSURVJVLMZVLNZCOZVMVJKWBVHMZVHNCOZPZWIPVQWNWJVJWAWBWCTVJW
        SWIVJWKWSVJWKWIWLRVHCSQZWMUSWHABCDEFGHIJUTVAVJKVTWRWIPZVQWPWJVJVSVTWDTV
        JWRWIVJWQWRWTTWMUSWHKVTPXAVQPPWOGUDCGVBABCDEFGHIJVCVDVAVLCSURBCGVEUR $.

      ${
        $d f g A $.  $d f g B $.
        sbthlem.4 $e |- B e. _V $.
        $( Lemma for ~ isbth .  (Contributed by NM, 28-Mar-1998.) $)
        sbthlemi10 $p |- ( ( EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B ) $=
          ( wem cdom wbr wa cv wf1 wex brdom cvv anbi12i eeanv bitr4i wcel wf1o
          cen w3a cuni cres ccnv cdif cun resex cnvex eqeltri sbthlemi9 f1oen3g
          vex unex sylancr 3expib exlimdvv biimtrid imp ) LBCMNZCBMNZOZBCUFNZVG
          BCEPZQZCBFPZQZOZFRERZLVHVGVJERZVLFRZOVNVEVOVFVPBCEKSCBFHSUAVJVLEFUBUC
          LVMVHEFLVJVLVHLVJVLUGGTUDBCGUEVHGVIDUHZUIZVKUJZBVQUKZUIZULTJVRWAVIVQE
          URUMVSVTVKFURUNUMUSUOABCDEFGHIJUPBCGTUQUTVAVBVCVD $.
      $}
    $}
  $}

  ${
    $d x y z w f g A $.  $d x y z w f g B $.
    $( Schroeder-Bernstein Theorem.  Theorem 18 of [Suppes] p. 95.  This
       theorem states that if set ` A ` is smaller (has lower cardinality) than
       ` B ` and vice-versa, then ` A ` and ` B ` are equinumerous (have the
       same cardinality).  The interesting thing is that this can be proved
       without invoking the Axiom of Choice, as we do here, but the proof as
       you can see is quite difficult.  (The theorem can be proved more easily
       if we allow AC.) The main proof consists of lemmas ~ sbthlem1 through
       ~ sbthlemi10 ; this final piece mainly changes bound variables to
       eliminate the hypotheses of ~ sbthlemi10 .  We follow closely the proof
       in Suppes, which you should consult to understand our proof at a higher
       level.  Note that Suppes' proof, which is credited to J. M. Whitaker,
       does not require the Axiom of Infinity.  The proof does require the law
       of the excluded middle which cannot be avoided as shown at
       ~ exmidsbthr .  (Contributed by NM, 8-Jun-1998.) $)
    isbth $p |- ( ( EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B ) $=
      ( vw vz vx vy wem cdom wbr wa cen cv cvv wceq breq2 breq1 wss cima cdif
      wi vg vf simprl simprr reldom brrelex1i syl wb anbi12d imbi12d adantl cab
      wcel cuni cres cun vex sseq1 imaeq2 difeq2d imaeq2d difeq2 sseq12d cbvabv
      ccnv eqid sbthlemi10 ex adantr vtocld mp2and ) GABHIZBAHIZJZJZVLVMABKIZGV
      LVMUCZGVLVMUDZVOACLZHIZVSAHIZJZAVSKIZTZVNVPTZCBMVOVMBMUMVRBAHUEUFUGVSBNZW
      DWEUHVOWFWBVNWCVPWFVTVLWAVMVSBAHOVSBAHPUIVSBAKOUJUKVODLZVSHIZVSWGHIZJZWGV
      SKIZTZWDDAMVOVLAMUMVQABHUEUFUGWGANZWLWDUHVOWMWJWBWKWCWMWHVTWIWAWGAVSHPWGA
      VSHOUIWGAVSKPUJUKGWLVNGWJWKEWGVSFLZWGQZUALZVSUBLZWNRZSZRZWGWNSZQZJZFULZUB
      UAWQXDUNZUOWPVEWGXESUOUPZDUQXCELZWGQZWPVSWQXGRZSZRZWGXGSZQZJFEWNXGNZWOXHX
      BXMWNXGWGURXNWTXKXAXLXNWSXJWPXNWRXIVSWNXGWQUSUTVAWNXGWGVBVCUIVDXFVFCUQVGV
      HVIVJVJVK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Finitely supported functions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c finSupp $.  $( Class for predicate "finitely supported function". $)

  $( Extend class definition to include the predicate to be a finitely
     supported function. $)
  cfsupp $a class finSupp $.

  ${
    $d r z $.
    $( Define the property of a function to be finitely supported (in relation
       to a given zero).  (Contributed by AV, 23-May-2019.) $)
    df-fsupp $a |- finSupp
                   = { <. r , z >. | ( Fun r /\ ( r supp z ) e. Fin ) } $.

    $( The property of a function to be finitely supported is a relation.
       (Contributed by AV, 7-Jun-2019.) $)
    relfsupp $p |- Rel finSupp $=
      ( vr vz cv wfun csupp co cfn wcel wa cfsupp df-fsupp relopabiv ) ACZDMBCE
      FGHIABJBAKL $.
  $}

  $( A proper class is never finitely supported.  (Contributed by AV,
     7-Jun-2019.) $)
  relprcnfsupp $p |- ( -. A e. _V -> -. A finSupp Z ) $=
    ( cfsupp wbr cvv wcel relfsupp brrelex1i con3i ) ABCDAEFABCGHI $.

  ${
    $d R r z $.  $d Z r z $.
    $( The property of a class to be a finitely supported function (in relation
       to a given zero).  (Contributed by AV, 23-May-2019.) $)
    isfsupp $p |- ( ( R e. V /\ Z e. W ) -> ( R finSupp Z
                                    <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) ) $=
      ( vr vz cv wfun csupp co cfn wcel wa cfsupp wb funeq adantr oveq12 eleq1d
      wceq anbi12d df-fsupp brabga ) EGZHZUDFGZIJZKLZMAHZADIJZKLZMEFADNBCUDATZU
      FDTZMZUEUIUHUKULUEUIOUMUDAPQUNUGUJKUDAUFDIRSUAFEUBUC $.
  $}

  ${
    isfsuppd.r $e |- ( ph -> R e. V ) $.
    isfsuppd.z $e |- ( ph -> Z e. W ) $.
    isfsuppd.1 $e |- ( ph -> Fun R ) $.
    isfsuppd.2 $e |- ( ph -> ( R supp Z ) e. Fin ) $.
    $( Deduction form of ~ isfsupp .  (Contributed by SN, 29-Jul-2024.) $)
    isfsuppd $p |- ( ph -> R finSupp Z ) $=
      ( cfsupp wbr wfun csupp co cfn wcel wa wb isfsupp syl2anc mpbir2and ) ABE
      JKZBLZBEMNOPZHIABCPEDPUBUCUDQRFGBCDESTUA $.
  $}

  $( The property of a function to be finitely supported (in relation to a
     given zero).  (Contributed by AV, 23-May-2019.) $)
  funisfsupp $p |- ( ( Fun R /\ R e. V /\ Z e. W )
                     -> ( R finSupp Z <-> ( R supp Z ) e. Fin ) ) $=
    ( wfun wcel w3a cfsupp wbr csupp co cfn wa wb isfsupp 3adant1 ibar 3ad2ant1
    bicomd bitrd ) AEZABFZDCFZGADHIZUAADJKLFZMZUEUBUCUDUFNUAABCDOPUAUBUFUENUCUA
    UEUFUAUEQSRT $.

  $( Implications of a class being a finitely supported function (in relation
     to a given zero).  (Contributed by AV, 26-May-2019.) $)
  fsuppimp $p |- ( R finSupp Z -> ( Fun R /\ ( R supp Z ) e. Fin ) ) $=
    ( cvv wcel wa cfsupp wbr wfun csupp co cfn relfsupp brrelex12i biimpd mpcom
    isfsupp ) ACDBCDEZABFGZAHABIJKDEZABFLMQRSACCBPNO $.

  ${
    fsuppimpd.f $e |- ( ph -> F finSupp Z ) $.
    $( A finitely supported function is a function with a finite support.
       (Contributed by AV, 6-Jun-2019.) $)
    fsuppimpd $p |- ( ph -> ( F supp Z ) e. Fin ) $=
      ( cfsupp wbr csupp co cfn wcel wfun fsuppimp simprd syl ) ABCEFZBCGHIJZDO
      BKPBCLMN $.
  $}

  ${
    fsuppfund.1 $e |- ( ph -> F finSupp Z ) $.
    $( A finitely supported function is a function.  (Contributed by SN,
       8-Mar-2025.) $)
    fsuppfund $p |- ( ph -> Fun F ) $=
      ( cfsupp wbr wfun csupp co cfn wcel fsuppimp simpld syl ) ABCEFZBGZDOPBCH
      IJKBCLMN $.
  $}

  $( If two functions have the same support, one function is finitely supported
     iff the other one is finitely supported.  (Contributed by AV,
     30-Jun-2019.) $)
  suppeqfsuppbi $p |- ( ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) )
                        -> ( ( F supp Z ) = ( G supp Z )
                             -> ( F finSupp Z <-> G finSupp Z ) ) ) $=
    ( wcel wfun wa csupp co cfsupp wbr wb cvv relfsupp brrelex2i a1i cfn adantr
    wi wceq simprlr simprll simpl funisfsupp syl3anc simpr adantl impcom bicomd
    ex eleq1 sylan9bb bitr4d expl com12 pm5.21ndd ) BAFZBGZHZCDFZCGZHZHZBEIJZCE
    IJZUAZBEKLZCEKLZMZVDVGHZENFZVHVIVHVLTVKBEKOPQVIVLTVKCEKOPQVLVKVJVLVDVGVJVLV
    DHZVGHVHVERFZVIVMVHVNMZVGVMUSURVLVOVLURUSVCUBVLURUSVCUCVLVDUDBANEUEUFSVMVIV
    FRFZVGVNVDVLVIVPMZVCVLVQTUTVCVLVQVCVLHVBVAVLVQVCVBVLVAVBUGSVCVAVLVAVBUDSVCV
    LUGCDNEUEUFUKUHUIVGVNVPVEVFRULUJUMUNUOUPUQUK $.

  $( The cartesian product of two finitely supported functions is finite.
     (Contributed by AV, 17-Jul-2019.) $)
  fsuppxpfi $p |- ( ( F finSupp Z /\ G finSupp Z )
                    -> ( ( F supp Z ) X. ( G supp Z ) ) e. Fin ) $=
    ( cfsupp wbr csupp co cfn wcel cxp id fsuppimpd xpfi syl2an ) ACDEZACFGZHIB
    CFGZHIPQJHIBCDEZOACOKLRBCRKLPQMN $.

  ${
    fczfsuppd.b $e |- ( ph -> B e. V ) $.
    fczfsuppd.z $e |- ( ph -> Z e. W ) $.
    $( A constant function with value zero is finitely supported.  (Contributed
       by AV, 30-Jun-2019.) $)
    fczfsuppd $p |- ( ph -> ( B X. { Z } ) finSupp Z ) $=
      ( csn cxp cvv wcel snexg syl xpexd wfn wfun fnconstg fnfun 3syl cfn csupp
      co c0 fczsupp0 0fi eqeltri a1i isfsuppd ) ABEHZIZJDEABUICJFAEDKZUIJKGEDLM
      NGAUKUJBOUJPGBEDQBUJRSUJEUAUBZTKAULUCTBEUDUEUFUGUH $.
  $}

  $( The empty set is a finitely supported function.  (Contributed by AV,
     19-Jul-2019.) $)
  0fsupp $p |- ( Z e. V -> (/) finSupp Z ) $=
    ( wcel c0 cfsupp wbr csupp co cfn supp0 0fi eqeltrdi wfun cvv wb funisfsupp
    fun0 0ex mp3an12 mpbird ) BACZDBEFZDBGHZICZUAUCDIABJKLDMDNCUAUBUDOQRDNABPST
    $.

  ${
    snopfsuppdc.x $e |- ( ph -> X e. V ) $.
    snopfsuppdc.y $e |- ( ph -> Y e. W ) $.
    snopfsuppdc.z $e |- ( ph -> Z e. U ) $.
    snopfsuppdc.dc $e |- ( ph -> DECID Y = Z ) $.
    $( A singleton containing an ordered pair is a finitely supported function.
       (Contributed by AV, 19-Jul-2019.) $)
    snopfsuppdc $p |- ( ph -> { <. X , Y >. } finSupp Z ) $=
      ( cop csn cvv wcel opexg syl2anc syl c0 cfn snexg wfun funsng co wceq cif
      csupp eqid suppsnopdc 0fi a1i snfig ifcldcd eqeltrd isfsuppd ) AEFLZMZNBG
      AUPNOZUQNOAECOZFDOZURHIEFCDPQUPNUARJAUSUTUQUBHIEFCDUCQAUQGUGUDFGUEZSEMZUF
      TABUQCDEFGUQUHHIJKUIAVASVBTSTOAUJUKAUSVBTOHECULRKUMUNUO $.
  $}

  $( Two ways of saying that a function with known codomain is finitely
     supported.  (Contributed by AV, 8-Jul-2019.) $)
  ffsuppbi $p |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S
                 -> ( F finSupp Z <-> ( `' F " ( S \ { Z } ) ) e. Fin ) ) ) $=
    ( wcel wa wf cfsupp wbr ccnv csn cdif cima cfn wb csupp cvv imp wfun adantl
    co ffun wi fex expcom adantr simplr funisfsupp syl3anc fsuppeq eleq1d bitrd
    wceq ex ) CDGZFEGZHZCABIZBFJKZBLAFMNOZPGZQUSUTHZVABFRUCZPGZVCVDBUAZBSGZURVA
    VFQUTVGUSCABUDUBUSUTVHUQUTVHUEURUTUQVHCADBUFUGUHTUQURUTUIBSEFUJUKVDVEVBPUSU
    TVEVBUOABCDEFULTUMUNUP $.

  ${
    fsuppco.f $e |- ( ph -> F finSupp Z ) $.
    fsuppco.g $e |- ( ph -> G : X -1-1-> Y ) $.
    fsuppco.z $e |- ( ph -> Z e. W ) $.
    fsuppco.v $e |- ( ph -> F e. V ) $.
    fsuppcorn.g $e |- ( ph -> G e. U ) $.
    fsuppcorn.rn $e |- ( ph -> ( F supp Z ) C_ ran G ) $.
    $( The composition of a 1-1 function with a finitely supported function is
       finitely supported.  The purpose of the ` ( F supp Z ) C_ ran G `
       condition is to ensure we don't subset the support of the function in
       such a way as to fun afoul of ~ exmidssfi .  (Other alternative
       conditions might also be sufficient).  (Contributed by AV, 28-May-2019.)
       (Revised by Jim Kingdon, 15-May-2026.) $)
    fsuppcorn $p |- ( ph -> ( F o. G ) finSupp Z ) $=
      ( wcel wfun syl syl2anc cfn ccom cvv ccnv wf1 simprbi cofunex2g fsuppfund
      df-f1 f1fun funco csupp cima wceq suppcofn syl22anc cen wbr fsuppimpd crn
      wf co wss wf1o f1cnv f1of1 f1imaeng syl3anc enfii eqeltrd isfsuppd ) ACDU
      AZUBFIACEPZDUCZQZVKUBPMAGHDUDZVNKVOGHDUTVNGHDUHUERCDEUFSLACQZDQZVKQACIJUG
      ZAVOVQKGHDUIRZCDUJSAVKIUKVAZVMCIUKVAZULZTAVLDBPVPVQVTWBUMMNVRVSCDEBIUNUOA
      WATPZWBWAUPUQZWBTPACIJURZADUSZGVMUDZWAWFVBWCWDAWFGVMVCZWGAVOWHKGHDVDRWFGV
      MVEROWEWFGWAVMTVFVGWBWAVHSVIVJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Finite intersections
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c fi $.
  $( Extend class notation with the function whose value is the class of finite
     intersections of the elements of a given set. $)
  cfi $a class fi $.

  ${
    $d x y z $.
    $( Function whose value is the class of finite intersections of the
       elements of the argument.  Note that the empty intersection being the
       universal class, hence a proper class, it cannot be an element of that
       class.  Therefore, the function value is the class of nonempty finite
       intersections of elements of the argument (see ~ elfi2 ).  (Contributed
       by FL, 27-Apr-2008.) $)
    df-fi $a
          |- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } ) $.
  $}

  ${
    $d x y z A $.  $d x y B $.  $d x V $.  $d x W $.
    $( The set of all the finite intersections of the elements of ` A ` .
       (Contributed by FL, 27-Apr-2008.)  (Revised by Mario Carneiro,
       24-Nov-2013.) $)
    fival $p |- ( A e. V -> ( fi ` A ) = { y |
                  E. x e. ( ~P A i^i Fin ) y = |^| x } ) $=
      ( vz wcel cv cint wceq cpw cfn cin wrex cab cvv cfi df-fi wss c0 syl pweq
      ineq1d rexeqdv abbidv elex cuni wa wex elinel1 elpwid wne eqvisset intexr
      adantl neneqd wo elinel2 adantr fin0or orcomd intssuni2m syl2an2r eqsstrd
      simpr ecased velpw sylibr rexlimiva abssi uniexg pwexd sylancr fvmptd3
      ssexg ) CDFZECBGZAGZHZIZAEGZJZKLZMZBNVSACJZKLZMZBNZOPOEABQVTCIZWCWFBWHVSA
      WBWEWHWAWDKVTCUAUBUCUDCDUEVOWGCUFZJZRWJOFWGOFWFBWJVSVPWJFZAWEVQWEFZVSUGZV
      PWIRWKWMVPVRWIWLVSVDWLVQCRVSVTVQFEUHZVRWIRWLVQCVQWDKUIUJWMWNVQSIZWMVQSVSV
      QSUKZWLVSVROFWPBVRULVQUMTUNUOWMVQKFZWNWOUPWLWQVSVQWDKUQURWQWOWNEVQUSUTTVE
      EVQCVAVBVCBWIVFVGVHVIVOWIOCDVJVKWGWJOVNVLVM $.

    $( Specific properties of an element of ` ( fi `` B ) ` .  (Contributed by
       FL, 27-Apr-2008.)  (Revised by Mario Carneiro, 24-Nov-2013.) $)
    elfi $p |- ( ( A e. V /\ B e. W ) -> ( A e. ( fi ` B ) <->
        E. x e. ( ~P B i^i Fin ) A = |^| x ) ) $=
      ( vy wcel cfi cfv cv cint wceq cpw cfn cin wrex cab fival eleq2d eqeq1
      rexbidv elabg sylan9bbr ) CEGZBCHIZGBFJZAJKZLZACMNOZPZFQZGBDGBUGLZAUIPZUD
      UEUKBAFCERSUJUMFBDUFBLUHULAUIUFBUGTUAUBUC $.

    $( The empty intersection need not be considered in the set of finite
       intersections.  (Contributed by Mario Carneiro, 21-Mar-2015.) $)
    elfi2 $p |- ( B e. V -> ( A e. ( fi ` B ) <->
        E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x ) ) $=
      ( vz wcel cvv cv cint cfn c0 wrex wi a1i wa simpr adantr wb syl ancom cfi
      cfv wceq cpw cin csn cdif elex eldifsni eldifi elin2d fin0 mpbid inteximm
      wex eqeltrd rexlimiva elfi wn vprc elsni inteqd int0 eqtrdi eleq1d mtbiri
      wne eqeltrrd nsyl3 biantrud eldif bitr4di pm5.32da 3bitr4g rexbidv2 bitrd
      simpll expcom pm5.21ndd ) CDFZBGFZBCUAUBZFZBAHZIZUCZACUDZJUEZKUFZUGZLZWCW
      AMVTBWBUHNWKWAMVTWFWAAWJWDWJFZWFOZBWEGWLWFPWMEHWDFEUOZWEGFZWMWDKVGZWNWLWP
      WFWDWHKUIQWMWDJFZWPWNRWLWQWFWLWGJWDWDWHWIUJUKQEWDULSUMEWDUNSUPUQNWAVTWCWK
      RWAVTOZWCWFAWHLWKABCGDURWRWFWFAWHWJWRWFWDWHFZOWFWLOWSWFOWMWRWFWSWLWRWFOZW
      SWSWDWIFZUSZOWLWTXBWSXAWOWTXAWOGGFUTXAWEGGXAWEKIGXAWDKWDKVAVBVCVDVEVFWTBW
      EGWRWFPWAVTWFVQVHVIVJWDWHWIVKVLVMWSWFTWLWFTVNVOVPVRVS $.

    $( Sufficient condition for an element of ` ( fi `` B ) ` .  (Contributed
       by Mario Carneiro, 24-Nov-2013.) $)
    elfir $p |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) ->
        |^| A e. ( fi ` B ) ) $=
      ( vx vz wcel wss c0 wne cfn w3a wa cint cfi cfv cv wceq cpw cvv wb elpw2g
      cin wrex simp1 imbitrrid imp simpr3 elind eqid inteq rspceeqv sylancl wex
      simp2 fin0 3ad2ant3 mpbid inteximm syl id elfi syl2anr mpbird ) BCFZABGZA
      HIZAJFZKZLZAMZBNOFZVJDPZMZQDBRZJUBZUCZVIAVOFVJVJQVPVIVNJAVDVHAVNFZVHVQVDV
      EVEVFVGUDABCUAUEUFVDVEVFVGUGUHVJUIDAVOVMVJVJVLAUJUKULVHVJSFZVDVKVPTVDVHEP
      AFEUMZVRVHVFVSVEVFVGUNVGVEVFVSTVFEAUOUPUQEAURUSVDUTDVJBSCVAVBVC $.
  $}

  ${
    $d A x y $.  $d x V $.
    $( Any element of a set ` A ` is the intersection of a finite subset of
       ` A ` .  (Contributed by FL, 27-Apr-2008.)  (Proof shortened by Mario
       Carneiro, 21-Mar-2015.) $)
    ssfii $p |- ( A e. V -> A C_ ( fi ` A ) ) $=
      ( vx wcel cfi cfv cv wa csn cint vex intsn wss c0 wne cfn simpl simpr a1i
      snssd snnz cvv snfig elv elfir syl13anc eqeltrrid ex ssrdv ) ABDZCAAEFZUJ
      CGZADZULUKDUJUMHZULULIZJZUKULCKZLUNUJUOAMUONOZUOPDZUPUKDUJUMQUNULAUJUMRTU
      RUNULUQUASUSUNUSCULUBUCUDSUOABUEUFUGUHUI $.

    $( The set of finite intersections of the empty set.  (Contributed by Mario
       Carneiro, 30-Aug-2015.) $)
    fi0 $p |- ( fi ` (/) ) = (/) $=
      ( vy vx c0 cfi cfv cv cint wceq cpw cfn cin wrex cab cvv wcel fival ax-mp
      0ex vprc id wss elinel1 elpwi 3syl inteqd int0 eqtrdi sylan9eqr rexlimiva
      ss0 vex eqeltrrdi mto abf eqtri ) CDEZAFZBFZGZHZBCIZJKZLZAMZCCNOUPVDHRBAC
      NPQVCAVCNNOSVCNUQNUTUQNHBVBUTURVBOZUQUSNUTTVEUSCGNVEURCVEURVAOURCUAURCHUR
      VAJUBURCUCURUJUDUEUFUGUHUIAUKULUMUNUO $.

    $( A set is empty iff the class of all the finite intersections of that set
       is empty.  (Contributed by FL, 27-Apr-2008.)  (Revised by Mario
       Carneiro, 24-Nov-2013.) $)
    fieq0 $p |- ( A e. V -> ( A = (/) <-> ( fi ` A ) = (/) ) ) $=
      ( wcel c0 wceq cfi cfv fveq2 fi0 eqtrdi wss ssfii sseq0 sylan ex impbid2
      ) ABCZADEZAFGZDEZRSDFGDADFHIJQTRQASKTRABLASMNOP $.
  $}

  ${
    $d A r x $.  $d B r x $.  $d V r x $.
    $( Subset relationship for function ` fi ` .  (Contributed by Jeff Hankins,
       7-Oct-2009.)  (Revised by Mario Carneiro, 24-Nov-2013.) $)
    fiss $p |- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) ) $=
      ( vr vx wcel wss wa cfi cfv cv cint wceq cpw cfn cin wrex cvv wb elfi vex
      wi simpr sspwb ssrin sylbi ssrexv 3syl simpl ssexd sylancr adantr 3imtr4d
      mpan ssrdv ) BCFZABGZHZDAIJZBIJZURDKZEKLMZEANZOPZQZVBEBNZOPZQZVAUSFZVAUTF
      ZURUQVDVGGZVEVHUBUPUQUCZUQVCVFGVKABUDVCVFOUEUFVBEVDVGUGUHURVARFZARFVIVESD
      UAZURABCUPUQUIVLUJEVAARRTUKUPVJVHSZUQVMUPVOVNEVABRCTUNULUMUO $.
  $}

  ${
    $d A x y z $.  $d V x y z $.  $d w z $.
    $( The union of the finite intersections of a set is simply the union of
       the set itself.  (Contributed by Jeff Hankins, 5-Sep-2009.)  (Revised by
       Mario Carneiro, 24-Nov-2013.) $)
    fiuni $p |- ( A e. V -> U. A = U. ( fi ` A ) ) $=
      ( vx vy vz vw wcel cuni cfi cfv cv wa wex cfn c0 simprr wb mpbid ad2antrl
      wss ssfii unissd eluni biimpi adantl cint wceq cpw cin cdif wrex ad2antrr
      csn elfi2 eldifi elin1d elpwid wne eldifsni elin2d syl intssuni2m syl2anc
      fin0 eqsstrd simplrl sseldd rexlimddv exlimddv eqelssd ) ABGZCAHZAIJZHZVK
      AVMABUAUBVKCKZVNGZLZVODKZGZVRVMGZLZVOVLGZDVPWADMZVKVPWCDVOVMUCUDUEVQWALZV
      REKZUFZUGZWBEAUHZNUIZOUMZUJZWDVTWGEWKUKZVQVSVTPVKVTWLQVPWAEVRABUNULRWDWEW
      KGZWGLZLZVRVLVOWOVRWFVLWDWMWGPWOWEATZFKWEGFMZWFVLTWMWPWDWGWMWEAWMWHNWEWEW
      IWJUOZUPUQSWMWQWDWGWMWEOURZWQWEWIOUSWMWENGWSWQQWMWHNWEWRUTFWEVDVARSFWEAVB
      VCVEVQVSVTWNVFVGVHVIVJ $.
  $}

  $( If a set is a family of subsets of some base set, then so is its finite
     intersection.  (Contributed by Stefan O'Rear, 2-Aug-2015.) $)
  fipwssg $p |- ( ( A e. V /\ A C_ ~P X ) -> ( fi ` A ) C_ ~P X ) $=
    ( wcel cpw wss cfi cfv cuni fiuni sseq1d sspwuni 3bitr4g biimpa ) ABDZACEZF
    ZAGHZPFZOAIZCFRIZCFQSOTUACABJKACLRCLMN $.

  ${
    $d x y A $.  $d x F $.  $d x y V $.  $d w y $.
    fifo.1 $e |- F = ( y e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| y ) $.
    $( Describe a surjection from nonempty finite sets to finite intersections.
       (Contributed by Mario Carneiro, 18-May-2015.) $)
    fifo $p |- ( A e. V ->
      F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) ) $=
      ( vw vx wcel cpw cfn cin c0 csn cdif wfo cv cvv wb syl wceq cfi cint wral
      cfv crn wfn wex wne eldifsni eldifi elin2d fin0 mpbid inteximm rgen fnmpt
      mp1i dffn4 sylib wrex elfi2 elrnmpt elv bitr4di eqrdv foeq3 mpbird ) BDHZ
      BIZJKZLMZNZBUAUDZCOZVLCUEZCOZVHCVLUFZVPAPZUBZQHZAVLUCVQVHVTAVLVRVLHZFPVRH
      FUGZVTWAVRLUHZWBVRVJLUIWAVRJHWCWBRWAVIJVRVRVJVKUJUKFVRULSUMFVRUNSUOAVLVSC
      QEUPUQVLCURUSVHVMVOTVNVPRVHGVMVOVHGPZVMHWDVSTAVLUTZWDVOHZAWDBDVAWFWERGAVL
      VSWDCQEVBVCVDVEVMVOVLCVFSVG $.
  $}

  ${
    $d A w x y z $.  $d ph w y z $.
    $( Decidability of a family of propositions indexed by a finite set.
       (Contributed by Jim Kingdon, 30-Sep-2024.) $)
    dcfi $p |- ( ( A e. Fin /\ A. x e. A DECID ph ) -> DECID A. x e. A ph ) $=
      ( vw vy vz cfn wcel wdc wral wa cv csn cun wceq raleq dcbid dcbii sylibr
      c0 wn wo ral0 orci df-dc mpbir a1i wss cdif simpr simplrr eldifad simp-4r
      wsbc nfsbc1v nfdc sbceq1a rspc wb cvv ralsns elv dcand ralunb findcard2sd
      sylc ex simpl ) CGHZAIZBCJZKZABDLZJZIABTJZIZABELZJZIZABVQFLZMZNZJZIZABCJZ
      IDEFCVMTOVNVOABVMTPQVMVQOVNVRABVMVQPQVMWBOVNWCABVMWBPQVMCOVNWEABVMCPQVPVL
      VPVOVOUAZUBVOWFABUCUDVOUEUFUGVLVQGHZKZVQCUHZVTCVQUIHZKZKZVSWDWLVSKZVRABWA
      JZKZIWDWMVRWNWLVSUJWMABVTUNZIZWNIWMVTCHVKWQWMVTCVQWHWIWJVSUKULVIVKWGWKVSU
      MVJWQBVTCWPBABVTUOUPBLVTOAWPABVTUQQURVFWNWPWNWPUSFABVTUTVAVBRSVCWCWOABVQW
      AVDRSVGVIVKVHVE $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d F x y $.
    $( It is decidable whether a function from a finite set into another finite
       set is one-to-one.  (Contributed by Jim Kingdon, 13-Jul-2026.) $)
    fdcf1 $p |- ( ( A e. Fin /\ B e. Fin /\ F : A --> B )
        -> DECID F : A -1-1-> B ) $=
      ( vx vy cfn wcel cfv wceq wral wdc sylibr adantr ffvelcdmd fidceq syl3anc
      cv wa ralrimiva dcfi wf w3a wi wf1 wn wo simp3 orcd df-dc simpll2 simpll3
      simp1 simplr simpr dcim sylc syl2anc dcand dff13 dcbii ) AFGZBFGZABCUAZUB
      ZVCDQZCHZEQZCHZIZVEVGIZUCZEAJZDAJZRZKABCUDZKVDVCVMVDVCVCUEZUFVCKVDVCVPVAV
      BVCUGUHVCUILVDVAVLKZDAJVMKVAVBVCULZVDVQDAVDVEAGZRZVAVKKZEAJVQVDVAVSVRMZVT
      WAEAVTVGAGZRZVIKZVJKZWAWDVBVFBGVHBGWEVAVBVCVSWCUJWDABVECVAVBVCVSWCUKZVDVS
      WCUMZNWDABVGCWGVTWCUNZNBVFVHOPWDVAVSWCWFVTVAWCWBMWHWIAVEVGOPVIVJUOUPSVKEA
      TUQSVLDATUQURVOVNDEABCUSUTL $.
  $}

  ${
    $d A f g $.  $d B f g $.
    $( The set of injections between two finite sets is finite.  (Contributed
       by Jim Kingdon, 13-Jul-2026.) $)
    f1setfi $p |- ( ( A e. Fin /\ B e. Fin )
        -> { f | f : A -1-1-> B } e. Fin ) $=
      ( vg cfn wcel wa cv wf cab wf1 wss wdc wral cmap wceq ancoms elab syl3anc
      co mapvalg mapfi eqeltrrd f1f ss2abi a1i simpll simplr bilani fdcf1 f1eq1
      vex feq1 dcbii sylibr ralrimiva ssfidc ) AEFZBEFZGZABCHZIZCJZEFABVAKZCJZV
      CLZDHZVEFZMZDVCNVEEFUTBAOTZVCEUSURVJVCPBAEECUAQUSURVJEFBAUBQUCVFUTVDVBCAB
      VAUDUEUFUTVIDVCUTVGVCFZGZABVGKZMZVIVLURUSABVGIZVNURUSVKUGURUSVKUHVKVOUTVB
      VOCVGDULZABVAVGUMRUIABVGUJSVHVMVDVMCVGVPABVAVGUKRUNUOUPDVCVEUQS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The sizes of sets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A s u w y z $.  $d A s w x y z $.  $d V s u w y z $.
    2omap.f $e |- F = ( s e. ( 2o ^m A ) |-> { z e. A | ( s ` z ) = 1o } ) $.
    $( Mapping between ` ( 2o ^m A ) ` and decidable subsets of ` A ` .
       (Contributed by Jim Kingdon, 12-Nov-2025.) $)
    2omap $p |- ( A e. V -> F : ( 2o ^m A ) -1-1-onto->
        { x e. ~P A | A. y e. A DECID y e. x } ) $=
      ( vw vu wcel c2o cv wdc c1o wceq c0 wa adantr simpr cmap co wral cpw crab
      cfv cif cmpt eleq2 dcbid ralbidv wss ssrab2 elpw2g mpbiri 2ssom wf elmapi
      com ad2antlr ffvelcdmd sselid 1onn nndceq sylancl wb adantl fveqeq2 elrab
      bitr4di mpbid ralrimiva elrabd weq 1lt2o a1i 0lt2o elequ1 simplrr rspcdva
      ibar ifcldcd fmpttd 2onn simpl elmapd mpbird sylan2b anbi2i simplr fveq1d
      wn eqid ifbid simprrr ad2antrr fvmptd3 eqtrd iftrued 2thd nesymi iffalsed
      1n0 eqeq1d mtbiri 2falsed exmiddc syl mpjaodan rabbidva elpwi dfss1 sylib
      wo cin dfin5 eqtr3di ad2antrl simplrl simpll feqmptd eleq2d bitrdi biimpa
      baibd eqtr4d mpbirand mtbid df2o3 eleqtrdi elpri ecased mpteq2dva impbida
      cpr f1o2d ) DFKZGILDUAUBZBMZAMZKZNZBDUCZADUDZUEZCMZGMZUFZOPZCDUEZJDJMZIMZ
      KZOQUGZUHZEHYQUUGYRKZRZUUCYSUUJKZNZBDUCAUUJUUDYTUUJPZUUBUUSBDUUTUUAUURYTU
      UJYSUIUJUKYQUUJUUDKZUUPYQUVAUUJDULUUICDUMUUJDFUNUOSUUQUUSBDUUQYSDKZRZYSUU
      GUFZOPZNZUUSUVCUVDUSKOUSKUVFUVCLUSUVDUPUVCDLYSUUGUUPDLUUGUQZYQUVBUUGLDURU
      TUUQUVBTVAVBVCUVDOVDVEUVCUVEUURUVCUVEUVBUVERZUURUVBUVEUVHVFUUQUVBUVEWAVGU
      UIUVECYSDUUFYSOUUGVHVIVJUJVKVLVMUULUUEKZYQUULUUDKZYSUULKZNZBDUCZRZUUOYRKZ
      UUCUVMAUULUUDAIVNZUUBUVLBDUVPUUAUVKYTUULYSUIUJUKVIZYQUVNRZUVODLUUOUQUVRJD
      UUNLUVRUUKDKZRZUUMOQLOLKZUVTVOVPQLKZUVTVQVPUVTUVLUUMNZBDUUKBJVNUVKUUMBJIV
      RUJZYQUVJUVMUVSVSUVRUVSTVTWBWCUVRLDUUOUSFLUSKZUVRWDVPYQUVNWEWFWGWHUUPUVIR
      YQUUPUVNRZUUGUUOPZUULUUJPZVFUVIUVNUUPUVQWIYQUWFRZUWGUWHUWIUWGRZUWHUUFUULK
      ZCDUEZUUJPZUWJUWKUUICDUWJUUFDKZRZUWKUWKUUIVFUWKWLZUWOUWKRZUWKUUIUWOUWKTZU
      WQUUHUWKOQUGZOUWOUUHUWSPZUWKUWOUUHUUFUUOUFUWSUWOUUFUUGUUOUWIUWGUWNWJWKUWO
      JUUFUUNUWSDUUOLUUOWMJCVNUUMUWKOQJCIVRWNUWJUWNTZUWOUWKOQLUWAUWOVOVPUWBUWOV
      QVPUWOUVLUWKNZBDUUFBCVNUVKUWKBCIVRUJUWIUVMUWGUWNYQUUPUVJUVMWOZWPUXAVTZWBW
      QWRZSUWQUWKOQUWRWSWRWTUWOUWPRZUWKUUIUWOUWPTZUXFUUIQOPOQXCXAUXFUUHQOUXFUUH
      UWSQUWOUWTUWPUXESUXFUWKOQUXGXBWRXDXEXFUWOUXBUWKUWPXNUXDUWKXGXHXIXJUWFUWHU
      WMVFZYQUWGUVJUXHUUPUVMUVJUULUWLUUJUVJDUULXOZUULUWLUVJUULDULUXIUULPUULDXKU
      ULDXLXMCDUULXPXQXDXRUTWGUWIUWHRZUUGJDUUKUUGUFZUHUUOUXJJDLUUGUXJUUPUVGYQUU
      PUVNUWHXSUXJLDUUGUSFUWEUXJWDVPYQUWFUWHXTWFVKZYAUXJJDUXKUUNUXJUVSRZUUMUXKU
      UNPUUMWLZUXMUUMRZUXKOUUNUXMUUMUXKOPZUXJUUMUVSUXPUXJUUMUUKUUJKZUVSUXPRZUXJ
      UULUUJUUKUWIUWHTYBUUIUXPCUUKDUUFUUKOUUGVHVIZYCYEYDUXOUUMOQUXMUUMTWSYFUXMU
      XNRZUXKQUUNUXTUXKQPZUXPUXTUUMUXPUXMUXNTZUXMUUMUXPVFUXNUXMUUMUVSUXPUXJUVST
      ZUXMUUMUXQUXRUXMUULUUJUUKUWIUWHUVSWJYBUXSYCYGSYHUXMUYAUXPXNZUXNUXMUXKQOYO
      ZKUYDUXMUXKLUYEUXMDLUUKUUGUXJUVGUVSUXLSUYCVAYIYJUXKQOYKXHSYLUXTUUMOQUYBXB
      YFUXMUWCUUMUXNXNUXMUVLUWCBDUUKUWDUWIUVMUWHUVSUXCWPUYCVTUUMXGXHXIYMWRYNWHY
      P $.
  $}

  ${
    $d A s x y z $.  $d V s y z $.
    $( Equinumerosity of ` ( 2o ^m A ) ` and the set of decidable subsets of
       ` A ` .  (Contributed by Jim Kingdon, 14-Nov-2025.) $)
    2omapen $p |- ( A e. V -> ( 2o ^m A ) ~~
        { x e. ~P A | A. y e. A DECID y e. x } ) $=
      ( vs vz wcel c2o cmap co cvv cv wdc wral cpw crab cfv c1o wceq cmpt fnmap
      wf1o cen wbr cxp wfn 2onn elexi elex fnovex mp3an12i 2omap f1oeng syl2anc
      com eqid ) CDGZHCIJZKGZURBLALGMBCNACOPZEURFLELQRSFCPTZUBURUTUCUDIKKUEUFHK
      GUQCKGUSUAHUOUGUHCDUIHCKKIUJUKABFCVADEVAUPULURUTKVAUMUN $.
  $}

  ${
    $d A x y $.
    $( The number of finite subsets of a finite set.  For a similar theorem
       with set size expressed using ` # ` ( ~ df-ihash ), see ~ hashpwfi .
       (Contributed by Jim Kingdon, 18-May-2026.) $)
    2omapfi $p |- ( A e. Fin -> ( 2o ^m A ) ~~ ( ~P A i^i Fin ) ) $=
      ( vy vx cfn wcel c2o cmap co cv wdc wral crab cen wbr cvv simpll ad2antlr
      wa simpr syl3anc cpw cin 2omapen wceq eqid pwexg rabexd wss ssfidc fissfi
      elpwi impbida rabbidva eqeng sylc dfin5 breqtrrdi entr syl2anc ) ADEZFAGH
      ZBICIZEJBAKZCAUAZLZMNVEVDDUBZMNVAVFMNCBADUCUTVEVBDEZCVDLZVFMUTVEOEVEVHUDV
      EVHMNUTVCCVDVEOVEUEADUFUGUTVCVGCVDUTVBVDEZRZVCVGVJVCRUTVBAUHZVCVGUTVIVCPV
      IVKUTVCVBAUKZQVJVCSBAVBUITVJVGRVKUTVGVCVIVKUTVGVLQUTVIVGPVJVGSBAVBUJTULUM
      VEVHOUNUOCVDDUPUQVAVEVFURUS $.
  $}

  $( The set of finite subsets of a finite set is finite.  (Contributed by Jim
     Kingdon, 19-May-2026.) $)
  fipwfi $p |- ( A e. Fin -> ( ~P A i^i Fin ) e. Fin ) $=
    ( cfn wcel c2o cmap co cpw cin cen wbr 2onn nnfi ax-mp mapfi 2omapfi ensymd
    com mpan enfii syl2anc ) ABCZDAEFZBCZAGBHZUBIJUDBCDBCZUAUCDQCUEKDLMDANRUAUB
    UDAOPUDUBST $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Supremum and infimum
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Add new constant symbols. $)
  $c sup $.  $( Supremum. $)
  $c inf $.  $( Infimum. $)

  $( Extend class notation to include supremum of class ` A ` .  Here ` R ` is
     ordinarily a relation that strictly orders class ` B ` .  For example,
     ` R ` could be 'less than' and ` B ` could be the set of real numbers. $)
  csup $a class sup ( A , B , R ) $.

  $( Extend class notation to include infimum of class ` A ` .  Here ` R ` is
     ordinarily a relation that strictly orders class ` B ` .  For example,
     ` R ` could be 'less than' and ` B ` could be the set of real numbers. $)
  cinf $a class inf ( A , B , R ) $.

  ${
    $d x y z R $.  $d x y z A $.  $d x y z B $.
    $( Define the supremum of class ` A ` .  It is meaningful when ` R ` is a
       relation that strictly orders ` B ` and when the supremum exists.
       (Contributed by NM, 22-May-1999.) $)
    df-sup $a |- sup ( A , B , R ) = U. { x e. B | ( A. y e. A -. x R y /\
                  A. y e. B ( y R x -> E. z e. A y R z ) ) } $.
  $}

  $( Define the infimum of class ` A ` .  It is meaningful when ` R ` is a
     relation that strictly orders ` B ` and when the infimum exists.  For
     example, ` R ` could be 'less than', ` B ` could be the set of real
     numbers, and ` A ` could be the set of all positive reals; in this case
     the infimum is 0.  The infimum is defined as the supremum using the
     converse ordering relation.  In the given example, 0 is the supremum of
     all reals (greatest real number) for which all positive reals are greater.
     (Contributed by AV, 2-Sep-2020.) $)
  df-inf $a |- inf ( A , B , R ) = sup ( A , B , `' R ) $.

  ${
    $d x y z R $.  $d x y z A $.  $d x y z B $.  $d x y z C $.
    $( Equality theorem for supremum.  (Contributed by NM, 22-May-1999.) $)
    supeq1 $p |- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) ) $=
      ( vx vy vz wceq cv wbr wn wral wrex wi wa crab cuni csup raleq df-sup
      rexeq imbi2d ralbidv anbi12d rabbidv unieqd 3eqtr4g ) BCHZEIZFIZDJKZFBLZU
      JUIDJZUJGIDJZGBMZNZFALZOZEAPZQUKFCLZUMUNGCMZNZFALZOZEAPZQBADRCADRUHUSVEUH
      URVDEAUHULUTUQVCUKFBCSUHUPVBFAUHUOVAUMUNGBCUAUBUCUDUEUFEFGBADTEFGCADTUG
      $.
  $}

  ${
    supeq1d.1 $e |- ( ph -> B = C ) $.
    $( Equality deduction for supremum.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    supeq1d $p |- ( ph -> sup ( B , A , R ) = sup ( C , A , R ) ) $=
      ( wceq csup supeq1 syl ) ACDGCBEHDBEHGFBCDEIJ $.
  $}

  ${
    supeq1i.1 $e |- B = C $.
    $( Equality inference for supremum.  (Contributed by Paul Chapman,
       22-Jun-2011.) $)
    supeq1i $p |- sup ( B , A , R ) = sup ( C , A , R ) $=
      ( wceq csup supeq1 ax-mp ) BCFBADGCADGFEABCDHI $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d R x y z $.
    $( Equality theorem for supremum.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    supeq2 $p |- ( B = C -> sup ( A , B , R ) = sup ( A , C , R ) ) $=
      ( vx vy vz wceq cv wbr wn wral wrex wi wa crab cuni csup rabeq df-sup
      raleq anbi2d rabbidv eqtrd unieqd 3eqtr4g ) BCHZEIZFIZDJKFALZUIUHDJUIGIDJ
      GAMNZFBLZOZEBPZQUJUKFCLZOZECPZQABDRACDRUGUNUQUGUNUMECPUQUMEBCSUGUMUPECUGU
      LUOUJUKFBCUAUBUCUDUEEFGABDTEFGACDTUF $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d R x y z $.  $d S x y z $.
    $( Equality theorem for supremum.  (Contributed by Scott Fenton,
       13-Jun-2018.) $)
    supeq3 $p |- ( R = S -> sup ( A , B , R ) = sup ( A , B , S ) ) $=
      ( vx vy vz cv wbr wn wral wrex wi wa crab cuni csup breq ralbidv df-sup
      wceq notbid rexbidv imbi12d anbi12d rabbidv unieqd 3eqtr4g ) CDUAZEHZFHZC
      IZJZFAKZUKUJCIZUKGHZCIZGALZMZFBKZNZEBOZPUJUKDIZJZFAKZUKUJDIZUKUPDIZGALZMZ
      FBKZNZEBOZPABCQABDQUIVBVLUIVAVKEBUIUNVEUTVJUIUMVDFAUIULVCUJUKCDRUBSUIUSVI
      FBUIUOVFURVHUKUJCDRUIUQVGGAUKUPCDRUCUDSUEUFUGEFGABCTEFGABDTUH $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d D x y z $.  $d E x y z $.
    $d F x y z $.  $d ph x y z $.
    supeq123d.a $e |- ( ph -> A = D ) $.
    supeq123d.b $e |- ( ph -> B = E ) $.
    supeq123d.c $e |- ( ph -> C = F ) $.
    $( Equality deduction for supremum.  (Contributed by Stefan O'Rear,
       20-Jan-2015.) $)
    supeq123d $p |- ( ph -> sup ( A , B , C ) = sup ( D , E , F ) ) $=
      ( vx vy vz cv wbr wn wral wrex wi breqd wa crab cuni csup imbi12d anbi12d
      notbid raleqbidv rexeqbidv rabeqbidv unieqd df-sup 3eqtr4g ) AKNZLNZDOZPZ
      LBQZUOUNDOZUOMNZDOZMBRZSZLCQZUAZKCUBZUCUNUOGOZPZLEQZUOUNGOZUOUTGOZMERZSZL
      FQZUAZKFUBZUCBCDUDEFGUDAVFVPAVEVOKCFIAURVIVDVNAUQVHLBEHAUPVGADGUNUOJTUGUH
      AVCVMLCFIAUSVJVBVLADGUOUNJTAVAVKMBEHADGUOUTJTUIUEUHUFUJUKKLMBCDULKLMEFGUL
      UM $.
  $}

  ${
    $d A u v w $.  $d B u v w $.  $d R u v w $.  $d u v w x $.
    nfsup.1 $e |- F/_ x A $.
    nfsup.2 $e |- F/_ x B $.
    nfsup.3 $e |- F/_ x R $.
    $( Hypothesis builder for supremum.  (Contributed by Mario Carneiro,
       20-Mar-2014.) $)
    nfsup $p |- F/_ x sup ( A , B , R ) $=
      ( vu vv vw csup cv wbr wn wral wrex wi nfcv nfbr nfralya crab cuni df-sup
      wa nfn nfrexya nfim nfan nfrabw nfuni nfcxfr ) ABCDKHLZILZDMZNZIBOZUMULDM
      ZUMJLZDMZJBPZQZICOZUDZHCUAZUBHIJBCDUCAVDVCAHCUPVBAUOAIBEUNAAULUMDAULRZGAU
      MRZSUETVAAICFUQUTAAUMULDVFGVESUSAJBEAUMURDVFGAURRSUFUGTUHFUIUJUK $.
  $}

  ${
    $d A u v x $.  $d A v w x $.  $d A w x y $.  $d A z $.  $d B w x y z $.
    $d R u v x $.  $d R v w x $.  $d R w x y z $.  $d ph u v x $.
    $d ph v w x $.
    supmoti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
        ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    $( Any class ` B ` has at most one supremum in ` A ` (where ` R ` is
       interpreted as 'less than').  The hypothesis is satisfied by real
       numbers (see ~ lttri3 ) or other orders which correspond to tight
       apartnesses.  (Contributed by Jim Kingdon, 23-Nov-2021.) $)
    supmoti $p |- ( ph -> E* x e. A ( A. y e. B -. x R y /\
                  A. y e. A ( y R x -> E. z e. B y R z ) ) ) $=
      ( vw cv wbr wn wral wrex wi wa weq breq1 wrmo wcel anbi2ci 3bitr4i ralnex
      ancom an42 rexbidv imbi12d breq2 cbvrexv imbitrrdi con3d biimtrid expimpd
      rspcva ad2antrl anim12d ralrimivva equequ1 notbid anbi12d bibi12d equequ2
      ad2antll wb rspc2v mpan9 sylibrd ralbidv imbi1d rmo4 sylibr ) ABLZCLZIMZN
      ZCHOZVOVNIMZVODLZIMZDHPZQZCGOZRZKLZVOIMZNZCHOZVOWFIMZWBQZCGOZRZRZBKSZQZKG
      OBGOWEBGUAAWPBKGGAVNGUBZWFGUBZRZRZWNVNWFIMZNZWFVNIMZNZRZWOWNWLVRRZWDWIRZR
      ZWTXEVRWIRZWLWDRZRXJWIVRRZRWNXHXIXKXJVRWIUFUCVRWDWIWLUGWLVRWDWIUGUDWTXFXB
      XGXDWQXFXBQAWRWQWLVRXBVRVPCHPZNWQWLRZXBVPCHUEXMXAXLXMXAVNVTIMZDHPZXLWKXAX
      OQCVNGCBSZWJXAWBXOVOVNWFITXPWAXNDHVOVNVTITUHUIUPVPXNCDHVOVTVNIUJUKULUMUNU
      OUQWRXGXDQAWQWRWDWIXDWIWGCHPZNWRWDRZXDWGCHUEXRXCXQXRXCWFVTIMZDHPZXQWCXCXT
      QCWFGCKSZVSXCWBXTVOWFVNITYAWAXSDHVOWFVTITUHUIUPWGXSCDHVOVTWFIUJUKULUMUNUO
      VEURUNAFESZFLZELZIMZNZYDYCIMZNZRZVFZEGOFGOWSWOXEVFZAYJFEGGJUSYJYKBESZVNYD
      IMZNZYDVNIMZNZRZVFFEVNWFGGFBSZYBYLYIYQFBEUTYRYFYNYHYPYRYEYMYCVNYDITVAYRYG
      YOYCVNYDIUJVAVBVCEKSZYLWOYQXEEKBVDYSYNXBYPXDYSYMXAYDWFVNIUJVAYSYOXCYDWFVN
      ITVAVBVCVGVHVIUSWEWMBKGWOVRWIWDWLWOVQWHCHWOVPWGVNWFVOITVAVJWOWCWKCGWOVSWJ
      WBVNWFVOIUJVKVJVBVLVM $.

    ${
      supeuti.2 $e |- ( ph -> E. x e. A ( A. y e. B -. x R y /\
                      A. y e. A ( y R x -> E. z e. B y R z ) ) ) $.
      $( A supremum is unique.  Similar to Theorem I.26 of [Apostol] p. 24 (but
         for suprema in general).  (Contributed by Jim Kingdon,
         23-Nov-2021.) $)
      supeuti $p |- ( ph ->
                    E! x e. A ( A. y e. B -. x R y /\
                    A. y e. A ( y R x -> E. z e. B y R z ) ) ) $=
        ( cv wbr wn wral wrex wi wa wrmo wreu supmoti reu5 sylanbrc ) ABLZCLZIM
        NCHOUEUDIMUEDLIMDHPQCGORZBGPUFBGSUFBGTKABCDEFGHIJUAUFBGUBUC $.
    $}

    ${
      supval2ti.ex $e |- ( ph -> E. x e. A ( A. y e. B -. x R y /\
                      A. y e. A ( y R x -> E. z e. B y R z ) ) ) $.
      $( Alternate expression for the supremum.  (Contributed by Jim Kingdon,
         23-Nov-2021.) $)
      supval2ti $p |- ( ph -> sup ( B , A , R ) =
          ( iota_ x e. A ( A. y e. B -. x R y /\
          A. y e. A ( y R x -> E. z e. B y R z ) ) ) ) $=
        ( csup cv wbr wn wral wrex wi wa crab cuni crio df-sup supeuti riotauni
        wreu wceq syl eqtr4id ) AHGILBMZCMZINOCHPUKUJINUKDMINDHQRCGPSZBGTUAZULB
        GUBZBCDHGIUCAULBGUFUNUMUGABCDEFGHIJKUDULBGUEUHUI $.
    $}

    ${
      $d x y u v C $.  $d B u v $.  $d u v z $.
      $( Sufficient condition for an element to be equal to the supremum.
         (Contributed by Jim Kingdon, 23-Nov-2021.) $)
      eqsupti $p |- ( ph -> ( ( C e. A /\ A. y e. B -. C R y /\
       A. y e. A ( y R C -> E. z e. B y R z ) ) -> sup ( B , A , R ) = C ) ) $=
        ( vx wcel cv wbr wn wral wrex wi wceq wa w3a csup crio wb adantlr breq1
        notbid ralbidv breq2 imbi1d anbi12d rspcev adantl supval2ti 3simpc wreu
        3impb simpr1 supeuti riota2 syl2anc mpbid eqtrd ex ) AHFLZHBMZINZOZBGPZ
        VFHINZVFCMINCGQZRZBFPZUAZGFIUBZHSAVNTZVOKMZVFINZOZBGPZVFVQINZVKRZBFPZTZ
        KFUCZHVPKBCDEFGIAEMZFLDMZFLTWFWGSWFWGINOWGWFINOTUDVNJUEZVNWDKFQZAVEVIVM
        WIWDVIVMTZKHFVQHSZVTVIWCVMWKVSVHBGWKVRVGVQHVFIUFUGUHWKWBVLBFWKWAVJVKVQH
        VFIUIUJUHUKZULUQUMZUNVPWJWEHSZVNWJAVEVIVMUOUMVPVEWDKFUPWJWNUDAVEVIVMURV
        PKBCDEFGIWHWMUSWDWJKFHWLUTVAVBVCVD $.

      $d y ph $.
      eqsuptid.2 $e |- ( ph -> C e. A ) $.
      eqsuptid.3 $e |- ( ( ph /\ y e. B ) -> -. C R y ) $.
      eqsuptid.4 $e |- ( ( ph /\ ( y e. A /\ y R C ) ) -> E. z e. B y R z ) $.
      $( Sufficient condition for an element to be equal to the supremum.
         (Contributed by Jim Kingdon, 24-Nov-2021.) $)
      eqsuptid $p |- ( ph -> sup ( B , A , R ) = C ) $=
        ( wcel cv wbr wn wral wrex ralrimiva wi csup wceq expr eqsupti mp3and )
        AHFNHBOZIPQZBGRUGHIPZUGCOIPCGSZUAZBFRGFIUBHUCKAUHBGLTAUKBFAUGFNUIUJMUDT
        ABCDEFGHIJUEUF $.
    $}

    supclti.2 $e |- ( ph -> E. x e. A ( A. y e. B -. x R y /\
                      A. y e. A ( y R x -> E. z e. B y R z ) ) ) $.
    $( A supremum belongs to its base class (closure law).  See also ~ supubti
       and ~ suplubti .  (Contributed by Jim Kingdon, 24-Nov-2021.) $)
    supclti $p |- ( ph -> sup ( B , A , R ) e. A ) $=
      ( csup cv wbr wn wral wrex wi wa crio supval2ti wreu wcel supeuti riotacl
      syl eqeltrd ) AHGILBMZCMZINOCHPUIUHINUIDMINDHQRCGPSZBGTZGABCDEFGHIJKUAAUJ
      BGUBUKGUCABCDEFGHIJKUDUJBGUEUFUG $.

    ${
      $d C w $.
      $( A supremum is an upper bound.  See also ~ supclti and ~ suplubti .

         This proof demonstrates how to expand an iota-based definition
         ( ~ df-iota ) using ~ riotacl2 .

         (Contributed by Jim Kingdon, 24-Nov-2021.) $)
      supubti $p |- ( ph -> ( C e. B -> -. sup ( B , A , R ) R C ) ) $=
        ( vw cv wbr wn wral wcel wi wceq csup crab wrex simpl ss2rabi supval2ti
        wa a1i crio wreu supeuti riotacl2 syl eqeltrd sselid breq2 notbid breq1
        cbvralv ralbidv bitrid elrab simprbi rspccv 3syl ) AHGJUAZBNZCNZJOZPZCH
        QZBGUBZRZVFMNZJOZPZMHQZIHRVFIJOZPZSAVKVHVGJOVHDNJODHUCSCGQZUGZBGUBZVLVF
        WAVKBGWAVKSVGGRVKVTUDUHUEAVFWABGUIZWBABCDEFGHJKLUFAWABGUJWCWBRABCDEFGHJ
        KLUKWABGULUMUNUOVMVFGRVQVKVQBVFGVKVGVNJOZPZMHQVGVFTZVQVJWECMHVHVNTVIWDV
        HVNVGJUPUQUSWFWEVPMHWFWDVOVGVFVNJURUQUTVAVBVCVPVSMIHVNITVOVRVNIVFJUPUQV
        DVE $.
    $}

    ${
      $d C w z $.
      $( A supremum is the least upper bound.  See also ~ supclti and
         ~ supubti .  (Contributed by Jim Kingdon, 24-Nov-2021.) $)
      suplubti $p |- ( ph -> ( ( C e. A /\ C R sup ( B , A , R ) ) ->
                      E. z e. B C R z ) ) $=
        ( vw cv wbr wrex wi wral wcel breq1 wa wn simpr rexbidv imbi12d cbvralv
        csup crab wceq a1i ss2rabi crio supval2ti wreu supeuti riotacl2 eqeltrd
        sylib syl sselid breq2 imbi1d ralbidv elrab simprbi rspccv impd 3syl )
        AHGJUGZMNZBNZJOZVJDNZJOZDHPZQZMGRZBGUHZSZVJVIJOZVOQZMGRZIGSZIVIJOZUAIVM
        JOZDHPZQAVKCNZJOUBCHRZWGVKJOZWGVMJOZDHPZQZCGRZUAZBGUHZVRVIWNVQBGWNVQQVK
        GSWNWMVQWHWMUCWLVPCMGWGVJUIZWIVLWKVOWGVJVKJTWPWJVNDHWGVJVMJTUDUEUFURUJU
        KAVIWNBGULZWOABCDEFGHJKLUMAWNBGUNWQWOSABCDEFGHJKLUOWNBGUPUSUQUTVSVIGSWB
        VQWBBVIGVKVIUIZVPWAMGWRVLVTVOVKVIVJJVAVBVCVDVEWBWCWDWFWAWDWFQMIGVJIUIZV
        TWDVOWFVJIVIJTWSVNWEDHVJIVMJTUDUEVFVGVH $.
    $}

    ${
      $d A v w x $.  $d A w x y z $.  $d B w x y z $.  $d C w z $.
      $d R v w x $.  $d R w x y z $.  $d ph v w x $.
      suplub2ti.or $e |- ( ph -> R Or A ) $.
      suplub2ti.3 $e |- ( ph -> B C_ A ) $.
      $( Bidirectional form of ~ suplubti .  (Contributed by Jim Kingdon,
         17-Jan-2022.) $)
      suplub2ti $p |- ( ( ph /\ C e. A ) ->
          ( C R sup ( B , A , R ) <-> E. z e. B C R z ) ) $=
        ( vw wcel wa wbr cv ad3antrrr suplubti expdimp breq2 cbvrexv wn simplll
        csup wrex simplr supubti sylc wo simpr wor wi simpllr wss sseldd sowlin
        supclti syl13anc mpd ecased ex rexlimdva biimtrid impbid ) AIGPZQZIHGJU
        GZJRZIDSZJRZDHUHZAVHVKVNABCDEFGHIJKLUAUBVNIOSZJRZOHUHVIVKVMVPDOHVLVOIJU
        CUDVIVPVKOHVIVOHPZQZVPVKVRVPQZVKVJVOJRZVSAVQVTUEAVHVQVPUFVIVQVPUIZABCDE
        FGHVOJKLUJUKVSVPVKVTULZVRVPUMVSGJUNZVHVOGPVJGPZVPWBUOAWCVHVQVPMTAVHVQVP
        UPVSHGVOAHGUQVHVQVPNTWAURAWDVHVQVPABCDEFGHJKLUTTGIVOVJJUSVAVBVCVDVEVFVG
        $.
    $}
  $}

  ${
    $d A u v x $.  $d A x y z $.  $d B x y z $.  $d C x $.  $d R u v x $.
    $d R x y z $.  $d ph u v x $.
    supelti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
        ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    supelti.ex $e |- ( ph -> E. x e. C ( A. y e. B -. x R y /\
      A. y e. A ( y R x -> E. z e. B y R z ) ) ) $.
    supelti.ss $e |- ( ph -> C C_ A ) $.
    $( Supremum membership in a set.  (Contributed by Jim Kingdon,
       16-Jan-2022.) $)
    supelti $p |- ( ph -> sup ( B , A , R ) e. C ) $=
      ( cv wceq wcel wex wbr wrex wa csup wss wn wral wi ssrexv supclti elisset
      sylc eqcom exbii sylib simpr crio supval2ti eqeq1d biimpa wb wreu supeuti
      syl riota1 adantr mpbird simpld simprd reupick syl2an2r eqeltrd exlimddv
      jca32 ) AHGJUAZBNZOZVLIPBAVMVLOZBQZVNBQAVLGPVPABCDEFGHJKAIGUBZVMCNZJRUCCH
      UDVRVMJRVRDNJRDHSUECGUDTZBISZVSBGSMLVSBIGUFUIZUGBVLGUHVAVOVNBVMVLUJUKULAV
      NTZVLVMIAVNUMWBVMIPZVMGPZWBWDVSWBWDVSTZVSBGUNZVMOZAVNWGAVLWFVMABCDEFGHJKW
      AUOUPUQAWEWGURZVNAVSBGUSZWHABCDEFGHJKWAUTZVSBGVBVAVCVDZVEAVQVTWITTVNVSWCW
      DURAVQVTWIMLWJVKWBWDVSWKVFVSBIGVGVHVDVIVJ $.
  $}

  ${
    $d B x y z $.  $d R x y z $.
    $( The supremum under an empty base set is always the empty set.
       (Contributed by AV, 4-Sep-2020.) $)
    sup00 $p |- sup ( B , (/) , R ) = (/) $=
      ( vx vy vz c0 csup cv wbr wn wral wrex wi wa crab cuni df-sup rab0 unieqi
      uni0 3eqtri ) AFBGCHZDHZBIJDAKUCUBBIUCEHBIEALMDFKNZCFOZPFPFCDEAFBQUEFUDCR
      STUA $.
  $}

  ${
    $d A u v x y $.  $d B u v x y $.  $d C u v x y $.  $d R u v x y $.
    $d ph u v y $.
    supmaxti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
      ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    supmaxti.2 $e |- ( ph -> C e. A ) $.
    supmaxti.3 $e |- ( ph -> C e. B ) $.
    supmaxti.4 $e |- ( ( ph /\ y e. B ) -> -. C R y ) $.
    $( The greatest element of a set is its supremum.  Note that the converse
       is not true; the supremum might not be an element of the set considered.
       (Contributed by Jim Kingdon, 24-Nov-2021.) $)
    supmaxti $p |- ( ph -> sup ( B , A , R ) = C ) $=
      ( vx wcel cv wbr wa wrex simprr breq2 rspcev syl2an2r eqsuptid ) ABMCDEFG
      HIJLAGFNBOZENZUDGHPZQUFUDMOZHPZMFRKAUEUFSUHUFMGFUGGUDHTUAUBUC $.
  $}

  ${
    $d A u v x $.  $d B u v x $.  $d R u v x $.  $d ph u v x $.
    supsnti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
      ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    supsnti.b $e |- ( ph -> B e. A ) $.
    $( The supremum of a singleton.  (Contributed by Jim Kingdon,
       26-Nov-2021.) $)
    supsnti $p |- ( ph -> sup ( { B } , A , R ) = B ) $=
      ( vx wcel cv wa wbr wn wceq wb wral breq1 notbid breq2 csn snidg syl eqid
      ralrimivva wi eqeq1 anbi12d bibi12d eqeq2 rspc2v syl2anc mpd mpbii simpld
      adantr elsni breq2d adantl mpbird supmaxti ) AIBCDEUAZEFGHAEDJZEVBJHEDUBU
      CAIKZVBJZLEVDFMZNZEEFMZNZAVIVEAVIVIAEEOZVIVILZEUDACKZBKZOZVLVMFMZNZVMVLFM
      ZNZLZPZBDQCDQZVJVKPZAVTCBDDGUEAVCVCWAWBUFHHVTWBEVMOZEVMFMZNZVMEFMZNZLZPCB
      EEDDVLEOZVNWCVSWHVLEVMUGWIVPWEVRWGWIVOWDVLEVMFRSWIVQWFVLEVMFTSUHUIVMEOZWC
      VJWHVKVMEEUJWJWEVIWGVIWJWDVHVMEEFTSWJWFVHVMEEFRSUHUIUKULUMUNUOUPVEVGVIPAV
      EVFVHVEVDEEFVDEUQURSUSUTVA $.
  $}

  ${
    $d A u v $.  $d B u v x y $.  $d F u v x y $.  $d R u v $.  $d S u v x y $.
    $( Lemma for ~ isoti .  (Contributed by Jim Kingdon, 26-Nov-2021.) $)
    isotilem $p |- ( F Isom R , S ( A , B ) -> (
        A. x e. B A. y e. B ( x = y <-> ( -. x S y /\ -. y S x ) ) ->
        A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) ) ) $=
      ( cv wceq wbr wn wa wb wral wcel notbid anbi12d bibi12d wiso wi wf isof1o
      cfv wf1o f1of ffvelcdm ex anim12d 3syl imp eqeq1 breq1 breq2 eqeq2 rspc2v
      syl wf1 f1of1 f1fveq sylan bicomd isorel ancom2s sylibrd ralrimdvva ) EFG
      HIUAZAJZBJZKZVIVJHLZMZVJVIHLZMZNZOZBFPAFPZDJZCJZKZVSVTGLZMZVTVSGLZMZNZOZD
      CEEVHVSEQZVTEQZNZNZVRVSIUEZVTIUEZKZWLWMHLZMZWMWLHLZMZNZOZWGWKWLFQZWMFQZNZ
      VRWTUBVHWJXCVHEFIUFZEFIUCZWJXCUBEFGHIUDZEFIUGXEWHXAWIXBXEWHXAEFVSIUHUIXEW
      IXBEFVTIUHUIUJUKULVQWTWLVJKZWLVJHLZMZVJWLHLZMZNZOABWLWMFFVIWLKZVKXGVPXLVI
      WLVJUMXMVMXIVOXKXMVLXHVIWLVJHUNRXMVNXJVIWLVJHUORSTVJWMKZXGWNXLWSVJWMWLUPX
      NXIWPXKWRXNXHWOVJWMWLHUORXNXJWQVJWMWLHUNRSTUQURWKWAWNWFWSWKWNWAVHEFIUSZWJ
      WNWAOVHXDXOXFEFIUTUREFVSVTIVAVBVCWKWCWPWEWRWKWBWOEFVSVTGHIVDRVHWIWHWEWROV
      HWIWHNNWDWQEFVTVSGHIVDRVESTVFVG $.
  $}

  ${
    $d A u v x y $.  $d B u v x y $.  $d F u v x y $.  $d R u v x y $.
    $d S u v x y $.
    $( An isomorphism preserves tightness.  (Contributed by Jim Kingdon,
       26-Nov-2021.) $)
    isoti $p |- ( F Isom R , S ( A , B ) -> (
        A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) <->
        A. u e. B A. v e. B ( u = v <-> ( -. u S v /\ -. v S u ) ) ) ) $=
      ( vx vy wiso weq cv wbr wn wa wb wral isotilem breq1 notbid wi isocnv syl
      ccnv impbid equequ1 breq2 anbi12d bibi12d equequ2 cbvral2v bitrdi ) CDEFG
      JZBAKZBLZALZEMNUPUOEMNOPACQBCQZHIKZHLZILZFMZNZUTUSFMZNZOZPZIDQHDQZUNUOUPF
      MZNZUPUOFMZNZOZPZADQBDQUMUQVGUMDCFEGUDZJUQVGUACDEFGUBBAIHDCFEVNRUCHIABCDE
      FGRUEVFVMBIKZUOUTFMZNZUTUOFMZNZOZPHIBADDHBKZURVOVEVTHBIUFWAVBVQVDVSWAVAVP
      USUOUTFSTWAVCVRUSUOUTFUGTUHUIIAKZVOUNVTVLIABUJWBVQVIVSVKWBVPVHUTUPUOFUGTW
      BVRVJUTUPUOFSTUHUIUKUL $.
  $}

  ${
    $d u v w x y z A $.  $d u v w x y z C $.  $d w y z D $.  $d u w ph $.
    $d u v w x y z F $.  $d u w x y z R $.  $d u v w x y z S $.
    $d u v w x y z B $.
    supiso.1 $e |- ( ph -> F Isom R , S ( A , B ) ) $.
    supiso.2 $e |- ( ph -> C C_ A ) $.
    $( Lemma for ~ supisoti .  (Contributed by Mario Carneiro, 24-Dec-2016.) $)
    supisolem $p |- ( ( ph /\ D e. A ) ->
      ( ( A. y e. C -. D R y /\ A. y e. A ( y R D -> E. z e. C y R z ) ) <->
        ( A. w e. ( F " C ) -. ( F ` D ) S w /\
          A. w e. B ( w S ( F ` D ) -> E. v e. ( F " C ) w S v ) ) ) ) $=
      ( wa wcel wbr wral wb adantr wiso wss cv wn wrex wi cfv jca simpll simplr
      cima sselda isorel syl12anc notbid ralbidva wfn wf1o syl f1ofn wceq breq2
      isof1o ralima syl2anc bitr4d simpr rexima imbi12d wfo f1ofo breq1 rexbidv
      rexbidva cbvfo 3syl bitrd anbi12d sylan ) AFGJKLUAZHFUBZOZIFPZIBUCZJQZUDZ
      BHRZWDIJQZWDCUCZJQZCHUEZUFZBFRZOILUGZDUCZKQZUDZDLHUKZRZWOWNKQZWOEUCZKQZEW
      RUEZUFZDGRZOSAVTWAMNUHWBWCOZWGWSWMXEXFWGWNWDLUGZKQZUDZBHRZWSXFWFXIBHXFWDH
      PZOZWEXHXLVTWCWDFPZWEXHSXFVTXKVTWAWCUIZTWBWCXKUJXFHFWDVTWAWCUJZULFGIWDJKL
      UMUNUOUPXFLFUQZWAWSXJSXFFGLURZXPXFVTXQXNFGJKLVCUSZFGLUTUSZXOWQXIDBFHLWOXG
      VAWPXHWOXGWNKVBUOVDVEVFXFWMXGWNKQZXGXAKQZEWRUEZUFZBFRZXEXFWLYCBFXFXMOZWHX
      TWKYBYEVTXMWCWHXTSXFVTXMXNTZXFXMVGWBWCXMUJFGWDIJKLUMUNYEWKXGWILUGZKQZCHUE
      ZYBYEWJYHCHYEWIHPZOVTXMWIFPWJYHSYEVTYJYFTXFXMYJUJYEHFWIXFWAXMXOTZULFGWDWI
      JKLUMUNVNYEXPWAYBYISXFXPXMXSTYKYAYHECFHLXAYGXGKVBVHVEVFVIUPXFXQFGLVJYDXES
      XRFGLVKYCXDBDFGLXGWOVAZXTWTYBXCXGWOWNKVLYLYAXBEWRXGWOXAKVLVMVIVOVPVQVRVS
      $.

    supisoex.3 $e |- ( ph -> E. x e. A ( A. y e. C -. x R y /\
                      A. y e. A ( y R x -> E. z e. C y R z ) ) ) $.
    $( Lemma for ~ supisoti .  (Contributed by Mario Carneiro, 24-Dec-2016.) $)
    supisoex $p |- ( ph -> E. u e. B ( A. w e. ( F " C ) -. u S w /\
          A. w e. B ( w S u -> E. v e. ( F " C ) w S v ) ) ) $=
      ( cv wbr wral wi wn wrex wa cima wiso wss wcel simpl simpr supisolem wf1o
      cfv wf isof1o f1of 3syl ffvelcdmda wceq breq1 notbid ralbidv breq2 imbi1d
      anbi12d rspcev ex syl sylbid rexlimdva syl2anc mpd ) ABQZCQZKRUACJSVMVLKR
      VMDQKRDJUBTCHSUCZBHUBZGQZEQZLRZUAZEMJUDZSZVQVPLRZVQFQLRFVTUBZTZEISZUCZGIU
      BZPAHIKLMUEZJHUFZVOWGTNOWHWIUCZVNWGBHWJVLHUGUCZVNVLMULZVQLRZUAZEVTSZVQWLL
      RZWCTZEISZUCZWGWJCDEFHIJVLKLMWHWIUHZWHWIUIUJWKWLIUGZWSWGTWJHIVLMWJWHHIMUK
      HIMUMWTHIKLMUNHIMUOUPUQXAWSWGWFWSGWLIVPWLURZWAWOWEWRXBVSWNEVTXBVRWMVPWLVQ
      LUSUTVAXBWDWQEIXBWBWPWCVPWLVQLVBVCVAVDVEVFVGVHVIVJVK $.

    $d A j k w z $.  $d B j k w z $.  $d C j k w z $.  $d F j k w z $.
    $d R j w z v $.  $d S j k w z $.  $d ph j k w v x $.  $d j k u v $.
    supisoti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
      ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    $( Image of a supremum under an isomorphism.  (Contributed by Jim Kingdon,
       26-Nov-2021.) $)
    supisoti $p |- ( ph ->
        sup ( ( F " C ) , B , S ) = ( F ` sup ( C , A , R ) ) ) $=
      ( vw vj wbr wral vk cima csup cfv cv wcel weq wn wa ralrimivva wiso isoti
      wb syl mpbid r19.21bi anasss wf1o isof1o f1of 3syl supclti ffvelcdmd wrex
      wf wi supubti ralrimiv suplubti expd supisolem mpdan mpbi2and simpld impr
      simprd eqsuptid ) AQUAEFHLIUBZIGJUCZLUDZKAFUEZHUFZEUEZHUFFEUGZWAWCKSUHWCW
      AKSUHUIUMZAWBUIWEEHAWEEHTZFHAWDWAWCJSUHWCWAJSUHUIUMZEGTFGTZWFFHTZAWGFEGGP
      UJAGHJKLUKZWHWIUMMEFGHJKLULUNUOUPUPUQAGHVSLAWJGHLURGHLVEMGHJKLUSGHLUTVAAB
      CDEFGIJPOVBZVCAVTQUEZKSUHZQVRAWMQVRTZWLVTKSZWLUAUEKSUAVRVDZVFZQHTZAVSRUEZ
      JSUHZRITZWSVSJSZWSDUEJSDIVDZVFZRGTZWNWRUIZAWTRIABCDEFGIWSJPOVGVHAXDRGAWSG
      UFXBXCABCDEFGIWSJPOVIVJVHAVSGUFXAXEUIXFUMWKARDQUAGHIVSJKLMNVKVLVMZVNUPAWL
      HUFWOWPAWQQHAWNWRXGVPUPVOVQ $.
  $}

  $( Equality theorem for infimum.  (Contributed by AV, 2-Sep-2020.) $)
  infeq1 $p |- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) ) $=
    ( wceq ccnv csup cinf supeq1 df-inf 3eqtr4g ) BCEBADFZGCALGBADHCADHABCLIBAD
    JCADJK $.

  ${
    infeq1d.1 $e |- ( ph -> B = C ) $.
    $( Equality deduction for infimum.  (Contributed by AV, 2-Sep-2020.) $)
    infeq1d $p |- ( ph -> inf ( B , A , R ) = inf ( C , A , R ) ) $=
      ( wceq cinf infeq1 syl ) ACDGCBEHDBEHGFBCDEIJ $.
  $}

  ${
    infeq1i.1 $e |- B = C $.
    $( Equality inference for infimum.  (Contributed by AV, 2-Sep-2020.) $)
    infeq1i $p |- inf ( B , A , R ) = inf ( C , A , R ) $=
      ( wceq cinf infeq1 ax-mp ) BCFBADGCADGFEABCDHI $.
  $}

  $( Equality theorem for infimum.  (Contributed by AV, 2-Sep-2020.) $)
  infeq2 $p |- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) ) $=
    ( wceq ccnv csup cinf supeq2 df-inf 3eqtr4g ) BCEABDFZGACLGABDHACDHABCLIABD
    JACDJK $.

  $( Equality theorem for infimum.  (Contributed by AV, 2-Sep-2020.) $)
  infeq3 $p |- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) ) $=
    ( wceq ccnv csup cinf cnveq supeq3 syl df-inf 3eqtr4g ) CDEZABCFZGZABDFZGZA
    BCHABDHNOQEPRECDIABOQJKABCLABDLM $.

  ${
    infeq123d.a $e |- ( ph -> A = D ) $.
    infeq123d.b $e |- ( ph -> B = E ) $.
    infeq123d.c $e |- ( ph -> C = F ) $.
    $( Equality deduction for infimum.  (Contributed by AV, 2-Sep-2020.) $)
    infeq123d $p |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) ) $=
      ( ccnv csup cinf cnveqd supeq123d df-inf 3eqtr4g ) ABCDKZLEFGKZLBCDMEFGMA
      BCREFSHIADGJNOBCDPEFGPQ $.
  $}

  ${
    nfinf.1 $e |- F/_ x A $.
    nfinf.2 $e |- F/_ x B $.
    nfinf.3 $e |- F/_ x R $.
    $( Hypothesis builder for infimum.  (Contributed by AV, 2-Sep-2020.) $)
    nfinf $p |- F/_ x inf ( A , B , R ) $=
      ( cinf ccnv csup df-inf nfcnv nfsup nfcxfr ) ABCDHBCDIZJBCDKABCOEFADGLMN
      $.
  $}

  ${
    $d ph x $.  $d ph y $.  $d ph z $.
    cnvinfex.ex $e |- ( ph -> E. x e. A ( A. y e. B -. y R x /\
                      A. y e. A ( x R y -> E. z e. B z R y ) ) ) $.
    $( Two ways of expressing existence of an infimum (one in terms of
       converse).  (Contributed by Jim Kingdon, 17-Dec-2021.) $)
    cnvinfex $p |- ( ph -> E. x e. A ( A. y e. B -. x `' R y
        /\ A. y e. A ( y `' R x -> E. z e. B y `' R z ) ) ) $=
      ( cv wbr wn wral wrex wi wa wb vex brcnv a1i ralbidv ccnv rexbidv imbi12d
      notbid anbi12d mpbird ) ABIZCIZGUAZJZKZCFLZUHUGUIJZUHDIZUIJZDFMZNZCELZOZB
      EMUHUGGJZKZCFLZUGUHGJZUNUHGJZDFMZNZCELZOZBEMHAUSVHBEAULVBURVGAUKVACFAUJUT
      UJUTPAUGUHGBQZCQZRSUDTAUQVFCEAUMVCUPVEUMVCPAUHUGGVJVIRSAUOVDDFUOVDPAUHUNG
      VJDQRSUBUCTUEUBUF $.
  $}

  ${
    eqinfti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
        ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    $d A u v y z $.  $d ph u v $.  $d R u v y z $.

    $( If a relation satisfies a condition corresponding to tightness of an
       apartness generated by an order, so does its converse.  (Contributed by
       Jim Kingdon, 17-Dec-2021.) $)
    cnvti $p |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
        ( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) ) $=
      ( cv wcel wa weq wbr wn ccnv ancom bitrdi wb brcnvg notbid ancoms anbi12d
      adantl bitr4d ) ACGZDHZBGZDHZIZIZCBJZUEUCEKZLZUCUEEKZLZIZUCUEEMZKZLZUEUCU
      OKZLZIZUHUIUMUKIUNFUMUKNOUGUTUNPAUGUQUKUSUMUGUPUJUCUEDDEQRUGURULUFUDURULP
      UEUCDDEQSRTUAUB $.

    $d B u v y z $.
    ${
      $d C u v y z $.
      $( Sufficient condition for an element to be equal to the infimum.
         (Contributed by Jim Kingdon, 16-Dec-2021.) $)
      eqinfti $p |- ( ph -> ( ( C e. A /\ A. y e. B -. y R C /\
       A. y e. A ( C R y -> E. z e. B z R y ) ) -> inf ( B , A , R ) = C ) ) $=
        ( wcel cv wbr wn wral wrex wa cvv wb bicomd wi w3a cinf wceq ccnv cnvti
        csup df-inf eqsupti vex brcnvg mpan2 ralbidv mpan brcnv rexbidv imbi12d
        notbid a1i anbi12d pm5.32i 3anass 3bitr4i biimpi impel eqtrid ex ) AHFK
        ZBLZHIMZNZBGOZHVIIMZCLZVIIMZCGPZUAZBFOZUBZGFIUCZHUDAVSQVTGFIUEZUGZHGFIU
        HAVHHVIWAMZNZBGOZVIHWAMZVIVNWAMZCGPZUAZBFOZUBZWBHUDVSABCDEFGHWAADEFIJUF
        UIVSWKVHVLVRQZQVHWEWJQZQVSWKVHWLWMVHVLWEVRWJVHVKWDBGVHVJWCVHVIRKZVJWCSB
        UJZVHWNQWCVJHVIFRIUKTULURUMVHVQWIBFVHVMWFVPWHVHWFVMWNVHWFVMSWOVIHRFIUKU
        NTVHVOWGCGVHWGVOWGVOSVHVIVNIWOCUJUOUSTUPUQUMUTVAVHVLVRVBVHWEWJVBVCVDVEV
        FVG $.

      $d ph y $.
      eqinftid.2 $e |- ( ph -> C e. A ) $.
      eqinftid.3 $e |- ( ( ph /\ y e. B ) -> -. y R C ) $.
      eqinftid.4 $e |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y ) $.
      $( Sufficient condition for an element to be equal to the infimum.
         (Contributed by Jim Kingdon, 16-Dec-2021.) $)
      eqinftid $p |- ( ph -> inf ( B , A , R ) = C ) $=
        ( wcel cv wbr wn wral wrex ralrimiva wi cinf wceq expr eqinfti mp3and )
        AHFNBOZHIPQZBGRHUGIPZCOUGIPCGSZUAZBFRGFIUBHUCKAUHBGLTAUKBFAUGFNUIUJMUDT
        ABCDEFGHIJUEUF $.
    $}

    infvalti.ex $e |- ( ph -> E. x e. A ( A. y e. B -. y R x /\
                      A. y e. A ( x R y -> E. z e. B z R y ) ) ) $.
    $d A x $.  $d B x $.  $d R x $.  $d ph u v x y z $.
    $( Alternate expression for the infimum.  (Contributed by Jim Kingdon,
       17-Dec-2021.) $)
    infvalti $p |- ( ph -> inf ( B , A , R ) =
      ( iota_ x e. A ( A. y e. B -. y R x /\
                  A. y e. A ( x R y -> E. z e. B z R y ) ) ) ) $=
      ( cv wbr wn wral wrex wb vex brcnv a1i cinf ccnv csup wi wa crio cnvinfex
      df-inf supval2ti notbid ralbidv rexbidv imbi12d anbi12d riotabidv eqtrid
      cnvti eqtrd ) AHGIUAHGIUBZUCZCLZBLZIMZNZCHOZVBVAIMZDLZVAIMZDHPZUDZCGOZUEZ
      BGUFZHGIUHAUTVBVAUSMZNZCHOZVAVBUSMZVAVGUSMZDHPZUDZCGOZUEZBGUFVMABCDEFGHUS
      AEFGIJUQABCDGHIKUGUIAWBVLBGAVPVEWAVKAVOVDCHAVNVCVNVCQAVBVAIBRZCRZSTUJUKAV
      TVJCGAVQVFVSVIVQVFQAVAVBIWDWCSTAVRVHDHVRVHQAVAVGIWDDRSTULUMUKUNUOURUP $.
  $}

  ${
    $d A u v x y z $.  $d B u v x y z $.  $d R u v x y z $.  $d ph u v x y z $.
    infclti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
        ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    infclti.ex $e |- ( ph -> E. x e. A ( A. y e. B -. y R x /\
                      A. y e. A ( x R y -> E. z e. B z R y ) ) ) $.
    $( An infimum belongs to its base class (closure law).  See also ~ inflbti
       and ~ infglbti .  (Contributed by Jim Kingdon, 17-Dec-2021.) $)
    infclti $p |- ( ph -> inf ( B , A , R ) e. A ) $=
      ( cinf ccnv csup df-inf cnvti cnvinfex supclti eqeltrid ) AHGILHGIMZNGHGI
      OABCDEFGHTAEFGIJPABCDGHIKQRS $.

    $( An infimum is a lower bound.  See also ~ infclti and ~ infglbti .
       (Contributed by Jim Kingdon, 18-Dec-2021.) $)
    inflbti $p |- ( ph -> ( C e. B -> -. C R inf ( B , A , R ) ) ) $=
      ( wcel cinf wbr wn wa ccnv csup cnvti cnvinfex supubti wceq df-inf breq2d
      imp a1i wb supclti brcnvg bicomd sylan bitrd mtbird ex ) AIHMZIHGJNZJOZPA
      UPQZURHGJRZSZIUTOZAUPVBPABCDEFGHIUTAEFGJKTZABCDGHJLUAZUBUFUSURIVAJOZVBUSU
      QVAIJUQVAUCUSHGJUDUGUEAVAGMZUPVEVBUHABCDEFGHUTVCVDUIVFUPQVBVEVAIGHJUJUKUL
      UMUNUO $.

    $d C z $.  $d ph z $.
    $( An infimum is the greatest lower bound.  See also ~ infclti and
       ~ inflbti .  (Contributed by Jim Kingdon, 18-Dec-2021.) $)
    infglbti $p |- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) ->
                    E. z e. B z R C ) ) $=
      ( wcel cinf wbr wrex wa wb brcnvg cvv ccnv csup df-inf breq1i simpr cnvti
      cv cnvinfex supclti adantr bicomd syl2anc bitrid suplubti expdimp sylancl
      vex rexbidv sylibd sylbid expimpd ) AIGMZHGJNZIJOZDUGZIJOZDHPZAVBQZVDIHGJ
      UAZUBZVIOZVGVDVJIJOZVHVKVCVJIJHGJUCUDVHVBVJGMZVLVKRAVBUEZAVMVBABCDEFGHVIA
      EFGJKUFZABCDGHJLUHZUIUJVBVMQVKVLIVJGGJSUKULUMVHVKIVEVIOZDHPZVGAVBVKVRABCD
      EFGHIVIVOVPUNUOVHVQVFDHVHVBVETMVQVFRVNDUQIVEGTJSUPURUSUTVA $.

    $( A lower bound is not greater than the infimum.  (Contributed by Jim
       Kingdon, 18-Dec-2021.) $)
    infnlbti $p |- ( ph -> ( ( C e. A /\ A. z e. B -. z R C ) ->
                    -. inf ( B , A , R ) R C ) ) $=
      ( wcel cv wbr wn wral cinf wa wrex infglbti expdimp rexalim con2d expimpd
      syl6 ) AIGMZDNIJOZPDHQZHGJRIJOZPAUGSZUJUIUKUJUHDHTZUIPAUGUJULABCDEFGHIJKL
      UAUBUHDHUCUFUDUE $.
  $}

  ${
    $d A u v y z $.  $d B u v y z $.  $d C u v y z $.  $d R u v y z $.
    $d u v y z ph $.
    infminti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
        ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    infminti.2 $e |- ( ph -> C e. A ) $.
    infminti.3 $e |- ( ph -> C e. B ) $.
    infminti.4 $e |- ( ( ph /\ y e. B ) -> -. y R C ) $.
    $( The smallest element of a set is its infimum.  Note that the converse is
       not true; the infimum might not be an element of the set considered.
       (Contributed by Jim Kingdon, 18-Dec-2021.) $)
    infminti $p |- ( ph -> inf ( B , A , R ) = C ) $=
      ( vz wcel cv wbr wa wrex simprr breq1 rspcev syl2an2r eqinftid ) ABMCDEFG
      HIJLAGFNBOZENZGUDHPZQUFMOZUDHPZMFRKAUEUFSUHUFMGFUGGUDHTUAUBUC $.
  $}

  ${
    $d u v x y z A $.  $d u v x y z R $.  $d u v x y z B $.  $d u v x ph $.
    infmoti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
        ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    $( Any class ` B ` has at most one infimum in ` A ` (where ` R ` is
       interpreted as 'less than').  (Contributed by Jim Kingdon,
       18-Dec-2021.) $)
    infmoti $p |- ( ph -> E* x e. A ( A. y e. B -. y R x /\
                  A. y e. A ( x R y -> E. z e. B z R y ) ) ) $=
      ( cv wbr wn wral wrex wi wa wrmo vex brcnv supmoti notbii imbi12i anbi12i
      ccnv cnvti ralbii rexbii rmobii sylib ) ABKZCKZIUEZLZMZCHNZULUKUMLZULDKZU
      MLZDHOZPZCGNZQZBGRULUKILZMZCHNZUKULILZURULILZDHOZPZCGNZQZBGRABCDEFGHUMAEF
      GIJUFUAVCVLBGUPVFVBVKUOVECHUNVDUKULIBSZCSZTUBUGVAVJCGUQVGUTVIULUKIVNVMTUS
      VHDHULURIVNDSTUHUCUGUDUIUJ $.

    infeuti.2 $e |- ( ph -> E. x e. A ( A. y e. B -. y R x /\
                      A. y e. A ( x R y -> E. z e. B z R y ) ) ) $.
    $( An infimum is unique.  (Contributed by Jim Kingdon, 19-Dec-2021.) $)
    infeuti $p |- ( ph ->
                    E! x e. A ( A. y e. B -. y R x /\
                    A. y e. A ( x R y -> E. z e. B z R y ) ) ) $=
      ( cv wbr wn wral wrex wi wa wrmo wreu infmoti reu5 sylanbrc ) ACLZBLZIMNC
      HOUEUDIMDLUDIMDHPQCGORZBGPUFBGSUFBGTKABCDEFGHIJUAUFBGUBUC $.
  $}

  ${
    $d A u v $.  $d B u v $.  $d R u v $.  $d ph u v $.
    infsnti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
      ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    infsnti.b $e |- ( ph -> B e. A ) $.
    $( The infimum of a singleton.  (Contributed by Jim Kingdon,
       19-Dec-2021.) $)
    infsnti $p |- ( ph -> inf ( { B } , A , R ) = B ) $=
      ( csn cinf ccnv csup df-inf cnvti supsnti eqtrid ) AEIZDFJQDFKZLEQDFMABCD
      ERABCDFGNHOP $.
  $}

  $( The infimum regarding an empty base set is always the empty set.
     (Contributed by AV, 4-Sep-2020.) $)
  inf00 $p |- inf ( B , (/) , R ) = (/) $=
    ( c0 cinf ccnv csup df-inf sup00 eqtri ) ACBDACBEZFCACBGAJHI $.

  ${
    $d A u v x y z $.  $d B u v x y z $.  $d C u v x y z $.  $d F u v x y z $.
    $d R u v x y z $.  $d S u v x y z $.  $d ph u v x y z $.
    infisoti.1 $e |- ( ph -> F Isom R , S ( A , B ) ) $.
    infisoti.2 $e |- ( ph -> C C_ A ) $.
    infisoti.3 $e |- ( ph -> E. x e. A ( A. y e. C -. y R x /\
                      A. y e. A ( x R y -> E. z e. C z R y ) ) ) $.
    infisoti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
      ( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
    $( Image of an infimum under an isomorphism.  (Contributed by Jim Kingdon,
       19-Dec-2021.) $)
    infisoti $p |- ( ph ->
                   inf ( ( F " C ) , B , S ) = ( F ` inf ( C , A , R ) ) ) $=
      ( ccnv csup cfv cinf cima wiso isocnv2 sylib cnvinfex cnvti df-inf fveq2i
      supisoti 3eqtr4g ) ALIUAZHKQZRIGJQZRZLSUKHKTIGJTZLSABCDEFGHIUMULLAGHJKLUB
      GHUMULLUBMGHJKLUCUDNABCDGIJOUEAEFGJPUFUIUKHKUGUOUNLIGJUGUHUJ $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d R x y z $.
    $( Existence of supremum.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    supex2g $p |- ( A e. C -> sup ( B , A , R ) e. _V ) $=
      ( vx vy vz wcel csup cv wbr wn wral wrex wi wa crab cuni cvv df-sup
      rabexg uniexd eqeltrid ) ACHZBADIEJZFJZDKLFBMUFUEDKUFGJDKGBNOFAMPZEAQZRSE
      FGBADTUDUHSUGEACUAUBUC $.
  $}

  ${
    $( Existence of infimum.  (Contributed by Jim Kingdon, 1-Oct-2024.) $)
    infex2g $p |- ( A e. C -> inf ( B , A , R ) e. _V ) $=
      ( wcel cinf ccnv csup cvv df-inf supex2g eqeltrid ) ACEBADFBADGZHIBADJABC
      MKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordinal isomorphism
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d f w x y z A $.  $d f w x y z B $.  $d w x y z F $.
    $( Generalize ~ ordiso to proper classes.  (Contributed by Mario Carneiro,
       24-Jun-2015.) $)
    ordiso2 $p |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) ->
                    A = B ) $=
      ( vx vy vz vw cep cv cfv wceq wcel wi weq fveq2 id eqeq12d wa wb syl wiso
      word w3a wral con0 wss ordsson sseld eleq1 imbi12d imbi2d r19.21v ordelss
      3ad2ant2 3ad2antl2 sselda pm5.5 ralbidva isof1o 3ad2ant1 ad2antrr simpll3
      ccnv wf1o simpr wf f1of simplrl ffvelcdmd jca ordtr1 sylc syl2anc eqeltrd
      f1ocnvfv2 wbr simpll1 f1ocnv 3syl isorel syl12anc vex epelc a1i cvv f1ofn
      wfn funfvex funfni sylan epelg 3bitr3d mpbird simplrr rspcv eqtr3d simprr
      rspccva epel biimpri adantl simpl2 simprl mpbid impbida eqrdv expr sylbid
      eqeltrrd ex com23 biimtrid tfis2 com3l mpdd ralrimiv ffvelcdmda 3ad2antl1
      a2i adantll adantlr wrex crn simpl1 wfo forn eleq2d adantr fvelrnb bitr3d
      f1ofo simpl simplr exp43 syldd imp rexlimdv impbid mpdan ) ABHHCUAZAUBZBU
      BZUCZDIZCJZUUDKZDAUDZABKUUCUUFDAUUCUUDALZUUDUELZUUFUUCAUEUUDUUAYTAUEUFUUB
      AUGUNUHUUIUUCUUHUUFUUCUUHUUFMZMZUUCEIZALZUULCJZUULKZMZMZDEDENZUUJUUPUUCUU
      RUUHUUMUUFUUOUUDUULAUIUURUUEUUNUUDUULUUDUULCOUURPQUJUKUUQEUUDUDUUCUUPEUUD
      UDZMZUUIUUKUUCUUPEUUDULUUTUUKMUUIUUCUUSUUJUUCUUHUUSUUFUUCUUHUUSUUFMUUCUUH
      RZUUSUUOEUUDUDZUUFUVAUUPUUOEUUDUVAUULUUDLRUUMUUPUUOSUVAUUDAUULUUAYTUUHUUD
      AUFZUUBAUUDUMZUOUPUUMUUOUQTURUUCUUHUVBUUFUUCUUHUVBRZRZFUUEUUDUVFFIZUUELZU
      VGUUDLZUVFUVHRZUVGUVGCVCZJZUUDUVJUVLCJZUVGUVLUVJABCVDZUVGBLZUVMUVGKUUCUVN
      UVEUVHYTUUAUVNUUBABHHCUSZUTVAZUVJUUBUVHUUEBLZRUVOYTUUAUUBUVEUVHVBUVJUVHUV
      RUVFUVHVEZUVJABUUDCUUCABCVFZUVEUVHYTUUAUVTUUBYTUVNUVTUVPABCVGTZUTVAUUCUUH
      UVBUVHVHZVIVJUVGUUEBVKVLZABUVGCVOVMZUVJUVLUUDLZUVBUVMUVLKZUVJUWEUVMUUELZU
      VJUVMUVGUUEUWDUVSVNUVJUVLUUDHVPZUVMUUEHVPZUWEUWGUVJYTUVLALUUHUWHUWISYTUUA
      UUBUVEUVHVQZUVJBAUVGUVKUVJUVNBAUVKVDBAUVKVFUVQABCVRBAUVKVGVSUWCVIUWBABUVL
      UUDHHCVTWAUWHUWESUVJUVLUUDDWBWCWDUVJUUEWELZUWIUWGSUVJYTUUHUWKUWJUWBYTCAWG
      ZUUHUWKYTUVNUWLUVPABCWFTZUWKAUUDCUUDCWHWIWJZVMUVMUUEWEWKTWLWMZUUCUUHUVBUV
      HWNUUOUWFEUVLUUDUULUVLKZUUNUVMUULUVLUULUVLCOUWPPQWOVLWPUWOVNUVFUVIRZUVGCJ
      ZUVGUUEUVFUVBUVIUWRUVGKZUUCUUHUVBWQUUOUWSEUVGUUDEFNZUUNUWRUULUVGUULUVGCOU
      WTPQWRWJUWQUWRUUEHVPZUWRUUELZUWQUVGUUDHVPZUXAUVIUXCUVFUXCUVIFDWSWTXAUWQYT
      UVGALZUUHUXCUXASYTUUAUUBUVEUVIVQZUVFUUDAUVGUVFUUAUUHUVCYTUUAUUBUVEXBUUCUU
      HUVBXCUVDVMUPUUCUUHUVBUVIVHZABUVGUUDHHCVTWAXDUWQUWKUXAUXBSUWQYTUUHUWKUXEU
      XFUWNVMUWRUUEWEWKTXDXIXEXFXGXHXJXKXSWDXLXMXNXOXPUUCUUGRZFABUXGUXDUVOUXGUX
      DUVOUXGUXDRUWRUVGBUUGUXDUWSUUCUUFUWSDUVGADFNZUUEUWRUUDUVGUUDUVGCOUXHPQWRX
      TUUCUXDUWRBLZUUGYTUUAUXDUXIUUBYTABUVGCUWAXQXRYAXIXJUXGUVOGIZCJZUVGKZGAYBZ
      UXDUXGUVGCYCZLZUVOUXMUXGUXNBUVGUXGYTUXNBKZYTUUAUUBUUGYDYTUVNABCYEUXPUVPAB
      CYKABCYFVSTYGUXGUWLUXOUXMSUUCUWLUUGYTUUAUWLUUBUWMUTYHGAUVGCYITYJUXGUXLUXD
      GAUUCUUGUXJALZUXLUXDMZMUUCUXQUUGUXRUUCUXQUUGUXKUXJKZUXRUXQUUGUXSMMUUCUUFU
      XSDUXJADGNZUUEUXKUUDUXJUUDUXJCOUXTPQWOWDUUCUXQUXSUXLUXDUUCUXQRZUXSUXLRZRU
      VGUXJAUYBFGNUYAUYBUXKUVGUXJUXSUXLVEUXSUXLYLWPXAUUCUXQUYBYMVNYNYOXKYPYQXHY
      RXFYS $.

    $( Order-isomorphic ordinal numbers are equal.  (Contributed by Jeff
       Hankins, 16-Oct-2009.)  (Proof shortened by Mario Carneiro,
       24-Jun-2015.) $)
    ordiso $p |- ( ( A e. On /\ B e. On ) -> ( A = B
                <-> E. f f Isom _E , _E ( A , B ) ) ) $=
      ( con0 wcel wa wceq cep wiso wex cid cres cvv resiexg isoid isoeq1 spcegv
      cv word eloni mpisyl adantr isoeq5 exbidv syl5ibcom ordiso2 3expia syl2an
      wi 3coml exlimdv impbid ) ADEZBDEZFZABGZABHHCRZIZCJZUOAAHHUQIZCJZUPUSUMVA
      UNUMKALZMEAAHHVBIZVAADNAHOUTVCCVBMAAHHVBUQPQUAUBUPUTURCAABHHUQUCUDUEUOURU
      PCUMASZBSZURUPUIUNATBTVDVEURUPURVDVEUPABUQUFUJUGUHUKUL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Disjoint union
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Disjoint union
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c |_| $.

  $( Extend class notation to include disjoint union of two classes. $)
  cdju $a class ( A |_| B ) $.

  $( Disjoint union of two classes.  This is a way of creating a class which
     contains elements corresponding to each element of ` A ` or ` B ` ,
     tagging each one with whether it came from ` A ` or ` B ` .  (Contributed
     by Jim Kingdon, 20-Jun-2022.) $)
  df-dju $a |- ( A |_| B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) $.

  $( Equality theorem for disjoint union.  (Contributed by Jim Kingdon,
     23-Jun-2022.) $)
  djueq12 $p |- ( ( A = B /\ C = D ) -> ( A |_| C ) = ( B |_| D ) ) $=
    ( wceq wa csn cxp c1o cun cdju xpeq2 adantr adantl uneq12d df-dju 3eqtr4g
    c0 ) ABEZCDEZFZRGZAHZIGZCHZJUBBHZUDDHZJACKBDKUAUCUFUEUGSUCUFETABUBLMTUEUGES
    CDUDLNOACPBDPQ $.

  $( Equality theorem for disjoint union.  (Contributed by Jim Kingdon,
     23-Jun-2022.) $)
  djueq1 $p |- ( A = B -> ( A |_| C ) = ( B |_| C ) ) $=
    ( wceq cdju eqid djueq12 mpan2 ) ABDCCDACEBCEDCFABCCGH $.

  $( Equality theorem for disjoint union.  (Contributed by Jim Kingdon,
     23-Jun-2022.) $)
  djueq2 $p |- ( A = B -> ( C |_| A ) = ( C |_| B ) ) $=
    ( wceq cdju eqid djueq12 mpan ) CCDABDCAECBEDCFCCABGH $.

  ${
    nfdju.1 $e |- F/_ x A $.
    nfdju.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for disjoint union.  (Contributed by
       Jim Kingdon, 23-Jun-2022.) $)
    nfdju $p |- F/_ x ( A |_| B ) $=
      ( cdju c0 csn cxp c1o cun df-dju nfcv nfxp nfun nfcxfr ) ABCFGHZBIZJHZCIZ
      KBCLARTAQBAQMDNASCASMENOP $.
  $}

  $( The disjoint union of sets is a set.  See also the more precise ~ djuss .
     (Contributed by AV, 28-Jun-2022.) $)
  djuex $p |- ( ( A e. V /\ B e. W ) -> ( A |_| B ) e. _V ) $=
    ( wcel wa cdju csn cxp c1o cun cvv df-dju p0ex a1i anim1i ancoms xpexg syl
    c0 con0 1on elexi snex sylan unexg syl2anc eqeltrid ) ACEZBDEZFZABGTHZAIZJH
    ZBIZKZLABMUKUMLEZUOLEZUPLEUKULLEZUIFZUQUJUIUTUJUSUIUSUJNOPQULALCRSUIUNLEZUJ
    URVAUIJJUAUBUCUDOUNBLDRUEUMUOLLUFUGUH $.

  $( The disjoint union of two classes is a set iff both classes are sets.
     (Contributed by Jim Kingdon, 6-Sep-2023.) $)
  djuexb $p |- ( ( A e. _V /\ B e. _V ) <-> ( A |_| B ) e. _V ) $=
    ( vx cvv wcel wa cdju djuex c0 csn cxp c1o crn wex wceq snm rnxpm eqeltrrid
    ax-mp rnexg cun df-dju eleq1i unexb bitr4i cv 0ex 1oex anim12i sylbi impbii
    ) ADEZBDEZFZABGZDEZABDDHUPIJZAKZDEZLJZBKZDEZFZUNUPURVAUAZDEVCUOVDDABUBUCURV
    AUDUEUSULVBUMUSAURMZDCUFZUQECNVEAOCIUGPCUQAQSURDTRVBBVAMZDVFUTECNVGBOCLUHPC
    UTBQSVADTRUIUJUK $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Left and right injections of a disjoint union
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-

  In this section, we define the left and right injections of a disjoint union
  and prove their main properties.  These injections are restrictions of the
  "template" functions ` inl ` and ` inr ` , which appear in most applications
  in the form ` ( inl |`` A ) ` and ` ( inr |`` B ) ` .

$)

  $c inl $.
  $c inr $.

  $( Extend class notation to include left injection of a disjoint union. $)
  cinl $a class inl $.

  $( Extend class notation to include right injection of a disjoint union. $)
  cinr $a class inr $.

  $( Left injection of a disjoint union.  (Contributed by Mario Carneiro,
     21-Jun-2022.) $)
  df-inl $a |- inl = ( x e. _V |-> <. (/) , x >. ) $.

  $( Right injection of a disjoint union.  (Contributed by Mario Carneiro,
     21-Jun-2022.) $)
  df-inr $a |- inr = ( x e. _V |-> <. 1o , x >. ) $.

  ${
    $d C x $.
    $( Left closure of disjoint union.  (Contributed by Jim Kingdon,
       21-Jun-2022.)  (Revised by BJ, 6-Jul-2022.) $)
    djulclr $p |- ( C e. A -> ( ( inl |` A ) ` C ) e. ( A |_| B ) ) $=
      ( vx wcel cinl cres cfv cdju fvres cop cvv csn cxp wceq elex snid eqeltrd
      c0 0ex opelxpi cv opeq2 df-inl fvmptg syl2anc c1o cun elun1 syl eleqtrrdi
      mpan df-dju ) CAEZCFAGHCFHZABIZCAFJUNUOSCKZUPUNCLEUQSMZANZEZUOUQOCAPSUREU
      NUTSTQSCURAUAULZDCSDUBZKUQLUSFVBCSUCDUDUEUFUNUQUSUGMBNZUHZUPUNUTUQVDEVAUQ
      USVCUIUJABUMUKRR $.
  $}

  ${
    $d C x $.
    $( Right closure of disjoint union.  (Contributed by Jim Kingdon,
       21-Jun-2022.)  (Revised by BJ, 6-Jul-2022.) $)
    djurclr $p |- ( C e. B -> ( ( inr |` B ) ` C ) e. ( A |_| B ) ) $=
      ( vx wcel cinr cres cfv cdju fvres c1o cop cvv csn wceq elex 1oex eqeltrd
      cxp snid opelxpi mpan cv opeq2 df-inr fvmptg syl2anc cun elun2 syl df-dju
      c0 eleqtrrdi ) CBEZCFBGHCFHZABIZCBFJUNUOKCLZUPUNCMEUQKNZBSZEZUOUQOCBPKURE
      UNUTKQTKCURBUAUBZDCKDUCZLUQMUSFVBCKUDDUEUFUGUNUQULNASZUSUHZUPUNUTUQVDEVAU
      QUSVCUIUJABUKUMRR $.
  $}

  ${
    $d C x $.
    $( Left closure of disjoint union.  (Contributed by Jim Kingdon,
       21-Jun-2022.) $)
    djulcl $p |- ( C e. A -> ( inl ` C ) e. ( A |_| B ) ) $=
      ( vx wcel cinl cfv c0 cop cdju cvv csn cxp wceq elex snid opelxpi mpan cv
      0ex opeq2 df-inl fvmptg syl2anc c1o cun elun1 df-dju eleqtrrdi eqeltrd
      syl ) CAEZCFGZHCIZABJZULCKEUNHLZAMZEZUMUNNCAOHUPEULURHTPHCUPAQRZDCHDSZIUN
      KUQFUTCHUADUBUCUDULUNUQUELBMZUFZUOULURUNVBEUSUNUQVAUGUKABUHUIUJ $.
  $}

  ${
    $d C x $.
    $( Right closure of disjoint union.  (Contributed by Jim Kingdon,
       21-Jun-2022.) $)
    djurcl $p |- ( C e. B -> ( inr ` C ) e. ( A |_| B ) ) $=
      ( wcel cinr cfv c1o cop cdju cvv csn cxp wceq elex 1oex snid opelxpi mpan
      vx cv opeq2 df-inr fvmptg syl2anc cun elun2 syl df-dju eleqtrrdi eqeltrd
      c0 ) CBDZCEFZGCHZABIZULCJDUNGKZBLZDZUMUNMCBNGUPDULURGOPGCUPBQRZSCGSTZHUNJ
      UQEUTCGUASUBUCUDULUNUKKALZUQUEZUOULURUNVBDUSUNUQVAUFUGABUHUIUJ $.
  $}

  ${
    $d X x y $.  $d A x y $.
    djuf1olem.1 $e |- X e. _V $.
    djuf1olem.2 $e |- F = ( x e. A |-> <. X , x >. ) $.
    $( Lemma for ~ djulf1o and ~ djurf1o .  (Contributed by BJ and Jim Kingdon,
       4-Jul-2022.) $)
    djuf1olem $p |- F : A -1-1-onto-> ( { X } X. A ) $=
      ( vy csn cxp wf1o wtru cv cop c2nd cfv wcel snid opelxpi adantl wceq mpan
      xp2nd wb c1st 1st2nd2 xp1st elsni syl opeq1d eqtrd eqeq2d eqcom eqid opth
      vex mpbiran 3bitr3g bicomd ad2antll f1o2d mptru ) BDHZBIZCJKAGBVCDALZMZGL
      ZNOZCFVDBPZVEVCPZKDVBPVHVIDEQDVDVBBRUASVFVCPZVGBPKVFVBBUBSVJVDVGTZVFVETZU
      CKVHVJVLVKVJVEVFTVEDVGMZTZVLVKVJVFVMVEVJVFVFUDOZVGMVMVFVBBUEVJVODVGVJVOVB
      PVODTVFVBBUFVODUGUHUIUJUKVEVFULVNDDTVKDUMDVDDVGEAUOUNUPUQURUSUTVA $.
  $}

  ${
    $d X x $.  $d A x $.
    djuf1olemr.1 $e |- X e. _V $.
    djuf1olemr.2 $e |- F = ( x e. _V |-> <. X , x >. ) $.
    $( Lemma for ~ djulf1or and ~ djurf1or .  For a version of this lemma with
       ` F ` defined on ` A ` and no restriction in the conclusion, see
       ~ djuf1olem .  (Contributed by BJ and Jim Kingdon, 4-Jul-2022.) $)
    djuf1olemr $p |- ( F |` A ) : A -1-1-onto-> ( { X } X. A ) $=
      ( cres cvv cv cop cmpt reseq1i wss wceq ssv resmpt ax-mp eqtri djuf1olem
      ) ABCBGZDETAHDAIJZKZBGZABUAKZCUBBFLBHMUCUDNBOAHBUAPQRS $.
  $}

  ${
    $d A x $.  $d C x $.
    $( Left biconditional closure of disjoint union.  (Contributed by Jim
       Kingdon, 2-Jul-2022.) $)
    djulclb $p |- ( C e. V -> ( C e. A <-> ( inl ` C ) e. ( A |_| B ) ) ) $=
      ( vx wcel cinl cfv cdju djulcl wa c0 csn cop cxp c1o 0ex opelxp cvv sylib
      wn 1n0 necomi elsn nemtbir intnanr mtbir wo wceq elex opexg mpan cv opeq2
      cun df-inl fvmptg adantr df-dju eleq2i biimpi adantl eqeltrrd elun orcomd
      syl2anc ord mpi simprd ex impbid2 ) CDFZCAFZCGHZABIZFZABCJVLVPVMVLVPKZLLM
      ZFZVMVQLCNZVRAOZFZVSVMKVQVTPMZBOZFZUAWBWELWCFZCBFZKWFWGWFLPPLUBUCLPQUDUEU
      FLCWCBRUGVQWEWBVQWBWEVQVTWAWDUOZFWBWEUHVQVNVTWHVLVNVTUIZVPVLCSFVTSFZWICDU
      JLSFVLWJQLCSDUKULECLEUMZNVTSSGWKCLUNEUPUQVFURVPVNWHFZVLVPWLVOWHVNABUSUTVA
      VBVCVTWAWDVDTVEVGVHLCVRARTVIVJVK $.
  $}

  ${
    $d A x $.
    $( The left injection function on all sets is one to one and onto.
       (Contributed by BJ and Jim Kingdon, 22-Jun-2022.) $)
    djulf1or $p |- ( inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) $=
      ( vx cinl c0 0ex df-inl djuf1olemr ) BACDEBFG $.

    $( The right injection function on all sets is one to one and onto.
       (Contributed by BJ and Jim Kingdon, 22-Jun-2022.) $)
    djurf1or $p |- ( inr |` A ) : A -1-1-onto-> ( { 1o } X. A ) $=
      ( vx cinr c1o 1oex df-inr djuf1olemr ) BACDEBFG $.
  $}

  $( The left injection function on all sets is one to one and onto.
     (Contributed by Jim Kingdon, 22-Jun-2022.) $)
  djulf1o $p |- inl : _V -1-1-onto-> ( { (/) } X. _V ) $=
    ( vx cvv cinl c0 0ex df-inl djuf1olem ) ABCDEAFG $.

  $( The right injection function on all sets is one to one and onto.
     (Contributed by Jim Kingdon, 22-Jun-2022.) $)
  djurf1o $p |- inr : _V -1-1-onto-> ( { 1o } X. _V ) $=
    ( vx cvv cinr c1o 1oex df-inr djuf1olem ) ABCDEAFG $.

  ${
    $d A x $.  $d B x $.  $d F x $.
    inresflem.1 $e |- F : A -1-1-onto-> ( { X } X. A ) $.
    inresflem.2 $e |- ( x e. A -> ( F ` x ) e. B ) $.
    $( Lemma for ~ inlresf1 and ~ inrresf1 .  (Contributed by BJ,
       4-Jul-2022.) $)
    inresflem $p |- F : A -1-1-> B $=
      ( csn cxp wf1 wf wf1o f1of1 ax-mp wfn crn wss f1ofn cv mp2an cfv fnfvrnss
      wcel wral rgen df-f mpbir2an f1ff1 ) BEHBIZDJZBCDKZBCDJBUIDLZUJFBUIDMNUKD
      BOZDPCQZULUMFBUIDRNZUMASDUACUCZABUDUNUOUPABGUEABCDUBTBCDUFUGBUICDUHT $.
  $}

  ${
    $d A x $.  $d B x $.
    $( The left injection restricted to the left class of a disjoint union is
       an injective function from the left class into the disjoint union.
       (Contributed by AV, 28-Jun-2022.) $)
    inlresf1 $p |- ( inl |` A ) : A -1-1-> ( A |_| B ) $=
      ( vx cdju cinl cres c0 djulf1or cv djulclr inresflem ) CAABDEAFGAHABCIJK
      $.

    $( The right injection restricted to the right class of a disjoint union is
       an injective function from the right class into the disjoint union.
       (Contributed by AV, 28-Jun-2022.) $)
    inrresf1 $p |- ( inr |` B ) : B -1-1-> ( A |_| B ) $=
      ( vx cdju cinr cres c1o djurf1or cv djurclr inresflem ) CBABDEBFGBHABCIJK
      $.
  $}

  $( The ranges of any left and right injections are disjoint.  Remark: the
     extra generality offered by the two restrictions makes the theorem more
     readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler
     statement ` |- ( ran inl i^i ran inr ) = (/) ` is easily recovered from it
     by substituting ` _V ` for both ` A ` and ` B ` as done in ~ casefun ).
     (Contributed by BJ and Jim Kingdon, 21-Jun-2022.) $)
  djuinr $p |- ( ran ( inl |` A ) i^i ran ( inr |` B ) ) = (/) $=
    ( cinl cres crn cinr cin c0 csn cxp c1o wf1o wceq djulf1or wf1 dff1o5 ax-mp
    simprbi djurf1or ineq12i wne 1n0 necomi disjsn2 xpdisj1 eqtri ) CADZEZFBDZE
    ZGHIZAJZKIZBJZGZHUHULUJUNAULUGLZUHULMZANUPAULUGOUQAULUGPRQBUNUILZUJUNMZBSUR
    BUNUIOUSBUNUIPRQTUKUMGHMZUOHMHKUAUTKHUBUCHKUDQUKUMABUEQUF $.

  $( The images of any classes under right and left injection produce disjoint
     sets.  (Contributed by Jim Kingdon, 21-Jun-2022.)  (Proof shortened by BJ,
     9-Jul-2023.) $)
  djuin $p |- ( ( inl " A ) i^i ( inr " B ) ) = (/) $=
    ( cinl cima cinr cin cres crn c0 df-ima ineq12i djuinr eqtri ) CADZEBDZFCAG
    HZEBGHZFINPOQCAJEBJKABLM $.

  ${
    $d A x $.  $d B x $.
    $( Left injection is one-to-one.  (Contributed by Jim Kingdon,
       12-Jul-2023.) $)
    inl11 $p |- ( ( A e. V /\ B e. W )
        -> ( ( inl ` A ) = ( inl ` B ) <-> A = B ) ) $=
      ( vx wcel wa cinl cfv wceq c0 cop cvv opeq2 elex adantr 0ex opexg fvmptd3
      wb cv df-inl simpl sylancr adantl a1i sylancom eqeq12d mpan eqid biantrur
      opthg bitr4di bitrd ) ACFZBDFZGZAHIZBHIZJKALZKBLZJZABJZUQURUTUSVAUQEAKEUA
      ZLZUTMHMEUBZVDAKNUOAMFUPACOPUQKMFZUOUTMFQUOUPUCKAMCRUDSUQEBVEVAMHMVFVDBKN
      UPBMFUOBDOUEUOUPVGVAMFVGUQQUFKBMDRUGSUHUOVBVCTUPUOVBKKJZVCGZVCVGUOVBVITQK
      AKBMCULUIVHVCKUJUKUMPUN $.
  $}

  $( The disjoint union of two classes is the union of the images of those two
     classes under right and left injection.  (Contributed by Jim Kingdon,
     22-Jun-2022.)  (Proof shortened by BJ, 6-Jul-2022.) $)
  djuunr $p |- ( ran ( inl |` A ) u. ran ( inr |` B ) ) = ( A |_| B ) $=
    ( cinl cres crn cinr cun csn cxp c1o cdju wf1o wfo wceq djulf1or f1ofo forn
    c0 mp2b djurf1or uneq12i df-dju eqtr4i ) CADZEZFBDZEZGRHAIZJHBIZGABKUEUHUGU
    IAUHUDLAUHUDMUEUHNAOAUHUDPAUHUDQSBUIUFLBUIUFMUGUINBTBUIUFPBUIUFQSUAABUBUC
    $.

  $( The disjoint union of two classes is the union of the images of those two
     classes under right and left injection.  (Contributed by Jim Kingdon,
     22-Jun-2022.)  (Proof shortened by BJ, 9-Jul-2023.) $)
  djuun $p |- ( ( inl " A ) u. ( inr " B ) ) = ( A |_| B ) $=
    ( cinl cima cinr cun cres crn cdju df-ima uneq12i djuunr eqtri ) CADZEBDZFC
    AGHZEBGHZFABINPOQCAJEBJKABLM $.

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Element of a disjoint union.  (Contributed by BJ and Jim Kingdon,
       23-Jun-2022.) $)
    eldju $p |- ( C e. ( A |_| B ) <-> ( E. x e. A C = ( ( inl |` A ) ` x )
        \/ E. x e. B C = ( ( inr |` B ) ` x ) ) ) $=
      ( wcel cres crn cfv wceq wrex wo csn cxp wf1o wb f1ofn fvelrnb mp2b bitri
      wfn cdju cinl cinr cun cv djuunr eqcomi eleq2i elun djulf1or eqcom rexbii
      c0 c1o djurf1or orbi12i ) DBCUAZEDUBBFZGZUCCFZGZUDZEZDAUEZURHZIZABJZDVDUT
      HZIZACJZKZUQVBDVBUQBCUFUGUHVCDUSEZDVAEZKVKDUSVAUIVLVGVMVJVLVEDIZABJZVGBUM
      LBMZURNURBTVLVOOBUJBVPURPABDURQRVNVFABVEDUKULSVMVHDIZACJZVJCUNLCMZUTNUTCT
      VMVROCUOCVSUTPACDUTQRVQVIACVHDUKULSUPSS $.
  $}

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( A member of a disjoint union can be mapped from one of the classes which
       produced it.  (Contributed by Jim Kingdon, 23-Jun-2022.)  Upgrade
       implication to biconditional and shorten proof.  (Revised by BJ,
       14-Jul-2023.) $)
    djur $p |- ( C e. ( A |_| B ) <-> ( E. x e. A C = ( inl ` x )
        \/ E. x e. B C = ( inr ` x ) ) ) $=
      ( cdju wcel cv cinl cres cfv wceq wrex cinr wo eldju fvres eqeq2d rexbiia
      orbi12i bitri ) DBCEFDAGZHBIJZKZABLZDUAMCIJZKZACLZNDUAHJZKZABLZDUAMJZKZAC
      LZNABCDOUDUJUGUMUCUIABUABFUBUHDUABHPQRUFULACUACFUEUKDUACMPQRST $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Universal property of the disjoint union
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d A x y $.  $d B x y $.
    $( A disjoint union is a subset of a Cartesian product.  (Contributed by
       AV, 25-Jun-2022.) $)
    djuss $p |- ( A |_| B ) C_ ( { (/) , 1o } X. ( A u. B ) ) $=
      ( vx vy c0 c1o cv wcel cinl cfv wceq wrex cinr cop simpr cvv opeq2 adantr
      wa elex cdju cpr cun cxp djur df-inl 0ex vex opex a1i fvmptd3 eqtrd elun1
      prid1 jctil opelxp sylibr eqeltrd rexlimiva df-inr 1oex elun2 prid2 sylbi
      wo jaoi ssriv ) CABUAZEFUBZABUCZUDZCGZVHHVLDGZIJZKZDALZVLVMMJZKZDBLZVEVLV
      KHZDABVLUEVPVTVSVOVTDAVMAHZVOSZVLEVMNZVKWBVLVNWCWAVOOWAVNWCKVOWACVMEVLNWC
      PIPCUFVLVMEQVMATWCPHWAEVMUGDUHZUIUJUKRULWBEVIHZVMVJHZSZWCVKHWAWGVOWAWFWEV
      MABUMEFUGUNUOREVMVIVJUPUQURUSVRVTDBVMBHZVRSZVLFVMNZVKWIVLVQWJWHVROWHVQWJK
      VRWHCVMFVLNWJPMPCUTVLVMFQVMBTWJPHWHFVMVAWDUIUJUKRULWIFVIHZWFSWJVKHWIWFWKW
      HWFVRVMBAVBREFVAVCUOFVMVIVJUPUQURUSVFVDVG $.
  $}

  $( The first component of an element of a disjoint union is either ` (/) ` or
     ` 1o ` .  (Contributed by AV, 26-Jun-2022.) $)
  eldju1st $p |- ( X e. ( A |_| B )
                   -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) ) $=
    ( cdju c0 c1o cpr cun cxp wss wcel c1st cfv wceq wo djuss ssel2 xp1st elpri
    wa 3syl mpan ) ABDZEFGZABHZIZJZCUCKZCLMZENUIFNOZABPUGUHTCUFKUIUDKUJUCUFCQCU
    DUERUIEFSUAUB $.

  $( The second component of an element of a disjoint union is an element of
     the left class of the disjoint union if its first component is the empty
     set.  (Contributed by AV, 26-Jun-2022.) $)
  eldju2ndl $p |- ( ( X e. ( A |_| B ) /\ ( 1st ` X ) = (/) )
                       -> ( 2nd ` X ) e. A ) $=
    ( cdju wcel c1st cfv c0 wceq c2nd csn cxp c1o wo wi cun wa elxp6 sylbi wne
    df-dju eleq2i elun bitri cop simprr a1d elsni 1n0 neeq1 eqneqall com12 3syl
    mpbiri ad2antrl jaoi imp ) CABDZEZCFGZHIZCJGZAEZUSCHKZALZEZCMKZBLZEZNZVAVCO
    ZUSCVEVHPZEVJURVLCABUAUBCVEVHUCUDVFVKVIVFCUTVBUEIZUTVDEZVCQQZVKCVDARVOVCVAV
    MVNVCUFUGSVIVMUTVGEZVBBEZQQVKCVGBRVPVKVMVQVPUTMIZUTHTZVKUTMUHVRVSMHTUIUTMHU
    JUNVAVSVCVCUTHUKULUMUOSUPSUQ $.

  $( The second component of an element of a disjoint union is an element of
     the right class of the disjoint union if its first component is not the
     empty set.  (Contributed by AV, 26-Jun-2022.) $)
  eldju2ndr $p |- ( ( X e. ( A |_| B ) /\ ( 1st ` X ) =/= (/) )
                       -> ( 2nd ` X ) e. B ) $=
    ( cdju wcel c1st cfv c0 wne c2nd csn cxp c1o wo wi cun wceq wa elxp6 sylbi
    df-dju eleq2i elun bitri cop elsni eqneqall syl ad2antrl simprr a1d jaoi
    imp ) CABDZEZCFGZHIZCJGZBEZUOCHKZALZEZCMKZBLZEZNZUQUSOZUOCVAVDPZEVFUNVHCABU
    AUBCVAVDUCUDVBVGVEVBCUPURUEQZUPUTEZURAEZRRVGCUTASVJVGVIVKVJUPHQVGUPHUFUSUPH
    UGUHUITVEVIUPVCEZUSRRZVGCVCBSVMUSUQVIVLUSUJUKTULTUM $.

  ${
    $d V x $.  $d X x $.
    $( The first component of the value of a left injection is the empty set.
       (Contributed by AV, 27-Jun-2022.) $)
    1stinl $p |- ( X e. V -> ( 1st ` ( inl ` X ) ) = (/) ) $=
      ( vx wcel cinl cfv c1st c0 cop cvv cmpt wceq df-inl a1i opeq2 adantl elex
      cv 0ex mpan opexg fvmptd fveq2d op1stg eqtrd ) BADZBEFZGFHBIZGFZHUFUGUHGU
      FCBHCRZIZUHJEJECJUKKLUFCMNUJBLUKUHLUFUJBHOPBAQHJDZUFUHJDSHBJAUATUBUCULUFU
      IHLSHBJAUDTUE $.

    $( The second component of the value of a left injection is its argument.
       (Contributed by AV, 27-Jun-2022.) $)
    2ndinl $p |- ( X e. V -> ( 2nd ` ( inl ` X ) ) = X ) $=
      ( vx wcel cinl cfv c2nd c0 cop cvv cmpt wceq df-inl a1i opeq2 adantl elex
      cv 0ex mpan opexg fvmptd fveq2d op2ndg eqtrd ) BADZBEFZGFHBIZGFZBUFUGUHGU
      FCBHCRZIZUHJEJECJUKKLUFCMNUJBLUKUHLUFUJBHOPBAQHJDZUFUHJDSHBJAUATUBUCULUFU
      IBLSHBJAUDTUE $.

    $( The first component of the value of a right injection is ` 1o ` .
       (Contributed by AV, 27-Jun-2022.) $)
    1stinr $p |- ( X e. V -> ( 1st ` ( inr ` X ) ) = 1o ) $=
      ( vx wcel cinr cfv c1st c1o cop cv cvv cmpt wceq df-inr opeq2 adantl con0
      a1i 1on mpan elex opexg fvmptd fveq2d op1stg eqtrd ) BADZBEFZGFHBIZGFZHUG
      UHUIGUGCBHCJZIZUIKEKECKULLMUGCNRUKBMULUIMUGUKBHOPBAUAHQDZUGUIKDSHBQAUBTUC
      UDUMUGUJHMSHBQAUETUF $.

    $( The second component of the value of a right injection is its argument.
       (Contributed by AV, 27-Jun-2022.) $)
    2ndinr $p |- ( X e. V -> ( 2nd ` ( inr ` X ) ) = X ) $=
      ( vx wcel cinr cfv c2nd c1o cop cv cvv cmpt wceq df-inr opeq2 adantl con0
      a1i 1on mpan elex opexg fvmptd fveq2d op2ndg eqtrd ) BADZBEFZGFHBIZGFZBUG
      UHUIGUGCBHCJZIZUIKEKECKULLMUGCNRUKBMULUIMUGUKBHOPBAUAHQDZUGUIKDSHBQAUBTUC
      UDUMUGUJBMSHBQAUETUF $.
  $}

  ${
    $( Left and right injection never produce equal values.  (Contributed by
       Jim Kingdon, 2-Jul-2022.) $)
    djune $p |- ( ( A e. V /\ B e. W ) -> ( inl ` A ) =/= ( inr ` B ) ) $=
      ( wcel wa cinl cfv cinr c1st c0 c1o nesymi 1stinl 1stinr eqeqan12d mtbiri
      wceq 1n0 fveq2 nsyl neqned ) ACEZBDEZFZAGHZBIHZUEUFJHZUGJHZRZUFUGRUEUJKLR
      LKSMUCUDUHKUILCANDBOPQUFUGJTUAUB $.
  $}

  ${
    updjud.f $e |- ( ph -> F : A --> C ) $.
    updjud.g $e |- ( ph -> G : B --> C ) $.
    ${
      $d A x $.  $d B x $.  $d C x $.  $d ph x $.
      updjudhf.h $e |- H = ( x e. ( A |_| B ) |-> if ( ( 1st ` x ) = (/) ,
                           ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) ) $.
      $( The mapping of an element of the disjoint union to the value of the
         corresponding function is a function.  (Contributed by AV,
         26-Jun-2022.) $)
      updjudhf $p |- ( ph -> H : ( A |_| B ) --> C ) $=
        ( cfv c0 wceq wcel ex wf wi syl c1o cdju cv c1st cif eldju2ndl ffvelcdm
        c2nd wa sylan9r imp wn wne df-ne eldju2ndr biimtrrid wdc eldju1st neeq1
        wo 1n0 mpbiri orim2i adantl dcne sylibr ifcldadc fmptd ) ABCDUAZBUBZUCL
        ZMNZVIUGLZFLZVLGLZUDEHAVIVHOZUHZVKVMVNEVPVKVMEOZVOVKVLCOZAVQVOVKVRCDVIU
        EPACEFQZVRVQRIVSVRVQCEVLFUFPSUIUJVPVKUKZVNEOZVTVJMULZVPWAVJMUMVOWBVLDOZ
        AWAVOWBWCCDVIUNPADEGQZWCWARJWDWCWADEVLGUFPSUIUOUJVPVKWBUSZVKUPVOWEAVOVK
        VJTNZUSWECDVIUQWFWBVKWFWBTMULUTVJTMURVAVBSVCVJMVDVEVFKVG $.

      $d A a x $.  $d F a x $.  $d H a $.  $d ph a $.
      $( The composition of the mapping of an element of the disjoint union to
         the value of the corresponding function and the left injection equals
         the first function.  (Contributed by AV, 27-Jun-2022.) $)
      updjudhcoinlf $p |- ( ph -> ( H o. ( inl |` A ) ) = F ) $=
        ( cinl wfn wf cfv c2nd wceq adantl fveq2d c1st va cres crn wss updjudhf
        ccom cdju ffn syl wf1 inlresf1 f1fn mp1i f1f ax-mp fnco syl3anc cv wcel
        frn wa fvco2 sylan fvres c0 cif cmpt a1i eqeq1d ifbieq12d 1stinl adantr
        fveq2 iftrued eqtrd djulcl 2ndinl simpr eqeltrd ffvelcdmd fvmptd 3eqtrd
        eqfnfvd ) AUACHLCUBZUFZFAHCDUGZMZWDCMZWDUCWFUDZWECMAWFEHNWGABCDEFGHIJKU
        EWFEHUHUICWFWDUJZWHACDUKZCWFWDULUMZCWFWDNZWIAWJWMWKCWFWDUNUOCWFWDUTUMWF
        CHWDUPUQACEFNZFCMICEFUHUIAUAURZCUSZVAZWOWEOZWOWDOZHOZWOLOZPOZFOZWOFOAWH
        WPWRWTQWLCHWDWOVBVCWQWTXAHOXCWQWSXAHWPWSXAQAWOCLVDRSWQBXABURZTOZVEQZXDP
        OZFOZXGGOZVFZXCWFHEHBWFXJVGQWQKVHWQXDXAQZVAZXJXATOZVEQZXCXBGOZVFZXCXKXJ
        XPQWQXKXFXNXHXIXCXOXKXEXMVEXDXATVMVIXKXGXBFXDXAPVMZSXKXGXBGXQSVJRXLXNXC
        XOWQXNXKWPXNACWOVKRVLVNVOWPXAWFUSACDWOVPRWQCEXBFAWNWPIVLWQXBWOCWPXBWOQA
        CWOVQRZAWPVRVSVTWAVOWQXBWOFXRSWBWC $.

      $d B b x $.  $d G b x $.  $d H b $.  $d ph b $.
      $( The composition of the mapping of an element of the disjoint union to
         the value of the corresponding function and the right injection equals
         the second function.  (Contributed by AV, 27-Jun-2022.) $)
      updjudhcoinrg $p |- ( ph -> ( H o. ( inr |` B ) ) = G ) $=
        ( cinr wfn wf cfv wceq adantl fveq2d c0 c1o cres ccom cdju crn updjudhf
        vb wss ffn syl wf1 inrresf1 f1fn mp1i f1f ax-mp fnco syl3anc cv wcel wa
        frn c2nd fvco2 sylan fvres c1st cif cmpt a1i eqeq1d ifbieq12d wn 1stinr
        fveq2 1n0 neii eqeq1 mtbiri adantr iffalsed eqtrd djurcl 2ndinr eqeltrd
        simpr ffvelcdmd fvmptd 3eqtrd eqfnfvd ) AUFDHLDUAZUBZGAHCDUCZMZWJDMZWJU
        DWLUGZWKDMAWLEHNWMABCDEFGHIJKUEWLEHUHUIDWLWJUJZWNACDUKZDWLWJULUMZDWLWJN
        ZWOAWPWSWQDWLWJUNUODWLWJVAUMWLDHWJUPUQADEGNZGDMJDEGUHUIAUFURZDUSZUTZXAW
        KOZXAWJOZHOZXALOZVBOZGOZXAGOAWNXBXDXFPWRDHWJXAVCVDXCXFXGHOXIXCXEXGHXBXE
        XGPAXADLVEQRXCBXGBURZVFOZSPZXJVBOZFOZXMGOZVGZXIWLHEHBWLXPVHPXCKVIXCXJXG
        PZUTZXPXGVFOZSPZXHFOZXIVGZXIXQXPYBPXCXQXLXTXNXOYAXIXQXKXSSXJXGVFVNVJXQX
        MXHFXJXGVBVNZRXQXMXHGYCRVKQXRXTYAXIXCXTVLZXQXBYDAXBXSTPZYDDXAVMYEXTTSPT
        SVOVPXSTSVQVRUIQVSVTWAXBXGWLUSACDXAWBQXCDEXHGAWTXBJVSXCXHXADXBXHXAPADXA
        WCQZAXBWEWDWFWGWAXCXHXAGYFRWHWI $.
    $}

    $d A h k x $.  $d A k x y z $.  $d B h k x $.  $d B k x y z $.
    $d C h k x $.  $d C k x y z $.  $d F h k x $.  $d F k x y z $.
    $d G h k x $.  $d G k x y z $.  $d h k ph x $.  $d ph x y z $.
    updjud.a $e |- ( ph -> A e. V ) $.
    updjud.b $e |- ( ph -> B e. W ) $.
    $( Universal property of the disjoint union.  (Proposed by BJ,
       25-Jun-2022.)  (Contributed by AV, 28-Jun-2022.) $)
    updjud $p |- ( ph -> E! h ( h : ( A |_| B ) --> C
                                /\ ( h o. ( inl |` A ) ) = F
                                /\ ( h o. ( inr |` B ) ) = G ) ) $=
      ( vk wceq cvv wi wa cfv wcel vx vy vz cdju cv wf cinl cres ccom cinr wreu
      w3a weu wral wrex c1st c0 c2nd cif cmpt jca djuex mptexg 3syl feq1 eqeq1d
      wb coeq1 3anbi123d eqeq1 imbi2d ralbidv anbi12d adantl eqid updjudhcoinlf
      updjudhf updjudhcoinrg simpr eqeq2 fvres eqcomd eqeq2d fveq1 ad2antrr wfn
      wo inlresf1 f1fn mp1i fvco2 sylan 3eqtr3d fveq2 eqeq12d syl5ibrcom sylbid
      expimpd ex eqcoms biimtrrdi com23 3ad2ant2 impcom com12 inrresf1 3ad2ant3
      wf1 rexlimiva jaoi djur biimpi ralrimiv ffn 3ad2ant1 eqfnfv syl2an mpbird
      syl11 ralrimivw mp3and rspcedvd reu8 sylibr reuv sylib ) ABCUDZDEUEZUFZYH
      UGBUHZUIZFOZYHUJCUHZUIZGOZULZEPUKZYPEUMAYPYGDNUEZUFZYRYJUIZFOZYRYMUIZGOZU
      LZYHYROZQZNPUNZRZEPUOYQAUUHYGDUAYGUAUEZUPSUQOUUIURSZFSUUJGSUSZUTZUFZUULYJ
      UIZFOZUULYMUIZGOZULZUUDUULYROZQZNPUNZRZEUULPABHTZCITZRYGPTUULPTAUVCUVDLMV
      ABCHIVBUAYGUUKPVCVDYHUULOZUUHUVBVGAUVEYPUURUUGUVAUVEYIUUMYLUUOYOUUQYGDYHU
      ULVEUVEYKUUNFYHUULYJVHVFUVEYNUUPGYHUULYMVHVFVIUVEUUFUUTNPUVEUUEUUSUUDYHUU
      LYRVJVKVLVMVNAUUMUUOUUQUVBAUABCDFGUULJKUULVOZVQAUABCDFGUULJKUVFVPAUABCDFG
      UULJKUVFVRAUURUVBAUURRZUURUVAAUURVSUVGUUTNPUVGUUDUUSUVGUUDRZUUSUBUEZUULSZ
      UVIYRSZOZUBYGUNZUVHUVLUBYGUVIUCUEZUGSZOZUCBUOZUVIUVNUJSZOZUCCUOZWGZUVHUVL
      UVIYGTZUVQUVHUVLQZUVTUVPUWCUCBUVHUVNBTZUVPRZUVLUUDUVGUWEUVLQZUUAYSUVGUWFQ
      UUCUVGUUAUWFUURAUUAUWFQZUUOUUMAUWGQUUQUUOUUAAUWFUUOUUAYTUUNOAUWFQZUUNFYTV
      TUWHUUNYTUUNYTOZAUWFUWIARZUWDUVPUVLUWJUWDRZUVPUVIUVNYJSZOZUVLUWDUVPUWMVGU
      WJUWDUVOUWLUVIUWDUWLUVOUVNBUGWAWBWCVNUWKUVLUWMUWLUULSZUWLYRSZOUWKUVNUUNSZ
      UVNYTSZUWNUWOUWIUWPUWQOAUWDUVNUUNYTWDWEUWJYJBWFZUWDUWPUWNOBYGYJXHUWRUWJBC
      WHBYGYJWIWJZBUULYJUVNWKWLUWJUWRUWDUWQUWOOUWSBYRYJUVNWKWLWMUWMUVJUWNUVKUWO
      UVIUWLUULWNUVIUWLYRWNWOWPWQWRWSWTXAXBXCXDXEXCXDXEXIUVSUWCUCCUVHUVNCTZUVSR
      ZUVLUUDUVGUXAUVLQZUUCYSUVGUXBQUUAUVGUUCUXBUURAUUCUXBQZUUQUUMAUXCQUUOUUQUU
      CAUXBUUQUUCUUBUUPOAUXBQZUUPGUUBVTUXDUUPUUBUUPUUBOZAUXBUXEARZUWTUVSUVLUXFU
      WTRZUVSUVIUVNYMSZOZUVLUWTUVSUXIVGUXFUWTUVRUXHUVIUWTUXHUVRUVNCUJWAWBWCVNUX
      GUVLUXIUXHUULSZUXHYRSZOUXGUVNUUPSZUVNUUBSZUXJUXKUXEUXLUXMOAUWTUVNUUPUUBWD
      WEUXFYMCWFZUWTUXLUXJOCYGYMXHUXNUXFBCXFCYGYMWIWJZCUULYMUVNWKWLUXFUXNUWTUXM
      UXKOUXOCYRYMUVNWKWLWMUXIUVJUXJUVKUXKUVIUXHUULWNUVIUXHYRWNWOWPWQWRWSWTXAXB
      XGXDXEXGXDXEXIXJUWBUWAUCBCUVIXKXLXSXMUVGUULYGWFZYRYGWFZUUSUVMVGUUDUURUXPA
      UUMUUOUXPUUQYGDUULXNXOVNYSUUAUXQUUCYGDYRXNXOUBYGUULYRXPXQXRWSXTVAWSYAYBYP
      UUDENPUUEYIYSYLUUAYOUUCYGDYHYRVEUUEYKYTFYHYRYJVHVFUUEYNUUBGYHYRYMVHVFVIYC
      YDYPEYEYF $.
  $}

  $c case $.

  $( Syntax for the "case" construction. $)
  cdjucase $a class case ( R , S ) $.

  $( The "case" construction: if ` F : A --> X ` and ` G : B --> X ` are
     functions, then ` case ( F , G ) : ( A |_| B ) --> X ` is the natural
     function obtained by a definition by cases, hence the name.  It is the
     unique function whose existence is asserted by the universal property of
     disjoint unions ~ updjud .  The definition is adapted to make sense also
     for binary relations (where the universal property also holds).
     (Contributed by MC and BJ, 10-Jul-2022.) $)
  df-case $a |- case ( R , S ) = ( ( R o. `' inl ) u. ( S o. `' inr ) ) $.

  ${
    casefun.f $e |- ( ph -> Fun F ) $.
    casefun.g $e |- ( ph -> Fun G ) $.
    $( The "case" construction of two functions is a function.  (Contributed by
       BJ, 10-Jul-2022.) $)
    casefun $p |- ( ph -> Fun case ( F , G ) ) $=
      ( cinl ccnv ccom cinr wfun cdm cin c0 wceq cvv csn cxp wf1 wss crn cun wf
      cdjucase wf1o djulf1o f1of1 ax-mp df-f1 simprbi funco syl2anc c1o djurf1o
      mp1i dmcoss df-rn sseqtrri ss2in mp2an rnresv eqcomi ineq12i djuinr eqtri
      cres a1i sseqtrid ss0 syl funun syl21anc df-case funeqi sylibr ) ABFGZHZC
      IGZHZUAZJZBCUCZJAVPJZVRJZVPKZVRKZLZMNZVTABJVOJZWBDOMPOQZFRZWHAOWIFUDWJUEO
      WIFUFUGWJOWIFUBWHOWIFUHUIUNBVOUJUKACJVQJZWCEOULPOQZIRZWKAOWLIUDWMUMOWLIUF
      UGWMOWLIUBWKOWLIUHUIUNCVQUJUKAWFMSWGAFTZITZLZWFMWDWNSWEWOSWFWPSWDVOKWNBVO
      UOFUPUQWEVQKWOCVQUOIUPUQWDWNWEWOURUSWPMNAWPFOVETZIOVETZLMWNWQWOWRWQWNFUTV
      AWRWOIUTVAVBOOVCVDVFVGWFVHVIVPVRVJVKWAVSBCVLVMVN $.
  $}

  $( The domain of the "case" construction is the disjoint union of the
     domains.  TODO (although less important):
     ` |- ran case ( F , G ) = ( ran F u. ran G ) ` .  (Contributed by BJ,
     10-Jul-2022.) $)
  casedm $p |- dom case ( F , G ) = ( dom F |_| dom G ) $=
    ( cdjucase cdm cinl ccnv ccom cinr cun cdju df-case dmun cres crn cima dmco
    dmeqi imacnvcnv df-ima 3eqtri uneq12i djuunr eqtri ) ABCZDAEFZGZBHFZGZIZDUF
    DZUHDZIZADZBDZJZUDUIABKQUFUHLULEUMMNZHUNMNZIUOUJUPUKUQUJUEFUMOEUMOUPAUEPEUM
    REUMSTUKUGFUNOHUNOUQBUGPHUNRHUNSTUAUMUNUBUCT $.

  $( The "case" construction of two relations is a relation, with bounds on its
     domain and codomain.  Typically, the "case" construction is used when both
     relations have a common codomain.  (Contributed by BJ, 10-Jul-2022.) $)
  caserel $p
          |- case ( R , S ) C_ ( ( dom R |_| dom S ) X. ( ran R u. ran S ) ) $=
  ( cinl ccnv ccom cinr cun cdm crn cxp cres cocnvss wss f1rn ax-mp resss sstri
  wf1 rnss xpss12 cdjucase df-case inlresf1 ssun1 mp2an inrresf1 ssun2 eqsstri
  cdju unssi ) ABUAACDEZBFDEZGAHZBHZUIZAIZBIZGZJZABUBUKULUSUKCUMKZIZACHZKZIZJZU
  SCALVAUOMZVDURMVEUSMUMUOUTRVFUMUNUCUMUOUTNOVDUPURVCAMVDUPMAVBPVCASOUPUQUDQVAU
  OVDURTUEQULFUNKZIZBFHZKZIZJZUSFBLVHUOMZVKURMVLUSMUNUOVGRVMUMUNUFUNUOVGNOVKUQU
  RVJBMVKUQMBVIPVJBSOUQUPUGQVHUOVKURTUEQUJUH $.

  ${
    casef.f $e |- ( ph -> F : A --> X ) $.
    casef.g $e |- ( ph -> G : B --> X ) $.
    $( The "case" construction of two functions is a function on the disjoint
       union of their domains.  (Contributed by BJ, 10-Jul-2022.) $)
    casef $p |- ( ph -> case ( F , G ) : ( A |_| B ) --> X ) $=
      ( cdm wf cdju wfun cxp wss ffun syl crn frn syl2anc wceq cdjucase casefun
      cun caserel unssd xpss12 sylancr sstrid wa funssxp simplbi casedm djueq12
      ssid fdm eqtrid feq2d mpbid ) ADEUAZIZFUSJZBCKZFUSJAUSLZUSDIZEIZKZFMZNZVA
      ADEABFDJZDLGBFDOPACFEJZELHCFEOPUBAUSVFDQZEQZUCZMZVGDEUDAVFVFNVMFNVNVGNVFU
      NAVKVLFAVIVKFNGBFDRPAVJVLFNHCFERPUEVFVFVMFUFUGUHVCVHUIVAUTVFNVFFUSUJUKSAU
      TVBFUSAUTVFVBDEULAVDBTZVECTZVFVBTAVIVOGBFDUOPAVJVPHCFEUOPVDBVECUMSUPUQUR
      $.
  $}

  ${
    caseinj.r $e |- ( ph -> Fun `' R ) $.
    caseinj.s $e |- ( ph -> Fun `' S ) $.
    caseinj.disj $e |- ( ph -> ( ran R i^i ran S ) = (/) ) $.
    $( The "case" construction of two injective relations with disjoint ranges
       is an injective relation.  (Contributed by BJ, 10-Jul-2022.) $)
    caseinj $p |- ( ph -> Fun `' case ( R , S ) ) $=
      ( vy vx cinl ccnv ccom cinr cun wfun cdm c0 funeqi sylibr wss crn cin cvv
      cdjucase cv cop df-inl funmpt2 funcnvcnv ax-mp funco sylancr cnvco df-inr
      wceq c1o df-rn rncoss eqsstrri ss2in mp2an sseqtrid ss0 syl funun df-case
      syl21anc cnveqi cnvun eqtri ) ABIJZKZJZCLJZKZJZMZNZBCUCZJZNAVLNZVONZVLOZV
      OOZUAZPUNZVQAVJJZBJZKZNZVTAWFNZWGNWIINWJGUBPGUDUEIGUFUGIUHUIDWFWGUJUKVLWH
      BVJULQRAVMJZCJZKZNZWAAWKNZWLNWNLNWOHUBUOHUDUELHUMUGLUHUIEWKWLUJUKVOWMCVMU
      LQRAWDPSWEABTZCTZUAZWDPWBWPSWCWQSWDWRSWBVKTWPVKUPBVJUQURWCVNTWQVNUPCVMUQU
      RWBWPWCWQUSUTFVAWDVBVCVLVOVDVFVSVPVSVKVNMZJVPVRWSBCVEVGVKVNVHVIQR $.
  $}

  ${
    casef1.f $e |- ( ph -> F : A -1-1-> X ) $.
    casef1.g $e |- ( ph -> G : B -1-1-> X ) $.
    casef1.disj $e |- ( ph -> ( ran F i^i ran G ) = (/) ) $.
    $( The "case" construction of two injective functions with disjoint ranges
       is an injective function.  (Contributed by BJ, 10-Jul-2022.) $)
    casef1 $p |- ( ph -> case ( F , G ) : ( A |_| B ) -1-1-> X ) $=
      ( cdju cdjucase wf ccnv wfun wf1 f1f syl casef df-f1 simprbi sylanbrc
      caseinj ) ABCJZFDEKZLUDMNUCFUDOABCDEFABFDOZBFDLZGBFDPQACFEOZCFELZHCFEPQRA
      DEAUEDMNZGUEUFUIBFDSTQAUGEMNZHUGUHUJCFESTQIUBUCFUDSUA $.
  $}

  ${
    caseinl.f $e |- ( ph -> F Fn B ) $.
    caseinl.g $e |- ( ph -> Fun G ) $.
    caseinl.a $e |- ( ph -> A e. B ) $.
    $( Applying the "case" construction to a left injection.  (Contributed by
       Jim Kingdon, 15-Mar-2023.) $)
    caseinl $p |- ( ph -> ( case ( F , G ) ` ( inl ` A ) ) = ( F ` A ) ) $=
      ( cinl cfv ccnv cinr cima wfn cdm wceq wcel wfun cvv ax-mp vx cun df-case
      cdjucase ccom fveq1i cin c0 fnfun syl csn cxp djulf1o f1ocnv f1ofun funco
      wf1o sylancl dmco wrel cop df-inl funmpt2 funrel dfrel2 mpbi a1i imaeq12d
      cv eqtrid df-fn sylanbrc c1o djurf1o imacnvcnv eqtri djuin wa elexd f1odm
      fndm eleqtrrdi jctil funfvima sylc fvun1 syl112anc f1of ffvelcdmd sylancr
      f1ofn wf fvco2 f1ocnvfv1 fveq2d 3eqtrd ) ABIJZDEUDZJZWQDIKZUEZJZWQWTJZDJZ
      BDJAWSWQXAELKZUEZUBZJZXBWQWRXGDEUCUFAXAICMZNZXFLEOZMZNZXIXLUGUHPZWQXIQZXH
      XBPAXARZXAOZXIPXJADRZWTRZXPADCNZXRFCDUIUJUHUKSULZSWTUQZXSSYAIUQZYBUMSYAIU
      NTZYASWTUOTDWTUPURAXQWTKZDOZMXIDWTUSAYEIYFCYEIPZAIUTZYGIRZYHUASUHUAVIVAIU
      AVBVCZIVDTIVEVFVGAXTYFCPFCDWAUJVHVJXAXIVKVLAXFRZXFOZXLPZXMAERXERZYKGVMUKS
      ULZSXEUQZYNSYOLUQYPVNSYOLUNTYOSXEUOTEXEUPURYMAYLXEKXKMXLEXEUSLXKVOVPVGXFX
      LVKVLXNACXKVQVGAYIBIOZQZVRBCQXOAYRYIABSYQABCHVSZYCYQSPUMSYAIVTTWBYJWCHCBI
      WDWEXIXLXAXFWQWFWGVJAWTYANZWQYAQXBXDPYBYTYDYASWTWKTASYABISYAIWLZAYCUUAUMS
      YAIWHTVGYSWIYADWTWQWMWJAXCBDAYCBSQXCBPUMYSSYABIWNWJWOWP $.
  $}

  ${
    caseinr.f $e |- ( ph -> Fun F ) $.
    caseinr.g $e |- ( ph -> G Fn B ) $.
    caseinr.a $e |- ( ph -> A e. B ) $.
    $( Applying the "case" construction to a right injection.  (Contributed by
       Jim Kingdon, 12-Jul-2023.) $)
    caseinr $p |- ( ph -> ( case ( F , G ) ` ( inr ` A ) ) = ( G ` A ) ) $=
      ( cinr cfv ccnv cinl cdm cima wfn wceq wcel wfun cvv ax-mp vx cun df-case
      cdjucase ccom fveq1i cin csn cxp wf1o djulf1o f1ocnv f1ofun funco sylancl
      c0 dmco imacnvcnv eqtri a1i df-fn sylanbrc fnfun syl c1o djurf1o wrel cop
      cv df-inr funmpt2 funrel dfrel2 mpbi imaeq12d eqtrid djuin wa elexd f1odm
      fndm eleqtrrdi jctil funfvima sylc fvun2 syl112anc f1of ffvelcdmd sylancr
      f1ofn wf fvco2 f1ocnvfv1 fveq2d 3eqtrd ) ABIJZDEUDZJZWQEIKZUEZJZWQWTJZEJZ
      BEJAWSWQDLKZUEZXAUBZJZXBWQWRXGDEUCUFAXFLDMZNZOZXAICNZOZXJXLUGUPPZWQXLQZXH
      XBPAXFRZXFMZXJPZXKADRXERZXPFUPUHSUIZSXEUJZXSSXTLUJYAUKSXTLULTXTSXEUMTDXEU
      NUOXRAXQXEKXINXJDXEUQLXIURUSUTXFXJVAVBAXARZXAMZXLPXMAERZWTRZYBAECOZYDGCEV
      CVDVEUHSUIZSWTUJZYESYGIUJZYHVFSYGIULTZYGSWTUMTEWTUNUOAYCWTKZEMZNXLEWTUQAY
      KIYLCYKIPZAIVGZYMIRZYNUASVEUAVIVHIUAVJVKZIVLTIVMVNUTAYFYLCPGCEWAVDVOVPXAX
      LVAVBXNAXICVQUTAYOBIMZQZVRBCQXOAYRYOABSYQABCHVSZYIYQSPVFSYGIVTTWBYPWCHCBI
      WDWEXJXLXFXAWQWFWGVPAWTYGOZWQYGQXBXDPYHYTYJYGSWTWKTASYGBISYGIWLZAYIUUAVFS
      YGIWHTUTYSWIYGEWTWQWMWJAXCBEAYIBSQXCBPVFYSSYGBIWNWJWOWP $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Dominance and equinumerosity properties of disjoint union
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d A f g h $.  $d B f g h $.  $d C f g h $.  $d D f g h $.
    $( Dominance law for disjoint union.  (Contributed by Jim Kingdon,
       25-Jul-2022.) $)
    djudom $p |- ( ( A ~<_ B /\ C ~<_ D )
        -> ( A |_| C ) ~<_ ( B |_| D ) ) $=
      ( vf vg vh cdom wbr wa cv wf1 wex cinl cres cinr cvv wcel crn reldom cdju
      brdomi adantr ad2antlr cdjucase wf inlresf1 simplr sylancr inrresf1 simpr
      ccom f1co cin c0 rnco wss wceq f1rn resabs1 syl rneqd eqtrid 3syl ineq12d
      djuinr eqtrdi casef1 f1f brrelex1i ad3antrrr ad3antlr djuex syl2anc f1eq1
      fex spcegv sylc wb brrelex2i brdomg mpbird exlimddv ) ABHIZCDHIZJZABEKZLZ
      ACUAZBDUAZHIZEWDWHEMWEABEUBUCWFWHJZCDFKZLZWKFWEWNFMWDWHCDFUBUDWLWNJZWKWIW
      JGKZLZGMZWONBOZWGULZPDOZWMULZUEZQRZWIWJXCLZWRWOWIWJXCUFZWIQRZXDWOXEXFWOAC
      WTXBWJWOBWJWSLWHAWJWTLBDUGWFWHWNUHABWJWSWGUMUIWODWJXALWNCWJXBLBDUJWLWNUKZ
      CDWJXAWMUMUIWOWTSZXBSZUNNWGSZOZSZPWMSZOZSZUNUOWOXIXMXJXPWOXIWSXKOZSXMWSWG
      UPWOXQXLWOXKBUQZXQXLURWHXRWFWNABWGUSUDNXKBUTVAVBVCWOXJXAXNOZSXPXAWMUPWOXS
      XOWOWNXNDUQXSXOURXHCDWMUSPXNDUTVDVBVCVEXKXNVFVGVHZWIWJXCVIVAWOAQRZCQRZXGW
      DYAWEWHWNABHTVJVKWEYBWDWHWNCDHTVJVLACQQVMVNWIWJQXCVPVNXTWQXEGXCQWIWJWPXCV
      OVQVRWOBQRZDQRZWKWRVSZWDYCWEWHWNABHTVTVKWEYDWDWHWNCDHTVTVLYCYDJWJQRYEBDQQ
      VMWIWJQGWAVAVNWBWCWC $.
  $}

  ${
    $d G x z $.  $d x y z $.
    omp1eom.f $e |- F = ( x e. _om |->
      if ( x = (/) , ( inr ` x ) , ( inl ` U. x ) ) ) $.
    omp1eom.s $e |- S = ( x e. _om |-> suc x ) $.
    omp1eom.g $e |- G = case ( S , ( _I |` 1o ) ) $.
    $( Lemma for ~ omp1eom .  (Contributed by Jim Kingdon, 11-Jul-2023.) $)
    omp1eomlem $p |- F : _om -1-1-onto-> ( _om |_| 1o ) $=
      ( vz com c1o wtru c0 wceq cfv wcel wa adantl a1i adantr cvv cdju wf1o cif
      vy cv cinr cuni cinl el1o biimpri djurcl syl nnpredcl ad2antlr djulcl wdc
      wn nndceq0 ifcldadc cid cres cdjucase wf csuc peano2 fmpti wss f1oi ax-mp
      f1of 1onn omelon onelssi fss mp2an casef feq1i sylibr ffvelcdmda wrex wne
      wb wfn ffn mp1i wfun ffun simpl caseinl eqcomi simpr eqcomd fveq12d suceq
      fvmptg mpdan 3eqtr3d peano3 necomd neneqd simplr eqeq1d mtbird djune elvd
      eqnetrd elv neii simprr iftrued eqeq12d mtbiri 2falsed rexlimdvaa funmpt2
      fnresi caseinr fvresi eqtrd sylib 3eqtrd eqtr4d fveq2d 3eqtr4rd 2thd djur
      biimpi mpjaod simplll neqned nnsucpred syl2anc eqeq2d eqcom bitrdi vuniex
      wo iffalsed con0 nnon inl11 ad2antrl ad3antrrr bitr4d 3bitr4rd eqeqan12rd
      mpan2 suc11 nesymi exmiddc mpjaodan f1o2d mptru ) IIJUAZCUBKAUDIUUNAUEZLM
      ZUUOUFNZUUOUGZUHNZUCZUDUEZDNZCEKUUOIOZPZUUPUUQUUSUUNUVDUUPPUUOJOZUUQUUNOU
      UPUVEUVDUVEUUPUUOUIUJQIJUUOUKULUVDUUPUQZPUURIOZUUSUUNOUVCUVGKUVFUUOUMZUNI
      JUURUOULUVCUUPUPZKUUOURZQUSKUUNIUVADKUUNIBUTJVAZVBZVCUUNIDVCKIJBUVKIIIBVC
      ZKAIIUUOVDZBFUUOVEVFZRJIUVKVCZKJJUVKVCZJIVGZUVPJJUVKUBUVQJVHJJUVKVJVIZJIO
      UVRVKIJVLVMVIJJIUVKVNVORVPUUNIDUVLGVQVRVSUVCUVAUUNOZPZUUOUVBMZUVAUUTMZWBZ
      KUWAUUPUWDUVFUWAUUPPZUVAHUEZUHNZMZHIVTZUWDUVAUWFUFNZMZHJVTZUWEUWHUWDHIUWE
      UWFIOZUWHPZPZUWBUWCUWOUWBLUVBMUWOLUVBUWOUVBLUWNUVBLWAUWEUWNUVBUWFVDZLUWNU
      WGUVLNUWFBNZUVBUWPUWNUWFIBUVKUVMBIWCUWNUVOIIBWDWEUVQUVKWFUWNUVSJJUVKWGWEU
      WMUWHWHWIUWNUWGUVAUVLDUVLDMUWNDUVLGWJRUWNUVAUWGUWMUWHWKWLWMUWMUWQUWPMZUWH
      UWMUWPIOUWRUWFVEAUWFUVNUWPIIBUUOUWFWNFWOWPSWQZUWMUWPLWAUWHUWFWRSXFQWSWTUW
      OUUOLUVBUWAUUPUWNXAXBXCUWOUWCUWGUUQMUWGUUQUWGUUQWAZHUWFTOZUWTAUWFUUOTTXDX
      EXGXHUWOUVAUWGUUTUUQUWEUWMUWHXIUWEUUTUUQMZUWNUWEUUPUUQUUSUWAUUPWKXJZSXKXL
      XMXNUWEUWKUWDHJUWEUWFJOZUWKPZPZUWBUWCUXFUUOLUVBUWAUUPUXEXAZUXEUVBLMZUWEUX
      EUVBUWJUVLNZUWFLUXEUVAUWJDUVLDUVLMUXEGRUXDUWKWKZWMUXEUXIUWFUVKNZUWFUXEUWF
      JBUVKBWFUXEAIUVNBFXORUVKJWCUXEJXPRUXDUWKWHZXQUXDUXKUWFMUWKJUWFXRSXSUXEUXD
      UWFLMZUXLUWFUIXTZYAZQYBUXFUUQUWJUUTUVAUXFUUOUWFUFUXFUUOLUWFUXGUXEUXMUWEUX
      NQYBYCUWEUXBUXEUXCSUWEUXDUWKXIYDYEXNUVTUWIUWLYQZUVCUUPUVTUXPHIJUVAYFYGZUN
      YHUWAUVFPZUWIUWDUWLUXRUWHUWDHIUXRUWNPZUWPUURVDZMZUUOUWPMZUWCUWBUXSUYAUWPU
      UOMUYBUXSUXTUUOUWPUXSUVCUUOLWAUXTUUOMUVCUVTUVFUWNYIUXSUUOLUWAUVFUWNXAYJUU
      OYKYLYMUWPUUOYNYOUXSUWCUWFUURMZUYAUXSUWCUWGUUSMZUYCUXSUVAUWGUUTUUSUXRUWMU
      WHXIUXRUUTUUSMUWNUXRUUPUUQUUSUWAUVFWKYRZSXKUYDUYCWBZHUXAUURTOZUYFAYPZUWFU
      URTTUUAUUGXGYOUXSUWFYSOZUURYSOZUYAUYCWBUWMUYIUXRUWHUWFYTUUBUXSUVGUYJUVCUV
      GUVTUVFUWNUVHUUCUURYTULUWFUURUUHYLUUDUXSUVBUWPUUOUWNUVBUWPMUXRUWSQYMUUEXN
      UXRUWKUWDHJUXRUXEPZUWBUWCUYKUWBUUPUWAUVFUXEXAUYKUVBLUUOUXEUXHUXRUXOQYMXCU
      YKUWCUWJUUSMUUSUWJUYGUUSUWJWAZUYHUYGUYLHUURUWFTTXDXEVIUUIUXEUXRUVAUWJUUTU
      USUXJUYEUUFXLXMXNUVTUXPUVCUVFUXQUNYHUVCUUPUVFYQZUVTUVCUVIUYMUVJUUPUUJULSU
      UKQUULUUM $.
  $}

  ${
    $d x y $.
    $( Adding one to ` _om ` .  (Contributed by Jim Kingdon, 10-Jul-2023.) $)
    omp1eom $p |- ( _om |_| 1o ) ~~ _om $=
      ( vy vx com c1o cdju cvv wcel cv c0 wceq cinr cfv cuni cinl cif cmpt wf1o
      cen csuc cbvmptv wbr omex cid cdjucase eqeq1 fveq2 unieq fveq2d ifbieq12d
      cres suceq eqid omp1eomlem f1oeng mp2an ensymi ) CCDEZCFGCUQACAHZIJZURKLZ
      URMZNLZOZPZQCUQRUAUBBACURSZPZVDVFUCDUJUDZABCVCBHZIJZVHKLZVHMZNLZOURVHJZUS
      VIUTVBVJVLURVHIUEURVHKUFVMVAVKNURVHUGUHUITABCVEVHSURVHUKTVGULUMCUQFVDUNUO
      UP $.
  $}

  $( Reversing right and left operands of a disjoint union produces an
     equinumerous result.  (Contributed by Jim Kingdon, 10-Jul-2023.) $)
  endjusym $p |- ( ( A e. V /\ B e. W )
      -> ( A |_| B ) ~~ ( B |_| A ) ) $=
    ( wcel cinl cima cinr cun cen wbr cin c0 cres cvv f1ores mp2an f1oeng mpan2
    wf1o wa cdju wceq csn cxp wf1 wss djulf1o f1of1 ssv ensymd c1o djurf1o entr
    ax-mp syl2anc adantr adantl djuin a1i incom unen syl22anc djuun uncom eqtri
    eqtr3i 3brtr3g ) ACEZBDEZUAZFAGZHBGZIZHAGZFBGZIZABUBBAUBZJVKVLVOJKZVMVPJKZV
    LVMLMUCZVOVPLZMUCZVNVQJKVIVSVJVIVLAJKAVOJKZVSVIAVLVIAVLFANZTZAVLJKOMUDOUEZF
    UFZAOUGZWFOWGFTWHUHOWGFUIUOZAUJZOWGAFPQAVLCWERSUKVIAVOHANZTZWDOULUDOUEZHUFZ
    WIWMOWNHTWOUMOWNHUIUOZWKOWNAHPQAVOCWLRSVLAVOUNUPUQVKVMBJKBVPJKZVTVKBVMVJBVM
    JKZVIVJBVMHBNZTZWRWOBOUGZWTWPBUJZOWNBHPQBVMDWSRSURUKVJWQVIVJBVPFBNZTZWQWHXA
    XDWJXBOWGBFPQBVPDXCRSURVMBVPUNUPWAVKABUSUTWCVKVPVOLWBMVPVOVABAUSVGUTVLVOVMV
    PVBVCABVDVQVPVOIVRVOVPVEBAVDVFVH $.

  $( Equinumerosity of a set and its image under left injection.  (Contributed
     by Jim Kingdon, 30-Jul-2023.) $)
  eninl $p |- ( A e. V -> ( inl " A ) ~~ A ) $=
    ( wcel cinl cima csn cxp cen cres wf1o wbr djulf1or f1oeng mpan2 crn df-ima
    c0 wf1 wceq wa dff1o5 mpbi simpri eqtri breqtrrdi ensymd ) ABCZADAEZUGAQFAG
    ZUHHUGAUIDAIZJZAUIHKALZAUIBUJMNUHUJOZUIDAPAUIUJRZUMUISZUKUNUOTULAUIUJUAUBUC
    UDUEUF $.

  $( Equinumerosity of a set and its image under right injection.  (Contributed
     by Jim Kingdon, 30-Jul-2023.) $)
  eninr $p |- ( A e. V -> ( inr " A ) ~~ A ) $=
    ( wcel cinr cima c1o csn cxp cen cres wf1o wbr djurf1or f1oeng mpan2 df-ima
    crn wf1 wceq wa dff1o5 mpbi simpri eqtri breqtrrdi ensymd ) ABCZADAEZUGAFGA
    HZUHIUGAUIDAJZKZAUIILAMZAUIBUJNOUHUJQZUIDAPAUIUJRZUMUISZUKUNUOTULAUIUJUAUBU
    CUDUEUF $.

  ${
    $d A m n x y $.  $d B m n x y $.  $d F m n x y $.  $d G m $.  $d m n ph $.
    difinfsnlem.dc $e |- ( ph -> A. x e. A A. y e. A DECID x = y ) $.
    difinfsnlem.b $e |- ( ph -> B e. A ) $.
    difinfsnlem.f $e |- ( ph -> F : ( _om |_| 1o ) -1-1-> A ) $.
    difinfsnlem.fb $e |- ( ph -> ( F ` ( inr ` (/) ) ) =/= B ) $.
    difinfsnlem.g $e |- G = ( n e. _om |-> if ( ( F ` ( inl ` n ) ) = B ,
      ( F ` ( inr ` (/) ) ) , ( F ` ( inl ` n ) ) ) ) $.
    $( Lemma for ~ difinfsn .  The case where we need to swap ` B ` and
       ` ( inr `` (/) ) ` in building the mapping ` G ` .  (Contributed by Jim
       Kingdon, 9-Aug-2023.) $)
    difinfsnlem $p |- ( ph -> G : _om -1-1-> ( A \ { B } ) ) $=
      ( wceq wcel com wa c1o ad2antrr adantr vm cinl cfv cinr cif csn cdif wral
      cv c0 wi wf1 cdju wf f1f syl 0lt1o djurcl ffvelcdmd wne wn elsni necon3ai
      mp1i eldifd djulcl adantl con3i wdc eqeq12 dcbid rspc2gv syl2anc ifcldadc
      ralrimiva simplr simpr eqtr4d ad3antrrr ad2antrl simprr f1veqaeq syl12anc
      mpd wb inl11 ad3antlr mpbid a1d djune necomd sylancl neneqd mtod iffalsed
      iftrued eqeq12d mtbird pm2.21d wo exmiddc simprl sylbid sylibd ralrimivva
      mpjaodan adantrr 2fveq3 eqeq1d ifbieq2d f1mpt sylanbrc ) AFUIZUBUCZGUCZEN
      ZUJUDUCZGUCZXOUEZDEUFZUGZOZFPUHXSUAUIZUBUCZGUCZENZXRYEUEZNZXMYCNZUKZUAPUH
      FPUHPYAHULAYBFPAXMPOZQZXPXRXOYAAXRYAOYKXPAXRDXTAPRUMZDXQGAYMDGULZYMDGUNZK
      YMDGUOUPZUJROZXQYMOZAUQPRUJURZVDUSAXREUTXRXTOZVALYTXREXREVBVCUPVESYLXPVAZ
      QXODXTYLXODOZUUAYLYMDXNGAYOYKYPTYKXNYMOZAPRXMVFZVGUSZTUUAXOXTOZVAYLUUFXPX
      OEVBVHVGVEYLBUIZCUIZNZVIZCDUHBDUHZXPVIZAUUKYKITYLUUBEDOZUUKUULUKUUEAUUMYK
      JTUUJUULBCXOEDDUUGXONUUHENZQUUIXPUUGXOUUHEVJVKVLVMWDZVNVOAYJFUAPPAYKYCPOZ
      QZQZXPYJUUAUURXPQZYFYJYFVAZUUSYFQZYIYHUVAXNYDNZYIUVAXOYENZUVBUVAXOEYEUURX
      PYFVPUUSYFVQVRUVAYNUUCYDYMOZUVCUVBUKZAYNUUQXPYFKVSUURUUCXPYFYKUUCAUUPUUDV
      TZSUVAUUPUVDUURUUPXPYFAYKUUPWAZSPRYCVFZUPYMDXNYDGWBZWCWDUUQUVBYIWEZAXPYFX
      MYCPPWFZWGWHWIUUSUUTQZYHYIUVLYHXRYENZUVLUVMXQYDNZUVLXQYDUVLUUPYQXQYDUTUUR
      UUPXPUUTUVGSUQUUPYQQYDXQYCUJPRWJWKWLWMUVLYNYRUVDUVMUVNUKAYNUUQXPUUTKVSYQY
      RUVLUQYSVDUURUVDXPUUTUURUUPUVDUVGUVHUPZSYMDXQYDGWBWCWNUVLXSXRYGYEUVLXPXRX
      OUURXPUUTVPWPUVLYFXRYEUUSUUTVQWOWQWRWSUURYFUUTWTZXPUURYFVIZUVPUURUUKUVQAU
      UKUUQITUURYEDOUUMUUKUVQUKUURYMDYDGAYOUUQYPTUVOUSAUUMUUQJTUUJUVQBCYEEDDUUG
      YENUUNQUUIYFUUGYEUUHEVJVKVLVMWDYFXAUPZTXFUURUUAQZYFYJUUTUVSYFQZYHYIUVTYHX
      OXRNZUVTUWAXNXQNZUVTXNXQUVTYKYQXNXQUTUURYKUUAYFAYKUUPXBSUQXMUJPRWJWLWMUVT
      YNUUCYRUWAUWBUKAYNUUQUUAYFKVSUURUUCUUAYFUVFSYQYRUVTUQYSVDYMDXNXQGWBWCWNUV
      TXSXOYGXRUVTXPXRXOUURUUAYFVPWOUVTYFXRYEUVSYFVQWPWQWRWSUVSUUTQZYHUVBYIUWCY
      HUVCUVBUWCXSXOYGYEUWCXPXRXOUURUUAUUTVPWOUWCYFXRYEUVSUUTVQWOWQUWCYNUUCUVDU
      VEAYNUUQUUAUUTKVSUURUUCUUAUUTUVFSUURUVDUUAUUTUVOSUVIWCXCUUQUVJAUUAUUTUVKW
      GXDUURUVPUUAUVRTXFAYKXPUUAWTZUUPYLUULUWDUUOXPXAUPXGXFXEFUAPYAXSYGHMYIXPYF
      XOYEXRYIXOYEEXMYCGUBXHZXIUWEXJXKXL $.
  $}

  ${
    $d A a f g $.  $d A f n s x y $.  $d B a f g $.  $d B f n s x y $.
    $d a f x y $.
    $( An infinite set minus one element is infinite.  We require that the set
       has decidable equality.  (Contributed by Jim Kingdon, 8-Aug-2023.) $)
    difinfsn $p |- ( ( A. x e. A A. y e. A DECID x = y /\ _om ~<_ A /\ B e. A )
        -> _om ~<_ ( A \ { B } ) ) $=
      ( vg vf vs vn va cv wceq com wcel wf1 c1o syl wa cfv adantr cvv wral cdom
      wdc wbr w3a csn cdif wex cdju cen omp1eom simp2 endomtr sylancr brdomi c0
      cinr wn cinl cres ccom crn wss inlresf1 f1co mpan2 ad2antlr wf f1f sselda
      frnd simpllr simpr ax-mp fvco3 fvresd fveq2d eqtrd 3eqtr2rd simp-4r 0lt1o
      djulcl djurcl a1i f1veqaeq syl12anc mpd wne djune sylancl neneqd pm2.65da
      wi ralrimiva wfn wb ffnd eqeq1 notbid ralrn mpbird r19.21bi velsn sylnibr
      eldifd ex ssrdv f1ssr syl2anc omex fex f1eq1 spcegv mpcom cif cmpt simpl1
      simpl3 neqned eqid difinfsnlem mptex spcev wo ffvelcdmd jca dcbid rspc2gv
      eqeq12 exmiddc mpjaodan exlimddv reldom brrelex2i difexg 3ad2ant2 brdomg
      sylc ) AJZBJZKZUCZBCUAACUAZLCUBUDZDCMZUEZLCDUFZUGZUBUDZLUUHEJZNZEUHZUUFLO
      UIZCFJZNZUULFUUFUUMCUBUDZUUOFUHUUFUUMLUJUDUUDUUPUKUUCUUDUUEULUUMLCUMUNUUM
      CFUOPUUFUUOQZUPUQRZUUNRZDKZUULUUTURZUUQUUTQZLUUHUUNUSLUTZVAZNZUULUVBLCUVD
      NZUVDVBZUUHVCUVEUUOUVFUUFUUTUUOLUUMUVCNZUVFLOVDZLUUMCUUNUVCVEVFVGZUVBGUVG
      UUHUVBGJZUVGMZUVKUUHMUVBUVLQZUVKCUUGUVBUVGCUVKUVBLCUVDUVBUVFLCUVDVHUVJLCU
      VDVIPZVKVJUVMUVKDKZUVKUUGMUVBUVOURZGUVGUVBUVPGUVGUAZHJZUVDRZDKZURZHLUAZUV
      BUWAHLUVBUVRLMZQZUVTUVRUSRZUURKZUWDUVTQZUWEUUNRZUUSKZUWFUWGUUSDUVSUWHUUQU
      UTUWCUVTVLUWDUVTVMUWDUVSUWHKUVTUWDUVSUVRUVCRZUUNRZUWHUWDLUUMUVCVHZUWCUVSU
      WKKUVHUWLUVILUUMUVCVIVNUVBUWCVMZLUUMUVRUUNUVCVOUNUWDUWJUWEUUNUWDUVRLUSUWM
      VPVQVRSVSUWGUUOUWEUUMMZUURUUMMZUWIUWFWMUUFUUOUUTUWCUVTVTUWCUWNUVBUVTLOUVR
      WBVGUWOUWGUPOMZUWOWALOUPWCVNZWDUUMCUWEUURUUNWEWFWGUWGUWEUURUWGUWCUWPUWEUU
      RWHUWDUWCUVTUWMSWAUVRUPLOWIWJWKWLWNUVBUVDLWOUVQUWBWPUVBLCUVDUVNWQUVPUWAGH
      LUVDUVKUVSKUVOUVTUVKUVSDWRWSWTPXAXBGDXCXDXEXFXGLCUUHUVDXHXIUVDTMZUVEUULUV
      ELUUHUVDVHLTMUWRLUUHUVDVIXJLUUHTUVDXKWJUUKUVEEUVDTLUUHUUJUVDXLXMXNPUUQUVA
      QZLUUHILIJUSRUUNRZDKUUSUWTXOZXPZNZUULUWSABCDIUUNUXBUUQUUCUVAUUCUUDUUEUUOX
      QZSUUQUUEUVAUUCUUDUUEUUOXRZSUUQUUOUVAUUFUUOVMZSUWSUUSDUUQUVAVMXSUXBXTYAUU
      KUXCEUXBILUXAXJYBLUUHUUJUXBXLYCPUUQUUTUCZUUTUVAYDUUQUUSCMZUUEQUUCUXGUUQUX
      HUUEUUQUUMCUURUUNUUQUUOUUMCUUNVHUXFUUMCUUNVIPUWOUUQUWQWDYEUXEYFUXDUUBUXGA
      BUUSDCCYSUUSKYTDKQUUAUUTYSUUSYTDYIYGYHYRUUTYJPYKYLUUFUUHTMZUUIUULWPUUDUUC
      UXIUUEUUDCTMUXILCUBYMYNCUUGTYOPYPLUUHTEYQPXA $.
  $}

  ${
    $d A u v w x y $.  $d B u v w $.
    $( An infinite set minus a finite subset is infinite.  We require that the
       set has decidable equality.  (Contributed by Jim Kingdon,
       8-Aug-2023.) $)
    difinfinf $p |- ( ( ( A. x e. A A. y e. A DECID x = y /\ _om ~<_ A )
        /\ ( B C_ A /\ B e. Fin ) ) -> _om ~<_ ( A \ B ) ) $=
      ( vw vu vv cv wceq wral com cdom wbr wa wss wcel cdif c0 difeq2 breq2d wi
      wdc cfn csn simplr dif0 breqtrrdi difss ssralv ax-mp ralimi mpsyl ad5antr
      cun simpr simprl ad3antrrr simplrr ssdif sseld difinfsn syl3anc difun1 ex
      sylc simprr findcard2sd ) AHBHIUBZBCJZACJZKCLMZNZDCOZDUCPZNZNZKCEHZQZLMKC
      RQZLMKCFHZQZLMZKCVTGHZUDZUNZQZLMZKCDQZLMEFGDVQRIVRVSKLVQRCSTVQVTIVRWAKLVQ
      VTCSTVQWEIVRWFKLVQWECSTVQDIVRWHKLVQDCSTVPKCVSLVJVKVOUECUFUGVPVTUCPZNZVTDO
      ZWCDVTQZPZNZNZWBWGWOWBNZKWAWDQZWFLWPVHBWAJZAWAJZWBWCWAPZKWQLMVJWSVKVOWIWN
      WBWACOZVJWRACJWSCVTUHZVIWRACXAVIWRUAXBVHBWACUIUJUKWRAWACUIULUMWOWBUOWPVMW
      MWTVPVMWIWNWBVLVMVNUPUQWJWKWMWBURVMWLWAWCDCVTUSUTVEABWAWCVAVBCVTWDVCUGVDV
      LVMVNVFVG $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Older definition temporarily kept for comparison, to be deleted
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c |_|d $.

  $( Syntax for the domain-disjoint-union of two relations. $)
  cdjud $a class ( R |_|d S ) $.

  $( The "domain-disjoint-union" of two relations: if ` R C_ ( A X. X ) ` and
     ` S C_ ( B X. X ) ` are two binary relations, then ` ( R |_|d S ) ` is the
     binary relation from ` ( A |_| B ) ` to ` X ` having the universal
     property of disjoint unions (see ~ updjud in the case of functions).

     Remark: the restrictions to ` dom R ` (resp. ` dom S ` ) are not necessary
     since extra stuff would be thrown away in the post-composition with ` R `
     (resp. ` S ` ), as in ~ df-case , but they are explicitly written for
     clarity.  (Contributed by MC and BJ, 10-Jul-2022.) $)
  df-djud $a |- ( R |_|d S ) =
            ( ( R o. `' ( inl |` dom R ) ) u. ( S o. `' ( inr |` dom S ) ) ) $.

  ${
    djufun.f $e |- ( ph -> Fun F ) $.
    djufun.g $e |- ( ph -> Fun G ) $.
    $( The "domain-disjoint-union" of two functions is a function.
       (Contributed by BJ, 10-Jul-2022.) $)
    djufun $p |- ( ph -> Fun ( F |_|d G ) ) $=
      ( cdm cres ccnv ccom wfun cin c0 wceq wf1 wf df-f1 simprbi mp1i funco wss
      cinl cinr cun cdjud cdju inlresf1 syl2anc inrresf1 crn dmcoss df-rn ss2in
      sseqtrri mp2an djuinr a1i sseqtrid ss0 syl syl21anc df-djud funeqi sylibr
      funun ) ABUABFZGZHZIZCUBCFZGZHZIZUCZJZBCUDZJAVHJZVLJZVHFZVLFZKZLMZVNABJVG
      JZVPDVEVEVIUEZVFNZWBAVEVIUFWDVEWCVFOWBVEWCVFPQRBVGSUGACJVKJZVQEVIWCVJNZWE
      AVEVIUHWFVIWCVJOWEVIWCVJPQRCVKSUGAVTLTWAAVFUIZVJUIZKZVTLVRWGTVSWHTVTWITVR
      VGFWGBVGUJVFUKUMVSVKFWHCVKUJVJUKUMVRWGVSWHULUNWILMAVEVIUOUPUQVTURUSVHVLVD
      UTVOVMBCVAVBVC $.
  $}

  $( The domain of the "domain-disjoint-union" is the disjoint union of the
     domains.  Remark: its range is the (standard) union of the ranges.
     (Contributed by BJ, 10-Jul-2022.) $)
  djudm $p |- dom ( F |_|d G ) = ( dom F |_| dom G ) $=
    ( cdjud cdm cinl cres ccnv ccom cinr cun cdju df-djud cima imacnvcnv resima
    crn dmco df-ima eqtri 3eqtri dmeqi dmun uneq12i djuunr ) ABCZDAEADZFZGZHZBI
    BDZFZGZHZJZDUIDZUMDZJZUFUJKZUEUNABLUAUIUMUBUQUGPZUKPZJURUOUSUPUTUOUHGUFMUGU
    FMZUSAUHQUGUFNVAEUFMUSEUFOEUFRSTUPULGUJMUKUJMZUTBULQUKUJNVBIUJMUTIUJOIUJRST
    UCUFUJUDST $.

  ${
    djuinj.r $e |- ( ph -> Fun `' R ) $.
    djuinj.s $e |- ( ph -> Fun `' S ) $.
    djuinj.disj $e |- ( ph -> ( ran R i^i ran S ) = (/) ) $.
    $( The "domain-disjoint-union" of two injective relations with disjoint
       ranges is an injective relation.  (Contributed by BJ, 10-Jul-2022.) $)
    djuinj $p |- ( ph -> Fun `' ( R |_|d S ) ) $=
      ( cA cdm cres ccnv ccom cun wfun cin c0 ax-mp funeqi sylibr wss crn cdjud
      cinl cinr wceq cdju inlresf1 f1fun funcnvcnv funco sylancr cnvco inrresf1
      wf1 df-rn rncoss eqsstrri ss2in mp2an sseqtrid ss0 funun syl21anc df-djud
      syl cnveqi cnvun eqtri ) ABUBBHZIZJZKZJZCUCCHZIZJZKZJZLZMZBCUAZJZMAVLMZVQ
      MZVLHZVQHZNZOUDZVSAVJJZBJZKZMZWBAWHMZWIMWKVIMZWLVHVHGUEZVIUMWMVHGUFVHWNVI
      UGPVIUHPDWHWIUIUJVLWJBVJUKQRAVOJZCJZKZMZWCAWOMZWPMWRVNMZWSVMGVMUEZVNUMWTG
      VMULVMXAVNUGPVNUHPEWOWPUIUJVQWQCVOUKQRAWFOSWGABTZCTZNZWFOWDXBSWEXCSWFXDSW
      DVKTXBVKUNBVJUOUPWEVPTXCVPUNCVOUOUPWDXBWEXCUQURFUSWFUTVDVLVQVAVBWAVRWAVKV
      PLZJVRVTXEBCVCVEVKVPVFVGQR $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Countable sets
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d f y z w $.
    $( The empty set is countable.  Remark of [BauerSwan], p. 14:3 which also
       has the definition of countable used here.  (Contributed by Jim Kingdon,
       13-Mar-2023.) $)
    0ct $p |- E. f f : _om -onto-> ( (/) |_| 1o ) $=
      ( vy vz vw com c0 c1o cdju cinr cfv csn wfo cv wceq wrex wcel ax-mp sylib
      peano1 fveq2 cxp wf wral 0lt1o djurcl fconst6 cinl wn rex0 wo djur biimpi
      wex ord mpi df1o2 rexeqi 0ex eqeq2d rexsn elexi fvconst2 eqtr4di rspceeqv
      sylancr rgen dffo3 mpbir2an omex snex xpex foeq1 spcev ) EFGHZEFIJZKZUAZL
      ZEVNAMZLZAUMVREVNVQUBBMZCMZVQJZNCEOZBVNUCEVOVNFGPVOVNPUDFGFUEQZUFWDBVNWAV
      NPZFEPZWAFVQJZNWDSWFWAVOWHWFWADMZIJZNZDFKZOZWAVONZWFWKDGOZWMWFWAWIUGJNZDF
      OZUHWOWPDUIWFWQWOWFWQWOUJDFGWAUKULUNUOWKDGWLUPUQRWKWNDFURWIFNWJVOWAWIFITU
      SUTRWGWHVONSEVOFVOVNWEVAZVBQVCCFEWCWHWAWBFVQTVDVEVFCBEVNVQVGVHVTVRAVQEVPV
      IVOWRVJVKEVNVSVQVLVMQ $.
  $}

  ${
    $d A f g n $.  $d A f n u w y z $.  $d f n u w x y $.
    $( Lemma for ~ ctm .  One of the directions of the biconditional.
       (Contributed by Jim Kingdon, 16-Mar-2023.) $)
    ctmlemr $p |- ( E. x x e. A -> ( E. f f : _om -onto-> A
        -> E. f f : _om -onto-> ( A |_| 1o ) ) ) $=
      ( vg vn vy vz vw vu cv wcel com c1o wa c0 wceq cfv cinl fveq2d eqtr4d wex
      wfo cdju cinr cuni cif cmpt wf wrex 0lt1o djurcl ax-mp a1i wn simpllr fof
      syl nnpredcl ad2antlr ffvelcdmd djulcl wdc nndceq0 adantl ifcldadc fmpttd
      wral simprl foelrn syl2anc csuc peano2 ad2antrl wtr word nnord ordtr 3syl
      simplrr wb unisucg mpbid simprr peano3 iffalsed eqid eqeq1 unieq ifbieq2d
      neneqd eqeltrrd fvmptd3 fveq2 rspceeqv rexlimdvaa peano1 el1o sylib eqtrd
      rexlimddv iftruei eqtr4di eqeltri fvmptg sylancr wo djur biimpi ralrimiva
      mpjaod dffo3 sylanbrc omex mptex foeq1 spcev ex exlimdv cbvexv imbitrrdi
      ) AJBKAUAZLBCJZUBZCUALBMUCZDJZUBZDUAZLYDYBUBZCUAYAYCYGCYAYCYGYAYCNZLYDELE
      JZOPZOUDQZYJUEZYBQZRQZUFZUGZUBZYGYILYDYQUHFJZGJZYQQZPGLUIZFYDVGYRYIELYPYD
      YIYJLKZNZYKYLYOYDYLYDKZUUDYKNOMKUUEUJBMOUKULZUMUUDYKUNZNZYNBKYOYDKUUHLBYM
      YBUUHYCLBYBUHYAYCUUCUUGUOLBYBUPUQUUCYMLKYIUUGYJURUSUTBMYNVAUQUUCYKVBYIYJV
      CVDVEVFYIUUBFYDYIYSYDKZNZYSHJZRQZPZHBUIZUUBYSUUKUDQZPZHMUIZUUJUUMUUBHBUUJ
      UUKBKZUUMNZNZUUKIJZYBQZPZUUBILUUTYCUURUVCILUIYAYCUUIUUSUOUUJUURUUMVHILBUU
      KYBVIVJUUTUVALKZUVCNZNZUVAVKZLKZYSUVGYQQZPUUBUVDUVHUUTUVCUVAVLVMZUVFYSUVG
      OPZYLUVGUEZYBQZRQZUFZUVIUVFYSUVNUVOUVFYSUULUVNUUJUURUUMUVEVSUVFUVMUUKRUVF
      UVMUVBUUKUVFUVLUVAYBUVFUVAVNZUVLUVAPZUVFUVDUVAVOUVPUUTUVDUVCVHUVAVPUVAVQV
      RUVDUVPUVQVTUUTUVCUVALWAVMWBSUUTUVDUVCWCTSTUVFUVKYLUVNUVDUVKUNUUTUVCUVDUV
      GOUVAWDWJVMWETZUVFEUVGYPUVOLYQYDYQWFZYJUVGPZYKUVKYOUVNYLYJUVGOWGUVTYNUVMR
      UVTYMUVLYBYJUVGWHSSWIUVJUVFYSUVOYDUVRYIUUIUUSUVEUOWKWLTGUVGLUUAUVIYSYTUVG
      YQWMWNVJWTWOUUJUUPUUBHMUUJUUKMKZUUPNNZOLKZYSOYQQZPUUBWPUWBYSOOPZYLOUEZYBQ
      ZRQZUFZUWDUWBYSYLUWIUWBYSUUOYLUUJUWAUUPWCUWBUUKOUDUWBUWAUUKOPUUJUWAUUPVHU
      UKWQWRSWSUWEYLUWHOWFXAZXBUWBUWCUWIYDKZUWDUWIPWPUWKUWBUWIYLYDUWJUUFXCUMEOY
      PUWILYDYQYKYKUWEYOUWHYLYJOOWGYKYNUWGRYKYMUWFYBYJOWHSSWIUVSXDXETGOLUUAUWDY
      SYTOYQWMWNXEWOUUIUUNUUQXFZYIUUIUWLHBMYSXGXHVDXJXIGFLYDYQXKXLYFYRDYQELYPXM
      XNLYDYEYQXOXPUQXQXRYHYFCDLYDYBYEXOXSXT $.
  $}

  ${
    $d A f g x $.  $d A f x y z $.
    $( Two equivalent definitions of countable for an inhabited set.  Remark of
       [BauerSwan], p. 14:3.  (Contributed by Jim Kingdon, 13-Mar-2023.) $)
    ctm $p |- ( E. x x e. A -> ( E. f f : _om -onto-> ( A |_| 1o )
        <-> E. f f : _om -onto-> A ) ) $=
      ( vg vy vz cv wcel wex com c1o wfo wa cvv wfun wf mp1i syl2anc cfv wceq
      cdju wi cid cres csn cxp cdjucase ccom wf1o f1oi f1of fconst6g casef ffun
      adantr syl vex a1i cofunexg wrex wral fof adantl fco simplr djulcl foelrn
      cinl wfn fofn ad4antlr fvco2 simpr fveq2d fconst6 simpllr caseinl 3eqtr2d
      fnresi fvresi eqtr2d ex reximdva mpd ralrimiva dffo3 sylanbrc spcegv sylc
      foeq1 exlimiv exlimdv cbvexv imbitrrdi ctmlemr impbid ) AGZBHZAIZJBKUAZCG
      ZLZCIZJBXALZCIZWSXCJBDGZLZDIZXEWSXBXHCWRXBXHUBAWRXBXHWRXBMZUCBUDZKWQUEUFZ
      UGZXAUHZNHZJBXMLZXHXIXLOZXANHZXNXIWTBXLPZXPXIBKXJXKBBBXJUIBBXJPXIBUJBBXJU
      KQWRKBXKPXBKWQBULUOUMZWTBXLUNUPXQXICUQURXLXANUSRXIJBXMPZEGZFGZXMSZTZFJUTZ
      EBVAXOXIXRJWTXAPZXTXSXBYFWRJWTXAVBVCJWTBXLXAVDRXIYEEBXIYABHZMZYAVHSZYBXAS
      ZTZFJUTZYEYHXBYIWTHZYLWRXBYGVEYGYMXIBKYAVFVCFJWTYIXAVGRYHYKYDFJYHYBJHZMZY
      KYDYOYKMZYCYAXJSZYAYPYCYJXLSZYIXLSYQYPXAJVIZYNYCYRTXBYSWRYGYNYKJWTXAVJVKY
      HYNYKVEJXLXAYBVLRYPYIYJXLYOYKVMVNYPYABXJXKXJBVIYPBVSURKNXKPXKOYPKWQNAUQVO
      KNXKUNQXIYGYNYKVPZVQVRYPYGYQYATYTBYAVTUPWAWBWCWDWEFEJBXMWFWGXGXODXMNJBXFX
      MWJWHWIWBWKWLXDXGCDJBXAXFWJWMWNABCWOWP $.
  $}

  ${
    $d A g m $.  $d A m x y z $.  $d F g m $.  $d F m x y z $.  $d S g m $.
    $d S m n $.  $d S m x y z $.  $d m ph x y z $.
    ctssdclemn0.ss $e |- ( ph -> S C_ _om ) $.
    ctssdclemn0.dc $e |- ( ph -> A. n e. _om DECID n e. S ) $.
    ctssdclemn0.f $e |- ( ph -> F : S -onto-> A ) $.
    ctssdclemn0.n0 $e |- ( ph -> -. (/) e. S ) $.
    $( Lemma for ~ ctssdc .  The ` -. (/) e. S ` case.  (Contributed by Jim
       Kingdon, 16-Aug-2023.) $)
    ctssdclemn0 $p |- ( ph -> E. g g : _om -onto-> ( A |_| 1o ) ) $=
      ( vm vy com c1o wcel cfv cinl c0 wceq wa vx vz cdju cinr cif cmpt wfo wex
      cv wf wrex wral ad2antrr fof simpr ffvelcdmd djulcl wn 0lt1o djurcl ax-mp
      syl a1i wdc eleq1 adantr rspcdva ifcldadc fmpttd ad3antrrr simplr syl2anc
      dcbid foelrn iftrued eqid 2fveq3 ifbieq1d ad5antr eqeltrd fvmptd3 simpllr
      wss sseldd fveq2d eqtrd 3eqtr4rd ex reximdva mpd ssrexv rexlimdva2 peano1
      wi iffalsed eqeltrdi el1o sylib fveq2 rspceeqv wo biimpi adantl ralrimiva
      djur mpjaod dffo3 sylanbrc omex mptex foeq1 spcev ) AMBNUCZKMKUIZCOZXNFPZ
      QPZRUDPZUEZUFZUGZMXMDUIZUGZDUHAMXMXTUJUAUIZLUIZXTPZSZLMUKZUAXMULYAAKMXSXM
      AXNMOZTZXOXQXRXMYJXOTZXPBOXQXMOYKCBXNFYKCBFUGZCBFUJZAYLYIXOIUMCBFUNZVBYJX
      OUOUPBNXPUQVBXRXMOZYJXOURTRNOYOUSBNRUTVAZVCYJEUIZCOZVDZXOVDEMXNYQXNSYRXOY
      QXNCVEVMAYSEMULYIHVFAYIUOVGVHVIAYHUAXMAYDXMOZTZYDUBUIZQPZSZUBBUKZYHYDUUBU
      DPZSZUBNUKZUUAUUDYHUBBUUAUUBBOZTZUUDTZYGLCUKZYHUUKUUBYEFPZSZLCUKZUULUUKYL
      UUIUUOAYLYTUUIUUDIVJUUAUUIUUDVKLCBUUBFVNVLUUKUUNYGLCUUKYECOZTZUUNYGUUQUUN
      TZUUPUUMQPZXRUEZUUSYFYDUURUUPUUSXRUUKUUPUUNVKZVOZUURKYEXSUUTMXTXMXTVPZXNY
      ESXOUUPXQUUSXRXNYECVEXNYEQFVQVRUURCMYEACMWCZYTUUIUUDUUPUUNGVSUVAWDUURUUTU
      USXMUVBUURUUMBOUUSXMOUURCBYEFAYMYTUUIUUDUUPUUNAYLYMIYNVBVSUVAUPBNUUMUQVBV
      TWAUURYDUUCUUSUUJUUDUUPUUNWBUURUUBUUMQUUQUUNUOWEWFWGWHWIWJAUULYHWNZYTUUIU
      UDAUVDUVEGYGLCMWKVBVJWJWLUUAUUGYHUBNUUAUUBNOZTZUUGTZRMOZYDRXTPZSYHUVIUVHW
      MVCZUVHRCOZRFPQPZXRUEZXRUVJYDUVHUVLUVMXRAUVLURYTUVFUUGJVJWOZUVHKRXSUVNMXT
      XMUVCXNRSXOUVLXQUVMXRXNRCVEXNRQFVQVRUVKUVHUVNXRXMUVOYPWPWAUVHYDUUFXRUVGUU
      GUOUVHUUBRUDUVHUVFUUBRSUUAUVFUUGVKUUBWQWRWEWFWGLRMYFUVJYDYERXTWSWTVLWLYTU
      UEUUHXAZAYTUVPUBBNYDXEXBXCXFXDLUAMXMXTXGXHYCYADXTKMXSXIXJMXMYBXTXKXLVB $.
  $}

  ${
    $d A m x y z $.  $d F m n x y z $.  $d G m n y $.  $d S m n y z $.
    $d ph m n y z $.
    ctssdccl.f $e |- ( ph -> F : _om -onto-> ( A |_| 1o ) ) $.
    ctssdccl.s $e |- S = { x e. _om | ( F ` x ) e. ( inl " A ) } $.
    ctssdccl.g $e |- G = ( `' inl o. F ) $.
    $( A mapping from a decidable subset of the natural numbers onto a
       countable set.  This is similar to one direction of ~ ctssdc but
       expressed in terms of classes rather than ` E. ` .  (Contributed by Jim
       Kingdon, 30-Oct-2023.) $)
    ctssdccl $p |- ( ph -> ( S C_ _om /\ G : S -onto-> A
        /\ A. n e. _om DECID n e. S ) ) $=
      ( vm vy com wcel cfv cinl wceq cvv c1o wa vz wss wfo wdc wral cima ssrab2
      cv crab eqsstri a1i wf wrex wfn wfun cdm ccnv csn cxp wf1o djulf1o f1ocnv
      c0 f1ofun mp2b cdju fofun syl ccom funco funeqi sylibr sylancr fof eleq2d
      fdmd anbi1d wb dmcoss sseli pm4.71ri dmfco pm5.32da bitrid cinr simpr crn
      imassrn adantl df-rn eleq2i sylib 2thd cin djuin disjel mpan con2i eleq2s
      wn dfrn4 sylnib 2falsed cun wo ffvelcdmda djuun elun mpjaodan 3bitr4d weq
      dmeqi fveq2 eleq1d elrab2 3bitr4g eqrdv df-fn fveq1i adantr biimpi simpld
      sylanbrc fvco3 syl2anc eqtrid ax-mp fvelima simprr fveq2d f1ocnvfv1 mp2an
      vex simprl eqeltrrd ralrimiva wex df-rex ex mpd simpl2im eqeltrid eqeltrd
      rexlimddv ffnfv djulcl foelrn sylan sylan2 f1odm eleqtrri funfvima elrabd
      ad2antlr eleqtrrdi jca eximdv eqtr3id ad2antrr eqtr4d dffo3 bitr3i orim2i
      wi reximdva df-dc ibar bitr4di dcbid mpbid 3jca ) ADMUBZDCGUCZEUHZDNZUDZE
      MUEUVLADBUHZFOZPCUFZNZBMUIZMIUVTBMUGUJZUKADCGULZKUHZLUHZGOZQZLDUMZKCUEUVM
      AGDUNZUWDGOZCNZKDUEUWCAGUOZGUPZDQUWIAPUQZUOZFUOZUWLRVCURRUSZPUTZUWQRUWNUT
      UWOVARUWQPVBUWQRUWNVDVEAMCSVFZFUCZUWPHMUWSFVGVHZUWOUWPTUWNFVIZUOUWLUWNFVJ
      GUXBJVKVLVMAEUWMDAUVNUXBUPZNZUVNMNZUVNFOZUVSNZTZUVNUWMNUVOAUVNFUPZNZUXFUW
      NUPZNZTZUXEUXLTUXDUXHAUXJUXEUXLAUXIMUVNAMUWSFAUWTMUWSFULZHMUWSFVNVHZVPVOV
      QAUWPUXDUXMVRUXAUXDUXJUXDTUWPUXMUXDUXJUXCUXIUVNUWNFVSVTWAUWPUXJUXDUXLUVNU
      WNFWBWCWDVHAUXEUXGUXLAUXETZUXGUXGUXLVRUXFWESUFZNZUXPUXGTZUXGUXLUXPUXGWFUX
      SUXFPWGZNZUXLUXGUYAUXPUVSUXTUXFPCWHVTWIUXTUXKUXFPWJWKZWLWMUXPUXRTZUXGUXLU
      XRUXGWTZUXPUXGUXRUVSUXQWNVCQUXGUXRWTZCSWOUVSUXQUXFWPWQWRZWIUYCUYAUXLUXRUY
      AWTUXPUYAUXRUYEUXFPRUFZUXTUYGUXQWNVCQUXFUYGNUYERSWOUYGUXQUXFWPWQPXAWSWRWI
      UYBXBXCUXPUXFUVSUXQXDZNZUXGUXRXEZUXPUXFUWSNZUYIAMUWSUVNFUXOXFZUYHUWSUXFCS
      XGWKZVLUXFUVSUXQXHZWLXIWCXJUWMUXCUVNGUXBJXLWKUVTUXGBUVNMDBEXKUVRUXFUVSUVQ
      UVNFXMXNIXOZXPXQGDXRYCAUWKKDAUWDDNZTZUWJUWDFOZUWNOZCUYQUWJUWDUXBOZUYSUWDG
      UXBJXSUYQUXNUWDMNZUYTUYSQAUXNUYPUXOXTUYQVUAUYRUVSNZUYPVUAVUBTZAUYPVUCUVTV
      UBBUWDMDBKXKUVRUYRUVSUVQUWDFXMXNIXOYAWIZYBMUWSUWDUWNFYDYEYFUYQUAUHZPOZUYR
      QZUYSCNUACUYQVUAVUBVUGUACUMZVUDPUOZVUBVUHUWRVUIVARUWQPVDYGZUAUYRCPYHWQUUA
      UYQVUECNZVUGTTZVUFUWNOZUYSCVULVUFUYRUWNUYQVUKVUGYIYJVULVUMVUECUWRVUERNVUM
      VUEQVAUAYMRUWQVUEPYKYLUYQVUKVUGYNUUBYOUUDUUCYPKDCGUUEYCAUWHKCAUWDCNZTZUWD
      POZUWEFOZQZLDUMZUWHVUOUWEDNZVURTZLYQZVUSVUOUWEMNZVURTZLYQZVVBVUNAVUPUWSNZ
      VVECSUWDUUFAVVFTVURLMUMZVVEAUWTVVFVVGHLMUWSVUPFUUGUUHVURLMYRWLUUIVUOVVDVV
      ALVUOVVDVVAVUOVVDTZVUTVURVVHUWEUWADVVHUVTVUQUVSNBUWEMBLXKUVRVUQUVSUVQUWEF
      XMXNVUOVVCVURYNVVHVUPVUQUVSVUOVVCVURYIZVUNVUPUVSNZAVVDVUIUWDPUPZNVUNVVJUV
      DVUJUWDRVVKKYMZUWRVVKRQVARUWQPUUJYGUUKCUWDPUULYLUUNYOUUMIUUOVVIUUPYSUUQYT
      VURLDYRVLVUOVURUWGLDVUOVUTTZVURUWGVVMVURTZUWDVUQUWNOZUWFVVNUWDVUPUWNOZVVO
      UWRUWDRNVVPUWDQVAVVLRUWQUWDPYKYLVVNVUPVUQUWNVVMVURWFYJUURVVMUWFVVOQVURVVM
      UWFUWEUXBOZVVOUWEGUXBJXSVVMUXNVVCVVQVVOQAUXNVUNVUTUXOUUSVUTVVCVUODMUWEUWB
      VTWIMUWSUWEUWNFYDYEYFXTUUTYSUVEYTYPLKDCGUVAYCAUVPEMUXPUXGUDZUVPUXPUXGUYDX
      EZVVRUXPUYJVVSUXPUYKUYJUYLUYKUYIUYJUYMUYNUVBWLUXRUYDUXGUYFUVCVHUXGUVFVLUX
      PUXGUVOUXPUXGUXHUVOUXEUXGUXHVRAUXEUXGUVGWIUYOUVHUVIUVJYPUVK $.
  $}

  ${
    $d A f g s t $.  $d A g n s t x y z $.
    $( Lemma for ~ ctssdc .  Showing that our usual definition of countable
       implies the alternate one.  (Contributed by Jim Kingdon,
       16-Aug-2023.) $)
    ctssdclemr $p |- ( E. f f : _om -onto-> ( A |_| 1o )
        -> E. s ( s C_ _om /\ E. f f : s -onto-> A
        /\ A. n e. _om DECID n e. s ) ) $=
      ( vg vt com cv wfo wex wss wcel wdc wral w3a foeq1 cinl eqid cvv wf1o c1o
      cdju cbvexv cfv cima crab ccnv ccom id ctssdccl c0 csn cxp djulf1o f1ocnv
      wfun f1ofun mp2b cofunexg mp2an spcev 3anim2i omex rabex wceq sseq1 foeq2
      vex syl exbidv eleq2 dcbid ralbidv 3anbi123d exlimiv sylbi ) GAUAUBZBHZIZ
      BJGVQEHZIZEJDHZGKZWBAVRIZBJZCHZWBLZMZCGNZOZDJZVSWABEGVQVRVTPUCWAWKEWAFHVT
      UDQAUELZFGUFZGKZWMAVRIZBJZWFWMLZMZCGNZOZWKWAWNWMAQUGZVTUHZIZWSOWTWAFAWMCV
      TXBWAUIWMRXBRUJXCWPWNWSWOXCBXBXAUPZVTSLXBSLSUKULSUMZQTXESXATXDUNSXEQUOXES
      XAUQUREVHXAVTSUSUTWMAVRXBPVAVBVIWJWTDWMWLFGVCVDWBWMVEZWCWNWEWPWIWSWBWMGVF
      XFWDWOBWBWMAVRVGVJXFWHWRCGXFWGWQWBWMWFVKVLVMVNVAVIVOVP $.
  $}

  ${
    $d A f g m s $.  $d A f g x $.  $d A f m s y z $.  $d f m n s y z $.
    $d A n $.
    $( A set is countable iff there is a surjection from a decidable subset of
       the natural numbers onto it.  The decidability condition is needed as
       shown at ~ ctssexmid .  (Contributed by Jim Kingdon, 15-Aug-2023.) $)
    ctssdc $p |- ( E. s ( s C_ _om /\ E. f f : s -onto-> A
        /\ A. n e. _om DECID n e. s )
        <-> E. f f : _om -onto-> ( A |_| 1o ) ) $=
      ( vg vm vz vx cv com wfo wex wcel wdc wa c0 wceq syl ad3antrrr simpr wral
      vy wss w3a c1o cdju wn cfv cif cmpt wfn crn wf simp3 fof ffvelcdmd elequ1
      simpllr dcbid simpll2 rspcdva ifcldadc fmpttd ffnd wrex wb fvelrnb foelrn
      ad2antrr sylancom simpll1 eqid ifbieq1d sselda ad4antr ffvelcdmda fvmptd3
      fveq2 simp2 iftrued adantr eqtr4d ex reximdva ssrexv sylsyld mpd eqeltrrd
      eqtrd eqeltrd rexlimdva2 impbid bitr4d eqrdv df-fo omex mptex foeq1 spcev
      sylanbrc elex2 ctm mpbird simpl1 ctssdclemn0 wo eleq1 peano1 a1i mpjaodan
      exmiddc 3expia exlimdv 3impia 3com23 exlimiv cbvexv ctssdclemr impbii
      sylib ) DIZJUCZYAABIZKZBLZCIZYAMZNZCJUAZUDZDLZJAUEUFZYCKZBLZYKJYLEIZKZELZ
      YNYJYQDYBYIYEYQYBYIYEYQYBYIOYDYQBYBYIYDYQYBYIYDUDZPYAMZYQYSUGZYRYSOZYQJAY
      OKZELZUUAJAFJFIZYAMZUUDYCUHZPYCUHZUIZUJZKZUUCUUAUUIJUKZUUIULZAQUUJUUAJAUU
      IUUAFJUUHAUUAUUDJMZOZUUEUUFUUGAUUNUUEOYAAUUDYCYRYAAYCUMZYSUUMUUEYRYDUUOYB
      YIYDUNZYAAYCUORZSUUNUUETUPUUNUUEUGZOYAAPYCYRUUOYSUUMUURUUQSYRYSUUMUURURUP
      UUNYHUUENCJUUDYFUUDQYGUUECFDUQUSYBYIYDYSUUMUTUUAUUMTVAVBVCVDZUUAUBUULAUUA
      UBIZUULMZGIZUUIUHZUUTQZGJVEZUUTAMZUUAUUKUVAUVEVFUUSGJUUTUUIVGRUUAUVFUVEUU
      AUVFUVEUUAUVFOZUUTUVBYCUHZQZGYAVEZUVEUUAUVFYDUVJYRYDYSUVFUUPVIGYAAUUTYCVH
      VJUVGYBUVJUVDGYAVEUVEYBYIYDYSUVFVKZUVGUVIUVDGYAUVGUVBYAMZOZUVIUVDUVMUVIOU
      VCUVHUUTUVMUVCUVHQUVIUVMUVCUVLUVHUUGUIZUVHUVMFUVBUUHUVNJUUIAUUIVLZUUDUVBQ
      UUEUVLUUFUVHUUGFGDUQUUDUVBYCVRVMZUVGYAJUVBUVKVNZUVMUVLUVHUUGAUVMUVLOYAAUV
      BYCYRUUOYSUVFUVLUVLUUQVOUVMUVLTUPUUAUUGAMZUVFUVLUVLUGZYRYAAPYCUUQVPZSUVMY
      HUVLNZCJUVBYFUVBQYGUVLCGDUQUSZYRYIYSUVFUVLYBYIYDVSZSUVQVAVBVQUVMUVLUVHUUG
      UVGUVLTVTWIWAUVMUVITWBWCWDUVDGYAJWEWFWGWCUUAUVDUVFGJUUAUVBJMZOZUVDOUVCUUT
      AUWEUVDTUWEUVCAMUVDUWEUVCUVNAUWEFUVBUUHUVNJUUIAUVOUVPUUAUWDTZUWEUVLUVHUUG
      AUWEUVLOYAAUVBYCYRUUOYSUWDUVLUUQSUWEUVLTUPUUAUVRUWDUVSUVTVIUWEYHUWACJUVBU
      WBYBYIYDYSUWDUTUWFVAVBZVQUWGWJWAWHWKWLWMWNJAUUIWOWTUUBUUJEUUIFJUUHWPWQJAY
      OUUIWRWSRUUAHIAMHLZYQUUCVFUUAUVRUWHUVTHUUGAXARHAEXBRXCYRYTOAYAECYCYBYIYDY
      TXDYRYIYTUWCWAYRYDYTUUPWAYRYTTXEYRYSNZYSYTXFYRYHUWICJPYFPQYGYSYFPYAXGUSUW
      CPJMYRXHXIVAYSXKRXJXLXMXNXOXPYPYMEBJYLYOYCWRXQXTABCDXRXS $.
  $}

  ${
    $d A k x $.  $d A x y $.  $d F k x $.  $d G x y $.  $d N k x $.
    $d k ph x $.  $d ph x y $.
    enumctlemm.f $e |- ( ph -> F : N -onto-> A ) $.
    enumctlemm.n $e |- ( ph -> N e. _om ) $.
    enumctlemm.n0 $e |- ( ph -> (/) e. N ) $.
    enumctlemm.g $e |- G = ( k e. _om |->
      if ( k e. N , ( F ` k ) , ( F ` (/) ) ) ) $.
    $( Lemma for ~ enumct .  The case where ` N ` is greater than zero.
       (Contributed by Jim Kingdon, 13-Mar-2023.) $)
    enumctlemm $p |- ( ph -> G : _om -onto-> A ) $=
      ( vy vx com cv cfv wceq wrex wcel wa adantr wf wral wfo c0 cif ffvelcdmda
      fof syl adantlr wn ffvelcdmd ad2antrr simpr nndcel syl2anc ifcldadc fmptd
      wdc foelrn sylan eleq1w fveq2 ifbieq1d elnn iftrued eqeltrd fvmptd3 eqtrd
      wb eqeq2d rexbidva mpbird wi wss omelon onelssi ssrexv 3syl mpd ralrimiva
      dffo3 sylanbrc ) AMBEUAKNZLNZEOZPZLMQZKBUBMBEUCACMCNZFRZWHDOZUDDOZUEZBEAW
      HMRZSZWIWJWKBAWIWJBRWMAFBWHDAFBDUCZFBDUAGFBDUGUHZUFUIAWKBRWMWIUJAFBUDDWPI
      UKULWNWMFMRZWIURAWMUMAWQWMHTWHFUNUOUPJUQAWGKBAWCBRZSZWFLFQZWGWSWTWCWDDOZP
      ZLFQZAWOWRXCGLFBWCDUSUTAWTXCVIWRAWFXBLFAWDFRZSZWEXAWCXEWEXDXAWKUEZXAXECWD
      WLXFMEBJWHWDPWIXDWJXAWKCLFVAWHWDDVBVCXEXDWQWDMRAXDUMZAWQXDHTWDFVDUOXEXFXA
      BXEXDXAWKXGVEZAFBWDDWPUFVFVGXHVHVJVKTVLAWTWGVMZWRAWQFMVNXIHMFVOVPWFLFMVQV
      RTVSVTLKMBEWAWB $.
  $}

  ${
    $d A f g n x k $.  $d A f n y z $.  $d w y $.
    $( A finitely enumerable set is countable.  Lemma 8.1.14 of [AczelRathjen],
       p. 73 (except that our definition of countable does not require the set
       to be inhabited).  "Finitely enumerable" is defined as
       ` E. n e. _om E. f f : n -onto-> A ` per Definition 8.1.4 of
       [AczelRathjen], p. 71 and "countable" is defined as
       ` E. g g : _om -onto-> ( A |_| 1o ) ` per Definition on page 14:3 of
       [BauerSwan], p. 3.  (Contributed by Jim Kingdon, 13-Mar-2023.) $)
    enumct $p |- ( E. n e. _om E. f f : n -onto-> A
        -> E. g g : _om -onto-> ( A |_| 1o ) ) $=
      ( vk vx cv wfo wex com c1o cdju wcel wa c0 wceq simpll wb adantl syl fo00
      wi foeq2 mpbid sylib 0ct djueq1 foeq3 exbidv mpbiri simpl2im cfv cif cmpt
      cvv omex mptex simplr simpr enumctlemm foeq1 spcegv mpsyl wf fof ad2antrr
      eqid ffvelcdmd eleq1 sylc ctm mpbird wo mpjaodan exlimiv impcom rexlimiva
      0elnn ex ) DGZABGZHZBIZJAKLZCGZHZCIZDJWCVTJMZWGWBWHWGUBBWBWHWGWBWHNZVTOPZ
      WGOVTMZWIWJNZWAOPZAOPZWGWLOAWAHZWMWNNWLWBWOWBWHWJQWJWBWORWIVTOAWAUCSUDAWA
      UAUEWNWGJOKLZWEHZCICUFWNWFWQCWNWDWPPWFWQRAOKUGWDWPJWEUHTUIUJUKWIWKNZWGJAW
      EHZCIZEJEGZVTMXAWAULOWAULZUMZUNZUOMWRJAXDHZWTEJXCUPUQWRAEWAXDVTWBWHWKQWBW
      HWKURWIWKUSZXDVGUTWSXECXDUOJAWEXDVAVBVCWRFGZAMZFIZWGWTRWRXBAMZXJXIWRVTAOW
      AWBVTAWAVDWHWKVTAWAVEVFXFVHZXKXHXJFXBAXGXBAVIVBVJFACVKTVLWHWJWKVMWBVTVRSV
      NVSVOVPVQ $.
  $}

  ${
    $d A f g n x y $.  $d g A $.
    $( A finite set is countable.  (Contributed by Jim Kingdon,
       17-Mar-2023.) $)
    finct $p |- ( A e. Fin -> E. g g : _om -onto-> ( A |_| 1o ) ) $=
      ( vx vy vn vf cfn wcel weq wdc wral cv wfo wex com wrex c1o cdju fidcenum
      enumct simplbiim ) AGHCDIJDAKCAKELAFLMFNEOPOAQRBLMBNCDAFESAFBETUA $.
  $}

  ${
    $d f x $.
    $( ` _om ` is countable.  (Contributed by Jim Kingdon, 23-Dec-2023.) $)
    omct $p |- E. f f : _om -onto-> ( _om |_| 1o ) $=
      ( vx com c1o cdju cv wfo wex cid cres wf1o f1oi f1ofo cvv wcel omex ax-mp
      resiexg mp2b c0 foeq1 spcev wb peano1 elex2 ctm mpbir ) CCDEAFZGAHZCCUHGZ
      AHZCCICJZKCCULGZUKCLCCULMUJUMAULCNOULNOPCNRQCCUHULUAUBSTCOBFCOBHUIUKUCUDB
      TCUEBCAUFSUG $.
  $}

  ${
    $d A f $.
    $( A countable class is a set.  (Contributed by Jim Kingdon,
       25-Dec-2023.) $)
    ctfoex $p |- ( E. f f : _om -onto-> ( A |_| 1o ) -> A e. _V ) $=
      ( com c1o cdju cv wfo cvv wcel wa omex focdmex ax-mp djuexb sylibr simpld
      wi exlimiv ) CADEZBFZGZAHIZBUAUBDHIZUASHIZUBUCJCHIUAUDQKCSHTLMADNOPR $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The one-point compactification of the natural numbers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This section introduces the one-point compactification of the set of natural
  numbers, introduced by Escardo as the set of nonincreasing sequences on
  ` _om ` with values in ` 2o ` .  The topological results justifying its name
  will be proved later.

$)

  $c NN+oo $. $( Set of nonincreasing sequences in ` 2o ^m _om ` $)

  $( Set of nonincreasing sequences in ` 2o ^m _om ` . $)
  xnninf $a class NN+oo $.

  ${
    $d f i $.
    $( Define the set of nonincreasing sequences in ` 2o ^m _om ` .  Definition
       in Section 3.1 of [Pierik], p. 15.  If we assumed excluded middle, this
       would be essentially the same as ` NN0* ` as defined at ~ df-xnn0 but in
       its absence the relationship between the two is more complicated.  This
       definition would function much the same whether we used ` _om ` or
       ` NN0 ` , but the former allows us to take advantage of
       ` 2o = { (/) , 1o } ` ( ~ df2o3 ) so we adopt it.  (Contributed by Jim
       Kingdon, 14-Jul-2022.) $)
    df-nninf $a |- NN+oo = { f e. ( 2o ^m _om ) |
      A. i e. _om ( f ` suc i ) C_ ( f ` i ) } $.
  $}

  ${
    $d f i $.
    $( ` NN+oo ` is a set.  (Contributed by Jim Kingdon, 10-Aug-2022.) $)
    nninfex $p |- NN+oo e. _V $=
      ( vi vf vg xnninf cv csuc cfv wss com wral c2o cmap co crab df-nninf omex
      cvv wf wcel eqeltri cab 2onn elexi mapval mapex mp2an rabex ) DAEZFBEZGUH
      UIGHAIJZBKILMZNQBAOUJBUKUKIKCERCUAZQKICKIUBUCZPUDIQSKQSULQSPUMIKQQCUEUFTU
      GT $.
  $}

  ${
    $d A f i $.
    $( An element of ` NN+oo ` is a sequence of zeroes and ones.  (Contributed
       by Jim Kingdon, 4-Aug-2022.) $)
    nninff $p |- ( A e. NN+oo -> A : _om --> 2o ) $=
      ( vi vf xnninf wcel c2o com cmap co wf cv csuc cfv wss wral fveq1 sseq12d
      wceq ralbidv df-nninf elrab2 simplbi elmapi syl ) ADEZAFGHIZEZGFAJUEUGBKZ
      LZAMZUHAMZNZBGOZUICKZMZUHUNMZNZBGOUMCAUFDUNARZUQULBGURUOUJUPUKUIUNAPUHUNA
      PQSCBTUAUBAFGUCUD $.
  $}

  ${
    $d A f i n k w $.  $d X k n w $.  $d Y n $.  $d ph n k w $.
    nninfninc.a $e |- ( ph -> A e. NN+oo ) $.
    nninfninc.x $e |- ( ph -> X e. _om ) $.
    nninfninc.y $e |- ( ph -> Y e. _om ) $.
    nninfninc.le $e |- ( ph -> X C_ Y ) $.
    nninfninc.z $e |- ( ph -> ( A ` X ) = (/) ) $.
    $( All values beyond a zero in an ` NN+oo ` sequence are zero.  This is
       another way of stating that elements of ` NN+oo ` are nonincreasing.
       (Contributed by Jim Kingdon, 12-Jul-2025.) $)
    nninfninc $p |- ( ph -> ( A ` Y ) = (/) ) $=
      ( vi wss cfv c0 wceq cv wi com sseq2 wcel wa vn vw fveqeq2 imbi12d anbi2d
      vk vf csuc ss0 adantl fveq2d adantr eqtr3d suceq sseq12d wral simplrl c2o
      fveq2 co xnninf fveq1 ralbidv df-nninf elrab2 sylib simprd simpll rspcdva
      cmap syl wb ad2antrl nnsssuc syl2anc mpbird simpllr mp2and sseqtrd simprr
      simpr wo peano2 mpbid elsuci mpjaodan exp31 finds impancom ralrimiva mpd
      com12 ) ACDKZDBLMNZHACUAOZKZWOBLMNZPZWMWNPUAQDWODNWPWMWQWNWODCRWODMBUCUDA
      WRUAQAWPWOQSZWQWSAWPTZWQACUBOZKZTZXABLMNZPACMKZTZMBLZMNZPACUFOZKZTZXIBLZM
      NZPZACXIUHZKZTZXOBLZMNZPWTWQPUBUFWOXAMNZXCXFXDXHXTXBXEAXAMCRUEXAMMBUCUDXA
      XINZXCXKXDXMYAXBXJAXAXICRUEXAXIMBUCUDXAXONZXCXQXDXSYBXBXPAXAXOCRUEXAXOMBU
      CUDXAWONZXCWTXDWQYCXBWPAXAWOCRUEXAWOMBUCUDXFCBLZXGMXFCMBXECMNACUIUJUKAYDM
      NZXEIULUMXIQSZXNXQXSYFXNTZXQTZCXOSZXSCXONZYHYITZXRMKXSYKXRXLMYKJOZUHZBLZY
      LBLZKZXRXLKJQXIYLXINZYNXRYOXLYQYMXOBYLXIUNUKYLXIBUSUOYKAYPJQUPZYGAXPYIUQZ
      ABURQVJUTZSZYRABVASUUAYRTEYMUGOZLZYLUUBLZKZJQUPYRUGBYTVAUUBBNZUUEYPJQUUFU
      UCYNUUDYOYMUUBBVBYLUUBBVBUOVCUGJVDVEVFVGVKYHYFYIYFXNXQVHZULZVIYKAXJXMYSYK
      XJYIYHYIWAYKCQSZYFXJYIVLYHUUIYIAUUIYGXPFVMZULUUHCXIVNVOVPYFXNXQYIVQVRVSXR
      UIVKYHYJTZYDXRMUUKCXOBYHYJWAUKUUKAYEYGAXPYJUQIVKUMYHCXOUHSZYIYJWBYHXPUULY
      GAXPVTYHUUIXOQSZXPUULVLUUJYHYFUUMUUGXIWCVKCXOVNVOWDCXOWEVKWFWGWHWLWIWJGVI
      WK $.
  $}

  ${
    $d f i j $.
    $( The point at infinity in ` NN+oo ` is the constant sequence equal to
       ` 1o ` .  Note that with our encoding of functions, that constant
       function can also be expressed as ` ( _om X. { 1o } ) ` , as ~ fconstmpt
       shows.  (Contributed by Jim Kingdon, 14-Jul-2022.)  Use maps-to
       notation.  (Revised by BJ, 10-Aug-2024.) $)
    infnninf $p |- ( i e. _om |-> 1o ) e. NN+oo $=
      ( vj vf com c1o xnninf wcel c2o cv cfv wss wral wtru con0 wceq eqidd 1oex
      cvv fvmpt syl cmpt cmap co csuc wf 1lt2o a1i fmpttd mptru 2on omex elmapg
      wa wb mp2an mpbir peano2 eqid eqtr4d eqimss rgen sseq12d ralbidv df-nninf
      weq fveq1 elrab2 mpbir2an ) ADEUAZFGVIHDUBUCZGZBIZUDZVIJZVLVIJZKZBDLZVKDH
      VIUEZVRMADEHEHGMAIZDGUMUFUGUHUIHNGDRGVKVRUNUJUKHDVINRULUOUPVPBDVLDGZVNVOO
      VPVTVNEVOVTVMDGVNEOVLUQAVMEEDVIVSVMOEPVIURZQSTAVLEEDVIABVEEPWAQSUSVNVOUTT
      VAVMCIZJZVLWBJZKZBDLVQCVIVJFWBVIOZWEVPBDWFWCVNWDVOVMWBVIVFVLWBVIVFVBVCCBV
      DVGVH $.
  $}

  ${
    $d f i $.
    $( Obsolete version of ~ infnninf as of 10-Aug-2024.  (Contributed by Jim
       Kingdon, 14-Jul-2022.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    infnninfOLD $p |- ( _om X. { 1o } ) e. NN+oo $=
      ( vi vf com c1o csn cxp xnninf wcel c2o cmap co cv cfv wss wral wceq 1oex
      fvconst2 syl fveq1 csuc 1lt2o fconst6 2onn elexi omex elmap peano2 eqtr4d
      wf mpbir eqimss rgen sseq12d ralbidv df-nninf elrab2 mpbir2an ) CDEFZGHUS
      ICJKZHZALZUAZUSMZVBUSMZNZACOZVACIUSUJCDIUBUCICUSICUDUEUFUGUKVFACVBCHZVDVE
      PVFVHVDDVEVHVCCHVDDPVBUHCDVCQRSCDVBQRUIVDVEULSUMVCBLZMZVBVIMZNZACOVGBUSUT
      GVIUSPZVLVFACVMVJVDVKVEVCVIUSTVBVIUSTUNUOBAUPUQUR $.
  $}

  ${
    $d N f i j $.
    $( Elements of ` NN+oo ` corresponding to natural numbers.  The natural
       number ` N ` corresponds to a sequence of ` N ` ones followed by zeroes.
       This can be strengthened to include infinity, see ~ nnnninf2 .
       (Contributed by Jim Kingdon, 14-Jul-2022.) $)
    nnnninf $p |- ( N e. _om ->
        ( i e. _om |-> if ( i e. N , 1o , (/) ) ) e. NN+oo ) $=
      ( vj vf com wcel cv c1o c0 cif c2o cfv wss a1i wdc nndcel adantl syl wceq
      wa cmpt cmap co csuc wral xnninf wf 1lt2o 0lt2o ancoms ifcldcd 2onn elexi
      fmpttd omex elmap sylibr wn wb iftrue sseq1d mpbiri 0ss iffalse wo peano2
      ssid simpl syl2anc exmiddc mpjaodan adantr sseqtrrd wtr nnord ordtr trsuc
      wi word sylan ex con3dimp 3sstr4d eleq1 ifbid eqid fvmptg simpr ralrimiva
      fveq1 sseq12d ralbidv df-nninf elrab2 sylanbrc ) BEFZAEAGZBFZHIJZUAZKEUBU
      CZFZCGZUDZWTLZXCWTLZMZCEUEZWTUFFWPEKWTUGXBWPAEWSKWPWQEFZTZWRHIKHKFZXJUHNI
      KFZXJUINXIWPWROWQBPUJUKUNKEWTKEULUMUOUPUQWPXGCEWPXCEFZTZXDBFZHIJZXCBFZHIJ
      ZXEXFXNXQXPXRMXQURZXNXQTXPHXRXNXPHMZXQXNXOXTXOURZXNXOTXTHHMZHVGXOXTYBUSXN
      XOXPHHXOHIUTVAQVBXNYATXTIHMZHVCYAXTYCUSXNYAXPIHXOHIVDZVAQVBXNXOOZXOYAVEXN
      XDEFZWPYEXMYFWPXCVFQZWPXMVHXDBPVIZXOVJRVKVLXQXRHSXNXQHIUTQVMXNXSTZIIXPXRI
      IMYIIVGNYIYAXPISXNXOXQWPXOXQVRXMWPXOXQWPBVNZXOXQWPBVSYJBVOBVPRBXCVQVTWAVL
      WBYDRXSXRISXNXQHIVDQWCXNXQOZXQXSVEXMWPYKXCBPUJZXQVJRVKXNYFXPKFXEXPSYGXNXO
      HIKXKXNUHNZXLXNUINZYHUKAXDWSXPEKWTWQXDSWRXOHIWQXDBWDWEWTWFZWGVIXNXMXRKFXF
      XRSWPXMWHXNXQHIKYMYNYLUKAXCWSXREKWTWQXCSWRXQHIWQXCBWDWEYOWGVIWCWIXDDGZLZX
      CYPLZMZCEUEXHDWTXAUFYPWTSZYSXGCEYTYQXEYRXFXDYPWTWJXCYPWTWJWKWLDCWMWNWO $.
  $}

  ${
    $d N i $.
    $( Canonical embedding of ` suc _om ` into ` NN+oo ` .  (Contributed by BJ,
       10-Aug-2024.) $)
    nnnninf2 $p |-
      ( N e. suc _om -> ( i e. _om |-> if ( i e. N , 1o , (/) ) ) e. NN+oo ) $=
      ( com csuc wcel wceq wo cv c1o c0 cif xnninf elsuci nnnninf iftrue eqcomd
      cmpt eleq2 ifbid sylan9eqr mpteq2dva infnninf eqeltrrdi jaoi syl ) BCDEBC
      EZBCFZGACAHZBEZIJKZQZLEZBCMUFULUGABNUGUKACIQLUGACIUJUHCEZUGIUMIJKZUJUMUNI
      UMIJOPUGUJUNUGUIUMIJBCUHRSPTUAAUBUCUDUE $.
  $}

  ${
    $d N i j $.  $d N j k w $.  $d N k x $.  $d P f j $.  $d P j k w $.
    $d P k x $.  $d i j ph $.  $d k ph w $.
    nnnninfeq.p $e |- ( ph -> P e. NN+oo ) $.
    nnnninfeq.n $e |- ( ph -> N e. _om ) $.
    nnnninfeq.1 $e |- ( ph -> A. x e. N ( P ` x ) = 1o ) $.
    nnnninfeq.0 $e |- ( ph -> ( P ` N ) = (/) ) $.
    $( Mapping of a natural number to an element of ` NN+oo ` .  (Contributed
       by Jim Kingdon, 4-Aug-2022.) $)
    nnnninfeq $p |- ( ph
        -> P = ( i e. _om |-> if ( i e. N , 1o , (/) ) ) ) $=
      ( vj com wcel c1o c0 c2o wa simpr cfv wceq fveq2 vw vk vf cif cmpt xnninf
      cv wf nninff syl ffnd wral wfn 1lt2o a1i 0lt2o wdc adantr syl2anc ifcldcd
      nndcel ralrimiva eqid fnmpt csuc eleq1 ifbid eqeq12d imbi2d eleq1w eleq2d
      wi mtbiri iffalsed fveq2d 3eqtr2rd eqeq1d rspcdva iftrued eqtr4d wo 0elnn
      noel mpjaodan ad3antlr wn word ad2antlr nnord ordirr eqneltrd 3eqtr4d wss
      3syl suceq sseq12d cmap co ralbidv df-nninf elrab2 simprbi simplll simplr
      fveq1 wtr ordtr trsucss sylc nntri1 mpbid eqtrd sseqtrd ss0 ordn2lp mtand
      jca w3o peano2 ad2antrr nntri3or mpjao3dan exp31 a2d finds impcom eqfnfvd
      wb fvmptg ) AJKCDKDUGZELZMNUDZUEZAKOCACUFLZKOCUHFCUIUJUKAYLOLZDKULYMKUMAY
      ODKAYJKLZPZYKMNOMOLZYQUNUONOLZYQUPUOYQYPEKLZYKUQAYPQAYTYPGURYJEVAUSUTVBDK
      YLYMOYMVCZVDUJAJUGZKLZPZUUBCRZUUBELZMNUDZUUBYMRZUUCAUUEUUGSZAUAUGZCRZUUJE
      LZMNUDZSZVLANCRZNELZMNUDZSZVLAUBUGZCRZUUSELZMNUDZSZVLAUUSVEZCRZUVDELZMNUD
      ZSZVLAUUIVLUAUBUUBUUJNSZUUNUURAUVIUUKUUOUUMUUQUUJNCTUVIUULUUPMNUUJNEVFVGV
      HVIUUJUUSSZUUNUVCAUVJUUKUUTUUMUVBUUJUUSCTUVJUULUVAMNUAUBEVJVGVHVIUUJUVDSZ
      UUNUVHAUVKUUKUVEUUMUVGUUJUVDCTUVKUULUVFMNUUJUVDEVFVGVHVIUUJUUBSZUUNUUIAUV
      LUUKUUEUUMUUGUUJUUBCTUVLUULUUFMNUAJEVJVGVHVIAENSZUURUUPAUVMPZUUQNECRZUUOU
      VNUUPMNUVNUUPNNLNWCUVNENNAUVMQZVKVMVNAUVONSZUVMIURUVNENCUVPVOVPAUUPPZUUOM
      UUQUVRBUGZCRZMSZUUOMSBENUVSNSUVTUUOMUVSNCTVQAUWABEULZUUPHURAUUPQZVRUVRUUP
      MNUWCVSVTAYTUVMUUPWAGEWBUJWDUUSKLZAUVCUVHUWDAUVCUVHUWDAPZUVCPZUVFUVHUVDES
      ZEUVDLZUWFUVFPZUVEMUVGUWIUWAUVEMSBEUVDUVSUVDSUVTUVEMUVSUVDCTVQAUWBUWDUVCU
      VFHWEUWFUVFQZVRUWIUVFMNUWJVSVTUWFUWGPZUVONUVEUVGAUVQUWDUVCUWGIWEUWKUVDECU
      WFUWGQZVOUWKUVFMNUWKUVDEEUWLUWFEELWFZUWGUWFYTEWGZUWMAYTUWDUVCGWHZEWIZEWJW
      NURWKVNWLUWFUWHPZUVENUVGUWQUVENWMUVENSUWQUVEUUTNUWQUUBVEZCRZUUEWMZUVEUUTW
      MJKUUSUUBUUSSZUWSUVEUUEUUTUXAUWRUVDCUUBUUSWOVOUUBUUSCTWPUWQYNUWTJKULZAYNU
      WDUVCUWHFWEYNCOKWQWRZLUXBUWRUCUGZRZUUBUXDRZWMZJKULUXBUCCUXCUFUXDCSZUXGUWT
      JKUXHUXEUWSUXFUUEUWRUXDCXEUUBUXDCXEWPWSUCJWTXAXBUJUWDAUVCUWHXCZVRUWQUUTUV
      BNUWEUVCUWHXDUWQUVAMNUWQEUUSWMZUVAWFZUWQUWDUWHUXJUXIUWFUWHQUWDUUSWGUUSXFU
      WHUXJVLUUSWIUUSXGUUSEXHWNXIUWQYTUWDUXJUXKYHUWFYTUWHUWOURZUXIEUUSXJUSXKVNX
      LXMUVEXNUJUWQUVFMNUWQUVFUWHUVFPZUWQYTUWNUXMWFUXLUWPEUVDXOWNUWQUVFPUWHUVFU
      WFUWHUVFXDUWQUVFQXQXPVNVTUWFUVDKLZYTUVFUWGUWHXRUWDUXNAUVCUUSXSXTUWOUVDEYA
      USYBYCYDYEYFUUDUUCUUGOLUUHUUGSAUUCQZUUDUUFMNOYRUUDUNUOYSUUDUPUOUUDUUCYTUU
      FUQUXOAYTUUCGURUUBEVAUSUTDUUBYLUUGKOYMYJUUBSYKUUFMNDJEVJVGUUAYIUSVTYG $.
  $}

  ${
    $d N i $.  $d N w x $.  $d P f j $.  $d P j k $.  $d P k w x $.  $d i ph $.
    $d k ph w $.
    nnnninfeq2.p $e |- ( ph -> P e. NN+oo ) $.
    nnnninfeq2.n $e |- ( ph -> N e. _om ) $.
    nnnninfeq2.1 $e |- ( ph -> ( P ` U. N ) = 1o ) $.
    nnnninfeq2.0 $e |- ( ph -> ( P ` N ) = (/) ) $.
    $( Mapping of a natural number to an element of ` NN+oo ` .  Similar to
       ~ nnnninfeq but if we have information about a single ` 1o ` digit, that
       gives information about all previous digits.  (Contributed by Jim
       Kingdon, 4-Aug-2022.) $)
    nnnninfeq2 $p |- ( ph
        -> P = ( i e. _om |-> if ( i e. N , 1o , (/) ) ) ) $=
      ( vx vj cuni cfv c1o wceq wral com wcel wa wi c0 vw vk vf cv adantr unieq
      csuc fveqeq2d anbi2d raleq imbi12d ral0 a1i csn wn uni0 id 3eqtr4a fveq2d
      cun wtr word nnord ordtr ad3antlr wb unisucg mpbid simpr eqtr3d sylan9eqr
      syl wss c2o wf xnninf nninff nnpredcl adantl ffvelcdmd el2oss1o ad3antrrr
      wne simp-4r neqned nnsucpred syl2anc eqtrd suceq fveq2 sseq12d cmap fveq1
      co ralbidv df-nninf elrab2 sylib simprd rspcdva eqsstrrd eqssd wo nndceq0
      wdc exmiddc mpjaodan simplr fveqeq2 ralunsn mpbir2and df-suc raleqi exp31
      mpd sylibr expcom a2d impexp 3imtr4g finds mpcom mpdan nnnninfeq ) AIBCDE
      FADKZBLMNZIUDZBLMNZIDOZGDPQZAYFRZYIAYJYFFUEAUAUDZKZBLMNZRZYHIYLOZSATKZBLM
      NZRZYHITOZSAUBUDZKZBLZMNZRZYHIUUAOZSZAUUAUGZKZBLZMNZRZYHIUUHOZSZYKYISUAUB
      DYLTNZYOYSYPYTUUOYNYRAUUOYMYQMBYLTUFUHUIYHIYLTUJUKYLUUANZYOUUEYPUUFUUPYNU
      UDAUUPYMUUBMBYLUUAUFUHUIYHIYLUUAUJUKYLUUHNZYOUULYPUUMUUQYNUUKAUUQYMUUIMBY
      LUUHUFUHUIYHIYLUUHUJUKYLDNZYOYKYPYIUURYNYFAUURYMYEMBYLDUFUHUIYHIYLDUJUKYT
      YSYHIULUMUUAPQZAUUDUUFSZSAUUKUUMSZSUUGUUNUUSAUUTUVAAUUSUUTUVASAUUSRZUUTUU
      KUUMUVBUUTRZUUKRZYHIUUAUUAUNUTZOZUUMUVDUVFUUFUUABLZMNZUVDUUDUUFUVDUUATNZU
      UDUVIUOZUVIUVDUUCUVGMUVIUUBUUABUVIYQTUUBUUAUPUUATUFUVIUQURUSUVDUUJUVGMUVD
      UUIUUABUVDUUAVAZUUIUUANZUUSUVKAUUTUUKUUSUUAVBUVKUUAVCUUAVDVLVEUUSUVKUVLVF
      AUUTUUKUUAPVGVEVHUSUVCUUKVIVJZVKUVDUVJRZUUCMUVBUUCMVMZUUTUUKUVJUVBUUCVNQU
      VOUVBPVNUUBBAPVNBVOZUUSABVPQZUVPEBVQVLUEUUSUUBPQZAUUAVRZVSVTUUCWAVLWBUVNM
      UUBUGZBLZUUCUVNUWAUVGMUVNUVTUUABUVNUUSUUATWCUVTUUANAUUSUUTUUKUVJWDUVNUUAT
      UVDUVJVIWEUUAWFWGUSUVDUVHUVJUVMUEWHUVDUWAUUCVMZUVJUVDJUDZUGZBLZUWCBLZVMZU
      WBJPUUBUWCUUBNZUWEUWAUWFUUCUWHUWDUVTBUWCUUBWIUSUWCUUBBWJWKAUWGJPOZUUSUUTU
      UKABVNPWLWNZQZUWIAUVQUWKUWIREUWDUCUDZLZUWCUWLLZVMZJPOUWIUCBUWJVPUWLBNZUWO
      UWGJPUWPUWMUWEUWNUWFUWDUWLBWMUWCUWLBWMWKWOUCJWPWQWRWSWBUUSUVRAUUTUUKUVSVE
      WTUEXAXBUUSUVIUVJXCZAUUTUUKUUSUVIXEUWQUUAXDUVIXFVLVEXGUVBUUTUUKXHXOUVMUUS
      UVFUUFUVHRVFAUUTUUKYHUVHIUUAUUAPYGUUAMBXIXJVEXKYHIUUHUVEUUAXLXMXPXNXQXRAU
      UDUUFXSAUUKUUMXSXTYAYBYCHYD $.
  $}

  ${
    nninfisol.x $e |- ( ph -> X e. NN+oo ) $.
    nninfisol.0 $e |- ( ph -> ( X ` N ) = (/) ) $.
    nninfisol.n $e |- ( ph -> N e. _om ) $.

    ${
      $d N i $.  $d N j $.  $d X j $.  $d i ph $.
      nninfisollem0.0 $e |- ( ph -> N = (/) ) $.
      $( Lemma for ~ nninfisol .  The case where ` N ` is zero.  (Contributed
         by Jim Kingdon, 13-Sep-2024.) $)
      nninfisollem0 $p |- ( ph
          -> DECID ( i e. _om |-> if ( i e. N , 1o , (/) ) ) = X ) $=
        ( vj com cv wcel c1o c0 cif cmpt wceq wn wo wral wdc cfv raleqdv mpbiri
        ral0 nnnninfeq eqcomd orcd df-dc sylibr ) ABJBKCLMNOPZDQZULRZSULUAAULUM
        ADUKAIDBCEGAIKDUBMQZICTUNINTUNIUEAUNICNHUCUDFUFUGUHULUIUJ $.
    $}

    ${
      $d N i $.
      nninfisollemne.s $e |- ( ph -> N =/= (/) ) $.
      nninfisollemne.0 $e |- ( ph -> ( X ` U. N ) = (/) ) $.
      $( Lemma for ~ nninfisol .  A case where ` N ` is a successor and ` N `
         and ` X ` are not equal.  (Contributed by Jim Kingdon,
         13-Sep-2024.) $)
      nninfisollemne $p |- ( ph
          -> DECID ( i e. _om |-> if ( i e. N , 1o , (/) ) ) = X ) $=
        ( com cv wcel c1o c0 cif wceq cfv adantr wne c2o cmpt wn wo wdc cuni wa
        simpr fveq1d eleq1 ifbid nnpredcl syl nnpredlt syl2anc iftrued eqeltrdi
        eqid 1lt2o fvmptd3 eqtrd eqtr3d 1n0 pm13.181 neneqd pm2.65da olcd df-dc
        sylancl sylibr ) ABJBKZCLZMNOZUAZDPZVNUBZUCVNUDAVOVNAVNCUEZDQZNPZAVRVNI
        RAVNUFZVQNVSVQMPMNSVQNSVSVPVMQZVQMVSVPVMDAVNUGUHAVTMPVNAVTVPCLZMNOZMABV
        PVLWBJVMTVMUQVJVPPVKWAMNVJVPCUIUJACJLZVPJLGCUKULAWBMTAWAMNAWCCNSWAGHCUM
        UNUOZURUPUSWDUTRVAVBVQMNVCVHVDVEVFVNVGVI $.
    $}

    ${
      $d N i $.  $d i ph $.
      nninfisollemeq.s $e |- ( ph -> N =/= (/) ) $.
      nninfisollemeq.0 $e |- ( ph -> ( X ` U. N ) = 1o ) $.
      $( Lemma for ~ nninfisol .  The case where ` N ` is a successor and ` N `
         and ` X ` are equal.  (Contributed by Jim Kingdon, 13-Sep-2024.) $)
      nninfisollemeq $p |- ( ph
          -> DECID ( i e. _om |-> if ( i e. N , 1o , (/) ) ) = X ) $=
        ( com cv wcel c1o c0 cif cmpt wceq wn wo wdc nnnninfeq2 eqcomd sylibr
        orcd df-dc ) ABJBKCLMNOPZDQZUGRZSUGTAUGUHADUFADBCEGIFUAUBUDUGUEUC $.
    $}
  $}

  ${
    $d N i j $.  $d X i j $.
    $( Finite elements of ` NN+oo ` are isolated.  That is, given a natural
       number and any element of ` NN+oo ` , it is decidable whether the
       natural number (when converted to an element of ` NN+oo ` ) is equal to
       the given element of ` NN+oo ` .  Stated in an online post by Martin
       Escardo.  One way to understand this theorem is that you do not need to
       look at an unbounded number of elements of the sequence ` X ` to decide
       whether it is equal to ` N ` (in fact, you only need to look at two
       elements and ` N ` tells you where to look).

       By contrast, the point at infinity being isolated is equivalent to the
       Weak Limited Principle of Omniscience (WLPO) ( ~ nninfinfwlpo ).
       (Contributed by BJ and Jim Kingdon, 12-Sep-2024.) $)
    nninfisol $p |- ( ( N e. _om /\ X e. NN+oo )
        -> DECID ( i e. _om |-> if ( i e. N , 1o , (/) ) ) = X ) $=
      ( com wcel wa cfv c0 wceq c1o cif wdc wn simpllr simpr adantr wo mpjaodan
      c2o syl xnninf cmpt simplr simplll nninfisollem0 cuni simp-4r simp-4l wne
      neqned nninfisollemne nninfisollemeq cpr nninff adantl nnpredcl ffvelcdmd
      cv wf df2o3 eleqtrdi elpri ad2antrr nndceq0 exmiddc 1n0 neii fveq1d eleq1
      eqid ifbid id word nnord ordirr iffalsed eqeltrdi fvmptd3 eqtrd ad3antrrr
      peano1 3eqtr3rd ex mtoi olcd df-dc sylibr simpl ) BDEZCUAEZFZBCGZHIZADAUR
      ZBEZJHKZUBZCIZLZWLJIZWKWMFZBHIZWSXBMZXAXBFABCWIWJWMXBNWKWMXBUCWIWJWMXBUDX
      AXBOUEXAXCFZBUFZCGZHIZWSXFJIZXDXGFABCWIWJWMXCXGUGWKWMXCXGNWIWJWMXCXGUHXDB
      HUIZXGXDBHXAXCOUJZPXDXGOUKXDXHFABCWIWJWMXCXHUGWKWMXCXHNWIWJWMXCXHUHXDXIXH
      XJPXDXHOULWKXGXHQZWMXCWKXFHJUMZEXKWKXFSXLWKDSXECWJDSCUSWICUNUOZWIXEDEWJBU
      PPUQUTVAXFHJVBTVCRWIXBXCQZWJWMWIXBLXNBVDXBVETVCRWKWTFZWRWRMZQWSXOXPWRXOWR
      JHIZJHVFVGXOWRXQXOWRFZBWQGZWLHJXRBWQCXOWROVHWIXSHIWJWTWRWIXSBBEZJHKZHWIAB
      WPYADWQDWQVJWNBIWOXTJHWNBBVIVKWIVLWIYAHDWIXTJHWIBVMXTMBVNBVOTVPZWAVQVRYBV
      SVTWKWTWRUCWBWCWDWEWRWFWGWKWLXLEWMWTQWKWLSXLWKDSBCXMWIWJWHUQUTVAWLHJVBTR
      $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Omniscient sets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c Omni $.  $( Class of all omniscient sets. $)

  $( Extend class definition to include the class of omniscient sets. $)
  comni $a class Omni $.

  ${
    $d f x y $.
    $( An omniscient set is one where we can decide whether a predicate (here
       represented by a function ` f ` ) holds (is equal to ` 1o ` ) for all
       elements or fails to hold (is equal to ` (/) ` ) for some element.
       Definition 3.1 of [Pierik], p. 14.

       In particular, ` _om e. Omni ` is known as the Limited Principle of
       Omniscience (LPO).  (Contributed by Jim Kingdon, 28-Jun-2022.) $)
    df-omni $a |- Omni = { y | A. f ( f : y --> 2o
        -> ( E. x e. y ( f ` x ) = (/) \/ A. x e. y ( f ` x ) = 1o ) ) } $.
  $}

  ${
    $d A f x y $.
    $( The predicate of being omniscient.  (Contributed by Jim Kingdon,
       28-Jun-2022.) $)
    isomni $p |- ( A e. V -> ( A e. Omni <-> A. f ( f : A --> 2o
        -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) ) ) $=
      ( vy cv c2o wf cfv c0 wceq wrex c1o wral wo wi wal comni feq2 rexeq raleq
      orbi12d imbi12d albidv df-omni elab2g ) EFZGCFZHZAFUHIZJKZAUGLZUJMKZAUGNZ
      OZPZCQBGUHHZUKABLZUMABNZOZPZCQEBRDUGBKZUPVACVBUIUQUOUTUGBGUHSVBULURUNUSUK
      AUGBTUMAUGBUAUBUCUDAECUEUF $.
  $}

  ${
    $d A f x $.  $d V f $.
    $( The predicate of being omniscient stated in terms of set exponentiation.
       (Contributed by Jim Kingdon, 13-Jul-2022.) $)
    isomnimap $p |- ( A e. V -> ( A e. Omni <-> A. f e. ( 2o ^m A )
        ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) ) $=
      ( wcel comni cv c2o cmap co cfv c0 wceq wrex c1o wral wo wi wal com wf wb
      isomni 2onn elmapg mpan imbi1d albidv bitr4d df-ral bitr4di ) BDEZBFEZCGZ
      HBIJZEZAGUNKZLMABNUQOMABPQZRZCSZURCUOPULUMBHUNUAZURRZCSUTABCDUCULUSVBCULU
      PVAURHTEULUPVAUBUDHBUNTDUEUFUGUHUIURCUOUJUK $.
  $}

  ${
    $d A f g h x $.  $d A g h x y $.  $d B g h x y $.
    $( Lemma for ~ enomni .  One direction of the biconditional.  (Contributed
       by Jim Kingdon, 13-Jul-2022.) $)
    enomnilem $p |- ( A ~~ B -> ( A e. Omni -> B e. Omni ) ) $=
      ( vy vg vh vx vf comni wcel wa cv cfv c0 wceq wrex c1o wral c2o eqeq1d wf
      cen wbr wo cmap co wf1o bren biimpi ad2antrr ccom rexbidv ralbidv orbi12d
      wex fveq1 isomnimap ibi ad3antlr simpr wb com 2onn relen brrelex2i elmapg
      cvv sylancr mpbid adantr f1of adantl syl2anc simpllr mpbird rspcdva f1ofn
      fco wfn ad2antlr sylancom ffvelcdmda fveq2d rspcedv sylbid rexlimdva ccnv
      fvco2 f1ocnv syl ffvelcdmd simplr f1ocnvfv2 3eqtr3rd ralrimiva ex orim12d
      fveq2 mpd exlimddv ) ABUAUBZAHIZBHIZWTXAJZXBCKZDKZLZMNZCBOZXFPNZCBQZUCZDR
      BUDUEZQZXCXKDXLXCXEXLIZJZABEKZUFZXKEWTXQEUNZXAXNWTXRABEUGUHUIXOXQJZFKZXEX
      PUJZLZMNZFAOZYBPNZFAQZUCZXKXSXTGKZLZMNZFAOZYIPNZFAQZUCZYGGRAUDUEZYAYHYANZ
      YKYDYMYFYPYJYCFAYPYIYBMXTYHYAUOZSUKYPYLYEFAYPYIYBPYQSULUMXAYNGYOQZWTXNXQX
      AYRFAGHUPUQURXSYAYOIZARYATZXSBRXETZABXPTZYTXOUUAXQXOXNUUAXCXNUSWTXNUUAUTZ
      XAXNWTRVAIZBVFIZUUCVBABUAVCVDZRBXEVAVFVEVGUIVHVIXQUUBXOABXPVJVKZABRXEXPVQ
      VLXSUUDXAYSYTUTVBWTXAXNXQVMRAYAVAHVEVGVNVOXSYDXHYFXJXSYCXHFAXSXTAIZJZYCXT
      XPLZXELZMNZXHUUIYBUUKMXSUUHXPAVRZYBUUKNXQUUMXOUUHABXPVPZVSAXEXPXTWGVTSUUI
      XGUULCUUJBXSABXTXPUUGWAUUIXDUUJNZJZXFUUKMUUPXDUUJXEUUIUUOUSWBSWCWDWEXSYFX
      JXSYFJZXICBUUQXDBIZJZXDXPWFZLZYALZUVAXPLZXELZPXFUUSUUMUVAAIUVBUVDNXQUUMXO
      YFUURUUNURUUSBAXDUUTXQBAUUTTZXOYFUURXQBAUUTUFUVEABXPWHBAUUTVJWIURUUQUURUS
      WJZAXEXPUVAWGVLUUSYEUVBPNFAUVAXTUVANYBUVBPXTUVAYAWQSXSYFUURWKUVFVOUUQUURX
      QUVDXFNXOXQYFUURVMXQUURJUVCXDXEABXDXPWLWBVTWMWNWOWPWRWSWNWTXBXMUTZXAWTUUE
      UVGUUFCBDVFUPWIVIVNWO $.
  $}

  $( Omniscience is invariant with respect to equinumerosity.  For example,
     this means that we can express the Limited Principle of Omniscience as
     either ` _om e. Omni ` or ` NN0 e. Omni ` .  The former is a better match
     to conventional notation in the sense that ~ df2o3 says that
     ` 2o = { (/) , 1o } ` whereas the corresponding relationship does not
     exist between ` 2 ` and ` { 0 , 1 } ` .  (Contributed by Jim Kingdon,
     13-Jul-2022.) $)
  enomni $p |- ( A ~~ B -> ( A e. Omni <-> B e. Omni ) ) $=
    ( cen wbr comni wcel enomnilem wi ensym syl impbid ) ABCDZAEFZBEFZABGLBACDN
    MHABIBAGJK $.

  ${
    $d A w y z $.  $d f g x y $.  $d g u x y z $.
    $( A finite set is omniscient.  Remark right after Definition 3.1 of
       [Pierik], p. 14.  (Contributed by Jim Kingdon, 28-Jun-2022.) $)
    finomni $p |- ( A e. Fin -> A e. Omni ) $=
      ( vz vf vx vg vu cv comni wcel c0 c2o cfv wceq c1o wral wo wi cvv eqeq1d
      wa vw vy csn cun eleq1 wf wrex wal wb 0ex isomni ral0 olci a1i mpgbir cfn
      ax-mp cres elun1 ad2antlr fvres simpr eqtr3d fveq2 rspcev syl2anc orcd ex
      rexlimdva vsnid elun2 sylan a1d cbvralv ralbiia bitri sylib simplr sylibr
      vex ralsn ralun olcd cpr ffvelcdmd df2o3 eleqtrdi elpri syl biimpi adantl
      mpjaodan resex feq1 fveq1 rexbidv ralbidv orbi12d imbi12d spcv wss fssres
      ssun1 mpan2 impel mpjaod alrimiv snex unex findcard2 ) UAGZHIJHIZUBGZHIZX
      MBGZUCZUDZHIZAHIUAUBBAXKJHUEXKXMHUEXKXQHUEXKAHUEXLJKCGZUFZDGZXSLZJMZDJUGZ
      YBNMZDJOZPZQZCJRIXLYHCUHUIUJDJCRUKUQYGXTYFYDYEDULUMUNUOXMUPIZXNXRYIXNTZXQ
      KEGZUFZFGZYKLZJMZFXQUGZYNNMZFXQOZPZQZEUHZXRYJYTEYJYLYSYJYLTZYAYKXMURZLZJM
      ZDXMUGZYSUUDNMZDXMOZUUBUUEYSDXMUUBYAXMIZTZUUEYSUUJUUETZYPYRUUKYAXQIZYAYKL
      ZJMZYPUUIUULUUBUUEYAXMXPUSUTUUKUUDUUMJUUIUUDUUMMUUBUUEYAXMYKVAUTUUJUUEVBV
      CYOUUNFYAXQYMYAMYNUUMJYMYAYKVDSVEVFVGVHVIUUBXOYKLZJMZUUHYSQUUONMZUUBUUPTZ
      YSUUHUURYPYRUUBXOXQIZUUPYPUUSUUBXOXPIUUSBVJXOXPXMVKUQUNZYOUUPFXOXQYMXOMZY
      NUUOJYMXOYKVDZSVEVLVGVMUUBUUQTZUUHYSUVCUUHTZYRYPUVDYQFXMOZYQFXPOZYRUVDUUH
      UVEUVCUUHVBUUHYMUUCLZNMZFXMOUVEUUGUVHDFXMYAYMMUUDUVGNYAYMUUCVDSVNUVHYQFXM
      YMXMIUVGYNNYMXMYKVASVOVPVQUVDUUQUVFUUBUUQUUHVRYQUUQFXOBVTZUVAYNUUONUVBSWA
      VSYQFXMXPWBVFWCVHUUBUUOJNWDZIUUPUUQPUUBUUOKUVJUUBXQKXOYKYJYLVBUUTWEWFWGUU
      OJNWHWIWLYJXMKUUCUFZUUFUUHPZYLYJXMKXSUFZYCDXMUGZYEDXMOZPZQZCUHZUVKUVLQZXN
      UVRYIXNUVRXMRIXNUVRUIUBVTZDXMCRUKUQWJWKUVQUVSCUUCYKXMEVTWMXSUUCMZUVMUVKUV
      PUVLXMKXSUUCWNUWAUVNUUFUVOUUHUWAYCUUEDXMUWAYBUUDJYAXSUUCWOZSWPUWAYEUUGDXM
      UWAYBUUDNUWBSWQWRWSWTWIYLXMXQXAUVKXMXPXCXQKXMYKXBXDXEXFVHXGXQRIXRUUAUIXMX
      PUVTXOUVIXHXIFXQERUKUQVSVHXJ $.
  $}

  ${
    $d f x y $.
    $( Given excluded middle, every set is omniscient.  Remark following
       Definition 3.1 of [Pierik], p. 14.  This is one direction of the
       biconditional ~ exmidomni .  (Contributed by Jim Kingdon,
       29-Jun-2022.) $)
    exmidomniim $p |- ( EXMID -> A. x x e. Omni ) $=
      ( vf vy wem cv wcel c2o c0 wceq c1o wo wi wa wn wdc exmidexmid syl adantr
      wral wb comni wf cfv wal exmiddc orcomd cpr ffvelcdm df2o3 eleqtrdi elpri
      wrex ord ralimdva con3d adantl dfrex2dc sylibrd orim1d mpd ex alrimiv cvv
      isomni elv sylibr ) DAEZUAFZADVGGBEZUBZCEZVIUCZHIZCVGULZVLJIZCVGSZKZLZBUD
      ZVHDVRBDVJVQDVJMZVPNZVPKZVQDWBVJDVPWADVPOVPWAKVPPVPUEQUFRVTWAVNVPVTWAVMNZ
      CVGSZNZVNVJWAWELDVJWDVPVJWCVOCVGVJVKVGFMZVMVOWFVLHJUGZFVMVOKWFVLGWGVGGVKV
      IUHUIUJVLHJUKQUMUNUOUPDVNWETZVJDVNOWHVNPVMCVGUQQRURUSUTVAVBVHVSTACVGBVCVD
      VEVFVB $.
  $}

  ${
    $d u f x y $.
    $( Excluded middle is equivalent to every set being omniscient.
       (Contributed by BJ and Jim Kingdon, 30-Jun-2022.) $)
    exmidomni $p |- ( EXMID <-> A. x x e. Omni ) $=
      ( vu vy vf cv comni wcel wal c0 wceq wo wi wa cfv wrex c1o wral eqeq1d ex
      c2o wem exmidomniim csn wss cxp eleq1w xpeq1 fveq1d rexeqbi1dv raleqbi1dv
      vex spcv orbi12d wf cvv wb isomni ax-mp biimpi cpr prid1 eleqtrri fconst6
      0ex df2o3 p0ex xpex feq1 fveq1 rexbidv ralbidv imbi12d mpisyl vtoclga syl
      adantr simplr wex rexm sssnm adantl mpbid nfv nfan nfcv 1n0 neii r19.21bi
      nfra1 simpr fvconst2 eqtr3d mtoi pm2.21d ssrd ss0 orim12d alrimiv exmid01
      mpd orcomd sylibr impbii ) UAAEZFGZAHZAUBXFBEZIUCZUDZXGIJZXGXHJZKZLZBHUAX
      FXMBXFXIXLXFXIMZXKXJXNCEZXGXHUEZNZIJZCXGOZXQPJZCXGQZKZXKXJKXFYBXIXFXGFGZY
      BXEYCAXGBUKABFUFULXOXDXHUEZNZIJZCXDOZYEPJZCXDQZKZYBAXGFXDXGJZYGXSYIYAYFXR
      CXDXGYKYEXQIYKXOYDXPXDXGXHUGUHZRUIYHXTCXDXGYKYEXQPYLRUJUMXEXDTDEZUNZXOYMN
      ZIJZCXDOZYOPJZCXDQZKZLZDHZXDTYDUNZYJXEUUBXDUOGXEUUBUPAUKZCXDDUOUQURUSXDIT
      IIPUTTIPVDVAVEVBVCUUAUUCYJLDYDXDXHUUDVFVGYMYDJZYNUUCYTYJXDTYMYDVHUUEYQYGY
      SYIUUEYPYFCXDUUEYOYEIXOYMYDVIZRVJUUEYRYHCXDUUEYOYEPUUFRVKUMVLULVMVNVOVPXN
      XSXKYAXJXNXSXKXNXSMXIXKXFXIXSVQXSXIXKUPZXNXSXOXGGZCVRUUGXRCXGVSCXGIVTVOWA
      WBSXNYAXJXNYAMZXGIUDXJUUICXGIXNYACXNCWCXTCXGWIWDCXGWECIWEUUIUUHXOIGUUIUUH
      PIJZPIWFWGUUIUUHUUJUUIUUHMXQPIUUIXTCXGXNYAWJWHUUHXRUUIXGIXOVDWKWAWLSWMWNW
      OXGWPVOSWQWTXASWRBWSXBXC $.
  $}

  $( Excluded middle implies the Limited Principle of Omniscience (LPO).
     (Contributed by Jim Kingdon, 29-Mar-2023.) $)
  exmidlpo $p |- ( EXMID -> _om e. Omni ) $=
    ( vx wem cv comni wcel wal com exmidomni omex eleq1 spcv sylbi ) BACZDEZAFG
    DEZAHNOAGIMGDJKL $.

  ${
    $d A z $.  $d B w z $.  $d F w z $.  $d O w z $.  $d X w z $.  $d ph w z $.
    fodjuomnilemdc.fo $e |- ( ph -> F : O -onto-> ( A |_| B ) ) $.
    $( Lemma for ~ fodjuomni .  Decidability of a condition we use in various
       lemmas.  (Contributed by Jim Kingdon, 27-Jul-2022.) $)
    fodjuomnilemdc $p |- ( ( ph /\ X e. O )
        -> DECID E. z e. A ( F ` X ) = ( inl ` z ) ) $=
      ( vw wcel wa cfv cv wceq wrex wn wo cinr sylib cvv cinl wdc wf ffvelcdmda
      cdju wfo fof syl djur wral nfv nfre1 nfan simpr eqeq2d cbvrexv wi wne vex
      fveq2 djune mp2an neeq2 mpbiri necomd neneqd a1i rexlimdvw mpd a1d ralnex
      ralrimi ex orim2d df-dc sylibr ) AGFJKZGELZBMZUALZNZBCOZWBPZQZWBUBVQWBVRV
      SRLZNZBDOZQZWDVQVRCDUEZJWHAFWIGEAFWIEUFFWIEUCHFWIEUGUHUDBCDVRUISVQWGWCWBV
      QWGWCVQWGKZWAPZBCUJWCWJWKBCVQWGBVQBUKWFBDULUMWJWKVSCJWJVRIMZRLZNZIDOZWKWJ
      WGWOVQWGUNWFWNBIDVSWLNWEWMVRVSWLRUTUOUPSWJWNWKIDWNWKUQWJWNVRVTWNVTVRWNVTV
      RURVTWMURZVSTJWLTJWPBUSIUSVSWLTTVAVBVRWMVTVCVDVEVFVGVHVIVJVLWABCVKSVMVNVI
      WBVOVP $.
  $}

  ${
    fodjuf.fo $e |- ( ph -> F : O -onto-> ( A |_| B ) ) $.
    $d ph y z $.  $d O y z $.  $d A z $.  $d B z $.  $d F z $.
    fodjuf.p $e |- P = ( y e. O |-> if (
      E. z e. A ( F ` y ) = ( inl ` z ) , (/) , 1o ) ) $.
    ${
      fodjuf.o $e |- ( ph -> O e. V ) $.
      $( Lemma for ~ fodjuomni and ~ fodjumkv .  Domain and range of ` P ` .
         (Contributed by Jim Kingdon, 27-Jul-2022.)  (Revised by Jim Kingdon,
         25-Mar-2023.) $)
      fodjuf $p |- ( ph -> P e. ( 2o ^m O ) ) $=
        ( c2o wcel cv cfv c0 c1o a1i com cmap co wf cinl wceq wrex cif wa 0lt2o
        1lt2o fodjuomnilemdc ifcldcd fmptd 2onn elmapd mpbird ) AFMHUAUBNHMFUCA
        BHBOZGPCOUDPUECDUFZQRUGMFAUQHNUHZURQRMQMNUSUISRMNUSUJSACDEGHUQJUKULKUMA
        MHFTIMTNAUNSLUOUP $.
    $}

    ${
      $d A w x z $.  $d A w y z $.  $d B z $.  $d F y z $.  $d O y z $.
      $d ph w y z $.
      fodjum.z $e |- ( ph -> E. w e. O ( P ` w ) = (/) ) $.
      $( Lemma for ~ fodjuomni and ~ fodjumkv .  A condition which shows that
         ` A ` is inhabited.  (Contributed by Jim Kingdon, 27-Jul-2022.)
         (Revised by Jim Kingdon, 25-Mar-2023.) $)
      fodjum $p |- ( ph -> E. x x e. A ) $=
        ( cv cfv c0 wceq wcel wa c1o wex cinl wrex wn 1n0 nesymi intnan a1i cif
        wo simprr com fveqeq2 rexbidv simprl peano1 1onn fodjuomnilemdc adantrr
        ifbid wdc ifcldcd fvmptd3 eqtr3d wb eqifdc syl mpbid ecased simpld rexm
        eleq1w cbvexv sylib rexlimddv ) AENZHOZPQZBNFRZBUAZEJMAVPJRZVRSSZDNZFRZ
        DUAZVTWBVPIOWCUBOZQZDFUCZWEWBWHPPQZWBWHWISZWHUDZPTQZSZWMUDWBWLWKTPUEUFU
        GUHWBPWHPTUIZQZWJWMUJZWBVQPWNAWAVRUKWBCVPCNZIOWFQZDFUCZPTUIWNJHULLWQVPQ
        ZWSWHPTWTWRWGDFWQVPWFIUMUNUTAWAVRUOWBWHPTULPULRWBUPUHTULRWBUQUHAWAWHVAZ
        VRADFGIJVPKURUSZVBVCVDWBXAWOWPVEXBWHPPTVFVGVHVIVJWGDFVKVGWDVSDBDBFVLVMV
        NVO $.
    $}

    ${
      $d A u v y z $.  $d B z $.  $d F v y z $.  $d O v w $.  $d O v y z $.
      $d P w $.  $d ph u v y z $.
      fodju0.1 $e |- ( ph -> A. w e. O ( P ` w ) = 1o ) $.
      $( Lemma for ~ fodjuomni and ~ fodjumkv .  A condition which shows that
         ` A ` is empty.  (Contributed by Jim Kingdon, 27-Jul-2022.)  (Revised
         by Jim Kingdon, 25-Mar-2023.) $)
      fodju0 $p |- ( ph -> A = (/) ) $=
        ( vv cv wcel c0 c1o wceq cfv com vu wa cinl cdju wfo wrex djulcl foelrn
        syl2an fveqeq2 rexbidv ifbid simprl peano1 1onn fodjuomnilemdc ad2ant2r
        cif a1i wdc ifcldcd fvmptd3 ad2antrr rspcdva simplr simprr eqcomd fveq2
        wral rspceeqv syl2anc iftrued 3eqtr3rd rexlimddv wn 1n0 nesymi pm2.65da
        eq0rdv ) AUAEAUANZEOZPQRZAWAUBZVTUCSZMNZHSZRZWBMIAIEFUDZHUEWDWHOWGMIUFW
        AJEFVTUGMIWHWDHUHUIWCWEIOZWGUBZUBZWEGSZWFCNZUCSZRZCEUFZPQURZQPWKBWEBNZH
        SWNRZCEUFZPQURWQIGTKWRWERZWTWPPQXAWSWOCEWRWEWNHUJUKULWCWIWGUMZWKWPPQTPT
        OWKUNUSQTOWKUOUSAWIWPUTWAWGACEFHIWEJUPUQVAVBWKDNZGSQRZWLQRDIWEXCWEQGUJA
        XDDIVIWAWJLVCXBVDWKWPPQWKWAWFWDRWPAWAWJVEWKWDWFWCWIWGVFVGCVTEWNWDWFWMVT
        UCVHVJVKVLVMVNWBVOWCQPVPVQUSVRVS $.
    $}
  $}

  ${
    fodjuomni.o $e |- ( ph -> O e. Omni ) $.
    fodjuomni.fo $e |- ( ph -> F : O -onto-> ( A |_| B ) ) $.

    ${
      $d ph y z $.  $d O y z $.  $d A z $.  $d B z $.  $d F z $.
      fodjuomni.p $e |- P = ( y e. O |-> if (
        E. z e. A ( F ` y ) = ( inl ` z ) , (/) , 1o ) ) $.

      $d A v x z $.  $d A v y z $.  $d B z $.  $d F y z $.  $d O f w $.
      $d O v w y z $.  $d P f w $.  $d P v w y z $.  $d ph v y z $.
      $( Lemma for ~ fodjuomni .  The final result with ` P ` expressed as a
         local definition.  (Contributed by Jim Kingdon, 29-Jul-2022.) $)
      fodjuomnilemres $p |- ( ph -> ( E. x x e. A \/ A = (/) ) ) $=
        ( vw vf vv cv cfv c0 wceq wrex c1o wral wo wcel wex c2o co fveq1 eqeq1d
        cmap rexbidv ralbidv orbi12d comni wb isomnimap mpbid fodjuf rspcdva wa
        syl wfo adantr simpr fveqeq2 cbvrexv sylib fodjum ex fodju0 orim12d mpd
        cdju ) AMPZGQZRSZMITZVOUASZMIUBZUCZBPEUDBUEZERSZUCAVNNPZQZRSZMITZWDUASZ
        MIUBZUCZVTNUFIUJUGZGWCGSZWFVQWHVSWKWEVPMIWKWDVORVNWCGUHZUIUKWKWGVRMIWKW
        DVOUAWLUIULUMAIUNUDZWINWJUBZJAWMWMWNUOJMINUNUPVAUQACDEFGHIUNKLJURUSAVQW
        AVSWBAVQWAAVQUTZBCDOEFGHIAIEFVMHVBZVQKVCLWOVQOPZGQRSZOITAVQVDVPWRMOIVNW
        QRGVEVFVGVHVIAVSWBAVSUTCDMEFGHIAWPVSKVCLAVSVDVJVIVKVL $.
    $}

    ${
      $d A a b x y z $.  $d B z $.  $d F a b y z $.  $d O a y z $.
      $d ph y z $.
      $( A condition which ensures ` A ` is either inhabited or empty.  Lemma
         3.2 of [PradicBrown2022], p. 4.  (Contributed by Jim Kingdon,
         27-Jul-2022.) $)
      fodjuomni $p |- ( ph -> ( E. x x e. A \/ A = (/) ) ) $=
        ( vy vz va vb cv cfv cinl wceq wrex c0 c1o cif cmpt eqeq2d cbvrexv ifbi
        fveq2 ax-mp mpteq2i eqeq1d rexbidv ifbid cbvmptv eqtri fodjuomnilemres
        wb ) ABIJCDKFKMZENZLMZONZPZLCQZRSTZUAZEFGHVBKFUPJMZONZPZJCQZRSTZUAIFIMZ
        ENZVDPZJCQZRSTZUAKFVAVGUTVFUNVAVGPUSVELJCUQVCPURVDUPUQVCOUEUBUCUTVFRSUD
        UFUGKIFVGVLUOVHPZVFVKRSVMVEVJJCVMUPVIVDUOVHEUEUHUIUJUKULUM $.
    $}
  $}

  ${
    $d f ph w z $.  $d f ph x y z $.
    ctssexmid.1 $e |- ( ( y C_ _om /\ E. f f : y -onto-> x )
      -> E. f f : _om -onto-> ( x |_| 1o ) ) $.
    ctssexmid.lpo $e |- _om e. Omni $.
    $( The decidability condition in ~ ctssdc is needed.  More specifically,
       ~ ctssdc minus that condition, plus the Limited Principle of Omniscience
       (LPO), implies excluded middle.  (Contributed by Jim Kingdon,
       15-Aug-2023.) $)
    ctssexmid $p |- ( ph \/ -. ph ) $=
      ( vz vw vu com c1o cv wfo wex wcel c0 wceq cvv mp2b wa crab wo wss ssrab2
      cdju cid cres wf1o f1oi f1ofo comni elexi rabex resiexg ax-mp foeq1 spcev
      wn wi simpr sseq1d eqidd simpl foeq123d exbidv anbi12d foeq3 3syl imbi12d
      wb djueq1 vtocl2 mp2an a1i id fodjuomni exlimiv biidd simprbi wral rabeq0
      elrab peano1 elex2 r19.3rmv sylbb2 orim12i ) JAGJUAZKUEZDLZMZDNZHLZWHOZHN
      ZWHPQZUBZAAURZUBWHJUCZWHWHWJMZDNZWLAGJUDWHWHUFWHUGZUHWHWHXBMZXAWHUIWHWHXB
      UJWTXCDXBWHROXBROAGJJUKFULUMZWHRUNUOWHWHWJXBUPUQSCLZJUCZXEBLZWJMZDNZTZJXG
      KUEZWJMZDNZUSWSXATZWLUSBCWHWHXDXDXGWHQZXEWHQZTZXJXNXMWLXQXFWSXIXAXQXEWHJX
      OXPUTZVAXQXHWTDXQXEWHXGWHWJWJXQWJVBXRXOXPVCZVDVEVFXQXLWKDXQXOXKWIQXLWKVJX
      SXGWHKVKXKWIJWJVGVHVEVIEVLVMWKWQDWKHWHKWJJJUKOWKFVNWKVOVPVQWOAWPWRWNAHWNW
      MJOAAAGWMJGLWMQAVRWBVSVQWPWRGJVTZWRAGJWAPJOILJOINWRXTVJWCIPJWDWRGIJWESWFW
      GS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Markov's principle
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c Markov $.  $( Class of all Markov sets. $)

  $( Extend class definition to include the class of Markov sets. $)
  cmarkov $a class Markov $.

  ${
    $d f x y $.
    $( A Markov set is one where if a predicate (here represented by a function
       ` f ` ) on that set does not hold (where hold means is equal to ` 1o ` )
       for all elements, then there exists an element where it fails (is equal
       to ` (/) ` ).  Generalization of definition 2.5 of [Pierik], p. 9.

       In particular, ` _om e. Markov ` is known as Markov's Principle (MP).
       (Contributed by Jim Kingdon, 18-Mar-2023.) $)
    df-markov $a |- Markov = { y | A. f ( f : y --> 2o
        -> ( -. A. x e. y ( f ` x ) = 1o -> E. x e. y ( f ` x ) = (/) ) ) } $.
  $}

  ${
    $d A f x y $.
    $( The predicate of being Markov.  (Contributed by Jim Kingdon,
       18-Mar-2023.) $)
    ismkv $p |- ( A e. V -> ( A e. Markov <-> A. f ( f : A --> 2o
        -> ( -. A. x e. A ( f ` x ) = 1o
          -> E. x e. A ( f ` x ) = (/) ) ) ) ) $=
      ( vy cv c2o wf cfv c1o wceq wral wn c0 wrex wi wal cmarkov feq2 imbi12d
      raleq notbid rexeq albidv df-markov elab2g ) EFZGCFZHZAFUHIZJKZAUGLZMZUJN
      KZAUGOZPZPZCQBGUHHZUKABLZMZUNABOZPZPZCQEBRDUGBKZUQVCCVDUIURUPVBUGBGUHSVDU
      MUTUOVAVDULUSUKAUGBUAUBUNAUGBUCTTUDAECUEUF $.
  $}

  ${
    $d A f x $.  $d V f $.
    $( The predicate of being Markov stated in terms of set exponentiation.
       (Contributed by Jim Kingdon, 18-Mar-2023.) $)
    ismkvmap $p |- ( A e. V -> ( A e. Markov <-> A. f e. ( 2o ^m A )
        ( -. A. x e. A ( f ` x ) = 1o
          -> E. x e. A ( f ` x ) = (/) ) ) ) $=
      ( wcel cmarkov cv c2o cmap co cfv c1o wceq wral wn c0 wrex wi wal com wf
      ismkv wb 2onn elmapg mpan imbi1d albidv bitr4d df-ral bitr4di ) BDEZBFEZC
      GZHBIJZEZAGUNKZLMABNOUQPMABQRZRZCSZURCUONULUMBHUNUAZURRZCSUTABCDUBULUSVBC
      ULUPVAURHTEULUPVAUCUDHBUNTDUEUFUGUHUIURCUOUJUK $.
  $}

  ${
    $d A f g x z $.  $d V f g x z $.
    $( The predicate of being Markov stated in terms of double negation and
       comparison with ` 1o ` .  (Contributed by Jim Kingdon, 29-Nov-2023.) $)
    ismkvnex $p |- ( A e. V -> ( A e. Markov <-> A. f e. ( 2o ^m A )
        ( -. -. E. x e. A ( f ` x ) = 1o
          -> E. x e. A ( f ` x ) = 1o ) ) ) $=
      ( vz vg wcel cfv c1o wceq wn c2o wral wa cdif c0 eqeq1d notbid cvv adantl
      cmarkov cv wrex wi cmap co cmpt fveq1 ralbidv rexbidv imbi12d elex biimpd
      ismkvmap mpcom adantr elmapi ffvelcdmda 2oconcl syl fmpttd com 2onn simpl
      wf a1i elmapd mpbird rspcdva eqid fveq2 difeq2d simpr 1oex difexg fvmptd3
      mp1i difeq2 dif0 eqtrdi 1n0 nesymi eqeq1 mtbiri difid eqeq1i mtbir notnot
      wb 2falsed cpr df2o3 eleqtrdi elpri mpjaodan bitrd ralbidva ralnex notbii
      2thd wo bitrdi nemtbir rexbidva 3imtr3d ralrimiva simplr simpll nfv nfre1
      nfcv nfim nfralxy nfan notnotd ad2antlr ffvelcdmd ralbida bitr3id rexbida
      nfn nsyl3 biimprd sylc ex impbid2 ) BDGZBUAGZAUBZCUBZHZIJZABUCZKZKZYMUDZC
      LBUEUFZMZYHYPCYQYHYJYQGZNZYIEBIEUBZYJHZOZUGZHZIJZABMZKZUUEPJZABUCZYOYMYTY
      IFUBZHZIJZABMZKZUULPJZABUCZUDZUUHUUJUDFYQUUDUUKUUDJZUUOUUHUUQUUJUUSUUNUUG
      UUSUUMUUFABUUSUULUUEIYIUUKUUDUHZQUIRUUSUUPUUIABUUSUULUUEPUUTQUJUKYHUURFYQ
      MZYSBSGZYHUVABUAULUVBYHUVAABFSUNZUMUOUPYTUUDYQGBLUUDVEYTEBUUCLYTUUABGZNUU
      BLGUUCLGYTBLUUAYJYSBLYJVEYHYJLBUQTZURUUBUSUTVAYTLBUUDVBUALVBGZYTVCVFYHYSV
      DVGVHVIYTUUHYLKZABMZKYOYTUUGUVHYTUUFUVGABYTYIBGZNZUUFIYKOZIJZUVGUVJUUEUVK
      IUVJEYIUUCUVKBUUDSUUDVJUUAYIJZUUBYKIUUAYIYJVKVLYTUVIVMISGZUVKSGUVJVNIYKSV
      OVQVPZQUVJYKPJZUVLUVGWIYLUVJUVPNZUVLUVGUVPUVLUVJUVPUVKIPOZIYKPIVRZIVSZVTT
      UVPUVGUVJUVPYLPIJZIPWAWBZYKPIWCWDTZWTUVJYLNZUVLUVGYLUVLKUVJYLUVLIIOZIJZUW
      FUWAUWBUWEPIIWEZWFWGZYLUVKUWEIYKIIVRZQWDTYLUVGKUVJYLWHTWJUVJYKPIWKZGUVPYL
      XAUVJYKLUWJYTBLYIYJUVEURWLWMYKPIWNUTZWOWPWQRUVHYNYLABWRWSXBYTUUIYLABUVJUU
      IUVKPJZYLUVJUUEUVKPUVOQUVJUVPUWLYLWIYLUVQUWLYLUVPUWLKUVJUVPUWLUVRPJZUWMIP
      WAUVRIPUVTWFXCUVPUVKUVRPUVSQWDTUWCWJUWDUWLYLYLUWLUVJYLUVKUWEPUWIUWGVTTUVJ
      YLVMWTUWKWOWPXDXEXFYGYRYHYGYRNZUVBUVAYHYGUVBYRBDULUPUWNUURFYQUWNUUKYQGZNZ
      YIEBIUUAUUKHZOZUGZHZIJZABUCZKZKZUXBUUOUUQUWPYPUXDUXBUDCYQUWSYJUWSJZYOUXDY
      MUXBUXEYNUXCUXEYMUXBUXEYLUXAABUXEYKUWTIYIYJUWSUHQUJZRRUXFUKYGYRUWOXGUWPUW
      SYQGBLUWSVEUWPEBUWRLUWPUVDNUWQLGUWRLGUWPBLUUAUUKUWOBLUUKVEZUWNUUKLBUQZTUR
      UWQUSUTVAUWPLBUWSVBDUVFUWPVCVFYGYRUWOXHVGVHVIUXDUXAKZABMZKUWPUUOUXJUXCUXA
      ABWRWSUWPUXJUUNUWPUXIUUMABUWNUWOAYGYRAYGAXIYPACYQAYQXKYOYMAYNAYMAYLABXJZY
      AYAUXKXLXMXNUWOAXIXNZUWPUVINZUXIIUULOZIJZKZUUMUXMUXAUXOUXMUWTUXNIUXMEYIUW
      RUXNBUWSSUWSVJUVMUWQUULIUUAYIUUKVKVLUWPUVIVMZUVNUXNSGUXMVNIUULSVOVQVPQZRU
      XMUUPUXPUUMWIUUMUXMUUPNZUXPUUMUXSUXOUUPUXOUXMUUPUXNUVRIUULPIVRUVTVTTZXOUU
      PUUMKUXMUUPUUMUWAUWBUULPIWCWDZTWJUXMUUMNZUXPUUMUUMUXPUXMUUMUXOUWFUWHUUMUX
      NUWEIUULIIVRQWDTZUXMUUMVMZWTUXMUULUWJGUUPUUMXAUXMUULLUWJUXMBLYIUUKUWOUXGU
      WNUVIUXHXPUXQXQWLWMUULPIWNUTZWOWPXRRXSUWPUXAUUPABUXLUXMUXAUXOUUPUXRUXMUUP
      UXOUUPWIUUMUXSUXOUUPUXTUXMUUPVMWTUYBUXOUUPUYCUUPUUMUYBUYAUYDYBWJUYEWOWPXT
      XEXFUVBYHUVAUVCYCYDYEYF $.
  $}

  ${
    $d A f x $.
    $( An omniscient set is Markov.  In particular, the case where ` A ` is
       ` _om ` means that the Limited Principle of Omniscience (LPO) implies
       Markov's Principle (MP).  (Contributed by Jim Kingdon, 18-Mar-2023.) $)
    omnimkv $p |- ( A e. Omni -> A e. Markov ) $=
      ( vf vx comni wcel cmarkov c2o cv wf cfv c1o wceq wral wn wrex wal isomni
      c0 wi wo ibi pm2.53 orcoms a1i imim2d alimdv mpd ismkv mpbird ) ADEZAFEAG
      BHZIZCHUKJZKLCAMZNUMRLCAOZSZSZBPZUJULUOUNTZSZBPZURUJVACABDQUAUJUTUQBUJUSU
      PULUSUPSUJUNUOUPUNUOUBUCUDUEUFUGCABDUHUI $.
  $}

  $( Excluded middle implies Markov's Principle (MP).  (Contributed by Jim
     Kingdon, 4-Apr-2023.) $)
  exmidmp $p |- ( EXMID -> _om e. Markov ) $=
    ( wem com comni wcel cmarkov exmidlpo omnimkv syl ) ABCDBEDFBGH $.

  ${
    $d A f n $.  $d f ph $.
    $( Markov's Principle expressed in terms of propositions (or more
       precisely, the ` A = _om ` case is Markov's Principle).  (Contributed by
       Jim Kingdon, 19-Mar-2023.) $)
    mkvprop $p |- (
        ( A e. Markov /\ A. n e. A DECID ph /\ -. A. n e. A -. ph )
        -> E. n e. A ph ) $=
      ( vf cmarkov wcel wral wn cv c0 c1o cfv wceq wa nfra1 c2o simpr eqeq1d wi
      wrex wdc w3a cif cmpt nfv nfan 0lt2o a1i 1lt2o rsp ifcldcd adantll fvmpt2
      imp syl2anc 1n0 nesymi iftrue mtbiri con2i biimtrdi ralimdaa con3d 3impia
      eqid cvv wf wal mptexg 3ad2ant1 ismkv ibi nfn nf3an 3ad2antl2 fmptdf feq1
      nfmpt1 nfeq2 fveq1 ralbid notbid rexbid imbi12d spcgv syl3c 3ad2ant2 neii
      mpd iffalsed ex con2d notnotrdc sylsyld impbid1 bitrd rexbida mpbid ) BEF
      ZAUAZCBGZAHZCBGZHZUBZCIZCBAJKUCZUDZLZJMZCBTZACBTXEXIKMZCBGZHZXKWSXAXDXNWS
      XANZXMXCXOXLXBCBWSXACWSCUEZWTCBOZUFXOXFBFZNZXLXGKMZXBXSXIXGKXSXRXGPFZXIXG
      MZXOXRQXAXRYAWSXAXRNZAJKPJPFYCUGUHKPFYCUIUHXAXRWTWTCBUJZUNUKZULCBXGPXHXHV
      EZUMZUORAXTAXTJKMKJUPUQAXGJKAJKURZRUSUTVAVBVCVDXEXHVFFZBPDIZVGZXFYJLZKMZC
      BGZHZYLJMZCBTZSZSZDVHZBPXHVGZXNXKSZWSXAYIXDCBXGEVIVJWSXAYTXDWSYTCBDEVKVLV
      JXECBXGPXHWSXAXDCXPXQXCCXBCBOVMVNZXAWSXRYAXDYEVOZYFVPYSUUAUUBSDXHVFYJXHMZ
      YKUUAYRUUBBPYJXHVQUUEYOXNYQXKUUEYNXMUUEYMXLCBCYJXHCBXGVRVSZUUEYLXIKXFYJXH
      VTZRWAWBUUEYPXJCBUUFUUEYLXIJUUGRWCWDWDWEWFWIXEXJACBUUCXEXRNZXJXGJMZAUUHXI
      XGJUUHXRYAYBXEXRQUUDYGUORUUHUUIAUUHWTUUIXBHAXEXRWTXAWSXRWTSXDYDWGUNUUHXBU
      UIUUHXBUUIHUUHXBNZUUIKJMKJUPWHUUJXGKJUUJAJKUUHXBQWJRUSWKWLAWMWNYHWOWPWQWR
      $.
  $}

  ${
    fodjumkv.o $e |- ( ph -> M e. Markov ) $.
    fodjumkv.fo $e |- ( ph -> F : M -onto-> ( A |_| B ) ) $.

    ${
      $d ph y z $.  $d M y z $.  $d A z $.  $d B z $.  $d F z $.
      fodjumkv.p $e |- P = ( y e. M |-> if (
        E. z e. A ( F ` y ) = ( inl ` z ) , (/) , 1o ) ) $.

      $d A v x z $.  $d A v y z $.  $d B z $.  $d F y z $.  $d M f w $.
      $d M v w y z $.  $d P f w $.  $d P v w y z $.  $d ph v y z $.
      $( Lemma for ~ fodjumkv .  The final result with ` P ` expressed as a
         local definition.  (Contributed by Jim Kingdon, 25-Mar-2023.) $)
      fodjumkvlemres $p |- ( ph -> ( A =/= (/) -> E. x x e. A ) ) $=
        ( vw vf vv c0 cv cfv c1o wceq wne wral wn wrex wcel wex cdju wfo adantr
        wa simpr fodju0 ex necon3ad wi c2o cmap co fveq1 eqeq1d ralbidv rexbidv
        notbid imbi12d cmarkov ismkvmap ibi fodjuf rspcdva fveqeq2 sylib fodjum
        syl cbvrexv 3syld ) AEPUAMQZGRZSTZMIUBZUCZVQPTZMIUDZBQEUEBUFZAVSEPAVSEP
        TAVSUJCDMEFGHIAIEFUGHUHZVSKUILAVSUKULUMUNAVPNQZRZSTZMIUBZUCZWFPTZMIUDZU
        OZVTWBUONUPIUQURZGWEGTZWIVTWKWBWNWHVSWNWGVRMIWNWFVQSVPWEGUSZUTVAVCWNWJW
        AMIWNWFVQPWOUTVBVDAIVEUEZWLNWMUBZJWPWQMINVEVFVGVMACDEFGHIVEKLJVHVIAWBWC
        AWBUJZBCDOEFGHIAWDWBKUILWRWBOQZGRPTZOIUDAWBUKWAWTMOIVPWSPGVJVNVKVLUMVO
        $.
    $}

    ${
      $d A a b x y z $.  $d B z $.  $d F a b y z $.  $d M a y z $.
      $d ph y z $.
      $( A condition which ensures that a nonempty set is inhabited.
         (Contributed by Jim Kingdon, 25-Mar-2023.) $)
      fodjumkv $p |- ( ph -> ( A =/= (/) -> E. x x e. A ) ) $=
        ( vy vz va vb cv cfv cinl wceq wrex c0 c1o cif cmpt eqeq2d cbvrexv ifbi
        fveq2 ax-mp mpteq2i fveqeq2 rexbidv ifbid cbvmptv eqtri fodjumkvlemres
        wb ) ABIJCDKFKMZENZLMZONZPZLCQZRSTZUAZEFGHVBKFUPJMZONZPZJCQZRSTZUAIFIMZ
        ENVDPZJCQZRSTZUAKFVAVGUTVFUNVAVGPUSVELJCUQVCPURVDUPUQVCOUEUBUCUTVFRSUDU
        FUGKIFVGVKUOVHPZVFVJRSVLVEVIJCUOVHVDEUHUIUJUKULUM $.
    $}
  $}

  ${
    $d A f g h x $.  $d A g h x y $.  $d B g h x y $.
    $( Lemma for ~ enmkv .  One direction of the biconditional.  (Contributed
       by Jim Kingdon, 25-Jun-2024.) $)
    enmkvlem $p |- ( A ~~ B -> ( A e. Markov -> B e. Markov ) ) $=
      ( vy vg vh vx vf cmarkov wcel wa cv cfv c1o wceq wral wn c0 wrex c2o wf
      cen wbr wi cmap co wf1o wex bren biimpi ad2antrr ccom ccnv f1ofn ad3antlr
      wfn f1ocnv f1of syl simpr ffvelcdmd fvco2 syl2anc fveqeq2 rspcdva simpllr
      simplr f1ocnvfv2 sylancom fveq2d 3eqtr3rd ralrimiva ex con3d fveq1 eqeq1d
      ralbidv notbid rexbidv imbi12d ismkvmap ibi com cvv 2onn brrelex2i elmapg
      relen sylancr mpbid adantr adantl fco mpbird ad2antlr ffvelcdmda fveqeq2d
      wb rspcedv sylbid rexlimdva 3syld exlimddv ) ABUAUBZAHIZBHIZXCXDJZXECKZDK
      ZLZMNZCBOZPZXIQNZCBRZUCZDSBUDUEZOZXFXODXPXFXHXPIZJZABEKZUFZXOEXCYAEUGZXDX
      RXCYBABEUHUIUJXSYAJZXLFKZXHXTUKZLZMNZFAOZPZYFQNZFARZXNYCYHXKYCYHXKYCYHJZX
      JCBYLXGBIZJZXGXTULZLZYELZYPXTLZXHLZMXIYNXTAUOZYPAIYQYSNYAYTXSYHYMABXTUMZU
      NYNBAXGYOYNBAYOUFZBAYOTYAUUBXSYHYMABXTUPUNBAYOUQURYLYMUSUTZAXHXTYPVAVBYNY
      GYQMNFAYPYDYPMYEVCYCYHYMVFUUCVDYNYRXGXHYLYMYAYRXGNXSYAYHYMVEABXGXTVGVHVIV
      JVKVLVMYCYDGKZLZMNZFAOZPZUUEQNZFARZUCZYIYKUCGSAUDUEZYEUUDYENZUUHYIUUJYKUU
      MUUGYHUUMUUFYGFAUUMUUEYFMYDUUDYEVNZVOVPVQUUMUUIYJFAUUMUUEYFQUUNVOVRVSXDUU
      KGUULOZXCXRYAXDUUOFAGHVTWAUNYCYEUULIZASYETZYCBSXHTZABXTTZUUQXSUURYAXSXRUU
      RXFXRUSXCXRUURWQZXDXRXCSWBIZBWCIZUUTWDABUAWGWEZSBXHWBWCWFWHUJWIWJYAUUSXSA
      BXTUQWKZABSXHXTWLVBYCUVAXDUUPUUQWQWDXCXDXRYAVESAYEWBHWFWHWMVDYCYJXNFAYCYD
      AIZJZYJYDXTLZXHLZQNZXNUVFYFUVHQYCUVEYTYFUVHNYAYTXSUVEUUAWNAXHXTYDVAVHVOUV
      FXMUVICUVGBYCABYDXTUVDWOUVFXGUVGNZJXGUVGQXHUVFUVJUSWPWRWSWTXAXBVKXCXEXQWQ
      ZXDXCUVBUVKUVCCBDWCVTURWJWMVL $.
  $}

  $( Being Markov is invariant with respect to equinumerosity.  For example,
     this means that we can express the Markov's Principle as either
     ` _om e. Markov ` or ` NN0 e. Markov ` .  The former is a better match to
     conventional notation in the sense that ~ df2o3 says that
     ` 2o = { (/) , 1o } ` whereas the corresponding relationship does not
     exist between ` 2 ` and ` { 0 , 1 } ` .  (Contributed by Jim Kingdon,
     24-Jun-2024.) $)
  enmkv $p |- ( A ~~ B -> ( A e. Markov <-> B e. Markov ) ) $=
    ( cen wbr cmarkov wcel enmkvlem wi ensym syl impbid ) ABCDZAEFZBEFZABGLBACD
    NMHABIBAGJK $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Weakly omniscient sets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
  $c WOmni $.  $( Class of weakly omniscient sets. $)

  $( Extend class definition to include the class of weakly omniscient sets. $)
  cwomni $a class WOmni $.

  ${
    $d f x y $.
    $( A weakly omniscient set is one where we can decide whether a predicate
       (here represented by a function ` f ` ) holds (is equal to ` 1o ` ) for
       all elements or not.  Generalization of definition 2.4 of [Pierik],
       p. 9.

       In particular, ` _om e. WOmni ` is known as the Weak Limited Principle
       of Omniscience (WLPO).

       The term WLPO is common in the literature; there appears to be no
       widespread term for what we are calling a weakly omniscient set.
       (Contributed by Jim Kingdon, 9-Jun-2024.) $)
    df-womni $a |- WOmni = { y | A. f ( f : y --> 2o
      -> DECID A. x e. y ( f ` x ) = 1o ) } $.
  $}

  ${
    $d A f x y $.
    $( The predicate of being weakly omniscient.  (Contributed by Jim Kingdon,
       9-Jun-2024.) $)
    iswomni $p |- ( A e. V -> ( A e. WOmni <-> A. f ( f : A --> 2o
        -> DECID A. x e. A ( f ` x ) = 1o ) ) ) $=
      ( vy cv c2o wf cfv c1o wceq wral wdc wal cwomni feq2 raleq dcbid imbi12d
      wi albidv df-womni elab2g ) EFZGCFZHZAFUEIJKZAUDLZMZTZCNBGUEHZUGABLZMZTZC
      NEBODUDBKZUJUNCUOUFUKUIUMUDBGUEPUOUHULUGAUDBQRSUAAECUBUC $.
  $}

  ${
    $d A f x $.  $d V f $.
    $( The predicate of being weakly omniscient stated in terms of set
       exponentiation.  (Contributed by Jim Kingdon, 9-Jun-2024.) $)
    iswomnimap $p |- ( A e. V -> ( A e. WOmni <-> A. f e. ( 2o ^m A )
        DECID A. x e. A ( f ` x ) = 1o ) ) $=
      ( wcel cwomni cv c2o cmap co cfv c1o wceq wral wdc wi wal wf iswomni com
      wb 2onn elmapg mpan imbi1d albidv bitr4d df-ral bitr4di ) BDEZBFEZCGZHBIJ
      ZEZAGULKLMABNOZPZCQZUOCUMNUJUKBHULRZUOPZCQUQABCDSUJUPUSCUJUNURUOHTEUJUNUR
      UAUBHBULTDUCUDUEUFUGUOCUMUHUI $.
  $}

  ${
    $d A f x $.
    $( A set is omniscient if and only if it is weakly omniscient and Markov.
       The case ` A = _om ` says that LPO ` <-> ` WLPO ` /\ ` MP which is a
       remark following Definition 2.5 of [Pierik], p. 9.  (Contributed by Jim
       Kingdon, 9-Jun-2024.) $)
    omniwomnimkv $p |- ( A e. Omni <-> ( A e. WOmni /\ A e. Markov ) ) $=
      ( vf vx comni wcel cvv cwomni cmarkov wa cv c0 wceq wrex c1o wo wi wal wn
      orcomd bitr4di elex simpl elexd c2o cfv wral wdc 1n0 nesymi mtbiri reximi
      eqeq1 rexnalim syl orim1i df-dc sylibr adantl simpr ord jca simprl simprr
      wf sylib orim2d impbida pm5.74da albidv isomni iswomni ismkv anbi12d jcab
      mpd 19.26 albii 3bitr4d pm5.21nii ) ADEZAFEZAGEZAHEZIZADUAWDAGWBWCUBUCWAA
      UDBJZVDZCJWEUEZKLZCAMZWGNLZCAUFZOZPZBQWFWKUGZWKRZWIPZIZPZBQZVTWDWAWMWRBWA
      WFWLWQWAWFIZWLWQWTWLIZWNWPWLWNWTWLWKWOOZWNWLWOWKWIWOWKWIWJRZCAMWOWHXCCAWH
      WJKNLNKUHUIWGKNULUJUKWJCAUMUNUOSWKUPZUQURXAWKWIXAWIWKWTWLUSSUTVAWTWQIZWKW
      IXEXBWKWIOXEWNXBWTWNWPVBXDVEXEWOWIWKWTWNWPVCVFVOSVGVHVICABFVJWAWDWFWNPZWF
      WPPZIZBQZWSWAWDXFBQZXGBQZIXIWAWBXJWCXKCABFVKCABFVLVMXFXGBVPTWRXHBWFWNWPVN
      VQTVRVS $.
  $}

  $( LPO implies WLPO. Easy corollary of the more general ~ omniwomnimkv .
     There is an analogue in terms of analytic omniscience principles at
     ~ tridceq .  (Contributed by Jim Kingdon, 24-Jul-2024.) $)
  lpowlpo $p |- ( _om e. Omni -> _om e. WOmni ) $=
    ( com comni wcel cwomni cmarkov omniwomnimkv simplbi ) ABCADCAECAFG $.

  ${
    $d A f g h x $.  $d A g h x y $.  $d B g h x y $.
    $( Lemma for ~ enwomni .  One direction of the biconditional.  (Contributed
       by Jim Kingdon, 20-Jun-2024.) $)
    enwomnilem $p |- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) ) $=
      ( vy vg vh vx vf cwomni wcel wa cv cfv c1o wceq wdc c2o ad3antlr wf simpr
      wral cen wbr cmap wf1o wex bren biimpi ad2antrr ccom fveq1 eqeq1d ralbidv
      co dcbid iswomnimap ibi com cvv 2onn relen brrelex2i elmapg sylancr mpbid
      wb adantr f1of adantl fco syl2anc simpllr mpbird rspcdva wfn f1ofn f1ocnv
      ccnv ffvelcdmd fvco2 fveqeq2 simplr f1ocnvfv2 ad4ant24 3eqtr3rd ralrimiva
      syl fveq2d ffvelcdmda adantlr eqtrd impbida exlimddv ex ) ABUAUBZAHIZBHIZ
      WNWOJZWPCKZDKZLZMNZCBTZOZDPBUCUMZTZWQXCDXDWQWSXDIZJZABEKZUDZXCEWNXIEUEZWO
      XFWNXJABEUFUGUHXGXIJZFKZWSXHUIZLZMNZFATZOZXCXKXLGKZLZMNZFATZOZXQGPAUCUMZX
      MXRXMNZYAXPYDXTXOFAYDXSXNMXLXRXMUJUKULUNWOYBGYCTZWNXFXIWOYEFAGHUOUPQXKXMY
      CIZAPXMRZXKBPWSRZABXHRZYGXGYHXIXGXFYHWQXFSWNXFYHVEZWOXFWNPUQIZBURIZYJUSAB
      UAUTVAZPBWSUQURVBVCUHVDVFXIYIXGABXHVGVHZABPWSXHVIVJXKYKWOYFYGVEUSWNWOXFXI
      VKPAXMUQHVBVCVLVMXKXPXBXKXPXBXKXPJZXACBYOWRBIZJZWRXHVQZLZXMLZYSXHLZWSLZMW
      TYQXHAVNZYSAIYTUUBNXIUUCXGXPYPABXHVOZQYQBAWRYRXIBAYRRZXGXPYPXIBAYRUDUUEAB
      XHVPBAYRVGWFQYOYPSVRZAWSXHYSVSVJYQXOYTMNFAYSXLYSMXMVTXKXPYPWAUUFVMXIYPUUB
      WTNXGXPXIYPJUUAWRWSABWRXHWBWGWCWDWEXKXBJZXOFAUUGXLAIZJZXNXLXHLZWSLZMUUIUU
      CUUHXNUUKNXIUUCXGXBUUHUUDQUUGUUHSAWSXHXLVSVJUUIXAUUKMNCBUUJWRUUJMWSVTXKXB
      UUHWAXKUUHUUJBIXBXKABXLXHYNWHWIVMWJWEWKUNVDWLWEWNWPXEVEZWOWNYLUULYMCBDURU
      OWFVFVLWM $.
  $}

  $( Weak omniscience is invariant with respect to equinumerosity.  For
     example, this means that we can express the Weak Limited Principle of
     Omniscience as either ` _om e. WOmni ` or ` NN0 e. WOmni ` .  The former
     is a better match to conventional notation in the sense that ~ df2o3 says
     that ` 2o = { (/) , 1o } ` whereas the corresponding relationship does not
     exist between ` 2 ` and ` { 0 , 1 } ` .  (Contributed by Jim Kingdon,
     20-Jun-2024.) $)
  enwomni $p |- ( A ~~ B -> ( A e. WOmni <-> B e. WOmni ) ) $=
    ( cen wbr cwomni wcel enwomnilem wi ensym syl impbid ) ABCDZAEFZBEFZABGLBAC
    DNMHABIBAGJK $.

  ${
    $d N f x $.  $d N i x $.  $d i ph x $.
    nninfdcinf.w $e |- ( ph -> _om e. WOmni ) $.
    nninfdcinf.n $e |- ( ph -> N e. NN+oo ) $.
    $( The Weak Limited Principle of Omniscience (WLPO) implies that it is
       decidable whether an element of ` NN+oo ` equals the point at infinity.
       (Contributed by Jim Kingdon, 3-Dec-2024.) $)
    nninfdcinf $p |- ( ph -> DECID N = ( i e. _om |-> 1o ) ) $=
      ( vx vf com c1o wceq wdc cv cfv wral c2o dcbid cwomni wcel cvv syl eqeq1d
      cmpt cmap co fveq1 ralbidv wb elexd iswomnimap mpbid wf xnninf 2onn elexi
      nninff omex elmap sylibr rspcdva ffnd eqidd wa 1onn a1i fnmptfvd mpbird )
      ACBHIUBJZKFLZCMZIJZFHNZKZAVHGLZMZIJZFHNZKZVLGOHUCUDZCVMCJZVPVKVSVOVJFHVSV
      NVIIVHVMCUEUAUFPAHQRZVQGVRNZDAHSRVTWAUGAHQDUHFHGSUITUJAHOCUKZCVRRACULRWBE
      CUOTZOHCOHUMUNUPUQURUSAVGVKAHIIHFCHBAHOCWCUTVHBLZJIVAIHRZAVHHRVBVCVDWEAWD
      HRVBVCVDVEPVF $.
  $}

  ${
    nninfwlporlem.x $e |- ( ph -> X : _om --> 2o ) $.
    nninfwlporlem.y $e |- ( ph -> Y : _om --> 2o ) $.
    nninfwlporlem.d $e |- D = ( i e. _om |->
      if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) ) $.

    ${
      $d D i j $.  $d X i j $.  $d Y i j $.  $d i j ph $.
      $( Given two countably infinite sequences of zeroes and ones, they are
         equal if and only if a sequence formed by pointwise comparing them is
         all ones.  (Contributed by Jim Kingdon, 6-Dec-2024.) $)
      nninfwlporlemd $p |- ( ph -> ( X = Y <-> D = ( i e. _om |-> 1o ) ) ) $=
        ( vj cfv wceq com wral c1o wcel wa c0 c2o fveq2 a1i cv cmpt wn 1n0 neii
        intnan biorfi eqid biantru orbi1i bitri eqcom cif eqeq12d ifbid cbvmptv
        wo eqtri simpr 1lt2o 0lt2o wdc ffvelcdmda sselid nndceq syl2anc ifcldcd
        2ssom fvmptd3 eqeq2d wb eqifdc bitr3id bitr4id ralbidva fveqeq2 cbvralv
        syl bitrd bitrdi wfn ffnd eqfnfv ralrimiva fnmpt eqidd fnmptfvd 3bitr4d
        1onn ) ACUAZDJZWJEJZKZCLMZIUAZBJNKZILMZDEKZBCLNUBKAWNWJBJZNKZCLMWQAWMWT
        CLAWJLOZPZWMWMNNKZPZWMUCZNQKZPZUQZWTWMWMXGUQXHXGWMXFXENQUDUEUFUGWMXDXGX
        CWMNUHUIUJUKWTNWSKZXBXHNWSULXBXINWMNQUMZKZXHXBWSXJNXBIWJWODJZWOEJZKZNQU
        MZXJLBRBCLXJUBILXOUBHCILXJXOWJWOKZWMXNNQXPWKXLWLXMWJWODSWJWOESUNUOUPURW
        OWJKZXNWMNQXQXLWKXMWLWOWJDSWOWJESUNUOAXAUSXBWMNQRNROXBUTTQROXBVATXBWKLO
        WLLOWMVBZXBRLWKVHALRWJDFVCVDXBRLWLVHALRWJEGVCVDWKWLVEVFZVGZVIVJXBXRXKXH
        VKXSWMNNQVLVRVSVMVNVOWTWPCILWJWONBVPVQVTADLWAELWAWRWNVKALRDFWBALREGWBCL
        DEWCVFALNNLIBLCAXJROZCLMBLWAAYACLXTWDCLXJBRHWEVRXQNWFNLOZAWOLOPWITYBXBW
        ITWGWH $.
    $}

    ${
      $d D f x $.  $d D i x $.  $d i ph x $.  $d X i $.  $d Y i $.
      nninfwlporlem.w $e |- ( ph -> _om e. WOmni ) $.
      $( Lemma for ~ nninfwlpor .  The result.  (Contributed by Jim Kingdon,
         7-Dec-2024.) $)
      nninfwlporlem $p |- ( ph -> DECID X = Y ) $=
        ( vx vf wceq wdc com c1o cv cfv c2o wcel a1i cmpt wral co fveq1 ralbidv
        cmap eqeq1d dcbid cwomni cvv wb omex iswomnimap ax-mp sylib wf c0 1lt2o
        cif wa 0lt2o 2ssom ffvelcdmda sselid nndceq syl2anc ifcldcd fmptd elexi
        2onn elmap sylibr rspcdva ffnd eqidd fnmptfvd mpbird nninfwlporlemd
        1onn ) ADELZMBCNOUALZMZAWBJPZBQZOLZJNUBZMZAWCKPZQZOLZJNUBZMZWGKRNUFUCZB
        WHBLZWKWFWNWJWEJNWNWIWDOWCWHBUDUGUEUHANUISZWLKWMUBZINUJSWOWPUKULJNKUJUM
        UNUOANRBUPBWMSACNCPZDQZWQEQZLZOUQUSRBAWQNSUTZWTOUQRORSXAURTUQRSXAVATXAW
        RNSWSNSWTMXARNWRVBANRWQDFVCVDXARNWSVBANRWQEGVCVDWRWSVEVFVGHVHZRNBRNVJVI
        ULVKVLVMAWAWFANOONJBNCANRBXBVNWCWQLOVOONSZAWCNSUTVSTXCXAVSTVPUHVQAVTWAA
        BCDEFGHVRUHVQ $.
    $}
  $}

  ${
    $d i j x y $.
    $( The Weak Limited Principle of Omniscience (WLPO) implies that equality
       for ` NN+oo ` is decidable.  (Contributed by Jim Kingdon,
       7-Dec-2024.) $)
    nninfwlpor $p |- ( _om e. WOmni
        -> A. x e. NN+oo A. y e. NN+oo DECID x = y ) $=
      ( vj vi com cwomni wcel cv wceq wdc xnninf wa cfv c1o c0 cif c2o wf fveq2
      nninff cmpt ad2antrl eqeq12d ifbid cbvmptv simpl nninfwlporlem ralrimivva
      ad2antll ) EFGZAHZBHZIJABKKUJUKKGZULKGZLZLCECHZUKMZUPULMZIZNOPZUADUKULUME
      QUKRUJUNUKTUBUNEQULRUJUMULTUICDEUTDHZUKMZVAULMZIZNOPUPVAIZUSVDNOVEUQVBURV
      CUPVAUKSUPVAULSUCUDUEUJUOUFUGUH $.
  $}

  ${
    nninfwlpoimlemg.f $e |- ( ph -> F : _om --> 2o ) $.
    nninfwlpoimlemg.g $e |- G = ( i e. _om |->
      if ( E. x e. suc i ( F ` x ) = (/) , (/) , 1o ) ) $.
    ${
      $d F i $.  $d G f j $.  $d i j ph x $.
      $( Lemma for ~ nninfwlpoim .  (Contributed by Jim Kingdon,
         8-Dec-2024.) $)
      nninfwlpoimlemg $p |- ( ph -> G e. NN+oo ) $=
        ( vj c2o com wcel c0 wceq c1o cif wa a1i syl adantr syl2anc vf cmap cfv
        co cv csuc wss wral xnninf wrex 0lt2o 1lt2o cfn wdc peano2 adantl 2ssom
        wf nnfi ad2antrr simpr ffvelcdmd sselid peano1 nndceq ralrimiva finexdc
        elnn ifcldcd fmptd 2onn elexi omex elmap sylibr wn csn iftrued wo suceq
        rexeqdv ifbid fvmptd3 cun df-suc rexeqi rexun bitri ax-mp eqtrdi ifordc
        wb ifbi eqtrd 3eqtr4d eqimss eqeltrd el2oss1o iffalsed sseqtrrd exmiddc
        mpjaodan fveq1 sseq12d ralbidv df-nninf elrab2 sylanbrc ) AEIJUBUDZKZHU
        EZUFZEUCZXKEUCZUGZHJUHZEUIKAJIEURXJACJBUEZDUCZLMZBCUEZUFZUJZLNOZIEAXTJK
        ZPZYBLNILIKZYEUKQNIKZYEULQYEYAUMKZXSUNZBYAUHYBUNYEYAJKZYHYDYJAXTUOUPZYA
        USRYEYIBYAYEXQYAKZPZXRJKZLJKZYIYMIJXRUQYMJIXQDAJIDURZYDYLFUTYMYLYJXQJKZ
        YEYLVAYEYJYLYKSXQYAVHTVBVCYOYMVDQXRLVEZTVFXSBYAVGTVIGVJIJEIJVKVLVMVNVOA
        XOHJAXKJKZPZXSBXLUJZXOUUAVPZYTUUAPZXMXNMXOUUCUUALXSBXLVQZUJZLNOZOZLXMXN
        UUCUUALUUFYTUUAVAZVRYTXMUUGMUUAYTXMUUAUUEVSZLNOZUUGYTXMXSBXLUFZUJZLNOZU
        UJYTCXLYCUUMJEIGXTXLMZYBUULLNUUNXSBYAUUKXTXLVTWAWBYSXLJKZAXKUOUPZYTUULL
        NIYFYTUKQZYGYTULQZYTUUKUMKZYIBUUKUHUULUNYTUUKJKZUUSYTUUOUUTUUPXLUORZUUK
        USRYTYIBUUKYTXQUUKKZPZYNYOYIUVCIJXRUQUVCJIXQDAYPYSUVBFUTUVCUVBUUTYQYTUV
        BVAYTUUTUVBUVASXQUUKVHTVBVCYOUVCVDQYRTVFXSBUUKVGTVIZWCZUULUUIWLUUMUUJMU
        ULXSBXLUUDWDZUJUUIXSBUUKUVFXLWEWFXSBXLUUDWGWHUULUUILNWMWIWJYTUUAUNZUUJU
        UGMYTXLUMKZYIBXLUHUVGYTUUOUVHUUPXLUSRYTYIBXLYTXQXLKZPZYNYOYIUVJIJXRUQUV
        JJIXQDAYPYSUVIFUTUVJUVIUUOYQYTUVIVAYTUUOUVIUUPSXQXLVHTVBVCYOUVJVDQYRTVF
        XSBXLVGTZUUAUUELNWKRWNSUUCXNUUALNOZLYTXNUVLMZUUAYTCXKYCUVLJEIGXTXKMZYBU
        UALNUVNXSBYAXLXTXKVTWAWBAYSVAYTUUALNIUUQUURUVKVIWCZSUUCUUALNUUHVRWNWOXM
        XNWPRYTUUBPZXMNXNYTXMNUGZUUBYTXMIKUVQYTXMUUMIUVEUVDWQXMWRRSUVPXNUVLNYTU
        VMUUBUVOSUVPUUALNYTUUBVAWSWNWTYTUVGUUAUUBVSUVKUUAXARXBVFXLUAUEZUCZXKUVR
        UCZUGZHJUHXPUAEXIUIUVREMZUWAXOHJUWBUVSXMUVTXNXLUVREXCXKUVREXCXDXEUAHXFX
        GXH $.
    $}

    ${
      $d F i n x $.  $d G f j $.  $d G n x $.  $d i j ph x $.  $d n ph x $.
      $( Lemma for ~ nninfwlpoim .  (Contributed by Jim Kingdon,
         8-Dec-2024.) $)
      nninfwlpoimlemginf $p |- ( ph -> (
          G = ( i e. _om |-> 1o ) <-> A. n e. _om ( F ` n ) = 1o ) ) $=
        ( com c1o wceq cfv wa wcel c0 wrex c2o a1i adantr syl2anc cmpt wral cif
        cv csuc suceq rexeqdv ifbid simpr 0lt2o 1lt2o cfn wdc peano2 adantl syl
        nnfi wf ad3antrrr elnn ffvelcdmd sselid peano1 nndceq ralrimiva finexdc
        2ssom ifcldcd fvmptd3 vex sucid fveqeq2 rspcev iftrued 1n0 neii simpllr
        eqtrd fveq1d eqidd eqeq1d mtbiri pm2.65da wo ffvelcdmda cpr elpri df2o3
        eqid eleq2s orcomd ecased wn eqeq1 ralimi ralnex sylib cbvrexv ad2antlr
        sylnib wss elomssom ssrexv 3syl mtod iffalsed mpteq2dva eqtrid impbida
        wi ) AFCIJUAZKZDUDZELZJKZDIUBZAXLMZXODIXQXMINZMZXOXNOKZXSXTXMFLZOKZXSXT
        MZYABUDZELZOKZBXMUEZPZOJUCZOXSYAYIKXTXSCXMYFBCUDZUEZPZOJUCZYIIFQHYJXMKZ
        YLYHOJYNYFBYKYGYJXMUFUGUHXQXRUIZXSYHOJQOQNXSUJRJQNZXSUKRXSYGULNZYFUMZBY
        GUBYHUMXSYGINZYQXRYSXQXMUNUOZYGUQUPXSYRBYGXSYDYGNZMZYEINOINZYRUUBQIYEVG
        UUBIQYDEAIQEURZXLXRUUAGUSUUBUUAYSYDINXSUUAUIXSYSUUAYTSYDYGUTTVAVBUUCUUB
        VCRYEOVDTVEYFBYGVFTVHVISYCYHOJYCXMYGNZXTYHUUEYCXMDVJVKRXSXTUIYFXTBXMYGY
        DXMOEVLVMTVNVRYCYBJOKZJOVOVPZYCYAJOYCYAXMXKLJYCXMFXKAXLXRXTVQVSYCCXMJJI
        XKQXKWIYNJVTXSXRXTYOSYPYCUKRVIVRWAWBWCXSXTXOXSXNQNXTXOWDZXQIQXMEAUUDXLG
        SWEUUHXNOJWFQXNOJWGWHWJUPWKWLVEAXPMZFCIYMUAXKHUUICIYMJUUIYJINZMZYLOJUUK
        YLYFBIPZXPUULWMAUUJXPXTDIPZUULXPXTWMZDIUBUUMWMXOUUNDIXOXTUUFUUGXNJOWNWB
        WOXTDIWPWQXTYFDBIXMYDOEVLWRWTWSUUKYKINZYKIXAYLUULXJUUJUUOUUIYJUNUOYKXBY
        FBYKIXCXDXEXFXGXHXI $.
    $}

    ${
      $d ph n x i $.  $d ph i $.  $d G n x i $.  $d G x y i $.  $d F n x i $.
      nninfwlpoilemdc.eq $e |- ( ph
        -> A. x e. NN+oo A. y e. NN+oo DECID x = y ) $.
      $( Lemma for ~ nninfwlpoim .  (Contributed by Jim Kingdon,
         8-Dec-2024.) $)
      nninfwlpoimlemdc $p |- ( ph ->
          DECID A. n e. _om ( F ` n ) = 1o ) $=
        ( com c1o cmpt wceq wdc cv wral xnninf dcbid rspcdva eqeq2 ralbidv wcel
        cfv eqeq1 nninfwlpoimlemg infnninf a1i nninfwlpoimlemginf mpbid ) AGDKL
        MZNZOZEPFUDLNEKQZOAGCPZNZOZUMCRUKUOUKNUPULUOUKGUASABPZUONZOZCRQUQCRQBRG
        URGNZUTUQCRVAUSUPURGUOUESUBJABDFGHIUFTUKRUCADUGUHTAULUNABDEFGHIUISUJ $.
    $}

    ${
      $d F i n x $.  $d G n x $.  $d i n ph x $.
      nninfinfwlpoimlem.eq $e |- ( ph
        -> A. x e. NN+oo DECID x = ( i e. _om |-> 1o ) ) $.
      $( Lemma for ~ nninfinfwlpo .  (Contributed by Jim Kingdon,
         8-Dec-2024.) $)
      nninfinfwlpolem $p |- ( ph ->
          DECID A. n e. _om ( F ` n ) = 1o ) $=
        ( com c1o cmpt wceq wdc cv cfv wral xnninf eqeq1 dcbid nninfwlpoimlemg
        rspcdva nninfwlpoimlemginf mpbid ) AFCJKLZMZNZDOEPKMDJQZNABOZUEMZNUGBRF
        UIFMUJUFUIFUESTIABCEFGHUAUBAUFUHABCDEFGHUCTUD $.
    $}
  $}

  ${
    $d f i j n q z $.  $d f i j q w z $.  $d f i n x y z $.  $d w y z $.
    $( Decidable equality for ` NN+oo ` implies the Weak Limited Principle of
       Omniscience (WLPO).  (Contributed by Jim Kingdon, 9-Dec-2024.) $)
    nninfwlpoim $p |- ( A. x e. NN+oo A. y e. NN+oo DECID x = y
        -> _om e. WOmni ) $=
      ( vn vf vz vw vi vj vq weq wdc xnninf wral cv cfv c1o wceq com c2o c0 cif
      cmap co cwomni wcel wa csuc wrex cmpt elmapi adantl fveqeq2 cbvrexv suceq
      rexeqdv bitrid ifbid cbvmptv simpl equequ1 dcbid equequ2 nninfwlpoimlemdc
      wf cbvral2v sylib ralrimiva cvv wb omex iswomnimap ax-mp sylibr ) ABJZKZB
      LMALMZCNDNZOPQCRMKZDSRUBUCZMZRUDUEZVPVRDVSVPVQVSUEZUFZEFGCVQHRINZVQOTQZIH
      NZUGZUHZTPUAZUIWBRSVQVDVPVQSRUJUKHGRWIENZVQOTQZEGNZUGZUHZTPUAHGJZWHWNTPWH
      WKEWGUHWOWNWEWKIEWGWDWJTVQULUMWOWKEWGWMWFWLUNUOUPUQURWCVPEFJZKZFLMELMVPWB
      USVOWQEBJZKABEFLLAEJVNWRAEBUTVABFJWRWPBFEVBVAVEVFVCVGRVHUEWAVTVIVJCRDVHVK
      VLVM $.
  $}

  ${
    $d f i k n x z $.  $d f j k n q z $.
    $( The point at infinity in ` NN+oo ` being isolated is equivalent to the
       Weak Limited Principle of Omniscience (WLPO).  By isolated, we mean that
       the equality of that point with every other element of ` NN+oo ` is
       decidable.  From an online post by Martin Escardo.  By contrast,
       elements of ` NN+oo ` corresponding to natural numbers are isolated
       ( ~ nninfisol ).  (Contributed by Jim Kingdon, 25-Nov-2025.) $)
    nninfinfwlpo $p |- ( A. x e. NN+oo DECID x = ( i e. _om |-> 1o )
        <-> _om e. WOmni ) $=
      ( vn vf vz vk vj vq cv com c1o cmpt wceq wdc xnninf wral wcel cfv c2o c0
      cwomni cmap co wa csuc cif wf elmapi adantl fveqeq2 cbvrexv suceq rexeqdv
      wrex bitrid ifbid cbvmptv simpl eqidd a1i eqeq12d cbvralv nninfinfwlpolem
      id dcbid sylib ralrimiva wb omex iswomnimap ax-mp sylibr simpr nninfdcinf
      cvv impbii ) AIZBJKLZMZNZAOPZJUAQZWACIDIZRKMCJPNZDSJUBUCZPZWBWAWDDWEWAWCW
      EQZUDZEFCWCGJHIZWCRTMZHGIZUEZUNZTKUFZLWGJSWCUGWAWCSJUHUIGFJWNEIZWCRTMZEFI
      ZUEZUNZTKUFWKWQMZWMWSTKWMWPEWLUNWTWSWJWPHEWLWIWOTWCUJUKWTWPEWLWRWKWQULUMU
      OUPUQWHWAWOFJKLZMZNZEOPWAWGURVTXCAEOVQWOMZVSXBXDVQWOVRXAXDVDVRXAMXDBFJKKB
      IWQMKUSUQUTVAVEVBVFVCVGJVOQWBWFVHVICJDVOVJVKVLWBVTAOWBVQOQZUDBVQWBXEURWBX
      EVMVNVGVP $.
  $}

  ${
    $d x y $.
    $( Decidability of equality for ` NN+oo ` is equivalent to the Weak Limited
       Principle of Omniscience (WLPO).  (Contributed by Jim Kingdon,
       3-Dec-2024.) $)
    nninfwlpo $p |- ( A. x e. NN+oo A. y e. NN+oo DECID x = y
        <-> _om e. WOmni ) $=
      ( weq wdc xnninf wral com cwomni wcel nninfwlpoim nninfwlpor impbii ) ABC
      DBEFAEFGHIABJABKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Cardinal numbers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c card $. $( Cardinal size function $)
  $c AC_ $.

  $( Extend class definition to include the cardinal size function. $)
  ccrd $a class card $.

  $( The axiom of choice for limited-length sequences. $)
  wacn $a class AC_ A $.

  ${
    $d x y $.
    $( Define the cardinal number function.  The cardinal number of a set is
       the least ordinal number equinumerous to it.  In other words, it is the
       "size" of the set.  Definition of [Enderton] p. 197.  Our notation is
       from Enderton.  Other textbooks often use a double bar over the set to
       express this function.  (Contributed by NM, 21-Oct-2003.) $)
    df-card $a |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } ) $.
  $}

  ${
    $d A x f z j g y $.
    $( Define a local and length-limited version of the axiom of choice.  The
       definition of the predicate ` X e. AC_ A ` is that for all families of
       inhabited subsets of ` X ` indexed on ` A ` (i.e. functions
       ` A --> { z e. ~P X | E. j j e. z } ` ), there is a function which
       selects an element from each set in the family.  (Contributed by Mario
       Carneiro, 31-Aug-2015.)  Change nonempty to inhabited.  (Revised by Jim
       Kingdon, 22-Nov-2025.) $)
    df-acnm $a |- AC_ A = { x | ( A e. _V /\
      A. f e. ( { z e. ~P x | E. j j e. z } ^m A )
      E. g A. y e. A ( g ` y ) e. ( f ` y ) ) } $.
  $}

  ${
    $d x y A $.
    $( The cardinality of a well-orderable set is an ordinal.  (Contributed by
       Jim Kingdon, 30-Aug-2021.) $)
    cardcl $p |- ( E. y e. On y ~~ A -> ( card ` A ) e. On ) $=
      ( vx cen wbr con0 wrex ccrd cfv crab cint cvv cmpt wceq df-card a1i breq2
      cv rabbidv wcel inteqd adantl simprd rexlimivw fvmptd onintrab2im eqeltrd
      encv intexrabim ) ARZBDEZAFGZBHIUKAFJZKZFULCBUJCRZDEZAFJZKZUNLHLHCLURMNUL
      CAOPUOBNZURUNNULUSUQUMUSUPUKAFUOBUJDQSUAUBUKBLTZAFUKUJLTUTUJBUHUCUDUKAFUI
      UEUKAUFUG $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( A set equinumerous to an ordinal is numerable.  (Contributed by Mario
       Carneiro, 29-Apr-2015.) $)
    isnumi $p |- ( ( A e. On /\ A ~~ B ) -> B e. dom card ) $=
      ( vy vx con0 wcel cen wbr wa crab cint cvv ccrd cdm wrex breq1 intexrabim
      cv rspcev syl encv simprd wceq rabbidv inteqd eleq1d elrab3 adantl mpbird
      wb breq2 df-card dmmpt eleqtrrdi ) AEFZABGHZIZBCRZDRZGHZCEJZKZLFZDLJZMNUQ
      BVDFZURBGHZCEJZKZLFZUQVFCEOVIVFUPCAEURABGPSVFCEQTUPVEVIUJZUOUPBLFZVJUPALF
      VKABUAUBVCVIDBLUSBUCZVBVHLVLVAVGVLUTVFCEUSBURGUKUDUEUFUGTUHUIDLVBMDCULUMU
      N $.
  $}

  ${
    $d x A $.
    $( Every finite set is numerable.  (Contributed by Mario Carneiro,
       4-Feb-2013.)  (Revised by Mario Carneiro, 29-Apr-2015.) $)
    finnum $p |- ( A e. Fin -> A e. dom card ) $=
      ( vx cfn wcel cv cen wbr com wrex ccrd isfi con0 nnon ensym isnumi syl2an
      cdm rexlimiva sylbi ) ACDABEZFGZBHIAJQDZBAKUAUBBHTHDTLDTAFGUBUATMATNTAOPR
      S $.

    $( Every ordinal number is numerable.  (Contributed by Mario Carneiro,
       29-Apr-2015.) $)
    onenon $p |- ( A e. On -> A e. dom card ) $=
      ( con0 wcel cen wbr ccrd cdm enrefg isnumi mpdan ) ABCAADEAFGCABHAAIJ $.
  $}

  ${
    $d A x y $.  $d A y z $.
    $( The value of ` ( card `` A ) ` .  (Contributed by Jim Kingdon,
       30-Aug-2021.) $)
    cardval3ex $p |- ( E. x e. On x ~~ A ->
        ( card ` A ) = |^| { y e. On | y ~~ A } ) $=
      ( vz cv cen wbr con0 wrex cvv wcel crab cint ccrd cfv wceq encv rexlimivw
      simprd breq1 cbvrexv sylbir breq2 rabbidv inteqd df-card fvmptg syl2anc
      intexrabim ) AEZCFGZAHIZCJKZBEZCFGZBHLZMZJKZCNOUQPUKUMAHUKUJJKUMUJCQSRULU
      OBHIURUOUKBAHUNUJCFTUAUOBHUIUBDCUNDEZFGZBHLZMUQJJNUSCPZVAUPVBUTUOBHUSCUNF
      UCUDUEDBUFUGUH $.
  $}

  ${
    $d x y A $.
    $( The value of the cardinal number function with an ordinal number as its
       argument.  (Contributed by NM, 24-Nov-2003.)  (Revised by Mario
       Carneiro, 13-Sep-2013.) $)
    oncardval $p |- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } ) $=
      ( vy con0 wcel cv cen wbr wrex ccrd cfv crab cint wceq enrefg breq1 mpdan
      rspcev cardval3ex syl ) BDEZCFZBGHZCDIZBJKAFBGHADLMNUABBGHZUDBDOUCUECBDUB
      BBGPRQCABST $.

    $( The cardinal of an ordinal number is less than or equal to the ordinal
       number.  Proposition 10.6(3) of [TakeutiZaring] p. 85.  (Contributed by
       NM, 22-Oct-2003.) $)
    cardonle $p |- ( A e. On -> ( card ` A ) C_ A ) $=
      ( vx con0 wcel ccrd cfv cv cen wbr crab cint oncardval wss breq1 intminss
      enrefg mpdan eqsstrd ) ACDZAEFBGZAHIZBCJKZABALSAAHIZUBAMACPUAUCBACTAAHNOQ
      R $.
  $}

  $( The cardinality of the empty set is the empty set.  (Contributed by NM,
     25-Oct-2003.) $)
  card0 $p |- ( card ` (/) ) = (/) $=
    ( c0 ccrd cfv wss wceq con0 wcel 0elon cardonle ax-mp ss0b mpbi ) ABCZADZMA
    EAFGNHAIJMKL $.

  ${
    $d A x $.
    $( The cardinal number of a finite set is an ordinal.  (Contributed by Jim
       Kingdon, 1-Nov-2025.) $)
    ficardon $p |- ( A e. Fin -> ( card ` A ) e. On ) $=
      ( vx cfn wcel cv cen wbr con0 wrex ccrd cfv com isfi wss wi omsson ssrexv
      ax-mp sylbi ensymb rexbii sylib cardcl syl ) ACDZBEZAFGZBHIZAJKHDUEAUFFGZ
      BHIZUHUEUIBLIZUJBAMLHNUKUJOPUIBLHQRSUIUGBHAUFTUAUBBAUCUD $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( If two numerable sets are equinumerous, then they have equal
       cardinalities.  (Contributed by Jim Kingdon, 30-Aug-2021.) $)
    carden2bex $p |- ( ( A ~~ B /\ E. x e. On x ~~ A ) ->
        ( card ` A ) = ( card ` B ) ) $=
      ( vy cen wbr cv con0 wrex wa crab cint ccrd cfv wceq enen2 rabbidv inteqd
      adantr cardval3ex adantl entr expcom reximdv imp syl 3eqtr4d ) BCEFZAGZBE
      FZAHIZJZDGZBEFZDHKZLZUMCEFZDHKZLZBMNZCMNZUHUPUSOUKUHUOURUHUNUQDHBCUMPQRSU
      KUTUPOUHADBTUAULUICEFZAHIZVAUSOUHUKVCUHUJVBAHUJUHVBUIBCUBUCUDUEADCTUFUG
      $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Theorem *54.43 of [WhiteheadRussell] p. 360.  (Contributed by NM,
       4-Apr-2007.) $)
    pm54.43 $p |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <->
                  ( A u. B ) ~~ 2o ) ) $=
      ( vx vy c1o cen wbr wa cin c0 wceq cun c2o csn wi 1on ensn1 entr wn ex cv
      con0 elexi ensymi mpan2 wcel onirri disjsn mpbir unen mpanr2 sylan2 df-2o
      csuc df-suc eqtri breq2i imbitrrdi wex en1 wne 1nen2 uneq2d unidm eqtr3di
      a1i vex eqbrtrdi ensymd sylan mtand necon2ai disjsn2 uneq12 breq1d ineq12
      sneq syl eqeq1d 3imtr4d exlimdv exlimiv imp syl2anb impbid ) AEFGZBEFGZHZ
      ABIZJKZABLZMFGZWHWJWKEENZLZFGZWLWGWFBWMFGZWJWOOWGEWMFGWPWMEEEUBPUCQUDBEWM
      RUEWFWPHZWJWOWQWJEWMIJKZWOWREEUFSEPUGEEUHUIAEBWMUJUKTULMWNWKFMEUNWNUMEUOU
      PUQURWFACUAZNZKZCUSZBDUAZNZKZDUSZWLWJOZWGCAUTDBUTXBXFXGXAXFXGOCXAXEXGDXAX
      EXGXAXEHZWTXDLZMFGZWTXDIZJKZWLWJXJXLOXHXJWSXCVAXLXJWSXCWSXCKZXJEMFGZXNSXM
      VBVFXMEXIFGXJXNXMXIEXMXIWTEFXMWTWTLXIWTXMWTXDWTWSXCVQVCWTVDVEWSCVGQVHVIEX
      IMRVJVKVLWSXCVMVRVFXHWKXIMFAWTBXDVNVOXHWIXKJAWTBXDVPVSVTTWAWBWCWDWE $.
  $}

  $( Lemma for ~ pr2ne .  (Contributed by FL, 17-Aug-2008.) $)
  pr2nelem $p |- ( ( A e. C /\ B e. D /\ A =/= B ) -> { A , B } ~~ 2o ) $=
    ( wcel wne cpr c2o cen wbr csn cin c0 wceq disjsn2 wi c1o ensn1g wa cun ex
    pm54.43 df-pr breq1i bitr4di biimpd syl2an syl7 3imp ) ACEZBDEZABFZABGZHIJZ
    ULAKZBKZLMNZUJUKUNABOUJUKUQUNPZUJUOQIJZUPQIJZURUKACRBDRUSUTSZUQUNVAUQUOUPTZ
    HIJUNUOUPUBUMVBHIABUCUDUEUFUGUAUHUI $.

  $( If an unordered pair has two elements they are different.  (Contributed by
     FL, 14-Feb-2010.) $)
  pr2ne $p |- ( ( A e. C /\ B e. D ) -> ( { A , B } ~~ 2o <-> A =/= B ) ) $=
    ( wcel wa cpr c2o cen wbr wne wceq wn preq2 c1o wi cvv syl entr ex mtoi a1d
    eqcoms enpr1g adantr prexg eqeng 1nen2 ensym syld mpid syl5 necon2ad 3expia
    com3r pr2nelem impbid ) ACEZBDEZFZABGZHIJZABKZUTVBABABLVAAAGZLZUTVBMZVEBABA
    ANUCUTVEVDOIJZVFURVGUSACUDUEUTVEVAVDIJZVGVFPUTVAQEVEVHPABCDUFVAVDQUGRVHVGUT
    VFVHVGUTVFPZVHVGFVAOIJZVIVAVDOSVJVFUTVJVBOHIJZUHVJOVAIJZVBVKPVAOUIVLVBVKOVA
    HSTRUAUBRTUOUJUKULUMURUSVCVBABCDUPUNUQ $.

  ${
    $d V a b $.
    $( A set of size two is an unordered pair of two different elements.
       (Contributed by Alexander van der Vekens, 8-Dec-2017.)  (Revised by Jim
       Kingdon, 11-Jan-2026.) $)
    en2prde $p |- ( V ~~ 2o -> E. a E. b ( a =/= b /\ V = { a , b } ) ) $=
      ( c2o cen wbr cv cpr wceq wex wne wa en2 breq1 wb cvv el2v bitrdi biimpcd
      pr2ne ancrd 2eximdv mpd ) ADEFZABGZCGZHZIZCJBJUEUFKZUHLZCJBJBCAMUDUHUJBCU
      DUHUIUHUDUIUHUDUGDEFZUIAUGDENUKUIOBCUEUFPPTQRSUAUBUC $.
  $}

  $( An unordered pair, with decidable equality for the specified elements, has
     either one or two elements.  (Contributed by Jim Kingdon, 7-Jan-2026.) $)
  pr1or2 $p |- ( ( A e. C /\ B e. D /\ DECID A = B )
      -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) ) $=
    ( wcel wceq wdc cpr c1o cen wbr c2o wo wa wne dcne enpr1g ad2antrr wb preq2
    breq1d adantl mpbid orcd pr2ne biimpar olcd jaodan sylan2b 3impa ) ACEZBDEZ
    ABFZGZABHZIJKZUOLJKZMZUNUKULNZUMABOZMURABPUSUMURUTUSUMNZUPUQVAAAHZIJKZUPUKV
    CULUMACQRUMVCUPSUSUMVBUOIJABATUAUBUCUDUSUTNUQUPUSUQUTABCDUEUFUGUHUIUJ $.

  ${
    $d A f $.  $d B f $.
    $( If an unordered pair is equinumerous to ordinal two, then a part is a
       set.  (Contributed by RP, 21-Oct-2023.) $)
    pr2cv1 $p |- ( { A , B } ~~ 2o -> A e. _V ) $=
      ( vf cpr c2o cen wbr c0 c1o cvv wcel cfv wceq 0ex fvex eleq1 mpbii adantl
      a1i syl cv wf1o wex df2o3 ensym eqbrtrrid bren sylib wa 1oex simplr simpr
      vex eqtr4d wf1 wi f1of1 ad2antrr prid1 prid2 f1veqaeq syl12anc mpd wn 1n0
      nesymi pm2.21dd wo f1of ffvelcdmd elpri adantr mpjaodan exlimiv ) ABDZEFG
      ZHIDZVOCUAZUBZCUCZAJKZVPVQVOFGVTVPVQEVOFUDVOEUEUFVQVOCUGUHVSWACVSHVRLZAMZ
      WAWBBMZWCWAVSWCWBJKWAHVRJJCUMZNOWBAJPQRVSWDUIZIVRLZAMZWAWGBMZWHWAWFWHWGJK
      WAIVRJJWEUJOWGAJPQRWFWIUIZHIMZWAWJWBWGMZWKWJWBBWGVSWDWIUKWFWIULUNWJVQVOVR
      UOZHVQKZIVQKZWLWKUPVSWMWDWIVQVOVRUQURWNWJHINUSZSWOWJHIUJUTZSVQVOHIVRVAVBV
      CWKVDWJIHVEVFSVGVSWHWIVHZWDVSWGVOKWRVSVQVOIVRVQVOVRVIZWOVSWQSVJWGABVKTVLV
      MVSWBVOKWCWDVHVSVQVOHVRWSWNVSWPSVJWBABVKTVMVNT $.
  $}

  $( If an unordered pair is equinumerous to ordinal two, then a part is a set.
     (Contributed by RP, 21-Oct-2023.) $)
  pr2cv2 $p |- ( { A , B } ~~ 2o -> B e. _V ) $=
    ( cpr c2o cen wbr cvv wcel prcom breq1i pr2cv1 sylbir ) ABCZDEFBACZDEFBGHNM
    DEBAIJBAKL $.

  $( If an unordered pair is equinumerous to ordinal two, then both parts are
     sets.  (Contributed by RP, 8-Oct-2023.) $)
  pr2cv $p |- ( { A , B } ~~ 2o -> ( A e. _V /\ B e. _V ) ) $=
    ( cpr c2o cen wbr cvv wcel pr2cv1 pr2cv2 jca ) ABCDEFAGHBGHABIABJK $.

  ${
    $d V s $.  $d j s x $.
    $( The set of subsets of a given set with one or two elements can be
       expressed as elements of the power set or as inhabited elements of the
       power set.  (Contributed by Jim Kingdon, 31-Mar-2026.) $)
    sspw1or2 $p |-
        { x e. { s e. ~P V | E. j j e. s } | ( x ~~ 1o \/ x ~~ 2o ) }
        = { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } $=
      ( cv c1o cen wbr c2o wo wel wex cpw crab wcel wa weq elequ2 exbidv elrab
      anbi1i en1m en2m jaoi biantrud pm5.32ri bitr4i rabbia2 ) AEZFGHZUIIGHZJZU
      LABDKZBLZDCMZNZUOUIUPOZULPUIUOOZBAKZBLZPZULPURULPUQVAULUNUTDUIUODAQUMUSBD
      ABRSTUAULURVAULUTURUJUTUKBUIUBBUIUCUDUEUFUGUH $.
  $}

  ${
    $d A r s $.  $d A z $.  $d ph x $.  $d s x $.
    exmidonfinlem.a $e |- A
      = { { x e. { (/) } | ph } , { x e. { (/) } | -. ph } } $.
    $( Lemma for ~ exmidonfin .  (Contributed by Andrew W Swan and Jim Kingdon,
       9-Mar-2024.) $)
    exmidonfinlem $p |- ( _om = ( On i^i Fin ) -> DECID ph ) $=
      ( vz vs vr vw vy com wceq wcel c0 cv wral eleq2 sylib wb cvv c2o con0 cfn
      cin csn crab wn wo wdc word wtr cpr elpri eleq2s elrabi velsn biidd elrab
      biimpcd simprbi notnotd wex 0ex r19.3rmv ax-mp rabeq0 sylibr eqtr4d rabex
      p0ex prid2 eleqtrri eqeltrdi prid1 mpan9 rgen2 dftr5 mpbir ordtriexmidlem
      snm syl6 jaod ontrci treq mpbiri jaoi syl rgen dford3 prexg mp2an eqeltri
      mpbir2an elon cen wbr 2onn nnfi wne pm5.19 mtbi rabbi pr2ne eqbrtri enfii
      neir elini c1o df1o2 1lt2o eqeltrri nneneq mpan2 eleqtrrid wa snid simprd
      mpbii orim12i df-dc 3syl ) JUAUBUCZKZCJLZMUDZABYDUEZKZYDAUFZBYDUEZKZUGZAU
      HZYBYCCYALCUAUBCUALCUIZYLCUJZENZUJZECOYMFNZCLZFGNZOGCOYQGFCYRYRCLYRYEKZYR
      YHKZUGZYPYRLZYQUUAYRYEYHUKZCYRYEYHULDUMUUBYSYQYTUUBYSYPYELZYQYSUUBUUDYRYE
      YPPURUUDYPYHCUUDYPMYHUUDYPYDLZYPMKZABYPYDUNFMUOZQUUDYGUFZBYDOZYHMKUUDUUHU
      UIUUDAUUDUUEAAABYPYDBNZYPKZAUPUQUSUTHNYDLHVAZUUHUUIRHMVBVSZUUHBHYDVCVDQYG
      BYDVEVFVGYHUUCCYEYHYGBYDVIVHZVJDVKVLVTUUBYTYPYHLZYQYTUUBUUOYRYHYPPURUUOYP
      YECUUOYPMYEUUOUUEUUFYGBYPYDUNUUGQUUOYGBYDOZYEMKUUOYGUUPUUOUUEYGYGYGBYPYDU
      UKYGUPUQUSUULYGUUPRUUMYGBHYDVCVDQABYDVEVFVGYEUUCCYEYHABYDVIVHZVMDVKVLVTWA
      VNVOGFCVPVQYOECYNCLYNYEKZYNYHKZUGZYOUUTYNUUCCYNYEYHULDUMUURYOUUSUURYOYEUJ
      YEABVRWBYNYEWCWDUUSYOYHUJYHYGBVRWBYNYHWCWDWEWFWGECWHWLCCUUCSDYESLZYHSLZUU
      CSLUUQUUNYEYHSSWIWJWKWMVQTUBLZCTWNWOZCUBLTJLZUVCWPTWQVDCUUCTWNDUUCTWNWOZY
      EYHWRZYEYHAYGRZBYDOZYEYHKUVHUVIAWSINYDLIVAUVHUVIRIMVBVSUVHBIYDVCVDWTAYGBY
      DXAWTXEUVAUVBUVFUVGRUUQUUNYEYHSSXBWJVQXCZCTXDWJXFJYACPWDYCYDCLYJYCYDTCXGY
      DTXHXIXJYCUVDCTKZUVJYCUVEUVDUVKRWPCTXKXLXQXMYJYDUUCCYDYEYHULDUMWFYJAYGUGY
      KYFAYIYGYFMYDLZAYFMYELZUVLAXNYFUVLUVMMVBXOZYDYEMPXQAABMYDUUJMKZAUPUQQXPYI
      UVLYGYIMYHLZUVLYGXNYIUVLUVPUVNYDYHMPXQYGYGBMYDUVOYGUPUQQXPXRAXSVFXT $.
  $}

  ${
    $d x z $.
    $( If a finite ordinal is a natural number, excluded middle follows.  That
       excluded middle implies that a finite ordinal is a natural number is
       proved in the Metamath Proof Explorer.  That a natural number is a
       finite ordinal is shown at ~ nnfi and ~ nnon .  (Contributed by Andrew W
       Swan and Jim Kingdon, 9-Mar-2024.) $)
    exmidonfin $p |- ( _om = ( On i^i Fin ) -> EXMID ) $=
      ( vz vx com con0 cfn cin wceq cv c0 csn wdc wss crab wn cpr exmidonfinlem
      eqid adantr exmid1dc ) CDEFGZATAHZIJZGZKUAUBLUCBUCBUBMUCNBUBMOZUDQPRS $.
  $}

  $( Express a set of pair cardinality as the unordered pair of a given element
     and the other element.  (Contributed by Stefan O'Rear, 22-Aug-2015.) $)
  en2eleq $p |- ( ( X e. P /\ P ~~ 2o ) -> P = { X , U. ( P \ { X } ) } ) $=
    ( wcel c2o cen wbr csn cdif cuni wne cpr wceq c1o com csuc 1onn simpr df-2o
    wa breqtrdi simpl dif1en mp3an2i en1uniel syl sylib simprd necomd wi simpld
    eldifsn en2eqpr syl3anc mpd ) BACZADEFZSZBABGHZIZJZABUSKLZUQUSBUQUSACZUSBJZ
    UQUSURCZVBVCSUQURMEFZVDMNCUQAMOZEFUOVEPUQADVFEUOUPQZRTUOUPUAZAMBUBUCURUDUEU
    SABUKUFZUGUHUQUPUOVBUTVAUIVGVHUQVBVCVIUJBUSAULUMUN $.

  $( Taking the other element twice in a pair gets back to the original
     element.  (Contributed by Stefan O'Rear, 22-Aug-2015.) $)
  en2other2 $p |- ( ( X e. P /\ P ~~ 2o ) ->
      U. ( P \ { U. ( P \ { X } ) } ) = X ) $=
    ( wcel c2o cen wbr csn cdif cuni cpr en2eleq prcom eqtrdi difeq1d difprsnss
    wa eqsstrdi wne simpl c1o com csuc 1onn simpr df-2o breqtrdi dif1en syl3anc
    a1i en1uniel eldifsni 3syl necomd eldifsn sylanbrc snssd unieqd wceq unisng
    eqssd adantr eqtrd ) BACZADEFZPZAABGZHZIZGZHZIVFIZBVEVJVFVEVJVFVEVJVHBJZVIH
    VFVEAVLVIVEABVHJVLABKBVHLMNVHBOQVEBVJVEVCBVHRBVJCVCVDSZVEVHBVEVGTEFZVHVGCVH
    BRVETUACZATUBZEFVCVNVOVEUCUIVEADVPEVCVDUDUEUFVMATBUGUHVGUJVHABUKULUMBAVHUNU
    OUPUTUQVCVKBURVDBAUSVAVB $.

  $( Disjoint union version of one plus one equals two.  (Contributed by Jim
     Kingdon, 1-Jul-2022.) $)
  dju1p1e2 $p |- ( 1o |_| 1o ) ~~ 2o $=
    ( cinl c1o cima cinr c2o cen c0 wbr cres cvv csn cxp wf1 f1of1 ax-mp f1ores
    wf1o mp2an 1oex f1oen cun cdju djuun cin wceq djuin wss djulf1o ssv djurf1o
    wb ensymi pm54.43 mpbi eqbrtrri ) ABCZDBCZUAZBBUBEFBBUCUPUQUDGUEZUREFHZBBUF
    UPBFHUQBFHUSUTUKBUPBUPABIZQZBUPFHJGKJLZAMZBJUGZVBJVCAQVDUHJVCANOBUIZJVCBAPR
    BUPVASTOULBUQBUQDBIZQZBUQFHJBKJLZDMZVEVHJVIDQVJUJJVIDNOVFJVIBDPRBUQVGSTOULU
    PUQUMRUNUO $.

  ${
    $d x y A $.
    $( The collection of finite subsets of a set dominates the set.  (We use
       the weaker sethood assumption ` ( ~P A i^i Fin ) e. _V ` because this
       theorem also implies that ` A ` is a set if ` ~P A i^i Fin ` is.)
       (Contributed by Mario Carneiro, 17-May-2015.) $)
    infpwfidom $p |- ( ( ~P A i^i Fin ) e. _V -> A ~<_ ( ~P A i^i Fin ) ) $=
      ( vx vy cpw cfn cin cv csn wcel snelpwi snfig elind wceq wb sneqbg adantr
      cvv dom2 ) BCAADZEFBGZHZCGZHZQTAIZSEUATAJTAKLUDUAUCMTUBMNUBAITUBAOPR $.
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.
    exmidfodomrlemeldju.a $e |- ( ph -> A C_ 1o ) $.
    exmidfodomrlemeldju.el $e |- ( ph -> B e. ( A |_| 1o ) ) $.
    $( Lemma for ~ exmidfodomr .  A variant of ~ djur .  (Contributed by Jim
       Kingdon, 2-Jul-2022.) $)
    exmidfodomrlemeldju $p |- ( ph
        -> ( B = ( inl ` (/) ) \/ B = ( inr ` (/) ) ) ) $=
      ( vx cinl cfv wceq wrex c0 cinr wo c1o wa wcel sylib fveq2d eqeq2d biimpd
      cv sselda el1o rexlimdva imp orcd simpr olcd cdju djur mpjaodan ) ACFUAZG
      HZIZFBJZCKGHZIZCKLHZIZMCULLHZIZFNJZAUOOUQUSAUOUQAUNUQFBAULBPOZUNUQVCUMUPC
      VCULKGVCULNPZULKIZABNULDUBULUCZQRSTUDUEUFAVBOUSUQAVBUSAVAUSFNAVDOZVAUSVGU
      TURCVGULKLVGVDVEAVDUGVFQRSTUDUEUHACBNUIPUOVBMEFBNCUJQUK $.
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.
    exmidfodomrlemreseldju.a $e |- ( ph -> A C_ 1o ) $.
    exmidfodomrlemreseldju.el $e |- ( ph -> B e. ( A |_| 1o ) ) $.
    $( Lemma for ~ exmidfodomrlemrALT .  A variant of ~ eldju .  (Contributed
       by Jim Kingdon, 9-Jul-2022.) $)
    exmidfodomrlemreseldju $p |- ( ph
        -> ( ( (/) e. A /\ B = ( ( inl |` A ) ` (/) ) )
          \/ B = ( ( inr |` 1o ) ` (/) ) ) ) $=
      ( vx cres cfv wceq wrex c0 wcel wa c1o sylib fveq2d eqeq2d simpr biimpd
      wo cinl cinr sselda el1o eqeltrrd biantrurd bitrd rexlimdva imp orcd olcd
      cv cdju eldju mpjaodan ) ACFULZUABGZHZIZFBJZKBLZCKUQHZIZMZCKUBNGZHZIZTCUP
      VEHZIZFNJZAUTMVDVGAUTVDAUSVDFBAUPBLZMZUSVDVLUSVCVDVLURVBCVLUPKUQVLUPNLZUP
      KIZABNUPDUCUPUDZOZPQVLVAVCVLUPKBVPAVKRUEUFUGSUHUIUJAVJMVGVDAVJVGAVIVGFNAV
      MMZVIVGVQVHVFCVQUPKVEVQVMVNAVMRVOOPQSUHUIUKACBNUMLUTVJTEFBNCUNOUO $.
  $}

  ${
    $d f g x z $.  $d f g y z $.  $d g u x $.  $d g v x $.  $d w z $.
    $( Excluded middle implies the existence of a mapping from any set onto any
       inhabited set that it dominates.  Proposition 1.1 of [PradicBrown2022],
       p. 2.  (Contributed by Jim Kingdon, 1-Jul-2022.) $)
    exmidfodomrlemim $p |- ( EXMID -> A. x A. y ( ( E. z z e. y /\ y ~<_ x )
        -> E. f f : x -onto-> y ) ) $=
      ( vg vw vu vv wem cv wcel wex wa crn cun wceq wfun cdm c0 vex cdom wbr wi
      wfo wf1 brdomi ad2antll ccnv cdif csn cxp wfn cin wf df-f1 simprbi fconst
      ax-mp jctir df-rn eqcomi dmxpm ineq12i disjdif eqtri funun sylancl adantl
      ffun snm dmun uneq1i uneq2i 3eqtr2i wss wdc wral f1rn exmidexmid ad2antrr
      ralrimivw undifdcss sylanbrc eqtr4id df-fn rnun dfdm4 f1dm eqtr3id uneq1d
      wn wo exmiddc syl rnxpm adantr snssi eqsstrd notm0 xpeq1 0xp eqtrdi rneqd
      ex rn0 0ss eqsstrdi a1d sylbi jaoi imp sylib sylan9eqr eqtrid df-fo cnvex
      ssequn2 difexg snex xpex unex foeq1 spcev an32s exlimdv adantlrr exlimddv
      cvv alrimivv ) ICJZBJZKZCLZYKAJZUAUBZMZYNYKDJZUDZDLZUCABIYPYSIYPMYKYNEJZU
      EZYSEYOUUAELIYMYKYNEUFUGIYMUUAYSYOIUUAYMYSIUUAMZYMYSUUBYLYSCUUBYLYSIYLUUA
      YSIYLMZUUAMZYNYKYTUHZYNYTNZUIZYJUJZUKZOZUDZYSUUDUUJYNULZUUJNZYKPUUKUUDUUJ
      QZUUJRZYNPUULUUAUUNUUCUUAUUEQZUUIQZMUUERZUUIRZUMZSPUUNUUAUUPUUQUUAYKYNYTU
      NUUPYKYNYTUOUPUUGUUHUUIUNUUQUUGYJCTZUQUUGUUHUUIVIURUSUUTUUFUUGUMSUURUUFUU
      SUUGUUFUURYTUTZVAFJUUHKFLUUSUUGPFYJUVAVJFUUGUUHVBURZVCUUFYNVDVEUUEUUIVFVG
      VHUUDUUOUUFUUGOZYNUUOUURUUSOUUFUUSOUVDUUEUUIVKUUFUURUUSUVBVLUUSUUGUUFUVCV
      MVNUUDUUFYNVOZGJUUFKZVPZGYNVQZYNUVDPUUAUVEUUCYKYNYTVRVHIUVHYLUUAIUVGGYNUV
      FVSWAVTGYNUUFWBWCWDUUJYNWEWCUUDUUMUUENZUUINZOZYKUUEUUIWFUUAUUCUVKYKUVJOZY
      KUUAUVIYKUVJUUAUVIYTRYKYTWGYKYNYTWHWIWJUUCUVJYKVOZUVLYKPIYLUVMIHJUUGKHLZU
      VNWKZWLZYLUVMUCZIUVNVPUVPUVNVSUVNWMWNUVNUVQUVOUVNYLUVMUVNYLMUVJUUHYKUVNUV
      JUUHPYLHUUGUUHWOWPYLUUHYKVOUVNYJYKWQVHWRXDUVOUUGSPZUVQHUUGWSUVRUVMYLUVRUV
      JSYKUVRUVJSNSUVRUUISUVRUUISUUHUKSUUGSUUHWTUUHXAXBXCXEXBYKXFXGXHXIXJWNXKUV
      JYKXQXLXMXNYNYKUUJXOWCYRUUKDUUJUUEUUIYTETXPUUGUUHYNYHKUUGYHKATYNUUFYHXRUR
      YJUVAXSXTYAYNYKYQUUJYBYCWNYDXDYEXKYDYFYGXDYI $.
  $}

  ${
    $d f u x y z $.  $d f u w x y z $.
    $( The existence of a mapping from any set onto any inhabited set that it
       dominates implies excluded middle.  Proposition 1.2 of
       [PradicBrown2022], p. 2.  (Contributed by Jim Kingdon, 1-Jul-2022.) $)
    exmidfodomrlemr $p |- ( A. x A. y ( ( E. z z e. y /\ y ~<_ x )
        -> E. f f : x -onto-> y ) -> EXMID ) $=
      ( vu cv wcel wex cdom wbr wa wi c0 c2o c1o cvv ax-mp com wceq sylibr cdju
      vw wfo wal csn wss wdc wem nfv nfe1 nfim nfal nfan simpl cen ssdomg df1o2
      p0ex breqtrrdi 1onn domrefg djudom sylancl dju1p1e2 adantl cinr cfv 0lt1o
      domentr djurcl elex2 jctil vex djuex mp2an 2onn breq2 anbi2d foeq2 exbidv
      imbi12d albidv spcgv eleq2 breq1 anbi12d foeq3 mpsyl sylc wn wo simpr fof
      cinl wf csuc elelsuc df-2o eleqtrri a1i ffvelcdmd adantr eqeltrrd djulclb
      wb 0ex orcd df-dc cpr 1oex prid2 df2o3 wrex simp-4r djulcl foelrn syl2anc
      simplrr fveq2d simp-5r 3eqtrd elpri ad2antrl mpjaodan rexlimddv wne djune
      eleq2s neii pm2.65da olcd simplr sseqtrrdi exmidfodomrlemeldju exlimdd ex
      alrimiv df-exmid ) CFZBFZGZCHZYTAFZIJZKZUUCYTDFZUCZDHZLZBUDZAUDZEFZMUEZUF
      ZMUULGZUGZLZEUDUHUUKUUQEUUKUUNUUPUUKUUNKZNUULOUAZUUFUCZUUPDUUKUUNDUUJDAUU
      IDBUUEUUHDUUEDUIUUGDUJUKULULUUNDUIUMUUPDUIUURUUKYSUUSGZCHZUUSNIJZKZUUTDHZ
      UUKUUNUNUURUVCUVBUUNUVCUUKUUNUUSOOUAZIJZUVFNUOJUVCUUNUULOIJOOIJZUVGUUNUUL
      UUMOIUUMPGUUNUULUUMIJLURUULUUMPUPQUQUSORGZUVHUTORVAQUULOOOVBVCVDUUSUVFNVI
      VCVEMVFVGZUUSGZUVBMOGZUVKVHUULOMVJQCUVJUUSVKQVLUUSPGZUUKUUBYTNIJZKZNYTUUF
      UCZDHZLZBUDZUVDUVELZUULPGUVIUVMEVMUTUULOPRVNVONRGUUKUVSLVPUUJUVSANRUUCNSZ
      UUIUVRBUWAUUEUVOUUHUVQUWAUUDUVNUUBUUCNYTIVQVRUWAUUGUVPDUUCNYTUUFVSVTWAWBW
      CQUVRUVTBUUSPYTUUSSZUVOUVDUVQUVEUWBUUBUVBUVNUVCUWBUUAUVACYTUUSYSWDVTYTUUS
      NIWEWFUWBUVPUUTDYTUUSNUUFWGVTWAWCWHWIUURUUTKZMUUFVGZMWNVGZSZUUPUWDUVJSZUW
      CUWFKZUUOUUOWJZWKZUUPUWHUUOUWIUWHUWEUUSGZUUOUWHUWDUWEUUSUWCUWFWLUWCUWDUUS
      GUWFUWCNUUSMUUFUUTNUUSUUFWOZUURNUUSUUFWMVEZMNGUWCMOWPZNUVLMUWNGVHMOWQQWRW
      SWTXAZXBXCMPGZUUOUWKXEXFUULOMPXDQZTXGUUOXHZTUWCUWGKZOUUFVGZUWESZUUPUWTUVJ
      SZUWSUXAKZUWJUUPUXCUUOUWIUXCUWKUUOUXCUWTUWEUUSUWSUXAWLUWSUWTUUSGUXAUWSNUU
      SOUUFUWCUWLUWGUWMXBONGUWSOMOXIZNMOXJXKXLWSWTXAZXBXCUWQTXGUWRTUWSUXBKZUWJU
      UPUXFUWIUUOUXFUUOUWEUVJSZUXFUUOKZUWEUBFZUUFVGZSZUXGUBNUXHUUTUWKUXKUBNXMUU
      RUUTUWGUXBUUOXNUUOUWKUXFUULOMXOVEUBNUUSUWEUUFXPXQUXHUXINGZUXKKZKZUXIMSZUX
      GUXIOSZUXNUXOKZUWEUXJUWDUVJUXHUXLUXKUXOXRUXQUXIMUUFUXNUXOWLXSUWCUWGUXBUUO
      UXMUXOXTYAUXNUXPKZUWEUXJUWTUVJUXHUXLUXKUXPXRUXRUXIOUUFUXNUXPWLXSUWSUXBUUO
      UXMUXPXNYAUXLUXOUXPWKZUXHUXKUXSUXIUXDNUXIMOYBXLYHYCYDYEUXGWJUXHUWEUVJUWPU
      WPUWEUVJYFXFXFMMPPYGVOYIWTYJYKUWRTUWSUULUWTUWCUULOUFUWGUWCUULUUMOUUKUUNUU
      TYLUQYMZXBUXEYNYDUWCUULUWDUXTUWOYNYDYOYPYQEYRT $.
  $}

  ${
    $d f u x y z $.  $d f u w x y z $.
    $( The existence of a mapping from any set onto any inhabited set that it
       dominates implies excluded middle.  Proposition 1.2 of
       [PradicBrown2022], p. 2.  An alternative proof of ~ exmidfodomrlemr .
       In particular, this proof uses ~ eldju instead of ~ djur and avoids
       ~ djulclb .  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Jim Kingdon,
       9-Jul-2022.) $)
    exmidfodomrlemrALT $p |- ( A. x A. y ( ( E. z z e. y /\ y ~<_ x )
        -> E. f f : x -onto-> y ) -> EXMID ) $=
      ( vu cv wcel wex cdom wbr wa wi c0 c2o c1o cvv ax-mp com cfv wceq wfo wal
      vw csn wss wdc wem cdju nfv nfe1 nfim nfal nfan cen p0ex ssdomg breqtrrdi
      simpl df1o2 1onn domrefg djudom sylancl dju1p1e2 adantl cinr 0lt1o djurcl
      domentr elex2 jctil vex djuex mp2an 2onn breq2 anbi2d foeq2 exbidv albidv
      imbi12d spcgv eleq2 breq1 anbi12d foeq3 mpsyl sylc cinl cres wn wo simprl
      orcd df-dc sylibr simp-4r djulcl foelrn syl2anc simprr wb eqeq1d ad2antlr
      wrex fvres mpbird adantr simpr fveq2d simp-5r 3eqtrd elpri df2o3 ad2antrl
      cpr eleq2s mpjaodan rexlimddv wne 0ex djune neii a1i mtbiri pm2.65da olcd
      eqeq12d simplr sseqtrrdi wf fof eleqtrri ffvelcdmd exmidfodomrlemreseldju
      1oex prid2 csuc elelsuc df-2o exlimdd ex alrimiv df-exmid ) CFZBFZGZCHZUU
      FAFZIJZKZUUIUUFDFZUAZDHZLZBUBZAUBZEFZMUDZUEZMUURGZUFZLZEUBUGUUQUVCEUUQUUT
      UVBUUQUUTKZNUUROUHZUULUAZUVBDUUQUUTDUUPDAUUODBUUKUUNDUUKDUIUUMDUJUKULULUU
      TDUIUMUVBDUIUVDUUQUUEUVEGZCHZUVENIJZKZUVFDHZUUQUUTURUVDUVIUVHUUTUVIUUQUUT
      UVEOOUHZIJZUVLNUNJUVIUUTUUROIJOOIJZUVMUUTUURUUSOIUUSPGUUTUURUUSIJLUOUURUU
      SPUPQUSUQORGZUVNUTORVAQUUROOOVBVCVDUVEUVLNVIVCVEMVFSZUVEGZUVHMOGZUVQVGUUR
      OMVHQCUVPUVEVJQVKUVEPGZUUQUUHUUFNIJZKZNUUFUULUAZDHZLZBUBZUVJUVKLZUURPGUVO
      UVSEVLUTUUROPRVMVNNRGUUQUWELVOUUPUWEANRUUINTZUUOUWDBUWGUUKUWAUUNUWCUWGUUJ
      UVTUUHUUINUUFIVPVQUWGUUMUWBDUUINUUFUULVRVSWAVTWBQUWDUWFBUVEPUUFUVETZUWAUV
      JUWCUVKUWHUUHUVHUVTUVIUWHUUGUVGCUUFUVEUUEWCVSUUFUVENIWDWEUWHUWBUVFDUUFUVE
      NUULWFVSWAWBWGWHUVDUVFKZUVAMUULSZMWIUURWJSZTZKZUVBUWJMVFOWJSZTZUWIUWMKZUV
      AUVAWKZWLZUVBUWPUVAUWQUWIUVAUWLWMWNUVAWOZWPUWIUWOKZUVAOUULSZUWKTZKZUVBUXA
      UWNTZUWTUXCKZUWRUVBUXEUVAUWQUWTUVAUXBWMWNUWSWPUWTUXDKZUWRUVBUXFUWQUVAUXFU
      VAUWKUWNTZUXFUVAKZMWISZUCFZUULSZTZUXGUCNUXHUVFUXIUVEGZUXLUCNXEUVDUVFUWOUX
      DUVAWQUVAUXMUXFUUROMWRVEUCNUVEUXIUULWSWTUXHUXJNGZUXLKZKZUXJMTZUXGUXJOTZUX
      PUXQKZUWKUXKUWJUWNUXPUWKUXKTZUXQUXPUXTUXLUXHUXNUXLXAUVAUXTUXLXBUXFUXOUVAU
      WKUXIUXKMUURWIXFZXCXDXGZXHUXSUXJMUULUXPUXQXIXJUWIUWOUXDUVAUXOUXQXKXLUXPUX
      RKZUWKUXKUXAUWNUXPUXTUXRUYBXHUYCUXJOUULUXPUXRXIXJUWTUXDUVAUXOUXRWQXLUXNUX
      QUXRWLZUXHUXLUYDUXJMOXPZNUXJMOXMXNXQXOXRXSUVAUXGWKUXFUVAUXGUXIUVPTUXIUVPM
      PGZUYFUXIUVPXTYAYAMMPPYBVNYCUVAUWKUXIUWNUVPUYAUWNUVPTZUVAUVRUYGVGMOVFXFQY
      DYHYEVEYFYGUWSWPUWTUURUXAUWIUUROUEUWOUWIUURUUSOUUQUUTUVFYIUSYJZXHUWTNUVEO
      UULUWINUVEUULYKZUWOUVFUYIUVDNUVEUULYLVEZXHONGUWTOUYENMOYPYQXNYMYDYNYOXRUW
      IUURUWJUYHUWINUVEMUULUYJMNGUWIMOYRZNUVRMUYKGVGMOYSQYTYMYDYNYOXRUUAUUBUUCE
      UUDWP $.
  $}

  ${
    $d f x y z $.
    $( Excluded middle is equivalent to the existence of a mapping from any set
       onto any inhabited set that it dominates.  (Contributed by Jim Kingdon,
       1-Jul-2022.) $)
    exmidfodomr $p |- ( EXMID <-> A. x A. y ( ( E. z z e. y /\ y ~<_ x )
        -> E. f f : x -onto-> y ) ) $=
      ( wem wel wex cv cdom wbr wfo wal exmidfodomrlemim exmidfodomrlemr impbii
      wa wi ) ECBFCGBHZAHZIJPSRDHKDGQBLALABCDMABCDNO $.
  $}

  ${
    $d f g h x y z j w A $.  $d g y z B $.  $d f g x F $.  $d g ph $.
    $d y ps $.  $d f g x y z j w C $.  $d f g h x y z w X $.
    $( Reverse closure for the choice set predicate.  (Contributed by Mario
       Carneiro, 31-Aug-2015.) $)
    acnrcl $p |- ( X e. AC_ A -> A e. _V ) $=
      ( vy vg vf vj vz vx vw cvv wcel cv cfv wral wex cpw crab cmap co wa elex2
      cab wacn abn0m simpl exlimiv sylbi syl df-acnm eleq2s ) AJKZBUKCLZDLMULEL
      MKCANDOEFLGLKFOGHLPQARSNZTZHUBZAUCBUOKILUOKIOZUKIBUOUAUPUNHOUKUNHIUDUNUKH
      UKUMUEUFUGUHHCGAEDFUIUJ $.

    $( Equality theorem for the choice set function.  (Contributed by Mario
       Carneiro, 31-Aug-2015.) $)
    acneq $p |- ( A = C -> AC_ A = AC_ C ) $=
      ( vy vg vf vj vz vx cvv wcel cv cfv wral wex cmap co wa cab wacn df-acnm
      wceq cpw crab eleq1 oveq2 raleq exbidv raleqbidv anbi12d abbidv 3eqtr4g )
      ABUAZAIJZCKZDKLUNEKLJZCAMZDNZEFKGKJFNGHKUBUCZAOPZMZQZHRBIJZUOCBMZDNZEURBO
      PZMZQZHRASBSULVAVGHULUMVBUTVFABIUDULUQVDEUSVEABUROUEULUPVCDUOCABUFUGUHUIU
      JHCGAEDFTHCGBEDFTUK $.

    $( The property of being a choice set of length ` A ` .  (Contributed by
       Mario Carneiro, 31-Aug-2015.) $)
    isacnm $p |- ( ( X e. V /\ A e. W ) -> ( X e. AC_ A <->
      A. f e. ( { z e. ~P X | E. j j e. z } ^m A )
        E. g A. x e. A ( g ` x ) e. ( f ` x ) ) ) $=
      ( vy wcel cv cfv wral wex cpw crab cmap co wa wacn rabeqdv oveq1d raleqdv
      cvv wceq pweq anbi2d df-acnm elab2g wb elex biid baib syl sylan9bb ) IGKI
      CUAZKCUEKZALZELMUSDLMKACNEOZDFLBLKFOZBIPZQZCRSZNZTZCHKZVEURUTDVABJLZPZQZC
      RSZNZTVFJIUQGVHIUFZVLVEURVMUTDVKVDVMVJVCCRVMVABVIVBVHIUGUBUCUDUHJABCDEFUI
      UJVGURVFVEUKCHULVFURVEVFUMUNUOUP $.

    $( Every set has finite choice sequences.  (Contributed by Mario Carneiro,
       31-Aug-2015.) $)
    finacn $p |- ( A e. Fin -> AC_ A = _V ) $=
      ( vx vy vg vf vj vw vz cfn wcel cvv cv cfv wral wex wa wf sylib ralrimiva
      syl wacn crab cmap co wrex elmapi adantl ffvelcdm wceq eleq2 exbidv elrab
      cpw simprd eleq1w cbvexv sylibr eleq1 ac6sfi syldan exsimpr wb vex isacnm
      rexv mpan mpbird a1i 2thd eqrdv ) AIJZBAUAZKVKBLZVLJZVMKJZVKVNCLZDLZMZVPE
      LZMZJZCANZDOZEFLZGLZJZFOZGVMUMZUBZAUCUDZNZVKWCEWJVKVSWJJZPZAKVQQZWBPDOZWC
      VKWLHLZVTJZHKUEZCANZWOWMAWIVSQZWSWLWTVKVSWIAUFUGWTWRCAWTVPAJPZWQHOZWRXAWD
      VTJZFOZXBXAVTWHJZXDXAVTWIJXEXDPAWIVPVSUHWGXDGVTWHWEVTUIWFXCFWEVTWDUJUKULR
      UNXCWQFHFHVTUOUPRWQHVEUQSTWQWACHAKDWPVRVTURUSUTWNWBDVATSVOVKVNWKVBBVCZCGA
      EDFKIVMVDVFVGVOVKXFVHVIVJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom of Choice equivalents
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c CHOICE $.

  $( Formula for an abbreviation of the axiom of choice. $)
  wac $a wff CHOICE $.

  ${
    $d f x $.
    $( The expression ` CHOICE ` will be used as a readable shorthand for any
       form of the axiom of choice; all concrete forms are long, cryptic, have
       dummy variables, or all three, making it useful to have a short name.
       Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

       There are some decisions about how to write this definition especially
       around whether ~ ax-setind is needed to show equivalence to other ways
       of stating choice, and about whether choice functions are available for
       nonempty sets or inhabited sets.  (Contributed by Mario Carneiro,
       22-Feb-2015.) $)
    df-ac $a |- ( CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) ) $.
  $}

  ${
    $d A f u v x $.  $d A f u v y $.  $d f ph u x $.  $d u v w x $.
    acfun.ac $e |- ( ph -> CHOICE ) $.
    acfun.a $e |- ( ph -> A e. V ) $.
    acfun.m $e |- ( ph -> A. x e. A E. w w e. x ) $.
    $( A convenient form of choice.  The goal here is to state choice as the
       existence of a choice function on a set of inhabited sets, while making
       full use of our notation around functions and function values.
       (Contributed by Jim Kingdon, 20-Nov-2023.) $)
    acfun $p |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) ) $=
      ( vu vv vy cv wcel wel wa wex wral cvv wceq copab wss cdm wfn cfv wal cab
      elexd abid2 vex eqeltri a1i opabex3d wac df-ac sylib sseq2 fneq2d anbi12d
      dmeq exbidv spcgv sylc simprr elequ2 cbvralv elequ1 cbvexv ralbii dmopab3
      wb bitri adantr mpbid cop simplrl fnopfv sylan sseldd fvex eleq1 opelopab
      anbi2d simprd ralrimiva jca ex eximdv mpd ) AEMZJMZDNZKJOZPZJKUAZUBZWJWOU
      CZUDZPZEQZWJDUDZBMZWJUEZXBNZBDRZPZEQAWOSNWJLMZUBZWJXGUCZUDZPZEQZLUFZWTAWM
      JKDADFHUHWMKUGZSNAWLPXNWKSKWKUIJUJUKULUMAUNXMGLEUOUPXLWTLWOSXGWOTZXKWSEXO
      XHWPXJWRXGWOWJUQXOXIWQWJXGWOUTURUSVAVBVCAWSXFEAWSXFAWSPZXAXEXPWRXAAWPWRVD
      AWRXAVKWSAWQDWJAWMKQZJDRZWQDTACBOZCQZBDRZXRIYACJOZCQZJDRXRXTYCBJDXBWKTXSY
      BCBJCVEVAVFYCXQJDYBWMCKCKJVGVHVIVLUPWMJKDVJUPURVMVNZXPXDBDXPXBDNZPZYEXDYF
      XBXCVOZWONYEXDPZYFWJWOYGAWPWRYEVPXPXAYEYGWJNYDDXBWJVQVRVSWNYEKBOZPYHJKXBX
      CBUJZXBWJSSEUJYJVTWKXBTWLYEWMYIWKXBDWAJBKVEUSKMZXCTYIXDYEYKXCXBWAWCWBUPWD
      WEWFWGWHWI $.
  $}

  ${
    $d A x $.  $d A z $.  $d B x $.  $d B z $.  $d C f z $.  $d f x y $.
    $d w z $.  $d y z $.
    exmidaclem.a $e |- A = { x e. { (/) , { (/) } }
      | ( x = (/) \/ y = { (/) } ) } $.
    exmidaclem.b $e |- B = { x e. { (/) , { (/) } }
      | ( x = { (/) } \/ y = { (/) } ) } $.
    exmidaclem.c $e |- C = { A , B } $.
    $( Lemma for ~ exmidac .  The result, with a few hypotheses to break out
       commonly used expressions.  (Contributed by Jim Kingdon,
       21-Nov-2023.) $)
    exmidaclem $p |- ( CHOICE -> EXMID ) $=
      ( vz vw cv c0 wa wcel wceq cvv wo simpr eqeq1 orbi1d wac csn wss wfn wral
      cfv wdc simpl cpr crab pp0ex rabex eqeltri prexg mp2an a1i eleqtrdi elpri
      wex 0ex prid1 eqid orci elrab2 mpbir2an eleq2 mpbiri elex2 syl p0ex prid2
      vf jaoi 3syl ralrimiva acfun wn 0nep0 simplr eqeq12d mtbiri olc ralrimivw
      neii rabid2 sylibr eqtr4di eqtr3d fveq2d nsyl olcd orcd id eleq12d simprr
      fveq2 eleqtrri rspcdva sylib simprd adantr mpjaodan exlimddv exmid1dc
      df-dc ) UABUABKZLUBZUCZMZVLKZEUDZIKZXJUFZXLNZIEUEZMZXFXGOZUGZVLXIIJEVLPUA
      XHUHEPNXIECDUIZPHCPNDPNXSPNCAKZLOZXQQZALXGUIZUJZPFYBAYCUKULUMZDXTXGOZXQQZ
      AYCUJZPGYGAYCUKULUMZCDPPUNUOUMUPXIJKXLNJUSZIEXIXLENZMZXLXSNXLCOZXLDOZQYJY
      LXLEXSXIYKRHUQXLCDURYMYJYNYMLXLNZYJYMYOLCNZYPLYCNLLOZXQQZLXGUTVAYQXQLVBVC
      YBYRALYCCYAYAYQXQXTLLSTFVDVEXLCLVFVGJLXLVHVIYNXGXLNZYJYNYSXGDNZYTXGYCNXGX
      GOZXQQZLXGVJVKUUAXQXGVBVCYGUUBAXGYCDYFYFUUAXQXTXGXGSTGVDVEXLDXGVFVGJXGXLV
      HVIVMVNVOVPXIXPMZCXJUFZLOZXRXQUUCUUEMZXQXQVQZQZXRUUFDXJUFZXGOZUUHXQUUFUUJ
      MZUUGXQUUKUUDUUIOZXQUUKUULLXGOLXGVRWDUUKUUDLUUIXGUUCUUEUUJVSUUFUUJRVTWAXQ
      CDXJXQYCCDXQYCYDCXQYBAYCUEYCYDOXQYBAYCXQYAWBWCYBAYCWEWFFWGXQYCYHDXQYGAYCU
      EYCYHOXQYGAYCXQYFWBWCYGAYCWEWFGWGWHWIWJWKUUFXQMXQUUGUUFXQRWLUUCUUJXQQZUUE
      UUCUUIYCNZUUMUUCUUIDNZUUNUUMMUUCXNUUOIEDYNXMUUIXLDXLDXJWPYNWMWNXIXKXOWOZD
      ENUUCDXSECDYIVKHWQUPWRYGUUMAUUIYCDXTUUIOYFUUJXQXTUUIXGSTGVDWSWTXAXBXQXEZW
      FUUCXQMZUUHXRUURXQUUGUUCXQRWLUUQWFUUCUUDYCNZUUEXQQZUUCUUDCNZUUSUUTMUUCXNU
      VAIECYMXMUUDXLCXLCXJWPYMWMWNUUPCENUUCCXSECDYEVAHWQUPWRYBUUTAUUDYCCXTUUDOY
      AUUEXQXTUUDLSTFVDWSWTXBXCXD $.
  $}

  ${
    $d u x y $.
    $( The axiom of choice implies excluded middle.  See ~ acexmid for more
       discussion of this theorem and a way of stating it without using
       ` CHOICE ` or ` EXMID ` .  (Contributed by Jim Kingdon, 21-Nov-2023.) $)
    exmidac $p |- ( CHOICE -> EXMID ) $=
      ( vx vy vu cv c0 wceq csn cpr crab eqeq1 orbi1d cbvrabv eqid exmidaclem
      wo ) ABCDZEFZBDEGZFZOZCERHZIZPRFZSOZCUAIZUBUEHZTADZEFZSOCAUAPUGFZQUHSPUGE
      JKLUDUGRFZSOCAUAUIUCUJSPUGRJKLUFMN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Cardinal number arithmetic
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Equinumerosity of a disjoint union and a union of two disjoint sets.
     (Contributed by Jim Kingdon, 30-Jul-2023.) $)
  endjudisj $p |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) )
      -> ( A |_| B ) ~~ ( A u. B ) ) $=
    ( wcel cin c0 wceq w3a cdju cinl cima cinr cun cen djuun wbr eninl 3ad2ant1
    eninr 3ad2ant2 djuin a1i simp3 unen syl22anc eqbrtrrid ) ACEZBDEZABFGHZIZAB
    JKALZMBLZNZABNZOABPUKULAOQZUMBOQZULUMFGHZUJUNUOOQUHUIUPUJACRSUIUHUQUJBDTUAU
    RUKABUBUCUHUIUJUDULAUMBUEUFUG $.

  $( Disjoint unions of equinumerous sets are equinumerous.  (Contributed by
     Jim Kingdon, 30-Jul-2023.) $)
  djuen $p |- ( ( A ~~ B /\ C ~~ D ) -> ( A |_| C ) ~~ ( B |_| D ) ) $=
    ( cen wbr wa cinl cima cinr cun cdju cin c0 wceq cvv wcel encv entr syl2anc
    adantr simpld eninl syl simpl simpl2im ensymd eninr sylancom djuin a1i unen
    adantl syl22anc djuun 3brtr3g ) ABEFZCDEFZGZHAIZJCIZKZHBIZJDIZKZACLBDLEUSUT
    VCEFZVAVDEFZUTVAMNOZVCVDMNOZVBVEEFUSUTBEFZBVCEFVFUSUTAEFZUQVJUSAPQZVKUSVLBP
    QZUQVLVMGURABRUAZUBAPUCUDUQURUEUTABSTUSVCBUSVLVMVCBEFVNBPUCUFUGUTBVCSTUSVAD
    EFZDVDEFVGUQURVACEFZVOUSCPQZVPUSVQDPQZURVQVRGUQCDRUMZUBCPUHUDVACDSUIUSVDDUS
    VQVRVDDEFVSDPUHUFUGVADVDSTVHUSACUJUKVIUSBDUJUKUTVCVAVDULUNACUOBDUOUP $.

  $( Disjoint union is equinumerous to union for disjoint sets.  (Contributed
     by Mario Carneiro, 29-Apr-2015.)  (Revised by Jim Kingdon,
     19-Aug-2023.) $)
  djuenun $p |- ( ( A ~~ B /\ C ~~ D /\ ( B i^i D ) = (/) ) ->
      ( A |_| C ) ~~ ( B u. D ) ) $=
    ( cen wbr cin wceq w3a cdju cun djuen 3adant3 cvv relen brrelex2i endjudisj
    c0 wcel id syl3an entr syl2anc ) ABEFZCDEFZBDGRHZIACJZBDJZEFZUHBDKZEFZUGUJE
    FUDUEUIUFABCDLMUDBNSUEDNSUFUFUKABEOPCDEOPUFTBDNNQUAUGUHUJUBUC $.

  $( Cardinal addition with cardinal one (which is the same as ordinal one).
     Used in proof of Theorem 6J of [Enderton] p. 143.  (Contributed by NM,
     28-Sep-2004.)  (Revised by Mario Carneiro, 29-Apr-2015.) $)
  dju1en $p |- ( ( A e. V /\ -. A e. A ) -> ( A |_| 1o ) ~~ suc A ) $=
    ( wcel wn wa c1o cdju csn cun csuc cen wbr cin c0 wceq enrefg adantr ensn1g
    ensymd simpr disjsn sylibr djuenun syl3anc df-suc breqtrrdi ) ABCZAACDZEZAF
    GZAAHZIZAJKUIAAKLZFUKKLZAUKMNOZUJULKLUGUMUHABPQUGUNUHUGUKFABRSQUIUHUOUGUHTA
    AUAUBAAFUKUCUDAUEUF $.

  $( Cardinal addition with cardinal zero (the empty set).  Part (a1) of proof
     of Theorem 6J of [Enderton] p. 143.  (Contributed by NM, 27-Sep-2004.)
     (Revised by Mario Carneiro, 29-Apr-2015.) $)
  dju0en $p |- ( A e. V -> ( A |_| (/) ) ~~ A ) $=
    ( wcel cdju cun cen cvv cin wceq wbr 0ex in0 endjudisj mp3an23 un0 breqtrdi
    c0 ) ABCZAQDZAQEZAFRQGCAQHQISTFJKALAQBGMNAOP $.

  $( Two times a cardinal number.  Exercise 4.56(g) of [Mendelson] p. 258.
     (Contributed by NM, 27-Sep-2004.)  (Revised by Mario Carneiro,
     29-Apr-2015.) $)
  xp2dju $p |- ( 2o X. A ) = ( A |_| A ) $=
    ( c0 csn c1o cun cxp c2o cdju xpundir cpr df2o3 df-pr xpeq1i df-dju 3eqtr4i
    eqtri ) BCZDCZEZAFQAFRAFEGAFAAHQRAIGSAGBDJSKBDLPMAANO $.

  $( Commutative law for cardinal addition.  Exercise 4.56(c) of [Mendelson]
     p. 258.  (Contributed by NM, 24-Sep-2004.)  (Revised by Mario Carneiro,
     29-Apr-2015.) $)
  djucomen $p |- ( ( A e. V /\ B e. W ) -> ( A |_| B ) ~~ ( B |_| A ) ) $=
    ( wcel wa cdju c1o csn cxp cun cen wbr cvv xpsnen2g mpan ensym cin syl2an
    c0 1oex wceq incom xp01disjl eqtri djuenun mp3an3 df-dju equncomi breqtrrdi
    0ex ) ACEZBDEZFABGZHIAJZTIBJZKZBAGZLULUOALMZUPBLMZUNUQLMZUMHNEULUSUAHANCOPT
    NEUMUTUKTBNDOPUSAUOLMZBUPLMZVAUTUOAQUPBQVBVCUOUPRZTUBVAVDUPUORTUOUPUCBAUDUE
    AUOBUPUFUGSSURUPUOBAUHUIUJ $.

  $( Associative law for cardinal addition.  Exercise 4.56(c) of [Mendelson]
     p. 258.  (Contributed by NM, 26-Sep-2004.)  (Revised by Mario Carneiro,
     29-Apr-2015.) $)
  djuassen $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
      ( ( A |_| B ) |_| C ) ~~ ( A |_| ( B |_| C ) ) ) $=
    ( wcel cdju c0 cxp c1o cun cen wbr cin cvv 0ex xpsnen2g sylancr 1oex ensymd
    w3a csn wceq simp1 snex simp2 xpexg syl2anc xp01disjl djuenun syl3anc simp3
    entr a1i indir xpeq2i xpindi xp0 3eqtr3i uneq12i eqtri df-dju xpundi uneq2i
    un0 unass 3eqtr4i breqtrrdi ) ADGZBEGZCFGZUBZABHZCHZIUCZAJZKUCZVPBJZJZLZVRV
    RCJZJZLZABCHZHZMVMVNWAMNZCWCMNWAWCOZIUDZVOWDMNVMAVQMNBVTMNVQVTOIUDZWGVMVQAV
    MIPGZVJVQAMNQVJVKVLUEIAPDRSUAVMVTBVMVTVSMNZVSBMNZVTBMNVMKPGZVSPGZWLTVMVPPGV
    KWOIQUFVJVKVLUGZVPBPEUHSKVSPPRSVMWKVKWMQWPIBPERSVTVSBUNUIUAWJVMAVSUJUOAVQBV
    TUKULVMWCCVMWCWBMNZWBCMNZWCCMNVMWNWBPGZWQTVMVRPGVLWSKTUFVJVKVLUMZVRCPFUHSKW
    BPPRSVMWNVLWRTWTKCPFRSWCWBCUNUIUAWIVMWHVQWCOZVTWCOZLZIVQVTWCUPXCIILIXAIXBIA
    WBUJVRVSWBOZJVRIJXBIXDIVRBCUJUQVRVSWBURVRUSUTVAIVFVBVBUOVNWACWCUKULVQVRWEJZ
    LVQVTWCLZLWFWDXEXFVQXEVRVSWBLZJXFWEXGVRBCVCUQVRVSWBVDVBVEAWEVCVQVTWCVGVHVI
    $.

  $( Cardinal multiplication distributes over cardinal addition.  Theorem 6I(3)
     of [Enderton] p. 142.  (Contributed by NM, 26-Sep-2004.)  (Revised by
     Mario Carneiro, 29-Apr-2015.) $)
  xpdjuen $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
      ( A X. ( B |_| C ) ) ~~ ( ( A X. B ) |_| ( A X. C ) ) ) $=
    ( wcel cxp cdju c0 csn c1o cun cen wbr cin cvv xpsnen2g sylancr ensymd wceq
    w3a enrefg 3ad2ant1 0ex simp2 xpen syl2anc con0 1on xp01disjl xpeq2i xpindi
    simp3 xp0 3eqtr3i a1i djuenun syl3anc df-dju xpundi eqtri breqtrrdi ) ADGZB
    EGZCFGZUBZABHZACHZIZABCIZHZVGVJAJKBHZHZALKCHZHZMZVLNVGVHVNNOZVIVPNOZVNVPPZJ
    UAZVJVQNOVGAANOZBVMNOVRVDVEWBVFADUCUDZVGVMBVGJQGVEVMBNOUEVDVEVFUFJBQERSTAAB
    VMUGUHVGWBCVONOVSWCVGVOCVGLUIGVFVOCNOUJVDVEVFUNLCUIFRSTAACVOUGUHWAVGAVMVOPZ
    HAJHVTJWDJABCUKULAVMVOUMAUOUPUQVHVNVIVPURUSVLAVMVOMZHVQVKWEABCUTULAVMVOVAVB
    VCT $.

  ${
    $d A f $.  $d B f $.
    $( A set is dominated by its disjoint union with another.  (Contributed by
       Jim Kingdon, 11-Jul-2023.) $)
    djudoml $p |- ( ( A e. V /\ B e. W ) -> A ~<_ ( A |_| B ) ) $=
      ( vf vx wcel wa cdju cdom wbr cv wf1 wex cinl cres cvv wfun c0 cop df-inl
      funmpt2 simpl resfunexg sylancr inlresf1 f1eq1 spcegv mpisyl djuex brdomg
      wb syl mpbird ) ACGZBDGZHZAABIZJKZAURELZMZENZUQOAPZQGZAURVCMZVBUQORUOVDFQ
      SFLTOFUAUBUOUPUCOACUDUEABUFVAVEEVCQAURUTVCUGUHUIUQURQGUSVBULABCDUJAURQEUK
      UMUN $.
  $}

  ${
    $d A f $.  $d B f $.
    $( A set is dominated by its disjoint union with another.  (Contributed by
       Jim Kingdon, 11-Jul-2023.) $)
    djudomr $p |- ( ( A e. V /\ B e. W ) -> B ~<_ ( A |_| B ) ) $=
      ( vf vx wcel wa cdju cdom wbr cv wf1 wex cinr cres cvv wfun c1o cop simpr
      df-inr funmpt2 resfunexg inrresf1 f1eq1 spcegv mpisyl wb djuex brdomg syl
      sylancr mpbird ) ACGZBDGZHZBABIZJKZBURELZMZENZUQOBPZQGZBURVCMZVBUQORUPVDF
      QSFLTOFUBUCUOUPUAOBDUDUMABUEVAVEEVCQBURUTVCUFUGUHUQURQGUSVBUIABCDUJBURQEU
      KULUN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordinal trichotomy
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Lemma for ~ exmidontriim .  A variation of ~ r19.30dc .  (Contributed by
     Jim Kingdon, 12-Aug-2024.) $)
  exmidontriimlem1 $p |- ( ( A. x e. A ( ph \/ ps \/ ch ) /\ EXMID )
      -> ( E. x e. A ph \/ E. x e. A ps \/ A. x e. A ch ) ) $=
    ( w3o wral wem wa wrex wo wdc 3orass biimpi orcomd ralimi exmidexmid syl2an
    r19.30dc simpr simplr orcom ralbii syl2anc ex orim2d mpd sylibr ) ABCFZDEGZ
    HIZADEJZBDEJZCDEGZKZKZULUMUNFUKULBCKZDEGZKUPUKURULUJUQAKZDEGULLURULKHUIUSDE
    UIAUQUIAUQKABCMNOPULQUQADESROUKURUOULUKURUOUKURIURHUOUKURTUJHURUAURHIUNUMUR
    CBKZDEGZUMLUNUMKHURVAUQUTDEBCUBUCNUMQCBDESROUDUEUFUGULUMUNMUH $.

  ${
    $d A y $.  $d B y $.  $d ph y $.
    exmidontriimlem2.b $e |- ( ph -> B e. On ) $.
    exmidontriimlem2.em $e |- ( ph -> EXMID ) $.
    exmidontriimlem2.hb $e |- ( ph -> A. y e. B
      ( A e. y \/ A = y \/ y e. A ) ) $.
    $( Lemma for ~ exmidontriim .  (Contributed by Jim Kingdon,
       12-Aug-2024.) $)
    exmidontriimlem2 $p |- ( ph -> ( A e. B \/ A. y e. B y e. A ) ) $=
      ( cv wcel wrex wral wo wceq wa con0 simpr simplr r19.29an orcd w3o ontr1
      ad2antrr jca sylc eqeltrd olcd wem exmidontriimlem1 syl2anc mpjao3dan ) A
      CBHZIZBDJZCDIZUKCIZBDKZLCUKMZBDJZUPAUMNUNUPAULUNBDAUKDIZNZULNZDOIZULUSNUN
      AVBUSULEUBVAULUSUTULPAUSULQUCCUKDUAUDRSAURNUNUPAUQUNBDUTUQNCUKDUTUQPAUSUQ
      QUERSAUPNUPUNAUPPUFAULUQUOTBDKUGUMURUPTGFULUQUOBDUHUIUJ $.
  $}

  ${
    $d A u w y z $.  $d B u w y $.  $d ph u w $.
    exmidontriimlem3.a $e |- ( ph -> A e. On ) $.
    exmidontriimlem3.b $e |- ( ph -> B e. On ) $.
    exmidontriimlem3.em $e |- ( ph -> EXMID ) $.
    exmidontriimlem3.ha $e |- ( ph -> A. z e. A A. y e. On
      ( z e. y \/ z = y \/ y e. z ) ) $.
    exmidontriimlem3.hb $e |- ( ph -> A. y e. B
      ( A e. y \/ A = y \/ y e. A ) ) $.
    $( Lemma for ~ exmidontriim .  What we get to do based on induction on both
       ` A ` and ` B ` .  (Contributed by Jim Kingdon, 10-Aug-2024.) $)
    exmidontriimlem3 $p |- ( ph -> ( A e. B \/ A = B \/ B e. A ) ) $=
      ( vw vu wcel wceq w3o cv wral con0 wel weq 3mix1 adantl 3mix3 simpr dfss3
      wa wss sylibr simplr eqssd 3mix2d eleq1 equequ1 3orbi123d ralbidv cbvralv
      eleq2 sylib eqeq2 rspcv ralimdv sylc biid 3orbi123i 3orcomb 3orrot 3bitri
      wo eqcom ralbii exmidontriimlem2 adantr mpjaodan ) ADEMZVNDENZEDMZOZKPZDM
      ZKEQZVNVQAVNVOVPUAUBAVTUFZVPVQLPZEMZLDQZVPVQWAVPVNVOUCUBWAWDUFZVOVNVPWEDE
      WEWDDEUGWAWDUDLDEUEUHWEVTEDUGAVTWDUIKEDUEUHUJUKAVPWDVHVTALEDFHAWCWBENZEWB
      MZOZLDQZWGEWBNZWCOZLDQAERMZLBSZLBTZBLSZOZBRQZLDQZWIGACBSZCBTZBCSZOZBRQZCD
      QWRIXCWQCLDCLTZXBWPBRXDWSWMWTWNXAWOCPZWBBPZULCLBUMXEWBXFUQUNUOUPURWLWQWHL
      DWPWHBERXFENWMWCWNWFWOWGXFEWBUQXFEWBUSXFEWBULUNUTVAVBWHWKLDWHWCWJWGOWCWGW
      JOWKWCWCWFWJWGWGWCVCWBEVIWGVCVDWCWJWGVEWCWGWJVFVGVJURVKVLVMAKDEGHADXFMZDX
      FNZXFDMZOZBEQDVRMZDVRNZVSOZKEQJXJXMBKEBKTXGXKXHXLXIVSXFVRDUQXFVRDUSXFVRDU
      LUNUPURVKVM $.
  $}

  ${
    $d A b v w y z $.  $d B b $.  $d a b ph u v w $.
    exmidontriimlem4.a $e |- ( ph -> A e. On ) $.
    exmidontriimlem4.b $e |- ( ph -> B e. On ) $.
    exmidontriimlem4.em $e |- ( ph -> EXMID ) $.
    exmidontriimlem4.h $e |- ( ph -> A. z e. A A. y e. On
      ( z e. y \/ z = y \/ y e. z ) ) $.
    $( Lemma for ~ exmidontriim .  The induction step for the induction on
       ` A ` .  (Contributed by Jim Kingdon, 10-Aug-2024.) $)
    exmidontriimlem4 $p |- ( ph -> ( A e. B \/ A = B \/ B e. A ) ) $=
      ( vb vw vv cv wcel wceq w3o con0 eqeq2 3orbi123d wral eleq2 wi weq eleq2w
      eleq1 eleq1w imbi2d wa adantl simpll wem wel simplr simpllr simpr rspcdva
      mpd ralrimiva cbvralv sylib exmidontriimlem3 exp31 tfis2 impcom ) ADJMZNZ
      DVEOZVEDNZPZDENZDEOZEDNZPJQEVEEOVFVJVGVKVHVLVEEDUAVEEDRVEEDUESAVIJQVEQNZA
      VIAVIUBADKMZNZDVNOZVNDNZPZUBZJKJKUCZVIVRAVTVFVOVGVPVHVQJKDUDVEVNDRJKDUFSU
      GVMVSKVETZAVIVMWAUHZAUHZBCDVEADQNWBFUIVMWAAUJAUKWBHUIACBULCBUCBCULPBQTCDT
      WBIUIWCDLMZNZDWDOZWDDNZPZLVETDBMZNZDWIOZWIDNZPZBVETWCWHLVEWCLJULZUHZAWHWB
      AWNUMWOVSAWHUBKVEWDKLUCZVRWHAWPVOWEVPWFVQWGKLDUDVNWDDRKLDUFSUGVMWAAWNUNWC
      WNUOUPUQURWHWMLBVELBUCWEWJWFWKWGWLLBDUDWDWIDRLBDUFSUSUTVAVBVCVDURGUP $.
  $}

  ${
    $d a x y z $.
    $( Excluded middle implies ordinal trichotomy.  Lemma 10.4.1 of [HoTT], p.
       (varies).  The proof follows the proof from the HoTT book fairly
       closely.  (Contributed by Jim Kingdon, 10-Aug-2024.) $)
    exmidontriim $p |- ( EXMID
        -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) ) $=
      ( vz va wem wel weq w3o con0 wral cv wcel wi eleq1w equequ1 eleq2 ralbidv
      3orbi123d wa ralrimiva imbi2d simplll simpr simplr simpllr pm2.27 ralimdv
      ad2antlr mpd exmidontriimlem4 equequ2 cbvralv sylib exp31 tfis2 impcom )
      EABFZABGZBAFZHZBIJZAIAKZILZEVAEVAMECBFZCBGZBCFZHZBIJZMZACACGZVAVHEVJUTVGB
      IVJUQVDURVEUSVFACBKZNACBOVBCKVKPRQUAVCVICVBJZEVAVCVLSZESZADFZADGZDAFZHZDI
      JVAVNVRDIVNDKZILZSZBCVBVSVCVLEVTUBVNVTUCVMEVTUDWAVLVHCVBJZVCVLEVTUEEVLWBM
      VMVTEVIVHCVBEVHUFUGUHUIUJTVRUTDBIDBGVOUQVPURVQUSVSVKVBPDBAUKDBVBNRULUMUNU
      OUPT $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Excluded middle and the power set of a singleton
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d ph x $.
    $( Using an ` if ` expression to represent a truth value by ` (/) ` or
       ` 1o ` .  Unlike some theorems using ` if ` , ` ph ` does not need to be
       decidable.  (Contributed by Jim Kingdon, 9-Jan-2026.) $)
    iftrueb01 $p |- ( if ( ph , 1o , (/) ) = 1o <-> ph ) $=
      ( vx c1o c0 cif wceq cv wcel wex 0lt1o elex2 ax-mp wa wb wi eleq2 wn elif
      wo noel intnan biorfi bitr4i bitr3di pm4.71r sylibr exlimdv iftrue impbii
      mpi ) ACDEZCFZAULBGZCHZBIZADCHUOJBDCKLULUNABULUNAUNMZNUNAOULUMUKHZUNUPUKC
      UMPUQUPAQZUMDHZMZSUPAUMCDRUTUPUSURUMTUAUBUCUDUNAUEUFUGUJACDUHUI $.
  $}

  ${
    $d A x $.
    $( A truth value which is inhabited is equal to true.  This is a variation
       of ~ pwntru and ~ pwtrufal .  (Contributed by Jim Kingdon,
       10-Jan-2026.) $)
    pw1m $p |- ( ( A e. ~P 1o /\ E. x x e. A ) -> A = 1o ) $=
      ( c1o cpw wcel cv wex wceq wa c0 csn elpwi df1o2 sseqtrdi adantr eleqtrdi
      wss sselda elsni syl simpr eqeltrrd snssd eqssd eqtr4di ex exlimdv imp )
      BCDEZAFZBEZAGBCHZUIUKULAUIUKULUIUKIZBJKZCUMBUNUIBUNQUKUIBCUNBCLZMNOUMJBUM
      UJJBUMUJUNEUJJHUMUJCUNUIBCUJUORMPUJJSTUIUKUAUBUCUDMUEUFUGUH $.
  $}

  ${
    $d A x y $.
    $( Expressing a truth value in terms of an ` if ` expression.  (Contributed
       by Jim Kingdon, 10-Jan-2026.) $)
    pw1if $p |- ( A e. ~P 1o -> if ( A = 1o , 1o , (/) ) = A ) $=
      ( vx vy c1o cpw wcel wceq c0 cif cv wa simpr wn elif intnan biorfi bitr4i
      wo noel sylib simprd simpld eleqtrrd pw1m sylan2 eleqtrd sylanbrc impbida
      wex elex2 eqrdv ) ADEFZBADGZDHIZAULBJZUNFZUOAFZULUPKZUODAURUMUODFZURUPUMU
      SKZULUPLUPUTUMMZUOHFZKZRUTUMUODHNVCUTVBVAUOSOPQZTZUAURUMUSVEUBUCULUQKZUMU
      SUPUQULCJAFCUIUMCUOAUJCAUDUEZVFUOADULUQLVGUFVDUGUHUK $.
  $}

  ${
    $d x y $.
    $( The power set of ` 1o ` is an ordinal.  (Contributed by Jim Kingdon,
       29-Jul-2024.) $)
    pw1on $p |- ~P 1o e. On $=
      ( vx vy c1o cpw con0 wcel word cvv wtr cv wss c0 csn df1o2 elsni eqeltrdi
      wral 0elpw ssriv mpbir2an eqsstri sspwb mpbi dftr4 mpbir wceq sselda el1o
      wa elpwi sylib 0ss eqsstrdi ralrimiva dftr3 sylibr rgen dford3 1oex elon2
      pwex ) CDZEFVBGZVBHFVCVBIZAJZIZAVBQVDVBVBDKZCVBKVGCLMZVBNAVHVBVEVHFVELVBV
      ELOCRPSUACVBUBUCVBUDUEVFAVBVEVBFZBJZVEKZBVEQVFVIVKBVEVIVJVEFUIZVJLVEVLVJC
      FVJLUFVIVECVJVECUJUGVJUHUKVEULUMUNBVEUOUPUQAVBURTCUSVAVBUTT $.
  $}

  $( The power set of ` 1o ` dominates ` 2o ` .  Also see ~ pwpw0ss which is
     similar.  (Contributed by Jim Kingdon, 21-Sep-2022.) $)
  pw1dom2 $p |- 2o ~<_ ~P 1o $=
    ( c2o c0 csn cpr cen wbr c1o cpw cdom wne 0nep0 cvv wcel wb 0ex pr2ne mp2an
    p0ex mpbir ensymi wss pwex pwpw0ss ssdomg mp2 df1o2 pweqi breqtrri endomtr
    ) ABBCZDZEFUKGHZIFAULIFUKAUKAEFZBUJJZKBLMUJLMUMUNNORBUJLLPQSTUKUJHZULIUOLMU
    KUOUAUKUOIFUJRUBUCUKUOLUDUEGUJUFUGUHAUKULUIQ $.

  $( The power set of ` 1o ` is not zero.  (Contributed by Jim Kingdon,
     30-Jul-2024.) $)
  pw1ne0 $p |- ~P 1o =/= (/) $=
    ( c0 c1o cpw 0elpw ne0ii ) ABCBDE $.

  $( The power set of ` 1o ` is not one.  (Contributed by Jim Kingdon,
     30-Jul-2024.) $)
  pw1ne1 $p |- ~P 1o =/= 1o $=
    ( c1o cpw wceq wcel pw1on onirri c0 csn df1o2 wa cpr pwpw0ss pweqi sseqtrri
    wss 0ex p0ex prss mpbir simpri eqeltri eleq1 mpbiri mto neir ) ABZAUFACZUFU
    FDZUFEFUGUHAUFDAGHZUFIGUFDZUIUFDZUJUKJGUIKZUFOULUIBUFLAUIIMNGUIUFPQRSTUAUFA
    UFUBUCUDUE $.

  $( The power set of ` 1o ` is not three.  (Contributed by James E. Hanson and
     Jim Kingdon, 30-Jul-2024.) $)
  pw1ne3 $p |- ~P 1o =/= 3o $=
    ( c1o cpw c3o wceq c2o wcel wss 1lt2o ssnel mt2 2onn elexi elpw mtbir sucid
    com csuc df-3o eleqtrri eleq2 mpbiri mto neir ) ABZCUDCDZEUDFZUFEAGZUGAEFHE
    AIJEAEPKLZMNUEUFECFEEQCEUHORSUDCETUAUBUC $.

  $( Negated excluded middle implies that the power set of ` 1o ` is not an
     element of ` 3o ` .  (Contributed by James E. Hanson and Jim Kingdon,
     30-Jul-2024.) $)
  pw1nel3 $p |- ( -. EXMID -> -. ~P 1o e. 3o ) $=
    ( wem wn c1o cpw c2o wcel csn c3o cpr pw1ne0 pw1ne1 nelpri a1i df2o3 eleq2i
    c0 sylnibr wceq notbii bitri exmidpweq 1oex pwex elsn sylbb2 wo wa cun csuc
    df-3o df-suc eqtri elun ioran sylanbrc ) ABZCDZEFZBZUQEGZFZBZUQHFZBZUPUQPCI
    ZFZURVFBUPUQPCJKLMEVEUQNOQUPUQERZBVBAVGUASVAVGUQECUBUCUDSUEVDURVAUFZBUSVBUG
    VCVHVCUQEUTUHZFVHHVIUQHEUIVIUJEUKULOUQEUTUMTSURVAUNTUO $.

  $( Negated excluded middle implies that the successor of the power set of
     ` 1o ` is not three .  (Contributed by James E. Hanson and Jim Kingdon,
     30-Jul-2024.) $)
  sucpw1ne3 $p |- ( -. EXMID -> suc ~P 1o =/= 3o ) $=
    ( wem wn c1o cpw c3o wcel csuc wne pw1nel3 wceq 1oex pwex sucid eleq2 mpbii
    necon3bi syl ) ABCDZEFZBRGZEHISTETEJRTFSRCKLMTERNOPQ $.

  $( The successor of the power set of ` 1o ` is not an element of ` 3o ` .
     (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) $)
  sucpw1nel3 $p |- -. suc ~P 1o e. 3o $=
    ( c1o cpw csuc c3o wcel wn c2o csn cpr wceq pw1ne0 eleq2i mtbir eleq2 mpbii
    c0 elsn mto nelpri df2o3 1oex pwex sucid ne0ii nemtbir df1o2 neir pw1ne1 wo
    sucex wa ioran cun df-3o df-suc eqtri elun bitri xchnxbir mpbir2an ) ABZCZD
    EZFVBGEZFZVBGHZEZFZVDVBPAIZEVBPAVAVBVAAUAUBZUCZUDVBAVBAJZVAAEZVMVAPHZEZVOVA
    PKVAPVJQUEAVNVAUFLMVLVAVBEZVMVKVBAVANORUGSGVIVBTLMVGVBGJZVQVAGEZVRVAVIEVAPA
    KUHSGVIVATLMVQVPVRVKVBGVANORVBGVAVJUJQMVDVGUIZVEVHUKVCVDVGULVCVBGVFUMZEVSDV
    TVBDGCVTUNGUOUPLVBGVFUQURUSUT $.

  $( Three is not an element of the successor of the power set of ` 1o ` .
     (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) $)
  3nelsucpw1 $p |- -. 3o e. suc ~P 1o $=
    ( c3o c1o cpw csuc wcel wss c2o 1lt2o elelsuc ax-mp df-3o eleqtrri mt2 wceq
    ssnel wn pw1ne3 nesymi a1i elsuci ecased elpwid mto ) ABCZDEZABFZUFBAEBGDZA
    BGEBUGEHBGIJKLABOMUEABUEAUDEAUDNZUHPUEUDAQRSAUDTUAUBUC $.

  $( Negated excluded middle implies that the successor of the power set of
     ` 1o ` is not a subset of ` 3o ` .  (Contributed by James E. Hanson and
     Jim Kingdon, 31-Jul-2024.) $)
  sucpw1nss3 $p |- ( -. EXMID -> -. suc ~P 1o C_ 3o ) $=
    ( wem wn c1o cpw c3o wcel csuc wss pw1nel3 con0 wi pw1on sucssel ax-mp nsyl
    ) ABCDZEFZPGEHZIPJFRQKLPEJMNO $.

  $( Negated excluded middle implies that ` 3o ` is not a subset of the
     successor of the power set of ` 1o ` .  (Contributed by James E. Hanson
     and Jim Kingdon, 31-Jul-2024.) $)
  3nsssucpw1 $p |- ( -. EXMID -> -. 3o C_ suc ~P 1o ) $=
    ( c3o c1o cpw csuc wss wem c2o wceq df-3o sseq1i wcel wn 1lt2o mt2 com 2onn
    ssnel elexi elpw mtbir a1i wo sucssel ax-mp elsuci syl orcomd ecased eqcomd
    wi sylbi exmidpweq sylibr con3i ) ABCZDZEZFUQUOGHFUQGUOUQGDZUPEZGUOHZAURUPI
    JUSUTGUOKZVALUSVAGBEZVBBGKMGBQNGBGOPRSTUAUSVAUTUSGUPKZVAUTUBGOKUSVCUJPGUPOU
    CUDGUOUEUFUGUHUKUIULUMUN $.

  ${
    $d x y $.
    $( Double negated ordinal trichotomy.

       There are five equivalent statements:  (1)
       ` -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) ` , (2)
       ` -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ) ` , (3)
       ` A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) ` , (4)
       ` A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) ` , and (5)
       ` -. -. EXMID ` .  That these are all equivalent is expressed by (1)
       implies (3) ( ~ onntri13 ), (3) implies (5) ( ~ onntri35 ), (5) implies
       (1) ( ~ onntri51 ), (2) implies (4) ( ~ onntri24 ), (4) implies (5)
       ( ~ onntri45 ), and (5) implies (2) ( ~ onntri52 ).

       Another way of stating this is that ` EXMID ` is equivalent to
       trichotomy, either the ` x e. y \/ x = y \/ y e. x ` or the
       ` x C_ y \/ y C_ x ` form, as shown in ~ exmidontri and
       ~ exmidontri2or , respectively.  Thus ` -. -. EXMID ` is equivalent to
       (1) or (2).  In addition, ` -. -. EXMID ` is equivalent to (3) by
       ~ onntri3or and (4) by ~ onntri2or .

       (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) $)
    onntri35 $p |- ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x )
        -> -. -. EXMID ) $=
      ( cv wcel wceq w3o wn con0 wral csuc c3o wem pw1on eleq1 3orbi123d notbid
      eleq2 mp2an a1i c2o c1o cpw w3a onsuci 3on eqeq1 rspc2v 3ioran sucpw1nel3
      wi eqeq2 sylnib 2on suc11 df-3o eqeq2i exmidpweq notbii biimpi 3nelsucpw1
      wb 3bitr4ri 3jca nsyl ) ACZBCZDZVEVFEZVFVEDZFZGZGZBHIAHIZUAUBZJZKDZGZVOKE
      ZGZKVODZGZUCZLGZVMVPVRVTFZGZWBVOHDKHDVMWEGZUJVNMUDUEVLWFVOVFDZVOVFEZVFVOD
      ZFZGZGABVOKHHVEVOEZVKWKWLVJWJWLVGWGVHWHVIWIVEVOVFNVEVOVFUFVEVOVFQOPPVFKEZ
      WKWEWMWJWDWMWGVPWHVRWIVTVFKVOQVFKVOUKVFKVONOPPUGRVPVRVTUHULWCVQVSWAVQWCUI
      SWCVSLVRVOTJZEZVNTEZVRLVNHDTHDWOWPVAMUMVNTUNRKWNVOUOUPUQVBURUSWAWCUTSVCVD
      $.
  $}

  $( Double negated ordinal trichotomy.  (Contributed by James E. Hanson and
     Jim Kingdon, 2-Aug-2024.) $)
  onntri13 $p |- ( -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x )
      -> A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) ) $=
    ( wel weq w3o con0 wral wn nnral ralimi syl ) ABCABDBACEZBFGZAFGHHMHHZAFGLH
    HBFGZAFGMAFINOAFLBFIJK $.

  ${
    $d x y z $.
    $( Ordinal trichotomy is equivalent to excluded middle.  (Contributed by
       Jim Kingdon, 26-Aug-2024.) $)
    exmidontri $p |- ( EXMID
        <-> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) ) $=
      ( vz wem wel cv wceq w3o con0 exmidontriim c0 csn wdc ontriexmidim adantr
      wral wss exmid1dc impbii ) DABEAFBFGBAEHBIPAIPZABJTCTCFZKLZGZMUAUBQUCABNO
      RS $.
  $}

  ${
    $d x y $.
    $( Double negated ordinal trichotomy.  (Contributed by James E. Hanson and
       Jim Kingdon, 2-Aug-2024.) $)
    onntri51 $p |- ( -. -. EXMID
        -> -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) ) $=
      ( wel weq w3o con0 wral wn wem exmidontriim con3i ) ABCABDBACEBFGAFGZHIHI
      LABJKK $.
  $}

  ${
    $d x y $.
    $( Double negated ordinal trichotomy.  (Contributed by James E. Hanson and
       Jim Kingdon, 2-Aug-2024.) $)
    onntri45 $p |- ( A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x )
        -> -. -. EXMID ) $=
      ( cv wss wo wn con0 wral c1o cpw csuc c3o wa wem wcel sseq1 sseq2 orbi12d
      wceq notbid wi pw1on onsuci 3on rspc2v mp2an sylnib sucpw1nss3 3nsssucpw1
      ioran jca nsyl ) ACZBCZDZUNUMDZEZFZFZBGHAGHZIJZKZLDZFZLVBDZFZMZNFZUTVCVEE
      ZFZVGVBGOLGOUTVJFZUAVAUBUCUDUSVKVBUNDZUNVBDZEZFZFABVBLGGUMVBSZURVOVPUQVNV
      PUOVLUPVMUMVBUNPUMVBUNQRTTUNLSZVOVJVQVNVIVQVLVCVMVEUNLVBQUNLVBPRTTUEUFVCV
      EUJUGVHVDVFUHUIUKUL $.
  $}

  $( Double negated ordinal trichotomy.  (Contributed by James E. Hanson and
     Jim Kingdon, 2-Aug-2024.) $)
  onntri24 $p |- ( -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x )
      -> A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) ) $=
    ( cv wss wo con0 wral wn nnral ralimi syl ) ACZBCZDMLDEZBFGZAFGHHOHHZAFGNHH
    BFGZAFGOAFIPQAFNBFIJK $.

  ${
    $d x y z $.
    $( Ordinal trichotomy is equivalent to excluded middle.  (Contributed by
       Jim Kingdon, 26-Aug-2024.) $)
    exmidontri2or $p |- ( EXMID
        <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) ) $=
      ( vz wem cv wss wo con0 wral wcel wceq w3o exmidontriim wa wi onelss syl6
      adantl syl adantr orc eqimss a1i olc ralimdva ralimia csn ontri2orexmidim
      3jaod c0 wdc exmid1dc impbii ) DAEZBEZFZUOUNFZGZBHIZAHIZDUNUOJZUNUOKZUOUN
      JZLZBHIZAHIUTABMVEUSAHUNHJZVDURBHVFUOHJZNZVAURVBVCVHVAUPURVGVAUPOVFUOUNPR
      UPUQUAZQVBUROVHVBUPURUNUOUBVISUCVHVCUQURVFVCUQOVGUNUOPTUQUPUDQUIUEUFSUTCU
      TCEZUJUGZKZUKVJVKFVLABUHTULUM $.
  $}

  ${
    $d x y $.
    $( Double negated ordinal trichotomy.  (Contributed by James E. Hanson and
       Jim Kingdon, 2-Aug-2024.) $)
    onntri52 $p |- ( -. -. EXMID
        -> -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ) ) $=
      ( cv wss wo con0 wral wn wem exmidontri2or biimpi con3i ) ACZBCZDNMDEBFGA
      FGZHIHIOIOABJKLL $.

    $( Double negated ordinal trichotomy.  (Contributed by Jim Kingdon,
       25-Aug-2024.) $)
    onntri3or $p |- ( -. -. EXMID
        <-> A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) ) $=
      ( wem wn wel weq w3o con0 wral onntri51 onntri13 syl onntri35 impbii ) CD
      DZABEABFBAEGZDDBHIAHIZOPBHIAHIDDQABJABKLABMN $.

    $( Double negated ordinal trichotomy.  (Contributed by Jim Kingdon,
       25-Aug-2024.) $)
    onntri2or $p |- ( -. -. EXMID
        <-> A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) ) $=
      ( wem wn cv wss wo con0 wral onntri52 onntri24 syl onntri45 impbii ) CDDZ
      AEZBEZFQPFGZDDBHIAHIZORBHIAHIDDSABJABKLABMN $.
  $}

  $( With a formula ` ph ` one can associate an element of ` ~P 1o ` , which
     can therefore be thought of as the set of "truth values" (but recall that
     there are no other genuine truth values than ` T. ` and ` F. ` , by
     ~ nndc , which translate to ` 1o ` and ` (/) ` respectively by ~ iftrue
     and ~ iffalse , giving ~ pwtrufal ).

     As proved in ~ if0ab , the associated element of ` ~P 1o ` is the
     extension, in ` ~P 1o ` , of the formula ` ph ` .  (Contributed by BJ,
     15-Aug-2024.)  (Proof shortened by BJ, 5-May-2026.) $)
  fmelpw1o $p |- if ( ph , 1o , (/) ) e. ~P 1o $=
    ( c1o cvv wcel c0 cif cpw 1oex if0elpw ax-mp ) BCDABEFBGDHABCIJ $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Apartness relations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c Ap $.

  $( Apartness predicate symbol. $)
  wap $a wff R Ap A $.

  ${
    $d x y z A $.  $d x y z R $.
    $( Apartness predicate.  A relation ` R ` is an apartness if it is
       irreflexive, symmetric, and cotransitive.  (Contributed by Jim Kingdon,
       14-Feb-2025.) $)
    df-pap $a |- ( R Ap A <-> ( ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) /\
      ( A. x e. A A. y e. A ( x R y -> y R x ) /\
      A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) ) ) ) $.
  $}

  ${
    $d A x y z $.  $d R x y z $.  $d S x y z $.
    $( Equality theorem for apartness predicate.  (Contributed by Jim Kingdon,
       3-Jun-2026.) $)
    papeq1 $p |- ( R = S -> ( R Ap A <-> S Ap A ) ) $=
      ( vx vy vz wss cv wbr wn wral wa wi breq ralbidv anbi12d imbi12d 2ralbidv
      wo wap wceq cxp sseq1 notbid orbi12d df-pap 3bitr4g ) BCUAZBAAUBZGZDHZUKB
      IZJZDAKZLZUKEHZBIZUPUKBIZMZEAKDAKZUQUKFHZBIZUPVABIZSZMZFAKZEAKDAKZLZLCUIG
      ZUKUKCIZJZDAKZLZUKUPCIZUPUKCIZMZEAKDAKZVNUKVACIZUPVACIZSZMZFAKZEAKDAKZLZL
      ABTACTUHUOVMVHWDUHUJVIUNVLBCUIUCUHUMVKDAUHULVJUKUKBCNUDOPUHUTVQVGWCUHUSVP
      DEAAUHUQVNURVOUKUPBCNZUPUKBCNQRUHVFWBDEAAUHVEWAFAUHUQVNVDVTWEUHVBVRVCVSUK
      VABCNUPVABCNUEQORPPDEFABUFDEFACUFUG $.
  $}

  ${
    $d R x y z $.  $d A x y z $.  $d B x y z $.
    $( Equality theorem for apartness predicate.  (Contributed by Jim Kingdon,
       3-Jun-2026.) $)
    papeq2 $p |- ( A = B -> ( R Ap A <-> R Ap B ) ) $=
      ( vx vy vz wceq cxp wss cv wbr wn wral wa wi wap raleq anbi12d raleqbi1dv
      df-pap wo id sqxpeqd sseq2d 3bitr4g ) ABGZCAAHZIZDJZUICKLZDAMZNZUIEJZCKZU
      MUICKOZEAMZDAMZUNUIFJZCKUMURCKUAOZFAMZEAMZDAMZNZNCBBHZIZUJDBMZNZUOEBMZDBM
      ZUSFBMZEBMZDBMZNZNACPBCPUFULVGVCVMUFUHVEUKVFUFUGVDCUFABUFUBUCUDUJDABQRUFU
      QVIVBVLUPVHDABUOEABQSVAVKDABUTVJEABUSFABQSSRRDEFACTDEFBCTUE $.
  $}

  ${
    $d A x y z $.  $d R x y z $.  $d X x $.
    $( An apartness is irreflexive.  (Contributed by Jim Kingdon,
       27-May-2026.) $)
    papirr $p |- ( ( R Ap A /\ X e. A ) -> -. X R X ) $=
      ( vx vy vz wap wcel wa cv wbr wn wceq id breq12d notbid wral cxp wss wi
      wo df-pap simplbi simprd adantr simpr rspcdva ) ABGZCAHZIDJZUJBKZLZCCBKZL
      DACUJCMZUKUMUNUJCUJCBUNNZUOOPUHULDAQZUIUHBAARSZUPUHUQUPIUJEJZBKZURUJBKTEA
      QDAQUSUJFJZBKURUTBKUATFAQEAQDAQIDEFABUBUCUDUEUHUIUFUG $.
  $}

  ${
    $d A x y z $.  $d R x y z $.  $d X x y $.  $d Y y $.
    papsym.r $e |- ( ph -> R Ap A ) $.
    papsym.x $e |- ( ph -> X e. A ) $.
    papsym.y $e |- ( ph -> Y e. A ) $.
    papsym.ap $e |- ( ph -> X R Y ) $.
    $( An apartness is symmetric.  (Contributed by Jim Kingdon,
       27-May-2026.) $)
    papsym $p |- ( ph -> Y R X ) $=
      ( vy vx vz wbr cv wi wceq breq2 breq1 wral wa imbi12d ralbidv cxp wss wap
      wn wo df-pap sylib simprld rspcdva mpd ) ADECMZEDCMZIADJNZCMZUODCMZOZUMUN
      OJBEUOEPUPUMUQUNUOEDCQUOEDCRUAAKNZUOCMZUOUSCMZOZJBSZURJBSKBDUSDPZVBURJBVD
      UTUPVAUQUSDUOCRUSDUOCQUAUBACBBUCUDUSUSCMUFKBSTZVCKBSZUTUSLNZCMUOVGCMUGOLB
      SJBSKBSZABCUEVEVFVHTTFKJLBCUHUIUJGUKHUKUL $.

    $d X x y z $.  $d Y y z $.  $d Z z $.
    papcotr.z $e |- ( ph -> Z e. A ) $.
    $( An apartness is cotransitive.  (Contributed by Jim Kingdon,
       28-May-2026.) $)
    papcotr $p |- ( ph -> ( X R Z \/ Y R Z ) ) $=
      ( vz vy vx wbr wo cv wi wceq wral orbi12d imbi2d imbi12d ralbidv 2ralbidv
      breq2 breq1 orbi2d orbi1d cxp wss wn wap df-pap sylib simprrd rspcdva mpd
      wa ) ADECOZDFCOZEFCOZPZJAUTDLQZCOZEVDCOZPZRZUTVCRLBFVDFSZVGVCUTVIVEVAVFVB
      VDFDCUFVDFECUFUAUBADMQZCOZVEVJVDCOZPZRZLBTZVHLBTMBEVJESZVNVHLBVPVKUTVMVGV
      JEDCUFVPVLVFVEVJEVDCUGUHUCUDANQZVJCOZVQVDCOZVLPZRZLBTMBTZVOMBTNBDVQDSZWAV
      NMLBBWCVRVKVTVMVQDVJCUGWCVSVEVLVQDVDCUGUIUCUEACBBUJUKVQVQCOULNBTUSZVRVJVQ
      CORMBTNBTZWBNBTZABCUMWDWEWFUSUSGNMLBCUNUOUPHUQIUQKUQUR $.
  $}

  $c TAp $.

  $( Tight apartness predicate symbol. $)
  wtap $a wff R TAp A $.

  ${
    $d x y A $.  $d x y R $.
    $( Tight apartness predicate.  A relation ` R ` is a tight apartness if it
       is irreflexive, symmetric, cotransitive, and tight.  (Contributed by Jim
       Kingdon, 5-Feb-2025.) $)
    df-tap $a |- ( R TAp A <-> ( R Ap A /\
      A. x e. A A. y e. A ( -. x R y -> x = y ) ) ) $.
  $}

  ${
    $d A x y $.  $d R x y $.
    $( A tight apartness is an apartness.  (Contributed by Jim Kingdon,
       29-May-2026.) $)
    tapap $p |- ( R TAp A -> R Ap A ) $=
      ( vx vy wtap wap cv wbr wn weq wi wral df-tap simplbi ) ABEABFCGDGBHICDJK
      DALCALCDABMN $.
  $}

  ${
    $d A x y z $.  $d R x y z $.
    $( Tight apartness with the apartness properties from ~ df-pap expanded.
       (Contributed by Jim Kingdon, 21-Feb-2025.) $)
    dftap2 $p |- ( R TAp A <-> ( R C_ ( A X. A ) /\ ( A. x e. A -. x R x /\
      A. x e. A A. y e. A ( x R y -> y R x ) ) /\
      ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) /\
      A. x e. A A. y e. A ( -. x R y -> x = y ) ) ) ) $=
      ( wtap cxp wss cv wbr wn wral wi wa wo weq simpld simprd jca sylanbrc w3a
      df-tap biimpi df-pap sylib 3jca simp1 simp2l simp2r simp3l simp3r impbii
      wap ) DEFZEDDGHZAIZUPEJKADLZUPBIZEJZURUPEJMBDLADLZNZUSUPCIZEJURVBEJOMCDLB
      DLADLZUSKABPMBDLADLZNZUAZUNUOVAVEUNUOUQUNUOUQNZUTVCNZUNDEUMZVGVHNUNVIVDUN
      VIVDNABDEUBZUCZQABCDEUDZUEZQZQUNUQUTUNUOUQVNRUNUTVCUNVGVHVMRZQSUNVCVDUNUT
      VCVORUNVIVDVKRSUFVFVIVDUNVFVGVHVIVFUOUQUOVAVEUGUOUQUTVEUHSVFUTVCUOUQUTVEU
      IUOVAVCVDUJSVLTUOVAVCVDUKVJTUL $.
  $}

  ${
    $d A x y z $.  $d R x y z $.  $d S x y z $.
    $( Equality theorem for tight apartness predicate.  (Contributed by Jim
       Kingdon, 8-Feb-2025.) $)
    tapeq1 $p |- ( R = S -> ( R TAp A <-> S TAp A ) ) $=
      ( vx vy vz wceq wss cv wbr wn wral wi wa wo w3a wtap breq notbid 2ralbidv
      cxp sseq1 ralbidv imbi12d anbi12d orbi12d imbi1d 3anbi123d dftap2 3bitr4g
      ) BCGZBAAUAZHZDIZUNBJZKZDALZUNEIZBJZURUNBJZMZEALDALZNZUSUNFIZBJZURVDBJZOZ
      MZFALZEALDALZUSKZUNURGZMZEALDALZNZPCULHZUNUNCJZKZDALZUNURCJZURUNCJZMZEALD
      ALZNZVTUNVDCJZURVDCJZOZMZFALZEALDALZVTKZVLMZEALDALZNZPABQACQUKUMVPVCWDVOW
      NBCULUBUKUQVSVBWCUKUPVRDAUKUOVQUNUNBCRSUCUKVAWBDEAAUKUSVTUTWAUNURBCRZURUN
      BCRUDTUEUKVJWJVNWMUKVIWIDEAAUKVHWHFAUKUSVTVGWGWOUKVEWEVFWFUNVDBCRURVDBCRU
      FUDUCTUKVMWLDEAAUKVKWKVLUKUSVTWOSUGTUEUHDEFABUIDEFACUIUJ $.
  $}

  ${
    $d R x y z $.  $d A x y z $.  $d B x y z $.
    $( Equality theorem for tight apartness predicate.  (Contributed by Jim
       Kingdon, 15-Feb-2025.) $)
    tapeq2 $p |- ( A = B -> ( R TAp A <-> R TAp B ) ) $=
      ( vx vy vz wceq cxp wss cv wbr wn wral wi wa w3a raleq raleqbi1dv anbi12d
      wtap wo xpeq12 anidms sseq2d 3anbi123d dftap2 3bitr4g ) ABGZCAAHZIZDJZUKC
      KLZDAMZUKEJZCKZUNUKCKNZEAMZDAMZOZUOUKFJZCKUNUTCKUANZFAMZEAMZDAMZUOLUKUNGN
      ZEAMZDAMZOZPCBBHZIZULDBMZUPEBMZDBMZOZVAFBMZEBMZDBMZVEEBMZDBMZOZPACTBCTUHU
      JVJUSVNVHVTUHUIVICUHUIVIGABABUBUCUDUHUMVKURVMULDABQUQVLDABUPEABQRSUHVDVQV
      GVSVCVPDABVBVOEABVAFABQRRVFVRDABVEEABQRSUEDEFACUFDEFBCUFUG $.
  $}

  ${
    $d A a b c u v $.  $d A a b c x y $.
    $( Negated equality on a set with decidable equality is a tight apartness.
       (Contributed by Jim Kingdon, 5-Feb-2025.) $)
    netap $p |- ( A. x e. A A. y e. A DECID x = y
        -> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A ) $=
      ( va vb vc weq wral cv wcel wa wne wbr wn cop df-br wb neeq2 wdc copab wi
      cxp wss wtap opabssxp a1i neirr neeq1 opelopab2 anidms bitrid mtbiri rgen
      wo ancoms necom bitrdi bitr4d biimpd rgen2 jca w3a 3adant3 simpr eqnetrrd
      adantl simplr necomd neqned equequ2 dcbid equequ1 ralbidv simplr1 rspcdva
      olcd simpll simplr3 exmiddc syl mpjaodan 3adant2 3adant1 orbi12d ad2antlr
      orcd mpbird ex sylbid ralrimivvva notbid df-ne notbii simpl simprl simprr
      notnotrdc ralrimivva dftap2 syl3anbrc ) ABIZUAZBEJZAEJZDKZELCKZELMXGXHNZM
      DCUBZEEUDUEZFKZXLXJOZPZFEJZXLGKZXJOZXPXLXJOZUCZGEJFEJZMXQXLHKZXJOZXPYAXJO
      ZUPZUCZHEJGEJFEJZXQPZFGIZUCZGEJFEJZMEXJUFXKXFXIDCEEUGUHXFXOXTXOXFXNFEXLEL
      ZXMXLXLNZXLUIXMXLXLQXJLZYKYLXLXLXJRYKYMYLSXIXLXHNZYLDCXLXLEEXGXLXHUJZXHXL
      XLTUKULUMUNUOUHXTXFXSFGEEYKXPELZMZXQXRYQXQXLXPNZXRXQXLXPQXJLYQYRXLXPXJRXI
      YNYRDCXLXPEEYOXHXPXLTUKUMZXRXPXLQXJLZYQYRXPXLXJRYQYTXPXLNZYRYPYKYTUUASXIX
      PXHNZUUADCXPXLEEXGXPXHUJZXHXLXPTUKUQXPXLURUSUMUTVAVBUHVCXFYFYJXFYEFGHEEEX
      FYKYPYAELZVDZMZXQYRYDUUEXQYRSZXFYKYPUUGUUDYSVEVHUUFYRYDUUFYRMZYDXLYANZXPY
      ANZUPZUUHFHIZUUKUULPZUUHUULMZUUJUUIUUNYAXPUUNXLYAXPUUHUULVFUUFYRUULVIVGVJ
      VRUUHUUMMZUUIUUJUUOXLYAUUHUUMVFVKWHUUHUULUAZUULUUMUPUUHFBIZUAZUUPBEYABHIU
      UQUULBHFVLVMUUHXEUURBEJZAEXLAFIZXDUURBEUUTXCUUQAFBVNVMVOZXFUUEYRVSYKYPUUD
      XFYRVPVQYKYPUUDXFYRVTVQUULWAWBWCUUEYDUUKSXFYRUUEYBUUIYCUUJYBXLYAQXJLZUUEU
      UIXLYAXJRYKUUDUVBUUISYPXIYNUUIDCXLYAEEYOXHYAXLTUKWDUMYCXPYAQXJLZUUEUUJXPY
      AXJRYPUUDUVCUUJSYKXIUUBUUJDCXPYAEEUUCXHYAXPTUKWEUMWFWGWIWJWKWLXFYIFGEEXFY
      QMZYGYHPZPZYHYQYGUVFSXFYQYGYRPUVFYQXQYRYSWMYRUVEXLXPWNWOUSVHUVDYHUAZUVFYH
      UCUVDUURUVGBEXPBGIUUQYHBGFVLVMUVDXEUUSAEXLUVAXFYQWPXFYKYPWQVQXFYKYPWRVQYH
      WSWBWKWTVCFGHEXJXAXB $.
  $}

  ${
    $d u v $.  $d x y $.
    $( Negated equality is a tight apartness on ` 2o ` .  (Contributed by Jim
       Kingdon, 6-Feb-2025.) $)
    2onetap $p |- { <. u , v >. |
        ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } TAp 2o $=
      ( vx vy weq wdc c2o wral cv wcel wa wne copab wtap 2onn elnn mpan2 nndceq
      com syl2an rgen2 netap ax-mp ) CDEFZDGHCGHGBIZGJAIZGJKUEUFLKBAMNUDCDGGCIZ
      GJZUGSJZDIZSJZUDUJGJZUHGSJZUIOUGGPQULUMUKOUJGPQUGUJRTUACDABGUBUC $.
  $}

  ${
    $d u v $.
    $( ` (/) ` and ` 1o ` are two unequal elements of ` 2o ` .  (Contributed by
       Jim Kingdon, 8-Feb-2025.) $)
    2oneel $p |- <. (/) , 1o >. e. { <. u , v >. |
        ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } $=
      ( c0 c1o cop cv c2o wcel wa wne copab 1n0 necomi wb 0lt2o 1lt2o opelopab2
      neeq1 neeq2 mp2an mpbir ) CDEBFZGHAFZGHIUBUCJZIBAKHZCDJZDCLMCGHDGHUEUFNOP
      UDCUCJUFBACDGGUBCUCRUCDCSQTUA $.
  $}

  ${
    $d a b c ph u v $.
    $( Lemma for ~ 2omotap .  (Contributed by Jim Kingdon, 6-Feb-2025.) $)
    2omotaplemap $p |- ( -. -. ph -> { <. u , v >. |
        ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o ) $=
      ( va vb vc wn cv c2o wcel wa wne wbr wral weq cop df-br anbi2d neeq2 wb
      copab cxp wss wi wo wtap opabssxp a1i neeq1 opelopab2 bitrid ancoms necom
      anbi2i bitrdi bitr4d biimpd rgen2 neirr intnan anidms mtbiri rgen 3adant3
      jctil w3a simpr simplrr eqnetrrd necomd olcd neqned orcd simpl1 2onn elnn
      wdc com sylancl simpl3 nndceq syl2anc exmiddc syl mpjaodan 3adant2 adantr
      ibar adantl 3adant1 orbi12d mpbird sylbid ralrimivvva notbid simpll ancom
      ex sylnib imnan sylibr mtod simplrl nnedc ralrimivva jca dftap2 syl3anbrc
      mpbid ) AGZGZCHZIJBHZIJKAXLXMLZKZKCBUAZIIUBUCZDHZXRXPMZGZDINZXREHZXPMZYBX
      RXPMZUDZEINDINZKYCXRFHZXPMZYBYGXPMZUEZUDZFINEINDINZYCGZDEOZUDZEINDINZKIXP
      UFXQXKXOCBIIUGUHXKYFYAYFXKYEDEIIXRIJZYBIJZKZYCYDYSYCAXRYBLZKZYDYCXRYBPXPJ
      YSUUAXRYBXPQXOAXRXMLZKZUUACBXRYBIICDOXNUUBAXLXRXMUIRZBEOUUBYTAXMYBXRSRUJU
      KZYDYBXRPXPJZYSUUAYBXRXPQYSUUFAYBXRLZKZUUAYRYQUUFUUHTXOAYBXMLZKZUUHCBYBXR
      IICEOXNUUIAXLYBXMUIRZBDOZUUIUUGAXMXRYBSRUJULUUGYTAYBXRUMUNUOUKUPUQURUHXTD
      IYQXSAXRXRLZKZUUMAXRUSUTXSXRXRPXPJZYQUUNXRXRXPQYQUUOUUNTXOUUCUUNCBXRXRIIU
      UDUULUUBUUMAXMXRXRSRUJVAUKVBVCVEXKYLYPXKYKDEFIIIYQYRYGIJZVFZYKXKUUQYCUUAY
      JYQYRYCUUATUUPUUEVDUUQUUAYJUUQUUAKZYJXRYGLZYBYGLZUEZUURDFOZUVAUVBGZUURUVB
      KZUUTUUSUVDYGYBUVDXRYGYBUURUVBVGUUQAYTUVBVHVIVJVKUURUVCKZUUSUUTUVEXRYGUUR
      UVCVGVLVMUURUVBVQZUVBUVCUEUURXRVRJZYGVRJZUVFUURYQIVRJZUVGYQYRUUPUUAVNVOXR
      IVPZVSUURUUPUVIUVHYQYRUUPUUAVTVOYGIVPVSXRYGWAWBUVBWCWDWEUURYHUUSYIUUTUURY
      HAUUSKZUUSUUQYHUVKTUUAYHXRYGPXPJZUUQUVKXRYGXPQYQUUPUVLUVKTYRXOUUCUVKCBXRY
      GIIUUDBFOZUUBUUSAXMYGXRSRUJWFUKWGUUAUUSUVKTZUUQAUVNYTAUUSWHWGWIUPUURYIAUU
      TKZUUTUUQYIUVOTUUAYIYBYGPXPJZUUQUVOYBYGXPQYRUUPUVPUVOTYQXOUUJUVOCBYBYGIIU
      UKUVMUUIUUTAXMYGYBSRUJWJUKWGUUAUUTUVOTZUUQAUVQYTAUUTWHWGWIUPWKWLWRWMWIWNX
      KYODEIIXKYSKZYMUUAGZYNYSYMUVSTXKYSYCUUAUUEWOWIUVRUVSYNUVRUVSKZYTGZYNUVTYT
      XJXKYSUVSWPUVTYTAKZGYTXJUDUVTUUAUWBUVRUVSVGAYTWQWSYTAWTXAXBUVTYNVQZUWAYNT
      UVTUVGYBVRJZUWCUVTYQUVIUVGXKYQYRUVSXCVOUVJVSUVTYRUVIUWDXKYQYRUVSVHVOYBIVP
      VSXRYBWAWBXRYBXDWDXIWRWMXEXFDEFIXPXGXH $.
  $}

  ${
    $d ph r u v $.
    $( Lemma for ~ 2omotap .  (Contributed by Jim Kingdon, 6-Feb-2025.) $)
    2omotaplemst $p |- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph ) $=
      ( vu vv c2o cv wtap wn wa c0 c1o wne wcel copab cvv wb opabssxp ssexi a1i
      wceq wmo cop 2oneel 2omotaplemap adantl cxp 2onn elexi xpex simpl 2onetap
      tapeq1 syl211anc mpbird eleqtrid 0lt2o 1lt2o neeq1 anbi2d neeq2 opelopab2
      com mob mp2an sylib simpld ) EBFZGZBUAZAHHZIZAJKLZVKJKUBZCFZEMDFZEMIZAVNV
      OLZIZICDNZMZAVLIZVKVMVPVQICDNZVSDCUCVKWBVSTZEVSGZVJWDVIADCUDUEVKWBOMZVSOM
      ZVIEWBGZWCWDPWEVKWBEEUFZEEEVBUGUHZWIUIZVQCDEEQRSWFVKVSWHWJVRCDEEQRSVIVJUJ
      WGVKDCUKSVHWGWDBWBVSOOEVGWBULEVGVSULVCUMUNUOJEMKEMVTWAPUPUQVRAJVOLZIWACDJ
      KEEVNJTVQWKAVNJVOURUSVOKTWKVLAVOKJUTUSVAVDVEVF $.
  $}

  ${
    $d r x $.
    $( If there is at most one tight apartness on ` 2o ` , excluded middle
       follows.  Based on online discussions by Tom de Jong, Andrew W Swan, and
       Martin Escardo.  (Contributed by Jim Kingdon, 6-Feb-2025.) $)
    2omotap $p |- ( E* r r TAp 2o -> EXMID ) $=
      ( vx c2o cv wtap wmo c0 csn wceq wstab wss wn 2omotaplemst df-stab sylibr
      wi ex adantr exmid1stab ) CADEAFZBTBDZGHZIZJZUAUBKTUCLLZUCPUDTUEUCUCAMQUC
      NORS $.
  $}

  ${
    $d A p u v x y $.  $d A x y z $.  $d R p x y z $.
    $( Excluded middle implies there is only one tight apartness on any class,
       namely negated equality.  (Contributed by Jim Kingdon, 14-Feb-2025.) $)
    exmidapne $p |- ( EXMID -> ( R TAp A <->
        R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) ) $=
      ( vx vy vz wem cv wcel wa wne wceq simpr wbr wn wral wi syl adantr vp cfv
      wtap copab c1st c2nd cop cxp simplr wss wo w3a dftap2 biimpi simp1d sseld
      sylc 1st2nd2 xp1st xp2nd eqeltrrd opeq2d id notbid simp2d simpld ad3antlr
      breq12d df-br sylnib eqneltrrd pm2.65da neqned jca31 eleq1 bi2anan9 simpl
      rspcdva wb neeq12d anbi12d opelopabga syl2anc mpbird eqeltrd wrel relopab
      1st2nd mpan adantl breq2 eqeq2 imbi12d breq1 eqeq1 simp3d simprd ad2antlr
      ralbidv anbi1d neeq1 anbi2d elopabi neneqd wdc exmidexmid con1dc ad2antrr
      neeq2 mp2d sylib impbida eqrdv ralrimivw netap tapeq1 ) HCDUCZDBIZCJZAIZC
      JZKZXRXTLZKZBAUDZMZHXQKZUADYEYGUAIZDJZYHYEJZYGYIKZYHYHUEUBZYHUFUBZUGZYEYK
      YHCCUHZJZYHYNMZYKXQYIYPHXQYIUIYGYINZXQDYOYHXQDYOUJZEIZYTDOZPZECQZYTFIZDOZ
      UUDYTDORFCQECQZKZUUEYTGIZDOUUDUUHDOUKRGCQFCQECQZUUEPZYTUUDMZRZFCQZECQZKZX
      QYSUUGUUOULEFGCDUMUNZUOUPUQZYHCCURSZYKYNYEJZYLCJZYMCJZKZYLYMLZKZYKUUTUVAU
      VCYKYPUUTUUQYHCCUSSZYKYPUVAUUQYHCCUTSZYKYLYMYKYLYMMZYNDJZYKUVHUVGYKYHYNDU
      URYRVATYKUVGKZYLYLUGZYNDUVIYLYMYLYKUVGNVBUVIYLYLDOZUVJDJUVIUUBUVKPECYLYTY
      LMZUUAUVKUVLYTYLYTYLDUVLVCZUVMVHVDXQUUCHYIUVGXQUUCUUFXQYSUUGUUOUUPVEVFVGY
      KUUTUVGUVETVRYLYLDVIVJVKVLVMVNYKUUTUVAUUSUVDVSUVEUVFYDUVDBAYLYMCCXRYLMZXT
      YMMZKZYBUVBYCUVCUVNXSUUTUVOYAUVAXRYLCVOZXTYMCVOZVPUVPXRYLXTYMUVNUVOVQUVNU
      VONVTWAWBWCWDWEYGYJKZYHYNDYJYQYGYEWFYJYQYDBAWGYHYEWHWIWJUVSYLYMDOZUVHUVSU
      VTPZUVGRZUVGPZUVTUVSYLUUDDOZPZYLUUDMZRZUWBFCYMUUDYMMZUWEUWAUWFUVGUWHUWDUV
      TUUDYMYLDWKVDUUDYMYLWLWMUVSUUMUWGFCQECYLUVLUULUWGFCUVLUUJUWEUUKUWFUVLUUEU
      WDYTYLUUDDWNVDYTYLUUDWOWMWSXQUUNHYJXQUUIUUNXQYSUUGUUOUUPWPWQWRUVSUUTUVAUV
      SUVBUVCYJUVDYGYDUUTYAKZYLXTLZKUVDBAYHUVNYBUWIYCUWJUVNXSUUTYAUVQWTXRYLXTXA
      WAUVOUWIUVBUWJUVCUVOYAUVAUUTUVRXBXTYMYLXIWAXCWJZVFZVFVRUVSUUTUVAUWLWQVRUV
      SYLYMUVSUVBUVCUWKWQXDHUWBUWCUVTRRZXQYJHUVTXEUWMUVTXFUVTUVGXGSXHXJYLYMDVIX
      KWEXLXMHYFKXQCYEUCZHUWNYFHUUKXEZFCQZECQUWNHUWPECHUWOFCUUKXFXNXNEFABCXOSTY
      FXQUWNVSHCDYEXPWJWDXL $.
  $}

  ${
    $d r s u v x $.
    $( The proposition that every class has at most one tight apartness is
       equivalent to excluded middle.  (Contributed by Jim Kingdon,
       14-Feb-2025.) $)
    exmidmotap $p |- ( EXMID <-> A. x E* r r TAp x ) $=
      ( vs vu vv wem cv wtap wmo wal wa wceq wi wel wb exmidapne adantr c2o com
      mpbid wne simprl simprr eqtr4d ex alrimivv tapeq1 mo4 sylibr alrimiv wcel
      copab 2onn tapeq2 mobidv spcgv ax-mp 2omotap syl impbii ) FAGZBGZHZBIZAJZ
      FVDAFVCVACGZHZKZVBVFLZMZCJBJVDFVJBCFVHVIFVHKZVBDANEANKDGEGUAKDEULZVFVKVCV
      BVLLZFVCVGUBFVCVMOVHEDVAVBPQTVKVGVFVLLZFVCVGUCFVGVNOVHEDVAVFPQTUDUEUFVCVG
      BCVAVBVFUGUHUIUJVERVBHZBIZFRSUKVEVPMUMVDVPARSVARLVCVOBVARVBUNUOUPUQBURUSU
      T $.
  $}


$(
###############################################################################
  CHOICE PRINCIPLES
###############################################################################

  We have already introduced the full Axiom of Choice ~ df-ac but since it
  implies excluded middle as shown at ~ exmidac , it is not especially
  relevant to us.  In this section we define countable choice and dependent
  choice, which are not as strong as thus often considered in mathematics
  which seeks to avoid full excluded middle.

$)


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Countable Choice and Dependent Choice
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce Countable Choice
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c CCHOICE $.

  $( Formula for an abbreviation of countable choice. $)
  wacc $a wff CCHOICE $.

  ${
    $d f x $.
    $( The expression ` CCHOICE ` will be used as a readable shorthand for any
       form of countable choice, analogous to ~ df-ac for full choice.
       (Contributed by Jim Kingdon, 27-Nov-2023.) $)
    df-cc $a |- ( CCHOICE
      <-> A. x ( dom x ~~ _om -> E. f ( f C_ x /\ f Fn dom x ) ) ) $.
  $}

  ${
    $d A f u v x $.  $d A f u v y $.  $d f ph u x $.  $d u v w x $.
    ccfunen.cc $e |- ( ph -> CCHOICE ) $.
    ccfunen.a $e |- ( ph -> A ~~ _om ) $.
    ccfunen.m $e |- ( ph -> A. x e. A E. w w e. x ) $.
    $( Existence of a choice function for a countably infinite set.
       (Contributed by Jim Kingdon, 28-Nov-2023.) $)
    ccfunen $p |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) ) $=
      ( vu vv vy cv wcel wel wa wex cvv com cen wceq copab wss cdm wfn cfv wral
      wbr wi wal encv syl simpld cab abid2 vex eqeltri a1i opabex3d df-cc sylib
      wacc elequ2 exbidv cbvralv elequ1 cbvexv ralbii bitri dmopab3 dmeq breq1d
      eqbrtrd sseq2 fneq2d anbi12d imbi12d spcgv syl3c simprr wb adantr simplrl
      mpbid cop fnopfv sylan sseldd fvex eleq1 anbi2d opelopab simprd ralrimiva
      jca ex eximdv mpd ) AELZILZDMZJINZOZIJUAZUBZWRXCUCZUDZOZEPZWRDUDZBLZWRUEZ
      XJMZBDUFZOZEPAXCQMKLZUCZRSUGZWRXOUBZWRXPUDZOZEPZUHZKUIZXERSUGZXHAXAIJDADQ
      MZRQMZADRSUGYEYFOGDRUJUKULXAJUMZQMAWTOYGWSQJWSUNIUOUPUQURAVAYCFKEUSUTAXED
      RSAXAJPZIDUFZXEDTACBNZCPZBDUFZYIHYLCINZCPZIDUFYIYKYNBIDXJWSTYJYMCBICVBVCV
      DYNYHIDYMXACJCJIVEVFVGVHUTXAIJDVIUTZGVLYBYDXHUHKXCQXOXCTZXQYDYAXHYPXPXERS
      XOXCVJZVKYPXTXGEYPXRXDXSXFXOXCWRVMYPXPXEWRYQVNVOVCVPVQVRAXGXNEAXGXNAXGOZX
      IXMYRXFXIAXDXFVSAXFXIVTXGAXEDWRYOVNWAWCZYRXLBDYRXJDMZOZYTXLUUAXJXKWDZXCMY
      TXLOZUUAWRXCUUBAXDXFYTWBYRXIYTUUBWRMYSDXJWRWEWFWGXBYTJBNZOUUCIJXJXKBUOZXJ
      WRQQEUOUUEWHWSXJTWTYTXAUUDWSXJDWIIBJVBVOJLZXKTUUDXLYTUUFXKXJWIWJWKUTWLWMW
      NWOWPWQ $.
  $}

  ${
    $d a f w z $.  $d a f x z $.  $d f x z $.
    $( Countable choice in terms of a choice function on a countably infinite
       set of inhabited sets.  (Contributed by Jim Kingdon, 27-Apr-2024.) $)
    cc1 $p |- ( CCHOICE -> A. x ( ( x ~~ _om /\ A. z e. x E. w w e. z )
         -> E. f A. z e. x ( f ` z ) e. z ) ) $=
      ( va wacc cv com cen wbr wel wex wral wa cfv wcel wi weq cbvralvw sylib
      wfn simpl simprl simprr elequ2 exbidv ccfunen exsimpr fveq2 eleq12d exbii
      syl id ex alrimiv ) FAGZHIJZCBKZCLZBUPMZNZBGZDGZOZVBPZBUPMZDLZQAFVAVGFVAN
      ZEGZVCOZVIPZEUPMZDLZVGVHVCUPUAZVLNDLVMVHECUPDFVAUBFUQUTUCVHUTCEKZCLZEUPMF
      UQUTUDUSVPBEUPBERURVOCBECUEUFSTUGVNVLDUHULVLVFDVKVEEBUPEBRZVJVDVIVBVIVBVC
      UIVQUMUJSUKTUNUO $.
  $}

  ${
    cc2.cc $e |- ( ph -> CCHOICE ) $.
    cc2.a $e |- ( ph -> F Fn _om ) $.
    cc2.m $e |- ( ph -> A. x e. _om E. w w e. ( F ` x ) ) $.

    ${
      $d A f n z $.  $d A k n $.  $d A n w z $.  $d F a n w $.  $d F f g n $.
      $d F n w x $.  $d G g n $.  $d a n ph w $.  $d f n ph z $.  $d k n ph $.
      cc2lem.a $e |- A = ( n e. _om |-> ( { n } X. ( F ` n ) ) ) $.
      cc2lem.g $e |- G = ( n e. _om |-> ( 2nd ` ( f ` ( A ` n ) ) ) ) $.
      $( Lemma for ~ cc2 .  (Contributed by Jim Kingdon, 27-Apr-2024.) $)
      cc2lem $p |- ( ph
          -> E. g ( g Fn _om /\ A. n e. _om ( g ` n ) e. ( F ` n ) ) ) $=
        ( cfv wcel wa com cvv wceq vz vk va cv crn wfn wral wex wf1 cen wbr csn
        wf wi cxp vex snex funfvex sylan xpexg sylancr fmptd sneq fveq2 xpeq12d
        funfni simprr adantr syl2anc fvmptd3 eqeq2d simpr fvmpt2 adantrr eqeq1d
        snm eleq2d exbidv simprl rspcdva xp11m sneqr biimtrdi sylbid ralrimivva
        wb dff13 sylanbrc wf1o f1f1orn omex f1oen ensym 3syl syl wfun ralrimiva
        cdm wrex fnmpt fnfun elrnrexdm sylc simpll fndm eleqtrd eleq1 sylib cop
        cbvexv vsnid opelxpi spcegv exlimddv mpbird rexlimddv c2nd fvexg 2ndexg
        ccfunen eleq12d simplrr fnfvelrn adantlr mpbid xp2nd eqeltrd fnex fneq1
        id a1i fveq1 eleq1d ralbidv anbi12d mp2and ) AEUDZDUEZUFZUAUDZYQOZYTPZU
        AYRUGZQZFUDZRUFZGUDZUUEOZUUGHOZPZGRUGZQZFUHZEAUACYREJARSDUIZYRRUJUKZARS
        DUMUUGDOZUBUDZDOZTZUUGUUQTZUNZUBRUGGRUGUUNAGRUUGULZUUIUOZSDAUUGRPZQZUVB
        SPUUISPZUVCSPZUUGGUPZUQAHRUFZUVDUVFKUVFRUUGHUUGHURVFUSUVBUUISSUTVAZMVBA
        UVAGUBRRAUVDUUQRPZQZQZUUSUUPUUQULZUUQHOZUOZTZUUTUVMUURUVPUUPUVMGUUQUVCU
        VPRDSMUUTUVBUVNUUIUVOUUGUUQVCUUGUUQHVDVEAUVDUVKVGZUVMUVNSPUVOSPZUVPSPUU
        QUBUPUQUVMUVIUVKUVSAUVIUVLKVHUVRUVSRUUQHUUQHURVFVIUVNUVOSSUTVAVJVKUVMUV
        QUVCUVPTZUUTUVMUUPUVCUVPAUVDUUPUVCTZUVKUVEUVDUVGUWAAUVDVLZUVJGRUVCSDMVM
        VIZVNVOUVMUVTUVBUVNTZUUIUVOTZQZUUTUVMCUDZUVBPCUHUWGUUIPZCUHZUVTUWFWFCUU
        GUVHVPUVMUWGBUDZHOZPZCUHZUWIBRUUGUWJUUGTZUWLUWHCUWNUWKUUIUWGUWJUUGHVDVQ
        VRZAUWMBRUGZUVLLVHAUVDUVKVSVTCCUVBUUIUVNUVOWAVAUWDUUTUWEUUGUUQUVHWBVHWC
        WDWDWEGUBRSDWGWHUUNRYRDWIRYRUJUKUUORSDWJRYRDWKWLRYRWMWNWOAUWGYTPZCUHZUA
        YRAYTYRPZQZYTUUPTZUWRGDWRZUWTDWPZUWSUXAGUXBWSUWTDRUFZUXCAUXDUWSAUVGGRUG
        UXDAUVGGRUVJWQGRUVCDSMWTWOZVHRDXAWOAUWSVLGDYTXBXCUWTUUGUXBPZUXAQZQZUWRU
        WGUUPPZCUHZUXHAUVDUXJAUWSUXGXDZUXHUUGUXBRUWTUXFUXAVSUXHAUXDUXBRTUXKUXER
        DXEWNXFUVEUXJUWGUVCPZCUHZUVEUCUDZUUIPZUXMUCUVEUWIUXOUCUHUVEUWMUWIBRUUGU
        WOAUWPUVDLVHUWBVTUWHUXOCUCUWGUXNUUIXGXJXHUVEUXOQZUUGUXNXIZUVCPZUXRUXMUX
        PUUGUVBPUXOUXRGXKUVEUXOVLUUGUXNUVBUUIXLVAZUXSUXLUXRCUXQUVCUWGUXQUVCXGXM
        XCXNUVEUXIUXLCUVEUUPUVCUWGUWCVQVRXOVIUXHUWQUXICUXHYTUUPUWGUWTUXFUXAVGVQ
        VRXOXPWQXTAUUDQZIRUFZUUGIOZUUIPZGRUGZUUMAUYAUUDAUUPYQOZXQOZSPZGRUGUYAAU
        YGGRUVEUYESPZUYGUVEYQSPUUPSPZUYHEUPAUXDUVDUYIUXEUYIRUUGDUUGDURVFUSUUPYQ
        SSXRVAUYESXSWOWQGRUYFISNWTWOZVHUXTUYCGRUXTUVDQZUYBUYFUUIUYKUVDUYFUUIPZU
        YBUYFTUXTUVDVLUYKUYEUVCPZUYLUYKUYEUUPPZUYMUYKUUBUYNUAYRUUPUXAUUAUYEYTUU
        PYTUUPYQVDUXAYJYAAYSUUCUVDYBAUVDUUPYRPZUUDAUXDUVDUYOUXERUUGDYCUSYDVTAUV
        DUYNUYMWFUUDUVEUUPUVCUYEUWCVQYDYEUYEUVBUUIYFWOZGRUYFUUIINVMVIUYPYGWQAUY
        AUYDQZUUMUNZUUDAISPZUYRAUYARSPZUYSUYJUYTAWKYKRSIYHVIUULUYQFISUUEITZUUFU
        YAUUKUYDRUUEIYIVUAUUJUYCGRVUAUUHUYBUUIUUGUUEIYLYMYNYOXMWOVHYPXN $.
    $}

    ${
      $d F f g m n $.  $d F m n v $.  $d F v w y $.  $d F w x y $.
      $d f n ph $.  $d n v y $.  $d ph v $.
      $( Countable choice using sequences instead of countable sets.
         (Contributed by Jim Kingdon, 27-Apr-2024.) $)
      cc2 $p |- ( ph
        -> E. g ( g Fn _om /\ A. n e. _om ( g ` n ) e. ( F ` n ) ) ) $=
        ( vy vv vm com cv cfv c2nd wcel wex wral nfcv vf csn cxp cmpt weq fveq2
        eleq2d exbidv cbvralv eleq1w cbvexv ralbii sneq xpeq12d cbvmpt nffvmpt1
        sylib nffv 2fveq3 fveq2d cc2lem ) AJKLMLNZUBZVBFOZUCZUDZUADEFLMVBVFOUAN
        ZOZPOZUDGHACNZJNZFOZQZCRZJMSZKNVLQZKRZJMSAVJBNZFOZQZCRZBMSVOIWAVNBJMBJU
        EZVTVMCWBVSVLVJVRVKFUFUGUHUIUQVNVQJMVMVPCKCKVLUJUKULUQLEMVEENZUBZWCFOZU
        CZEVETLWFTLEUEZVCWDVDWEVBWCUMVBWCFUFUNUOLEMVIWCVFOZVGOZPOEVITLWIPLPTLWH
        VGLVGTLMVEWCUPURURWGVHWIPVBWCVGVFUSUTUOVA $.
    $}
  $}

  ${
    $d F f g h k m $.  $d F h k p w $.  $d N f g h k m n $.  $d N h k n p w $.
    $d g h k m n ph $.  $d p ph w $.
    cc3.cc $e |- ( ph -> CCHOICE ) $.
    cc3.f $e |- ( ph -> A. n e. N F e. _V ) $.
    cc3.m $e |- ( ph -> A. n e. N E. w w e. F ) $.
    cc3.n $e |- ( ph -> N ~~ _om ) $.
    $( Countable choice using a sequence F(n) .  (Contributed by Mario
       Carneiro, 8-Feb-2013.)  (Revised by Jim Kingdon, 29-Apr-2024.) $)
    cc3 $p |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. F ) ) $=
      ( vh cvv wcel cv wfn cfv wa com wceq syl2anc vk vg vm vp cmpt wex cen wbr
      wral relen brrelex1i mptexg 3syl wi wf1o bren sylib adantr ccnv ccom wacc
      ad2antrr eqid mptfng simpr fneq1d mpbird f1ocnv adantl f1of syl fnfco csb
      wf ffvelcdmda ad3antrrr nfcsb1v nfcri nfex csbeq1a eleq2d rspc sylc fvco3
      exbidv sylan fveq1d nfel1 eleq1d fvmpts eqtrd ralrimiva cc2 simprl nfmpt1
      simpllr nfv nfeq2 fveq2 eleq12d simplrr rspcdva eqeltrd eleqtrd f1ocnvfv1
      nfan fveq2d r19.21bi ad5ant15 fvmpt2 ralrimi vex coex fneq1 fveq1 ralbidv
      ex anbi12d spcev exlimddv expcom vtocleg mpcom ) DFEUEZLMZACNZFOZDNZYFPZE
      MZDFUIZQZCUFZAFRUGUHZFLMYEJFRUGUJUKDFELULUMAYMUNUAYDLAUANZYDSZYMAYPQZFRKN
      ZUOZYMKAYSKUFZYPAYNYTJFRKUPUQURYQYSQZUBNZROZUCNZUUBPZUUDYOYRUSZUTZPZMZUCR
      UIZQZYMUBUUAUDBUBUCUUGAVAYPYSGVBUUAYOFOZRFUUFVNZUUGROYQUULYSYQUULYDFOZAUU
      NYPAELMZDFUIZUUNHDFEYDYDVCZVDUQURYQFYOYDAYPVEZVFVGURUUARFUUFUOZUUMYSUUSYQ
      FRYRVHVIRFUUFVJVKZFRYOUUFVLTUUABNZUDNZUUGPZMZBUFZUDRUUAUVBRMZQZUVEUVADUVB
      UUFPZEVMZMZBUFZUVGUVHFMZUVAEMZBUFZDFUIZUVKUUARFUVBUUFUUTVOZAUVOYPYSUVFIVP
      UVNUVKDUVHFUVJDBDBUVIDUVHEVQZVRVSYHUVHSZUVMUVJBUVREUVIUVADUVHEVTZWAWEWBWC
      UVGUVDUVJBUVGUVCUVIUVAUVGUVCUVHYOPZUVIUUAUUMUVFUVCUVTSUUTRFUVBYOUUFWDWFUV
      GUVTUVHYDPZUVIUVGUVHYOYDAYPYSUVFWPWGUVGUVLUVILMZUWAUVISUVPUVGUVLUUPUWBUVP
      AUUPYPYSUVFHVPUUOUWBDUVHFDUVILUVQWHUVREUVILUVSWIWBWCDUVHEFYDLUUQWJTWKWKWA
      WEVGWLWMUUAUUKQZUUBYRUTZFOZYHUWDPZEMZDFUIZYMUWCUUCFRYRVNZUWEUUAUUCUUJWNUU
      AUWIUUKYSUWIYQFRYRVJVIURZRFUUBYRVLTUWCUWGDFUUAUUKDYQYSDAYPDADWQDYOYDDFEWO
      WRXFYSDWQXFUUKDWQXFUWCYHFMZUWGUWCUWKQZUWFYHYOPZEUWLUWFYHYRPZUUFPZYOPZUWMU
      WLUWFUWNUUGPZUWPUWLUWFUWNUUBPZUWQUWCUWIUWKUWFUWRSUWJFRYHUUBYRWDWFUWLUUIUW
      RUWQMUCRUWNUUDUWNSUUEUWRUUHUWQUUDUWNUUBWSUUDUWNUUGWSWTUUAUUCUUJUWKXAUWCFR
      YHYRUWJVOZXBXCUWLUUMUWNRMUWQUWPSUUAUUMUUKUWKUUTVBUWSRFUWNYOUUFWDTXDUWLUWO
      YHYOUWLYSUWKUWOYHSYQYSUUKUWKWPUWCUWKVEZFRYHYRXETXGXDUWLUWMYHYDPZEUWLYHYOY
      DYQYPYSUUKUWKUURVPWGUWLUWKUUOUXAESUWTAUWKUUOYPYSUUKAUUODFHXHXIDFELYDUUQXJ
      TWKXDXQXKYLUWEUWHQCUWDUUBYRUBXLKXLXMYFUWDSZYGUWEYKUWHFYFUWDXNUXBYJUWGDFUX
      BYIUWFEYHYFUWDXOWIXPXRXSTXTXTYAYBYC $.
  $}

  ${
    $d A f x $.  $d A w x $.  $d N f n $.  $d N n w $.  $d ch x $.
    $d f n ph $.  $d f ps $.  $d ph w $.  $d ps w $.  $d n x $.
    cc4f.cc $e |- ( ph -> CCHOICE ) $.
    cc4f.1 $e |- ( ph -> A e. V ) $.
    cc4f.a $e |- F/_ n A $.
    cc4f.2 $e |- ( ph -> N ~~ _om ) $.
    cc4f.3 $e |- ( x = ( f ` n ) -> ( ps <-> ch ) ) $.
    cc4f.m $e |- ( ph -> A. n e. N E. x e. A ps ) $.
    $( Countable choice by showing the existence of a function ` f ` which can
       choose a value at each index ` n ` such that ` ch ` holds.  (Contributed
       by Mario Carneiro, 7-Apr-2013.)  (Revised by Jim Kingdon,
       3-May-2024.) $)
    cc4f $p |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) ) $=
      ( vw cv wcel wral wa wfn cfv crab wex wf cvv rabexg ralrimivw wrex rabn0m
      syl ralbii sylibr cc3 simprl elrabi ralimi ad2antll ffnfvf sylanbrc elrab
      nfcv simprbi jca ex eximdv mpd ) AFQZHUAZGQVHUBZBDEUCZRZGHSZTZFUDHEVHUEZC
      GHSZTZFUDAPFGVKHJAVKUFRZGHAEIRVRKBDEIUGUKUHABDEUIZGHSPQVKRPUDZGHSOVTVSGHB
      DPEUJULUMMUNAVNVQFAVNVQAVNTZVOVPWAVIVJERZGHSZVOAVIVMUOVMWCAVIVLWBGHBDVJEU
      PUQURGHEVHGHVBLGVHVBUSUTVMVPAVIVLCGHVLWBCBCDVJENVAVCUQURVDVEVFVG $.
  $}

  ${
    $d A f n x $.  $d A n w x $.  $d N f n $.  $d N n w $.  $d ch x $.
    $d f n ph $.  $d f ps $.  $d ph w $.  $d ps w $.
    cc4.cc $e |- ( ph -> CCHOICE ) $.
    cc4.1 $e |- ( ph -> A e. V ) $.
    cc4.2 $e |- ( ph -> N ~~ _om ) $.
    cc4.3 $e |- ( x = ( f ` n ) -> ( ps <-> ch ) ) $.
    cc4.m $e |- ( ph -> A. n e. N E. x e. A ps ) $.
    $( Countable choice by showing the existence of a function ` f ` which can
       choose a value at each index ` n ` such that ` ch ` holds.  (Contributed
       by Mario Carneiro, 7-Apr-2013.)  (Revised by Jim Kingdon,
       1-May-2024.) $)
    cc4 $p |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) ) $=
      ( nfcv cc4f ) ABCDEFGHIJKGEOLMNP $.
  $}

  ${
    $d A f x w $.  $d N f n w $.  $d ch x $.  $d f n w ph $.  $d f w ps $.
    $d n x w $.
    cc4n.cc $e |- ( ph -> CCHOICE ) $.
    cc4n.1 $e |- ( ph -> A. n e. N { x e. A | ps } e. V ) $.
    cc4n.2 $e |- ( ph -> N ~~ _om ) $.
    cc4n.3 $e |- ( x = ( f ` n ) -> ( ps <-> ch ) ) $.
    cc4n.m $e |- ( ph -> A. n e. N E. x e. A ps ) $.
    $( Countable choice with a simpler restriction on how every set in the
       countable collection needs to be inhabited.  That is, compared with
       ~ cc4 , the hypotheses only require an A(n) for each value of ` n ` ,
       not a single set ` A ` which suffices for every ` n e. _om ` .
       (Contributed by Mario Carneiro, 7-Apr-2013.)  (Revised by Jim Kingdon,
       3-May-2024.) $)
    cc4n $p |- ( ph -> E. f ( f Fn N /\ A. n e. N ch ) ) $=
      ( vw cv wcel wral wa wex wfn cfv crab cvv ralimi syl rabn0m ralbii sylibr
      elex wrex cc3 simprl elrab simprbi ad2antll jca ex eximdv mpd ) AFPZHUAZG
      PVAUBZBDEUCZQZGHRZSZFTVBCGHRZSZFTAOFGVDHJAVDIQZGHRVDUDQZGHRKVJVKGHVDIUJUE
      UFABDEUKZGHROPVDQOTZGHRNVMVLGHBDOEUGUHUILULAVGVIFAVGVIAVGSVBVHAVBVFUMVFVH
      AVBVECGHVEVCEQCBCDVCEMUNUOUEUPUQURUSUT $.
  $}

  ${
    $d f g j y z $.  $d f j n z $.  $d f g x y z $.  $d n x z $.
    $( Given countable choice, every set has choice sets of length ` _om ` .
       (Contributed by Mario Carneiro, 31-Aug-2015.) $)
    acnccim $p |- ( CCHOICE -> AC_ _om = _V ) $=
      ( vx vy vg vf vj vz vn wacc com wacn cvv cv wcel cfv wex cpw wa ralrimiva
      wral wfn crab cmap co simpl elmapfn adantl elmapi ad2antlr ffvelcdmd wceq
      wf simpr eleq2 exbidv elrab sylib simprd cc2 exsimpr syl vex isacnm mp2an
      wb omex sylibr a1i 2thd eqrdv ) HAIJZKHALZVJMZVKKMZHBLZCLZNVNDLZNMBISZCOZ
      DELZFLZMZEOZFVKPZUAZIUBUCZSZVLHVRDWEHVPWEMZQZVOITZVQQCOVRWHGECBVPHWGUDWGV
      PITHVPWDIUEUFWHVSGLZVPNZMZEOZGIWHWJIMZQZWKWCMZWMWOWKWDMWPWMQWOIWDWJVPWGIW
      DVPUKHWNVPWDIUGUHWHWNULUIWBWMFWKWCVTWKUJWAWLEVTWKVSUMUNUOUPUQRURWIVQCUSUT
      RVMIKMVLWFVDAVAZVEBFIDCEKKVKVBVCVFVMHWQVGVHVI $.
  $}


$(
###############################################################################
  REAL AND COMPLEX NUMBERS
###############################################################################

  This section derives the basics of real and complex numbers.

  To construct the real numbers constructively, we follow two main sources.
  The first is Metamath Proof Explorer, which has the advantage of being
  already formalized in metamath.  Its disadvantage, for our purposes, is that
  it assumes the law of the excluded middle throughout.  Since we have already
  developed natural numbers ( for example, ~ nna0 and similar theorems ), going
  from there to positive integers ( ~ df-ni ) and then positive rational
  numbers ( ~ df-nqqs ) does not involve a major change in approach compared
  with the Metamath Proof Explorer.

  It is when we proceed to Dedekind cuts that we bring in more material from
  Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which
  are different without excluded middle or choice principles.  With excluded
  middle, it is natural to define a cut as the lower set only (as Metamath
  Proof Explorer does), but here we define the cut as a pair of both the lower
  and upper sets, as [HoTT] does.  There are also differences in how we handle
  order and replacing "not equal to zero" with "apart from zero".

  When working constructively, there are several possible definitions of real
  numbers.  Here we adopt the most common definition, as two-sided Dedekind
  cuts with the properties described at ~ df-inp .  The Cauchy reals (without
  countable choice) fail to satisfy ~ ax-caucvg and the MacNeille reals fail to
  satisfy ~ axltwlin , and we do not develop them here.  For more on differing
  definitions of the reals, see the introduction to Chapter 11 in [HoTT] or
  Section 1.2 of [BauerHanson].

$)


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Construction and axiomatization of real and complex numbers
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Dedekind-cut construction of real and complex numbers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)

  $c N. $. $( Set of positive integers $)

  $c +N $. $( Positive integer addition $)
  $c .N $. $( Positive integer multiplication $)
  $c <N $. $( Positive integer ordering relation $)

  $c +pQ $. $( Positive pre-fraction addition $)
  $c .pQ $. $( Positive pre-fraction multiplication $)
  $c <pQ $. $( Positive pre-fraction ordering relation $)

  $c ~Q $. $( Equivalence relation to construct positive fractions $)
  $c Q. $. $( Set of positive fractions $)
  $c 1Q $. $( Positive fraction constant 1 $)
  $c +Q $. $( Positive fraction addition $)
  $c .Q $. $( Positive fraction multiplication $)
  $c *Q $. $( Positive fraction reciprocal $)
  $c <Q $. $( Positive fraction ordering relation $)

  $c ~Q0 $. $( Equivalence relation to construct nonnegative fractions $)
  $c Q0. $. $( Set of nonnegative fractions $)
  $c 0Q0 $. $( Nonnegative fraction constant 0 $)
  $c +Q0 $. $( Nonnegative fraction addition $)
  $c .Q0 $. $( Nonnegative fraction multiplication $)

  $c P. $. $( Set of positive reals $)
  $c 1P $. $( Positive real constant 1 $)
  $c +P. $. $( Positive real addition $)
  $c .P. $. $( Positive real multiplication $)
  $c <P $. $( Positive real ordering relation $)

  $c ~R $. $( Equivalence relation to construct signed reals $)
  $c R. $. $( Set of signed reals $)
  $c 0R $. $( Signed real constant 0 $)
  $c 1R $. $( Signed real constant 1 $)
  $c -1R $. $( Signed real constant minus 1 $)
  $c +R $. $( Signed real addition $)
  $c .R $. $( Signed real multiplication $)
  $c <R $. $( Signed real ordering relation $)
  $c <RR $. $( 'Less than' relation (restricted to non-extended reals) $)

  $c CC $. $( The set of complex numbers (blackboard bold C) $)
  $c RR $. $( The set of real numbers (blackboard bold R) $)
  $c 0 $. $( The real number 0. $)
  $c 1 $. $( The real number 1. $)
  $c _i $. $( Letter i (the imaginary unit = square root of -1) $)
  $c + $. $( Plus (addition) symbol $)
  $c x. $. $( Multiplication symbol (center dot) $)

  $( Extend class notation with new sets (constants) used for positive
     integers: $)

  $( The set of positive integers, which is the set of natural numbers ` _om `
     with 0 removed.

     _Note:  This is the start of the Dedekind-cut construction of real and
     complex numbers_. $)
  cnpi $a class N. $.
  $( Positive integer addition. $)
  cpli $a class +N $.
  $( Positive integer multiplication. $)
  cmi $a class .N $.
  $( Positive integer ordering relation. $)
  clti $a class <N $.

  $( Extend class notation with new sets (constants) used in construction of
    positive fractions: $)

  $( Positive pre-fraction addition. $)
  cplpq $a class +pQ $.
  $( Positive pre-fraction multiplication. $)
  cmpq $a class .pQ $.
  $( Positive pre-fraction ordering relation. $)
  cltpq $a class <pQ $.
  $( Equivalence class used to construct positive fractions. $)
  ceq $a class ~Q $.
  $( Set of positive fractions. $)
  cnq $a class Q. $.
  $( The positive fraction constant 1. $)
  c1q $a class 1Q $.
  $( Positive fraction addition. $)
  cplq $a class +Q $.
  $( Positive fraction multiplication. $)
  cmq $a class .Q $.
  $( Positive fraction reciprocal operation. $)
  crq $a class *Q $.
  $( Positive fraction ordering relation. $)
  cltq $a class <Q $.

  $( Equivalence class used to construct nonnegative fractions. $)
  ceq0 $a class ~Q0 $.
  $( Set of nonnegative fractions. $)
  cnq0 $a class Q0. $.
  $( The nonnegative fraction constant 0. $)
  c0q0 $a class 0Q0 $.
  $( Nonnegative fraction addition. $)
  cplq0 $a class +Q0 $.
  $( Nonnegative fraction multiplication. $)
  cmq0 $a class .Q0 $.

  $( Extend class notation with new sets (constants) used in construction of
    positive real numbers: $)

  $( Set of positive reals. $)
  cnp $a class P. $.
  $( Positive real constant 1. $)
  c1p $a class 1P $.
  $( Positive real addition. $)
  cpp $a class +P. $.
  $( Positive real multiplication. $)
  cmp $a class .P. $.
  $( Positive real ordering relation. $)
  cltp $a class <P $.

  $( Extend class notation with new sets (constants) used in construction of
    signed real numbers: $)

  $( Equivalence class used to construct signed reals. $)
  cer $a class ~R $.
  $( Set of signed reals. $)
  cnr $a class R. $.
  $( The signed real constant 0. $)
  c0r $a class 0R $.
  $( The signed real constant 1. $)
  c1r $a class 1R $.
  $( The signed real constant -1. $)
  cm1r $a class -1R $.
  $( Signed real addition. $)
  cplr $a class +R $.
  $( Signed real multiplication. $)
  cmr $a class .R $.
  $( Signed real ordering relation. $)
  cltr $a class <R $.

  $( Define the class of positive integers.  This is a "temporary" set used in
     the construction of complex numbers, and is intended to be used only by
     the construction.  (Contributed by NM, 15-Aug-1995.) $)
  df-ni $a |- N. = ( _om \ { (/) } ) $.

  $( Define addition on positive integers.  This is a "temporary" set used in
     the construction of complex numbers, and is intended to be used only by
     the construction.  (Contributed by NM, 26-Aug-1995.) $)
  df-pli $a |- +N = ( +o |` ( N. X. N. ) ) $.

  $( Define multiplication on positive integers.  This is a "temporary" set
     used in the construction of complex numbers and is intended to be used
     only by the construction.  (Contributed by NM, 26-Aug-1995.) $)
  df-mi $a |- .N = ( .o |` ( N. X. N. ) ) $.

  $( Define 'less than' on positive integers.  This is a "temporary" set used
     in the construction of complex numbers, and is intended to be used only by
     the construction.  (Contributed by NM, 6-Feb-1996.) $)
  df-lti $a |- <N = ( _E i^i ( N. X. N. ) ) $.

  $( Membership in the class of positive integers.  (Contributed by NM,
     15-Aug-1995.) $)
  elni $p |- ( A e. N. <-> ( A e. _om /\ A =/= (/) ) ) $=
    ( cnpi wcel com c0 csn cdif wne wa df-ni eleq2i eldifsn bitri ) ABCADEFGZCA
    DCAEHIBNAJKADELM $.

  $( A positive integer is a natural number.  (Contributed by NM,
     15-Aug-1995.) $)
  pinn $p |- ( A e. N. -> A e. _om ) $=
    ( cnpi com c0 csn cdif df-ni difss eqsstri sseli ) BCABCDEZFCGCKHIJ $.

  $( A positive integer is an ordinal number.  (Contributed by NM,
     23-Mar-1996.) $)
  pion $p |- ( A e. N. -> A e. On ) $=
    ( cnpi wcel com con0 pinn nnon syl ) ABCADCAECAFAGH $.

  $( A positive integer is ordinal.  (Contributed by NM, 29-Jan-1996.) $)
  piord $p |- ( A e. N. -> Ord A ) $=
    ( cnpi wcel com word pinn nnord syl ) ABCADCAEAFAGH $.

  $( The class of positive integers is a set.  (Contributed by NM,
     15-Aug-1995.) $)
  niex $p |- N. e. _V $=
    ( cnpi com omex c0 csn cdif df-ni difss eqsstri ssexi ) ABCABDEZFBGBKHIJ $.

  $( The empty set is not a positive integer.  (Contributed by NM,
     26-Aug-1995.) $)
  0npi $p |- -. (/) e. N. $=
    ( c0 wceq cnpi wcel wn eqid com wne elni simprbi necon2bi ax-mp ) AABACDZEA
    FMAAMAGDAAHAIJKL $.

  $( Membership in the class of positive integers.  (Contributed by NM,
     27-Nov-1995.) $)
  elni2 $p |- ( A e. N. <-> ( A e. _om /\ (/) e. A ) ) $=
    ( cnpi wcel com c0 wa pinn wceq wn 0npi eleq1 mtbiri con2i wo 0elnn syl ord
    mpd sylib wi jca nndceq0 df-dc anim1i ancom andi bitr3i noel pm2.21d impcom
    wdc eleq2 a1i wne df-ne elni simplbi2 biimtrrid adantld jaod adantr impbii
    ) ABCZADCZEACZFZVCVDVEAGZVCAEHZIZVEVHVCVHVCEBCJAEBKLMVCVHVEVCVDVHVENVGAOPQR
    UAVFVEVHFZVEVIFZNZVCVFVHVINZVEFZVLVDVMVEVDVHUKVMAUBVHUCSUDVNVEVMFVLVEVMUEVE
    VHVIUFUGSVDVLVCTVEVDVJVCVKVJVCTVDVHVEVCVHVEVCVHVEEECEUHAEEULLUIUJUMVDVIVCVE
    VIAEUNZVDVCAEUOVCVDVOAUPUQURUSUTVARVB $.

  $( Ordinal 'one' is a positive integer.  (Contributed by NM, 29-Oct-1995.) $)
  1pi $p |- 1o e. N. $=
    ( c1o cnpi wcel com c0 wne 1onn 1n0 elni mpbir2an ) ABCADCAEFGHAIJ $.

  $( Positive integer addition in terms of ordinal addition.  (Contributed by
     NM, 27-Aug-1995.) $)
  addpiord $p |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( A +o B ) ) $=
    ( cnpi wcel wa cop cxp cpli co coa wceq opelxpi cres cfv fvres df-ov df-pli
    fveq1i eqtri 3eqtr4g syl ) ACDBCDEABFZCCGZDZABHIZABJIZKABCCLUDUBJUCMZNZUBJN
    UEUFUBUCJOUEUBHNUHABHPUBHUGQRSABJPTUA $.

  $( Positive integer multiplication in terms of ordinal multiplication.
     (Contributed by NM, 27-Aug-1995.) $)
  mulpiord $p |- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( A .o B ) ) $=
    ( cnpi wcel wa cop cxp cmi co comu wceq opelxpi cres cfv fvres df-ov fveq1i
    df-mi eqtri 3eqtr4g syl ) ACDBCDEABFZCCGZDZABHIZABJIZKABCCLUDUBJUCMZNZUBJNU
    EUFUBUCJOUEUBHNUHABHPUBHUGRQSABJPTUA $.

  $( 1 is an identity element for multiplication on positive integers.
     (Contributed by NM, 4-Mar-1996.)  (Revised by Mario Carneiro,
     17-Nov-2014.) $)
  mulidpi $p |- ( A e. N. -> ( A .N 1o ) = A ) $=
    ( cnpi wcel c1o cmi co comu wceq 1pi mulpiord mpan2 com pinn nnm1 syl eqtrd
    ) ABCZADEFZADGFZAQDBCRSHIADJKQALCSAHAMANOP $.

  $( Positive integer 'less than' in terms of ordinal membership.  (Contributed
     by NM, 6-Feb-1996.)  (Revised by Mario Carneiro, 28-Apr-2015.) $)
  ltpiord $p |- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) ) $=
    ( clti wbr cep cnpi cxp cin wcel wa df-lti breqi brinxp epelg adantl bitr3d
    wb bitrid ) ABCDABEFFGHZDZAFIZBFIZJZABIZABCSKLUCABEDZTUDABFFEMUBUEUDQUAABFN
    OPR $.

  ${
    $d x y z $.
    $( Positive integer 'less than' is a strict ordering.  (Contributed by NM,
       8-Feb-1996.)  (Proof shortened by Mario Carneiro, 10-Jul-2014.) $)
    ltsopi $p |- <N Or N. $=
      ( vx vy vz cnpi clti wor wtru cv wcel wbr wn wel wb ltpiord adantl wa w3o
      wi com pinn elirrv anidms mtbiri w3a con0 pion ontr1 syl 3ad2ant3 3adant3
      3adant1 anbi12d 3adant2 3imtr4d ispod weq nntri3or syl2an biidd 3orbi123d
      ancoms mpbird issod mptru ) DEFGABDEGABCDEAHZDIZVEVEEJZKGVFVGAALZAUAVFVGV
      HMVEVENUBUCOVFBHZDIZCHZDIZUDZVEVIEJZVIVKEJZPZVEVKEJZRGVMABLZBCLZPZACLZVPV
      QVLVFVTWARZVJVLVKUEIWBVKUFVEVIVKUGUHUIVMVNVRVOVSVFVJVNVRMVLVEVINZUJVJVLVO
      VSMVFVIVKNUKULVFVLVQWAMVJVEVKNUMUNOUOVFVJPZVNABUPZVIVEEJZQZGWDWGVRWEBALZQ
      ZVFVESIVISIWIVJVETVITVEVIUQURWDVNVRWEWEWFWHWCWDWEUSVJVFWFWHMVIVENVAUTVBOV
      CVD $.
  $}

  $( Trichotomy for positive integers.  (Contributed by Jim Kingdon,
     21-Sep-2019.) $)
  pitric $p |- ( ( A e. N. /\ B e. N. ) ->
      ( A <N B <-> -. ( A = B \/ B <N A ) ) ) $=
    ( cnpi wcel wa wceq wo wn clti wbr com wb pinn nntri2 syl2an ltpiord ancoms
    orbi2d notbid 3bitr4d ) ACDZBCDZEZABDZABFZBADZGZHZABIJUEBAIJZGZHUAAKDBKDUDU
    HLUBAMBMABNOABPUCUJUGUCUIUFUEUBUAUIUFLBAPQRST $.

  $( Trichotomy for positive integers.  (Contributed by Jim Kingdon,
     21-Sep-2019.) $)
  pitri3or $p |- ( ( A e. N. /\ B e. N. ) ->
      ( A <N B \/ A = B \/ B <N A ) ) $=
    ( cnpi wcel clti wbr wceq w3o com pinn nntri3or syl2an ltpiord biidd ancoms
    wa wb 3orbi123d mpbird ) ACDZBCDZPZABEFZABGZBAEFZHABDZUDBADZHZTAIDBIDUHUAAJ
    BJABKLUBUCUFUDUDUEUGABMUBUDNUATUEUGQBAMORS $.

  $( Less-than for positive integers is decidable.  (Contributed by Jim
     Kingdon, 12-Dec-2019.) $)
  ltdcpi $p |- ( ( A e. N. /\ B e. N. ) -> DECID A <N B ) $=
    ( cnpi wcel wa clti wbr wdc com pinn nndcel syl2an ltpiord dcbid mpbird ) A
    CDZBCDZEZABFGZHABDZHZPAIDBIDUAQAJBJABKLRSTABMNO $.

  $( Positive integer 'less than' is a relation on positive integers.
     (Contributed by NM, 8-Feb-1996.) $)
  ltrelpi $p |- <N C_ ( N. X. N. ) $=
    ( clti cep cnpi cxp cin df-lti inss2 eqsstri ) ABCCDZEIFBIGH $.

  $( Domain of addition on positive integers.  (Contributed by NM,
     26-Aug-1995.) $)
  dmaddpi $p |- dom +N = ( N. X. N. ) $=
    ( coa cnpi cxp cres cdm con0 cin cpli dmres wfn wceq fnoa fndm ax-mp ineq2i
    eqtri df-pli dmeqi wss com wa c0 csn df-ni difss eqsstri omsson sstri anidm
    cdif mpbir xpss12 dfss mpbi 3eqtr4i ) ABBCZDZEZUPFFCZGZHEUPURUPAEZGUTAUPIVA
    USUPAUSJVAUSKLUSAMNOPHUQQRUPUSSZUPUTKBFSZVCUAZVBVDVCBTFBTUBUCZUJTUDTVEUEUFU
    GUHVCUIUKBFBFULNUPUSUMUNUO $.

  $( Domain of multiplication on positive integers.  (Contributed by NM,
     26-Aug-1995.) $)
  dmmulpi $p |- dom .N = ( N. X. N. ) $=
    ( comu cnpi cxp cres cdm con0 cin cmi dmres wfn wceq fnom fndm ax-mp ineq2i
    eqtri df-mi dmeqi wss com wa c0 cdif df-ni difss eqsstri omsson sstri anidm
    csn mpbir xpss12 dfss mpbi 3eqtr4i ) ABBCZDZEZUPFFCZGZHEUPURUPAEZGUTAUPIVAU
    SUPAUSJVAUSKLUSAMNOPHUQQRUPUSSZUPUTKBFSZVCUAZVBVDVCBTFBTUBUJZUCTUDTVEUEUFUG
    UHVCUIUKBFBFULNUPUSUMUNUO $.

  $( Closure of addition of positive integers.  (Contributed by NM,
     18-Oct-1995.) $)
  addclpi $p |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) e. N. ) $=
    ( cnpi wcel wa cpli co coa addpiord com pinn wne nnacl sylan2 elni2 nnaordi
    c0 wi ne0i syl6 expcom imp32 sylan2b elni sylanbrc sylan eqeltrd ) ACDZBCDZ
    EABFGABHGZCABIUHAJDZUIUJCDZAKUKUIEUJJDZUJQLZULUIUKBJDZUMBKABMNUIUKUOQBDZEUN
    BOUKUOUPUNUOUKUPUNRUOUKEUPAQHGZUJDUNQBAPUJUQSTUAUBUCUJUDUEUFUG $.

  $( Closure of multiplication of positive integers.  (Contributed by NM,
     18-Oct-1995.) $)
  mulclpi $p |- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) e. N. ) $=
    ( cnpi wcel wa cmi co comu mulpiord com wne pinn nnmcl syl2an elni2 simprbi
    c0 adantl wi adantr nnmordi syl21anc mpd ne0i syl elni sylanbrc eqeltrd ) A
    CDZBCDZEZABFGABHGZCABIUKULJDZULQKZULCDUIAJDZBJDZUMUJALZBLZABMNUKAQHGZULDZUN
    UKQBDZUTUJVAUIUJUPVABOPRUKUPUOQADZVAUTSUJUPUIURRUIUOUJUQTUIVBUJUIUOVBAOPTQB
    AUAUBUCULUSUDUEULUFUGUH $.

  $( Addition of positive integers is commutative.  (Contributed by Jim
     Kingdon, 26-Aug-2019.) $)
  addcompig $p |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( B +N A ) ) $=
    ( cnpi wcel wa coa cpli com wceq pinn nnacom syl2an addpiord ancoms 3eqtr4d
    co ) ACDZBCDZEABFPZBAFPZABGPBAGPZQAHDBHDSTIRAJBJABKLABMRQUATIBAMNO $.

  $( Addition of positive integers is associative.  (Contributed by Jim
     Kingdon, 26-Aug-2019.) $)
  addasspig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
      ( ( A +N B ) +N C ) = ( A +N ( B +N C ) ) ) $=
    ( cnpi wcel w3a coa co cpli com wceq pinn nnaass syl3an wa addclpi addpiord
    sylan oveq1d eqtrd adantr 3impa sylan2 oveq2d adantl 3impb 3eqtr4d ) ADEZBD
    EZCDEZFABGHZCGHZABCGHZGHZABIHZCIHZABCIHZIHZUHAJEUIBJEUJCJEULUNKALBLCLABCMNU
    HUIUJUPULKUHUIOZUJOUPUOCGHZULUSUODEUJUPUTKABPUOCQRUSUTULKUJUSUOUKCGABQSUATU
    BUHUIUJURUNKUHUIUJOZOURAUQGHZUNVAUHUQDEURVBKBCPAUQQUCVAVBUNKUHVAUQUMAGBCQUD
    UETUFUG $.

  $( Multiplication of positive integers is commutative.  (Contributed by Jim
     Kingdon, 26-Aug-2019.) $)
  mulcompig $p |- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( B .N A ) ) $=
    ( cnpi wcel wa comu cmi com wceq pinn nnmcom syl2an mulpiord ancoms 3eqtr4d
    co ) ACDZBCDZEABFPZBAFPZABGPBAGPZQAHDBHDSTIRAJBJABKLABMRQUATIBAMNO $.

  $( Multiplication of positive integers is associative.  (Contributed by Jim
     Kingdon, 26-Aug-2019.) $)
  mulasspig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. )
      -> ( ( A .N B ) .N C ) = ( A .N ( B .N C ) ) ) $=
    ( cnpi wcel w3a comu co cmi com wceq pinn nnmass syl3an wa mulclpi mulpiord
    sylan oveq1d eqtrd adantr 3impa sylan2 oveq2d adantl 3impb 3eqtr4d ) ADEZBD
    EZCDEZFABGHZCGHZABCGHZGHZABIHZCIHZABCIHZIHZUHAJEUIBJEUJCJEULUNKALBLCLABCMNU
    HUIUJUPULKUHUIOZUJOUPUOCGHZULUSUODEUJUPUTKABPUOCQRUSUTULKUJUSUOUKCGABQSUATU
    BUHUIUJURUNKUHUIUJOZOURAUQGHZUNVAUHUQDEURVBKBCPAUQQUCVAVBUNKUHVAUQUMAGBCQUD
    UETUFUG $.

  $( Multiplication of positive integers is distributive.  (Contributed by Jim
     Kingdon, 26-Aug-2019.) $)
  distrpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. )
      -> ( A .N ( B +N C ) ) = ( ( A .N B ) +N ( A .N C ) ) ) $=
    ( cnpi wcel w3a coa co comu cpli cmi wceq pinn nndi mulpiord addpiord eqtrd
    com wa mulclpi syl3an addclpi sylan2 oveq2d adantl syl2an oveqan12d 3eqtr4d
    3impb 3impdi ) ADEZBDEZCDEZFABCGHZIHZABIHZACIHZGHZABCJHZKHZABKHZACKHZJHZUKA
    REULBREUMCREUOURLAMBMCMABCNUAUKULUMUTUOLUKULUMSZSUTAUSIHZUOVDUKUSDEUTVELBCU
    BAUSOUCVDVEUOLUKVDUSUNAIBCPUDUEQUIUKULUMVCURLUKULSZUKUMSZSVCVAVBGHZURVFVADE
    VBDEVCVHLVGABTACTVAVBPUFVFVGVAUPVBUQGABOACOUGQUJUH $.

  $( Addition cancellation law for positive integers.  (Contributed by Jim
     Kingdon, 27-Aug-2019.) $)
  addcanpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
                 ( ( A +N B ) = ( A +N C ) <-> B = C ) ) $=
    ( cnpi wcel w3a cpli co wceq coa addpiord 3adant3 3adant2 eqeq12d wi nnacan
    com pinn biimpd syl3an sylbid oveq2 impbid1 ) ADEZBDEZCDEZFZABGHZACGHZIZBCI
    ZUGUJABJHZACJHZIZUKUGUHULUIUMUDUEUHULIUFABKLUDUFUIUMIUEACKMNUDAQEZUEBQEZUFC
    QEZUNUKOARBRCRUOUPUQFUNUKABCPSTUABCAGUBUC $.

  $( Multiplication cancellation law for positive integers.  (Contributed by
     Jim Kingdon, 29-Aug-2019.) $)
  mulcanpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
                 ( ( A .N B ) = ( A .N C ) <-> B = C ) ) $=
    ( cnpi wcel w3a cmi co wceq wi wa comu mulpiord adantr adantlr eqeq12d pinn
    com c0 elni2 simprbi nnmcan biimpd sylan2 ex syl3an 3exp com4r imp31 sylbid
    pm2.43i 3impa oveq2 impbid1 ) ADEZBDEZCDEZFABGHZACGHZIZBCIZUOUPUQUTVAJUOUPK
    ZUQKZUTABLHZACLHZIZVAVCURVDUSVEVBURVDIUQABMNUOUQUSVEIUPACMOPUOUPUQVFVAJZUOU
    PUQVGJJUOUPUQUOVGUOUPUQUOVGJZUOAREZUPBREZUQCREZVHAQBQCQVIVJVKFZUOVGUOVLSAEZ
    VGUOVIVMATUAVLVMKVFVAABCUBUCUDUEUFUGUHUKUIUJULBCAGUMUN $.

  $( There is no identity element for addition on positive integers.
     (Contributed by NM, 28-Nov-1995.) $)
  addnidpig $p |- ( ( A e. N. /\ B e. N. ) -> -. ( A +N B ) = A ) $=
    ( cnpi wcel wa cpli co wceq coa com wn pinn c0 elni2 wi nnaordi nna0 eleq1d
    word nnord ordirr eleq2 notbid syl5ibrcom sylbid adantl syld expcom sylan2b
    syl con2d imp32 sylan addpiord eqeq1d mtbird ) ACDZBCDZEZABFGZAHABIGZAHZUQA
    JDZURVBKZALURVCBJDZMBDZEVDBNVCVEVFVDVEVCVFVDOVEVCEVFAMIGZVADZVDMBAPVCVHVDOV
    EVCVHAVADZVDVCVGAVAAQRVCVBVIVCVIKVBAADZKZVCASVKATAUAUJVBVIVJVAAAUBUCUDUKUEU
    FUGUHULUIUMUSUTVAAABUNUOUP $.

  ${
    $d x A $.  $d x B $.
    $( Ordering on positive integers in terms of existence of sum.
       (Contributed by NM, 15-Mar-1996.)  (Revised by Mario Carneiro,
       14-Jun-2013.) $)
    ltexpi $p |- ( ( A e. N. /\ B e. N. ) ->
               ( A <N B <-> E. x e. N. ( A +N x ) = B ) ) $=
      ( cnpi wcel wa c0 cv coa wceq com wrex clti wbr cpli pinn nnaordex syl2an
      co wb ltpiord addpiord eqeq1d pm5.32da elni2 anbi1i anass bitrdi rexbidv2
      bitri adantr 3bitr4d ) BDEZCDEZFBCEZGAHZEZBUPISZCJZFZAKLZBCMNBUPOSZCJZADL
      ZUMBKECKEUOVATUNBPCPABCQRBCUAUMVDVATUNUMVCUTADKUMUPDEZVCFVEUSFZUPKEZUTFZU
      MVEVCUSUMVEFVBURCBUPUBUCUDVFVGUQFZUSFVHVEVIUSUPUEUFVGUQUSUGUJUHUIUKUL $.
  $}

  $( Ordering property of addition for positive integers.  (Contributed by Jim
     Kingdon, 31-Aug-2019.) $)
  ltapig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
      ( A <N B <-> ( C +N A ) <N ( C +N B ) ) ) $=
    ( cnpi wcel clti wbr cpli co wb wa coa com pinn nnaord ltpiord addclpi wceq
    adantr addpiord syl3an 3expa syl2an adantl eleq12d anandis ancoms 3bitr4d
    bitrd 3impa ) ADEZBDEZCDEZABFGZCAHIZCBHIZFGZJUKULKZUMKABEZCALIZCBLIZEZUNUQU
    KULUMUSVBJZUKAMEULBMEUMCMEVCANBNCNABCOUAUBURUNUSJUMABPSUMURUQVBJZUMUKULVDUM
    UKKZUMULKZKZUQUOUPEZVBVEUODEUPDEUQVHJVFCAQCBQUOUPPUCVGUOUTUPVAVEUOUTRVFCATS
    VFUPVARVECBTUDUEUIUFUGUHUJ $.

  $( Ordering property of multiplication for positive integers.  (Contributed
     by Jim Kingdon, 31-Aug-2019.) $)
  ltmpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
      ( A <N B <-> ( C .N A ) <N ( C .N B ) ) ) $=
    ( cnpi wcel clti wbr cmi co wb wa comu com pinn syl2an ltpiord mulclpi wceq
    adantr mulpiord wi c0 elni2 w3a iba nnmord sylan9bbr 3exp1 imp4b imp adantl
    biimtrid eleq12d bitrd anandis ancoms 3bitr4d 3impa ) ADEZBDEZCDEZABFGZCAHI
    ZCBHIZFGZJUSUTKZVAKABEZCALIZCBLIZEZVBVEVFVAVGVJJZUSAMEZBMEZVAVKUAUTANBNVACM
    EZUBCEZKVLVMKVKCUCVLVMVNVOVKVLVMVNVOVKVOVGVGVOKVLVMVNUDVJVOVGUEABCUFUGUHUIU
    LOUJVFVBVGJVAABPSVAVFVEVJJZVAUSUTVPVAUSKZVAUTKZKZVEVCVDEZVJVQVCDEVDDEVEVTJV
    RCAQCBQVCVDPOVSVCVHVDVIVQVCVHRVRCATSVRVDVIRVQCBTUKUMUNUOUPUQUR $.

  $( One is less than two (one plus one).  (Contributed by NM, 13-Mar-1996.) $)
  1lt2pi $p |- 1o <N ( 1o +N 1o ) $=
    ( c1o cpli co clti wbr wcel coa c0 com wceq 1onn nna0 ax-mp 0lt1o wb peano1
    nnaord cnpi 1pi mp2an mp3an mpbi eqeltrri addpiord eleqtrri addclpi ltpiord
    mpbir ) AAABCZDEZAUIFZAAAGCZUIAHGCZAULAIFZUMAJKALMHAFZUMULFZNHIFUNUNUOUPOPK
    KHAAQUAUBUCARFZUQUIULJSSAAUDTUEUQUIRFZUJUKOSUQUQURSSAAUFTAUIUGTUH $.

  $( No positive integer is less than one.  (Contributed by Jim Kingdon,
     31-Aug-2019.) $)
  nlt1pig $p |- ( A e. N. -> -. A <N 1o ) $=
    ( cnpi wcel c0 wne c1o clti wbr wn com elni simprbi wceq wa noel wo ltpiord
    wb 1pi mpan2 csuc df-1o eleq2i elsucg bitrid biimpa ord mpi ex necon3ad mpd
    bitrd ) ABCZADEZAFGHZIUMAJCUNAKLUMUOADUMUOADMZUMUONZADCZIUPAOUQURUPUMUOURUP
    PZUMUOAFCZUSUMFBCUOUTRSAFQTUTADUAZCUMUSFVAAUBUCADBUDUEULUFUGUHUIUJUK $.

  ${
    $d x y $.  $d x A $.  $d x ps $.  $d x ch $.  $d x th $.  $d x ta $.
    $d y ph $.
    $( Substitutions. $)
    indpi.1 $e |- ( x = 1o -> ( ph <-> ps ) ) $.
    indpi.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    indpi.3 $e |- ( x = ( y +N 1o ) -> ( ph <-> th ) ) $.
    indpi.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Basis. $)
    indpi.5 $e |- ps $.
    $( Induction step. $)
    indpi.6 $e |- ( y e. N. -> ( ch -> th ) ) $.
    $( Principle of Finite Induction on positive integers.  (Contributed by NM,
       23-Mar-1996.) $)
    indpi $p |- ( A e. N. -> ta ) $=
      ( wcel com c0 wceq wo c1o cnpi cv wne wa elni cab csuc wral wss wsbc eqid
      orci nfv nfsbc1v nfor 0ex eqeq1 sbceq1a orbi12d elabf mpbir suceq eqtr4di
      wi df-1o olci 1oex elab eqeltrdi a1d a1i wn wb simprbi neneqd syl bitr4di
      biorf vex cpli co 1pi addclpi mpan2 sylib simprd elabg addpiord con0 pion
      coa oa1suc eqtrd eleq1d 3bitr2d 3imtr3d wdc nndceq0 df-dc necon3bd anc2li
      idd imbitrrdi orim2d mpd mpjaod rgen peano5 mp2an sseli abid adantr df-ne
      sylbi adantl mpbird vtoclga ) AEFHUALFUBZUAOXRPOZXRQUCZUDZAXRUEYAAXRQRZAS
      ZXSYCXTXSXRYCFUFZOYCPYDXRQYDOZGUBZYDOZYFUGZYDOZVDZGPUHPYDUIYEQQRZAFQUJZSZ
      YKYLQUKULYCYMFQYKYLFYKFUMAFQUNUOUPYBYBYKAYLXRQQUQAFQURUSUTVAYJGPYFPOZYFQR
      ZYJYFUAOZYOYJVDYNYOYIYGYOYHTYDYOYHQUGTYFQVBVEVCTYDOTQRZBSZBYQMVFYCYRFTVGX
      RTRYBYQABXRTQUQIUSVHVAVIVJVKYPYJVDYNYPCDYGYINYPCYOCSZYGYPYOVLZCYSVMYPYFQY
      PYNYFQUCZYFUEZVNVOYOCVRVPYCYSFYFGVSXRYFRYBYOACXRYFQUQJUSVHVQYPDYFTVTWAZQR
      ZDSZUUCYDOZYIYPUUDVLDUUEVMYPUUCQYPUUCPOZUUCQUCZYPUUCUAOZUUGUUHUDYPTUAOZUU
      IWBYFTWCWDZUUCUEWEWFVOUUDDVRVPYPUUIUUFUUEVMUUKYCUUEFUUCUAXRUUCRYBUUDADXRU
      UCQUQKUSWGVPYPUUCYHYDYPUUCYFTWKWAZYHYPUUJUUCUULRWBYFTWHWDYPYFWIOUULYHRYFW
      JYFWLVPWMWNWOWPVKYNYOYTSZYOYPSYNYOWQUUMYFWRYOWSWEYNYTYPYOYNYTYNUUAUDYPYNY
      TUUAYNYOYFQYNYOXBWTXAUUBXCXDXEXFXGGYDXHXIXJYCFXKWEXLXTAYCVMZXSXTYBVLUUNXR
      QXMYBAVRXNXOXPXNXQ $.
  $}

  $( A natural number plus a positive integer is a positive integer.
     (Contributed by Jim Kingdon, 10-Nov-2019.) $)
  nnppipi $p |- ( ( A e. _om /\ B e. N. ) -> ( A +o B ) e. N. ) $=
    ( com wcel cnpi wa coa co pinn nnacl sylan2 wss nnaword2 sylan ancoms elni2
    c0 simprbi adantl sseldd sylanbrc ) ACDZBEDZFZABGHZCDZQUEDUEEDUCUBBCDZUFBIZ
    ABJKUDBUEQUCUBBUELZUCUGUBUIUHBAMNOUCQBDZUBUCUGUJBPRSTUEPUA $.

  ${
    $d x y z w v u f $.
    $( Define pre-addition on positive fractions.  This is a "temporary" set
       used in the construction of complex numbers, and is intended to be used
       only by the construction.  This "pre-addition" operation works directly
       with ordered pairs of integers.  The actual positive fraction addition
       ` +Q ` ( ~ df-plqqs ) works with the equivalence classes of these
       ordered pairs determined by the equivalence relation ` ~Q `
       ( ~ df-enq ).  (Analogous remarks apply to the other "pre-" operations
       in the complex number construction that follows.)  From Proposition
       9-2.3 of [Gleason] p. 117.  (Contributed by NM, 28-Aug-1995.) $)
    df-plpq $a |- +pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |->
      <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N
           ( ( 1st ` y ) .N ( 2nd ` x ) ) ) ,
           ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) $.

    $( Define pre-multiplication on positive fractions.  This is a "temporary"
       set used in the construction of complex numbers, and is intended to be
       used only by the construction.  From Proposition 9-2.4 of [Gleason]
       p. 119.  (Contributed by NM, 28-Aug-1995.) $)
    df-mpq $a |- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |->
        <. ( ( 1st ` x ) .N ( 1st ` y ) ) ,
           ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) $.

    $( Define pre-ordering relation on positive fractions.  This is a
       "temporary" set used in the construction of complex numbers, and is
       intended to be used only by the construction.  Similar to Definition 5
       of [Suppes] p. 162.  (Contributed by NM, 28-Aug-1995.) $)
    df-ltpq $a |- <pQ = { <. x , y >. |
      ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\
        ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) } $.

    $( Define equivalence relation for positive fractions.  This is a
       "temporary" set used in the construction of complex numbers, and is
       intended to be used only by the construction.  From Proposition 9-2.1 of
       [Gleason] p. 117.  (Contributed by NM, 27-Aug-1995.) $)
    df-enq $a |- ~Q = { <. x , y >. | ( ( x e. ( N. X. N. ) /\
                   y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u
                   ( ( x = <. z , w >. /\ y = <. v , u >. ) /\
                   ( z .N u ) = ( w .N v ) ) ) } $.

    $( Define class of positive fractions.  This is a "temporary" set used in
       the construction of complex numbers, and is intended to be used only by
       the construction.  From Proposition 9-2.2 of [Gleason] p. 117.
       (Contributed by NM, 16-Aug-1995.) $)
    df-nqqs $a |- Q. = ( ( N. X. N. ) /. ~Q ) $.

    $( Define addition on positive fractions.  This is a "temporary" set used
       in the construction of complex numbers, and is intended to be used only
       by the construction.  From Proposition 9-2.3 of [Gleason] p. 117.
       (Contributed by NM, 24-Aug-1995.) $)
    df-plqqs $a |- +Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\
      E. w E. v E. u E. f (
         ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\
               z = [ ( <. w , v >. +pQ <. u , f >. ) ] ~Q ) ) } $.

    $( Define multiplication on positive fractions.  This is a "temporary" set
       used in the construction of complex numbers, and is intended to be used
       only by the construction.  From Proposition 9-2.4 of [Gleason] p. 119.
       (Contributed by NM, 24-Aug-1995.) $)
    df-mqqs $a |- .Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\
      E. w E. v E. u E. f (
         ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\
               z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q ) ) } $.

    $( Define positive fraction constant 1.  This is a "temporary" set used in
       the construction of complex numbers, and is intended to be used only by
       the construction.  From Proposition 9-2.2 of [Gleason] p. 117.
       (Contributed by NM, 29-Oct-1995.) $)
    df-1nqqs $a |- 1Q = [ <. 1o , 1o >. ] ~Q $.

    $( Define reciprocal on positive fractions.  It means the same thing as one
       divided by the argument (although we don't define full division since we
       will never need it).  This is a "temporary" set used in the construction
       of complex numbers, and is intended to be used only by the construction.
       From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to
       denote this unary operation.  (Contributed by Jim Kingdon,
       20-Sep-2019.) $)
    df-rq $a |- *Q = { <. x , y >. |
        ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) } $.

    $( Define ordering relation on positive fractions.  This is a "temporary"
       set used in the construction of complex numbers, and is intended to be
       used only by the construction.  Similar to Definition 5 of [Suppes]
       p. 162.  (Contributed by NM, 13-Feb-1996.) $)
    df-ltnqqs $a |- <Q = { <. x , y >. | ( ( x e. Q. /\ y e. Q. ) /\
       E. z E. w E. v E. u
       ( ( x = [ <. z , w >. ] ~Q /\ y = [ <. v , u >. ] ~Q ) /\
                ( z .N u ) <N ( w .N v ) ) ) } $.
  $}

  ${
    $d x y z w v u f $.
    $( Alternate definition of pre-addition on positive fractions.
       (Contributed by Jim Kingdon, 12-Sep-2019.) $)
    dfplpq2 $p |- +pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\
        y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\
        y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) ,
        ( v .N f ) >. ) ) } $=
      ( cnpi cv cfv cmi co cpli cop wcel wa wceq wex vex oveq12d c1st c2nd cmpo
      coprab cplpq df-mpo df-plpq 1st2nd2 eqeq1d bi2anan9 anbi1d xp1st ad2antlr
      cxp wb biimpa simprd opth2 simplbi eleq1d syl mpbid xp2nd ad2antrr simpld
      simprbi mulcompig syl2anc oveq2d opeq1d eqeq2d pm5.32da bitr3d jca simpll
      4exbidv simprr simprl simplr opeq12d copsex4g pm5.32i oprabbii 3eqtr4i
      syl2an ) ABHHUNZWFAIZUAJZBIZUBJZKLZWIUAJZWGUBJZKLZMLZWMWJKLZNZUCWGWFOZWIW
      FOZPZCIZWQQZPZABCUDUEWTWGDIZEIZNZQZWIFIZGIZNZQZPZXAXDXIKLZXEXHKLZMLZXEXIK
      LZNZQZPZGRFRERDRZPZABCUDABCWFWFWQUFABUGYAXCABCWTXTXBWTWHWMNZXFQZWLWJNZXJQ
      ZPZXAXMXHXEKLZMLZXPNZQZPZGRFRERDRZXTXBWTYKXSDEFGWTXLYJPYKXSWTXLYFYJWRXGYC
      WSXKYEWRWGYBXFWGHHUHUIWSWIYDXJWIHHUHUIUJZUKWTXLYJXRWTXLPZYIXQXAYNYHXOXPYN
      YGXNXMMYNXHHOZXEHOZYGXNQYNWLHOZYOWSYQWRXLWIHHULZUMYNYEYQYOUOYNYCYEWTXLYFY
      MUPZUQYEWLXHHYEWLXHQWJXIQWLWJXHXIFSGSURUSUTVAVBYNWMHOZYPWRYTWSXLWGHHVCZVD
      YNYCYTYPUOYNYCYEYSVEYCWMXEHYCWHXDQWMXEQWHWMXDXEDSESURVFUTVAVBXHXEVGVHVIVJ
      VKVLVMVPWRWHHOZYTPYQWJHOZPYLXBUOWSWRUUBYTWGHHULUUAVNWSYQUUCYRWIHHVCVNYJXB
      DEFGWHWMWLWJHHXDWHQZXEWMQZPZXHWLQZXIWJQZPZPZYIWQXAUUJYHWOXPWPUUJXMWKYGWNM
      UUJXDWHXIWJKUUDUUEUUIVOUUFUUGUUHVQZTUUJXHWLXEWMKUUFUUGUUHVRUUDUUEUUIVSZTT
      UUJXEWMXIWJKUULUUKTVTVKWAWEVMWBWCWD $.
  $}

  ${
    $d x y z w v u f $.
    $( Alternate definition of pre-multiplication on positive fractions.
       (Contributed by Jim Kingdon, 13-Sep-2019.) $)
    dfmpq2 $p |- .pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y
        e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\
        y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) } $=
      ( cnpi cv c1st cfv cmi co c2nd cop wcel wa wceq coprab wex df-mpo 1st2nd2
      cxp cmpo cmpq df-mpq eqeq1d bi2anan9 anbi1d bicomd 4exbidv wb xp1st xp2nd
      simpll simprl oveq12d simplr simprr opeq12d eqeq2d copsex4g syl2an bitr3d
      jca pm5.32i oprabbii 3eqtr4i ) ABHHUCZVIAIZJKZBIZJKZLMZVJNKZVLNKZLMZOZUDV
      JVIPZVLVIPZQZCIZVRRZQZABCSUEWAVJDIZEIZOZRZVLFIZGIZOZRZQZWBWEWILMZWFWJLMZO
      ZRZQZGTFTETDTZQZABCSABCVIVIVRUAABUFWTWDABCWAWSWCWAVKVOOZWGRZVMVPOZWKRZQZW
      QQZGTFTETDTZWSWCWAXFWRDEFGWAWRXFWAWMXEWQVSWHXBVTWLXDVSVJXAWGVJHHUBUGVTVLX
      CWKVLHHUBUGUHUIUJUKVSVKHPZVOHPZQVMHPZVPHPZQXGWCULVTVSXHXIVJHHUMVJHHUNVEVT
      XJXKVLHHUMVLHHUNVEWQWCDEFGVKVOVMVPHHWEVKRZWFVORZQZWIVMRZWJVPRZQZQZWPVRWBX
      RWNVNWOVQXRWEVKWIVMLXLXMXQUOXNXOXPUPUQXRWFVOWJVPLXLXMXQURXNXOXPUSUQUTVAVB
      VCVDVFVGVH $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u D $.
    $( Equivalence relation for positive fractions in terms of positive
       integers.  (Contributed by NM, 27-Aug-1995.) $)
    enqbreq $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
          ( <. A , B >. ~Q <. C , D >. <-> ( A .N D ) = ( B .N C ) ) ) $=
      ( vx vy vz vw vv vu cmi ceq cnpi df-enq ecopoveq ) EFGHIJABCDKLMEFGHIJNO
      $.
  $}

  $( Equivalence relation for positive fractions in terms of positive integers.
     (Contributed by Mario Carneiro, 8-May-2013.) $)
  enqbreq2 $p |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) ->
        ( A ~Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) =
                     ( ( 1st ` B ) .N ( 2nd ` A ) ) ) ) $=
    ( cnpi cxp wcel wa ceq wbr c1st cfv c2nd cop cmi co 1st2nd2 xp1st xp2nd jca
    wceq syl2an breqan12d wb enqbreq mulcompig eqeq2d 3bitrd ) ACCDZEZBUGEZFZAB
    GHAIJZAKJZLZBIJZBKJZLZGHZUKUOMNZULUNMNZSZURUNULMNZSUHUIAUMBUPGACCOBCCOUAUHU
    KCEZULCEZFUNCEZUOCEZFUQUTUBUIUHVBVCACCPACCQZRUIVDVEBCCPZBCCQRUKULUNUOUCTUJU
    SVAURUHVCVDUSVASUIVFVGULUNUDTUEUF $.

  ${
    $d x y z w v u $.
    $( The equivalence relation for positive fractions is an equivalence
       relation.  Proposition 9-2.1 of [Gleason] p. 117.  (Contributed by NM,
       27-Aug-1995.)  (Revised by Mario Carneiro, 6-Jul-2015.) $)
    enqer $p |- ~Q Er ( N. X. N. ) $=
      ( vx vy vz vw vv cmi ceq cnpi df-enq mulcompig mulclpi mulasspig wcel w3a
      vu cv co wceq mulcanpig biimpd ecopoverg ) ABCDEOFGHABCDEOIAPZBPZJUBUCKUB
      UCCPZLUBHMUCHMUDHMNUBUCFQUBUDFQRUCUDRUBUCUDSTUA $.
  $}

  $( Equivalence class equality of positive fractions in terms of positive
     integers.  (Contributed by NM, 29-Nov-1995.) $)
  enqeceq $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
                ( [ <. A , B >. ] ~Q = [ <. C , D >. ] ~Q <->
                ( A .N D ) = ( B .N C ) ) ) $=
    ( cnpi wcel wa cop ceq wbr cec wceq cmi co cxp wer enqer a1i opelxpi adantr
    erth enqbreq bitr3d ) AEFBEFGZCEFDEFGZGZABHZCDHZIJUGIKUHIKLADMNBCMNLUFUGUHI
    EEOZUIIPUFQRUDUGUIFUEABEESTUAABCDUBUC $.

  ${
    $d x y z w v u $.
    $( The equivalence relation for positive fractions exists.  (Contributed by
       NM, 3-Sep-1995.) $)
    enqex $p |- ~Q e. _V $=
      ( vx vy vz vw vv vu ceq cnpi cxp niex xpex cv wcel wa cop wceq cmi co wex
      copab df-enq opabssxp eqsstri ssexi ) GHHIZUEIZUEUEHHJJKZUGKGALZUEMBLZUEM
      NUHCLZDLZOPUIELZFLZOPNUJUMQRUKULQRPNFSESDSCSZNABTUFABCDEFUAUNABUEUEUBUCUD
      $.
  $}

  $( The equivalence relation for positive fractions is decidable.
     (Contributed by Jim Kingdon, 7-Sep-2019.) $)
  enqdc $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
      DECID <. A , B >. ~Q <. C , D >. ) $=
    ( cnpi wcel wa cop ceq wbr wdc cmi co wceq mulclpi pinn nndceq syl2an an42s
    com enqbreq dcbid mpbird ) AEFZBEFZGCEFZDEFZGGZABHCDHIJZKADLMZBCLMZNZKZUDUG
    UEUFUMUDUGGUJEFZUKEFZUMUEUFGADOBCOUNUJTFUKTFUMUOUJPUKPUJUKQRRSUHUIULABCDUAU
    BUC $.

  $( The equivalence relation for positive fractions is decidable.
     (Contributed by Jim Kingdon, 7-Sep-2019.) $)
  enqdc1 $p |- ( ( ( A e. N. /\ B e. N. ) /\ C e. ( N. X. N. ) ) ->
      DECID <. A , B >. ~Q C ) $=
    ( cnpi wcel wa cxp cop ceq wbr wdc c1st cfv xp1st xp2nd jca enqdc sylan2 wb
    c2nd 1st2nd2 breq2d dcbid adantl mpbird ) ADEBDEFZCDDGEZFABHZCIJZKZUHCLMZCT
    MZHZIJZKZUGUFUKDEZULDEZFUOUGUPUQCDDNCDDOPABUKULQRUGUJUOSUFUGUIUNUGCUMUHICDD
    UAUBUCUDUE $.

  ${
    $d x y z w u v $.
    $( The class of positive fractions exists.  (Contributed by NM,
       16-Aug-1995.)  (Revised by Mario Carneiro, 27-Apr-2013.) $)
    nqex $p |- Q. e. _V $=
      ( cnq cnpi cxp ceq cqs cvv df-nqqs niex xpex qsex eqeltri ) ABBCZDEFGLDBB
      HHIJK $.

    $( The empty set is not a positive fraction.  (Contributed by NM,
       24-Aug-1995.)  (Revised by Mario Carneiro, 27-Apr-2013.) $)
    0nnq $p |- -. (/) e. Q. $=
      ( c0 cnq wcel cnpi cxp ceq cqs wne neirr cdm wceq enqer erdm ax-mp elqsn0
      wer mpan mto df-nqqs eleq2i mtbir ) ABCADDEZFGZCZUDAAHZAIFJUBKZUDUEUBFPUF
      LUBFMNUBAFOQRBUCASTUA $.

    $( Positive fraction 'less than' is a relation on positive fractions.
       (Contributed by NM, 14-Feb-1996.)  (Revised by Mario Carneiro,
       27-Apr-2013.) $)
    ltrelnq $p |- <Q C_ ( Q. X. Q. ) $=
      ( vx vy vz vw vv vu cltq cv cnq wcel wa cop ceq cec wceq cmi clti wbr wex
      co copab cxp df-ltnqqs opabssxp eqsstri ) GAHZIJBHZIJKUFCHZDHZLMNOUGEHZFH
      ZLMNOKUHUKPTUIUJPTQRKFSESDSCSZKABUAIIUBABCDEFUCULABIIUDUE $.
  $}

  $( The positive fraction 'one'.  (Contributed by NM, 29-Oct-1995.) $)
  1nq $p |- 1Q e. Q. $=
    ( c1o cop ceq cec cnpi cxp cqs c1q cnq wcel 1pi opelxpi mp2an enqex ecelqsi
    ax-mp df-1nqqs df-nqqs 3eltr4i ) AABZCDZEEFZCGZHITUBJZUAUCJAEJZUEUDKKAAEELM
    UBTCNOPQRS $.

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.  $d x y z F $.
    $d x y z G $.  $d x y z R $.  $d x y z S $.

    $( Lemma showing compatibility of addition.  (Contributed by NM,
       27-Aug-1995.) $)
    addcmpblnq $p |- ( ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. )
             ) /\ ( ( F e. N. /\ G e. N. ) /\ ( R e. N. /\ S e. N. ) ) ) ->
                ( ( ( A .N D ) = ( B .N C ) /\ ( F .N S ) = ( G .N R ) ) ->
                 <. ( ( A .N G ) +N ( B .N F ) ) , ( B .N G ) >. ~Q
                 <. ( ( C .N S ) +N ( D .N R ) ) , ( D .N S ) >. ) ) $=
      ( vx vy vz cnpi wcel wa cmi co wceq cpli adantl mulclpi distrpig ad2ant2l
      cop ceq wbr w3a simplll simprlr syl2anc simpllr simprll addclpi mulcompig
      cv caovdir2d simplrr mulasspig simprrr caov4d eqtrd oveq1 oveq2 oveqan12d
      oveq12d sylan9eq simplrl simprrl syl3anc adantr eqtr4d syl2an jca anim12i
      wb an42s an4s enqbreq syl mpbird ex ) ALMZBLMZNZCLMZDLMZNZNZGLMZHLMZNZELM
      ZFLMZNZNZNZADOPZBCOPZQZGFOPZHEOPZQZNZAHOPZBGOPZRPZBHOPZUCCFOPZDEOPZRPZDFO
      PZUCUDUEZWOXBNZXKXEXJOPZXFXIOPZQZXLXMWQHFOPZOPZBDOPZWTOPZRPZXNWOXBXMWPXPO
      PZXRWSOPZRPZXTWOXMXCXJOPZXDXJOPZRPYCWOIJKXCXDXJLROIUNZLMZJUNZLMZKUNZLMUFZ
      YFYHYJRPOPYFYHOPZYFYJOPRPQWOYFYHYJUASWOWAWIXCLMZWAWBWFWNUGZWGWHWIWMUHZAHT
      ZUIWOWBWHXDLMZWAWBWFWNUJZWGWHWIWMUKZBGTZUIWFWMXJLMZWCWJWEWLUUAWDWKDFTUBZU
      BYGYINZYFYHRPLMWOYFYHULSUUCYLYHYFOPQWOYFYHUMSZUOWOYDYAYEYBRWOIJKAHDFLOYNY
      OWCWDWEWNUPZUUDYKYLYJOPYFYHYJOPOPQWOYFYHYJUQSZWGWJWKWLURZUUCYLLMWOYFYHTSZ
      USWOIJKBGDFLOYRYSUUEUUDUUFUUGUUHUSVDUTWRXAYAXQYBXSRWPWQXPOVAWSWTXROVBVCVE
      WOXNXTQXBWOXNXFXGOPZXFXHOPZRPZXTWOXFLMZXGLMZXHLMZXNUUKQWOWBWIUULYRYOBHTZU
      IWOWDWLUUMWCWDWEWNVFZUUGCFTZUIWOWEWKUUNUUEWGWJWKWLVGZDETZUIXFXGXHUAVHWOUU
      IXQUUJXSRWOIJKBHCFLOYRYOUUPUUDUUFUUGUUHUSWOIJKBHDELOYRYOUUEUUDUUFUURUUHUS
      VDUTVIVJWOXKXOVNZXBWOXELMZUULNZXILMZUUANZNZUUTWCWJWFWMUVEWCWJNZUVBWFWMNZU
      VDUVFUVAUULWAWIWBWHUVAWAWINYMYQUVAWBWHNYPYTXCXDULVKVOWBWIUULWAWHUUOUBVLUV
      GUVCUUAWDWLWEWKUVCWDWLNUUMUUNUVCWEWKNUUQUUSXGXHULVKVOUUBVLVMVPXEXFXIXJVQV
      RVIVSVT $.

    $( Lemma showing compatibility of multiplication.  (Contributed by NM,
       27-Aug-1995.) $)
    mulcmpblnq $p |- ( ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. )
             ) /\ ( ( F e. N. /\ G e. N. ) /\ ( R e. N. /\ S e. N. ) ) ) ->
                ( ( ( A .N D ) = ( B .N C ) /\ ( F .N S ) = ( G .N R ) ) ->
     <. ( A .N F ) , ( B .N G ) >. ~Q <. ( C .N R ) , ( D .N S ) >. ) ) $=
      ( vx vy vz cmi co wceq wa cnpi wcel mulclpi anim12i an4s cop oveq12 wb cv
      ceq wbr enqbreq simplll simprll simplrr mulcompig adantl mulasspig caov4d
      syl w3a simprrr simpllr simprlr simplrl simprrl eqeq12d bitrd imbitrrid )
      ADLMZBCLMZNGFLMZHELMZNOAGLMZBHLMZUACELMZDFLMZUAUEUFZAPQZBPQZOZCPQZDPQZOZO
      ZGPQZHPQZOZEPQZFPQZOZOZOZVEVGLMZVFVHLMZNZVEVFVGVHLUBWHVMVIVLLMZVJVKLMZNZW
      KWHVIPQZVJPQZOZVKPQZVLPQZOZOZVMWNUCVPWCVSWFXAVPWCOWQVSWFOWTVNWAVOWBWQVNWA
      OWOVOWBOWPAGRBHRSTVQWDVRWEWTVQWDOWRVRWEOWSCERDFRSTSTVIVJVKVLUGUOWHWLWIWMW
      JWHIJKAGDFPLVNVOVSWGUHVTWAWBWFUIVPVQVRWGUJIUDZPQZJUDZPQZOZXBXDLMZXDXBLMNW
      HXBXDUKULZXCXEKUDZPQUPXGXILMXBXDXILMLMNWHXBXDXIUMULZVTWCWDWEUQXFXGPQWHXBX
      DRULZUNWHIJKBHCEPLVNVOVSWGURVTWAWBWFUSVPVQVRWGUTXHXJVTWCWDWEVAXKUNVBVCVD
      $.
  $}

  $( Lemma for ~ addpipqqs .  (Contributed by Jim Kingdon, 11-Sep-2019.) $)
  addpipqqslem $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
      <. ( ( A .N D ) +N ( B .N C ) ) , ( B .N D ) >. e. ( N. X. N. ) ) $=
    ( cnpi wcel wa cmi co cop cxp mulclpi addclpi syl2an an42s ad2ant2l opelxpi
    cpli syl2anc ) AEFZBEFZGCEFZDEFZGGADHIZBCHIZRIZEFZBDHIZEFZUFUHJEEKFTUCUAUBU
    GTUCGUDEFUEEFUGUAUBGADLBCLUDUEMNOUAUCUITUBBDLPUFUHEEQS $.

  ${
    $d x y z w v u t s f g h a b c d A $.
    $d x y z w v u t s f g h a b c d B $.
    $d x y z w v u t s f g h a b c d C $.
    $d x y z w v u t s f g h a b c d D $.
    $( Addition of positive fractions in terms of positive integers.
       (Contributed by NM, 28-Aug-1995.) $)
    addpipqqs $p |- ( ( ( A e. N. /\ B e. N. ) /\
                     ( C e. N. /\ D e. N. ) ) ->
                ( [ <. A , B >. ] ~Q +Q [ <. C , D >. ] ~Q ) =
                   [ <. ( ( A .N D ) +N ( B .N C ) ) ,
                              ( B .N D ) >. ] ~Q ) $=
      ( vz vu vw vv va vd vb vc cv cmi co wceq cpli weq wa oveq12 vg vs vh cplq
      vt vx vy cplpq cnq ceq cnpi cop addpipqqslem enqex enqer df-enq eqeqan12d
      vf wb an42s dfplpq2 oveqan12d ad2ant2l opeq12d df-plqqs addcmpblnq oviec
      df-nqqs ) EMZFMZNOZGMZHMZNOZPZIMZJMZNOZKMZLMZNOZPZUAMZUBMZNOZUCMZUEMZNOZP
      ZUFUGEGHFUEABCDUHUDUIUJUKURUAUCADNOZBCNOZQOZBDNOZULVLURMZNOZVMVJNOZQOZVMW
      NNOZULVPWFNOZVSWCNOZQOZVSWFNOZULVTWDNOZVQWGNOZQOZVQWDNOZULUBIKLJABCDUMVPV
      SWCWFUMVTVQWGWDUMUNUOUFUGEGHFUPEIRZFJRZGKRZHLRZVOWBUSXGXHSXIXJSVKVRVNWAVI
      VPVJVQNTVLVSVMVTNTUQUTEUARZFUBRZGUCRZHUERZVOWIUSXKXLSXMXNSVKWEVNWHVIWCVJW
      DNTVLWFVMWGNTUQUTUFUGEGHFURVAGIRZHKRZSFUARZURUCRZSSWQXAWRXBXOXRXPXQWQXAPX
      OXRSXPXQSWOWSWPWTQVLVPWNWFNTVMVSVJWCNTVBUTXPXRWRXBPXOXQVMVSWNWFNTVCVDGLRZ
      HJRZSFUERZURUBRZSSWQXEWRXFXSYBXTYAWQXEPXSYBSXTYASWOXCWPXDQVLVTWNWDNTVMVQV
      JWGNTVBUTXTYBWRXFPXSYAVMVQWNWDNTVCVDVLAPZVMBPZSVJCPZWNDPZSSWQWLWRWMYCYFYD
      YEWQWLPYCYFSYDYESWOWJWPWKQVLAWNDNTVMBVJCNTVBUTYDYFWRWMPYCYEVMBWNDNTVCVDUF
      UGEIKLJVEVHVPVSVTVQWGWDWCWFVFVG $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C y $.  $d D y $.
    $( Multiplication of positive fractions in terms of positive integers.
       (Contributed by Mario Carneiro, 8-May-2013.) $)
    mulpipq2 $p |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) ->
      ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) ,
                       ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) $=
      ( vx vy cnpi wcel c1st cfv cmi co c2nd cop cmpq wceq xp1st mulclpi syl2an
      cvv xp2nd fveq2 cxp wa opexg syl2anc oveq1d opeq12d oveq2d df-mpq mpd3an3
      cv ovmpog ) AEEUAZFZBULFZAGHZBGHZIJZAKHZBKHZIJZLZRFZABMJVANUMUNUBUQEFZUTE
      FZVBUMUOEFUPEFVCUNAEEOBEEOUOUPPQUMUREFUSEFVDUNAEESBEESURUSPQUQUTEEUCUDCDA
      BULULCUJZGHZDUJZGHZIJZVEKHZVGKHZIJZLVAMUOVHIJZURVKIJZLRVEANZVIVMVLVNVOVFU
      OVHIVEAGTUEVOVJURVKIVEAKTUEUFVGBNZVMUQVNUTVPVHUPUOIVGBGTUGVPVKUSURIVGBKTU
      GUFCDUHUKUI $.

    $( Multiplication of positive fractions in terms of positive integers.
       (Contributed by NM, 28-Aug-1995.)  (Revised by Mario Carneiro,
       8-May-2013.) $)
    mulpipq $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
          ( <. A , B >. .pQ <. C , D >. ) = <. ( A .N C ) , ( B .N D ) >. ) $=
      ( cnpi wcel wa cop cmpq co c1st cfv cmi c2nd cxp opelxpi op1stg oveqan12d
      wceq op2ndg mulpipq2 syl2an opeq12d eqtrd ) AEFBEFGZCEFDEFGZGZABHZCDHZIJZ
      UHKLZUIKLZMJZUHNLZUINLZMJZHZACMJZBDMJZHUEUHEEOZFUIUTFUJUQSUFABEEPCDEEPUHU
      IUAUBUGUMURUPUSUEUFUKAULCMABEEQCDEEQRUEUFUNBUODMABEETCDEETRUCUD $.
  $}

  ${
    $d x y z w v u t s f g h a b c d A $.
    $d x y z w v u t s f g h a b c d B $.
    $d x y z w v u t s f g h a b c d C $.
    $d x y z w v u t s f g h a b c d D $.
    $( Multiplication of positive fractions in terms of positive integers.
       (Contributed by NM, 28-Aug-1995.) $)
    mulpipqqs $p |- ( ( ( A e. N. /\ B e. N. ) /\
                     ( C e. N. /\ D e. N. ) ) ->
                ( [ <. A , B >. ] ~Q .Q [ <. C , D >. ] ~Q ) =
                   [ <. ( A .N C ) , ( B .N D ) >. ] ~Q ) $=
      ( vz vu vw vv va vd cv cmi co wceq cnpi wcel wa mulclpi weq oveq12d vb vc
      vg vs vh vt vx vy vf cmpq cmq cnq ceq cop opelxpi syl2an an4s enqex enqer
      df-enq simpll simprr simplr simprl eqeq12d dfmpq2 opeq12d df-mqqs df-nqqs
      cxp mulcmpblnq oviec ) EKZFKZLMZGKZHKZLMZNIKZJKZLMZUAKZUBKZLMZNUCKZUDKZLM
      ZUEKZUFKZLMZNUGUHEGHFUFABCDUJUKULUMOUIUCUEACLMZBDLMZUNZVPVNLMZVQUIKZLMZUN
      VSWELMZWBWHLMZUNZWCWILMZVTWFLMZUNZUDIUAUBJAOPZCOPZBOPZDOPZWMOOVJZPZXCXDQW
      KOPWLOPXHXEXFQACRBDRWKWLOOUOUPUQVSOPZWEOPZWBOPZWHOPZWSXGPZXIXJQWQOPWROPXM
      XKXLQVSWERWBWHRWQWROOUOUPUQWCOPZWIOPZVTOPZWFOPZXBXGPZXNXOQWTOPXAOPXRXPXQQ
      WCWIRVTWFRWTXAOOUOUPUQURUSUGUHEGHFUTEISZGUASZQZHUBSZFJSZQZQZVOWAVRWDYEVMV
      SVNVTLXSXTYDVAYAYBYCVBTYEVPWBVQWCLXSXTYDVCYAYBYCVDTVEEUCSZGUESZQZHUFSZFUD
      SZQZQZVOWGVRWJYLVMWEVNWFLYFYGYKVAYHYIYJVBTYLVPWHVQWILYFYGYKVCYHYIYJVDTVEU
      GUHEGHFUIVFGISZHUASZQZFUCSZUIUESZQZQZWNWQWPWRYSVPVSVNWELYMYNYRVAYOYPYQVDT
      YSVQWBWOWHLYMYNYRVCYOYPYQVBTVGGUBSZHJSZQZFUFSZUIUDSZQZQZWNWTWPXAUUFVPWCVN
      WILYTUUAUUEVAUUBUUCUUDVDTUUFVQVTWOWFLYTUUAUUEVCUUBUUCUUDVBTVGVPANZVQBNZQZ
      VNCNZWODNZQZQZWNWKWPWLUUMVPAVNCLUUGUUHUULVAUUIUUJUUKVDTUUMVQBWODLUUGUUHUU
      LVCUUIUUJUUKVBTVGUGUHEIUAUBJVHVIVSWBWCVTWIWFWEWHVKVL $.
  $}

  ${
    $d A x y z w v u f $.  $d B x y z w v u f $.  $d C x y z w v u f $.
    $d D x y z w v u f $.
    $( Ordering of positive fractions in terms of positive integers.
       (Contributed by Jim Kingdon, 14-Sep-2019.) $)
    ordpipqqs $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
        ( [ <. A , B >. ] ~Q <Q [ <. C , D >. ] ~Q <->
          ( A .N D ) <N ( B .N C ) ) ) $=
      ( vx vy vf cv cmi co clti wbr ceq cnpi wcel wa cop cec wceq adantl vz cnq
      vu vw vv cltq enqex enqer df-nqqs df-ltnqqs enqeceq eqcom bitrdi bi2anan9
      wb oveq12 simplll simprlr simplrr mulcompig w3a mulasspig simprrl mulclpi
      caov4d simpllr simprll simplrl simprrr eqeq12d imbitrrid sylbid caovord3d
      ltmpig caovcld syld brecop ) UAHZUCHZIJZUDHZUEHZIJZKLZADIJZBCIJZKLZEFUAUD
      UEUCABCDMNUBUFUGUHUIEFUAUDUEUCUJVRNOZWANOZPZANOZBNOZPZPZWBNOZVSNOZPZCNOZD
      NOZPZPZPZVRWAQMRABQMRSZWBVSQMRCDQMRSZPZVTWFIJZWCWEIJZSZWDWGUOXBXEVRBIJZWA
      AIJZSZVSCIJZWBDIJZSZPZXHWNXCXKXAXDXNVRWAABUKXAXDXMXLSXNWBVSCDUKXMXLULUMUN
      XOXHXBXIXLIJZXJXMIJZSXIXJXLXMIUPXBXFXPXGXQXBEFGVRVSBCNIWHWIWMXAUQZWNWOWPW
      TURZWJWKWLXAUSZEHZNOZFHZNOZPZYAYCIJZYCYAIJSXBYAYCUTTZYBYDGHZNOVAZYFYHIJYA
      YCYHIJIJSXBYAYCYHVBTZWNWQWRWSVCZYEYFNOXBYAYCVDTZVEXBEFGWAWBADNIWHWIWMXAVF
      ZWNWOWPWTVGZWJWKWLXAVHZYGYJWNWQWRWSVIZYLVEVJVKVLXBEFGVTWFWCWEKNIYIYAYCKLY
      HYAIJYHYCIJKLUOXBYAYCYHVNTXBEFVRVSNNNIYLXRXSVOXBEFBCNNNIYLXTYKVOXBEFWAWBN
      NNIYLYMYNVOYGXBEFADNNNIYLYOYPVOVMVPVQ $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Closure of addition on positive fractions.  (Contributed by NM,
       29-Aug-1995.) $)
    addclnq $p |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. ) $=
      ( vx vy vz vw cnq wcel wa cplq co cnpi ceq cv cop cec df-nqqs cmi mulclpi
      wceq cxp cqs oveq1 eleq1d oveq2 cpli addpipqqs addclpi an42s ad2ant2l jca
      syl2an opelxpi enqex ecelqsi 3syl eqeltrd 2ecoptocl eleqtrrdi ) AGHBGHIAB
      JKZLLUAZMUBZGCNZDNZOMPZENZFNZOMPZJKZVBHAVHJKZVBHUTVBHCDEFABLLMGQVEATVIVJV
      BVEAVHJUCUDVHBTVJUTVBVHBAJUEUDVCLHZVDLHZIVFLHZVGLHZIIZVIVCVGRKZVDVFRKZUFK
      ZVDVGRKZOZMPZVBVCVDVFVGUGVOVRLHZVSLHZIVTVAHWAVBHVOWBWCVKVNVLVMWBVKVNIVPLH
      VQLHWBVLVMIVCVGSVDVFSVPVQUHULUIVLVNWCVKVMVDVGSUJUKVRVSLLUMVAVTMUNUOUPUQUR
      QUS $.

    $( Closure of multiplication on positive fractions.  (Contributed by NM,
       29-Aug-1995.) $)
    mulclnq $p |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) e. Q. ) $=
      ( vx vy vz vw cnq wcel wa cmq co cnpi ceq cop cec df-nqqs wceq eleq1d cmi
      cv cxp oveq1 oveq2 mulpipqqs mulclpi anim12i an4s opelxpi ecelqsi eqeltrd
      cqs enqex 3syl 2ecoptocl eleqtrrdi ) AGHBGHIABJKZLLUAZMUKZGCTZDTZNMOZETZF
      TZNMOZJKZURHAVDJKZURHUPURHCDEFABLLMGPVAAQVEVFURVAAVDJUBRVDBQVFUPURVDBAJUC
      RUSLHZUTLHZIVBLHZVCLHZIIZVEUSVBSKZUTVCSKZNZMOZURUSUTVBVCUDVKVLLHZVMLHZIZV
      NUQHVOURHVGVIVHVJVRVGVIIVPVHVJIVQUSVBUEUTVCUEUFUGVLVMLLUHUQVNMULUIUMUJUNP
      UO $.
  $}

  ${
    $d A a v w x $.
    $( Decomposition of a positive fraction into numerator and denominator.
       Lemma for ~ dmaddpq .  (Contributed by Jim Kingdon, 15-Sep-2019.) $)
    dmaddpqlem $p |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q ) $=
      ( va cv cop ceq cec wceq wex cnpi cxp cqs cnq wcel wrex elqsi 2eximi syl
      wa elxpi simpl anim1i 19.41vv sylibr simpr eceq1 adantr rexlimiva df-nqqs
      eqtrd eleq2s ) AEZBEZCEZFZGHZIZCJBJZUMKKLZGMZNUMVAOUMDEZGHZIZDUTPUSDUTUMG
      QVDUSDUTVBUTOZVDTZVBUPIZVDTZCJBJZUSVFVGCJBJZVDTVIVEVJVDVEVGUNKOUOKOTZTZCJ
      BJVJBCVBKKUAVLVGBCVGVKUBRSUCVGVDBCUDUEVHURBCVHUMVCUQVGVDUFVGVCUQIVDVBUPGU
      GUHUKRSUISUJUL $.

    $( Decomposition of a positive fraction into numerator and denominator.
       Similar to ~ dmaddpqlem but also shows that the numerator and
       denominator are positive integers.  (Contributed by Jim Kingdon,
       20-Sep-2019.) $)
    nqpi $p |- ( A e. Q. ->
        E. w E. v ( ( w e. N. /\ v e. N. ) /\ A = [ <. w , v >. ] ~Q ) ) $=
      ( va cv cnpi wcel cop ceq cec wceq wex cxp cqs cnq wrex elqsi elxpi syl
      wa anim1i 19.41vv sylibr simplr simpr eceq1 ad2antrr jca 2eximi rexlimiva
      eqtrd df-nqqs eleq2s ) AEZFGBEZFGTZCUNUOHZIJZKZTZBLALZCFFMZINZOCVCGCDEZIJ
      ZKZDVBPVADVBCIQVFVADVBVDVBGZVFTZVDUQKZUPTZVFTZBLALZVAVHVJBLALZVFTVLVGVMVF
      ABVDFFRUAVJVFABUBUCVKUTABVKUPUSVIUPVFUDVKCVEURVJVFUEVIVEURKUPVFVDUQIUFUGU
      KUHUISUJSULUM $.
  $}

  ${
    $d x y z v w u f $.
    $( Domain of addition on positive fractions.  (Contributed by NM,
       24-Aug-1995.) $)
    dmaddpq $p |- dom +Q = ( Q. X. Q. ) $=
      ( vx vy vw vv vu vf vz cplq cdm cv cnq wcel copab ceq cec wceq wex sylibr
      wa cvv cxp cop cplpq co coprab dmoprab df-plqqs dmaddpqlem anim12i ee4anv
      dmeqi enqex ecexg ax-mp isseti ax-ia3 eximdv mpi 2eximi exrot3 syl 19.42v
      pm4.71i bitr4i opabbii 3eqtr4i df-xp eqtr4i ) HIZAJZKLZBJZKLZSZABMZKKUAVN
      VJCJDJUBZNOPZVLEJFJUBZNOPZSZGJVPVRUCUDZNOZPZSZFQEQZDQCQZSZABGUEZIWGGQZABM
      VIVOWGABGUFHWHABGCDEFUGUKVNWIABVNVNWFGQZSWIVNWJVNVTFQEQZDQCQZWJVNVQDQCQZV
      SFQEQZSWLVKWMVMWNACDUHBEFUHUIVQVSCDEFUJRWLWEGQZDQCQWJWKWOCDWKWDGQZFQEQWOV
      TWPEFVTWCGQWPGWBNTLWBTLULWATNUMUNUOVTWCWDGVTWCUPUQURUSWDGEFUTRUSWEGCDUTRV
      AVCVNWFGVBVDVEVFABKKVGVH $.

    $( Domain of multiplication on positive fractions.  (Contributed by NM,
       24-Aug-1995.) $)
    dmmulpq $p |- dom .Q = ( Q. X. Q. ) $=
      ( vx vy vw vv vu vf vz cmq cdm cv cnq wcel wa ceq cec wceq wex sylibr cvv
      copab cxp cop cmpq coprab dmoprab df-mqqs dmeqi dmaddpqlem anim12i ee4anv
      co enqex ecexg ax-mp isseti ax-ia3 eximdv mpi 2eximi exrot3 19.42v bitr4i
      syl pm4.71i opabbii 3eqtr4i df-xp eqtr4i ) HIZAJZKLZBJZKLZMZABTZKKUAVNVJC
      JDJUBZNOPZVLEJFJUBZNOPZMZGJVPVRUCUKZNOZPZMZFQEQZDQCQZMZABGUDZIWGGQZABTVIV
      OWGABGUEHWHABGCDEFUFUGVNWIABVNVNWFGQZMWIVNWJVNVTFQEQZDQCQZWJVNVQDQCQZVSFQ
      EQZMWLVKWMVMWNACDUHBEFUHUIVQVSCDEFUJRWLWEGQZDQCQWJWKWOCDWKWDGQZFQEQWOVTWP
      EFVTWCGQWPGWBNSLWBSLULWASNUMUNUOVTWCWDGVTWCUPUQURUSWDGEFUTRUSWEGCDUTRVCVD
      VNWFGVAVBVEVFABKKVGVH $.
  $}

  ${
    $d A w x y z $.  $d B w z $.
    $( Addition of positive fractions is commutative.  (Contributed by Jim
       Kingdon, 15-Sep-2019.) $)
    addcomnqg $p |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( B +Q A ) ) $=
      ( vx vy vz vw cnq cv cmi co cpli cplq cnpi addpipqqs wcel wa wceq mulclpi
      mulcompig ancoms ceq df-nqqs oveqan12d an42s ad2ant2rl ad2ant2lr ad2ant2l
      addcompig syl2anc eqtrd ecovicom ) CDEFABGCHZFHZIJZDHZEHZIJZKJZLUAMUOUMIJ
      ZUPUOIJZUMULIJZKJZUMUOIJZUBULUOUPUMNUPUMULUONULMOZUOMOZPUPMOZUMMOZPPZURVA
      UTKJZVBVDVGVEVFURVIQVDVGPVEVFPUNVAUQUTKULUMSUOUPSUCUDVHVAMOZUTMOZVIVBQVDV
      GVJVEVFVGVDVJUMULRTUEVEVFVKVDVGVFVEVKUPUORTUFVAUTUHUIUJVEVGUSVCQVDVFUOUMS
      UGUK $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u f g h $.
    $( Addition of positive fractions is associative.  (Contributed by Jim
       Kingdon, 16-Sep-2019.) $)
    addassnqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
        ( ( A +Q B ) +Q C ) = ( A +Q ( B +Q C ) ) ) $=
      ( vx cv cmi co cnpi cpli addpipqqs wcel mulclpi addclpi syl2anc mulcompig
      wa wceq adantl ancoms mulasspig vy vz vw vv vu vf vg cnq cplq ceq df-nqqs
      vh ad2ant2rl ad2ant2lr ad2ant2l simp1l simp2r simp3r simp1r simp2l simp3l
      jca addasspig syl3anc distrpig 3coml sylan2 3impa 3adant2 3adant1 oveq12d
      w3a 3eqtr3d caovdilemd 3adant1l 3adant2l 3adant3r oveq2d 3eqtr4d 3adant3l
      ecoviass ) DUAUBUCUDUEABCUHUIUCEZUEEZFGZUJHDEZWBFGZUAEZUBEZFGZIGZWGWBFGZW
      JWCFGZWKUDEZFGZIGZWKWCFGZWEWDFGZWGWHWCFGZWBWMFGZIGZFGZIGZWGWDFGZWTUKWEWGW
      HWBJWHWBWMWCJWJWKWMWCJWEWGWTWDJWEHKZWGHKZPZWHHKZWBHKZPZPZWJHKZWKHKZXJWFHK
      ZWIHKZXKXDXHXMXEXGWEWBLUMXEXGXNXDXHWGWHLUNWFWIMNXEXHXLXDXGWGWBLUOVBXIWMHK
      ZWCHKZPZPZWTHKZWDHKZXRWRHKZWSHKZXSXGXPYAXHXOWHWCLZUMXHXOYBXGXPWBWMLZUNWRW
      SMNXHXPXTXGXOWBWCLZUOVBXFXIXQVLZWQWGWRFGZIGZWGWSFGZIGZWQYGYIIGZIGZWOXBYFW
      QHKZYGHKZYIHKZYJYLQYFXDXTYMXDXEXIXQUPZYFXHXPXTXFXGXHXQUQZXFXIXOXPURZYENWE
      WDLNYFXEYAYNXDXEXIXQUSZYFXGXPYAXFXGXHXQUTZYRYCNZWGWRLNYFXEYBYOYSYFXHXOYBY
      QXFXIXOXPVAYDNZWGWSLNWQYGYIVCVDYFWLYHWNYIIYFUFUGULWEWGWBWHHIFWCUFEZHKZUGE
      ZHKZPZUUCUUEFGZUUEUUCFGQYFUUCUUEORUUDUUFULEZHKZVLZUUCUUEIGZUUIFGZUUCUUIFG
      ZUUEUUIFGZIGZQYFUUKUUIUULFGZUUIUUCFGZUUIUUEFGZIGZUUMUUPUUJUUDUUFUUQUUTQUU
      IUUCUUEVEVFUUDUUFUUJUUQUUMQZUUJUUGUVAUUGUUJUULHKUVAUUCUUEMUUIUULOVGSVHUUK
      UURUUNUUSUUOIUUDUUJUURUUNQZUUFUUJUUDUVBUUIUUCOSVIUUFUUJUUSUUOQZUUDUUJUUFU
      VCUUIUUEOSVJVKVMRUUKUUHUUIFGUUCUUOFGQYFUUCUUEUUITRUUGUUHHKYFUUCUUELRYPYSY
      QYTYRVNXFXIXOWNYIQZXPXFXHXOUVDXGXEXHXOUVDXDWGWBWMTVOVPVQVKYFXAYKWQIYFXEYA
      YBXAYKQYSUUAUUBWGWRWSVEVDVRVSXFXIXPWPXCQZXOXFXHXPUVEXGXEXHXPUVEXDWGWBWCTV
      OVPVTWA $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Multiplication of positive fractions is commutative.  (Contributed by
       Jim Kingdon, 17-Sep-2019.) $)
    mulcomnqg $p |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( B .Q A ) ) $=
      ( vx vy vz vw cnq cv cmi co cmq ceq cnpi df-nqqs mulpipqqs wcel mulcompig
      wceq ad2ant2r ad2ant2l ecovicom ) CDEFABGCHZEHZIJZKLMDHZFHZIJZUCUBIJZUFUE
      IJZNUBUEUCUFOUCUFUBUEOUBMPZUCMPZUDUHRUEMPZUFMPZUBUCQSULUMUGUIRUJUKUEUFQTU
      A $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $( Multiplication of positive fractions is associative.  (Contributed by
       Jim Kingdon, 17-Sep-2019.) $)
    mulassnqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
        ( ( A .Q B ) .Q C ) = ( A .Q ( B .Q C ) ) ) $=
      ( vx vy vz vw vv vu cv cmi co cnpi mulpipqqs wa mulclpi ad2ant2r ad2ant2l
      wcel jca cnq cmq ceq df-nqqs 3adant1r 3adant2r 3adant3r 3adant1l 3adant2l
      wceq mulasspig 3adant3l ecoviass ) DEFGHIABCUAUBGJZIJZKLZUCMDJZFJZKLZEJZU
      NKLZUSHJZKLZVAUOKLZUQURVBKLZKLZUTUPKLZVEUDUQUTURUNNURUNVBUONUSVAVBUONUQUT
      VEUPNUQMSZUTMSZOZURMSZUNMSZOZOUSMSZVAMSZVHVKVNVIVLUQURPQVIVLVOVHVKUTUNPRT
      VMVBMSZUOMSZOOVEMSZUPMSZVKVPVRVLVQURVBPQVLVQVSVKVPUNUOPRTVJVMVPVCVFUJZVQV
      JVKVPVTVLVHVKVPVTVIUQURVBUKUEUFUGVJVMVQVDVGUJZVPVJVLVQWAVKVIVLVQWAVHUTUNU
      OUKUHUIULUM $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.
    $( Lemma for distributive law: cancellation of common factor.  (Contributed
       by NM, 2-Sep-1995.)  (Revised by Mario Carneiro, 8-May-2013.) $)
    mulcanenq $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
      <. ( A .N B ) , ( A .N C ) >. ~Q <. B , C >. ) $=
      ( vx vy vz cnpi wcel w3a cmi co cop ceq wbr wceq simp1 cv adantl mulclpi
      wa simp2 simp3 mulcompig mulasspig caov32d anim12i simpr an4s jca enqbreq
      wb 3impdi syl mpbird ) AGHZBGHZCGHZIZABJKZACJKZLBCLMNZUSCJKUTBJKOZURDEFAB
      CGJUOUPUQPUOUPUQUAUOUPUQUBDQZGHZEQZGHZTVCVEJKZVEVCJKOURVCVEUCRVDVFFQZGHIV
      GVHJKVCVEVHJKJKOURVCVEVHUDRUEURUSGHZUTGHZTZUPUQTZTZVAVBUKUOUPUQVMUOUPTZUO
      UQTZTVKVLVNVIVOVJABSACSUFUOUOUPUQVLUOUOTVLUGUHUIULUSUTBCUJUMUN $.
  $}

  $( Lemma for distributive law: cancellation of common factor.  (Contributed
     by Jim Kingdon, 17-Sep-2019.) $)
  mulcanenqec $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
      [ <. ( A .N B ) , ( A .N C ) >. ] ~Q = [ <. B , C >. ] ~Q ) $=
    ( cnpi wcel w3a cmi co cop ceq cxp wer enqer a1i mulcanenq erthi ) ADEBDECD
    EFZABGHACGHIBCIJDDKZRJLQMNABCOP $.

  ${
    $d A u v w x y z $.  $d B u v w x y z $.  $d C u v w x y z $.
    $d f g h u v w x y z $.
    $( Multiplication of positive fractions is distributive.  (Contributed by
       Jim Kingdon, 17-Sep-2019.) $)
    distrnqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
        ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) ) ) $=
      ( vf vg vh ceq cnpi cv cmi co cpli wcel cop cec w3a wceq mulclpi adantl
      wa vx vy vz vw vv vu cnq cplq cmq df-nqqs addpipqqs mulpipqqs jca anim12i
      simpl an12 3anass bitr4i sylib an4s mulcanenqec syl eqtr4d addclpi syl2an
      an42s ad2ant2l an42 anbi2i 3bitr4i ancoms distrpig syl3an simp1r 3ad2ant2
      simp1l 3ad2ant3 syl2anc syl3anc oveq1d adantr simpll simplr simprl simprr
      mulasspig mulcompig caov4d eqtr3d 3adant3 3adant2 oveq12d 3eqtr3d ecovidi
      sylbir ) UAUBUCUDUEUFABCUGUHGHUIUBIZUAIZUCIZUFIZJKZUDIZUEIZJKZLKZJKZJKZWP
      WPXAWSJKZJKZJKZWQWRJKZWPWSJKZJKZWPXAJKZWQXBJKZJKZLKZXMXKJKZXDXGXJXMXNXKUJ
      WRXAXBWSUKWQHMZWPHMZTZXDHMZXGHMZTTZWQWPNGOXDXGNGOUIKXEXHNGOZXFXINGOZWQWPX
      DXGULYCXSXEHMZXHHMZPZYEYDQXRYAXSYBYHXRYATZXSYBTZTYFXSYGTZTZYHYIYFYJYKWQXD
      RYJXSYGXSYBUOWPXGRUMUNYLXSYFYGTTYHYFXSYGUPXSYFYGUQURUSUTWPXEXHVAVBVCWQWPW
      RXAULWQWPXBWSULXJXMXNXKUKWRHMZXAHMZTZXBHMZWSHMZTZTZYAYBYMYQYNYPYAYMYQTZWT
      HMZXCHMZYAYNYPTZWRWSRZXAXBRZWTXCVDZVEVFYNYQYBYMYPXAWSRZVGUMXRYMXSYNXJHMZX
      MHMZTXRYMTUUHXSYNTUUIWQWRRWPXARUNUTXRYPXSYQXNHMZXKHMZTXRYPTUUJXSYQTUUKWQX
      BRWPWSRUNUTXTYOYRPZXTYTUUCPZXFXPQXTYTUUCTZTXTYSTUUMUULUUNYSXTYMYQYNYPVHVI
      XTYTUUCUQXTYOYRUQVJZUUMWPWQJKZXDJKZUUPWTJKZUUPXCJKZLKZXFXPXTUUPHMZYTUUAUU
      CUUBUUQUUTQXSXRUVAWPWQRVKUUDUUEUUPWTXCVLVMUUMXSXRYAUUQXFQXRXSYTUUCVNZXRXS
      YTUUCVPUUMUUAUUBYAYTXTUUAUUCUUDVOUUCXTUUBYTUUEVQUUFVRWPWQXDWFVSUUMUURXLUU
      SXOLXTYTUURXLQUUCXTYTTZWQWPJKZWTJKZUURXLXTUVEUURQYTXTUVDUUPWTJWQWPWGVTWAU
      VCDEFWQWPWRWSHJXRXSYTWBXRXSYTWCXTYMYQWDDIZHMZEIZHMZTZUVFUVHJKZUVHUVFJKQZU
      VCUVFUVHWGZSUVGUVIFIZHMPZUVKUVNJKUVFUVHUVNJKJKQZUVCUVFUVHUVNWFZSXTYMYQWEZ
      UVJUVKHMZUVCUVFUVHRZSWHWIWJXTUUCUUSXOQYTXTUUCTZDEFWPWQXAXBHJXRXSUUCWCXRXS
      UUCWBXTYNYPWDZUVJUVLUWAUVMSUVOUVPUWAUVQSXTYNYPWEUVJUVSUWAUVTSWHWKWLWMWOUU
      LUUMXIXQQUUOUUMWPWPJKXGJKZXIXQUUMXSXSYBUWCXIQUVBUVBUUMYNYQYBXTUUCYNYTUWBW
      KZXTYTYQUUCUVRWJZUUGVRWPWPXGWFVSUUMDEFWPWPXAWSHJUVBUVBUWDUVJUVLUUMUVMSUVO
      UVPUUMUVQSUWEUVJUVSUUMUVTSWHWIWOWN $.
  $}

  $( The equivalence class of ratio 1.  (Contributed by NM, 4-Mar-1996.) $)
  1qec $p |- ( A e. N. -> 1Q = [ <. A , A >. ] ~Q ) $=
    ( cnpi wcel c1q c1o cop ceq cec df-1nqqs cmi co wceq mulcanenqec mp3an23 wa
    1pi mulidpi jca opeq12 eceq1 3syl eqtr3d eqtrid ) ABCZDEEFGHZAAFZGHZIUDAEJK
    ZUHFZGHZUEUGUDEBCZUKUJUELPPAEEMNUDUHALZULOUIUFLUJUGLUDULULAQZUMRUHUHAASUIUF
    GTUAUBUC $.

  ${
    $d x y A $.
    $( Multiplication identity element for positive fractions.  (Contributed by
       NM, 3-Mar-1996.) $)
    mulidnq $p |- ( A e. Q. -> ( A .Q 1Q ) = A ) $=
      ( vx vy cv cop ceq cec c1q cmq co wceq cnpi cnq df-nqqs c1o cmi mulcompig
      wcel 1pi mpan oveq1 id eqeq12d wa df-1nqqs oveq2i mulpipqqs eqtrid adantr
      mpanr12 adantl opeq12d eceq1d mulcanenqec mp3an1 3eqtr2d ecoptocl ) BDZCD
      ZEFGZHIJZUTKAHIJZAKBCALLFMNUTAKZVAVBUTAUTAHIUAVCUBUCURLRZUSLRZUDZVAUROPJZ
      USOPJZEZFGZOURPJZOUSPJZEZFGZUTVFVAUTOOEFGZIJZVJHVOUTIUEUFVFOLRZVQVPVJKSSU
      RUSOOUGUJUHVFVMVIFVFVKVGVLVHVDVKVGKZVEVQVDVRSOURQTUIVEVLVHKZVDVQVEVSSOUSQ
      TUKULUMVQVDVEVNUTKSOURUSUNUOUPUQ $.
  $}

  ${
    $d x y z A $.
    $( Existence of positive fraction reciprocal.  (Contributed by Jim Kingdon,
       20-Sep-2019.) $)
    recexnq $p |- ( A e. Q. -> E. y ( y e. Q. /\ ( A .Q y ) = 1Q ) ) $=
      ( vx vz cv cnq wcel cop ceq cec cmq co c1q wceq wa wex df-nqqs eqeq1d syl
      cnpi oveq1 anbi2d exbidv cxp cqs opelxpi ancoms enqex eleqtrrdi mulcompig
      ecelqsi cmi opeq2d eceq1d mulclpi 1qec mulpipqqs an42s 3eqtr4rd jca eleq1
      anidms oveq2 anbi12d spcegv sylc ecoptocl ) AEZFGZCEZDEZHIJZVHKLZMNZOZAPZ
      VIBVHKLZMNZOZAPCDBTTIFQVLBNZVOVSAVTVNVRVIVTVMVQMVLBVHKUARUBUCVJTGZVKTGZOZ
      VKVJHZIJZFGZWFVLWEKLZMNZOZVPWCWETTUDZIUEZFWCWDWJGZWEWKGWBWAWLVKVJTTUFUGWJ
      WDIUHUKSQUIZWCWFWHWMWCVJVKULLZWNHZIJZWNVKVJULLZHZIJZMWGWCWOWRIWCWNWQWNVJV
      KUJUMUNWCWNTGMWPNVJVKUOWNUPSWCWGWSNZWAWBWBWAWTVJVKVKVJUQURVBUSUTVOWIAWEFV
      HWENZVIWFVNWHVHWEFVAXAVMWGMVHWEVLKVCRVDVEVFVG $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y z w v $.
    $( Relationship between reciprocal and multiplication on positive
       fractions.  (Contributed by Jim Kingdon, 19-Sep-2019.) $)
    recmulnqg $p |- ( ( A e. Q. /\ B e. Q. ) ->
      ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) ) $=
      ( vy vx vz vw vv cnq wcel wa crq cfv wceq cmq co c1q oveq1 eqeq1d copab
      cv anbi2d eleq1 oveq2 anbi12d wex wmo weu recexnq 1nq mulcomnqg mulassnqg
      mulidnq caovimo eu5 sylanbrc w3a df-rq 3anass opabbii eqtri fvopab3g ibar
      wb adantl bitr4d ) AHIZBHIZJAKLBMVGABNOZPMZJZVICTZHIZDTZVKNOZPMZJZVLAVKNO
      ZPMZJVJDCABHHKVMAMZVOVRVLVSVNVQPVMAVKNQRUAVKBMZVLVGVRVIVKBHUBVTVQVHPVKBAN
      UCRUDVMHIZVPCUEVPCUFVPCUGCVMUHEFGCVMPHNUIETZFTZUJWBWCGTUKWBULUMVPCUNUOKWA
      VLVOUPZDCSWAVPJZDCSDCUQWDWEDCWAVLVOURUSUTVAVGVIVJVCVFVGVIVBVDVE $.
  $}

  ${
    $d A y $.
    $( Closure law for positive fraction reciprocal.  (Contributed by NM,
       6-Mar-1996.)  (Revised by Mario Carneiro, 8-May-2013.) $)
    recclnq $p |- ( A e. Q. -> ( *Q ` A ) e. Q. ) $=
      ( vy cnq wcel cv cmq co c1q wceq wa wex crq cfv recexnq recmulnqg biimpar
      wi eleq1a ad2antlr mpd expl exlimdv ) ACDZBEZCDZAUDFGHIZJZBKALMZCDZBANUCU
      GUIBUCUEUFUIUCUEJZUFJUHUDIZUIUJUKUFAUDOPUEUKUIQUCUFUDCUHRSTUAUBT $.
  $}

  $( A positive fraction times its reciprocal is 1.  (Contributed by NM,
     6-Mar-1996.)  (Revised by Mario Carneiro, 8-May-2013.) $)
  recidnq $p |- ( A e. Q. -> ( A .Q ( *Q ` A ) ) = 1Q ) $=
    ( cnq wcel crq cfv cmq co c1q wceq recclnq wa eqid recmulnqg mpbii mpdan )
    ABCZADEZBCZAQFGHIZAJPRKQQISQLAQMNO $.

  $( Reciprocal of reciprocal of positive fraction.  (Contributed by NM,
     26-Apr-1996.)  (Revised by Mario Carneiro, 29-Apr-2013.) $)
  recrecnq $p |- ( A e. Q. -> ( *Q ` ( *Q ` A ) ) = A ) $=
    ( cnq wcel crq cfv wceq cmq c1q recclnq mulcomnqg mpancom recidnq recmulnqg
    co eqtrd wb mpbird ) ABCZADEZDEAFZSAGNZHFZRUAASGNZHSBCZRUAUCFAIZSAJKALOUDRT
    UBPUESAMKQ $.

  $( Reciprocal of positive fraction one.  (Contributed by Jim Kingdon,
     29-Dec-2019.) $)
  rec1nq $p |- ( *Q ` 1Q ) = 1Q $=
    ( c1q crq cfv cmq co cnq wcel wceq 1nq recclnq ax-mp mulcomnqg mulidnq mp2b
    mp2an recidnq 3eqtr3i ) ABCZADEZARDEZRARFGZAFGZSTHUBUAIAJZKIRALOUBUASRHIUCR
    MNUBTAHIAPKQ $.

  ${
    $d A u v w z $.  $d B u v $.
    $( Trichotomy for positive fractions.  (Contributed by Jim Kingdon,
       21-Sep-2019.) $)
    nqtri3or $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( A <Q B \/ A = B \/ B <Q A ) ) $=
      ( vz vw vu vv cv cop ceq cltq wbr wceq w3o cnpi 3orbi123d wcel wa co clti
      cmi cec cnq df-nqqs breq1 eqeq1 breq2 mulclpi ad2ant2rl ad2ant2lr syl2anc
      eqeq2 pitri3or ordpipqqs enqeceq ancoms mulcompig bitr4d mpbird 2ecoptocl
      wb breq12d ) CGZDGZHIUAZEGZFGZHIUAZJKZVDVGLZVGVDJKZMZAVGJKZAVGLZVGAJKZMAB
      JKZABLZBAJKZMCDEFABNNIUBUCVDALVHVLVIVMVJVNVDAVGJUDVDAVGUEVDAVGJUFOVGBLVLV
      OVMVPVNVQVGBAJUFVGBAUKVGBAJUDOVBNPZVCNPZQZVENPZVFNPZQZQZVKVBVFTRZVCVETRZS
      KZWEWFLZWFWESKZMZWDWENPZWFNPZWJVRWBWKVSWAVBVFUGUHVSWAWLVRWBVCVEUGUIWEWFUL
      UJWDVHWGVIWHVJWIVBVCVEVFUMVBVCVEVFUNWDVJVEVCTRZVFVBTRZSKZWIWCVTVJWOUTVEVF
      VBVCUMUOWDWFWMWEWNSVSWAWFWMLVRWBVCVEUPUIVRWBWEWNLVSWAVBVFUPUHVAUQOURUS $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.
    $( Less-than for positive fractions is decidable.  (Contributed by Jim
       Kingdon, 12-Dec-2019.) $)
    ltdcnq $p |- ( ( A e. Q. /\ B e. Q. ) -> DECID A <Q B ) $=
      ( vx vy vz vw cnq wcel wa cv cnpi cop ceq cec wceq wex cltq wbr wdc nqpi
      anim12i ee4anv sylibr co clti mulclpi ltdcpi syl2an an42s ordpipqqs dcbid
      cmi mpbird ad2ant2r wb breq12 ad2ant2l exlimivv syl ) AGHZBGHZIZCJZKHZDJZ
      KHZIZAVCVELMNZOZIZEJZKHZFJZKHZIZBVKVMLMNZOZIZIZFPEPZDPCPZABQRZSZVBVJDPCPZ
      VRFPEPZIWAUTWDVAWECDATEFBTUAVJVRCDEFUBUCVTWCCDVSWCEFVSWCVHVPQRZSZVGVOWGVI
      VQVGVOIZWGVCVMULUDZVEVKULUDZUERZSZVDVNVFVLWLVDVNIWIKHWJKHWLVFVLIVCVMUFVEV
      KUFWIWJUGUHUIWHWFWKVCVEVKVMUJUKUMUNVSWBWFVIVQWBWFUOVGVOAVHBVPQUPUQUKUMURU
      RUS $.
  $}

  ${
    $d a b c d e f x y z w $.
    $( 'Less than' is a strict ordering on positive fractions.  (Contributed by
       NM, 19-Feb-1996.)  (Revised by Mario Carneiro, 4-May-2013.) $)
    ltsonq $p |- <Q Or Q. $=
      ( vx vy vz cnq cltq wtru cv wcel wbr ceq cnpi wceq wa cmi co clti mulclpi
      wb adantl syl2anc vw va vb vc vd ve vf wor cop cec df-nqqs breq12d notbid
      wn ltsopi ltrelpi soirri ordpipqqs anidms mulcompig breq1d bitrd ecoptocl
      id mtbiri w3a wi breq1 anbi1d imbi12d breq2 anbi12d imbi1d anbi2d 3adant3
      simp1l simp2r simp1r simp2l simp3r ltmpig syl3anc adantrr breqtrd 3adant1
      biimpa adantr simp3l adantrl sotri mulasspig caov411d eqtrd caov4d oveq1d
      caovcomd mpbird 3adant2 ex 3ecoptocl ispod w3o nqtri3or issod mptru ) DEU
      HFABDEFABCDEAGZDHZXFXFEIZUNZFCGZUAGZUIJUJZXLEIZUNXICUAXFKKJDUKXLXFLZXMXHX
      NXLXFXLXFEXNVDZXOULUMXJKHZXKKHMZXMXKXJNOZXRPIZXRPKUOUPUQXQXMXJXKNOZXRPIZX
      SXQXMYARXJXKXJXKURUSXQXTXRXRPXJXKUTVAVBVEVCSXGBGZDHZXJDHVFXFYBEIZYBXJEIZM
      ZXFXJEIZVGZFUBGZUCGZUIJUJZUDGZUEGZUIJUJZEIZYNUFGZUGGZUIJUJZEIZMZYKYREIZVG
      XFYNEIZYSMZXFYREIZVGYDYBYREIZMZUUDVGYHUBUCUDUEUFUGXFYBXJKJDUKYKXFLZYTUUCU
      UAUUDUUGYOUUBYSYKXFYNEVHVIYKXFYREVHVJYNYBLZUUCUUFUUDUUHUUBYDYSUUEYNYBXFEV
      KYNYBYREVHVLVMYRXJLZUUFYFUUDYGUUIUUEYEYDYRXJYBEVKVNYRXJXFEVKVJYIKHZYJKHZM
      ZYLKHZYMKHZMZYPKHZYQKHZMZVFZYTUUAUUSYTMZUUAYIYQNOZYJYPNOZPIZUUTUVCYLYMNOZ
      UVANOZUVDUVBNOZPIZUUTUVGYLYQNOZYIYMNOZNOZYJYLNOZYMYPNOZNOZPIZUUTUVJUVKUVH
      NOZPIUVOUVMPIZUVNUUTUVJUVHUVKNOZUVOPUUSYOUVJUVQPIZYSUUSYOUVRUUSYOUVIUVKPI
      ZUVRUULUUOYOUVSRUURYIYJYLYMURVOUUSUVIKHZUVKKHZUVHKHZUVSUVRRUUSUUJUUNUVTUU
      JUUKUUOUURVPZUULUUMUUNUURVQZYIYMQTZUUSUUKUUMUWAUUJUUKUUOUURVRZUULUUMUUNUU
      RVSZYJYLQTZUUSUUMUUQUWBUWGUULUUOUUPUUQVTZYLYQQTZUVIUVKUVHWAWBVBWFWCUUSUVQ
      UVOLZYTUUSUWBUWAUWKUWJUWHUVHUVKUTTWGWDUUSYSUVPYOUUSYSUVPUUSYSUVHUVLPIZUVP
      UUOUURYSUWLRUULYLYMYPYQURWEUUSUWBUVLKHZUWAUWLUVPRUWJUUSUUNUUPUWMUWDUULUUO
      UUPUUQWHZYMYPQTUWHUVHUVLUVKWAWBVBWFWIUVJUVOUVMPKUOUPWJTUUSUVGUVNRYTUUSUVE
      UVJUVFUVMPUUSUVEUVIUVHNOUVJUUSABCYLYMYIYQKNUWGUWDUWCXFKHZYBKHZMZXFYBNOZYB
      XFNOLUUSXFYBUTSZUWOUWPXPVFUWRXJNOXFYBXJNONOLUUSXFYBXJWKSZUWIUWQUWRKHUUSXF
      YBQSZWLUUSABUVIUVHKNUWSUWEUWJWPWMUUSUVFYLYJNOZUVLNOUVMUUSABCYLYMYJYPKNUWG
      UWDUWFUWSUWTUWNUXAWNUUSUXBUVKUVLNUUSABYLYJKNUWSUWGUWFWPWOWMULWGWQUUSUVCUV
      GRZYTUUSUVAKHZUVBKHZUVDKHZUXCUUSUUJUUQUXDUWCUWIYIYQQTUUSUUKUUPUXEUWFUWNYJ
      YPQTUUSUUMUUNUXFUWGUWDYLYMQTUVAUVBUVDWAWBWGWQUUSUUAUVCRZYTUULUURUXGUUOYIY
      JYPYQURWRWGWQWSWTSXAXGYCMYDXFYBLYBXFEIXBFXFYBXCSXDXE $.
  $}

  $( Trichotomy for positive fractions.  (Contributed by Jim Kingdon,
     21-Sep-2019.) $)
  nqtric $p |- ( ( A e. Q. /\ B e. Q. ) ->
      ( A <Q B <-> -. ( A = B \/ B <Q A ) ) ) $=
    ( cnq cltq ltsonq nqtri3or sotritric ) CABDEABFG $.

  ${
    $d A x y z w v u f g h $.  $d B x y z w v u f g h $.
    $d C x y z w v u f g h $.
    $( Ordering property of addition for positive fractions.  Proposition
       9-2.6(ii) of [Gleason] p. 120.  (Contributed by Jim Kingdon,
       22-Sep-2019.) $)
    ltanqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
        ( A <Q B <-> ( C +Q A ) <Q ( C +Q B ) ) ) $=
      ( vf vg vh cv cltq wbr cplq co wb cnpi wceq wcel cmi cpli mulclpi syl2anc
      clti vx vy vz vw vv vu cop ceq cec cnq df-nqqs breq1 oveq2 breq1d bibi12d
      breq2 breq2d oveq1 breq12d bibi2d w3a addclpi adantl simp3l simp1r simp3r
      wa simp1l caovcld simp2r ordpipqqs syl22anc simp3 simp1 addpipqqs 3adant3
      simp2l simp2 ltmpig syl3anc ltapig mulcompig mulasspig distrpig caovdir2d
      caov4d caovdid caov411d oveq1d eqtrd 3bitr4d 3bitrd 3bitr4rd 3ecoptocl )
      UAGZUBGZUGUHUIZUCGZUDGZUGUHUIZHIZUEGZUFGZUGUHUIZWQJKZXDWTJKZHIZLAWTHIZXDA
      JKZXFHIZLABHIZXIXDBJKZHIZLXKCAJKZCBJKZHIZLUAUBUCUDUEUFABCMUHUJUKWQANZXAXH
      XGXJWQAWTHULXQXEXIXFHWQAXDJUMUNUOWTBNZXHXKXJXMWTBAHUPXRXFXLXIHWTBXDJUMUQU
      OXDCNZXMXPXKXSXIXNXLXOHXDCAJURXDCBJURUSUTWOMOZWPMOZVGZWRMOZWSMOZVGZXBMOZX
      CMOZVGZVAZXBWPPKZXCWOPKZQKZXCWPPKZUGUHUIZXBWSPKZXCWRPKZQKZXCWSPKZUGUHUIZH
      IZYLYRPKZYMYQPKZTIZXGXAYIYLMOYMMOZYQMOYRMOZYTUUCLYIDEYJYKMMMQDGZMOZEGZMOZ
      VGZUUFUUHQKMOYIUUFUUHVBVCZYIYFYAYJMOZYBYEYFYGVDZXTYAYEYHVEZXBWPRSZYIYGXTY
      KMOZYBYEYFYGVFZXTYAYEYHVHZXCWORSZVIYIYGYAUUDUUQUUNXCWPRSZYIDEYOYPMMMQUUKY
      IDEXBWSMMMPUUJUUFUUHPKZMOYIUUFUUHRVCZUUMYBYCYDYHVJZVIZYIYGYCYPMOZUUQYBYCY
      DYHVQZXCWRRSZVIYIYGYDUUEUUQUVCXCWSRSZYLYMYQYRVKVLYIXEYNXFYSHYIYHYBXEYNNYB
      YEYHVMZYBYEYHVNXBXCWOWPVOSYIYHYEXFYSNUVIYBYEYHVRXBXCWRWSVOSUSYIXAWOWSPKZW
      PWRPKZTIZXCXCPKZUVJPKZUVMUVKPKZTIZUUCYBYEXAUVLLYHWOWPWRWSVKVPYIUVJMOZUVKM
      OZUVMMOZUVLUVPLYIXTYDUVQUURUVCWOWSRSYIYAYCUVRUUNUVFWPWRRSYIYGYGUVSUUQUUQX
      CXCRSUVJUVKUVMVSVTYIYKYRPKZYMYPPKZTIZYJYRPKZUVTQKZUWCUWAQKZTIZUVPUUCYIUVT
      MOZUWAMOZUWCMOZUWBUWFLYIUUPUUEUWGUUSUVHYKYRRSYIUUDUVEUWHUUTUVGYMYPRSYIUUL
      UUEUWIUUOUVHYJYRRSUVTUWAUWCWAVTYIUVNUVTUVOUWATYIDEFXCXCWOWSMPUUQUUQUURUUJ
      UVAUUHUUFPKNYIUUFUUHWBVCZUUGUUIFGZMOVAZUVAUWKPKUUFUUHUWKPKPKNYIUUFUUHUWKW
      CVCZUVCUVBWFYIDEFXCXCWPWRMPUUQUUQUUNUWJUWMUVFUVBWFUSYIUUAUWDUUBUWETYIDEFY
      JYKYRMQPUWLUUFUUHUWKQKPKUVAUUFUWKPKQKNYIUUFUUHUWKWDVCZUUOUUSUVHUUKUWJWEYI
      UUBYMYOPKZUWAQKUWEYIDEFYMYOYPMQPQMUWNUUTUVDUVGWGYIUWOUWCUWAQYIDEFXCWPXBWS
      MPUUQUUNUUMUWJUWMUVCUVBWHWIWJUSWKWLWMWN $.

    $( Ordering property of multiplication for positive fractions.  Proposition
       9-2.6(iii) of [Gleason] p. 120.  (Contributed by Jim Kingdon,
       22-Sep-2019.) $)
    ltmnqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
      ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) ) $=
      ( vf vg cv cop ceq cec cltq wbr cmq co wb cnpi wceq wcel wa cmi caovcld
      vx vy vz vw vv vu vh cnq df-nqqs breq1 oveq2 breq1d bibi12d breq2 breq12d
      breq2d oveq1 bibi2d w3a clti mulclpi adantl simp1l simp2r simp1r 3ad2ant3
      simp2l ltmpig syl3anc simp3l simp3r mulcompig caov4d oveq1d ancoms bitr4d
      mulasspig ordpipqqs 3adant3 syl22anc 3bitr4d mulpipqqs 3adant2 3ecoptocl
      eqtrd 3adant1 ) UAFZUBFZGHIZUCFZUDFZGHIZJKZUEFZUFFZGHIZWILMZWPWLLMZJKZNAW
      LJKZWPALMZWRJKZNABJKZXAWPBLMZJKZNXCCALMZCBLMZJKZNUAUBUCUDUEUFABCOHUHUIWIA
      PZWMWTWSXBWIAWLJUJXIWQXAWRJWIAWPLUKULUMWLBPZWTXCXBXEWLBAJUNXJWRXDXAJWLBWP
      LUKUPUMWPCPZXEXHXCXKXAXFXDXGJWPCALUQWPCBLUQUOURWGOQZWHOQZRZWJOQZWKOQZRZWN
      OQZWOOQZRZUSZWMWNWGSMZWOWHSMZGHIZWNWJSMZWOWKSMZGHIZJKZWSYAWGWKSMZWHWJSMZU
      TKZYBYFSMZYCYESMZUTKZWMYHYAYKWNWOSMZYISMZYOYJSMZUTKZYNYAYIOQYJOQYOOQZYKYR
      NYADEWGWKOOOSDFZOQZEFZOQZRZYTUUBSMZOQYAYTUUBVAVBZXLXMXQXTVCZXNXOXPXTVDZTY
      ADEWHWJOOOSUUFXLXMXQXTVEZXNXOXPXTVGZTXTXNYSXQWNWOVAVFYIYJYOVHVIYAYLYPYMYQ
      UTYADEUGWNWGWOWKOSXNXQXRXSVJZUUGXNXQXRXSVKZUUDUUEUUBYTSMPYAYTUUBVLVBZUUAU
      UCUGFZOQUSUUEUUNSMYTUUBUUNSMSMPYAYTUUBUUNVQVBZUUHUUFVMYAYMWOWNSMZYJSMZYQY
      ADEUGWOWHWNWJOSUULUUIUUKUUMUUOUUJUUFVMXTXNUUQYQPZXQXSXRUURXSXRRUUPYOYJSWO
      WNVLVNVOVFWEUOVPXNXQWMYKNXTWGWHWJWKVRVSYAYBOQYCOQYEOQYFOQYHYNNYADEWNWGOOO
      SUUFUUKUUGTYADEWOWHOOOSUUFUULUUITYADEWNWJOOOSUUFUUKUUJTYADEWOWKOOOSUUFUUL
      UUHTYBYCYEYFVRVTWAYAWQYDWRYGJXNXTWQYDPZXQXTXNUUSWNWOWGWHWBVOWCXQXTWRYGPZX
      NXTXQUUTWNWOWJWKWBVOWFUOVPWD $.
  $}

  $( Ordering property of addition for positive fractions.  One direction of
     ~ ltanqg .  (Contributed by Jim Kingdon, 9-Dec-2019.) $)
  ltanqi $p |- ( ( A <Q B /\ C e. Q. ) -> ( C +Q A ) <Q ( C +Q B ) ) $=
    ( cltq wbr cnq wcel wa cplq co simpl ltrelnq brel ltanqg 3expa sylan mpbid
    wb ) ABDEZCFGZHSCAIJCBIJDEZSTKSAFGZBFGZHTSUARZABFFDLMUBUCTUDABCNOPQ $.

  $( Ordering property of multiplication for positive fractions.  One direction
     of ~ ltmnqg .  (Contributed by Jim Kingdon, 9-Dec-2019.) $)
  ltmnqi $p |- ( ( A <Q B /\ C e. Q. ) -> ( C .Q A ) <Q ( C .Q B ) ) $=
    ( cltq wbr cnq wcel wa cmq co simpl ltrelnq brel ltmnqg 3expa sylan mpbid
    wb ) ABDEZCFGZHSCAIJCBIJDEZSTKSAFGZBFGZHTSUARZABFFDLMUBUCTUDABCNOPQ $.

  $( Ordering property of addition for positive fractions.  (Contributed by Jim
     Kingdon, 7-Dec-2019.) $)
  lt2addnq $p |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) ->
      ( ( A <Q B /\ C <Q D ) -> ( A +Q C ) <Q ( B +Q D ) ) ) $=
    ( cnq wcel wa cltq wbr cplq co ltanqg 3expa adantrr wceq addcomnqg ad2ant2r
    wb ancoms ad2ant2lr breq12d bitrd adantll anbi12d ltsonq ltrelnq biimtrdi
    sotri ) AEFZBEFZGZCEFZDEFZGZGZABHIZCDHIZGACJKZBCJKZHIZUSBDJKZHIZGURVAHIUOUP
    UTUQVBUOUPCAJKZCBJKZHIZUTUKULUPVERZUMUIUJULVFABCLMNUOVCURVDUSHUIULVCUROZUJU
    MULUIVGCAPSQUJULVDUSOZUIUMULUJVHCBPSTUAUBUJUNUQVBRZUIUNUJVIULUMUJVICDBLMSUC
    UDURUSVAHEUEUFUHUG $.

  $( Ordering property of multiplication for positive fractions.  (Contributed
     by Jim Kingdon, 18-Jul-2021.) $)
  lt2mulnq $p |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) ->
      ( ( A <Q B /\ C <Q D ) -> ( A .Q C ) <Q ( B .Q D ) ) ) $=
    ( cnq wcel wa cltq wbr cmq co wb ltmnqg 3expa adantrr wceq mulcomnqg ancoms
    ad2ant2r ad2ant2lr breq12d bitrd adantll anbi12d ltsonq ltrelnq biimtrdi
    sotri ) AEFZBEFZGZCEFZDEFZGZGZABHIZCDHIZGACJKZBCJKZHIZUSBDJKZHIZGURVAHIUOUP
    UTUQVBUOUPCAJKZCBJKZHIZUTUKULUPVELZUMUIUJULVFABCMNOUOVCURVDUSHUIULVCURPZUJU
    MULUIVGCAQRSUJULVDUSPZUIUMULUJVHCBQRTUAUBUJUNUQVBLZUIUNUJVIULUMUJVICDBMNRUC
    UDURUSVAHEUEUFUHUG $.

  $( One is less than two (one plus one).  (Contributed by NM, 13-Mar-1996.)
     (Revised by Mario Carneiro, 10-May-2013.) $)
  1lt2nq $p |- 1Q <Q ( 1Q +Q 1Q ) $=
    ( c1o cop ceq cec cmi co cpli c1q cplq cltq wbr clti cnpi wcel wceq oveq12i
    1pi 3brtr4i wb df-1nqqs 1lt2pi mulidpi ax-mp mulclpi mp2an ltmpig ordpipqqs
    addclpi mp3an mpbi mp4an mpbir addpipqqs eqtri ) AABCDZAAEFZUPGFZUPBCDZHHHI
    FZJUOURJKZAUPEFAUQEFLKZUPUQLKZVAAAAGFUPUQLUAAMNZUPAOQAUBUCZUPAUPAGVDVDPRUPM
    NZUQMNZVCVBVASVCVCVEQQAAUDUEZVEVEVFVGVGUPUPUHUEZQUPUQAUFUIUJVCVCVFVEUTVASQQ
    VHVGAAUQUPUGUKULTUSUOUOIFZURHUOHUOITTPVCVCVCVCVIUROQQQQAAAAUMUKUNR $.

  ${
    $d A r s t $.  $d B r s t $.
    $( The sum of two fractions is greater than one of them.  (Contributed by
       NM, 14-Mar-1996.)  (Revised by Mario Carneiro, 10-May-2013.) $)
    ltaddnq $p |- ( ( A e. Q. /\ B e. Q. ) -> A <Q ( A +Q B ) ) $=
      ( vr vs vt cnq wcel wa cplq co cltq wbr c1q cmq wb addclnq wceq adantl cv
      1nq 1lt2nq mp2an ltmnqg mp3an12 mpbii mulidnq mp3an23 oveq12d eqtrd simpr
      3brtr3d anidms simpl ltanqg syl3anc mpbid addcomnqg w3a addassnqg caov12d
      distrnqg mpbird ) AFGZBFGZHZAABIJZKLZBAIJZBVFIJZKLZVEVFABBIJZIJZVHVIKVEBV
      KKLZVFVLKLZVDVMVCVDBMNJZBMMIJZNJZBVKKVDMVPKLZVOVQKLZUAMFGZVPFGZVDVRVSOTVT
      VTWATTMMPUBMVPBUCUDUEBUFZVDVQVOVOIJZVKVDVTVTVQWCQTTBMMVAUGVDVOBVOBIWBWBUH
      UIUKRVEVDVKFGZVCVMVNOVCVDUJZVDWDVCVDWDBBPULRVCVDUMZBVKAUNUOUPABUQVECDEABB
      FIWFWEWECSZFGZDSZFGZHWGWIIJZWIWGIJQVEWGWIUQRWHWJESZFGURWKWLIJWGWIWLIJIJQV
      EWGWIWLUSRUTUKVEVCVFFGVDVGVJOWFABPWEAVFBUNUOVB $.
  $}

  ${
    $d f g h x y z w v u A $.  $d x y z w v u B $.
    $( Ordering on positive fractions in terms of existence of sum.  Definition
       in Proposition 9-2.6 of [Gleason] p. 119.  (Contributed by Jim Kingdon,
       23-Sep-2019.) $)
    ltexnqq $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( A <Q B <-> E. x e. Q. ( A +Q x ) = B ) ) $=
      ( vu cnq wcel wa cltq wbr cv cplq co wceq wrex cop ceq cec cnpi cmi syl
      vy vz vw vv vf vg vh df-nqqs breq1 oveq1 rexbidv imbi12d breq2 eqeq2 clti
      wi eqeq1d ordpipqqs cpli wex wb mulclpi an42s ltexpi df-rex bitrdi simpll
      anim12i adantr adantrr addpipqqs simplll simpllr simplrr mulcompig adantl
      simpr w3a mulasspig caov12d oveq1d syl2anc simprl distrpig syl3anc eqtr4d
      jca32 opeq1d eceq1d ad2ant2rl addclpi sylan 3jca mulcanenqec eqtrd 3anass
      biimpri adantll anim1i adantrl opeq1 sylan9eqr 3eqtrd cxp opelxpi ecelqsi
      cqs enqex eleqtrrdi oveq2 rspcedv mpd ex exlimdv sylbid 2ecoptocl ltaddnq
      syl5ibcom rexlimdva impbid ) BEFZCEFZGBCHIZBAJZKLZCMZAENZUAJZUBJZOPQZUCJZ
      UDJZOPQZHIZYJYDKLZYMMZAENZUPBYMHIZYEYMMZAENZUPYCYGUPUAUBUCUDBCRRPEUHYJBMZ
      YNYRYQYTYJBYMHUIUUAYPYSAEUUAYOYEYMYJBYDKUJUQUKULYMCMZYRYCYTYGYMCBHUMUUBYS
      YFAEYMCYEUNUKULYHRFZYIRFZGZYKRFZYLRFZGZGZYNYHYLSLZYIYKSLZUOIZYQYHYIYKYLUR
      UUIUULDJZRFZUUJUUMUSLZUUKMZGZDUTZYQUUIUULUUPDRNZUURUUIUUJRFZUUKRFZGZUULUU
      SVAUUCUUGUUDUUFUVBUUCUUGGUUTUUDUUFGUVAYHYLVBZYIYKVBVHVCDUUJUUKVDTUUPDRVEV
      FUUIUUQYQDUUIUUQYQUUIUUQGZYJUUMYIYLSLZOZPQZKLZYMMZYQUVDUVHYHUVESLZYIUUMSL
      ZUSLZYIUVESLZOZPQZUUOUVEOZPQZYMUVDUUEUUNUVERFZGZGZUVHUVOMUUIUUNUVTUUPUUIU
      UNGZUUEUUNUVRUUEUUHUUNVGUUIUUNVQUWAUUDUUGGZUVRUUIUWBUUNUUEUUDUUHUUGUUCUUD
      VQUUFUUGVQVHVIYIYLVBZTZWGVJYHYIUUMUVEVKTUVDUVOYIUUOSLZUVMOZPQZUVQUVDUVNUW
      FPUVDUVLUWEUVMUVDUVLYIUUJSLZUVKUSLZUWEUVDUVJUWHUVKUSUVDUEUFUGYHYIYLRSUUCU
      UDUUHUUQVLZUUCUUDUUHUUQVMZUUEUUFUUGUUQVNZUEJZRFZUFJZRFZGUWMUWOSLZUWOUWMSL
      MUVDUWMUWOVOVPUWNUWPUGJZRFVRUWQUWRSLUWMUWOUWRSLSLMUVDUWMUWOUWRVSVPVTWAUVD
      UUDUUTUUNUWEUWIMUWKUVDUUCUUGUUTUWJUWLUVCWBUUIUUNUUPWCZYIUUJUUMWDWEWFWHWIU
      VDUUDUUORFZUVRVRZUWGUVQMUUIUUNUXAUUPUWAUUDUWTUVRUUCUUDUUHUUNVMUUIUUTUUNUW
      TUUCUUGUUTUUDUUFUVCWJUUJUUMWKWLUWDWMVJYIUUOUVEWNTWOUVDUUDUUFUUGVRZUUPGZUV
      QYMMUUIUUPUXCUUNUUIUXBUUPUUDUUHUXBUUCUXBUUDUUHGUUDUUFUUGWPWQWRWSWTUUPUXBU
      VQUUKUVEOZPQYMUUPUVPUXDPUUOUUKUVEXAWIYIYKYLWNXBTXCUVDYPUVIAUVGEUVDUVGRRXD
      ZPXGZEUVDUUNUVRUVGUXFFZUWSUVDUUDUUGUVRUWKUWLUWCWBUVSUVFUXEFUXGUUMUVERRXEU
      XEUVFPXHXFTWBUHXIYDUVGMZYPUVIVAUVDUXHYOUVHYMYDUVGYJKXJUQVPXKXLXMXNXOXOXPY
      AYGYCUPYBYAYFYCAEYAYDEFGBYEHIYFYCBYDXQYECBHUMXRXSVIXT $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Ordering on positive fractions in terms of existence of sum.
       (Contributed by Jim Kingdon, 30-Apr-2020.) $)
    ltexnqi $p |- ( A <Q B -> E. x e. Q. ( A +Q x ) = B ) $=
      ( cnq wcel wa cltq cv cplq co wceq wrex ltrelnq brel ltexnqq biimpd mpcom
      wbr ) BDECDEFZBCGRZBAHIJCKADLZBCDDGMNSTUAABCOPQ $.
  $}

  ${
    $d A x $.
    $( One-half of any positive fraction is a fraction.  (Contributed by Jim
       Kingdon, 23-Sep-2019.) $)
    halfnqq $p |- ( A e. Q. -> E. x e. Q. ( x +Q x ) = A ) $=
      ( cnq wcel c1q cplq co crq cmq wceq 1nq mp2an ax-mp distrnqg mp3an oveq1i
      addclnq oveq2i mulidnq 3eqtr3i cfv wrex recclnq recidnq oveq12i mulassnqg
      eqtri mulcomnqg eqtr3i syl2anc mp2b mp3an23 3eqtr3a mulclnq mpan2 oveq12d
      cv wb id eqeq1d adantl rspcedv mpd ) BCDZBEEFGZHUAZIGZVGFGZBJZAUQZVJFGZBJ
      ZACUBVDBVFVFFGZIGZBEIGVHBVMEBIVEVMIGZVFIGZVEVFIGZVMEVOVEVFIVOVQVQFGZVEVEC
      DZVFCDZVTVOVRJECDZWAVSKKEEQLZVSVTWBVEUCZMZWDVEVFVFNOVQEVQEFVSVQEJWBVEUDMZ
      WEUEUGPVMVQIGZVMEIGZVPVMVQEVMIWERVMVEIGZVFIGZWFVPVMCDZVSVTWIWFJVTVTWJWDWD
      VFVFQZLZWBWDVMVEVFUFOWHVOVFIWJVSWHVOJWLWBVMVEUHLPUIVSWJWGVMJWBVSVTVTWJWCW
      CWKUJVMSUKTWETRVDVTVTVNVHJWDWDBVFVFNULBSUMVDVLVIAVGCVDVTVGCDWDBVFUNUOVJVG
      JZVLVIURVDWMVKVHBWMVJVGVJVGFWMUSZWNUPUTVAVBVC $.

    $( One-half of any positive fraction exists.  Lemma for Proposition
       9-2.6(i) of [Gleason] p. 120.  (Contributed by NM, 16-Mar-1996.)
       (Revised by Mario Carneiro, 10-May-2013.) $)
    halfnq $p |- ( A e. Q. -> E. x ( x +Q x ) = A ) $=
      ( cnq wcel cv cplq co wceq wrex wex halfnqq rexex syl ) BCDAEZNFGBHZACIOA
      JABKOACLM $.
  $}

  ${
    $d x A $.
    $( There is no smallest positive fraction.  (Contributed by Jim Kingdon,
       24-Sep-2019.) $)
    nsmallnqq $p |- ( A e. Q. -> E. x e. Q. x <Q A ) $=
      ( cnq wcel cv cplq co wceq wrex cltq wbr halfnqq ltaddnq anidms syl5ibcom
      breq2 reximia syl ) BCDAEZSFGZBHZACISBJKZACIABLUAUBACSCDZSTJKZUAUBUCUDSSM
      NTBSJPOQR $.

    $( There is no smallest positive fraction.  (Contributed by NM,
       26-Apr-1996.)  (Revised by Mario Carneiro, 10-May-2013.) $)
    nsmallnq $p |- ( A e. Q. -> E. x x <Q A ) $=
      ( cnq wcel cv cltq wbr wrex wex nsmallnqq rexex syl ) BCDAEBFGZACHMAIABJM
      ACKL $.
  $}

  ${
    $d A x y $.
    $( There is a number which is less than half of any positive fraction.  The
       case where ` A ` is one is Lemma 11.4 of [BauerTaylor], p. 50, and they
       use the word "approximate half" for such a number (since there may be
       constructions, for some structures other than the rationals themselves,
       which rely on such an approximate half but do not require division by
       two as seen at ~ halfnqq ).  (Contributed by Jim Kingdon,
       25-Nov-2019.) $)
    subhalfnqq $p |- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A ) $=
      ( vy cnq wcel cv cplq co cltq wbr wa wex wrex halfnqq df-rex eximi sylibr
      wceq syl wb adantr ancli sylbi anbi2i exbii exdistr simprl simpll ltaddnq
      sylib syl2anc breq2 ad2antlr mpbid breq1 ad2antll mpbird jca exlimiv ) BD
      EZAFZDEZVAVAGHZBIJZKZALZVDADMUTCFZDEZVGVGGHZBRZKZVBVCVGRZKZKZALZCLZVFUTVK
      VMALZKZCLZVPUTVKVLADMZKZCLZVSUTVJCDMZWBCBNWCVKCLWBVJCDOVKWACVKVTVHVTVJAVG
      NUAUBPUCSWAVRCVTVQVKVLADOUDUEUJVKVMCAUFQVOVFCVNVEAVNVBVDVKVBVLUGVNVDVGBIJ
      ZVNVGVIIJZWDVNVHVHWEVHVJVMUHZWFVGVGUIUKVJWEWDTVHVMVIBVGIULUMUNVLVDWDTVKVB
      VCVGBIUOUPUQURPUSSVDADOQ $.
  $}

  ${
    $d A x y z $.  $d B x y z $.
    $( There exists a number between any two positive fractions.  Proposition
       9-2.6(i) of [Gleason] p. 120.  (Contributed by Jim Kingdon,
       24-Sep-2019.) $)
    ltbtwnnqq $p |- ( A <Q B <-> E. x e. Q. ( A <Q x /\ x <Q B ) ) $=
      ( vy vz cltq wbr cv wa cnq wrex wcel cplq wceq ltrelnq brel simpld sylan2
      co adantr ltexnqi wex nsmallnq ltaddnq ancoms ltanqi breq2 adantl addclnq
      wb mpbid breq1 anbi12d rspcedv mp2and 3impa 3coml 3expia exlimdv impancom
      syl5 rexlimdva sylc ltsonq sotri rexlimivw impbii ) BCFGZBAHZFGZVICFGZIZA
      JKZVHBJLZBDHZMSZCNZDJKVMVHVNCJLBCJJFOPQDBCUAVNVQVMDJVNVQVOJLZVMVREHZVOFGZ
      EUBVNVQIZVMEVOUCWAVTVMEVNVQVTVMVTVNVQVMVTVNVQVMVTVNIZVQIZBBVSMSZFGZWDCFGZ
      VMWBWEVQVNVTWEVTVNVSJLZWEVTWGVRVSVOJJFOPQZBVSUDRUETWCWDVPFGZWFWBWIVQVSVOB
      UFTVQWIWFUJWBVPCWDFUGUHUKWCVLWEWFIZAWDJWBWDJLZVQVNVTWKVTVNWGWKWHBVSUIRUET
      VIWDNZVLWJUJWCWLVJWEVKWFVIWDBFUGVIWDCFULUMUHUNUOUPUQURUSVAUTVBVCVLVHAJBVI
      CFJVDOVEVFVG $.

    $( There exists a number between any two positive fractions.  Proposition
       9-2.6(i) of [Gleason] p. 120.  (Contributed by NM, 17-Mar-1996.)
       (Revised by Mario Carneiro, 10-May-2013.) $)
    ltbtwnnq $p |- ( A <Q B <-> E. x ( A <Q x /\ x <Q B ) ) $=
      ( cv cltq wbr wa cnq wrex wcel wex df-rex ltbtwnnqq ltrelnq simprd adantr
      brel pm4.71ri exbii 3bitr4i ) BADZEFZUACEFZGZAHIUAHJZUDGZAKBCEFUDAKUDAHLA
      BCMUDUFAUDUEUBUEUCUBBHJUEBUAHHENQOPRST $.
  $}

  ${
    $d A w x z $.
    $( For any fraction, there is an integer that is greater than it.  This is
       also known as the "archimedean property".  (Contributed by Jim Kingdon,
       1-Dec-2019.) $)
    archnqq $p |- ( A e. Q. -> E. x e. N. A <Q [ <. x , 1o >. ] ~Q ) $=
      ( vz vw wcel cnpi wa ceq wceq c1o cltq wbr co adantr clti comu 1onn sylan
      com c0 cnq cv cop cec wex wrex nqpi cpli 1pi addclpi mpan2 cmi pinn nnacl
      coa sylancl nnm1 syl wss elni2 word nnord ordgt0ge1 biimpa sylbi nnaword1
      adantl wb simprbi sseldd nnmword mp3anl1 syl21anc mpbid eqsstrrd 0lt1o wi
      nna0 nnaordi mpi eqeltrrd mulclpi ltpiord syldan mulpiord addpiord oveq1d
      eqtrd eleq2d bitrd mpbird mulcompig breq2d jctir ordpipqqs mulidpi breq1d
      mpan mpdan breq1 opeq1 eceq1d rspcev syl2anc exlimivv ) BUAECUBZFEZDUBZFE
      ZGZBXFXHUCHUDZIZGZDUECUEBAUBZJUCZHUDZKLZAFUFZCDBUGXMXRCDXMXFJUHMZFEZBXSJU
      CZHUDZKLZXRXJXTXLXGXTXIXGJFEZXTUIXFJUJUKZNZNXMYCXKYBKLZXJYGXLXJYGXFXHXSUL
      MZOLZXJXFXSXHULMZOLZYIXJYKXFXFJUOMZXHPMZEZXJYLYMXFXJYLYLJPMZYMXJYLSEZYOYL
      IXGYPXIXGXFSEZJSEZYPXFUMZQXFJUNUPNZYLUQURXJJXHUSZYOYMUSZXIUUAXGXIXHSEZTXH
      EZGUUAXHUTUUCXHVAZUUDUUAXHVBUUEUUDUUAXHVCVDRVEVGXJUUCYPTYLEZUUAUUBVHZXIUU
      CXGXHUMVGYTXGUUFXIXGXFYLTXGYQYRXFYLUSYSQXFJVFUPXGYQTXFEXFUTVIVJNYRUUCYPUU
      FUUGQJXHYLVKVLVMVNVOXGXFYLEZXIXGYQUUHYSYQXFTUOMZXFYLXFVRYQTJEZUUIYLEZVPYR
      YQUUJUUKVQQTJXFVSWRVTWAURNVJXJYKXFYJEZYNXGXIYJFEZYKUULVHXGXTXIUUMYEXSXHWB
      RXFYJWCWDXJYJYMXFXJYJXSXHPMZYMXGXTXIYJUUNIYEXSXHWERXJXSYLXHPXGXSYLIZXIXGY
      DUUOUIXFJWFUKNWGWHWIWJWKXJYJYHXFOXGXTXIYJYHIYEXSXHWLRWMVNXJYGXFJULMZYHOLZ
      YIXJXTYDGYGUUQVHXJXTYDYFUIWNXFXHXSJWOWSXJUUPXFYHOXGUUPXFIXIXFWPNWQWJWKNXL
      YCYGVHXJBXKYBKWTVGWKXQYCAXSFXNXSIZXPYBBKUURXOYAHXNXSJXAXBWMXCXDXEUR $.
  $}

  ${
    $d A x $.  $d B x $.
    $( A version of the Archimedean property.  This variation is "stronger"
       than ~ archnqq in the sense that we provide an integer which is larger
       than a given rational ` A ` even after being multiplied by a second
       rational ` B ` .  (Contributed by Jim Kingdon, 30-Nov-2019.) $)
    prarloclemarch $p |- ( ( A e. Q. /\ B e. Q. ) ->
        E. x e. N. A <Q ( [ <. x , 1o >. ] ~Q .Q B ) ) $=
      ( cnq wcel wa cmq c1o ceq cltq wbr cnpi wrex mulclnq syl2anc syl3anc wceq
      co syl mulcomnqg crq cfv cv cop cec recclnq sylan2 archnqq simpll cxp cqs
      wb 1pi opelxpi mpan2 enqex ecelqsi df-nqqs eleqtrrdi adantl simplr ltmnqg
      c1q mulassnqg recidnq oveq2d mulidnq eqtrd breq12d bitrd biimprd reximdva
      3eqtrd mpd ) BDEZCDEZFZBCUAUBZGRZAUCZHUDZIUEZJKZALMZBWBCGRZJKZALMVQVSDEZW
      DVPVOVRDEZWGCUFZBVRNUGAVSUHSVQWCWFALVQVTLEZFZWFWCWKWFVRBGRZVRWEGRZJKZWCWK
      VOWEDEZWHWFWNULVOVPWJUIZWKWBDEZVPWOWJWQVQWJWBLLUJZIUKZDWJWAWREZWBWSEWJHLE
      WTUMVTHLLUNUOWRWAIUPUQSURUSUTZVOVPWJVAZWBCNOZWKVPWHXBWISZBWEVRVBPWKWLVSWM
      WBJWKWHVOWLVSQXDWPVRBTOWKWMWEVRGRZWBWKWHWOWMXEQXDXCVRWETOWKXEWBCVRGRZGRZW
      BVCGRZWBWKWQVPWHXEXGQXAXBXDWBCVRVDPWKVPXGXHQXBVPXFVCWBGCVEVFSWKWQXHWBQXAW
      BVGSVMVHVIVJVKVLVN $.
  $}

  ${
    $d A x z $.  $d B x z $.  $d C x z $.
    $( Like ~ prarloclemarch but the integer must be at least two, and there is
       also ` B ` added to the right hand side.  These details follow
       straightforwardly but are chosen to be helpful in the proof of
       ~ prarloc .  (Contributed by Jim Kingdon, 25-Nov-2019.) $)
    prarloclemarch2 $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
        E. x e. N. ( 1o <N x /\
          A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) ) $=
      ( cnq wcel c1o ceq cmq co cltq wbr clti wa cnpi c2o 1pi syl wceq c0 vz cv
      w3a cop cec cplq wrex prarloclemarch 3adant2 cpli coa com pinn csuc elexi
      sucid df-2o eleqtrri wss 2onn nnaword2 mpan sseld o1p1e2 addpiord addclpi
      mpi mp2an eqeltrri mpan2 eleqtrrd ltpiord mpbird adantl adantrr cmi 0lt1o
      wb nna0 con0 wi 1on onsuci ontr1 ax-mp nnaordi eqeltrrd mulidpi mulcompig
      mpdan eqtr3d 3brtr4d ordpipqqs mpanl2 mpanr2 cxp cqs opelxpi ecelqsi 3syl
      df-nqqs eleqtrrdi ltmnqg syl3an1 syl3an2 3anidm12 ancoms mulcomnqg sylan2
      enqex mpbid 3brtr3d 3ad2antl3 ltsonq ltrelnq sotri ex mpd mulclnq ltaddnq
      sylan simpl2 syl2anc breq2d simpr breq2 opeq1 eceq1d oveq1d oveq2d rspcev
      addcomnqg anbi12d mp2and rexlimddv ) BEFZCEFZDEFZUCZBUAUBZGUDZHUEZDIJZKLZ
      GAUBZMLZBCUUEGUDZHUEZDIJZUFJZKLZNZAOUGZUAOYPYRUUDUAOUGYQUABDUHUIYSYTOFZUU
      DNZNZGYTPUJJZMLZBCUUQGUDZHUEZDIJZUFJZKLZUUMYSUUNUURUUDUUNUURYSUUNUURGUUQF
      ZUUNGYTPUKJZUUQUUNYTULFZGUVEFZYTUMZUVFGPFUVGGGUNZPGGOQUOUPZUQURUVFPUVEGPU
      LFZUVFPUVEUSUTPYTVAVBVCVGRUUNPOFZUUQUVESGGUKJZPOVDGGUJJZUVMOGOFZUVOUVNUVM
      SQQGGVEVHUVOUVOUVNOFQQGGVFVHVIVIZYTPVEVJZVKUUNUUQOFZUURUVDVRZUUNUVLUVRUVP
      YTPVFVJZUVOUVRUVSQGUUQVLVBRVMVNVOUUPBUVACUFJZKLZUVCUUPBUVAKLZUVAUWAKLZUWB
      UUPUUCUVAKLZUWCYSUUNUWEUUDYRYPUUNUWEYQYRUUNNZDUUBIJZDUUTIJZUUCUVAKUWFUUBU
      UTKLZUWGUWHKLZUUNUWIYRUUNUWIYTGVPJZGUUQVPJZMLZUUNYTUUQUWKUWLMUUNYTUUQMLZY
      TUUQFZUUNYTUVEUUQUUNUVFYTUVEFUVHUVFYTTUKJZYTUVEYTVSUVFTPFZUWPUVEFZTUVIPTG
      FZGUVIFZTUVIFZVQUVJUVIVTFUWSUWTNUXAWAGWBWCTGUVIWDWEVHUQURUVKUVFUWQUWRWAUT
      TPYTWFVBVGWGRUVQVKUUNUVRUWNUWOVRUVTYTUUQVLWJVMYTWHUUNUUQGVPJZUWLUUQUUNUVR
      UXBUWLSZUVTUVRUVOUXCQUUQGWIVJRUUNUVRUXBUUQSUVTUUQWHRWKWLUUNUVRUWIUWMVRZUV
      TUUNUVRUVOUXDQUUNUVOUVRUVONUXDQYTGUUQGWMWNWOWJVMVNUUNYRUWIUWJVRZUUNYRUXEU
      UNUUNUUTEFZYRUXEUUNUUTOOWPZHWQZEUUNUVRUUSUXGFZUUTUXHFUVTUVRUVOUXIQUUQGOOW
      RVJUXGUUSHXJWSWTXAXBZUUNUUBEFZUXFYRUXEUUNUUBUXHEUUNUUAUXGFZUUBUXHFUUNUVOU
      XLQYTGOOWRVJUXGUUAHXJWSRXAXBZUUBUUTDXCXDXEXFXGXKUUNYRUXKUWGUUCSUXMDUUBXHX
      IUUNYRUXFUWHUVASUXJDUUTXHXIXLXMVOUUOUWEUWCWAZYSUUDUXNUUNUUDUWEUWCBUUCUVAK
      EXNXOXPXQVNVNXRYSUUNUWDUUDYSUUNNZUVAEFZYQUWDYRYPUUNUXPYQUUNYRUXPUUNUXFYRU
      XPUXJUUTDXSYAXGXMZYPYQYRUUNYBZUVACXTYCVOBUVAUWAKEXNXOXPYCYSUUNUWBUVCVRUUD
      UXOUWAUVBBKUXOUXPYQUWAUVBSUXQUXRUVACYLYCYDVOXKYSUUNUURUVCNZUUMWAZUUDUXOUU
      NUVRUXTYSUUNYEUVTUVRUXSUUMUULUXSAUUQOUUEUUQSZUUFUURUUKUVCUUEUUQGMYFUYAUUJ
      UVBBKUYAUUIUVACUFUYAUUHUUTDIUYAUUGUUSHUUEUUQGYGYHYIYJYDYMYKXQWTVOYNYO $.
  $}

  $( Ordering property of reciprocal for positive fractions.  For a simplified
     version of the forward implication, see ~ ltrnqi .  (Contributed by Jim
     Kingdon, 29-Dec-2019.) $)
  ltrnqg $p |- ( ( A e. Q. /\ B e. Q. ) ->
      ( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) ) ) $=
    ( cnq wcel cltq wbr crq cfv cmq co recclnq mulclnq wceq mulcomnqg mulassnqg
    syl2anc syl3anc c1q recidnq oveq2d wa wb syl2an ltmnqg mpd3an3 simpl adantr
    adantl 3eqtr2d mulidnq sylan9eq eqtrd simpr sylan9eqr 3eqtrd breq12d bitrd
    syl ) ACDZBCDZUAZABEFZAGHZBGHZIJZAIJZVEBIJZEFZVDVCEFUSUTVECDZVBVHUBUSVCCDZV
    DCDZVIUTAKZBKZVCVDLUCZABVEUDUEVAVFVDVGVCEVAVFVDAVCIJZIJZVDVAVFAVEIJZVOVDIJZ
    VPVAVIUSVFVQMVNUSUTUFZVEANPVAUSVJVKVRVQMVSUSVJUTVLUGZUTVKUSVMUHZAVCVDOQVAVO
    CDZVKVRVPMVAUSVJWBVSVTAVCLPWAVOVDNPUIUSUTVPVDRIJZVDUSVORVDIASTUTVKWCVDMVMVD
    UJURUKULVAVGVCVDBIJZIJZVCBVDIJZIJZVCVAVJVKUTVGWEMVTWAUSUTUMZVCVDBOQVAWDWFVC
    IVAVKUTWDWFMWAWHVDBNPTUTUSWGVCRIJZVCUTWFRVCIBSTUSVJWIVCMVLVCUJURUNUOUPUQ $.

  $( Ordering property of reciprocal for positive fractions.  For the converse,
     see ~ ltrnqg .  (Contributed by Jim Kingdon, 24-Sep-2019.) $)
  ltrnqi $p |- ( A <Q B -> ( *Q ` B ) <Q ( *Q ` A ) ) $=
    ( cltq wbr crq cfv cnq wcel wa wb ltrelnq brel ltrnqg syl ibi ) ABCDZBEFAEF
    CDZPAGHBGHIPQJABGGCKLABMNO $.

  $( The canonical embedding of positive integers into positive fractions.
     (Contributed by Jim Kingdon, 26-Apr-2020.) $)
  nnnq $p |- ( A e. N. -> [ <. A , 1o >. ] ~Q e. Q. ) $=
    ( cnpi wcel c1o cop ceq cec cxp cqs cnq 1pi opelxpi mpan2 enqex ecelqsi syl
    df-nqqs eleqtrrdi ) ABCZADEZFGZBBHZFIZJSTUBCZUAUCCSDBCUDKADBBLMUBTFNOPQR $.

  $( Ordering of positive integers via ` <N ` or ` <Q ` is equivalent.
     (Contributed by Jim Kingdon, 3-Oct-2020.) $)
  ltnnnq $p |- ( ( A e. N. /\ B e. N. ) -> ( A <N B <->
      [ <. A , 1o >. ] ~Q <Q [ <. B , 1o >. ] ~Q ) ) $=
    ( cnpi wcel wa c1o cop ceq cec cltq wbr cmi co clti wb 1pi wceq mulidpi syl
    simpl a1i simpr ordpipqqs syl22anc mulcompig sylancr eqtrd breq12d bitr2d )
    ACDZBCDZEZAFGHIBFGHIJKZAFLMZFBLMZNKZABNKULUJFCDZUKUQUMUPOUJUKTZUQULPUAZUJUK
    UBZUSAFBFUCUDULUNAUOBNULUJUNAQURARSULUOBFLMZBULUQUKUOVAQPUTFBUEUFULUKVABQUT
    BRSUGUHUI $.

  ${
    $d x y z w v u f $.
    $( Define equivalence relation for nonnegative fractions.  This is a
       "temporary" set used in the construction of complex numbers, and is
       intended to be used only by the construction.  (Contributed by Jim
       Kingdon, 2-Nov-2019.) $)
    df-enq0 $a |- ~Q0 = { <. x , y >. | ( ( x e. ( _om X. N. ) /\
                   y e. ( _om X. N. ) ) /\ E. z E. w E. v E. u
                   ( ( x = <. z , w >. /\ y = <. v , u >. ) /\
                   ( z .o u ) = ( w .o v ) ) ) } $.

    $( Define class of nonnegative fractions.  This is a "temporary" set used
       in the construction of complex numbers, and is intended to be used only
       by the construction.  (Contributed by Jim Kingdon, 2-Nov-2019.) $)
    df-nq0 $a |- Q0. = ( ( _om X. N. ) /. ~Q0 ) $.

    $( Define nonnegative fraction constant 0.  This is a "temporary" set used
       in the construction of complex numbers, and is intended to be used only
       by the construction.  (Contributed by Jim Kingdon, 5-Nov-2019.) $)
    df-0nq0 $a |- 0Q0 = [ <. (/) , 1o >. ] ~Q0 $.

    $( Define addition on nonnegative fractions.  This is a "temporary" set
       used in the construction of complex numbers, and is intended to be used
       only by the construction.  (Contributed by Jim Kingdon, 2-Nov-2019.) $)
    df-plq0 $a |- +Q0 = { <. <. x , y >. , z >. | ( ( x e. Q0. /\ y e. Q0. ) /\
      E. w E. v E. u E. f (
         ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\
           z = [ <. ( ( w .o f ) +o ( v .o u ) ) ,
             ( v .o f ) >. ] ~Q0 ) ) } $.

    $( Define multiplication on nonnegative fractions.  This is a "temporary"
       set used in the construction of complex numbers, and is intended to be
       used only by the construction.  (Contributed by Jim Kingdon,
       2-Nov-2019.) $)
    df-mq0 $a |- .Q0 = { <. <. x , y >. , z >. | ( ( x e. Q0. /\ y e. Q0. ) /\
      E. w E. v E. u E. f (
         ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\
               z = [ <. ( w .o u ) , ( v .o f ) >. ] ~Q0 ) ) } $.
  $}

  ${
    $d x y z w v u f $.
    $( Multiplication on nonnegative fractions.  This definition is similar to
       ~ df-mq0 but expands ` Q0. ` .  (Contributed by Jim Kingdon,
       22-Nov-2019.) $)
    dfmq0qs $p |- .Q0 = { <. <. x , y >. , z >. |
      ( ( x e. ( ( _om X. N. ) /. ~Q0 ) /\ y e. ( ( _om X. N. ) /. ~Q0 ) ) /\
      E. w E. v E. u E. f (
         ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\
               z = [ <. ( w .o u ) , ( v .o f ) >. ] ~Q0 ) ) } $=
      ( cv cnq0 wcel wa cop ceq0 cec wceq comu co wex coprab df-nq0 com cxp cqs
      cmq0 cnpi df-mq0 eleq2i anbi12i anbi1i oprabbii eqtri ) UDAHZIJZBHZIJZKZU
      LDHZEHZLMNOUNFHZGHZLMNOKCHUQUSPQURUTPQLMNOKGRFRERDRZKZABCSULUAUEUBMUCZJZU
      NVCJZKZVAKZABCSABCDEFGUFVBVGABCUPVFVAUMVDUOVEIVCULTUGIVCUNTUGUHUIUJUK $.

    $( Addition on nonnegative fractions.  This definition is similar to
       ~ df-plq0 but expands ` Q0. ` .  (Contributed by Jim Kingdon,
       24-Nov-2019.) $)
    dfplq0qs $p |- +Q0 = { <. <. x , y >. , z >. |
      ( ( x e. ( ( _om X. N. ) /. ~Q0 ) /\ y e. ( ( _om X. N. ) /. ~Q0 ) ) /\
      E. w E. v E. u E. f (
         ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\
           z = [ <. ( ( w .o f ) +o ( v .o u ) ) ,
             ( v .o f ) >. ] ~Q0 ) ) } $=
      ( cv cnq0 wcel wa cop ceq0 cec wceq comu co wex coprab df-nq0 coa com cxp
      cplq0 cnpi cqs df-plq0 eleq2i anbi12i anbi1i oprabbii eqtri ) UDAHZIJZBHZ
      IJZKZUMDHZEHZLMNOUOFHZGHZLMNOKCHURVAPQUSUTPQUAQUSVAPQLMNOKGRFRERDRZKZABCS
      UMUBUEUCMUFZJZUOVDJZKZVBKZABCSABCDEFGUGVCVHABCUQVGVBUNVEUPVFIVDUMTUHIVDUO
      TUHUIUJUKUL $.
  $}

  ${
    $d u v w x y z $.
    $( Equivalence on positive fractions in terms of equivalence on nonnegative
       fractions.  (Contributed by Jim Kingdon, 12-Nov-2019.) $)
    enq0enq $p |- ~Q = ( ~Q0 i^i ( ( N. X. N. ) X. ( N. X. N. ) ) ) $=
      ( vx vy vz vw vv vu cnpi cv com wcel wa wceq co copab bitr4i anbi1i anass
      wex 3bitr3i 19.42vv ceq0 cxp cin cop ceq df-enq0 df-xp ineq12i inopab cmi
      comu an32 an4 wss pinn ssriv xpss1 ax-mp sseli anbi12i bitri eleq1 opelxp
      pm4.71ri bitrdi bi2anan9 pm5.32i wb mulpiord eqeqan12d an42s anbi2i ancom
      2exbii opabbii df-enq eqtr4i 3eqtrri ) UAGGUBZVSUBZUCAHZIGUBZJZBHZWBJZKZW
      ACHZDHZUDZLZWDEHZFHZUDZLZKZWGWLUKMZWHWKUKMZLZKZFRERZDRCRZKZABNZWAVSJZWDVS
      JZKZABNZUCXBXFKZABNZUEUAXCVTXGABCDEFUFABVSVSUGUHXBXFABUIXIXFWOWGWLUJMZWHW
      KUJMZLZKZFRERZDRCRKZABNUEXHXOABXHXFXAKZXOXHWFXFKZXAKXPWFXAXFULXQXFXAXQWCX
      DKZWEXEKZKXFWCWEXDXEUMXDXRXEXSXDWCVSWBWAGIUNVSWBUNAGIWAUOUPGIGUQURZUSVDXE
      WEVSWBWDXTUSVDUTOPVAXFWTKZDRCRXFXNKZDRCRXPXOYAYBCDXFWSKZFRERXFXMKZFRERYAY
      BYCYDEFXFWOKZWRKZYEXLKZYCYDWOXFKZWRKZYHXLKZYFYGYIWOWGGJZWHGJZKZWKGJZWLGJZ
      KZKZKZXLKZYJYIWOYQXLKZKZYSYIWOYQWRKZKZUUAYIYRWRKUUCYHYRWRWOXFYQWJXDYMWNXE
      YPWJXDWIVSJYMWAWIVSVBWGWHGGVCVEWNXEWMVSJYPWDWMVSVBWKWLGGVCVEVFVGZPWOYQWRQ
      VAYTUUBWOYQXLWRYKYOYLYNXLWRVHYKYOKYLYNKXJWPXKWQWGWLVIWHWKVIVJVKVGVLOWOYQX
      LQOYHYRXLUUDPOYHYEWRWOXFVMZPYHYEXLUUEPSXFWOWRQXFWOXLQSVNXFWSEFTXFXMEFTSVN
      XFWTCDTXFXNCDTSVAVOABCDEFVPVQVR $.
  $}

  ${
    $d f a b c d u v w x y z $.  $d g a b c d u v w x y z $.
    $( The equivalence relation for nonnegative fractions is symmetric.  Lemma
       for ~ enq0er .  (Contributed by Jim Kingdon, 14-Nov-2019.) $)
    enq0sym $p |- ( f ~Q0 g -> g ~Q0 f ) $=
      ( vc vd va vb vz vw vv vu cv com wcel wa wceq comu co wex anbi1d anbi12d
      vx vy ceq0 wbr cnpi cxp cop vex eleq1 eqeq1 4exbidv anbi2d df-enq0 biimpi
      brab opeq12 eqeq2d simpl oveq1d simpr eqeq12d oveq2d cbvex4v anbi2i sylib
      19.42vv sylibr 2exbii pm3.22 adantr ad2antrl simprr opelxp bitrdi biimpcd
      wb im2anan9 adantrr pinn nnmcom sylan2 sylan eqeqan12d an42s mpbid eqcomd
      imp syl jca32 2eximi exrot4 biimpri sylan2br sylbi ) AKZBKZUCUDZWPLUEUFZM
      ZWOWRMZNZWPCKZDKZUGZOZWOEKZFKZUGZOZNZXBXGPQZXCXFPQZOZNZNZDRCRZFRERZWPWOUC
      UDZWQWTWSNZXIXENZXFXCPQZXGXBPQZOZNZNZDRCRZFRERZXQWQXSYDDRCRZNZFRERZYGWQXS
      YHFRERZNZYJWQXSWOGKZHKZUGZOZWPIKZJKZUGZOZNZYMYRPQZYNYQPQZOZNZJRIRHRGRZNZY
      LWQUUGUAKZWRMZUBKZWRMZNZUUHYOOZUUJYSOZNZUUDNZJRIRHRGRZNZWTUUKNZYPUUNNZUUD
      NZJRIRHRGRZNUUGUAUBWOWPUCAUHZBUHZUUHWOOZUULUUSUUQUVBUVEUUIWTUUKUUHWOWRUIS
      UVEUUPUVAGHIJUVEUUOUUTUUDUVEUUMYPUUNUUHWOYOUJSSUKTUUJWPOZUUSXSUVBUUFUVFUU
      KWSWTUUJWPWRUIULUVFUVAUUEGHIJUVFUUTUUAUUDUVFUUNYTYPUUJWPYSUJULSUKTUAUBGHI
      JUMZUOUNUUFYKXSUUEXIYTNZXFYRPQZXGYQPQZOZNYDGHIJEFCDYMXFOZYNXGOZNZUUAUVHUU
      DUVKUVNYPXIYTUVNYOXHWOYMYNXFXGUPUQSUVNUUBUVIUUCUVJUVNYMXFYRPUVLUVMURUSUVN
      YNXGYQPUVLUVMUTUSVATYQXBOZYRXCOZNZUVHXTUVKYCUVQYTXEXIUVQYSXDWPYQYRXBXCUPU
      QULUVQUVIYAUVJYBUVQYRXCXFPUVOUVPUTVBUVQYQXBXGPUVOUVPURVBVATVCVDVEXSYHEFVF
      VGYFYIEFXSYDCDVFVHVGYFXPEFYEXOCDYEXAXJXMXSXAYDWTWSVIVJXTXJXSYCXIXEVIVKYEX
      LXKYEYCXLXKOZXSXTYCVLYEXFLMZXGUEMZNZXBLMZXCUEMZNZNZYCUVRVPZXSXTUWEYCXSXTU
      WEWTXIUWAWSXEUWDXIWTUWAXIWTXHWRMUWAWOXHWRUIXFXGLUEVMVNVOXEWSUWDXEWSXDWRMU
      WDWPXDWRUIXBXCLUEVMVNVOVQWGVRUVSUWCUVTUWBUWFUVSUWCNUVTUWBNYAXLYBXKUWCUVSX
      CLMYAXLOXCVSXFXCVTWAUVTXGLMUWBYBXKOXGVSXGXBVTWBWCWDWHWEWFWIWJWJWHXQXOFRER
      ZDRCRZXRXOEFCDWKUWHXAXNFRERZNZDRCRZXRUWGUWJCDXAXNEFVFVHUWKXAUWIDRCRZNXRXA
      UWICDVFUWLXAWPYOOZWOYSOZNZUUDNZJRIRHRGRZXRUWPXEUWNNZXBYRPQZXCYQPQZOZNXNGH
      IJCDEFYMXBOZYNXCOZNZUWOUWRUUDUXAUXDUWMXEUWNUXDYOXDWPYMYNXBXCUPUQSUXDUUBUW
      SUUCUWTUXDYMXBYRPUXBUXCURUSUXDYNXCYQPUXBUXCUTUSVATYQXFOZYRXGOZNZUWRXJUXAX
      MUXGUWNXIXEUXGYSXHWOYQYRXFXGUPUQULUXGUWSXKUWTXLUXGYRXGXBPUXEUXFUTVBUXGYQX
      FXCPUXEUXFURVBVATVCXRXAUWQNZUURWSUUKNZUWMUUNNZUUDNZJRIRHRGRZNUXHUAUBWPWOU
      CUVDUVCUUHWPOZUULUXIUUQUXLUXMUUIWSUUKUUHWPWRUISUXMUUPUXKGHIJUXMUUOUXJUUDU
      XMUUMUWMUUNUUHWPYOUJSSUKTUUJWOOZUXIXAUXLUWQUXNUUKWTWSUUJWOWRUIULUXNUXKUWP
      GHIJUXNUXJUWOUUDUXNUUNUWNUWMUUJWOYSUJULSUKTUVGUOWLWMWNWNWNWH $.
  $}

  ${
    $d f u v w x y z $.
    $( The equivalence relation for nonnegative fractions is reflexive.  Lemma
       for ~ enq0er .  (Contributed by Jim Kingdon, 14-Nov-2019.) $)
    enq0ref $p |- ( f e. ( _om X. N. ) <-> f ~Q0 f ) $=
      ( vz vw vv vu vx vy cv com cnpi wcel ceq0 wceq comu wex vex anbi2d anbi1d
      wa co cxp wbr elxpi ee4anv sylanbrc eqtr2 opth sylib oveq1 oveq2 sylan9eq
      cop equcoms syl ancli ad2ant2r pinn nnmcom sylan2 eqeq2d ancoms ad2ant2lr
      wb ad2ant2l mpbid 2eximi eleq1 eqeq1 4exbidv anbi12d df-enq0 anidm anbi1i
      brab bitri sylibr simplbi impbii ) AHZIJUAZKZVSVSLUBZWAWAVSBHZCHZULZMZVSD
      HZEHZULZMZSZWCWHNTZWDWGNTZMZSZEODOZCOBOZSZWBWAWQWAWFWCIKZWDJKZSZSZWJWGIKZ
      WHJKZSZSZSZEODOZCOBOZWQWAXBCOBOXFEODOXIBCVSIJUCDEVSIJUCXBXFBCDEUDUEXHWPBC
      XGWODEXGWKWLWGWDNTZMZSZWOWFWJXLXAXEWKXKWKWCWGMZWDWHMZSZXKWKWEWIMXOVSWEWIU
      FWCWDWGWHBPCPUGUHXMXNWLWGWHNTZXJWCWGWHNUIXPXJMECWHWDWGNUJUMUKUNUOUPXGXKWN
      WKXAXEXKWNVCZWFWJWTXCXQWSXDXCWTXQXCWTSXJWMWLWTXCWDIKXJWMMWDUQWGWDURUSUTVA
      VBVDQVEVFVFUNUOWBWAWASZWQSZWRFHZVTKZGHZVTKZSZXTWEMZYBWIMZSZWNSZEODOCOBOZS
      WAYCSZWFYFSZWNSZEODOCOBOZSXSFGVSVSLAPZYNXTVSMZYDYJYIYMYOYAWAYCXTVSVTVGRYO
      YHYLBCDEYOYGYKWNYOYEWFYFXTVSWEVHRRVIVJYBVSMZYJXRYMWQYPYCWAWAYBVSVTVGQYPYL
      WOBCDEYPYKWKWNYPYFWJWFYBVSWIVHQRVIVJFGBCDEVKVNXRWAWQWAVLVMVOZVPWBWAWQYQVQ
      VR $.
  $}

  ${
    $d a b c d e f s t u v w x y z $.  $d a b c d e g s t u v w x y z $.
    $d a b c d e h s t u v w x y z $.
    $( The equivalence relation for nonnegative fractions is transitive.  Lemma
       for ~ enq0er .  (Contributed by Jim Kingdon, 14-Nov-2019.) $)
    enq0tr $p |- ( ( f ~Q0 g /\ g ~Q0 h ) -> f ~Q0 h ) $=
      ( vz vw vs vt vv vu va vb cv wa com wcel wceq comu co wex c0 vx vy vc wbr
      vd ve ceq0 cnpi cxp cop w3a vex eleq1 anbi1d eqeq1 4exbidv anbi12d anbi2d
      df-enq0 anbi12i biimpi an4 sylib 3anass anass anbi2i anidm anbi1i 3bitr2i
      bitr4i sylibr ee8anv 19.42vvvv 2exbii bitri 3simpb adantr simplll simprlr
      brab jca adantl oveq1 simpl3 eqeltrrd opelxp simprd pinn syl nnm0r eqeq2d
      wo imbitrid wb simprr eqtr2 opth eqeqan12d ad2ant2lr mpbird eqeq1d sylibd
      ad2ant2r simpllr simpl2 simpld nnm00 syl2anc elni2 simprbi n0i biorf 3syl
      wn sylibrd simpl1 nnm0 oveq2 syl5ibrcom syld orcom bitrdi jcad eqtr3 syl6
      eqcomd simplr oveq12d nnmcl nnmcom nnmass caov13d caovassd 3eqtr3d nnmcan
      syl31anc mpbid caov12d 2eximi exlimivv 3jca sylan ex 0elnn mpjaod jca32 )
      ALZBLZUGUDZUUHCLZUGUDZMZUUGNUHUIZOZUUJUUMOZMZUUGDLZELZUJZPZUUJFLZGLZUJZPZ
      MZUUQUVBQRZUURUVAQRZPZMZMZGSFSZESDSZUUGUUJUGUDZUULUUNUUHUUMOZUUOUKZUUTUUH
      HLZILZUJZPZMZUUQUVQQRZUURUVPQRZPZMZUUHJLZKLZUJZPZUVDMZUWEUVBQRZUWFUVAQRZP
      ZMZMZMZGSFSZKSJSZISHSZESDSZUVLUULUVOUWNGSFSKSJSZISHSESDSZMZUWSUULUVOUWDIS
      HSESDSZUWMGSFSKSJSZMZMZUXBUULUUNUVNMZUVNUUOMZMZUXEMZUXFUULUXGUXCMZUXHUXDM
      ZMZUXJUULUXMUUIUXKUUKUXLUALZUUMOZUBLZUUMOZMZUXNUUSPZUXPUVRPZMZUWCMZISHSES
      DSZMUUNUXQMZUUTUXTMZUWCMZISHSESDSZMUXKUAUBUUGUUHUGAULZBULZUXNUUGPZUXRUYDU
      YCUYGUYJUXOUUNUXQUXNUUGUUMUMUNZUYJUYBUYFDEHIUYJUYAUYEUWCUYJUXSUUTUXTUXNUU
      GUUSUOZUNUNUPUQUXPUUHPZUYDUXGUYGUXCUYMUXQUVNUUNUXPUUHUUMUMURUYMUYFUWDDEHI
      UYMUYEUVTUWCUYMUXTUVSUUTUXPUUHUVRUOURUNUPUQUAUBDEHIUSVTUXRUXNUWGPZUXPUVCP
      ZMZUWLMZGSFSKSJSZMUVNUXQMZUWHUYOMZUWLMZGSFSKSJSZMUXLUAUBUUHUUJUGUYICULZUX
      NUUHPZUXRUYSUYRVUBVUDUXOUVNUXQUXNUUHUUMUMUNVUDUYQVUAJKFGVUDUYPUYTUWLVUDUY
      NUWHUYOUXNUUHUWGUOUNUNUPUQUXPUUJPZUYSUXHVUBUXDVUEUXQUUOUVNUXPUUJUUMUMZURV
      UEVUAUWMJKFGVUEUYTUWIUWLVUEUYOUVDUWHUXPUUJUVCUOZURUNUPUQUAUBJKFGUSVTUTVAU
      XGUXCUXHUXDVBVCUVOUXIUXEUVOUUNUXHMZUXIUUNUVNUUOVDUXIUUNUVNUXHMZMUUNUVNUVN
      MZUUOMZMVUHUUNUVNUXHVEVUKVUIUUNUVNUVNUUOVEVFVUKUXHUUNVUJUVNUUOUVNVGVHVFVI
      VJVHVKUXAUXEUVOUWDUWMDEHIJKFGVLVFVKUWSUVOUWTMZISHSZESDSUXBUWRVUMDEUWQVULH
      IUVOUWNKFGJVMVNVNUVOUWTEHIDVMVOVKUWRUVKDEUWQUVKHIUWPUVKJKUWOUVJFGUWOUUPUV
      EUVHUVOUUPUWNUUNUVNUUOVPVQUWNUVEUVOUWNUUTUVDUUTUVSUWCUWMVRZUWDUWHUVDUWLVS
      ZWAWBUWOUVPTPZUVHTUVPOZUWOVUPUVGTPZUVFTPZMZUVHUWOVUPVURVUSUWOVUPUVATPZVUR
      UWOVUPUVQTPZVVAWLZVVAUWOVUPUVQUVAQRZTPZVVCUWOVUPUVPUVBQRZTPZVVEVUPVVFTUVB
      QRZPUWOVVGUVPTUVBQWCUWOVVHTVVFUWOUVBNOZVVHTPZUWOUVBUHOZVVIUWOUVANOZVVKUWO
      UVCUUMOVVLVVKMUWOUUJUVCUUMUWNUVDUVOVUOWBUUNUVNUUOUWNWDWEUVAUVBNUHWFVCZWGU
      VBWHWIZUVBWJWIZWKWMUWNVVGVVEWNUVOUWNVVFVVDTUWNVVFVVDPZUWLUWDUWIUWLWOUVTUW
      IVVPUWLWNZUWCUWLUVSUWHVVQUUTUVDUVSUWHMZUVPUWEPZUVQUWFPZMZVVQVVRUVRUWGPVWA
      UUHUVRUWGWPUVPUVQUWEUWFHULIULWQVCVVSVVTVVFUWJVVDUWKUVPUWEUVBQWCUVQUWFUVAQ
      WCWRWIWSXCWTZXAWBXBUWOUVQNOZVVLVVEVVCWNUWOUVQUHOZVWCUWOUVPNOZVWDUWOUVRUUM
      OVWEVWDMUWOUUHUVRUUMUWNUVSUVOUUTUVSUWCUWMXDWBUUNUVNUUOUWNXEWEUVPUVQNUHWFV
      CZWGZUVQWHWIZUWOVVLVVKVVMXFZUVQUVAXGXHXBUWOTUVQOZVVBXNZVVAVVCWNUWOVWDVWJV
      WGVWDVWCVWJUVQXIXJWIZUVQTXKZVVBVVAXLXMXOUWOVURVVAUURTQRZTPZUWOUURNOZVWOUW
      OUURUHOZVWPUWOUUQNOZVWQUWOUUSUUMOVWRVWQMUWOUUGUUSUUMUWNUUTUVOVUNWBUUNUVNU
      UOUWNXPWEUUQUURNUHWFVCZWGUURWHWIZUURXQWIZVVAUVGVWNTUVATUURQXRXAXSXTUWOVUP
      UUQTPZVUSUWOVUPVXBVVBWLZVXBUWOVUPUWATPZVXCUWOVUPUWBTPZVXDVUPUWBVWNPUWOVXE
      UVPTUURQXRUWOVWNTUWBVXAWKWMUWOUWAUWBTUVOUVTUWCUWMVSXAXOUWOVWRVWCVXDVXCWNU
      WOVWRVWQVWSXFZVWHUUQUVQXGXHXBUWOVWJVWKVXBVXCWNVWLVWMVWKVXBVVBVXBWLVXCVVBV
      XBXLVVBVXBYAYBXMXOUWOVUSVXBVVJVVOVXBUVFVVHTUUQTUVBQWCXAXSXTYCVUTUVGUVFUVG
      UVFTYDYFYEUWOVUQUVHUWOVUQMUVPUVFQRZUVPUVGQRZPZUVHUWOVXIVUQUWOUUQVVFQRZUVA
      UWBQRZVXGVXHUWOUVQVXJQRZUVQVXKQRZPZVXJVXKPZUWOVVFUWAQRZVVDUWBQRZVXLVXMUWN
      VXPVXQPUVOUWNVVFVVDUWAUWBQVWBUVTUWCUWMYGYHWBUWOUCUEUFVVFUUQUVQNQUWOVWEVVI
      VVFNOZUWOVWEVWDVWFXFZVVNUVPUVBYIXHZVXFVWHUCLZNOZUELZNOZMVYAVYCQRZVYCVYAQR
      PUWOVYAVYCYJWBZVYBVYDUFLZNOUKVYEVYGQRVYAVYCVYGQRQRPUWOVYAVYCVYGYKWBZYLUWO
      UCUEUFUVQUVAUWBNQVYHVWHVWIUWOVWPVWEUWBNOZVWTVXSUURUVPYIXHZYMYNUWOVWCVXJNO
      ZVXKNOZVWJVXNVXOWNVWHUWOVWRVXRVYKVXFVXTUUQVVFYIXHUWOVVLVYIVYLVWIVYJUVAUWB
      YIXHVWLUVQVXJVXKYOYPYQUWOUCUEUFUUQUVPUVBNQVXFVXSVVNVYFVYHYRUWOUCUEUFUVAUU
      RUVPNQVWIVWTVXSVYFVYHYLYNVQUWOVWEUVFNOZUVGNOZUKVUQVXIUVHWNUWOVWEVYMVYNVXS
      UWOVWRVVIVYMVXFVVNUUQUVBYIXHUWOVWPVVLVYNVWTVWIUURUVAYIXHUUAUVPUVFUVGYOUUB
      YQUUCUWOVWEVUPVUQWLVXSUVPUUDWIUUEUUFYSYTYTYSWIUVLUUPUVIGSFSESDSZMZUVMUUPU
      VIEFGDVMUXRUXSUYOMZUVHMZGSFSESDSZMUYDUUTUYOMZUVHMZGSFSESDSZMVYPUAUBUUGUUJ
      UGUYHVUCUYJUXRUYDVYSWUBUYKUYJVYRWUADEFGUYJVYQVYTUVHUYJUXSUUTUYOUYLUNUNUPU
      QVUEUYDUUPWUBVYOVUEUXQUUOUUNVUFURVUEWUAUVIDEFGVUEVYTUVEUVHVUEUYOUVDUUTVUG
      URUNUPUQUAUBDEFGUSVTVJVC $.
  $}

  ${
    $d f g h x y z w v u $.
    $( The equivalence relation for nonnegative fractions is an equivalence
       relation.  (Contributed by Jim Kingdon, 12-Nov-2019.) $)
    enq0er $p |- ~Q0 Er ( _om X. N. ) $=
      ( vf vg vh vx vy vz vw vv vu ceq0 wtru cv wcel wa cop wceq comu wex wbr
      co com cnpi cxp wer df-enq0 relopabi a1i enq0sym adantl enq0tr wb enq0ref
      wrel iserd mptru ) UAUBUCZJUDKABCUPJJUMKDLZUPMELZUPMNUQFLZGLZOPURHLZILZOP
      NUSVBQTUTVAQTPNIRHRGRFRNDEJDEFGHIUEUFUGALZBLZJSZVDVCJSKABUHUIVEVDCLZJSNVC
      VFJSKABCUJUIVCUPMVCVCJSUKKAULUGUNUO $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u D $.
    $( Equivalence relation for nonnegative fractions in terms of natural
       numbers.  (Contributed by NM, 27-Aug-1995.) $)
    enq0breq $p |- ( ( ( A e. _om /\ B e. N. ) /\ ( C e. _om /\ D e. N. ) ) ->
          ( <. A , B >. ~Q0 <. C , D >. <-> ( A .o D ) = ( B .o C ) ) ) $=
      ( vz vw vv vu com wcel cnpi wa cop cv wceq comu co wex cvv anbi1d vx ceq0
      vy cxp wbr wb oveq12 eqeqan12d an42s copsex4g opexg eleq1 4exbidv anbi12d
      anbi2d eqeq1 df-enq0 brabg syl2an opelxpi anim12i biantrurd 3bitr4d ) AIJ
      BKJLZCIJDKJLZLZABMZIKUDZJZCDMZVHJZLZVGENZFNZMZOZVJGNZHNZMZOZLZVMVRPQZVNVQ
      PQZOZLZHRGRFRERZLZVLADPQZBCPQZOZLVGVJUBUEZWJVFWFWJVLWDWJEFGHABCDIKVMAOZVR
      DOZVNBOZVQCOZWDWJUFWLWMLWNWOLWBWHWCWIVMAVRDPUGVNBVQCPUGUHUIUJUOVDVGSJVJSJ
      WKWGUFVEABIKUKCDIKUKUANZVHJZUCNZVHJZLZWPVOOZWRVSOZLZWDLZHRGRFRERZLVIWSLZV
      PXBLZWDLZHRGRFRERZLWGUAUCVGVJSSUBWPVGOZWTXFXEXIXJWQVIWSWPVGVHULTXJXDXHEFG
      HXJXCXGWDXJXAVPXBWPVGVOUPTTUMUNWRVJOZXFVLXIWFXKWSVKVIWRVJVHULUOXKXHWEEFGH
      XKXGWAWDXKXBVTVPWRVJVSUPUOTUMUNUAUCEFGHUQURUSVFVLWJVDVIVEVKABIKUTCDIKUTVA
      VBVC $.
  $}

  $( Equivalence class equality of nonnegative fractions in terms of natural
     numbers.  (Contributed by Jim Kingdon, 24-Nov-2019.) $)
  enq0eceq $p |- ( ( ( A e. _om /\ B e. N. ) /\ ( C e. _om /\ D e. N. ) ) ->
                ( [ <. A , B >. ] ~Q0 = [ <. C , D >. ] ~Q0 <->
                ( A .o D ) = ( B .o C ) ) ) $=
    ( com wcel cnpi wa cop ceq0 wbr cec wceq comu co cxp wer enq0er a1i opelxpi
    adantr erth enq0breq bitr3d ) AEFBGFHZCEFDGFHZHZABIZCDIZJKUHJLUIJLMADNOBCNO
    MUGUHUIJEGPZUJJQUGRSUEUHUJFUFABEGTUAUBABCDUCUD $.

  ${
    $d u v w x y z $.
    $( A nonnegative fraction is a positive fraction if its numerator and
       denominator are positive integers.  (Contributed by Jim Kingdon,
       10-Nov-2019.) $)
    nqnq0pi $p |- ( ( A e. N. /\ B e. N. ) ->
        [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q ) $=
      ( vy vx vz vw vu vv cnpi wcel wa cop ceq0 wceq cv wex com wb c0 wn opelxp
      cec cxp cin ceq cima wss wbr vex elima2 elxp anbi1i 19.41vv bitr4i simplr
      breq1 adantr biimpa wer enq0er simpr ercl2 sylib 19.42vv sylanbrc syl2anc
      id a1i wne simprrl wo comu co simprbi neneqd ad2antrr ad2antll jca pm4.56
      elni pinn nnm00 mtbird ad2ant2rl breq2 biimpac ad2ant2lr enq0breq sylanl1
      mpbid eqeq1d mtbid sylan sylibr simprd neneqad simprrr ad2antrl mpbir2and
      eleq1 bitrdi exlimivv syl sylbi exlimiv ssriv ecinxp sylbir enq0enq eceq2
      mpan ax-mp eqtr4di ) AIJBIJKZABLZMUBZXOMIIUCZXQUCUDZUBZXOUEUBZXNXOXQJZXPX
      SNZABIIUAMXQUFZXQUGYAYBCYCXQCOZYCJDOZXQJZYEYDMUHZKZDPYDXQJZDYDMXQCUIUJYHY
      IDYHYEEOZFOZLZNZYJIJZYKIJZKZKZYGKZFPEPZYIYHYQFPEPZYGKYSYFYTYGEFYEIIUKULYQ
      YGEFUMUNYRYIEFYRYPYLYDMUHZKZYDGOZHOZLZNZUUCQJZUUDIJZKZKZKZHPGPZYIYRYPUUAU
      ULYMYPYGUOYQYGUUAYMYGUUARYPYEYLYDMUPUQURUUBUUBUUJHPGPZUULUUBVGUUBYDQIUCZJ
      UUMUUBYLYDMUUNUUNMUSUUBUTVHYPUUAVAVBGHYDQIUKVCUUBUUJGHVDVEVFUUKYIGHUUKYIU
      UCIJZUUHUUKUUGUUCSVIUUOUUBUUFUUGUUHVJUUKUUCSUUKYKSNZTZUUCSNZTZUUKUUPUURVK
      ZTUUQUUSKUUKYKUUCVLVMZSNZUUTUUKYJUUDVLVMZSNZUVBYPUUIUVDTUUAUUFYPUUIKZUVDY
      JSNZUUDSNZVKZUVEUVFTZUVGTZKUVHTUVEUVIUVJYNUVIYOUUIYNYJSYNYJQJZYJSVIYJVTVN
      VOVPUUHUVJYPUUGUUHUUDSUUHUUDQJZUUDSVIUUDVTVNVOVQVRUVFUVGVSVCUVEUVKUVLUVDU
      VHRYNUVKYOUUIYJWAZVPUUHUVLYPUUGUUDWAVQYJUUDWBVFWCWDUUKUVCUVASUUKYLUUEMUHZ
      UVCUVANZUUAUUFUVNYPUUIUUFUUAUVNYDUUEYLMWEWFWGYPUUIUVNUVORZUUAUUFYNUVKYOUU
      IUVPUVMYJYKUUCUUDWHWIWDWJWKWLYPUUIUVBUUTRZUUAUUFYOUUGUVQYNUUHYOYKQJUUGUVQ
      YKWAYKUUCWBWMWGWDWLUUPUURVSWNWOWPUUCVTVEUUBUUFUUGUUHWQUUFYIUUOUUHKZRUUBUU
      IUUFYIUUEXQJUVRYDUUEXQWTUUCUUDIIUAXAWRWSXBXCXBXDXEXDXFXQXOMXGXKXHUEXRNXTX
      SNXIUEXRXOXJXLXM $.
  $}

  ${
    $d x y z w v u $.
    $( The equivalence relation for positive fractions exists.  (Contributed by
       Jim Kingdon, 18-Nov-2019.) $)
    enq0ex $p |- ~Q0 e. _V $=
      ( vv vu vx vy vz vw ceq0 com cnpi cxp omex xpex cv wcel cop wceq comu wex
      wa co niex copab df-enq0 opabssxp eqsstri ssexi ) GHIJZUGJZUGUGHIKUALZUIL
      GAMZUGNBMZUGNSUJCMZDMZOPUKEMZFMZOPSULUOQTUMUNQTPSFRERDRCRZSABUBUHABCDEFUC
      UPABUGUGUDUEUF $.
  $}

  $( The class of positive fractions exists.  (Contributed by Jim Kingdon,
     18-Nov-2019.) $)
  nq0ex $p |- Q0. e. _V $=
    ( cnq0 com cnpi cxp ceq0 cqs cvv df-nq0 omex niex xpex qsex eqeltri ) ABCDZ
    EFGHNEBCIJKLM $.

  ${
    $d u v x y $.
    $( A positive fraction is a nonnegative fraction.  (Contributed by Jim
       Kingdon, 18-Nov-2019.) $)
    nqnq0 $p |- Q. C_ Q0. $=
      ( vy vx vu vv cnq cnq0 cv wcel cnpi ceq cec wceq wa ceq0 adantl wb adantr
      wex mpbird com cxp cqs wrex df-nqqs eleq2i vex elqs df-rex 3bitri nqnq0pi
      cop elxpi eceq1 eqeq12d pinn opelxpi sylan eleq1 enq0ex ecelqsi eleqtrrdi
      df-nq0 syl eqeltrrd exlimivv exlimiv sylbi ssriv ) AEFAGZEHZBGZIIUAZHZVIV
      KJKZLZMZBRZVIFHZVJVIVLJUBZHVOBVLUCVQEVSVIUDUEBVLVIJAUFUGVOBVLUHUIVPVRBVPV
      RVNFHZVMVTVOVMVKCGZDGZUKZLZWAIHZWBIHZMZMZDRCRVTCDVKIIULWHVTCDWHVKNKZVNFWH
      WIVNLZWCNKZWCJKZLZWGWMWDWAWBUJOWDWJWMPWGWDWIWKVNWLVKWCNUMVKWCJUMUNQSWHVKT
      IUAZHZWIFHWHWOWCWNHZWGWPWDWEWATHWFWPWAUOWAWBTIUPUQOWDWOWPPWGVKWCWNURQSWOW
      IWNNUBFWNVKNUSUTVBVAVCVDVEVCQVOVRVTPVMVIVNFUROSVFVGVH $.
  $}

  ${
    $d A a v w $.
    $( Decomposition of a nonnegative fraction into numerator and denominator.
       (Contributed by Jim Kingdon, 24-Nov-2019.) $)
    nq0nn $p |- ( A e. Q0. ->
        E. w E. v ( ( w e. _om /\ v e. N. ) /\ A = [ <. w , v >. ] ~Q0 ) ) $=
      ( va cv com wcel cnpi cop ceq0 cec wceq wex cxp cqs cnq0 wrex elqsi syl
      wa elxpi anim1i sylibr simplr simpr eceq1 ad2antrr eqtrd 2eximi rexlimiva
      19.41vv jca df-nq0 eleq2s ) AEZFGBEZHGTZCUOUPIZJKZLZTZBMAMZCFHNZJOZPCVDGC
      DEZJKZLZDVCQVBDVCCJRVGVBDVCVEVCGZVGTZVEURLZUQTZVGTZBMAMZVBVIVKBMAMZVGTVMV
      HVNVGABVEFHUAUBVKVGABUKUCVLVAABVLUQUTVJUQVGUDVLCVFUSVKVGUEVJVFUSLUQVGVEUR
      JUFUGUHULUISUJSUMUN $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d D x y z $.  $d F x y z $.
    $d G x y z $.  $d R x y z $.  $d S x y z $.
    $( Lemma showing compatibility of addition on nonnegative fractions.
       (Contributed by Jim Kingdon, 23-Nov-2019.) $)
    addcmpblnq0 $p |- ( ( ( ( A e. _om /\ B e. N. ) /\ ( C e. _om /\ D e. N. )
             ) /\ ( ( F e. _om /\ G e. N. ) /\ ( R e. _om /\ S e. N. ) ) ) ->
                ( ( ( A .o D ) = ( B .o C ) /\ ( F .o S ) = ( G .o R ) ) ->
        <. ( ( A .o G ) +o ( B .o F ) ) , ( B .o G ) >. ~Q0
         <. ( ( C .o S ) +o ( D .o R ) ) , ( D .o S ) >. ) ) $=
      ( vx vy com wcel cnpi wa comu co wceq coa nnmcl syl2anc vz cop wbr cv w3a
      ceq0 nndi adantl simplll simprlr pinn syl simpllr simprll simplrr simprrr
      nnacl nnmcom caovdir2d nnmass caov4d eqtrd oveq1 oveq2 oveqan12d sylan9eq
      oveq12d simplrl simprrl syl3anc adantr mulpiord mulclpi eqeltrrd ad2ant2l
      eqtr4d wb cmi ad2ant2r enq0breq syl22anc mpbird ex ) AKLZBMLZNZCKLZDMLZNZ
      NZGKLZHMLZNZEKLZFMLZNZNZNZADOPZBCOPZQZGFOPZHEOPZQZNZAHOPZBGOPZRPZBHOPZUBC
      FOPZDEOPZRPZDFOPZUBUFUCZWRXENZXNXHXMOPZXIXLOPZQZXOXPWTHFOPZOPZBDOPZXCOPZR
      PZXQWRXEXPWSXSOPZYAXBOPZRPZYCWRXPXFXMOPZXGXMOPZRPYFWRIJUAXFXGXMKROIUDZKLZ
      JUDZKLZUAUDZKLUEZYIYKYMRPOPYIYKOPZYIYMOPRPQWRYIYKYMUGUHWRWDHKLZXFKLZWDWEW
      IWQUIZWRWLYPWJWKWLWPUJHUKULZAHSTZWRBKLZWKXGKLZWRWEUUAWDWEWIWQUMBUKULZWJWK
      WLWPUNZBGSTZWRDKLZFKLZXMKLWRWHUUFWFWGWHWQUODUKULZWRWOUUGWJWMWNWOUPFUKULZD
      FSTYJYLNZYIYKRPKLWRYIYKUQUHUUJYOYKYIOPQWRYIYKURUHZUSWRYGYDYHYERWRIJUAAHDF
      KOYRYSUUHUUKYNYOYMOPYIYKYMOPOPQWRYIYKYMUTUHZUUIUUJYOKLWRYIYKSUHZVAWRIJUAB
      GDFKOUUCUUDUUHUUKUULUUIUUMVAVGVBXAXDYDXTYEYBRWSWTXSOVCXBXCYAOVDVEVFWRXQYC
      QXEWRXQXIXJOPZXIXKOPZRPZYCWRXIKLZXJKLZXKKLZXQUUPQWRUUAYPUUQUUCYSBHSTWRWGU
      UGUURWFWGWHWQVHZUUICFSTZWRUUFWNUUSUUHWJWMWNWOVIZDESTZXIXJXKUGVJWRUUNXTUUO
      YBRWRIJUABHCFKOUUCYSUUTUUKUULUUIUUMVAWRIJUABHDEKOUUCYSUUHUUKUULUVBUUMVAVG
      VBVKVPWRXNXRVQZXEWRXHKLZXIMLZXLKLZXMMLZUVDWRYQUUBUVEYTUUEXFXGUQTWFWMUVFWI
      WPWEWLUVFWDWKWEWLNBHVRPXIMBHVLBHVMVNVOVSWRUURUUSUVGUVAUVCXJXKUQTWIWPUVHWF
      WMWHWOUVHWGWNWHWONDFVRPXMMDFVLDFVMVNVOVOXHXIXLXMVTWAVKWBWC $.

    $( Lemma showing compatibility of multiplication on nonnegative fractions.
       (Contributed by Jim Kingdon, 20-Nov-2019.) $)
    mulcmpblnq0 $p |- ( ( ( ( A e. _om /\ B e. N. ) /\ ( C e. _om /\ D e. N. )
             ) /\ ( ( F e. _om /\ G e. N. ) /\ ( R e. _om /\ S e. N. ) ) ) ->
                ( ( ( A .o D ) = ( B .o C ) /\ ( F .o S ) = ( G .o R ) ) ->
     <. ( A .o F ) , ( B .o G ) >. ~Q0 <. ( C .o R ) , ( D .o S ) >. ) ) $=
      ( vx vy vz comu co wceq wa com wcel cnpi syl pinn cop wbr oveq12 wb nnmcl
      cmi mulpiord mulclpi eqeltrrd anim12i enq0breq simplll simprll simplrr cv
      ceq0 an4s nnmcom adantl w3a nnmass simprrr caov4d simpllr simprlr simplrl
      simprrl eqeq12d bitrd imbitrrid ) ADLMZBCLMZNGFLMZHELMZNOAGLMZBHLMZUACELM
      ZDFLMZUAUPUBZAPQZBRQZOZCPQZDRQZOZOZGPQZHRQZOZEPQZFRQZOZOZOZVKVMLMZVLVNLMZ
      NZVKVLVMVNLUCWNVSVOVRLMZVPVQLMZNZWQWNVOPQZVPRQZOZVQPQZVRRQZOZOZVSWTUDWBWI
      WEWLXGWBWIOXCWEWLOXFVTWGWAWHXCVTWGOXAWAWHOZXBAGUEXHBHUFMVPRBHUGBHUHUIUJUQ
      WCWJWDWKXFWCWJOXDWDWKOZXECEUEXIDFUFMVRRDFUGDFUHUIUJUQUJUQVOVPVQVRUKSWNWRW
      OWSWPWNIJKAGDFPLVTWAWEWMULWFWGWHWLUMWNWDDPQWBWCWDWMUNDTSIUOZPQZJUOZPQZOZX
      JXLLMZXLXJLMNWNXJXLURUSZXKXMKUOZPQUTXOXQLMXJXLXQLMLMNWNXJXLXQVAUSZWNWKFPQ
      WFWIWJWKVBFTSXNXOPQWNXJXLUEUSZVCWNIJKBHCEPLWNWABPQVTWAWEWMVDBTSWNWHHPQWFW
      GWHWLVEHTSWBWCWDWMVFXPXRWFWIWJWKVGXSVCVHVIVJ $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.
    $( Lemma for distributive law: cancellation of common factor.  (Contributed
       by Jim Kingdon, 29-Nov-2019.) $)
    mulcanenq0ec $p |- ( ( A e. N. /\ B e. _om /\ C e. N. ) ->
        [ <. ( A .o B ) , ( A .o C ) >. ] ~Q0 = [ <. B , C >. ] ~Q0 ) $=
      ( vx vy vz cnpi wcel com w3a comu co cop ceq0 cxp wceq pinn cv wa adantl
      wer enq0er a1i wbr 3ad2ant1 simp2 3ad2ant3 nnmcom nnmass caov32d wb nnmcl
      sylan cmi mulpiord mulclpi eqeltrrd anim12i simpr jca 3impdi enq0breq syl
      an4s mpbird erthi ) AGHZBIHZCGHZJZABKLZACKLZMZBCMZNIGOZVONUAVJUBUCVJVMVNN
      UDZVKCKLVLBKLPZVJDEFABCIKVGVHAIHZVIAQZUEVGVHVIUFVIVGCIHVHCQUGDRZIHZERZIHZ
      SVTWBKLZWBVTKLPVJVTWBUHTWAWCFRZIHJWDWEKLVTWBWEKLKLPVJVTWBWEUITUJVJVKIHZVL
      GHZSZVHVISZSZVPVQUKVGVHVIWJVGVHSZVGVISZSWHWIWKWFWLWGVGVRVHWFVSABULUMWLACU
      NLVLGACUOACUPUQURVGVGVHVIWIVGVGSWIUSVDUTVAVKVLBCVBVCVEVF $.
  $}

  ${
    $d z w v u t s q f g h A $.  $d z w v u t s q f g h B $.

    $( Decomposing nonnegative fractions into natural numbers.  Lemma for
       ~ addnnnq0 and ~ mulnnnq0 .  (Contributed by Jim Kingdon,
       23-Nov-2019.) $)
    nnnq0lem1 $p |- ( (
        ( A e. ( ( _om X. N. ) /. ~Q0 ) /\ B e. ( ( _om X. N. ) /. ~Q0 ) )
        /\ ( ( ( A = [ <. w , v >. ] ~Q0 /\ B = [ <. u , t >. ] ~Q0 )
          /\ z = [ C ] ~Q0 )
        /\ ( ( A = [ <. s , f >. ] ~Q0 /\ B = [ <. g , h >. ] ~Q0 )
          /\ q = [ D ] ~Q0 ) ) ) -> (
        ( ( ( w e. _om /\ v e. N. ) /\ ( s e. _om /\ f e. N. ) )
          /\ ( ( u e. _om /\ t e. N. ) /\ ( g e. _om /\ h e. N. ) ) )
        /\ ( ( w .o f ) = ( v .o s ) /\ ( u .o h ) = ( t .o g ) ) ) ) $=
      ( com ceq0 wcel wa cv wceq cnpi cxp cqs cop cec comu cdm wer enq0er ax-mp
      co erdm simpll simplll eleq1d adantl mpbid ecelqsdm sylancr sylib simprll
      wb opelxp jca simplr simpllr simprlr wbr eqtr3d a1i erth enq0breq syl2anc
      mpbird jca31 ) FOUAUBZPUCZQZGVQQZRZFBSZCSZUDZPUEZTZGDSZESZUDZPUEZTZRASHPU
      ETZRZFMSZJSZUDZPUEZTZGKSZLSZUDZPUEZTZRNSIPUETZRZRZRZWAOQWBUAQRZWMOQWNUAQR
      ZRWFOQWGUAQRZWROQWSUAQRZRWAWNUFUKWBWMUFUKTZWFWSUFUKWGWRUFUKTZRXFXGXHXFWCV
      PQZXGXFPUGVPTZWDVQQZXMVPPUHZXNUIVPPULUJZXFVRXOVRVSXEUMZXEVRXOVBVTXEFWDVQW
      EWJWKXDUNZUOUPUQVPWCPURUSZWAWBOUAVCUTZXFWOVPQZXHXFXNWPVQQZYBXQXFVRYCXRXEV
      RYCVBVTXEFWPVQWLWQXBXCVAZUOUPUQVPWOPURUSWMWNOUAVCUTZVDXFXIXJXFWHVPQZXIXFX
      NWIVQQZYFXQXFVSYGVRVSXEVEZXEVSYGVBVTXEGWIVQWEWJWKXDVFZUOUPUQVPWHPURUSZWFW
      GOUAVCUTZXFWTVPQZXJXFXNXAVQQZYLXQXFVSYMYHXEVSYMVBVTXEGXAVQWLWQXBXCVGZUOUP
      UQVPWTPURUSWRWSOUAVCUTZVDXFXKXLXFWCWOPVHZXKXFYPWDWPTZXEYQVTXEFWDWPXSYDVIU
      PXFWCWOPVPXPXFUIVJZXTVKVNXFXGXHYPXKVBYAYEWAWBWMWNVLVMUQXFWHWTPVHZXLXFYSWI
      XATZXEYTVTXEGWIXAYIYNVIUPXFWHWTPVPYRYJVKVNXFXIXJYSXLVBYKYOWFWGWRWSVLVMUQV
      DVO $.
  $}

  ${
    $d A f g h q s t u v w z $.  $d B f g h q s t u v w z $.
    $( There is at most one result from adding nonnegative fractions.
       (Contributed by Jim Kingdon, 23-Nov-2019.) $)
    addnq0mo $p |- ( ( A e. ( ( _om X. N. ) /. ~Q0 ) /\
        B e. ( ( _om X. N. ) /. ~Q0 ) ) -> E* z E. w E. v E. u E. t
        ( ( A = [ <. w , v >. ] ~Q0 /\ B = [ <. u , t >. ] ~Q0 )
        /\ z = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) ) $=
      ( vq vs vf ceq0 wcel wa cv cop cec wceq comu co wex vg vh com cxp cqs coa
      wi wal wmo wer enq0er a1i wbr nnnq0lem1 addcmpblnq0 imp syl erthi simprlr
      cnpi simprrr 3eqtr4d expr exlimdvv ex alrimivv opeq12 eceq1d eqeq2d simpl
      anbi1d oveq1d oveq12d opeq12d anbi12d anbi2d oveq2d cbvex4v anbi2i imbi1i
      impd simpr 2albii sylibr eqeq1 4exbidv mo4 ) FUCUTUDZKUEZLGWILMZFBNZCNZOZ
      KPZQZGDNZENZOZKPZQZMZANZWKWQRSZWLWPRSZUFSZWLWQRSZOZKPZQZMZETDTZCTBTZXAHNZ
      XHQZMZETDTCTBTZMZXBXMQZUGZHUHAUHZXLAUIWJXLFINZJNZOZKPZQZGUANZUBNZOZKPZQZM
      ZXMYAYGRSZYBYFRSZUFSZYBYGRSZOZKPZQZMZUBTUATZJTITZMZXRUGZHUHAUHXTWJUUCAHWJ
      XLUUAXRWJXKUUAXRUGZBCWJXJUUDDEWJXJUUDWJXJMZYTXRIJUUEYSXRUAUBWJXJYSXRWJXJY
      SMMZXHYQXBXMUUFXGYPKWHWHKUJUUFUKULUUFWKUCLWLUTLMYAUCLYBUTLMMWPUCLWQUTLMYF
      UCLYGUTLMMMZWKYBRSWLYARSQWPYGRSWQYFRSQMZMXGYPKUMZABCDEFGXGYPJUAUBIHUNUUGU
      UHUUIWKWLYAYBYFYGWPWQUOUPUQURWJXAXIYSUSWJXJYKYRVAVBVCVDVDVEVDVDWAVFXSUUCA
      HXQUUBXRXPUUAXLXOYEWTMZXMYAWQRSZYBWPRSZUFSZYBWQRSZOZKPZQZMYSBCDEIJUAUBWKY
      AQZWLYBQZMZXAUUJXNUUQUUTWOYEWTUUTWNYDFUUTWMYCKWKWLYAYBVGVHVIVKUUTXHUUPXMU
      UTXGUUOKUUTXEUUMXFUUNUUTXCUUKXDUULUFUUTWKYAWQRUURUUSVJVLUUTWLYBWPRUURUUSW
      BZVLVMUUTWLYBWQRUVAVLVNVHVIVOWPYFQZWQYGQZMZUUJYKUUQYRUVDWTYJYEUVDWSYIGUVD
      WRYHKWPWQYFYGVGVHVIVPUVDUUPYQXMUVDUUOYPKUVDUUMYNUUNYOUVDUUKYLUULYMUFUVDWQ
      YGYARUVBUVCWBZVQUVDWPYFYBRUVBUVCVJVQVMUVDWQYGYBRUVEVQVNVHVIVOVRVSVTWCWDXL
      XPAHXRXJXOBCDEXRXIXNXAXBXMXHWEVPWFWGWD $.

    $( There is at most one result from multiplying nonnegative fractions.
       (Contributed by Jim Kingdon, 20-Nov-2019.) $)
    mulnq0mo $p |- ( ( A e. ( ( _om X. N. ) /. ~Q0 ) /\
        B e. ( ( _om X. N. ) /. ~Q0 ) ) -> E* z E. w E. v E. u E. t
        ( ( A = [ <. w , v >. ] ~Q0 /\ B = [ <. u , t >. ] ~Q0 )
        /\ z = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) ) $=
      ( vq vs vf ceq0 wcel wa cv cop cec wceq comu co wex vg vh com cnpi cxp wi
      cqs wal wmo wer a1i wbr nnnq0lem1 mulcmpblnq0 imp simprlr simprrr 3eqtr4d
      enq0er syl erthi expr exlimdvv alrimivv opeq12 eceq1d eqeq2d anbi1d simpl
      ex impd oveq1d opeq12d anbi12d anbi2d oveq2d cbvex4v anbi2i imbi1i 2albii
      simpr sylibr eqeq1 4exbidv mo4 ) FUCUDUEZKUGZLGWGLMZFBNZCNZOZKPZQZGDNZENZ
      OZKPZQZMZANZWIWNRSZWJWORSZOZKPZQZMZETDTZCTBTZWSHNZXDQZMZETDTCTBTZMZWTXIQZ
      UFZHUHAUHZXHAUIWHXHFINZJNZOZKPZQZGUANZUBNZOZKPZQZMZXIXQYBRSZXRYCRSZOZKPZQ
      ZMZUBTUATZJTITZMZXNUFZHUHAUHXPWHYQAHWHXHYOXNWHXGYOXNUFZBCWHXFYRDEWHXFYRWH
      XFMZYNXNIJYSYMXNUAUBWHXFYMXNWHXFYMMMZXDYKWTXIYTXCYJKWFWFKUJYTUSUKYTWIUCLW
      JUDLMXQUCLXRUDLMMWNUCLWOUDLMYBUCLYCUDLMMMZWIXRRSWJXQRSQWNYCRSWOYBRSQMZMXC
      YJKULZABCDEFGXCYJJUAUBIHUMUUAUUBUUCWIWJXQXRYBYCWNWOUNUOUTVAWHWSXEYMUPWHXF
      YGYLUQURVBVCVCVJVCVCVKVDXOYQAHXMYPXNXLYOXHXKYAWRMZXIXQWNRSZXRWORSZOZKPZQZ
      MYMBCDEIJUAUBWIXQQZWJXRQZMZWSUUDXJUUIUULWMYAWRUULWLXTFUULWKXSKWIWJXQXRVEV
      FVGVHUULXDUUHXIUULXCUUGKUULXAUUEXBUUFUULWIXQWNRUUJUUKVIVLUULWJXRWORUUJUUK
      WAVLVMVFVGVNWNYBQZWOYCQZMZUUDYGUUIYLUUOWRYFYAUUOWQYEGUUOWPYDKWNWOYBYCVEVF
      VGVOUUOUUHYKXIUUOUUGYJKUUOUUEYHUUFYIUUOWNYBXQRUUMUUNVIVPUUOWOYCXRRUUMUUNW
      AVPVMVFVGVNVQVRVSVTWBXHXLAHXNXFXKBCDEXNXEXJWSWTXIXDWCVOWDWEWB $.
  $}

  ${
    $d x y z w v u t A $.  $d x y z w v u t B $.  $d x y z w v u t C $.
    $d x y z w v u t D $.

    $( Addition of nonnegative fractions in terms of natural numbers.
       (Contributed by Jim Kingdon, 22-Nov-2019.) $)
    addnnnq0 $p |- ( ( ( A e. _om /\ B e. N. ) /\
                     ( C e. _om /\ D e. N. ) ) ->
                ( [ <. A , B >. ] ~Q0 +Q0 [ <. C , D >. ] ~Q0 ) =
                   [ <. ( ( A .o D ) +o ( B .o C ) ) ,
                              ( B .o D ) >. ] ~Q0 ) $=
      ( vw vv vu vt com wcel cnpi wa cop ceq0 cec cv wceq comu co wex vx vy cxp
      vz cqs coa cplq0 opelxpi enq0ex ecelqsi anim12i eqid pm3.2i opeq12 eceq1d
      eqeq2d anbi1d simpl oveq1d oveq12d opeq12d anbi12d spc2egv anbi2d 2eximdv
      syl simpr oveq2d sylan9 mp2ani cvv wi ecexg ax-mp w3a simp1 simp2 4exbidv
      eqeq1d simp3 addnq0mo dfplq0qs ovig mp3an3 sylc ) AIJBKJLZCIJDKJLZLZABMZN
      OZIKUCZNUEZJZCDMZNOZWLJZLWJEPZFPZMZNOZQZWOGPZHPZMZNOZQZLZADRSZBCRSZUFSZBD
      RSZMZNOZWQXCRSZWRXBRSZUFSZWRXCRSZMZNOZQZLZHTGTZFTETZWJWOUGSXMQZWFWMWGWPWF
      WIWKJWMABIKUHWKWINUIUJVFWGWNWKJWPCDIKUHWKWNNUIUJVFUKWHWJWJQZWOWOQZLZXMXMQ
      ZYCYEYFWJULWOULUMXMULWFYGYHLZXAYFLZXMWQDRSZWRCRSZUFSZWRDRSZMZNOZQZLZFTETW
      GYCYRYIEFABIKWQAQZWRBQZLZYJYGYQYHUUAXAYEYFUUAWTWJWJUUAWSWINWQWRABUNUOUPUQ
      UUAYPXMXMUUAYOXLNUUAYMXJYNXKUUAYKXHYLXIUFUUAWQADRYSYTURUSUUAWRBCRYSYTVGZU
      SUTUUAWRBDRUUBUSVAUOUPVBVCWGYRYBEFYAYRGHCDIKXBCQZXCDQZLZXGYJXTYQUUEXFYFXA
      UUEXEWOWOUUEXDWNNXBXCCDUNUOUPVDUUEXSYPXMUUEXRYONUUEXPYMXQYNUUEXNYKXOYLUFU
      UEXCDWQRUUCUUDVGZVHUUEXBCWRRUUCUUDURVHUTUUEXCDWRRUUFVHVAUOUPVBVCVEVIVJWMW
      PXMVKJZYCYDVLNVKJUUGUIXLVKNVMVNUAPZWTQZUBPZXEQZLZUDPZXSQZLZHTGTFTETYCUAUB
      UDWJWOXMVKWLWLUGUUHWJQZUUJWOQZUUMXMQZVOZUUOYAEFGHUUSUULXGUUNXTUUSUUIXAUUK
      XFUUSUUHWJWTUUPUUQUURVPVSUUSUUJWOXEUUPUUQUURVQVSVBUUSUUMXMXSUUPUUQUURVTVS
      VBVRUDEFGHUUHUUJWAUAUBUDEFGHWBWCWDWE $.

    $( Multiplication of nonnegative fractions in terms of natural numbers.
       (Contributed by Jim Kingdon, 19-Nov-2019.) $)
    mulnnnq0 $p |- ( ( ( A e. _om /\ B e. N. ) /\
                     ( C e. _om /\ D e. N. ) ) ->
                ( [ <. A , B >. ] ~Q0 .Q0 [ <. C , D >. ] ~Q0 ) =
                   [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) $=
      ( vw vv vu vt com wcel cnpi wa cop ceq0 cec cv wceq comu co wex vx vy cxp
      vz cqs cmq0 opelxpi enq0ex ecelqsi syl anim12i pm3.2i opeq12 eceq1 eqeq2d
      eqid wb anbi1d vex opth oveq1 adantr adantl opeq12d sylbi anbi12d spc2egv
      eceq1d anbi2d oveq2 2eximdv sylan9 mp2ani cvv wi ecexg ax-mp w3a bi2anan9
      eqeq1 3impa 4exbidv mulnq0mo dfmq0qs ovig mp3an3 sylc ) AIJBKJLZCIJDKJLZL
      ZABMZNOZIKUCZNUEZJZCDMZNOZWNJZLWLEPZFPZMZNOZQZWQGPZHPZMZNOZQZLZACRSZBDRSZ
      MZNOZWSXDRSZWTXERSZMZNOZQZLZHTGTZFTETZWLWQUFSXMQZWHWOWIWRWHWKWMJWOABIKUGW
      MWKNUHUIUJWIWPWMJWRCDIKUGWMWPNUHUIUJUKWJWLWLQZWQWQQZLZXMXMQZYAYCYDWLUPWQU
      PULXMUPWHYEYFLZXCYDLZXMWSCRSZWTDRSZMZNOZQZLZFTETWIYAYNYGEFABIKWSAQZWTBQZL
      ZXAWKQZYNYGUQWSWTABUMYRYHYEYMYFYRXCYCYDYRXBWLWLXAWKNUNUOURYRYLXMXMYRYKXLN
      YRYQYKXLQWSWTABEUSFUSUTYQYIXJYJXKYOYIXJQYPWSACRVAVBYPYJXKQYOWTBDRVAVCVDVE
      VHUOVFUJVGWIYNXTEFXSYNGHCDIKXDCQZXEDQZLZXFWPQZXSYNUQXDXECDUMUUBXIYHXRYMUU
      BXHYDXCUUBXGWQWQXFWPNUNUOVIUUBXQYLXMUUBXPYKNUUBUUAXPYKQXDXECDGUSHUSUTUUAX
      NYIXOYJYSXNYIQYTXDCWSRVJVBYTXOYJQYSXEDWTRVJVCVDVEVHUOVFUJVGVKVLVMWOWRXMVN
      JZYAYBVONVNJUUCUHXLVNNVPVQUAPZXBQZUBPZXGQZLZUDPZXQQZLZHTGTFTETYAUAUBUDWLW
      QXMVNWNWNUFUUDWLQZUUFWQQZUUIXMQZVRUUKXSEFGHUULUUMUUNUUKXSUQUULUUMLUUHXIUU
      NUUJXRUULUUEXCUUMUUGXHUUDWLXBVTUUFWQXGVTVSUUIXMXQVTVSWAWBUDEFGHUUDUUFWCUA
      UBUDEFGHWDWEWFWG $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Closure of addition on nonnegative fractions.  (Contributed by Jim
       Kingdon, 29-Nov-2019.) $)
    addclnq0 $p |- ( ( A e. Q0. /\ B e. Q0. ) -> ( A +Q0 B ) e. Q0. ) $=
      ( vx vy vz vw cnq0 wcel wa cplq0 co com cnpi ceq0 cv cop df-nq0 wceq comu
      cec cxp cqs oveq1 eleq1d oveq2 coa addnnnq0 pinn nnmcl sylan2 sylan nnacl
      syl2an cmi mulpiord mulclpi eqeltrrd ad2ant2l opelxpi enq0ex ecelqsi 3syl
      an42s jca eqeltrd 2ecoptocl eleqtrrdi ) AGHBGHIABJKZLMUAZNUBZGCOZDOZPNTZE
      OZFOZPNTZJKZVJHAVPJKZVJHVHVJHCDEFABLMNGQVMARVQVRVJVMAVPJUCUDVPBRVRVHVJVPB
      AJUEUDVKLHZVLMHZIVNLHZVOMHZIIZVQVKVOSKZVLVNSKZUFKZVLVOSKZPZNTZVJVKVLVNVOU
      GWCWFLHZWGMHZIWHVIHWIVJHWCWJWKVSWBVTWAWJVSWBIWDLHZWELHZWJVTWAIWBVSVOLHWLV
      OUHVKVOUIUJVTVLLHWAWMVLUHVLVNUIUKWDWEULUMVCVTWBWKVSWAVTWBIVLVOUNKWGMVLVOU
      OVLVOUPUQURVDWFWGLMUSVIWHNUTVAVBVEVFQVG $.

    $( Closure of multiplication on nonnegative fractions.  (Contributed by Jim
       Kingdon, 30-Nov-2019.) $)
    mulclnq0 $p |- ( ( A e. Q0. /\ B e. Q0. ) -> ( A .Q0 B ) e. Q0. ) $=
      ( vx vy vz vw cnq0 wcel wa cmq0 co com cnpi ceq0 cv cop cec df-nq0 eleq1d
      wceq cxp cqs oveq1 oveq2 comu mulnnnq0 nnmcl cmi mulpiord mulclpi anim12i
      eqeltrrd an4s opelxpi enq0ex ecelqsi 3syl eqeltrd 2ecoptocl eleqtrrdi ) A
      GHBGHIABJKZLMUAZNUBZGCOZDOZPNQZEOZFOZPNQZJKZVCHAVIJKZVCHVAVCHCDEFABLMNGRV
      FATVJVKVCVFAVIJUCSVIBTVKVAVCVIBAJUDSVDLHZVEMHZIVGLHZVHMHZIIZVJVDVGUEKZVEV
      HUEKZPZNQZVCVDVEVGVHUFVPVQLHZVRMHZIZVSVBHVTVCHVLVNVMVOWCVLVNIWAVMVOIZWBVD
      VGUGWDVEVHUHKVRMVEVHUIVEVHUJULUKUMVQVRLMUNVBVSNUOUPUQURUSRUT $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.

    $( A positive fraction plus a nonnegative fraction is a positive fraction.
       (Contributed by Jim Kingdon, 30-Nov-2019.) $)
    nqpnq0nq $p |- ( ( A e. Q. /\ B e. Q0. ) -> ( A +Q0 B ) e. Q. ) $=
      ( vx vy vz vw cnq wcel wa cv cnpi cop ceq cec wceq com wex cplq0 syl2anc
      co cnq0 ceq0 nqpi nq0nn anim12i ee4anv sylibr coa oveq12 ad2ant2l nqnq0pi
      comu oveq1d adantr addnnnq0 sylanl1 eqtr3d ad2ant2r eqtrd cxp nnmcl sylan
      pinn cqs ad2ant2lr cmi mulpiord mulclpi eqeltrrd ad2ant2rl nnacom nnppipi
      sylan2 opelxpi enqex ecelqsi syl df-nqqs eleqtrrdi eleq1d mpbird exlimivv
      wb eqeltrd ) AGHZBUAHZIZCJZKHZDJZKHZIZAWHWJLZMNZOZIZEJZPHZFJZKHZIZBWQWSLU
      BNZOZIZIZFQEQZDQCQZABRTZGHZWGWPDQCQZXDFQEQZIXGWEXJWFXKCDAUCEFBUDUEWPXDCDE
      FUFUGXFXICDXEXIEFXEXHWHWSULTZWJWQULTZUHTZWJWSULTZLZUBNZGXEXHWNXBRTZXQWOXC
      XHXROWLXAAWNBXBRUIUJWLXAXRXQOWOXCWLXAIZWMUBNZXBRTZXRXQWLYAXROXAWLXTWNXBRW
      HWJUKUMUNWIWHPHWKXAYAXQOWHVCWHWJWQWSUOUPUQURUSWLXAXQGHZWOXCXSYBXPMNZGHZXS
      YCKKUTZMVDZGXSXPYEHZYCYFHXSXNKHZXOKHZYGXSXMXLUHTZXNKXSXMPHZXLKHZYJXNOZWKW
      RYKWIWTWKWJPHWRYKWJVCWJWQVAVBVEZWIWTYLWKWRWIWTIWHWSVFTXLKWHWSVGWHWSVHVIVJ
      ZYLYKXLPHYMXLVCXMXLVKVMSXSYKYLYJKHYNYOXMXLVLSVIZWKWTYIWIWRWKWTIWJWSVFTXOK
      WJWSVGWJWSVHVIUJZXNXOKKVNSYEXPMVOVPVQVRVSXSYHYIYBYDWCYPYQYHYIIXQYCGXNXOUK
      VTSWAURWDWBWBVQ $.
  $}

  ${
    $d A u v w z $.  $d B u v w z $.

    $( Addition of positive fractions is equal with ` +Q ` or ` +Q0 ` .
       (Contributed by Jim Kingdon, 10-Nov-2019.) $)
    nqnq0a $p |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( A +Q0 B ) ) $=
      ( vz vw vv vu wcel wa cv cnpi cop ceq cec wceq wex cplq co cplq0 ceq0 cmi
      cnq nqpi anim12i ee4anv sylibr oveq12 cpli coa mulclpi ad2ant2rl addpiord
      comu ad2ant2lr syl2anc mulpiord oveq12d ad2ant2l opeq12d eceq1d addpipqqs
      eqtrd addclpi nqnq0pi eqtr4d com anim1i addnnnq0 syl2an 3eqtr4d sylan9eqr
      pinn adantr eqeq2d adantl anbi12d pm5.32i sylbir an4s exlimivv syl ) AUAG
      ZBUAGZHZCIZJGZDIZJGZHZAWDWFKZLMZNZHZEIZJGZFIZJGZHZBWMWOKZLMZNZHZHZFOEOZDO
      COZABPQZABRQZNZWCWLDOCOZXAFOEOZHXDWAXHWBXICDAUBEFBUBUCWLXACDEFUDUEXCXGCDX
      BXGEFWHWQWKWTXGWHWQHZWKWTHZHZXEWISMZWRSMZRQZXFXKXJXEWJWSPQZXOAWJBWSPUFXJW
      DWOTQZWFWMTQZUGQZWFWOTQZKZSMZWDWOULQZWFWMULQZUHQZWFWOULQZKZSMZXPXOXJYAYGS
      XJXSYEXTYFXJXSXQXRUHQZYEXJXQJGZXRJGZXSYINWEWPYJWGWNWDWOUIUJZWGWNYKWEWPWFW
      MUIUMZXQXRUKUNXJXQYCXRYDUHWEWPXQYCNWGWNWDWOUOUJWGWNXRYDNWEWPWFWMUOUMUPVAW
      GWPXTYFNWEWNWFWOUOUQURUSXJXPYALMZYBWDWFWMWOUTXJXSJGZXTJGZYBYNNXJYJYKYOYLY
      MXQXRVBUNWGWPYPWEWNWFWOUIUQXSXTVCUNVDWHWDVEGZWGHWMVEGZWPHXOYHNWQWEYQWGWDV
      KVFWNYRWPWMVKVFWDWFWMWOVGVHVIVJXLXJAXMNZBXNNZHZHXFXONZXJUUAXKXJYSWKYTWTXJ
      XMWJAWHXMWJNWQWDWFVCVLVMXJXNWSBWQXNWSNWHWMWOVCVNVMVOVPUUAUUBXJAXMBXNRUFVN
      VQVDVRVSVSVT $.
  $}

  ${
    $d A z w v u $.  $d B z w v u $.
    $( Multiplication of positive fractions is equal with ` .Q ` or ` .Q0 ` .
       (Contributed by Jim Kingdon, 10-Nov-2019.) $)
    nqnq0m $p |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( A .Q0 B ) ) $=
      ( vz vw vv vu wcel wa cv cnpi cop ceq cec wceq wex cmq cmq0 ceq0 nqnq0pi
      co cnq nqpi anim12i ee4anv sylibr comu mulpiord ad2ant2r ad2ant2l opeq12d
      oveq12 cmi eceq1d mulpipqqs mulclpi syl2anc eqtr4d com pinn anim1i syl2an
      mulnnnq0 3eqtr4d adantr eqeq2d adantl anbi12d pm5.32i sylbir exlimivv syl
      sylan9eqr an4s ) AUAGZBUAGZHZCIZJGZDIZJGZHZAVQVSKZLMZNZHZEIZJGZFIZJGZHZBW
      FWHKZLMZNZHZHZFOEOZDOCOZABPTZABQTZNZVPWEDOCOZWNFOEOZHWQVNXAVOXBCDAUBEFBUB
      UCWEWNCDEFUDUEWPWTCDWOWTEFWAWJWDWMWTWAWJHZWDWMHZHZWRWBRMZWKRMZQTZWSXDXCWR
      WCWLPTZXHAWCBWLPUKXCVQWFULTZVSWHULTZKZRMZVQWFUFTZVSWHUFTZKZRMZXIXHXCXLXPR
      XCXJXNXKXOVRWGXJXNNVTWIVQWFUGUHVTWIXKXONVRWGVSWHUGUIUJUMXCXIXLLMZXMVQVSWF
      WHUNXCXJJGZXKJGZXMXRNVRWGXSVTWIVQWFUOUHVTWIXTVRWGVSWHUOUIXJXKSUPUQWAVQURG
      ZVTHWFURGZWIHXHXQNWJVRYAVTVQUSUTWGYBWIWFUSUTVQVSWFWHVBVAVCVLXEXCAXFNZBXGN
      ZHZHWSXHNZXCYEXDXCYCWDYDWMXCXFWCAWAXFWCNWJVQVSSVDVEXCXGWLBWJXGWLNWAWFWHSV
      FVEVGVHYEYFXCAXFBXGQUKVFVIUQVMVJVJVK $.
  $}

  ${
    $d A v w $.
    $( Multiplication with zero for nonnegative fractions.  (Contributed by Jim
       Kingdon, 5-Nov-2019.) $)
    nq0m0r $p |- ( A e. Q0. -> ( 0Q0 .Q0 A ) = 0Q0 ) $=
      ( vw vv wcel cv com cnpi wa cop ceq0 cec wceq c0q0 cmq0 co c0 comu peano1
      c1o 1pi cnq0 wex nq0nn df-0nq0 oveq12 mpan mpanl12 sylan9eqr nnm0r oveq1d
      mulnnnq0 1onn ax-mp eqtrdi adantr cmi mulpiord mulclpi eqeltrrd pinn nnm0
      adantl eqtr4d wb eqeltrdi enq0eceq mpanr12 syl2an mpbird eqtr4di exlimivv
      3syl eqtrd syl ) AUADBEZFDZCEZGDZHZAVOVQIJKZLZHZCUBBUBMANOZMLZBCAUCWBWDBC
      WBWCPVOQOZSVQQOZIJKZMWAVSWCPSIJKZVTNOZWGMWHLWAWCWILUDMWHAVTNUEUFPFDZSGDZV
      SWIWGLRTPSVOVQUKUGUHVSWGMLWAVSWGWHMVSWGWHLZWESQOZWFPQOZLZVSWMPWNVPWMPLVRV
      PWMPSQOZPVPWEPSQVOUIZUJSFDWPPLULSUIUMUNUOVRWNPLZVPVRWFGDZWFFDWRWKVRWSTWKV
      RHSVQUPOWFGSVQUQSVQURUSUFZWFUTWFVAVLVBVCVPWEFDZWSWLWOVDZVRVPWEPFWQRVEWTXA
      WSHWJWKXBRTWEWFPSVFVGVHVIUDVJUOVMVKVN $.

    $( Addition with zero for nonnegative fractions.  (Contributed by Jim
       Kingdon, 5-Nov-2019.) $)
    nq0a0 $p |- ( A e. Q0. -> ( A +Q0 0Q0 ) = A ) $=
      ( vw vv wcel cv com cnpi wa cop ceq0 cec wceq c0q0 cplq0 c1o comu coa syl
      co c0 cnq0 wex nq0nn df-0nq0 oveq12 peano1 1pi addnnnq0 mpanr12 sylan9eqr
      mpan2 pinn nnm0 oveq2d nnm1 oveq1d nna0 eqtrd adantl eceq1d eqeq2d eqtr4d
      opeq12d biimpar exlimivv ) AUADBEZFDZCEZGDZHZAVFVHIZJKZLZHZCUBBUBAMNSZALZ
      BCAUCVNVPBCVNVOVFOPSZVHTPSZQSZVHOPSZIZJKZAVMVJVOVLTOIJKZNSZWBVMMWCLVOWDLU
      DAVLMWCNUEUKVJTFDOGDWDWBLUFUGVFVHTOUHUIUJVJAWBLVMVJWBVLAVJWAVKJVJVSVFVTVH
      VIVGVSVQTQSZVFVIVHFDZVSWELVHULZWFVRTVQQVHUMUNRVGWEVFTQSVFVGVQVFTQVFUOUPVF
      UQURUJVIVTVHLZVGVIWFWHWGVHUORUSVCUTVAVDVBVER $.
  $}

  $( Addition of nonnegative fractions with a common denominator.  You can add
     two fractions with the same denominator by adding their numerators and
     keeping the same denominator.  (Contributed by Jim Kingdon,
     1-Dec-2019.) $)
  nnanq0 $p |- ( ( N e. _om /\ M e. _om /\ A e. N. ) ->
      [ <. ( N +o M ) , A >. ] ~Q0 =
      ( [ <. N , A >. ] ~Q0 +Q0 [ <. M , A >. ] ~Q0 ) ) $=
    ( com wcel cnpi w3a cop ceq0 cec cplq0 co comu wceq addnnnq0 3impdir nnmcom
    coa pinn sylan2 3adant2 oveq1d nndi 3coml syl3an3 eqtr4d opeq1d simp3 nnacl
    eceq1d 3adant3 mulcanenq0ec syl3anc 3eqtrrd ) CDEZBDEZAFEZGZCAHIJBAHIJKLZCA
    MLZABMLZRLZAAMLZHZIJZACBRLZMLZVCHZIJZVFAHIJZUOUQUPUSVENCABAOPURVDVHIURVBVGV
    CURVBACMLZVARLZVGURUTVKVARUOUQUTVKNZUPUQUOADEZVMASZCAQTUAUBUQUOUPVNVGVLNZVO
    VNUOUPVPACBUCUDUEUFUGUJURUQVFDEZUQVIVJNUOUPUQUHZUOUPVQUQCBUIUKVRAVFAULUMUN
    $.

  ${
    $d A u v w x y z $.  $d B u v w x y z $.  $d C u v w x y z $.
    $d f g h u v w x y z $.
    $( Multiplication of nonnegative fractions is distributive.  (Contributed
       by Jim Kingdon, 27-Nov-2019.) $)
    distrnq0 $p |- ( ( A e. Q0. /\ B e. Q0. /\ C e. Q0. ) ->
        ( A .Q0 ( B +Q0 C ) ) = ( ( A .Q0 B ) +Q0 ( A .Q0 C ) ) ) $=
      ( cnq0 wcel cplq0 co cmq0 wceq wa cop ceq0 cec com cnpi comu nnmcl adantl
      cv syl vz vw vv vu vx vy vf vg vh df-nq0 oveq1 oveq2d oveq2 oveq1d imbi2d
      wi eqeq12d oveq12d w3a coa an42 anbi2i 3anass 3bitr4i sylan ancoms sylan2
      pinn syl3an simp1r simp1l 3ad2ant2 3ad2ant3 nnacl syl2anc syl3an1 syl3anc
      nnmass nnmcom adantr simpll ad2antlr simprl simprr caov4d 3adant3 3adant2
      nndi eqtr3d 3eqtr3d sylbir cmi mulpiord ad2ant2lr 3adant1 eqeltrrd opeq12
      mulclpi eceq1d addnnnq0 syl2an an42s ad2ant2l eleq1d mpbid mulnnnq0 simpl
      wb jca anim12i an12 bitr4i sylib an4s mulcanenq0ec eqtr4d eqtrd oveqan12d
      3impb 3impdi 3eqtr4d 3expib ecoptocl com12 2ecoptocl 3impib ) ADEZBDEZCDE
      ZABCFGZHGZABHGZACHGZFGZIZYHYIJYGYOYGAUASZUBSZKLMZUCSZUDSZKLMZFGZHGZAYRHGZ
      AUUAHGZFGZIZUPYGABUUAFGZHGZYLUUEFGZIZUPYGYOUPUAUBUCUDBCNOLDUJYRBIZUUGUUKY
      GUULUUCUUIUUFUUJUULUUBUUHAHYRBUUAFUKULUULUUDYLUUEFYRBAHUMUNUQUOUUACIZUUKY
      OYGUUMUUIYKUUJYNUUMUUHYJAHUUACBFUMULUUMUUEYMYLFUUACAHUMULUQUOYGYPNEZYQOEZ
      JZYSNEZYTOEZJZJZUUGUUTUESZUFSZKLMZUUBHGZUVCYRHGZUVCUUAHGZFGZIZUPUUTUUGUPU
      EUFANOLDUJUVCAIZUVHUUGUUTUVIUVDUUCUVGUUFUVCAUUBHUKUVIUVEUUDUVFUUEFUVCAYRH
      UKUVCAUUAHUKURUQUOUVANEZUVBOEZJZUUPUUSUVHUVLUUPUUSUSZUVBUVAYPYTPGZYQYSPGZ
      UTGZPGZPGZUVBUVBYQYTPGZPGZPGZKZLMZUVAYPPGZUVBYTPGZPGZUVBYQPGZUVAYSPGZPGZU
      TGZUWGUWEPGZKZLMZUVDUVGUVMUVRUWJIZUWAUWKIZUWCUWMIUVMUVLUUNUURJZUUOUUQJZUS
      ZUWNUVLUWPUWQJZJUVLUUTJZUWRUVMUWSUUTUVLUUNUURUUOUUQVAVBUVLUWPUWQVCUVLUUPU
      USVCVDZUWRUVBUVAPGZUVPPGZUXBUVNPGZUXBUVOPGZUTGZUVRUWJUVLUXBNEZUWPUVNNEZUW
      QUVONEZUXCUXFIUVKUVJUXGUVKUVBNEZUVJUXGUVBVHZUVBUVAQVEVFUURUUNYTNEZUXHYTVH
      ZYPYTQVGZUUOYQNEZUUQUXIYQVHZYQYSQVEZUXBUVNUVOWHVIUWRUVKUVJUVPNEZUXCUVRIZU
      VJUVKUWPUWQVJZUVJUVKUWPUWQVKUWRUXHUXIUXRUWPUVLUXHUWQUXNVLUWQUVLUXIUWPUXQV
      MUVNUVOVNZVOUVKUXJUVJUXRUXSUXKUVBUVAUVPVRVPVQUWRUXDUWFUXEUWIUTUVLUWPUXDUW
      FIUWQUVLUWPJZUVAUVBPGZUVNPGZUXDUWFUVLUYDUXDIUWPUVLUYCUXBUVNPUVKUVJUXJUYCU
      XBIUXKUVAUVBVSVGUNVTUYBUGUHUIUVAUVBYPYTNPUVJUVKUWPWAUVKUXJUVJUWPUXKWBUVLU
      UNUURWCUGSZNEZUHSZNEZJZUYEUYGPGZUYGUYEPGIZUYBUYEUYGVSZRUYFUYHUISZNEUSZUYJ
      UYMPGUYEUYGUYMPGPGIZUYBUYEUYGUYMVRZRUYBUURUXLUVLUUNUURWDZUXMTUYIUYJNEZUYB
      UYEUYGQZRWEWIWFUVLUWQUXEUWIIUWPUVLUWQJZUGUHUIUVBUVAYQYSNPUVKUXJUVJUWQUXKW
      BUVJUVKUWQWAUYTUUOUXOUVLUUOUUQWCZUXPTUYIUYKUYTUYLRUYNUYOUYTUYPRUVLUUOUUQW
      DUYIUYRUYTUYSRWEWGURWJWKUVMUWRUWOUXAUWRUVBUVBPGUVSPGZUWAUWKUWRUXJUXJUVSNE
      VUBUWAIUWRUVKUXJUXTUXKTZVUCUWRYQYTWLGZUVSNUWPUWQVUDUVSIZUVLUURUUOVUEUUNUU
      QUUOUURVUEYQYTWMZVFWNWOUWRVUDOEZVUDNEUWRUUOUURVUGUVLUWQUUOUWPVUAWGZUVLUWP
      UURUWQUYQWFZYQYTWRZVOVUDVHTWPUVBUVBUVSVRVQUWRUGUHUIUVBUVBYQYTNPVUCVUCUWRU
      UOUXOVUHUXPTUYIUYKUWRUYLRUYNUYOUWRUYPRUWRUURUXLVUIUXMTUYIUYRUWRUYSRWEWIWK
      UWNUWOJUWBUWLLUVRUWAUWJUWKWQWSVOUVLUUPUUSUVDUWCIUWTUVDUVCUVPUVSKLMZHGZUWC
      UUTUVDVULIUVLUUTUUBVUKUVCHYPYQYSYTWTULRUUTUVLUXRUVSOEZJZVULUWCIUUTUXRVUMU
      UNUURUUOUUQUXRUWPUXHUXIUXRUWQUXNUXQUYAXAXBUUTVUGVUMUUOUURVUGUUNUUQVUJXCUU
      OUURVUGVUMXHUUNUUQUUOUURJVUDUVSOVUFXDXCXEXIUVLVUNJZVULUVQUVTKLMZUWCUVAUVB
      UVPUVSXFVUOUVKUVQNEZUVTOEZUSZUWCVUPIUVJUXRUVKVUMVUSUVJUXRJZUVKVUMJZJVUQUV
      KVURJZJZVUSVUTVUQVVAVVBUVAUVPQVVAUVKVURUVKVUMXGVVAUVBUVSWLGUVTOUVBUVSWMUV
      BUVSWRWPXIXJVVCUVKVUQVURJJVUSVUQUVKVURXKUVKVUQVURVCXLXMXNUVBUVQUVTXOTXPVG
      XQXSUVLUUPUUSUVGUWMIUVLUUPJZUVLUUSJZJUVGUWDUWGKLMZUWHUWEKLMZFGZUWMVVDVVEU
      VEVVFUVFVVGFUVAUVBYPYQXFUVAUVBYSYTXFXRVVDUWDNEZUWGOEZJZUWHNEZUWEOEZJZVVHU
      WMIVVEUVJUUNUVKUUOVVKUVJUUNJVVIUVKUUOJZVVJUVAYPQVVOUVBYQWLGUWGOUVBYQWMUVB
      YQWRWPXJXNUVJUUQUVKUURVVNUVJUUQJVVLUVKUURJZVVMUVAYSQVVPUVBYTWLGUWEOUVBYTW
      MUVBYTWRWPXJXNUWDUWGUWHUWEWTXAXQXTYAYBYCYDYEYDYF $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Multiplication of nonnegative fractions is commutative.  (Contributed by
       Jim Kingdon, 27-Nov-2019.) $)
    mulcomnq0 $p |- ( ( A e. Q0. /\ B e. Q0. ) ->
        ( A .Q0 B ) = ( B .Q0 A ) ) $=
      ( vx vy vz vw cv cop ceq0 cec cmq0 co wceq com cnpi oveq1 oveq2 wcel comu
      wa cnq0 df-nq0 eqeq12d nnmcom ad2ant2r pinn syl2an ad2ant2l opeq12 eceq1d
      syl2anc mulnnnq0 ancoms 3eqtr4d 2ecoptocl ) CGZDGZHIJZEGZFGZHIJZKLZVAURKL
      ZMAVAKLZVAAKLZMABKLZBAKLZMCDEFABNOIUAUBURAMVBVDVCVEURAVAKPURAVAKQUCVABMVD
      VFVEVGVABAKQVABAKPUCUPNRZUQORZTZUSNRZUTORZTZTZUPUSSLZUQUTSLZHZIJZUSUPSLZU
      TUQSLZHZIJZVBVCVNVOVSMZVPVTMZVRWBMVHVKWCVIVLUPUSUDUEVIVLWDVHVKVIUQNRUTNRW
      DVLUQUFUTUFUQUTUDUGUHWCWDTVQWAIVOVPVSVTUIUJUKUPUQUSUTULVMVJVCWBMUSUTUPUQU
      LUMUNUO $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u f g h $.

    $( A natural number closure law.  Lemma for ~ addassnq0 .  (Contributed by
       Jim Kingdon, 3-Dec-2019.) $)
    addassnq0lemcl $p |- (
        ( ( I e. _om /\ J e. N. ) /\ ( K e. _om /\ L e. N. ) ) ->
        ( ( ( I .o L ) +o ( J .o K ) ) e. _om /\ ( J .o L ) e. N. ) ) $=
      ( com wcel cnpi wa comu co coa pinn nnmcl ad2ant2rl sylan ad2ant2lr nnacl
      sylan2 syl2anc cmi mulpiord mulclpi eqeltrrd ad2ant2l jca ) AEFZBGFZHCEFZ
      DGFZHHZADIJZBCIJZKJEFZBDIJZGFZUJUKEFZULEFZUMUFUIUPUGUHUIUFDEFUPDLADMRNUGU
      HUQUFUIUGBEFUHUQBLBCMOPUKULQSUGUIUOUFUHUGUIHBDTJUNGBDUABDUBUCUDUE $.

    $( Addition of nonnegative fractions is associative.  (Contributed by Jim
       Kingdon, 29-Nov-2019.) $)
    addassnq0 $p |- ( ( A e. Q0. /\ B e. Q0. /\ C e. Q0. ) ->
        ( ( A +Q0 B ) +Q0 C ) = ( A +Q0 ( B +Q0 C ) ) ) $=
      ( cnq0 wcel cplq0 co wceq wa cv cop ceq0 cec wi com cnpi comu coa syl2anc
      nnmcl vz vw vv vu vx vy vf vg vh df-nq0 oveq2 oveq1d oveq1 oveq2d eqeq12d
      imbi2d w3a simp1l simp2r simp3r simp1r simp2l simp3l nnaass nnmcom adantl
      pinn syl3anc nndir nnmass caovdilemd oveq12d nndi 3eqtr4d opeq12 addnnnq0
      syl eceq1d adantr addassnq0lemcl sylan eqtrd 3impa sylan2 3expib ecoptocl
      3impb com12 2ecoptocl 3impib ) ADEZBDEZCDEZABFGZCFGZABCFGZFGZHZWLWMIWKWRW
      KAUAJZUBJZKLMZFGZUCJZUDJZKLMZFGZAXAXEFGZFGZHZNWKWNXEFGZABXEFGZFGZHZNWKWRN
      UAUBUCUDBCOPLDUJXABHZXIXMWKXNXFXJXHXLXNXBWNXEFXABAFUKULXNXGXKAFXABXEFUMUN
      UOUPXECHZXMWRWKXOXJWOXLWQXECWNFUKXOXKWPAFXECBFUKUNUOUPWKWSOEZWTPEZIZXCOEZ
      XDPEZIZIZXIYBUEJZUFJZKLMZXAFGZXEFGZYEXGFGZHZNYBXINUEUFAOPLDUJYEAHZYIXIYBY
      JYGXFYHXHYJYFXBXEFYEAXAFUMULYEAXGFUMUOUPYCOEZYDPEZIZXRYAYIYMXRYAUQZYCWTQG
      YDWSQGRGZXDQGZYDWTQGZXCQGZRGZYQXDQGZKZLMZYCWTXDQGZQGZYDWSXDQGZWTXCQGZRGZQ
      GZRGZYDUUCQGZKZLMZYGYHYNYSUUIHZYTUUJHZUUBUULHYNUUDYDUUEQGZRGZYDUUFQGZRGZU
      UDUUOUUQRGZRGZYSUUIYNUUDOEZUUOOEZUUQOEZUURUUTHYNYKUUCOEZUVAYKYLXRYAURZYNW
      TOEZXDOEZUVDYNXQUVFYMXPXQYAUSWTVGVQZYNXTUVGYMXRXSXTUTXDVGVQZWTXDTSYCUUCTS
      YNYDOEZUUEOEZUVBYNYLUVJYKYLXRYAVAYDVGVQZYNXPUVGUVKYMXPXQYAVBZUVIWSXDTSZYD
      UUETSYNUVJUUFOEZUVCUVLYNUVFXSUVOUVHYMXRXSXTVCZWTXCTSZYDUUFTSUUDUUOUUQVDVH
      YNYPUUPYRUUQRYNUGUHUIYCYDWTWSORQXDUGJZOEZUHJZOEZIZUVRUVTQGZUVTUVRQGHYNUVR
      UVTVEVFUVSUWAUIJZOEUQZUVRUVTRGUWDQGUVRUWDQGUVTUWDQGZRGHYNUVRUVTUWDVIVFUWE
      UWCUWDQGUVRUWFQGHYNUVRUVTUWDVJVFUWBUWCOEYNUVRUVTTVFUVEUVLUVHUVMUVIVKYNUVJ
      UVFXSYRUUQHUVLUVHUVPYDWTXCVJVHVLYNUUHUUSUUDRYNUVJUVKUVOUUHUUSHUVLUVNUVQYD
      UUEUUFVMVHUNVNYNUVJUVFUVGUUNUVLUVHUVIYDWTXDVJVHUUMUUNIUUAUUKLYSYTUUIUUJVO
      VRSYMXRYAYGUUBHYMXRIZYAIYGYOYQKLMZXEFGZUUBUWGYGUWIHYAUWGYFUWHXEFYCYDWSWTV
      PULVSUWGYOOEYQPEIYAUWIUUBHYCYDWSWTVTYOYQXCXDVPWAWBWCYMXRYAYHUULHYMYBIYHYE
      UUGUUCKLMZFGZUULYBYHUWKHYMYBXGUWJYEFWSWTXCXDVPUNVFYBYMUUGOEUUCPEIUWKUULHW
      SWTXCXDVTYCYDUUGUUCVPWDWBWGVNWEWFWHWIWHWJ $.
  $}

  $( Multiplication of nonnegative fractions is distributive.  Version of
     ~ distrnq0 with the multiplications commuted.  (Contributed by Jim
     Kingdon, 29-Nov-2019.) $)
  distnq0r $p |- ( ( A e. Q0. /\ B e. Q0. /\ C e. Q0. ) ->
      ( ( B +Q0 C ) .Q0 A ) = ( ( B .Q0 A ) +Q0 ( C .Q0 A ) ) ) $=
    ( cnq0 wcel w3a cplq0 co cmq0 distrnq0 wceq addclnq0 mulcomnq0 sylan2 3impb
    wa 3adant3 3adant2 oveq12d 3eqtr3d ) ADEZBDEZCDEZFZABCGHZIHZABIHZACIHZGHUEA
    IHZBAIHZCAIHZGHABCJUAUBUCUFUIKZUBUCPUAUEDEULBCLAUEMNOUDUGUJUHUKGUAUBUGUJKUC
    ABMQUAUCUHUKKUBACMRST $.

  $( Addition of one to the numerator of a fraction whose denominator is one.
     (Contributed by Jim Kingdon, 26-Apr-2020.) $)
  addpinq1 $p |- ( A e. N. ->
      [ <. ( A +N 1o ) , 1o >. ] ~Q = ( [ <. A , 1o >. ] ~Q +Q 1Q ) ) $=
    ( cnpi wcel c1o cop ceq cec c1q cplq co cpli df-1nqqs oveq2i wceq addpipqqs
    cmi 1pi wa mulidpi ax-mp mpanr12 eqtrid opeq12i eqtrdi oveq1d opeq1d eceq1d
    mpan2 eceq1 eqtr2d ) ABCZADEFGZHIJZADPJZDKJZDEZFGZADKJZDEZFGUKUMUNDDPJZKJZU
    TEZFGZUQUKUMULDDEFGZIJZVCHVDULILMUKDBCZVEVCNZQUKVFRVFVFVGQQADDDOUAUHUBVBUPN
    VCUQNVAUOUTDUTDUNKVFUTDNQDSTZMVHUCVBUPFUITUDUKUPUSFUKUOURDUKUNADKASUEUFUGUJ
    $.

  ${
    $d A w z $.
    $( Multiply a nonnegative fraction by two.  (Contributed by Jim Kingdon,
       29-Nov-2019.) $)
    nq02m $p |- ( A e. Q0. ->
        ( [ <. 2o , 1o >. ] ~Q0 .Q0 A ) = ( A +Q0 A ) ) $=
      ( vz vw wcel cv com cnpi wa cop ceq0 cec wceq wex c2o c1o cmq0 cplq0 comu
      co adantr cnq0 nq0nn coa 2onn mulnnnq0 mpanl12 nn2m pinn 1onn nnmcom mpan
      1pi nnm1 eqtrd adantl opeq12d eceq1d nnanq0 3anidm12 3eqtrd oveq2 oveq12d
      syl wb id eqeq12d mpbird exlimivv ) AUADBEZFDZCEZGDZHZAVIVKIJKZLZHZCMBMNO
      IJKZAPSZAAQSZLZBCAUBVPVTBCVPVTVQVNPSZVNVNQSZLZVMWCVOVMWANVIRSZOVKRSZIZJKZ
      VIVIUCSZVKIZJKZWBNFDOGDVMWAWGLUDULNOVIVKUEUFVMWFWIJVMWDWHWEVKVJWDWHLVLVIU
      GTVLWEVKLZVJVLVKFDZWKVKUHWLWEVKORSZVKOFDWLWEWMLUIOVKUJUKVKUMUNVCUOUPUQVJV
      LWJWBLVKVIVIURUSUTTVOVTWCVDVMVOVRWAVSWBAVNVQPVAVOAVNAVNQVOVEZWNVBVFUOVGVH
      VC $.
  $}

  ${
    $d l u q r $.
    $( Define the set of positive reals.  A "Dedekind cut" is a partition of
       the positive rational numbers into two classes such that all the numbers
       of one class are less than all the numbers of the other.

       Here we follow the definition of a Dedekind cut from Definition 11.2.1
       of [HoTT], p.  (varies) with the one exception that we define it over
       positive rational numbers rather than all rational numbers.

       A Dedekind cut is an ordered pair of a lower set ` l ` and an upper set
       ` u ` which is inhabited ( ` E. q e. Q. q e. l /\ E. r e. Q. r e. u ` ),
       rounded ( ` A. q e. Q. ( q e. l <-> E. r e. Q. ( q <Q r /\ r e. l ) ) `
       and likewise for ` u ` ), disjoint
       ( ` A. q e. Q. -. ( q e. l /\ q e. u ) ` ) and located
       ( ` A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. l \/ r e. u ) ) ` ).  See
       HoTT for more discussion of those terms and different ways of defining
       Dedekind cuts.

       (Note:  This is a "temporary" definition used in the construction of
       complex numbers, and is intended to be used only by the construction.)
       (Contributed by Jim Kingdon, 25-Sep-2019.) $)
    df-inp $a |- P. = { <. l , u >. | ( ( ( l C_ Q. /\ u C_ Q. ) /\
      ( E. q e. Q. q e. l /\ E. r e. Q. r e. u ) ) /\ (
      ( A. q e. Q. ( q e. l <-> E. r e. Q. ( q <Q r /\ r e. l ) ) /\
        A. r e. Q. ( r e. u <-> E. q e. Q. ( q <Q r /\ q e. u ) ) ) /\
      A. q e. Q. -. ( q e. l /\ q e. u ) /\
      A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. l \/ r e. u ) )
      ) ) } $.
  $}

  ${
    $d l u $.
    $( Define the positive real constant 1.  This is a "temporary" set used in
       the construction of complex numbers and is intended to be used only by
       the construction.  (Contributed by Jim Kingdon, 25-Sep-2019.) $)
    df-i1p $a |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >. $.
  $}

  ${
    $d x y q r s $.
    $( Define addition on positive reals.  From Section 11.2.1 of [HoTT], p.
       (varies).  We write this definition to closely resemble the definition
       in HoTT although some of the conditions are redundant (for example,
       ` r e. ( 1st `` x ) ` implies ` r e. Q. ` ) and can be simplified as
       shown at ~ genpdf .

       This is a "temporary" set used in the construction of complex numbers,
       and is intended to be used only by the construction.  (Contributed by
       Jim Kingdon, 26-Sep-2019.) $)
    df-iplp $a |- +P. = ( x e. P. , y e. P. |->
      <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\
        s e. ( 1st ` y ) /\ q = ( r +Q s ) ) } ,
        { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\
        s e. ( 2nd ` y ) /\ q = ( r +Q s ) ) } >. ) $.

    $( Define multiplication on positive reals.  Here we use a simple
       definition which is similar to ~ df-iplp or the definition of
       multiplication on positive reals in Metamath Proof Explorer.  This is as
       opposed to the more complicated definition of multiplication given in
       Section 11.2.1 of [HoTT], p.  (varies), which appears to be motivated by
       handling negative numbers or handling modified Dedekind cuts in which
       locatedness is omitted.

       This is a "temporary" set used in the construction of complex numbers,
       and is intended to be used only by the construction.  (Contributed by
       Jim Kingdon, 29-Sep-2019.) $)
    df-imp $a |- .P. = ( x e. P. , y e. P. |->
      <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\
        s e. ( 1st ` y ) /\ q = ( r .Q s ) ) } ,
        { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\
        s e. ( 2nd ` y ) /\ q = ( r .Q s ) ) } >. ) $.
  $}

  ${
    $d x y q $.
    $( Define ordering on positive reals.  We define ` x <P y ` if there is a
       positive fraction ` q ` which is an element of the upper cut of ` x `
       and the lower cut of ` y ` .  From the definition of < in Section 11.2.1
       of [HoTT], p.  (varies).

       This is a "temporary" set used in the construction of complex numbers,
       and is intended to be used only by the construction.  (Contributed by
       Jim Kingdon, 29-Sep-2019.) $)
    df-iltp $a |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\
      E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) } $.
  $}

  ${
    $d l u q r $.
    $( Lemma for proving existence of reals.  (Contributed by Jim Kingdon,
       27-Sep-2019.) $)
    npsspw $p |- P. C_ ( ~P Q. X. ~P Q. ) $=
      ( vl vu vq vr cv cnq wss wa wcel wrex cltq wbr wb wral wn wo wi w3a copab
      velpw cpw cnp cxp simpll anbi12i sylibr ssopab2i df-inp df-xp 3sstr4i ) A
      EZFGZBEZFGZHZCEZUKIZCFJDEZUMIZDFJHZHUQUPURKLZURUKIHDFJMCFNUSVAUPUMIZHCFJM
      DFNHUQVBHOCFNVAUQUSPQDFNCFNRZHZABSUKFUAZIZUMVEIZHZABSUBVEVEUCVDVHABVDUOVH
      UOUTVCUDVFULVGUNAFTBFTUEUFUGBDCAUHABVEVEUIUJ $.
  $}

  $( Two reals are equal if and only if their lower and upper cuts are.
     (Contributed by Jim Kingdon, 11-Dec-2019.) $)
  preqlu $p |- ( ( A e. P. /\ B e. P. ) -> ( A = B <->
      ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) ) $=
    ( cnp wcel wa wceq c1st cfv c2nd cop cnq cpw npsspw sseli 1st2nd2 eqeqan12d
    cxp syl wb xp1st xp2nd opthg syl2anc adantr bitrd ) ACDZBCDZEABFAGHZAIHZJZB
    GHZBIHZJZFZUHUKFUIULFEZUFUGAUJBUMUFAKLZUPQZDZAUJFCUQAMNZAUPUPORUGBUQDBUMFCU
    QBMNBUPUPORPUFUNUOSZUGUFUHUPDZUIUPDZUTUFURVAUSAUPUPTRUFURVBUSAUPUPUARUHUIUK
    ULUPUPUBUCUDUE $.

  $( The class of positive reals is a set.  (Contributed by NM,
     31-Oct-1995.) $)
  npex $p |- P. e. _V $=
    ( cnp cnq cpw cxp nqex pwex xpex npsspw ssexi ) ABCZJDJJBEFZKGHI $.

  ${
    $d l u q r L $.  $d l u q r U $.
    $( Membership in positive reals.  (Contributed by Jim Kingdon,
       27-Sep-2019.) $)
    elinp $p |- ( <. L , U >. e. P. <-> ( ( ( L C_ Q. /\ U C_ Q. ) /\
        ( E. q e. Q. q e. L /\ E. r e. Q. r e. U ) ) /\ (
        ( A. q e. Q. ( q e. L <-> E. r e. Q. ( q <Q r /\ r e. L ) ) /\
          A. r e. Q. ( r e. U <-> E. q e. Q. ( q <Q r /\ q e. U ) ) ) /\
        A. q e. Q. -. ( q e. L /\ q e. U ) /\
        A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. L \/ r e. U ) )
        ) ) ) $=
      ( vl vu wcel cvv wa cnq wss cv wrex wb wral anbi1d rexbidv anbi12d anbi2d
      eleq2 cop cnp cltq wbr wn wo wi w3a cpw npsspw sseli opelxp sylib anim12i
      cxp elex syl nqex ssex ad2antrr copab df-inp eleq2i sseq1 bibi12d ralbidv
      notbid orbi1d imbi2d 2ralbidv 3anbi123d orbi2d opelopabg bitrid pm5.21nii
      wceq ) BAUAZUBGZBHGZAHGZIZBJKZAJKZIZDLZBGZDJMZCLZAGZCJMZIZIZWFWEWHUCUDZWH
      BGZIZCJMZNZDJOZWIWMWEAGZIZDJMZNZCJOZIZWFWSIZUEZDJOZWMWFWIUFZUGZCJODJOZUHZ
      IZVRBJUIZGZAXMGZIZWAVRVQXMXMUOZGXPUBXQVQUJUKBAXMXMULUMXNVSXOVTBXMUPAXMUPU
      NUQWDWAWKXKWBVSWCVTBJURUSAJURUSUNUTVRVQELZJKZFLZJKZIZWEXRGZDJMZWHXTGZCJMZ
      IZIZYCWMWHXRGZIZCJMZNZDJOZYEWMWEXTGZIZDJMZNZCJOZIZYCYNIZUEZDJOZWMYCYEUFZU
      GZCJODJOZUHZIZEFVAZGWAXLUBUUHVQFCDEVBVCUUGWBYAIZWGYFIZIZWRYRIZWFYNIZUEZDJ
      OZWMWFYEUFZUGZCJODJOZUHZIXLEFBAHHXRBVPZYHUUKUUFUUSUUTYBUUIYGUUJUUTXSWBYAX
      RBJVDPUUTYDWGYFUUTYCWFDJXRBWETZQPRUUTYSUULUUBUUOUUEUURUUTYMWRYRUUTYLWQDJU
      UTYCWFYKWPUVAUUTYJWOCJUUTYIWNWMXRBWHTSQVEVFPUUTUUAUUNDJUUTYTUUMUUTYCWFYNU
      VAPVGVFUUTUUDUUQDCJJUUTUUCUUPWMUUTYCWFYEUVAVHVIVJVKRXTAVPZUUKWLUUSXKUVBUU
      IWDUUJWKUVBYAWCWBXTAJVDSUVBYFWJWGUVBYEWICJXTAWHTZQSRUVBUULXDUUOXGUURXJUVB
      YRXCWRUVBYQXBCJUVBYEWIYPXAUVCUVBYOWTDJUVBYNWSWMXTAWETZSQVEVFSUVBUUNXFDJUV
      BUUMXEUVBYNWSWFUVDSVGVFUVBUUQXIDCJJUVBUUPXHWMUVBYEWIWFUVCVLVIVJVKRVMVNVO
      $.
  $}

  $( A positive real is an ordered pair of a lower cut and an upper cut.
     (Contributed by Jim Kingdon, 27-Sep-2019.) $)
  prop $p |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. ) $=
    ( cnp wcel c1st cfv c2nd cop cnq cpw cxp npsspw sseli 1st2nd2 eleq1 biimpcd
    wceq syl mpd ) ABCZAADEAFEGZPZTBCZSAHIZUCJZCUABUDAKLAUCUCMQUASUBATBNOR $.

  ${
    $d q r A $.
    $( Membership in positive reals, using ` 1st ` and ` 2nd ` to refer to the
       lower and upper cut.  (Contributed by Jim Kingdon, 3-Oct-2019.) $)
    elnp1st2nd $p |- ( A e. P. <-> ( (
        A e. ( ~P Q. X. ~P Q. ) /\
        ( E. q e. Q. q e. ( 1st ` A ) /\ E. r e. Q. r e. ( 2nd ` A ) ) ) /\ (
        ( A. q e. Q. ( q e. ( 1st ` A ) <->
            E. r e. Q. ( q <Q r /\ r e. ( 1st ` A ) ) ) /\
          A. r e. Q. ( r e. ( 2nd ` A ) <->
            E. q e. Q. ( q <Q r /\ q e. ( 2nd ` A ) ) ) ) /\
        A. q e. Q. -. ( q e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) /\
        A. q e. Q. A. r e. Q. ( q <Q r ->
          ( q e. ( 1st ` A ) \/ r e. ( 2nd ` A ) ) )
        ) ) ) $=
      ( cnp wcel cnq cpw cxp cv c1st cfv wrex wral wss simprd jca elpwid anim1i
      wa wb c2nd cltq wbr wn wo wi w3a npsspw sseli cop prop elinp sylib simpld
      wceq 1st2nd2 ad2antrr xp1st xp2nd sylibr eqeltrd impbii ) ADEZAFGZVDHZEZC
      IZAJKZEZCFLBIZAUAKZEZBFLSZSZVIVGVJUBUCZVJVHESBFLTCFMVLVOVGVKEZSCFLTBFMSVI
      VPSUDCFMVOVIVLUEUFBFMCFMUGZSZVCVNVQVCVFVMDVEAUHUIVCVHFNZVKFNZSZVMVCWAVMSZ
      VQVCVHVKUJZDEZWBVQSZAUKVKVHBCULZUMZUNOPVCWBVQWGOPVRAWCDVFAWCUOVMVQAVDVDUP
      UQVRWEWDVNWBVQVFWAVMVFVSVTVFVHFAVDVDURQVFVKFAVDVDUSQPRRWFUTVAVB $.
  $}

  ${
    $d x y L $.  $d U x y $.
    $( A positive real's lower cut is inhabited.  (Contributed by Jim Kingdon,
       27-Sep-2019.) $)
    prml $p |- ( <. L , U >. e. P. -> E. x e. Q. x e. L ) $=
      ( vy cop cnp wcel cnq wss wa cv wrex cltq wbr wb wral wn wo wi w3a elinp
      simplrl sylbi ) CBEFGCHIBHIJZAKZCGZAHLZDKZBGZDHLZJJUFUEUHMNZUHCGJDHLOAHPU
      IUKUEBGZJAHLODHPJUFULJQAHPUKUFUIRSDHPAHPTZJUGBCDAUAUDUGUJUMUBUC $.

    $( A positive real's upper cut is inhabited.  (Contributed by Jim Kingdon,
       27-Sep-2019.) $)
    prmu $p |- ( <. L , U >. e. P. -> E. x e. Q. x e. U ) $=
      ( vy cop cnp wcel cnq wss wa cv wrex cltq wbr wb wral wn wo wi w3a elinp
      simplrr sylbi ) CBEFGCHIBHIJZDKZCGZDHLZAKZBGZAHLZJJUFUEUHMNZUHCGJAHLODHPU
      IUKUEBGZJDHLOAHPJUFULJQDHPUKUFUIRSAHPDHPTZJUJBCADUAUDUGUJUMUBUC $.

    $( The lower cut of a positive real is a subset of the positive fractions.
       (Contributed by Jim Kingdon, 28-Sep-2019.) $)
    prssnql $p |- ( <. L , U >. e. P. -> L C_ Q. ) $=
      ( vx vy cop cnp wcel cnq wss wa cv wrex cltq wbr wb wral wn wo wi w3a
      elinp simplll sylbi ) BAEFGBHIZAHIZJCKZBGZCHLDKZAGZDHLJZJUGUFUHMNZUHBGJDH
      LOCHPUIUKUFAGZJCHLODHPJUGULJQCHPUKUGUIRSDHPCHPTZJUDABDCUAUDUEUJUMUBUC $.

    $( The upper cut of a positive real is a subset of the positive fractions.
       (Contributed by Jim Kingdon, 28-Sep-2019.) $)
    prssnqu $p |- ( <. L , U >. e. P. -> U C_ Q. ) $=
      ( vx vy cop cnp wcel cnq wss wa cv wrex cltq wbr wb wral wn wo wi w3a
      elinp simpllr sylbi ) BAEFGBHIZAHIZJCKZBGZCHLDKZAGZDHLJZJUGUFUHMNZUHBGJDH
      LOCHPUIUKUFAGZJCHLODHPJUGULJQCHPUKUGUIRSDHPCHPTZJUEABDCUAUDUEUJUMUBUC $.
  $}

  $( An element of a positive real's lower cut is a positive fraction.
     (Contributed by Jim Kingdon, 28-Sep-2019.) $)
  elprnql $p |- ( ( <. L , U >. e. P. /\ B e. L ) -> B e. Q. ) $=
    ( cop cnp wcel cnq prssnql sselda ) CBDEFCGABCHI $.

  $( An element of a positive real's upper cut is a positive fraction.
     (Contributed by Jim Kingdon, 28-Sep-2019.) $)
  elprnqu $p |- ( ( <. L , U >. e. P. /\ B e. U ) -> B e. Q. ) $=
    ( cop cnp wcel cnq prssnqu sselda ) CBDEFBGABCHI $.

  $( The empty set is not a positive real.  (Contributed by NM,
     15-Nov-1995.) $)
  0npr $p |- -. (/) e. P. $=
    ( vx c0 cnp wcel c1st cfv c2nd cop cv cnq wrex noel 1st0 eleq2i mtbir rexex
    wex nex mto prml prop ) BCDBEFZBGFZHCDZUDAIZUBDZAJKZUGUFAQUFAUFUEBDUELUBBUE
    MNORUFAJPSAUCUBTSBUAS $.

  ${
    $d b c L $.  $d b c U $.  $d b c B $.  $d b c C $.
    $( A lower cut is closed downwards under the positive fractions.
       (Contributed by Jim Kingdon, 28-Sep-2019.) $)
    prcdnql $p |- ( ( <. L , U >. e. P. /\ B e. L ) ->
        ( C <Q B -> C e. L ) ) $=
      ( vc vb wcel wa cltq wbr cnq ltrelnq brel cv wi wceq eleq1 wrex wb wral
      cop cnp simpld adantl breq1 imbi12d imbi2d ancomd an42 anbi12d rspcev wss
      breq2 wn wo w3a elinp simpr1l sylbi r19.21bi syl5ibrcom 3impb 3com12 impd
      3expib biimtrid mpand com12 ancoms vtoclg mpcom ex ) DCUAUBGZADGZHZBAIJZB
      DGZBKGZVOVPHVQVPVRVOVPVRAKGZBAKKILMUCUDVRVOVPVQVOENZAIJZVTDGZOZOVOVPVQOZO
      EBKVTBPZWCWDVOWEWAVPWBVQVTBAIUEVTBDQUFUGVNVMWCWAVNVMHZWBWAVSVTKGZHZWFWBWA
      WGVSVTAKKILMUHWHWFHVSVNHZVMWGHZHWAWBVSWGVNVMUIWAWIWJWBWAVSVNWJWBOZVSWAVNW
      KVSWAVNWKVSWAVNHZHWBWJVTFNZIJZWMDGZHZFKRZWPWLFAKWMAPWNWAWOVNWMAVTIUMWMADQ
      UJUKVMWBWQSZEKVMDKULCKULHWBEKRWMCGZFKRHHZWREKTZWSWNVTCGZHEKRSFKTZHWBXBHUN
      EKTZWNWBWSUOOFKTEKTZUPHXACDFEUQXAXCXDXEWTURUSUTVAVBVCVEVDVFVGVHVIVJVDVKVL
      $.
  $}

  ${
    $d b c L $.  $d b c U $.  $d b c B $.  $d b c C $.
    $( An upper cut is closed upwards under the positive fractions.
       (Contributed by Jim Kingdon, 25-Nov-2019.) $)
    prcunqu $p |- ( ( <. L , U >. e. P. /\ C e. U ) ->
        ( C <Q B -> B e. U ) ) $=
      ( vb vc wcel wa cltq wbr cnq ltrelnq brel cv wi wceq eleq1 wrex wb wral
      cop cnp simprd adantl breq2 imbi12d imbi2d breq1 anbi12d rspcev wss wn wo
      an42 w3a elinp simpr1r sylbi r19.21bi syl5ibrcom 3impb 3com12 3expib impd
      biimtrid mpand com12 ancoms vtoclg mpcom ex ) DCUAUBGZBCGZHZBAIJZACGZAKGZ
      VNVOHVPVOVQVNVOBKGZVQBAKKILMUCUDVQVNVOVPVNBENZIJZVSCGZOZOVNVOVPOZOEAKVSAP
      ZWBWCVNWDVTVOWAVPVSABIUEVSACQUFUGVMVLWBVTVMVLHZWAVTVRVSKGZHZWEWABVSKKILMW
      GWEHVRVMHZVLWFHZHVTWAVRWFVMVLUNVTWHWIWAVTVRVMWIWAOZVRVTVMWJVRVTVMWJVRVTVM
      HZHWAWIFNZVSIJZWLCGZHZFKRZWOWKFBKWLBPWMVTWNVMWLBVSIUHWLBCQUIUJVLWAWPSZEKV
      LDKUKCKUKHWLDGZFKRWAEKRHHZWRWMVSDGHEKRSFKTZWQEKTZHWRWNHULFKTZWMWRWAUMOEKT
      FKTZUOHXACDEFUPWTXAXBXCWSUQURUSUTVAVBVCVDVEVFVGVHVIVDVJVK $.
  $}

  $( A positive fraction not in a lower cut is an upper bound.  (Contributed by
     Jim Kingdon, 29-Sep-2019.) $)
  prubl $p |- ( ( ( <. L , U >. e. P. /\ B e. L ) /\ C e. Q. ) ->
      ( -. C e. L -> B <Q C ) ) $=
    ( cop cnp wcel wa cnq wn wceq cltq wbr wo eleq1 biimpcd adantl prcdnql jaod
    wi con3d adantr wb elprnql nqtric sylan sylibrd ) DCEFGZADGZHZBIGZHBDGZJZAB
    KZBALMZNZJZABLMZUJUMUQTUKUJUPULUJUNULUOUIUNULTUHUNUIULABDOPQABCDRSUAUBUJAIG
    UKURUQUCACDUDABUEUFUG $.

  ${
    $d C q $.  $d L q r $.  $d U q r $.
    $( An element of a lower cut is less than an element of the corresponding
       upper cut.  (Contributed by Jim Kingdon, 15-Oct-2019.) $)
    prltlu $p |- ( ( <. L , U >. e. P. /\ B e. L /\ C e. U ) -> B <Q C ) $=
      ( vq vr wcel w3a wn cltq wbr wa wi cv cnq eleq1 wral wss wrex wb cop wceq
      cnp simp3 anbi12d notbid wo elinp simpr2 3ad2ant1 elprnqu 3adant2 rspcdva
      sylbi ancom notbii imnan bitr4i sylib mpd 3simpa prubl syl2anc ) DCUAUCGZ
      ADGZBCGZHZBDGZIZABJKZVGVFVIVDVEVFUDVGVHVFLZIZVFVIMZVGENZDGZVNCGZLZIZVLEOB
      VNBUBZVQVKVSVOVHVPVFVNBDPVNBCPUEUFVDVEVREOQZVFVDDORCORLVOEOSFNZCGZFOSLLZV
      OVNWAJKZWADGLFOSTEOQWBWDVPLEOSTFOQLZVTWDVOWBUGMFOQEOQZHLVTCDFEUHWCWEVTWFU
      IUNUJVDVFBOGZVEBCDUKULZUMVLVFVHLZIVMVKWIVHVFUOUPVFVHUQURUSUTVGVDVELWGVIVJ
      MVDVEVFVAWHABCDVBVCUT $.
  $}

  ${
    $d x y B $.  $d x y L $.  $d x y U $.
    $( A lower cut has no largest member.  (Contributed by Jim Kingdon,
       29-Sep-2019.) $)
    prnmaxl $p |- ( ( <. L , U >. e. P. /\ B e. L ) -> E. x e. L B <Q x ) $=
      ( vy cop wcel wa cv cltq wbr wex wrex cnq wb wral wi wss df-rex sylibr wn
      cnp elprnql wo w3a elinp simpr1l sylbi eleq1 breq1 anbi1d rexbidv bibi12d
      wceq rspcv biimp syl56 impd mpcom sylib brel simprd pm4.71ri anbi1i ancom
      ltrelnq anass 3bitr3i exbii ) DCFUBGZBDGZHZAIZDGZBVMJKZHZALZVOADMVLVMNGZV
      OVNHZHZALZVQVLVSANMZWABNGZVLWBBCDUCWCVJVKWBVJEIZDGZWDVMJKZVNHZANMZOZENPZW
      CVKWBOZVKWBQVJDNRCNRHWEENMVMCGZANMHHZWJWLWFWDCGZHENMOANPZHWEWNHUAENPZWFWE
      WLUDQANPENPZUEHWJCDAEUFWJWOWPWQWMUGUHWIWKEBNWDBUNZWEVKWHWBWDBDUIWRWGVSANW
      RWFVOVNWDBVMJUJUKULUMUOVKWBUPUQURUSVSANSUTVPVTAVSVRVOHZVNHVPVTVOWSVNVOVRV
      OWCVRBVMNNJVFVAVBVCVDVOVNVEVRVOVNVGVHVITVOADST $.

    $( An upper cut has no smallest member.  (Contributed by Jim Kingdon,
       7-Nov-2019.) $)
    prnminu $p |- ( ( <. L , U >. e. P. /\ B e. U ) -> E. x e. U x <Q B ) $=
      ( vy cop wcel wa cv cltq wbr wex wrex cnq wb wral wi wss df-rex sylibr wn
      cnp elprnqu wo w3a elinp simpr1r sylbi eleq1 breq2 anbi1d rexbidv bibi12d
      wceq rspcv biimp syl56 impd mpcom sylib brel simpld pm4.71ri anbi1i ancom
      ltrelnq anass 3bitr3i exbii ) DCFUBGZBCGZHZAIZCGZVMBJKZHZALZVOACMVLVMNGZV
      OVNHZHZALZVQVLVSANMZWABNGZVLWBBCDUCWCVJVKWBVJEIZCGZVMWDJKZVNHZANMZOZENPZW
      CVKWBOZVKWBQVJDNRCNRHVMDGZANMWEENMHHZWLWFWDDGHENMOANPZWJHWLVNHUAANPZWFWLW
      EUDQENPANPZUEHWJCDEAUFWNWJWOWPWMUGUHWIWKEBNWDBUNZWEVKWHWBWDBCUIWQWGVSANWQ
      WFVOVNWDBVMJUJUKULUMUOVKWBUPUQURUSVSANSUTVPVTAVSVRVOHZVNHVPVTVOWRVNVOVRVO
      VRWCVMBNNJVFVAVBVCVDVOVNVEVRVOVNVGVHVITVOACST $.
  $}

  ${
    $d x y B $.  $d x y L $.  $d x y U $.
    $( A lower cut has no largest member.  Addition version.  (Contributed by
       Jim Kingdon, 29-Sep-2019.) $)
    prnmaddl $p |- ( ( <. L , U >. e. P. /\ B e. L ) ->
        E. x e. Q. ( B +Q x ) e. L ) $=
      ( vy cop cnp wcel wa cv cltq wbr wrex cplq co prnmaxl wceq ltexnqi eleq1a
      cnq reximdv syl5 rexlimiv syl ) DCFGHBDHIBEJZKLZEDMBAJNOZDHZATMZEBCDPUFUI
      EDUFUGUEQZATMUEDHZUIABUERUKUJUHATUEDUGSUAUBUCUD $.
  $}

  ${
    $d A q r $.  $d B q r $.  $d L q r $.  $d U q r $.
    $( A Dedekind cut is located.  (Contributed by Jim Kingdon,
       23-Oct-2019.) $)
    prloc $p |- ( ( <. L , U >. e. P. /\ A <Q B ) -> ( A e. L \/ B e. U ) ) $=
      ( vq vr wcel cltq wbr wa cv wo wi cnq wral wss wrex wb simpr adantl elinp
      cop cnp wn w3a simpr3 sylbi adantr ltrelnq brel simpld wceq breq1d eleq1d
      orbi1d imbi12d ralbidv rspcdv simprd breq2d orbi2d syld mp2d ) DCUBUCGZAB
      HIZJZEKZFKZHIZVGDGZVHCGZLZMZFNOZENOZVEADGZBCGZLZVDVOVEVDDNPCNPJVJENQVKFNQ
      JJZVJVIVHDGJFNQRENOVKVIVGCGZJENQRFNOJZVJVTJUDENOZVOUEJVOCDFEUAVSWAWBVOUFU
      GUHVDVESVFVOAVHHIZVPVKLZMZFNOZVEVRMZVFVNWFEANVEANGZVDVEWHBNGZABNNHUIUJZUK
      TVFVGAULZJZVMWEFNWLVIWCVLWDWLVGAVHHVFWKSZUMWLVJVPVKWLVGADWMUNUOUPUQURVFWE
      WGFBNVEWIVDVEWHWIWJUSTVFVHBULZJZWCVEWDVRWOVHBAHVFWNSZUTWOVKVQVPWOVHBCWPUN
      VAUPURVBVC $.
  $}

  ${
    $d A q r $.  $d L q r $.  $d U q r $.
    $( A Dedekind cut is disjoint.  (Contributed by Jim Kingdon,
       15-Dec-2019.) $)
    prdisj $p |- ( ( <. L , U >. e. P. /\ A e. Q. ) ->
        -. ( A e. L /\ A e. U ) ) $=
      ( vq vr cop cnp wcel cnq wa wn cv wi wceq eleq1 anbi2d wss wrex wb wral
      anbi12d notbid imbi12d cltq wbr wo w3a elinp simpr2 sylbi r19.21bi vtoclg
      anabsi7 ) CBFGHZAIHZACHZABHZJZKZUNDLZIHZJZUTCHZUTBHZJZKZMUNUOJZUSMDAIUTAN
      ZVBVGVFUSVHVAUOUNUTAIOPVHVEURVHVCUPVDUQUTACOUTABOUAUBUCUNVFDIUNCIQBIQJVCD
      IRELZBHZEIRJJZVCUTVIUDUEZVICHJEIRSDITVJVLVDJDIRSEITJZVFDITZVLVCVJUFMEITDI
      TZUGJVNBCEDUHVKVMVNVOUIUJUKULUM $.
  $}

  $( Two possible ways of contracting an interval which straddles a Dedekind
     cut.  Lemma for ~ prarloc .  (Contributed by Jim Kingdon, 10-Nov-2019.) $)
  prarloclemlt $p |- ( ( ( X e. _om
      /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) )
      /\ y e. _om ) ->
      ( A +Q ( [ <. ( y +o 1o ) , 1o >. ] ~Q .Q P ) ) <Q
      ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) ) $=
    ( com wcel cnq wa c1o coa co ceq c2o wbr cnpi 1pi wceq syl2anc cop cnp cltq
    w3a cv cec cmq cplq cmi clti wss 2onn nnacl mpan2 nnaword1 sylan csuc sucid
    1oex df-2o eleqtrri wi nnaordi mpan mpi adantr sseldd ancoms nnppipi adantl
    wb o1p1e2 1onn mp2an eqeltrri pinn syl nnacom sylan2 ltpiord mpbird mulidpi
    eqeltrrd mulcompig sylancl eqtr3d breq12d simpr ordpipqqs mpanl2 mpanr2 cxp
    adantlr cqs opelxpi enqex ecelqsi df-nqqs eleqtrrdi simplr3 mpbid mulcomnqg
    ltmnqg syl3anc mulclnq simplr1 simplr2 elprnql ltanqg ) FGHZEDUAUBHZBEHZCIH
    ZUDZJAUEZGHZJZXOKLMZKUAZNUFZCUGMZXOOLMZFLMZKUAZNUFZCUGMZUCPZBYAUHMBYFUHMUCP
    ZXQCXTUGMZCYEUGMZUCPZYGXQXTYEUCPZYKXJXPYLXNXJXPJZYLXRKUIMZKYCUIMZUJPZYMYPXR
    YCUJPZYMYQXRYCHZXPXJYRXPXJJYBYCXRXPYBGHZXJYBYCUKXPOGHZYSULXOOUMUNYBFUOUPXPX
    RYBHZXJXPKOHZUUAKKUQOKUSURUTVAYTXPUUBUUAVBULKOXOVCVDVEVFVGVHYMXRQHZYCQHZYQY
    RVKXPUUCXJXPKQHZUUCRXOKVIUNZVJZYMFYBLMZYCQXPXJYSUUHYCSXPYBQHZYSXPOQHUUIKKLM
    ZOQVLKGHUUEUUJQHVMRKKVIVNVOXOOVIUNZYBVPVQFYBVRVSXPXJUUIUUHQHUUKFYBVIVSWCZXR
    YCVTTWAYMYNXRYOYCUJYMUUCYNXRSUUGXRWBVQYMYCKUIMZYOYCYMUUDUUEUUMYOSUULRYCKWDW
    EYMUUDUUMYCSUULYCWBVQWFWGWAYMXPUUDYLYPVKZXJXPWHUULXPUUCUUDUUNUUFUUCUUDUUEUU
    NRUUCUUEUUDUUEJUUNRXRKYCKWIWJWKUPTWAWMXQXTIHZYEIHZXMYLYKVKXJXPUUOXNYMXTQQWL
    ZNWNZIYMXSUUQHZXTUURHYMUUCUUEUUSUUGRXRKQQWOWEUUQXSNWPWQVQWRWSWMZXJXPUUPXNYM
    YEUURIYMYDUUQHZYEUURHYMUUDUUEUVAUULRYCKQQWOWEUUQYDNWPWQVQWRWSWMZXKXLXMXJXPW
    TZXTYECXCXDXAXQYIYAYJYFUCXQXMUUOYIYASUVCUUTCXTXBTXQXMUUPYJYFSUVCUVBCYEXBTWG
    XAXQYAIHZYFIHZBIHZYGYHVKXQUUOXMUVDUUTUVCXTCXETXQUUPXMUVEUVBUVCYECXETXQXKXLU
    VFXKXLXMXJXPXFXKXLXMXJXPXGBDEXHTYAYFBXIXDXA $.

  ${
    $d A f g h y z $.  $d L f g h y z $.  $d P f g h y z $.  $d U f g h y z $.
    $d X f g h y z $.
    $( Contracting the lower side of an interval which straddles a Dedekind
       cut.  Lemma for ~ prarloc .  (Contributed by Jim Kingdon,
       10-Nov-2019.) $)
    prarloclemlo $p |- ( ( ( X e. _om
      /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) )
      /\ y e. _om ) -> (
      ( A +Q ( [ <. ( y +o 1o ) , 1o >. ] ~Q .Q P ) ) e. L -> (
        ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
          /\ ( A +Q
            ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) ->
        E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
        /\ ( A +Q
          ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ) ) $=
      ( com wcel cop wa c1o coa co ceq cec cmq cplq c2o wceq oveq1d vf vg vh vz
      cnp cnq w3a cv ceq0 cmq0 cplq0 csuc wrex nnaass adantl simpr 1onn sylancl
      wi nnacl 2onn a1i simpll caovassd nnacom caov4d caovcomd con0 nnon oa1suc
      eqtrd oveq2d 3eqtr2rd opeq1d eceq1d eleq1d biimpd simplr1 simplr2 elprnql
      3syl syl2anc cnpi cxp 1pi nnppipi opelxpi mpan2 cqs enqex ecelqsi df-nqqs
      eleqtrrdi simplr3 mulclnq nqnq0a nqnq0m nqnq0pi eqtr4d anbi1d opeq1 oveq1
      anbi12d rspcev ex syl sylbid cbvrexv imbitrdi sylan2d expdimp adantld ) F
      GHZEDIUEHZBEHZCUFHZUGZJZAUHZGHZJZBXSKLMZKIZNOZCPMZQMZEHZBXSKIZUIOZCUJMZUK
      MZEHZBXSRLMZFULZLMZKIZNOZCPMZQMZDHZJYLBYMFLMZKIZNOZCPMZQMZDHZJZAGUMZUSYAY
      GJYTUUHYLYAYGYTUUHYAYTBYBRLMZFLMZKIZNOZCPMZQMZDHZYGUUHYAYTUUOYAYSUUNDYAYR
      UUMBQYAYQUULCPYAYPUUKNYAYOUUJKYAUUJYBRFLMLMYMKFLMZLMYOYAUAUBUCYBRFGLUAUHZ
      GHZUBUHZGHZUCUHZGHUGUUQUUSLMZUVALMUUQUUSUVALMLMSYAUUQUUSUVAUNUOZYAXTKGHZY
      BGHZXRXTUPZUQXSKUTURZRGHYAVAVBZXMXQXTVCZVDYAUAUBUCXSRKFGLUVFUVHUVDYAUQVBZ
      UURUUTJZUVBUUSUUQLMSYAUUQUUSVEUOZUVCUVIUVKUVBGHYAUUQUUSUTUOVFYAUUPYNYMLYA
      UUPFKLMZYNYAUAUBKFGLUVLUVJUVIVGYAXMFVHHUVMYNSUVIFVIFVJWAVKVLVMVNVOTVLVPVQ
      YAYGUUOJZBUDUHZKIZUIOZCUJMZUKMZEHZBUVORLMZFLMZKIZNOZCPMZQMZDHZJZUDGUMZUUH
      YAUVNBYCUIOZCUJMZUKMZEHZUUOJZUWIYAYGUWMUUOYAYFUWLEYAYFBYEUKMZUWLYABUFHZYE
      UFHZYFUWOSYAXNXOUWPXNXOXPXMXTVRXNXOXPXMXTVSBDEVTWBYAYDUFHZXPUWQYAYBWCHZYC
      WCWCWDZHZUWRYAXTKWCHZUWSUVFWEXSKWFURZUWSUXBUXAWEYBKWCWCWGWHUXAYDUWTNWIUFU
      WTYCNWJWKWLWMWAZXNXOXPXMXTWNZYDCWOWBBYEWPWBYAYEUWKBUKYAYEYDCUJMZUWKYAUWRX
      PYEUXFSUXDUXEYDCWQWBYAUWJYDCUJYAUWSUXBUWJYDSUXCWEYBKWRURTWSVLVKVPWTYAUVEU
      WNUWIUSUVGUVEUWNUWIUWHUWNUDYBGUVOYBSZUVTUWMUWGUUOUXGUVSUWLEUXGUVRUWKBUKUX
      GUVQUWJCUJUXGUVPYCUIUVOYBKXAVOTVLVPUXGUWFUUNDUXGUWEUUMBQUXGUWDUULCPUXGUWC
      UUKNUXGUWBUUJKUXGUWAUUIFLUVOYBRLXBTVNVOTVLVPXCXDXEXFXGUWHUUGUDAGUVOXSSZUV
      TYLUWGUUFUXHUVSYKEUXHUVRYJBUKUXHUVQYICUJUXHUVPYHUIUVOXSKXAVOTVLVPUXHUWFUU
      EDUXHUWEUUDBQUXHUWDUUCCPUXHUWCUUBNUXHUWBUUAKUXHUWAYMFLUVOXSRLXBTVNVOTVLVP
      XCXHXIXJXKXLXE $.
  $}

  $( Contracting the upper side of an interval which straddles a Dedekind cut.
     Lemma for ~ prarloc .  (Contributed by Jim Kingdon, 10-Nov-2019.) $)
  prarloclemup $p |- ( ( ( X e. _om
      /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) )
      /\ y e. _om ) -> (
      ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U -> (
        ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
          /\ ( A +Q
            ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) ->
        E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
        /\ ( A +Q
          ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ) ) $=
    ( com wcel cop cnp cnq w3a wa coa co c1o ceq cec cmq cplq cv c2o ceq0 cplq0
    cmq0 csuc wrex simpllr simprl simplr rspe syl12anc exp31 ) FGHEDIJHBEHCKHLM
    ZAUAZGHZMZBUOUBNOZFNOPIQRCSOTODHZBUOPIUCRCUEOUDOEHZBURFUFNOPIQRCSOTODHZMZUT
    USMZAGUGZUQUSMZVBMUPUTUSVDUNUPUSVBUHVEUTVAUIUQUSVBUJVCAGUKULUM $.

  ${
    $d A y $.  $d L y $.  $d P y $.  $d U y $.  $d X y $.
    $( Induction step for ~ prarloclem3 .  (Contributed by Jim Kingdon,
       9-Nov-2019.) $)
    prarloclem3step $p |- (
      ( ( X e. _om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) )
      /\ E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
        /\ ( A +Q
          ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ->
      E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
        /\ ( A +Q
          ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) $=
      ( com wcel cop wa c1o cec co coa ceq cmq cplq wrex wi ex cnp cnq w3a ceq0
      cmq0 cplq0 c2o csuc nfv nfre1 prarloclemlo prarloclemup cltq prarloclemlt
      cv wbr wo prloc 3ad2ant1 ad2antlr mpd mpjaod rexlimd imp ) FGHZEDIUAHZBEH
      ZCUBHZUCZJZBAUOZKIUDLCUEMUFMEHZBVKUGNMZFUHNMKIOLCPMQMDHJZAGRVLBVMFNMKIOLC
      PMQMZDHZJZAGRZVJVNVRAGVJAUIVQAGUJVJVKGHZVNVRSZVJVSJZBVKKNMKIOLCPMQMZEHZVT
      VPABCDEFUKABCDEFULWAWBVOUMUPZWCVPUQZABCDEFUNVIWDWESZVEVSVFVGWFVHVFWDWEWBV
      ODEURTUSUTVAVBTVCVD $.
  $}

  ${
    $d A j x y z $.  $d L j x y z $.  $d P j x y z $.  $d U j x y z $.
    $d X x y $.
    $( Contracting an interval which straddles a Dedekind cut.  Lemma for
       ~ prarloc .  (Contributed by Jim Kingdon, 27-Oct-2019.) $)
    prarloclem3 $p |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\
      ( X e. _om /\ P e. Q. ) /\
      E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
        /\ ( A +Q
          ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ->
      E. j e. _om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 .Q0 P ) ) e. L /\
        ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) ) $=
      ( cop wcel wa com c1o cec co coa ceq cmq cplq wrex wi vx vz cnp ceq0 cmq0
      cnq cv cplq0 c2o simprl simpll simplr simprr w3a wceq oveq2 opeq1d eceq1d
      oveq1d oveq2d eleq1d anbi2d rexbidv imbi1d imbi2d c0 csuc 2onn nnacl nna0
      mpan2 rexbiia opeq1 oveq1 anbi12d cbvrexv bitri biimpi prarloclem3step ex
      syl a1i imim1d finds2 vtoclga imp syl13anc 3impia ) FDHUCIZBFIZJZGKIZCUFI
      ZJZBAUGZLHZUDMZCUENZUHNZFIZBWOUIONZGONZLHZPMZCQNZRNZDIZJZAKSZBEUGZLHZUDMZ
      CUENZUHNZFIZBXJUIONZLHZPMZCQNZRNZDIZJZEKSZWKWNJWLWIWJWMXIYCTZWKWLWMUJWIWJ
      WNUKWIWJWNULWKWLWMUMWLWIWJWMUNZYDYEWTBXAUAUGZONZLHZPMZCQNZRNZDIZJZAKSZYCT
      ZTYEYDTUAGKYFGUOZYOYDYEYPYNXIYCYPYMXHAKYPYLXGWTYPYKXFDYPYJXEBRYPYIXDCQYPY
      HXCPYPYGXBLYFGXAOUPUQURUSUTVAVBVCVDVEYOWTBXAVFONZLHZPMZCQNZRNZDIZJZAKSZYC
      TZWTBXAUBUGZONZLHZPMZCQNZRNZDIZJZAKSZYCTZWTBXAUUFVGZONZLHZPMZCQNZRNZDIZJZ
      AKSZYCTZYEUAUBYFVFUOZYNUUDYCUVFYMUUCAKUVFYLUUBWTUVFYKUUADUVFYJYTBRUVFYIYS
      CQUVFYHYRPUVFYGYQLYFVFXAOUPUQURUSUTVAVBVCVDYFUUFUOZYNUUNYCUVGYMUUMAKUVGYL
      UULWTUVGYKUUKDUVGYJUUJBRUVGYIUUICQUVGYHUUHPUVGYGUUGLYFUUFXAOUPUQURUSUTVAV
      BVCVDYFUUPUOZYNUVDYCUVHYMUVCAKUVHYLUVBWTUVHYKUVADUVHYJUUTBRUVHYIUUSCQUVHY
      HUURPUVHYGUUQLYFUUPXAOUPUQURUSUTVAVBVCVDUUEYEUUDYCUUDWTBXALHZPMZCQNZRNZDI
      ZJZAKSYCUUCUVNAKWOKIZUUBUVMWTUVOUUAUVLDUVOYTUVKBRUVOYSUVJCQUVOYRUVIPUVOYQ
      XALUVOUIKIZYQXAUOZVHUVOUVPJXAKIUVQWOUIVIXAVJWAVKUQURUSUTVAVBVLUVNYBAEKWOX
      JUOZWTXOUVMYAUVRWSXNFUVRWRXMBUHUVRWQXLCUEUVRWPXKUDWOXJLVMURUSUTVAUVRUVLXT
      DUVRUVKXSBRUVRUVJXRCQUVRUVIXQPUVRXAXPLWOXJUIOVNUQURUSUTVAVOVPVQVRWBUUFKIZ
      YEUUOUVETUVSYEJZUVDUUNYCUVTUVDUUNABCDFUUFVSVTWCVTWDWEWFWGWH $.
  $}

  ${
    $d A j x y $.  $d L j x y $.  $d P j x y $.  $d U j x y $.
    $( A slight rearrangement of ~ prarloclem3 .  Lemma for ~ prarloc .
       (Contributed by Jim Kingdon, 4-Nov-2019.) $)
    prarloclem4 $p |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ P e. Q. ) -> (
      E. x e. _om E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
        /\ ( A +Q
          ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) ->
      E. j e. _om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 .Q0 P ) ) e. L /\
        ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U )
      ) ) $=
      ( cop wcel wa cv c1o ceq0 cec cmq0 co cplq0 c2o coa com cnp cnq cplq wrex
      ceq cmq wi prarloclem3 3expia ancom2s anassrs rexlimdva ) GEHUAICGIJZDUBI
      ZJCBKZLHMNDOPQPGICUORSPAKZSPLHUENDUFPUCPEIJBTUDZCFKZLHMNDOPQPGICURRSPLHUE
      NDUFPUCPEIJFTUDZATUMUNUPTIZUQUSUGZUMUTUNVAUMUTUNJUQUSBCDEFGUPUHUIUJUKUL
      $.
  $}

  ${
    $d N x $.
    $( Subtracting two from a positive integer.  Lemma for ~ prarloc .
       (Contributed by Jim Kingdon, 5-Nov-2019.) $)
    prarloclemn $p |- ( ( N e. N. /\ 1o <N N ) ->
      E. x e. _om ( 2o +o x ) = N ) $=
      ( cnpi wcel c1o clti wbr wa csuc wss c2o cv coa co wceq com wrex mpan syl
      wb simpl 1pi ltpiord biimpa word wi piord ordsucss sylc df-2o sseq1i pinn
      2onn nnawordex adantr bitr3id mpbid ) BCDZEBFGZHZEIZBJZKALMNBOAPQZUTUREBD
      ZVBURUSUAURUSVDECDURUSVDTUBEBUCRUDURBUEVDVBUFBUGEBUHSUIVBKBJZUTVCKVABUJUK
      URVEVCTZUSURBPDZVFBULKPDVGVFUMAKBUNRSUOUPUQ $.
  $}

  ${
    $d A x y $.  $d L x y $.  $d N x $.  $d P x y $.  $d U x y $.
    $( A substitution of zero for ` y ` and ` N ` minus two for ` x ` .  Lemma
       for ~ prarloc .  (Contributed by Jim Kingdon, 4-Nov-2019.) $)
    prarloclem5 $p |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\
      ( N e. N. /\ P e. Q. /\ 1o <N N ) /\
      ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) ->
      E. x e. _om E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
        /\ ( A +Q
          ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) ) $=
      ( cop wcel c1o ceq cec cmq co c2o coa wceq com c0 c0q0 cnp wa cnq wbr w3a
      cnpi clti cplq wrex ceq0 cmq0 cplq0 prarloclemn 3adant2 3ad2ant2 3ad2ant1
      cv elprnql simp22 cnq0 nqnq0 sseli nq0a0 syl nq0m0r df-0nq0 oveq1i oveq2d
      eqtr3di sylan9req syl2anc simp1r eqeltrrd 2onn nna0r ax-mp eqeq1i biimpri
      opeq1d eceq1d oveq1d eleq1d biimprcd 3ad2ant3 peano1 opeq1 anbi12d rspcev
      wi oveq1 mpan syl6an reximdv mpd ) FEHUAIZCFIZUBZGUFIZDUCIZJGUGUDZUEZCGJH
      ZKLZDMNZUHNZEIZUEZOAUQZPNZGQZARUIZCBUQZJHZUJLZDUKNZULNZFIZCXLOPNZXHPNZJHZ
      KLZDMNZUHNZEIZUBZBRUIZARUIXAWQXKXFWRWTXKWSAGUMUNUOXGXJYFARXGCSJHZUJLZDUKN
      ZULNZFIZXJCSOPNZXHPNZJHZKLZDMNZUHNZEIZYFXGCYJFXGCUCIZWSCYJQWQXAYSXFCEFURU
      PWQWRWSWTXFUSYSWSCCTULNZYJYSCUTIYTCQUCUTCVAVBCVCVDWSTYICULWSTDUKNZTYIWSDU
      TIUUATQUCUTDVAVBDVEVDTYHDUKVFVGVIVHVJVKWOWPXAXFVLVMXFWQXJYRWIXAXJYRXFXJYQ
      XEEXJYPXDCUHXJYOXCDMXJYNXBKXJYMGJYMGQXJYMXIGYLOXHPORIYLOQVNOVOVPVGVQVRVSV
      TWAVHWBWCWDSRIYKYRUBZYFWEYEUUBBSRXLSQZXQYKYDYRUUCXPYJFUUCXOYICULUUCXNYHDU
      KUUCXMYGUJXLSJWFVTWAVHWBUUCYCYQEUUCYBYPCUHUUCYAYODMUUCXTYNKUUCXSYMJUUCXRY
      LXHPXLSOPWJWAVSVTWAVHWBWGWHWKWLWMWN $.
  $}

  ${
    $d A j x y $.  $d L j x y $.  $d N j x $.  $d P j x y $.  $d U j x y $.
    $( A special case of Lemma 6.16 from [BauerTaylor], p. 32.  Given evenly
       spaced rational numbers from ` A ` to ` A +Q ( N .Q P ) ` (which are in
       the lower and upper cuts, respectively, of a real number), there are a
       pair of numbers, two positions apart in the even spacing, which straddle
       the cut.  (Contributed by Jim Kingdon, 22-Oct-2019.) $)
    prarloclem $p |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\
      ( N e. N. /\ P e. Q. /\ 1o <N N ) /\
      ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) ->
      E. j e. _om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 .Q0 P ) ) e. L /\
        ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) ) $=
      ( vy vx cop wcel wa c1o ceq cec cmq co cplq cv coa com cnp cnpi clti ceq0
      cnq wbr w3a cmq0 cplq0 c2o wrex prarloclem5 prarloclem4 3ad2antr2 3adant3
      wi mpd ) ECIUAJAEJKZFUBJZBUEJZLFUCUFZUGZAFLIMNBOPQPCJZUGAGRZLIUDNBUHPUIPE
      JAVDUJSPHRSPLIMNBOPQPCJKGTUKHTUKZADRZLIUDNBUHPUIPEJAVFUJSPLIMNBOPQPCJKDTU
      KZHGABCEFULURVBVEVGUPZVCURUSUTVHVAHGABCDEUMUNUOUQ $.
  $}

  $( Some calculations for ~ prarloc .  (Contributed by Jim Kingdon,
     26-Oct-2019.) $)
  prarloclemcalc $p |- ( ( (
      A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 .Q0 Q ) )
      /\
      B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) )
    /\ (
      ( Q e. Q. /\ ( Q +Q Q ) <Q P )
      /\ ( X e. Q. /\ M e. _om )
    ) ) -> B <Q ( A +Q P ) ) $=
    ( c1o ceq0 cmq0 co cplq0 wceq c2o wa cnq wcel syl2anc cnq0 cnpi syl cop cec
    coa ceq cmq cplq cltq wbr com simprll nqnq0a oveq2d simprrl simprrr cxp cqs
    simpll 1pi opelxpi enq0ex ecelqsi df-nq0 eleqtrrdi sselid mulclnq0 nqpnq0nq
    mpan2 eqeltrd addclnq simplr c0 wne 2onn 2on0 elni mpbir2an nnppipi sylancl
    nqnq0 enqex df-nqqs mulclnq nqnq0m oveq1d eqtr4d 3eqtrd nnanq0 mp3an3 mp2an
    nqnq0pi ax-mp eleqtrri distnq0r eqtrd nq02m addclnq0 addassnq0 oveq1 eqeq2d
    syl3anc wb mpbird 3eqtr4rd simprlr ltrelnq brel ltanqg 3expa mpbid eqbrtrd
    ) AFEGUAZHUBZDIJZKJZLZBFEMUCJZGUAZUDUBZDUEJZUFJZLZNZDOPZDDUFJZCUGUHZNZFOPZE
    UIPZNZNZNZBAYDUFJZACUFJZUGYKAYDKJZADDKJZKJZYLBYKYDYOAKYKYCYCYDYOLYBYCYEYIUJ
    ZYQDDUKQULYKAOPZYDOPZYLYNLYKAXNOXOYAYJUQZYKYGXMRPZXNOPYBYFYGYHUMZYKXLRPZDRP
    ZUUAYKYHUUCYBYFYGYHUNZYHXLUISUOZHUPZRYHXKUUFPZXLUUGPYHGSPZUUHUREGUISUSVGUUF
    XKHUTVATVBVCTZYKORDVSYQVDZXLDVEQZFXMVFQVHZYKYCYCYSYQYQDDVIQAYDUKQYKBYPLZBXN
    YOKJZLZYKBFXQHUBZDIJZKJZFXMMGUAZHUBZDIJZKJZKJZUUOYKBXTFXSKJZUUSXOYAYJVJYKYG
    XSOPZXTUVELUUBYKXROPZYCUVFYKYHUVGUUEYHXRSSUOZUDUPZOYHXQUVHPZXRUVIPYHXPSPZUU
    IUVJYHMSPZUVKUVLMUIPZMVKVLVMVNMVOVPEMVQVGZURXPGSSUSVRUVHXQUDVTVATWAVCTZYQXR
    DWBQFXSUKQYKXSUURFKYKXSXRDIJZUURYKUVGYCXSUVPLUVOYQXRDWCQYKUUQXRDIYKYHUUQXRL
    ZUUEYHUVKUUIUVQUVNURXPGWJVRTWDWEULWFYKUURUVCFKYKUURXLUVAKJZDIJZUVCYKUUQUVRD
    IYKYHUVMUUQUVRLZUUEVMYHUVMUUIUVTURGMEWGWHVRWDYKUUDUUCUVSUVCLZUUKUUJUUDUUCUV
    ARPUWAUVAUUGRUUTUUFPZUVAUUGPUVMUUIUWBVMURMGUISUSWIUUFUUTHUTVAWKVBWLDXLUVAWM
    WHQWNULYKUVDFXMYOKJZKJZUUOYKUUDUVDUWDLUUKUUDUVCUWCFKUUDUVBYOXMKDWOULULTYKFR
    PUUAYORPZUUOUWDLYKORFVSUUBVDUULYKUUDUUDUWEUUKUUKDDWPQFXMYOWQWTWEWFYKXOUUNUU
    PXAYTXOYPUUOBAXNYOKWRWSTXBXCYKYEYLYMUGUHZYBYCYEYIXDZYKYSCOPZNZYRYEUWFXAZYKY
    EUWIUWGYDCOOUGXEXFTUUMYSUWHYRUWJYDCAXGXHQXIXJ $.

  ${
    $d L a b m n q x y $.  $d P a b m n q x y $.  $d U a b m n q x y $.
    $( A Dedekind cut is arithmetically located.  Part of Proposition 11.15 of
       [BauerTaylor], p. 52, slightly modified.  It states that given a
       tolerance ` P ` , there are elements of the lower and upper cut which
       are within that tolerance of each other.

       Usually, proofs will be shorter if they use ~ prarloc2 instead.
       (Contributed by Jim Kingdon, 22-Oct-2019.) $)
    prarloc $p |- ( ( <. L , U >. e. P. /\ P e. Q. ) ->
        E. a e. L E. b e. U b <Q ( a +Q P ) ) $=
      ( vx vy vq vn vm wcel cnq wa cv co wex wrex df-rex sylib anbi2d cplq cltq
      cop cnp wbr w3a cnpi c1o clti ceq cec prml adantr subhalfnqq adantl ancli
      cmq prmu 19.42v sylibr eeeanv syl3anbrc prarloclemarch2 3adant1r 3adant2r
      3com12 3adant3r 3adant3l 2eximi eximi com ceq0 cmq0 cplq0 c2o coa simpl1l
      simp3rl simp3rr simp3ll simpl1r simprl simprrl simprrr wi simpl2r prcunqu
      3jca syl2anc mpd prarloclem syl231anc wceq eleq1 anbi1d ceqsexgv biimprcd
      exbidv sylc exbii simprl2 simprl3 simprl1 prarloclemcalc syl12anc ancom1s
      jca simpl anasss syl exlimiv exlimivv 3syl excom anbi2i bitr4i ) CBUCUDKZ
      ALKZMZDNZCKZENZBKZYBXTAUAOUBUEZMZMZEPZDPZYDEBQZDCQZXSYFDPEPZYHXSFNZLKZYLC
      KZMZGNZLKZYPBKZMZXSHNZLKZYTYTUAOAUBUEZMZMZUFZHPGPZFPZUUEINZUGKZUHUUHUIUEZ
      YPYLUUHUHUCUJUKYTUQOUAOZUBUEZMZMZMZIPZHPZGPZFPYKXSYOFPZYSGPZUUDHPZUUGXQUU
      SXRXQYNFLQUUSFBCULYNFLRSUMXQUUTXRXQYRGLQUUTGBCURYRGLRSUMXSXSUUCHPZMUVAXSU
      VBXSUUBHLQZUVBXRUVCXQHAUNUOUUBHLRSUPXSUUCHUSUTYOYSUUDFGHVAVBUUFUURFUUEUUP
      GHUUEUUEUUNIPZMUUPUUEUVDYOYSUUCUVDXSYOYSUUAUVDUUBYOYQUUAUVDYRYMYQUUAUVDYN
      YQYMUUAUVDYQYMUUAUFUUMIUGQUVDIYPYLYTVCUUMIUGRSVFVDVEVGVHUPUUEUUNIUSUTVIVJ
      UUQYKFGUUOYKHIUUOYMUUAUUBUFZJNZVKKZYLUVFUHUCVLUKYTVMOVNOZCKZYLUVFVOVPOUHU
      CUJUKYTUQOUAOZBKZMZMZMZJPZYKUUOUVEUVMJPZMUVOUUOUVEUVPUUOYMUUAUUBYMYNYSUUD
      UUNVQUUEUUAUUNUUAUUBXSYOYSVRUMZUUEUUBUUNUUAUUBXSYOYSVSUMWHUUOUVLJVKQZUVPU
      UOXQYNUUIUUAUUJUUKBKZUVRUUEXQUUNXQXRUUCYOYSVTUMZYMYNYSUUDUUNWAUUEUUIUUMWB
      UVQUUEUUIUUJUULWCUUOUULUVSUUEUUIUUJUULWDUUOXQYRUULUVSWEUVTYQYRYOUUDUUNWFU
      UKYPBCWGWIWJYLYTBJCUUHWKWLUVLJVKRSXGUVEUVMJUSUTUVNYKJUVNYBUVJWMZXTUVHWMZU
      VEUVGYAYCMZMZMZMZMZDPZEPZYKUVNUWAUWFDPZMZEPZUWIUVNUWBUVEUVGYAUVKMZMZMZMZD
      PZUVKUWLUVNUVIUWQUVEUVGUVIUVKWCUVIUWQUVNUWOUVNDUVHCUWBUWNUVMUVEUWBUWMUVLU
      VGUWBYAUVIUVKXTUVHCWNWOTTWPWQWJUVEUVGUVIUVKWDUVKUWLUWQUWJUWQEUVJBUWAUWFUW
      PDUWAUWEUWOUWBUWAUWDUWNUVEUWAUWCUWMUVGUWAYCUVKYAYBUVJBWNTTTTWRWPWQWSUWHUW
      KEUWAUWFDUSWTUTUWGYFEDUWAUWBUWEYFUWBUWAUWEYFUWBUWAMZUWEMZYAYEUWEYAUWRUVEU
      VGYAYCWCUOUWSYCYDUWEYCUWRUVEUVGYAYCWDUOUWSUWRUUCYMUVGMYDUWRUWEXHUWSUUAUUB
      YMUUAUUBUWDUWRXAYMUUAUUBUWDUWRXBXGUWSYMUVGYMUUAUUBUWDUWRXCUWRUVEUVGUWCWCX
      GXTYBAYTUVFYLXDXEXGXGXFXIVIXJXKXJXLXLXMYFEDXNSYHYAYIMZDPYJYGUWTDYGYAYEEPZ
      MUWTYAYEEUSYIUXAYAYDEBRXOXPWTYIDCRXPS $.
  $}

  ${
    $d L a b $.  $d P a b $.  $d U a b $.
    $( A Dedekind cut is arithmetically located.  This is a variation of
       ~ prarloc which only constructs one (named) point and is therefore often
       easier to work with.  It states that given a tolerance ` P ` , there are
       elements of the lower and upper cut which are exactly that tolerance
       from each other.  (Contributed by Jim Kingdon, 26-Dec-2019.) $)
    prarloc2 $p |- ( ( <. L , U >. e. P. /\ P e. Q. ) ->
        E. a e. L ( a +Q P ) e. U ) $=
      ( vb cop cnp wcel cnq wa cv cplq co cltq wbr prarloc wi prcunqu rexlimdva
      wrex reximdv adantr mpd ) CBFGHZAIHZJEKZDKALMZNOZEBTZDCTZUGBHZDCTZABCDEPU
      DUJULQUEUDUIUKDCUDUHUKEBUGUFBCRSUAUBUC $.
  $}

  ${
    $d q x y $.
    $( Positive real 'less than' is a relation on positive reals.  (Contributed
       by NM, 14-Feb-1996.) $)
    ltrelpr $p |- <P C_ ( P. X. P. ) $=
      ( vx vy vq cltp cv cnp wcel c2nd cfv c1st cnq wrex copab df-iltp opabssxp
      wa cxp eqsstri ) DAEZFGBEZFGPCEZSHIGUATJIGPCKLZPABMFFQABCNUBABFFOR $.
  $}

  ${
    $d A q x y $.  $d B q x y $.
    $( More convenient form of ~ df-iltp .  (Contributed by Jim Kingdon,
       15-Dec-2019.) $)
    ltdfpr $p |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <->
        E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) $=
      ( vx vy cltp wbr cop cv cnp wcel wa c2nd cfv c1st wrex wceq fveq2d eleq2d
      cnq copab df-br df-iltp eleq2i bitri simpl simpr anbi12d rexbidv bitrid
      opelopab2a ) ABFGZABHZDIZJKEIZJKLCIZUNMNZKZUPUOONZKZLZCTPZLDEUAZKZAJKBJKL
      UPAMNZKZUPBONZKZLZCTPZULUMFKVDABFUBFVCUMDECUCUDUEVBVJDEABJJUNAQZUOBQZLZVA
      VICTVMURVFUTVHVMUQVEUPVMUNAMVKVLUFRSVMUSVGUPVMUOBOVKVLUGRSUHUIUKUJ $.
  $}

  ${
    $d A s $.  $d ph q r s $.
    genpdflem.r $e |- ( ( ph /\ r e. A ) -> r e. Q. ) $.
    genpdflem.s $e |- ( ( ph /\ s e. B ) -> s e. Q. ) $.
    $( Simplification of upper or lower cut expression.  Lemma for ~ genpdf .
       (Contributed by Jim Kingdon, 30-Sep-2019.) $)
    genpdflem $p |- ( ph -> { q e. Q. | E. r e. Q. E. s e. Q.
        ( r e. A /\ s e. B /\ q = ( r G s ) ) } =
        { q e. Q. | E. r e. A E. s e. B q = ( r G s ) } ) $=
      ( cv wcel cnq wrex wa wex rexbii bitri df-rex anass exbii wceq w3a 3anass
      co r19.42v bitr4i ex pm4.71rd anbi1d exbidv bitr4id bitr4di rexbidv bitrd
      rabbidv ) AFJZBKZEJZCKZGJUPURDUDUAZUBZELMZFLMZUTECMZFBMZGLAVCUSUTNZELMZFB
      MZVEAVCUQVGNZFOZVHAVCUPLKZUQNZVGNZFOZVJVCVKVINZFOZVNVCVIFLMVPVBVIFLVBUQVF
      NZELMVIVAVQELUQUSUTUCPUQVFELUEQPVIFLRQVMVOFVKUQVGSTUFAVIVMFAUQVLVGAUQVKAU
      QVKHUGUHUIUJUKVGFBRULAVGVDFBAVGVFEOZVDAVGURLKZUSNZUTNZEOZVRVGVSVFNZEOWBVF
      ELRWAWCEVSUSUTSTUFAVFWAEAUSVTUTAUSVSAUSVSIUGUHUIUJUKUTECRULUMUNUO $.
  $}

  ${
    $d q r s v w $.
    genpdf.1 $e |- F = ( w e. P. , v e. P. |->
      <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\
        s e. ( 1st ` v ) /\ q = ( r G s ) ) } ,
        { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\
        s e. ( 2nd ` v ) /\ q = ( r G s ) ) } >. ) $.
    $( Simplified definition of addition or multiplication on positive reals.
       (Contributed by Jim Kingdon, 30-Sep-2019.) $)
    genpdf $p |- F = ( w e. P. , v e. P. |-> <.
        { q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } ,
        { q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) }
        >. ) $=
      ( cnp cv c1st cfv wcel w3a cnq wrex crab c2nd cop sylan co wceq cmpo prop
      wa elprnql adantlr adantll genpdflem elprnqu opeq12d mpoeq3ia eqtri ) CAB
      IIFJZAJZKLZMZEJZBJZKLZMZGJUNURDUAUBZNEOPFOPGOQZUNUORLZMZURUSRLZMZVBNEOPFO
      PGOQZSZUCABIIVBEUTPFUPPGOQZVBEVFPFVDPGOQZSZUCHABIIVIVLUOIMZUSIMZUEZVCVJVH
      VKVOUPUTDEFGVMUQUNOMZVNVMUPVDSIMZUQVPUOUDZUNVDUPUFTUGVNVAUROMZVMVNUTVFSIM
      ZVAVSUSUDZURVFUTUFTUHUIVOVDVFDEFGVMVEVPVNVMVQVEVPVRUNVDUPUJTUGVNVGVSVMVNV
      TVGVSWAURVFUTUJTUHUIUKULUM $.
  $}

  ${
    $d x y z f g q r s A $.  $d x y z f g q r s B $.
    $d x y z f g w v q r s G $.  $d f g F $.
    genp.1 $e |- F = ( w e. P. , v e. P. |->
      <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\
        z e. ( 1st ` v ) /\ x = ( y G z ) ) } ,
        { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\
        z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. ) $.
    genp.2 $e |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. ) $.
    $( Value of general operation (addition or multiplication) on positive
       reals.  (Contributed by Jim Kingon, 3-Oct-2019.) $)
    genipv $p |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = <.
        { q e. Q. | E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) } ,
        { q e. Q. | E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) }
        >. ) $=
      ( wcel wceq cfv wrex cnq crab vf vg cnp wa co c1st c2nd cop oveq1 rexeqdv
      cv fveq2 rabbidv opeq12d eqeq12d oveq2 rexbidv cvv cxp a1i cab rabssab wi
      nqex elprnql sylan eleq1 syl5ibrcom syl2an rexlimdvva abssdv sstrid ssexd
      prop an4s elprnqu opelxp sylanbrc genpdf ovmpog mpd3an3 vtocl2ga 2rexbidv
      eqeq1 eqeq2d cbvrex2v bitrdi cbvrabv opeq12i eqtrdi ) FUCOGUCOUDFGHUEZAUK
      ZBUKZCUKZIUEZPZCGUFQZRZBFUFQZRZASTZWPCGUGQZRZBFUGQZRZASTZUHZLUKZKUKZJUKZI
      UEZPZJWQRKWSRZLSTZXLJXBRKXDRZLSTZUHUAUKZUBUKZHUEZWPCXRUFQZRZBXQUFQZRZASTZ
      WPCXRUGQZRZBXQUGQZRZASTZUHZPZFXRHUEZYABWSRZASTZYFBXDRZASTZUHZPWKXGPUAUBFG
      UCUCXQFPZXSYLYJYQXQFXRHUIYRYDYNYIYPYRYCYMASYRYABYBWSXQFUFULUJUMYRYHYOASYR
      YFBYGXDXQFUGULUJUMUNUOXRGPZYLWKYQXGXRGFHUPYSYNXAYPXFYSYMWTASYSYAWRBWSYSWP
      CXTWQXRGUFULUJUQUMYSYOXEASYSYFXCBXDYSWPCYEXBXRGUGULUJUQUMUNUOXQUCOZXRUCOZ
      YJURURUSZOZYKYTUUAUDZYDUROYIUROUUCUUDYDSURSUROUUDVDUTZUUDYDYCAVASYCASVBUU
      DYCASUUDWPWLSOZBCYBXTYTWMYBOZUUAWNXTOZWPUUFVCZYTUUGUDWMSOZWNSOZUUIUUAUUHU
      DYTYBYGUHUCOZUUGUUJXQVNZWMYGYBVEVFUUAXTYEUHUCOZUUHUUKXRVNZWNYEXTVEVFUUJUU
      KUDUUFWPWOSONWLWOSVGVHZVIVOVJVKVLVMUUDYISURUUEUUDYIYHAVASYHASVBUUDYHASUUD
      WPUUFBCYGYEYTWMYGOZUUAWNYEOZUUIYTUUQUDUUJUUKUUIUUAUURUDYTUULUUQUUJUUMWMYG
      YBVPVFUUAUUNUURUUKUUOWNYEXTVPVFUUPVIVOVJVKVLVMYDYIURURVQVRDEXQXRUCUCWPCEU
      KZUFQZRZBDUKZUFQZRZASTZWPCUUSUGQZRZBUVBUGQZRZASTZUHYJHUVABYBRZASTZUVGBYGR
      ZASTZUHUUBUVBXQPZUVEUVLUVJUVNUVOUVDUVKASUVOUVABUVCYBUVBXQUFULUJUMUVOUVIUV
      MASUVOUVGBUVHYGUVBXQUGULUJUMUNUUSXRPZUVLYDUVNYIUVPUVKYCASUVPUVAYABYBUVPWP
      CUUTXTUUSXRUFULUJUQUMUVPUVMYHASUVPUVGYFBYGUVPWPCUVFYEUUSXRUGULUJUQUMUNDEH
      ICBAMVSVTWAWBXAXNXFXPWTXMALSWLXHPZWTXHWOPZCWQRBWSRXMUVQWPUVRBCWSWQWLXHWOW
      DZWCUVRXLXHXIWNIUEZPZBCKJWSWQWMXIPWOUVTXHWMXIWNIUIWEZWNXJPUVTXKXHWNXJXIIU
      PWEZWFWGWHXEXOALSUVQXEUVRCXBRBXDRXOUVQWPUVRBCXDXBUVSWCUVRXLUWABCKJXDXBUWB
      UWCWFWGWHWIWJ $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d D x y z $.  $d G x y z $.
    genplt2i.ord $e |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) ->
      ( x <Q y <-> ( z G x ) <Q ( z G y ) ) ) $.
    genplt2i.com $e |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) ) $.
    $( Operating on both sides of two inequalities, when the operation is
       consistent with ` <Q ` .  (Contributed by Jim Kingdon, 6-Oct-2019.) $)
    genplt2i $p |- ( ( A <Q B /\ C <Q D ) -> ( A G C ) <Q ( B G D ) ) $=
      ( cltq wbr wa co cnq cv wcel adantl ltrelnq syl2an simpl wb simpll simplr
      w3a brel simprl wceq caovord2d mpbid simpr simprr caovordd ltsonq syl2anc
      sotri ) DEKLZFGKLZMZDFHNZEFHNZKLZVAEGHNZKLZUTVCKLUSUQVBUQURUAUSABCDEFKOHA
      PZOQZBPZOQZCPZOQUEVEVGKLVIVEHNVIVGHNKLUBUSIRZUQDOQZEOQZMZFOQZGOQZMZVKURDE
      OOKSUFZFGOOKSUFZVKVLVPUCTUQVMVPVLURVQVRVKVLVPUDTZUQVMVPVNURVQVRVMVNVOUGTZ
      VFVHMVEVGHNVGVEHNUHUSJRUIUJUSURVDUQURUKUSABCFGEKOHVJVTUQVMVPVOURVQVRVMVNV
      OULTVSUMUJUTVAVCKOUNSUPUO $.
  $}

  ${
    $d x y z f g h w v q A $.  $d x y z f g h w v q B $.
    $d x y z f g h w v q G $.  $d f g q F $.  $d f g h C $.  $d f g h D $.
    genpelvl.1 $e |- F = ( w e. P. , v e. P. |->
      <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\
        z e. ( 1st ` v ) /\ x = ( y G z ) ) } ,
        { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\
        z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. ) $.
    $( Set containing the result of adding or multiplying positive reals.
       (Contributed by Jim Kingdon, 5-Dec-2019.) $)
    genpelxp $p |- ( ( A e. P. /\ B e. P. ) ->
        ( A F B ) e. ( ~P Q. X. ~P Q. ) ) $=
      ( cnp wcel cv c1st cfv w3a cnq wrex crab c2nd wceq cop cpw cxp wss ssrab2
      wa co nqex elpw2 mpbir opelxpi mp2an fveq2 eleq2d 3anbi1d rabbidv opeq12d
      2rexbidv 3anbi2d ovmpog mp3an3 eqeltrdi ) FKLZGKLZUGFGHUHZBMZFNOZLZCMZGNO
      ZLZAMVGVJIUHUAZPZCQRBQRZAQSZVGFTOZLZVJGTOZLZVMPZCQRBQRZAQSZUBZQUCZWEUDZVD
      VEWDWFLZVFWDUAVPWELZWCWELZWGWHVPQUEVOAQUFVPQUIUJUKWIWCQUEWBAQUFWCQUIUJUKV
      PWCWEWEULUMZDEFGKKVGDMZNOZLZVJEMZNOZLZVMPZCQRBQRZAQSZVGWKTOZLZVJWNTOZLZVM
      PZCQRBQRZAQSZUBWDHVIWPVMPZCQRBQRZAQSZVRXCVMPZCQRBQRZAQSZUBWFWKFUAZWSXIXFX
      LXMWRXHAQXMWQXGBCQQXMWMVIWPVMXMWLVHVGWKFNUNUOUPUSUQXMXEXKAQXMXDXJBCQQXMXA
      VRXCVMXMWTVQVGWKFTUNUOUPUSUQURWNGUAZXIVPXLWCXNXHVOAQXNXGVNBCQQXNWPVLVIVMX
      NWOVKVJWNGNUNUOUTUSUQXNXKWBAQXNXJWABCQQXNXCVTVRVMXNXBVSVJWNGTUNUOUTUSUQUR
      JVAVBWJVC $.

    genpelvl.2 $e |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. ) $.
    $( Membership in lower cut of general operation (addition or
       multiplication) on positive reals.  (Contributed by Jim Kingdon,
       2-Oct-2019.) $)
    genpelvl $p |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 1st ` ( A F B ) ) <->
        E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) ) ) $=
      ( vf wcel wa cnq cfv wrex cnp co c1st cv wceq crab c2nd cop genipv fveq2d
      nqex rabex op1st eqtrdi eleq2d elrabi biimtrdi prop elprnql caovcl syl2an
      sylan an4s eleq1 syl5ibrcom rexlimdvva wb eqeq1 2rexbidv elrab3 pm5.21ndd
      sylan9bb ex ) FUAPZGUAPZQZHRPZHFGKUBZUCSZPZHIUDZJUDZLUBZUEZJGUCSZTIFUCSZT
      ZVPVTHOUDZWCUEZJWETIWFTZORUFZPZVQVPVSWKHVPVSWKWIJGUGSZTIFUGSZTZORUFZUHZUC
      SWKVPVRWQUCABCDEFGKLJIOMNUIUJWKWPWJORUKULWOORUKULUMUNUOZWJOHRUPUQVPWDVQIJ
      WFWEVPWAWFPZWBWEPZQQVQWDWCRPZVNWSVOWTXAVNWSQWARPZWBRPZXAVOWTQVNWFWNUHUAPW
      SXBFURWAWNWFUSVBVOWEWMUHUAPWTXCGURWBWMWEUSVBBCWAWBRLNUTVAVCHWCRVDVEVFVPVQ
      VTWGVGVPVTWLVQWGWRWJWGOHRWHHUEWIWDIJWFWEWHHWCVHVIVJVLVMVK $.

    $( Membership in upper cut of general operation (addition or
       multiplication) on positive reals.  (Contributed by Jim Kingdon,
       15-Oct-2019.) $)
    genpelvu $p |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 2nd ` ( A F B ) ) <->
        E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) ) $=
      ( vf wcel wa cnq cfv wrex cnp co c2nd cv wceq crab c1st cop genipv fveq2d
      nqex rabex op2nd eqtrdi eleq2d elrabi biimtrdi prop elprnqu caovcl syl2an
      sylan an4s eleq1 syl5ibrcom rexlimdvva wb eqeq1 2rexbidv elrab3 pm5.21ndd
      sylan9bb ex ) FUAPZGUAPZQZHRPZHFGKUBZUCSZPZHIUDZJUDZLUBZUEZJGUCSZTIFUCSZT
      ZVPVTHOUDZWCUEZJWETIWFTZORUFZPZVQVPVSWKHVPVSWIJGUGSZTIFUGSZTZORUFZWKUHZUC
      SWKVPVRWQUCABCDEFGKLJIOMNUIUJWPWKWOORUKULWJORUKULUMUNUOZWJOHRUPUQVPWDVQIJ
      WFWEVPWAWFPZWBWEPZQQVQWDWCRPZVNWSVOWTXAVNWSQWARPZWBRPZXAVOWTQVNWNWFUHUAPW
      SXBFURWAWFWNUSVBVOWMWEUHUAPWTXCGURWBWEWMUSVBBCWAWBRLNUTVAVCHWCRVDVEVFVPVQ
      VTWGVGVPVTWLVQWGWRWJWGOHRWHHUEWIWDIJWFWEWHHWCVHVIVJVLVMVK $.

    $( Pre-closure law for general operation on lower cuts.  (Contributed by
       Jim Kingdon, 2-Oct-2019.) $)
    genpprecll $p |- ( ( A e. P. /\ B e. P. ) ->
        ( ( C e. ( 1st ` A ) /\ D e. ( 1st ` B ) ) ->
        ( C G D ) e. ( 1st ` ( A F B ) ) ) ) $=
      ( vg vh c1st cfv wcel wa co cnp cv wceq wrex eqid rspceov mp3an3 genpelvl
      imbitrrid ) HFPQZRZIGPQZRZSHIKTZFGJTPQRFUARGUARSUNNUBOUBKTUCOULUDNUJUDZUK
      UMUNUNUCUOUNUENOUJULHIUNKUFUGABCDEFGUNNOJKLMUHUI $.

    $( Pre-closure law for general operation on upper cuts.  (Contributed by
       Jim Kingdon, 7-Nov-2019.) $)
    genppreclu $p |- ( ( A e. P. /\ B e. P. ) ->
        ( ( C e. ( 2nd ` A ) /\ D e. ( 2nd ` B ) ) ->
        ( C G D ) e. ( 2nd ` ( A F B ) ) ) ) $=
      ( vg vh c2nd cfv wcel wa co cnp cv wceq wrex eqid rspceov mp3an3 genpelvu
      imbitrrid ) HFPQZRZIGPQZRZSHIKTZFGJTPQRFUARGUARSUNNUBOUBKTUCOULUDNUJUDZUK
      UMUNUNUCUOUNUENOUJULHIUNKUFUGABCDEFGUNNOJKLMUHUI $.

    $( Domain of general operation on positive reals.  (Contributed by Jim
       Kingdon, 2-Oct-2019.) $)
    genipdm $p |- dom F = ( P. X. P. ) $=
      ( cnp cv c1st cfv wcel w3a cnq wrex crab c2nd nqex co wceq cop rabex opex
      dmmpo ) DEJJBKZDKZLMNCKZEKZLMNAKUGUIGUAUBZOCPQBPQZAPRZUGUHSMNUIUJSMNUKOCP
      QBPQZAPRZUCFHUMUOULAPTUDUNAPTUDUEUF $.

    $( The lower cut produced by addition or multiplication on positive reals
       is inhabited.  (Contributed by Jim Kingdon, 5-Oct-2019.) $)
    genpml $p |- ( ( A e. P. /\ B e. P. ) ->
        E. q e. Q. q e. ( 1st ` ( A F B ) ) ) $=
      ( vf vg cnp wcel wa cv cfv cnq c1st co wrex wex c2nd prop prml rexex 3syl
      cop adantr ad2antlr genpprecll imp elprnql sylan anim12i an4s caovcl wceq
      syl simpr eleq1d rspcedv mpd anassrs exlimddv ) FOPZGOPZQZMRZFUASZPZJRZFG
      HUBUASZPZJTUCZMVHVMMUDZVIVHVLFUESZUJOPZVMMTUCVRFUFZMVSVLUGVMMTUHUIUKVJVMQ
      NRZGUASZPZVQNVIWDNUDZVHVMVIWCGUESZUJOPZWDNTUCWEGUFZNWFWCUGWDNTUHUIULVJVMW
      DVQVJVMWDQZQZVKWBIUBZVOPZVQVJWIWLABCDEFGVKWBHIKLUMUNWJVPWLJWKTWJVKTPZWBTP
      ZQZWKTPVHVMVIWDWOVHVMQWMVIWDQWNVHVTVMWMWAVKVSVLUOUPVIWGWDWNWHWBWFWCUOUPUQ
      URBCVKWBTILUSVAWJVNWKUTZQVNWKVOWJWPVBVCVDVEVFVGVG $.

    $( The upper cut produced by addition or multiplication on positive reals
       is inhabited.  (Contributed by Jim Kingdon, 5-Dec-2019.) $)
    genpmu $p |- ( ( A e. P. /\ B e. P. ) ->
        E. q e. Q. q e. ( 2nd ` ( A F B ) ) ) $=
      ( vf vg cnp wcel wa cv cfv cnq c2nd co wrex wex c1st prop prmu rexex 3syl
      cop adantr ad2antlr genppreclu imp elprnqu sylan anim12i an4s caovcl wceq
      syl simpr eleq1d rspcedv mpd anassrs exlimddv ) FOPZGOPZQZMRZFUASZPZJRZFG
      HUBUASZPZJTUCZMVHVMMUDZVIVHFUESZVLUJOPZVMMTUCVRFUFZMVLVSUGVMMTUHUIUKVJVMQ
      NRZGUASZPZVQNVIWDNUDZVHVMVIGUESZWCUJOPZWDNTUCWEGUFZNWCWFUGWDNTUHUIULVJVMW
      DVQVJVMWDQZQZVKWBIUBZVOPZVQVJWIWLABCDEFGVKWBHIKLUMUNWJVPWLJWKTWJVKTPZWBTP
      ZQZWKTPVHVMVIWDWOVHVMQWMVIWDQWNVHVTVMWMWAVKVLVSUOUPVIWGWDWNWHWBWCWFUOUPUQ
      URBCVKWBTILUSVAWJVNWKUTZQVNWKVOWJWPVBVCVDVEVFVGVG $.

    ${
      $d F h $.
      genpcdl.2 $e |- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\
                       ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) ->
                       ( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) ) $.
      $( Downward closure of an operation on positive reals.  (Contributed by
         Jim Kingdon, 14-Oct-2019.) $)
      genpcdl $p |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 1st ` ( A F B ) ) ->
                   ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) ) $=
        ( wcel wa cv cltq wi cnp wbr c1st cfv cnq ltrelnq brel simpld wceq wrex
        co genpelvl adantr breq2 biimpd sylan9r exp31 impancom rexlimdvv sylbid
        wb an4s ex syl5 com34 pm2.43d com23 ) FUAPZGUAPZQZARZHRZSUBZVLFGKUKUCUD
        ZPZVKVNPZVJVMVOVPTVJVMVOVMVPVMVKUEPZVJVOVMVPTZTZVMVQVLUEPVKVLUEUESUFUGU
        HVJVQVSVJVQQZVOVLIRZJRZLUKZUIZJGUCUDZUJIFUCUDZUJZVRVJVOWGVAVQABCDEFGVLI
        JKLMNULUMVTWDVRIJWFWEVJWAWFPZWBWEPZQVQWDVRTZVHWHVIWIVQWJTVHWHQVIWIQQZVQ
        WDVRWDVMVKWCSUBZWKVQQVPWDVMWLVLWCVKSUNUOOUPUQVBURUSUTVCVDVEVFVG $.
    $}

    ${
      $d F h $.
      genpcuu.2 $e |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\
                       ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) ->
                       ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) $.
      $( Upward closure of an operation on positive reals.  (Contributed by Jim
         Kingdon, 8-Nov-2019.) $)
      genpcuu $p |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) ->
                   ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) $=
        ( wcel wa cv cltq wi cnp wbr c2nd cfv cnq ltrelnq brel simprd wceq wrex
        co genpelvu adantr breq1 biimpd sylan9r exp31 impancom rexlimdvv sylbid
        wb an4s ex syl5 com34 pm2.43d com23 ) FUAPZGUAPZQZHRZARZSUBZVKFGKUKUCUD
        ZPZVLVNPZVJVMVOVPTVJVMVOVMVPVMVLUEPZVJVOVMVPTZTZVMVKUEPVQVKVLUEUESUFUGU
        HVJVQVSVJVQQZVOVKIRZJRZLUKZUIZJGUCUDZUJIFUCUDZUJZVRVJVOWGVAVQABCDEFGVKI
        JKLMNULUMVTWDVRIJWFWEVJWAWFPZWBWEPZQVQWDVRTZVHWHVIWIVQWJTVHWHQVIWIQQZVQ
        WDVRWDVMWCVLSUBZWKVQQVPWDVMWLVKWCVLSUNUOOUPUQVBURUSUTVCVDVEVFVG $.
    $}

    ${
      $d A a b c d q r v w x y z $.  $d B a b c d g h q r v w x y z $.
      $d F a b c d g h q r v w x y z $.  $d G a b c d g h q r v w x y z $.
      genprndl.ord $e |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) ->
        ( x <Q y <-> ( z G x ) <Q ( z G y ) ) ) $.
      genprndl.com $e |- ( ( x e. Q. /\ y e. Q. ) ->
        ( x G y ) = ( y G x ) ) $.
      genprndl.lower $e |- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\
                       ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) ->
                       ( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) ) $.
      $( The lower cut produced by addition or multiplication on positive reals
         is rounded.  (Contributed by Jim Kingdon, 7-Oct-2019.) $)
      genprndl $p |- ( ( A e. P. /\ B e. P. ) ->
          A. q e. Q. ( q e. ( 1st ` ( A F B ) ) <->
            E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) ) $=
        ( wcel wa vc vd va vb cnp cv co c1st cfv cltq wbr wrex wb wceq genpelvl
        cnq wex r2ex bitrdi biimpa adantrl c2nd prop prnmaxl sylan anim12i an4s
        wi cop reeanv sylibr genplt2i reximi syl adantrr breq1 biimprd ad2antll
        reximdv mpd exlimdvv adantr genpprecll imp elprnql caovcl breq2 anbi12d
        eleq1 adantl rspcedv mpan2d rexlimdvva expr wal genpcdl alrimdv imbi12d
        ex cbvalv imbitrdi sp syl6 ancomsd ad2antrr rexlimdva impbid ralrimiva
        impd ) FUESZGUESZTZMUFZFGJUGUHUIZSZXMLUFZUJUKZXPXNSZTZLUPULZUMMUPXLXMUP
        SZTZXOXTXLYAXOXTXLYAXOTZTZXMUAUFZUBUFZKUGZUJUKZUBGUHUIZULZUAFUHUIZULZXT
        YDUCUFZYKSZUDUFZYISZTZXMYMYOKUGZUNZTZUDUQUCUQZYLXLXOUUAYAXLXOUUAXLXOYSU
        DYIULUCYKULUUAABCDEFGXMUCUDJKNOUOYSUCUDYKYIURUSUTVAXLUUAYLVHYCXLYTYLUCU
        DXLYTYLXLYTTYRYGUJUKZUBYIULZUAYKULZYLXLYQUUDYSXLYQTZYMYEUJUKZYOYFUJUKZT
        ZUBYIULZUAYKULZUUDUUEUUFUAYKULZUUGUBYIULZTZUUJXJYNXKYPUUMXJYNTUUKXKYPTU
        ULXJYKFVBUIZVIUESZYNUUKFVCZUAYMUUNYKVDVEXKYIGVBUIZVIUESZYPUULGVCZUBYOUU
        QYIVDVEVFVGUUFUUGUAUBYKYIVJVKUUIUUCUAYKUUHUUBUBYIABCYMYEYOYFKPQVLVMVMVN
        VOYSUUDYLVHXLYQYSUUCYJUAYKYSUUBYHUBYIYSYHUUBXMYRYGUJVPVQVSVSVRVTWSWAWBV
        TXLYLXTVHYCXLYHXTUAUBYKYIXLYEYKSZYFYISZTZTZYHYGXNSZXTXLUVBUVDABCDEFGYEY
        FJKNOWCWDUVCXSYHUVDTZLYGUPUVCYEUPSZYFUPSZTZYGUPSXJUUTXKUVAUVHXJUUTTUVFX
        KUVATUVGXJUUOUUTUVFUUPYEUUNYKWEVEXKUURUVAUVGUUSYFUUQYIWEVEVFVGBCYEYFUPK
        OWFVNXPYGUNZXSUVEUMUVCUVIXQYHXRUVDXPYGXMUJWGXPYGXNWIWHWJWKWLWMWBVTWNYBX
        SXOLUPXLXSXOVHYAXPUPSXLXRXQXOXLXRXQXOXLXRXQXOVHZMWOZUVJXLXRAUFZXPUJUKZU
        VLXNSZVHZAWOUVKXLXRUVOAABCDEFGLHIJKNORWPWQUVOUVJAMUVLXMUNUVMXQUVNXOUVLX
        MXPUJVPUVLXMXNWIWRWTXAUVJMXBXCXIXDXEXFXGXH $.
    $}

    ${
      $d A a b c d q r v w x y z $.  $d B a b c d g h q r v w x y z $.
      $d F a b c d g h q r v w x y z $.  $d G a b c d g h q r v w x y z $.
      genprndu.ord $e |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) ->
        ( x <Q y <-> ( z G x ) <Q ( z G y ) ) ) $.
      genprndu.com $e |- ( ( x e. Q. /\ y e. Q. ) ->
        ( x G y ) = ( y G x ) ) $.
      genprndu.upper $e |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\
                       ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) ->
                       ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) $.
      $( The upper cut produced by addition or multiplication on positive reals
         is rounded.  (Contributed by Jim Kingdon, 7-Oct-2019.) $)
      genprndu $p |- ( ( A e. P. /\ B e. P. ) ->
          A. r e. Q. ( r e. ( 2nd ` ( A F B ) ) <->
            E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) ) $=
        ( wcel wa vc vd va vb cnp cv co c2nd cfv cltq wbr wrex wb wceq genpelvu
        cnq wex r2ex bitrdi biimpa adantrl c1st prop prnminu sylan anim12i an4s
        wi cop reeanv sylibr genplt2i reximi syl adantrr breq2 biimprd ad2antll
        reximdv mpd exlimdvv adantr genppreclu imp elprnqu caovcl breq1 anbi12d
        eleq1 adantl rspcedv mpan2d rexlimdvva expr wal genpcuu alrimdv imbi12d
        ex cbvalv imbitrdi sp syl6 ancomsd ad2antrr rexlimdva impbid ralrimiva
        impd ) FUESZGUESZTZLUFZFGJUGUHUIZSZMUFZXMUJUKZXPXNSZTZMUPULZUMLUPXLXMUP
        SZTZXOXTXLYAXOXTXLYAXOTZTZUAUFZUBUFZKUGZXMUJUKZUBGUHUIZULZUAFUHUIZULZXT
        YDUCUFZYKSZUDUFZYISZTZXMYMYOKUGZUNZTZUDUQUCUQZYLXLXOUUAYAXLXOUUAXLXOYSU
        DYIULUCYKULUUAABCDEFGXMUCUDJKNOUOYSUCUDYKYIURUSUTVAXLUUAYLVHYCXLYTYLUCU
        DXLYTYLXLYTTYGYRUJUKZUBYIULZUAYKULZYLXLYQUUDYSXLYQTZYEYMUJUKZYFYOUJUKZT
        ZUBYIULZUAYKULZUUDUUEUUFUAYKULZUUGUBYIULZTZUUJXJYNXKYPUUMXJYNTUUKXKYPTU
        ULXJFVBUIZYKVIUESZYNUUKFVCZUAYMYKUUNVDVEXKGVBUIZYIVIUESZYPUULGVCZUBYOYI
        UUQVDVEVFVGUUFUUGUAUBYKYIVJVKUUIUUCUAYKUUHUUBUBYIABCYEYMYFYOKPQVLVMVMVN
        VOYSUUDYLVHXLYQYSUUCYJUAYKYSUUBYHUBYIYSYHUUBXMYRYGUJVPVQVSVSVRVTWSWAWBV
        TXLYLXTVHYCXLYHXTUAUBYKYIXLYEYKSZYFYISZTZTZYHYGXNSZXTXLUVBUVDABCDEFGYEY
        FJKNOWCWDUVCXSYHUVDTZMYGUPUVCYEUPSZYFUPSZTZYGUPSXJUUTXKUVAUVHXJUUTTUVFX
        KUVATUVGXJUUOUUTUVFUUPYEYKUUNWEVEXKUURUVAUVGUUSYFYIUUQWEVEVFVGBCYEYFUPK
        OWFVNXPYGUNZXSUVEUMUVCUVIXQYHXRUVDXPYGXMUJWGXPYGXNWIWHWJWKWLWMWBVTWNYBX
        SXOMUPXLXSXOVHYAXPUPSXLXRXQXOXLXRXQXOXLXRXQXOVHZLWOZUVJXLXRXPAUFZUJUKZU
        VLXNSZVHZAWOUVKXLXRUVOAABCDEFGMHIJKNORWPWQUVOUVJALUVLXMUNUVMXQUVNXOUVLX
        MXPUJVPUVLXMXNWIWRWTXAUVJLXBXCXIXDXEXFXGXH $.
    $}

    ${
      $d A a b c d q v w x y z $.  $d B a b c d q v w x y z $.
      $d F a b c d q $.  $d G a b c d q v w x y z $.
      genpdisj.ord $e |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) ->
        ( x <Q y <-> ( z G x ) <Q ( z G y ) ) ) $.
      genpdisj.com $e |-
        ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) ) $.
      $( The lower and upper cuts produced by addition or multiplication on
         positive reals are disjoint.  (Contributed by Jim Kingdon,
         15-Oct-2019.) $)
      genpdisj $p |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q.
          -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) $=
        ( va vb vc vd wcel wa cnp cv co c1st cfv c2nd wn cnq wceq wex wfal wrex
        genpelvl r2ex bitrdi genpelvu anbi12d ee4anv bitr4di biimpa wbr an4 cop
        cltq wi prop prltlu 3expib syl im2anan9 genplt2i syl6 biimtrrid adantlr
        imp adantrlr adantrrr eqtr2 ad2ant2l adantl ltsonq ltrelnq soirri breq2
        mtbii pm2.21fal ex exlimdvv mpd inegd ralrimivw ) FUASZGUASZTZJUBZFGHUC
        ZUDUESZWOWPUFUESZTZUGJUHWNWSWNWSTZOUBZFUDUEZSZPUBZGUDUEZSZTZWOXAXDIUCZU
        IZTZQUBZFUFUEZSZRUBZGUFUEZSZTZWOXKXNIUCZUIZTZTZRUJQUJZPUJOUJZUKWNWSYCWN
        WSXJPUJOUJZXTRUJQUJZTYCWNWQYDWRYEWNWQXIPXEULOXBULYDABCDEFGWOOPHIKLUMXIO
        PXBXEUNUOWNWRXSRXOULQXLULYEABCDEFGWOQRHIKLUPXSQRXLXOUNUOUQXJXTOPQRURUSU
        TWTYBUKOPWTYAUKQRWTYAUKWTYATZXHXRVDVAZWTXJXQYGXSWTXGXQYGXIWNXGXQTZYGWSW
        NYHYGYHXCXMTZXFXPTZTZWNYGXCXMXFXPVBWNYKXAXKVDVAZXDXNVDVAZTYGWLYIYLWMYJY
        MWLXBXLVCUASZYIYLVEFVFYNXCXMYLXAXKXLXBVGVHVIWMXEXOVCUASZYJYMVEGVFYOXFXP
        YMXDXNXOXEVGVHVIVJABCXAXKXDXNIMNVKVLVMVOVNVPVQYFXHXRUIZYGUGYAYPWTXIXSYP
        XGXQWOXHXRVRVSVTYPXHXHVDVAYGXHVDUHWAWBWCXHXRXHVDWDWEVIWFWGWHWHWIWJWK $.
    $}

    ${
      $d A f g h t v w x y z $.  $d B f g h t v w x y z $.
      $d C f g h t v w x y z $.  $d F f g h t v w x y z $.
      $d G f g h t v w x y z $.
      genpassg.4 $e |- dom F = ( P. X. P. ) $.
      genpassg.5 $e |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. ) $.
      genpassg.6 $e |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) ->
          ( ( f G g ) G h ) = ( f G ( g G h ) ) ) $.
      $( Associativity of lower cuts.  Lemma for ~ genpassg .  (Contributed by
         Jim Kingdon, 11-Dec-2019.) $)
      genpassl $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
          ( 1st ` ( ( A F B ) F C ) ) = ( 1st ` ( A F ( B F C ) ) ) ) $=
        ( vt wa cnp wcel w3a co c1st cfv cv wceq wrex cnq c2nd cop prop elprnql
        sylan wex oveq2 adantr adantl eqtr4d eqeq2d expcom pm5.32d 3expa sylan2
        anassrs rexbidva r19.41v bitr3di an32s bitrdi exbidv caovcl elisset syl
        wb an31s biantrurd 19.41v bitr4di rexcom4 ancoms df-rex genpelvl anbi1d
        bitrid 3bitr4rd 3impb 3adant3 3impa oveq1 rexbidv pm5.32i exbii 3bitr4d
        bitri rexbii eqrdv ) FUAUBZGUAUBZHUAUBZUCZAFGLUDZHLUDUEUFZFGHLUDZLUDUEU
        FZXBAUGZIUGZSUGZMUDZUHZSXEUEUFZUIZIFUEUFZUIZXGXHJUGZMUDZKUGZMUDZUHZKHUE
        UFZUIZJGUEUFZUIZIXNUIZXGXFUBZXGXDUBZWSWTXAXOYEVPZWTXATZWSYHYIWSTXMYDIXN
        YIWSXHXNUBZXMYDVPZWSYJTZYIXHUJUBZYKWSXNFUKUFZULUAUBYJYMFUMXHYNXNUNUOZYI
        YMTZXIXPXRMUDZUHZXTTZKYAUIZJYCUIZSUPZYRKYAUIZJYCUIZXKTZSUPZYDXMYPUUAUUE
        SYMXAWTUUAUUEVPYMXATZWTTZUUAUUCXKTZJYCUIUUEUUHYTUUIJYCUUGWTXPYCUBZYTUUI
        VPZWTUUJTZUUGXPUJUBZUUKWTYCGUKUFZULUAUBUUJUUMGUMXPUUNYCUNUOZYMUUMXAUUKY
        MUUMTZXATZYRXKTZKYAUIYTUUIUUQUURYSKYAUUPXAXRYAUBZUURYSVPZXAUUSTZUUPXRUJ
        UBZUUTXAYAHUKUFZULUAUBUUSUVBHUMXRUVCYAUNUOZYMUUMUVBUUTYMUUMUVBUCZYRXKXT
        YRUVEXKXTVPYRUVETZXJXSXGUVFXJXHYQMUDZXSYRXJUVGUHUVEXIYQXHMUQURUVEXSUVGU
        HYRRUSUTVAVBVCVDVEVFVGYRXKKYAVHVIVJVEVFVGUUCXKJYCVHVKVQVLYIYDUUBVPYMYIY
        DYTSUPZJYCUIZUUBXAWTYDUVIVPXAWTTYBUVHJYCXAWTUUJYBUVHVPZUULXAUUMUVJUUOUU
        MXAUVJUUMXATZYBYSSUPZKYAUIUVHUVKXTUVLKYAUUMXAUUSXTUVLVPZUVAUUMUVBUVMUVD
        UUMUVBTZXTYRSUPZXTTUVLUVNUVOXTUVNYQUJUBUVOBCXPXRUJMOVMSYQUJVNVOVRYRXTSV
        SVTVEVFVGYSKSYAWAVKWBVEVFVGWBYTJSYCWAVKURYIXMUUFVPYMXMXIXLUBZXKTZSUPYIU
        UFXKSXLWCYIUVQUUESYIUVPUUDXKABCDEGHXIJKLMNOWDWEVLWFURWGVEVFVGWBWHWSWTXA
        YFXOVPZYIWSXEUAUBUVRIJGHUALQVMABCDEFXEXGISLMNOWDVEWHXBXGXIXRMUDZUHZKYAU
        IZSXCUEUFZUIZXIXQUHZJYCUIZIXNUIZUWATZSUPZYGYEUWCXIUWBUBZUWATZSUPXBUWHUW
        ASUWBWCXBUWJUWGSXBUWIUWFUWAWSWTUWIUWFVPXAABCDEFGXIIJLMNOWDWIWEVLWFWSWTX
        AYGUWCVPZWSWTTXCUAUBXAUWKIJFGUALQVMABCDEXCHXGSKLMNOWDUOWJWSWTYEUWHVPZXA
        WTWSUWLWTWSTZYEUWDUWATZJYCUIZIXNUIZSUPZUWHUWMYEUWOSUPZIXNUIUWQUWMYDUWRI
        XNWTWSYJYDUWRVPZYLWTYMUWSYOYMWTUWSYMWTTZYDUWNSUPZJYCUIUWRUWTYBUXAJYCYMW
        TUUJYBUXAVPZUULYMUUMUXBUUOUUPYBUWDSUPZYBTZUXAUUPUXCYBUUPXQUJUBUXCBCXHXP
        UJMOVMSXQUJVNVOVRUXAUWDYBTZSUPUXDUWNUXESUWDUWAYBUWDUVTXTKYAUWDUVSXSXGXI
        XQXRMWKVAWLWMWNUWDYBSVSWPVTVEVFVGUWNJSYCWAVKWBVEVFVGUWOISXNWAVKUWPUWGSU
        WPUWEUWATZIXNUIUWGUWOUXFIXNUWDUWAJYCVHWQUWEUWAIXNVHWPWNVKWBWIWOWGWR $.

      $( Associativity of upper cuts.  Lemma for ~ genpassg .  (Contributed by
         Jim Kingdon, 11-Dec-2019.) $)
      genpassu $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
          ( 2nd ` ( ( A F B ) F C ) ) = ( 2nd ` ( A F ( B F C ) ) ) ) $=
        ( vt wa cnp wcel w3a co c2nd cfv cv wceq wrex cnq c1st cop prop elprnqu
        sylan wex oveq2 adantr adantl eqtr4d eqeq2d expcom pm5.32d 3expa sylan2
        anassrs rexbidva r19.41v bitr3di an32s bitrdi exbidv caovcl elisset syl
        wb an31s biantrurd 19.41v bitr4di rexcom4 ancoms df-rex genpelvu anbi1d
        bitrid 3bitr4rd 3impb 3adant3 3impa oveq1 rexbidv pm5.32i exbii 3bitr4d
        bitri rexbii eqrdv ) FUAUBZGUAUBZHUAUBZUCZAFGLUDZHLUDUEUFZFGHLUDZLUDUEU
        FZXBAUGZIUGZSUGZMUDZUHZSXEUEUFZUIZIFUEUFZUIZXGXHJUGZMUDZKUGZMUDZUHZKHUE
        UFZUIZJGUEUFZUIZIXNUIZXGXFUBZXGXDUBZWSWTXAXOYEVPZWTXATZWSYHYIWSTXMYDIXN
        YIWSXHXNUBZXMYDVPZWSYJTZYIXHUJUBZYKWSFUKUFZXNULUAUBYJYMFUMXHXNYNUNUOZYI
        YMTZXIXPXRMUDZUHZXTTZKYAUIZJYCUIZSUPZYRKYAUIZJYCUIZXKTZSUPZYDXMYPUUAUUE
        SYMXAWTUUAUUEVPYMXATZWTTZUUAUUCXKTZJYCUIUUEUUHYTUUIJYCUUGWTXPYCUBZYTUUI
        VPZWTUUJTZUUGXPUJUBZUUKWTGUKUFZYCULUAUBUUJUUMGUMXPYCUUNUNUOZYMUUMXAUUKY
        MUUMTZXATZYRXKTZKYAUIYTUUIUUQUURYSKYAUUPXAXRYAUBZUURYSVPZXAUUSTZUUPXRUJ
        UBZUUTXAHUKUFZYAULUAUBUUSUVBHUMXRYAUVCUNUOZYMUUMUVBUUTYMUUMUVBUCZYRXKXT
        YRUVEXKXTVPYRUVETZXJXSXGUVFXJXHYQMUDZXSYRXJUVGUHUVEXIYQXHMUQURUVEXSUVGU
        HYRRUSUTVAVBVCVDVEVFVGYRXKKYAVHVIVJVEVFVGUUCXKJYCVHVKVQVLYIYDUUBVPYMYIY
        DYTSUPZJYCUIZUUBXAWTYDUVIVPXAWTTYBUVHJYCXAWTUUJYBUVHVPZUULXAUUMUVJUUOUU
        MXAUVJUUMXATZYBYSSUPZKYAUIUVHUVKXTUVLKYAUUMXAUUSXTUVLVPZUVAUUMUVBUVMUVD
        UUMUVBTZXTYRSUPZXTTUVLUVNUVOXTUVNYQUJUBUVOBCXPXRUJMOVMSYQUJVNVOVRYRXTSV
        SVTVEVFVGYSKSYAWAVKWBVEVFVGWBYTJSYCWAVKURYIXMUUFVPYMXMXIXLUBZXKTZSUPYIU
        UFXKSXLWCYIUVQUUESYIUVPUUDXKABCDEGHXIJKLMNOWDWEVLWFURWGVEVFVGWBWHWSWTXA
        YFXOVPZYIWSXEUAUBUVRIJGHUALQVMABCDEFXEXGISLMNOWDVEWHXBXGXIXRMUDZUHZKYAU
        IZSXCUEUFZUIZXIXQUHZJYCUIZIXNUIZUWATZSUPZYGYEUWCXIUWBUBZUWATZSUPXBUWHUW
        ASUWBWCXBUWJUWGSXBUWIUWFUWAWSWTUWIUWFVPXAABCDEFGXIIJLMNOWDWIWEVLWFWSWTX
        AYGUWCVPZWSWTTXCUAUBXAUWKIJFGUALQVMABCDEXCHXGSKLMNOWDUOWJWSWTYEUWHVPZXA
        WTWSUWLWTWSTZYEUWDUWATZJYCUIZIXNUIZSUPZUWHUWMYEUWOSUPZIXNUIUWQUWMYDUWRI
        XNWTWSYJYDUWRVPZYLWTYMUWSYOYMWTUWSYMWTTZYDUWNSUPZJYCUIUWRUWTYBUXAJYCYMW
        TUUJYBUXAVPZUULYMUUMUXBUUOUUPYBUWDSUPZYBTZUXAUUPUXCYBUUPXQUJUBUXCBCXHXP
        UJMOVMSXQUJVNVOVRUXAUWDYBTZSUPUXDUWNUXESUWDUWAYBUWDUVTXTKYAUWDUVSXSXGXI
        XQXRMWKVAWLWMWNUWDYBSVSWPVTVEVFVGUWNJSYCWAVKWBVEVFVGUWOISXNWAVKUWPUWGSU
        WPUWEUWATZIXNUIUWGUWOUXFIXNUWDUWAJYCVHWQUWEUWAIXNVHWPWNVKWBWIWOWGWR $.

      $( Associativity of an operation on reals.  (Contributed by Jim Kingdon,
         11-Dec-2019.) $)
      genpassg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
          ( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
        ( cnp wcel w3a co wceq c1st cfv genpassl genpassu wa caovcl sylan 3impa
        c2nd wb sylan2 3impb preqlu syl2anc mpbir2and ) FSTZGSTZHSTZUAZFGLUBZHL
        UBZFGHLUBZLUBZUCZVDUDUEVFUDUEUCZVDULUEVFULUEUCZABCDEFGHIJKLMNOPQRUFABCD
        EFGHIJKLMNOPQRUGVBVDSTZVFSTZVGVHVIUHUMUSUTVAVJUSUTUHVCSTVAVJIJFGSLQUIIJ
        VCHSLQUIUJUKUSUTVAVKUTVAUHUSVESTVKIJGHSLQUIIJFVESLQUIUNUOVDVFUPUQUR $.
    $}
  $}

  ${
    $d G w y z $.  $d L w y z $.  $d S w y z $.  $d U w y z $.  $d X w y z $.
    $( Lemma to prove downward closure in positive real addition.  (Contributed
       by Jim Kingdon, 7-Dec-2019.) $)
    addnqprllem $p |- ( ( ( <. L , U >. e. P. /\ G e. L ) /\ X e. Q. ) ->
        ( X <Q S -> ( ( X .Q ( *Q ` S ) ) .Q G ) e. L ) ) $=
      ( vy vz vw wcel wa cnq cltq wbr cmq co wb adantl cv syl2anc c1q cop simpr
      cnp crq cfv ltrnqi ltrelnq brel simprd recclnq syl simplr syl3anc mulclnq
      ltmnqg w3a elprnql ad2antrr wceq mulcomnqg caovord2d imbitrid mpd recidnq
      bitrd oveq1d 1nq mpan mulidnq eqtrd sylan9eqr breq2d mpbid wi prcdnql ex
      ) DBUAUCICDIJZEKIZJZEALMZEAUDUEZNOZCNOZDIZVSVTJZWCCLMZWDWEWCEEUDUEZNOZCNO
      ZLMZWFWEVTWJVSVTUBVTWAWGLMZWEWJEAUFWEWKWBWHLMZWJWEWAKIZWGKIZVRWKWLPWEAKIZ
      WMWEVRWOVTVRWOJVSEAKKLUGUHQUIAUJUKZWEVRWNVQVRVTULZEUJUKZWQWAWGEUOUMWEFGHW
      BWHCLKNFRZKIZGRZKIZHRZKIUPWSXALMXCWSNOXCXANOLMPWEWSXAXCUOQWEVRWMWBKIWQWPE
      WAUNSWEVRWNWHKIWQWREWGUNSVQCKIZVRVTCBDUQURZWTXBJWSXANOXAWSNOUSWEWSXAUTQVA
      VEVBVCWEXDVRWJWFPXEWQXDVRJWICWCLVRXDWITCNOZCVRWHTCNEVDVFXDXFCTNOZCTKIXDXF
      XGUSVGTCUTVHCVIVJVKVLSVMVQWFWDVNVRVTCWCBDVOURVCVP $.

    $( Lemma to prove upward closure in positive real addition.  (Contributed
       by Jim Kingdon, 7-Dec-2019.) $)
    addnqprulem $p |- ( ( ( <. L , U >. e. P. /\ G e. U ) /\ X e. Q. ) ->
        ( S <Q X -> ( ( X .Q ( *Q ` S ) ) .Q G ) e. U ) ) $=
      ( vy vz vw wcel wa cnq cltq wbr cmq co wb adantl cv syl2anc c1q cop simpr
      cnp crq cfv ltrnqi simplr recclnq syl ltrelnq brel simpld syl3anc mulclnq
      ltmnqg w3a elprnqu ad2antrr wceq mulcomnqg caovord2d imbitrid mpd recidnq
      bitrd oveq1d 1nq mpan mulidnq eqtrd sylan9eqr breq1d mpbid wi prcunqu ex
      ) DBUAUCICBIJZEKIZJZAELMZEAUDUEZNOZCNOZBIZVSVTJZCWCLMZWDWEEEUDUEZNOZCNOZW
      CLMZWFWEVTWJVSVTUBVTWGWALMZWEWJAEUFWEWKWHWBLMZWJWEWGKIZWAKIZVRWKWLPWEVRWM
      VQVRVTUGZEUHUIZWEAKIZWNWEWQVRVTWQVRJVSAEKKLUJUKQULAUHUIZWOWGWAEUOUMWEFGHW
      HWBCLKNFRZKIZGRZKIZHRZKIUPWSXALMXCWSNOXCXANOLMPWEWSXAXCUOQWEVRWMWHKIWOWPE
      WGUNSWEVRWNWBKIWOWREWAUNSVQCKIZVRVTCBDUQURZWTXBJWSXANOXAWSNOUSWEWSXAUTQVA
      VEVBVCWEXDVRWJWFPXEWOXDVRJWICWCLVRXDWITCNOZCVRWHTCNEVDVFXDXFCTNOZCTKIXDXF
      XGUSVGTCUTVHCVIVJVKVLSVMVQWFWDVNVRVTWCCBDVOURVCVP $.
  $}

  ${
    $d q r s x y A $.  $d q r s x y B $.
    $( Lemma to prove downward closure in positive real addition.  (Contributed
       by Jim Kingdon, 5-Dec-2019.) $)
    addnqprl $p |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\
                       ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) ->
                       ( X <Q ( G +Q H ) -> X e. ( 1st ` ( A +P. B ) ) ) ) $=
      ( vr vs cnp wcel c1st cfv wa cnq cplq co cmq wi simpl wceq c1q vq vx cltq
      vy wbr crq cpp c2nd prop addnqprllem sylanl1 adantlr adantll jcad anim12i
      cop df-iplp cv addclnq genpprecll 3syl syld simpr sylan ad2antrr ad2antlr
      elprnql syl2anc recclnq mulassnqg syl3anc mulclnq mulcomnqg recidnq eqtrd
      syl distrnqg oveq2d mulidnq adantl 3eqtr3d eleq1d sylibd ) AHIZCAJKZIZLZB
      HIZDBJKZIZLZLZEMIZLZECDNOZUCUEZEWOUFKZPOZCPOZWRDPOZNOZABUGOJKZIZEXBIWNWPW
      SWEIZWTWIIZLZXCWNWPXDXEWGWMWPXDQZWKWDWEAUHKZUPHIZWFWMXGAUIZWOXHCWEEUJUKUL
      WKWMWPXEQZWGWHWIBUHKZUPHIZWJWMXKBUIZWOXLDWIEUJUKUMUNWNWLWDWHLXFXCQWLWMRWG
      WDWKWHWDWFRWHWJRUOUAFGUBUDABWSWTUGNUBUDGFUAUQFURGURUSUTVAVBWNXAEXBWNWRWOP
      OZEWQWOPOZPOZXAEWNWMWQMIZWOMIZXOXQSWLWMVCZWNXSXRWNCMIZDMIZXSWGYAWKWMWDXIW
      FYAXJCXHWEVGVDVEZWKYBWGWMWHXMWJYBXNDXLWIVGVDVFZCDUSVHZWOVIVPZYEEWQWOVJVKW
      NWRMIZYAYBXOXASWNWMXRYGXTYFEWQVLVHYCYDWRCDVQVKWNXQETPOZEWNXPTEPWNXPWOWQPO
      ZTWNXRXSXPYISYFYEWQWOVMVHWNXSYITSYEWOVNVPVOVRWMYHESWLEVSVTVOWAWBWC $.

    $( Lemma to prove upward closure in positive real addition.  (Contributed
       by Jim Kingdon, 5-Dec-2019.) $)
    addnqpru $p |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\
                       ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) ->
                       ( ( G +Q H ) <Q X -> X e. ( 2nd ` ( A +P. B ) ) ) ) $=
      ( vr vs cnp wcel c2nd cfv wa cnq cplq co cmq wi simpl wceq c1q vq vx cltq
      vy wbr crq cpp c1st prop addnqprulem sylanl1 adantlr adantll jcad anim12i
      cop df-iplp cv addclnq genppreclu 3syl syld simpr sylan ad2antrr ad2antlr
      elprnqu syl2anc recclnq mulassnqg syl3anc mulclnq mulcomnqg recidnq eqtrd
      syl distrnqg oveq2d mulidnq adantl 3eqtr3d eleq1d sylibd ) AHIZCAJKZIZLZB
      HIZDBJKZIZLZLZEMIZLZCDNOZEUCUEZEWOUFKZPOZCPOZWRDPOZNOZABUGOJKZIZEXBIWNWPW
      SWEIZWTWIIZLZXCWNWPXDXEWGWMWPXDQZWKWDAUHKZWEUPHIZWFWMXGAUIZWOWECXHEUJUKUL
      WKWMWPXEQZWGWHBUHKZWIUPHIZWJWMXKBUIZWOWIDXLEUJUKUMUNWNWLWDWHLXFXCQWLWMRWG
      WDWKWHWDWFRWHWJRUOUAFGUBUDABWSWTUGNUBUDGFUAUQFURGURUSUTVAVBWNXAEXBWNWRWOP
      OZEWQWOPOZPOZXAEWNWMWQMIZWOMIZXOXQSWLWMVCZWNXSXRWNCMIZDMIZXSWGYAWKWMWDXIW
      FYAXJCWEXHVGVDVEZWKYBWGWMWHXMWJYBXNDWIXLVGVDVFZCDUSVHZWOVIVPZYEEWQWOVJVKW
      NWRMIZYAYBXOXASWNWMXRYGXTYFEWQVLVHYCYDWRCDVQVKWNXQETPOZEWNXPTEPWNXPWOWQPO
      ZTWNXRXSXPYISYFYEWQWOVMVHWNXSYITSYEWOVNVPVOVRWMYHESWLEVSVTVOWAWBWC $.
  $}

  ${
    $d D f g h $.  $d E f g h $.  $d P f g h $.  $d f g h ph $.
    addlocprlem.a $e |- ( ph -> A e. P. ) $.
    addlocprlem.b $e |- ( ph -> B e. P. ) $.
    addlocprlem.qr $e |- ( ph -> Q <Q R ) $.
    addlocprlem.p $e |- ( ph -> P e. Q. ) $.
    addlocprlem.qppr $e |- ( ph -> ( Q +Q ( P +Q P ) ) = R ) $.
    addlocprlem.dlo $e |- ( ph -> D e. ( 1st ` A ) ) $.
    addlocprlem.uup $e |- ( ph -> U e. ( 2nd ` A ) ) $.
    addlocprlem.du $e |- ( ph -> U <Q ( D +Q P ) ) $.
    addlocprlem.elo $e |- ( ph -> E e. ( 1st ` B ) ) $.
    addlocprlem.tup $e |- ( ph -> T e. ( 2nd ` B ) ) $.
    addlocprlem.et $e |- ( ph -> T <Q ( E +Q P ) ) $.
    $( Lemma for ~ addlocpr .  The ` Q <Q ( D +Q E ) ` case.  (Contributed by
       Jim Kingdon, 6-Dec-2019.) $)
    addlocprlemlt $p |- ( ph ->
        ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) ) $=
      ( cnp wcel c1st cfv wa cnq cplq co cltq wbr cpp wi jca ltrelnq simpld syl
      brel addnqprl syl21anc ) ABUBUCZDBUDUEUCZUFCUBUCZJCUDUEUCZUFFUGUCZFDJUHUI
      UJUKFBCULUIUDUEUCUMAVAVBKPUNAVCVDLSUNAFGUJUKZVEMVFVEGUGUCFGUGUGUJUOURUPUQ
      BCDJFUSUT $.

    $( Lemma for ~ addlocpr .  This is a step used in both the
       ` Q = ( D +Q E ) ` and ` ( D +Q E ) <Q Q ` cases.  (Contributed by Jim
       Kingdon, 7-Dec-2019.) $)
    addlocprlemeqgt $p |- ( ph ->
        ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) ) $=
      ( vf vg vh cplq co cltq wbr cnq wcel wa wi c1st cfv c2nd cop cnp prop syl
      elprnqu syl2anc elprnql addclnq lt2addnq syl22anc mp2and addcomnqg adantl
      cv wceq w3a addassnqg caov4d breqtrd ) AIHUEUFZDEUEUFZJEUEUFZUEUFZDJUEUFE
      EUEUFUEUFUGAIVPUGUHZHVQUGUHZVOVRUGUHZRUAAIUIUJZVPUIUJZHUIUJZVQUIUJZVSVTUK
      WAULABUMUNZBUOUNZUPUQUJZIWGUJWBABUQUJWHKBURUSZQIWGWFUTVAADUIUJZEUIUJZWCAW
      HDWFUJWJWIPDWGWFVBVAZNDEVCVAACUMUNZCUOUNZUPUQUJZHWNUJWDACUQUJWOLCURUSZTHW
      NWMUTVAAJUIUJZWKWEAWOJWMUJWQWPSJWNWMVBVAZNJEVCVAIVPHVQVDVEVFAUBUCUDDEJEUI
      UEWLNWRUBVIZUIUJZUCVIZUIUJZUKZWSXAUEUFZXAWSUEUFVJAWSXAVGVHWTXBUDVIZUIUJVK
      XDXEUEUFWSXAXEUEUFUEUFVJAWSXAXEVLVHNXCXDUIUJAWSXAVCVHVMVN $.

    $( Lemma for ~ addlocpr .  The ` Q = ( D +Q E ) ` case.  (Contributed by
       Jim Kingdon, 6-Dec-2019.) $)
    addlocprlemeq $p |- ( ph ->
        ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) ) $=
      ( cplq co wceq cpp c2nd cfv wcel wa cltq wbr addlocprlemeqgt adantr oveq1
      sylan9req breqtrrd wi cnp cnq jca ltrelnq simprd syl addnqpru syl21anc ex
      brel mpd ) AFDJUBUCZUDZGBCUEUCUFUGUHZAVJUIZIHUBUCZGUJUKZVKVLVMVIEEUBUCZUB
      UCZGUJAVMVPUJUKVJABCDEFGHIJKLMNOPQRSTUAULUMAVJGFVOUBUCVPOFVIVOUBUNUOUPAVN
      VKUQZVJABURUHZIBUFUGUHZUICURUHZHCUFUGUHZUIGUSUHZVQAVRVSKQUTAVTWALTUTAFGUJ
      UKZWBMWCFUSUHWBFGUSUSUJVAVGVBVCBCIHGVDVEUMVHVF $.

    $( Lemma for ~ addlocpr .  The ` ( D +Q E ) <Q Q ` case.  (Contributed by
       Jim Kingdon, 6-Dec-2019.) $)
    addlocprlemgt $p |- ( ph ->
        ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) ) $=
      ( cplq co cltq wbr cpp c2nd cfv wcel wa addlocprlemeqgt adantr cnq wb cop
      c1st cnp prop syl elprnql syl2anc addclnq ltrelnq brel simpld ltanqg wceq
      syl3anc addcomnqg breq12d bitrd biimpa breq2d mpbid ltsonq sotri addnqpru
      jca wi simprd syl21anc mpd ex ) ADJUBUCZFUDUEZGBCUFUCUGUHUIZAWEUJZIHUBUCZ
      GUDUEZWFWGWHWDEEUBUCZUBUCZUDUEZWKGUDUEZUJWIWGWLWMAWLWEABCDEFGHIJKLMNOPQRS
      TUAUKULWGWKFWJUBUCZUDUEZWMAWEWOAWEWJWDUBUCZWJFUBUCZUDUEZWOAWDUMUIZFUMUIZW
      JUMUIZWEWRUNADUMUIZJUMUIZWSABUPUHZBUGUHZUOUQUIZDXDUIXBABUQUIZXFKBURUSPDXE
      XDUTVAACUPUHZCUGUHZUOUQUIZJXHUIXCACUQUIZXJLCURUSSJXIXHUTVADJVBVAZAWTGUMUI
      ZAFGUDUEWTXMUJMFGUMUMUDVCVDUSZVEZAEUMUIZXPXANNEEVBVAZWDFWJVFVHAWPWKWQWNUD
      AXAWSWPWKVGXQXLWJWDVIVAAXAWTWQWNVGXQXOWJFVIVAVJVKVLAWOWMUNWEAWNGWKUDOVMUL
      VNVRWHWKGUDUMVOVCVPUSAWIWFVSZWEAXGIXEUIZUJXKHXIUIZUJXMXRAXGXSKQVRAXKXTLTV
      RAWTXMXNVTBCIHGVQWAULWBWC $.

    $( Lemma for ~ addlocpr .  The result, in deduction form.  (Contributed by
       Jim Kingdon, 6-Dec-2019.) $)
    addlocprlem $p |- ( ph ->
        ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) ) $=
      ( cplq co cltq wbr wceq w3o cpp c1st cfv wcel c2nd wo ltrelnq brel simpld
      cnq syl cop cnp prop elprnql syl2anc nqtri3or addlocprlemlt addlocprlemeq
      addclnq orc syl6 olc addlocprlemgt 3jaod mpd ) AFDJUBUCZUDUEZFVNUFZVNFUDU
      EZUGZFBCUHUCZUIUJUKZGVSULUJUKZUMZAFUQUKZVNUQUKZVRAFGUDUEZWCMWEWCGUQUKFGUQ
      UQUDUNUOUPURADUQUKZJUQUKZWDABUIUJZBULUJZUSUTUKZDWHUKWFABUTUKWJKBVAURPDWIW
      HVBVCACUIUJZCULUJZUSUTUKZJWKUKWGACUTUKWMLCVAURSJWLWKVBVCDJVGVCFVNVDVCAVOW
      BVPVQAVOVTWBABCDEFGHIJKLMNOPQRSTUAVEVTWAVHVIAVPWAWBABCDEFGHIJKLMNOPQRSTUA
      VFWAVTVJZVIAVQWAWBABCDEFGHIJKLMNOPQRSTUAVKWNVIVLVM $.
  $}

  ${
    $d A d e h p q r t u $.  $d B d e h p q r t u $.
    $( Locatedness of addition on positive reals.  Lemma 11.16 in
       [BauerTaylor], p. 53.  The proof in BauerTaylor relies on signed
       rationals, so we replace it with another proof which applies ~ prarloc
       to both ` A ` and ` B ` , and uses ~ nqtri3or rather than ~ prloc to
       decide whether ` q ` is too big to be in the lower cut of ` A +P. B `
       (and deduce that if it is, then ` r ` must be in the upper cut).  What
       the two proofs have in common is that they take the difference between
       ` q ` and ` r ` to determine how tight a range they need around the real
       numbers.  (Contributed by Jim Kingdon, 5-Dec-2019.) $)
    addlocpr $p |- ( ( A e. P. /\ B e. P. ) ->
        A. q e. Q. A. r e. Q. ( q <Q r ->
          ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) ) $=
      ( vp vh vu vd vt ve cnp wcel wa cv co cfv cnq cplq wrex adantr wbr cpp wo
      cltq c1st c2nd wi w3a ltexnqq biimpa 3adant1 halfnqq ad2antrl cop prarloc
      wceq prop sylan adantlr 3ad2antl1 ad2ant2r adantll ad2antrr simpld simprd
      simpll1 simpll3 simplrl simplrr wb eqeq1d ad2antll mpbird simprll simprlr
      oveq2 simprr addlocprlem expr rexlimdvva mpd rexlimddv 3expia ralrimivva
      ) AKLZBKLZMZDNZCNZUDUAZWHABUBOZUEPLWIWKUFPLUCZUGDCQQWGWHQLWIQLMZWJWLWGWMW
      JUHZWHENZROZWIUPZWLEQWMWJWQEQSZWGWMWJWREWHWIUIUJUKWNWOQLZWQMZMZFNZXBROZWO
      UPZWLFQWSXDFQSWNWQFWOULUMXAXBQLZXDMZMZGNZHNZXBROUDUAZGAUFPZSHAUEPZSZWLWNX
      EXMWTXDWGWMXEXMWJWEXEXMWFWEXLXKUNKLXEXMAUQXBXKXLHGUOURUSUTVAXGXJWLHGXLXKX
      GXIXLLZXHXKLZMZXJWLXGXPXJMZMZINZJNZXBROUDUAZIBUFPZSJBUEPZSZWLXGYDXQWNXEYD
      WTXDWGWMXEYDWJWFXEYDWEWFYCYBUNKLXEYDBUQXBYBYCJIUOURVBUTVATXRYAWLJIYCYBXRX
      TYCLZXSYBLZMZYAWLXRYGYAMZMZABXIXBWHWIXSXHXTYIWEWFXGWGXQYHWGWMWJWTXFVFVCZV
      DYIWEWFYJVEXGWJXQYHWGWMWJWTXFVGVCXRXEYHXAXEXDXQVHTXGWHXCROZWIUPZXQYHXGYLW
      QWNWSWQXFVIXDYLWQVJXAXEXDYKWPWIXCWOWHRVPVKVLVMVCXRXNYHXGXNXOXJVNTXRXOYHXG
      XNXOXJVOTXGXPXJYHVIXRYEYFYAVNXRYEYFYAVOXRYGYAVQVRVSVTWAVSVTWAWBWBWCWD $.
  $}

  ${
    $d x y z w v g h q r A $.  $d x y z w v g h q r B $.
    $( Closure of addition on positive reals.  First statement of Proposition
       9-3.5 of [Gleason] p. 123.  Combination of Lemma 11.13 and Lemma 11.16
       in [BauerTaylor], p. 53.  (Contributed by NM, 13-Mar-1996.) $)
    addclpr $p |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. ) $=
      ( vq vr vx vy vz vw vv vg vh cnp wcel wa cpp cnq cv wrex wral cplq co cpw
      cxp c1st cfv c2nd cltq wbr wb wn wo wi w3a df-iplp genpelxp genpml genpmu
      addclnq ltanqg addcomnqg addnqprl genprndl addnqpru genprndu jca genpdisj
      jca32 addlocpr 3jca elnp1st2nd sylanbrc ) ALMBLMNZABOUAZPUBZVNUCMZCQZVMUD
      UEZMZCPRZDQZVMUFUEZMZDPRZNNVRVPVTUGUHZVTVQMNDPRUICPSZWBWDVPWAMZNCPRUIDPSZ
      NZVRWFNUJCPSZWDVRWBUKULDPSCPSZUMVMLMVLVOVSWCEFGHIABOTHIGFEUNZUOEFGHIABOTC
      WKFQZGQZURZUPEFGHIABOTDWKWNUQVGVLWHWIWJVLWEWGEFGHIABJKOTDCWKWNEQZWLWMUSZW
      OWLUTZABJQZKQZWOVAVBEFGHIABJKOTDCWKWNWPWQABWRWSWOVCVDVEEFGHIABOTCWKWNWPWQ
      VFABDCVHVIVMDCVJVK $.
  $}

  ${
    $d x y z f g h A $.  $d x y z f g h B $.  $d x y z w v f g h $.
    $( Value of addition on positive reals.  (Contributed by Jim Kingdon,
       8-Dec-2019.) $)
    plpvlu $p |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = <.
        { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y +Q z ) } ,
        { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y +Q z ) }
        >. ) $=
      ( vf vg vh vw vv cpp cplq df-iplp cv addclnq genipv ) FGHIJDEKLCBAIJHGFMG
      NHNOP $.

    $( Value of multiplication on positive reals.  (Contributed by Jim Kingdon,
       8-Dec-2019.) $)
    mpvlu $p |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = <.
        { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y .Q z ) } ,
        { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y .Q z ) }
        >. ) $=
      ( vf vg vh vw vv cmp cmq df-imp cv mulclnq genipv ) FGHIJDEKLCBAIJHGFMGNH
      NOP $.

    $( Domain of addition on positive reals.  (Contributed by NM,
       18-Nov-1995.) $)
    dmplp $p |- dom +P. = ( P. X. P. ) $=
      ( vv vw vz vx vy cpp cplq df-iplp cv addclnq genipdm ) ABCDEFGDECBAHBICIJ
      K $.

    $( Domain of multiplication on positive reals.  (Contributed by NM,
       18-Nov-1995.) $)
    dmmp $p |- dom .P. = ( P. X. P. ) $=
      ( vv vw vz vx vy cmp cmq df-imp cv mulclnq genipdm ) ABCDEFGDECBAHBICIJK
      $.
  $}

  ${
    $d A x n r q $.
    $( A cut produced from a rational is inhabited.  Lemma for ~ nqprlu .
       (Contributed by Jim Kingdon, 8-Dec-2019.) $)
    nqprm $p |- ( A e. Q. -> ( E. q e. Q. q e. { x | x <Q A }
        /\ E. r e. Q. r e. { x | A <Q x } ) ) $=
      ( vn cnq wcel cv cltq wbr cab wrex vex elab rexbii sylibr cnpi c1o ceq wa
      nsmallnqq breq1 cop cec wex archnqq sylib cxp cqs 1pi opelxpi ecelqsi syl
      df-rex enqex mpan2 df-nqqs eleqtrrdi breq2 rspcev sylan exlimiv jca ) BFG
      ZDHZAHZBIJZAKGZDFLZCHZBVFIJZAKGZCFLZVDVEBIJZDFLVIDBUAVHVNDFVGVNAVEDMVFVEB
      IUBNOPVDBVJIJZCFLZVMVDEHZQGZBVQRUCZSUDZIJZTZEUEZVPVDWAEQLWCEBUFWAEQUNUGWB
      VPEVRVTFGWAVPVRVTQQUHZSUIZFVRRQGZVTWEGZUJVRWFTVSWDGWGVQRQQUKWDVSSUOULUMUP
      UQURVOWACVTFVJVTBIUSUTVAVBUMVLVOCFVKVOAVJCMVFVJBIUSNOPVC $.

    $( A cut produced from a rational is rounded.  Lemma for ~ nqprlu .
       (Contributed by Jim Kingdon, 8-Dec-2019.) $)
    nqprrnd $p |- ( A e. Q. -> ( A. q e. Q. ( q e. { x | x <Q A }
        <-> E. r e. Q. ( q <Q r /\ r e. { x | x <Q A } ) )
        /\ A. r e. Q. ( r e. { x | A <Q x }
        <-> E. q e. Q. ( q <Q r /\ q e. { x | A <Q x } ) ) ) ) $=
      ( cnq wcel cv cltq wbr cab wa wrex wral ltbtwnnqq rexbii vex breq2 anbi2i
      wb elab ancom bitri 3bitr4i rgenw a1i breq1 jctil ) BEFZCGZBAGZHIZAJZFZDG
      ZUIHIZUNULFZKZDELZSZCEMZUNUJBHIZAJZFZUOUIVBFZKZCELZSZDEMUTUHUSCEBUIHIZUOB
      UNHIZKZDELZUMURVHVIUOKZDELVKDBUINVLVJDEVIUOUAOUBUKVHAUICPZUJUIBHQTUQVJDEU
      PVIUOUKVIAUNDPZUJUNBHQTROUCUDUEVGDEUNBHIZUOUIBHIZKZCELVCVFCUNBNVAVOAUNVNU
      JUNBHUFTVEVQCEVDVPUOVAVPAUIVMUJUIBHUFTROUCUDUG $.

    $( A cut produced from a rational is disjoint.  Lemma for ~ nqprlu .
       (Contributed by Jim Kingdon, 8-Dec-2019.) $)
    nqprdisj $p |- ( A e. Q. -> A. q e. Q.
        -. ( q e. { x | x <Q A } /\ q e. { x | A <Q x } ) ) $=
      ( cv cltq wbr cab wcel wa cnq wral ltsonq ltrelnq son2lpi vex breq1 breq2
      wn elab anbi12i mtbir rgenw a1i ) CDZADZBEFZAGHZUDBUEEFZAGHZIZRZCJKBJHUKC
      JUJUDBEFZBUDEFZIUDBEJLMNUGULUIUMUFULAUDCOZUEUDBEPSUHUMAUDUNUEUDBEQSTUAUBU
      C $.

    $( A cut produced from a rational is located.  Lemma for ~ nqprlu .
       (Contributed by Jim Kingdon, 8-Dec-2019.) $)
    nqprloc $p |- ( A e. Q. -> A. q e. Q. A. r e. Q. ( q <Q r
        -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) ) $=
      ( cnq wcel cv cltq wbr cab wo wi wral wceq w3o vex breq1 elab ralrimiva
      wa nqtri3or ancoms ad2antrr biimpri orcd a1i simpr syl5ibcom breq2 sylbir
      olc syl6 ltsonq ltrelnq sotri syl expcom adantl 3jaod mpd ex ) BEFZDGZCGZ
      HIZVCAGZBHIZAJFZVDBVFHIZAJFZKZLZCEMDEVBVCEFZTZVLCEVNVDEFZTZVEVKVPVETZVCBH
      IZVCBNZBVCHIZOZVKVNWAVOVEVMVBWAVCBUAUBUCVQVRVKVSVTVRVKLVQVRVHVJVHVRVGVRAV
      CDPVFVCBHQRUDUEUFVQVSBVDHIZVKVQVEVSWBVPVEUGVCBVDHQUHWBVJVKVIWBAVDCPVFVDBH
      UIRVJVHUKUJZULVEVTVKLVPVTVEVKVTVETWBVKBVCVDHEUMUNUOWCUPUQURUSUTVASS $.

    $( The canonical embedding of the rationals into the reals, expressed with
       the same variable for the lower and upper cuts.  (Contributed by Jim
       Kingdon, 8-Dec-2019.) $)
    nqprxx $p |- ( A e. Q. -> <. { x | x <Q A } , { x | A <Q x } >. e. P. ) $=
      ( vq vr cnq wcel cv cltq wbr cab wa wrex wb wral wn wo ltrelnq brel abssi
      wss w3a cop cnp nqprm simpld simprd pm3.2i jctil nqprrnd nqprdisj nqprloc
      wi 3jca elinp sylanbrc ) BEFZAGZBHIZAJZETZBUQHIZAJZETZKZCGZUSFZCELDGZVBFZ
      DELKZKVFVEVGHIZVGUSFKDELMCENVHVJVEVBFZKCELMDENKZVFVKKOCENZVJVFVHPULDENCEN
      ZUAUSVBUBUCFUPVIVDABDCUDUTVCURAEURUQEFZUPUQBEEHQRUESVAAEVAUPVOBUQEEHQRUFS
      UGUHUPVLVMVNABDCUIABCUJABDCUKUMVBUSDCUNUO $.
  $}

  ${
    $d A l a $.  $d A u a $.
    $( The canonical embedding of the rationals into the reals.  (Contributed
       by Jim Kingdon, 24-Jun-2020.) $)
    nqprlu $p |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. ) $=
      ( va cnq wcel cv cltq wbr cab breq2 cbvabv eqtr4i opeq2i nqprxx eqeltrrid
      cop cnp ) BEFCGZBHICJZBAGZHIZAJZQTBSHIZCJZQRUEUCTUEBDGZHIZDJUCUDUGCDSUFBH
      KLUBUGADUAUFBHKLMNCBOP $.
  $}

  ${
    $d A l u $.
    $( The reciprocal of a positive integer, as a positive real.  (Contributed
       by Jim Kingdon, 27-Feb-2021.) $)
    recnnpr $p |- ( A e. N. ->
        <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } ,
        { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. e. P. ) $=
      ( cnpi wcel c1o cop ceq cec cnq crq cfv cv cltq wbr cab cnp nnnq recclnq
      nqprlu 3syl ) BDEBFGHIZJEUBKLZJECMUCNOCPUCAMNOAPGQEBRUBSAUCCTUA $.
  $}

  $( The class of rationals less than a given rational is a set.  (Contributed
     by Jim Kingdon, 13-Dec-2019.) $)
  ltnqex $p |- { x | x <Q A } e. _V $=
    ( cv cltq wbr cab cnq nqex wcel ltrelnq brel simpld abssi ssexi ) ACZBDEZAF
    GHPAGPOGIBGIOBGGDJKLMN $.

  $( The class of rationals greater than a given rational is a set.
     (Contributed by Jim Kingdon, 13-Dec-2019.) $)
  gtnqex $p |- { x | A <Q x } e. _V $=
    ( cv cltq wbr cab cnq nqex wcel ltrelnq brel simprd abssi ssexi ) BACZDEZAF
    GHPAGPBGIOGIBOGGDJKLMN $.

  ${
    $d A l u x $.  $d B x $.
    $( Comparing a fraction to a real can be done by whether it is an element
       of the lower cut, or by ` <P ` .  (Contributed by Jim Kingdon,
       8-Jul-2020.) $)
    nqprl $p |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) <->
        <. { l | l <Q A } , { u | A <Q u } >. <P B ) ) $=
      ( vx cnq wcel cnp wa cfv cv cltq wbr cab cop wi c2nd wrex sylan elprnql
      c1st cltp prop prnmaxl wex ad2ant2r vex breq2 biimpri ltnqex gtnqex op2nd
      elab eleq2i sylibr ad2antll simprl 19.8a syl12anc df-rex wb nqprlu ltdfpr
      simpl syl2anc adantr mpbird rexlimddv adantl biimpa bitri biimpi ad2antrl
      ex simpllr simprrr prcdnql mpd impbid ) BFGZCHGZIZBCUAJZGZDKBLMDNZBAKZLMZ
      ANZOZCUBMZWAWDWJPVTWAWDWJWAWDIZBEKZLMZWJEWCWAWCCQJZOHGZWDWMEWCRCUCZEBWNWC
      UDSWKWLWCGZWMIZIZWJWLWIQJZGZWQIZEFRZWSWLFGZXBIZEUEZXCWSXDXAWQXFWAWQXDWDWM
      WAWOWQXDWPWLWNWCTSUFWMXAWKWQWMWLWHGZXAXGWMWGWMAWLEUGWFWLBLUHUMZUIWTWHWLWE
      WHDBUJABUKULUNZUOUPWKWQWMUQXEEURUSXBEFUTUOWKWJXCVAZWRWKVTWAXJWAWOWDVTWPBW
      NWCTSWAWDVDVTWIHGWAXJABDVBWICEVCSZVEVFVGVHVNVIWBWJWDWBWJIZXBWDEFWBWJXCXKV
      JXLXEIZWMWDXEWMXLXAWMXDWQXAWMXAXGWMXIXHVKVLVMVIXMWAWQWMWDPZVTWAWJXEVOXLXD
      XAWQVPWAWOWQXNWPWLBWNWCVQSVEVRVHVNVS $.
  $}

  ${
    $d A l u x $.  $d B x $.
    $( Comparing a fraction to a real can be done by whether it is an element
       of the upper cut, or by ` <P ` .  (Contributed by Jim Kingdon,
       29-Nov-2020.) $)
    nqpru $p |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 2nd ` B ) <->
        B <P <. { l | l <Q A } , { u | A <Q u } >. ) ) $=
      ( vx cnq wcel cnp wa cfv cv cltq wbr cab cop wi c1st wrex sylan elprnqu
      c2nd cltp prop prnminu wex ad2ant2r simprl vex elab biimpri ltnqex gtnqex
      breq1 op1st eleq2i sylibr ad2antll 19.8a syl12anc df-rex wb nqprlu ltdfpr
      sylan2 syldan adantr mpbird rexlimddv adantl ancoms biimpa biimpi simpllr
      ex bitri simprrl prcunqu syl2anc mpd impbid ) BFGZCHGZIZBCUAJZGZCDKZBLMZD
      NZBAKLMANZOZUBMZWBWEWKPWAWBWEWKWBWEIZEKZBLMZWKEWDWBCQJZWDOHGZWEWNEWDRCUCZ
      EBWDWOUDSWLWMWDGZWNIZIZWKWRWMWJQJZGZIZEFRZWTWMFGZXCIZEUEZXDWTXEWRXBXGWBWR
      XEWEWNWBWPWRXEWQWMWDWOTSUFWLWRWNUGWNXBWLWRWNWMWHGZXBXHWNWGWNDWMEUHWFWMBLU
      MUIZUJXAWHWMWHWIDBUKABULUNUOZUPUQXFEURUSXCEFUTUPWLWKXDVAZWSWBWEWAXKWBWPWE
      WAWQBWDWOTSWAWBWJHGXKABDVBCWJEVCVDZVEVFVGVHVNVIWCWKWEWCWKIZXCWEEFWCWKXDWB
      WAXKXLVJVKXMXFIZWNWEXFWNXMXBWNXEWRXBWNXBXHWNXJXIVOVLUQVIXNWBWRWNWEPZWAWBW
      KXFVMXMXEWRXBVPWBWPWRXOWQBWMWDWOVQSVRVSVHVNVT $.
  $}

  ${
    $d A l u $.
    $( The canonical embedding of positive integers into the positive reals.
       (Contributed by Jim Kingdon, 23-Apr-2020.) $)
    nnprlu $p |- ( A e. N. -> <. { l | l <Q [ <. A , 1o >. ] ~Q } ,
        { u | [ <. A , 1o >. ] ~Q <Q u } >. e. P. ) $=
      ( cnpi wcel c1o cop ceq cec cnq cv cltq wbr cab cnp nnnq nqprlu syl ) BDE
      BFGHIZJECKSLMCNSAKLMANGOEBPASCQR $.
  $}

  ${
    $d x y $.
    $( The positive real number 'one'.  (Contributed by NM, 13-Mar-1996.)
       (Revised by Mario Carneiro, 12-Jun-2013.) $)
    1pr $p |- 1P e. P. $=
      ( vx vy c1p cv c1q cltq wbr cab cop cnp df-i1p cnq wcel 1nq ax-mp eqeltri
      nqprlu ) CADEFGAHEBDFGBHIZJBAKELMRJMNBEAQOP $.

    $( The lower cut of the positive real number 'one'.  (Contributed by Jim
       Kingdon, 28-Dec-2019.) $)
    1prl $p |- ( 1st ` 1P ) = { x | x <Q 1Q } $=
      ( vy c1p c1st cfv cv c1q cltq wbr df-i1p fveq2i ltnqex gtnqex op1st eqtri
      cab cop ) CDEAFGHIAPZGBFHIBPZQZDERCTDBAJKRSAGLBGMNO $.

    $( The upper cut of the positive real number 'one'.  (Contributed by Jim
       Kingdon, 28-Dec-2019.) $)
    1pru $p |- ( 2nd ` 1P ) = { x | 1Q <Q x } $=
      ( vy c1p c2nd cfv cv c1q cltq wbr df-i1p fveq2i ltnqex gtnqex op2nd eqtri
      cab cop ) CDEBFGHIBPZGAFHIAPZQZDESCTDABJKRSBGLAGMNO $.
  $}

  ${
    $d A f g h l r s t u x y $.  $d B f g h l r s t u x y $.
    $( Lemma for ~ addnqpr .  The reverse subset relationship for the lower
       cut.  (Contributed by Jim Kingdon, 19-Aug-2020.) $)
    addnqprlemrl $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P.
        <. { l | l <Q B } , { u | B <Q u } >. ) ) C_
        ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. )
        ) $=
      ( vr vs vt vg vh cnq wcel wa cv cltq wbr cab cop c1st cfv breq1 vf vx cpp
      vy co cplq wceq wrex cnp wb nqprlu df-iplp addclnq genpelvl syl2an biimpa
      ltnqex gtnqex op1st elab2 biimpi ad2antrl adantr ad2antll wi ltrelnq brel
      vex syl lt2addnq syl2anc mp2and adantl mpbird sylibr rexlimdvva mpd ssrdv
      ex ) BJKZCJKZLZEDMZBNOZDPZBAMZNOAPZQZWCCNOZDPZCWFNOAPZQZUCUERSZWCBCUFUEZN
      OZDPZWNWFNOAPZQRSZWBEMZWMKZWSWRKZWBWTLZWSFMZGMZUFUEZUGZGWLRSZUHFWHRSZUHZX
      AWBWTXIVTWHUIKWLUIKWTXIUJWAABDUKACDUKUAHIUBUDWHWLWSFGUCUFUBUDIHUAULHMIMUM
      UNUOUPXBXFXAFGXHXGXBXCXHKZXDXGKZLLZXFXAXLXFLZWSWNNOZXAXMXNXEWNNOZXMXCBNOZ
      XDCNOZXOXLXPXFXJXPXBXKXJXPWDXPDXCXHFVHWCXCBNTWEWGDBUQABURUSUTVAVBVCZXLXQX
      FXKXQXBXJXKXQWIXQDXDXGGVHWCXDCNTWJWKDCUQACURUSUTVAVDVCZXMXCJKVTLZXDJKWALZ
      XPXQLXOVEXMXPXTXRXCBJJNVFVGVIXMXQYAXSXDCJJNVFVGVIXCBXDCVJVKVLXFXNXOUJXLWS
      XEWNNTVMVNWOXNDWSWREVHWCWSWNNTWPWQDWNUQAWNURUSUTVOVSVPVQVSVR $.

    $( Lemma for ~ addnqpr .  The reverse subset relationship for the upper
       cut.  (Contributed by Jim Kingdon, 19-Aug-2020.) $)
    addnqprlemru $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P.
        <. { l | l <Q B } , { u | B <Q u } >. ) ) C_
        ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. )
        ) $=
      ( vr vs vt vg vh cnq wcel wa cv cltq wbr cab cop c2nd cfv breq2 vf vx cpp
      vy co cplq wceq wrex cnp wb nqprlu df-iplp addclnq genpelvu syl2an biimpa
      ltnqex gtnqex op2nd elab2 biimpi ad2antrl adantr ad2antll wi ltrelnq brel
      vex syl lt2addnq syl2anc mp2and adantl mpbird sylibr rexlimdvva mpd ssrdv
      ex ) BJKZCJKZLZEDMZBNODPZBAMZNOZAPZQZWCCNODPZCWENOZAPZQZUCUERSZWCBCUFUEZN
      ODPZWNWENOZAPZQRSZWBEMZWMKZWSWRKZWBWTLZWSFMZGMZUFUEZUGZGWLRSZUHFWHRSZUHZX
      AWBWTXIVTWHUIKWLUIKWTXIUJWAABDUKACDUKUAHIUBUDWHWLWSFGUCUFUBUDIHUAULHMIMUM
      UNUOUPXBXFXAFGXHXGXBXCXHKZXDXGKZLLZXFXAXLXFLZWNWSNOZXAXMXNWNXENOZXMBXCNOZ
      CXDNOZXOXLXPXFXJXPXBXKXJXPWFXPAXCXHFVHWEXCBNTWDWGDBUQABURUSUTVAVBVCZXLXQX
      FXKXQXBXJXKXQWJXQAXDXGGVHWEXDCNTWIWKDCUQACURUSUTVAVDVCZXMVTXCJKLZWAXDJKLZ
      XPXQLXOVEXMXPXTXRBXCJJNVFVGVIXMXQYAXSCXDJJNVFVGVIBXCCXDVJVKVLXFXNXOUJXLWS
      XEWNNTVMVNWPXNAWSWREVHWEWSWNNTWOWQDWNUQAWNURUSUTVOVSVPVQVSVR $.

    $( Lemma for ~ addnqpr .  The forward subset relationship for the lower
       cut.  (Contributed by Jim Kingdon, 19-Aug-2020.) $)
    addnqprlemfl $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. )
        C_ ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P.
        <. { l | l <Q B } , { u | B <Q u } >. ) ) ) $=
      ( vr cnq wcel wa cv co cltq wbr cab cop c1st cfv c2nd wn syl cnp cplq cpp
      addnqprlemru wor ltsonq addclnq sonr sylancr cvv ltrelnq brel simpld elex
      breq2 elab3 sylnibr ltnqex gtnqex eleq2i ssneldd adantr wo nqprlu addclpr
      op2nd syl2an prop vex breq1 op1st elab2 biimpi prloc ecased ex ssrdv ) BF
      GZCFGZHZEDIZBCUAJZKLZDMZWAAIZKLZAMZNZOPZVTBKLDMBWDKLAMNZVTCKLDMCWDKLAMNZU
      BJZOPZVSEIZWHGZWMWLGZVSWNHWOWAWKQPZGZVSWQRWNVSWPWGQPZWAABCDUCVSWAWFGZWAWR
      GVSWAWAKLZWSVSFKUDWAFGZWTRUEBCUFFWAKUGUHWEWTAWAWTXAWAUIGWTXAXAWAWAFFKUJUK
      ULWAFUMSWDWAWAKUNUOUPWRWFWAWCWFDWAUQZAWAURZVEUSUPUTVAVSWLWPNTGZWMWAKLZWOW
      QVBWNVSWKTGZXDVQWITGWJTGXFVRABDVCACDVCWIWJVDVFWKVGSWNXEWBXEDWMWHEVHVTWMWA
      KVIWCWFXBXCVJVKVLWMWAWPWLVMVFVNVOVP $.

    $( Lemma for ~ addnqpr .  The forward subset relationship for the upper
       cut.  (Contributed by Jim Kingdon, 19-Aug-2020.) $)
    addnqprlemfu $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. )
        C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P.
        <. { l | l <Q B } , { u | B <Q u } >. ) ) ) $=
      ( vr cnq wcel wa cv co cltq wbr cab cop c2nd cfv c1st wn syl cnp cplq cpp
      addnqprlemrl wor ltsonq addclnq sonr sylancr cvv ltrelnq brel simpld elex
      breq1 elab3 sylnibr ltnqex gtnqex eleq2i ssneldd adantr wo nqprlu addclpr
      op1st syl2an prop vex breq2 op2nd elab2 biimpi prloc orcomd ecased ssrdv
      ex ) BFGZCFGZHZEDIZBCUAJZKLZDMZWBAIZKLZAMZNZOPZWABKLDMBWEKLAMNZWACKLDMCWE
      KLAMNZUBJZOPZVTEIZWIGZWNWMGZVTWOHZWPWBWLQPZGZVTWSRWOVTWRWHQPZWBABCDUCVTWB
      WDGZWBWTGVTWBWBKLZXAVTFKUDWBFGZXBRUEBCUFFWBKUGUHWCXBDWBXBXCWBUIGXBXCXCWBW
      BFFKUJUKULWBFUMSWAWBWBKUNUOUPWTWDWBWDWGDWBUQZAWBURZVEUSUPUTVAWQWSWPVTWRWM
      NTGZWBWNKLZWSWPVBWOVTWLTGZXFVRWJTGWKTGXHVSABDVCACDVCWJWKVDVFWLVGSWOXGWFXG
      AWNWIEVHWEWNWBKVIWDWGXDXEVJVKVLWBWNWMWRVMVFVNVOVQVP $.

    $( Addition of fractions embedded into positive reals.  One can either add
       the fractions as fractions, or embed them into positive reals and add
       them as positive reals, and get the same result.  (Contributed by Jim
       Kingdon, 19-Aug-2020.) $)
    addnqpr $p |- ( ( A e. Q. /\ B e. Q. ) ->
        <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. =
        ( <. { l | l <Q A } , { u | A <Q u } >. +P.
          <. { l | l <Q B } , { u | B <Q u } >. ) ) $=
      ( cnq wcel wa cv co cltq wbr cab cop wceq c1st cfv c2nd eqssd cnp nqprlu
      cpp addnqprlemfl addnqprlemrl addnqprlemfu addnqprlemru wb addclnq syl2an
      cplq syl addclpr preqlu syl2anc mpbir2and ) BEFZCEFZGZDHZBCUIIZJKDLUSAHZJ
      KALMZURBJKDLBUTJKALMZURCJKDLCUTJKALMZUAIZNZVAOPZVDOPZNZVAQPZVDQPZNZUQVFVG
      ABCDUBABCDUCRUQVIVJABCDUDABCDUERUQVASFZVDSFZVEVHVKGUFUQUSEFVLBCUGAUSDTUJU
      OVBSFVCSFVMUPABDTACDTVBVCUKUHVAVDULUMUN $.
  $}

  ${
    $d A l u $.
    $( Addition of one to a fraction embedded into a positive real.  One can
       either add the fraction one to the fraction, or the positive real one to
       the positive real, and get the same result.  Special case of ~ addnqpr .
       (Contributed by Jim Kingdon, 26-Apr-2020.) $)
    addnqpr1 $p |- ( A e. Q. ->
        <. { l | l <Q ( A +Q 1Q ) } , { u | ( A +Q 1Q ) <Q u } >. =
        ( <. { l | l <Q A } , { u | A <Q u } >. +P. 1P ) ) $=
      ( cnq wcel cv c1q cplq co cltq wbr cab cop cpp c1p wceq 1nq addnqpr mpan2
      df-i1p oveq2i eqtr4di ) BDEZCFZBGHIZJKCLUEAFZJKALMZUDBJKCLBUFJKALMZUDGJKC
      LGUFJKALMZNIZUHONIUCGDEUGUJPQABGCRSOUIUHNACTUAUB $.
  $}

  ${
    $d A m $.  $d B m $.  $d C m $.
    $( Approximate division for positive rationals.  Proposition 12.7 of
       [BauerTaylor], p. 55 (a special case where ` A ` and ` B ` are positive,
       as well as ` C ` ).  Our proof is simpler than the one in BauerTaylor
       because we have reciprocals.  (Contributed by Jim Kingdon,
       8-Dec-2019.) $)
    appdivnq $p |- ( ( A <Q B /\ C e. Q. ) ->
        E. m e. Q. ( A <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) ) $=
      ( cltq wbr cnq wcel wa cmq co adantr ltmnqg syl3anc syl2anc c1q mulcomnqg
      wb wceq bitrd crq cv wrex simpl ltrelnq brel simpld simprd recclnq adantl
      cfv mpbid ltbtwnnqq sylib mulclnq simpr simplr recidnq ad2antlr mulassnqg
      oveq1d 1nq mpan mulidnq syl 3eqtr3d breq1d breq2d anbi12d biimpd reximdva
      eqtrd mpd ) ABEFZCGHZIZCUAUKZAJKZDUBZEFZVSVQBJKZEFZIZDGUCZAVSCJKZEFZWEBEF
      ZIZDGUCVPVRWAEFZWDVPVNWIVNVOUDVPAGHZBGHZVQGHZVNWIRVPWJWKVNWJWKIVOABGGEUEU
      FLZUGZVPWJWKWMUHZVOWLVNCUIUJZABVQMNULDVRWAUMUNVPWCWHDGVPVSGHZIZWCWHWRWCAC
      VSJKZEFZWSBEFZIWHWRVTWTWBXAWRVTCVRJKZWSEFZWTWRVRGHZWQVOVTXCRWRWLWJXDVPWLW
      QWPLZVPWJWQWNLZVQAUOOVPWQUPZVNVOWQUQZVRVSCMNWRXBAWSEWRCVQJKZAJKZPAJKZXBAV
      OXJXKSVNWQVOXIPAJCURZVAUSWRVOWLWJXJXBSXHXEXFCVQAUTNWRWJXKASXFWJXKAPJKZAPG
      HZWJXKXMSVBPAQVCAVDVLVEVFVGTWRWBWSCWAJKZEFZXAWRWQWAGHZVOWBXPRXGWRWLWKXQXE
      VPWKWQWOLZVQBUOOXHVSWACMNWRXOBWSEWRXIBJKZPBJKZXOBVOXSXTSVNWQVOXIPBJXLVAUS
      WRVOWLWKXSXOSXHXEXRCVQBUTNWRWKXTBSXRWKXTBPJKZBXNWKXTYASVBPBQVCBVDVLVEVFVH
      TVIWRWTWFXAWGWRWSWEAEWRVOWQWSWESXHXGCVSQOZVHWRWSWEBEYBVGVITVJVKVM $.
  $}

  ${
    $d B m x $.  $d C m x $.
    $( Approximate division for positive rationals.  This can be thought of as
       a variation of ~ appdivnq in which ` A ` is zero, although it can be
       stated and proved in terms of positive rationals alone, without zero as
       such.  (Contributed by Jim Kingdon, 9-Dec-2019.) $)
    appdiv0nq $p |- ( ( B e. Q. /\ C e. Q. ) ->
        E. m e. Q. ( m .Q C ) <Q B ) $=
      ( vx cnq wcel wa cv cltq wbr cmq co wrex nsmallnqq adantr appdivnq reximi
      simpr syl ancoms ad2ant2l rexlimddv ) AEFZBEFZGDHZAIJZCHBKLZAIJZCEMZDEUCU
      FDEMUDDANOUDUFUIUCUEEFUFUDUIUFUDGUEUGIJZUHGZCEMUIUEABCPUKUHCEUJUHRQSTUAUB
      $.
  $}

  ${
    prmuloclemcalc.ru $e |- ( ph -> R <Q U ) $.
    prmuloclemcalc.udp $e |- ( ph -> U <Q ( D +Q P ) ) $.
    prmuloclemcalc.axb $e |- ( ph -> ( A +Q X ) = B ) $.
    prmuloclemcalc.pbrx $e |- ( ph -> ( P .Q B ) <Q ( R .Q X ) ) $.
    prmuloclemcalc.a $e |- ( ph -> A e. Q. ) $.
    prmuloclemcalc.b $e |- ( ph -> B e. Q. ) $.
    prmuloclemcalc.d $e |- ( ph -> D e. Q. ) $.
    prmuloclemcalc.p $e |- ( ph -> P e. Q. ) $.
    prmuloclemcalc.x $e |- ( ph -> X e. Q. ) $.
    $( Calculations for ~ prmuloc .  (Contributed by Jim Kingdon,
       9-Dec-2019.) $)
    prmuloclemcalc $p |- ( ph -> ( U .Q A ) <Q ( D .Q B ) ) $=
      ( co cltq syl2anc cmq wbr cplq oveq2d cnq wcel wceq wa ltrelnq syl simprd
      brel distrnqg syl3anc eqtr3d ltmnqi breqtrd eqbrtrrd mulclnq ltanqi sotri
      mulcomnqg ltsonq simpld 3brtr3d addcomnqg wb ltanqg mpbird ) AGBUARZCDUAR
      ZDCUARZSAVJVKSUBZGHUARZVJUCRZVNVKUCRZSUBZAVJVNUCRZVKVNUCRZVOVPSAGCUARZVRV
      SSAGBHUCRZUARZVTVRAWACGUAKUDAGUEUFZBUEUFZHUEUFZWBVRUGAFUEUFZWCAFGSUBZWFWC
      UHIFGUEUESUIULUJZUKZMQGBHUMUNUOACGUARZVTVSSACUEUFZWCWJVTUGNWICGVBTAWJVKFH
      UARZUCRZSUBZWMVSSUBZWJVSSUBAWJVKCEUARZUCRZSUBWQWMSUBZWNAWJCDEUCRZUARZWQSA
      GWSSUBWKWJWTSUBJNGWSCUPTAWKDUEUFZEUEUFZWTWQUGNOPCDEUMUNUQAWPWLSUBVKUEUFZW
      RAECUARZWPWLSAXBWKXDWPUGPNECVBTLURAWKXAXCNOCDUSTZWPWLVKUTTWJWQWMSUEVCUIVA
      TAWLVNSUBXCWOAHFUARZHGUARZWLVNSAWGWEXFXGSUBIQFGHUPTAWEWFXFWLUGQAWFWCWHVDH
      FVBTAWEWCXGVNUGQWIHGVBTVEXEWLVNVKUTTWJWMVSSUEVCUIVATURURAVJUEUFZVNUEUFZVR
      VOUGAWCWDXHWIMGBUSTZAWCWEXIWIQGHUSTZVJVNVFTAXCXIVSVPUGXEXKVKVNVFTVEAXHXCX
      IVMVQVGXJXEXKVJVKVNVHUNVIAWKXAVKVLUGNOCDVBTUQ $.
  $}

  ${
    $d A d p r u x $.  $d B d p r u x $.  $d L d p r u x $.  $d U d p r u x $.
    $( Positive reals are multiplicatively located.  Lemma 12.8 of
       [BauerTaylor], p. 56.  (Contributed by Jim Kingdon, 8-Dec-2019.) $)
    prmuloc $p |- ( ( <. L , U >. e. P. /\ A <Q B ) -> E. d e. Q. E. u e. Q.
        ( d e. L /\ u e. U /\ ( u .Q A ) <Q ( d .Q B ) ) ) $=
      ( vx vr vp wcel cltq wbr wa cv co cmq cnq wrex ad2antrr adantlr cplq wceq
      cop cnp ltexnqi adantl prml simprl simplrl mulclnq syl2anc ltrelnq simprd
      w3a brel ad3antlr appdiv0nq prarloc ad2ant2r r2ex elprnql adantrr simplll
      wex sylib elprnqu prltlu 3adant1r 3adant2l 3adant3l simprr simplrr simpld
      3expa prmuloclemcalc df-3an sylanbrc jca31 2eximdv mpd sylibr rexlimddv
      ex ) EDUCUDJZBCKLZMZBGNZUAOCUBZFNZEJZANZDJZWKBPOWICPOKLZUNZAQRFQRZGQWEWHG
      QRWDGBCUEUFWFWGQJZWHMZMZHNZEJZWOHQWDWTHQRWEWQHDEUGSWRWSQJZWTMZMZINZCPOWSW
      GPOZKLZWOIQXCXEQJZCQJZXFIQRXCXAWPXGWRXAWTUHWFWPWHXBUIZWSWGUJUKWEXHWDWQXBW
      EBQJZXHBCQQKULUOZUMUPZXECIUQUKXCXDQJZXFMZMZWIQJZWKQJZMWNMZAVDFVDZWOXOWJWL
      MZWKWIXDUAOKLZMZAVDFVDZXSXOYAADRFERZYCWRXMYDXBXFWFXMYDWQWDXMYDWEXDDEFAURT
      TUSYAFAEDUTVEXOYBXRFAXOYBXRXOYBMZXPXQWNXOXTXPYAXCWJXPXNWLWRWJXPXBWFWJXPWQ
      WDWJXPWEWIDEVATTTUSVBZYEWDWLXQXCWDXNYBWDWEWQXBVCSYEWJWLXOXTYAUHZUMWKDEVFU
      KYEXTWMWNYGYEBCWIXDWSWKWGXCXTWSWKKLZXNYAWRXBXTYHWFXBXTYHWQWFXBWLYHWJWFWTW
      LYHXAWDWTWLYHWEWSWKDEVGVHVIVJVHVNUSXOXTYAVKXCWHXNYBWFWPWHXBVLSXCXMXFYBVLX
      CXJXNYBWEXJWDWQXBWEXJXHXKVMUPSXCXHXNYBXLSYFXCXMXFYBUIXCWPXNYBXISVOWJWLWMV
      PVQVRWCVSVTWNFAQQUTWAWBWBWB $.
  $}

  ${
    $d B x y $.  $d L x y $.  $d U x y $.
    $( Positive reals are multiplicatively located.  This is a variation of
       ~ prmuloc which only constructs one (named) point and is therefore often
       easier to work with.  It states that given a ratio ` B ` , there are
       elements of the lower and upper cut which have exactly that ratio
       between them.  (Contributed by Jim Kingdon, 28-Dec-2019.) $)
    prmuloc2 $p |- ( ( <. L , U >. e. P. /\ 1Q <Q B ) ->
        E. x e. L ( x .Q B ) e. U ) $=
      ( vy cop wcel c1q cltq wbr wa cv cmq co cnq wrex wi syl2anc mpd ex simpr1
      cnp w3a prmuloc nfv nfre1 simpr3 wceq wb simplrr mulidnq breq1 3syl mpbid
      simplll simpr2 prcunqu rspe anassrs rexlimdva rexlimd ) DCFUBGZHBIJZKZALZ
      DGZELZCGZVGHMNZVEBMNZIJZUCZEOPZAOPVJCGZADPZEHBCDAUDVDVMVOAOVDAUEVNADUFVDV
      EOGZVMVOQVDVPKVLVOEOVDVPVGOGZVLVOQVDVPVQKZKZVLVOVSVLKZVFVNVOVSVFVHVKUAVTV
      GVJIJZVNVTVKWAVSVFVHVKUGVTVQVIVGUHVKWAUIVDVPVQVLUJVGUKVIVGVJIULUMUNVTVBVH
      WAVNQVBVCVRVLUOVSVFVHVKUPVJVGCDUQRSVNADURRTUSUTTVAS $.
  $}

  ${
    $d A v w x y z $.  $d B v w x y z $.  $d G w y z $.  $d H w y z $.
    $d X w y z $.
    $( Lemma to prove downward closure in positive real multiplication.
       (Contributed by Jim Kingdon, 10-Dec-2019.) $)
    mulnqprl $p |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\
                       ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) ->
                       ( X <Q ( G .Q H ) -> X e. ( 1st ` ( A .P. B ) ) ) ) $=
      ( vy vz vw cnp wcel cfv wa cnq cmq co cltq wbr syl wceq c1q vx vv crq cmp
      c1st cv w3a ltmnqg adantl simpr c2nd prop elprnql sylan ad2antrr ad2antlr
      cop mulclnq syl2anc recclnq mulcomnqg caovord2d mulassnqg syl3anc recidnq
      wb oveq2d mulidnq 3eqtrd breq2d bitrd wi prcdnql sylbid df-imp genpprecll
      exp4b com34 imp32 adantlr adantr syld eqtrd eleq1d sylibd ) AIJZCAUEKZJZL
      ZBIJZDBUEKZJZLZLZEMJZLZECDNOZPQZEDUCKZNOZDNOZABUDOUEKZJZEXBJWPWRWTWGJZXCW
      PWRWTCPQZXDWPWRWTWQWSNOZPQXEWPFGHEWQWSPMNFUFZMJZGUFZMJZHUFZMJUGXGXIPQXKXG
      NOXKXINOPQVFWPXGXIXKUHUIWNWOUJZWPCMJZDMJZWQMJWIXMWMWOWFWGAUKKZUQIJZWHXMAU
      LZCXOWGUMUNUOZWMXNWIWOWJWKBUKKZUQIJWLXNBULDXSWKUMUNUPZCDURUSWPXNWSMJZXTDU
      TRZXHXJLXGXINOXIXGNOSWPXGXIVAUIVBWPXFCWTPWPXFCDWSNOZNOZCTNOZCWPXMXNYAXFYD
      SXRXTYBCDWSVCVDWPXNYDYESXTXNYCTCNDVEZVGRWPXMYECSXRCVHRVIVJVKWIXEXDVLZWMWO
      WFXPWHYGXQCWTXOWGVMUNUOVNWNXDXCVLZWOWFWMYHWHWFWJWLYHWFWJXDWLXCWFWJXDWLXCU
      AFGHUBABWTDUDNHUBGFUAVOXGXIURVPVQVRVSVTWAWBWPXAEXBWPXAEWSDNOZNOZETNOZEWPW
      OYAXNXAYJSXLYBXTEWSDVCVDWPYITENWPYIYCTWPYAXNYIYCSYBXTWSDVAUSWPXNYCTSXTYFR
      WCVGWOYKESWNEVHUIVIWDWE $.

    $( Lemma to prove upward closure in positive real multiplication.
       (Contributed by Jim Kingdon, 10-Dec-2019.) $)
    mulnqpru $p |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\
                       ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) ->
                       ( ( G .Q H ) <Q X -> X e. ( 2nd ` ( A .P. B ) ) ) ) $=
      ( vy vz vw cnp wcel cfv wa cnq cmq co cltq wbr syl wceq c1q vx vv crq cmp
      c2nd cv w3a wb ltmnqg adantl c1st cop prop elprnqu sylan ad2antrr mulclnq
      ad2antlr syl2anc simpr recclnq mulcomnqg caovord2d syl3anc recidnq oveq2d
      mulassnqg mulidnq 3eqtrd breq1d bitrd wi prcunqu sylbid df-imp genppreclu
      exp4b com34 imp32 adantlr adantr syld eqtrd eleq1d sylibd ) AIJZCAUEKZJZL
      ZBIJZDBUEKZJZLZLZEMJZLZCDNOZEPQZEDUCKZNOZDNOZABUDOUEKZJZEXBJWPWRWTWGJZXCW
      PWRCWTPQZXDWPWRWQWSNOZWTPQXEWPFGHWQEWSPMNFUFZMJZGUFZMJZHUFZMJUGXGXIPQXKXG
      NOXKXINOPQUHWPXGXIXKUIUJWPCMJZDMJZWQMJWIXLWMWOWFAUKKZWGULIJZWHXLAUMZCWGXN
      UNUOUPZWMXMWIWOWJBUKKZWKULIJWLXMBUMDWKXRUNUOURZCDUQUSWNWOUTZWPXMWSMJZXSDV
      ARZXHXJLXGXINOXIXGNOSWPXGXIVBUJVCWPXFCWTPWPXFCDWSNOZNOZCTNOZCWPXLXMYAXFYD
      SXQXSYBCDWSVGVDWPXMYDYESXSXMYCTCNDVEZVFRWPXLYECSXQCVHRVIVJVKWIXEXDVLZWMWO
      WFXOWHYGXPWTCWGXNVMUOUPVNWNXDXCVLZWOWFWMYHWHWFWJWLYHWFWJXDWLXCWFWJXDWLXCU
      AFGHUBABWTDUDNHUBGFUAVOXGXIUQVPVQVRVSVTWAWBWPXAEXBWPXAEWSDNOZNOZETNOZEWPW
      OYAXMXAYJSXTYBXSEWSDVGVDWPYITENWPYIYCTWPYAXMYIYCSYBXSWSDVBUSWPXMYCTSXSYFR
      WCVFWOYKESWNEVHUJVIWDWE $.
  $}

  ${
    $d D x y z $.  $d E x y z $.  $d T x y z $.  $d U x y z $.  $d ph x y z $.
    mullocprlem.ab $e |- ( ph -> ( A e. P. /\ B e. P. ) ) $.
    mullocprlem.uqedu $e |- ( ph -> ( U .Q Q ) <Q ( E .Q ( D .Q U ) ) ) $.
    mullocprlem.edutdu $e |- ( ph ->
      ( E .Q ( D .Q U ) ) <Q ( T .Q ( D .Q U ) ) ) $.
    mullocprlem.tdudr $e |- ( ph -> ( T .Q ( D .Q U ) ) <Q ( D .Q R ) ) $.
    mullocprlem.qr $e |- ( ph -> ( Q e. Q. /\ R e. Q. ) ) $.
    mullocprlem.duq $e |- ( ph -> ( D e. Q. /\ U e. Q. ) ) $.
    mullocprlem.du $e |- ( ph -> ( D e. ( 1st ` A ) /\ U e. ( 2nd ` A ) ) ) $.
    mullocprlem.et $e |- ( ph -> ( E e. Q. /\ T e. Q. ) ) $.
    $( Calculations for ~ mullocpr .  (Contributed by Jim Kingdon,
       10-Dec-2019.) $)
    mullocprlem $p |- ( ph -> ( Q e. ( 1st ` ( A .P. B ) ) \/
        R e. ( 2nd ` ( A .P. B ) ) ) ) $=
      ( wcel co cmq vx vy vz c1st cfv cmp c2nd wo wa cltq wbr cnq simpld simprd
      cv wceq mulcomnqg adantl w3a mulassnqg caov13d breqtrd wb mulclnq syl2anc
      ltmnqg syl3anc mpbird adantr cnp wi jca anim1i mulnqprl syl21anc mpd orcd
      caov12d breq1d bitr4d eqbrtrrd mulnqpru olcd syl breq12d bitrd prop prloc
      cop sylan mpjaodan ) AICUDUEZRZEBCUFSZUDUERZFWNUGUERZUHGCUGUEZRZAWMUIZWOW
      PWSEDITSZUJUKZWOAXAWMAXAHETSZHWTTSZUJUKZAXBIDHTSZTSZXCUJKAUAUBUCIDHULTAIU
      LRZGULRZQUMZADULRZHULRZOUMZAXJXKOUNZUAUOZULRZUBUOZULRZUIXNXPTSZXPXNTSUPAX
      NXPUQURZXOXQUCUOZULRUSXRXTTSXNXPXTTSTSUPAXNXPXTUTURZVAVBAEULRZWTULRZXKXAX
      DVCAYBFULRZNUMZAXJXGYCXLXIDIVDVEXMEWTHVFVGVHVIWSBVJRZDBUDUERZUIZCVJRZWMUI
      YBXAWOVKAYHWMAYFYGAYFYIJUMZAYGHBUGUERZPUMVLVIAYIWMAYFYIJUNZVMAYBWMYEVIBCD
      IEVNVOVPVQAWRUIZWPWOYMHGTSZFUJUKZWPAYOWRAGHTSZYNFUJAXHXKYPYNUPAXGXHQUNZXM
      GHUQVEAYPFUJUKZGXETSZDFTSZUJUKZMAYRDYPTSZYTUJUKZUUAAYPULRZYDXJYRUUCVCAXHX
      KUUDYQXMGHVDVEAYBYDNUNZXLYPFDVFVGAYSUUBYTUJAUAUBUCGDHULTYQXLXMXSYAVRVSVTV
      HWAVIYMYFYKUIZYIWRUIYDYOWPVKAUUFWRAYFYKYJAYGYKPUNVLVIAYIWRYLVMAYDWRUUEVIB
      CHGFWBVOVPWCAYIIGUJUKZWMWRUHZYLAUUGXFYSUJUKZLAUUGXEITSZXEGTSZUJUKZUUIAXGX
      HXEULRZUUGUULVCXIYQAXJXKUIUUMODHVDWDZIGXEVFVGAUUJXFUUKYSUJAUUMXGUUJXFUPUU
      NXIXEIUQVEAUUMXHUUKYSUPUUNYQXEGUQVEWEWFVHYIWLWQWIVJRUUGUUHCWGIGWQWLWHWJVE
      WK $.
  $}

  ${
    $d A d e q r t u $.  $d B d e q r t u $.
    $( Locatedness of multiplication on positive reals.  Lemma 12.9 in
       [BauerTaylor], p. 56 (but where both ` A ` and ` B ` are positive, not
       just ` A ` ).  (Contributed by Jim Kingdon, 8-Dec-2019.) $)
    mullocpr $p |- ( ( A e. P. /\ B e. P. ) ->
        A. q e. Q. A. r e. Q. ( q <Q r ->
          ( q e. ( 1st ` ( A .P. B ) ) \/ r e. ( 2nd ` ( A .P. B ) ) ) ) ) $=
      ( vd vu ve vt cnp wcel wa cv cltq wbr co cfv cnq cmq wrex ad2antrr cmp wo
      c1st c2nd wi w3a wex prop prmuloc sylan r2ex sylib adantlr simprr3 simprl
      cop mulclnq syl appdivnq syl2anc simprrr simplll ad2antlr simprrl simpllr
      adantr ad2antll simplrl jca mullocprlem rexlimddv exlimdvv mpd ralrimivva
      3simpa ex ) AIJZBIJZKZDLZCLZMNZVTABUAOZUCPJWAWCUDPJUBZUEDCQQVSVTQJWAQJKZK
      ZWBWDWFWBKZELZQJFLZQJKZWHAUCPZJZWIAUDPZJZWIVTROZWHWAROZMNZUFZKZFUGEUGZWDV
      SWBWTWEVQWBWTVRVQWBKWRFQSEQSZWTVQWKWMUPIJWBXAAUHFVTWAWMWKEUIUJWREFQQUKULU
      MUMWGWSWDEFWGWSWDWGWSKZWOGLZWHWIROZROZMNZXEWPMNZKZWDGQXBWQXDQJZXHGQSWLWNW
      QWJWGUNXBWJXIWGWJWRUOZWHWIUQURZWOWPXDGUSUTXBXCQJZXHKZKZXEHLZXDROZMNZXPWPM
      NZKZWDHQXNXGXIXSHQSXBXLXFXGVAXBXIXMXKVFXEWPXDHUSUTXNXOQJZXSKZKZABWHVTWAXO
      WIXCXBVSXMYAVSWEWBWSVBTXMXFXBYAXLXFXGUOVCXNXTXQXRVDXNXTXQXRVAXBWEXMYAVSWE
      WBWSVETXBWJXMYAXJTXBWLWNKZXMYAWRYCWGWJWLWNWQVOVGTYBXLXTXBXLXHYAVHXNXTXSUO
      VIVJVKVKVPVLVMVPVN $.
  $}

  ${
    $d A q r t u v w x y z $.  $d B q r t u v w x y z $.
    $( Closure of multiplication on positive reals.  First statement of
       Proposition 9-3.7 of [Gleason] p. 124.  (Contributed by NM,
       13-Mar-1996.) $)
    mulclpr $p |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. ) $=
      ( vq vr vx vy vz vw vv vu vt cnp wcel wa cmp cnq cv wrex wral cmq cpw cxp
      co c1st cfv c2nd cltq wbr wb wn w3a df-imp genpelxp mulclnq genpml genpmu
      wo wi jca32 ltmnqg mulcomnqg mulnqprl genprndl mulnqpru genprndu genpdisj
      jca mullocpr 3jca elnp1st2nd sylanbrc ) ALMBLMNZABOUCZPUAZVNUBMZCQZVMUDUE
      ZMZCPRZDQZVMUFUEZMZDPRZNNVRVPVTUGUHZVTVQMNDPRUICPSZWBWDVPWAMZNCPRUIDPSZNZ
      VRWFNUJCPSZWDVRWBUQURDPSCPSZUKVMLMVLVOVSWCEFGHIABOTHIGFEULZUMEFGHIABOTCWK
      FQZGQZUNZUOEFGHIABOTDWKWNUPUSVLWHWIWJVLWEWGEFGHIABJKOTDCWKWNEQZWLWMUTZWOW
      LVAZABJQZKQZWOVBVCEFGHIABJKOTDCWKWNWPWQABWRWSWOVDVEVGEFGHIABOTCWKWNWPWQVF
      ABDCVHVIVMDCVJVK $.
  $}

  ${
    $d A f g h l r s t u x y $.  $d B f g h l r s t u x y $.
    $( Lemma for ~ mulnqpr .  The reverse subset relationship for the lower
       cut.  (Contributed by Jim Kingdon, 18-Jul-2021.) $)
    mulnqprlemrl $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P.
        <. { l | l <Q B } , { u | B <Q u } >. ) ) C_
        ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. )
        ) $=
      ( vr vs vt vg vh cnq wcel wa cv cltq wbr cab cop c1st cfv breq1 vf vx cmp
      vy co cmq wceq cnp wb nqprlu df-imp mulclnq genpelvl syl2an biimpa ltnqex
      wrex vex gtnqex op1st biimpi ad2antrl adantr ad2antll wi ltrelnq brel syl
      elab2 lt2mulnq syl2anc mp2and adantl mpbird sylibr rexlimdvva mpd ssrdv
      ex ) BJKZCJKZLZEDMZBNOZDPZBAMZNOAPZQZWCCNOZDPZCWFNOAPZQZUCUERSZWCBCUFUEZN
      OZDPZWNWFNOAPZQRSZWBEMZWMKZWSWRKZWBWTLZWSFMZGMZUFUEZUGZGWLRSZUQFWHRSZUQZX
      AWBWTXIVTWHUHKWLUHKWTXIUIWAABDUJACDUJUAHIUBUDWHWLWSFGUCUFUBUDIHUAUKHMIMUL
      UMUNUOXBXFXAFGXHXGXBXCXHKZXDXGKZLLZXFXAXLXFLZWSWNNOZXAXMXNXEWNNOZXMXCBNOZ
      XDCNOZXOXLXPXFXJXPXBXKXJXPWDXPDXCXHFURWCXCBNTWEWGDBUPABUSUTVIVAVBVCZXLXQX
      FXKXQXBXJXKXQWIXQDXDXGGURWCXDCNTWJWKDCUPACUSUTVIVAVDVCZXMXCJKVTLZXDJKWALZ
      XPXQLXOVEXMXPXTXRXCBJJNVFVGVHXMXQYAXSXDCJJNVFVGVHXCBXDCVJVKVLXFXNXOUIXLWS
      XEWNNTVMVNWOXNDWSWREURWCWSWNNTWPWQDWNUPAWNUSUTVIVOVSVPVQVSVR $.

    $( Lemma for ~ mulnqpr .  The reverse subset relationship for the upper
       cut.  (Contributed by Jim Kingdon, 18-Jul-2021.) $)
    mulnqprlemru $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P.
        <. { l | l <Q B } , { u | B <Q u } >. ) ) C_
        ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. )
        ) $=
      ( vr vs vt vg vh cnq wcel wa cv cltq wbr cab cop c2nd cfv breq2 vf vx cmp
      vy co cmq wceq cnp wb nqprlu df-imp mulclnq genpelvu syl2an biimpa ltnqex
      wrex vex gtnqex op2nd biimpi ad2antrl adantr ad2antll wi ltrelnq brel syl
      elab2 lt2mulnq syl2anc mp2and adantl mpbird sylibr rexlimdvva mpd ssrdv
      ex ) BJKZCJKZLZEDMZBNODPZBAMZNOZAPZQZWCCNODPZCWENOZAPZQZUCUERSZWCBCUFUEZN
      ODPZWNWENOZAPZQRSZWBEMZWMKZWSWRKZWBWTLZWSFMZGMZUFUEZUGZGWLRSZUQFWHRSZUQZX
      AWBWTXIVTWHUHKWLUHKWTXIUIWAABDUJACDUJUAHIUBUDWHWLWSFGUCUFUBUDIHUAUKHMIMUL
      UMUNUOXBXFXAFGXHXGXBXCXHKZXDXGKZLLZXFXAXLXFLZWNWSNOZXAXMXNWNXENOZXMBXCNOZ
      CXDNOZXOXLXPXFXJXPXBXKXJXPWFXPAXCXHFURWEXCBNTWDWGDBUPABUSUTVIVAVBVCZXLXQX
      FXKXQXBXJXKXQWJXQAXDXGGURWEXDCNTWIWKDCUPACUSUTVIVAVDVCZXMVTXCJKLZWAXDJKLZ
      XPXQLXOVEXMXPXTXRBXCJJNVFVGVHXMXQYAXSCXDJJNVFVGVHBXCCXDVJVKVLXFXNXOUIXLWS
      XEWNNTVMVNWPXNAWSWREURWEWSWNNTWOWQDWNUPAWNUSUTVIVOVSVPVQVSVR $.

    $( Lemma for ~ mulnqpr .  The forward subset relationship for the lower
       cut.  (Contributed by Jim Kingdon, 18-Jul-2021.) $)
    mulnqprlemfl $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. )
        C_ ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P.
        <. { l | l <Q B } , { u | B <Q u } >. ) ) ) $=
      ( vr cnq wcel wa cv co cltq wbr cab cop c1st cfv c2nd wn syl cnp cmq sonr
      cmp mulnqprlemru wor ltsonq mulclnq sylancr cvv ltrelnq brel simpld breq2
      elab3 sylnibr ltnqex gtnqex op2nd eleq2i ssneldd adantr wo nqprlu mulclpr
      elex syl2an prop vex breq1 op1st elab2 biimpi prloc ecased ex ssrdv ) BFG
      ZCFGZHZEDIZBCUAJZKLZDMZWAAIZKLZAMZNZOPZVTBKLDMBWDKLAMNZVTCKLDMCWDKLAMNZUC
      JZOPZVSEIZWHGZWMWLGZVSWNHWOWAWKQPZGZVSWQRWNVSWPWGQPZWAABCDUDVSWAWFGZWAWRG
      VSWAWAKLZWSVSFKUEWAFGZWTRUFBCUGFWAKUBUHWEWTAWAWTXAWAUIGWTXAXAWAWAFFKUJUKU
      LWAFVESWDWAWAKUMUNUOWRWFWAWCWFDWAUPZAWAUQZURUSUOUTVAVSWLWPNTGZWMWAKLZWOWQ
      VBWNVSWKTGZXDVQWITGWJTGXFVRABDVCACDVCWIWJVDVFWKVGSWNXEWBXEDWMWHEVHVTWMWAK
      VIWCWFXBXCVJVKVLWMWAWPWLVMVFVNVOVP $.

    $( Lemma for ~ mulnqpr .  The forward subset relationship for the upper
       cut.  (Contributed by Jim Kingdon, 18-Jul-2021.) $)
    mulnqprlemfu $p |- ( ( A e. Q. /\ B e. Q. ) ->
        ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. )
        C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P.
        <. { l | l <Q B } , { u | B <Q u } >. ) ) ) $=
      ( vr cnq wcel wa cv co cltq wbr cab cop c2nd cfv c1st wn syl cnp cmq sonr
      cmp mulnqprlemrl wor ltsonq mulclnq sylancr cvv ltrelnq brel simpld breq1
      elab3 sylnibr ltnqex gtnqex op1st eleq2i ssneldd adantr wo nqprlu mulclpr
      elex syl2an prop vex breq2 op2nd elab2 biimpi prloc orcomd ecased ssrdv
      ex ) BFGZCFGZHZEDIZBCUAJZKLZDMZWBAIZKLZAMZNZOPZWABKLDMBWEKLAMNZWACKLDMCWE
      KLAMNZUCJZOPZVTEIZWIGZWNWMGZVTWOHZWPWBWLQPZGZVTWSRWOVTWRWHQPZWBABCDUDVTWB
      WDGZWBWTGVTWBWBKLZXAVTFKUEWBFGZXBRUFBCUGFWBKUBUHWCXBDWBXBXCWBUIGXBXCXCWBW
      BFFKUJUKULWBFVESWAWBWBKUMUNUOWTWDWBWDWGDWBUPZAWBUQZURUSUOUTVAWQWSWPVTWRWM
      NTGZWBWNKLZWSWPVBWOVTWLTGZXFVRWJTGWKTGXHVSABDVCACDVCWJWKVDVFWLVGSWOXGWFXG
      AWNWIEVHWEWNWBKVIWDWGXDXEVJVKVLWBWNWMWRVMVFVNVOVQVP $.

    $( Multiplication of fractions embedded into positive reals.  One can
       either multiply the fractions as fractions, or embed them into positive
       reals and multiply them as positive reals, and get the same result.
       (Contributed by Jim Kingdon, 18-Jul-2021.) $)
    mulnqpr $p |- ( ( A e. Q. /\ B e. Q. ) ->
        <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. =
        ( <. { l | l <Q A } , { u | A <Q u } >. .P.
          <. { l | l <Q B } , { u | B <Q u } >. ) ) $=
      ( cnq wcel wa cv co cltq wbr cab cop wceq c1st cfv c2nd eqssd cnp nqprlu
      cmq mulnqprlemfl mulnqprlemrl mulnqprlemfu mulnqprlemru wb mulclnq syl2an
      cmp syl mulclpr preqlu syl2anc mpbir2and ) BEFZCEFZGZDHZBCUAIZJKDLUSAHZJK
      ALMZURBJKDLBUTJKALMZURCJKDLCUTJKALMZUIIZNZVAOPZVDOPZNZVAQPZVDQPZNZUQVFVGA
      BCDUBABCDUCRUQVIVJABCDUDABCDUERUQVASFZVDSFZVEVHVKGUFUQUSEFVLBCUGAUSDTUJUO
      VBSFVCSFVMUPABDTACDTVBVCUKUHVAVDULUMUN $.
  $}

  ${
    $d f g h v w x y z A $.  $d f g h v w x y z B $.  $d f g h v w x y z C $.
    $( Addition of positive reals is commutative.  Proposition 9-3.5(ii) of
       [Gleason] p. 123.  (Contributed by Jim Kingdon, 11-Dec-2019.) $)
    addcomprg $p |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = ( B +P. A ) ) $=
      ( vx vy vz cnp wcel wa cv wceq cfv wrex cnq crab cop sylan sylan2 anassrs
      co rexbidva cplq c1st c2nd wb prop elprnql addcomnqg eqeq2d ancoms rexcom
      cpp bitrdi rabbidv elprnqu opeq12d plpvlu 3eqtr4rd ) AFGZBFGZHZCIZDIZEIZU
      ASZJZEAUBKZLZDBUBKZLZCMNZVEEAUCKZLZDBUCKZLZCMNZOZVAVCVBUASZJZDVHLEVFLZCMN
      ZVRDVMLEVKLZCMNZOBAUKSZABUKSUTVJVTVOWBUTVIVSCMUTVIVREVFLZDVHLVSUTVGWDDVHU
      RUSVBVHGZVGWDUDZUSWEHURVBMGZWFUSVHVMOFGZWEWGBUEZVBVMVHUFPWGURWFWGURHZVEVR
      EVFWGURVCVFGZVEVRUDZURWKHWGVCMGZWLURVFVKOFGZWKWMAUEZVCVKVFUFPWGWMHVDVQVAV
      BVCUGUHZQRTUIQRTVRDEVHVFUJULUMUTVNWACMUTVNVREVKLZDVMLWAUTVLWQDVMURUSVBVMG
      ZVLWQUDZUSWRHURWGWSUSWHWRWGWIVBVMVHUNPWGURWSWJVEVREVKWGURVCVKGZWLURWTHWGW
      MWLURWNWTWMWOVCVKVFUNPWPQRTUIQRTVRDEVMVKUJULUMUOUSURWCVPJCDEBAUPUICEDABUP
      UQ $.

    $( Addition of positive reals is associative.  Proposition 9-3.5(i) of
       [Gleason] p. 123.  (Contributed by Jim Kingdon, 11-Dec-2019.) $)
    addassprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
        ( ( A +P. B ) +P. C ) = ( A +P. ( B +P. C ) ) ) $=
      ( vx vy vz vw vv vf vg vh cplq df-iplp cv addclnq dmplp addclpr addassnqg
      cpp genpassg ) DEFGHABCIJKSLGHFEDMENFNOPINZJNZQUAUBKNRT $.

    $( Multiplication of positive reals is commutative.  Proposition 9-3.7(ii)
       of [Gleason] p. 124.  (Contributed by Jim Kingdon, 11-Dec-2019.) $)
    mulcomprg $p |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) ) $=
      ( vx vz vy cnp wcel wa cv wceq cfv wrex cnq crab cop sylan sylan2 anassrs
      co rexbidva cmq c1st c2nd cmp prop elprnql mulcomnqg eqeq2d ancoms rexcom
      wb bitrdi rabbidv elprnqu opeq12d mpvlu 3eqtr4rd ) AFGZBFGZHZCIZDIZEIZUAS
      ZJZEAUBKZLZDBUBKZLZCMNZVEEAUCKZLZDBUCKZLZCMNZOZVAVCVBUASZJZDVHLEVFLZCMNZV
      RDVMLEVKLZCMNZOBAUDSZABUDSUTVJVTVOWBUTVIVSCMUTVIVREVFLZDVHLVSUTVGWDDVHURU
      SVBVHGZVGWDUKZUSWEHURVBMGZWFUSVHVMOFGZWEWGBUEZVBVMVHUFPWGURWFWGURHZVEVREV
      FWGURVCVFGZVEVRUKZURWKHWGVCMGZWLURVFVKOFGZWKWMAUEZVCVKVFUFPWGWMHVDVQVAVBV
      CUGUHZQRTUIQRTVRDEVHVFUJULUMUTVNWACMUTVNVREVKLZDVMLWAUTVLWQDVMURUSVBVMGZV
      LWQUKZUSWRHURWGWSUSWHWRWGWIVBVMVHUNPWGURWSWJVEVREVKWGURVCVKGZWLURWTHWGWMW
      LURWNWTWMWOVCVKVFUNPWPQRTUIQRTVRDEVMVKUJULUMUOUSURWCVPJCDEBAUPUICEDABUPUQ
      $.

    $( Multiplication of positive reals is associative.  Proposition 9-3.7(i)
       of [Gleason] p. 124.  (Contributed by Jim Kingdon, 11-Dec-2019.) $)
    mulassprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
        ( ( A .P. B ) .P. C ) = ( A .P. ( B .P. C ) ) ) $=
      ( vx vy vz vw vv vf vg vh cmp cmq df-imp cv mulclnq dmmp mulclpr genpassg
      mulassnqg ) DEFGHABCIJKLMGHFEDNEOFOPQIOZJOZRUAUBKOTS $.
  $}

  ${
    $d x y z w v u f g h A $.  $d x y z w v u f g h B $.
    $d x y z w v u f g h C $.
    $( Lemma for distributive law for positive reals.  (Contributed by Jim
       Kingdon, 12-Dec-2019.) $)
    distrlem1prl $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
        ( 1st ` ( A .P. ( B +P. C ) ) ) C_
          ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) $=
      ( vw vx vv vf vg vh vy vz cnp wcel co cmp c1st cfv cv cmq wa w3a cpp wceq
      wrex wb addclpr df-imp mulclnq genpelvl sylan2 3impb cplq df-iplp addclnq
      3adant1 adantr cnq c2nd prop elprnql sylan 3ad2antl1 adantrr anim12i an4s
      cop 3adantl1 ad2ant2r 3anass sylanbrc simprr simpr oveq2 biimpac distrnqg
      eqeq2d imbitrid sylc mulclpr 3adant3 ad2antrr 3adant2 genpprecll adantlrr
      simpll impl simplr imp syl22anc eqeltrd exp32 rexlimdvv sylbid com34 impd
      wi ssrdv ) ALMZBLMZCLMZUAZDABCUBNZONPQZABONZACONZUBNPQZXADRZXCMZXGERZFRZS
      NZUCZFXBPQZUDEAPQZUDZXGXFMZWRWSWTXHXOUEZWSWTTWRXBLMXQBCUFGHIJKAXBXGEFOSJK
      IHGUGZHRZIRZUHZUIUJUKXAXLXPEFXNXMXAXIXNMZXJXMMZXLXPWPXAYBXLYCXPXAYBXLYCXP
      WPXAYBXLTZTZYCXJJRZKRZULNZUCZKCPQZUDJBPQZUDZXPXAYCYLUEZYDWSWTYMWRGHIDEBCX
      JJKUBULDEIHGUMZXSXTUNZUIUOUPYEYIXPJKYKYJYEYFYKMZYGYJMZTZYIXPYEYRYITZTZXGX
      IYFSNZXIYGSNZULNZXFYTXIUQMZYFUQMZYGUQMZUAZXLYITZXGUUCUCZYTUUDUUEUUFTZUUGY
      EUUDYSXAYBUUDXLWRWSYBUUDWTWRXNAURQZVFLMYBUUDAUSXIUUKXNUTVAVBVCUPXAYRUUJYD
      YIWSWTYRUUJWRWSYPWTYQUUJWSYPTUUEWTYQTUUFWSYKBURQZVFLMYPUUEBUSYFUULYKUTVAW
      TYJCURQZVFLMYQUUFCUSYGUUMYJUTVAVDVEVGVHUUDUUEUUFVIVJYEXLYSYIXAYBXLVKYRYIV
      LVDUUHXGXIYHSNZUCZUUGUUIYIXLUUOYIXKUUNXGXJYHXISVMVPVNUUGUUNUUCXGXIYFYGVOV
      PVQVRYTXDLMZXELMZUUAXDPQMZUUBXEPQMZUUCXFMZXAUUPYDYSWRWSUUPWTABVSVTWAXAUUQ
      YDYSWRWTUUQWSACVSWBWAYSYEYPUURYPYQYIWEXAYBYPUURXLXAYBYPUURWRWSYBYPTUURWPW
      TGHIJKABXIYFOSXRYAWCVTWFWDUJYSYEYQUUSYPYQYIWGXAYBYQUUSXLXAYBYQUUSWRWTYBYQ
      TUUSWPWSGHIJKACXIYGOSXRYAWCWBWFWDUJUUPUUQTUURUUSTUUTGHIDEXDXEUUAUUBUBULYN
      YOWCWHWIWJWKWLWMWKWNWOWLWMWQ $.

    $( Lemma for distributive law for positive reals.  (Contributed by Jim
       Kingdon, 12-Dec-2019.) $)
    distrlem1pru $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
        ( 2nd ` ( A .P. ( B +P. C ) ) ) C_
          ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) $=
      ( vw vx vv vf vg vh vy vz cnp wcel co cmp c2nd cfv cv cmq wa w3a cpp wceq
      wrex wb addclpr df-imp mulclnq genpelvu sylan2 3impb cplq df-iplp addclnq
      3adant1 adantr cnq c1st prop elprnqu sylan 3ad2antl1 adantrr anim12i an4s
      cop 3adantl1 ad2ant2r 3anass sylanbrc simprr simpr oveq2 biimpac distrnqg
      eqeq2d imbitrid sylc mulclpr 3adant3 ad2antrr 3adant2 genppreclu adantlrr
      simpll impl simplr imp syl22anc eqeltrd exp32 rexlimdvv sylbid com34 impd
      wi ssrdv ) ALMZBLMZCLMZUAZDABCUBNZONPQZABONZACONZUBNPQZXADRZXCMZXGERZFRZS
      NZUCZFXBPQZUDEAPQZUDZXGXFMZWRWSWTXHXOUEZWSWTTWRXBLMXQBCUFGHIJKAXBXGEFOSJK
      IHGUGZHRZIRZUHZUIUJUKXAXLXPEFXNXMXAXIXNMZXJXMMZXLXPWPXAYBXLYCXPXAYBXLYCXP
      WPXAYBXLTZTZYCXJJRZKRZULNZUCZKCPQZUDJBPQZUDZXPXAYCYLUEZYDWSWTYMWRGHIDEBCX
      JJKUBULDEIHGUMZXSXTUNZUIUOUPYEYIXPJKYKYJYEYFYKMZYGYJMZTZYIXPYEYRYITZTZXGX
      IYFSNZXIYGSNZULNZXFYTXIUQMZYFUQMZYGUQMZUAZXLYITZXGUUCUCZYTUUDUUEUUFTZUUGY
      EUUDYSXAYBUUDXLWRWSYBUUDWTWRAURQZXNVFLMYBUUDAUSXIXNUUKUTVAVBVCUPXAYRUUJYD
      YIWSWTYRUUJWRWSYPWTYQUUJWSYPTUUEWTYQTUUFWSBURQZYKVFLMYPUUEBUSYFYKUULUTVAW
      TCURQZYJVFLMYQUUFCUSYGYJUUMUTVAVDVEVGVHUUDUUEUUFVIVJYEXLYSYIXAYBXLVKYRYIV
      LVDUUHXGXIYHSNZUCZUUGUUIYIXLUUOYIXKUUNXGXJYHXISVMVPVNUUGUUNUUCXGXIYFYGVOV
      PVQVRYTXDLMZXELMZUUAXDPQMZUUBXEPQMZUUCXFMZXAUUPYDYSWRWSUUPWTABVSVTWAXAUUQ
      YDYSWRWTUUQWSACVSWBWAYSYEYPUURYPYQYIWEXAYBYPUURXLXAYBYPUURWRWSYBYPTUURWPW
      TGHIJKABXIYFOSXRYAWCVTWFWDUJYSYEYQUUSYPYQYIWGXAYBYQUUSXLXAYBYQUUSWRWTYBYQ
      TUUSWPWSGHIJKACXIYGOSXRYAWCWBWFWDUJUUPUUQTUURUUSTUUTGHIDEXDXEUUAUUBUBULYN
      YOWCWHWIWJWKWLWMWKWNWOWLWMWQ $.

    $( Lemma for distributive law for positive reals.  (Contributed by Jim
       Kingdon, 12-Dec-2019.) $)
    distrlem4prl $p |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\
         ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\
           ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) ->
         ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) $=
      ( vw vv vu cnp wcel cv wa cltq cmq co cplq cnq syl2anc vg vh w3a c1st cfv
      wbr wo cpp cmp wn ltmnqg adantl simp1 simpll c2nd cop prop elprnql syl2an
      wb sylan simprl simpl2 simprlr mulcomnqg caovord2d ltanqg mulclnq simprrr
      wceq simpl3 addcomnqg bitrd simpl1 addclpr 3adant1 adantr mulclpr syl3anc
      distrnqg simprrl df-iplp addclnq genpprecll imp syl22anc eqeltrrd prcdnql
      wi df-imp sylbid simprll ltsonq nqtri3or sotritrieq oveq1 oveq2d sylan9eq
      jaod ex sylbird wdc ltdcnq ancoms dcor sylc df-dc sylib mpjaod ) DKLZEKLZ
      FKLZUCZAMZDUDUEZLZBMZEUDUEZLZNZGMZXOLZCMZFUDUEZLZNZNZNZXNYAOUFZYAXNOUFZUG
      ZXNXQPQZYAYCPQZRQZDEFUHQZUIQZUDUEZLZYKUJZYHYIYRYJYHYIYNYAXQPQZYMRQZOUFZYR
      YHYIYLYTOUFUUBYHHIJXNYAXQOSPHMZSLZIMZSLZJMZSLUCZUUCUUEOUFZUUGUUCPQUUGUUEP
      QOUFUTYHUUCUUEUUGUKULZXMXJXPXNSLZYGXJXKXLUMZXPXSYFUNXJXODUOUEZUPKLZXPUUKD
      UQZXNUUMXOURVAUSZXMXJYBYASLZYGUULXTYBYEVBXJUUNYBUUQUUOYAUUMXOURVAUSZYHXKX
      SXQSLZXJXKXLYGVCZXMXPXSYFVDZXKXREUOUEZUPKLXSUUSEUQXQUVBXRURVATZUUDUUFNZUU
      CUUEPQUUEUUCPQVJYHUUCUUEVEULZVFYHHIJYLYTYMOSRUUHUUIUUGUUCRQUUGUUERQOUFUTY
      HUUCUUEUUGVGULYHUUKUUSYLSLZUUPUVCXNXQVHTZYHUUQUUSYTSLUURUVCYAXQVHTYHUUQYC
      SLZYMSLZUURYHXLYEUVHXJXKXLYGVKZXMXTYBYEVIZXLYDFUOUEZUPKLYEUVHFUQYCUVLYDUR
      VATZYAYCVHTZUVDUUCUUERQUUEUUCRQVJYHUUCUUEVLULVFVMYHYPKLZUUAYQLZUUBYRWIZYH
      XJYOKLZUVOXJXKXLYGVNZXMUVRYGXKXLUVRXJEFVOVPVQZDYOVRTZYHYAXQYCRQZPQZUUAYQY
      HUUQUUSUVHUWCUUAVJUURUVCUVMYAXQYCVTVSYHXJUVRYBUWBYOUDUELZUWCYQLZUVSUVTXMX
      TYBYEWAYHXKXLXSYEUWDUUTUVJUVAUVKXKXLNXSYENUWDHUAUBJIEFXQYCUHRJIUBUAHWBUAM
      ZUBMZWCWDWEWFZXJUVRNZYBUWDNUWEHUAUBJIDYOYAUWBUIPJIUBUAHWJZUWFUWGVHZWDWEWF
      WGUVOYQYPUOUEZUPKLZUVPUVQYPUQZUUAYNUWLYQWHVATWKYHYJYNYLXNYCPQZRQZOUFZYRYH
      YJYMUWOOUFZUWQYHHIJYAXNYCOSPUUJUURUUPUVMUVEVFYHUVIUWOSLZUVFUWRUWQUTUVNYHU
      UKUVHUWSUUPUVMXNYCVHTUVGYMUWOYLVGVSVMYHUVOUWPYQLZUWQYRWIZUWAYHXNUWBPQZUWP
      YQYHUUKUUSUVHUXBUWPVJUUPUVCUVMXNXQYCVTVSZYHXJUVRXPUWDUXBYQLZUVSUVTXMXPXSY
      FWLUWHUWIXPUWDNUXDHUAUBJIDYOXNUWBUIPUWJUWKWDWEWFZWGUVOUWMUWTUXAUWNUWPYNUW
      LYQWHVATWKWSYHYSXNYAVJZYRYHUUKUUQUXFYSUTUUPUURSXNYAOWMXNYAWNWOTYHUXFYRYHU
      XFNUXBYNYQYHUXFUXBUWPYNUXCUXFUWOYMYLRXNYAYCPWPWQWRYHUXDUXFUXEVQWGWTXAYHYK
      XBZYKYSUGYHUUKUUQUXGUUPUURUUKUUQNYIXBYJXBZUXGXNYAXCUUQUUKUXHYAXNXCXDYIYJX
      EXFTYKXGXHXI $.

    $( Lemma for distributive law for positive reals.  (Contributed by Jim
       Kingdon, 12-Dec-2019.) $)
    distrlem4pru $p |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\
         ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\
           ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) ->
         ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) $=
      ( vw vv vu cnp wcel cv wa cltq cmq co cplq cnq syl2anc vg vh w3a c2nd cfv
      wbr wo cpp cmp wn ltmnqg adantl simp1 simpll c1st cop prop elprnqu syl2an
      wb sylan simprl simpl3 simprrr mulcomnqg caovord2d mulclnq simpl2 simprlr
      wceq ltanqg syl3anc bitrd simpl1 addclpr 3adant1 mulclpr distrnqg simprll
      wi adantr df-iplp addclnq genppreclu imp syl22anc df-imp eqeltrrd prcunqu
      sylbid addcomnqg simprrl ltsonq nqtri3or sotritrieq oveq1 oveq2d sylan9eq
      jaod ex sylbird wdc ltdcnq ancoms dcor sylc df-dc sylib mpjaod ) DKLZEKLZ
      FKLZUCZAMZDUDUEZLZBMZEUDUEZLZNZGMZXOLZCMZFUDUEZLZNZNZNZXNYAOUFZYAXNOUFZUG
      ZXNXQPQZYAYCPQZRQZDEFUHQZUIQZUDUEZLZYKUJZYHYIYRYJYHYIYLXNYCPQZRQZYNOUFZYR
      YHYIYTYMOUFZUUBYHHIJXNYAYCOSPHMZSLZIMZSLZJMZSLUCZUUDUUFOUFZUUHUUDPQUUHUUF
      PQOUFUTYHUUDUUFUUHUKULZXMXJXPXNSLZYGXJXKXLUMZXPXSYFUNXJDUOUEZXOUPKLZXPUUL
      DUQZXNXOUUNURVAUSZXMXJYBYASLZYGUUMXTYBYEVBXJUUOYBUURUUPYAXOUUNURVAUSZYHXL
      YEYCSLZXJXKXLYGVCZXMXTYBYEVDZXLFUOUEZYDUPKLYEUUTFUQYCYDUVCURVATZUUEUUGNZU
      UDUUFPQUUFUUDPQVJYHUUDUUFVEULZVFYHYTSLZYMSLZYLSLZUUCUUBUTYHUULUUTUVGUUQUV
      DXNYCVGTYHUURUUTUVHUUSUVDYAYCVGTZYHUULXQSLZUVIUUQYHXKXSUVKXJXKXLYGVHZXMXP
      XSYFVIZXKEUOUEZXRUPKLXSUVKEUQXQXRUVNURVATZXNXQVGTZYTYMYLVKVLVMYHYPKLZUUAY
      QLZUUBYRVTZYHXJYOKLZUVQXJXKXLYGVNZXMUVTYGXKXLUVTXJEFVOVPWAZDYOVQTZYHXNXQY
      CRQZPQZUUAYQYHUULUVKUUTUWEUUAVJUUQUVOUVDXNXQYCVRVLZYHXJUVTXPUWDYOUDUELZUW
      EYQLZUWAUWBXMXPXSYFVSYHXKXLXSYEUWGUVLUVAUVMUVBXKXLNXSYENUWGHUAUBJIEFXQYCU
      HRJIUBUAHWBUAMZUBMZWCWDWEWFZXJUVTNZXPUWGNUWHHUAUBJIDYOXNUWDUIPJIUBUAHWGZU
      WIUWJVGZWDWEWFZWHUVQYPUOUEZYQUPKLZUVRUVSYPUQZYNUUAYQUWPWIVATWJYHYJYAXQPQZ
      YMRQZYNOUFZYRYHYJUWSYLOUFUXAYHHIJYAXNXQOSPUUKUUSUUQUVOUVFVFYHHIJUWSYLYMOS
      RUUIUUJUUHUUDRQUUHUUFRQOUFUTYHUUDUUFUUHVKULYHUURUVKUWSSLUUSUVOYAXQVGTUVPU
      VJUVEUUDUUFRQUUFUUDRQVJYHUUDUUFWKULVFVMYHUVQUWTYQLZUXAYRVTZUWCYHYAUWDPQZU
      WTYQYHUURUVKUUTUXDUWTVJUUSUVOUVDYAXQYCVRVLYHXJUVTYBUWGUXDYQLZUWAUWBXMXTYB
      YEWLUWKUWLYBUWGNUXEHUAUBJIDYOYAUWDUIPUWMUWNWDWEWFWHUVQUWQUXBUXCUWRYNUWTYQ
      UWPWIVATWJWSYHYSXNYAVJZYRYHUULUURUXFYSUTUUQUUSSXNYAOWMXNYAWNWOTYHUXFYRYHU
      XFNUWEYNYQYHUXFUWEUUAYNUWFUXFYTYMYLRXNYAYCPWPWQWRYHUWHUXFUWOWAWHWTXAYHYKX
      BZYKYSUGYHUULUURUXGUUQUUSUULUURNYIXBYJXBZUXGXNYAXCUURUULUXHYAXNXCXDYIYJXE
      XFTYKXGXHXI $.

    $( Lemma for distributive law for positive reals.  (Contributed by Jim
       Kingdon, 12-Dec-2019.) $)
    distrlem5prl $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
        ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_
        ( 1st ` ( A .P. ( B +P. C ) ) ) ) $=
      ( vw vv vf vg vh vx vy vz cnp wcel cmp co c1st cfv cv wrex wa vu w3a cplq
      cpp wb mulclpr 3adant3 3adant2 df-iplp addclnq genpelvl syl2anc wi df-imp
      wceq cmq mulclnq anbi2d distrlem4prl oveq12 eleq1 biimtrdi imp syl5ibrcom
      eqeq2d exp4b com3l com23 rexlimivv rexlimdvv com3r sylbid impd ssrdv ) AL
      MZBLMZCLMZUBZDABNOZACNOZUDOPQZABCUDONOPQZVRDRZWAMZWCERZUARZUCOZUOZUAVTPQZ
      SEVSPQZSZWCWBMZVRVSLMZVTLMZWDWKUEVOVPWMVQABUFUGVOVQWNVPACUFUHFGHIJVSVTWCE
      UAUDUCIJHGFUIGRZHRZUJUKULVRWHWLEUAWJWIVRWEWJMZWFWIMZTWQWFFRZKRZUPOZUOZKCP
      QZSFAPQZSZTWHWLUMZVRWRXEWQVOVQWRXEUEVPIGHDEACWFFKNUPDEHGIUNWOWPUQZUKUHURV
      RWQXEXFVRWQWEIRZJRZUPOZUOZJBPQZSIXDSZXEXFUMVOVPWQXMUEVQFGHDEABWEIJNUPDEHG
      FUNXGUKUGXMXEVRXFXMXBVRXFUMZFKXDXCXKWSXDMWTXCMTZXBXNUMZUMIJXDXLXHXDMXIXLM
      TZXOXKXPXQXOXKXBXNVRXQXOTZXKXBTZXFVRXRXSWHWLVRXRTWLXSWHTXJXAUCOZWBMZIJKAB
      CFUSXSWHWLYAUEZXSWHWCXTUOYBXSWGXTWCWEXJWFXAUCUTVEWCXTWBVAVBVCVDVFVGVFVHVI
      VJVKVLVMVLVJVLVN $.

    $( Lemma for distributive law for positive reals.  (Contributed by Jim
       Kingdon, 12-Dec-2019.) $)
    distrlem5pru $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
        ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_
        ( 2nd ` ( A .P. ( B +P. C ) ) ) ) $=
      ( vw vv vf vg vh vx vy vz cnp wcel cmp co c2nd cfv cv wrex wa vu w3a cplq
      cpp wb mulclpr 3adant3 3adant2 df-iplp addclnq genpelvu syl2anc wi df-imp
      wceq cmq mulclnq anbi2d distrlem4pru oveq12 eleq1 biimtrdi imp syl5ibrcom
      eqeq2d exp4b com3l com23 rexlimivv rexlimdvv com3r sylbid impd ssrdv ) AL
      MZBLMZCLMZUBZDABNOZACNOZUDOPQZABCUDONOPQZVRDRZWAMZWCERZUARZUCOZUOZUAVTPQZ
      SEVSPQZSZWCWBMZVRVSLMZVTLMZWDWKUEVOVPWMVQABUFUGVOVQWNVPACUFUHFGHIJVSVTWCE
      UAUDUCIJHGFUIGRZHRZUJUKULVRWHWLEUAWJWIVRWEWJMZWFWIMZTWQWFFRZKRZUPOZUOZKCP
      QZSFAPQZSZTWHWLUMZVRWRXEWQVOVQWRXEUEVPIGHDEACWFFKNUPDEHGIUNWOWPUQZUKUHURV
      RWQXEXFVRWQWEIRZJRZUPOZUOZJBPQZSIXDSZXEXFUMVOVPWQXMUEVQFGHDEABWEIJNUPDEHG
      FUNXGUKUGXMXEVRXFXMXBVRXFUMZFKXDXCXKWSXDMWTXCMTZXBXNUMZUMIJXDXLXHXDMXIXLM
      TZXOXKXPXQXOXKXBXNVRXQXOTZXKXBTZXFVRXRXSWHWLVRXRTWLXSWHTXJXAUCOZWBMZIJKAB
      CFUSXSWHWLYAUEZXSWHWCXTUOYBXSWGXTWCWEXJWFXAUCUTVEWCXTWBVAVBVCVDVFVGVFVHVI
      VJVKVLVMVLVJVLVN $.
  $}

  $( Multiplication of positive reals is distributive.  Proposition 9-3.7(iii)
     of [Gleason] p. 124.  (Contributed by Jim Kingdon, 12-Dec-2019.) $)
  distrprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
      ( A .P. ( B +P. C ) ) = ( ( A .P. B ) +P. ( A .P. C ) ) ) $=
    ( cnp wcel w3a cpp cmp wceq c1st cfv distrlem1prl distrlem5prl distrlem1pru
    co c2nd eqssd addclpr syl2anc mulclpr distrlem5pru wa wb simp1 simp2 preqlu
    simp3 mpbir2and ) ADEZBDEZCDEZFZABCGOZHOZABHOZACHOZGOZIZUNJKZUQJKZIZUNPKZUQ
    PKZIZULUSUTABCLABCMQULVBVCABCNABCUAQULUNDEZUQDEZURVAVDUBUCULUIUMDEZVEUIUJUK
    UDZULUJUKVGUIUJUKUEZUIUJUKUGZBCRSAUMTSULUODEZUPDEZVFULUIUJVKVHVIABTSULUIUKV
    LVHVJACTSUOUPRSUNUQUFSUH $.

  ${
    $d A x y $.  $d B x y $.
    $( If a positive real is less than a second positive real, its lower cut is
       a subset of the second's lower cut.  (Contributed by Jim Kingdon,
       23-Dec-2019.) $)
    ltprordil $p |- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) ) $=
      ( vx vy cltp wbr cv c2nd cfv wcel c1st wa wss cnq cnp wrex adantr cop syl
      prop ltrelpr brel ltdfpr biimpd mpcom simpll simpr simprrl simpld syl3an1
      cltq prltlu syl3anc wi simprrr simprd prcdnql sylan syl2anc mpd rexlimddv
      ex ssrdv ) ABEFZCGZAHIZJZVEBKIZJZLZAKIZVHMCNAOJZBOJZLZVDVJCNPZABOOEUAUBZV
      NVDVOABCUCUDUEVDVENJZVJLZLZDVKVHVSDGZVKJZVTVHJZVSWALZVTVEUKFZWBWCVDWAVGWD
      VDVRWAUFZVSWAUGVSVGWAVDVQVGVIUHQVDVKVFROJZWAVGWDVDVLWFVDVLVMVPUIATSVTVEVF
      VKULUJUMWCVDVIWDWBUNZWEVSVIWAVDVQVGVIUOQVDVHBHIZROJZVIWGVDVMWIVDVLVMVPUPB
      TSVEVTWHVHUQURUSUTVBVCVA $.
  $}

  ${
    $d x y z w v u f g h A $.
    $( Lemma for ~ 1idpr .  (Contributed by Jim Kingdon, 13-Dec-2019.) $)
    1idprl $p |- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) ) $=
      ( vf vg vw vy wcel c1p co cfv cv cmq wceq wrex cltq wbr wa cnq c1q adantl
      wb vx vu vv cnp cmp c1st wss ssid rexss ax-mp 1pr c2nd prop elprnql sylan
      vz cop mpan w3a breq1 3ad2ant3 abeq2i ltmnqg mp3an2 ancoms mulidnq breq2d
      1nq adantr bitrd bitr2id 3adant3 syl3an1 syl3an2 3expia pm5.32rd rexbidva
      1prl r19.42v bitr3di bitrid df-imp mulclnq genpelvl mpan2 prnmaxl ltrelnq
      wi crq brel simpl simpr recclnq mulcomnqg caovord2d recidnq biimpd sylan2
      mpcom elab2g mpbird mulassnqg caov12d oveq2d 3eqtrrd oveq2 eqeq2d syl2anc
      syl rspcev a1i ancld reximia ex prcdnql adantrd rexlimdva impbid 3bitr4d
      eqrdv ) AUDFZUAAGUEHUFIZAUFIZYAUAJZBJZCJZKHZLZCGUFIZMZBYCMZYDYENOZYJPZBYC
      MZYDYBFZYDYCFZYAYJYMBYCYJYFYIFZYHPZCYIMZYAYEYCFZPZYMYIYIUGYJYSTYIUHYHCYIY
      IUIUJUUAYLYHPZCYIMYSYMUUAUUBYRCYIUUAYQPYHYLYQUUAYQYHYLYQTZYQUUAYFQFZYHUUC
      GUDFZYQUUDUKUUEYIGULIZUQUDFYQUUDGUMYFUUFYIUNUOURUUAYEQFZUUDYHUUCYAYCAULIZ
      UQUDFZYTUUGAUMZYEUUHYCUNUOUUGUUDYHUSYLYGYENOZYQYHUUGYLUUKTUUDYDYGYENUTVAU
      UGUUDUUKYQTYHYQYFRNOZUUGUUDPZUUKUULCYICVRZVBUUMUULYGYERKHZNOZUUKUUDUUGUUL
      UUPTZUUDRQFUUGUUQVHYFRYEVCVDVEUUGUUPUUKTUUDUUGUUOYEYGNYEVFVGVIVJVKVLVJVMV
      NVOVPVQYLYHCYIVSVTWAVQYAUUEYOYKTUKDUBUCEUPAGYDBCUEKEUPUCUBDWBUBJUCJWCWDWE
      YAYPYNYAYPYNYAYPPYLBYCMZYNYAUUIYPUURUUJBYDUUHYCWFUOYLYMBYCYTYLYJYLYJWHYTY
      LYDYEWIIZKHZYIFZYDYEUUTKHZLZYJYLUVAUUTRNOZYDQFZUUGPZYLUVDYDYEQQNWGWJZUVFY
      LUVDUVFYLUUTYEUUSKHZNOZUVDUVFEUPDYDYEUUSNQKEJZQFZUPJZQFZDJZQFUSZUVJUVLNOU
      VNUVJKHUVNUVLKHNOTUVFUVJUVLUVNVCSUVEUUGWKZUVEUUGWLZUUGUUSQFZUVEYEWMZSZUVK
      UVMPUVJUVLKHZUVLUVJKHLUVFUVJUVLWNSZWOUUGUVIUVDTUVEUUGUVHRUUTNYEWPZVGSVJWQ
      WSYLUUTQFZUVAUVDTYLUVFUWDUVGUUGUVEUVRUWDUVSYDUUSWCWRXIUULUVDCUUTYIQYFUUTR
      NUTUUNWTXIXAYLUVFUVCUVGUVFUVBYDUVHKHZYDRKHZYDUVFEUPDYEYDUUSQKUVQUVPUVTUWB
      UVOUWAUVNKHUVJUVLUVNKHKHLUVFUVJUVLUVNXBSXCUUGUWEUWFLUVEUUGUVHRYDKUWCXDSUV
      EUWFYDLUUGYDVFVIXEXIYHUVCCUUTYIYFUUTLYGUVBYDYFUUTYEKXFXGXJXHXKXLXMXIXNYAY
      MYPBYCUUAYLYPYJYAUUIYTYLYPWHUUJYEYDUUHYCXOUOXPXQXRXSXT $.

    $( Lemma for ~ 1idpr .  (Contributed by Jim Kingdon, 13-Dec-2019.) $)
    1idpru $p |- ( A e. P. -> ( 2nd ` ( A .P. 1P ) ) = ( 2nd ` A ) ) $=
      ( vf vh vw vy wcel c1p co cfv cv cmq wceq wrex cltq wbr wa cnq c1q adantl
      wb vx vu vv cnp cmp c2nd wss ssid rexss ax-mp 1pr c1st prop elprnqu sylan
      vz cop mpan w3a breq2 3ad2ant3 abeq2i ltmnqg mp3an1 ancoms mulidnq breq1d
      1nq adantr bitrd bitr2id 3adant3 syl3an1 syl3an2 3expia pm5.32rd rexbidva
      1pru r19.42v bitr3di bitrid df-imp mulclnq genpelvu mpan2 prnminu ltrelnq
      wi crq brel ancomd simpr simpl recclnq mulcomnqg caovord2d recidnq biimpd
      mpcom sylan2 elab2g 3syl mpbird mulassnqg caov12d oveq2d syl oveq2 eqeq2d
      3eqtrrd rspcev syl2anc a1i ancld reximia prcunqu adantrd rexlimdva impbid
      ex 3bitr4d eqrdv ) AUDFZUAAGUEHUFIZAUFIZYCUAJZBJZCJZKHZLZCGUFIZMZBYEMZYGY
      FNOZYLPZBYEMZYFYDFZYFYEFZYCYLYOBYEYLYHYKFZYJPZCYKMZYCYGYEFZPZYOYKYKUGYLUU
      ATYKUHYJCYKYKUIUJUUCYNYJPZCYKMUUAYOUUCUUDYTCYKUUCYSPYJYNYSUUCYSYJYNYSTZYS
      UUCYHQFZYJUUEGUDFZYSUUFUKUUGGULIZYKUQUDFYSUUFGUMYHYKUUHUNUOURUUCYGQFZUUFY
      JUUEYCAULIZYEUQUDFZUUBUUIAUMZYGYEUUJUNUOUUIUUFYJUSYNYGYINOZYSYJUUIYNUUMTU
      UFYFYIYGNUTVAUUIUUFUUMYSTYJYSRYHNOZUUIUUFPZUUMUUNCYKCVRZVBUUOUUNYGRKHZYIN
      OZUUMUUFUUIUUNUURTZRQFUUFUUIUUSVHRYHYGVCVDVEUUIUURUUMTUUFUUIUUQYGYINYGVFV
      GVIVJVKVLVJVMVNVOVPVQYNYJCYKVSVTWAVQYCUUGYQYMTUKDUBUCEUPAGYFBCUEKEUPUCUBD
      WBUBJUCJWCWDWEYCYRYPYCYRYPYCYRPYNBYEMZYPYCUUKYRUUTUULBYFYEUUJWFUOYNYOBYEU
      UBYNYLYNYLWHUUBYNYFYGWIIZKHZYKFZYFYGUVBKHZLZYLYNUVCRUVBNOZYFQFZUUIPZYNUVF
      YNUUIUVGYGYFQQNWGWJWKZUVHYNUVFUVHYNYGUVAKHZUVBNOZUVFUVHEUPDYGYFUVANQKEJZQ
      FZUPJZQFZDJZQFUSZUVLUVNNOUVPUVLKHUVPUVNKHNOTUVHUVLUVNUVPVCSUVGUUIWLZUVGUU
      IWMZUUIUVAQFZUVGYGWNZSZUVMUVOPUVLUVNKHZUVNUVLKHLUVHUVLUVNWOSZWPUUIUVKUVFT
      UVGUUIUVJRUVBNYGWQZVGSVJWRWSYNUVHUVBQFZUVCUVFTUVIUUIUVGUVTUWFUWAYFUVAWCWT
      UUNUVFCUVBYKQYHUVBRNUTUUPXAXBXCYNUVHUVEUVIUVHUVDYFUVJKHZYFRKHZYFUVHEUPDYG
      YFUVAQKUVRUVSUWBUWDUVQUWCUVPKHUVLUVNUVPKHKHLUVHUVLUVNUVPXDSXEUUIUWGUWHLUV
      GUUIUVJRYFKUWEXFSUVGUWHYFLUUIYFVFVIXJXGYJUVECUVBYKYHUVBLYIUVDYFYHUVBYGKXH
      XIXKXLXMXNXOXGXTYCYOYRBYEUUCYNYRYLYCUUKUUBYNYRWHUULYFYGYEUUJXPUOXQXRXSYAY
      B $.

    $( 1 is an identity element for positive real multiplication.  Theorem
       9-3.7(iv) of [Gleason] p. 124.  (Contributed by NM, 2-Apr-1996.) $)
    1idpr $p |- ( A e. P. -> ( A .P. 1P ) = A ) $=
      ( cnp wcel c1p cmp co wceq c1st cfv 1idprl 1idpru wa wb 1pr mulclpr mpan2
      c2nd preqlu mpancom mpbir2and ) ABCZADEFZAGZUBHIAHIGZUBQIAQIGZAJAKUBBCZUA
      UCUDUELMUADBCUFNADOPUBARST $.
  $}

  ${
    $d A l x $.  $d A u x $.  $d B l x $.  $d B u x $.
    $( We can order fractions via ` <Q ` or ` <P ` .  (Contributed by Jim
       Kingdon, 19-Jun-2021.) $)
    ltnqpr $p |- ( ( A e. Q. /\ B e. Q. ) -> (
        A <Q B <-> <. { l | l <Q A } , { u | A <Q u } >. <P
        <. { l | l <Q B } , { u | B <Q u } >. ) ) $=
      ( vx cnq wcel wa cltq wbr wrex cab cop cfv cnp nqprlu ltnqex gtnqex elab2
      cv cltp ltbtwnnqq c2nd c1st ltdfpr syl2an breq2 op2nd breq1 op1st anbi12i
      wb vex rexbii bitrdi bitr4id ) BFGZCFGZHZBCIJBETZIJZUTCIJZHZEFKZDTZBIJDLZ
      BATZIJZALZMZVECIJZDLZCVGIJALZMZUAJZEBCUBUSVOUTVJUCNZGZUTVNUDNZGZHZEFKZVDU
      QVJOGVNOGVOWAULURABDPACDPVJVNEUEUFVTVCEFVQVAVSVBVHVAAUTVPEUMZVGUTBIUGVFVI
      DBQABRUHSVKVBDUTVRWBVEUTCIUIVLVMDCQACRUJSUKUNUOUP $.
  $}

  ${
    $d A l x $.  $d A u x $.  $d B l x $.  $d B u x $.
    $( We can order fractions via ` <Q ` or ` <P ` .  (Contributed by Jim
       Kingdon, 8-Jan-2021.) $)
    ltnqpri $p |- ( A <Q B -> <. { l | l <Q A } , { u | A <Q u } >. <P
        <. { l | l <Q B } , { u | B <Q u } >. ) $=
      ( vx cltq wbr cv cab cop cnq wrex cfv wcel cnp nqprlu syl ltnqex gtnqex
      wa cltp c2nd c1st wb ltrelnq simpld simprd ltdfpr syl2anc vex breq2 op2nd
      brel elab2 breq1 op1st anbi12i rexbii bitrdi ltbtwnnqq bitr4di ibir ) BCF
      GZDHZBFGDIZBAHZFGZAIZJZVDCFGZDIZCVFFGAIZJZUAGZVCVNBEHZFGZVOCFGZTZEKLZVCVC
      VNVOVIUBMZNZVOVMUCMZNZTZEKLZVSVCVIONZVMONZVNWEUDVCBKNZWFVCWHCKNZBCKKFUEUM
      ZUFABDPQVCWIWGVCWHWIWJUGACDPQVIVMEUHUIWDVREKWAVPWCVQVGVPAVOVTEUJZVFVOBFUK
      VEVHDBRABSULUNVJVQDVOWBWKVDVOCFUOVKVLDCRACSUPUNUQURUSEBCUTVAVB $.
  $}

  ${
    $d q r s t u $.
    $( Positive real 'less than' is a partial ordering.  Remark ("< is
       transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p.
       (varies).  Lemma for ~ ltsopr .  (Contributed by Jim Kingdon,
       15-Dec-2019.) $)
    ltpopr $p |- <P Po P. $=
      ( vq vr cnp cltp wtru cv wcel wbr c2nd cfv c1st wa cnq wrex cop wb ltdfpr
      prop wi mpd vs vt vu wpo wn prdisj sylnib nrexdv anidms mtbird adantl w3a
      sylan ancom 3adant3 3adant1 anbi12d reeanv bitr4di biimpa simprll syl3an1
      cltq prltlu 3adant3r 3adant2l 3ad2antl2 adantlr prcdnql adantrl 3ad2antl3
      3expb jca ex rexlimdvw reximdv 3adant2 biimprd adantr ispod mptru ) CDUDE
      UAUBUCCDUAFZCGZWBWBDHZUEEWCWDAFZWBIJZGZWEWBKJZGZLZAMNZWCWJAMWCWEMGZLWIWGL
      ZWJWCWHWFOCGWLWMUEWBRWEWFWHUFUMWIWGUNUGUHWCWDWKPWBWBAQUIUJUKWCUBFZCGZUCFZ
      CGZULZWBWNDHZWNWPDHZLZWBWPDHZSEWRXAXBWRXALZWGWEWPKJZGZLZAMNZXBXCWGWEWNKJZ
      GZLZBFZWNIJZGZXKXDGZLZLZBMNZAMNZXGWRXAXRWRXAXJAMNZXOBMNZLXRWRWSXSWTXTWCWO
      WSXSPWQWBWNAQUOWOWQWTXTPWCWNWPBQUPUQXJXOABMMURUSUTXCXQXFAMXCXPXFBMXCXPXFX
      CXPLZWGXEXCWGXIXOVAYAWEXKVCHZXEWRXPYBXAWOWCXPYBWQWOXJXOYBWOXIXOYBWGWOXIXM
      YBXNWOXHXLOCGXIXMYBWNRWEXKXLXHVDVBVEVFVLVGVHWRXPYBXESZXAWQWCXPYCWOWQXOYCX
      JWQXNYCXMWQXDWPIJZOCGXNYCWPRXKWEYDXDVIUMVJVJVKVHTVMVNVOVPTWRXGXBSXAWRXBXG
      WCWQXBXGPWOWBWPAQVQVRVSTVNUKVTWA $.
  $}

  ${
    $d q r s x y z $.
    $( Positive real 'less than' is a weak linear order (in the sense of
       ~ df-iso ).  Proposition 11.2.3 of [HoTT], p.  (varies).  (Contributed
       by Jim Kingdon, 16-Dec-2019.) $)
    ltsopr $p |- <P Or P. $=
      ( vx vy vz vq vr vs cnp cltp cv wbr wi wcel cfv wa cnq wrex cltq sylan ex
      mpd wor wpo wo wral ltpopr w3a c2nd wb ltdfpr 3adant3 cop prnminu prnmaxl
      c1st prop anim12i an4s reeanv sylibr 3adantl3 ltsonq ltrelnq sotri adantl
      prloc 3ad2antl3 adantr ad2antrr wex elprnqu ax-ia3 19.8a syl6an 3ad2antl1
      imp df-rex biimprd 3adant2 adantrr elprnql pm3.21 3adant1 adantrl orim12d
      3ad2antl2 ancoms adantlr rexlimdvva rexlimdvw sylbid df-iso mpbir2an
      rgen3 ) GHUAGHUBAIZBIZHJZWNCIZHJZWQWOHJZUCZKZCGUDBGUDAGUDUEXAABCGGGWNGLZW
      OGLZWQGLZUFZWPDIZWNUGMZLZXFWOUNMZLZNZDOPZWTXBXCWPXLUHXDWNWODUIUJXEXKWTDOX
      EXKWTXEXKNZEIZXFQJZXFFIZQJZNZFXIPEXGPZWTXBXCXKXSXDXBXCNXKNXOEXGPZXQFXIPZN
      ZXSXBXHXCXJYBXBXHNXTXCXJNYAXBWNUNMZXGUKGLZXHXTWNUOZEXFXGYCULRXCXIWOUGMZUK
      GLZXJYAWOUOZFXFYFXIUMRUPUQXOXQEFXGXIURUSUTXMXRWTEFXGXIXMXNXGLZXPXILZNZNZX
      RWTYLXRNZXNWQUNMZLZXPWQUGMZLZUCZWTYMXNXPQJZYRXRYSYLXNXFXPQOVAVBVCVDXMYSYR
      KZYKXRXEYTXKXEYSYRXDXBYSYRXCXDYNYPUKGLYSYRWQUOXNXPYPYNVERVFSVGVHTYLYRWTKZ
      XRXEYKUUAXKXEYKNYOWRYQWSXEYIYOWRKYJXEYINZYOWRUUBYONZYIYONZEOPZWRUUCXNOLZU
      UDNZEVIZUUEUUBYOUUHXBXCYIYOUUHKXDXBYINUUFYOUUDUUHXBYDYIUUFYEXNXGYCVJRYIYO
      UUDKXBYIYOVKVDUUGEVLVMVNVOUUDEOVPUSXEUUEWRKZYIYOXBXDUUIXCXBXDNWRUUEWNWQEU
      IVQVRVHTSVSXEYJYQWSKYIXEYJNZYQWSUUJYQNZYQYJNZFOPZWSUUKXPOLZUULNZFVIZUUMUU
      JYQUUPXCXBYJYQUUPKXDXCYJNUUNYQUULUUPXCYGYJUUNYHXPYFXIVTRYJYQUULKXCYJYQWAV
      DUUOFVLVMWEVOUULFOVPUSXEUUMWSKZYJYQXCXDUUQXBXDXCUUQXDXCNWSUUMWQWOFUIVQWFW
      BVHTSWCWDWGVGTSWHTSWIWJWMABCGHWKWL $.
  $}

  ${
    $d A f g h x y p q r $.  $d B f g h x y p q r $.
    $( The sum of two positive reals is greater than one of them.  Proposition
       9-3.5(iii) of [Gleason] p. 123.  (Contributed by NM, 26-Mar-1996.)
       (Revised by Mario Carneiro, 12-Jun-2013.) $)
    ltaddpr $p |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) ) $=
      ( vp vq vr vf vg vh cnp wcel wa cv c1st cfv cnq wrex c2nd cop prop sylan
      vx vy cpp co cltp wbr prml syl cplq cltq prarloc ad2ant2r elprnqu adantlr
      adantl ad2ant2rl adantr simplrr wi simprl simplr jca df-iplp addclnq syl5
      genpprecll imdistani addclpr prcdnql anassrs imp rspe wb ltdfpr ad3antrrr
      syl12anc syldan mpbird ex rexlimdvva mpd rexlimddv ) AIJZBIJZKZCLZBMNZJZA
      ABUCUDZUEUFZCOWDWHCOPZWCWDWGBQNZRIJWKBSCWLWGUGUHUOWEWFOJZWHKZKZDLZELZWFUI
      UDZUJUFZDAQNZPEAMNZPZWJWCWMXBWDWHWCXAWTRIJZWMXBASZWFWTXAEDUKTULWOWSWJEDXA
      WTWOWQXAJZWPWTJZKZKZWSWJXHWSKZWJXFWPWIMNZJZKZDOPZXIWPOJZXFXKXMXHXNWSWEXFX
      NWNXEWCXFXNWDWCXCXFXNXDWPWTXAUMTUNUPUQWOXEXFWSURXHWSXKWEWNXGWSXKUSZWEWNXG
      KZKWEWRXJJZKXOWEXPXQXPXEWHKWEXQXPXEWHWNXEXFUTWMWHXGVAVBFGHUAUBABWQWFUCUIU
      AUBHGFVCGLHLVDVFVEVGWEWIIJZXQXOABVHZXRXJWIQNZRIJXQXOWISWRWPXTXJVITTUHVJVK
      XLDOVLVPWEWJXMVMZWNXGWSWCWDXRYAXSAWIDVNVQVOVRVSVTWAWB $.
  $}

  ${
    $d x y z w u v f g h q r s t A $.  $d x y z w u v f g h q r s t B $.
    $d x y z w u v f g h q r s t C $.
    ltexprlem.1 $e |- C = <. { x e. Q. | E. y (
      y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y (
      y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >. $.
    $( Element in lower cut of the constructed difference.  Lemma for
       ~ ltexpri .  (Contributed by Jim Kingdon, 21-Dec-2019.) $)
    ltexprlemell $p |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\
        E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) $=
      ( cv c2nd cfv wcel cplq co c1st wa wex cnq crab nqex rabex wceq oveq2 cop
      eleq1d anbi2d exbidv fveq2i op1st eqtri elrab2 ) BHZCIJKZUKAHZLMZDNJZKZOZ
      BPZULUKFHZLMZUOKZOZBPAUSQENJZUMUSUAZUQVBBVDUPVAULVDUNUTUOUMUSUKLUBUDUEUFV
      CURAQRZUKCNJKUNDIJKOBPZAQRZUCZNJVEEVHNGUGVEVGURAQSTVFAQSTUHUIUJ $.

    $( Element in upper cut of the constructed difference.  Lemma for
       ~ ltexpri .  (Contributed by Jim Kingdon, 21-Dec-2019.) $)
    ltexprlemelu $p |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\
        E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) $=
      ( cv c1st cfv wcel cplq co c2nd wa wex cnq crab nqex rabex wceq oveq2 cop
      eleq1d anbi2d exbidv fveq2i op2nd eqtri elrab2 ) BHZCIJKZUKAHZLMZDNJZKZOZ
      BPZULUKFHZLMZUOKZOZBPAUSQENJZUMUSUAZUQVBBVDUPVAULVDUNUTUOUMUSUKLUBUDUEUFV
      CUKCNJKUNDIJKOBPZAQRZURAQRZUCZNJVGEVHNGUGVFVGVEAQSTURAQSTUHUIUJ $.

    $( Our constructed difference is inhabited.  Lemma for ~ ltexpri .
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlemm $p |- ( A <P B -> ( E. q e. Q. q e. ( 1st ` C ) /\
        E. r e. Q. r e. ( 2nd ` C ) ) ) $=
      ( vw cv cfv wcel cnq wrex wa wex cnp sylan syl sylibr cltp c1st c2nd cplq
      wbr co ltrelpr brel ltdfpr biimpd mpcom simprrl simprd wceq cltq cop prop
      prnmaxl ltexnqi reximi sylib r19.42v exbii eleq1 biimparc exlimiv adantrl
      df-rex jca expr reximdva mpd rexbii rexcom simpld elprnqu pm4.71rd anbi1d
      ex anass bitrdi exbidv rexbidv bitr4di mpbird ltexprlemell wss ssid rexss
      wb ax-mp bitr4i nfv nfre1 prmu rexex 3syl prml adantr w3a 3adant3 elprnql
      simp3 3adant2 addcomnqg syl2anc ltaddnq eqeltrrd 19.8a ltexprlemelu 3expa
      wi prcunqu exlimddv exlimdd ) CDUAUEZGJZEUBKLZGMNZFJZEUCKLZFMNZXPBJZCUCKZ
      LZYCXQUDUFZDUBKZLZOZBPZGMNZXSXPYKYIBMNZGMNZXPYIGMNZBMNZYMXPYEYHGMNZOZBMNZ
      YOXPYEYCYGLZOZBMNZYRCQLZDQLZOZXPUUACDQQUAUGUHZUUDXPUUACDBUIUJUKXPYTYQBMXP
      YCMLZYTYQXPUUFYTOOYEYPXPUUFYEYSULXPYTYPUUFXPYSYPYEXPUUCYSYPXPUUBUUCUUEUMZ
      UUCYSOZIJZYGLZYFUUIUNZOZGMNZIPZYPUUHUUJUUKGMNZOZIPZUUNUUHUUOIYGNZUUQUUHYC
      UUIUOUEZIYGNZUURUUCYGDUCKZUPQLZYSUUTDUQZIYCUVAYGURRUUSUUOIYGGYCUUIUSUTSUU
      OIYGVHVAUUMUUPIUUJUUKGMVBVCTUUMYPIUULYHGMUUKYHUUJYFUUIYGVDVEUTVFSRVGVGVIV
      JVKVLYNYQBMYEYHGMVBVMTYIBGMMVNVAXPYKUUFYIOZBPZGMNYMXPYJUVEGMXPYIUVDBXPYIU
      UFYEOZYHOUVDXPYEUVFYHXPYEUUFXPYEUUFXPUUBYEUUFXPUUBUUCUUEVOZUUBCUBKZYDUPQL
      ZYEUUFCUQZYCYDUVHVPRRVSVQVRUUFYEYHVTWAWBWCYLUVEGMYIBMVHVMWDWEXSXQMLYJOZGM
      NZYKXRUVKGMABCDEGHWFVMMMWGYKUVLWJMWHYJGMMWIWKWLTXPXTUVALZYBFXPFWMYAFMWNXP
      UUCUVBUVMFPZUUGUVCUVBUVMFMNUVNFUVAYGWOUVMFMWPSWQXPUVMOZXTMLZYAOZFPZYBUVOU
      VPYAUVRXPUUCUVMUVPUUGUUCUVBUVMUVPUVCXTUVAYGVPRRZUVOYCUVHLZYABXPUVTBPZUVMX
      PUUBUVTBMNZUWAUVGUUBUVIUWBUVJBYDUVHWRSUVTBMWPWQWSXPUVMUVTYAXPUVMUVTWTZUVP
      UVTYCXTUDUFZUVALZOZBPZOYAUWCUVPUWGXPUVMUVPUVTUVSXAZUWCUVTUWEUWGXPUVMUVTXC
      UWCXTYCUDUFZUWDUVAUWCUVPUUFUWIUWDUNUWHXPUVTUUFUVMXPUUBUVTUUFUVGUUBUVIUVTU
      UFUVJYCYDUVHXBRRXDZXTYCXEXFUWCXTUWIUOUEZUWIUVALZUWCUVPUUFUWKUWHUWJXTYCXGX
      FXPUVMUWKUWLXLZUVTXPUUCUVMUWMUUGUUCUVBUVMUWMUVCUWIXTUVAYGXMRRXAVLXHUWFBXI
      XFVIABCDEFHXJTXKXNUVQFXIXFYAFMVHTXOVI $.

    $( The lower cut of our constructed difference is open.  Lemma for
       ~ ltexpri .  (Contributed by Jim Kingdon, 21-Dec-2019.) $)
    ltexprlemopl $p |- ( ( A <P B /\ q e. Q. /\ q e. ( 1st ` C ) )
        -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) $=
      ( vs wbr cv cnq wcel cfv cltq wa wrex wex cnp syl2anc cltp c1st c2nd cplq
      ltexprlemell simprbi 19.42v anbi2i bitri ltrelpr brel simprd prop prnmaxl
      co cop sylan adantrl wceq simpld ad2antrr simplrr elprnqu simplrl ltaddnq
      simprr ltsonq ltrelnq sotri wb simprl elprnql ltexnqq mpbid adantr ltanqg
      breqtrrd syl3anc mpbird eqeltrd jca jca32 reximdva rexlimddv eximi sylbir
      expr mpd rexcom4 sylibr rexbii sylib sylanr2 3impb ) CDUAJZGKZLMZWPEUBNZM
      ZWPFKZOJZWTWRMZPZFLQZWSWOWQBKZCUCNZMZXEWPUDUOZDUBNZMZPZBRZXDWSWQXLABCDEGH
      UEUFWOWQXLPZPZXAWTLMZXGXEWTUDUOZXIMZPZBRPZPZFLQZXDXNXAXOXRPZPZBRZFLQZYAXN
      YCFLQZBRZYEXNWOWQXKPZPZBRZYGYJWOYHBRZPXNWOYHBUGYKXMWOWQXKBUGUHUIYIYFBYIXH
      IKZOJZYFIXIWOXKYMIXIQZWQWOXJYNXGWODSMZXJYNWOCSMZYOCDSSUAUJUKZULZYOXIDUCNZ
      UPSMZXJYNDUMZIXHYSXIUNUQUQURURYIYLXIMZYMPZPZXPYLUSZFLQZYFUUDXEYLOJZUUFUUD
      XEXHOJZYMUUGUUDXELMZWQUUHUUDYPXGUUIWOYPYHUUCWOYPYOYQUTVAUUDXGXJWOWQXKUUCV
      BUTZYPCUBNZXFUPSMXGUUICUMXEXFUUKVCUQTZWOWQXKUUCVDZXEWPVETYIUUBYMVFXEXHYLO
      LVGVHVITUUDUUIYLLMZUUGUUFVJUULUUDYOUUBUUNWOYOYHUUCYRVAYIUUBYMVKYOYTUUBUUN
      UUAYLYSXIVLUQTFXEYLVMTVNUUDUUEYCFLUUDXOUUEYCUUDXOUUEPZPZXAXOXRUUPXAXHXPOJ
      ZUUPXHYLXPOYIUUBYMUUOVBUUDXOUUEVFZVQUUPWQXOUUIXAUUQVJUUDWQUUOUUMVOUUDXOUU
      EVKZUUDUUIUUOUULVOWPWTXEVPVRVSUUSUUPXGXQUUDXGUUOUUJVOUUPXPYLXIUURYIUUBYMU
      UOVDVTWAWBWGWCWHWDWEWFYCFBLWIWJYDXTFLYDXAYBBRZPXTXAYBBUGUUTXSXAXOXRBUGUHU
      IWKWLXCXTFLXBXSXAABCDEFHUEUHWKWJWMWN $.

    $( The lower cut of our constructed difference is lower.  Lemma for
       ~ ltexpri .  (Contributed by Jim Kingdon, 21-Dec-2019.) $)
    ltexprlemlol $p |- ( ( A <P B /\ q e. Q. ) ->
        ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) ) $=
      ( wbr cv cnq wcel wa c1st cfv wex cnp sylan 19.42v bitr4i cltp co simprrr
      cltq c2nd cplq simplr simpld simprl ltrelpr brel cop prop elprnqu syl2anc
      simpll ltanqi wi simprd syl prcdnql mpd jca32 ltexprlemell anbi2i 3imtr4i
      eximi ex rexlimdvw ) CDUAIZGJZKLZMZVKFJZUDIZVNENOZLZMZVKVPLZFKVMVRVSVMVOV
      NKLZBJZCUEOZLZWAVNUFUBZDNOZLZMZMZMZMZBPZVLWCWAVKUFUBZWELZMZMZBPZVMVRMZVSW
      JWOBWJVLWCWMVJVLWIUGWJWCWFVMVOVTWGUCZUHZWJWLWDUDIZWMWJVOWAKLZWTVMVOWHUIWJ
      VJWCXAVJVLWIUPZWSVJCQLZWCXAVJXCDQLZCDQQUAUJUKZUHXCCNOZWBULQLWCXACUMWAWBXF
      UNRRUOVKVNWAUQUOWJXDWFWTWMURZWJVJXDXBVJXCXDXEUSUTWJWCWFWRUSXDWEDUEOZULQLW
      FXGDUMWDWLXHWEVARUOVBVCVGWQVMWIBPZMWKVRXIVMVRVOWHBPZMXIVQXJVOVQVTWGBPMXJA
      BCDEFHVDVTWGBSTVEVOWHBSTVEVMWIBSTVSVLWNBPMWPABCDEGHVDVLWNBSTVFVHVI $.

    $( The upper cut of our constructed difference is open.  Lemma for
       ~ ltexpri .  (Contributed by Jim Kingdon, 21-Dec-2019.) $)
    ltexprlemopu $p |- ( ( A <P B /\ r e. Q. /\ r e. ( 2nd ` C ) )
        -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) $=
      ( vs wbr cv cnq wcel cfv cltq wa wrex wex cnp adantr vt cltp c2nd c1st co
      cplq ltexprlemelu simprbi 19.42v anbi2i bitri cop ltrelpr brel simprd syl
      prnminu sylan adantrl ltdfpr biimpd mpcom ad2antrr simpld simplrr simprrl
      prop prltlu syl3an1 syl3anc simplll simprrr simplrl ltrelnq sotri syl2anc
      ltsonq rexlimddv wb elprnql elprnqu ad2ant2r ltexnqq mpbid simprr eqbrtrd
      wceq simprl ltanqg mpbird eqeltrd jca jca32 reximdva eximi sylbir rexcom4
      expr mpd sylibr rexbii sylib sylanr2 3impb ) CDUBJZFKZLMZXFEUCNZMZGKZXFOJ
      ZXJXHMZPZGLQZXIXEXGBKZCUDNZMZXOXFUFUEZDUCNZMZPZBRZXNXIXGYBABCDEFHUGUHXEXG
      YBPZPZXKXJLMZXQXOXJUFUEZXSMZPZBRPZPZGLQZXNYDXKYEYHPZPZBRZGLQZYKYDYMGLQZBR
      ZYOYDXEXGYAPZPZBRZYQYTXEYRBRZPYDXEYRBUIUUAYCXEXGYABUIUJUKYSYPBYSIKZXROJZY
      PIXSXEYAUUCIXSQZXGXEXTUUDXQXEDUDNZXSULSMZXTUUDXEDSMZUUFXECSMZUUGCDSSUBUMU
      NZUODVGUPZIXRXSUUEUQURUSUSYSUUBXSMZUUCPZPZYFUUBWGZGLQZYPUUMXOUUBOJZUUOUUM
      UAKZCUCNZMZUUQUUEMZPZUUPUALXEUVAUALQZYRUULUUHUUGPZXEUVBUUIUVCXEUVBCDUAUTV
      AVBVCUUMUUQLMZUVAPZPZXOUUQOJZUUQUUBOJZUUPUVFUUHXQUUSUVGUUMUUHUVEXEUUHYRUU
      LXEUUHUUGUUIVDVCZTUUMXQUVEUUMXQXTXEXGYAUULVEVDZTUUMUVDUUSUUTVFUUHXPUURULS
      MZXQUUSUVGCVGZXOUUQUURXPVHVIVJUVFXEUUTUUKUVHXEYRUULUVEVKUUMUVDUUSUUTVLYSU
      UKUUCUVEVMXEUUFUUTUUKUVHUUJUUQUUBXSUUEVHVIVJXOUUQUUBOLVQVNVOVPVRUUMXOLMZU
      UBLMZUUPUUOVSUUMUUHXQUVMUVIUVJUUHUVKXQUVMUVLXOUURXPVTURVPZXEUUKUVNYRUUCXE
      UUFUUKUVNUUJUUBXSUUEWAURWBGXOUUBWCVPWDUUMUUNYMGLUUMYEUUNYMUUMYEUUNPZPZXKY
      EYHUVQXKYFXROJZUVQYFUUBXROUUMYEUUNWEZYSUUKUUCUVPVEWFUVQYEXGUVMXKUVRVSUUMY
      EUUNWHZUUMXGUVPXEXGYAUULVMTUUMUVMUVPUVOTXJXFXOWIVJWJUVTUVQXQYGUUMXQUVPUVJ
      TUVQYFUUBXSUVSYSUUKUUCUVPVMWKWLWMWRWNWSVRWOWPYMGBLWQWTYNYJGLYNXKYLBRZPYJX
      KYLBUIUWAYIXKYEYHBUIUJUKXAXBXMYJGLXLYIXKABCDEGHUGUJXAWTXCXD $.

    $( The upper cut of our constructed difference is upper.  Lemma for
       ~ ltexpri .  (Contributed by Jim Kingdon, 21-Dec-2019.) $)
    ltexprlemupu $p |- ( ( A <P B /\ r e. Q. ) -> (
        E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ->
        r e. ( 2nd ` C ) ) ) $=
      ( wbr cv cnq wcel wa c2nd cfv wex cnp syl2anc 19.42v bitr4i simplr simpld
      cltp cltq c1st cplq simprrr simprl simpll simprrl adantl cop ltrelpr brel
      co prop syl elprnql sylan ltanqi wi simprd prcunqu mpd jca32 ltexprlemelu
      eximi anbi2i 3imtr4i ex rexlimdvw ) CDUCIZFJZKLZMZGJZVMUDIZVPENOZLZMZVMVR
      LZGKVOVTWAVOVQVPKLZBJZCUEOZLZWCVPUFUOZDNOZLZMZMZMZMZBPZVNWEWCVMUFUOZWGLZM
      ZMZBPZVOVTMZWAWLWQBWLVNWEWOVLVNWKUAWLWEWHVOVQWBWIUGZUBWLWFWNUDIZWOWLVQWCK
      LZXAVOVQWJUHWLVLWEXBVLVNWKUIZWKWEVOVQWBWEWHUJUKVLWDCNOZULQLZWEXBVLCQLZXEV
      LXFDQLZCDQQUCUMUNZUBCUPUQWCXDWDURUSRVPVMWCUTRWLXGWHXAWOVAZWLVLXGXCVLXFXGX
      HVBUQWLWEWHWTVBXGDUEOZWGULQLWHXIDUPWNWFWGXJVCUSRVDVEVGWSVOWKBPZMWMVTXKVOV
      TVQWJBPZMXKVSXLVQVSWBWIBPMXLABCDEGHVFWBWIBSTVHVQWJBSTVHVOWKBSTWAVNWPBPMWR
      ABCDEFHVFVNWPBSTVIVJVK $.

    $( Our constructed difference is rounded.  Lemma for ~ ltexpri .
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlemrnd $p |- ( A <P B -> ( A. q e. Q. ( q e. ( 1st ` C )
        <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )
        /\ A. r e. Q. ( r e. ( 2nd ` C )
        <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) ) ) $=
      ( wbr cv cfv wcel wa cnq wrex wb wral 3expia impbid ralrimiva cltp c1st
      cltq c2nd ltexprlemopl ltexprlemlol ltexprlemopu ltexprlemupu jca ) CDUAI
      ZGJZEUBKZLZUKFJZUCIZUNULLMFNOZPZGNQUNEUDKZLZUOUKURLMGNOZPZFNQUJUQGNUJUKNL
      ZMUMUPUJVBUMUPABCDEFGHUERABCDEFGHUFSTUJVAFNUJUNNLZMUSUTUJVCUSUTABCDEFGHUG
      RABCDEFGHUHSTUI $.

    $( Our constructed difference is disjoint.  Lemma for ~ ltexpri .
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlemdisj $p |- ( A <P B -> A. q e. Q.
        -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) ) $=
      ( vz wbr cv cfv wcel wa wn cnq cplq co cltq cnp wex vf cltp c1st c2nd wal
      vg ltsonq ltrelnq son2lpi cop ltrelpr brel simprd prop syl prltlu syl3an1
      vh 3expb adantlr adantrll adantrrl w3a ltanqg adantl simpld elprnqu sylan
      ad2ant2r adantrr elprnql adantrl simplr addcomnqg caovord2d mpbird 3com23
      wb wceq adantrlr adantrrr mtoi alrimivv ltexprlemell ltexprlemelu anbi12i
      jca ex anandi bitr4i baib eleq1 oveq1 eleq1d anbi12d cbvexv anbi2i bitrdi
      eeanv bitr4di notbid alnex albii bitri ralrimiva ) CDUBIZFJZEUCKLZXGEUDKL
      ZMZNZFOXFXGOLZMZXKBJZCUDKZLZXNXGPQZDUCKZLZMZHJZCUCKZLZYAXGPQZDUDKZLZMZMZN
      ZHUEZBUEZXMYIBHXMYHXNYARIZYAXNRIZMZXNYAROUGUHUIXMYHYNXMYHMZYLYMYOYLXQYDRI
      ZXMXTYFYPYCXMXSYFYPXPXFXSYFMYPXLXFXSYFYPXFXRYEUJSLZXSYFYPXFDSLZYQXFCSLZYR
      CDSSUBUKULZUMDUNUOXQYDYEXRUPUQUSUTVAVBYOUAUFURXNYAXGROPUAJZOLZUFJZOLZURJZ
      OLVCUUAUUCRIUUEUUAPQUUEUUCPQRIVRYOUUAUUCUUEVDVEXMXTXNOLZYGXFXPUUFXLXSXFYB
      XOUJSLZXPUUFXFYSUUGXFYSYRYTVFCUNUOZXNXOYBVGVHVIVJXMYGYAOLZXTXFYCUUIXLYFXF
      UUGYCUUIUUHYAXOYBVKVHVIVLXFXLYHVMUUBUUDMUUAUUCPQUUCUUAPQVSYOUUAUUCVNVEVOV
      PXMXTYCYMYFXMXPYCYMXSXFXPYCMYMXLXFXPYCYMXFYCXPYMXFUUGYCXPYMUUHYAXNXOYBUPU
      QVQUSUTVTWAWGWHWBWCXLXKYKVRXFXLXKYHHTZBTZNZYKXLXJUUKXLXJXTBTZYGHTZMZUUKXL
      XJUUMXNYBLZXQYELZMZBTZMZUUOXJXLUUTXJXLUUMMZXLUUSMZMXLUUTMXHUVAXIUVBABCDEF
      GWDABCDEFGWEWFXLUUMUUSWIWJWKUUSUUNUUMUURYGBHXNYAVSZUUPYCUUQYFXNYAYBWLUVCX
      QYDYEXNYAXGPWMWNWOWPWQWRXTYGBHWSWTXAYKUUJNZBUEUULYJUVDBYHHXBXCUUJBXBXDWTV
      EVPXE $.

    $( Our constructed difference is located.  Lemma for ~ ltexpri .
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlemloc $p |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r
        -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) ) $=
      ( vz wbr cv cltq wcel wo cnq wa cplq co wrex wb vw vf vg vh cltp c1st cfv
      c2nd wi wral wceq ltexnqi adantl cnp ltrelpr brel simpld cop prop prarloc
      sylan ad2ant2r simprd ad2antrr w3a ltanqg elprnqu adantlr elprnql simplrl
      ad2ant2rl addclnq syl2anc ltrelnq addcomnqg syl3anc oveq2d simplrr 3eqtrd
      caovord2d addassnqg breq2d bitrd biimpa prloc anassrs reximdva prml rexex
      wex 3syl r19.45mv adantr prmu r19.43 r19.9rmv orbi2d bitr4id ltexprlemell
      rexbidv ltexprlemelu eleq1 oveq1 eleq1d anbi12d cbvexv bitri orbi12i ibar
      ex anbi2i orbi12d syl df-rex bitr4di 3bitr4rd sylibrd rexlimddv ralrimivw
      mpd ) CDUEJZGKZFKZLJZYBEUFUGMZYCEUHUGMZNZUIZFOUJGOYAYHFOYAYDYGYAYDPZYBUAK
      ZQRZYCUKZYGUAOYDYLUAOSYAUAYBYCULUMYIYJOMZYLPZPZBKZIKZYJQRZLJZBCUHUGZSZICU
      FUGZSZYGYAYMUUCYDYLYACUNMZYMUUCYAUUDDUNMZCDUNUNUEUOUPZUQZUUDUUBYTURUNMZYM
      UUCCUSZYJYTUUBIBUTVAVAVBYOUUCYPYBQRZDUFUGZMZYQYCQRZDUHUGZMZNZBYTSZIUUBSZY
      GYOUUAUUQIUUBYOYQUUBMZPYSUUPBYTYOUUSYPYTMZYSUUPUIYOUUSUUTPZPZYSUUPUVBYSPU
      UEUUJUUMLJZUUPYOUUEUVAYSYAUUEYDYNYAUUDUUEUUFVCVDVDUVBYSUVCUVBYSUUJYRYBQRZ
      LJUVCUVBUBUCUDYPYRYBLOQUBKZOMZUCKZOMZUDKZOMVEUVEUVGLJUVIUVEQRUVIUVGQRLJTU
      VBUVEUVGUVIVFUMYIUUTYPOMZYNUUSYAUUTUVJYDYAUUDUUTUVJUUGUUDUUHUUTUVJUUIYPYT
      UUBVGVAVAVHVKUVBYQOMZYMYROMYIUUSUVKYNUUTYAUUSUVKYDYAUUDUUSUVKUUGUUDUUHUUS
      UVKUUIYQYTUUBVIVAVAVHVBZYIYMYLUVAVJZYQYJVLVMYIYBOMZYNUVAYDUVNYAYDUVNYCOMZ
      YBYCOOLVNUPZUQUMVDZUVFUVHPUVEUVGQRUVGUVEQRUKUVBUVEUVGVOUMVTUVBUVDUUMUUJLU
      VBUVDYQYJYBQRZQRZYQYKQRUUMUVBUVKYMUVNUVDUVSUKUVLUVMUVQYQYJYBWAVPUVBUVRYKY
      QQUVBYMUVNUVRYKUKUVMUVQYJYBVOVMVQUVBYKYCYQQYIYMYLUVAVRVQVSWBWCWDUUEUUKUUN
      URUNMUVCUUPDUSUUJUUMUUNUUKWEVAVMXJWFWGWGYIYGUURTYNYIUULBYTSZUUONZIUUBSZUV
      TUUOIUUBSZNZUURYGYAUWBUWDTZYDYAUUDUUSIWJZUWEUUGUUDUUHUUSIOSUWFUUIIYTUUBWH
      UUSIOWIWKUVTUUOIUUBWLWKWMYAUURUWBTYDYAUUQUWAIUUBYAUUDAKYTMZAWJZUUQUWATUUG
      UUDUUHUWGAOSUWHUUIAYTUUBWNUWGAOWIWKUWHUUQUVTUUOBYTSZNUWAUULUUOBYTWOUWHUUO
      UWIUVTUUOBAYTWPWQWRWKWTWMYDYGUWDTYAYDYGUUTUULPBWJZUUSUUOPZIWJZNZUWDYDYGUV
      NUWJPZUVOUWLPZNZUWMYEUWNYFUWOABCDEGHWSYFUVOYPUUBMZYPYCQRZUUNMZPZBWJZPUWOA
      BCDEFHXAUXAUWLUVOUWTUWKBIYPYQUKZUWQUUSUWSUUOYPYQUUBXBUXBUWRUUMUUNYPYQYCQX
      CXDXEXFXKXGXHYDUVNUVOPZUWMUWPTUVPUXCUWJUWNUWLUWOUVNUWJUWNTUVOUVNUWJXIWMUV
      OUWLUWOTUVNUVOUWLXIUMXLXMWRUVTUWJUWCUWLUULBYTXNUUOIUUBXNXHXOUMXPWMXQXTXRX
      JXSXS $.

    $( Our constructed difference is a positive real.  Lemma for ~ ltexpri .
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlempr $p |- ( A <P B -> C e. P. ) $=
      ( vq vr wbr cnq wcel cv c1st cfv wrex c2nd wa wb wral wex cltp cpw cxp wn
      cltq wo wi w3a cnp ltexprlemm cplq crab cop wss ssrab2 nqex elpw2 opelxpi
      co mpbir mp2an eqeltri jctil ltexprlemrnd ltexprlemdisj ltexprlemloc 3jca
      elnp1st2nd sylanbrc ) CDUAIZEJUBZVKUCZKZGLZEMNZKZGJOHLZEPNZKZHJOQZQVPVNVQ
      UEIZVQVOKQHJORGJSVSWAVNVRKZQGJORHJSQZVPWBQUDGJSZWAVPVSUFUGHJSGJSZUHEUIKVJ
      VTVMABCDEHGFUJEBLZCPNKWFALUKUSZDMNKQBTZAJULZWFCMNKWGDPNKQBTZAJULZUMZVLFWI
      VKKZWKVKKZWLVLKWMWIJUNWHAJUOWIJUPUQUTWNWKJUNWJAJUOWKJUPUQUTWIWKVKVKURVAVB
      VCVJWCWDWEABCDEHGFVDABCDEGFVEABCDEHGFVFVGEHGVHVI $.

    $( Lemma for ~ ltexpri .  One direction of our result for lower cuts.
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlemfl $p |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) ) $=
      ( vz vw vu vf wbr co cfv cv wcel cplq cnp wa cnq cltq vg vh cltp cpp c1st
      wceq wrex ltrelpr brel simpld ltexprlempr df-iplp addclnq genpelvl simprr
      wb syl2anc c2nd wex ltexprlemell biimpi ad2antlr adantl simprd w3a prltlu
      cop prop syl3an1 3adant2r 3adant3r ltanqg ltrelnq elprnql adantrl adantrr
      syl sylan 3adant3 addcomnqg caovord2d wi prcdnql 3adant2 sylbid mpd 3expa
      exlimddv eqeltrd expr rexlimdvva ssrdv ) CDUCKZGCEUDLUEMZDUEMZWMGNZWNOZWP
      HNZINZPLZUFZIEUEMZUGHCUEMZUGZWPWOOZWMCQOZEQOZWQXDUPWMXFDQOZCDQQUCUHUIZUJZ
      ABCDEFUKZJUAUBGBCEWPHIUDPGBUBUAJULUANZUBNZUMUNUQWMXAXEHIXCXBWMWRXCOZWSXBO
      ZRZXAXEWMXPXARZRZWPWTWOWMXPXAUOXRBNZCURMZOZXSWSPLZWOOZRZWTWOOZBXRWSSOZYDB
      USZXQYFYGRZWMXOYHXNXAXOYHABCDEIFUTVAVBVCVDWMXQYDYEWMXQYDVEZWRXSTKZYEWMXQY
      AYJYCWMXPYAYJXAWMXNYAYJXOWMXCXTVGQOZXNYAYJWMXFYKXJCVHVQWRXSXTXCVFVIVJVJVK
      ZYIYJWTYBTKZYEYIJUAUBWRXSWSTSPJNZSOZXLSOZXMSOVEYNXLTKXMYNPLXMXLPLTKUPYIYN
      XLXMVLVCYIWRSOZXSSOZYIYJYQYRRYLWRXSSSTVMUIVQZUJYIYQYRYSVDWMXQYFYDWMXPYFXA
      WMXOYFXNWMXBEURMZVGQOZXOYFWMXGUUAXKEVHVQWSYTXBVNVRVOVPVSYOYPRYNXLPLXLYNPL
      UFYIYNXLVTVCWAWMYDYMYEWBZXQWMYCUUBYAWMWODURMZVGQOZYCUUBWMXHUUDWMXFXHXIVDD
      VHVQYBWTUUCWOWCVRVOWDWEWFWGWHWIWJWKWEWL $.

    $( Lemma for ~ ltexpri .  Reverse direction of our result for lower cuts.
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlemrl $p |- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. C ) ) ) $=
      ( vv vz cfv co cv wcel wa cplq cnq cnp syl sylan syl2anc ad2antrr vw cltp
      vu vs vf vg vh wbr c1st cpp c2nd cop wrex ltrelpr brel prop prnmaddl cltq
      simprd simpld prarloc ad2ant2r wceq w3o simplll simplrl elprnql ad3antrrr
      adantr nqtri3or wb ltexnqq biimpa simprr wex simprl simpr simplrr prcunqu
      wi mpd addcomnqg adantl w3a addassnqg caov32d oveq1 eleq1d mpbird eqeltrd
      eleq1 anbi12d spcegv anabsi5 ltexprlemell sylanbrc ltexprlempr genpprecll
      df-iplp addclnq mp2and eqeltrrd rexlimddv ex ltaddpr ltprordil sseld sylc
      prcdnql sylsyld 3jaod rexlimdvva ssrdv ) CDUBUHZUADUIIZCEUJJZUIIZXNUAKZXO
      LZXRXQLZXNXSMZXRGKZNJZXOLZXTGOXNXODUKIZULPLZXSYDGOUMXNDPLZYFXNCPLZYGCDPPU
      BUNUOZUSDUPQZGXRYEXOUQRYAYBOLZYDMZMZUCKZHKZYBNJZURUHZUCCUKIZUMHCUIIZUMZXT
      XNYKYTXSYDXNYSYRULPLZYKYTXNYHUUAXNYHYGYIUTZCUPQZYBYRYSHUCVARVBYMYQXTHUCYS
      YRYMYOYSLZYNYRLZMZMZYQXTUUGYQMZYOXRURUHZYOXRVCZXRYOURUHZVDZXTUUHYOOLZXROL
      ZUULUUHXNUUDUUMUUGXNYQXNXSYLUUFVEZVIZYMUUDUUEYQVFZXNUUAUUDUUMUUCYOYRYSVGR
      SZYAUUNYLUUFYQXNYFXSUUNYJXRYEXOVGRVHZYOXRVJSUUHUUIXTUUJUUKUUHUUIXTUUHUUIM
      ZYOUDKZNJZXRVCZXTUDOUUHUUIUVCUDOUMZUUHUUMUUNUUIUVDVKUURUUSUDYOXRVLSVMUUTU
      VAOLZUVCMZMZUVBXRXQUUTUVEUVCVNZUVGUUDUVAEUIILZUVBXQLZUUHUUDUUIUVFUUQTUVGU
      VEBKZYRLZUVKUVANJZXOLZMZBVOZUVIUUTUVEUVCVPZUVGYPYRLZYPUVANJZXOLZUVPUUHUVR
      UUIUVFUUHYQUVRUUGYQVQUUHXNUUEYQUVRVTZUUPYMUUDUUEYQVRXNUUAUUEUWAUUCYPYNYRY
      SVSRSWATUVGUVSUVBYBNJZXOUVGUEUFUGYOYBUVAONUUHUUMUUIUVFUURTUUGYKYQUUIUVFYA
      YKYDUUFVFVHUVQUEKZOLZUFKZOLZMUWCUWENJZUWEUWCNJVCUVGUWCUWEWBWCUWDUWFUGKZOL
      WDUWGUWHNJUWCUWEUWHNJNJVCUVGUWCUWEUWHWEWCWFUVGUWBXOLZYDUUGYDYQUUIUVFYAYKY
      DUUFVRVHUVGUVCUWIYDVKUVHUVCUWBYCXOUVBXRYBNWGWHQWIWJUVRUVTUVPUVOUVRUVTMBYP
      YRUVKYPVCZUVLUVRUVNUVTUVKYPYRWKUWJUVMUVSXOUVKYPUVANWGWHWLWMWNSABCDEUDFWOW
      PUVGYHEPLZUUDUVIMUVJVTUUHYHUUIUVFUUHXNYHUUPUUBQTUUHUWKUUIUVFUUHXNUWKUUPAB
      CDEFWQZQTHUEGAUACEYOUVAUJNAUAGUEHWSUWCYBWTWRSXAXBXCXDUUHUUJXTUUHUUJMZXNXR
      YSLZXTUUGXNYQUUJUUOTUWMYOXRYSUUHUUJVQUUHUUDUUJUUQVIXBXNCXPUBUHZUWNXTVTXNY
      HUWKUWOUUBUWLCEXESUWOYSXQXRCXPXFXGQZXHXDUUHXNUUKUWNXTUUPUUHXNUUDUUKUWNVTZ
      UUPUUQXNUUAUUDUWQUUCYOXRYRYSXIRSUWPXJXKWAXDXLWAXCXDXM $.

    $( Lemma for ~ ltexpri .  One direction of our result for upper cuts.
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlemfu $p |- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) ) $=
      ( vz vw vu vf wbr co cfv cv wcel cplq cnp wa cnq cltq vg vh cltp cpp c2nd
      wceq wrex ltrelpr brel simpld ltexprlempr df-iplp addclnq genpelvu simprr
      wb syl2anc c1st wex ltexprlemelu biimpi ad2antlr simprd adantl w3a prltlu
      cop prop syl syl3an1 3com23 3adant2r ltanqg elprnql sylan adantrr 3adant2
      3adant3r elprnqu 3adant3 adantrl addcomnqg caovord2d prcunqu sylbid 3expa
      wi mpd exlimddv eqeltrd expr rexlimdvva ssrdv ) CDUCKZGCEUDLUEMZDUEMZWNGN
      ZWOOZWQHNZINZPLZUFZIEUEMZUGHCUEMZUGZWQWPOZWNCQOZEQOZWRXEUPWNXGDQOZCDQQUCU
      HUIZUJZABCDEFUKZJUAUBGBCEWQHIUDPGBUBUAJULUANZUBNZUMUNUQWNXBXFHIXDXCWNWSXD
      OZWTXCOZRZXBXFWNXQXBRZRZWQXAWPWNXQXBUOXSBNZCURMZOZXTWTPLZWPOZRZXAWPOZBXRY
      EBUSZWNXRWTSOZYGXPYHYGRZXOXBXPYIABCDEIFUTVAVBVCVDWNXRYEYFWNXRYEVEZXTWSTKZ
      YFWNXRYBYKYDWNXQYBYKXBWNXOYBYKXPWNYBXOYKWNYAXDVGQOZYBXOYKWNXGYLXKCVHVIZXT
      WSXDYAVFVJVKVLVLVRYJYKYCXATKZYFYJJUAUBXTWSWTTSPJNZSOZXMSOZXNSOVEYOXMTKXNY
      OPLXNXMPLTKUPYJYOXMXNVMVDWNYEXTSOZXRWNYBYRYDWNYLYBYRYMXTXDYAVNVOVPVQWNXRW
      SSOZYEWNXQYSXBWNXOYSXPWNYLXOYSYMWSXDYAVSVOVPVPVTWNXRYHYEWNXQYHXBWNXPYHXOW
      NEURMZXCVGQOZXPYHWNXHUUAXLEVHVIWTXCYTVSVOWAVPVTYPYQRYOXMPLXMYOPLUFYJYOXMW
      BVDWCWNYEYNYFWGZXRWNYDUUBYBWNDURMZWPVGQOZYDUUBWNXIUUDWNXGXIXJVCDVHVIXAYCW
      PUUCWDVOWAVQWEWHWFWIWJWKWLWEWM $.

    $( Lemma for ~ ltexpri .  One direction of our result for upper cuts.
       (Contributed by Jim Kingdon, 17-Dec-2019.) $)
    ltexprlemru $p |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) ) $=
      ( vv vz wbr cfv co cv wcel wa cltq cnp syl cplq cnq ad2antrr vw cltp c2nd
      vt vu vs vq vf vg vh cpp c1st wrex ltrelpr brel simprd prop prnminu sylan
      cop wceq simprr wb elprnqu ad2ant2r adantr ltexnqq syl2anc simpld prarloc
      mpbid adantlr simplll ltdfpr biimpd mpcom simplrl simprrl syl3an1 syl3anc
      prltlu simprrr ltsonq ltrelnq sotri rexlimddv ltexnqi oveq1 eqeq1d mpbird
      simplrr elprnql simprl addcomnqg adantl w3a addassnqg caov32d prcunqu mpd
      simpr wi wex eqeltrd eleq1 eleq1d anbi12d spcegv ltexprlemelu ltexprlempr
      anabsi5 sylanbrc df-iplp addclnq genppreclu mp2and eqeltrrd ex rexlimdvva
      ssrdv ) CDUBIZUADUCJZCEUKKUCJZYAUALZYBMZYDYCMZYAYENZUDLZYDOIZYFUDYBYADULJ
      ZYBUTPMZYEYIUDYBUMYADPMZYKYACPMZYLCDPPUBUNUOZUPDUQQZUDYDYBYJURUSYGYHYBMZY
      INZNZYHGLZRKZYDVAZYFGSYRYIUUAGSUMZYGYPYIVBYRYHSMZYDSMZYIUUBVCYAYPUUCYEYIY
      AYKYPUUCYOYHYBYJVDUSVEYGUUDYQYAYKYEUUDYOYDYBYJVDUSVFGYHYDVGVHVKYRYSSMZUUA
      NZNZUELZHLZYSRKZOIZUECUCJZUMHCULJZUMZYFYGUUEUUNYQUUAYAUUEUUNYEYAUUMUULUTP
      MZUUEUUNYAYMUUOYAYMYLYNVIZCUQQZYSUULUUMHUEVJUSVLVEUUGUUKYFHUEUUMUULUUGUUI
      UUMMZUUHUULMZNZNZUUKYFUVAUUKNZUUIUFLZRKZYHVAZYFUFSUVBUUIYHOIZUVEUFSUMUVBU
      GLZUULMZUVGYJMZNZUVFUGSUVBYAUVJUGSUMZUUGYAUUTUUKYAYEYQUUFVMZTZYMYLNZYAUVK
      YNUVNYAUVKCDUGVNVOVPQUVBUVGSMZUVJNZNZUUIUVGOIZUVGYHOIZUVFUVQYAUURUVHUVRUV
      AYAUUKUVPUUGYAUUTUVLVFZTZUVBUURUVPUUGUURUUSUUKVQZVFUVBUVOUVHUVIVRYAUUOUUR
      UVHUVRUUQUUIUVGUULUUMWAVSVTUVQYAUVIYPUVSUWAUVBUVOUVHUVIWBUVAYPUUKUVPUUGYP
      UUTYGYPYIUUFVQVFZTYAYKUVIYPUVSYOUVGYHYBYJWAVSVTUUIUVGYHOSWCWDWEVHWFUFUUIY
      HWGQUVBUVCSMZUVENZNZUVDYSRKZYDYCUWFUWGYDVAZUUAUVAUUAUUKUWEYRUUEUUAUUTWKTU
      WFUVEUWHUUAVCUVBUWDUVEVBZUVEUWGYTYDUVDYHYSRWHWIQWJUWFUUJUVCRKZUWGYCUWFUHU
      IUJUUIYSUVCSRUVAUUISMZUUKUWEYRUUTUWKUUFYGUURUWKYQUUSYAUURUWKYEYAUUOUURUWK
      UUQUUIUULUUMWLUSVLVEVLTUVAUUEUUKUWEYRUUEUUAUUTVQTUVBUWDUVEWMZUHLZSMZUILZS
      MZNUWMUWORKZUWOUWMRKVAUWFUWMUWOWNWOUWNUWPUJLZSMWPUWQUWRRKUWMUWOUWRRKRKVAU
      WFUWMUWOUWRWQWOWRUWFUUJUULMZUVCEUCJMZUWJYCMZUVBUWSUWEUVBUUKUWSUVAUUKXAUVB
      YAUUSUUKUWSXBZUVMUUGUURUUSUUKWKYAUUOUUSUXBUUQUUJUUHUULUUMWSUSVHWTVFUWFUWD
      BLZUUMMZUXCUVCRKZYBMZNZBXCZUWTUWLUWFUURUVDYBMZUXHUVBUURUWEUWBVFUWFUVDYHYB
      UWIUVAYPUUKUWEUWCTXDUURUXIUXHUXGUURUXINBUUIUUMUXCUUIVAZUXDUURUXFUXIUXCUUI
      UUMXEUXJUXEUVDYBUXCUUIUVCRWHXFXGXHXKVHABCDEUFFXIXLUWFYMEPMZUWSUWTNUXAXBUW
      FYAYMUVAYAUUKUWEUVTTZUUPQUWFYAUXKUXLABCDEFXJQHUHGAUACEUUJUVCUKRAUAGUHHXMU
      WMYSXNXOVHXPXQXQWFXRXSWTWFWFXRXT $.
  $}

  ${
    $d x y z u v A $.  $d x y z u v B $.
    $( Proposition 9-3.5(iv) of [Gleason] p. 123.  (Contributed by NM,
       13-May-1996.)  (Revised by Mario Carneiro, 14-Jun-2013.) $)
    ltexpri $p |- ( A <P B -> E. x e. P. ( A +P. x ) = B ) $=
      ( vz vy vu vv cv c2nd cfv wcel cplq co wa wex cnq crab cnp wceq eleq1d wb
      cltp wbr c1st cop cpp wrex simpr oveq12d anbi12d cbvexdva cbvrabv opeq12i
      simpl ltexprlempr ltexprlemfl ltexprlemrl ltexprlemfu ltexprlemru ltrelpr
      eqssd simpld addclpr syl2anc simprd preqlu mpbir2and oveq2 eqeq1d rspcev
      brel ) BCUBUCZDHZBIJZKZVMEHZLMZCUDJZKZNZDOZEPQZVMBUDJZKZVQCIJZKZNZDOZEPQZ
      UEZRKZBWJUFMZCSZBAHZUFMZCSZARUGFGBCWJWBGHZVNKZWQFHZLMZVRKZNZGOZFPQWIWQWCK
      ZWTWEKZNZGOZFPQWAXCEFPVPWSSZVTXBDGXHVMWQSZNZVOWRVSXAXJVMWQVNXHXIUHZTXJVQW
      TVRXJVMWQVPWSLXKXHXIUNUIZTUJUKULWHXGEFPXHWGXFDGXJWDXDWFXEXJVMWQWCXKTXJVQW
      TWEXLTUJUKULUMZUOZVLWMWLUDJZVRSZWLIJZWESZVLXOVRFGBCWJXMUPFGBCWJXMUQVAVLXQ
      WEFGBCWJXMURFGBCWJXMUSVAVLWLRKZCRKZWMXPXRNUAVLBRKZWKXSVLYAXTBCRRUBUTVKZVB
      XNBWJVCVDVLYAXTYBVEWLCVFVDVGWPWMAWJRWNWJSWOWLCWNWJBUFVHVIVJVD $.
  $}

  ${
    $d A f g h q r s t u v w $.  $d B f g h q r s t u v w $.
    $d C f g h q r s t u v w $.
    $( Lemma for ~ addcanprg .  (Contributed by Jim Kingdon, 25-Dec-2019.) $)
    addcanprleml $p |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\
        ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) ) $=
      ( vf vg vh cnp wcel co wceq cfv cplq cnq c2nd syl adantr syl2anc ad2antrr
      wa cv vv vw vt vu vr vs w3a cpp c1st wrex cop prop prnmaddl sylan adantlr
      3ad2antl2 simprl halfnqq simplll simp1d prarloc2 wn simp2d addclpr simplr
      elprnql simplrl addclnq simplrr df-iplp genpprecll imp syl22anc addcomnqg
      prdisj adantl addassnqg caov4d simprr oveq2d syl3anc 3eqtrd wi genppreclu
      simpr simp3d mp2and eqeltrrd ad3antrrr fveq2 eleq2d mpbird jca mtand cltq
      wb wbr wo ltaddnq prloc ecased rexlimddv ex ssrdv ) AGHZBGHZCGHZUGZABUHIZ
      ACUHIZJZSZUABUIKZCUIKZXLUATZXMHZXOXNHZXLXPSZXOUBTZLIZXMHZXQUBMXHXPYAUBMUJ
      ZXKXFXEXPYBXGXFXMBNKZUKGHZXPYBBULZUBXOYCXMUMUNUPUOXRXSMHZYASZSZUCTZYILIZX
      SJZXQUCMYHYFYKUCMUJXRYFYAUQUCXSUROYHYIMHZYKSZSZUDTZYILIZANKZHZXQUDAUIKZYN
      YSYQUKGHZYLYRUDYSUJYNXEYTYNXEXFXGYHXHYMXHXKXPYGUSZPUTAULZOYHYLYKUQZYIYQYS
      UDVAQYNYOYSHZYRSZSZXQXOYILIZCNKZHZUUFUUIYOXTLIZXIUIKZHZUUJXINKZHZSZUUFUUK
      UUMUKGHZUUJMHZUUOVBUUFXIGHZUUPUUFXEXFUURUUFXEXFXGYHXHYMUUEUUARZUTZUUFXEXF
      XGUUSVCZABVDQXIULOUUFYOMHZXTMHZUUQUUFYTUUDUVBUUFXEYTUUTUUBOYNUUDYRUQYOYQY
      SVFQZUUFXOMHZYFUVCUUFYDXPUVEUUFXFYDUVAYEOYHXPYMUUEXLXPYGVERXOYCXMVFQZYNYF
      UUEXRYFYAYMVGZPXOXSVHQYOXTVHQUUJUUMUUKVOQUUFUUISZUULUUNUVHXEXFUUDYAUULUUF
      XEUUIUUTPZUUFXFUUIUVAPYNUUDYRUUIVGYNYAUUEUUIXRYFYAYMVIRXEXFSUUDYASUULDEFU
      EUFABYOXTUHLUEUFFEDVJZETZFTZVHZVKVLVMUVHUUNUUJXJNKZHZUVHYPUUGLIZUUJUVNUVH
      UVPYOXOLIZYJLIUVQXSLIZUUJUVHDEFYOYIXOYIMLUUFUVBUUIUVDPZYNYLUUEUUIUUCRZUUF
      UVEUUIUVFPZDTZMHZUVKMHZSZUWBUVKLIZUVKUWBLIJUVHUWBUVKVNVPUWCUWDUVLMHUGUWFU
      VLLIUWBUVKUVLLILIJUVHUWBUVKUVLVQVPUVTUWEUWFMHUVHUWBUVKVHVPVRUVHYJXSUVQLYN
      YKUUEUUIYHYLYKVSRVTUVHUVBUVEYFUVRUUJJUVSUWAYNYFUUEUUIUVGRYOXOXSVQWAWBUVHY
      RUUIUVPUVNHZYNUUDYRUUIVIUUFUUIWEUVHXEXGYRUUISUWGWCUVIUUFXGUUIUUFXEXFXGUUS
      WFZPDEFUEUFACYPUUGUHLUVJUVMWDQWGWHUVHXKUUNUVOWPYNXKUUEUUIXLXKXPYGYMXHXKWE
      WIRXKUUMUVNUUJXIXJNWJWKOWLWMWNUUFXNUUHUKGHZXOUUGWOWQZXQUUIWRUUFXGUWIUWHCU
      LOUUFUVEYLUWJUVFYHYLYKUUEVGXOYIWSQXOUUGUUHXNWTQXAXBXBXBXCXD $.

    $( Lemma for ~ addcanprg .  (Contributed by Jim Kingdon, 25-Dec-2019.) $)
    addcanprlemu $p |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\
        ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) ) $=
      ( vr vw cnp wcel co wceq wa cfv cv cltq wbr c1st adantlr cplq cnq syl2anc
      syl vv vt vu vf vg vh vq vs w3a cpp c2nd wrex cop prnminu sylan 3ad2antl2
      prop simprr ltexnqi simprl halfnqq adantrr 3ad2antl1 wn simplll ad3antrrr
      prarloc2 simp1d addclpr elprnql simplrl addclnq elprnqu addassnqg syl3anc
      simp2d prdisj addcomnqg oveq2d adantr wi simp3d df-iplp genpprecll mp2and
      eqtrd simpr eqeltrd wb eleq2d ad7antlr mpbird genppreclu ancomsd ad2antrr
      fveq2 3adant3 imp adantrlr anassrs ad2ant2rl jca mtand wo ltaddnq simplrr
      breqtrd ltanqi prloc orcomd ecased rexlimddv ex ssrdv ) AFGZBFGZCFGZUIZAB
      UJHZACUJHZIZJZUABUKKZCUKKZYBUALZYCGZYEYDGZYBYFJZDLZYEMNZYGDYCXRYFYJDYCULZ
      YAXPXOYFYKXQXPBOKZYCUMFGZYFYKBUQZDYEYCYLUNUOUPPYHYIYCGZYJJZJZYIELZQHZYEIZ
      YGERYQYJYTERULYHYOYJUREYIYEUSTYQYRRGZYTJZJZUBLZUUDQHZYRIZYGUBRUUCUUAUUFUB
      RULYQUUAYTUTUBYRVATUUCUUDRGZUUFJZJZUCLZUUDQHZAUKKZGZYGUCAOKZYQUUHUUMUCUUN
      ULZUUBYHUUHUUOYPYBUUHUUOYFXRUUHUUOYAXOXPUUHUUOXQXOUUGUUOUUFXOUUNUULUMFGZU
      UGUUOAUQZUUDUULUUNUCVGUOVBVCPPPPUUIUUJUUNGZUUMJZJZYGYIUUDQHZCOKZGZUUTUVCU
      UKYIQHZXSOKZGZUVDXSUKKZGZJZUUTUVEUVGUMFGZUVDRGZUVIVDUUTXSFGZUVJUUTXOXPUVL
      UUTXOXPXQYQXRUUBUUHUUSXRYAYFYPVEVFZVHZUUTXOXPXQUVMVPZABVISXSUQTUUTUUKRGZY
      IRGZUVKUUTUUJRGZUUGUVPUUTUUPUURUVRUUTXOUUPUVNUUQTUUIUURUUMUTUUJUULUUNVJSZ
      UUCUUGUUFUUSVKZUUJUUDVLSUUTYMYOUVQUUTXPYMUVOYNTYQYOUUBUUHUUSYHYOYJUTVFYIY
      CYLVMSZUUKYIVLSUVDUVGUVEVQSUUTUVCJZUVFUVHUWBUVFUVDXTOKZGZUWBUVDUUJUVAQHZU
      WCUUTUVDUWEIUVCUUTUVDUUJUUDYIQHZQHZUWEUUTUVRUUGUVQUVDUWGIUVSUVTUWAUUJUUDY
      IVNVOUUTUUGUVQUWGUWEIUVTUWAUUGUVQJUWFUVAUUJQUUDYIVRVSSWFVTUWBUURUVCUWEUWC
      GZUUIUURUUMUVCVKUUTUVCWGUWBXOXQUURUVCJUWHWAUUTXOUVCUVNVTUUTXQUVCUUTXOXPXQ
      UVMWBZVTUDUEUFUGUHACUUJUVAUJQUGUHUFUEUDWCZUELUFLVLZWDSWEWHYAUVFUWDWIXRYFY
      PUUBUUHUUSUVCYAUVEUWCUVDXSXTOWPWJWKWLUUTUVHUVCUUCUUSUVHUUHYQUUMUVHUUBUURY
      HYPUUMUVHYHYOUUMUVHYJYHYOUUMJZUVHXRUWLUVHWAZYAYFXOXPUWMXQXOXPJUUMYOUVHUDU
      EUFUGUHABUUKYIUJQUWJUWKWMWNWQWOWRWSWTXAPVTXBXCUUTUVCYGUUTUVBYDUMFGZUVAYEM
      NUVCYGXDUUTXQUWNUWICUQTUUTUVAYSYEMUUTUUDYRMNUVQUVAYSMNUUTUUDUUEYRMUUTUUGU
      UGUUDUUEMNUVTUVTUUDUUDXESUUCUUGUUFUUSXFXGUWAUUDYRYIXHSUUCYTUUHUUSYQUUAYTU
      RWOXGUVAYEYDUVBXISXJXKXLXLXLXLXMXN $.
  $}

  $( Addition cancellation law for positive reals.  Proposition 9-3.5(vi) of
     [Gleason] p. 123.  (Contributed by Jim Kingdon, 24-Dec-2019.) $)
  addcanprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
                 ( ( A +P. B ) = ( A +P. C ) -> B = C ) ) $=
    ( cnp wcel w3a cpp co wceq wa c1st cfv addcanprleml wss 3ancomb eqcom sylbi
    c2nd eqssd addcanprlemu anbi12i jca wb preqlu 3adant1 adantr mpbird ex ) AD
    EZBDEZCDEZFZABGHZACGHZIZBCIZULUOJZUPBKLZCKLZIZBRLZCRLZIZJZUQUTVCUQURUSABCMU
    QUIUKUJFZUNUMIZJZUSURNULVEUOVFUIUJUKOUMUNPUAZACBMQSUQVAVBABCTUQVGVBVANVHACB
    TQSUBULUPVDUCZUOUJUKVIUIBCUDUEUFUGUH $.

  ${
    $d A x y $.  $d B x y $.
    $( The difference from ~ ltexpri is unique.  (Contributed by Jim Kingdon,
       7-Jul-2021.) $)
    lteupri $p |- ( A <P B -> E! x e. P. ( A +P. x ) = B ) $=
      ( vy cltp wbr cv cpp co wceq cnp wrex wrmo wreu ltexpri wcel ltrelpr wral
      wa ralrimiva brel simpld w3a eqtr3 addcanprg syl5 3expa oveq2 eqeq1d rmo4
      wi sylibr syl reu5 sylanbrc ) BCEFZBAGZHIZCJZAKLUSAKMZUSAKNABCOUPBKPZUTUP
      VACKPBCKKEQUAUBVAUSBDGZHIZCJZSZUQVBJZUKZDKRZAKRUTVAVHAKVAUQKPZSVGDKVAVIVB
      KPZVGVEURVCJVAVIVJUCVFURVCCUDBUQVBUEUFUGTTUSVDADKVFURVCCUQVBBHUHUIUJULUMU
      SAKUNUO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Lemma for Proposition 9-3.5(v) of [Gleason] p. 123.  (Contributed by NM,
       8-Apr-1996.) $)
    ltaprlem $p |- ( C e. P. -> ( A <P B ->
                    ( C +P. A ) <P ( C +P. B ) ) ) $=
      ( vx cltp wbr cnp wcel cpp co wa cv wceq wrex ltexpri adantr ltrelpr brel
      simplr syl2anc simpld addclpr simprl ltaddpr syl3anc oveq2 ad2antll eqtrd
      addassprg breqtrd rexlimddv expcom ) ABEFZCGHZCAIJZCBIJZEFZUMUNKZADLZIJZB
      MZUQDGUMVADGNUNDABOPURUSGHZVAKZKZUOUOUSIJZUPEVDUOGHZVBUOVEEFVDUNAGHZVFUMU
      NVCSZURVGVCUMVGUNUMVGBGHABGGEQRUAPPZCAUBTURVBVAUCZUOUSUDTVDVECUTIJZUPVDUN
      VGVBVEVKMVHVIVJCAUSUIUEVAVKUPMURVBUTBCIUFUGUHUJUKUL $.

    $( Ordering property of addition.  Proposition 9-3.5(v) of [Gleason]
       p. 123.  (Contributed by Jim Kingdon, 26-Dec-2019.) $)
    ltaprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
        ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) ) $=
      ( vx cnp wcel w3a cltp wbr cpp co wi ltaprlem 3ad2ant3 wa cv wceq ltexpri
      wrex syl2anc adantl simpl1 simprl ltaddpr addassprg 3com12 adantrr simprr
      3expa eqtr3d 3adantl2 simpl3 addclpr simpl2 addcanprg syl3anc mpd breqtrd
      adantlr rexlimddv ex impbid ) AEFZBEFZCEFZGZABHIZCAJKZCBJKZHIZVEVCVGVJLVD
      ABCMNVFVJVGVFVJOVHDPZJKZVIQZVGDEVJVMDESVFDVHVIRUAVFVKEFZVMOZVGVJVFVOOZAAV
      KJKZBHVPVCVNAVQHIVCVDVEVOUBZVFVNVMUCZAVKUDTVPCVQJKZVIQZVQBQZVCVEVOWAVDVCV
      EOZVOOVLVTVIWCVNVLVTQZVMVCVEVNWDVEVCVNWDCAVKUEUFUIUGWCVNVMUHUJUKVPVEVQEFZ
      VDWAWBLVCVDVEVOULVPVCVNWEVRVSAVKUMTVCVDVEVOUNCVQBUOUPUQURUSUTVAVB $.
  $}

  ${
    $d A f g h l u z $.  $d A y z $.  $d Q f g h l u z $.  $d Q y z $.
    $d X f g h z $.  $d X y z $.  $d f g h ph z $.
    prplnqu.x $e |- ( ph -> X e. P. ) $.
    prplnqu.q $e |- ( ph -> Q e. Q. ) $.
    prplnqu.sum $e |- ( ph -> A e.
      ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) $.
    $( Membership in the upper cut of a sum of a positive real and a fraction.
       (Contributed by Jim Kingdon, 16-Jun-2021.) $)
    prplnqu $p |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A ) $=
      ( co wceq cltq wbr cab cltp cpp cnp wcel syl2anc vz vf vg vh cv cplq c2nd
      cfv wrex cnq cop nqprlu ltaddpr addcomprg breqtrd wb addclpr c1st elprnqu
      syl prop sylan nqpru mpbid ltsopr ltrelpr ltnqpr mpbird ltexnqi wa adantr
      sotri simprl addcomnqg simprr eqtr3d abbidv opeq12d addnqpr ltaprg adantl
      breq2 breq1 w3a caovord2d oveq1 eqeq1d rspcev rexlimddv ) AEUAUEZUFKZDLZB
      UEZEUFKZDLZBFUGUHZUIZUAUJAEDMNZWLUAUJUIAWRGUEZEMNGOECUEZMNCOUKZWSDMNZGOZD
      WTMNZCOZUKZPNZAXAFXAQKZPNXHXFPNZXGAXAXAFQKZXHPAXARSZFRSZXAXJPNAEUJSZXKICE
      GULZUTZHXAFUMTAXKXLXJXHLXOHXAFUNTUOADXHUGUHZSZXIJADUJSZXHRSZXQXIUPAXSXQXR
      AXLXKXSHXOFXAUQTZJXSXHURUHZXPUKRSXQXRXHVADXPYAUSVBTZXTCDXHGVCTVDZXAXHXFPR
      VEVFVLTAXMXRWRXGUPIYBCEDGVGTVHUAEDVIUTAWJUJSZWLVJZVJZWJWPSZWJEUFKZDLZWQYF
      YGFWSWJMNGOWJWTMNCOUKZPNZYFYKXHYJXAQKZPNYFXHXFYLPAXIYEYCVKYFWSYHMNZGOZYHW
      TMNZCOZUKZXFYLYFYIYQXFLYFWKYHDYFXMYDWKYHLAXMYEIVKZAYDWLVMZEWJVNTAYDWLVOVP
      ZYIYNXCYPXEYIYMXBGYHDWSMWBVQYIYOXDCYHDWTMWCVQVRUTYFYDXMYQYLLYSYRCWJEGVSTV
      PUOYFUBUCUDFYJXAPRQUBUEZRSZUCUEZRSZUDUEZRSWDUUAUUCPNUUEUUAQKUUEUUCQKPNUPY
      FUUAUUCUUEVTWAAXLYEHVKZYFYDYJRSYSCWJGULUTYFXMXKYRXNUTUUBUUDVJUUAUUCQKUUCU
      UAQKLYFUUAUUCUNWAWEVHYFYDXLYGYKUPYSUUFCWJFGVCTVHYTWOYIBWJWPWMWJLWNYHDWMWJ
      EUFWFWGWHTWI $.
  $}

  ${
    $d A f g $.  $d B f g $.  $d C f g $.  $d D f g $.
    $( Strong extensionality of addition (ordering version).  This is similar
       to ~ addext but for positive reals and based on less-than rather than
       apartness.  (Contributed by Jim Kingdon, 17-Feb-2020.) $)
    addextpr $p |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ->
        ( ( A +P. B ) <P ( C +P. D ) -> ( A <P C \/ B <P D ) ) ) $=
      ( vf vg cnp wcel wa cpp co cltp wbr wo addclpr adantl syl3anc wb ltaprg
      cv wi adantr simprl simplr syl2anc wor w3a ltsopr sowlin mpan simpll wceq
      addcomprg caovcomd breq12d bitr4d simprr orbi12d sylibrd ) AGHZBGHZIZCGHZ
      DGHZIZIZABJKZCDJKZLMZVGCBJKZLMZVJVHLMZNZACLMZBDLMZNVFVGGHZVHGHZVJGHZVIVMU
      AZVBVPVEABOUBVEVQVBCDOPVFVCVAVRVBVCVDUCZUTVAVEUDZCBOUEGLUFVPVQVRUGVSUHGVG
      VHVJLUIUJQVFVNVKVOVLVFVNBAJKZBCJKZLMZVKVFUTVCVAVNWDRUTVAVEUKZVTWAACBSQVFV
      GWBVJWCLVFEFABGJETZGHFTZGHIWFWGJKWGWFJKULVFWFWGUMPZWEWAUNVFEFCBGJWHVTWAUN
      UOUPVFVAVDVCVOVLRWAVBVCVDUQVTBDCSQURUS $.
  $}

  ${
    $d q r x y z w v u f g h A $.  $d q r x y z w v u f g B $.  $d x y C $.
    recexpr.1 $e |- B = <.
      { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } ,
      { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. $.
    $( Membership in the lower cut of ` B ` .  Lemma for ~ recexpr .
       (Contributed by Jim Kingdon, 27-Dec-2019.) $)
    recexprlemell $p |- ( C e. ( 1st ` B ) <->
        E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) $=
      ( c1st cfv wcel cvv cv cltq wbr wa wex cnq ltrelnq brel adantr exlimiv
      crq c2nd elex simpld syl wceq breq1 anbi1d exbidv fveq2i nqex abssi ssexi
      cab cop simprd op1st eqtri elab2g pm5.21nii ) EDGHZIEJIZEBKZLMZVCUAHZCUBH
      IZNZBOZEVAUCVGVBBVDVBVFVDEPIZVBVDVIVCPIZEVCPPLQRUDEPUCUESTAKZVCLMZVFNZBOZ
      VHAEVAJVKEUFZVMVGBVOVLVDVFVKEVCLUGUHUIVAVNAUNZVCVKLMZVECGHIZNZBOZAUNZUOZG
      HVPDWBGFUJVPWAVPPUKVNAPVMVKPIZBVLWCVFVLWCVJVKVCPPLQRUDSTULUMWAPUKVTAPVSWC
      BVQWCVRVQVJWCVCVKPPLQRUPSTULUMUQURUSUT $.

    $( Membership in the upper cut of ` B ` .  Lemma for ~ recexpr .
       (Contributed by Jim Kingdon, 27-Dec-2019.) $)
    recexprlemelu $p |- ( C e. ( 2nd ` B ) <->
        E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) $=
      ( c2nd cfv wcel cvv cv cltq wbr wa wex cnq ltrelnq brel adantr exlimiv
      crq c1st elex simprd syl wceq breq2 anbi1d exbidv cab fveq2i simpld abssi
      cop nqex ssexi op2nd eqtri elab2g pm5.21nii ) EDGHZIEJIZBKZELMZVCUAHZCUBH
      IZNZBOZEVAUCVGVBBVDVBVFVDEPIZVBVDVCPIZVIVCEPPLQRUDEPUCUESTVCAKZLMZVFNZBOZ
      VHAEVAJVKEUFZVMVGBVOVLVDVFVKEVCLUGUHUIVAVKVCLMZVECGHIZNZBOZAUJZVNAUJZUNZG
      HWADWBGFUKVTWAVTPUOVSAPVRVKPIZBVPWCVQVPWCVJVKVCPPLQRULSTUMUPWAPUOVNAPVMWC
      BVLWCVFVLVJWCVCVKPPLQRUDSTUMUPUQURUSUT $.

    $( ` B ` is inhabited.  Lemma for ~ recexpr .  (Contributed by Jim Kingdon,
       27-Dec-2019.) $)
    recexprlemm $p |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\
        E. r e. Q. r e. ( 2nd ` B ) ) ) $=
      ( wcel cv cfv cnq wrex wa crq cltq wbr syl wi eleq1d c1q cnp c1st recclnq
      c2nd cop prop prmu nsmallnqq adantr wex recrecnq wceq breq2 fveq2 anbi12d
      anbi2d sylbird recexprlemell imbitrrdi expcomd imp reximdv rexlimiva 3syl
      spcegv mpd prml cplq co 1nq addclnq sylancl ltaddnq syl2anc recexprlemelu
      rspcev breq1 jca ) CUAHZFIZDUBJHZFKLZEIZDUDJHZEKLZVSCUBJZCUDJZUEUAHZAIZWG
      HZAKLWBCUFZAWGWFUGWJWBAKWIKHZWJMZVTWINJZOPZFKLZWBWLWPWJWLWNKHZWPWIUCZFWNU
      HQUIWMWOWAFKWLWJWOWARWLWOWJWAWLWOWJMZVTBIZOPZWTNJZWGHZMZBUJZWAWLWSWOWNNJZ
      WGHZMZXEWLXGWJWOWLXFWIWGWIUKZSUPWLWQXHXERWRXDXHBWNKWTWNULZXAWOXCXGWTWNVTO
      UMXJXBXFWGWTWNNUNZSUOVEQUQABCDVTGURUSUTVAVBVFVCVDVSWHWIWFHZAKLWEWKAWGWFVG
      XLWEAKWLXLMZWNWCOPZEKLZWEWLXOXLWLWNTVHVIZKHZWNXPOPZXOWLWQTKHZXQWRVJWNTVKV
      LWLWQXSXRWRVJWNTVMVLXNXREXPKWCXPWNOUMVPVNUIXMXNWDEKWLXLXNWDRWLXNXLWDWLXNX
      LMZWTWCOPZXBWFHZMZBUJZWDWLXTXNXFWFHZMZYDWLYEXLXNWLXFWIWFXISUPWLWQYFYDRWRY
      CYFBWNKXJYAXNYBYEWTWNWCOVQXJXBXFWFXKSUOVEQUQABCDWCGVOUSUTVAVBVFVCVDVR $.

    $( The lower cut of ` B ` is open.  Lemma for ~ recexpr .  (Contributed by
       Jim Kingdon, 28-Dec-2019.) $)
    recexprlemopl $p |- ( ( A e. P. /\ q e. Q. /\ q e. ( 1st ` B ) )
        -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) ) $=
      ( cv c1st cfv wcel cnp cltq wbr wa cnq wrex crq wex recexprlemell adantll
      c2nd ltbtwnnqq biimpi simpll 19.8a sylibr jca reximdv mpan9 exlimiv sylbi
      expcom 3ad2ant3 ) FHZDIJZKZCLKUOEHZMNZURUPKZOZEPQZUOPKUQUOBHZMNZVCRJCUBJK
      ZOZBSVBABCDUOGTVFVBBVDUSURVCMNZOZEPQZVEVBVDVIEUOVCUCUDVEVHVAEPVHVEVAVHVEO
      USUTUSVGVEUEVGVEUTUSVGVEOZVJBSUTVJBUFABCDURGTUGUAUHUMUIUJUKULUN $.

    $( The lower cut of ` B ` is lower.  Lemma for ~ recexpr .  (Contributed by
       Jim Kingdon, 28-Dec-2019.) $)
    recexprlemlol $p |- ( ( A e. P. /\ q e. Q. ) ->
        ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) -> q e. ( 1st ` B ) ) ) $=
      ( cv cltq wbr c1st cfv wcel wa cnq wrex wi cnp wex recexprlemell crq c2nd
      ltsonq ltrelnq sotri ex anim1d eximdv 3imtr4g imp rexlimivw a1i ) FHZEHZI
      JZUNDKLZMZNZEOPUMUPMZQCRMUMOMNURUSEOUOUQUSUOUNBHZIJZUTUALCUBLMZNZBSUMUTIJ
      ZVBNZBSUQUSUOVCVEBUOVAVDVBUOVAVDUMUNUTIOUCUDUEUFUGUHABCDUNGTABCDUMGTUIUJU
      KUL $.

    $( The upper cut of ` B ` is open.  Lemma for ~ recexpr .  (Contributed by
       Jim Kingdon, 28-Dec-2019.) $)
    recexprlemopu $p |- ( ( A e. P. /\ r e. Q. /\ r e. ( 2nd ` B ) )
        -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) $=
      ( cv c2nd cfv wcel cnp cltq wbr wa cnq wrex crq wex recexprlemelu adantlr
      c1st ltbtwnnqq biimpi simplr 19.8a sylibr jca reximdv mpan9 exlimiv sylbi
      expcom 3ad2ant3 ) EHZDIJZKZCLKFHZUOMNZURUPKZOZFPQZUOPKUQBHZUOMNZVCRJCUBJK
      ZOZBSVBABCDUOGTVFVBBVDVCURMNZUSOZFPQZVEVBVDVIFVCUOUCUDVEVHVAFPVHVEVAVHVEO
      USUTVGUSVEUEVGVEUTUSVGVEOZVJBSUTVJBUFABCDURGTUGUAUHUMUIUJUKULUN $.

    $( The upper cut of ` B ` is upper.  Lemma for ~ recexpr .  (Contributed by
       Jim Kingdon, 28-Dec-2019.) $)
    recexprlemupu $p |- ( ( A e. P. /\ r e. Q. ) -> (
        E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) ) ) $=
      ( cv cltq wbr c2nd cfv wcel wa cnq wrex wi cnp wex recexprlemelu crq c1st
      ltsonq ltrelnq sotri expcom anim1d eximdv 3imtr4g imp rexlimivw a1i ) FHZ
      EHZIJZUMDKLZMZNZFOPUNUPMZQCRMUNOMNURUSFOUOUQUSUOBHZUMIJZUTUALCUBLMZNZBSUT
      UNIJZVBNZBSUQUSUOVCVEBUOVAVDVBVAUOVDUTUMUNIOUCUDUEUFUGUHABCDUMGTABCDUNGTU
      IUJUKUL $.

    $( ` B ` is rounded.  Lemma for ~ recexpr .  (Contributed by Jim Kingdon,
       27-Dec-2019.) $)
    recexprlemrnd $p |- ( A e. P. -> ( A. q e. Q. ( q e. ( 1st ` B )
        <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
        /\ A. r e. Q. ( r e. ( 2nd ` B )
        <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) ) $=
      ( cnp wcel cv c1st cfv wa cnq wrex wb wral 3expia impbid ralrimiva cltq
      wbr c2nd recexprlemopl recexprlemlol recexprlemopu recexprlemupu jca ) CH
      IZFJZDKLZIZUJEJZUAUBZUMUKIMENOZPZFNQUMDUCLZIZUNUJUQIMFNOZPZENQUIUPFNUIUJN
      IZMULUOUIVAULUOABCDEFGUDRABCDEFGUESTUIUTENUIUMNIZMURUSUIVBURUSABCDEFGUFRA
      BCDEFGUGSTUH $.

    $( ` B ` is disjoint.  Lemma for ~ recexpr .  (Contributed by Jim Kingdon,
       27-Dec-2019.) $)
    recexprlemdisj $p |- ( A e. P. -> A. q e. Q.
        -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) ) $=
      ( vz cnp wcel cv c1st cfv wa wn cnq cltq wbr crq wal wex ltsonq wi simprr
      c2nd ltrelnq son2lpi simplr jca prltlu syl3an1 3expb sylan2 simprl simpll
      cop prop sotri syl2anc ltrnqi syl adantl ex adantr alrimivv recexprlemell
      mtoi recexprlemelu anbi12i breq1 fveq2 eleq1d anbi12d cbvexv anbi2i bitri
      wceq eeanv bitr4i notbii alnex albii sylibr ralrimiva ) CHIZEJZDKLIZWEDUD
      LIZMZNZEOWDWEOIZMZWEBJZPQZWLRLZCUDLZIZMZGJZWEPQZWRRLZCKLZIZMZMZNZGSZBSZWI
      WKXEBGWKXDWTWNPQZWNWTPQZMZWTWNPOUAUEUFWDXDXJUBWJWDXDXJWDXDMXHXIXDWDXBWPMX
      HXDXBWPWQWSXBUCWMWPXCUGUHWDXBWPXHWDXAWOUOHIXBWPXHCUPWTWNWOXAUIUJUKULXDXIW
      DXDWRWLPQZXIXDWSWMXKWQWSXBUMWMWPXCUNWRWEWLPOUAUEUQURWRWLUSUTVAUHVBVCVFVDW
      IXDGTZBTZNZXGWHXMWHWQBTZXCGTZMZXMWHXOWLWEPQZWNXAIZMZBTZMXQWFXOWGYAABCDWEF
      VEABCDWEFVGVHYAXPXOXTXCBGWLWRVPZXRWSXSXBWLWRWEPVIYBWNWTXAWLWRRVJVKVLVMVNV
      OWQXCBGVQVRVSXGXLNZBSXNXFYCBXDGVTWAXLBVTVOVRWBWC $.

    $( ` B ` is located.  Lemma for ~ recexpr .  (Contributed by Jim Kingdon,
       27-Dec-2019.) $)
    recexprlemloc $p |- ( A e. P. -> A. q e. Q. A. r e. Q. ( q <Q r
        -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) ) $=
      ( vu vv wcel cv cltq wbr cfv cnq wa crq sylan adantlr syl c1st c2nd wo wi
      cnp wral wrex prop prnmaxl simprr wceq elprnql ad2ant2r recrecnq breqtrrd
      cop wb recclnq ltrelnq brel adantl ad2antrr simprd ltrnqg syl2anc eqeltrd
      mpbird simprl breq1 fveq2 eleq1d spcegv recexprlemelu imbitrrdi rexlimddv
      wex anbi12d mp2and olcd prnminu elprnqu eqbrtrd simpld recexprlemell orcd
      breq2 ltrnqi prloc syl2an mpjaodan ex ralrimivw ) CUEJZFKZEKZLMZWNDUANJZW
      ODUBNJZUCZUDZEOUFFOWMWTEOWMWPWSWMWPPZWOQNZCUANZJZWSWNQNZCUBNZJZXAXDPZWRWQ
      XHXBHKZLMZWRHXCWMXDXJHXCUGZWPWMXCXFUPUEJZXDXKCUHZHXBXFXCUIRSXHXIXCJZXJPZP
      ZXIQNZWOLMZXQQNZXCJZWRXPXRXBXSLMZXPXBXIXSLXHXNXJUJXPXIOJZXSXIUKXAXOYBXDWM
      XNYBWPXJWMXLXNYBXMXIXFXCULRUMSZXIUNTZUOXPXQOJZWOOJZXRYAUQXPYBYEYCXIURTZXP
      WNOJZYFXAYHYFPZXDXOWPYIWMWNWOOOLUSUTVAZVBVCXQWOVDVEVGXPXSXIXCYDXHXNXJVHVF
      XPYEXRXTPZWRUDYGYEYKBKZWOLMZYLQNZXCJZPZBVPWRYPYKBXQOYLXQUKZYMXRYOXTYLXQWO
      LVIYQYNXSXCYLXQQVJVKVQVLABCDWOGVMVNTVRVOVSXAXGPZWQWRYRIKZXELMZWQIXFWMXGYT
      IXFUGZWPWMXLXGUUAXMIXEXFXCVTRSYRYSXFJZYTPZPZWNYSQNZLMZUUEQNZXFJZWQUUDUUFU
      UGXELMZUUDUUGYSXELUUDYSOJZUUGYSUKXAUUBUUJXGYTWMUUBUUJWPWMXLUUBUUJXMYSXFXC
      WARSUMZYSUNTZYRUUBYTUJWBUUDYHUUEOJZUUFUUIUQUUDYHYFXAYIXGUUCYJVBWCUUDUUJUU
      MUUKYSURTZWNUUEVDVEVGUUDUUGYSXFUULYRUUBYTVHVFUUDUUMUUFUUHPZWQUDUUNUUMUUOW
      NYLLMZYNXFJZPZBVPWQUURUUOBUUEOYLUUEUKZUUPUUFUUQUUHYLUUEWNLWFUUSYNUUGXFYLU
      UEQVJVKVQVLABCDWNGWDVNTVRVOWEWMXLXBXELMXDXGUCWPXMWNWOWGXBXEXFXCWHWIWJWKWL
      WL $.

    $( ` B ` is a positive real.  Lemma for ~ recexpr .  (Contributed by Jim
       Kingdon, 27-Dec-2019.) $)
    recexprlempr $p |- ( A e. P. -> B e. P. ) $=
      ( vq vr cnp wcel cnq cv c1st cfv wrex c2nd wa cltq wbr wb wral cpw cxp wn
      wo w3a recexprlemm crq wex cab cop wss ltrelnq brel simpld adantr exlimiv
      abssi nqex elpw2 mpbir opelxpi mp2an eqeltri recexprlemrnd recexprlemdisj
      wi simprd jctil recexprlemloc 3jca elnp1st2nd sylanbrc ) CHIZDJUAZVNUBZIZ
      FKZDLMZIZFJNGKZDOMZIZGJNPZPVSVQVTQRZVTVRIPGJNSFJTWBWDVQWAIZPFJNSGJTPZVSWE
      PUCFJTZWDVSWBUDVFGJTFJTZUEDHIVMWCVPABCDGFEUFDAKZBKZQRZWJUGMZCOMIZPZBUHZAU
      IZWJWIQRZWLCLMIZPZBUHZAUIZUJZVOEWPVNIZXAVNIZXBVOIXCWPJUKWOAJWNWIJIZBWKXEW
      MWKXEWJJIZWIWJJJQULUMUNUOUPUQWPJURUSUTXDXAJUKWTAJWSXEBWQXEWRWQXFXEWJWIJJQ
      ULUMVGUOUPUQXAJURUSUTWPXAVNVNVAVBVCVHVMWFWGWHABCDGFEVDABCDFEVEABCDGFEVIVJ
      DGFVKVL $.

    $( The lower cut of one is a subset of the lower cut of ` A .P. B ` .
       Lemma for ~ recexpr .  (Contributed by Jim Kingdon, 27-Dec-2019.) $)
    recexprlem1ssl $p |- ( A e. P. ->
        ( 1st ` 1P ) C_ ( 1st ` ( A .P. B ) ) ) $=
      ( vw vz vf wcel cfv co cv c1q cltq wbr wa crq cmq wceq cnq vv vg cnp c1st
      vh vu c1p cmp 1prl abeq2i c2nd wrex rec1nq ltrnqi eqbrtrrid prop prmuloc2
      cop sylan sylan2 prnmaxl ad2ant2r w3a elprnql 3adant3 simp1r ltrelnq brel
      simpld syl simp3 simp2r simpr wb ltmnqg adantl simprl simprd recclnq 3syl
      ad2antrr simplr mulcomnqg caovord2d mpdan recidnq eqtr3d oveqan12d adantr
      imbitrid simpll mulassnqg mulclnq mulidnq ax-mp eqtr3di recmulnqg syl2anc
      caov4d 1nq mpbird eleq1d biimprd wi wex breq2 fveq2 anbi12d recexprlemell
      spcegv imbitrrdi syl2and syl22anc 3ad2ant3 mpan2 syl3an2 3anidm12 3eqtr2d
      mpd oveq1d oveq2 eqeq2d rspcev 3expia reximdv recexprlempr df-imp sylibrd
      genpelvl rexlimddv ex biimtrid ssrdv ) CUCIZFUGUDJZCDUHKUDJZFLZYOIYQMNOZY
      NYQYPIZYRFYOFUIUJYNYRYSYNYRPZUALZYQQJZRKZCUKJZIZYSUACUDJZYRYNMUUBNOZUUEUA
      UUFULZYRMMQJUUBNUMYQMUNUOYNUUFUUDURUCIZUUGUUHCUPZUAUUBUUDUUFUQUSUTYTUUAUU
      FIZUUEPZPZUUAGLZNOZGUUFULZYSYNUUKUUPYRUUEYNUUIUUKUUPUUJGUUAUUDUUFVAUSVBUU
      MUUPYQUUNALZRKZSZADUDJZULZGUUFULZYSUUMUUOUVAGUUFYTUULUUOUVAYTUULUUOVCZUUN
      QJZYQRKZUUTIZYQUUNUVERKZSZUVAUVCUUATIZYQTIZUUOUUEUVFYTUULUVIUUOYNUUKUVIYR
      UUEYNUUIUUKUVIUUJUUAUUDUUFVDUSVBVEUVCYRUVJYNYRUULUUOVFYRUVJMTIZYQMTTNVGVH
      VIVJZYTUULUUOVKYTUUKUUEUUOVLUVIUVJPZUUOUUEPZPZUVNUVFUVMUVNVMUVOUUOUVEUUAQ
      JZYQRKZNOZUUEUVQQJZUUDIZUVFUUOUVDUVPNOUVOUVRUUAUUNUNUVOHUBUEUVDUVPYQNTRHL
      ZTIZUBLZTIZUELZTIVCZUWAUWCNOUWEUWARKUWEUWCRKNOVNUVOUWAUWCUWEVOVPUVOUUOUUN
      TIZUVDTIZUVMUUOUUEVQUUOUVIUWGUUAUUNTTNVGVHVRZUUNVSZVTUVIUVPTIZUVJUVNUUAVS
      ZWAZUVIUVJUVNWBZUWBUWDPZUWAUWCRKZUWCUWARKSUVOUWAUWCWCVPZWDWJUVOUVTUUEUVOU
      VSUUCUUDUVOUVSUUCSZUVQUUCRKZMSZUVOMMRKZUWSMUVOUVPUUARKZYQUUBRKZRKZUXAUWSU
      VMUXDUXASUVNUVIUVJUXBMUXCMRUVIUUAUVPRKZUXBMUVIUWKUXEUXBSUWLUUAUVPWCWEUUAW
      FWGYQWFWHWIUVOHUBUEUVPUUAYQUUBTRUWMUVIUVJUVNWKUWNUWQUWFUWPUWERKUWAUWCUWER
      KRKSUVOUWAUWCUWEWLVPUVOUVJUUBTIZUWNYQVSZVJUWOUWPTIUVOUWAUWCWMZVPWSWGUVKUX
      AMSWTMWNWOWPUVMUWRUWTVNZUVNUVMUVQTIZUUCTIZUXIUVIUWKUVJUXJUWLUVPYQWMUSZUVJ
      UVIUXFUXKUXGUUAUUBWMUTUVQUUCWQWRWIXAXBXCUVMUVRUVTPZUVFXDUVNUVMUXMUVEBLZNO
      ZUXNQJZUUDIZPZBXEZUVFUVMUXJUXMUXSXDUXLUXRUXMBUVQTUXNUVQSZUXOUVRUXQUVTUXNU
      VQUVENXFUXTUXPUVSUUDUXNUVQQXGXBXHXJVJABCDUVEEXIXKWIXLXSXMUVCUWGUVJUVHUUOY
      TUWGUULUWIXNUVLUWGUVJPYQMYQRKZUUNUVDRKZYQRKZUVGUVJYQUYASUWGUVJYQMRKZYQUYA
      YQWNUVJUVKUYDUYASWTYQMWCXOWGVPUWGUYCUYASUVJUWGUYBMYQRUUNWFXTWIUWGUVJUYCUV
      GSZUWGUWGUWHUVJUYEUWJUUNUVDYQWLXPXQXRWRUUSUVHAUVEUUTUUQUVESUURUVGYQUUQUVE
      UUNRYAYBYCWRYDYEYNYSUVBVNZYRUULYNDUCIUYFABCDEYFUFHUBBFCDYQGAUHRBFUBHUFYGU
      XHYIWEWAYHXSYJYKYLYM $.

    $( The upper cut of one is a subset of the upper cut of ` A .P. B ` .
       Lemma for ~ recexpr .  (Contributed by Jim Kingdon, 27-Dec-2019.) $)
    recexprlem1ssu $p |- ( A e. P. ->
        ( 2nd ` 1P ) C_ ( 2nd ` ( A .P. B ) ) ) $=
      ( vw vf vg wcel cfv co cv c1q cltq wbr wa cmq wceq cnq syl vv vz cnp c2nd
      vh vu c1p cmp 1pru abeq2i c1st wrex prop prmuloc2 sylan prnminu ad2ant2rl
      cop w3a simp3 simp2l elprnql ad2ant2r 3adant3 simp1r ltrelnq brel recclnq
      crq simprd mulassnqg syl3anc oveq2d 3eqtrd eleq1d mpbird ltrnqi wb ltmnqg
      recidnq mulidnq adantl mulclnq syl2anc mulcomnqg caovord2d imbitrid mpdan
      simpld eqtr3d oveqan12d caov4d ax-mp eqtr3di caovcld recmulnqg biimprd wi
      1nq wex breq1 fveq2 anbi12d spcegv recexprlemelu imbitrrdi syl2and mp2and
      mpan2 oveq1d adantr syl3an2 3anidm12 3eqtr2d eqeq2d rspcev 3expia reximdv
      oveq2 recexprlempr df-imp genpelvu ad2antrr sylibrd rexlimddv ex biimtrid
      mpd ssrdv ) CUCIZFUGUDJZCDUHKUDJZFLZYKIMYMNOZYJYMYLIZYNFYKFUIUJYJYNYOYJYN
      PZUALZYMQKZCUDJZIZYOUACUKJZYJUUAYSURUCIZYNYTUAUUAULCUMZUAYMYSUUAUNUOYPYQU
      UAIZYTPZPZUBLZYRNOZUBYSULZYOYJYTUUIYNUUDYJUUBYTUUIUUCUBYRYSUUAUPUOUQUUFUU
      IYMUUGALZQKZRZADUDJZULZUBYSULZYOUUFUUHUUNUBYSYPUUEUUHUUNYPUUEUUHUSZUUGVIJ
      ZYMQKZUUMIZYMUUGUURQKZRZUUNUUPUUHYRYMVIJZQKZUUAIZUUSYPUUEUUHUTZUUPUVDUUDY
      PUUDYTUUHVAUUPUVCYQUUAUUPUVCYQYMUVBQKZQKZYQMQKZYQUUPYQSIZYMSIZUVBSIZUVCUV
      GRYPUUEUVIUUHYJUUDUVIYNYTYJUUBUUDUVIUUCYQYSUUAVBUOVCVDZUUPYNUVJYJYNUUEUUH
      VEYNMSIZUVJMYMSSNVFVGVJTZUUPUVJUVKUVNYMVHTZYQYMUVBVKVLUUPUVFMYQQUUPUVJUVF
      MRUVNYMVTZTVMUUPUVIUVHYQRUVLYQWATVNVOVPUUPUUHYRVIJZYMQKZUURNOZUVDUVRVIJZU
      UAIZUUSUUHUVQUUQNOUUPUVSUUGYRVQUUPGHUEUVQUUQYMNSQGLZSIZHLZSIZUELZSIUSZUWB
      UWDNOUWFUWBQKUWFUWDQKNOVRUUPUWBUWDUWFVSWBUUPYRSIZUVQSIZUUPUVIUVJUWHUVLUVN
      YQYMWCWDZYRVHZTZUUPUUGSIZUUQSIZUUPUUHUWMUVEUUHUWMUWHUUGYRSSNVFVGWITZUUGVH
      ZTUVNUWCUWEPZUWBUWDQKZUWDUWBQKRUUPUWBUWDWEWBZWFWGUUPUWAUVDUUPUVTUVCUUAUUP
      UVTUVCRZUVRUVCQKZMRZUUPMMQKZUXAMUUPUVQYRQKZUVFQKZUXCUXAUUPUWHUVJUXEUXCRUW
      JUVNUWHUVJUXDMUVFMQUWHYRUVQQKZUXDMUWHUWIUXFUXDRUWKYRUVQWEWHYRVTWJUVPWKWDU
      UPGHUEUVQYRYMUVBSQUWLUWJUVNUWSUWGUWRUWFQKUWBUWDUWFQKQKRUUPUWBUWDUWFVKWBUV
      OUWQUWRSIUUPUWBUWDWCZWBZWLWJUVMUXCMRWSMWAWMWNUUPUVRSIZUVCSIUWTUXBVRUUPGHU
      VQYMSSSQUXHUWLUVNWOZUUPGHYRUVBSSSQUXHUWJUVOWOUVRUVCWPWDVPVOWQUUPUVSUWAPZB
      LZUURNOZUXLVIJZUUAIZPZBWTZUUSUUPUXIUXKUXQWRUXJUXPUXKBUVRSUXLUVRRZUXMUVSUX
      OUWAUXLUVRUURNXAUXRUXNUVTUUAUXLUVRVIXBVOXCXDTABCDUUREXEXFXGXHUUPUWMUVJUVA
      UWOUVNUWMUVJPYMMYMQKZUUGUUQQKZYMQKZUUTUVJYMUXSRUWMUVJYMMQKZYMUXSYMWAUVJUV
      MUYBUXSRWSYMMWEXIWJWBUWMUYAUXSRUVJUWMUXTMYMQUUGVTXJXKUWMUVJUYAUUTRZUWMUWM
      UWNUVJUYCUWPUUGUUQYMVKXLXMXNWDUULUVAAUURUUMUUJUURRUUKUUTYMUUJUURUUGQXSXOX
      PWDXQXRYJYOUUOVRZYNUUEYJDUCIUYDABCDEXTUFGHBFCDYMUBAUHQBFHGUFYAUXGYBWHYCYD
      YHYEYFYGYI $.

    $( The lower cut of ` A .P. B ` is a subset of the lower cut of one.  Lemma
       for ~ recexpr .  (Contributed by Jim Kingdon, 27-Dec-2019.) $)
    recexprlemss1l $p |- ( A e. P. ->
        ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) ) $=
      ( vw vz vq wcel co c1st cfv cv c1q cltq wbr cmq wi wa cnq vu cnp cmp wceq
      vf vg c1p wrex wb recexprlempr df-imp mulclnq genpelvl mpdan crq c2nd wex
      recexprlemell ltrelnq brel simprd prop elprnql sylan ltmnqi expcom adantr
      cop prltlu syl3an1 3expia mulcomnqg recidnq breq12d sylibd ancoms anim12d
      syl syld ltsonq sotri syl6 exp4b syl5 pm2.43d impd exlimdv biimtrid breq1
      biimprcd expimpd rexlimdvv sylbid 1prl abeq2i imbitrrdi ssrdv ) CUBIZFCDU
      CJKLZUGKLZWRFMZWSIZXANOPZXAWTIWRXBXAGMZHMZQJZUDZHDKLZUHGCKLZUHZXCWRDUBIXB
      XJUIABCDEUJUAUEUFBFCDXAGHUCQBFUFUEUAUKUEMUFMULUMUNWRXGXCGHXIXHWRXDXIIZXEX
      HIZXGXCRZWRXKSZXLXFNOPZXMXLXEBMZOPZXPUOLZCUPLZIZSZBUQXNXOABCDXEEURXNYAXOB
      XNXQXTXOXNXQXTXORZXQXPTIZXNXQYBRXQXETIYCXEXPTTOUSUTVAXNYCXQXTXOXNYCSZYAXF
      XDXPQJZOPZYENOPZSXOYDXQYFXTYGXNXQYFRZYCXNXDTIZYHWRXIXSVHUBIZXKYICVBZXDXSX
      IVCVDZXQYIYFXEXPXDVEVFVRVGYDXTXDXROPZYGXNXTYMRYCWRXKXTYMWRYJXKXTYMYKXDXRX
      SXIVIVJVKVGXNYIYCYMYGRZYLYCYIYNYCYISZYMXPXDQJZXPXRQJZOPZYGYCYMYRRYIYMYCYR
      XDXRXPVEVFVGYOYPYEYQNOXPXDVLYCYQNUDYIXPVMVGVNVOVPVDVSVQXFYENOTVTUSWAWBWCW
      DWEWFWGWHXGXCXOXAXFNOWIWJWBWKWLWMXCFWTFWNWOWPWQ $.

    $( The upper cut of ` A .P. B ` is a subset of the upper cut of one.  Lemma
       for ~ recexpr .  (Contributed by Jim Kingdon, 27-Dec-2019.) $)
    recexprlemss1u $p |- ( A e. P. ->
        ( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) ) $=
      ( vw vz vq wcel co c2nd cfv cv c1q cltq wbr cmq wi wa cnq vu cnp cmp wceq
      vf vg c1p wrex wb recexprlempr df-imp mulclnq genpelvu mpdan crq c1st wex
      recexprlemelu ltrelnq brel simpld prop elprnqu sylan ltmnqi expcom adantr
      cop syl prltlu syl3an1 3com23 3expia recidnq mulcomnqg sylibd ancoms syld
      breq12d anim12d sotri syl6 exp4b syl5 pm2.43d impd exlimdv biimtrid breq2
      ltsonq biimprcd expimpd rexlimdvv sylbid 1pru abeq2i imbitrrdi ssrdv ) CU
      BIZFCDUCJKLZUGKLZWSFMZWTIZNXBOPZXBXAIWSXCXBGMZHMZQJZUDZHDKLZUHGCKLZUHZXDW
      SDUBIXCXKUIABCDEUJUAUEUFBFCDXBGHUCQBFUFUEUAUKUEMUFMULUMUNWSXHXDGHXJXIWSXE
      XJIZXFXIIZXHXDRZWSXLSZXMNXGOPZXNXMBMZXFOPZXQUOLZCUPLZIZSZBUQXOXPABCDXFEUR
      XOYBXPBXOXRYAXPXOXRYAXPRZXRXQTIZXOXRYCRXRYDXFTIXQXFTTOUSUTVAXOYDXRYAXPXOY
      DSZYBXEXQQJZXGOPZNYFOPZSXPYEXRYGYAYHXOXRYGRZYDXOXETIZYIWSXTXJVHUBIZXLYJCV
      BZXEXJXTVCVDZXRYJYGXQXFXEVEVFVIVGYEYAXSXEOPZYHXOYAYNRYDWSXLYAYNWSYAXLYNWS
      YKYAXLYNYLXSXEXJXTVJVKVLVMVGXOYJYDYNYHRZYMYDYJYOYDYJSZYNXQXSQJZXQXEQJZOPZ
      YHYDYNYSRYJYNYDYSXSXEXQVEVFVGYPYQNYRYFOYDYQNUDYJXQVNVGXQXEVOVSVPVQVDVRVTY
      HYGXPNYFXGOTWJUSWAVQWBWCWDWEWFWGWHXHXDXPXBXGNOWIWKWBWLWMWNXDFXAFWOWPWQWR
      $.

    $( ` B ` is the reciprocal of ` A ` .  Lemma for ~ recexpr .  (Contributed
       by Jim Kingdon, 27-Dec-2019.) $)
    recexprlemex $p |- ( A e. P. -> ( A .P. B ) = 1P ) $=
      ( cnp wcel cmp c1p wceq c1st cfv c2nd recexprlemss1l recexprlem1ssl eqssd
      co recexprlemss1u recexprlem1ssu wa recexprlempr mulclpr mpdan 1pr preqlu
      wb sylancl mpbir2and ) CFGZCDHQZIJZUJKLZIKLZJZUJMLZIMLZJZUIULUMABCDENABCD
      EOPUIUOUPABCDERABCDESPUIUJFGZIFGUKUNUQTUFUIDFGURABCDEUACDUBUCUDUJIUEUGUH
      $.
  $}

  ${
    $d A x u v w z $.
    $( The reciprocal of a positive real exists.  Part of Proposition 9-3.7(v)
       of [Gleason] p. 124.  (Contributed by NM, 15-May-1996.)  (Revised by
       Mario Carneiro, 12-Jun-2013.) $)
    recexpr $p |- ( A e. P. -> E. x e. P. ( A .P. x ) = 1P ) $=
      ( vz vw vu vv cnp wcel cv cltq wbr crq cfv wa wex cab cmp co c1p wceq cop
      c2nd c1st wrex breq12 simpr fveq2d eleq1d anbi12d cbvexdva cbvabv breq12d
      simpl opeq12i recexprlempr recexprlemex oveq2 eqeq1d rspcev syl2anc ) BGH
      CIZDIZJKZVBLMZBUBMZHZNZDOZCPZVBVAJKZVDBUCMZHZNZDOZCPZUAZGHBVPQRZSTZBAIZQR
      ZSTZAGUDEFBVPVIEIZFIZJKZWCLMZVEHZNZFOZEPVOWCWBJKZWEVKHZNZFOZEPVHWHCEVAWBT
      ZVGWGDFWMVBWCTZNZVCWDVFWFVAWBVBWCJUEWOVDWEVEWOVBWCLWMWNUFZUGZUHUIUJUKVNWL
      CEWMVMWKDFWOVJWIVLWJWOVBWCVAWBJWPWMWNUMULWOVDWEVKWQUHUIUJUKUNZUOEFBVPWRUP
      WAVRAVPGVSVPTVTVQSVSVPBQUQURUSUT $.
  $}

  ${
    $d A f g h s t u v x $.  $d B f g h s t u v x $.
    $( Lemma for ~ aptipr .  (Contributed by Jim Kingdon, 28-Jan-2020.) $)
    aptiprleml $p |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) ->
        ( 1st ` A ) C_ ( 1st ` B ) ) $=
      ( vs vt vu vv cnp wcel wbr cfv cv wa cltq wrex sylan cplq co wceq syl2anc
      cnq vx vf vg vh cltp wn w3a c1st c2nd cop prop prnmaxl ad2ant2rl ad2antll
      wi ltexnqi simplr ad2antrr simprl prarloc2 adantr simpll ad3antrrr simprr
      w3o elprnql nqtri3or simplrl addclnq wb ltanqg adantl addcomnqg caovord2d
      simplrr eqeltrd prcdnql sylbid jctild eleq1 rspcev ltdfpr mpbird pm2.21dd
      imp anbi12d ex syl5ibcom 3jaod mpd rexlimddv expr 3impa ssrdv ) AGHZBGHZB
      AUEIZUFZUGUAAUHJZBUHJZWOWPWRUAKZWSHZXAWTHZUOWOWPLZWRXBXCXDWRXBLZLZXACKZMI
      ZXCCWSWOXBXHCWSNZWPWRWOWSAUIJZUJGHZXBXIAUKZCXAXJWSULOUMXFXGWSHZXHLZLZXADK
      ZPQZXGRZXCDTXHXRDTNXFXMDXAXGUPUNXOXPTHZXRLZLZEKZXPPQZBUIJZHZXCEWTYAWPXSYE
      EWTNZXFWPXNXTWOWPXEUQURZXOXSXRUSWPWTYDUJGHZXSYFBUKZXPYDWTEUTOSYAYBWTHZYEL
      ZLZYBXAMIZYBXARZXAYBMIZVEZXCYLYBTHZXATHZYPYLWPYJYQYAWPYKYGVAZYAYJYEUSZWPY
      HYJYQYIYBYDWTVFOSZYLWOXBYRXFWOXNXTYKWOWPXEVBVCZXFXBXNXTYKXDWRXBVDVCWOXKXB
      YRXLXAXJWSVFOSZYBXAVGSYLYMXCYNYOYLYMXCYLYMLZWQXCUUDWQFKZYDHZUUEWSHZLZFTNZ
      UUDYCTHZYEYCWSHZLZUUIUUDYQXSUUJYLYQYMUUAVAYLXSYMXOXSXRYKVHZVAYBXPVISYLYMU
      ULYLYMUUKYEYLYMYCXQMIZUUKYLUBUCUDYBXAXPMTPUBKZTHZUCKZTHZUDKZTHUGUUOUUQMIU
      USUUOPQUUSUUQPQMIVJYLUUOUUQUUSVKVLUUAUUCUUMUUPUURLUUOUUQPQUUQUUOPQRYLUUOU
      UQVMVLVNYLWOXQWSHZUUNUUKUOZUUBYLXQXGWSXOXSXRYKVOXOXMXTYKXFXMXHUSURVPWOXKU
      UTUVAXLXQYCXJWSVQOSVRYAYJYEVDVSWEUUHUULFYCTUUEYCRUUFYEUUGUUKUUEYCYDVTUUEY
      CWSVTWFWASYLWQUUIVJZYMYLWPWOUVBYSUUBBAFWBSVAWCXOWRXTYKYMXDWRXBXNVHVCWDWGY
      LYJYNXCYTYBXAWTVTWHYLWPYJYOXCUOZYSYTWPYHYJUVCYIYBXAYDWTVQOSWIWJWKWKWKWLWM
      WN $.

    $( Lemma for ~ aptipr .  (Contributed by Jim Kingdon, 28-Jan-2020.) $)
    aptiprlemu $p |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) ->
        ( 2nd ` B ) C_ ( 2nd ` A ) ) $=
      ( vs vt vu cnp wcel wbr cfv cv cltq wrex sylan cplq cnq syl2anc ad3antrrr
      wa co adantr vx vf vg vh cltp wn w3a c2nd c1st cop prop prnminu 3ad2antl2
      wceq simprr ltexnqi syl simpl1 ad2antrr simprl prarloc2 w3o simpr elprnqu
      simpl2 elprnql addclnq nqtri3or wi simplrr breq1 biimprd wb ltanqg adantl
      imp addcomnqg caovord2d mpbird prcdnql mpd rspe syl12anc simpll3 pm2.21dd
      ltdfpr ex eleq1 syl5ibrcom prcunqu 3jaod rexlimddv ssrdv ) AFGZBFGZBAUEHZ
      UFZUGZUABUHIZAUHIZWRUAJZWSGZXAWTGZWRXBRZCJZXAKHZXCCWSWOWNXBXFCWSLZWQWOBUI
      IZWSUJFGZXBXGBUKZCXAWSXHULMUMXDXEWSGZXFRZRZXEDJZNSZXAUNZXCDOXMXFXPDOLXDXK
      XFUODXEXAUPUQXMXNOGZXPRZRZEJZXNNSZWTGZXCEAUIIZXSWNXQYBEYCLZXDWNXLXRWNWOWQ
      XBURZUSXMXQXPUTZWNYCWTUJFGZXQYDAUKZXNWTYCEVAMPXSXTYCGZYBRZRZXAYAKHZXAYAUN
      ZYAXAKHZVBZXCYKXAOGZYAOGZYOYKWOXBYPXDWOXLXRYJWNWOWQXBVEZQZXDXBXLXRYJWRXBV
      CQWOXIXBYPXJXAWSXHVDMPYKXTOGZXQYQYKWNYIYTXDWNXLXRYJYEQZXSYIYBUTZWNYGYIYTY
      HXTWTYCVFMPZXSXQYJYFTZXTXNVGPXAYAVHPYKYLXCYMYNYKYLXCYKYLRZWPXCUUEWPXKXEYC
      GZRZCOLZUUEXEOGZXKUUFUUHXMUUIXRYJYLXMWOXKUUIXDWOXLYRTXDXKXFUTZWOXIXKUUIXJ
      XEWSXHVDMPQZXMXKXRYJYLUUJQUUEXEXTKHZUUFUUEUULXOYAKHZYKYLUUMYKXPYLUUMVIXMX
      QXPYJVJXPUUMYLXOXAYAKVKVLUQVPUUEUBUCUDXEXTXNKONUBJZOGZUCJZOGZUDJZOGUGUUNU
      UPKHUURUUNNSUURUUPNSKHVMUUEUUNUUPUURVNVOUUKYKYTYLUUCTYKXQYLUUDTUUOUUQRUUN
      UUPNSUUPUUNNSUNUUEUUNUUPVQVOVRVSUUEWNYIUULUUFVIZYKWNYLUUATZYKYIYLUUBTWNYG
      YIUUSYHXTXEWTYCVTMPWAUUGCOWBWCUUEWOWNWPUUHVMYKWOYLYSTUUTBACWFPVSXMWQXRYJY
      LWNWOWQXBXLWDQWEWGYKXCYMYBXSYIYBUOZXAYAWTWHWIYKWNYBYNXCVIZUUAUVAWNYGYBUVB
      YHXAYAWTYCWJMPWKWAWLWLWLWGWM $.

    $( Apartness of positive reals is tight.  (Contributed by Jim Kingdon,
       28-Jan-2020.) $)
    aptipr $p |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) )
        -> A = B ) $=
      ( cnp wcel cltp wbr wo wn w3a wceq c1st cfv c2nd simp1 aptiprleml syl3anc
      wss wa eqssd aptiprlemu simp2 ioran biimpi 3ad2ant3 simprd simpld 3adant3
      wb preqlu mpbir2and ) ACDZBCDZABEFZBAEFZGHZIZABJZAKLZBKLZJZAMLZBMLZJZUPUR
      USUPUKULUNHZURUSQUKULUONZUKULUOUAZUPUMHZVDUOUKVGVDRZULUOVHUMUNUBUCUDZUEZA
      BOPUPULUKVGUSURQVFVEUPVGVDVIUFZBAOPSUPVAVBUPULUKVGVAVBQVFVEVKBATPUPUKULVD
      VBVAQVEVFVJABTPSUKULUQUTVCRUHUOABUIUGUJ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Ordering property of multiplication.  (Contributed by Jim Kingdon,
       18-Feb-2020.) $)
    ltmprr $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
        ( ( C .P. A ) <P ( C .P. B ) -> A <P B ) ) $=
      ( vy vx cnp wcel cmp co cltp wbr wa cv c1p wceq syl2anc syl3anc mulcomprg
      cpp oveq1d w3a wrex recexpr 3ad2ant3 adantr ltexpri simplll simp1d simprl
      ad2antlr simplrl mulclpr ltaddpr simprr oveq2d distrprg mulassprg simplrr
      simp3d eqtrd 1pr mpan2 1idpr eqtr3d syl 3eqtr2d 3eqtr3d breqtrd rexlimddv
      simp2d ex ) AFGZBFGZCFGZUAZCAHIZCBHIZJKZABJKZVOVRLZCDMZHIZNOZVSDFVOWCDFUB
      ZVRVNVLWDVMDCUCUDUEVTWAFGZWCLZLZVPEMZSIZVQOZVSEFVRWJEFUBVOWFEVPVQUFUJWGWH
      FGZWJLZLZAAWAWHHIZSIZBJWMVLWNFGZAWOJKWMVLVMVNVOVRWFWLUGZUHZWMWEWKWPVTWEWC
      WLUKZWGWKWJUIZWAWHULPAWNUMPWMWAWIHIZWAVQHIZWOBWMWIVQWAHWGWKWJUNUOWMXAWAVP
      HIZWNSIZWACHIZAHIZWNSIWOWMWEVPFGZWKXAXDOWSWMVNVLXGWMVLVMVNWQUSZWRCAULPWTW
      AVPWHUPQWMXFXCWNSWMWEVNVLXFXCOWSXHWRWACAUQQTWMXFAWNSWMXFNAHIZAWMXENAHWMXE
      WBNWMWEVNXEWBOWSXHWACRPVTWEWCWLURUTZTWMVLXIAOWRVLANHIZXIAVLNFGZXKXIOVAANR
      VBAVCVDVEUTTVFWMXEBHIZNBHIZXBBWMXENBHXJTWMWEVNVMXMXBOWSXHWMVLVMVNWQVJZWAC
      BUQQWMVMXNBOXOVMBNHIZXNBVMXLXPXNOVABNRVBBVCVDVEVGVGVHVIVIVK $.
  $}

  ${
    $d A l u w x z $.
    $( For any positive real, there is an integer that is greater than it.
       This is also known as the "archimedean property".  The integer ` x ` is
       embedded into the reals as described at ~ nnprlu .  (Contributed by Jim
       Kingdon, 22-Apr-2020.) $)
    archpr $p |- ( A e. P. -> E. x e. N. A <P
        <. { l | l <Q [ <. x , 1o >. ] ~Q } ,
          { u | [ <. x , 1o >. ] ~Q <Q u } >. ) $=
      ( vz vw cnp wcel cv cfv cop cltq wbr cab cnpi wrex cnq c1st wa ad2antrr
      c2nd c1o ceq cec cltp prop syl archnqq ad2antrl simprl simprr simpr breq1
      vex ltnqex gtnqex op1st elab2 sylibr weq eleq1 anbi12d rspcev syl12anc wb
      prmu simplll nnprlu ad2antlr ltdfpr syl2anc mpbird reximdva mpd rexlimddv
      ex ) CGHZEIZCUAJZHZCDIZAIZUBKUCUDZLMZDNZWCBILMBNZKZUEMZAOPZEQVQCRJZVSKGHV
      TEQPCUFEVSWJVFUGVQVRQHZVTSZSZVRWCLMZAOPZWIWKWOVQVTAVRUHUIWMWNWHAOWMWBOHZS
      ZWNWHWQWNSZWHFIZVSHZWSWGRJZHZSZFQPZWRWKVTVRXAHZXDWMWKWPWNVQWKVTUJTWMVTWPW
      NVQWKVTUKTWRWNXEWQWNULWDWNDVRXAEUNWAVRWCLUMWEWFDWCUOBWCUPUQURUSXCVTXESFVR
      QFEUTWTVTXBXEWSVRVSVAWSVRXAVAVBVCVDWRVQWGGHZWHXDVEVQWLWPWNVGWPXFWMWNBWBDV
      HVICWGFVJVKVLVPVMVNVO $.
  $}

  ${
    $d L f g h $.  $d Q f g h l u $.  $d R f g h l u $.  $d S f g h l u $.
    $d f g h ph $.
    caucvgprlemcanl.l $e |- ( ph -> L e. P. ) $.
    caucvgprlemcanl.s $e |- ( ph -> S e. Q. ) $.
    caucvgprlemcanl.r $e |- ( ph -> R e. Q. ) $.
    caucvgprlemcanl.q $e |- ( ph -> Q e. Q. ) $.
    $( Lemma for ~ cauappcvgprlemladdrl .  Cancelling a term from both sides.
       (Contributed by Jim Kingdon, 15-Aug-2020.) $)
    caucvgprlemcanl $p |- ( ph -> (
        ( R +Q Q ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q Q ) } ,
          { u | ( S +Q Q ) <Q u } >. ) ) <->
        R e. ( 1st ` ( L +P. <. { l | l <Q S } ,
          { u | S <Q u } >. ) ) ) ) $=
      ( cltq wbr cab cpp co cltp wcel cnp syl2anc vf vg vh cv cop c1st cfv cplq
      w3a wb ltaprg adantl cnq nqprlu addclpr wa wceq addcomprg caovord2d nqprl
      addnqpr oveq2d breq12d addclnq addassprg syl3anc breq2d 3bitr4d 3bitr4rd
      syl ) AGUDZDLMGNDBUDZLMBNUEZFVKELMGNEVLLMBNUEZOPZQMZVMVKCLMGNCVLLMBNUEZOP
      ZVOVQOPZQMZDVOUFUGRZDCUHPZFVKECUHPZLMGNWCVLLMBNUEZOPZUFUGRZAUAUBUCVMVOVQQ
      SOUAUDZSRZUBUDZSRZUCUDZSRUIWGWIQMWKWGOPWKWIOPQMUJAWGWIWKUKULADUMRZVMSRJBD
      GUNVJAFSRZVNSRZVOSRZHAEUMRZWNIBEGUNVJZFVNUOTZACUMRZVQSRZKBCGUNVJZWHWJUPWG
      WIOPWIWGOPUQAWGWIURULUSAWLWOWAVPUJJWRBDVOGUTTAVKWBLMGNWBVLLMBNUEZWEQMZVRF
      VNVQOPZOPZQMWFVTAXBVRWEXEQAWLWSXBVRUQJKBDCGVATAWDXDFOAWPWSWDXDUQIKBECGVAT
      VBVCAWBUMRZWESRZWFXCUJAWLWSXFJKDCVDTAWMWDSRZXGHAWCUMRZXHAWPWSXIIKECVDTBWC
      GUNVJFWDUOTBWBWEGUTTAVSXEVRQAWMWNWTVSXEUQHWQXAFVNVQVEVFVGVHVI $.
  $}

  ${
    $d A p $.
    cauappcvgpr.f $e |- ( ph -> F : Q. --> Q. ) $.
    cauappcvgpr.app $e |- ( ph -> A. p e. Q. A. q e. Q. (
      ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\
      ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) ) $.
    cauappcvgpr.bnd $e |- ( ph -> A. p e. Q. A <Q ( F ` p ) ) $.
    ${
      $d L p q $.  $d ph p q $.
      cauappcvgpr.lim $e |- L = <.
        { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } ,
        { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
        >. $.

      ${
        $d L r s t f g h x y $.  $d A p s x $.  $d F l u p q r s t f g h x y $.
        $d ph r s t f g h x y $.

        $( Lemma for ~ cauappcvgpr .  The putative limit is inhabited.
           (Contributed by Jim Kingdon, 18-Jul-2020.) $)
        cauappcvgprlemm $p |- ( ph -> ( E. s e. Q. s e. ( 1st ` L )
            /\ E. r e. Q. r e. ( 2nd ` L ) ) ) $=
          ( wcel cnq cplq c1q cltq wbr cv c1st cfv wrex c2nd co wceq breq2d 1nq
          fveq2 a1i rspcdva ltrelnq brel simpld halfnqq wa simplr wral ad2antrr
          syl wi rspcv ad2antlr mpd wb breq1 adantl mpbird oveq2 breq12d rspcev
          syl2anc oveq1 breq1d rexbidv crab cop fveq2i rabex op1st eqtri elrab2
          sylanbrc ex reximdva ffvelcdmd addclnq ltaddnq id oveq12d breq2 op2nd
          nqex eleq1 jca ) AFUAZEUBUCZOZFPUDZGUAZEUEUCZOZGPUDZAWQWQQUFZCUGZFPUD
          ZWTACPOZXGACRDUCZSTZXHACIUAZDUCZSTZXJIPRXKRUGXLXICSXKRDUJUHMRPOZAUIUK
          ZULXJXHXIPOZCXIPPSUMUNUOVAFCUPVAAXFWSFPAWQPOZUQZXFWSXRXFUQZXQWQHUAZQU
          FZXTDUCZSTZHPUDZWSAXQXFURZXSXQXEWQDUCZSTZYDYEXSYGCYFSTZXSXMIPUSZYHAYI
          XQXFMUTXQYIYHVBAXFXMYHIWQPXKWQUGXLYFCSXKWQDUJUHVCVDVEXFYGYHVFXRXECYFS
          VGVHVIYCYGHWQPXTWQUGYAXEYBYFSXTWQWQQVJXTWQDUJVKVLVMJUAZXTQUFZYBSTZHPU
          DZYDJWQPWRYJWQUGZYLYCHPYNYKYAYBSYJWQXTQVNVOVPWRYMJPVQZYBXTQUFZBUAZSTZ
          HPUDZBPVQZVRZUBUCYOEUUAUBNVSYOYTYMJPWNVTZYSBPWNVTZWAWBWCWDWEWFVEAXIRQ
          UFZRQUFZPOZUUEXBOZXDAUUDPOZXNUUFAXPXNUUHAPPRDKXOWGXOXIRWHVMZXOUUDRWHV
          MZAUUFYPUUESTZHPUDZUUGUUJAXNUUDUUESTZUULXOAUUHXNUUMUUIXOUUDRWIVMUUKUU
          MHRPXTRUGZYPUUDUUESUUNYBXIXTRQXTRDUJUUNWJWKVOVLVMYSUULBUUEPXBYQUUEUGY
          RUUKHPYQUUEYPSWLVPXBUUAUEUCYTEUUAUENVSYOYTUUBUUCWMWBWCWDXCUUGGUUEPXAU
          UEXBWOVLVMWP $.

        $( Lemma for ~ cauappcvgpr .  The lower cut of the putative limit is
           open.  (Contributed by Jim Kingdon, 4-Aug-2020.) $)
        cauappcvgprlemopl $p |- ( ( ph /\ s e. ( 1st ` L ) ) ->
            E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) $=
          ( wa cplq cltq wbr cnq wrex vt cv c1st cfv wcel co wceq oveq1 rexbidv
          breq1d crab cop fveq2i rabex op1st eqtri elrab2 simprbi adantl simprr
          nqex ltbtwnnqq sylib simplrl simplbi ad3antlr ltaddnq syl2anc breqtrd
          addcomnqg simprrl ltsonq ltrelnq sotri wb simprl mpbid ad2antrr simpr
          ltexnqq breqtrrd simplr ltanqg syl3anc mpbird simprrr eqtr3d sylanbrc
          rspe jca ex reximdva mpd rexlimddv ) AFUBZEUCUDZUEZOZWOHUBZPUFZWSDUDZ
          QRZWOGUBZQRZXCWPUEZOZGSTZHSWQXBHSTZAWQWOSUEZXHJUBZWSPUFZXAQRZHSTZXHJW
          OSWPXJWOUGZXLXBHSXNXKWTXAQXJWOWSPUHUJUIWPXMJSUKZXAWSPUFBUBQRHSTZBSUKZ
          ULZUCUDXOEXRUCNUMXOXQXMJSVAUNXPBSVAUNUOUPZUQZURUSWRWSSUEZXBOZOZWTUAUB
          ZQRZYDXAQRZOZXGUASYCXBYGUASTWRYAXBUTUAWTXAVBVCYCYDSUEZYGOZOZWSXCPUFZY
          DUGZGSTZXGYJWSYDQRZYMYJWSWTQRYEYNYJWSWSWOPUFZWTQYJYAXIWSYOQRWRYAXBYIV
          DZWQXIAYBYIWQXIXHXTVEVFZWSWOVGVHYJYAXIYOWTUGYPYQWSWOVJVHVIYCYHYEYFVKZ
          WSWTYDQSVLVMVNVHYJYAYHYNYMVOYPYCYHYGVPGWSYDVTVHVQYJYLXFGSYJXCSUEZOZYL
          XFYTYLOZXDXEUUAXDYOYKQRZUUAYOYDYKQUUAYEYOYDQRYJYEYSYLYRVRUUAWTYOYDQUU
          AXIYAWTYOUGYJXIYSYLYQVRZYJYAYSYLYPVRZWOWSVJVHUJVQYTYLVSZWAUUAXIYSYAXD
          UUBVOUUCYJYSYLWBZUUDWOXCWSWCWDWEUUAYSXCWSPUFZXAQRZHSTZXEUUFUUAYAUUHUU
          IUUDUUAUUHYFYJYFYSYLYCYHYEYFWFVRUUAUUGYDXAQUUAYKUUGYDUUAYAYSYKUUGUGUU
          DUUFWSXCVJVHUUEWGUJWEUUHHSWIVHXMUUIJXCSWPXJXCUGZXLUUHHSUUJXKUUGXAQXJX
          CWSPUHUJUIXSUQWHWJWKWLWMWNWN $.

        $( Lemma for ~ cauappcvgpr .  The lower cut of the putative limit is
           lower.  (Contributed by Jim Kingdon, 4-Aug-2020.) $)
        cauappcvgprlemlol $p |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) ->
            s e. ( 1st ` L ) ) $=
          ( cv cltq wbr wcel cnq cplq vf vg vh c1st cfv w3a wrex ltrelnq simpld
          co brel 3ad2ant2 weq oveq1 breq1d rexbidv crab cop fveq2i rabex op1st
          nqex eqtri elrab2 simprbi 3ad2ant3 wa simpll2 wb ltanqg adantl simprd
          ad2antrr simplr addcomnqg caovord2d mpbid ltsonq sylancom ex reximdva
          wceq sotri mpd sylanbrc ) AFOZGOZPQZWGEUDUEZRZUFZWFSRZWFHOZTUJZWMDUEZ
          PQZHSUGZWFWIRWHAWLWJWHWLWGSRZWFWGSSPUHUKZUIULZWKWGWMTUJZWOPQZHSUGZWQW
          JAXCWHWJWRXCJOZWMTUJZWOPQZHSUGZXCJWGSWIJGUMZXFXBHSXHXEXAWOPXDWGWMTUNU
          OUPWIXGJSUQZWOWMTUJBOPQHSUGZBSUQZURZUDUEXIEXLUDNUSXIXKXGJSVBUTXJBSVBU
          TVAVCZVDVEVFWKXBWPHSWKWMSRZVGZXBWPXOXBWNXAPQZWPXOXBVGZWHXPAWHWJXNXBVH
          XQUAUBUCWFWGWMPSTUAOZSRZUBOZSRZUCOZSRUFXRXTPQYBXRTUJYBXTTUJPQVIXQXRXT
          YBVJVKWKWLXNXBWTVMWKWRXNXBWHAWRWJWHWLWRWSVLULVMWKXNXBVNXSYAVGXRXTTUJX
          TXRTUJWBXQXRXTVOVKVPVQWNXAWOPSVRUHWCVSVTWAWDXGWQJWFSWIJFUMZXFWPHSYCXE
          WNWOPXDWFWMTUNUOUPXMVDWE $.

        $( Lemma for ~ cauappcvgpr .  The upper cut of the putative limit is
           open.  (Contributed by Jim Kingdon, 4-Aug-2020.) $)
        cauappcvgprlemopu $p |- ( ( ph /\ r e. ( 2nd ` L ) ) ->
            E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) $=
          ( cv wcel wa cltq cnq wrex c2nd cfv cplq co wbr weq breq2 rexbidv cop
          crab fveq2i nqex rabex op2nd eqtri elrab2 simprbi adantl simprr sylib
          ltbtwnnqq simplr simplrl adantr simprl rspe syl2anc sylanbrc reximdva
          jca ex mpd rexlimddv ) AGOZEUAUBZPZQZHOZDUBZVRUCUDZVNRUEZFOZVNRUEZWBV
          OPZQZFSTZHSVPWAHSTZAVPVNSPWGVTBOZRUEZHSTZWGBVNSVOBGUFWIWAHSWHVNVTRUGU
          HVOJOVRUCUDVSRUEHSTZJSUJZWJBSUJZUIZUAUBWMEWNUANUKWLWMWKJSULUMWJBSULUM
          UNUOZUPUQURVQVRSPZWAQQZVTWBRUEZWCQZFSTZWFWQWAWTVQWPWAUSFVTVNVAUTWQWSW
          EFSWQWBSPZQZWSWEXBWSQZWCWDXBWRWCUSXCXAWRHSTZWDWQXAWSVBXCWPWRXDXBWPWSV
          QWPWAXAVCVDXBWRWCVEWRHSVFVGWJXDBWBSVOBFUFWIWRHSWHWBVTRUGUHWOUPVHVJVKV
          IVLVM $.

        $( Lemma for ~ cauappcvgpr .  The upper cut of the putative limit is
           upper.  (Contributed by Jim Kingdon, 4-Aug-2020.) $)
        cauappcvgprlemupu $p |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) ->
            r e. ( 2nd ` L ) ) $=
          ( cv cltq wbr wcel cnq wrex c2nd cfv w3a cplq ltrelnq simprd 3ad2ant2
          brel weq breq2 rexbidv crab cop fveq2i nqex rabex op2nd eqtri simprbi
          co elrab2 3ad2ant3 wi ltsonq sotri expcom reximdv mpd sylanbrc ) AFOZ
          GOZPQZVJEUAUBZRZUCZVKSRZHOZDUBZVQUDUTZVKPQZHSTZVKVMRVLAVPVNVLVJSRZVPV
          JVKSSPUEUHUFUGVOVSVJPQZHSTZWAVNAWDVLVNWBWDVSBOZPQZHSTZWDBVJSVMBFUIWFW
          CHSWEVJVSPUJUKVMJOVQUDUTVRPQHSTZJSULZWGBSULZUMZUAUBWJEWKUANUNWIWJWHJS
          UOUPWGBSUOUPUQURZVAUSVBVOWCVTHSVLAWCVTVCVNWCVLVTVSVJVKPSVDUEVEVFUGVGV
          HWGWABVKSVMBGUIWFVTHSWEVKVSPUJUKWLVAVI $.

        $( Lemma for ~ cauappcvgpr .  The putative limit is rounded.
           (Contributed by Jim Kingdon, 18-Jul-2020.) $)
        cauappcvgprlemrnd $p |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <->
              E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\
            A. r e. Q. ( r e. ( 2nd ` L ) <->
              E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) ) $=
          ( cv cfv wcel wa cnq wrex c1st cltq wb wral c2nd cauappcvgprlemopl ex
          cauappcvgprlemlol 3expib rexlimdvw impbid ralrimivw cauappcvgprlemopu
          wbr cauappcvgprlemupu jca ) AFOZEUAPZQZUQGOZUBUNZUTURQZRZGSTZUCZFSUDU
          TEUEPZQZVAUQVFQZRZFSTZUCZGSUDAVEFSAUSVDAUSVDABCDEFGHIJKLMNUFUGAVCUSGS
          AVAVBUSABCDEFGHIJKLMNUHUIUJUKULAVKGSAVGVJAVGVJABCDEFGHIJKLMNUMUGAVIVG
          FSAVAVHVGABCDEFGHIJKLMNUOUIUJUKULUP $.

        $( Lemma for ~ cauappcvgpr .  The putative limit is disjoint.
           (Contributed by Jim Kingdon, 18-Jul-2020.) $)
        cauappcvgprlemdisj $p |- ( ph -> A. s e. Q.
            -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) $=
          ( wcel wa cnq cplq co cltq wbr vf vg vh cv c1st cfv c2nd wn wrex wfal
          wral simpl ralimi syl adantr weq oveq1 breq1d rexbidv crab cop fveq2i
          nqex rabex op1st eqtri elrab2 simprbi oveq2 fveq2 breq12d sylib breq2
          cbvrexv anim12i reeanv sylibr adantl r19.29d2r simprl jca wi ad3antlr
          op2nd simplbi simplr addclnq syl2anc wf ad3antrrr ffvelcdmd simpr wor
          w3a ltsonq sotr mpan syl3anc syl5 simprr a1i jcad addcomnqg addassnqg
          wceq caov12d anbi1d sylibd ltanqg sylibrd so2nr pm2.21d rexlimdva mpd
          wb syld inegd ralrimivw ) AFUDZEUEUFZNZXSEUGUFZNZOZUHFPAYDAYDOZHUDZDU
          FZGUDZDUFZYFYHQRZQRZSTZXSYFQRZYGSTZYIYHQRZXSSTZOZOZGPUIZHPUIUJYEYLYQH
          GPPAYLGPUKZHPUKZYDAYLYIYGYJQRSTZOZGPUKZHPUKUUAKUUDYTHPUUCYLGPYLUUBULU
          MUMUNUOYDYQGPUIHPUIZAYDYNHPUIZYPGPUIZOUUEYAUUFYCUUGYAXSYHQRZYISTZGPUI
          ZUUFYAXSPNZUUJIUDZYHQRZYISTZGPUIZUUJIXSPXTIFUPZUUNUUIGPUUPUUMUUHYISUU
          LXSYHQUQURUSXTUUOIPUTZYOBUDZSTZGPUIZBPUTZVAZUEUFUUQEUVBUEMVBUUQUVAUUO
          IPVCVDZUUTBPVCVDZVEVFVGZVHUUIYNGHPGHUPUUHYMYIYGSYHYFXSQVIYHYFDVJVKVNV
          LYCUUKUUGUUTUUGBXSPYBBFUPUUSYPGPUURXSYOSVMUSYBUVBUGUFUVAEUVBUGMVBUUQU
          VAUVCUVDWDVFVGVHVOYNYPHGPPVPVQVRVSYEYSUJHPYEYFPNZOZYRUJGPUVGYHPNZOZYR
          XSYOSTZYPOZUJUVIYRYFXSQRZYFYOQRZSTZYPOZUVKUVIYRYMYKSTZYPOUVOUVIYRUVPY
          PYRYNYLOZUVIUVPYRYNYLYLYNYPVTYLYQULWAUVIYMPNZYGPNZYKPNZUVQUVPWBZUVIUU
          KUVFUVRYDUUKAUVFUVHYAUUKYCYAUUKUUJUVEWEUOWCZYEUVFUVHWFZXSYFWGWHUVIPPY
          FDAPPDWIYDUVFUVHJWJZUWCWKUVIYIPNZYJPNZUVTUVIPPYHDUWDUVGUVHWLZWKZUVIUV
          FUVHUWFUWCUWGYFYHWGWHYIYJWGWHPSWMZUVRUVSUVTWNUWAWOPYMYGYKSWPWQWRWSYRY
          PWBUVIYLYNYPWTXAXBUVIUVPUVNYPUVIYMUVLYKUVMSUVIUUKUVFYMUVLXEUWBUWCXSYF
          XCWHUVIUAUBUCYIYFYHPQUWHUWCUWGUAUDZPNZUBUDZPNZOUWJUWLQRZUWLUWJQRXEUVI
          UWJUWLXCVRUWKUWMUCUDZPNWNUWNUWOQRUWJUWLUWOQRQRXEUVIUWJUWLUWOXDVRXFVKX
          GXHUVIUVJUVNYPUVIUUKYOPNZUVFUVJUVNXOUWBUVIUWEUVHUWPUWHUWGYIYHWGWHZUWC
          XSYOYFXIWRXGXJUVIUVKUJUVIUUKUWPUVKUHZUWBUWQUWIUUKUWPOUWRWOPXSYOSXKWQW
          HXLXPXMXMXNXQXR $.

        $( Lemma for ~ cauappcvgpr .  The putative limit is located.
           (Contributed by Jim Kingdon, 18-Jul-2020.) $)
        cauappcvgprlemloc $p |- ( ph -> A. s e. Q. A. r e. Q.
            ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) $=
          ( cltq wbr wcel cnq cplq co vy vx vf vg vh cv c1st c2nd wo wi wa wceq
          wrex ltexnqi adantl subhalfnqq ad2antrl simprr simplrl adantr syl2anc
          cfv ltanqi simplrr breqtrd simprl addclnq wf ad4antr ffvelcdmd ltsonq
          wor w3a sowlin syl3anc mpd simpr addassnqg breq1d mpbird wb addcomnqg
          mpan ltanqg caovord2d oveq2 fveq2 breq12d rspcev oveq1 rexbidv fveq2i
          crab nqex rabex op1st eqtri elrab2 sylanbrc ex id oveq12d sylan breq2
          cop op2nd orim12d rexlimddv ralrimivva ) AFUFZGUFZOPZXJEUGVBZQZXKEUHV
          BZQZUIZUJFGRRAXJRQZXKRQZUKZUKZXLXQYAXLUKZXJUAUFZSTZXKULZXQUARXLYEUARU
          MYAUAXJXKUNUOYBYCRQZYEUKZUKZUBUFZYISTZYCOPZXQUBRYFYKUBRUMYBYEUBYCUPUQ
          YHYIRQZYKUKZUKZXJYJSTZYIDVBZYISTZOPZYQXKOPZUIZXQYNYOXKOPZYTYNYOYDXKOY
          NYKXRYOYDOPYHYLYKURYHXRYMYBXRYGAXRXSXLUSUTUTZYJYCXJVCVAYBYFYEYMVDVEYN
          YORQZXSYQRQZUUAYTUJZYNXRYJRQZUUCUUBYNYLYLUUFYHYLYKVFZUUGYIYIVGVAXJYJV
          GVAYHXSYMYBXSYGAXRXSXLVDUTUTZYNYPRQZYLUUDYNRRYIDARRDVHXTXLYGYMKVIUUGV
          JZUUGYPYIVGVAROVLUUCXSUUDVMUUEVKRYOXKYQOVNWCVOVPYNYRXNYSXPYNYRXNYNYRU
          KZXRXJHUFZSTZUULDVBZOPZHRUMZXNYNXRYRUUBUTZUUKYLXJYISTZYPOPZUUPYHYLYKY
          RUSZUUKUUSUURYISTZYQOPZUUKUVBYRYNYRVQUUKUVAYOYQOUUKXRYLYLUVAYOULUUQUU
          TUUTXJYIYIVRVOVSVTUUKUCUDUEUURYPYIORSUCUFZRQZUDUFZRQZUEUFZRQVMUVCUVEO
          PUVGUVCSTUVGUVESTOPWAUUKUVCUVEUVGWDUOUUKXRYLUURRQUUQUUTXJYIVGVAYNUUIY
          RUUJUTUUTUVDUVFUKUVCUVESTUVEUVCSTULUUKUVCUVEWBUOWEVTUUOUUSHYIRUULYIUL
          ZUUMUURUUNYPOUULYIXJSWFUULYIDWGZWHWIVAJUFZUULSTZUUNOPZHRUMZUUPJXJRXMU
          VJXJULZUVLUUOHRUVNUVKUUMUUNOUVJXJUULSWJVSWKXMUVMJRWMZUUNUULSTZBUFZOPZ
          HRUMZBRWMZXEZUGVBUVOEUWAUGNWLUVOUVTUVMJRWNWOZUVSBRWNWOZWPWQWRWSWTYNYS
          XPYNYSUKXSUVPXKOPZHRUMZXPYNXSYSUUHUTYNYLYSUWEUUGUWDYSHYIRUVHUVPYQXKOU
          VHUUNYPUULYISUVIUVHXAXBVSWIXCUVSUWEBXKRXOUVQXKULUVRUWDHRUVQXKUVPOXDWK
          XOUWAUHVBUVTEUWAUHNWLUVOUVTUWBUWCXFWQWRWSWTXGVPXHXHWTXI $.

        $( Lemma for ~ cauappcvgpr .  The putative limit is a positive real.
           (Contributed by Jim Kingdon, 20-Jun-2020.) $)
        cauappcvgprlemcl $p |- ( ph -> L e. P. ) $=
          ( vs vr cnq wcel cv wrex wa wral cpw cxp c1st cfv c2nd cltq wbr wb wn
          wo wi w3a cnp cauappcvgprlemm cplq co crab cop wss ssrab2 elpw2 mpbir
          nqex opelxpi mp2an eqeltri jctil cauappcvgprlemrnd cauappcvgprlemdisj
          cauappcvgprlemloc 3jca elnp1st2nd sylanbrc ) AEOUAZVNUBZPZMQZEUCUDZPZ
          MORNQZEUEUDZPZNORSZSVSVQVTUFUGZVTVRPSNORUHMOTWBWDVQWAPZSMORUHNOTSZVSW
          ESUIMOTZWDVSWBUJUKNOTMOTZULEUMPAWCVPABCDEMNFGHIJKLUNEHQFQZUOUPWIDUDZU
          FUGFORZHOUQZWJWIUOUPBQUFUGFORZBOUQZURZVOLWLVNPZWNVNPZWOVOPWPWLOUSWKHO
          UTWLOVCVAVBWQWNOUSWMBOUTWNOVCVAVBWLWNVNVNVDVEVFVGAWFWGWHABCDEMNFGHIJK
          LVHABCDEMFGHIJKLVIABCDEMNFGHIJKLVJVKENMVLVM $.
      $}

      ${
        $d F f g h l u p q r s t v $.  $d L f g h q r s t $.
        $d L f g h s t v x y $.  $d S f g h l q r s t u $.
        $d S f g h l s t u v x y $.  $d ph f g h q r s t v $.
        cauappcvgprlemladd.s $e |- ( ph -> S e. Q. ) $.
        $( Lemma for ~ cauappcvgprlemladd .  The forward subset relationship
           for the upper cut.  (Contributed by Jim Kingdon, 11-Jul-2020.) $)
        cauappcvgprlemladdfu $p |- ( ph ->
            ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_
            ( 2nd ` <.
            { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
            { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
            >. ) ) $=
          ( cltq wbr co cplq cnq wcel vr vs vt vf vg vh vx cab cop cpp c2nd cfv
          vy cv wrex crab wa cnp wb cauappcvgprlemcl nqprlu syl df-iplp addclnq
          genpelvu syl2anc biimpa breq2 rexbidv fveq2i rabex op2nd eqtri elrab2
          wceq nqex biimpi adantr adantl simpld vex ltnqex gtnqex elab2 ltrelnq
          brel sylbi simprd ad2antll eleq1 mpbird ad2antrr ad5antr wf ffvelcdmd
          simplr ltanqg syl3anc mpbid simpr addcomnqg caovord2d ltsonq breqtrrd
          w3a sotri simpllr ex reximdva mpd sylanbrc rexlimdvva ssrdv ) AUAFIUN
          ZDOPIUHZDBUNZOPZBUHZUIZUJQUKULZXNGUNZRQZYAEULZDRQOPGSUOZISUPZYCYARQZD
          RQZXPOPZGSUOZBSUPZUIUKULZAUAUNZXTTZYLYKTZAYMUQZYLUBUNZUCUNZRQZVOZUCXS
          UKULZUOUBFUKULZUOZYNAYMUUBAFURTXSURTZYMUUBUSABCEFGHIJKLMUTADSTZUUCNBD
          IVAVBUDUEUFUGUMFXSYLUBUCUJRUGUMUFUEUDVCUEUNZUFUNZVDVEVFVGYOYSYNUBUCUU
          AYTYOYPUUATZYQYTTZUQZUQZYSYNUUJYSUQZYLSTZYGYLOPZGSUOZYNUUKUULYRSTZUUK
          YPSTZYQSTZUUOUUKUUPYFYPOPZGSUOZUUJUUPUUSUQZYSUUIUUTYOUUGUUTUUHUUGUUTY
          FXPOPZGSUOZUUSBYPSUUAXPYPVOUVAUURGSXPYPYFOVHVIUUAYBYCOPGSUOZISUPZUVBB
          SUPZUIZUKULUVEFUVFUKMVJUVDUVEUVCISVPVKUVBBSVPVKVLVMVNVQVRVSVRZVTZUUJU
          UQYSUUHUUQYOUUGUUHUUDUUQUUHDYQOPZUUDUUQUQXQUVIBYQYTUCWAXPYQDOVHXOXRID
          WBBDWCVLWDZDYQSSOWEWFWGWHWIVRZYPYQVDVFYSUULUUOUSUUJYLYRSWJVSWKUUKUUSU
          UNUUKUUPUUSUVGWHUUKUURUUMGSUUKYASTZUQZUURUUMUVMUURUQZYGYRYLOUVNYGYFYQ
          RQZOPZUVOYROPZYGYROPUVNUVIUVPUUKUVIUVLUURUUJUVIYSUUHUVIYOUUGUUHUVIUVJ
          VQWIVRWLUVNUUDUUQYFSTZUVIUVPUSAUUDYMUUIYSUVLUURNWMUUKUUQUVLUURUVKWLZU
          VNYCSTUVLUVRUVNSSYAEASSEWNYMUUIYSUVLUURJWMUUKUVLUURWPZWOUVTYCYAVDVFZD
          YQYFWQWRWSUVNUURUVQUVMUURWTUVNUDUEUFYFYPYQOSRUDUNZSTZUUESTZUUFSTXEUWB
          UUEOPUUFUWBRQUUFUUERQOPUSUVNUWBUUEUUFWQVSUWAUUKUUPUVLUURUVHWLUVSUWCUW
          DUQUWBUUERQUUEUWBRQVOUVNUWBUUEXAVSXBWSYGUVOYROSXCWEXFVFUUJYSUVLUURXGX
          DXHXIXJYIUUNBYLSYKXPYLVOYHUUMGSXPYLYGOVHVIYEYJYDISVPVKYIBSVPVKVLVNXKX
          HXLXJXHXM $.

        $( Lemma for ~ cauappcvgprlemladd .  The forward subset relationship
           for the lower cut.  (Contributed by Jim Kingdon, 11-Jul-2020.) $)
        cauappcvgprlemladdfl $p |- ( ph ->
            ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_
            ( 1st ` <.
            { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
            { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
            >. ) ) $=
          ( cltq co cplq cnq wcel wa vr vs vt vf vg vh vx vy cv wbr cab cop cpp
          c1st cfv wrex crab cnp wb cauappcvgprlemcl nqprlu syl df-iplp addclnq
          genpelvl syl2anc biimpa oveq1 breq1d rexbidv fveq2i rabex op1st eqtri
          wceq nqex elrab2 biimpi ad2antrl adantr simpld vex breq1 ltnqex elab2
          gtnqex ad2antll ltrelnq eleq1 adantl mpbird simprd ad2antrr addcomnqg
          simplr w3a addassnqg caov32d simpr wi lt2addnq syl2anr mp2and adantlr
          brel eqbrtrrd ad3antlr ex reximdva mpd sylanbrc rexlimdvva ssrdv ) AU
          AFIUIZDOUJZIUKZDBUIZOUJBUKZULZUMPUNUOZXNGUIZQPZYAEUOZDQPZOUJZGRUPZIRU
          QZYCYAQPZDQPXQOUJGRUPZBRUQZULUNUOZAUAUIZXTSZYLYKSZAYMTZYLUBUIZUCUIZQP
          ZVOZUCXSUNUOZUPUBFUNUOZUPZYNAYMUUBAFURSXSURSZYMUUBUSABCEFGHIJKLMUTADR
          SZUUCNBDIVAVBUDUEUFUGUHFXSYLUBUCUMQUGUHUFUEUDVCUEUIZUFUIZVDVEVFVGYOYS
          YNUBUCUUAYTYOYPUUASZYQYTSZTTZYSYNUUIYSTZYLRSZYLYAQPZYDOUJZGRUPZYNUUJU
          UKYRRSZUUJYPRSZYQRSZUUOUUJUUPYPYAQPZYCOUJZGRUPZUUIUUPUUTTZYSUUGUVAYOU
          UHUUGUVAYBYCOUJZGRUPZUUTIYPRUUAXNYPVOZUVBUUSGRUVDYBUURYCOXNYPYAQVHVIV
          JUUAUVCIRUQZYHXQOUJGRUPZBRUQZULZUNUOUVEFUVHUNMVKUVEUVGUVCIRVPVLUVFBRV
          PVLVMVNVQVRVSVTZWAZUUJUUQUUDUUJYQDOUJZUUQUUDTZUUIUVKYSUUHUVKYOUUGUUHU
          VKXOUVKIYQYTUCWBXNYQDOWCXPXRIDWDBDWFVMWEVRWGZVTYQDRROWHXEVBZWAZYPYQVD
          VFYSUUKUUOUSUUIYLYRRWIWJWKUUJUUTUUNUUJUUPUUTUVIWLUUJUUSUUMGRUUJYARSZT
          ZUUSUUMUVQUUSTZUUMYRYAQPZYDOUJZUVRUURYQQPZUVSYDOUVRUDUEUFYPYAYQRQUUJU
          UPUVPUUSUVJWMUUJUVPUUSWOUUJUUQUVPUUSUVOWMUDUIZRSZUUERSZTUWBUUEQPZUUEU
          WBQPVOUVRUWBUUEWNWJUWCUWDUUFRSWPUWEUUFQPUWBUUEUUFQPQPVOUVRUWBUUEUUFWQ
          WJWRUUJUUSUWAYDOUJZUVPUUJUUSTUUSUVKUWFUUJUUSWSUUIUVKYSUUSUVMWMUUSUURR
          SYCRSTUVLUUSUVKTUWFWTUUJUURYCRROWHXEUVNUURYCYQDXAXBXCXDXFYSUUMUVTUSUU
          IUVPUUSYSUULUVSYDOYLYRYAQVHVIXGWKXHXIXJYFUUNIYLRYKXNYLVOZYEUUMGRUWGYB
          UULYDOXNYLYAQVHVIVJYGYJYFIRVPVLYIBRVPVLVMVQXKXHXLXJXHXM $.

        $( Lemma for ~ cauappcvgprlemladd .  The reverse subset relationship
           for the upper cut.  (Contributed by Jim Kingdon, 11-Jul-2020.) $)
        cauappcvgprlemladdru $p |- ( ph ->
            ( 2nd ` <.
            { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
            { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
            >. ) C_
            ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) $=
          ( cplq co cltq wbr cnq wcel vr vv vf vg vh cfv wrex crab cop c2nd cab
          cv cpp weq breq2 rexbidv nqex rabex op2nd elrab2 cauappcvgprlemladdfl
          wa c1st wn wss oveq2 fveq2 oveq1d breq12d cbvrexv a1i rabbiia oveq12d
          wb id breq1d opeq12i fveq2i sseqtrdi adantr ad2antrr simplr ffvelcdmd
          wceq wf simpr addassnqg syl3anc addclnq syl2anc sylancom eqeltrrd wor
          ltsonq so2nr addcomnqg adantl caov32d ltanqg caovord2d 3bitr2d biimpd
          mpan w3a wral wi oveq1 oveq2d breq2d anbi12d ralbidv rspcv mpd simpld
          sylc breqtrd jctird mtod nrexdv ffvelcdmda elrab3 mtbird op1st eleq2i
          rsp syl sylnibr ssneldd adantlr wo cnp cauappcvgprlemcl addclpr prloc
          nqprlu prop sylan orcomd ecased ex rexlimdva expimpd biimtrid ssrdv )
          AUAIULZGULZOPZUUFEUFZDOPZQRZGSUGZISUHZUUHUUFOPZDOPZBULZQRZGSUGZBSUHZU
          IZUJUFZFUUEDQRIUKDUUOQRBUKUIZUMPZUJUFZUAULZUUTTUVDSTZUUNUVDQRZGSUGZVB
          AUVDUVCTZUUQUVGBUVDSUUTBUAUNUUPUVFGSUUOUVDUUNQUOUPUULUURUUKISUQURUUQB
          SUQURUSUTAUVEUVGUVHAUVEVBZUVFUVHGSUVIUUFSTZVBZUVFUVHUVKUVFVBZUVHUUNUV
          BVCUFZTZUVKUVNVDZUVFAUVJUVOUVEAUVJVBZUVMUUEUBULZOPZUVQEUFZDOPZQRZUBSU
          GZISUHZUVSUVQOPZDOPZUUOQRZUBSUGZBSUHZUIZVCUFZUUNAUVMUWJVEUVJAUVMUUSVC
          UFUWJABCDEFGHIJKLMNVAUUSUWIVCUULUWCUURUWHUUKUWBISUUKUWBVNUUESTUUJUWAG
          UBSGUBUNZUUGUVRUUIUVTQUUFUVQUUEOVFUWKUUHUVSDOUUFUVQEVGZVHVIVJVKVLUUQU
          WGBSUUQUWGVNUUOSTUUPUWFGUBSUWKUUNUWEUUOQUWKUUMUWDDOUWKUUHUVSUUFUVQOUW
          LUWKVOVMVHVPVJVKVLVQVRVSVTUVPUUNUWCTZUUNUWJTUVPUWMUUNUVQOPZUVTQRZUBSU
          GZUVPUWOUBSUVPUVQSTZVBZUWOUUHUUFUVQOPZOPZUVSQRZUVSUWTQRZVBZUWRUWTSTZU
          VSSTZUXCVDZUWRUUMUVQOPZUWTSUWRUUHSTZUVJUWQUXGUWTWDUWRSSUUFEASSEWEUVJU
          WQJWAZAUVJUWQWBZWCZUXJUVPUWQWFZUUHUUFUVQWGWHZUVPUWQUUMSTZUXGSTUWRUXHU
          VJUXNUXKUXJUUHUUFWIZWJZUUMUVQWIWKZWLUWRSSUVQEUXIUXLWCZSQWMUXDUXEVBUXF
          WNSUWTUVSQWOXCWJUWRUWOUXAUXBUWRUWOUXAUWRUWOUXGDOPZUVTQRUXGUVSQRUXAUWR
          UWNUXSUVTQUWRUCUDUEUUMDUVQSOUXPADSTZUVJUWQNWAZUXLUCULZSTZUDULZSTZVBUY
          BUYDOPZUYDUYBOPWDUWRUYBUYDWPWQZUYCUYEUEULZSTXDZUYFUYHOPUYBUYDUYHOPOPW
          DUWRUYBUYDUYHWGWQWRVPUWRUCUDUEUXGUVSDQSOUYIUYBUYDQRUYHUYBOPUYHUYDOPQR
          VNUWRUYBUYDUYHWSWQUXQUXRUYAUYGWTUWRUXGUWTUVSQUXMVPXAXBUWRUVSUUHUVQUUF
          OPZOPZUWTQUWRUVSUYKQRZUUHUVSUYJOPZQRZUWRUYLUYNVBZGSXEZUVJUYOUWRHULZEU
          FZUUHUYQUUFOPZOPZQRZUUHUYRUYSOPZQRZVBZGSXEZHSXEZUYPAVUFUVJUWQKWAUWQVU
          FUYPXFUVPVUEUYPHUVQSHUBUNZVUDUYOGSVUGVUAUYLVUCUYNVUGUYRUVSUYTUYKQUYQU
          VQEVGZVUGUYSUYJUUHOUYQUVQUUFOXGZXHVIVUGVUBUYMUUHQVUGUYRUVSUYSUYJOVUHV
          UIVMXIXJXKXLWQXMUXJUYOGSYEXOXNUWRUYJUWSUUHOUWRUWQUVJUYJUWSWDUXLUXJUVQ
          UUFWPWJXHXPXQXRXSUVPUUNSTZUWMUWPVNUVPUXNUXTVUJAUVJUXHUXNASSUUFEJXTUXO
          WKAUXTUVJNVTUUMDWIWJUWBUWPIUUNSUUEUUNWDZUWAUWOUBSVUKUVRUWNUVTQUUEUUNU
          VQOXGVPUPYAYFYBUWJUWCUUNUWCUWHUWBISUQURUWGBSUQURYCYDYGYHYIVTUVLUVNUVH
          UVIUVFUVNUVHYJZUVJAUVFVULUVEAUVMUVCUIYKTZUVFVULAUVBYKTZVUMAFYKTUVAYKT
          ZVUNABCEFGHIJKLMYLAUXTVUONBDIYOYFFUVAYMWJUVBYPYFUUNUVDUVCUVMYNYQYIYIY
          RYSYTUUAUUBUUCUUD $.

        ${
          $d F f g h l p q r u b $.  $d L f g h b $.  $d S f g h l p u b $.
          $d b f g h p ph q r $.
          $( Lemma for ~ cauappcvgprlemladd .  The forward subset relationship
             for the lower cut.  (Contributed by Jim Kingdon, 11-Jul-2020.) $)
          cauappcvgprlemladdrl $p |- ( ph ->
              ( 1st ` <.
              { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
              { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
              >. ) C_
              ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) $=
            ( cplq co cltq wbr cnq wcel vr vb vf vg vh cv cfv wrex crab cop cab
            c1st cpp wa oveq1 breq1d rexbidv nqex rabex op1st elrab2 c2nd wn wf
            ad3antrrr ffvelcdmda simplr addclnq sylan syl2anc simpllr ffvelcdmd
            weq adantr wor ltsonq so2nr mpan wceq sylancom addassnqg syl3anc wb
            ltanqg adantl addcomnqg caovord2d oveq2d 3bitr4rd simpr eqtrd bitrd
            w3a biimpd fveq2 breq12d oveq12d breq2d anbi12d ralbidv rspcv mpan9
            wral sylc simprd breqtrd jctird nrexdv intnand oveq2 cbvralv ralbii
            rsp mtod sylibr cbvrexv a1i rabbiia id opeq12i cauappcvgprlemladdfu
            eqtri sseld breq2 op2nd imbitrdi cnp wo cauappcvgprlemcl nqprlu syl
            addclpr prop prloc ecased caucvgprlemcanl mpbid rexlimdva expimpd
            ex biimtrid ssrdv ) AUAIUFZGUFZOPZUUDEUGZDOPZQRZGSUHZISUIZUUFUUDOPZ
            DOPBUFZQRGSUHZBSUIZUJULUGZFUUCDQRIUKDUULQRBUKUJUMPULUGZUAUFZUUOTUUQ
            STZUUQUUDOPZUUGQRZGSUHZUNAUUQUUPTZUUIUVAIUUQSUUOIUAVMZUUHUUTGSUVCUU
            EUUSUUGQUUCUUQUUDOUOUPUQUUJUUNUUIISURUSUUMBSURUSUTVAAUURUVAUVBAUURU
            NZUUTUVBGSUVDUUDSTZUNZUUTUVBUVFUUTUNZUUSFUUCDUUDOPZQRIUKUVHUULQRBUK
            UJZUMPZULUGZTZUVBUVGUVLUUGUVJVBUGZTZUVGUVNUUGSTZUBUFZEUGZUVPOPZUVHO
            PZUUGQRZUBSUHZUNZUVGUWAUVOUVGUVTUBSUVGUVPSTZUNZUVTUVQUUDUVPOPZOPZUU
            FQRZUUFUWFQRZUNZUWDUWFSTZUUFSTZUWIVCZUWDUVQSTZUWESTZUWJUVGSSUVPEASS
            EVDZUURUVEUUTJVEZVFZUVGUVEUWCUWNUVDUVEUUTVGZUUDUVPVHVIUVQUWEVHVJUWD
            SSUUDEUVGUWOUWCUWPVNUVDUVEUUTUWCVKZVLZSQVOUWJUWKUNUWLVPSUWFUUFQVQVR
            VJUWDUVTUWGUWHUWDUVTUWGUWDUVTUVRUUDOPZUUFQRZUWGUWDUXADOPZUUGQRUVRUU
            DDOPZOPZUUGQRUXBUVTUWDUXCUXEUUGQUWDUVRSTZUVEDSTZUXCUXEVSUVGUWCUWMUX
            FUWQUVQUVPVHVTZUWSUVGUXGUWCAUXGUURUVEUUTNVEZVNZUVRUUDDWAWBUPUWDUCUD
            UEUXAUUFDQSOUCUFZSTZUDUFZSTZUEUFZSTWMUXKUXMQRUXOUXKOPUXOUXMOPQRWCUW
            DUXKUXMUXOWDWEUWDUXFUVEUXASTUXHUWSUVRUUDVHVJUWTUXJUXLUXNUNUXKUXMOPU
            XMUXKOPVSUWDUXKUXMWFWEWGUWDUVSUXEUUGQUWDUVHUXDUVROUWDUXGUVEUVHUXDVS
            UXJUWSDUUDWFVJWHUPWIUWDUXAUWFUUFQUWDUXAUVQUVPUUDOPZOPZUWFUWDUWMUWCU
            VEUXAUXQVSUWQUVGUWCWJZUWSUVQUVPUUDWAWBUWDUXPUWEUVQOUWDUWCUVEUXPUWEV
            SUXRUWSUVPUUDWFVJWHZWKUPWLWNUWDUUFUXQUWFQUWDUVQUUFUXPOPZQRZUUFUXQQR
            ZUWDUYAUYBUNZGSXCZUVEUYCUVGHUFZEUGZUUFUYEUUDOPZOPZQRZUUFUYFUYGOPZQR
            ZUNZGSXCZHSXCZUWCUYDAUYNUURUVEUUTKVEZUYMUYDHUVPSHUBVMZUYLUYCGSUYPUY
            IUYAUYKUYBUYPUYFUVQUYHUXTQUYEUVPEWOZUYPUYGUXPUUFOUYEUVPUUDOUOZWHWPU
            YPUYJUXQUUFQUYPUYFUVQUYGUXPOUYQUYRWQWRWSWTXAXBUWSUYCGSXMXDXEUXSXFXG
            XNXHXIUVGUVNUUGUUCUVPOPZUVQUVHOPQRUBSUHZISUIZUVSUULQRZUBSUHZBSUIZUJ
            VBUGZTUWBUVGUVMVUEUUGUVGBCUVHEFUBHIUWPUVGUYNUYFUVQUYEUVPOPZOPZQRZUV
            QUYFVUFOPZQRZUNZUBSXCZHSXCUYOVULUYMHSVUKUYLUBGSUBGVMZVUHUYIVUJUYKVU
            MVUGUYHUYFQVUMUVQUUFVUFUYGOUVPUUDEWOZUVPUUDUYEOXJZWQWRVUMUVQUUFVUIU
            YJQVUNVUMVUFUYGUYFOVUOWHWPWSXKXLXOACUYFQRHSXCUURUVEUUTLVEFUUEUUFQRZ
            GSUHZISUIZUUKUULQRZGSUHZBSUIZUJUYSUVQQRZUBSUHZISUIZUVRUULQRZUBSUHZB
            SUIZUJMVURVVDVVAVVGVUQVVCISVUQVVCWCUUCSTVUPVVBGUBSGUBVMZUUEUYSUUFUV
            QQUUDUVPUUCOXJUUDUVPEWOZWPXPXQXRVUTVVFBSVUTVVFWCUULSTVUSVVEGUBSVVHU
            UKUVRUULQVVHUUFUVQUUDUVPOVVIVVHXSWQUPXPXQXRXTYBUVGUXGUVEUVHSTZUXIUW
            RDUUDVHVJZYAYCVUCUWABUUGSVUEUULUUGVSVUBUVTUBSUULUUGUVSQYDUQVUAVUDUY
            TISURUSVUCBSURUSYEVAYFXNUVFUUTUVJYGTZUVLUVNYHZUVGFYGTZUVIYGTZVVLAVV
            NUURUVEUUTABCEFGHIJKLMYIVEZUVGVVJVVOVVKBUVHIYJYKFUVIYLVJVVLUVKUVMUJ
            YGTUUTVVMUVJYMUUSUUGUVMUVKYNVIVTYOUVGBUUDUUQDFIVVPUXIAUURUVEUUTVKUW
            RYPYQYTYRYSUUAUUB $.
        $}

        ${
          $d S p $.

          $( Lemma for ~ cauappcvgpr .  This takes ` L ` and offsets it by the
             positive fraction ` S ` .  (Contributed by Jim Kingdon,
             23-Jun-2020.) $)
          cauappcvgprlemladd $p |- ( ph ->
            ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <.
            { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
            { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
            >. ) $=
            ( co cfv cnq wceq wcel syl cltq wbr cab cop cpp cplq wrex crab c1st
            c2nd cauappcvgprlemladdfl cauappcvgprlemladdrl cauappcvgprlemladdfu
            cv eqssd cauappcvgprlemladdru wa wb cauappcvgprlemcl nqprlu addclpr
            cnp syl2anc cpw cxp npsspw sseli 1st2nd2 ssrab2 elpw2 mpbir opelxpi
            wss nqex mp2an mp1i eqeq12d xp1st xp2nd opthg bitrd mpbir2and ) AFI
            UNZDUAUBIUCDBUNZUAUBBUCUDZUEOZWCGUNZUFOWGEPZDUFOUAUBGQUGZIQUHZWHWGU
            FODUFOWDUAUBGQUGZBQUHZUDZRZWFUIPZWMUIPZRZWFUJPZWMUJPZRZAWOWPABCDEFG
            HIJKLMNUKABCDEFGHIJKLMNULUOAWRWSABCDEFGHIJKLMNUMABCDEFGHIJKLMNUPUOA
            WFVBSZWNWQWTUQZURAFVBSWEVBSZXAABCEFGHIJKLMUSADQSXCNBDIUTTFWEVAVCXAW
            NWOWRUDZWPWSUDZRZXBXAWFXDWMXEXAWFQVDZXGVEZSZWFXDRVBXHWFVFVGZWFXGXGV
            HTWMXHSZWMXERXAWJXGSZWLXGSZXKXLWJQVMWIIQVIWJQVNVJVKXMWLQVMWKBQVIWLQ
            VNVJVKWJWLXGXGVLVOWMXGXGVHVPVQXAWOXGSZWRXGSZXFXBURXAXIXNXJWFXGXGVRT
            XAXIXOXJWFXGXGVSTWOWRWPWSXGXGVTVCWATWB $.
        $}
      $}

      ${
        $d F f g h p q l u x $.  $d L q x $.  $d Q f g h p q l u x $.
        $d R f g h p q l u x $.  $d ph f g h q x $.
        cauappcvgprlem.q $e |- ( ph -> Q e. Q. ) $.
        cauappcvgprlem.r $e |- ( ph -> R e. Q. ) $.
        $( Lemma for ~ cauappcvgpr .  Part of showing the putative limit to be
           a limit.  (Contributed by Jim Kingdon, 23-Jun-2020.) $)
        cauappcvgprlem1 $p |- ( ph ->
            <. { l | l <Q ( F ` Q ) } , { u | ( F ` Q ) <Q u } >. <P
            ( L +P.
            <. { l | l <Q ( Q +Q R ) } , { u | ( Q +Q R ) <Q u } >. ) ) $=
          ( cplq co cltq cnq vx vf vg vh cfv cv wbr cab cop c1st wcel cltp wrex
          cpp wceq halfnqq syl wa simprl wral adantr fveq2 oveq1 oveq2d breq12d
          wi oveq12d breq2d anbi12d oveq2 rspc2v mpd simpld ffvelcdmd addassnqg
          syl2anc wf syl3anc breqtrrd w3a wb ltanqg addclnq addcomnqg caovord2d
          adantl mpbid simprr 3eqtrd oveq1d rspcev rexlimddv cauappcvgprlemladd
          breqtrd fveq2d rabex op1st eqtrdi eleq2d breq1d rexbidv elrab3 mpbird
          crab nqex bitrd cnp cauappcvgprlemcl nqprlu addclpr nqprl ) ADFUEZGJU
          FZDEQRZSUGJUHXNBUFZSUGBUHUIZUNRZUJUEZUKZXMXLSUGJUHXLXOSUGBUHUIXQULUGZ
          AXSXLHUFZQRZYAFUEZXNQRZSUGZHTUMZAUAUFZYGQRZEUOZYFUATAETUKZYIUATUMPUAE
          UPUQAYGTUKZYIURZURZYKXLYGQRZYGFUEZXNQRZSUGZYFAYKYIUSZYMYNYODQRZYGQRZY
          GQRZYPSYMXLYTSUGYNUUASUGYMXLYODYGQRZQRZYTSYMXLUUCSUGZYOXLUUBQRZSUGZYM
          IUFZFUEZYCUUGYAQRZQRZSUGZYCUUHUUIQRZSUGZURZHTUTITUTZUUDUUFURZAUUOYLLV
          AYMDTUKZYKUUOUUPVFAUUQYLOVAZYRUUNUUPXLYCDYAQRZQRZSUGZYCXLUUSQRZSUGZUR
          IHDYGTTUUGDUOZUUKUVAUUMUVCUVDUUHXLUUJUUTSUUGDFVBZUVDUUIUUSYCQUUGDYAQV
          CZVDVEUVDUULUVBYCSUVDUUHXLUUIUUSQUVEUVFVGVHVIYAYGUOZUVAUUDUVCUUFUVGUU
          TUUCXLSUVGYCYOUUSUUBQYAYGFVBZYAYGDQVJZVGVHUVGYCYOUVBUUESUVHUVGUUSUUBX
          LQUVIVDVEVIVKVPVLVMYMYOTUKZUUQYKYTUUCUOYMTTYGFATTFVQYLKVAYRVNZUURYRYO
          DYGVOVRVSYMUBUCUDXLYTYGSTQUBUFZTUKZUCUFZTUKZUDUFZTUKVTUVLUVNSUGUVPUVL
          QRUVPUVNQRSUGWAYMUVLUVNUVPWBWFAXLTUKZYLATTDFKOVNZVAYMYSTUKZYKYTTUKYMU
          VJUUQUVSUVKUURYODWCVPZYRYSYGWCVPYRUVMUVOURUVLUVNQRUVNUVLQRUOYMUVLUVNW
          DWFWEWGYMUUAYSYHQRZYSEQRZYPYMUVSYKYKUUAUWAUOUVTYRYRYSYGYGVOVRYMYHEYSQ
          AYKYIWHVDYMUVJUUQYJUWBYPUOUVKUURAYJYLPVAYODEVOVRWIWNYEYQHYGTUVGYBYNYD
          YPSYAYGXLQVJUVGYCYOXNQUVHWJVEWKVPWLAXSXLXMYAQRZYDSUGZHTUMZJTXDZUKZYFA
          XRUWFXLAXRUWFYCYAQRXNQRXOSUGHTUMZBTXDZUIZUJUEUWFAXQUWJUJABCXNFGHIJKLM
          NAUUQYJXNTUKZOPDEWCVPZWMWOUWFUWIUWEJTXEWPUWHBTXEWPWQWRWSAUVQUWGYFWAUV
          RUWEYFJXLTXMXLUOZUWDYEHTUWMUWCYBYDSXMXLYAQVCWTXAXBUQXFXCAUVQXQXGUKZXS
          XTWAUVRAGXGUKXPXGUKZUWNABCFGHIJKLMNXHAUWKUWOUWLBXNJXIUQGXPXJVPBXLXQJX
          KVPWG $.

        $( Lemma for ~ cauappcvgpr .  Part of showing the putative limit to be
           a limit.  (Contributed by Jim Kingdon, 23-Jun-2020.) $)
        cauappcvgprlem2 $p |- ( ph ->
          L <P <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } ,
          { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) $=
          ( cltq wbr cnq wcel vx cfv cplq co cv wa cab cop cltp ltaddnq syl2anc
          wrex ffvelcdmd ltanqi ltbtwnnqq sylib c2nd c1st simprl adantr simprrl
          wceq fveq2 oveq12d breq1d rspcev breq2 rexbidv crab fveq2i nqex rabex
          id op2nd eqtri elrab2 sylanbrc simprrr vex breq1 sylibr ltnqex gtnqex
          elab op1st eleqtrrdi rspe syl12anc wb cauappcvgprlemcl addclnq nqprlu
          cnp syl ltdfpr mpbird rexlimddv ) ADFUBZDUCUDZUAUEZQRZWTWRDEUCUDZUCUD
          ZQRZUFZGJUEZXCQRZJUGZXCBUEZQRBUGZUHZUIRZUASAWSXCQRZXEUASULADXBQRZWRST
          ZXMADSTZESTZXNOPDEUJUKASSDFKOUMZDXBWRUNUKUAWSXCUOUPAWTSTZXEUFZUFZXLWT
          GUQUBZTZWTXKURUBZTZUFZUASULZYAXSYCYEYGAXSXEUSZYAXSHUEZFUBZYIUCUDZWTQR
          ZHSULZYCYHYAXPXAYMAXPXTOUTAXSXAXDVAYLXAHDSYIDVBZYKWSWTQYNYJWRYIDUCYID
          FVCYNVMVDVEVFUKYKXIQRZHSULZYMBWTSYBXIWTVBYOYLHSXIWTYKQVGVHYBXFYIUCUDY
          JQRHSULZJSVIZYPBSVIZUHZUQUBYSGYTUQNVJYRYSYQJSVKVLYPBSVKVLVNVOVPVQYAWT
          XHYDYAXDWTXHTAXSXAXDVRXGXDJWTUAVSXFWTXCQVTWDWAXHXJJXCWBBXCWCWEWFYFUAS
          WGWHYAGWMTZXKWMTZXLYGWIAUUAXTABCFGHIJKLMNWJUTAUUBXTAXCSTZUUBAXOXBSTZU
          UCXRAXPXQUUDOPDEWKUKWRXBWKUKBXCJWLWNUTGXKUAWOUKWPWQ $.
      $}

      ${
        $d F l p q r u x y $.  $d L q r x y $.  $d ph x y q $.
        $( Lemma for ~ cauappcvgpr .  The putative limit is a limit.
           (Contributed by Jim Kingdon, 20-Jun-2020.) $)
        cauappcvgprlemlim $p |- ( ph -> A. q e. Q. A. r e. Q. (
            <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P
              ( L +P. <. { l | l <Q ( q +Q r ) } ,
                { u | ( q +Q r ) <Q u } >. ) /\
            L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } ,
              { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) ) $=
          ( cltq wbr cab cplq co cltp cnq vx vy cv cfv cop cpp wral wcel adantr
          wa simprl simprr cauappcvgprlem1 cauappcvgprlem2 jca ralrimivva fveq2
          weq breq2d abbidv breq1d opeq12d oveq1 oveq2d breq12d oveq12d anbi12d
          wf oveq2 cbvral2v sylib ) AIUCZUAUCZDUDZNOZIPZVNBUCZNOZBPZUEZEVLVMUBU
          CZQRZNOZIPZWBVQNOZBPZUEZUFRZSOZEVLVNWBQRZNOZIPZWJVQNOZBPZUEZSOZUJZUBT
          UGUATUGVLGUCZDUDZNOZIPZWSVQNOZBPZUEZEVLWRFUCZQRZNOZIPZXFVQNOZBPZUEZUF
          RZSOZEVLWSXFQRZNOZIPZXNVQNOZBPZUEZSOZUJZFTUGGTUGAWQUAUBTTAVMTUHZWATUH
          ZUJZUJZWIWPYEBCVMWADEGHIATTDVHYDJUIZAHUCZDUDZWSYGWRQRZQRNOWSYHYIQRNOU
          JGTUGHTUGYDKUIZACYHNOHTUGYDLUIZMAYBYCUKZAYBYCULZUMYEBCVMWADEGHIYFYJYK
          MYLYMUNUOUPWQYAXDEVLWRWAQRZNOZIPZYNVQNOZBPZUEZUFRZSOZEVLWSYNQRZNOZIPZ
          UUBVQNOZBPZUEZSOZUJUAUBGFTTUAGURZWIUUAWPUUHUUIVTXDWHYTSUUIVPXAVSXCUUI
          VOWTIUUIVNWSVLNVMWRDUQZUSUTUUIVRXBBUUIVNWSVQNUUJVAUTVBUUIWGYSEUFUUIWD
          YPWFYRUUIWCYOIUUIWBYNVLNVMWRWAQVCZUSUTUUIWEYQBUUIWBYNVQNUUKVAUTVBVDVE
          UUIWOUUGESUUIWLUUDWNUUFUUIWKUUCIUUIWJUUBVLNUUIVNWSWBYNQUUJUUKVFZUSUTU
          UIWMUUEBUUIWJUUBVQNUULVAUTVBUSVGUBFURZUUAXMUUHXTUUMYTXLXDSUUMYSXKEUFU
          UMYPXHYRXJUUMYOXGIUUMYNXFVLNWAXEWRQVIZUSUTUUMYQXIBUUMYNXFVQNUUNVAUTVB
          VDUSUUMUUGXSESUUMUUDXPUUFXRUUMUUCXOIUUMUUBXNVLNUUMYNXFWSQUUNVDZUSUTUU
          MUUEXQBUUMUUBXNVQNUUOVAUTVBUSVGVJVK $.
      $}
    $}

    ${
      $d F q y z $.  $d F r y z $.  $d F u $.  $d F p $.  $d F l $.
      $d l q y z $.  $d l r y z $.  $d q u y z $.  $d r u y z $.  $d q r $.
      $d p u $.  $d p r $.  $d p q $.  $d l u $.  $d l p $.  $d ph q $.
      $d ph p $.  $d p z $.  $d q z $.
      $( A Cauchy approximation has a limit.  A Cauchy approximation, here
         ` F ` , is similar to a Cauchy sequence but is indexed by the desired
         tolerance (that is, how close together terms needs to be) rather than
         by natural numbers.  This is basically Theorem 11.2.12 of [HoTT], p.
         (varies) with a few differences such as that we are proving the
         existence of a limit without anything about how fast it converges
         (that is, mere existence instead of existence, in HoTT terms), and
         that the codomain of ` F ` is ` Q. ` rather than ` P. ` .  We also
         specify that every term needs to be larger than a fraction ` A ` , to
         avoid the case where we have positive terms which "converge" to zero
         (which is not a positive real).

         This proof (including its lemmas) is similar to the proofs of
         ~ caucvgpr and ~ caucvgprpr but is somewhat simpler, so reading this
         one first may help understanding the other two.

         (Contributed by Jim Kingdon, 19-Jun-2020.) $)
      cauappcvgpr $p |- ( ph -> E. y e. P. A. q e. Q. A. r e. Q. (
          <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P
            ( y +P. <. { l | l <Q ( q +Q r ) } ,
            { u | ( q +Q r ) <Q u } >. ) /\
          y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } ,
            { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) ) $=
        ( cv cplq co cltq wbr cnq cab cltp vz cfv wrex crab cop cnp wcel cpp wa
        wral wb wceq oveq2 fveq2 breq12d cbvrexv a1i rabbiia id oveq12d opeq12i
        breq1d cauappcvgprlemcl cauappcvgprlemlim oveq1 breq2d anbi12d 2ralbidv
        breq1 rspcev syl2anc ) AIMZUAMZNOZVMEUBZPQZUARUCZIRUDZVOVMNOZCMZPQZUARU
        CZCRUDZUEZUFUGVLGMZEUBZPQISWFVTPQCSUEZWDVLWEFMNOZPQISWHVTPQCSUEZUHOZTQZ
        WDVLWFWHNOZPQISWLVTPQCSUEZTQZUIZFRUJGRUJZWGBMZWIUHOZTQZWQWMTQZUIZFRUJGR
        UJZBUFUCACDEWDGHIJKLVRVLWENOZWFPQZGRUCZIRUDWCWFWENOZVTPQZGRUCZCRUDVQXEI
        RVQXEUKVLRUGVPXDUAGRVMWEULZVNXCVOWFPVMWEVLNUMVMWEEUNZUOUPUQURWBXHCRWBXH
        UKVTRUGWAXGUAGRXIVSXFVTPXIVOWFVMWENXJXIUSUTVBUPUQURVAZVCACDEWDFGHIJKLXK
        VDXBWPBWDUFWQWDULZXAWOGFRRXLWSWKWTWNXLWRWJWGTWQWDWIUHVEVFWQWDWMTVIVGVHV
        JVK $.
    $}
  $}

  ${
    $d A j $.
    $( Archimedean principle for fractions (reciprocal version).  (Contributed
       by Jim Kingdon, 27-Sep-2020.) $)
    archrecnq $p |- ( A e. Q. -> E. j e. N.
        ( *Q ` [ <. j , 1o >. ] ~Q ) <Q A ) $=
      ( cnq wcel crq cfv c1o cop ceq cec cltq wbr cnpi wrex recclnq archnqq syl
      cv wa wb nnnq ltrnqg syl2an recrecnq breq2d adantr bitrd rexbidva mpbid )
      ACDZAEFZBRZGHIJZKLZBMNZUMEFZAKLZBMNUJUKCDZUOAOZBUKPQUJUNUQBMUJULMDZSUNUPU
      KEFZKLZUQUJURUMCDUNVBTUTUSULUAUKUMUBUCUJVBUQTUTUJVAAUPKAUDUEUFUGUHUI $.
  $}

  ${
    $d A j x $.  $d j l u $.
    $( Archimedean principle for positive reals (reciprocal version).
       (Contributed by Jim Kingdon, 25-Nov-2020.) $)
    archrecpr $p |- ( A e. P. -> E. j e. N.
        <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } ,
          { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u }
        >. <P A ) $=
      ( vx cnp wcel cv c1o cop cfv cnpi wrex cltq wbr cab cnq syl wa syl2anc wi
      ceq cec crq c1st cltp c2nd prop prml archrecnq ad2antrl ad2antrr reximdva
      simplrr prcdnql mpd rexlimddv wb nnnq recclnq adantl simpl nqprl rexbidva
      mpbid ) BFGZCHZIJUBUCZUDKZBUEKZGZCLMZDHVINODPVIAHNOAPJBUFOZCLMVFEHZVJGZVL
      EQVFVJBUGKZJFGZVOEQMBUHZEVPVJUIRVFVNQGZVOSZSZVIVNNOZCLMZVLVSWCVFVOVNCUJUK
      WAWBVKCLWAVGLGZSVQVOWBVKUAVFVQVTWDVRULVFVSVOWDUNVNVIVPVJUOTUMUPUQVFVKVMCL
      VFWDSVIQGZVFVKVMURWDWEVFWDVHQGWEVGUSVHUTRVAVFWDVBAVIBDVCTVDVE $.
  $}

  ${
    caucvgprlemk.jk $e |- ( ph -> J <N K ) $.
    caucvgprlemk.jkq $e |- ( ph ->
      ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q ) $.
    $( Lemma for ~ caucvgpr .  Reciprocals of positive integers decrease as the
       positive integers increase.  (Contributed by Jim Kingdon,
       9-Oct-2020.) $)
    caucvgprlemk $p |- ( ph ->
        ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q ) $=
      ( c1o cop ceq cec crq cfv cltq wbr clti cnpi wcel wa wb syl ltrelpi mpbid
      brel ltnnnq ltrnqi cnq ltsonq ltrelnq sotri syl2anc ) ADGHIJZKLZCGHIJZKLZ
      MNZUNBMNULBMNAUMUKMNZUOACDONZUPEACPQDPQRZUQUPSAUQURECDPPOUAUCTCDUDTUBUMUK
      UETFULUNBMUFUGUHUIUJ $.
  $}

  ${
    $d A j $.
    caucvgpr.f $e |- ( ph -> F : N. --> Q. ) $.
    caucvgpr.cau $e |- ( ph ->
      A. n e. N. A. k e. N. ( n <N k ->
      ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\
        ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) ) $.
    ${
      $d F a b k $.  $d F f g h $.  $d F a k n $.  $d J a b $.  $d J f g h $.
      $d K a b $.  $d K f g h $.  $d S f g h $.  $d f g h ph $.
      caucvgprlemnkj.k $e |- ( ph -> K e. N. ) $.
      caucvgprlemnkj.j $e |- ( ph -> J e. N. ) $.
      caucvgprlemnkj.s $e |- ( ph -> S e. Q. ) $.
      $( Lemma for ~ caucvgpr .  Part of disjointness.  (Contributed by Jim
         Kingdon, 23-Oct-2020.) $)
      caucvgprlemnkj $p |- ( ph -> -.
          ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\
          ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) $=
        ( clti wbr cplq co cltq wa cnq cnpi va vb vf vg c1o cop ceq cec crq cfv
        vh wn wceq ltsonq ltrelnq son2lpi simprl cv wi breq1 fveq2 opeq1 eceq1d
        wral fveq2d oveq2d breq12d oveq12d breq2d anbi12d imbi12d oveq1d breq1d
        breq2 cbvral2v sylib wcel rspc2v syl2anc mpd imp simpld adantr sotri wb
        w3a ltanqg adantl ad2antrr ffvelcdmd nnnq recclnq 3syl addcomnqg mpbird
        caovord2d ltaddnq simprr jca ex mtoi anbi1d addclnq bitrd jctird simprd
        biimtrdi w3o pitri3or mpjao3dan ) AGFMNZBGUEUFZUGUHZUIUJZOPZGEUJZQNZFEU
        JZFUEUFZUGUHZUIUJZOPZBQNZRZULGFUMZFGMNZAXKRZYDBXRQNZXRBQNZRZBXRQSUNUOUP
        YGYDYJYGYDRZYHYIYKYHXOXRXNOPZQNZYKXQXPYLQNZYMYGXQYCUQYGYNYDYGYNXRXPXNOP
        ZQNZAXKYNYPRZAUAURZUBURZMNZYREUJZYSEUJZYRUEUFZUGUHZUIUJZOPZQNZUUBUUAUUE
        OPZQNZRZUSZUBTVDUATVDZXKYQUSZADURZCURZMNZUUNEUJZUUOEUJZUUNUEUFZUGUHZUIU
        JZOPZQNZUURUUQUVAOPZQNZRZUSZCTVDDTVDUULIUVGUUKYRUUOMNZUUAUURUUEOPZQNZUU
        RUUHQNZRZUSDCUAUBTTUUNYRUMZUUPUVHUVFUVLUUNYRUUOMUTUVMUVCUVJUVEUVKUVMUUQ
        UUAUVBUVIQUUNYREVAZUVMUVAUUEUUROUVMUUTUUDUIUVMUUSUUCUGUUNYRUEVBVCVEZVFV
        GUVMUVDUUHUURQUVMUUQUUAUVAUUEOUVNUVOVHVIVJVKUUOYSUMZUVHYTUVLUUJUUOYSYRM
        VNUVPUVJUUGUVKUUIUVPUVIUUFUUAQUVPUURUUBUUEOUUOYSEVAZVLVIUVPUURUUBUUHQUV
        QVMVJVKVOVPZAGTVQZFTVQZUULUUMUSJKUUKUUMGYSMNZXPUUBXNOPZQNZUUBYOQNZRZUSU
        AUBGFTTYRGUMZYTUWAUUJUWEYRGYSMUTUWFUUGUWCUUIUWDUWFUUAXPUUFUWBQYRGEVAZUW
        FUUEXNUUBOUWFUUDXMUIUWFUUCXLUGYRGUEVBVCVEZVFVGUWFUUHYOUUBQUWFUUAXPUUEXN
        OUWGUWHVHVIVJVKYSFUMZUWAXKUWEYQYSFGMVNUWIUWCYNUWDYPUWIUWBYLXPQUWIUUBXRX
        NOYSFEVAZVLVIUWIUUBXRYOQUWJVMVJVKVRVSVTWAWBWCXOXPYLQSUNUOWDVSYKUCUDUKBX
        RXNQSOUCURZSVQZUDURZSVQZUKURZSVQWFZUWKUWMQNUWOUWKOPUWOUWMOPQNWEZYKUWKUW
        MUWOWGZWHABSVQZXKYDLWIAXRSVQZXKYDATSFEHKWJZWIZAXNSVQZXKYDAUVSXMSVQUXCJG
        WKXMWLWMZWIUWLUWNRZUWKUWMOPUWMUWKOPUMZYKUWKUWMWNZWHWPWOYKXRYBQNZYCYIYKU
        WTYASVQZUXHUXBAUXIXKYDAUVTXTSVQUXIKFWKXTWLWMZWIXRYAWQVSYGXQYCWRXRYBBQSU
        NUOWDVSWSWTXAAYERZYDBYAOPZYAOPZBQNZBUXMQNZRUXMBQSUNUOUPUXKYDUXNUXOUXKYD
        UXMYBQNZYCRZUXNUXKYDUXLXRQNZYCRZUXQYEYDUXSWEAYEXQUXRYCYEXOUXLXPXRQYEXNY
        ABOYEXMXTUIYEXLXSUGGFUEVBVCVEVFGFEVAVGXBWHUXKUXRUXPYCAUXRUXPWEYEAUCUDUK
        UXLXRYAQSOUWPUWQAUWRWHAUWSUXIUXLSVQZLUXJBYAXCVSZUXAUXJUXEUXFAUXGWHWPWCX
        BXDUXMYBBQSUNUOWDXGAUXOYEABUXLQNZUXLUXMQNZUXOAUWSUXIUYBLUXJBYAWQVSAUXTU
        XIUYCUYAUXJUXLYAWQVSBUXLUXMQSUNUOWDVSWCXEXAAYFRZYDBYBQNZYCRZBYBQSUNUOUP
        UYDYDUYFUYDYDRZUYEYCUYGBXOQNZXOYBQNZUYEUYGUWSUXCUYHAUWSYFYDLWIAUXCYFYDU
        XDWIBXNWQVSUYGXQXPYBQNZUYIUYDXQYCUQUYDUYJYDUYDXRXPYAOPZQNZUYJAYFUYLUYJR
        ZAUULYFUYMUSZUVRAUVTUVSUULUYNUSKJUUKUYNFYSMNZXRUUBYAOPZQNZUUBYBQNZRZUSU
        AUBFGTTYRFUMZYTUYOUUJUYSYRFYSMUTUYTUUGUYQUUIUYRUYTUUAXRUUFUYPQYRFEVAZUY
        TUUEYAUUBOUYTUUDXTUIUYTUUCXSUGYRFUEVBVCVEZVFVGUYTUUHYBUUBQUYTUUAXRUUEYA
        OVUAVUBVHVIVJVKYSGUMZUYOYFUYSUYMYSGFMVNVUCUYQUYLUYRUYJVUCUYPUYKXRQVUCUU
        BXPYAOYSGEVAZVLVIVUCUUBXPYBQVUDVMVJVKVRVSVTWAXFWCXOXPYBQSUNUOWDVSBXOYBQ
        SUNUOWDVSUYDXQYCWRWSWTXAAUVSUVTXKYEYFXHJKGFXIVSXJ $.
    $}

    ${
      $d B f g h $.  $d B k n $.  $d F f g h $.  $d F k n $.  $d J f g h $.
      $d J k n $.  $d f g h ph $.
      caucvgprlemnbj.b $e |- ( ph -> B e. N. ) $.
      caucvgprlemnbj.j $e |- ( ph -> J e. N. ) $.
      $( Lemma for ~ caucvgpr .  Non-existence of two elements of the sequence
         which are too far from each other.  (Contributed by Jim Kingdon,
         18-Oct-2020.) $)
      caucvgprlemnbj $p |- ( ph ->
          -. ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) )
            +Q ( *Q ` [ <. J , 1o >. ] ~Q ) )
            <Q ( F ` J ) ) $=
        ( cplq co cltq wbr clti wa cnpi wcel syl2anc cnq vf cfv c1o cop ceq cec
        vg vh wn wceq cv wi wral breq1 fveq2 opeq1 eceq1d fveq2d oveq2d breq12d
        crq oveq12d breq2d anbi12d imbi12d breq2 oveq1d breq1d rspc2v ffvelcdmd
        mpd imp simprd nnnq recclnq addclnq ltaddnq adantr ltsonq ltrelnq sotri
        3syl wb adantl mpbid simpld w3a ltanqg addcomnqg caovord2d w3o pitri3or
        mpjao3dan wor so2nr mpan imnan sylibr ) AFEUBZBEUBZBUCUDZUEUFZVAUBZKLZF
        UCUDZUEUFZVAUBZKLZMNZXHWSMNZUIZABFONZXIBFUJZFBONZAXLPZWSXDMNZXDXHMNZXIX
        OWTWSXCKLZMNZXPAXLXSXPPZADUKZCUKZONZYAEUBZYBEUBZYAUCUDZUEUFZVAUBZKLZMNZ
        YEYDYHKLZMNZPZULZCQUMDQUMZXLXTULZHABQRZFQRZYOYPULIJYNYPBYBONZWTYEXCKLZM
        NZYEXDMNZPZULDCBFQQYABUJZYCYSYMUUCYABYBOUNUUDYJUUAYLUUBUUDYDWTYIYTMYABE
        UOZUUDYHXCYEKUUDYGXBVAUUDYFXAUEYABUCUPUQURZUSUTUUDYKXDYEMUUDYDWTYHXCKUU
        EUUFVBVCVDVEYBFUJZYSXLUUCXTYBFBOVFUUGUUAXSUUBXPUUGYTXRWTMUUGYEWSXCKYBFE
        UOZVGVCUUGYEWSXDMUUHVHVDVEVISVKVLVMAXQXLAXDTRZXGTRZXQAWTTRZXCTRZUUIAQTB
        EGIVJZAYQXBTRUULIBVNXBVOWBZWTXCVPSZAYRXFTRUUJJFVNXFVOWBZXDXGVQSZVRWSXDX
        HMTVSVTWAZSAXMPZXPXQXIUUSWTXDMNZXPAUUTXMAUUKUULUUTUUMUUNWTXCVQSZVRXMUUT
        XPWCAXMWTWSXDMBFEUOVHWDWEAXQXMUUQVRUURSAXNPZWSWTXGKLZMNZUVCXHMNZXIUVBUV
        DWTWSXGKLZMNZAXNUVDUVGPZAYOXNUVHULZHAYRYQYOUVIULJIYNUVIFYBONZWSYEXGKLZM
        NZYEUVFMNZPZULDCFBQQYAFUJZYCUVJYMUVNYAFYBOUNUVOYJUVLYLUVMUVOYDWSYIUVKMY
        AFEUOZUVOYHXGYEKUVOYGXFVAUVOYFXEUEYAFUCUPUQURZUSUTUVOYKUVFYEMUVOYDWSYHX
        GKUVPUVQVBVCVDVEYBBUJZUVJXNUVNUVHYBBFOVFUVRUVLUVDUVMUVGUVRUVKUVCWSMUVRY
        EWTXGKYBBEUOZVGVCUVRYEWTUVFMUVSVHVDVEVISVKVLWFAUVEXNAUUTUVEUVAAUAUGUHWT
        XDXGMTKUAUKZTRZUGUKZTRZUHUKZTRWGUVTUWBMNUWDUVTKLUWDUWBKLMNWCAUVTUWBUWDW
        HWDUUMUUOUUPUWAUWCPUVTUWBKLUWBUVTKLUJAUVTUWBWIWDWJWEVRWSUVCXHMTVSVTWASA
        YQYRXLXMXNWKIJBFWLSWMAXIXJPUIZXIXKULAWSTRZXHTRZUWEAQTFEGJVJAUUIUUJUWGUU
        OUUPXDXGVPSTMWNUWFUWGPUWEVSTWSXHMWOWPSXIXJWQWRVK $.
    $}

    caucvgpr.bnd $e |- ( ph -> A. j e. N. A <Q ( F ` j ) ) $.

    ${
      caucvgpr.lim $e |- L = <.
        { l e. Q. | E. j e. N.
          ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } ,
        { u e. Q. | E. j e. N.
          ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
        >. $.

      ${
        $d A s $.  $d F j l $.  $d F r $.  $d F j u $.  $d L r $.  $d j ph s $.
        $d l s $.
        $( Lemma for ~ caucvgpr .  The putative limit is inhabited.
           (Contributed by Jim Kingdon, 27-Sep-2020.) $)
        caucvgprlemm $p |- ( ph -> ( E. s e. Q. s e. ( 1st ` L )
              /\ E. r e. Q. r e. ( 2nd ` L ) ) ) $=
          ( wcel cnq c1o cltq cnpi cv c1st cfv wrex c2nd cplq co wceq wbr fveq2
          breq2d 1pi a1i rspcdva ltrelnq brel simpld halfnqq wa cop ceq cec crq
          3syl simplr archrecnq syl simpr wb nnnq recclnq ad2antrr ltanqg mpbid
          syl3anc simpllr breqtrd wi wral rsp ad4antr mpd ltsonq sotri reximdva
          syl2anc ex oveq1 breq1d rexbidv crab fveq2i nqex rabex op1st sylanbrc
          eqtri elrab2 c1q ffvelcdmd 1nq addclnq sylancl df-1nqqs rec1nq eqtr3i
          oveq2i ltaddnq eqbrtrid opeq1 eceq1d fveq2d oveq12d breq2 op2nd eleq1
          rspcev jca ) AIUAZHUBUCZPZIQUDZJUAZHUEUCZPZJQUDZAXSXSUFUGZCUHZIQUDZYB
          ACRGUCZSUIZCQPZYIACDUAZGUCZSUIZYKDTRYMRUHZYNYJCSYMRGUJZUKNRTPZAULUMZU
          NYKYLYJQPZCYJQQSUOUPUQICURVDAYHYAIQAXSQPZUSZYHYAUUBYHUSZUUAXSYMRUTZVA
          VBZVCUCZUFUGZYNSUIZDTUDZYAAUUAYHVEZUUCUUFXSSUIZDTUDZUUIUUCUUAUULUUJXS
          DVFVGUUCUUKUUHDTUUCYMTPZUSZUUKUUHUUNUUKUSZUUGCSUIYOUUHUUOUUGYGCSUUOUU
          KUUGYGSUIZUUNUUKVHUUOUUFQPZUUAUUAUUKUUPVIUUOUUMUUEQPUUQUUCUUMUUKVEZYM
          VJUUEVKVDUUCUUAUUMUUKUUJVLZUUSUUFXSXSVMVOVNUUBYHUUMUUKVPVQUUOUUMYOUUR
          AUUMYOVRZUUAYHUUMUUKAYODTVSUUTNYODTVTVGWAWBUUGCYNSQWCUOWDWFWGWEWBKUAZ
          UUFUFUGZYNSUIZDTUDZUUIKXSQXTUVAXSUHZUVCUUHDTUVEUVBUUGYNSUVAXSUUFUFWHW
          IWJXTUVDKQWKZYNUUFUFUGZBUAZSUIZDTUDZBQWKZUTZUBUCUVFHUVLUBOWLUVFUVKUVD
          KQWMWNZUVJBQWMWNZWOWQWRWPWGWEWBAYJWSUFUGZWSUFUGZQPZUVPYDPZYFAUVOQPZWS
          QPZUVQAYTUVTUVSATQRGLYSWTXAYJWSXBXCZXAUVOWSXBXCZAUVQUVGUVPSUIZDTUDZUV
          RUWBAYRYJRRUTZVAVBZVCUCZUFUGZUVPSUIZUWDYSAUWHUVOUVPSUWGWSYJUFWSVCUCUW
          GWSWSUWFVCXDWLXEXFXGAUVSUVTUVOUVPSUIUWAXAUVOWSXHXCXIUWCUWIDRTYPUVGUWH
          UVPSYPYNYJUUFUWGUFYQYPUUEUWFVCYPUUDUWEVAYMRRXJXKXLXMWIXQWFUVJUWDBUVPQ
          YDUVHUVPUHUVIUWCDTUVHUVPUVGSXNWJYDUVLUEUCUVKHUVLUEOWLUVFUVKUVMUVNXOWQ
          WRWPYEUVRJUVPQYCUVPYDXPXQWFXR $.
      $}

      ${
        $d F f g h l r $.  $d F r s t $.  $d F u $.  $d L f g h j r t $.
        $d L s $.  $d j l r s $.  $d j ph f g h r t $.  $d ph s $.
        $d f g h s t $.  $d j u r s $.
        $( Lemma for ~ caucvgpr .  The lower cut of the putative limit is open.
           (Contributed by Jim Kingdon, 20-Oct-2020.) $)
        caucvgprlemopl $p |- ( ( ph /\ s e. ( 1st ` L ) ) ->
            E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) $=
          ( wa cltq wbr cnq cnpi vt c1st cfv wcel c1o cop ceq cec crq cplq wrex
          cv co wceq oveq1 breq1d rexbidv crab fveq2i rabex op1st eqtri simprbi
          nqex elrab2 adantl simprr ltbtwnnqq simplrl nnnq recclnq 3syl simplbi
          sylib ad3antlr ltaddnq syl2anc addcomnqg breqtrd simprrl ltsonq sotri
          ltrelnq wb simprl ltexnqq mpbid ad2antrr eqbrtrrd simpr simplr ltanqg
          breqtrrd syl3anc mpbird simprrr eqtr3d rspe sylanbrc jca reximdva mpd
          ex rexlimddv ) AIULZHUBUCZUDZPZXEDULZUEUFUGUHZUIUCZUJUMZXIGUCZQRZXEJU
          LZQRZXOXFUDZPZJSUKZDTXGXNDTUKZAXGXESUDZXTKULZXKUJUMZXMQRZDTUKZXTKXESX
          FYBXEUNZYDXNDTYFYCXLXMQYBXEXKUJUOUPUQXFYEKSURZXMXKUJUMBULQRDTUKZBSURZ
          UFZUBUCYGHYJUBOUSYGYIYEKSVDUTYHBSVDUTVAVBZVEZVCVFXHXITUDZXNPZPZXLUAUL
          ZQRZYPXMQRZPZXSUASYOXNYSUASUKXHYMXNVGUAXLXMVHVNYOYPSUDZYSPZPZXKXOUJUM
          ZYPUNZJSUKZXSUUBXKYPQRZUUEUUBXKXLQRYQUUFUUBXKXKXEUJUMZXLQUUBXKSUDZYAX
          KUUGQRUUBYMXJSUDUUHXHYMXNUUAVIZXIVJXJVKVLZXGYAAYNUUAXGYAXTYLVMVOZXKXE
          VPVQUUBUUHYAUUGXLUNUUJUUKXKXEVRVQVSYOYTYQYRVTZXKXLYPQSWAWCWBVQUUBUUHY
          TUUFUUEWDUUJYOYTYSWEJXKYPWFVQWGUUBUUDXRJSUUBXOSUDZPZUUDXRUUNUUDPZXPXQ
          UUOXPUUGUUCQRZUUOUUGYPUUCQUUOXLUUGYPQUUOYAUUHXLUUGUNUUBYAUUMUUDUUKWHZ
          UUBUUHUUMUUDUUJWHZXEXKVRVQUUBYQUUMUUDUULWHWIUUNUUDWJZWMUUOYAUUMUUHXPU
          UPWDUUQUUBUUMUUDWKZUURXEXOXKWLWNWOUUOUUMXOXKUJUMZXMQRZDTUKZXQUUTUUOYM
          UVBUVCUUBYMUUMUUDUUIWHUUOUVBYRUUBYRUUMUUDYOYTYQYRWPWHUUOUVAYPXMQUUOUU
          CUVAYPUUOUUHUUMUUCUVAUNUURUUTXKXOVRVQUUSWQUPWOUVBDTWRVQYEUVCKXOSXFYBX
          OUNZYDUVBDTUVDYCUVAXMQYBXOXKUJUOUPUQYKVEWSWTXCXAXBXDXD $.

        $( Lemma for ~ caucvgpr .  The lower cut of the putative limit is
           lower.  (Contributed by Jim Kingdon, 20-Oct-2020.) $)
        caucvgprlemlol $p |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) ->
            s e. ( 1st ` L ) ) $=
          ( cv cltq wcel cnq cplq vf vg vh wbr c1st cfv w3a c1o cop ceq cec crq
          co cnpi wrex ltrelnq brel simpld 3ad2ant2 oveq1 breq1d rexbidv fveq2i
          weq crab nqex rabex op1st eqtri elrab2 simprbi 3ad2ant3 wa simpll2 wb
          ltanqg adantl ad2antrr simprd simplr nnnq recclnq 3syl wceq addcomnqg
          caovord2d mpbid ltsonq sotri sylancom ex reximdva mpd sylanbrc ) AIPZ
          JPZQUDZWPHUEUFZRZUGZWOSRZWODPZUHUIUJUKZULUFZTUMZXBGUFZQUDZDUNUOZWOWRR
          WQAXAWSWQXAWPSRZWOWPSSQUPUQZURUSZWTWPXDTUMZXFQUDZDUNUOZXHWSAXNWQWSXIX
          NKPZXDTUMZXFQUDZDUNUOZXNKWPSWRKJVDZXQXMDUNXSXPXLXFQXOWPXDTUTVAVBWRXRK
          SVEZXFXDTUMBPQUDDUNUOZBSVEZUIZUEUFXTHYCUEOVCXTYBXRKSVFVGYABSVFVGVHVIZ
          VJVKVLWTXMXGDUNWTXBUNRZVMZXMXGYFXMXEXLQUDZXGYFXMVMZWQYGAWQWSYEXMVNYHU
          AUBUCWOWPXDQSTUAPZSRZUBPZSRZUCPZSRUGYIYKQUDYMYITUMYMYKTUMQUDVOYHYIYKY
          MVPVQWTXAYEXMXKVRWTXIYEXMWQAXIWSWQXAXIXJVSUSVRYHYEXCSRXDSRWTYEXMVTXBW
          AXCWBWCYJYLVMYIYKTUMYKYITUMWDYHYIYKWEVQWFWGXEXLXFQSWHUPWIWJWKWLWMXRXH
          KWOSWRKIVDZXQXGDUNYNXPXEXFQXOWOXDTUTVAVBYDVJWN $.

        $( Lemma for ~ caucvgpr .  The upper cut of the putative limit is open.
           (Contributed by Jim Kingdon, 20-Oct-2020.) $)
        caucvgprlemopu $p |- ( ( ph /\ r e. ( 2nd ` L ) ) ->
            E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) $=
          ( wa cltq cnq wrex cnpi cv c2nd cfv wcel c1o cop ceq cec crq cplq wbr
          co weq breq2 rexbidv crab fveq2i nqex rabex op2nd eqtri elrab2 adantl
          simprbi simprr ltbtwnnqq sylib simplr simplrl adantr syl2anc sylanbrc
          simprl rspe jca ex reximdva mpd rexlimddv ) AJUAZHUBUCZUDZPZDUAZGUCZW
          DUEUFUGUHUIUCZUJULZVTQUKZIUAZVTQUKZWIWAUDZPZIRSZDTWBWHDTSZAWBVTRUDWNW
          GBUAZQUKZDTSZWNBVTRWABJUMWPWHDTWOVTWGQUNUOWAKUAWFUJULWEQUKDTSZKRUPZWQ
          BRUPZUFZUBUCWTHXAUBOUQWSWTWRKRURUSWQBRURUSUTVAZVBVDVCWCWDTUDZWHPPZWGW
          IQUKZWJPZIRSZWMXDWHXGWCXCWHVEIWGVTVFVGXDXFWLIRXDWIRUDZPZXFWLXIXFPZWJW
          KXIXEWJVEXJXHXEDTSZWKXDXHXFVHXJXCXEXKXIXCXFWCXCWHXHVIVJXIXEWJVMXEDTVN
          VKWQXKBWIRWABIUMWPXEDTWOWIWGQUNUOXBVBVLVOVPVQVRVS $.

        $( Lemma for ~ caucvgpr .  The upper cut of the putative limit is
           upper.  (Contributed by Jim Kingdon, 20-Oct-2020.) $)
        caucvgprlemupu $p |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) ->
            r e. ( 2nd ` L ) ) $=
          ( cv cltq wbr cnq cnpi c2nd cfv wcel w3a c1o cop ceq cec cplq co wrex
          crq ltrelnq brel simprd 3ad2ant2 breq2 rexbidv crab fveq2i nqex rabex
          weq op2nd eqtri elrab2 simprbi 3ad2ant3 ltsonq sotri reximdv sylanbrc
          wi expcom mpd ) AIPZJPZQRZVPHUAUBZUCZUDZVQSUCZDPZGUBZWCUEUFUGUHULUBZU
          IUJZVQQRZDTUKZVQVSUCVRAWBVTVRVPSUCZWBVPVQSSQUMUNUOUPWAWFVPQRZDTUKZWHV
          TAWKVRVTWIWKWFBPZQRZDTUKZWKBVPSVSBIVCWMWJDTWLVPWFQUQURVSKPWEUIUJWDQRD
          TUKZKSUSZWNBSUSZUFZUAUBWQHWRUAOUTWPWQWOKSVAVBWNBSVAVBVDVEZVFVGVHWAWJW
          GDTVRAWJWGVMVTWJVRWGWFVPVQQSVIUMVJVNUPVKVOWNWHBVQSVSBJVCWMWGDTWLVQWFQ
          UQURWSVFVL $.
      $}

      ${
        $d L r s $.  $d ph r s $.  $d F l r $.  $d F s u $.  $d L j r $.
        $d j l r s $.  $d j ph r $.  $d j r u $.
        $( Lemma for ~ caucvgpr .  The putative limit is rounded.  (Contributed
           by Jim Kingdon, 27-Sep-2020.) $)
        caucvgprlemrnd $p |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <->
              E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\
            A. r e. Q. ( r e. ( 2nd ` L ) <->
              E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) ) $=
          ( cv cfv wcel wa cnq c1st cltq wbr wrex caucvgprlemopl caucvgprlemlol
          wb wral c2nd ex 3expib rexlimdvw impbid caucvgprlemopu caucvgprlemupu
          ralrimivw jca ) AIPZHUAQZRZURJPZUBUCZVAUSRZSZJTUDZUGZITUHVAHUIQZRZVBU
          RVGRZSZITUDZUGZJTUHAVFITAUTVEAUTVEABCDEFGHIJKLMNOUEUJAVDUTJTAVBVCUTAB
          CDEFGHIJKLMNOUFUKULUMUPAVLJTAVHVKAVHVKABCDEFGHIJKLMNOUNUJAVJVHITAVBVI
          VHABCDEFGHIJKLMNOUOUKULUMUPUQ $.
      $}

      ${
        $d F j k $.  $d F j l $.  $d F j u $.  $d F n $.  $d L j k $.
        $d j k ph s $.  $d l s $.  $d s u $.  $d k n $.
        $( Lemma for ~ caucvgpr .  The putative limit is disjoint.
           (Contributed by Jim Kingdon, 27-Sep-2020.) $)
        caucvgprlemdisj $p |- ( ph -> A. s e. Q.
            -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) $=
          ( cfv cnq cplq cltq wbr cnpi cv c1st wcel c2nd wa c1o cop ceq cec crq
          wn co wrex wfal weq oveq1 breq1d rexbidv crab fveq2i nqex rabex op1st
          eqtri elrab2 simprbi opeq1 eceq1d fveq2d oveq2d fveq2 breq12d cbvrexv
          sylib breq2 anim12i reeanv sylibr adantl wf ad2antrr clti wral simprl
          op2nd wi simprr simplbi ad2antrl adantr caucvgprlemnkj rexlimdvva mpd
          pm2.21d inegd ralrimivw ) AIUAZHUBOZUCZWQHUDOZUCZUEZUKIPAXBAXBUEZWQEU
          AZUFUGZUHUIZUJOZQULZXDGOZRSZDUAZGOZXKUFUGZUHUIZUJOZQULZWQRSZUEZDTUMET
          UMZUNXBXSAXBXJETUMZXQDTUMZUEXSWSXTXAYAWSWQXOQULZXLRSZDTUMZXTWSWQPUCZY
          DJUAZXOQULZXLRSZDTUMZYDJWQPWRJIUOZYHYCDTYJYGYBXLRYFWQXOQUPUQURWRYIJPU
          SZXPBUAZRSZDTUMZBPUSZUGZUBOYKHYPUBNUTYKYOYIJPVAVBZYNBPVAVBZVCVDVEZVFY
          CXJDETDEUOZYBXHXLXIRYTXOXGWQQYTXNXFUJYTXMXEUHXKXDUFVGVHVIVJXKXDGVKVLV
          MVNXAYEYAYNYABWQPWTBIUOYMXQDTYLWQXPRVOURWTYPUDOYOHYPUDNUTYKYOYQYRWEVD
          VEVFVPXJXQEDTTVQVRVSXCXRUNEDTTXCXDTUCZXKTUCZUEZUEZXRUNUUDWQEFGXKXDATP
          GVTXBUUCKWAAFUAZXDWBSUUEGOZXIUUEUFUGUHUIUJOZQULRSXIUUFUUGQULRSUEWFETW
          CFTWCXBUUCLWAXCUUAUUBWDXCUUAUUBWGXCYEUUCWSYEAXAWSYEYDYSWHWIWJWKWNWLWM
          WOWP $.
      $}

      ${
        $d F f g h $.  $d F j l $.  $d F u $.  $d f g h m j ph r s x y $.
        $d m j l s $.  $d m j r u $.  $d L m x y $.

        $( Lemma for ~ caucvgpr .  The putative limit is located.  (Contributed
           by Jim Kingdon, 27-Sep-2020.) $)
        caucvgprlemloc $p |- ( ph -> A. s e. Q. A. r e. Q.
            ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) $=
          ( cltq wbr wcel cnq cplq vy vx vm vf vg vh cv c1st c2nd wo wi wa wceq
          cfv wrex ltexnqi adantl subhalfnqq ad2antrl c1o cop ceq cec archrecnq
          co crq cnpi simprr nnnq recclnq syl simplrl lt2addnq syl22anc simplrr
          mp2and ltsonq ltrelnq sotri syl2anc ad3antrrr ltanqi ad2antrr breqtrd
          addclnq wf ad5antr simprl ffvelcdmd wor w3a sowlin syl3anc mpd adantr
          simpr addassnqg breq1d mpbird wb ltanqg addcomnqg caovord2d weq opeq1
          mpan eceq1d fveq2d oveq2d fveq2 breq12d rspcev oveq1 crab fveq2i nqex
          rexbidv rabex op1st eqtri elrab2 sylanbrc oveq12d sylan breq2 orim12d
          ex op2nd rexlimddv ralrimivva ) AIUGZJUGZPQZYKHUHUNZRZYLHUIUNZRZUJZUK
          IJSSAYKSRZYLSRZULZULZYMYRUUBYMULZYKUAUGZTVEZYLUMZYRUASYMUUFUASUOUUBUA
          YKYLUPUQUUCUUDSRZUUFULZULZUBUGZUUJTVEZUUDPQZYRUBSUUGUULUBSUOUUCUUFUBU
          UDURUSUUIUUJSRZUULULZULZUCUGZUTVAZVBVCZVFUNZUUJPQZYRUCVGUUMUUTUCVGUOU
          UIUULUUJUCVDUSUUOUUPVGRZUUTULZULZYKUUSUUSTVEZTVEZUUPGUNZUUSTVEZPQZUVG
          YLPQZUJZYRUVCUVEYLPQZUVJUVCUVEUUEYLPUVCUVDUUDPQZYSUVEUUEPQUVCUVDUUKPQ
          ZUULUVLUVCUUTUUTUVMUUOUVAUUTVHZUVNUVCUUSSRZUUMUVOUUMUUTUUTULUVMUKUVAU
          VOUUOUUTUVAUURSRUVOUUPVIUURVJVKUSZUUIUUMUULUVBVLZUVPUVQUUSUUJUUSUUJVM
          VNVPUUIUUMUULUVBVOUVDUUKUUDPSVQVRVSVTUUCYSUUHUUNUVBAYSYTYMVLWAZUVDUUD
          YKWBVTUUIUUFUUNUVBUUCUUGUUFVHWCWDUVCUVESRZYTUVGSRZUVKUVJUKZUVCYSUVDSR
          ZUVSUVRUVCUVOUVOUWBUVPUVPUUSUUSWEVTYKUVDWEVTUUCYTUUHUUNUVBAYSYTYMVOWA
          ZUVCUVFSRZUVOUVTUVCVGSUUPGAVGSGWFUUAYMUUHUUNUVBLWGUUOUVAUUTWHZWIZUVPU
          VFUUSWEVTSPWJUVSYTUVTWKUWAVQSUVEYLUVGPWLXFWMWNUVCUVHYOUVIYQUVCUVHYOUV
          CUVHULZYSYKDUGZUTVAZVBVCZVFUNZTVEZUWHGUNZPQZDVGUOZYOUVCYSUVHUVRWOZUWG
          UVAYKUUSTVEZUVFPQZUWOUUOUVAUUTUVHVLUWGUWRUWQUUSTVEZUVGPQZUWGUWTUVHUVC
          UVHWPUWGUWSUVEUVGPUWGYSUVOUVOUWSUVEUMUWPUVCUVOUVHUVPWOZUXAYKUUSUUSWQW
          MWRWSUWGUDUEUFUWQUVFUUSPSTUDUGZSRZUEUGZSRZUFUGZSRWKUXBUXDPQUXFUXBTVEU
          XFUXDTVEPQWTUWGUXBUXDUXFXAUQUWGYSUVOUWQSRUWPUXAYKUUSWEVTUVCUWDUVHUWFW
          OUXAUXCUXEULUXBUXDTVEUXDUXBTVEUMUWGUXBUXDXBUQXCWSUWNUWRDUUPVGDUCXDZUW
          LUWQUWMUVFPUXGUWKUUSYKTUXGUWJUURVFUXGUWIUUQVBUWHUUPUTXEXGXHZXIUWHUUPG
          XJZXKXLVTKUGZUWKTVEZUWMPQZDVGUOZUWOKYKSYNKIXDZUXLUWNDVGUXNUXKUWLUWMPU
          XJYKUWKTXMWRXQYNUXMKSXNZUWMUWKTVEZBUGZPQZDVGUOZBSXNZVAZUHUNUXOHUYAUHO
          XOUXOUXTUXMKSXPXRZUXSBSXPXRZXSXTYAYBYGUVCUVIYQUVCUVIULYTUXPYLPQZDVGUO
          ZYQUVCYTUVIUWCWOUVCUVAUVIUYEUWEUYDUVIDUUPVGUXGUXPUVGYLPUXGUWMUVFUWKUU
          STUXIUXHYCWRXLYDUXSUYEBYLSYPBJXDUXRUYDDVGUXQYLUXPPYEXQYPUYAUIUNUXTHUY
          AUIOXOUXOUXTUYBUYCYHXTYAYBYGYFWNYIYIYIYGYJ $.
      $}

      ${
        $d A s a c d $.  $d F j l r a c d $.  $d F j s r u a c d $.  $d F n $.
        $d F j k $.  $d L j r s a c d $.  $d L k $.  $d ph r s a c d $.
        $d l r s $.  $d u r s $.  $d j k r s a c d $.  $d n k a c d $.
        $( Lemma for ~ caucvgpr .  The putative limit is a positive real.
           (Contributed by Jim Kingdon, 26-Sep-2020.) $)
        caucvgprlemcl $p |- ( ph -> L e. P. ) $=
          ( va cnq cfv cltq wbr cnpi cplq vs vr vd vc cpw cxp wcel cv c1st wrex
          c2nd wa wb wral wn wo wi w3a cnp weq fveq2 breq2d cbvralv c1o cop ceq
          sylib cec crq crab opeq1 eceq1d fveq2d oveq2d breq12d cbvrexv rabbiia
          co a1i oveq12d breq1d opeq12i eqtri caucvgprlemm wss nqex elpw2 mpbir
          opelxpi mp2an eqeltri jctil caucvgprlemrnd clti breq1 anbi12d imbi12d
          ssrab2 oveq1d cbvral2v caucvgprlemdisj caucvgprlemloc 3jca elnp1st2nd
          breq2 sylanbrc ) AHOUEZXGUFZUGZUAUHZHUIPZUGZUAOUJUBUHZHUKPZUGZUBOUJUL
          ZULXLXJXMQRZXMXKUGULUBOUJUMUAOUNXOXQXJXNUGZULUAOUJUMUBOUNULZXLXRULUOU
          AOUNZXQXLXOUPUQUBOUNUAOUNZURHUSUGAXPXIABCNEFGHUAUBIJKACDUHZGPZQRZDSUN
          CNUHZGPZQRZNSUNLYDYGDNSDNUTZYCYFCQYBYEGVAZVBVCVGZHIUHZYBVDVEZVFVHZVIP
          ZTVRZYCQRZDSUJZIOVJZYCYNTVRZBUHZQRZDSUJZBOVJZVEZYKYEVDVEZVFVHZVIPZTVR
          ZYFQRZNSUJZIOVJZYFUUGTVRZYTQRZNSUJZBOVJZVEMYRUUKUUCUUOYQUUJIOYQUUJUMY
          KOUGYPUUIDNSYHYOUUHYCYFQYHYNUUGYKTYHYMUUFVIYHYLUUEVFYBYEVDVKVLVMZVNYI
          VOVPVSVQUUBUUNBOUUBUUNUMYTOUGUUAUUMDNSYHYSUULYTQYHYCYFYNUUGTYIUUPVTWA
          VPVSVQWBWCZWDHUUDXHMYRXGUGZUUCXGUGZUUDXHUGUURYROWEYQIOWRYROWFWGWHUUSU
          UCOWEUUBBOWRUUCOWFWGWHYRUUCXGXGWIWJWKWLAXSXTYAABCNEFGHUAUBIJKYJUUQWMA
          BCNUCUDGHUAIJAFUHZEUHZWNRZUUTGPZUVAGPZUUTVDVEZVFVHZVIPZTVRZQRZUVDUVCU
          VGTVRZQRZULZUQZESUNFSUNUDUHZUCUHZWNRZUVNGPZUVOGPZUVNVDVEZVFVHZVIPZTVR
          ZQRZUVRUVQUWATVRZQRZULZUQZUCSUNUDSUNKUVMUWGUVNUVAWNRZUVQUVDUWATVRZQRZ
          UVDUWDQRZULZUQFEUDUCSSFUDUTZUVBUWHUVLUWLUUTUVNUVAWNWOUWMUVIUWJUVKUWKU
          WMUVCUVQUVHUWIQUUTUVNGVAZUWMUVGUWAUVDTUWMUVFUVTVIUWMUVEUVSVFUUTUVNVDV
          KVLVMZVNVOUWMUVJUWDUVDQUWMUVCUVQUVGUWATUWNUWOVTVBWPWQEUCUTZUWHUVPUWLU
          WFUVAUVOUVNWNXEUWPUWJUWCUWKUWEUWPUWIUWBUVQQUWPUVDUVRUWATUVAUVOGVAZWSV
          BUWPUVDUVRUWDQUWQWAWPWQWTVGYJUUQXAABCNEFGHUAUBIJKYJUUQXBXCHUBUAXDXF
          $.
      $}

      ${
        $d F m j r s t u v w z $.  $d F l $.  $d F n k $.
        $d L f g h s t x y $.  $d L k $.  $d L m j r s t v w z $.
        $d S f g h l s t u x y $.  $d S m j l r s t u v w z $.
        $d ph m r s t v w z $.  $d j k $.
        caucvgprlemladd.s $e |- ( ph -> S e. Q. ) $.
        $( Lemma for ~ caucvgpr .  Adding ` S ` after embedding in positive
           reals, or adding it as a rational.  (Contributed by Jim Kingdon,
           9-Oct-2020.) $)
        caucvgprlemladdfu $p |- ( ph ->
            ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_
            { u e. Q. | E. j e. N.
              ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S )
              <Q u } ) $=
          ( cltq wbr cplq cnq wcel vr vs vt vf vg vh vx vy vm vz vw cab cop cpp
          vv cv co c2nd cfv c1o ceq cec crq cnpi wrex crab wa cnp caucvgprlemcl
          wceq nqprlu syl df-iplp addclnq genpelvu syl2anc biimpa breq2 rexbidv
          wb fveq2i rabex op2nd eqtri elrab2 biimpi adantr adantl simpld ltnqex
          nqex vex gtnqex elab2 ltrelnq brel sylbi simprd ad2antll eleq1 mpbird
          fveq2 opeq1 eceq1d fveq2d oveq12d breq1d cbvrexv sylib ad5antr simplr
          ad2antrr wf ffvelcdmd nnnq recclnq 3syl syl3anc mpbid simpr addcomnqg
          ltanqg w3a caovord2d ltsonq sotri simpllr breqtrrd ex reximdva oveq1d
          mpd sylibr elrab sylanbrc rexlimdvva ssrdv ) AUAIJUPZDPQJULZDBUPZPQZB
          ULZUMZUNUQURUSZEUPZHUSZUUEUTUMZVAVBZVCUSZRUQZDRUQZYTPQZEVDVEZBSVFZAUA
          UPZUUDTZUUOUUNTZAUUPVGZUUOUBUPZUCUPZRUQZVJZUCUUCURUSZVEUBIURUSZVEZUUQ
          AUUPUVEAIVHTUUCVHTZUUPUVEVTABCEFGHIJKLMNVIADSTZUVFOBDJVKVLUDUEUFUGUHI
          UUCUUOUBUCUNRUGUHUFUEUDVMUEUPUFUPVNVOVPVQUURUVBUUQUBUCUVDUVCUURUUSUVD
          TZUUTUVCTZVGZVGZUVBUUQUVKUVBVGZUUOSTZUUKUUOPQZEVDVEZUUQUVLUVMUVASTZUV
          LUUSSTZUUTSTZUVPUVLUVQUUJUUSPQZEVDVEZUVKUVQUVTVGZUVBUVJUWAUURUVHUWAUV
          IUVHUWAUUJYTPQZEVDVEZUVTBUUSSUVDYTUUSVJUWBUVSEVDYTUUSUUJPVRVSUVDYRUUI
          RUQUUFPQEVDVEZJSVFZUWCBSVFZUMZURUSUWFIUWGURNWAUWEUWFUWDJSWKWBUWCBSWKW
          BWCWDWEWFWGWHWGZWIZUVKUVRUVBUVIUVRUURUVHUVIUVGUVRUVIDUUTPQZUVGUVRVGUU
          AUWJBUUTUVCUCWLYTUUTDPVRYSUUBJDWJBDWMWCWNZDUUTSSPWOWPWQWRWSWGZUUSUUTV
          NVPUVBUVMUVPVTUVKUUOUVASWTWHXAUVLUIUPZHUSZUWMUTUMZVAVBZVCUSZRUQZDRUQZ
          UUOPQZUIVDVEZUVOUVLUWRUUSPQZUIVDVEZUXAUVLUVTUXCUVLUVQUVTUWHWRUVSUXBEU
          IVDUUEUWMVJZUUJUWRUUSPUXDUUFUWNUUIUWQRUUEUWMHXBUXDUUHUWPVCUXDUUGUWOVA
          UUEUWMUTXCXDXEXFZXGXHXIUVLUXBUWTUIVDUVLUWMVDTZVGZUXBUWTUXGUXBVGZUWSUV
          AUUOPUXHUWSUWRUUTRUQZPQZUXIUVAPQZUWSUVAPQUXHUWJUXJUVLUWJUXFUXBUVKUWJU
          VBUVIUWJUURUVHUVIUWJUWKWFWSWGXLUXHUVGUVRUWRSTZUWJUXJVTAUVGUUPUVJUVBUX
          FUXBOXJUVLUVRUXFUXBUWLXLZUXHUWNSTUWQSTZUXLUXHVDSUWMHAVDSHXMUUPUVJUVBU
          XFUXBKXJUVLUXFUXBXKZXNUXHUXFUWPSTUXNUXOUWMXOUWPXPXQUWNUWQVNVPZDUUTUWR
          YBXRXSUXHUXBUXKUXGUXBXTUXHUJUKUOUWRUUSUUTPSRUJUPZSTZUKUPZSTZUOUPZSTYC
          UXQUXSPQUYAUXQRUQUYAUXSRUQPQVTUXHUXQUXSUYAYBWHUXPUVLUVQUXFUXBUWIXLUXM
          UXRUXTVGUXQUXSRUQUXSUXQRUQVJUXHUXQUXSYAWHYDXSUWSUXIUVAPSYEWOYFVPUVKUV
          BUXFUXBYGYHYIYJYLUVNUWTEUIVDUXDUUKUWSUUOPUXDUUJUWRDRUXEYKXGXHYMUUMUVO
          BUUOSYTUUOVJUULUVNEVDYTUUOUUKPVRVSYNYOYIYPYLYIYQ $.

        $d a b f g h l u $.  $d S a b $.  $d S k $.  $d a b j $.
        $d F a b f g h k n $.  $d L a b $.  $d A b $.  $d b a r $.
        $d f g h j $.  $d f g h r $.  $d f g h a b ph $.  $d F l $.
        $d r f g h a b k $.
        $( Lemma for ~ caucvgpr .  Adding ` S ` after embedding in positive
           reals, or adding it as a rational.  (Contributed by Jim Kingdon,
           8-Oct-2020.) $)
        caucvgprlemladdrl $p |- ( ph ->
            { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q
            ( ( F ` j ) +Q S ) } C_
            ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) $=
          ( cplq cltq cnpi cnq wcel va vr vb vf vg vh cv c1o cop ceq cec crq co
          cfv wbr wrex crab cab c1st wb opeq1 eceq1d fveq2d oveq2d fveq2 oveq1d
          cpp weq breq12d cbvrexv rabbiia wa oveq1 breq1d rexbidv elrab c2nd wf
          a1i ad4antr clti wi wral simpr simpllr caucvgprlemnbj wceq ffvelcdmda
          ad3antrrr nnnq recclnq 3syl addclnq syl2anc syl3anc w3a ltanqg adantl
          addassnqg ffvelcdmd addcomnqg caovord2d 3bitr4rd mtbird nrexdv breq2d
          intnand cbvralv sylib oveq12d eqtri syl caucvgprlemladdfu sseld breq2
          opeq12i ad2antlr imbitrdi mtod cnp caucvgprlemcl nqprlu addclpr prloc
          wo prop sylan sylancom ecased caucvgprlemcanl mpbid rexlimdva expimpd
          ex biimtrid ssrdv eqsstrid ) AJUGZEUGZUHUIZUJUKZULUNZPUMZYSHUNZDPUMZQ
          UOZERUPZJSUQYRUAUGZUHUIZUJUKZULUNZPUMZUUHHUNZDPUMZQUOZUARUPZJSUQZIYRD
          QUOJURDBUGZQUOBURUIVGUMUSUNZUUGUUPJSUUGUUPUTYRSTZUUFUUOEUAREUAVHZUUCU
          ULUUEUUNQUVAUUBUUKYRPUVAUUAUUJULUVAYTUUIUJYSUUHUHVAVBVCVDUVAUUDUUMDPY
          SUUHHVEVFVIVJVSVKAUBUUQUUSUBUGZUUQTUVBSTZUVBUUKPUMZUUNQUOZUARUPZVLAUV
          BUUSTZUUPUVFJUVBSJUBVHZUUOUVEUARUVHUULUVDUUNQYRUVBUUKPVMVNVOVPAUVCUVF
          UVGAUVCVLZUVEUVGUARUVIUUHRTZVLZUVEUVGUVKUVEVLZUVDIYRDUUKPUMZQUOJURUVM
          UURQUOBURUIZVGUMZUSUNZTZUVGUVLUVQUUNUVOVQUNZTZUVLUVSUUNSTZUCUGZHUNZUW
          AUHUIZUJUKZULUNZPUMZUVMPUMZUUNQUOZUCRUPZVLZUVLUWIUVTUVLUWHUCRUVLUWART
          ZVLZUWHUWFUUKPUMZUUMQUOZUWLUWAFGHUUHARSHVRZUVCUVJUVEUWKKVTZAGUGZFUGZW
          AUOUWQHUNZUWRHUNZUWQUHUIUJUKULUNZPUMQUOUWTUWSUXAPUMQUOVLWBFRWCGRWCZUV
          CUVJUVEUWKLVTUVLUWKWDZUVIUVJUVEUWKWEZWFUWLUWMDPUMZUUNQUOUWFUUKDPUMZPU
          MZUUNQUOUWNUWHUWLUXEUXGUUNQUWLUWFSTZUUKSTZDSTZUXEUXGWGUWLUWBSTUWESTZU
          XHUVLRSUWAHAUWOUVCUVJUVEKWIZWHUWLUWKUWDSTUXKUXCUWAWJUWDWKWLUWBUWEWMWN
          ZUWLUVJUUJSTZUXIUXDUUHWJZUUJWKZWLZAUXJUVCUVJUVEUWKOVTZUWFUUKDWSWOVNUW
          LUDUEUFUWMUUMDQSPUDUGZSTZUEUGZSTZUFUGZSTWPUXSUYAQUOUYCUXSPUMUYCUYAPUM
          QUOUTUWLUXSUYAUYCWQWRUWLUXHUXIUWMSTUXMUXQUWFUUKWMWNUWLRSUUHHUWPUXDWTU
          XRUXTUYBVLUXSUYAPUMUYAUXSPUMWGUWLUXSUYAXAWRXBUWLUWGUXGUUNQUWLUVMUXFUW
          FPUWLUXJUXIUVMUXFWGUXRUXQDUUKXAWNVDVNXCXDXEXGUVLUVSUUNUWGUURQUOZUCRUP
          ZBSUQZTUWJUVLUVRUYFUUNUVLBCUVMUCFGHIJUXLAUXBUVCUVJUVELWIACUWBQUOZUCRW
          CZUVCUVJUVEACUUDQUOZERWCUYHMUYIUYGEUCREUCVHZUUDUWBCQYSUWAHVEZXFXHXIWI
          IUUCUUDQUOZERUPZJSUQZUUDUUBPUMZUURQUOZERUPZBSUQZUIYRUWEPUMZUWBQUOZUCR
          UPZJSUQZUWFUURQUOZUCRUPZBSUQZUINUYNVUBUYRVUEUYMVUAJSUYMVUAUTUUTUYLUYT
          EUCRUYJUUCUYSUUDUWBQUYJUUBUWEYRPUYJUUAUWDULUYJYTUWCUJYSUWAUHVAVBVCZVD
          UYKVIVJVSVKUYQVUDBSUYQVUDUTUURSTUYPVUCEUCRUYJUYOUWFUURQUYJUUDUWBUUBUW
          EPUYKVUFXJVNVJVSVKXPXKUVLUXJUXIUVMSTZAUXJUVCUVJUVEOWIZUVJUXIUVIUVEUVJ
          UXNUXIUXOUXPXLXQZDUUKWMWNZXMXNUYEUWIBUUNSUURUUNWGUYDUWHUCRUURUUNUWGQX
          OVOVPXRXSUVKUVEUVOXTTZUVQUVSYEZUVLIXTTZUVNXTTZVUKAVUMUVCUVJUVEABCEFGH
          IJKLMNYAWIZUVLVUGVUNVUJBUVMJYBXLIUVNYCWNVUKUVPUVRUIXTTUVEVULUVOYFUVDU
          UNUVRUVPYDYGYHYIUVLBUUKUVBDIJVUOVUHAUVCUVJUVEWEVUIYJYKYNYLYMYOYPYQ $.
      $}

      ${
        caucvgprlemlim.q $e |- ( ph -> Q e. Q. ) $.
        caucvgprlemlim.jk $e |- ( ph -> J <N K ) $.
        caucvgprlemlim.jkq $e |- ( ph ->
          ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q ) $.
        ${
          $d F j l $.  $d F l u $.  $d K j l $.  $d K l u $.  $d Q j l $.
          $d Q l u $.  $d Q k $.  $d L j $.  $d L k $.  $d j l $.  $d j u $.
          $d F k n $.  $d j k $.
          $( Lemma for ~ caucvgpr .  Part of showing the putative limit to be a
             limit.  (Contributed by Jim Kingdon, 3-Oct-2020.) $)
          caucvgprlem1 $p |- ( ph ->
             <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >.
             <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) $=
            ( cltq cfv wbr cab cop cpp c1st wcel cltp c1o ceq cec crq cplq cnpi
            cv co wrex clti wa ltrelpi syl simprd caucvgprlemk ffvelcdmd ltanqi
            brel syl2anc opeq1 eceq1d fveq2d oveq2d fveq2 oveq1d breq12d rspcev
            cnq wceq crab oveq1 breq1d rexbidv elrab3 caucvgprlemladdrl sylbird
            wb sseld mpd cnp caucvgprlemcl nqprlu addclpr nqprl mpbid ) AJHUAZK
            LUOZDTUBLUCDBUOZTUBBUCUDZUEUPZUFUAZUGZWOWNTUBLUCWNWPTUBBUCUDWRUHUBZ
            AWNEUOZUIUDZUJUKZULUAZUMUPZXBHUAZDUMUPZTUBZEUNUQZWTAJUNUGZWNJUIUDZU
            JUKZULUAZUMUPZWNDUMUPZTUBZXJAIUNUGZXKAIJURUBXRXKUSRIJUNUNURUTVFVAVB
            ZAXNDTUBWNVPUGZXQADIJRSVCAUNVPJHMXSVDZXNDWNVEVGXIXQEJUNXBJVQZXFXOXH
            XPTYBXEXNWNUMYBXDXMULYBXCXLUJXBJUIVHVIVJVKYBXGWNDUMXBJHVLVMVNVOVGAX
            JWNWOXEUMUPZXHTUBZEUNUQZLVPVRZUGZWTAXTYGXJWEYAYEXJLWNVPWOWNVQZYDXIE
            UNYHYCXFXHTWOWNXEUMVSVTWAWBVAAYFWSWNABCDEFGHKLMNOPQWCWFWDWGAXTWRWHU
            GZWTXAWEYAAKWHUGWQWHUGZYIABCEFGHKLMNOPWIADVPUGYJQBDLWJVAKWQWKVGBWNW
            RLWLVGWM $.
        $}

        ${
          $d A j $.  $d F j u x $.  $d F l u x $.  $d F n $.  $d F k $.
          $d K j u x $.  $d K l u x $.  $d L x $.  $d L j $.  $d L k $.
          $d Q l u x $.  $d ph x $.  $d j l $.  $d j k $.  $d k n $.
          $( Lemma for ~ caucvgpr .  Part of showing the putative limit to be a
             limit.  (Contributed by Jim Kingdon, 3-Oct-2020.) $)
          caucvgprlem2 $p |- ( ph ->
              L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } ,
              { u | ( ( F ` K ) +Q Q ) <Q u } >. ) $=
            ( cnq vx cfv c1o cop ceq cec crq cplq co cv cltq wbr cltp wrex wcel
            cab caucvgprlemk cnpi clti ltrelpi brel syl simprd ffvelcdmd ltanqi
            syl2anc ltbtwnnqq sylib c2nd c1st simprl adantr simprrl fveq2 opeq1
            wa wceq eceq1d fveq2d oveq12d breq1d rspcev breq2 rexbidv crab nqex
            fveq2i rabex op2nd eqtri elrab2 sylanbrc simprrr elab sylibr ltnqex
            vex breq1 gtnqex op1st eleqtrrdi rspe syl12anc cnp wb caucvgprlemcl
            addclnq nqprlu ltdfpr mpbird rexlimddv ) AJHUBZJUCUDZUEUFZUGUBZUHUI
            ZUAUJZUKULZXQXLDUHUIZUKULZVPZKLUJZXSUKULZLUPZXSBUJZUKULBUPZUDZUMULZ
            UATAXPXSUKULZYAUATUNAXODUKULXLTUOZYIADIJRSUQAURTJHMAIURUOZJURUOZAIJ
            USULYKYLVPRIJURURUSUTVAVBVCZVDZXODXLVEVFUAXPXSVGVHAXQTUOZYAVPZVPZYH
            XQKVIUBZUOZXQYGVJUBZUOZVPZUATUNZYQYOYSUUAUUCAYOYAVKZYQYOEUJZHUBZUUE
            UCUDZUEUFZUGUBZUHUIZXQUKULZEURUNZYSUUDYQYLXRUULAYLYPYMVLAYOXRXTVMUU
            KXREJURUUEJVQZUUJXPXQUKUUMUUFXLUUIXOUHUUEJHVNUUMUUHXNUGUUMUUGXMUEUU
            EJUCVOVRVSVTWAWBVFUUJYEUKULZEURUNZUULBXQTYRYEXQVQUUNUUKEURYEXQUUJUK
            WCWDYRYBUUIUHUIUUFUKULEURUNZLTWEZUUOBTWEZUDZVIUBUURKUUSVIPWGUUQUURU
            UPLTWFWHUUOBTWFWHWIWJWKWLYQXQYDYTYQXTXQYDUOAYOXRXTWMYCXTLXQUAWQYBXQ
            XSUKWRWNWOYDYFLXSWPBXSWSWTXAUUBUATXBXCYQKXDUOZYGXDUOZYHUUCXEAUUTYPA
            BCEFGHKLMNOPXFVLAUVAYPAXSTUOZUVAAYJDTUOUVBYNQXLDXGVFBXSLXHVBVLKYGUA
            XIVFXJXK $.
        $}
      $}

      ${
        $d A j $.  $d F j u $.  $d F l u $.  $d F k n $.  $d j k ph x $.
        $d k l u x $.  $d j l $.  $d L j $.  $d L k $.
        $( Lemma for ~ caucvgpr .  The putative limit is a limit.  (Contributed
           by Jim Kingdon, 1-Oct-2020.) $)
        caucvgprlemlim $p |- ( ph ->
          A. x e. Q. E. j e. N. A. k e. N. ( j <N k ->
          ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >.
              <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\
            L <P <. { l | l <Q ( ( F ` k ) +Q x ) } ,
              { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) $=
          ( cv wbr cltq cab wa cnpi clti cfv cop cpp co cltp cplq wral wrex cnq
          wi wcel c1o ceq cec archrecnq adantl wf ad5antr simpr ad4antr simpllr
          crq caucvgprlem1 caucvgprlem2 jca ex ralrimiva reximdva mpd ) AEOZFOZ
          UAPZJOZVLHUBZQPJRVOCOZQPCRUCIVNBOZQPJRVQVPQPCRUCUDUEUFPZIVNVOVQUGUEZQ
          PJRVSVPQPCRUCUFPZSZUKZFTUHZETUIZBUJAVQUJULZSZVKUMUCUNUOVCUBVQQPZETUIZ
          WDWEWHAVQEUPUQWFWGWCETWFVKTULZSZWGWCWJWGSZWBFTWKVLTULZSZVMWAWMVMSZVRV
          TWNCDVQEFGHVKVLIJATUJHURWEWIWGWLVMKUSZAGOZVLUAPWPHUBZVOWPUMUCUNUOVCUB
          ZUGUEQPVOWQWRUGUEQPSUKFTUHGTUHWEWIWGWLVMLUSZADVKHUBQPETUHWEWIWGWLVMMU
          SZNWFWEWIWGWLVMAWEUTVAZWMVMUTZWJWGWLVMVBZVDWNCDVQEFGHVKVLIJWOWSWTNXAX
          BXCVEVFVGVHVGVIVJVH $.
      $}
    $}

    ${
      $d A j $.  $d F j k n l u x y z $.  $d j k ph x $.
      $( A Cauchy sequence of positive fractions with a modulus of convergence
         converges to a positive real.  This is basically Corollary 11.2.13 of
         [HoTT], p.  (varies) (one key difference being that this is for
         positive reals rather than signed reals).  Also, the HoTT book theorem
         has a modulus of convergence (that is, a rate of convergence)
         specified by (11.2.9) in HoTT whereas this theorem fixes the rate of
         convergence to say that all terms after the nth term must be within
         ` 1 / n ` of the nth term (it should later be able to prove versions
         of this theorem with a different fixed rate or a modulus of
         convergence supplied as a hypothesis).  We also specify that every
         term needs to be larger than a fraction ` A ` , to avoid the case
         where we have positive terms which "converge" to zero (which is not a
         positive real).

         This proof (including its lemmas) is similar to the proofs of
         ~ cauappcvgpr and ~ caucvgprpr .  Reading ~ cauappcvgpr first (the
         simplest of the three) might help understanding the other two.

         (Contributed by Jim Kingdon, 18-Jun-2020.) $)
      caucvgpr $p |- ( ph ->
          E. y e. P. A. x e. Q. E. j e. N. A. k e. N. ( j <N k ->
          ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >.
              <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\
            y <P <. { l | l <Q ( ( F ` k ) +Q x ) } ,
              { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) $=
        ( cv cplq co cltq wbr cnpi cnq vz c1o cop ceq cec crq cfv wrex crab cnp
        wcel clti cab cpp cltp wa wi wral wceq opeq1 eceq1d fveq2d oveq2d fveq2
        breq12d cbvrexv a1i rabbiia oveq12d breq1d caucvgprlemcl caucvgprlemlim
        wb opeq12i oveq1 breq2d breq1 anbi12d imbi2d rexralbidv ralbidv syl2anc
        rspcev ) AJNZUANZUBUCZUDUEZUFUGZOPZWEIUGZQRZUASUHZJTUIZWJWHOPZDNZQRZUAS
        UHZDTUIZUCZUJUKFNZGNZULRZWDXAIUGZQRJUMXCWOQRDUMUCZWSWDBNZQRJUMXEWOQRDUM
        UCZUNPZUORZWSWDXCXEOPZQRJUMXIWOQRDUMUCZUORZUPZUQZGSURFSUHZBTURZXBXDCNZX
        FUNPZUORZXPXJUORZUPZUQZGSURFSUHZBTURZCUJUHADEFGHIWSJKLMWMWDWTUBUCZUDUEZ
        UFUGZOPZWTIUGZQRZFSUHZJTUIWRYHYFOPZWOQRZFSUHZDTUIWLYJJTWLYJVMWDTUKWKYIU
        AFSWEWTUSZWIYGWJYHQYNWHYFWDOYNWGYEUFYNWFYDUDWEWTUBUTVAVBZVCWEWTIVDZVEVF
        VGVHWQYMDTWQYMVMWOTUKWPYLUAFSYNWNYKWOQYNWJYHWHYFOYPYOVIVJVFVGVHVNZVKABD
        EFGHIWSJKLMYQVLYCXOCWSUJXPWSUSZYBXNBTYRYAXMFGSSYRXTXLXBYRXRXHXSXKYRXQXG
        XDUOXPWSXFUNVOVPXPWSXJUOVQVRVSVTWAWCWB $.
    $}
  $}

  ${
    $d J l $.  $d J u $.  $d K l $.  $d K u $.
    caucvgprprlemk.jk $e |- ( ph -> J <N K ) $.
    caucvgprprlemk.jkq $e |- ( ph ->
      <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
        { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >.
      <P Q ) $.
    $( Lemma for ~ caucvgprpr .  Reciprocals of positive integers decrease as
       the positive integers increase.  (Contributed by Jim Kingdon,
       28-Nov-2020.) $)
    caucvgprprlemk $p |- ( ph ->
        <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } ,
          { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >.
        <P Q ) $=
      ( cv c1o cop ceq cec crq cfv cltq wbr cab cltp cnpi clti wcel brel ltnnnq
      wa wb ltrelpi 3syl mpbid ltrnqi ltnqpri cnp ltsopr ltrelpr sotri syl2anc
      ) AFIZEJKLMZNOZPQFRUSBIZPQBRKZUQDJKLMZNOZPQFRVCUTPQBRKZSQZVDCSQVACSQAVBUR
      PQZUSVCPQVEADEUAQZVFGAVGDTUBETUBUEVGVFUFGDETTUAUGUCDEUDUHUIVBURUJBUSVCFUK
      UHHVAVDCSULUMUNUOUP $.
  $}

  ${
    $d M l u $.  $d S l u $.  $d T l u $.
    caucvgprprlemloccalc.st $e |- ( ph -> S <Q T ) $.
    caucvgprprlemloccalc.y $e |- ( ph -> Y e. Q. ) $.
    caucvgprprlemloccalc.syt $e |- ( ph -> ( S +Q Y ) = T ) $.
    caucvgprprlemloccalc.x $e |- ( ph -> X e. Q. ) $.
    caucvgprprlemloccalc.xxy $e |- ( ph -> ( X +Q X ) <Q Y ) $.
    caucvgprprlemloccalc.m $e |- ( ph -> M e. N. ) $.
    caucvgprprlemloccalc.mx $e |- ( ph ->
      ( *Q ` [ <. M , 1o >. ] ~Q ) <Q X ) $.
    $( Lemma for ~ caucvgprpr .  Rearranging some expressions for
       ~ caucvgprprlemloc .  (Contributed by Jim Kingdon, 8-Feb-2021.) $)
    caucvgprprlemloccalc $p |- ( ph ->
        ( <. { l | l <Q ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } ,
          { u | ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >.
          +P. <. { l | l <Q ( *Q ` [ <. M , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. M , 1o >. ] ~Q ) <Q u } >. )
        <P <. { l | l <Q T } , { u | T <Q u } >. ) $=
      ( co cltq wbr cab cnq c1o cop ceq cec crq cfv cplq cltp wcel wceq ltrelnq
      cv cpp wa brel simpld cnpi nnnq recclnq addclnq syl2anc addnqpr addassnqg
      syl 3syl syl3anc wi lt2addnq syl22anc mp2and ltsonq sotri breqtrd eqbrtrd
      ltanqi ltnqpri eqbrtrrd ) AHULZCEUAUBUCUDZUEUFZUGPZVTUGPZQRHSWBBULZQRBSUB
      ZVRWAQRHSWAWCQRBSUBVRVTQRHSVTWCQRBSUBUMPZVRDQRHSDWCQRBSUBZUHAWATUIZVTTUIZ
      WDWEUJACTUIZWHWGAWIDTUIZACDQRWIWJUNICDTTQUKUOVDUPZAEUQUIVSTUIWHNEURVSUSVE
      ZCVTUTVAWLBWAVTHVBVAAWBDQRWDWFUHRAWBCVTVTUGPZUGPZDQAWIWHWHWBWNUJWKWLWLCVT
      VTVCVFAWNCGUGPZDQAWMGQRZWIWNWOQRAWMFFUGPZQRZWQGQRWPAVTFQRZWSWROOAWHFTUIZW
      HWTWSWSUNWRVGWLLWLLVTFVTFVHVIVJMWMWQGQTVKUKVLVAWKWMGCVOVAKVMVNBWBDHVPVDVQ
      $.
  $}

  ${
    $d F a b $.  $d F l r $.  $d F u r $.  $d X a b p $.  $d X l p r $.
    $d X p r u $.  $d X a b q $.  $d a p r $.  $d l q r $.  $d q r u $.
    caucvgprprlemell.lim $e |- L = <.
      { l e. Q. | E. r e. N.
        <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } ,
          { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q }
        >. <P ( F ` r ) } ,
      { u e. Q. | E. r e. N.
        ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } ,
          { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P
        <. { p | p <Q u } , { q | u <Q q } >. }
      >. $.
    $( Lemma for ~ caucvgprpr .  Membership in the lower cut of the putative
       limit.  (Contributed by Jim Kingdon, 21-Jan-2021.) $)
    caucvgprprlemell $p |- ( X e. ( 1st ` L ) <-> ( X e. Q. /\
        E. b e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } ,
          { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q }
        >. <P ( F ` b ) ) ) $=
      ( cfv cnq cv cop cplq cltq wbr cab cltp cnpi va c1st wcel c1o ceq cec crq
      co wrex wa wceq breq2d abbidv breq1d opeq12d rexbidv crab cpp fveq2i nqex
      oveq1 rabex op1st eqtri elrab2 opeq1 eceq1d fveq2d oveq2d breq12d cbvrexv
      fveq2 bitri anbi2i ) DCUBKZUCDLUCZGMZDEMZUDNZUEUFZUGKZOUHZPQZGRZWBFMZPQZF
      RZNZVRBKZSQZETUIZUJVPVQDHMZUDNZUEUFZUGKZOUHZPQZGRZWPWEPQZFRZNZWLBKZSQZHTU
      IZUJVQIMZWAOUHZPQZGRZXFWEPQZFRZNZWISQZETUIZWKIDLVOXEDUKZXLWJETXNXKWHWISXN
      XHWDXJWGXNXGWCGXNXFWBVQPXEDWAOVAZULUMXNXIWFFXNXFWBWEPXOUNUMUOUNUPVOXMILUQ
      ZWIVQWAPQGRWAWEPQFRNURUHVQAMZPQGRXQWEPQFRNSQETUIZALUQZNZUBKXPCXTUBJUSXPXS
      XMILUTVBXRALUTVBVCVDVEWKXDVPWKVQDUAMZUDNZUEUFZUGKZOUHZPQZGRZYEWEPQZFRZNZY
      ABKZSQZUATUIXDWJYLEUATVRYAUKZWHYJWIYKSYMWDYGWGYIYMWCYFGYMWBYEVQPYMWAYDDOY
      MVTYCUGYMVSYBUEVRYAUDVFVGVHVIZULUMYMWFYHFYMWBYEWEPYNUNUMUOVRYABVLVJVKYLXC
      UAHTYAWLUKZYJXAYKXBSYOYGWRYIWTYOYFWQGYOYEWPVQPYOYDWODOYOYCWNUGYOYBWMUEYAW
      LUDVFVGVHVIZULUMYOYHWSFYOYEWPWEPYPUNUMUOYAWLBVLVJVKVMVNVM $.

    $( Lemma for ~ caucvgprpr .  Membership in the upper cut of the putative
       limit.  (Contributed by Jim Kingdon, 28-Jan-2021.) $)
    caucvgprprlemelu $p |- ( X e. ( 2nd ` L ) <-> ( X e. Q. /\ E. b e. N.
        ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } ,
          { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P
        <. { p | p <Q X } , { q | X <Q q } >. ) ) $=
      ( cfv cnq cv cop cltq wbr cab cltp cnpi abbidv va c2nd c1o ceq cec crq co
      wcel cpp wrex wa wceq breq2 breq1 opeq12d breq2d rexbidv cplq crab fveq2i
      rabex op2nd eqtri elrab2 fveq2 opeq1 eceq1d fveq2d breq1d oveq12d cbvrexv
      nqex bitri anbi2i ) DCUBKZUHDLUHZEMZBKZGMZVQUCNZUDUEZUFKZOPZGQZWBFMZOPZFQ
      ZNZUIUGZVSDOPZGQZDWEOPZFQZNZRPZESUJZUKVPHMZBKZVSWQUCNZUDUEZUFKZOPZGQZXAWE
      OPZFQZNZUIUGZWNRPZHSUJZUKWIVSAMZOPZGQZXJWEOPZFQZNZRPZESUJZWPADLVOXJDULZXP
      WOESXRXOWNWIRXRXLWKXNWMXRXKWJGXJDVSOUMTXRXMWLFXJDWEOUNTUOUPUQVOVSIMWBURUG
      ZOPGQXSWEOPFQNVRRPESUJZILUSZXQALUSZNZUBKYBCYCUBJUTYAYBXTILVLVAXQALVLVAVBV
      CVDWPXIVPWPUAMZBKZVSYDUCNZUDUEZUFKZOPZGQZYHWEOPZFQZNZUIUGZWNRPZUASUJXIWOY
      OEUASVQYDULZWIYNWNRYPVRYEWHYMUIVQYDBVEYPWDYJWGYLYPWCYIGYPWBYHVSOYPWAYGUFY
      PVTYFUDVQYDUCVFVGVHZUPTYPWFYKFYPWBYHWEOYQVITUOVJVIVKYOXHUAHSYDWQULZYNXGWN
      RYRYEWRYMXFUIYDWQBVEYRYJXCYLXEYRYIXBGYRYHXAVSOYRYGWTUFYRYFWSUDYDWQUCVFVGV
      HZUPTYRYKXDFYRYHXAWEOYSVITUOVJVIVKVMVNVM $.
  $}

  ${
    caucvgprpr.f $e |- ( ph -> F : N. --> P. ) $.
    caucvgprpr.cau $e |- ( ph ->
      A. n e. N. A. k e. N. ( n <N k ->
        ( ( F ` n ) <P ( ( F ` k ) +P.
          <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
        ) /\
        ( F ` k ) <P ( ( F ` n ) +P.
          <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
        ) ) ) ) $.
    ${
      $d F a b k $.  $d F a k n $.  $d a b k l $.  $d a b k u $.  $d l n $.
      $d n u $.
      $( Lemma for ~ caucvgprpr .  Change bound variables in Cauchy condition.
         (Contributed by Jim Kingdon, 12-Feb-2021.) $)
      caucvgprprlemcbv $p |- ( ph ->
        A. a e. N. A. b e. N. ( a <N b ->
          ( ( F ` a ) <P ( ( F ` b ) +P.
            <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } ,
              { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >.
          ) /\
          ( F ` b ) <P ( ( F ` a ) +P.
            <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } ,
              { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >.
          ) ) ) ) $=
        ( cv clti wbr cfv cop cltq cpp co cltp cnpi c1o ceq cec crq cab wa wral
        weq breq1 fveq2 opeq1 eceq1d fveq2d breq2d abbidv breq1d opeq12d oveq2d
        wi breq12d oveq12d anbi12d imbi12d breq2 oveq1d cbvral2v sylib ) ADKZCK
        ZLMZVHENZVIENZHKZVHUAOZUBUCZUDNZPMZHUEZVPBKZPMZBUEZOZQRZSMZVLVKWBQRZSMZ
        UFZUSZCTUGDTUGFKZGKZLMZWIENZWJENZVMWIUAOZUBUCZUDNZPMZHUEZWPVSPMZBUEZOZQ
        RZSMZWMWLXAQRZSMZUFZUSZGTUGFTUGJWHXGWIVILMZWLVLXAQRZSMZVLXDSMZUFZUSDCFG
        TTDFUHZVJXHWGXLVHWIVILUIXMWDXJWFXKXMVKWLWCXISVHWIEUJZXMWBXAVLQXMVRWRWAW
        TXMVQWQHXMVPWPVMPXMVOWOUDXMVNWNUBVHWIUAUKULUMZUNUOXMVTWSBXMVPWPVSPXOUPU
        OUQZURUTXMWEXDVLSXMVKWLWBXAQXNXPVAUNVBVCCGUHZXHWKXLXFVIWJWILVDXQXJXCXKX
        EXQXIXBWLSXQVLWMXAQVIWJEUJZVEUNXQVLWMXDSXRUPVBVCVFVG $.
    $}

    ${
      $d A a b l $.  $d A a b u $.  $d A l p $.  $d A q u $.  $d B b $.
      $d F a b k $.  $d F a k n $.  $d k l n $.  $d k n u $.  $d a b k u $.
      $( Lemma for ~ caucvgprpr .  Cauchy condition expressed in terms of
         classes.  (Contributed by Jim Kingdon, 3-Mar-2021.) $)
      caucvgprprlemval $p |- ( ( ph /\ A <N B ) ->
          ( ( F ` A ) <P ( ( F ` B ) +P.
            <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } ,
              { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >.
          ) /\
          ( F ` B ) <P ( ( F ` A ) +P.
            <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } ,
              { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >.
          ) ) ) $=
        ( clti wbr wa cltq cpp co cltp cnpi va cfv c1o cop ceq cec crq cab wcel
        vb cv wral ltrelpi brel adantl caucvgprprlemcbv adantr simpr wceq breq1
        fveq2 eceq1d fveq2d breq2d abbidv breq1d opeq12d oveq2d breq12d oveq12d
        wi opeq1 anbi12d imbi12d breq2 oveq1d rspc2v syl3c cbvabv oveq2i breq2i
        opeq12i anbi12i sylib ) ACDMNZOZCGUBZDGUBZJUKZCUCUDZUEUFZUGUBZPNZJUHZWL
        BUKZPNZBUHZUDZQRZSNZWHWGWRQRZSNZOZWGWHIUKZWLPNZIUHZWLHUKZPNZHUHZUDZQRZS
        NZWHWGXJQRZSNZOWFCTUIDTUIOZUAUKZUJUKZMNZXPGUBZXQGUBZWIXPUCUDZUEUFZUGUBZ
        PNZJUHZYCWOPNZBUHZUDZQRZSNZXTXSYHQRZSNZOZVKZUJTULUATULZWEXCWEXOACDTTMUM
        UNUOAYOWEABEFGUAUJJKLUPUQAWEURYNWEXCVKCXQMNZWGXTWRQRZSNZXTXASNZOZVKUAUJ
        CDTTXPCUSZXRYPYMYTXPCXQMUTUUAYJYRYLYSUUAXSWGYIYQSXPCGVAZUUAYHWRXTQUUAYE
        WNYGWQUUAYDWMJUUAYCWLWIPUUAYBWKUGUUAYAWJUEXPCUCVLVBVCZVDVEUUAYFWPBUUAYC
        WLWOPUUCVFVEVGZVHVIUUAYKXAXTSUUAXSWGYHWRQUUBUUDVJVDVMVNXQDUSZYPWEYTXCXQ
        DCMVOUUEYRWTYSXBUUEYQWSWGSUUEXTWHWRQXQDGVAZVPVDUUEXTWHXASUUFVFVMVNVQVRW
        TXLXBXNWSXKWGSWRXJWHQWNXFWQXIWMXEJIWIXDWLPUTVSWPXHBHWOXGWLPVOVSWBZVTWAX
        AXMWHSWRXJWGQUUGVTWAWCWD $.
    $}

    ${
      $d F f g h $.  $d F k n $.  $d J f g h $.  $d K f g h $.  $d S f g h $.
      $d f g h ph $.
      caucvgprprlemnkj.k $e |- ( ph -> K e. N. ) $.
      caucvgprprlemnkj.j $e |- ( ph -> J e. N. ) $.
      caucvgprprlemnkj.s $e |- ( ph -> S e. Q. ) $.
      ${
        $d J f g h p q $.  $d K f g h p q $.  $d K l p $.  $d K q u $.
        $d S f g h p q $.  $d k l n $.  $d k n u $.
        $( Lemma for ~ caucvgprpr .  Part of disjointness.  (Contributed by Jim
           Kingdon, 12-Feb-2021.) $)
        caucvgprprlemnkltj $p |- ( ( ph /\ K <N J ) -> -.
            ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } ,
                { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >.
              <P ( F ` K ) /\
            ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
              { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P
              <. { p | p <Q S } , { q | S <Q q } >. ) ) $=
          ( wbr cltp cnp wcel vf vg vh clti wa cv c1o cop ceq cec crq cplq cltq
          cfv co cab ltsopr ltrelpr simprl caucvgprprlemval simpld adantr sotri
          cpp son2lpi syl2anc w3a wb ltaprg adantl cnq ad2antrr nqprlu syl cnpi
          ffvelcdmd recnnpr wceq addcomprg caovord2d mpbird ltaddpr simprr mtoi
          jca ex nnnq recclnq 3syl addnqpr breq1d anbi1d mtbird ) AHGUDQZUEZJUF
          ZCHUGUHUIUJZUKUNZULUOZUMQJUPWSIUFZUMQIUPUHZHFUNZRQZGFUNZWPGUGUHUIUJUK
          UNZUMQJUPXEWTUMQIUPUHZVDUOZWPCUMQJUPCWTUMQIUPUHZRQZUEXHWPWRUMQJUPWRWT
          UMQIUPUHZVDUOZXBRQZXIUEZWOXMXHXDRQZXDXHRQZUEZXHXDRSUQURVEWOXMXPWOXMUE
          ZXNXOXQXNXKXDXJVDUOZRQZXQXLXBXRRQZXSWOXLXIUSWOXTXMWOXTXDXBXJVDUORQABH
          GDEFIJKLMUTVAVBXKXBXRRSUQURVCVFXQUAUBUCXHXDXJRSVDUAUFZSTZUBUFZSTZUCUF
          ZSTVGYAYCRQYEYAVDUOYEYCVDUORQVHXQYAYCYEVIVJXQCVKTZXHSTAYFWNXMPVLICJVM
          VNAXDSTZWNXMAVOSGFLOVPVLZAXJSTZWNXMAHVOTZYINIHJVQVNVLYBYDUEYAYCVDUOYC
          YAVDUOVRXQYAYCVSVJVTWAXQXDXGRQZXIXOXQYGXFSTZYKYHAYLWNXMAGVOTYLOIGJVQV
          NVLXDXFWBVFWOXLXIWCXDXGXHRSUQURVCVFWEWFWDWOXCXLXIWOXAXKXBRWOYFWRVKTZX
          AXKVRAYFWNPVBAYMWNAYJWQVKTYMNHWGWQWHWIVBICWRJWJVFWKWLWM $.
      $}

      ${
        $d J p q $.  $d K p q $.  $d S p q $.
        $( Lemma for ~ caucvgprpr .  Part of disjointness.  (Contributed by Jim
           Kingdon, 12-Feb-2021.) $)
        caucvgprprlemnkeqj $p |- ( ( ph /\ K = J ) -> -.
            ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } ,
                { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >.
              <P ( F ` K ) /\
            ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
              { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P
              <. { p | p <Q S } , { q | S <Q q } >. ) ) $=
          ( cltq wbr cab cltp wceq wa cv c1o cop ceq cec crq cfv cplq co ltsopr
          cpp cnp ltrelpr son2lpi wcel cnpi ffvelcdmd ad2antrr cnq nnnq recclnq
          adantr syl nqprlu ltaddpr syl2anc simprr sotri simprl wb breq1d mpbid
          addnqpr jca ex opeq1 eceq1d fveq2d oveq2d breq2d abbidv opeq12d fveq2
          mtoi breq12d anbi1d adantl mtbird ) AHGUAZUBZJUCZCHUDUEZUFUGZUHUIZUJU
          KZQRZJSZWQIUCZQRZISZUEZHFUIZTRZGFUIZWMGUDUEZUFUGZUHUIZQRJSXIWTQRISUEZ
          UMUKZWMCQRJSCWTQRISUEZTRZUBZWMCXIUJUKZQRZJSZXOWTQRZISZUEZXFTRZXMUBZWL
          YBXFXLTRZXLXFTRZUBZXFXLTUNULUOUPWLYBYEWLYBUBZYCYDYFXFXKTRZXMYCYFXFUNU
          QZXJUNUQZYGAYHWKYBAURUNGFLOUSUTWLYIYBWLXIVAUQZYIWLXHVAUQZYJWLGURUQZYK
          AYLWKOVDGVBVEXHVCVEZIXIJVFVEZVDXFXJVGVHWLYAXMVIXFXKXLTUNULUOVJVHYFXLX
          LXJUMUKZTRZYOXFTRZYDWLYPYBWLXLUNUQZYIYPWLCVAUQZYRAYSWKPVDZICJVFVEYNXL
          XJVGVHVDYFYAYQWLYAXMVKWLYAYQVLYBWLXTYOXFTWLYSYJXTYOUAYTYMICXIJVOVHVMV
          DVNXLYOXFTUNULUOVJVHVPVQWFWKXNYBVLAWKXEYAXMWKXCXTXDXFTWKWSXQXBXSWKWRX
          PJWKWQXOWMQWKWPXICUJWKWOXHUHWKWNXGUFHGUDVRVSVTWAZWBWCWKXAXRIWKWQXOWTQ
          UUAVMWCWDHGFWEWGWHWIWJ $.
      $}

      ${
        $d J l p $.  $d J q u $.  $d K p $.  $d K q $.  $d S p $.  $d S q $.
        $d k u $.  $d k l n $.  $d n u $.
        $( Lemma for ~ caucvgprpr .  Part of disjointness.  (Contributed by Jim
           Kingdon, 12-Feb-2021.) $)
        caucvgprprlemnjltk $p |- ( ( ph /\ J <N K ) -> -.
            ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } ,
                { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >.
              <P ( F ` K ) /\
            ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
              { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P
              <. { p | p <Q S } , { q | S <Q q } >. ) ) $=
          ( wbr cltq cab cltp clti wa cv c1o cop ceq cec crq cfv cplq co ltsopr
          cpp cnp ltrelpr son2lpi wcel ad2antrr recclnq ltaddnq syl2anc ltnqpri
          cnq cnpi nnnq 3syl simprl caucvgprprlemval simprd adantr sotri simprr
          syl jca ex mtoi ) AGHUAQZUBZJUCZCHUDUEUFUGZUHUIZUJUKZRQJSWBIUCZRQISUE
          ZHFUIZTQZGFUIZVSGUDUEUFUGUHUIZRQJSWHWCRQISUEZUMUKZVSCRQJSCWCRQISUEZTQ
          ZUBZWKWJTQZWLUBZWKWJTUNULUOUPVRWMWOVRWMUBZWNWLWPWKWDTQZWDWJTQZWNWPCWB
          RQZWQWPCVCUQZWAVCUQZWSAWTVQWMPURWPHVDUQZVTVCUQXAAXBVQWMNURHVEVTUSVFCW
          AUTVAICWBJVBVMWPWFWEWJTQZWRVRWFWLVGVRXCWMVRWGWEWIUMUKTQXCABGHDEFIJKLM
          VHVIVJWDWEWJTUNULUOVKVAWKWDWJTUNULUOVKVAVRWFWLVLVNVOVP $.
      $}

      ${
        $d J p q $.  $d J u $.  $d J l $.  $d K p q $.  $d K l $.  $d K u $.
        $d S p q $.  $d n u $.  $d l n $.  $d k l $.  $d k u $.  $d q u $.
        $d p l $.
        $( Lemma for ~ caucvgprpr .  Part of disjointness.  (Contributed by Jim
           Kingdon, 20-Jan-2021.) $)
        caucvgprprlemnkj $p |- ( ph -> -.
            ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } ,
                { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >.
              <P ( F ` K ) /\
            ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
              { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P
              <. { p | p <Q S } , { q | S <Q q } >. ) ) $=
          ( wbr cop cltq cab clti cv c1o ceq cec crq cfv cplq co cltp cpp wa wn
          caucvgprprlemnkltj caucvgprprlemnkeqj caucvgprprlemnjltk w3o pitri3or
          wceq cnpi wcel syl2anc mpjao3dan ) AHGUAQZJUBZCHUCRUDUEUFUGUHUIZSQJTV
          FIUBZSQITRHFUGUJQGFUGVEGUCRUDUEUFUGZSQJTVHVGSQITRUKUIVECSQJTCVGSQITRU
          JQULUMHGUSZGHUAQZABCDEFGHIJKLMNOPUNABCDEFGHIJKLMNOPUOABCDEFGHIJKLMNOP
          UPAHUTVAGUTVAVDVIVJUQNOHGURVBVC $.
      $}
    $}

    ${
      $d B k l n $.  $d B l p $.  $d B p q x y z $.  $d B k n u $.  $d F k n $.
      $d F x y z $.  $d J k l n $.  $d J l p $.  $d J p q x y z $.
      $d J k n u $.  $d ph x y z $.  $d q u $.
      caucvgprprlemnbj.b $e |- ( ph -> B e. N. ) $.
      caucvgprprlemnbj.j $e |- ( ph -> J e. N. ) $.
      $( Lemma for ~ caucvgprpr .  Non-existence of two elements of the
         sequence which are too far from each other.  (Contributed by Jim
         Kingdon, 17-Jun-2021.) $)
      caucvgprprlemnbj $p |- ( ph ->
          -. ( ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >. )
            +P. <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. )
            <P ( F ` J ) ) $=
        ( vp cltq wbr cpp co cltp cnp wcel vq vx vy cfv c1o cop ceq cec crq cab
        vz cv wn clti wceq wa caucvgprprlemval simprd ffvelcdmd recnnpr addclpr
        cnpi syl2anc ltaddpr adantr ltsopr ltrelpr sotri wb fveq2 breq1d adantl
        syl simpld w3a ltaprg addcomprg caovord2d w3o pitri3or mpjao3dan wi wor
        mpbid so2nr mpan imnan sylibr breq1 cbvabv breq2 opeq12i oveq2i oveq12i
        mpd breq1i sylnib ) ACFUDZMULZCUEUFUGUHUIUDZNOZMUJZWTUAULZNOZUAUJZUFZPQ
        ZWSGUEUFUGUHUIUDZNOZMUJZXHXCNOZUAUJZUFZPQZGFUDZROZWRHULZWTNOZHUJZWTBULZ
        NOZBUJZUFZPQZXQXHNOZHUJZXHXTNOZBUJZUFZPQZXOROAXOXNROZXPUMZACGUNOZYKCGUO
        ZGCUNOZAYMUPZXOXGROZXGXNROZYKYPWRXOXFPQROYQABCGDEFUAMHIJUQURAYRYMAXGSTZ
        XMSTZYRAWRSTZXFSTZYSAVBSCFIKUSZACVBTZUUBKUACMUTVMZWRXFVAVCZAGVBTZYTLUAG
        MUTVMZXGXMVDVCZVEXOXGXNRSVFVGVHZVCAYNUPZYQYRYKUUKWRXGROZYQAUULYNAUUAUUB
        UULUUCUUEWRXFVDVCZVEYNUULYQVIAYNWRXOXGRCGFVJVKVLWDAYRYNUUIVEUUJVCAYOUPZ
        XOWRXMPQZROZUUOXNROZYKUUNUUPWRXOXMPQROABGCDEFUAMHIJUQVNAUUQYOAUULUUQUUM
        AUBUCUKWRXGXMRSPUBULZSTZUCULZSTZUKULZSTVOUURUUTROUVBUURPQUVBUUTPQROVIAU
        URUUTUVBVPVLUUCUUFUUHUUSUVAUPUURUUTPQUUTUURPQUOAUURUUTVQVLVRWDVEXOUUOXN
        RSVFVGVHVCAUUDUUGYMYNYOVSKLCGVTVCWAAYKXPUPUMZYKYLWBAXOSTZXNSTZUVCAVBSGF
        ILUSAYSYTUVEUUFUUHXGXMVAVCSRWCUVDUVEUPUVCVFSXOXNRWEWFVCYKXPWGWHWOXNYJXO
        RXGYDXMYIPXFYCWRPXBXSXEYBXAXRMHWSXQWTNWIWJXDYAUABXCXTWTNWKWJWLWMXJYFXLY
        HXIYEMHWSXQXHNWIWJXKYGUABXCXTXHNWKWJWLWNWPWQ $.
    $}

    $d A m $.  $d F m $.
    caucvgprpr.bnd $e |- ( ph -> A. m e. N. A <P ( F ` m ) ) $.

    ${
      caucvgprpr.lim $e |- L = <.
        { l e. Q. | E. r e. N.
          <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } ,
            { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q }
          >. <P ( F ` r ) } ,
        { u e. Q. | E. r e. N.
          ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } ,
            { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P
          <. { p | p <Q u } , { q | u <Q q } >. }
        >. $.

      ${
        $d A m r $.  $d A r s x $.  $d F l $.  $d F m $.  $d L x $.
        $d l p q r s $.  $d l u x $.  $d ph r s x $.
        $( Lemma for ~ caucvgprpr .  The lower cut of the putative limit is
           inhabited.  (Contributed by Jim Kingdon, 29-Dec-2020.) $)
        caucvgprprlemml $p |- ( ph -> E. s e. Q. s e. ( 1st ` L ) ) $=
          ( cnq wbr cltq vx cv c1st cfv wcel wrex cnp c1o cltp cnpi wceq breq2d
          fveq2 1pi a1i rspcdva ltrelpr brel simpld syl c2nd cop prop prml cplq
          wa co subhalfnqq ad2antrl ceq cec crq cab simplr archrecnq simpr nnnq
          wb recclnq ad2antrr ltanqg syl3anc mpbid simpllr ltsonq ltrelnq sotri
          3syl syl2anc wi ad5antr simprr ad4antr prcdnql sylan mpd addclnq wral
          nqprl ltsopr ex reximdva oveq1 abbidv breq1d opeq12d rexbidv crab cpp
          fveq2i nqex rabex op1st eqtri elrab2 sylanbrc rexlimddv ) AUAUBZCUCUD
          ZUEZIUBZHUCUDZUEZIRUFZUARACUGUEZXTUARUFZACUHGUDZUISZYEACEUBZGUDZUISZY
          HEUJUHYIUHUKYJYGCUIYIUHGUMULPUHUJUEAUNUOUPYHYEYGUGUECYGUGUGUIUQURUSUT
          ZYEXSCVAUDZVBUGUEZYFCVCZUAYMXSVDUTUTAXRRUEZXTVFZVFZYAYAVEVGZXRTSZIRUF
          ZYDYPUUAAXTIXRVHVIYRYTYCIRYRYARUEZVFZYTYCUUCYTVFZUUBLUBZYAJUBZUHVBVJV
          KZVLUDZVEVGZTSZLVMZUUIKUBZTSZKVMZVBZUUFGUDZUISZJUJUFZYCYRUUBYTVNZUUDU
          UHYATSZJUJUFZUURUUDUUBUVAUUSYAJVOUTUUDUUTUUQJUJUUDUUFUJUEZVFZUUTUUQUV
          CUUTVFZUUOCUISZCUUPUISZUUQUVDUUIXSUEZUVEUVDUUIXRTSZUVGUVDUUIYSTSZYTUV
          HUVDUUTUVIUVCUUTVPUVDUUHRUEZUUBUUBUUTUVIVRUVDUVBUUGRUEUVJUUDUVBUUTVNZ
          UUFVQUUGVSWHZUUDUUBUVBUUTUUSVTZUVMUUHYAYAWAWBWCUUCYTUVBUUTWDUUIYSXRTR
          WEWFWGWIUVDYEXTUVHUVGWJZAYEYQUUBYTUVBUUTYLWKZYRXTUUBYTUVBUUTAYPXTWLWM
          YEYNXTUVNYOXRUUIYMXSWNWOWIWPUVDUUIRUEZYEUVGUVEVRUVDUUBUVJUVPUVMUVLYAU
          UHWQWIUVOKUUICLWSWIWCUVDYKUVFEUJUUFYIUUFUKYJUUPCUIYIUUFGUMULAYKEUJWRY
          QUUBYTUVBUUTPWKUVKUPUUOCUUPUIUGWTUQWGWIXAXBWPUUEMUBZUUHVEVGZTSZLVMZUV
          RUULTSZKVMZVBZUUPUISZJUJUFZUURMYARYBUVQYAUKZUWDUUQJUJUWFUWCUUOUUPUIUW
          FUVTUUKUWBUUNUWFUVSUUJLUWFUVRUUIUUETUVQYAUUHVEXCZULXDUWFUWAUUMKUWFUVR
          UUIUULTUWGXEXDXFXEXGYBUWEMRXHZUUPUUEUUHTSLVMUUHUULTSKVMVBXIVGUUEBUBZT
          SLVMUWIUULTSKVMVBUISJUJUFZBRXHZVBZUCUDUWHHUWLUCQXJUWHUWKUWEMRXKXLUWJB
          RXKXLXMXNXOXPXAXBWPXQ $.
      $}

      ${
        $d A m r $.  $d F f g h x $.  $d F m $.  $d F r u x $.  $d L t x $.
        $d f g h p q x $.  $d f g h ph x $.  $d p q r u x $.
        $( Lemma for ~ caucvgprpr .  The upper cut of the putative limit is
           inhabited.  (Contributed by Jim Kingdon, 29-Dec-2020.) $)
        caucvgprprlemmu $p |- ( ph -> E. t e. Q. t e. ( 2nd ` L ) ) $=
          ( wcel cltq wbr vx vf vg vh cv c1o cfv c2nd cnq wrex cnp c1st cop 1pi
          cnpi a1i ffvelcdmd prop prmu 3syl c1q cplq simprl 1nq addclnq sylancl
          wa co ceq cec crq cab cpp simprr wb adantr nqpru syl2anc mpbid ltaprg
          cltp w3a adantl nqprlu syl mp1i wceq addcomprg df-1nqqs fveq2i rec1nq
          eqtr3i breq2i abbii breq1i opeq12i oveq2i addnqpr 3brtr4d fveq2 opeq1
          eceq1d fveq2d breq2d abbidv breq1d opeq12d oveq12d rspcev breq2 breq1
          caovord2d crab nqex rabex op2nd eqtri elrab2 sylanbrc eleq1 rexlimddv
          rexbidv ) AUAUEZUFHUGZUHUGZRZCUEZIUHUGZRZCUIUJZUAUIAYDUKRZYDULUGZYEUM
          UKRYFUAUIUJAUOUKUFHNUFUORZAUNUPUQZYDURUAYEYLUSUTAYCUIRZYFVGZVGZYCVAVB
          VHZUIRZYRYHRZYJYQYOVAUIRZYSAYOYFVCZVDYCVAVEVFZYQYSJUEZHUGZLUEZUUDUFUM
          ZVIVJZVKUGZSTZLVLZUUIKUEZSTZKVLZUMZVMVHZUUFYRSTZLVLZYRUULSTZKVLZUMZWA
          TZJUOUJZYTUUCYQYMYDUUFUFUFUMZVIVJZVKUGZSTZLVLZUVFUULSTZKVLZUMZVMVHZUV
          AWATZUVCYMYQUNUPYQYDUUFVASTZLVLZVAUULSTZKVLZUMZVMVHZUUFYCSTLVLYCUULST
          KVLUMZUVRVMVHZUVLUVAWAYQYDUVTWATZUVSUWAWATYQYFUWBAYOYFVNYQYOYKYFUWBVO
          UUBAYKYPYNVPZKYCYDLVQVRVSYQUBUCUDYDUVTUVRWAUKVMUBUEZUKRZUCUEZUKRZUDUE
          ZUKRWBUWDUWFWATUWHUWDVMVHUWHUWFVMVHWATVOYQUWDUWFUWHVTWCUWCYQYOUVTUKRU
          UBKYCLWDWEUUAUVRUKRYQVDKVALWDWFUWEUWGVGUWDUWFVMVHUWFUWDVMVHWGYQUWDUWF
          WHWCXLVSUVLUVSWGYQUVKUVRYDVMUVHUVOUVJUVQUVGUVNLUVFVAUUFSVAVKUGUVFVAVA
          UVEVKWIWJWKWLZWMWNUVIUVPKUVFVAUULSUWIWOWNWPWQUPYQYOUUAUVAUWAWGUUBVDKY
          CVALWRVFWSUVBUVMJUFUOUUDUFWGZUUPUVLUVAWAUWJUUEYDUUOUVKVMUUDUFHWTUWJUU
          KUVHUUNUVJUWJUUJUVGLUWJUUIUVFUUFSUWJUUHUVEVKUWJUUGUVDVIUUDUFUFXAXBXCZ
          XDXEUWJUUMUVIKUWJUUIUVFUULSUWKXFXEXGXHXFXIVRUUPUUFBUEZSTZLVLZUWLUULST
          ZKVLZUMZWATZJUOUJZUVCBYRUIYHUWLYRWGZUWRUVBJUOUWTUWQUVAUUPWAUWTUWNUURU
          WPUUTUWTUWMUUQLUWLYRUUFSXJXEUWTUWOUUSKUWLYRUULSXKXEXGXDYBYHUUFMUEUUIV
          BVHZSTLVLUXAUULSTKVLUMUUEWATJUOUJZMUIXMZUWSBUIXMZUMZUHUGUXDIUXEUHQWJU
          XCUXDUXBMUIXNXOUWSBUIXNXOXPXQXRXSYIYTCYRUIYGYRYHXTXIVRYA $.
      $}

      ${
        $d A r $.  $d A s $.  $d F l $.  $d F u r $.  $d L t $.  $d ph r s $.
        $d l u p q r s $.  $d m r $.
        $( Lemma for ~ caucvgprpr .  The putative limit is inhabited.
           (Contributed by Jim Kingdon, 21-Dec-2020.) $)
        caucvgprprlemm $p |- ( ph -> ( E. s e. Q. s e. ( 1st ` L )
            /\ E. t e. Q. t e. ( 2nd ` L ) ) ) $=
          ( cv cfv c1st wcel cnq wrex c2nd caucvgprprlemml caucvgprprlemmu jca
          ) AJSIUATUBJUCUDCSIUETUBCUCUDABDEFGHIJKLMNOPQRUFABCDEFGHIKLMNOPQRUGUH
          $.
      $}

      ${
        $d A m $.  $d F a b l t $.  $d F m $.  $d F r $.  $d F t u $.
        $d L a b t $.  $d b r $.  $d a b l p q r s t $.  $d p q r s t u $.
        $d ph a b r t $.
        $( Lemma for ~ caucvgprpr .  The lower cut of the putative limit is
           open.  (Contributed by Jim Kingdon, 21-Dec-2020.) $)
        caucvgprprlemopl $p |- ( ( ph /\ s e. ( 1st ` L ) ) ->
            E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) $=
          ( cltq wbr vb va cv c1st cfv wcel wa c1o cop ceq cec crq cplq co cltp
          cab cnq wrex cnpi caucvgprprlemell simprbi adantl cnp ad2antrr simprl
          c2nd ffvelcdmd prop syl simprr simplbi ad2antlr nnnq recclnq ad2antrl
          wf addclnq syl2anc nqprl mpbird prnmaxl wceq adantr ltaddnq addcomnqg
          wb breqtrd ltsonq ltrelnq sotri elprnql mpbid eqbrtrrd simpr breqtrrd
          ltexnqq simplr ltanqg eqtr3d opeq1 eceq1d fveq2d oveq2d breq2d abbidv
          syl3anc ad3antrrr eqeltrd breq1d opeq12d fveq2 rspcev sylanbrc jca ex
          breq12d reximdva mpd rexlimddv ) AJUCZIUDUEZUFZUGZMUCZXTUAUCZUHUIZUJU
          KZULUEZUMUNZSTMUPYILUCZSTLUPUIYEHUEZUOTZXTCUCZSTZYMYAUFZUGZCUQURZUAUS
          YBYLUAUSURZAYBXTUQUFZYRBHIXTKLMUANRUTVAVBYCYEUSUFZYLUGZUGZYIUBUCZSTZY
          QUBYKUDUEZUUBUUEYKVFUEZUIVCUFZYIUUEUFZUUDUBUUEURUUBYKVCUFZUUGUUBUSVCY
          EHAUSVCHVPYBUUAOVDYCYTYLVEZVGZYKVHVIZUUBUUHYLYCYTYLVJUUBYIUQUFZUUIUUH
          YLWFUUBYSYHUQUFZUUMYBYSAUUAYBYSYDXTKUCZUHUIZUJUKZULUEZUMUNZSTMUPUUSYJ
          STLUPUIUUOHUEZUOTKUSURBHIXTKLMKNRUTVKVLZYTUUNYCYLYTYGUQUFUUNYEVMYGVNV
          IVOZXTYHVQVRUUKLYIYKMVSVRVTUBYIUUFUUEWAVRUUBUUCUUEUFZUUDUGZUGZYHYMUMU
          NZUUCWBZCUQURZYQUVEYHUUCSTZUVHUVEYHYISTUUDUVIUVEYHYHXTUMUNZYISUVEUUNY
          SYHUVJSTUUBUUNUVDUVBWCZUUBYSUVDUVAWCZYHXTWDVRUVEUUNYSUVJYIWBUVKUVLYHX
          TWEVRWGUUBUVCUUDVJZYHYIUUCSUQWHWIWJVRUVEUUNUUCUQUFZUVIUVHWFUVKUVEUUGU
          VCUVNUUBUUGUVDUULWCUUBUVCUUDVEZUUCUUFUUEWKVRCYHUUCWPVRWLUVEUVGYPCUQUV
          EYMUQUFZUGZUVGYPUVQUVGUGZYNYOUVRYNUVJUVFSTZUVRUVJUUCUVFSUVRYIUVJUUCSU
          VRYSUUNYIUVJWBUVEYSUVPUVGUVLVDZUVEUUNUVPUVGUVKVDZXTYHWEVRUVEUUDUVPUVG
          UVMVDWMUVQUVGWNZWOUVRYSUVPUUNYNUVSWFUVTUVEUVPUVGWQZUWAXTYMYHWRXFVTUVR
          UVPYDYMUURUMUNZSTZMUPZUWDYJSTZLUPZUIZUUTUOTZKUSURZYOUWCUVRYTYDYMYHUMU
          NZSTZMUPZUWLYJSTZLUPZUIZYKUOTZUWKUUBYTUVDUVPUVGUUJXGUVRUWLUUEUFZUWRUV
          RUWLUUCUUEUVRUVFUWLUUCUVRUUNUVPUVFUWLWBUWAUWCYHYMWEVRUWBWSUVEUVCUVPUV
          GUVOVDXHUVRUWLUQUFZUUIUWSUWRWFUVRUVPUUNUWTUWCUWAYMYHVQVRUUBUUIUVDUVPU
          VGUUKXGLUWLYKMVSVRWLUWJUWRKYEUSUUOYEWBZUWIUWQUUTYKUOUXAUWFUWNUWHUWPUX
          AUWEUWMMUXAUWDUWLYDSUXAUURYHYMUMUXAUUQYGULUXAUUPYFUJUUOYEUHWTXAXBXCZX
          DXEUXAUWGUWOLUXAUWDUWLYJSUXBXIXEXJUUOYEHXKXPXLVRBHIYMKLMKNRUTXMXNXOXQ
          XRXSXS $.
      $}

      ${
        $d F l $.  $d F u $.  $d F b x y z $.  $d F r $.  $d L b x y z $.
        $d l p s $.  $d l q s $.  $d l r s $.  $d l p t $.  $d u q s $.
        $d u r s $.  $d u p t $.  $d b r p s x y z $.  $d ph b r x y z $.
        $d b r q t x y z $.  $d b r t x y z $.
        $( Lemma for ~ caucvgprpr .  The lower cut of the putative limit is
           lower.  (Contributed by Jim Kingdon, 21-Dec-2020.) $)
        caucvgprprlemlol $p |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) ->
            s e. ( 1st ` L ) ) $=
          ( cltq wbr vb vx vy vz c1st cfv wcel w3a cnq c1o cop ceq cec crq cplq
          cv co cab cltp cnpi wrex ltrelnq brel simpld caucvgprprlemell simprbi
          3ad2ant2 3ad2ant3 wa simpll2 wb ltanqg adantl ad2antrr simprd recclnq
          simplr nnnq 3syl addcomnqg caovord2d mpbid ltnqpri syl ltsopr ltrelpr
          wceq cnp sotri sylancom ex reximdva opeq1 eceq1d fveq2d oveq2d breq2d
          mpd abbidv breq1d opeq12d fveq2 breq12d cbvrexv sylib sylanbrc ) AJUP
          ZCUPZSTZXHIUEUFZUGZUHZXGUIUGZMUPZXGKUPZUJUKZULUMZUNUFZUOUQZSTZMURZXSL
          UPZSTZLURZUKZXOHUFZUSTZKUTVAZXGXJUGXIAXMXKXIXMXHUIUGZXGXHUIUISVBVCZVD
          VGZXLXNXGUAUPZUJUKZULUMZUNUFZUOUQZSTZMURZYPYBSTZLURZUKZYLHUFZUSTZUAUT
          VAZYHXLXNXHYOUOUQZSTMURUUEYBSTLURUKZUUBUSTZUAUTVAZUUDXKAUUHXIXKYIUUHB
          HIXHKLMUANRVEVFVHXLUUGUUCUAUTXLYLUTUGZVIZUUGUUCUUJUUGUUAUUFUSTZUUCUUJ
          UUGVIZYPUUESTZUUKUULXIUUMAXIXKUUIUUGVJUULUBUCUDXGXHYOSUIUOUBUPZUIUGZU
          CUPZUIUGZUDUPZUIUGUHUUNUUPSTUURUUNUOUQUURUUPUOUQSTVKUULUUNUUPUURVLVMX
          LXMUUIUUGYKVNXLYIUUIUUGXIAYIXKXIXMYIYJVOVGVNUULUUIYNUIUGYOUIUGXLUUIUU
          GVQYLVRYNVPVSUUOUUQVIUUNUUPUOUQUUPUUNUOUQWGUULUUNUUPVTVMWAWBLYPUUEMWC
          WDUUAUUFUUBUSWHWEWFWIWJWKWLWRUUCYGUAKUTYLXOWGZUUAYEUUBYFUSUUSYRYAYTYD
          UUSYQXTMUUSYPXSXNSUUSYOXRXGUOUUSYNXQUNUUSYMXPULYLXOUJWMWNWOWPZWQWSUUS
          YSYCLUUSYPXSYBSUUTWTWSXAYLXOHXBXCXDXEBHIXGKLMKNRVEXF $.
      $}

      ${
        $d A m $.  $d F l r s $.  $d F m $.  $d F b r s u $.  $d L b s $.
        $d b l p q r s t $.  $d p q r s t u $.  $d b ph r s $.
        $( Lemma for ~ caucvgprpr .  The upper cut of the putative limit is
           open.  (Contributed by Jim Kingdon, 21-Dec-2020.) $)
        caucvgprprlemopu $p |- ( ( ph /\ t e. ( 2nd ` L ) ) ->
            E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) $=
          ( cltq wbr vb cv c2nd cfv wcel wa c1o cop ceq cec crq cab cpp co cltp
          cnq wrex cnpi caucvgprprlemelu simprbi adantl c1st simprr cnp recnnpr
          wb ffvelcdmda addclpr syl2anc ad2ant2r simplbi ad2antlr nqprlu ltdfpr
          syl mpbid simpr adantr nqpru vex breq1 ltnqex op1st elab2 a1i anbi12d
          gtnqex biimpd mpd simplr simplrl simprl weq fveq2 opeq1 eceq1d fveq2d
          reximdva breq2d abbidv breq1d opeq12d oveq12d rspcev jca ex rexlimddv
          sylanbrc ) ACUBZIUCUDZUEZUFZUAUBZHUDZMUBZXMUGUHZUIUJZUKUDZSTZMULZXRLU
          BZSTZLULZUHZUMUNZXOXISTZMULZXIYASTLULZUHZUOTZJUBZXISTZYKXJUEZUFZJUPUQ
          ZUAURXKYJUAURUQZAXKXIUPUEZYPBHIXIKLMUANRUSZUTVAXLXMURUEZYJUFZUFZYEXOY
          KSTMULYKYASTLULUHZUOTZYLUFZJUPUQZYOUUAYKYEUCUDUEZYKYIVBUDZUEZUFZJUPUQ
          ZUUEUUAYJUUJXLYSYJVCUUAYEVDUEZYIVDUEZYJUUJVFAYSUUKXKYJAYSUFXNVDUEYDVD
          UEZUUKAURVDXMHOVGYSUUMALXMMVEVAXNYDVHVIVJZUUAYQUULXKYQAYTXKYQYPYRVKVL
          LXIMVMVOYEYIJVNVIVPUUAUUIUUDJUPUUAYKUPUEZUFZUUIUUDUUPUUFUUCUUHYLUUPUU
          OUUKUUFUUCVFUUAUUOVQUUAUUKUUOUUNVRLYKYEMVSVIUUHYLVFUUPYFYLMYKUUGJVTXO
          YKXISWAYGYHMXIWBLXIWGWCWDWEWFWHWRWIUUAUUDYNJUPUUPUUDYNUUPUUDUFZYLYMUU
          PUUCYLVCUUQUUOKUBZHUDZXOUURUGUHZUIUJZUKUDZSTZMULZUVBYASTZLULZUHZUMUNZ
          UUBUOTZKURUQZYMUUAUUOUUDWJUUQYSUUCUVJUUPYSUUDXLYSYJUUOWKVRUUPUUCYLWLU
          VIUUCKXMURKUAWMZUVHYEUUBUOUVKUUSXNUVGYDUMUURXMHWNUVKUVDXTUVFYCUVKUVCX
          SMUVKUVBXRXOSUVKUVAXQUKUVKUUTXPUIUURXMUGWOWPWQZWSWTUVKUVEYBLUVKUVBXRY
          ASUVLXAWTXBXCXAXDVIBHIYKKLMKNRUSXHXEXFWRWIXG $.

        $( Lemma for ~ caucvgprpr .  The upper cut of the putative limit is
           upper.  (Contributed by Jim Kingdon, 21-Dec-2020.) $)
        caucvgprprlemupu $p |- ( ( ph /\ s <Q t /\ s e. ( 2nd ` L ) ) ->
            t e. ( 2nd ` L ) ) $=
          ( cltq wbr vb cv c2nd cfv wcel w3a cnq c1o cop ceq cec crq cab cpp co
          cltp cnpi wrex ltrelnq brel simprd 3ad2ant2 caucvgprprlemelu 3ad2ant3
          simprbi wi ltnqpri cnp ltsopr ltrelpr expcom syl reximdv mpd sylanbrc
          sotri ) AJUBZCUBZSTZVQIUCUDZUEZUFZVRUGUEZUAUBZHUDMUBZWDUHUIUJUKULUDZS
          TMUMWFLUBZSTLUMUIUNUOZWEVRSTMUMVRWGSTLUMUIZUPTZUAUQURZVRVTUEVSAWCWAVS
          VQUGUEZWCVQVRUGUGSUSUTVAVBWBWHWEVQSTMUMVQWGSTLUMUIZUPTZUAUQURZWKWAAWO
          VSWAWLWOBHIVQKLMUANRVCVEVDWBWNWJUAUQWBWMWIUPTZWNWJVFVSAWPWALVQVRMVGVB
          WNWPWJWHWMWIUPVHVIVJVPVKVLVMVNBHIVRKLMUANRVCVO $.
      $}

      ${
        $d F l t $.  $d F t u $.  $d F r $.  $d F s $.  $d L s t $.
        $d l p q r s t $.  $d p q r s t u $.  $d ph r s t $.
        $( Lemma for ~ caucvgprpr .  The putative limit is rounded.
           (Contributed by Jim Kingdon, 21-Dec-2020.) $)
        caucvgprprlemrnd $p |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <->
              E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) /\
            A. t e. Q. ( t e. ( 2nd ` L ) <->
              E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) ) ) $=
          ( wcel cnq cv c1st cfv cltq wbr wa wrex wb wral c2nd caucvgprprlemopl
          caucvgprprlemlol 3expib impbid caucvgprprlemopu caucvgprprlemupu jca
          ex rexlimdvw ralrimivw ) AJUAZIUBUCZSZVACUAZUDUEZVDVBSZUFZCTUGZUHZJTU
          IVDIUJUCZSZVEVAVJSZUFZJTUGZUHZCTUIAVIJTAVCVHAVCVHABCDEFGHIJKLMNOPQRUK
          URAVGVCCTAVEVFVCABCDEFGHIJKLMNOPQRULUMUSUNUTAVOCTAVKVNAVKVNABCDEFGHIJ
          KLMNOPQRUOURAVMVKJTAVEVLVKABCDEFGHIJKLMNOPQRUPUMUSUNUTUQ $.
      $}

      ${
        $d F a b k n $.  $d F l r $.  $d F m $.  $d F r u $.  $d L a b k $.
        $d a b l $.  $d a u $.  $d a b k p r s $.  $d ph a b r s $.
        $d a b k q r s $.  $d l p r s $.  $d l q r s $.  $d p r s u $.
        $d b q r s u $.  $d p q $.  $d n u $.  $d l n $.  $d k l $.  $d k u $.
        $( Lemma for ~ caucvgprpr .  The putative limit is disjoint.
           (Contributed by Jim Kingdon, 21-Dec-2020.) $)
        caucvgprprlemdisj $p |- ( ph -> A. s e. Q.
            -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) $=
          ( cfv wbr cnpi va vb cv c1st wcel c2nd wa wn cnq c1o cop ceq cec cplq
          crq cltq cab cltp wrex wfal caucvgprprlemell simprbi caucvgprprlemelu
          co cpp anim12i reeanv sylibr adantl wf ad2antrr clti wi simprl simprr
          cnp simplbi ad2antrl adantr caucvgprprlemnkj pm2.21d rexlimdvva inegd
          wral mpd ralrimivw ) AIUCZHUDRUEZWGHUFRUEZUGZUHIUIAWJAWJUGZLUCZWGUAUC
          ZUJUKULUMUORUNVDZUPSLUQWNKUCZUPSKUQUKWMGRURSZUBUCZGRWLWQUJUKULUMUORZU
          PSLUQWRWOUPSKUQUKVEVDWLWGUPSLUQWGWOUPSKUQUKURSZUGZUBTUSUATUSZUTWJXAAW
          JWPUATUSZWSUBTUSZUGXAWHXBWIXCWHWGUIUEZXBBGHWGJKLUAMQVAVBWIXDXCBGHWGJK
          LUBMQVCVBVFWPWSUAUBTTVGVHVIWKWTUTUAUBTTWKWMTUEZWQTUEZUGZUGZWTUTXHBWGD
          FGWQWMKLMATVPGVJWJXGNVKAFUCZDUCZVLSXIGRZXJGRZMUCXIUJUKULUMUORZUPSMUQX
          MBUCUPSBUQUKZVEVDURSXLXKXNVEVDURSUGVMDTWDFTWDWJXGOVKWKXEXFVNWKXEXFVOW
          KXDXGWHXDAWIWHXDWLWGJUCZUJUKULUMUORUNVDZUPSLUQXPWOUPSKUQUKXOGRURSJTUS
          BGHWGJKLJMQVAVQVRVSVTWAWBWEWCWF $.
      $}

      ${
        $d A m $.  $d F a $.  $d F b $.  $d F f g h $.  $d F l r $.
        $d F u r $.  $d F m $.  $d L c x y $.  $d a c p q s $.  $d b c p q t $.
        $d c f g h p q s t $.  $d c f g h ph s t x y $.  $d l p q r s t $.
        $d u p q r s t $.
        $( Lemma for ~ caucvgprpr .  The putative limit is located.
           (Contributed by Jim Kingdon, 21-Dec-2020.) $)
        caucvgprprlemloc $p |- ( ph -> A. s e. Q. A. t e. Q.
            ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) ) $=
          ( wbr wcel vy vx vc va vf vg vh vb cv cltq c1st cfv c2nd wo wi cnq wa
          cplq wceq wrex ltexnqi adantl subhalfnqq ad2antrl c1o cop ceq cec crq
          co cnpi archrecnq cab cpp simpllr adantr simplrl caucvgprprlemloccalc
          cltp simplrr simprl simprr cnp ad3antrrr nnnq recclnq addclnq syl2anc
          syl nqprlu addclpr ad5antr ffvelcdmd ltrelnq brel simpld ad2antll wor
          wf w3a ltsopr sowlin mpan syl3anc simpr wb ltaprg addcomprg caovord2d
          mpbird opeq1 eceq1d fveq2d oveq2d breq2d abbidv opeq12d fveq2 breq12d
          mpd breq1d rspcev caucvgprprlemell sylanbrc ex sylan caucvgprprlemelu
          oveq12d orim12d rexlimddv ralrimivva ) AJUIZCUIZUJSZYLIUKULTZYMIUMULT
          ZUNZUOJCUPUPAYLUPTZYMUPTZUQZUQZYNYQUUAYNUQZYLUAUIZURVJYMUSZYQUAUPYNUU
          DUAUPUTUUAUAYLYMVAVBUUBUUCUPTZUUDUQZUQZUBUIZUUHURVJUUCUJSZYQUBUPUUEUU
          IUBUPUTUUBUUDUBUUCVCVDUUGUUHUPTZUUIUQZUQZUCUIZVEVFZVGVHZVIULZUUHUJSZY
          QUCVKUUJUUQUCVKUTUUGUUIUUHUCVLVDUULUUMVKTZUUQUQZUQZMUIZYLUUPURVJZUJSZ
          MVMZUVBLUIZUJSZLVMZVFZUVAUUPUJSZMVMZUUPUVEUJSZLVMZVFZVNVJZUUMHULZUVMV
          NVJZVSSZUVPUVAYMUJSMVMYMUVEUJSLVMVFZVSSZUNZYQUUTUVNUVRVSSZUVTUUTLYLYM
          UUMUUHUUCMUULYNUUSUUAYNUUFUUKVOVPUULUUEUUSUUBUUEUUDUUKVQVPUULUUDUUSUU
          BUUEUUDUUKVTVPUUGUUJUUIUUSVQUUGUUJUUIUUSVTUULUURUUQWAZUULUURUUQWBVRUU
          TUVNWCTZUVRWCTZUVPWCTZUWAUVTUOZUUTUVHWCTZUVMWCTZUWCUUTUVBUPTZUWGUUTYR
          UUPUPTZUWIUUBYRUUFUUKUUSAYRYSYNVQWDZUUTUUOUPTZUWJUURUWLUULUUQUUMWEVDU
          UOWFWIZYLUUPWGZWHLUVBMWJZWIUUTUWJUWHUWMLUUPMWJZWIUVHUVMWKWHUUTYSUWDUU
          BYSUUFUUKUUSAYRYSYNVTWDZLYMMWJWIUUTUVOWCTZUWHUWEUUTVKWCUUMHAVKWCHWSYT
          YNUUFUUKUUSOWLUWBWMZUUTUWJUWHUUQUWJUULUURUUQUWJUUJUUPUUHUPUPUJWNWOWPW
          QZUWPWIUVOUVMWKWHWCVSWRUWCUWDUWEWTUWFXAWCUVNUVRUVPVSXBXCXDXTUUTUVQYOU
          VSYPUUTUVQYOUUTUVQUQZYRUVAYLUDUIZVEVFZVGVHZVIULZURVJZUJSZMVMZUXFUVEUJ
          SZLVMZVFZUXBHULZVSSZUDVKUTZYOUUTYRUVQUWKVPZUXAUURUVHUVOVSSZUXNUULUURU
          UQUVQVQUXAUXPUVQUUTUVQXEUXAUEUFUGUVHUVOUVMVSWCVNUEUIZWCTZUFUIZWCTZUGU
          IZWCTWTUXQUXSVSSUYAUXQVNVJUYAUXSVNVJVSSXFUXAUXQUXSUYAXGVBUXAUWIUWGUXA
          YRUWJUWIUXOUUTUWJUVQUWTVPZUWNWHUWOWIUUTUWRUVQUWSVPUXAUWJUWHUYBUWPWIUX
          RUXTUQUXQUXSVNVJUXSUXQVNVJUSUXAUXQUXSXHVBXIXJUXMUXPUDUUMVKUXBUUMUSZUX
          KUVHUXLUVOVSUYCUXHUVDUXJUVGUYCUXGUVCMUYCUXFUVBUVAUJUYCUXEUUPYLURUYCUX
          DUUOVIUYCUXCUUNVGUXBUUMVEXKXLXMXNZXOXPUYCUXIUVFLUYCUXFUVBUVEUJUYDYAXP
          XQUXBUUMHXRXSYBWHBHIYLKLMUDNRYCYDYEUUTUVSYPUUTUVSUQYSUHUIZHULZUVAUYEV
          EVFZVGVHZVIULZUJSZMVMZUYIUVEUJSZLVMZVFZVNVJZUVRVSSZUHVKUTZYPUUTYSUVSU
          WQVPUUTUURUVSUYQUWBUYPUVSUHUUMVKUYEUUMUSZUYOUVPUVRVSUYRUYFUVOUYNUVMVN
          UYEUUMHXRUYRUYKUVJUYMUVLUYRUYJUVIMUYRUYIUUPUVAUJUYRUYHUUOVIUYRUYGUUNV
          GUYEUUMVEXKXLXMZXOXPUYRUYLUVKLUYRUYIUUPUVEUJUYSYAXPXQYHYAYBYFBHIYMKLM
          UHNRYGYDYEYIXTYJYJYJYEYK $.
      $}

      ${
        $d A r $.  $d A s $.  $d F l t $.  $d F u r $.  $d F k n $.  $d F s $.
        $d L s t $.  $d L k $.  $d ph t r s $.  $d l u p q r s t $.  $d m r $.
        $d k p q r s $.  $d n u $.  $d l n $.  $d k l $.  $d k u $.
        $( Lemma for ~ caucvgprpr .  The putative limit is a positive real.
           (Contributed by Jim Kingdon, 21-Nov-2020.) $)
        caucvgprprlemcl $p |- ( ph -> L e. P. ) $=
          ( vs vt cnq wcel cpw cxp cv c1st cfv wrex c2nd wa cltq wbr wb wral wn
          wo w3a cnp caucvgprprlemm c1o cop ceq cec crq cplq cab cltp cnpi crab
          wi co cpp wss nqex elpw2 mpbir opelxpi mp2an eqeltri caucvgprprlemrnd
          ssrab2 jctil caucvgprprlemdisj caucvgprprlemloc elnp1st2nd sylanbrc
          3jca ) AHSUAZWFUBZTZQUCZHUDUEZTZQSUFRUCZHUGUEZTZRSUFUHZUHWKWIWLUIUJZW
          LWJTUHRSUFUKQSULWNWPWIWMTZUHQSUFUKRSULUHZWKWQUHUMQSULZWPWKWNUNVHRSULQ
          SULZUOHUPTAWOWHABRCDEFGHQIJKLMNOPUQHKUCZLUCIUCZURUSUTVAVBUEZVCVIZUIUJ
          KVDXDJUCZUIUJJVDUSXBGUEZVEUJIVFUFZLSVGZXFXAXCUIUJKVDXCXEUIUJJVDUSVJVI
          XABUCZUIUJKVDXIXEUIUJJVDUSVEUJIVFUFZBSVGZUSZWGPXHWFTZXKWFTZXLWGTXMXHS
          VKXGLSVSXHSVLVMVNXNXKSVKXJBSVSXKSVLVMVNXHXKWFWFVOVPVQVTAWRWSWTABRCDEF
          GHQIJKLMNOPVRABCDEFGHQIJKLMNOPWAABRCDEFGHQIJKLMNOPWBWEHRQWCWD $.
      $}

      ${
        $d A r $.  $d A s $.  $d F l $.  $d F u r $.  $d F k n $.  $d F s $.
        $d L s $.  $d L k $.  $d ph s $.  $d l u p q r s $.  $d m r $.
        $d m s $.  $d k p q r s $.  $d n u $.  $d l n $.  $d k l $.  $d k u $.
        $( Lemma for ~ caucvgprpr .  The putative limit is a positive real.
           Like ~ caucvgprprlemcl but without a disjoint variable condition
           between ` ph ` and ` r ` .  (Contributed by Jim Kingdon,
           19-Jun-2021.) $)
        caucvgprprlemclphr $p |- ( ph -> L e. P. ) $=
          ( cop cltq wbr cab vs cv c1o ceq cec crq cfv cplq cltp cnpi wrex crab
          co cnq cpp wcel weq eceq1d fveq2d oveq2d breq2d abbidv breq1d opeq12d
          opeq1 fveq2 breq12d cbvrexv a1i rabbiia oveq12d eqtri caucvgprprlemcl
          wb opeq12i ) ABCDEFGHUAJKLMNOHKUBZLUBZIUBZUCQZUDUEZUFUGZUHUMZRSZKTZWB
          JUBZRSZJTZQZVRGUGZUISZIUJUKZLUNULZWIVPWARSZKTZWAWERSZJTZQZUOUMZVPBUBZ
          RSKTWSWERSJTQZUISZIUJUKZBUNULZQVPVQUAUBZUCQZUDUEZUFUGZUHUMZRSZKTZXHWE
          RSZJTZQZXDGUGZUISZUAUJUKZLUNULZXNVPXGRSZKTZXGWERSZJTZQZUOUMZWTUISZUAU
          JUKZBUNULZQPWLXQXCYFWKXPLUNWKXPVNVQUNUPWJXOIUAUJIUAUQZWHXMWIXNUIYGWDX
          JWGXLYGWCXIKYGWBXHVPRYGWAXGVQUHYGVTXFUFYGVSXEUDVRXDUCVEURUSZUTZVAVBYG
          WFXKJYGWBXHWERYIVCVBVDVRXDGVFZVGVHVIVJXBYEBUNXBYEVNWSUNUPXAYDIUAUJYGW
          RYCWTUIYGWIXNWQYBUOYJYGWNXSWPYAYGWMXRKYGWAXGVPRYHVAVBYGWOXTJYGWAXGWER
          YHVCVBVDVKVCVHVIVJVOVLVM $.
      $}

      ${
        $d A m r $.  $d F b f g h x y $.  $d F k l n u $.  $d F m r $.
        $d L b f g h x y $.  $d L k $.  $d Q b f g h p q x y $.
        $d T b f g h x y $.  $d b f g h ph x y $.  $d b p q r y $.
        $d k l p q r u $.  $d ph y $.  $d u y $.
        caucvgprprlemexbt.q $e |- ( ph -> Q e. Q. ) $.
        caucvgprprlemexbt.t $e |- ( ph -> T e. P. ) $.
        caucvgprprlemexbt.lt $e |- ( ph -> ( L +P.
          <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T ) $.
        $( Lemma for ~ caucvgprpr .  Part of showing the putative limit to be a
           limit.  (Contributed by Jim Kingdon, 16-Jun-2021.) $)
        caucvgprprlemexbt $p |- ( ph -> E. b e. N. ( ( ( F ` b ) +P.
          <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } ,
            { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q }
          >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >.
          ) <P T ) $=
          ( vx vy vf vg vh cv cfv c1o cop ceq cec crq cltq wbr cab co cltp cnpi
          cpp wrex c2nd wcel c1st cnq cnp caucvgprprlemclphr nqprlu syl addclpr
          wa syl2anc ltdfpr mpbid cplq wceq adantr simprrl prplnqu simprl breq2
          weq abbidv breq1 opeq12d breq2d rexbidv crab fveq2i rabex op2nd eqtri
          wb nqex elrab2 biimpi simprd fveq2 opeq1 eceq1d fveq2d breq1d oveq12d
          cbvrexv sylib simpr ltaprg adantl wf ad4antr simplr ffvelcdmd recnnpr
          w3a ad2antrr simpld addcomprg caovord2d addnqpr simplrr simplrl nqpru
          breqtrrd mpbird simprrr ad3antrrr jca ex reximdva mpd expr ffvelcdmda
          rexlimddv rexcom rexbidva ) ANUHZIUIZMUHZYQUJUKZULUMZUNUIZUOUPZMUQZUU
          BLUHZUOUPZLUQZUKZVAURZYSDUOUPMUQDUUEUOUPLUQUKZVAURZEUSUPZNUTVBUCUHZUU
          KVCUIVDZUUMEVEUIVDZVLZUCVFVBZNUTVBZAUUPNUTVBZUCVFVBZUURAUUMJUUJVAURZV
          CUIVDZUUOVLZUCVFVBZUUTAUVAEUSUPZUVDUBAUVAVGVDZEVGVDZUVEUVDWNAJVGVDZUU
          JVGVDZUVFABCFGHIJKLMOPQRSVHZADVFVDZUVITLDMVIVJZJUUJVKVMUAUVAEUCVNVMVO
          AUVCUUSUCVFAUUMVFVDZUVCUUSAUVMUVCVLZVLZUDUHZDVPURZUUMVQZUUSUDJVCUIZUV
          OUDLUUMDJMAUVHUVNUVJVRAUVKUVNTVRAUVMUVBUUOVSVTUVOUVPUVSVDZUVRVLZVLZUU
          IYSUVPUOUPZMUQZUVPUUEUOUPZLUQZUKZUSUPZNUTVBZUUSUWBKUHZIUIZYSUWJUJUKZU
          LUMZUNUIZUOUPZMUQZUWNUUEUOUPZLUQZUKZVAURZUWGUSUPZKUTVBZUWIUWBUVTUXBUV
          OUVTUVRWAZUVTUVPVFVDZUXBUVTUXDUXBVLUWTYSBUHZUOUPZMUQZUXEUUEUOUPZLUQZU
          KZUSUPZKUTVBZUXBBUVPVFUVSBUDWCZUXKUXAKUTUXMUXJUWGUWTUSUXMUXGUWDUXIUWF
          UXMUXFUWCMUXEUVPYSUOWBWDUXMUXHUWELUXEUVPUUEUOWEWDWFWGWHUVSYSOUHUWNVPU
          RZUOUPMUQUXNUUEUOUPLUQUKUWKUSUPKUTVBZOVFWIZUXLBVFWIZUKZVCUIUXQJUXRVCS
          WJUXPUXQUXOOVFWOWKUXLBVFWOWKWLWMWPWQZWRVJUXAUWHKNUTKNWCZUWTUUIUWGUSUX
          TUWKYRUWSUUHVAUWJYQIWSUXTUWPUUDUWRUUGUXTUWOUUCMUXTUWNUUBYSUOUXTUWMUUA
          UNUXTUWLYTULUWJYQUJWTXAXBZWGWDUXTUWQUUFLUXTUWNUUBUUEUOUYAXCWDWFXDXCXE
          XFUWBUWHUUPNUTUWBYQUTVDZVLZUWHUUPUYCUWHVLZUUNUUOUYDUUNUUKYSUUMUOUPZMU
          QZUUMUUEUOUPZLUQZUKZUSUPZUYDUUKYSUVQUOUPZMUQZUVQUUEUOUPZLUQZUKZUSUPZU
          YJUYDUUKUWGUUJVAURZUYOUSUYDUWHUUKUYQUSUPUYCUWHXGUYDUEUFUGUUIUWGUUJUSV
          GVAUEUHZVGVDZUFUHZVGVDZUGUHZVGVDXOUYRUYTUSUPVUBUYRVAURVUBUYTVAURUSUPW
          NUYDUYRUYTVUBXHXIUYDYRVGVDZUUHVGVDZUUIVGVDZUYDUTVGYQIAUTVGIXJUVNUWAUY
          BUWHPXKUWBUYBUWHXLZXMUYDUYBVUDVUFLYQMXNZVJYRUUHVKZVMZUYDUXDUWGVGVDUYD
          UVTUXDUWBUVTUYBUWHUXCXPUVTUXDUXBUXSXQVJZLUVPMVIVJAUVIUVNUWAUYBUWHUVLX
          KZUYSVUAVLUYRUYTVAURUYTUYRVAURVQUYDUYRUYTXRXIXSVOUYDUXDUVKUYOUYQVQVUJ
          AUVKUVNUWAUYBUWHTXKLUVPDMXTVMYDUYDUVRUYPUYJWNUYCUVRUWHUVOUVTUVRUYBYAV
          RUVRUYOUYIUUKUSUVRUYLUYFUYNUYHUVRUYKUYEMUVQUUMYSUOWBWDUVRUYMUYGLUVQUU
          MUUEUOWEWDWFWGVJVOUYDUVMUUKVGVDZUUNUYJWNUWBUVMUYBUWHAUVMUVCUWAYBXPUYD
          VUEUVIVULVUIVUKUUIUUJVKZVMLUUMUUKMYCVMYEUVOUUOUWAUYBUWHAUVMUVBUUOYFYG
          YHYIYJYKYNYLYJYKUUPUCNVFUTYOXFAUULUUQNUTAUYBVLZVULUVGUULUUQWNVUNVUEUV
          IVULVUNVUCVUDVUEAUTVGYQIPYMUYBVUDAVUGXIVUHVMAUVIUYBUVLVRVUMVMAUVGUYBU
          AVRUUKEUCVNVMYPYE $.
      $}

      ${
        $d A m r $.  $d F b f g h $.  $d F k l n u $.  $d F m r $.
        $d L b f g h $.  $d L k $.  $d Q f g h $.  $d R b f g h p q $.
        $d R b p q $.  $d b f g h ph $.  $d k l p q r u $.  $d b r $.
        caucvgprprlemexb.q $e |- ( ph -> Q e. P. ) $.
        caucvgprprlemexb.r $e |- ( ph -> R e. N. ) $.
        $( Lemma for ~ caucvgprpr .  Part of showing the putative limit to be a
           limit.  (Contributed by Jim Kingdon, 15-Jun-2021.) $)
        caucvgprprlemexb $p |- ( ph -> (
          ( ( L +P. Q ) +P.
          <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } ,
            { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q }
          >. ) <P ( ( F ` R ) +P. Q )
          -> E. b e. N. ( ( ( F ` b ) +P.
          <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } ,
            { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q }
          >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } ,
            { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q }
          >. ) ) <P ( ( F ` R ) +P. Q ) ) ) $=
          ( vf vg vh cpp co c1o cop ceq cec crq cfv cltq wbr cab cltp cnpi wrex
          cv cnp wb caucvgprprlemclphr recnnpr addclpr syl2anc ffvelcdmd ltaprg
          wcel syl syl3anc wceq addassprg addcomprg oveq1d eqtr3d breq12d bitrd
          wa wf adantr clti wi wral cnq nnnq recclnq 3syl caucvgprprlemexbt w3a
          simpr adantl ffvelcdmda ad2antrr caovord2d breq1d oveq2d 3bitrd mpbid
          rexbidva ex sylbird ) AJDUEUFZMUSZEUGUHUIUJZUKULZUMUNMUOXELUSZUMUNLUO
          UHZUEUFZEIULZDUEUFZUPUNZJXGUEUFZXIUPUNZNUSZIULZXCXNUGUHUIUJUKULZUMUNM
          UOXPXFUMUNLUOUHZUEUFZDXGUEUFZUEUFZXJUPUNZNUQURZAXMDXLUEUFZDXIUEUFZUPU
          NZXKAXLUTVHZXIUTVHZDUTVHZXMYEVAAJUTVHZXGUTVHZYFABCFGHIJKLMOPQRSVBZAEU
          QVHZYJUALEMVCZVIZJXGVDVEAUQUTEIPUAVFZTXLXIDVGVJAYCXHYDXJUPADJUEUFZXGU
          EUFZYCXHAYHYIYJYQYCVKTYKYNDJXGVLVJAYPXBXGUEAYHYIYPXBVKTYKDJVMVEVNVOAY
          HYGYDXJVKTYODXIVMVEVPVQAXMYBAXMVRZXRXGUEUFZXIUPUNZNUQURYBYRBCXEXIFGHI
          JKLMNOAUQUTIVSXMPVTZAHUSZFUSZWAUNUUBIULZUUCIULZOUSUUBUGUHUIUJUKULZUMU
          NOUOUUFBUSUMUNBUOUHZUEUFUPUNUUEUUDUUGUEUFUPUNVRWBFUQWCHUQWCXMQVTACGUS
          IULUPUNGUQWCXMRVTSAXEWDVHZXMAYLXDWDVHUUHUAEWEXDWFWGVTAYGXMYOVTAXMWJWH
          YRYTYANUQYRXNUQVHZVRZYTYSDUEUFZXJUPUNXRXGDUEUFZUEUFZXJUPUNYAUUJUBUCUD
          YSXIDUPUTUEUBUSZUTVHZUCUSZUTVHZUDUSZUTVHWIUUNUUPUPUNUURUUNUEUFUURUUPU
          EUFUPUNVAUUJUUNUUPUURVGWKUUJXRUTVHZYJYSUTVHUUJXOUTVHXQUTVHZUUSYRUQUTX
          NIUUAWLUUIUUTYRLXNMVCWKXOXQVDVEZUUJYLYJAYLXMUUIUAWMYMVIZXRXGVDVEAYGXM
          UUIYOWMAYHXMUUITWMZUUOUUQVRUUNUUPUEUFUUPUUNUEUFVKUUJUUNUUPVMWKWNUUJUU
          KUUMXJUPUUJUUSYJYHUUKUUMVKUVAUVBUVCXRXGDVLVJWOUUJUUMXTXJUPUUJUULXSXRU
          EUUJYJYHUULXSVKUVBUVCXGDVMVEWPWOWQWSWRWTXA $.
      $}

      ${
        $d A m r $.  $d F b f g h l r u $.  $d F b k l n r u $.  $d F m r $.
        $d L f g h $.  $d L k $.  $d Q b f g h r $.  $d X b f g h r $.
        $d b f g h l p q r u $.  $d b f g h ph r $.  $d k l p q r u $.
        $d L b $.
        caucvgprprlemaddq.x $e |- ( ph -> X e. P. ) $.
        caucvgprprlemaddq.q $e |- ( ph -> Q e. P. ) $.
        caucvgprprlemaddq.ex $e |- ( ph -> E. r e. N.
          ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } ,
            { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q }
          >. ) <P ( ( F ` r ) +P. Q ) ) $.
        $( Lemma for ~ caucvgprpr .  Part of showing the putative limit to be a
           limit.  (Contributed by Jim Kingdon, 5-Jun-2021.) $)
        caucvgprprlemaddq $p |- ( ph -> X <P ( L +P. Q ) ) $=
          ( vb vf vg vh c1o cop ceq cec crq cfv cltq wbr cab cpp cltp cnpi wrex
          cv nfv nfcv cplq cnq crab nfre1 nfrabw nfop nfcxfr nfov nfbr wcel cnp
          co wa wf ad2antrr clti wi wral simpr simplrl caucvgprprlemnbj recnnpr
          wceq ffvelcdmd adantl addclpr syl2anc syl addassprg syl3anc breq1d wb
          ltaprg addcomprg caovord2d oveq2d 3bitr4rd mtbird nrexdv breq1 cbvabv
          w3a breq2 opeq12i oveq2i breq1i rexbii adantr simprl caucvgprprlemexb
          oveq12i sylnibr mtod wo simprr caucvgprprlemcl wor ltsopr sowlin mpan
          mpd ecased mpbird exp32 rexlimd ) AJMUSZKUSZUFUGUHUIUJUKZULUMZMUNZYIL
          USZULUMZLUNZUGZUOVMZYHHUKZDUOVMZUPUMZKUQURJIDUOVMZUPUMZUAAYSUUAKUQAKU
          TKJYTUPKJVAKUPVAKIDUOKIYGNUSZYIVBVMZULUMMUNUUCYLULUMLUNUGYQUPUMZKUQUR
          ZNVCVDZYQYOUOVMYGBUSZULUMMUNUUGYLULUMLUNUGUPUMZKUQURZBVCVDZUGRKUUFUUJ
          UUEKNVCUUDKUQVEKVCVAZVFUUIKBVCUUHKUQVEUUKVFVGVHKUOVAKDVAVIVJAYHUQVKZY
          SUUAAUULYSVNZVNZUUAYPYTYOUOVMZUPUMZUUNUUPUUOYRUPUMZUUNUUQUBUSZHUKZYGU
          URUFUGUHUIUJUKZULUMZMUNZUUTYLULUMZLUNZUGZUOVMZDYOUOVMZUOVMZYRUPUMZUBU
          QURZUUNUUSUUBUUTULUMZNUNZUUTUUGULUMZBUNZUGZUOVMZDUUBYIULUMZNUNZYIUUGU
          LUMZBUNZUGZUOVMZUOVMZYRUPUMZUBUQURUVJUUNUWDUBUQUUNUURUQVKZVNZUWDUVPUW
          AUOVMZYQUPUMZUWFBUUREGHYHNAUQVLHVOZUUMUWEOVPZAGUSZEUSZVQUMUWKHUKZUWLH
          UKZUUBUWKUFUGUHUIUJUKZULUMNUNUWOUUGULUMBUNUGZUOVMUPUMUWNUWMUWPUOVMUPU
          MVNVREUQVSGUQVSZUUMUWEPVPUUNUWEVTZAUULYSUWEWAZWBUWFUWGDUOVMZYRUPUMUVP
          UWADUOVMZUOVMZYRUPUMUWHUWDUWFUWTUXBYRUPUWFUVPVLVKZUWAVLVKZDVLVKZUWTUX
          BWDUWFUUSVLVKUVOVLVKZUXCUWFUQVLUURHUWJUWRWEUWEUXFUUNBUURNWCWFUUSUVOWG
          WHZUWFUULUXDUWSBYHNWCWIZAUXEUUMUWETVPZUVPUWADWJWKWLUWFUCUDUEUWGYQDUPV
          LUOUCUSZVLVKZUDUSZVLVKZUEUSZVLVKXCZUXJUXLUPUMUXNUXJUOVMUXNUXLUOVMUPUM
          WMZUWFUXJUXLUXNWNZWFUWFUXCUXDUWGVLVKUXGUXHUVPUWAWGWHUWFUQVLYHHUWJUWSW
          EUXIUXKUXMVNZUXJUXLUOVMUXLUXJUOVMWDZUWFUXJUXLWOZWFWPUWFUWCUXBYRUPUWFU
          WBUXAUVPUOUWFUXEUXDUWBUXAWDUXIUXHDUWAWOWHWQWLWRWSWTUVIUWDUBUQUVHUWCYR
          UPUVFUVPUVGUWBUOUVEUVOUUSUOUVBUVLUVDUVNUVAUVKMNYGUUBUUTULXAXBUVCUVMLB
          YLUUGUUTULXDXBXEXFYOUWADUOYKUVRYNUVTYJUVQMNYGUUBYIULXAXBYMUVSLBYLUUGY
          IULXDXBXEXFXLXGXHXMUUNBCDYHEFGHIKLMUBNAUWIUUMOXIZAUWQUUMPXIACFUSHUKUP
          UMFUQVSUUMQXIRAUXEUUMTXIZAUULYSXJZXKXNUUNYSUUPUUQXOZAUULYSXPUUNYPVLVK
          ZYRVLVKZUUOVLVKZYSUYDVRZUUNJVLVKZYOVLVKZUYEAUYIUUMSXIZUUNUULUYJUYCLYH
          MWCWIZJYOWGWHUUNYQVLVKUXEUYFUUNUQVLYHHUYAUYCWEUYBYQDWGWHUUNYTVLVKZUYJ
          UYGUUNIVLVKZUXEUYMAUYNUUMABCEFGHIKLMNOPQRXQXIUYBIDWGWHZUYLYTYOWGWHVLU
          PXRUYEUYFUYGXCUYHXSVLYPYRUUOUPXTYAWKYBYCUUNUCUDUEJYTYOUPVLUOUXOUXPUUN
          UXQWFUYKUYOUYLUXRUXSUUNUXTWFWPYDYEYFYB $.
      $}

      ${
        caucvgprprlemlim.q $e |- ( ph -> Q e. P. ) $.
        caucvgprprlemlim.jk $e |- ( ph -> J <N K ) $.
        caucvgprprlemlim.jkq $e |- ( ph ->
          <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >.
          <P Q ) $.

        ${
          $d A m $.  $d A r $.  $d F m $.  $d F r $.  $d F l u $.  $d F n k $.
          $d J l $.  $d J u $.  $d K l r $.  $d K r u $.  $d Q r $.  $d L k $.
          $d ph r $.  $d p q r l u $.  $d m r $.  $d k l u $.  $d k r $.
          $d k p q $.  $d n l u $.  $d n r $.
          $( Lemma for ~ caucvgprpr .  Part of showing the putative limit to be
             a limit.  (Contributed by Jim Kingdon, 25-Nov-2020.) $)
          caucvgprprlem1 $p |- ( ph -> ( F ` K ) <P ( L +P. Q ) ) $=
            ( cfv cnpi cnp wcel clti wbr wa ltrelpi syl simprd ffvelcdmd cv c1o
            brel cop ceq cec crq cltq cab cpp cltp wrex caucvgprprlemk cnq nnnq
            co wb recclnq nqprlu 3syl ltaprg syl3anc mpbid eceq1d fveq2d breq2d
            wceq opeq1 abbidv breq1d opeq12d oveq2d fveq2 oveq1d breq12d rspcev
            syl2anc breq1 cbvabv opeq12i oveq2i breq1i rexbii caucvgprprlemaddq
            breq2 sylib ) ABCDEFGHKJHUCZLMNOPQRSAUDUEJHPAIUDUFZJUDUFZAIJUGUHXAX
            BUIUAIJUDUDUGUJUPUKULZUMZTAWTOUNZLUNZUOUQZURUSZUTUCZVAUHZOVBZXIBUNZ
            VAUHZBVBZUQZVCVIZXFHUCZDVCVIZVDUHZLUDVEZWTNUNZXIVAUHZNVBZXIMUNZVAUH
            ZMVBZUQZVCVIZXRVDUHZLUDVEAXBWTXEJUOUQZURUSZUTUCZVAUHZOVBZYLXLVAUHZB
            VBZUQZVCVIZWTDVCVIZVDUHZXTXCAYQDVDUHZYTABDIJOUAUBVFAYQUEUFZDUEUFWTU
            EUFUUAYTVJAYKVGUFZYLVGUFUUBAXBUUCXCJVHUKYKVKBYLOVLVMTXDYQDWTVNVOVPX
            SYTLJUDXFJVTZXPYRXRYSVDUUDXOYQWTVCUUDXKYNXNYPUUDXJYMOUUDXIYLXEVAUUD
            XHYKUTUUDXGYJURXFJUOWAVQVRZVSWBUUDXMYOBUUDXIYLXLVAUUEWCWBWDWEUUDXQW
            TDVCXFJHWFWGWHWIWJXSYILUDXPYHXRVDXOYGWTVCXKYCXNYFXJYBONXEYAXIVAWKWL
            XMYEBMXLYDXIVAWRWLWMWNWOWPWSWQ $.
        $}

        ${
          $d A r $.  $d F r u x $.  $d F l $.  $d F k $.  $d F n $.
          $d K l p u x $.  $d K p q r u x $.  $d J l $.  $d J u $.  $d L x $.
          $d L k $.  $d Q x $.  $d ph x $.  $d ph r $.  $d k n $.  $d k r $.
          $d l u p q r $.  $d m r $.  $d k p q $.  $d n u $.  $d l n $.
          $d k l $.  $d k u $.
          $( Lemma for ~ caucvgprpr .  Part of showing the putative limit to be
             a limit.  (Contributed by Jim Kingdon, 25-Nov-2020.) $)
          caucvgprprlem2 $p |- ( ph -> L <P ( ( F ` K ) +P. Q ) ) $=
            ( vx cv cfv c1o cop ceq cec crq cltq wbr cab co c2nd wcel c1st cltp
            cpp wa cnq wrex caucvgprprlemk cnp wb cnpi clti ltrelpi brel simprd
            nnnq recclnq nqprlu ffvelcdmd ltaprg syl3anc addclpr syl2anc ltdfpr
            syl mpbid simprl adantr simprrl breq1 cbvabv breq2 opeq12i eleqtrdi
            oveq2i fveq2i nqpru fveq2 opeq1 eceq1d fveq2d breq2d abbidv opeq12d
            wceq breq1d oveq12d rspcev rexbidv cplq crab nqex rabex op2nd eqtri
            elrab2 sylanbrc simprrr syl12anc caucvgprprlemcl mpbird rexlimddv
            rspe ) AUCUDZJHUEZOUDZJUFUGZUHUIZUJUEZUKULZOUMZYDBUDZUKULZBUMZUGZUS
            UNZUOUEZUPZXSXTDUSUNZUQUEUPZUTZKYNURULZUCVAAYKYNURULZYPUCVAVBZAYJDU
            RULZYRABDIJOUAUBVCAYJVDUPZDVDUPZXTVDUPZYTYRVEAYDVAUPZUUAAJVFUPZUUDA
            IVFUPZUUEAIJVGULUUFUUEUTUAIJVFVFVGVHVIVTVJZUUEYCVAUPUUDJVKYCVLVTVTZ
            BYDOVMVTZTAVFVDJHPUUGVNZYJDXTVOVPWAAYKVDUPZYNVDUPZYRYSVEAUUCUUAUUKU
            UJUUIXTYJVQVRAUUCUUBUULUUJTXTDVQVRZYKYNUCVSVRWAAXSVAUPZYPUTZUTZYQXS
            KUOUEZUPZYOUTZUCVAVBZUUPUUNUURYOUUTAUUNYPWBZUUPUUNLUDZHUEZNUDZUVBUF
            UGZUHUIZUJUEZUKULZNUMZUVGMUDZUKULZMUMZUGZUSUNZUVDXSUKULZNUMZXSUVJUK
            ULZMUMZUGZURULZLVFVBZUURUVAUUPUUEXTUVDYDUKULZNUMZYDUVJUKULZMUMZUGZU
            SUNZUVSURULZUWAAUUEUUOUUGWCUUPXSUWGUOUEZUPZUWHUUPXSYLUWIAUUNYMYOWDY
            KUWGUOYJUWFXTUSYFUWCYIUWEYEUWBONYAUVDYDUKWEWFYHUWDBMYGUVJYDUKWGWFWH
            WJWKWIUUPUUNUWGVDUPZUWJUWHVEUVAAUWKUUOAUUCUWFVDUPZUWKUUJAUUDUWLUUHM
            YDNVMVTXTUWFVQVRWCMXSUWGNWLVRWAUVTUWHLJVFUVBJWTZUVNUWGUVSURUWMUVCXT
            UVMUWFUSUVBJHWMUWMUVIUWCUVLUWEUWMUVHUWBNUWMUVGYDUVDUKUWMUVFYCUJUWMU
            VEYBUHUVBJUFWNWOWPZWQWRUWMUVKUWDMUWMUVGYDUVJUKUWNXAWRWSXBXAXCVRUVNU
            VDYGUKULZNUMZYGUVJUKULZMUMZUGZURULZLVFVBZUWABXSVAUUQYGXSWTZUWTUVTLV
            FUXBUWSUVSUVNURUXBUWPUVPUWRUVRUXBUWOUVONYGXSUVDUKWGWRUXBUWQUVQMYGXS
            UVJUKWEWRWSWQXDUUQUVDYAUVGXEUNZUKULNUMUXCUVJUKULMUMUGUVCURULLVFVBZO
            VAXFZUXABVAXFZUGZUOUEUXFKUXGUOSWKUXEUXFUXDOVAXGXHUXABVAXGXHXIXJXKXL
            AUUNYMYOXMUUSUCVAXRXNUUPKVDUPZUULYQUUTVEAUXHUUOABCEFGHKLMNOPQRSXOWC
            AUULUUOUUMWCKYNUCVSVRXPXQ $.
        $}
      $}

      ${
        $d A r $.  $d A j $.  $d F u r $.  $d F l $.  $d F k $.  $d F n $.
        $d ph k r $.  $d L k $.  $d j k ph x $.  $d k l u $.  $d k l p $.
        $d k q r $.  $d j r $.  $d r x $.  $d l u p q r $.  $d m r $.
        $d k n $.  $d n u $.  $d l n $.  $d j l u $.  $d n r $.
        $( Lemma for ~ caucvgprpr .  The putative limit is a limit.
           (Contributed by Jim Kingdon, 21-Nov-2020.) $)
        caucvgprprlemlim $p |- ( ph ->
            A. x e. P. E. j e. N. A. k e. N. ( j <N k ->
            ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) ) $=
          ( wbr cnpi cv clti cfv cpp co cltp wral wrex cnp wcel c1o cop ceq cec
          wa wi crq cltq cab archrecpr adantl wf ad5antr ad4antr caucvgprprlem1
          simpr simpllr caucvgprprlem2 jca ex ralrimiva reximdva mpd ) AEUAZFUA
          ZUBSZVOIUCZJBUAZUDUEUFSZJVQVRUDUEUFSZUOZUPZFTUGZETUHZBUIAVRUIUJZUOZNU
          AZVNUKULUMUNUQUCZURSNUSWHCUAZURSCUSULVRUFSZETUHZWDWEWKACVRENUTVAWFWJW
          CETWFVNTUJZUOZWJWCWMWJUOZWBFTWNVOTUJZUOZVPWAWPVPUOZVSVTWQCDVRFGHIVNVO
          JKLMNATUIIVBWEWLWJWOVPOVCZAHUAZVOUBSWSIUCZVQWGWSUKULUMUNUQUCZURSNUSXA
          WIURSCUSULZUDUEUFSVQWTXBUDUEUFSUOUPFTUGHTUGWEWLWJWOVPPVCZADGUAIUCUFSG
          TUGWEWLWJWOVPQVCZRWFWEWLWJWOVPAWEVFVDZWPVPVFZWMWJWOVPVGZVEWQCDVRFGHIV
          NVOJKLMNWRXCXDRXEXFXGVHVIVJVKVJVLVMVK $.
      $}
    $}

    ${
      $d A r $.  $d A j $.  $d F j y $.  $d F k y $.  $d F x y $.  $d F u r $.
      $d F l $.  $d F n $.  $d j l y $.  $d j p y $.  $d j q y $.  $d j r y $.
      $d j u y $.  $d k l y $.  $d k p y $.  $d k q y $.  $d k r y $.
      $d k u y $.  $d k l u $.  $d l x y $.  $d l p $.  $d p x y $.
      $d q x y $.  $d r x y $.  $d u x y $.  $d p q r u $.  $d k q r $.
      $d l u p q r $.  $d ph x k j $.  $d ph r $.  $d m r $.  $d k n $.
      $d n u $.  $d l n $.  $d n r $.
      $( A Cauchy sequence of positive reals with a modulus of convergence
         converges to a positive real.  This is basically Corollary 11.2.13 of
         [HoTT], p.  (varies) (one key difference being that this is for
         positive reals rather than signed reals).  Also, the HoTT book theorem
         has a modulus of convergence (that is, a rate of convergence)
         specified by (11.2.9) in HoTT whereas this theorem fixes the rate of
         convergence to say that all terms after the nth term must be within
         ` 1 / n ` of the nth term (it should later be able to prove versions
         of this theorem with a different fixed rate or a modulus of
         convergence supplied as a hypothesis).  We also specify that every
         term needs to be larger than a given value ` A ` , to avoid the case
         where we have positive terms which "converge" to zero (which is not a
         positive real).

         This is similar to ~ caucvgpr except that values of the sequence are
         positive reals rather than positive fractions.  Reading that proof
         first (or ~ cauappcvgpr ) might help in understanding this one, as
         they are slightly simpler but similarly structured.  (Contributed by
         Jim Kingdon, 14-Nov-2020.) $)
      caucvgprpr $p |- ( ph ->
          E. y e. P. A. x e. P. E. j e. N. A. k e. N. ( j <N k ->
          ( ( F ` k ) <P ( y +P. x ) /\
            y <P ( ( F ` k ) +P. x ) ) ) ) $=
        ( vp vq cv wbr cltp cnpi c1o cop ceq cec crq cfv cplq cltq cab wrex cnq
        vr co crab cpp cnp wcel clti wral eqid caucvgprprlemcl caucvgprprlemlim
        wa wi wceq oveq1 breq2d breq1 anbi12d imbi2d rexralbidv ralbidv syl2anc
        rspcev ) AOQZKQULQZUAUBUCUDUEUFZUGUMZUHROUIVRPQZUHRPUIUBVPJUFZSRULTUJKU
        KUNVTVOVQUHROUIVQVSUHRPUIUBUOUMVODQZUHROUIWAVSUHRPUIUBSRULTUJDUKUNUBZUP
        UQFQGQZURRZWCJUFZWBBQZUOUMZSRZWBWEWFUOUMZSRZVCZVDZGTUSFTUJZBUPUSZWDWECQ
        ZWFUOUMZSRZWOWISRZVCZVDZGTUSFTUJZBUPUSZCUPUJADEGHIJWBULPOKLMNWBUTZVAABD
        EFGHIJWBULPOKLMNXCVBXBWNCWBUPWOWBVEZXAWMBUPXDWTWLFGTTXDWSWKWDXDWQWHWRWJ
        XDWPWGWESWOWBWFUOVFVGWOWBWISVHVIVJVKVLVNVM $.
    $}
  $}

  ${
    $d A x $.  $d B x $.
    $( Lemma for ~ suplocexpr .  Membership in the lower cut of the putative
       supremum.  (Contributed by Jim Kingdon, 9-Jan-2024.) $)
    suplocexprlemell $p |- ( B e. U. ( 1st " A )
        <-> E. x e. A B e. ( 1st ` x ) ) $=
      ( c1st cres cima cuni wcel cv cfv wrex wfn wb cvv wss wfo fo1st ax-mp ssv
      fofn fnssres mp2an eluniimadm resima unieqi eleq2i eleq2d rexbiia 3bitr3i
      fvres ) CDBEZBFZGZHZCAIZUKJZHZABKZCDBFZGZHCUODJZHZABKUKBLZUNURMDNLZBNOVCN
      NDPVDQNNDTRBSNBDUAUBABCUKUCRUMUTCULUSDBUDUEUFUQVBABUOBHUPVACUOBDUJUGUHUI
      $.
  $}

  ${
    suplocexprlem2b.b $e |- B = <. U. ( 1st " A ) ,
      { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. $.
    $( Lemma for ~ suplocexpr .  Expression for the lower cut of the putative
       supremum.  (Contributed by Jim Kingdon, 9-Jan-2024.) $)
    suplocexprlem2b $p |- ( A C_ P. -> ( 2nd ` B )
        = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) $=
      ( cnp wss c2nd cfv c1st cima cuni cv cltq wbr cint wrex cnq cvv wcel crab
      cop fveq2i wceq wfun fo1st fofun ax-mp npex ssex funimaexg sylancr uniexg
      wfo syl nqex rabex op2ndg sylancl eqtrid ) CFGZDHIJCKZLZAMBMNOAHCKPQZBRUA
      ZUBZHIZVEDVFHEUCVAVCSTZVESTVGVEUDVAVBSTZVHVAJUEZCSTVISSJUNVJUFSSJUGUHCFUI
      UJJCSUKULVBSUMUOVDBRUPUQVCVESSURUSUT $.
  $}

  ${
    suplocexpr.m $e |- ( ph -> E. x x e. A ) $.
    suplocexpr.ub $e |- ( ph -> E. x e. P. A. y e. A y <P x ) $.
    suplocexpr.loc $e |- ( ph -> A. x e. P. A. y e. P. ( x <P y
        -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) ) $.

    ${
      $d A x y $.  $d ph x y $.
      $( Lemma for ~ suplocexpr . ` A ` is a set of positive reals.
         (Contributed by Jim Kingdon, 7-Jan-2024.) $)
      suplocexprlemss $p |- ( ph -> A C_ P. ) $=
        ( cnp cv cltp wbr wral wrex wcel wi rsp ltrelpr brel simpld syl6 ssrdv
        a1i rexlimdvw mpd ) ACEIACJZBJZKLZCEMZBINUFEOZUFIOZPZGAUIULBIUIULPAUIUJ
        UHUKUHCEQUHUKUGIOUFUGIIKRSTUAUCUDUEUB $.
    $}

    ${
      $d A s x y $.  $d ph s x y $.
      $( Lemma for ~ suplocexpr .  The lower cut of the putative supremum is
         inhabited.  (Contributed by Jim Kingdon, 7-Jan-2024.) $)
      suplocexprlemml $p |- ( ph -> E. s e. Q. s e. U. ( 1st " A ) ) $=
        ( cv c1st cfv wcel cnq wrex cima cuni wex wral cnp c2nd suplocexprlemss
        wa cop sselda prop prml ralrimiva r19.2m suplocexprlemell rexbii rexcom
        3syl syl2anc bitri sylibr ) AFJZBJZKLZMZFNOZBEOZUQKEPQMZFNOZAUREMZBRVAB
        ESVBGAVABEAVEUCURTMUSURUALZUDTMVAAETURABCDEGHIUBUEURUFFVFUSUGUMUHVABBEU
        IUNVDUTBEOZFNOVBVCVGFNBEUQUJUKUTFBNEULUOUP $.
    $}

    ${
      $d A r s $.  $d A x y $.  $d ph q r s $.  $d ph x y $.
      $( Lemma for ~ suplocexpr .  The lower cut of the putative supremum is
         rounded.  (Contributed by Jim Kingdon, 9-Jan-2024.) $)
      suplocexprlemrl $p |- ( ph -> A. q e. Q. ( q e. U. ( 1st " A )
          <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) ) $=
        ( vs cv wcel wa cnq wrex cnp simprl simprr syl2anc c1st cima wbr wb cfv
        cuni cltq suplocexprlemell biimpi adantl c2nd suplocexprlemss ad3antrrr
        cop wss sseldd prop prnmaxl ltrelnq brel simprd ad2antll simplrl sylibr
        syl jca reximssdv rexlimddv ex sylib wex wi prcdnql mpd 19.8a rexlimdvw
        rspe df-rex impbid ralrimiva ) AGLZUAEUBUFZMZWAFLZUGUCZWDWBMZNZFOPZUDGO
        AWAOMZNZWCWHWJWCWHWJWCNZWAKLZUAUEZMZWHKEWCWNKEPZWJWCWOKEWAUHZUIUJWKWLEM
        ZWNNZNZWEWGFOWMWSWMWLUKUEZUNQMZWNWEFWMPWSWLQMZXAWSEQWLAEQUOZWIWCWRABCDE
        HIJULZUMWKWQWNRUPWLUQZVEWKWQWNSFWAWTWMURTWEWDOMZWSWDWMMZWEWIXFWAWDOOUGU
        SUTVAVBWSXGWENZNZWEWFWSXGWESXIXGKEPZWFXIWQXGXJWKWQWNXHVCWSXGWERXGKEVQTK
        EWDUHZVDVFVGVHVIWJWGWCFOWJWGWCWJWGNZXGWCKEXLWFXJWJWEWFSXKVJXLWQXGNZNZWO
        WCXNWRKVKZWOXNWQWNXOXLWQXGRZXNWEWNWJWEWFXMVCXNXAXGWEWNVLXNXBXAXNEQWLAXC
        WIWGXMXDUMXPUPXEVEXLWQXGSWDWAWTWMVMTVNWRKVOTWNKEVRVDWPVDVHVIVPVSVT $.
    $}

    ${
      suplocexpr.b $e |- B = <. U. ( 1st " A ) ,
        { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. $.

      ${
        $d A j s u w $.  $d A s t u x y $.  $d B j s $.  $d j ph s u $.
        $d ph s t u x y $.
        $( Lemma for ~ suplocexpr .  The upper cut of the putative supremum is
           inhabited.  (Contributed by Jim Kingdon, 7-Jan-2024.) $)
        suplocexprlemmu $p |- ( ph -> E. s e. Q. s e. ( 2nd ` B ) ) $=
          ( vj c2nd wcel cnq wrex cnp wa vt cv cfv cima cint cltp wbr wral c1st
          cop prop prmu syl ad2antrl wceq wfun cvv wfo fo2nd fofun fvelima mpan
          ax-mp adantl wss suplocexprlemss ad5antr simprl sseldd ad4antr simprr
          wn breq1 rspcdva wor ltsopr so2nr syl2anc imnan sylibr mpd aptiprlemu
          wi syl3anc simpllr eleqtrd rexlimddv ralrimiva vex elint2 ex reximdva
          cltq c1q cplq co 1nq addclnq sylancl ltaddnq breq2 rspcev weq rexbidv
          rexcom sylib wb ssid rexss suplocexprlem2b eleq2d elrab bitrdi adantr
          crab mpbird eleq1w cbvrexv ) ANUBZHOUCZPZNQRZIUBZXTPZIQRAYCOGUDZUEZPZ
          YBIQACUBZBUBZUFUGZCGUHZYGIQRZBSKAYISPZYKTZTZYCYIOUCZPZIQRZYLYMYRAYKYM
          YIUIUCZYPUJSPYRYIUKIYPYSULUMUNYOYQYGIQYOYCQPZTZYQYGUUAYQTZYCUAUBZPZUA
          YEUHYGUUBUUDUAYEUUBUUCYEPZTZFUBZOUCZUUCUOZUUDFGUUEUUIFGRZUUBOUPZUUEUU
          JUQUQOURUUKUSUQUQOUTVCFUUCGOVAVBVDUUFUUGGPZUUITZTZYCUUHUUCUUNYPUUHYCU
          UNUUGSPZYMYIUUGUFUGZVLZYPUUHVEUUNGSUUGAGSVEZYNYTYQUUEUUMABCDGJKLVFZVG
          UUFUULUUIVHZVIZYOYMYTYQUUEUUMAYMYKVHVJZUUNUUGYIUFUGZUUQUUNYJUVCCGUUGY
          HUUGYIUFVMYOYKYTYQUUEUUMAYMYKVKVJUUTVNUUNUVCUUPTVLZUVCUUQWCUUNUUOYMUV
          DUVAUVBSUFVOUUOYMTUVDVPSUUGYIUFVQVBVRUVCUUPVSVTWAUUGYIWBWDUUAYQUUEUUM
          WEVIUUFUULUUIVKWFWGWHUAYCYEIWIWJVTWKWLWAWGAYTYGTZTZYBXSQPEUBZXSWMUGZE
          YFRZTZNQRZUVFUVINQRZUVKUVFUVHNQRZEYFRZUVLUVFYGYCXSWMUGZNQRZUVNAYTYGVK
          UVFYCWNWOWPZQPZYCUVQWMUGZUVPUVFYTWNQPZUVRAYTYGVHZWQYCWNWRWSUVFYTUVTUV
          SUWAWQYCWNWTWSUVOUVSNUVQQXSUVQYCWMXAXBVRUVMUVPEYCYFEIXCUVHUVONQUVGYCX
          SWMVMXDXBVRUVHENYFQXEXFQQVEUVLUVKXGQXHUVINQQXIVCXFAYBUVKXGUVEAYAUVJNQ
          AYAXSUVGUUGWMUGZEYFRZFQXOZPUVJAXTUWDXSAUURXTUWDUOUUSEFGHMXJUMXKUWCUVI
          FXSQFNXCUWBUVHEYFUUGXSUVGWMXAXDXLXMXDXNXPWGYAYDNIQNIXTXQXRXF $.
      $}

      ${
        $d A q t $.  $d A s $.  $d A q u $.  $d A x y $.  $d B q w $.
        $d ph q r w $.  $d ph x y $.  $d r s $.  $d r u w $.
        $( Lemma for ~ suplocexpr .  The upper cut of the putative supremum is
           rounded.  (Contributed by Jim Kingdon, 9-Jan-2024.) $)
        suplocexprlemru $p |- ( ph -> A. r e. Q. ( r e. ( 2nd ` B )
            <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) $=
          ( cv wcel cltq wa cnq wrex c2nd cfv wb cima cint crab suplocexprlemss
          wbr cnp wss wceq suplocexprlem2b syl eleq2d adantr biimpa breq2 elrab
          rexbidv sylib simprd ltbtwnnqq biimpi ad2antll simprr simplr ad2antrr
          simprl jca rspe elrabd ad5antr mpbird ex reximdva rexlimddv ad3antrrr
          mpd simpllr mpbid simpr wi ltrelnq brel simpld adantl simp-4r wor w3a
          ltsonq sotr mpan syl3anc rexlimdva2 impbid ralrimiva ) AIOZHUAUBZPZJO
          ZWQQUHZWTWRPZRZJSTZUCISAWQSPZRZWSXDXFWSXDXFWSRZEOZWQQUHZXDEUAGUDUEZXG
          XEXIEXJTZXGWQXHFOZQUHZEXJTZFSUFZPZXEXKRXFWSXPAWSXPUCZXEAWRXOWQAGUIUJW
          RXOUKABCDGKLMUGEFGHNULUMZUNZUOUPXNXKFWQSXLWQUKXMXIEXJXLWQXHQUQUSZURUT
          VAXGXHXJPZXIRZRZXHWTQUHZXARZJSTZXDXIYFXGYAXIYFJXHWQVBVCVDYCYEXCJSYCWT
          SPZRZYEXCYHYERZXAXBYHYDXAVEYIXBWTXOPZYIXNYDEXJTZFWTSXLWTUKXMYDEXJXLWT
          XHQUQUSZYCYGYEVFYIYAYDRYKYIYAYDYCYAYGYEXGYAXIVHVGYHYDXAVHVIYDEXJVJUMV
          KAXBYJUCZXEWSYBYGYEAWRXOWTXRUNZVLVMVIVNVOVRVPVNXFXCWSJSXFYGRZXCRZWSXP
          YPXNXKFWQSXTAXEYGXCVSZYPYKXKYPYGYKYPYJYGYKRYPXBYJYOXAXBVEAYMXEYGXCYNV
          QVTXNYKFWTSYLURUTVAYPYDXIEXJYPYARZYDXIYRYDRZYEXIYSYDXAYRYDWAYPXAYAYDY
          OXAXBVHVGVIYSXHSPZYGXEYEXIWBZYDYTYRYDYTYGXHWTSSQWCWDWEWFXFYGXCYAYDWGY
          PXEYAYDYQVGSQWHYTYGXEWIUUAWJSXHWTWQQWKWLWMVRVNVOVRVKAXQXEYGXCXSVQVMWN
          WOWP $.
      $}

      ${
        $d A s t w $.  $d A u w $.  $d A x y $.  $d B s w $.  $d ph q s w $.
        $d ph x y $.  $d q s t w $.  $d q u w $.
        $( Lemma for ~ suplocexpr .  The putative supremum is disjoint.
           (Contributed by Jim Kingdon, 9-Jan-2024.) $)
        suplocexprlemdisj $p |- ( ph -> A. q e. Q.
            -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) $=
          ( vs cv wcel c2nd wa cnp cvv vt c1st cima cuni cfv wn cnq wfal simprl
          wrex suplocexprlemell sylib simprr cltq wbr cint crab suplocexprlemss
          simplrr wss wb ad3antrrr suplocexprlem2b eleq2d syl weq breq2 rexbidv
          elrab bitrdi mpbid simprd cop wi adantr simplrl sseldd prop eleq2 vex
          wral elint2 wfun cdm wfo fo2nd fofun ax-mp fof fdmi eleqtrri funfvima
          wf mp2an ad2antrl rspcdva prcunqu syl2anc mpd rexlimddv jca pm2.21fal
          simpllr prdisj inegd ralrimiva ) AIOZUBGUCUDPZXGHQUEZPZRZUFIUGAXGUGPZ
          RZXKXMXKRZXGNOZUBUEZPZUHNGXNXHXQNGUJXMXHXJUINGXGUKULXNXOGPZXQRZRZXQXG
          XOQUEZPZRZXTXQYBXNXRXQUMXTEOZXGUNUOZYBEQGUCZUPZXTXLYEEYGUJZXTXJXLYHRZ
          XMXHXJXSUSXTXJXGYDFOZUNUOZEYGUJZFUGUQZPZYIXTGSUTZXJYNVAAYOXLXKXSABCDG
          JKLURVBZYOXIYMXGEFGHMVCVDVEYLYHFXGUGFIVFYKYEEYGYJXGYDUNVGVHVIVJVKVLXT
          YDYGPZYERZRZYEYBXTYQYEUMYSXPYAVMSPZYDYAPZYEYBVNYSXOSPZYTYSGSXOXTYOYRY
          PVOXNXRXQYRVPVQXOVRZVEYSYDUAOZPZUUAUAYFYAUUDYAYDVSYSYQUUEUAYFWAXTYQYE
          UIUAYDYFEVTWBULXTYAYFPZYRXRUUFXNXQQWCZXOQWDZPXRUUFVNTTQWEZUUGWFTTQWGW
          HXOTUUHNVTTTQUUITTQWMWFTTQWIWHWJWKGXOQWLWNWOVOWPXGYDYAXPWQWRWSWTXAXTY
          TXLYCUFXTUUBYTXTGSXOYPXNXRXQUIVQUUCVEAXLXKXSXCXGYAXPXDWRXBWTXEXF $.
      $}

      ${
        $d A s t u v z $.  $d A u v w z $.  $d A u x y z $.  $d B v $.
        $d l q s t u v z $.  $d l q u v w z $.  $d l q u x y z $.
        $d ph q r v w $.  $d ph q s t v z $.  $d ph q x y z $.  $d r u v w $.
        $d v y z $.
        $( Lemma for ~ suplocexpr .  The putative supremum is located.
           (Contributed by Jim Kingdon, 9-Jan-2024.) $)
        suplocexprlemloc $p |- ( ph -> A. q e. Q. A. r e. Q.
            ( q <Q r -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) ) $=
          ( cltq wbr wcel cnq wa cnp vv vl vs vt cv c1st cima cuni c2nd wo wrex
          cfv wi simpr ltbtwnnqq sylib cint simplll simprl ad2antrr simprrl cab
          jca32 cop cltp wral ltnqpri adantl breq2 ralbidv orbi2d imbi12d breq1
          rexbidv orbi1d simplrl nqprlu syl rspcdva simplrr mpd suplocexprlemss
          wceq wss ad4antr simplr sseldd ltdfpr syl2anc mpbid vex ltnqex gtnqex
          op2nd elab2 anbi1i rexbii wex simpllr adantr prop simprrr prcdnql jca
          wb 19.8ad df-rex sylibr suplocexprlemell rexlimddv rexlimdva2 cvv wfo
          wfun fo2nd fofun ax-mp fvelima ad3antrrr ad5antr nqpru mpbird eleqtrd
          mpan simprr ralrimiva elint2 ex orim12d rspcev elrabd suplocexprlem2b
          crab eleq2d orim2d ralrimivva ) AJUEZIUEZOPZYQUFGUGUHQZYRHUIULZQZUJZU
          MJIRRAYQRQZYRRQZSZSZYSUUCUUGYSSZYQUAUEZOPZUUIYROPZSZUUCUARUUHYSUULUAR
          UKUUGYSUNUAYQYRUOUPUUHUUIRQZUULSZSZYTUUIUIGUGZUQZQZUJZUUCUUOAUUDUUMSZ
          SZUUJUUSUUOAUUDUUMAUUFYSUUNURUUGUUDYSUUNAUUDUUEUSUTUUHUUMUULUSVCUUHUU
          MUUJUUKVAUVAUUJSZUBUEZYQOPUBVBZYQFUEZOPZFVBZVDZDUEZVEPZDGUKZUVIUVCUUI
          OPUBVBUUIUVEOPFVBVDZVEPZDGVFZUJZUUSUVBUVHUVLVEPZUVOUUJUVPUVAFYQUUIUBV
          GVHUVBUVHCUEZVEPZUVKUVIUVQVEPZDGVFZUJZUMZUVPUVOUMCTUVLUVQUVLWCZUVRUVP
          UWAUVOUVQUVLUVHVEVIUWCUVTUVNUVKUWCUVSUVMDGUVQUVLUVIVEVIVJVKVLUVBBUEZU
          VQVEPZUWDUVIVEPZDGUKZUVTUJZUMZCTVFZUWBCTVFBTUVHUWDUVHWCZUWIUWBCTUWKUW
          EUVRUWHUWAUWDUVHUVQVEVMUWKUWGUVKUVTUWKUWFUVJDGUWDUVHUVIVEVMVNVOVLVJAU
          WJBTVFUUTUUJMUTUVBUUDUVHTQZAUUDUUMUUJVPFYQUBVQVRZVSUVBUUMUVLTQAUUDUUM
          UUJVTZFUUIUBVQVRVSWAUVBUVKYTUVNUURUVBUVJYTDGUVBUVIGQZSZUVJSZYQEUEZOPZ
          UWRUVIUFULZQZSZYTERUWQUWRUVHUIULZQZUXASZERUKZUXBERUKUWQUVJUXFUWPUVJUN
          UWQUWLUVITQZUVJUXFXEUVBUWLUWOUVJUWMUTUWQGTUVIAGTWDZUUTUUJUWOUVJABCDGK
          LMWBZWEUVBUWOUVJWFWGZUVHUVIEWHWIWJUXEUXBERUXDUWSUXAUVFUWSFUWRUXCEWKUV
          EUWRYQOVIUVDUVGUBYQWLFYQWMWNWOWPWQUPUWQUWRRQZUXBSZSZYQUWTQZDGUKZYTUXM
          UWOUXNSZDWRUXOUXMUXPDUXMUWOUXNUVBUWOUVJUXLWSUXMUWSUXNUWQUXKUWSUXAVAUX
          MUWTUVIUIULZVDTQZUXAUWSUXNUMUXMUXGUXRUWQUXGUXLUXJWTUVIXAVRUWQUXKUWSUX
          AXBUWRYQUXQUWTXCWIWAXDXFUXNDGXGXHDGYQXIXHXJXKUVBUVNUURUVBUVNSZUUIUCUE
          ZQZUCUUPVFUURUXSUYAUCUUPUXSUXTUUPQZSZUDUEZUIULZUXTWCZUYAUDGUYBUYFUDGU
          KZUXSUIXNZUYBUYGXLXLUIXMUYHXOXLXLUIXPXQUDUXTGUIXRYDVHUYCUYDGQZUYFSZSZ
          UUIUYEUXTUYKUUIUYEQZUYDUVLVEPZUYKUVMUYMDGUYDUVIUYDUVLVEVMUVBUVNUYBUYJ
          WSUYCUYIUYFUSZVSUYKUUMUYDTQUYLUYMXEUVBUUMUVNUYBUYJUWNXSUYKGTUYDAUXHUU
          TUUJUVNUYBUYJUXIXTUYNWGFUUIUYDUBYAWIYBUYCUYIUYFYEYCXJYFUCUUIUUPUAWKYG
          XHYHYIWAWIUUOUURUUBYTUUOUURUUBUUOUURSZUUBYRUWRUVEOPZEUUQUKZFRYMZQZUYO
          UYQUWRYROPZEUUQUKZFYRRUVEYRWCUYPUYTEUUQUVEYRUWROVIVNUUGUUEYSUUNUURAUU
          DUUEYEXSUYOUURUUKVUAUUOUURUNUUOUUKUURUUHUUMUUJUUKXBWTUYTUUKEUUIUUQUWR
          UUIYROVMYJWIYKAUUBUYSXEUUFYSUUNUURAUUAUYRYRAUXHUUAUYRWCUXIEFGHNYLVRYN
          WEYBYHYOWAXJYHYP $.
      $}

      ${
        $d A q r u w z $.  $d A s t u x $.  $d A q r u x y z $.  $d B q r w $.
        $d ph q r u w z $.  $d ph s t u x $.  $d ph q r u x y z $.
        $( Lemma for ~ suplocexpr .  The putative supremum is a positive real.
           (Contributed by Jim Kingdon, 7-Jan-2024.) $)
        suplocexprlemex $p |- ( ph -> B e. P. ) $=
          ( vq vr cnp cv wrex cnq wa wcel vs vt c1st cima cuni c2nd cfv cop wbr
          cltq cint crab wss suplocexprlemss suplocexprlem2b syl opeq2d eqtr4id
          wceq wb wral wn wo w3a suplocexprlemell biimpi adantl ad2antrr simprl
          wi sseldd prop simprr elprnql syl2anc rexlimddv ssrdv ssrab2 eqsstrdi
          suplocexprlemml suplocexprlemmu jca31 suplocexprlemrl suplocexprlemru
          ex jca suplocexprlemdisj suplocexprlemloc 3jca elinp sylanbrc eqeltrd
          ) AHUCGUDUEZHUFUGZUHZOAHWMEPFPUJUIEUFGUDUKQZFRULZUHWOLAWNWQWMAGOUMZWN
          WQUSABCDGIJKUNZEFGHLUOUPZUQURAWMRUMZWNRUMZSMPZWMTZMRQZNPZWNTZNRQZSZSX
          DXCXFUJUIZXFWMTSNRQUTMRVAZXGXJXCWNTZSMRQUTNRVAZSZXDXLSVBMRVAZXJXDXGVC
          VJNRVAMRVAZVDWOOTAXAXBXIAUAWMRAUAPZWMTZXQRTZAXRSZXQUBPZUCUGZTZXSUBGXR
          YCUBGQZAXRYDUBGXQVEVFVGXTYAGTZYCSZSZYBYAUFUGZUHOTZYCXSYGYAOTYIYGGOYAA
          WRXRYFWSVHXTYEYCVIVKYAVLUPXTYEYCVMXQYHYBVNVOVPWEVQAWNWQRWTWPFRVRVSAXE
          XHABCDGMIJKVTABCDEFGHNIJKLWAWFWBAXNXOXPAXKXMABCDGNMIJKWCABCDEFGHNMIJK
          LWDWFABCDEFGHMIJKLWGABCDEFGHNMIJKLWHWIWNWMNMWJWKWL $.
      $}

      ${
        $d A s u w y $.  $d A t w y $.  $d A u x y z $.  $d B s w $.
        $d ph s u w y $.  $d ph u x y z $.  $d w y z $.
        $( Lemma for ~ suplocexpr .  The putative supremum is an upper bound.
           (Contributed by Jim Kingdon, 14-Jan-2024.) $)
        suplocexprlemub $p |- ( ph -> A. y e. A -. B <P y ) $=
          ( vs cv wcel wa c2nd cnq cnp cvv vt cltp wbr cfv c1st wfal wrex simpr
          wn suplocexprlemex ad2antrr wss suplocexprlemss simplr sseldd syl2anc
          wb ltdfpr cltq cima cint crab simprrl wceq suplocexprlem2b syl eleq2d
          mpbid ad3antrrr breq2 rexbidv elrab sylib simprd adantr simprr cop wi
          simprrr prop eleq2 wral simprl vex elint2 wfun cdm wfo fo2nd fofun wf
          ax-mp fof fdmi eleqtrri funfvima ad4antlr rspcdva prcunqu mpd simplrl
          mp2an jca prdisj pm2.21fal rexlimddv inegd ralrimiva ) AHCNZUBUCZUICG
          AXIGOZPZXJXLXJPZMNZHQUDZOZXNXIUEUDZOZPZUFMRXMXJXSMRUGZXLXJUHXMHSOZXIS
          OZXJXTUQAYAXKXJABCDEFGHIJKLUJUKXMGSXIAGSULZXKXJABCDGIJKUMZUKAXKXJUNUO
          ZHXIMURUPVHXMXNROZXSPZPZENZXNUSUCZUFEQGUTZVAZYHYFYJEYLUGZYHXNYIFNZUSU
          CZEYLUGZFRVBZOZYFYMPYHXPYRXMYFXPXRVCAXPYRUQXKXJYGAXOYQXNAYCXOYQVDYDEF
          GHLVEVFVGVIVHYPYMFXNRYNXNVDYOYJEYLYNXNYIUSVJVKVLVMVNYHYIYLOZYJPZPZXRX
          NXIQUDZOZPZUUAXRUUCYHXRYTXMYFXPXRVSVOUUAYJUUCYHYSYJVPUUAXQUUBVQSOZYIU
          UBOZYJUUCVRUUAYBUUEXMYBYGYTYEUKXIVTVFZUUAYIUANZOZUUFUAYKUUBUUHUUBYIWA
          UUAYSUUIUAYKWBYHYSYJWCUAYIYKEWDWEVMXKUUBYKOZAXJYGYTQWFZXIQWGZOXKUUJVR
          TTQWHZUUKWITTQWJWLXITUULCWDTTQUUMTTQWKWITTQWMWLWNWOGXIQWPXBWQWRXNYIUU
          BXQWSUPWTXCUUAUUEYFUUDUIUUGXMYFXSYTXAXNUUBXQXDUPXEXFXFXGXH $.
      $}

      ${
        $d A s y z $.  $d A x y $.  $d B s z $.  $d ph s y z $.  $d ph x y $.
        $( Lemma for ~ suplocexpr .  The putative supremum is a least upper
           bound.  (Contributed by Jim Kingdon, 14-Jan-2024.) $)
        suplocexprlemlub $p |- ( ph -> ( y <P B -> E. z e. A y <P z ) ) $=
          ( vs cv wa wcel c1st cnq cnp cvv cltp wbr wrex c2nd cfv simpr ltrelpr
          brel simpld adantl simprd ltdfpr syl2anc mpbid cima cuni simprrr cltq
          wb cint crab cop fveq2i wceq npex a1i suplocexprlemss ssexd wfo fo1st
          wfun fofun ax-mp funimaexg mpan uniexg 3syl nqex rabex op1stg sylancl
          eqtrid eleq2d ad2antrr suplocexprlemell sylib simprl simprrl syl12anc
          rspe ad4antlr wss ad4antr simplr sseldd mpbird reximdva mpd rexlimddv
          ex ) ACNZHUAUBZXADNZUAUBZDGUCZAXBOZMNZXAUDUEPZXGHQUEZPZOZXEMRXFXBXKMR
          UCZAXBUFXFXASPZHSPZXBXLUSXBXMAXBXMXNXAHSSUAUGUHZUIZUJXBXNAXBXMXNXOUKU
          JXAHMULUMUNXFXGRPZXKOZOZXGXCQUEPZDGUCZXEXSXGQGUOZUPZPZYAXSXJYDXFXQXHX
          JUQAXJYDUSXBXRAXIYCXGAXIYCENFNURUBEUDGUOUTUCZFRVAZVBZQUEZYCHYGQLVCAYC
          TPZYFTPYHYCVDAGTPZYBTPZYIAGSTSTPAVEVFABCDGIJKVGZVHQVKZYJYKTTQVIYMVJTT
          QVLVMQGTVNVOYBTVPVQYEFRVRVSYCYFTTVTWAWBWCWDUNDGXGWEWFXSXTXDDGXSXCGPZO
          ZXTXDYOXTOZXDXHXTOZMRUCZYPXQXHXTYRXSXQYNXTXFXQXKWGWDXSXHYNXTXFXQXHXJW
          HWDYOXTUFYQMRWJWIYPXMXCSPXDYRUSXBXMAXRYNXTXPWKYPGSXCAGSWLXBXRYNXTYLWM
          XSYNXTWNWOXAXCMULUMWPWTWQWRWSWT $.
      $}
    $}

    $d A a u w y z $.  $d A a u x y z $.  $d ph u w y z $.  $d ph u x y z $.
    $( An inhabited, bounded-above, located set of positive reals has a
       supremum.  (Contributed by Jim Kingdon, 7-Jan-2024.) $)
    suplocexpr $p |- ( ph -> E. x e. P. ( A. y e. A -. x <P y /\
        A. y e. P. ( y <P x -> E. z e. A y <P z ) ) ) $=
      ( va vu vw cima cv cltq wbr wrex cnq cnp cltp wral c1st cuni c2nd cint wn
      crab cop wcel wi wa cbvrexv rabbii opeq2i suplocexprlemex suplocexprlemub
      breq1 suplocexprlemlub ralrimivw wceq notbid ralbidv breq2 imbi1d anbi12d
      rspcev syl12anc ) AUAELUBZIMZJMZNOZIUCELUDZPZJQUFZUGZRUHVNCMZSOZUEZCETZVO
      VNSOZVODMSODEPZUIZCRTZBMZVOSOZUEZCETZVOWCSOZVTUIZCRTZUJZBRPABCDKJEVNFGHVM
      KMZVINOZKVKPZJQUFVGVLWMJQVJWLIKVKVHWKVINUPUKULUMZUNABCDKJEVNFGHWNUOAWACRA
      BCDKJEVNFGHWNUQURWJVRWBUJBVNRWCVNUSZWFVRWIWBWOWEVQCEWOWDVPWCVNVOSUPUTVAWO
      WHWACRWOWGVSVTWCVNVOSVBVCVAVDVEVF $.
  $}

  ${
    $d x y z w v u f $.
    $( Define equivalence relation for signed reals.  This is a "temporary" set
       used in the construction of complex numbers, and is intended to be used
       only by the construction.  From Proposition 9-4.1 of [Gleason] p. 126.
       (Contributed by NM, 25-Jul-1995.) $)
    df-enr $a |- ~R = { <. x , y >. | ( ( x e. ( P. X. P. ) /\
                   y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u
                   ( ( x = <. z , w >. /\ y = <. v , u >. ) /\
                   ( z +P. u ) = ( w +P. v ) ) ) } $.

    $( Define class of signed reals.  This is a "temporary" set used in the
       construction of complex numbers, and is intended to be used only by the
       construction.  From Proposition 9-4.2 of [Gleason] p. 126.  (Contributed
       by NM, 25-Jul-1995.) $)
    df-nr $a |- R. = ( ( P. X. P. ) /. ~R ) $.

    $( Define addition on signed reals.  This is a "temporary" set used in the
       construction of complex numbers, and is intended to be used only by the
       construction.  From Proposition 9-4.3 of [Gleason] p. 126.  (Contributed
       by NM, 25-Aug-1995.) $)
    df-plr $a |- +R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\
      E. w E. v E. u E. f (
         ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\
               z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R ) ) } $.

    $( Define multiplication on signed reals.  This is a "temporary" set used
       in the construction of complex numbers, and is intended to be used only
       by the construction.  From Proposition 9-4.3 of [Gleason] p. 126.
       (Contributed by NM, 25-Aug-1995.) $)
    df-mr $a |- .R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\
      E. w E. v E. u E. f (
         ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\
           z = [ <. ( ( w .P. u ) +P. ( v .P. f ) ) ,
             ( ( w .P. f ) +P. ( v .P. u ) ) >. ] ~R ) ) } $.

    $( Define ordering relation on signed reals.  This is a "temporary" set
       used in the construction of complex numbers, and is intended to be used
       only by the construction.  From Proposition 9-4.4 of [Gleason] p. 127.
       (Contributed by NM, 14-Feb-1996.) $)
    df-ltr $a |- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\
       E. z E. w E. v E. u
       ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\
                ( z +P. u ) <P ( w +P. v ) ) ) } $.

    $( Define signed real constant 0.  This is a "temporary" set used in the
       construction of complex numbers, and is intended to be used only by the
       construction.  From Proposition 9-4.2 of [Gleason] p. 126.  (Contributed
       by NM, 9-Aug-1995.) $)
    df-0r $a |- 0R = [ <. 1P , 1P >. ] ~R $.

    $( Define signed real constant 1.  This is a "temporary" set used in the
       construction of complex numbers, and is intended to be used only by the
       construction.  From Proposition 9-4.2 of [Gleason] p. 126.  (Contributed
       by NM, 9-Aug-1995.) $)
    df-1r $a |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R $.

    $( Define signed real constant -1.  This is a "temporary" set used in the
       construction of complex numbers, and is intended to be used only by the
       construction.  (Contributed by NM, 9-Aug-1995.) $)
    df-m1r $a |- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u D $.
    $( Equivalence relation for signed reals in terms of positive reals.
       (Contributed by NM, 3-Sep-1995.) $)
    enrbreq $p |- ( ( ( A e. P. /\ B e. P. ) /\
                     ( C e. P. /\ D e. P. ) ) ->
          ( <. A , B >. ~R <. C , D >. <-> ( A +P. D ) = ( B +P. C ) ) ) $=
      ( vx vy vz vw vv vu cpp cer cnp df-enr ecopoveq ) EFGHIJABCDKLMEFGHIJNO
      $.
  $}

  ${
    $d x y z w v u $.
    $( The equivalence relation for signed reals is an equivalence relation.
       Proposition 9-4.1 of [Gleason] p. 126.  (Contributed by NM, 3-Sep-1995.)
       (Revised by Mario Carneiro, 6-Jul-2015.) $)
    enrer $p |- ~R Er ( P. X. P. ) $=
      ( vx vy vz vw vv cpp cer cnp df-enr addcomprg addclpr addassprg addcanprg
      vu cv ecopoverg ) ABCDENFGHABCDENIAOZBOZJQRKQRCOZLQRSMP $.
  $}

  $( Equivalence class equality of positive fractions in terms of positive
     integers.  (Contributed by NM, 29-Nov-1995.) $)
  enreceq $p |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ->
                ( [ <. A , B >. ] ~R = [ <. C , D >. ] ~R <->
                ( A +P. D ) = ( B +P. C ) ) ) $=
    ( cnp wcel wa cop cer wbr cec wceq cpp co cxp wer enrer a1i opelxpi adantr
    erth enrbreq bitr3d ) AEFBEFGZCEFDEFGZGZABHZCDHZIJUGIKUHIKLADMNBCMNLUFUGUHI
    EEOZUIIPUFQRUDUGUIFUEABEESTUAABCDUBUC $.

  ${
    $d x y z w v u $.
    $( The equivalence relation for signed reals exists.  (Contributed by NM,
       25-Jul-1995.) $)
    enrex $p |- ~R e. _V $=
      ( vx vy vz vw vv vu cer cnp cxp npex xpex cv wcel wa cop cpp co wex copab
      wceq df-enr opabssxp eqsstri ssexi ) GHHIZUEIZUEUEHHJJKZUGKGALZUEMBLZUEMN
      UHCLZDLZOTUIELZFLZOTNUJUMPQUKULPQTNFRERDRCRZNABSUFABCDEFUAUNABUEUEUBUCUD
      $.
  $}

  ${
    $d x y z w v u $.
    $( Signed real 'less than' is a relation on signed reals.  (Contributed by
       NM, 14-Feb-1996.) $)
    ltrelsr $p |- <R C_ ( R. X. R. ) $=
      ( vx vy vz vw vv vu cltr cv cnr wcel wa cop cer cec wceq cpp cltp wbr wex
      co copab cxp df-ltr opabssxp eqsstri ) GAHZIJBHZIJKUFCHZDHZLMNOUGEHZFHZLM
      NOKUHUKPTUIUJPTQRKFSESDSCSZKABUAIIUBABCDEFUCULABIIUDUE $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.  $d x y z F $.
    $d x y z G $.  $d x y z R $.  $d x y z S $.
    $( Lemma showing compatibility of addition.  (Contributed by NM,
       3-Sep-1995.) $)
    addcmpblnr $p |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) )
               /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) ->
                ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) ->
     <. ( A +P. F ) , ( B +P. G ) >. ~R <. ( C +P. R ) , ( D +P. S ) >. ) ) $=
      ( cpp co wceq wa cnp wcel addclpr anim12i an4s syl2anc addassprg syl3anc
      cop cer wbr oveq12 wb enrbreq syl simprll simplrr addcomprg oveq1d oveq2d
      simprrr 3eqtr3d simplll 3eqtr4d simprlr simplrl simprrl simpllr imbitrrid
      eqeq12d bitrd ) ADIJZBCIJZKGFIJZHEIJZKLAGIJZBHIJZUACEIJZDFIJZUAUBUCZAMNZB
      MNZLZCMNZDMNZLZLZGMNZHMNZLZEMNZFMNZLZLZLZVDVFIJZVEVGIJZKZVDVEVFVGIUDWGVLV
      HVKIJZVIVJIJZKZWJWGVHMNZVIMNZLZVJMNZVKMNZLZLZVLWMUEVOWBVRWEWTVOWBLWPVRWEL
      WSVMVTVNWAWPVMVTLWNVNWALWOAGOBHOPQVPWCVQWDWSVPWCLWQVQWDLWRCEOZDFOZPQPQVHV
      IVJVKUFUGWGWKWHWLWIWGAGVKIJZIJZADVFIJZIJZWKWHWGXCXEAIWGGDIJZFIJZDGIJZFIJZ
      XCXEWGXGXIFIWGVTVQXGXIKVSVTWAWEUHZVOVPVQWFUIZGDUJRUKWGVTVQWDXHXCKXKXLVSWB
      WCWDUMZGDFSTWGVQVTWDXJXEKXLXKXMDGFSTUNULWGVMVTWRWKXDKVMVNVRWFUOZXKWGVQWDW
      RXLXMXBRAGVKSTWGVMVQVFMNZWHXFKXNXLWGVTWDXOXKXMGFORADVFSTUPWGBHVJIJZIJZBCV
      GIJZIJZWLWIWGXPXRBIWGHCIJZEIJZCHIJZEIJZXPXRWGXTYBEIWGWAVPXTYBKVSVTWAWEUQZ
      VOVPVQWFURZHCUJRUKWGWAVPWCYAXPKYDYEVSWBWCWDUSZHCESTWGVPWAWCYCXRKYEYDYFCHE
      STUNULWGVNWAWQWLXQKVMVNVRWFUTZYDWGVPWCWQYEYFXARBHVJSTWGVNVPVGMNZWIXSKYGYE
      WGWAWCYHYDYFHEORBCVGSTUPVBVCVA $.

    $( Lemma used in lemma showing compatibility of multiplication.
       (Contributed by Jim Kingdon, 1-Jan-2020.) $)
    mulcmpblnrlemg $p |- (
        ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) )
        /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) ->
          ( ( ( A +P. D ) = ( B +P. C ) /\
              ( F +P. S ) = ( G +P. R ) ) ->
            ( ( D .P. F ) +P. ( ( ( A .P. F ) +P. ( B .P. G ) ) +P.
              ( ( C .P. S ) +P. ( D .P. R ) ) ) ) =
              ( ( D .P. F ) +P. ( ( ( A .P. G ) +P. ( B .P. F ) ) +P.
              ( ( C .P. R ) +P. ( D .P. S ) ) ) ) ) ) $=
      ( cnp wcel wa cpp co wceq cmp mulclpr syl2anc syl3anc adantr mulcomprg vx
      vy simpllr simprlr simplrr simprrl addclpr simplrl simprll oveq2 ad2antll
      addassprg simprrr distrprg 3eqtr3d simplll oveq1 ad2antrl oveq12d 3eqtr4d
      vz oveq2d oveq1d 3eqtr2d addcomprg adantl caov12d caov32d adantrr adantrl
      cv w3a eqtrd 3eqtr4rd caov13d caov4d caov42d ex ) AIJZBIJZKZCIJZDIJZKZKZG
      IJZHIJZKZEIJZFIJZKZKZKZADLMZBCLMZNZGFLMZHELMZNZKZDGOMZAGOMZBHOMZLMCFOMZDE
      OMZLMLMZLMZXAAHOMZBGOMZLMCEOMZDFOMZLMLMZLMZNWMWTKZXAXBXDLMZXCXELMZLMZLMZX
      AXHXKLMZXIXJLMZLMZLMZXGXMXNXPXOXALMZLMZXAXSLMZXTLMZXRYBXNXPCHOMZLMZXTLMZX
      PYGXTLMZLMZYFYDWMYIYKNZWTWMXPIJZYGIJZXTIJZYLWMXCIJZXEIJZYMWMVTWGYPVSVTWDW
      LUCZWEWFWGWKUDZBHPQZWMWCWIYQWAWBWCWLUEZWEWHWIWJUFZDEPQZXCXEUGQZWMWBWGYNWA
      WBWCWLUHZYSCHPQZWMXIIJZXJIJZYOWMVTWFUUGYRWEWFWGWKUIZBGPQZWMWBWIUUHUUEUUBC
      EPQZXIXJUGQZXPYGXTULRSXNYEYHXTLXNXHXAXKLMZLMZXCYGLMZXELMZYEYHXNUUNXHDHOMZ
      XELMZLMZXHUUQLMZXELMZUUPXNUUMUURXHLXNDWQOMZDWROMZUUMUURWSUVBUVCNWMWPWQWRD
      OUJUKWMUVBUUMNZWTWMWCWFWJUVDUUAUUIWEWHWIWJUMZDGFUNRSWMUVCUURNZWTWMWCWGWIU
      VFUUAYSUUBDHEUNRSUOVBWMUVAUUSNZWTWMXHIJZUUQIJZYQUVGWMVSWGUVHVSVTWDWLUPZYS
      AHPQZWMWCWGUVIUUAYSDHPQUUCXHUUQXEULRSXNUUTUUOXELXNWNHOMZWOHOMZUUTUUOWPUVL
      UVMNWMWSWNWOHOUQURWMUVLUUTNWTWMHWNOMZHAOMZHDOMZLMZUVLUUTWMWGVSWCUVNUVQNYS
      UVJUUAHADUNRWMWNIJZWGUVLUVNNWMVSWCUVRUVJUUAADUGQZYSWNHTQWMXHUVOUUQUVPLWMV
      SWGXHUVONUVJYSAHTQWMWCWGUUQUVPNUUAYSDHTQUSUTSWMUVMUUONWTWMHWOOMZHBOMZHCOM
      ZLMZUVMUUOWMWGVTWBUVTUWCNYSYRUUEHBCUNRWMWOIJZWGUVMUVTNWMVTWBUWDYRUUEBCUGQ
      ZYSWOHTQWMXCUWAYGUWBLWMVTWGXCUWANYRYSBHTQWMWBWGYGUWBNUUEYSCHTQUSUTSUOVCVD
      WMUUNYENWTWMUAUBVAXHXAXKILUVKWMWCWFXAIJZUUAUUIDGPQZWMWCWJXKIJZUUAUVEDFPQZ
      UAVKZIJZUBVKZIJZKZUWJUWLLMZUWLUWJLMNWMUWJUWLVEVFZUWKUWMVAVKZIJVLUWOUWQLMU
      WJUWLUWQLMLMNWMUWJUWLUWQULVFZVGSWMUUPYHNWTWMUAUBVAXCYGXEILYTUUFUUCUWPUWRV
      HSUOVCXNYCYJXPLXNXBXALMZXDLMZXIYGXJLMZLMZYCYJXNUWTXICGOMZLMZXDLMZUXBWMWPU
      WTUXENWSWMWPKZUWSUXDXDLUXFWNGOMZWOGOMZUWSUXDWPUXGUXHNWMWNWOGOUQVFWMUXGUWS
      NWPWMGWNOMZGAOMZGDOMZLMZUXGUWSWMWFVSWCUXIUXLNUUIUVJUUAGADUNRWMUVRWFUXGUXI
      NUVSUUIWNGTQWMXBUXJXAUXKLWMVSWFXBUXJNUVJUUIAGTQWMWCWFXAUXKNUUAUUIDGTQUSUT
      SWMUXHUXDNWPWMGWOOMZGBOMZGCOMZLMZUXHUXDWMWFVTWBUXMUXPNUUIYRUUEGBCUNRWMUWD
      WFUXHUXMNUWEUUIWOGTQWMXIUXNUXCUXOLWMVTWFXIUXNNYRUUIBGTQWMWBWFUXCUXONUUEUU
      ICGTQUSUTSUOVCVIWMWSUXEUXBNWPWMWSKZUXEXIUXCXDLMZLMZUXBWMUXEUXSNZWSWMUUGUX
      CIJZXDIJZUXTUUJWMWBWFUYAUUEUUICGPQWMWBWJUYBUUEUVECFPQZXIUXCXDULRSUXQUXRUX
      AXILUXQCWQOMZCWROMZUXRUXAWSUYDUYENWMWQWRCOUJVFWMUYDUXRNZWSWMWBWFWJUYFUUEU
      UIUVECGFUNRSWMUYEUXANZWSWMWBWGWIUYGUUEYSUUBCHEUNRSUOVBVMVJVMWMUWTYCNWTWMU
      AUBVAXBXAXDILWMVSWFXBIJZUVJUUIAGPQZUWGUYCUWPUWRVHSWMUXBYJNWTWMUAUBVAXIYGX
      JILUUJUUFUUKUWPUWRVGSUOVBVNWMYDXRNWTWMUAUBVAXPXOXAILUUDWMUYHUYBXOIJUYIUYC
      XBXDUGQUWGUWPUWRVOSWMYFYBNZWTWMUWFXSIJZYOUYJUWGWMUVHUWHUYKUVKUWIXHXKUGQUU
      LXAXSXTULRSUOWMXRXGNWTWMXQXFXALWMUAUBVAXBXDXCXEILUYIUYCYTUWPUWRUUCUWNUWOI
      JWMUWJUWLUGVFZVPVBSWMYBXMNWTWMYAXLXALWMUAUBVAXHXKXIXJILUVKUWIUUJUWPUWRUUK
      UYLVQVBSUOVR $.

    $( Lemma showing compatibility of multiplication.  (Contributed by NM,
       5-Sep-1995.) $)
    mulcmpblnr $p |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) )
               /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) ->
                ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) ->
    <. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R
 <. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. ) ) $=
      ( cnp wcel wa cpp co wceq cmp cop cer mulclpr syl2anc addclpr wbr simplrr
      mulcmpblnrlemg wi simprll simplll simpllr simprlr simplrl simprrr simprrl
      addcanprg syl3anc syld wb enrbreq syl22anc sylibrd ) AIJZBIJZKZCIJZDIJZKZ
      KZGIJZHIJZKZEIJZFIJZKZKZKZADLMBCLMNGFLMHELMNKZAGOMZBHOMZLMZCFOMZDEOMZLMZL
      MZAHOMZBGOMZLMZCEOMZDFOMZLMZLMZNZVQWDPWGVTPQUAZVMVNDGOMZWALMWKWHLMNZWIABC
      DEFGHUCVMWKIJZWAIJZWHIJZWLWIUDVMVCVFWMVAVBVCVLUBZVEVFVGVKUEZDGRSVMVQIJZVT
      IJZWNVMVOIJZVPIJZWRVMUSVFWTUSUTVDVLUFZWQAGRSVMUTVGXAUSUTVDVLUGZVEVFVGVKUH
      ZBHRSVOVPTSZVMVRIJZVSIJZWSVMVBVJXFVAVBVCVLUIZVEVHVIVJUJZCFRSVMVCVIXGWPVEV
      HVIVJUKZDERSVRVSTSZVQVTTSVMWDIJZWGIJZWOVMWBIJZWCIJZXLVMUSVGXNXBXDAHRSVMUT
      VFXOXCWQBGRSWBWCTSZVMWEIJZWFIJZXMVMVBVIXQXHXJCERSVMVCVJXRWPXIDFRSWEWFTSZW
      DWGTSWKWAWHULUMUNVMWRXLXMWSWJWIUOXEXPXSXKVQWDWGVTUPUQUR $.
  $}

  ${
    $d a b c d f g h s t u v w x y $.
    $( Decomposing signed reals into positive reals.  Lemma for ~ addsrpr and
       ~ mulsrpr .  (Contributed by Jim Kingdon, 30-Dec-2019.) $)
    prsrlem1 $p |- ( (
        ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) )
        /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R )
        /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> (
        ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) )
          /\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) )
        /\ ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) ) ) $=
      ( cnp cer wcel wa cv cop cec wceq cpp co vx vy va vb vc cxp cqs cdm enrer
      vd wer erdm ax-mp simprll simpll eqeltrrd ecelqsdm sylancr opelxp simprrl
      sylib jca simprlr simplr simprrr wbr eqtr3d a1i mpbird wb df-enr ecopoveq
      erth syl2anc mpbid jca31 ) EKKUFZLUGZMZFVRMZNZEAOZBOZPZLQZRZFCOZDOZPZLQZR
      ZNZEJOZGOZPZLQZRZFHOZIOZPZLQZRZNZNZNZWBKMWCKMNZWMKMWNKMNZNWGKMWHKMNZWRKMW
      SKMNZNWBWNSTWCWMSTRZWGWSSTWHWRSTRZNXEXFXGXEWDVQMZXFXELUHVQRZWEVRMXLVQLUKZ
      XMUIVQLULUMZXEEWEVRWAWFWKXCUNZVSVTXDUOZUPVQWDLUQURZWBWCKKUSVAZXEWOVQMZXGX
      EXMWPVRMXTXOXEEWPVRWAWLWQXBUTZXQUPVQWOLUQURWMWNKKUSVAZVBXEXHXIXEWIVQMZXHX
      EXMWJVRMYCXOXEFWJVRWAWFWKXCVCZVSVTXDVDZUPVQWILUQURZWGWHKKUSVAZXEWTVQMZXIX
      EXMXAVRMYHXOXEFXAVRWAWLWQXBVEZYEUPVQWTLUQURWRWSKKUSVAZVBXEXJXKXEWDWOLVFZX
      JXEYKWEWPRXEEWEWPXPYAVGXEWDWOLVQXNXEUIVHZXRVMVIXEXFXGYKXJVJXSYBUAUBUCUDUE
      UJWBWCWMWNSLKUAUBUCUDUEUJVKZVLVNVOXEWIWTLVFZXKXEYNWJXARXEFWJXAYDYIVGXEWIW
      TLVQYLYFVMVIXEXHXIYNXKVJYGYJUAUBUCUDUEUJWGWHWRWSSLKYMVLVNVOVBVP $.
  $}

  ${
    $d A f g h q s t u v w z $.  $d B f g h q s t u v w z $.
    $( There is at most one result from adding signed reals.  (Contributed by
       Jim Kingdon, 30-Dec-2019.) $)
    addsrmo $p |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\
        B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t
        ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R )
        /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) $=
      ( vq vs cnp cer wcel wa cv cop cec wceq cpp co wex vf cxp cqs wal wmo wer
      vg vh wi enrer a1i wbr prsrlem1 addcmpblnr imp syl erthi adantrlr simprlr
      adantrrr simprrr 3eqtr4d expr exlimdvv impd alrimivv opeq12 eceq1d eqeq2d
      ex anbi1d simpl oveq1d simpr opeq12d anbi12d anbi2d oveq2d cbvex4v anbi2i
      imbi1i 2albii sylibr eqeq1 4exbidv mo4 ) FJJUBZKUCZLGWHLMZFBNZCNZOZKPZQZG
      DNZENZOZKPZQZMZANZWJWORSZWKWPRSZOZKPZQZMZETDTZCTBTZWTHNZXEQZMZETDTCTBTZMZ
      XAXJQZUIZHUDAUDZXIAUEWIXIFINZUANZOZKPZQZGUGNZUHNZOZKPZQZMZXJXRYCRSZXSYDRS
      ZOZKPZQZMZUHTUGTZUATITZMZXOUIZHUDAUDXQWIYRAHWIXIYPXOWIXHYPXOUIZBCWIXGYSDE
      WIXGYSWIXGMZYOXOIUAYTYNXOUGUHWIXGYNXOWIXGYNMMXEYLXAXJWIXGYHXEYLQZYMWIWTYH
      UUAXFWIWTYHMMZXDYKKWGWGKUFUUBUJUKUUBWJJLWKJLMXRJLXSJLMMWOJLWPJLMYCJLYDJLM
      MMZWJXSRSWKXRRSQWOYDRSWPYCRSQMZMXDYKKULZBCDEFGUAUGUHIUMUUCUUDUUEWJWKXRXSY
      CYDWOWPUNUOUPUQURUTWIWTXFYNUSWIXGYHYMVAVBVCVDVDVJVDVDVEVFXPYRAHXNYQXOXMYP
      XIXLYBWSMZXJXRWORSZXSWPRSZOZKPZQZMYNBCDEIUAUGUHWJXRQZWKXSQZMZWTUUFXKUUKUU
      NWNYBWSUUNWMYAFUUNWLXTKWJWKXRXSVGVHVIVKUUNXEUUJXJUUNXDUUIKUUNXBUUGXCUUHUU
      NWJXRWORUULUUMVLVMUUNWKXSWPRUULUUMVNVMVOVHVIVPWOYCQZWPYDQZMZUUFYHUUKYMUUQ
      WSYGYBUUQWRYFGUUQWQYEKWOWPYCYDVGVHVIVQUUQUUJYLXJUUQUUIYKKUUQUUGYIUUHYJUUQ
      WOYCXRRUUOUUPVLVRUUQWPYDXSRUUOUUPVNVRVOVHVIVPVSVTWAWBWCXIXMAHXOXGXLBCDEXO
      XFXKWTXAXJXEWDVQWEWFWC $.

    $( There is at most one result from multiplying signed reals.  (Contributed
       by Jim Kingdon, 30-Dec-2019.) $)
    mulsrmo $p |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\
        B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t
        ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R )
        /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) ,
          ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) $=
      ( vq cnp cer wcel wa cv cop cec wceq cmp co cpp wex vs vf cxp cqs wal wmo
      vg vh wi wer enrer a1i wbr prsrlem1 mulcmpblnr imp erthi adantrlr simprlr
      syl adantrrr simprrr 3eqtr4d expr exlimdvv ex impd alrimivv opeq12 eceq1d
      eqeq2d anbi1d oveq1d oveq12d opeq12d anbi12d anbi2d oveq2d cbvex4v anbi2i
      simpl simpr imbi1i 2albii sylibr eqeq1 4exbidv mo4 ) FIIUCZJUDZKGWJKLZFBM
      ZCMZNZJOZPZGDMZEMZNZJOZPZLZAMZWLWQQRZWMWRQRZSRZWLWRQRZWMWQQRZSRZNZJOZPZLZ
      ETDTZCTBTZXBHMZXKPZLZETDTCTBTZLZXCXPPZUIZHUEAUEZXOAUFWKXOFUAMZUBMZNZJOZPZ
      GUGMZUHMZNZJOZPZLZXPYDYIQRZYEYJQRZSRZYDYJQRZYEYIQRZSRZNZJOZPZLZUHTUGTZUBT
      UATZLZYAUIZHUEAUEYCWKUUHAHWKXOUUFYAWKXNUUFYAUIZBCWKXMUUIDEWKXMUUIWKXMLZUU
      EYAUAUBUUJUUDYAUGUHWKXMUUDYAWKXMUUDLLXKUUBXCXPWKXMYNXKUUBPZUUCWKXBYNUUKXL
      WKXBYNLLZXJUUAJWIWIJUJUULUKULUULWLIKWMIKLYDIKYEIKLLWQIKWRIKLYIIKYJIKLLLZW
      LYESRWMYDSRPWQYJSRWRYISRPLZLXJUUAJUMZBCDEFGUBUGUHUAUNUUMUUNUUOWLWMYDYEYIY
      JWQWRUOUPUTUQURVAWKXBXLUUDUSWKXMYNUUCVBVCVDVEVEVFVEVEVGVHYBUUHAHXTUUGYAXS
      UUFXOXRYHXALZXPYDWQQRZYEWRQRZSRZYDWRQRZYEWQQRZSRZNZJOZPZLUUDBCDEUAUBUGUHW
      LYDPZWMYEPZLZXBUUPXQUVEUVHWPYHXAUVHWOYGFUVHWNYFJWLWMYDYEVIVJVKVLUVHXKUVDX
      PUVHXJUVCJUVHXFUUSXIUVBUVHXDUUQXEUURSUVHWLYDWQQUVFUVGWAZVMUVHWMYEWRQUVFUV
      GWBZVMVNUVHXGUUTXHUVASUVHWLYDWRQUVIVMUVHWMYEWQQUVJVMVNVOVJVKVPWQYIPZWRYJP
      ZLZUUPYNUVEUUCUVMXAYMYHUVMWTYLGUVMWSYKJWQWRYIYJVIVJVKVQUVMUVDUUBXPUVMUVCU
      UAJUVMUUSYQUVBYTUVMUUQYOUURYPSUVMWQYIYDQUVKUVLWAZVRUVMWRYJYEQUVKUVLWBZVRV
      NUVMUUTYRUVAYSSUVMWRYJYDQUVOVRUVMWQYIYEQUVNVRVNVOVJVKVPVSVTWCWDWEXOXSAHYA
      XMXRBCDEYAXLXQXBXCXPXKWFVQWGWHWE $.
  $}

  ${
    $d x y z w v u t A $.  $d x y z w v u t B $.  $d x y z w v u t C $.
    $d x y z w v u t D $.
    $( Addition of signed reals in terms of positive reals.  (Contributed by
       NM, 3-Sep-1995.)  (Revised by Mario Carneiro, 12-Aug-2015.) $)
    addsrpr $p |- ( ( ( A e. P. /\ B e. P. ) /\
                     ( C e. P. /\ D e. P. ) ) ->
                ( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) =
                   [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) $=
      ( vw vv vu vt vz cnp wcel wa cop cer cec cv wceq cpp co wex vx vy cxp cqs
      cplr opelxpi enrex ecelqsi syl anim12i pm3.2i opeq12 eceq1d eqeq2d anbi1d
      eqid simpl oveq1d simpr opeq12d anbi12d spc2egv anbi2d oveq2d 2eximdv cvv
      sylan9 mp2ani wi ecexg ax-mp w3a simp1 eqeq1d simp2 simp3 4exbidv addsrmo
      coprab df-plr df-nr eleq2i anbi12i anbi1i oprabbii eqtri ovig mp3an3 sylc
      cnr ) AJKBJKLZCJKDJKLZLZABMZNOZJJUCZNUDZKZCDMZNOZWQKZLWOEPZFPZMZNOZQZWTGP
      ZHPZMZNOZQZLZACRSZBDRSZMZNOZXBXGRSZXCXHRSZMZNOZQZLZHTGTZFTETZWOWTUESXPQZW
      KWRWLXAWKWNWPKWRABJJUFWPWNNUGUHUIWLWSWPKXACDJJUFWPWSNUGUHUIUJWMWOWOQZWTWT
      QZLZXPXPQZYDYFYGWOUPWTUPUKXPUPWKYHYILZXFYGLZXPXBCRSZXCDRSZMZNOZQZLZFTETWL
      YDYQYJEFABJJXBAQZXCBQZLZYKYHYPYIYTXFYFYGYTXEWOWOYTXDWNNXBXCABULUMUNUOYTYO
      XPXPYTYNXONYTYLXMYMXNYTXBACRYRYSUQURYTXCBDRYRYSUSURUTUMUNVAVBWLYQYCEFYBYQ
      GHCDJJXGCQZXHDQZLZXLYKYAYPUUCXKYGXFUUCXJWTWTUUCXIWSNXGXHCDULUMUNVCUUCXTYO
      XPUUCXSYNNUUCXQYLXRYMUUCXGCXBRUUAUUBUQVDUUCXHDXCRUUAUUBUSVDUTUMUNVAVBVEVG
      VHWRXAXPVFKZYDYEVINVFKUUDUGXOVFNVJVKUAPZXEQZUBPZXJQZLZIPZXTQZLZHTGTFTETZY
      DUAUBIWOWTXPVFWQWQUEUUEWOQZUUGWTQZUUJXPQZVLZUULYBEFGHUUQUUIXLUUKYAUUQUUFX
      FUUHXKUUQUUEWOXEUUNUUOUUPVMVNUUQUUGWTXJUUNUUOUUPVOVNVAUUQUUJXPXTUUNUUOUUP
      VPVNVAVQIEFGHUUEUUGVRUEUUEWJKZUUGWJKZLZUUMLZUAUBIVSUUEWQKZUUGWQKZLZUUMLZU
      AUBIVSUAUBIEFGHVTUVAUVEUAUBIUUTUVDUUMUURUVBUUSUVCWJWQUUEWAWBWJWQUUGWAWBWC
      WDWEWFWGWHWI $.

    $( Multiplication of signed reals in terms of positive reals.  (Contributed
       by NM, 3-Sep-1995.)  (Revised by Mario Carneiro, 12-Aug-2015.) $)
    mulsrpr $p |- ( ( ( A e. P. /\ B e. P. ) /\
                     ( C e. P. /\ D e. P. ) ) ->
                ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) =
                   [ <. ( ( A .P. C ) +P. ( B .P. D ) ) ,
                              ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) $=
      ( vw vv vu vt vz cnp wcel wa cop cer cec wceq cmp co cpp wex vx vy cxp cv
      cqs cmr opelxpi enrex ecelqsi syl eqid pm3.2i opeq12 eceq1d eqeq2d anbi1d
      anim12i simpl oveq1d simpr oveq12d opeq12d anbi12d spc2egv anbi2d 2eximdv
      oveq2d sylan9 mp2ani cvv ecexg ax-mp w3a simp1 eqeq1d simp2 simp3 4exbidv
      wi mulsrmo coprab df-mr df-nr eleq2i anbi12i anbi1i oprabbii eqtri mp3an3
      cnr ovig sylc ) AJKBJKLZCJKDJKLZLZABMZNOZJJUCZNUEZKZCDMZNOZWSKZLWQEUDZFUD
      ZMZNOZPZXBGUDZHUDZMZNOZPZLZACQRZBDQRZSRZADQRZBCQRZSRZMZNOZXDXIQRZXEXJQRZS
      RZXDXJQRZXEXIQRZSRZMZNOZPZLZHTGTZFTETZWQXBUFRYBPZWMWTWNXCWMWPWRKWTABJJUGW
      RWPNUHUIUJWNXAWRKXCCDJJUGWRXANUHUIUJUQWOWQWQPZXBXBPZLZYBYBPZYNYPYQWQUKXBU
      KULYBUKWMYRYSLZXHYQLZYBXDCQRZXEDQRZSRZXDDQRZXECQRZSRZMZNOZPZLZFTETWNYNUUK
      YTEFABJJXDAPZXEBPZLZUUAYRUUJYSUUNXHYPYQUUNXGWQWQUUNXFWPNXDXEABUMUNUOUPUUN
      UUIYBYBUUNUUHYANUUNUUDXQUUGXTUUNUUBXOUUCXPSUUNXDACQUULUUMURZUSUUNXEBDQUUL
      UUMUTZUSVAUUNUUEXRUUFXSSUUNXDADQUUOUSUUNXEBCQUUPUSVAVBUNUOVCVDWNUUKYMEFYL
      UUKGHCDJJXICPZXJDPZLZXNUUAYKUUJUUSXMYQXHUUSXLXBXBUUSXKXANXIXJCDUMUNUOVEUU
      SYJUUIYBUUSYIUUHNUUSYEUUDYHUUGUUSYCUUBYDUUCSUUSXICXDQUUQUURURZVGUUSXJDXEQ
      UUQUURUTZVGVAUUSYFUUEYGUUFSUUSXJDXDQUVAVGUUSXICXEQUUTVGVAVBUNUOVCVDVFVHVI
      WTXCYBVJKZYNYOVSNVJKUVBUHYAVJNVKVLUAUDZXGPZUBUDZXLPZLZIUDZYJPZLZHTGTFTETZ
      YNUAUBIWQXBYBVJWSWSUFUVCWQPZUVEXBPZUVHYBPZVMZUVJYLEFGHUVOUVGXNUVIYKUVOUVD
      XHUVFXMUVOUVCWQXGUVLUVMUVNVNVOUVOUVEXBXLUVLUVMUVNVPVOVCUVOUVHYBYJUVLUVMUV
      NVQVOVCVRIEFGHUVCUVEVTUFUVCWJKZUVEWJKZLZUVKLZUAUBIWAUVCWSKZUVEWSKZLZUVKLZ
      UAUBIWAUAUBIEFGHWBUVSUWCUAUBIUVRUWBUVKUVPUVTUVQUWAWJWSUVCWCWDWJWSUVEWCWDW
      EWFWGWHWKWIWL $.
  $}

  ${
    $d A x y z w v u f $.  $d B x y z w v u f $.  $d C x y z w v u f $.
    $d D x y z w v u f $.
    $( Ordering of signed reals in terms of positive reals.  (Contributed by
       Jim Kingdon, 2-Jan-2019.) $)
    ltsrprg $p |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ->
        ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <->
        ( A +P. D ) <P ( B +P. C ) ) ) $=
      ( vx vy cv cpp co cltp wbr cer cnp wcel wa wceq syl2anc addassprg syl3anc
      addclpr vz vu vw vv vf cnr cltr enrex enrer df-nr df-ltr cop cec wb eqcom
      enreceq bitrdi bi2anan9 simprlr simplrr addcomprg simprrl 3eqtr3d simplll
      oveq12 adantl oveq1d oveq2d ad2ant2lr anim12ci an4s simpld 3eqtr4d adantr
      simprll simplrl simprrr simpllr sylbid w3a ltaprg simprd caovord3d brecop
      ex syld ) UAGZUBGZHIZUCGZUDGZHIZJKZADHIZBCHIZJKZEFUAUCUDUBABCDLMUFUGUHUIU
      JEFUAUCUDUBUKWGMNZWJMNZOZAMNZBMNZOZOZWKMNZWHMNZOZCMNZDMNZOZOZOZWGWJULLUMA
      BULLUMPZWKWHULLUMCDULLUMPZOZWIWOHIZWLWNHIZPZWMWPUNXKXNWGBHIZWJAHIZPZWHCHI
      ZWKDHIZPZOZXQXCXLXTXJXMYCWGWJABUPXJXMYBYAPYCWKWHCDUPYBYAUOUQURXKYDXQXKYDO
      XRYAHIZXSYBHIZXOXPYDYEYFPXKXRXSYAYBHVEVFXKXOYEPYDXKWGWHWOHIZHIZWGBYAHIZHI
      ZXOYEXKYGYIWGHXKWHBHIZCHIZBWHHIZCHIZYGYIXKXEXAYLYNPXCXDXEXIUSZWSWTXAXJUTZ
      XEXAOYKYMCHWHBVAVGQXKXEXAXGYLYGPYOYPXCXFXGXHVBZWHBCRSXKXAXEXGYNYIPYPYOYQB
      WHCRSVCVHXKWQXEWOMNZXOYHPWQWRXBXJVDZYOXKYRWLMNZWSXFXBXIYRYTOWSXFOYTXBXIOY
      RWRXDYTWQXEWJWKTVIXAXGYRWTXHBCTVIVJVKZVLZWGWHWORSXKWQXAYAMNZYEYJPYSYPXKXE
      XGUUCYOYQWHCTQWGBYARSVMVNXKXPYFPYDXKWJWKWNHIZHIZWJAYBHIZHIZXPYFXKUUDUUFWJ
      HXKWKAHIZDHIZAWKHIZDHIZUUDUUFXKUUHUUJDHXKXDWTUUHUUJPXCXDXEXIVOZWSWTXAXJVP
      ZWKAVAQVGXKXDWTXHUUIUUDPUULUUMXCXFXGXHVQZWKADRSXKWTXDXHUUKUUFPUUMUULUUNAW
      KDRSVCVHXKWRXDWNMNZXPUUEPWQWRXBXJVRZUULXKWTXHUUOUUMUUNADTQZWJWKWNRSXKWRWT
      YBMNZYFUUGPUUPUUMXKXDXHUURUULUUNWKDTQWJAYBRSVMVNVMWEVSXKEFUEWIWOWLWNJMHEG
      ZMNZFGZMNZUEGZMNVTUUSUVAJKUVCUUSHIUVCUVAHIJKUNXKUUSUVAUVCWAVFXKWQXEWIMNYS
      YOWGWHTQUUBXKYRYTUUAWBUUTUVBOUUSUVAHIUVAUUSHIPXKUUSUVAVAVFUUQWCWFWD $.
  $}

  $( Greater than zero in terms of positive reals.  (Contributed by NM,
     13-May-1996.) $)
  gt0srpr $p |- ( 0R <R [ <. A , B >. ] ~R <-> B <P A ) $=
    ( c0r cop cer cec cltr wbr cnp wcel wa cltp cxp cnr brel c1p cpp co wb 1pr
    cdm wceq cqs wer enrer ax-mp ltrelsr simprd df-nr eleqtrdi ecelqsdm sylancr
    erdm opelxp sylib ltrelpr ancomd df-0r breq1i ltsrprg mpanl12 bitrid ltaprg
    mp3an3 ancoms bitr4d pm5.21nii ) CABDZEFZGHZAIJZBIJZKZBALHZVJVHIIMZJZVMVJEU
    AVOUBZVIVOEUCZJVPVOEUDVQUEVOEUMUFVJVINVRVJCNJVINJCVINNGUGOUHUIUJVOVHEUKULAB
    IIUNUOVNVLVKBAIILUPOUQVMVJPBQRPAQRLHZVNVJPPDEFZVIGHZVMVSCVTVIGURUSPIJZWBVMW
    AVSSTTPPABUTVAVBVLVKVNVSSZVLVKWBWCTBAPVCVDVEVFVG $.

  $( The empty set is not a signed real.  (Contributed by NM, 25-Aug-1995.)
     (Revised by Mario Carneiro, 10-Jul-2014.) $)
  0nsr $p |- -. (/) e. R. $=
    ( c0 wceq cnr wcel eqid wne cnp cxp cer cqs cdm wer enrer erdm ax-mp elqsn0
    wn mpan df-nr eleq2s necon2bi ) AABACDZQAEUBAAAAFZAGGHZIJZCIKUDBZAUEDUCUDIL
    UFMUDINOUDAIPRSTUAO $.

  $( The constant ` 0R ` is a signed real.  (Contributed by NM, 9-Aug-1995.) $)
  0r $p |- 0R e. R. $=
    ( c1p cop cer cec cnp cxp cqs c0r cnr 1pr opelxpi mp2an enrex ecelqsi ax-mp
    wcel df-0r df-nr 3eltr4i ) AABZCDZEEFZCGZHITUBPZUAUCPAEPZUEUDJJAAEEKLUBTCMN
    OQRS $.

  $( The constant ` 1R ` is a signed real.  (Contributed by NM, 9-Aug-1995.) $)
  1sr $p |- 1R e. R. $=
    ( c1p cpp co cop cer cec cnp cxp cqs c1r cnr wcel 1pr addclpr mp2an opelxpi
    enrex ecelqsi ax-mp df-1r df-nr 3eltr4i ) AABCZADZEFZGGHZEIZJKUDUFLZUEUGLUC
    GLZAGLZUHUJUJUIMMAANOMUCAGGPOUFUDEQRSTUAUB $.

  $( The constant ` -1R ` is a signed real.  (Contributed by NM,
     9-Aug-1995.) $)
  m1r $p |- -1R e. R. $=
    ( c1p cpp cop cer cec cnp cxp cqs cm1r cnr wcel addclpr mp2an opelxpi enrex
    co 1pr ecelqsi ax-mp df-m1r df-nr 3eltr4i ) AAABPZCZDEZFFGZDHZIJUDUFKZUEUGK
    AFKZUCFKZUHQUIUIUJQQAALMAUCFFNMUFUDDORSTUAUB $.

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Closure of addition on signed reals.  (Contributed by NM,
       25-Jul-1995.) $)
    addclsr $p |- ( ( A e. R. /\ B e. R. ) -> ( A +R B ) e. R. ) $=
      ( vx vy vz vw cnr wcel wa cplr co cnp cer cop cec df-nr wceq eleq1d cpp
      cv cxp cqs oveq1 oveq2 addsrpr addclpr anim12i an4s opelxpi enrex ecelqsi
      3syl eqeltrd 2ecoptocl eleqtrrdi ) AGHBGHIABJKZLLUAZMUBZGCTZDTZNMOZETZFTZ
      NMOZJKZURHAVDJKZURHUPURHCDEFABLLMGPVAAQVEVFURVAAVDJUCRVDBQVFUPURVDBAJUDRU
      SLHZUTLHZIVBLHZVCLHZIIZVEUSVBSKZUTVCSKZNZMOZURUSUTVBVCUEVKVLLHZVMLHZIZVNU
      QHVOURHVGVIVHVJVRVGVIIVPVHVJIVQUSVBUFUTVCUFUGUHVLVMLLUIUQVNMUJUKULUMUNPUO
      $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Closure of multiplication on signed reals.  (Contributed by NM,
       10-Aug-1995.) $)
    mulclsr $p |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) e. R. ) $=
      ( vx vy vz vw cnr wcel wa cmr co cnp cer cop cec df-nr wceq cmp mulclpr
      cv cxp cqs oveq1 eleq1d oveq2 cpp mulsrpr addclpr syl2an an4s jca opelxpi
      an42s enrex ecelqsi 3syl eqeltrd 2ecoptocl eleqtrrdi ) AGHBGHIABJKZLLUAZM
      UBZGCTZDTZNMOZETZFTZNMOZJKZVBHAVHJKZVBHUTVBHCDEFABLLMGPVEAQVIVJVBVEAVHJUC
      UDVHBQVJUTVBVHBAJUEUDVCLHZVDLHZIVFLHZVGLHZIIZVIVCVFRKZVDVGRKZUFKZVCVGRKZV
      DVFRKZUFKZNZMOZVBVCVDVFVGUGVOVRLHZWALHZIWBVAHWCVBHVOWDWEVKVMVLVNWDVKVMIVP
      LHVQLHWDVLVNIVCVFSVDVGSVPVQUHUIUJVKVNVLVMWEVKVNIVSLHVTLHWEVLVMIVCVGSVDVFS
      VSVTUHUIUMUKVRWALLULVAWBMUNUOUPUQURPUS $.
  $}

  ${
    $d f g h r s t u v w x y z A $.  $d u v w z B $.  $d u v w x y z C $.
    $( Addition of signed reals is commutative.  (Contributed by Jim Kingdon,
       3-Jan-2020.) $)
    addcomsrg $p |- ( ( A e. R. /\ B e. R. ) -> ( A +R B ) = ( B +R A ) ) $=
      ( vx vy vz vw cnr cv cpp co cplr cer cnp df-nr addsrpr wcel wceq ad2ant2r
      addcomprg ad2ant2l ecovicom ) CDEFABGCHZEHZIJZKLMDHZFHZIJZUCUBIJZUFUEIJZN
      UBUEUCUFOUCUFUBUEOUBMPZUCMPZUDUHQUEMPZUFMPZUBUCSRULUMUGUIQUJUKUEUFSTUA $.

    $( Addition of signed reals is associative.  (Contributed by Jim Kingdon,
       3-Jan-2020.) $)
    addasssrg $p |- ( ( A e. R. /\ B e. R. /\ C e. R. ) ->
        ( ( A +R B ) +R C ) = ( A +R ( B +R C ) ) ) $=
      ( vx vy vz vw vv vu cv cpp co cnp addsrpr wcel addclpr anim12i an4s wceq
      wa cnr cer 3adant1r 3adant2r 3adant3r 3adant1l 3adant2l 3adant3l ecoviass
      cplr df-nr addassprg ) DEFGHIABCUAUJGJZIJZKLZUBMDJZFJZKLZEJZUMKLZURHJZKLZ
      UTUNKLZUPUQVAKLZKLZUSUOKLZVDUKUPUSUQUMNUQUMVAUNNURUTVAUNNUPUSVDUONUPMOZUQ
      MOZUSMOZUMMOZURMOZUTMOZTVGVHTVKVIVJTVLUPUQPUSUMPQRVHVAMOZVJUNMOZVDMOZUOMO
      ZTVHVMTVOVJVNTVPUQVAPUMUNPQRVGVITZVHVJTZVMVBVESZVNVQVHVMVSVJVGVHVMVSVIUPU
      QVAULUCUDUEVQVRVNVCVFSZVMVQVJVNVTVHVIVJVNVTVGUSUMUNULUFUGUHUI $.

    $( Multiplication of signed reals is commutative.  (Contributed by Jim
       Kingdon, 3-Jan-2020.) $)
    mulcomsrg $p |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) = ( B .R A ) ) $=
      ( vx vy vz vw cv cmp co cpp mulsrpr wcel wceq mulcomprg oveq12d ad2ant2rl
      cnp wa ad2ant2lr mulclpr cnr cmr df-nr ad2ant2r ad2ant2l ancoms addcomprg
      cer syl2anc eqtrd ecovicom ) CDEFABUACGZEGZHIZDGZFGZHIZJIUBUHQULUPHIZUOUM
      HIZJIZUMULHIZUPUOHIZJIUMUOHIZUPULHIZJIZUCULUOUMUPKUMUPULUOKULQLZUOQLZRUMQ
      LZUPQLZRRZUNVAUQVBJVFVHUNVAMVGVIULUMNUDVGVIUQVBMVFVHUOUPNUEOVJUTVDVCJIZVE
      VJURVDUSVCJVFVIURVDMVGVHULUPNPVGVHUSVCMVFVIUOUMNSOVJVDQLZVCQLZVKVEMVFVIVL
      VGVHVIVFVLUPULTUFPVGVHVMVFVIVHVGVMUMUOTUFSVDVCUGUIUJUK $.

    $( Multiplication of signed reals is associative.  (Contributed by Jim
       Kingdon, 3-Jan-2020.) $)
    mulasssrg $p |- ( ( A e. R. /\ B e. R. /\ C e. R. ) ->
        ( ( A .R B ) .R C ) = ( A .R ( B .R C ) ) ) $=
      ( vf vg vh cv cmp co cpp cnp mulsrpr wcel wa mulclpr addclpr syl2anc wceq
      w3a adantl vx vy vz vw vv vu vr vs vt cnr cmr cer df-nr ad2ant2r ad2ant2l
      ad2ant2rl ad2ant2lr mulcomprg distrprg simp1 simp2 simp3 caovdir2d simp1l
      mulassprg simp1r simp2l simp2r simp3l simp3r addcomprg addassprg ecoviass
      jca caovlem2d ) UAUBUCUDUEUFABCUJUKUCGZUFGZHIZUDGZUEGZHIZJIZULKUAGZVPHIZU
      BGZVSHIZJIZWCVSHIZWEVPHIZJIZWGVTHIWJVQHIJIWGVQHIWJVTHIJIWCVPVTHIZVSVQHIZJ
      IZHIWEWBHIJIWCWBHIWEWMHIJIWMUMWCWEVPVSLVPVSVTVQLWGWJVTVQLWCWEWMWBLWCKMZWE
      KMZNZVPKMZVSKMZNZNZWGKMZWJKMZWTWDKMZWFKMZXAWNWQXCWOWRWCVPOUNWOWRXDWNWQWEV
      SOUOWDWFPQWTWHKMZWIKMZXBWNWRXEWOWQWCVSOUPWOWQXFWNWRWEVPOUQWHWIPQVNWSVTKMZ
      VQKMZNZNZWMKMZWBKMZXJWKKMZWLKMZXKWQXGXMWRXHVPVTOUNWRXHXNWQXGVSVQOUOWKWLPQ
      XJVRKMZWAKMZXLWQXHXOWRXGVPVQOUPWRXGXPWQXHVSVTOUQVRWAPQVNWPWSXISZDEFWCWEVP
      VSVQKJHVTDGZKMZEGZKMZNZXRXTHIZXTXRHIRXQXRXTURTZXSYAFGZKMZSZXRXTJIZYEHIXRY
      EHIXTYEHIZJIRXQYGUGUHUIXRXTYEKJHUGGZKMZUHGZKMZUIGZKMSYJYLYNJIHIYJYLHIZYJY
      NHIJIRYGYJYLYNUSTXSYAYFUTXSYAYFVAXSYAYFVBYKYMNZYJYLJIKMYGYJYLPTYPYOYLYJHI
      RYGYJYLURTVCTZYGYCYEHIXRYIHIRXQXRXTYEVETZYBYCKMXQXRXTOTZWNWOWSXIVDZWNWOWS
      XIVFZWPWQWRXIVGZWPWQWRXIVHZWPWSXGXHVIZWPWSXGXHVJZYBYHXTXRJIRXQXRXTVKTZYGY
      HYEJIXRXTYEJIJIRXQXRXTYEVLTZYBYHKMXQXRXTPTZVOXQDEFWCWEVPVSVTKJHVQYDYQYRYS
      YTUUAUUBUUCUUEUUDUUFUUGUUHVOVM $.

    $( Multiplication of signed reals is distributive.  (Contributed by Jim
       Kingdon, 4-Jan-2020.) $)
    distrsrg $p |- ( ( A e. R. /\ B e. R. /\ C e. R. ) ->
        ( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) ) ) $=
      ( vf vg vh cnp cv cpp co cmp mulsrpr wcel wa addclpr mulclpr syl2anc wceq
      distrprg syl3anc vx vy vz vw vv vu cnr cplr cer addsrpr ad2ant2r ad2ant2l
      cmr jca ad2ant2rl ad2ant2lr w3a simp1l simp2l simp3l simp1r simp2r simp3r
      df-nr oveq12d addcomprg adantl addassprg caov4d eqtrd ecovidi ) UAUBUCUDU
      EUFABCUGUHUIGUMUAHZUCHZUEHZIJZKJZUBHZUDHZUFHZIJZKJZIJZVLVTKJZVQVOKJZIJZVL
      VMKJZVQVRKJZIJZVLVNKJZVQVSKJZIJZIJZVLVRKJZVQVMKJZIJZVLVSKJZVQVNKJZIJZIJZV
      OVTWHWOWKWRVDVMVRVNVSUJVLVQVOVTLVLVQVMVRLVLVQVNVSLWHWOWKWRUJVMGMZVRGMZNZV
      NGMZVSGMZNZNVOGMZVTGMZWTXCXFXAXDVMVNOUKXAXDXGWTXCVRVSOULUNVLGMZVQGMZNZXBN
      ZWHGMZWOGMZXKWFGMZWGGMZXLXHWTXNXIXAVLVMPZUKXIXAXOXHWTVQVRPZULWFWGOQXKWMGM
      ZWNGMZXMXHXAXRXIWTVLVRPZUOXIWTXSXHXAVQVMPZUPWMWNOQUNXJXENZWKGMZWRGMZYBWIG
      MZWJGMZYCXHXCYEXIXDVLVNPZUKXIXDYFXHXCVQVSPZULWIWJOQYBWPGMZWQGMZYDXHXDYIXI
      XCVLVSPZUOXIXCYJXHXDVQVNPZUPWPWQOQUNXJXBXEUQZWBWFWIIJZWGWJIJZIJWLYMVPYNWA
      YOIYMXHWTXCVPYNRXHXIXBXEURZXJWTXAXEUSZXJXBXCXDUTZVLVMVNSTYMXIXAXDWAYORXHX
      IXBXEVAZXJWTXAXEVBZXJXBXCXDVCZVQVRVSSTVEYMDEFWFWIWGWJGIYMXHWTXNYPYQXPQYMX
      HXCYEYPYRYGQYMXIXAXOYSYTXQQDHZGMZEHZGMZNZUUBUUDIJZUUDUUBIJRYMUUBUUDVFVGZU
      UCUUEFHZGMUQUUGUUIIJUUBUUDUUIIJIJRYMUUBUUDUUIVHVGZYMXIXDYFYSUUAYHQUUFUUGG
      MYMUUBUUDOVGZVIVJYMWEWMWPIJZWNWQIJZIJWSYMWCUULWDUUMIYMXHXAXDWCUULRYPYTUUA
      VLVRVSSTYMXIWTXCWDUUMRYSYQYRVQVMVNSTVEYMDEFWMWPWNWQGIYMXHXAXRYPYTXTQYMXHX
      DYIYPUUAYKQYMXIWTXSYSYQYAQUUHUUJYMXIXCYJYSYRYLQUUKVIVJVK $.
  $}

  $( Minus one plus one is zero for signed reals.  (Contributed by NM,
     5-May-1996.) $)
  m1p1sr $p |- ( -1R +R 1R ) = 0R $=
    ( cm1r c1r cplr co c1p cpp cop cer cec c0r df-m1r cnp wcel wceq 1pr addclpr
    df-1r mp2an mp4an eqtr4i oveq12i df-0r addsrpr addassprg mp3an oveq2i mpbir
    wb enreceq ) ABCDEEEFDZGHIZUJEGHIZCDZJAUKBULCKQUAJEEGHIZUMUBUMEUJFDZUJEFDZG
    HIZUNELMZUJLMZUSURUMUQNOURURUSOOEEPRZUTOEUJUJEUCSUNUQNZEUPFDEUOFDNZUPUOEFUR
    URURUPUONOOOEEEUDUEUFURURUOLMZUPLMZVAVBUHOOURUSVCOUTEUJPRUSURVDUTOUJEPREEUO
    UPUISUGTTT $.

  $( Minus one times minus one is plus one for signed reals.  (Contributed by
     NM, 14-May-1996.) $)
  m1m1sr $p |- ( -1R .R -1R ) = 1R $=
    ( cm1r cmr c1p cpp cop cer cec c1r df-m1r oveq12i cmp cnp wcel wceq addclpr
    co 1pr mp2an mulclpr eqtr4i df-1r mulsrpr mp4an addassprg mp3an 1idpr ax-mp
    distrprg mulcomprg oveq1i oveq2i wb enreceq mpbir ) AABPCCCDPZEFGZUPBPZHAUP
    AUPBIIJHUOCEFGZUQUAUQCCKPZUOUOKPZDPZCUOKPZUOCKPZDPZEFGZURCLMZUOLMZVFVGUQVEN
    QVFVFVGQQCCORZQVHCUOCUOUBUCURVENZUOVDDPZCVADPZNZVJCCVDDPZDPZVKVFVFVDLMZVJVN
    NQQVBLMZVCLMZVOVFVGVPQVHCUOSRVGVFVQVHQUOCSRVBVCORZCCVDUDUEVAVMCDUSCUTVDDVFU
    SCNQCUFUGUTVCVCDPZVDVGVFVFUTVSNVHQQUOCCUHUEVBVCVCDVFVGVBVCNQVHCUOUIRUJTJUKT
    VGVFVALMZVOVIVLULVHQUSLMZUTLMZVTVFVFWAQQCCSRVGVGWBVHVHUOUOSRUSUTORVRUOCVAVD
    UMUCUNTTT $.

  ${
    $d f g h r s t x y z w v u $.
    $( Signed real 'less than' is a transitive relation.  (Contributed by Jim
       Kingdon, 4-Jan-2019.) $)
    lttrsr $p |- ( ( f e. R. /\ g e. R. /\ h e. R. ) ->
        ( ( f <R g /\ g <R h ) -> f <R h ) ) $=
      ( vr vs vt cv cer cltr wbr wa cnp wceq wcel cpp co cltp addclpr syl2anc
      wb vx vy vz vw vv vu cop cec cnr df-nr breq1 anbi1d imbi12d breq2 anbi12d
      wi imbi1d anbi2d ltsrprg ltaprg adantl simp1l simp2r simp1r simp2l simp3r
      3adant3 addcomprg caovord2d addassprg syl3anc breq12d bitrd simp3l ltsopr
      w3a 3adant1 ltrelpr biimprd caov12d 3adant2 3imtr4d syl5 sylbid 3ecoptocl
      sotri ) UAGZUBGZUGHUHZUCGZUDGZUGHUHZIJZWLUEGZUFGZUGHUHZIJZKZWIWPIJZUPAGZW
      LIJZWQKZWTWPIJZUPWTBGZIJZXDWPIJZKZXCUPXEXDCGZIJZKZWTXHIJZUPUAUBUCUDUEUFWT
      XDXHLHUIUJWIWTMZWRXBWSXCXLWMXAWQWIWTWLIUKULWIWTWPIUKUMWLXDMZXBXGXCXMXAXEW
      QXFWLXDWTIUNWLXDWPIUKUOUQWPXHMZXGXJXCXKXNXFXIXEWPXHXDIUNURWPXHWTIUNUMWGLN
      ZWHLNZKZWJLNZWKLNZKZWNLNZWOLNZKZVPZWRWGWKWOOPOPZWHWJWOOPZOPZQJZYGWHWKWNOP
      ZOPZQJZKZWSYDWMYHWQYKYDWMWGWKOPZWHWJOPZQJZYHXQXTWMYOTYCWGWHWJWKUSVGYDYOYM
      WOOPZYNWOOPZQJYHYDDEFYMYNWOQLODGZLNZEGZLNZFGZLNVPZYRYTQJUUBYROPUUBYTOPQJT
      YDYRYTUUBUTVAYDXOXSYMLNXOXPXTYCVBZXQXRXSYCVCZWGWKRSYDXPXRYNLNXOXPXTYCVDZX
      QXRXSYCVEZWHWJRSXQXTYAYBVFZYSUUAKYRYTOPZYTYROPMYDYRYTVHVAZVIYDYPYEYQYGQYD
      XOXSYBYPYEMUUDUUEUUHWGWKWOVJVKYDXPXRYBYQYGMUUFUUGUUHWHWJWOVJVKVLVMVMYDWQY
      FYIQJZYKXTYCWQUUKTXQWJWKWNWOUSVQYDYFLNZYILNZXPUUKYKTYDXRYBUULUUGUUHWJWORS
      YDXSYAUUMUUEXQXTYAYBVNZWKWNRSUUFYFYIWHUTVKVMUOYLYEYJQJZYDWSYEYGYJQLVOVRWF
      YDWKWGWOOPZOPZWKWHWNOPZOPZQJZUUPUURQJZUUOWSYDUVAUUTYDUUPLNZUURLNZXSUVAUUT
      TYDXOYBUVBUUDUUHWGWORSYDXPYAUVCUUFUUNWHWNRSUUEUUPUURWKUTVKVSYDYEUUQYJUUSQ
      YDDEFWGWKWOLOUUDUUEUUHUUJUUCUUIUUBOPYRYTUUBOPOPMYDYRYTUUBVJVAZVTYDDEFWHWK
      WNLOUUFUUEUUNUUJUVDVTVLXQYCWSUVATXTWGWHWNWOUSWAWBWCWDWE $.
  $}

  ${
    $d x y f g h $.
    $( Signed real 'less than' is a partial order.  (Contributed by Jim
       Kingdon, 4-Jan-2019.) $)
    ltposr $p |- <R Po R. $=
      ( vf vg vh vx vy cnr cltr wpo wtru cv wcel wbr wn cer cnp cpp cltp adantl
      wa co cop cec df-nr wceq id breq12d notbid ltsopr soirri addcomprg breq2d
      ltrelpr mtbii wb ltsrprg anidms mtbird ecoptocl w3a wi lttrsr ispod mptru
      ) FGHIABCFGAJZFKZVDVDGLZMZIDJZEJZUANUBZVJGLZMVGDEVDOONFUCVJVDUDZVKVFVLVJV
      DVJVDGVLUEZVMUFUGVHOKVIOKSZVKVHVIPTZVIVHPTZQLZVNVOVOQLVQVOQOUHULUIVNVOVPV
      OQVHVIUJUKUMVNVKVQUNVHVIVHVIUOUPUQURRVEBJZFKCJZFKUSVDVRGLVRVSGLSVDVSGLUTI
      ABCVARVBVC $.
  $}

  ${
    $d a b c d e f r s t x y z $.
    $( Signed real 'less than' is a strict ordering.  (Contributed by NM,
       19-Feb-1996.) $)
    ltsosr $p |- <R Or R. $=
      ( vx vy vr vs vt cnr cltr cv wbr wo wi cnp wceq wcel cltp addclpr syl2anc
      cpp co wb vz va vb vc vd ve vf wor wpo wral ltposr cop df-nr breq1 orbi1d
      cer cec imbi12d breq2 orbi2d imbi2d wa simp1l simp3r simp2r simp2l simp1r
      orbi12d simp3l ltsopr sowlin mpan syl3anc ltaprg addcomprg adantl caov12d
      addassprg caovassd eqtr4d caov32d eqtr3d breq12d bitrd caovord2d caovcomd
      w3a eqtrd breq1d bitr4d 3imtr4d ltsrprg 3adant3 3adant2 3adant1 3ecoptocl
      ancoms rgen3 df-iso mpbir2an ) FGUHFGUIAHZBHZGIZXAUAHZGIZXDXBGIZJZKZUAFUJ
      BFUJAFUJUKXHABUAFFFUBHZUCHZULUPUQZUDHZUEHZULUPUQZGIZXKUFHZUGHZULUPUQZGIZX
      RXNGIZJZKXAXNGIZXAXRGIZXTJZKXCYCXRXBGIZJZKXHUBUCUDUEUFUGXAXBXDLUPFUMXKXAM
      ZXOYBYAYDXKXAXNGUNYGXSYCXTXKXAXRGUNUOURXNXBMZYBXCYDYFXNXBXAGUSYHXTYEYCXNX
      BXRGUSUTURXRXDMZYFXGXCYIYCXEYEXFXRXDXAGUSXRXDXBGUNVHVAXILNZXJLNZVBZXLLNZX
      MLNZVBZXPLNZXQLNZVBZWGZXIXMRSZXJXLRSZOIZXIXQRSZXJXPRSZOIZXPXMRSZXQXLRSZOI
      ZJZXOYAYSUUCXMRSZUUGXJRSZOIZUUJUUDXMRSZOIZUUMUUKOIZJZUUBUUIYSUUJLNZUUKLNZ
      UUMLNZUULUUPKZYSUUCLNZYNUUQYSYJYQUVAYJYKYOYRVCZYLYOYPYQVDZXIXQPQZYLYMYNYR
      VEZUUCXMPQYSUUGLNZYKUURYSYQYMUVFUVCYLYMYNYRVFZXQXLPQZYJYKYOYRVGZUUGXJPQYS
      UUDLNZYNUUSYSYKYPUVJUVIYLYOYPYQVIZXJXPPQZUVEUUDXMPQLOUHUUQUURUUSWGUUTVJLU
      UJUUKUUMOVKVLVMYSUUBXQYTRSZXQUUARSZOIZUULYSYTLNZUUALNZYQUUBUVOTYSYJYNUVPU
      VBUVEXIXMPQYSYKYMUVQUVIUVGXJXLPQUVCYTUUAXQVNVMYSUVMUUJUVNUUKOYSUVMXIXQXMR
      SRSUUJYSCDEXQXIXMLRUVCUVBUVECHZLNZDHZLNZVBUVRUVTRSZUVTUVRRSMYSUVRUVTVOVPZ
      UVSUWAEHZLNWGZUWBUWDRSUVRUVTUWDRSRSMYSUVRUVTUWDVRVPZVQYSCDEXIXQXMLRUWFUVB
      UVCUVEVSVTYSXQXJRSXLRSUVNUUKYSCDEXQXJXLLRUWFUVCUVIUVGVSYSCDEXQXJXLLRUVCUV
      IUVGUWCUWFWAWBWCWDYSUUEUUNUUHUUOYSCDEUUCUUDXMOLRUWEUVRUVTOIUWDUVRRSUWDUVT
      RSOITYSUVRUVTUWDVNVPZUVDUVLUVEUWCWEYSUUHUUFXJRSZUUKOIUUOYSCDEUUFUUGXJOLRU
      WGYSYPYNUUFLNUVKUVEXPXMPQZUVHUVIUWCWEYSUUMUWHUUKOYSUUMXJUUFRSUWHYSCDEXJXP
      XMLRUWFUVIUVKUVEVSYSCDXJUUFLRUWCUVIUWIWFWHWIWJVHWKYLYOXOUUBTYRXIXJXLXMWLW
      MYSXSUUEXTUUHYLYRXSUUETYOXIXJXPXQWLWNYOYRXTUUHTZYLYRYOUWJXPXQXLXMWLWQWOVH
      WKWPWRABUAFGWSWT $.
  $}

  $( 0 is less than 1 for signed reals.  (Contributed by NM, 26-Mar-1996.) $)
  0lt1sr $p |- 0R <R 1R $=
    ( c1p cop cer cec cpp co c0r c1r cltr wbr cltp cnp wcel 1pr addclpr ltaddpr
    mp2an wceq addcomprg breqtrri wb ltsrprg mp4an mpbir df-0r df-1r 3brtr4i )
    AABCDZAAEFZABCDZGHIUHUJIJZUIAUIEFZKJZUIUIAEFZULKUILMZALMZUIUNKJUPUPUONNAAOQ
    ZNUIAPQUPUOULUNRNUQAUISQTUPUPUOUPUKUMUANNUQNAAUIAUBUCUDUEUFUG $.

  $( 1 and 0 are distinct for signed reals.  (Contributed by NM,
     26-Mar-1996.) $)
  1ne0sr $p |- -. 1R = 0R $=
    ( c1r c0r wceq cltr wbr cnr wor wcel wn ltsosr 1sr sonr mp2an 0lt1sr mpbiri
    breq1 mto ) ABCZAADEZFDGAFHSIJKFADLMRSBADENABADPOQ $.

  ${
    $d x y z w v $.  $d x y A $.
    $( The signed real number 0 is an identity element for addition of signed
       reals.  (Contributed by NM, 10-Apr-1996.) $)
    0idsr $p |- ( A e. R. -> ( A +R 0R ) = A ) $=
      ( vx vy vz vw vv cv cop cer cec c0r cplr co wceq cnp wcel wa c1p cpp 1pr
      df-nr oveq1 id eqeq12d df-0r oveq2i addsrpr mpanr12 simpl simpr addcomprg
      cnr a1i adantl w3a addassprg caov12d addclpr mpan2 anim12i enreceq mpbird
      wb mpdan eqtr4d eqtrid ecoptocl ) BGZCGZHIJZKLMZVJNAKLMZANBCAOOIULUAVJANZ
      VKVLVJAVJAKLUBVMUCUDVHOPZVIOPZQZVKVJRRHIJZLMZVJKVQVJLUEUFVPVRVHRSMZVIRSMZ
      HIJZVJVPROPZWBVRWANTTVHVIRRUGUHVPVJWANZVHVTSMVIVSSMNZVPDEFVHVIROSVNVOUIVN
      VOUJWBVPTUMDGZOPZEGZOPZQWEWGSMZWGWESMNVPWEWGUKUNWFWHFGZOPUOWIWJSMWEWGWJSM
      SMNVPWEWGWJUPUNUQVPVSOPZVTOPZQWCWDVCVNWKVOWLVNWBWKTVHRURUSVOWBWLTVIRURUSU
      TVHVIVSVTVAVDVBVEVFVG $.
  $}

  ${
    $d x y A $.  $d x y z w v $.
    $( 1 is an identity element for multiplication.  (Contributed by Jim
       Kingdon, 5-Jan-2020.) $)
    1idsr $p |- ( A e. R. -> ( A .R 1R ) = A ) $=
      ( cv cop cer cec c1r cmr co wceq cnp wcel c1p cpp cmp 1pr addclpr mulclpr
      wa sylancl mpan2 vx vy vz vw vv df-nr oveq1 id eqeq12d df-1r oveq2i mp2an
      cnr mulsrpr mpanr12 distrprg mp3an23 1idpr oveq1d eqtr2d oveqan12d adantl
      eqtrd simpl addassprg syl3anc simpr addcomprg w3a caov12d 3eqtr3d anim12i
      wb syl2an enreceq syldan anidms mpbird eqtr4d eqtrid ecoptocl ) UABZUBBZC
      DEZFGHZWDIAFGHZAIUAUBAJJDUMUFWDAIZWEWFWDAWDAFGUGWGUHUIWBJKZWCJKZRZWEWDLLM
      HZLCDEZGHZWDFWLWDGUJUKWJWMWBWKNHZWCLNHZMHZWBLNHZWCWKNHZMHZCDEZWDWJWKJKZLJ
      KZWMWTIXBXBXAOOLLPULZOWBWCWKLUNUOWJWDWTIZWBWSMHZWCWPMHZIZWJWBWQMHZWRMHZWN
      WCWOMHZMHXEXFWHWIXHWNWRXJMWHWNWQWQMHZXHWHXBXBWNXKIOOWBLLUPUQWHWQWBWQMWBUR
      USUTWIWRWOWOMHZXJWIXBXBWRXLIOOWCLLUPUQWIWOWCWOMWCURUSVCVAWJWHWQJKZWRJKZXI
      XEIWHWIVDZWJWHXBXMXOOWBLQZSWIXNWHWIXAXNXCWCWKQTZVBWBWQWRVEVFWJUCUDUEWNWCW
      OJMWJWHXAWNJKZXOXCWBWKQZSWHWIVGZWJWIXBWOJKZXTOWCLQZSUCBZJKZUDBZJKZRYCYEMH
      ZYEYCMHIWJYCYEVHVBYDYFUEBZJKVIYGYHMHYCYEYHMHMHIWJYCYEYHVEVBVJVKWJXDXGVMZW
      JWJWPJKZWSJKZRYIWJYJWJYKWHXRYAYJWIWHXAXRXCXSTWIXBYAOYBTWNWOPVNWHXMXNYKWIW
      HXBXMOXPTXQWQWRPVNVLWBWCWPWSVOVPVQVRVSVTWA $.
  $}

  ${
    $d x y A $.
    $( A signed real times 0 is 0.  (Contributed by NM, 10-Apr-1996.) $)
    00sr $p |- ( A e. R. -> ( A .R 0R ) = 0R ) $=
      ( vx vy cv cop cer cec c0r cmr co wceq cnp wa c1p cmp cpp mpanr12 mulclpr
      wcel 1pr cnr df-nr oveq1 eqeq1d mulsrpr mpan2 addclpr syl2an anim12i eqid
      enreceq mpbiri sylan anidms eqtrd df-0r oveq2i 3eqtr4g ecoptocl ) BDZCDZE
      FGZHIJZHKAHIJZHKBCALLFUAUBVBAKVCVDHVBAHIUCUDUTLSZVALSZMZVBNNEFGZIJZVHVCHV
      GVIUTNOJZVANOJZPJZVLEFGZVHVGNLSZVNVIVMKTTUTVANNUEQVGVMVHKZVGVGMZVNVNVOTTV
      PVLLSZVQMZVNVNMZVOVGVQVGVQVEVJLSZVKLSZVQVFVEVNVTTUTNRUFVFVNWATVANRUFVJVKU
      GUHZWBUIVRVSMVOVLNPJZWCKWCUJVLVLNNUKULUMQUNUOHVHVBIUPUQUPURUS $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d r s t x y z w v u $.
    $( Ordering property of addition.  (Contributed by NM, 10-May-1996.) $)
    ltasrg $p |- ( ( A e. R. /\ B e. R. /\ C e. R. ) ->
        ( A <R B <-> ( C +R A ) <R ( C +R B ) ) ) $=
      ( vr vs wcel cltr wbr cplr co wb cop cer cec cnp wceq cpp addclpr syl2anc
      cv vx vy vz vw vv vu vt cnr df-nr oveq1 breq12d bibi2d breq1 oveq2 breq1d
      bibi12d breq2 breq2d w3a cltp simp2l simp3r simp2r simp3l 3ad2ant1 ltaprg
      syl3anc ltsrprg 3adant1 simp1l simp1r syl22anc addcomprg adantl addassprg
      wa caovcomd caov42d eqtrd bitrd 3bitr4d addsrpr 3adant3 3adant2 3ecoptocl
      caov4d bitr4d 3coml ) CUHFAUHFBUHFABGHZCAIJZCBIJZGHZKZUATZUBTZLMNZUCTZUDT
      ZLMNZGHZUETZUFTZLMNZWPIJZXCWSIJZGHZKWTCWPIJZCWSIJZGHZKAWSGHZWJXHGHZKWMUEU
      FUAUBUCUDCABOMUHUIXCCPZXFXIWTXLXDXGXEXHGXCCWPIUJXCCWSIUJUKULWPAPZWTXJXIXK
      WPAWSGUMXMXGWJXHGWPACIUNUOUPWSBPZXJWIXKWLWSBAGUQXNXHWKWJGWSBCIUNURUPXAOFZ
      XBOFZVPZWNOFZWOOFZVPZWQOFZWROFZVPZUSZWTXAWNQJZXBWOQJZLMNZXAWQQJZXBWRQJZLM
      NZGHZXFYDWNWRQJZWOWQQJZUTHZXAXBQJZYLQJZYOYMQJZUTHZWTYKYDYLOFZYMOFZYOOFZYN
      YRKYDXRYBYSXQXRXSYCVAZXQXTYAYBVBZWNWRRSYDXSYAYTXQXRXSYCVCZXQXTYAYBVDZWOWQ
      RSXQXTUUAYCXAXBRVEYLYMYOVFVGXTYCWTYNKXQWNWOWQWRVHVIYDYKYEYIQJZYFYHQJZUTHZ
      YRYDYEOFZYFOFZYHOFZYIOFZYKUUHKYDXOXRUUIXOXPXTYCVJZUUBXAWNRSYDXPXSUUJXOXPX
      TYCVKZUUDXBWORSZYDXOYAUUKUUMUUEXAWQRSZYDXPYBUULUUNUUCXBWRRSYEYFYHYIVHVLYD
      UUFYPUUGYQUTYDDEUGXAWNXBWROQUUMUUBUUNDTZOFZETZOFZVPZUUQUUSQJZUUSUUQQJPYDU
      UQUUSVMVNZUURUUTUGTZOFUSUVBUVDQJUUQUUSUVDQJQJPYDUUQUUSUVDVOVNZUUCUVAUVBOF
      YDUUQUUSRVNZWFYDUUGYHYFQJYQYDDEYFYHOQUVCUUOUUPVQYDDEUGXAWQXBWOOQUUMUUEUUN
      UVCUVEUUDUVFVRVSUKVTWAYDXDYGXEYJGXQXTXDYGPYCXAXBWNWOWBWCXQYCXEYJPXTXAXBWQ
      WRWBWDUKWGWEWH $.
  $}

  $( A signed real plus its negative is zero.  (Contributed by NM,
     14-May-1996.) $)
  pn0sr $p |- ( A e. R. -> ( A +R ( A .R -1R ) ) = 0R ) $=
    ( cnr wcel c0r cmr co c1r cm1r cplr wceq m1r distrsrg mp3an23 m1p1sr oveq2i
    1sr a1i mulclsr mpan2 addcomsrg syl2anc 3eqtr3d 00sr 1idsr oveq1d 3eqtr3rd
    ) ABCZADEFZAGEFZAHEFZIFZDAUJIFUGAHGIFZEFZUJUIIFZUHUKUGHBCZGBCZUMUNJKPAHGLMU
    MUHJUGULDAENOQUGUJBCZUIBCZUNUKJUGUOUQKAHRSUGUPURPAGRSUJUITUAUBAUCUGUIAUJIAU
    DUEUF $.

  ${
    $d x A $.
    $( Existence of negative signed real.  Part of Proposition 9-4.3 of
       [Gleason] p. 126.  (Contributed by NM, 2-May-1996.) $)
    negexsr $p |- ( A e. R. -> E. x e. R. ( A +R x ) = 0R ) $=
      ( cnr wcel cm1r cmr co cplr c0r wceq cv wrex m1r mpan2 pn0sr oveq2 eqeq1d
      mulclsr rspcev syl2anc ) BCDZBEFGZCDZBUBHGZIJZBAKZHGZIJZACLUAECDUCMBERNBO
      UHUEAUBCUFUBJUGUDIUFUBBHPQST $.
  $}

  ${
    $d x y z A $.  $d x y z w v f g h $.
    $( The reciprocal of a positive signed real exists and is positive.
       (Contributed by Jim Kingdon, 6-Feb-2020.) $)
    recexgt0sr $p |- ( 0R <R A ->
        E. x e. R. ( 0R <R x /\ ( A .R x ) = 1R ) ) $=
      ( cnr wcel c0r cltr wbr cv co c1r wceq wa cer cnp cpp cmp c1p 1pr syl2anc
      adantr vy vz vw vv vf vg vh cmr wrex ltrelsr brel simprd cop cec wi df-nr
      breq2 oveq1 eqeq1d anbi2d rexbidv imbi12d gt0srpr ltexpri recexpr addclpr
      cltp sylbi adantl mpan2 ecopqsi sylancl ad2antlr simprr ltaddpr addcomprg
      enrex mpan breqtrd sylibr syl anim2i mulsrpr adantlrl eqcomd ad2antll w3a
      jctir mulcomprg 3adant2 3adant1 oveq12d 3coml simp3 3adant3 simplr simprl
      distrprg 3eqtr2rd caovdird oveq2 sylan9eq adantrr eqtrd oveq1d a1i simpll
      mulclpr addassprg caov32d syl3anc oveq2d caov12d 3eqtr4d wb eqeltrd mp2an
      enreceq syl22anc mpbird eqtr4di anbi12d rspcev syl12anc anassrs rexlimdva
      df-1r exp32 mpd syl5 ecoptocl mpcom ) BCDZEBFGZEAHZFGZBYOUHIZJKZLZACUIZYN
      ECDYMEBCCFUJUKULEUAHZUBHZUMMUNZFGZYPUUCYOUHIZJKZLZACUIZUOYNYTUOUAUBBNNMCU
      PUUCBKZUUDYNUUHYTUUCBEFUQUUIUUGYSACUUIUUFYRYPUUIUUEYQJUUCBYOUHURUSUTVAVBU
      UDUUBUCHZOIZUUAKZUCNUIZUUANDZUUBNDZLZUUHUUDUUBUUAVGGUUMUUAUUBVCUCUUBUUAVD
      VHUUPUULUUHUCNUUPUUJNDZLZUUJUDHZPIZQKZUDNUIZUULUUHUOZUUQUVBUUPUDUUJVEVIUU
      RUVAUVCUDNUUPUUQUUSNDZUVAUVCUOUUPUUQUVDLZLZUVAUULUUHUVFUVAUULLZLZUUSQOIZQ
      UMMUNZCDZEUVJFGZUUCUVJUHIZJKZUUHUVEUVKUUPUVGUVDUVKUUQUVDUVINDZQNDZUVKUVDU
      VPUVORUUSQVFZVJZRNUVIQMCVQUPVKVLVIVMUVHUVDUVLUVFUVDUVGUUPUUQUVDVNZTUVDQUV
      IVGGUVLUVDQQUUSOIZUVIVGUVPUVDQUVTVGGRQUUSVOVRUVPUVDUVTUVIKRQUUSVPVRVSUVIQ
      VCVTWAUVHUVMQQOIZQUMMUNZJUVHUVMUUAUVIPIZUUBQPIZOIZUUAQPIZUUBUVIPIZOIZUMMU
      NZUWBUUPUVDUVGUVMUWIKZUUQUUPUVDLZUVGLUUPUVOUVPLZLZUWJUWKUWMUVGUVDUWLUUPUV
      DUVOUVPUVRRWHWBTUUAUUBUVIQWCWAWDUVHUWIUWBKZUWEQOIZUWHUWAOIZKZUVHUUAUUSPIZ
      UWFUWDOIZOIZQOIZUUBUUSPIZUWSOIZUWAOIZUWOUWPUVHUXAUXCQOIZQOIZUXDUVHUWTUXEQ
      OUVHUWTUXBQOIZUWSOIZUXEUVHUWRUXGUWSOUVHUWRUUKUUSPIZUXGUULUWRUXIKUVFUVAUUL
      UXIUWRUUKUUAUUSPURWEWFUVFUVAUXIUXGKUULUVFUVAUXIUXBUUTOIUXGUVFUEUFUGUUBUUJ
      UUSNOPONUEHZNDZUFHZNDZUGHZNDZWGZUXJUXLOIZUXNPIZUXJUXNPIZUXLUXNPIZOIZKUVFU
      XPUYAUXNUXJPIZUXNUXLPIZOIZUXNUXQPIZUXRUXPUXSUYBUXTUYCOUXKUXOUXSUYBKUXMUXJ
      UXNWIWJUXMUXOUXTUYCKUXKUXLUXNWIWKWLUXOUXKUXMUYEUYDKUXNUXJUXLWRWMUXPUXOUXQ
      NDZUYEUXRKUXKUXMUXOWNUXKUXMUYFUXOUXJUXLVFWOUXNUXQWISWSVIUUNUUOUVEWPZUUPUU
      QUVDWQUVSWTUUTQUXBOXAXBXCXDXEUVFUXHUXEKUVGUVFUEUFUGUXBQUWSNOUVFUUOUVDUXBN
      DZUYGUVSUUBUUSXHSZUVPUVFRXFZUVFUWFNDZUWDNDZUWSNDZUVFUUNUVPUYKUUNUUOUVEXGZ
      RUUAQXHVLZUVFUUOUVPUYLUYGRUUBQXHVLZUWFUWDVFSZUXKUXMLUXQUXLUXJOIKUVFUXJUXL
      VPVIZUXPUXQUXNOIUXJUXLUXNOIOIKUVFUXJUXLUXNXIVIZXJTXDXEUVHUXCNDZUVPUVPUXFU
      XDKUVFUYTUVGUVFUYHUYMUYTUYIUYQUXBUWSVFSZTUVPUVHRXFZVUBUXCQQXIXKXDUVFUWOUX
      AKUVGUVFUWEUWTQOUVFUWEUWRUWFOIZUWDOIZUWTUVFUWCVUCUWDOUVFUUNUVDUVPUWCVUCKU
      YNUVSUYJUUAUUSQWRXKXEUVFUWRNDZUYKUYLVUDUWTKUVFUUNUVDVUEUYNUVSUUAUUSXHSUYO
      UYPUWRUWFUWDXIXKXDXETUVFUWPUXDKUVGUVFUWHUXCUWAOUVFUWHUWFUXBUWDOIZOIUXCUVF
      UWGVUFUWFOUVFUUOUVDUVPUWGVUFKUYGUVSUYJUUBUUSQWRXKXLUVFUEUFUGUWFUXBUWDNOUY
      OUYIUYPUYRUYSXMXDZXETXNUVFUWNUWQXOZUVGUVFUWENDZUWHNDUWANDZUVPVUHUVFUWCNDZ
      UYLVUIUVFUUNUVOVUKUYNUVFUVDUVPUVOUVSRUVQVLUUAUVIXHSUYPUWCUWDVFSUVFUWHUXCN
      VUGVUAXPVUJUVFUVPUVPVUJRRQQVFXQXFUYJUWEUWHUWAQXRXSTXTXDYGYAUUGUVLUVNLAUVJ
      CYOUVJKZYPUVLUUFUVNYOUVJEFUQVULUUEUVMJYOUVJUUCUHXAUSYBYCYDYHYEYFYIYFYJYKY
      L $.
  $}

  ${
    $d x A $.
    $( The reciprocal of a positive signed real exists.  Part of Proposition
       9-4.3 of [Gleason] p. 126.  (Contributed by NM, 15-May-1996.) $)
    recexsrlem $p |- ( 0R <R A -> E. x e. R. ( A .R x ) = 1R ) $=
      ( c0r cltr wbr cv cmr co c1r wceq wa cnr wrex recexgt0sr simpr reximi syl
      ) CBDECAFZDEZBRGHIJZKZALMTALMABNUATALSTOPQ $.
  $}

  $( The sum of two positive signed reals is positive.  (Contributed by NM,
     14-May-1996.) $)
  addgt0sr $p |- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A +R B ) ) $=
    ( c0r cltr wbr cplr co wa simpr cnr wcel wb ltrelsr simprd 0r ltasrg mp3an1
    brel syl2anr mpbid adantr 0idsr breq1d syl ltsosr sotri syldan ) CADEZCBDEZ
    AABFGZDEZCUJDEUHUIHZACFGZUJDEZUKULUIUNUHUIIUIBJKZAJKZUIUNLZUHUICJKZUOCBJJDM
    RNUHURUPCAJJDMRNZURUOUPUQOCBAPQSTULUPUNUKLUHUPUIUSUAUPUMAUJDAUBUCUDTCAUJDJU
    EMUFUG $.

  $( Adding one to a signed real yields a larger signed real.  (Contributed by
     Jim Kingdon, 7-Jul-2021.) $)
  ltadd1sr $p |- ( A e. R. -> A <R ( A +R 1R ) ) $=
    ( cnr wcel c0r cplr co c1r cltr 0idsr wbr 0lt1sr wb 0r ltasrg mp3an12 mpbii
    1sr eqbrtrrd ) ABCZADEFZAAGEFZHAISDGHJZTUAHJZKDBCGBCSUBUCLMQDGANOPR $.

  $( Adding minus one to a signed real yields a smaller signed real.
     (Contributed by Jim Kingdon, 21-Jan-2024.) $)
  ltm1sr $p |- ( A e. R. -> ( A +R -1R ) <R A ) $=
    ( cnr wcel cm1r cplr co c0r cltr c1r wbr m1r addclsr mpan2 ltadd1sr syl 1sr
    wceq addasssrg mp3an23 breqtrd m1p1sr oveq2i breqtrdi 0idsr ) ABCZADEFZAGEF
    ZAHUEUFADIEFZEFZUGHUEUFUFIEFZUIHUEUFBCZUFUJHJUEDBCZUKKADLMUFNOUEULIBCUJUIQK
    PADIRSTUHGAEUAUBUCAUDT $.

  ${
    $d x y z w A $.  $d x y z w B $.  $d x y z w v u f g h $.
    $( The product of two positive signed reals is positive.  (Contributed by
       NM, 13-May-1996.) $)
    mulgt0sr $p |- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) ) $=
      ( vf vg vh cnr wcel wa c0r cltr wbr co cv cnp wceq cmp cpp adantl mulclpr
      syl2anc vx vy vz vw vv vu cmr ltrelsr simprd anim12i cop cer cec wi df-nr
      brel breq2 anbi1d oveq1 breq2d imbi12d oveq2 cltp gt0srpr anbi12i ltexpri
      anbi2d wrex addclpr simplrr simplr ad2antrr simplrl caovcld adantr simprl
      eqeltrrd ltaddpr simprr oveq12 oveq1d distrprg eqtr3d mulcomprg caovdir2d
      syl3anc oveq12d addcomprg addassprg caov4d 3eqtr4d caov12d caov32d oveq2d
      w3a ad2antlr 3eqtr3d addcanprg breqtrrd rexlimdvaa mulsrpr bitrdi sylibrd
      mpd syl5 impd biimtrid 2ecoptocl mpcom ) AFGZBFGZHIAJKZIBJKZHZIABUGLZJKZX
      LXJXMXKXLIFGZXJIAFFJUHUPUIXMXQXKIBFFJUHUPUIUJIUAMZUBMZUKULUMZJKZIUCMZUDMZ
      UKULUMZJKZHZIXTYDUGLZJKZUNXLYEHZIAYDUGLZJKZUNXNXPUNUAUBUCUDABNNULFUOXTAOZ
      YFYIYHYKYLYAXLYEXTAIJUQURYLYGYJIJXTAYDUGUSUTVAYDBOZYIXNYKXPYMYEXMXLYDBIJU
      QVGYMYJXOIJYDBAUGVBUTVAYFXSXRVCKZYCYBVCKZHZXRNGZXSNGZHZYBNGZYCNGZHZHZYHYA
      YNYEYOXRXSVDYBYCVDVEUUCYPXRYCPLZXSYBPLZQLZXRYBPLZXSYCPLZQLZVCKZYHUUCYNYOU
      UJYNXSUEMZQLZXROZUENVHUUCYOUUJUNZUEXSXRVFUUCUUMUUNUENYOYCUFMZQLZYBOZUFNVH
      UUCUUKNGZUUMHZHZUUJUFYCYBVFUUTUUQUUJUFNUUTUUONGZUUQHZHZUUFUUFUUKUUOPLZQLZ
      UUIVCUVCUUFNGUVDNGZUUFUVEVCKUVCCDUUDUUENNNQCMZNGZDMZNGZHZUVGUVIQLZNGUVCUV
      GUVIVIRZUVCYQUUAUUDNGZUVCUULXRNUUCUURUUMUVBVJZUVCCDXSUUKNNNQUVMUUCYRUUSUV
      BYQYRUUBVKVLZUUCUURUUMUVBVMZVNZVQZUUTUUAUVBYSYTUUAUUSVJVOZXRYCSTZUVCYRYTU
      UENGZUVPUUTYTUVBYSYTUUAUUSVMVOZXSYBSTZVNZUVCUURUVAUVFUVQUUTUVAUUQVPZUUKUU
      OSTZUUFUVDVRTUVCUUKYCPLZUUIQLZUWHUVEQLZOZUUIUVEOZUVCUUGUUHUWHQLZQLZUWHUUE
      UVDQLZQLZUUDQLZUWIUWJUVCUULUUPPLZUWMQLZUWNUWQUVCUUMUUQUWSUWNOUVOUUTUVAUUQ
      VSUUMUUQHUWRUUGUWMQUULXRUUPYBPVTWATUVCUWRUWPUWMUUDQUVCUUHXSUUOPLZQLZUWHUV
      DQLZQLZUUEUXBQLUWRUWPUVCUXAUUEUXBQUVCXSUUPPLZUXAUUEUVCYRUUAUVAUXDUXAOUVPU
      VTUWFXSYCUUOWBWFUVBUXDUUEOZUUTUUQUXEUVAUUPYBXSPVBRRWCWAUVCUULYCPLZUULUUOP
      LZQLZUWMUWTUVDQLZQLUWRUXCUVCUXFUWMUXGUXIQUVCCDEXSUUKYCNQPUVHUVJEMZNGWOZUV
      GUVIUXJQLZPLUVGUVIPLZUVGUXJPLQLOUVCUVGUVIUXJWBRZUVPUVQUVTUVMUVKUXMUVIUVGP
      LOUVCUVGUVIWDRZWEZUVCCDEXSUUKUUONQPUXNUVPUVQUWFUVMUXOWEWGUVCUULNGUUAUVAUW
      RUXHOUVRUVTUWFUULYCUUOWBWFUVCCDEUUHUWTUWHUVDNQUVCYRUUAUUHNGZUVPUVTXSYCSTZ
      UVCYRUVAUWTNGUVPUWFXSUUOSTUVCUURUUAUWHNGZUVQUVTUUKYCSTZUVKUVLUVIUVGQLOUVC
      UVGUVIWHRZUXKUVLUXJQLUVGUXLQLOUVCUVGUVIUXJWIRZUWGUVMWJWKUVCCDEUWHUUEUVDNQ
      UXTUWDUWGUYAUYBWLWKUVCUXFUWMUUDUXPUUSUXFUUDOZUUCUVBUUMUYCUURUULXRYCPUSRWP
      WCWGWCUVCUUIUWHQLZUWNUWIUVCUUGNGZUXQUXSUYDUWNOUVCYQYTUYEUVSUWCXRYBSTZUXRU
      XTUUGUUHUWHWIWFUVCUUINGZUXSUYDUWIOUVCUYEUXQUYGUYFUXRUUGUUHVITZUXTUUIUWHWH
      TWCUVCUWHUUDQLUWOQLZUWHUUDUWOQLZQLZUWQUWJUVCUXSUVNUWONGUYIUYKOUXTUWAUVCCD
      UUEUVDNNNQUVMUWDUWGVNZUWHUUDUWOWIWFUVCCDEUWHUWOUUDNQUXTUYLUWAUYAUYBWMUVCU
      VEUYJUWHQUVCUVNUWBUVFUVEUYJOUWAUWDUWGUUDUUEUVDWIWFWNWKWQUVCUXSUYGUVENGUWK
      UWLUNUXTUYHUVCCDUUFUVDNNNQUVMUWEUWGVNUWHUUIUVEWRWFXDWSWTXEWTXEXFUUCYHIUUI
      UUFUKULUMZJKUUJUUCYGUYMIJXRXSYBYCXAUTUUIUUFVDXBXCXGXHXI $.
  $}

  ${
    $d A w x y z $.  $d B w z $.
    $( Apartness of signed reals is tight.  (Contributed by Jim Kingdon,
       29-Jan-2020.) $)
    aptisr $p |- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) )
        -> A = B ) $=
      ( vx cnr wcel cltr wbr wo wn wceq cv cer wi cnp orbi12d notbid wa co cltp
      cpp vy vz cop cec df-nr breq1 breq2 imbi12d addcomprg ad2ant2lr ad2ant2rl
      eqeq1 breq12d orbi2d addclpr aptipr 3expia syl2anc sylbird ltsrprg ancoms
      vw eqeq2 wb enreceq 3imtr4d 2ecoptocl 3impia ) ADEBDEABFGZBAFGZHZIZABJZCK
      ZUAKZUCLUDZUBKZVBKZUCLUDZFGZVSVPFGZHZIZVPVSJZMAVSFGZVSAFGZHZIZAVSJZMVLVMM
      CUAUBVBABNNLDUEVPAJZWCWHWDWIWJWBWGWJVTWEWAWFVPAVSFUFVPAVSFUGOPVPAVSULUHVS
      BJZWHVLWIVMWKWGVKWKWEVIWFVJVSBAFUGVSBAFUFOPVSBAVCUHVNNEZVONEZQZVQNEZVRNEZ
      QZQZVNVRTRZVOVQTRZSGZVQVOTRZVRVNTRZSGZHZIZWSWTJZWCWDWRXFXAWTWSSGZHZIZXGWR
      XIXEWRXHXDXAWRWTXBWSXCSWMWOWTXBJWLWPVOVQUIUJWLWPWSXCJWMWOVNVRUIUKUMUNPWRW
      SNEZWTNEZXJXGMWLWPXKWMWOVNVRUOUKWMWOXLWLWPVOVQUOUJXKXLXJXGWSWTUPUQURUSWRW
      BXEWRVTXAWAXDVNVOVQVRUTWQWNWAXDVDVQVRVNVOUTVAOPVNVOVQVRVEVFVGVH $.
  $}

  ${
    $d U f g h $.  $d V f g h $.  $d W f g h $.  $d X f g h $.  $d Y f g h $.
    $d Z f g h $.
    $( Lemma for ~ mulextsr1 .  (Contributed by Jim Kingdon, 17-Feb-2020.) $)
    mulextsr1lem $p |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. )
        /\ ( U e. P. /\ V e. P. ) ) -> (
        ( ( ( X .P. U ) +P. ( Y .P. V ) ) +P.
          ( ( Z .P. V ) +P. ( W .P. U ) ) )
        <P ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P.
          ( ( Z .P. U ) +P. ( W .P. V ) ) ) ->
        ( ( X +P. W ) <P ( Y +P. Z ) \/ ( Z +P. Y ) <P ( W +P. X ) ) ) ) $=
      ( vf vg vh cnp wcel cmp co cpp cltp wbr wceq adantl mulclpr syl2anc wa wo
      w3a addcomprg addclpr simp2l simp3r simp1r caovcld simp1l simp3l caovcomd
      cv simp2r addassprg caov411d distrprg mulcomprg caovdir2d oveq12d 3eqtr4d
      eqtr4d breq12d wi eqeltrd addextpr syl22anc 3adant2 3adant1 ltmprr sylbid
      syl3anc orim12d syld breq2d orbi12d sylibd ) DJKZEJKZUAZFJKZCJKZUAZAJKZBJ
      KZUAZUCZDALMZEBLMZNMFBLMZCALMZNMNMZDBLMZEALMZNMFALMZCBLMZNMNMZOPZDCNMZFEN
      MZOPZWTWSOPZUBZWSEFNMZOPZWTCDNMZOPZUBWGWRWSALMZWTBLMZNMZWTALMZWSBLMZNMZOP
      ZXCWGWLXJWQXMOWGWJWINMZWHWKNMZNMXPXONMWLXJWGGHXOXPJNGUMZJKZHUMZJKZUAZXQXS
      NMZXSXQNMQWGXQXSUDRZWGGHWJWIJJJNYAYBJKWGXQXSUERZWGWAWEWJJKVTWAWBWFUFZVTWC
      WDWEUGZFBSTZWGVSWEWIJKVRVSWCWFUHZYFEBSTZUIZWGGHWHWKJJJNYDWGVRWDWHJKVRVSWC
      WFUJZVTWCWDWEUKZDASTZWGWBWDWKJKVTWAWBWFUNZYLCASTZUIZULWGGHIWHWIWJWKJNYMYI
      YGYCXRXTIUMZJKUCZYBYQNMXQXSYQNMZNMQWGXQXSYQUORZYOYDUPWGXHXPXIXONWGGHIDCAJ
      NLYRXQYSLMXQXSLMZXQYQLMNMQWGXQXSYQUQRZYKYNYLYDYAUUAXSXQLMQWGXQXSURRZUSZWG
      GHIFEBJNLUUBYEYHYFYDUUCUSZUTVAWGWQWOWNNMZWMWPNMZNMXMWGGHIWMWNWOWPJNWGVRWE
      WMJKYKYFDBSTWGVSWDWNJKYHYLEASTWGWAWDWOJKYEYLFASTYCYTWGWBWEWPJKYNYFCBSTYDU
      PWGXKUUFXLUUGNWGGHIFEAJNLUUBYEYHYLYDUUCUSWGGHIDCBJNLUUBYKYNYFYDUUCUSUTVBV
      CWGXNXHXKOPZXIXLOPZUBZXCWGXHJKXIJKXKJKZXLJKZXNUUJVDWGXHXPJUUDYPVEWGXIXOJU
      UEYJVEWGWTJKZWDUUKWGWAVSUUMYEYHFEUETZYLWTASTWGWSJKZWEUULWGVRWBUUOYKYNDCUE
      TZYFWSBSTXHXIXKXLVFVGWGUUHXAUUIXBWGUUOUUMWDUUHXAVDUUPUUNYLUUOUUMWDUCZUUHA
      WSLMZAWTLMZOPXAUUQXHUURXKUUSOUUOWDXHUURQUUMWSAURVHUUMWDXKUUSQUUOWTAURVIVC
      WSWTAVJVKVLWGUUIBWTLMZBWSLMZOPZXBWGXIUUTXLUVAOWGUUMWEXIUUTQUUNYFWTBURTWGU
      UOWEXLUVAQUUPYFWSBURTVCWGUUMUUOWEUVBXBVDUUNUUPYFWTWSBVJVLVKVMVNVKWGXAXEXB
      XGWGWTXDWSOWGWAVSWTXDQYEYHFEUDTVOWGWSXFWTOWGVRWBWSXFQYKYNDCUDTVOVPVQ $.
  $}

  ${
    $d A u v w x y z $.  $d B u v w z $.  $d C u v $.
    $( Strong extensionality of multiplication of signed reals.  (Contributed
       by Jim Kingdon, 18-Feb-2020.) $)
    mulextsr1 $p |- ( ( A e. R. /\ B e. R. /\ C e. R. )
        -> ( ( A .R C ) <R ( B .R C ) -> ( A <R B \/ B <R A ) ) ) $=
      ( cv cop cer cec cmr co cltr wbr wo cnp wceq wcel cmp cpp mulclpr syl2anc
      wi vx vy vu vv vz vw df-nr oveq1 breq1d breq1 breq2 orbi12d imbi12d oveq2
      cnr breq2d breq12d imbi1d wa cltp mulextsr1lem mulsrpr 3adant2 3adant1 wb
      simp1l simp3l simp1r simp3r addclpr simp2l ltsrprg syl22anc bitrd 3adant3
      w3a simp2r ancoms 3imtr4d 3ecoptocl ) UADZUBDZEFGZUCDZUDDZEFGZHIZUEDZUFDZ
      EFGZWFHIZJKZWCWJJKZWJWCJKZLZTAWFHIZWKJKZAWJJKZWJAJKZLZTWPBWFHIZJKZABJKZBA
      JKZLZTACHIZBCHIZJKZXETUAUBUEUFUCUDABCMFUOUGWCANZWLWQWOWTXIWGWPWKJWCAWFHUH
      UIXIWMWRWNWSWCAWJJUJWCAWJJUKULUMWJBNZWQXBWTXEXJWKXAWPJWJBWFHUHUPXJWRXCWSX
      DWJBAJUKWJBAJUJULUMWFCNZXBXHXEXKWPXFXAXGJWFCAHUNWFCBHUNUQURWAMOZWBMOZUSZW
      HMOZWIMOZUSZWDMOZWEMOZUSZVPZWAWDPIZWBWEPIZQIZWHWEPIZWIWDPIZQIZQIWAWEPIZWB
      WDPIZQIZWHWDPIZWIWEPIZQIZQIUTKZWAWIQIWBWHQIUTKZWHWBQIWIWAQIUTKZLWLWOWDWEW
      IWAWBWHVAYAWLYDYJEFGZYMYGEFGZJKZYNYAWGYQWKYRJXNXTWGYQNXQWAWBWDWEVBVCXQXTW
      KYRNXNWHWIWDWEVBVDUQYAYDMOZYJMOZYMMOZYGMOZYSYNVEYAYBMOZYCMOZYTYAXLXRUUDXL
      XMXQXTVFZXNXQXRXSVGZWAWDRSYAXMXSUUEXLXMXQXTVHZXNXQXRXSVIZWBWERSYBYCVJSYAY
      HMOZYIMOZUUAYAXLXSUUJUUFUUIWAWERSYAXMXRUUKUUHUUGWBWDRSYHYIVJSYAYKMOZYLMOZ
      UUBYAXOXRUULXNXOXPXTVKZUUGWHWDRSYAXPXSUUMXNXOXPXTVQZUUIWIWERSYKYLVJSYAYEM
      OZYFMOZUUCYAXOXSUUPUUNUUIWHWERSYAXPXRUUQUUOUUGWIWDRSYEYFVJSYDYJYMYGVLVMVN
      YAWMYOWNYPXNXQWMYOVEXTWAWBWHWIVLVOXNXQWNYPVEZXTXQXNUURWHWIWAWBVLVRVOULVSV
      T $.
  $}

  ${
    $d A l u w x z $.
    $( For any signed real, there is an integer that is greater than it.  This
       is also known as the "archimedean property".  The expression
       ` [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } ` ,
       ` { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ` is the
       embedding of the positive integer ` x ` into the signed reals.
       (Contributed by Jim Kingdon, 23-Apr-2020.) $)
    archsr $p |- ( A e. R. -> E. x e. N. A <R
        [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } ,
          { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) $=
      ( vz cv cop cer cec wbr c1p cpp co cltr cnpi wrex cnp wcel cltp 1pr df-nr
      vw c1o ceq cltq cab cnr wceq breq1 rexbidv wa addclpr mpan2 archpr adantr
      nnprlu adantl sylancl simplr ltaddpr syl2anc addcomprg breqtrrd wi ltsopr
      syl ltrelpr sotri expcom sylsyld reximdva mpd simpl a1i syl12anc rexbidva
      wb ltsrprg mpbird ecoptocl ) EFZUBFZGHIZDFAFZUCGUDIZUEJDUFWEBFUEJBUFGZKLM
      ZKGHIZNJZAOPZCWHNJZAOPEUBCQQHUGUAWCCUHWIWKAOWCCWHNUIUJWAQRZWBQRZUKZWJWAKL
      MZWBWGLMZSJZAOPZWNWOWFSJZAOPZWRWLWTWMWLWOQRZWTWLKQRZXATWAKULUMABWODUNVFUO
      WNWSWQAOWNWDORZUKZWGWPSJZWSWOWGSJZWQXDWGWGWBLMZWPSXDWGQRZWMWGXGSJXDWFQRZX
      BXHXCXIWNBWDDUPUQZTWFKULURZWLWMXCUSZWGWBUTVAXDWMXHWPXGUHXLXKWBWGVBVAVCXDW
      FWGSJZWSXFVDXDXIXBXMXJTWFKUTURWSXMXFWOWFWGSQVEVGVHVIVFXFXEWQWOWGWPSQVEVGV
      HVIVJVKVLWNWIWQAOXDWNXHXBWIWQVQWNXCVMXKXBXDTVNWAWBWGKVRVOVPVSVT $.
  $}

  ${
    $d A a b x $.
    $( Mapping from a signed real greater than zero to a positive real.
       (Contributed by Jim Kingdon, 25-Jun-2021.) $)
    srpospr $p |- ( ( A e. R. /\ 0R <R A ) ->
        E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) $=
      ( va vb cnr wcel c0r cltr wbr cv c1p cpp co cer wceq cnp wreu wi syl2anc
      wa cop cec df-nr breq2 reubidv imbi12d cltp gt0srpr biimpi adantl lteupri
      eqeq2 syl wb simpr 1pr addclpr simplll simpllr enreceq syl22anc addcomprg
      a1i oveq1d addassprg syl3anc eqtrd eqeq1d addcanprg oveq2 impbid1 3bitrrd
      bitrd reubidva mpbid ex ecoptocl imp ) BEFGBHIZAJZKLMZKUANUBZBOZAPQZGCJZD
      JZUANUBZHIZWBWGOZAPQZRVSWDRCDBPPNEUCWGBOZWHVSWJWDWGBGHUDWKWIWCAPWGBWBULUE
      UFWEPFZWFPFZTZWHWJWNWHTZWFVTLMZWEOZAPQZWJWOWFWEUGIZWRWHWSWNWHWSWEWFUHUIUJ
      AWFWEUKUMWOWQWIAPWOVTPFZTZWIWAWFLMZKWELMZOZVTWFLMZWEOZWQXAWAPFZKPFZWLWMWI
      XDUNXAWTXHXGWOWTUOZXHXAUPVCZVTKUQSXJWLWMWHWTURZWLWMWHWTUSZWAKWEWFUTVAXAXD
      KXELMZXCOZXFXAXBXMXCXAXBKVTLMZWFLMZXMXAWAXOWFLXAWTXHWAXOOXIXJVTKVBSVDXAXH
      WTWMXPXMOXJXIXLKVTWFVEVFVGVHXAXNXFXAXHXEPFZWLXNXFRXJXAWTWMXQXIXLVTWFUQSXK
      KXEWEVIVFXEWEKLVJVKVMXAXEWPWEXAWTWMXEWPOXIXLVTWFVBSVHVLVNVOVPVQVR $.
  $}

  $( Mapping from a positive real to a signed real.  (Contributed by Jim
     Kingdon, 25-Jun-2021.) $)
  prsrcl $p |- ( A e. P. ->
      [ <. ( A +P. 1P ) , 1P >. ] ~R e. R. ) $=
    ( cnp wcel c1p cpp co cop cer cec cxp cqs cnr addclpr mpan2 opelxpi ecelqsi
    1pr enrex 3syl df-nr eleqtrrdi ) ABCZADEFZDGZHIZBBJZHKZLUBUCBCZUDUFCZUEUGCU
    BDBCZUHQADMNUHUJUIQUCDBBONUFUDHRPSTUA $.

  $( Mapping from a positive real to a signed real yields a result greater than
     zero.  (Contributed by Jim Kingdon, 25-Jun-2021.) $)
  prsrpos $p |- ( A e. P. -> 0R <R [ <. ( A +P. 1P ) , 1P >. ] ~R ) $=
    ( cnp wcel c1p cpp cltp wbr c0r cop cer cec cltr 1pr ltaddpr mpan addcomprg
    co wceq breqtrd gt0srpr sylibr ) ABCZDADEQZFGHUCDIJKLGUBDDAEQZUCFDBCZUBDUDF
    GMDANOUEUBUDUCRMDAPOSUCDTUA $.

  ${
    $d A f g h $.  $d B f g h $.
    $( Mapping from positive real addition to signed real addition.
       (Contributed by Jim Kingdon, 29-Jun-2021.) $)
    prsradd $p |- ( ( A e. P. /\ B e. P. ) ->
        [ <. ( ( A +P. B ) +P. 1P ) , 1P >. ] ~R =
        ( [ <. ( A +P. 1P ) , 1P >. ] ~R +R
          [ <. ( B +P. 1P ) , 1P >. ] ~R ) ) $=
      ( vf vg vh cnp wcel wa c1p cpp co cop cer cec wceq 1pr addclpr cv syl2anc
      adantl mpan2 addsrpr mpanl2 mpanr2 syl2an simpl simpr addcomprg addassprg
      cplr a1i w3a caov4d mp2an eqtrd oveq1d syl3anc wb enreceq mpbird eqtr2d )
      AFGZBFGZHZAIJKZILMNBIJKZILMNUJKZVEVFJKZIIJKZLMNZABJKZIJKZILMNZVBVEFGZVFFG
      ZVGVJOZVCVBIFGZVNPAIQUAZVCVQVOPBIQUAZVNVOVQVPPVNVQVOVQHVPPVEIVFIUBUCUDUEV
      DVJVMOZVHIJKZVIVLJKZOZVDWAVIVKJKZIJKZWBVDVHWDIJVDVHVKVIJKZWDVDCDEAIBIFJVB
      VCUFVQVDPUKZVBVCUGCRZFGZDRZFGZHZWHWJJKZWJWHJKOVDWHWJUHTWIWKERZFGULWMWNJKW
      HWJWNJKJKOVDWHWJWNUITWGWLWMFGVDWHWJQTUMVDVKFGZVIFGZWFWDOABQZWPVDVQVQWPPPI
      IQUNZUKZVKVIUHSUOUPVDWPWOVQWEWBOWSWQWGVIVKIUIUQUOVDVHFGZVLFGZVTWCURZVBVNV
      OWTVCVRVSVEVFQUEVDWOVQXAWQWGVKIQSWTWPXAXBWRWTWPHXAVQXBPVHVIVLIUSUDUCSUTVA
      $.
  $}

  ${
    $d A f g h $.  $d B f g h $.
    $( Mapping from positive real ordering to signed real ordering.
       (Contributed by Jim Kingdon, 29-Jun-2021.) $)
    prsrlt $p |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <->
        [ <. ( A +P. 1P ) , 1P >. ] ~R <R [ <. ( B +P. 1P ) , 1P >. ] ~R ) ) $=
      ( vf vg vh cnp wcel wa c1p cpp co cltp wbr cop wceq wb syl2anc cv addclpr
      cer cec cltr 1pr a1i simpr addassprg breq2d simpl ltaprg addcomprg breq1d
      syl3anc w3a adantl caovord2d 3bitr2d ltsrprg syl22anc 3bitr4d ) AFGZBFGZH
      ZAIJKZIJKZIBJKZIJKZLMZVDIBIJKZJKZLMZABLMZVCINTUAVHINTUAUBMZVBVFVIVDLVBIFG
      ZVAVMVFVIOVMVBUCUDZUTVAUEZVNIBIUFULUGVBVKIAJKZVELMZVCVELMVGVBUTVAVMVKVQPU
      TVAUHZVOVNABIUIULVBVCVPVELVBUTVMVCVPOVRVNAIUJQUKVBCDEVCVEILFJCRZFGZDRZFGZ
      ERZFGUMVSWALMWCVSJKWCWAJKLMPVBVSWAWCUIUNVBUTVMVCFGZVRVNAISQZVBVMVAVEFGVNV
      OIBSQVNVTWBHVSWAJKWAVSJKOVBVSWAUJUNUOUPVBWDVMVHFGZVMVLVJPWEVNVBVAVMWFVOVN
      BISQVNVCIVHIUQURUS $.
  $}

  ${
    $d A x y $.
    $( Mapping a restricted iota from a positive real to a signed real.
       (Contributed by Jim Kingdon, 29-Jun-2021.) $)
    prsrriota $p |- ( ( A e. R. /\ 0R <R A ) -> [ <. ( ( iota_ x e. P.
        [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. ] ~R = A ) $=
      ( vy wcel wa cv c1p cpp co cop cer cec wceq cnp wreu srpospr oveq1 opeq1d
      syl eceq1d cnr c0r cltr crio reurex simprr wb simprl adantr eqeq1d riota2
      wbr wrex syl2anc mpbid eqtrd rexlimddv ) BUADUBBUCULEZCFZGHIZGJZKLZBMZAFZ
      GHIZGJZKLZBMZANUDZGHIZGJZKLZBMCNURVCCNOVCCNUMCBPVCCNUESURUSNDZVCEZEZVLVBB
      VOVIUSMZVLVBMVOVCVPURVMVCUFZVOVMVHANOZVCVPUGURVMVCUHURVRVNABPUIVHVCANUSVD
      USMZVGVBBVSVFVAKVSVEUTGVDUSGHQRTUJUKUNUOVPVKVAKVPVJUTGVIUSGHQRTSVQUPUQ $.
  $}

  ${
    $d A m $.  $d A y $.  $d F m $.  $d F y $.
    caucvgsrlemcl.f $e |- ( ph -> F : N. --> R. ) $.
    caucvgsrlemcl.gt1 $e |- ( ph -> A. m e. N. 1R <R ( F ` m ) ) $.
    $( Lemma for ~ caucvgsr .  Terms of the sequence from ~ caucvgsrlemgt1 can
       be mapped to positive reals.  (Contributed by Jim Kingdon,
       2-Jul-2021.) $)
    caucvgsrlemcl $p |- ( ( ph /\ A e. N. ) -> ( iota_ y e. P.
        ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) e. P. ) $=
      ( cnpi wcel cfv cv c1p wceq cnp wreu cnr c0r cltr wbr c1r cpp cop cer cec
      wa co crio ffvelcdmda 0lt1sr wral fveq2 breq2d rspcv mpan9 ltsosr ltrelsr
      sotri sylancr srpospr syl2anc eqcom reubii sylib riotacl syl ) ACHIZUEZCE
      JZBKLUAUFLUBUCUDZMZBNOZVJBNUGNIVGVIVHMZBNOZVKVGVHPIQVHRSZVMAHPCEFUHVGQTRS
      TVHRSZVNUIATDKZEJZRSZDHUJVFVOGVRVODCHVPCMVQVHTRVPCEUKULUMUNQTVHRPUOUPUQUR
      BVHUSUTVLVJBNVIVHVAVBVCVJBNVDVE $.
  $}

  ${
    $d A m $.
    caucvgsrlemasr.bnd $e |- ( ph -> A. m e. N. A <R ( F ` m ) ) $.
    $( Lemma for ~ caucvgsr .  The lower bound is a signed real.  (Contributed
       by Jim Kingdon, 4-Jul-2021.) $)
    caucvgsrlemasr $p |- ( ph -> A e. R. ) $=
      ( vx cnr wcel cnpi wral cfv cltr wbr ltrelsr brel simpld ralimi syl c1o
      cv wex wb 1pi elex2 r19.3rmv mp2b sylibr ) ABGHZCIJZUHABCTDKZLMZCIJUIEUKU
      HCIUKUHUJGHBUJGGLNOPQRSIHFTIHFUAUHUIUBUCFSIUDUHCFIUEUFUG $.
  $}

  ${
    caucvgsr.f $e |- ( ph -> F : N. --> R. ) $.
    caucvgsr.cau $e |- ( ph ->
      A. n e. N. A. k e. N. ( n <N k ->
        ( ( F ` n ) <R ( ( F ` k ) +R
          [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R
        ) /\
        ( F ` k ) <R ( ( F ` n ) +R
          [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R
        ) ) ) ) $.

    ${
      caucvgsrlemgt1.gt1 $e |- ( ph -> A. m e. N. 1R <R ( F ` m ) ) $.

      ${
        caucvgsrlemf.xfr $e |- G =
          ( x e. N. |-> ( iota_ y e. P.
            ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) ) $.
        ${
          $d A m $.  $d A x y $.  $d F m $.  $d F x y $.  $d ph x $.
          $( Lemma for ~ caucvgsr .  Coercing sequence value from a positive
             real to a signed real.  (Contributed by Jim Kingdon,
             29-Jun-2021.) $)
          caucvgsrlemfv $p |- ( ( ph /\ A e. N. ) ->
              [ <. ( ( G ` A ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) $=
            ( cnpi c1p cpp wceq cnp wcel wa cfv co cop cer cec cv crio cmpt a1i
            fveq2 eqeq1d adantl simpr caucvgsrlemcl fvmptd oveq1d opeq1d eceq1d
            riotabidv eqcom riotabiia oveq1i opeq1i eceq1 mp1i cnr c0r cltr wbr
            wb ffvelcdmda 0lt1sr wral breq2d rspcv mpan9 ltsosr ltrelsr sylancr
            c1r sotri prsrriota syl2anc 3eqtrd ) AEPUAZUBZEJUCZQRUDZQUEZUFUGEIU
            CZCUHZQRUDQUEUFUGZSZCTUIZQRUDZQUEZUFUGZWNWLSZCTUIZQRUDZQUEZUFUGZWLW
            HWKWRUFWHWJWQQWHWIWPQRWHBEBUHZIUCZWNSZCTUIZWPPJTJBPXHUJSWHOUKXEESZX
            HWPSWHXIXGWOCTXIXFWLWNXEEIULUMVAUNAWGUOACEGILNUPUQURUSUTWRXCSWSXDSW
            HWQXBQWPXAQRWOWTCTWOWTVLWMTUAWLWNVBUKVCVDVEWRXCUFVFVGWHWLVHUAVIWLVJ
            VKZXDWLSAPVHEILVMWHVIWBVJVKWBWLVJVKZXJVNAWBGUHZIUCZVJVKZGPVOWGXKNXN
            XKGEPXLESXMWLWBVJXLEIULVPVQVRVIWBWLVJVHVSVTWCWACWLWDWEWF $.
        $}

        ${
          $d F m $.  $d F y $.  $d m x $.  $d ph x $.  $d x y $.
          $( Lemma for ~ caucvgsr .  Defining the sequence in terms of positive
             reals.  (Contributed by Jim Kingdon, 23-Jun-2021.) $)
          caucvgsrlemf $p |- ( ph -> G : N. --> P. ) $=
            ( cnpi cv cfv c1p cpp cnp cop cer cec wceq crio caucvgsrlemcl fmptd
            co ) ABOBPZHQCPRSUHRUAUBUCUDCTUETIACUIFHKMUFNUG $.
        $}

        ${
          $d F m x $.  $d F x y $.  $d k m n x $.  $d k n ph x $.
          $d k n x y $.  $d l n u $.
          $( Lemma for ~ caucvgsr .  Defining the Cauchy condition in terms of
             positive reals.  (Contributed by Jim Kingdon, 23-Jun-2021.) $)
          caucvgsrlemcau $p |- ( ph ->
            A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <P ( ( G ` k ) +P.
              <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
              { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\
              ( G ` k ) <P ( ( G ` n ) +P.
              <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
              { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) ) $=
            ( wbr cop co cnpi c1p cnp cv clti cfv c1o ceq cec crq cltq cab cltp
            cpp wa wi wral cplr cltr wcel wb wf caucvgsrlemf ad2antrr ffvelcdmd
            cer simplr adantr ffvelcdmda recnnpr syl addclpr wceq caucvgsrlemfv
            syl2anc prsrlt prsradd adantlr eqtrd breq12d bitrd anbi12d ralbidva
            oveq1d imbi2d mpbird ) AGUAZEUAZUBOZWDIUCZWEIUCZJUAWDUDPUEUFUGUCZUH
            OJUIWIDUAUHODUIPZUKQZUJOZWHWGWJUKQZUJOZULZUMZERUNZGRUNWFWDHUCZWEHUC
            ZWJSUKQSPVCUFZUOQZUPOZWSWRWTUOQZUPOZULZUMZERUNZGRUNLAWQXGGRAWDRUQZU
            LZWPXFERXIWERUQZULZWOXEWFXKWLXBWNXDXKWLWGSUKQSPVCUFZWKSUKQSPVCUFZUP
            OZXBXKWGTUQZWKTUQZWLXNURXKRTWDIARTIUSZXHXJABCDEFGHIJKLMNUTZVAAXHXJV
            DZVBZXKWHTUQZWJTUQZXPXIRTWEIAXQXHXRVEVFZXKXHYBXSDWDJVGVHZWHWJVIVLWG
            WKVMVLXKXLWRXMXAUPXIXLWRVJXJABCDWDEFGHIJKLMNVKVEZXKXMWHSUKQSPVCUFZW
            TUOQZXAXKYAYBXMYGVJYCYDWHWJVNVLXKYFWSWTUOAXJYFWSVJXHABCDWEEFGHIJKLM
            NVKVOZWAVPVQVRXKWNYFWMSUKQSPVCUFZUPOZXDXKYAWMTUQZWNYJURYCXKXOYBYKXT
            YDWGWJVIVLWHWMVMVLXKYFWSYIXCUPYHXKYIXLWTUOQZXCXKXOYBYIYLVJXTYDWGWJV
            NVLXKXLWRWTUOYEWAVPVQVRVSWBVTVTWC $.
        $}

        ${
          $d F m x y w $.  $d ph x w $.  $d G m w $.
          $( Lemma for ~ caucvgsr .  Defining the boundedness condition in
             terms of positive reals.  (Contributed by Jim Kingdon,
             25-Jun-2021.) $)
          caucvgsrlembound $p |- ( ph -> A. m e. N. 1P <P ( G ` m ) ) $=
            ( vw c1p wbr cnpi cltr c1r cv cfv cltp wral wcel wa cpp cop cer cec
            co weq fveq2 breq2d cbvralv sylib r19.21bi wceq df-1r caucvgsrlemfv
            eqcomi a1i 3brtr4d cnp wb 1pr caucvgsrlemf ffvelcdmda prsrlt mpbird
            sylancr ralrimiva ) APOUAZIUBZUCQZORUDPFUAZIUBZUCQZFRUDAVOORAVMRUEU
            FZVOPPUGUKPUHUIUJZVNPUGUKPUHUIUJZSQZVSTVMHUBZVTWASATWCSQZORATVPHUBZ
            SQZFRUDWDORUDMWFWDFORFOULWEWCTSVPVMHUMUNUOUPUQVTTURVSTVTUSVAVBABCDV
            MEFGHIJKLMNUTVCVSPVDUEVNVDUEVOWBVEVFARVDVMIABCDEFGHIJKLMNVGVHPVNVIV
            KVJVLVOVROFROFULVNVQPUCVMVPIUMUNUOUP $.
        $}
      $}

      ${
        $d F a b j k l u $.  $d F a i j k x $.  $d F m $.  $d F w z $.
        $d F k l n u $.  $d F a i j x y $.  $d a b j k ph x $.  $d ph z $.
        $d ph n $.  $d a b j k l u w $.  $d a b j k l u z $.  $d b c j k x $.
        $d m w $.  $d m z $.  $d n u w $.  $d n u z $.  $d w x $.  $d x z $.
        $d k m $.  $d m n $.
        $( Lemma for ~ caucvgsr .  A Cauchy sequence whose terms are greater
           than one converges.  (Contributed by Jim Kingdon, 22-Jun-2021.) $)
        caucvgsrlemgt1 $p |- ( ph ->
            E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. (
              j <N i ->
              ( ( F ` i ) <R ( y +R x ) /\
                y <R ( ( F ` i ) +R x ) ) ) ) ) $=
          ( wbr cnpi c1p co cnp cltr vz vw va vb vc cv clti cfv cpp cop cer cec
          wceq crio cmpt cltp wa wral wrex c0r cplr caucvgsrlemf caucvgsrlemcau
          wi eqid caucvgsrlembound caucvgprpr wcel prsrcl ad2antrl oveq2 breq2d
          cnr anbi12d imbi2d rexralbidv simplrr adantr wreu srpospr riotacl syl
          adantll rspcdva nfv nfcv nfre1 nfralya nfan nfra1 nfrexya wb wf simpr
          ad4antr ffvelcdmd simplrl addclpr syl2anc prsrlt caucvgsrlemfv oveq2d
          adantlr prsradd prsrriota eqtrd breq12d oveq12d 3bitrd ralbida rexbid
          bitrd mpbid breq2 fveq2 breq1d oveq1d imbi12d cbvralv rexbii sylib ex
          ralrimiva oveq1 breq1 ralbidv rspcev rexlimddv ) AFUFZGUFZUGOZYJUAPUA
          UFJUHUBUFQUIRQUJUKULUMUBSUNUOZUHZUCUFZUDUFZUIRZUPOZYNYMYOUIRZUPOZUQZV
          DZGPURZFPUSZUDSURZUTBUFZTOZYIEUFZUGOZUUGJUHZCUFZUUEVARZTOZUUJUUIUUEVA
          RZTOZUQZVDZEPURFPUSZVDZBVMURZCVMUSZUCSAUDUCDQFGHIYLKAUAUBDGHIJYLKLMNY
          LVEZVBZAUAUBDGHIJYLKLMNUVAVCAUAUBDGHIJYLKLMNUVAVFVGAYNSVHZUUDUQZUQZYN
          QUIRQUJUKULZVMVHZUUFUUHUUIUVFUUEVARZTOZUVFUUMTOZUQZVDZEPURZFPUSZVDZBV
          MURZUUTUVCUVGAUUDYNVIVJUVEUVOBVMUVEUUEVMVHZUQZUUFUVNUVRUUFUQZYKYJJUHZ
          UVHTOZUVFUVTUUEVARZTOZUQZVDZGPURZFPUSZUVNUVSYKYMYNUEUFQUIRQUJUKULUUEU
          MZUESUNZUIRZUPOZYNYMUWIUIRZUPOZUQZVDZGPURZFPUSZUWGUVSUUCUWQUDSUWIYOUW
          IUMZUUAUWOFGPPUWRYTUWNYKUWRYQUWKYSUWMUWRYPUWJYMUPYOUWIYNUIVKVLUWRYRUW
          LYNUPYOUWIYMUIVKVLVNVOVPUVRUUDUUFAUVCUUDUVQVQVRUVQUUFUWISVHZUVEUVQUUF
          UQZUWHUESVSUWSUEUUEVTUWHUESWAWBWCZWDUVSUWPUWFFPUVRUUFFUVEUVQFAUVDFAFW
          EUVCUUDFUVCFWEUUCFUDSFSWFUUBFPWGWHWIWIUVQFWEWIUUFFWEWIUVSUWOUWEGPUVRU
          UFGUVEUVQGAUVDGAGWEUVCUUDGUVCGWEUUCGUDSGSWFUUBGFPGPWFUUAGPWJWKWHWIWIU
          VQGWEWIUUFGWEWIUVSYJPVHZUQZUWNUWDYKUXCUWKUWAUWMUWCUXCUWKYMQUIRQUJUKUL
          ZUWJQUIRQUJUKULZTOZUWAUXCYMSVHZUWJSVHZUWKUXFWLUXCPSYJYLAPSYLWMUVDUVQU
          UFUXBUVBWOUVSUXBWNWPZUVSUXHUXBUVSUVCUWSUXHUVRUVCUUFAUVCUUDUVQWQVRZUXA
          YNUWIWRWSVRYMUWJWTWSUXCUXDUVTUXEUVHTUVRUXBUXDUVTUMZUUFUVEUXBUXKUVQAUX
          BUXKUVDAUAUBDYJGHIJYLKLMNUVAXAXCXCXCZUVSUXEUVHUMUXBUVSUXEUVFUWIQUIRQU
          JUKULZVARZUVHUVSUVCUWSUXEUXNUMUXJUXAYNUWIXDWSUVQUUFUXNUVHUMUVEUWTUXMU
          UEUVFVAUEUUEXEZXBWCXFVRXGXLUXCUWMUVFUWLQUIRQUJUKULZTOZUVFUXDUXMVARZTO
          UWCUXCUVCUWLSVHZUWMUXQWLUVSUVCUXBUXJVRUXCUXGUWSUXSUXIUVSUWSUXBUXAVRZY
          MUWIWRWSYNUWLWTWSUXCUXPUXRUVFTUXCUXGUWSUXPUXRUMUXIUXTYMUWIXDWSVLUXCUX
          RUWBUVFTUXCUXDUVTUXMUUEVAUXLUVSUXMUUEUMZUXBUVQUUFUYAUVEUXOWCVRXHVLXIV
          NVOXJXKXMUWFUVMFPUWEUVLGEPYJUUGUMZYKUUHUWDUVKYJUUGYIUGXNUYBUWAUVIUWCU
          VJUYBUVTUUIUVHTYJUUGJXOZXPUYBUWBUUMUVFTUYBUVTUUIUUEVAUYCXQVLVNXRXSXTY
          AYBYCUUSUVPCUVFVMUUJUVFUMZUURUVOBVMUYDUUQUVNUUFUYDUUPUVLFEPPUYDUUOUVK
          UUHUYDUULUVIUUNUVJUYDUUKUVHUUITUUJUVFUUEVAYDVLUUJUVFUUMTYEVNVOVPVOYFY
          GWSYH $.
      $}
    $}

    ${
      caucvgsrlembnd.bnd $e |- ( ph -> A. m e. N. A <R ( F ` m ) ) $.
      ${
        caucvgsrlembnd.offset $e |- G = ( a e. N. |->
          ( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) ) $.

        ${
          $d A a $.  $d A m $.  $d F a $.  $d J a $.  $d a ph $.
          $( Lemma for ~ caucvgsr .  Offsetting the values of the sequence so
             they are greater than one.  (Contributed by Jim Kingdon,
             3-Jul-2021.) $)
          caucvgsrlemoffval $p |- ( ( ph /\ J e. N. ) ->
              ( ( G ` J ) +R A ) = ( ( F ` J ) +R 1R ) ) $=
            ( wcel cplr co cnr wceq cnpi wa cfv c1r cm1r cmr cv cmpt a1i oveq1d
            fveq2 adantl simpr ffvelcdmda addclsr sylancl caucvgsrlemasr adantr
            1sr m1r mulclsr syl2anc addasssrg syl3anc c0r addcomsrg pn0sr eqtrd
            fvmptd syl oveq2d 0idsr 3eqtrd ) AIUAPZUBZIHUCZCQRIGUCZUDQRZCUEUFRZ
            QRZCQRZVRVSCQRZQRZVRVOVPVTCQVOJIJUGZGUCZUDQRZVSQRZVTUAHSHJUAWGUHTVO
            OUIWDITZWGVTTVOWHWFVRVSQWHWEVQUDQWDIGUKUJUJULAVNUMVOVRSPZVSSPZVTSPV
            OVQSPUDSPWIAUASIGLUNUSVQUDUOUPZVOCSPZUESPWJAWLVNACEGNUQURZUTCUEVAUP
            ZVRVSUOVBVIUJVOWIWJWLWAWCTWKWNWMVRVSCVCVDVOWCVRVEQRZVRVOWBVEVRQVOWB
            CVSQRZVEVOWJWLWBWPTWNWMVSCVFVBVOWLWPVETWMCVGVJVHVKVOWIWOVRTWKVRVLVJ
            VHVM $.
        $}

        ${
          $d A m $.  $d a ph $.
          $( Lemma for ~ caucvgsr .  Offsetting the values of the sequence so
             they are greater than one.  (Contributed by Jim Kingdon,
             3-Jul-2021.) $)
          caucvgsrlemofff $p |- ( ph -> G : N. --> R. ) $=
            ( cnpi c1r co cm1r cnr wcel cfv cplr cmr ffvelcdmda addclsr sylancl
            cv wa 1sr caucvgsrlemasr adantr m1r mulclsr syl2anc fmptd ) AIOIUGZ
            GUAZPUBQZCRUCQZUBQZSHAUPOTZUHZURSTZUSSTZUTSTVBUQSTPSTVCAOSUPGKUDUIU
            QPUEUFVBCSTZRSTVDAVEVAACEGMUJUKULCRUMUFURUSUEUNNUO $.
        $}

        ${
          $d A a $.  $d A f g h $.  $d A m $.  $d F a $.  $d F f g h $.
          $d G f g h $.  $d a k n $.  $d a k n ph $.  $d f g h k n ph $.
          $d f g h l n u $.
          $( Lemma for ~ caucvgsr .  Offsetting the values of the sequence so
             they are greater than one.  (Contributed by Jim Kingdon,
             3-Jul-2021.) $)
          caucvgsrlemoffcau $p |- ( ph ->
            A. n e. N. A. k e. N. ( n <N k ->
              ( ( G ` n ) <R ( ( G ` k ) +R
                [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
                  { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
                  +P. 1P ) , 1P >. ] ~R
              ) /\
              ( G ` k ) <R ( ( G ` n ) +R
                [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
                  { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
                  +P. 1P ) , 1P >. ] ~R
              ) ) ) ) $=
            ( co cplr cltr cnpi wcel cnr vf vg clti wbr cfv c1o cop ceq cec crq
            vh cv cltq cab c1p cpp cer wa wi wral wceq caucvgsrlemoffval adantr
            c1r eqcomd wf ad2antrr ffvelcdmd cnp simplr recnnpr prsrcl 3syl 1sr
            simpr a1i addcomsrg adantl addasssrg caov32d oveq1d caucvgsrlemofff
            w3a adantlr caucvgsrlemasr 3eqtr2d breq12d ltasrg addclsr caovord2d
            wb syl2anc 3bitr4d anbi12d biimpd imim2d ralimdva mpd ) AFULZDULZUC
            UDZWSGUEZWTGUEZJULWSUFUGUHUIUJUEZUMUDJUNXDBULUMUDBUNUGZUOUPOUOUGUQU
            IZPOZQUDZXCXBXFPOZQUDZURZUSZDRUTZFRUTXAWSHUEZWTHUEZXFPOZQUDZXOXNXFP
            OZQUDZURZUSZDRUTZFRUTLAXMYBFRAWSRSZURZXLYADRYDWTRSZURZXKXTXAYFXKXTY
            FXHXQXJXSYFXBVDPOZXGVDPOZQUDXNCPOZXPCPOZQUDXHXQYFYGYIYHYJQYFYIYGYDY
            IYGVAYEABCDEFGHWSIJKLMNVBVCZVEYFYHXCVDPOZXFPOXOCPOZXFPOYJYFUAUBUKXC
            XFVDTPYFRTWTGARTGVFYCYEKVGZYDYEVOZVHZYFYCXEVISXFTSZAYCYEVJZBWSJVKXE
            VLVMZVDTSYFVNVPZUAULZTSZUBULZTSZURUUAUUCPOZUUCUUAPOVAYFUUAUUCVQVRZU
            UBUUDUKULZTSWCZUUEUUGPOUUAUUCUUGPOPOVAYFUUAUUCUUGVSVRZVTYFYMYLXFPAY
            EYMYLVAYCABCDEFGHWTIJKLMNVBWDZWAYFUAUBUKXOCXFTPYFRTWTHARTHVFYCYEABC
            DEFGHIJKLMNWBVGZYOVHZACTSYCYEACEGMWEVGZYSUUFUUIVTWFWGYFUAUBUKXBXGVD
            QTPUUHUUAUUCQUDUUGUUAPOUUGUUCPOQUDWKYFUUAUUCUUGWHVRZYFRTWSGYNYRVHZY
            FXCTSYQXGTSYPYSXCXFWIWLYTUUFWJYFUAUBUKXNXPCQTPUUNYFRTWSHUUKYRVHZYFX
            OTSYQXPTSUULYSXOXFWIWLUUMUUFWJWMYFYLXIVDPOZQUDYMXRCPOZQUDXJXSYFYLYM
            UUQUURQYFYMYLUUJVEYFUUQYGXFPOYIXFPOUURYFUAUBUKXBXFVDTPUUOYSYTUUFUUI
            VTYFYIYGXFPYKWAYFUAUBUKXNCXFTPUUPUUMYSUUFUUIVTWFWGYFUAUBUKXCXIVDQTP
            UUNYPYFXBTSYQXITSUUOYSXBXFWIWLYTUUFWJYFUAUBUKXOXRCQTPUUNUULYFXNTSYQ
            XRTSUUPYSXNXFWIWLUUMUUFWJWMWNWOWPWQWQWR $.
        $}

        ${
          $d A a m $.  $d A f g h m $.  $d F a $.  $d F f g h $.  $d a m ph $.
          $d f g h m ph $.
          $( Lemma for ~ caucvgsr .  Offsetting the values of the sequence so
             they are greater than one.  (Contributed by Jim Kingdon,
             3-Jul-2021.) $)
          caucvgsrlemoffgt1 $p |- ( ph -> A. m e. N. 1R <R ( G ` m ) ) $=
            ( c1r cltr wcel cplr co cnr vf vg vh cv cfv wbr cnpi wa r19.21bi wb
            w3a ltasrg adantl caucvgsrlemasr ffvelcdmda 1sr addcomsrg caovord2d
            adantr a1i mpbid caucvgsrlemoffval breqtrrd caucvgsrlemofff syl2anc
            wceq breqtrd syl3anc mpbird ralrimiva ) AOEUDZHUEZPUFZEUGAVKUGQZUHZ
            VMCORSZCVLRSZPUFZVOVPVLCRSZVQPVOVPVKGUEZORSZVSPVOCVTPUFZVPWAPUFAWBE
            UGMUIVOUAUBUCCVTOPTRUAUDZTQZUBUDZTQZUCUDZTQUKWCWEPUFWGWCRSWGWERSPUF
            UJVOWCWEWGULUMACTQZVNACEGMUNUSZAUGTVKGKUOOTQZVOUPUTZWDWFUHWCWERSWEW
            CRSVFVOWCWEUQUMURVAABCDEFGHVKIJKLMNVBVCVOVLTQZWHVSVQVFAUGTVKHABCDEF
            GHIJKLMNVDUOZWIVLCUQVEVGVOWJWLWHVMVRUJWKWMWIOVLCULVHVIVJ $.
        $}

        ${
          $d A a i k $.  $d A f g h i x $.  $d A i j k x $.  $d A k m $.
          $d A j k x y $.  $d F a i k $.  $d F f g h i z $.  $d F k y z $.
          $d G f g h i x z $.  $d G i j k x z $.  $d G j k l u $.  $d G k m $.
          $d G k l n u $.  $d a k n ph $.  $d f g h i ph x z $.
          $d j k ph x z $.  $d ph m $.  $d m n $.  $d a m $.
          $( Lemma for ~ caucvgsr .  Offsetting the values of the sequence so
             they are greater than one.  (Contributed by Jim Kingdon,
             3-Jul-2021.) $)
          caucvgsrlemoffres $p |- ( ph ->
            E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. (
            j <N k -> ( ( F ` k ) <R ( y +R x ) /\
              y <R ( ( F ` k ) +R x ) ) ) ) ) $=
            ( cplr co cnr vi vz vf vg vh c0r cv cltr wbr clti wa cnpi wral wrex
            cfv wi caucvgsrlemoffcau caucvgsrlemoffgt1 caucvgsrlemgt1 wcel cm1r
            caucvgsrlemofff simprl caucvgsrlemasr adantr addclsr syl2anc m1r wb
            sylancl c1r w3a ltasrg adantl wf ad3antrrr ffvelcdmd simpllr simplr
            wceq addcomsrg caovord2d caucvgsrlemoffval adantlr breq1d addasssrg
            simpr bitrd caov32d breq2d ad2antrr ffvelcdmda 1sr a1i mp2an m1p1sr
            syl3anc eqtri oveq2i 0idsr eqtrid eqtrd 3bitrd biimpd oveq1d eqtrdi
            breq12d anim12d imim2d ralimdva breq2 fveq2 anbi12d imbi12d cbvralv
            syl imbitrdi reximdv impr oveq1 imbi2d rexralbidv ralbidv rexlimddv
            breq1 rspcev ) AUFBUGZUHUIZFUGZUAUGZUJUIZYJKUOZUBUGZYGRSZUHUIZYMYLY
            GRSZUHUIZUKZUPZUAULUMZFULUNZUPZBTUMZYHYIGUGZUJUIZUUDJUOZCUGZYGRSZUH
            UIZUUGUUFYGRSZUHUIZUKZUPZGULUMFULUNZUPZBTUMZCTUNZUBTABUBDUAFGHIKMAD
            EGHIJKLMNOPQVBZADEGHIJKLMNOPQUQADEGHIJKLMNOPQURUSAYMTUTZUUCUKZUKZYM
            ERSZVARSZTUTZYHUUEUUFUVCYGRSZUHUIZUVCUUJUHUIZUKZUPZGULUMZFULUNZUPZB
            TUMZUUQUVAUVBTUTZVATUTZUVDUVAUUSETUTZUVNAUUSUUCVCAUVPUUTAEHJPVDZVEY
            MEVFZVGVHUVBVAVFVJAUUSUUCUVMAUUSUKZUUBUVLBTUVSYGTUTZUKZUUAUVKYHUWAY
            TUVJFULUWAYTYKYJJUOZUVEUHUIZUVCUWBYGRSZUHUIZUKZUPZUAULUMUVJUWAYSUWG
            UAULUWAYJULUTZUKZYRUWFYKUWIYOUWCYQUWEUWIYOUWCUWIYOUWBVKRSZYNERSZUHU
            IZUWJUVBYGRSZUHUIZUWCUWIYOYLERSZUWKUHUIUWLUWIUCUDUEYLYNEUHTRUCUGZTU
            TZUDUGZTUTZUEUGZTUTVLZUWPUWRUHUIUWTUWPRSUWTUWRRSUHUIVIUWIUWPUWRUWTV
            MVNZUWIULTYJKAULTKVOUUSUVTUWHUURVPUWAUWHWGVQZUWIUUSUVTYNTUTAUUSUVTU
            WHVRZUVSUVTUWHVSZYMYGVFVGAUVPUUSUVTUWHUVQVPZUWQUWSUKUWPUWRRSZUWRUWP
            RSVTUWIUWPUWRWAVNZWBUWIUWOUWJUWKUHUVSUWHUWOUWJVTZUVTAUWHUXIUUSADEGH
            IJKYJLMNOPQWCWDWDZWEWHUWIUWKUWMUWJUHUWIUCUDUEYMYGETRUXDUXEUXFUXHUXA
            UXGUWTRSUWPUWRUWTRSRSVTUWIUWPUWRUWTWFVNZWIWJUWIUWNUWJVARSZUWMVARSZU
            HUIUWCUWIUCUDUEUWJUWMVAUHTRUXBUWIUWBTUTZVKTUTZUWJTUTZUWAULTYJJAULTJ
            VOUUSUVTNWKWLZWMUWBVKVFVJZUWIUVNUVTUWMTUTUWIUUSUVPUVNUXDUXFUVRVGZUX
            EUVBYGVFVGUVOUWIVHWNZUXHWBUWIUXLUWBUXMUVEUHUWIUXLUWBVKVARSZRSZUWBUW
            IUXNUXOUVOUXLUYBVTUXQUXOUWIWMWNZUXTUWBVKVAWFWQUWIUYBUWBUFRSZUWBUYAU
            FUWBRUYAVAVKRSZUFUXOUVOUYAUYEVTWMVHVKVAWAWOWPWRZWSUWIUXNUYDUWBVTUXQ
            UWBWTXPXAXBUWIUCUDUEUVBYGVATRUXSUXEUXTUXHUXKWIXGWHXCXDUWIYQUWEUWIYQ
            UVBUWJYGRSZUHUIZUVCUYGVARSZUHUIUWEUWIYQUVBYPERSZUHUIUYHUWIUCUDUEYMY
            PEUHTRUXBUXDUWIYLTUTUVTYPTUTUXCUXEYLYGVFVGUXFUXHWBUWIUYJUYGUVBUHUWI
            UYJUWOYGRSUYGUWIUCUDUEYLYGETRUXCUXEUXFUXHUXKWIUWIUWOUWJYGRUXJXEXBWJ
            WHUWIUCUDUEUVBUYGVAUHTRUXBUXSUWIUXPUVTUYGTUTUXRUXEUWJYGVFVGUXTUXHWB
            UWIUYIUWDUVCUHUWIUYIUWDUFRSZUWDUWIUYIUWDUYARSZUYKUWIUYIUWDVKRSZVARS
            ZUYLUWIUYGUYMVARUWIUCUDUEUWBVKYGTRUXQUYCUXEUXHUXKWIXEUWIUWDTUTZUXOU
            VOUYNUYLVTUWIUXNUVTUYOUXQUXEUWBYGVFVGZUYCUXTUWDVKVAWFWQXBUYAUFUWDRU
            YFWSXFUWIUYOUYKUWDVTUYPUWDWTXPXBWJXCXDXHXIXJUWGUVIUAGULYJUUDVTZYKUU
            EUWFUVHYJUUDYIUJXKUYQUWCUVFUWEUVGUYQUWBUUFUVEUHYJUUDJXLZWEUYQUWDUUJ
            UVCUHUYQUWBUUFYGRUYRXEWJXMXNXOXQXRXIXJXSUUPUVMCUVCTUUGUVCVTZUUOUVLB
            TUYSUUNUVKYHUYSUUMUVIFGULULUYSUULUVHUUEUYSUUIUVFUUKUVGUYSUUHUVEUUFU
            HUUGUVCYGRXTWJUUGUVCUUJUHYEXMYAYBYAYCYFVGYD $.
        $}
      $}

      ${
        $d A a b k n $.  $d A a j k l u $.  $d A a k m $.  $d A a j k x $.
        $d A j k x y $.  $d F a b k n $.  $d F a j k l u $.  $d F a k m $.
        $d F a j k x $.  $d F j k x y $.  $d b k n ph $.  $d j k ph x $.
        $d ph m $.  $d l n u $.  $d m n $.  $d b m $.
        $( Lemma for ~ caucvgsr .  A Cauchy sequence with a lower bound
           converges.  (Contributed by Jim Kingdon, 19-Jun-2021.) $)
        caucvgsrlembnd $p |- ( ph ->
            E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. (
              j <N k -> ( ( F ` k ) <R ( y +R x ) /\
                y <R ( ( F ` k ) +R x ) ) ) ) ) $=
          ( va vb cnpi c1r cplr co cv cfv cm1r cmr cmpt fveq2 caucvgsrlemoffres
          weq oveq1d cbvmptv ) ABCDEFGHIJOQOUAZJUBZRSTZEUCUDTZSTZUEPKLMNOPQUOPU
          AZJUBZRSTZUNSTOPUHZUMURUNSUSULUQRSUKUPJUFUIUIUJUG $.
      $}
    $}

    ${
      $d F f g h m $.  $d F j k l u $.  $d F k l n u $.  $d F j k x y $.
      $d f g h m ph $.  $d j k ph x $.  $d k m ph $.  $d n ph $.  $d m n $.
      $( A Cauchy sequence of signed reals with a modulus of convergence
         converges to a signed real.  This is basically Corollary 11.2.13 of
         [HoTT], p.  (varies).  The HoTT book theorem has a modulus of
         convergence (that is, a rate of convergence) specified by (11.2.9) in
         HoTT whereas this theorem fixes the rate of convergence to say that
         all terms after the nth term must be within ` 1 / n ` of the nth term
         (it should later be able to prove versions of this theorem with a
         different fixed rate or a modulus of convergence supplied as a
         hypothesis).

         This is similar to ~ caucvgprpr but is for signed reals rather than
         positive reals.

         Here is an outline of how we prove it:

         1.  Choose a lower bound for the sequence (see ~ caucvgsrlembnd ).

         2.  Offset each element of the sequence so that each element of the
         resulting sequence is greater than one (greater than zero would not
         suffice, because the limit as well as the elements of the sequence
         need to be positive) (see ~ caucvgsrlemofff ).

         3.  Since a signed real (element of ` R. ` ) which is greater than
         zero can be mapped to a positive real (element of ` P. ` ), perform
         that mapping on each element of the sequence and invoke ~ caucvgprpr
         to get a limit (see ~ caucvgsrlemgt1 ).

         4.  Map the resulting limit from positive reals back to signed reals
         (see ~ caucvgsrlemgt1 ).

         5.  Offset that limit so that we get the limit of the original
         sequence rather than the limit of the offsetted sequence (see
         ~ caucvgsrlemoffres ).  (Contributed by Jim Kingdon, 20-Jun-2021.) $)
      caucvgsr $p |- ( ph ->
          E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. ( j <N k ->
          ( ( F ` k ) <R ( y +R x ) /\
            y <R ( ( F ` k ) +R x ) ) ) ) ) $=
        ( c1o cplr co cltr wbr cnpi c1r c1p cnr vm vf vg vh cfv cm1r cv wcel wa
        clti wceq cop ceq cec crq cltq cab cpp wi wral breq1 fveq2 opeq1 eceq1d
        cer fveq2d breq2d abbidv breq1d opeq12d breq12d oveq12d anbi12d imbi12d
        oveq1d opeq1d oveq2d ralbidv 1pi rspcdva simpl imim2i breq2 rspcv mpan9
        a1i ralimi syl df-1nqqs fveq2i rec1nq eqtr3i breq2i abbii breq1i df-i1p
        c1q opeq12i eqtr4i oveq1i opeq1i eceq1 ax-mp oveq2i imbitrdi imp adantr
        df-1r wf ffvelcdmd ltadd1sr adantl breqtrd nlt1pig pm2.21d w3o pitri3or
        wn mpjao3dan w3a wb ltasrg ffvelcdmda 1sr addclsr sylancl m1r addcomsrg
        caovord2d mpbid addasssrg syl3anc mp2an m1p1sr eqtri 0idsr eqtrid eqtrd
        mpan c0r ralrimiva caucvgsrlembnd ) ABCDLHUEZUFMNZEFUAGHIJKAUUDUAUGZHUE
        ZOPUAQAUUEQUHZUIZUUDUUFRMNZUFMNZUUFOUUHUUCUUIOPZUUDUUJOPUUHLUUEUJPZUUKL
        UUEUKZUUELUJPZUUHUULUUKUUHUULUUCUUFIUGZLLULZUMUNZUOUEZUPPZIUQZUURDUGZUP
        PZDUQZULZSURNZSULZVEUNZMNZOPZUUKALFUGZUJPZUUCUVJHUEZUVGMNZOPZUSZFQUTZUU
        GUULUVIUSZAUVKUVNUVLUUCUVGMNZOPZUIZUSZFQUTZUVPAGUGZUVJUJPZUWCHUEZUVLUUO
        UWCLULZUMUNZUOUEZUPPZIUQZUWHUVAUPPZDUQZULZSURNZSULZVEUNZMNZOPZUVLUWEUWP
        MNZOPZUIZUSZFQUTUWBGQLUWCLUKZUXBUWAFQUXCUWDUVKUXAUVTUWCLUVJUJVAUXCUWRUV
        NUWTUVSUXCUWEUUCUWQUVMOUWCLHVBZUXCUWPUVGUVLMUXCUWOUVFVEUXCUWNUVESUXCUWM
        UVDSURUXCUWJUUTUWLUVCUXCUWIUUSIUXCUWHUURUUOUPUXCUWGUUQUOUXCUWFUUPUMUWCL
        LVCVDVFZVGVHUXCUWKUVBDUXCUWHUURUVAUPUXEVIVHVJVOVPVDZVQVKUXCUWSUVRUVLOUX
        CUWEUUCUWPUVGMUXDUXFVLVGVMVNVRKLQUHZAVSWFVTUWAUVOFQUVTUVNUVKUVNUVSWAWBW
        GWHUVOUVQFUUEQUVJUUEUKZUVKUULUVNUVIUVJUUELUJWCUXHUVMUVHUUCOUXHUVLUUFUVG
        MUVJUUEHVBVOVGVNWDWEUVHUUIUUCOUVGRUUFMUVGSSURNZSULZVEUNZRUVFUXJUKUVGUXK
        UKUVEUXISUVDSSURUVDUUOWQUPPZIUQZWQUVAUPPZDUQZULSUUTUXMUVCUXOUUSUXLIUURW
        QUUOUPWQUOUEUURWQWQUUQUOWIWJWKWLZWMWNUVBUXNDUURWQUVAUPUXPWOWNWRDIWPWSWT
        XAUVFUXJVEXBXCXHWSXDWMXEXFUUHUUMUIUUCUUCRMNZUUIOUUHUUCUXQOPZUUMUUHUUCTU
        HUXRUUHQTLHAQTHXIUUGJXGUXGUUHVSWFXJZUUCXKWHXGUUMUXQUUIUKUUHUUMUUCUUFRML
        UUEHVBVOXLXMUUHUUNUUKUUHUUNUUKUUGUUNXRAUUEXNXLXOXFUUGUULUUMUUNXPZAUXGUU
        GUXTVSLUUEXQYSXLXSUUHUBUCUDUUCUUIUFOTMUBUGZTUHZUCUGZTUHZUDUGZTUHXTUYAUY
        COPUYEUYAMNUYEUYCMNOPYAUUHUYAUYCUYEYBXLUXSUUHUUFTUHZRTUHZUUITUHAQTUUEHJ
        YCZYDUUFRYEYFUFTUHZUUHYGWFZUYBUYDUIUYAUYCMNUYCUYAMNUKUUHUYAUYCYHXLYIYJU
        UHUUJUUFRUFMNZMNZUUFUUHUYFUYGUYIUUJUYLUKUYHUYGUUHYDWFUYJUUFRUFYKYLUUHUY
        LUUFYTMNZUUFUYKYTUUFMUYKUFRMNZYTUYGUYIUYKUYNUKYDYGRUFYHYMYNYOXDUUHUYFUY
        MUUFUKUYHUUFYPWHYQYRXMUUAUUB $.
    $}
  $}

  ${
    $( Mapping of order from positive signed reals to positive reals.
       (Contributed by NM, 17-May-1996.)  (Revised by Mario Carneiro,
       15-Jun-2013.) $)
    ltpsrprg $p |- ( ( A e. P. /\ B e. P. /\ C e. R. )
        -> ( ( C +R [ <. A , 1P >. ] ~R ) <R
        ( C +R [ <. B , 1P >. ] ~R ) <-> A <P B ) ) $=
      ( cnp wcel cnr c1p cop cer cec cltr wbr cplr co cltp wb 1pr enrex sylancl
      cpp w3a simp1 df-nr ecopqsi simp2 simp3 ltasrg syl3anc wceq addcomprg a1i
      breq1d ltsrprg syl22anc ltaprg 3bitr4d bitr3d ) ADEZBDEZCFEZUAZAGHIJZBGHI
      JZKLZCVBMNCVCMNKLZABOLZVAVBFEZVCFEZUTVDVEPVAURGDEZVGURUSUTUBZQDAGIFRUCUDS
      VAUSVIVHURUSUTUEZQDBGIFRUCUDSURUSUTUFVBVCCUGUHVAAGTNZGBTNZOLZGATNZVMOLZVD
      VFVAVLVOVMOVAURVIVLVOUIVJQAGUJSULVAURVIUSVIVDVNPVJVIVAQUKZVKVQAGBGUMUNVAU
      RUSVIVFVPPVJVKVQABGUOUHUPUQ $.
  $}

  ${
    $( Mapping from positive signed reals to positive reals.  (Contributed by
       NM, 17-May-1996.)  (Revised by Mario Carneiro, 15-Jun-2013.) $)
    mappsrprg $p |- ( ( A e. P. /\ C e. R. )
        -> ( C +R -1R ) <R ( C +R [ <. A , 1P >. ] ~R ) ) $=
      ( cnp wcel cnr wa c1p cpp co wbr cm1r cplr cop cer cec cltr 1pr adantr wb
      a1i cltp addclpr mp2an ltaddpr mpan df-m1r breq1i ltsrprg syl22anc bitrid
      id m1r cxp cqs opelxpi enrex ecelqsi syl mpan2 eleqtrrdi ltasrg mp3an2ani
      df-nr simpr bitr3d mpbid ) ACDZBEDZFZGGHIZVJAHIUAJZBKLIBAGMZNOZLIPJZVGVKV
      HVJCDZVGVKGCDZVPVOQQGGUBUCZVJAUDUERVIKVMPJZVKVNVGVRVKSVHVRGVJMNOZVMPJZVGV
      KKVSVMPUFUGVGVPVOVGVPVTVKSVPVGQTZVOVGVQTVGUKWAGVJAGUHUIUJRKEDVGVMEDVHVHVR
      VNSULVGVMCCUMZNUNZEVGVPVMWCDZQVGVPFVLWBDWDAGCCUOWBVLNUPUQURUSVCUTVGVHVDKV
      MBVAVBVEVF $.
  $}

  ${
    $d x y z A $.  $d x y z C $.
    $( Equivalence for positive signed real.  (Contributed by NM, 17-May-1996.)
       (Revised by Mario Carneiro, 15-Jun-2013.) $)
    map2psrprg $p |- ( C e. R. -> ( ( C +R -1R ) <R A <->
        E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A ) ) $=
      ( cnr wcel cm1r cplr co cltr wbr c1p cer wceq cnp wa syl2anc c0r cpp 1pr
      wb vy vz cv cop cec wrex ltrelsr brel simprd anim2i simpr cmr m1r mulclsr
      a1i simpl addclsr ltasrg syl3anc pn0sr oveq1d adantr addasssrg 0r 3eqtr3d
      addcomsrg 0idsr adantl eqtrd breq2d bitrd mpan2 sylan df-nr breq2 rexbidv
      wi eqeq2 imbi12d cltp df-m1r breq1i mp3an12 addclpr mp2an ltsrprg mpanl12
      addassprg ltaprg 3bitr4d bitrid ltexpri biimtrdi enreceq mpanl2 addcomprg
      eqeq1d bitr4d ancoms rexbidva sylibrd ecoptocl oveq2 sylan9eqr ex reximdv
      syl syld sylbird sylc mappsrprg syl5ibcom rexlimdva impbid ) CDEZCFGHZBIJ
      ZCAUCZKUDLUEZGHZBMZANUFZXOXQYBXOXQOXOBDEZOZXQYBXQYCXOXQXPDEYCXPBDDIUGUHUI
      UJXOXQUKYDXQFCFULHZBGHZIJZYBYDYGXPCYFGHZIJZXQYDFDEZYFDEZXOYGYITYJYDUMUOZY
      DYEDEZYCYKYDXOYJYMXOYCUPZYLCFUNZPZXOYCUKZYEBUQZPYNFYFCURUSYDYHBXPIYDYHBQG
      HZBYDCYEGHZBGHZQBGHZYHYSXOUUAUUBMYCXOYTQBGCUTVAVBYDXOYMYCUUAYHMYNYPYQCYEB
      VCUSYDQDEZYCUUBYSMUUCYDVDUOYQQBVFPVEYCYSBMXOBVGVHVIZVJVKYDYGXSYFMZANUFZYB
      YDYKYGUUFVQZXOYMYCYKXOYJYMUMYOVLYRVMFUAUCZUBUCZUDLUEZIJZXSUUJMZANUFZVQUUG
      UAUBYFNNLDVNUUJYFMZUUKYGUUMUUFUUJYFFIVOUUNUULUUEANUUJYFXSVRVPVSUUHNEZUUIN
      EZOZUUKUUIXRRHZKUUHRHZMZANUFZUUMUUQUUKUUIUUSVTJZUVAUUKKKKRHZUDLUEZUUJIJZU
      UQUVBFUVDUUJIWAWBUUQKUUIRHZUVCUUHRHZVTJZUVFKUUSRHZVTJZUVEUVBUUOUVHUVJTUUP
      UUOUVGUVIUVFVTKNEZUVKUUOUVGUVIMSSKKUUHWHWCVJVBUVKUVCNEZUUQUVEUVHTSUVKUVKU
      VLSSKKWDWEKUVCUUHUUIWFWGUUQUUPUUSNEZUVKUVBUVJTUUOUUPUKZUUQUVKUUOUVMUVKUUQ
      SUOZUUOUUPUPKUUHWDPUVOUUIUUSKWIUSWJWKAUUIUUSWLWMUUQUULUUTANXRNEZUUQUULUUT
      TUVPUUQOZUULXRUUIRHZUUSMZUUTUVPUVKUUQUULUVSTSXRKUUHUUIWNWOUVQUURUVRUUSUVQ
      UUPUVPUURUVRMUUQUUPUVPUVNVHUVPUUQUPUUIXRWPPWQWRWSWTXAXBXGYDUUEYAANYDUUEYA
      UUEYDXTYHBXSYFCGXCUUDXDXEXFXHXIXJXEXOYAXQANUVPXOYAXQVQUVPXOOXPXTIJYAXQXRC
      XKXTBXPIVOXLWSXMXN $.
  $}

  ${
    suplocsrlem.b $e |- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A } $.
    suplocsrlem.ss $e |- ( ph -> A C_ R. ) $.
    suplocsrlem.c $e |- ( ph -> C e. A ) $.
    suplocsrlem.ub $e |- ( ph -> E. x e. R. A. y e. A y <R x ) $.
    suplocsrlem.loc $e |- ( ph -> A. x e. R. A. y e. R. ( x <R y ->
      ( E. z e. A x <R z \/ A. z e. A z <R y ) ) ) $.

    ${
      $d A q w $.  $d A x y z $.  $d B z $.  $d C q w $.  $d C x y z $.
      $d ph q u v z $.  $d u x y z $.  $d v y z $.
      $( Lemma for ~ suplocsr .  The set ` B ` is located.  (Contributed by Jim
         Kingdon, 18-Jan-2024.) $)
      suplocsrlemb $p |- ( ph -> A. u e. P. A. v e. P. ( u <P v ->
          ( E. q e. B u <P q \/ A. q e. B q <P v ) ) ) $=
        ( cnp wcel cltr cnr cv cltp wbr wrex wral wo wi wa c1p cop cer cec cplr
        co simpr wb simplrl simplrr sseldd ad2antrr ltpsrprg syl3anc wceq breq2
        mpbird ralbidv orbi2d imbi12d breq1 rexbidv cxp cqs 1pr opelxpd ecelqsi
        orbi1d a1i enrex df-nr eleqtrrdi addclsr syl2anc rspcdva cm1r mappsrprg
        syl mpd ltrelsr sotri map2psrprg mpbid simp-4r eqeltrd simpllr breqtrrd
        ltsosr ad4antr simplr jca ex reximdva opeq1 eceq1d oveq2d eleq1d elrab2
        anbi1i anass bitri rexbii2 sylibr sylib simprd simpld ralrimiva orim12d
        rexlimdva2 ralrimivva ) AGUAZFUAZUBUCZXSKUAZUBUCZKIUDZYBXTUBUCZKIUEZUFZ
        UGGFQQAXSQRZXTQRZUHZUHZYAYGYKYAUHZJXSUIUJZUKULZUMUNZDUAZSUCZDHUDZYPJXTU
        IUJZUKULZUMUNZSUCZDHUEZUFZYGYLYOUUASUCZUUDYLUUEYAYKYAUOYLYHYIJTRZUUEYAU
        PAYHYIYAUQZAYHYIYAURZAUUFYJYAAHTJMNUSUTZXSXTJVAVBVEYLYOCUAZSUCZYRYPUUJS
        UCZDHUEZUFZUGZUUEUUDUGCTUUAUUJUUAVCZUUKUUEUUNUUDUUJUUAYOSVDUUPUUMUUCYRU
        UPUULUUBDHUUJUUAYPSVDVFVGVHYLBUAZUUJSUCZUUQYPSUCZDHUDZUUMUFZUGZCTUEZUUO
        CTUEBTYOUUQYOVCZUVBUUOCTUVDUURUUKUVAUUNUUQYOUUJSVIUVDUUTYRUUMUVDUUSYQDH
        UUQYOYPSVIVJVPVHVFAUVCBTUEYJYAPUTYLUUFYNTRYOTRUUIYLYNQQVKZUKVLZTYLYMUVE
        RYNUVFRYLXSUIQQUUGUIQRYLVMVQZVNUVEYMUKVRVOWFVSVTJYNWAWBWCYLUUFYTTRZUUAT
        RUUIYLYSUVERZUVHYLXTUIQQUUHUVGVNUVIYTUVFTUVEYSUKVRVOVSVTWFJYTWAWBWCWGYL
        YRYDUUCYFYLYQYDDHYLYPHRZUHZYQUHZJYBUIUJZUKULZUMUNZHRZYCUHZKQUDZYDUVLUVO
        YPVCZKQUDZUVRUVLJWDUMUNZYPSUCZUVTUVLUWAYOSUCZYQUWBUVLYHUUFUWCYLYHUVJYQU
        UGUTYLUUFUVJYQUUIUTZXSJWEWBUVKYQUOUWAYOYPSTWPWHWIWBUVLUUFUWBUVTUPUWDKYP
        JWJWFWKUVLUVSUVQKQUVLYBQRZUHZUVSUVQUWFUVSUHZUVPYCUWGUVOYPHUWFUVSUOZYLUV
        JYQUWEUVSWLWMUWGYOUVOSUCZYCUWGYOYPUVOSUVKYQUWEUVSWNUWHWOUWGYHUWEUUFUWIY
        CUPYLYHUVJYQUWEUVSUUGWQUVLUWEUVSWRUVLUUFUWEUVSUWDUTXSYBJVAVBWKWSWTXAWGY
        CUVQKIQYBIRZYCUHUWEUVPUHZYCUHUWEUVQUHUWJUWKYCJEUAZUIUJZUKULZUMUNZHRUVPE
        YBQIUWLYBVCZUWOUVOHUWPUWNUVNJUMUWPUWMUVMUKUWLYBUIXBXCXDXELXFZXGUWEUVPYC
        XHXIXJXKXQYLUUCYFYLUUCUHZYEKIUWRUWJUHZUVOUUASUCZYEUWSUUBUWTDHUVOYPUVOUU
        ASVIYLUUCUWJWRUWSUWEUVPUWSUWJUWKUWRUWJUOUWQXLZXMWCUWSUWEYIUUFUWTYEUPUWS
        UWEUVPUXAXNYLYIUUCUWJUUHUTYLUUFUUCUWJUUIUTYBXTJVAVBWKXOWTXPWGWTXR $.
    $}

    ${
      $d A a w $.  $d A v w y $.  $d A u $.  $d A x z $.  $d B u v w $.
      $d B v w x $.  $d B z $.  $d C a w $.  $d C v w x y $.  $d C u z $.
      $d ph u v w x z $.  $d y z $.
      $( Lemma for ~ suplocsr .  The set ` B ` has a least upper bound.
         (Contributed by Jim Kingdon, 19-Jan-2024.) $)
      suplocsrlempr $p |- ( ph -> E. v e. P. ( A. w e. B -. v <P w /\
        A. w e. P. ( w <P v -> E. u e. B w <P u ) ) ) $=
        ( c1p wcel cnp cltr wbr va cv wex c0r cplr co cnr wceq sseldd 0idsr syl
        eqeltrd 1pr jctil cop cer cec opeq1 eceq1d eqtr4di oveq2d eleq1d elrab2
        wa df-0r sylibr elex2 wral wrex cltp breq1 rspccv mpan9 cm1r c1r 0lt1sr
        wb 0r 1sr ltasrg mp3an mpbi ax-mp m1p1sr 3brtr3i mp3an12i mpbii breqtrd
        ltsosr ltrelsr sotri sylan map2psrprg adantr mpbid syldan simpllr breq2
        m1r ralbidv adantl mpbird rabeq2i simprbi rspcdva ralrimiva ex reximdva
        mpd rexlimdvw crab elrabi cbvrabv eqtri eleq2s simplr ad2antrr ltpsrprg
        syl3anc ralbidva rexbidva suplocsrlemb suplocexpr ) AFEGIAPIQZFUBZIQFUC
        APRQZJUDUEUFZHQZVDYDAYHYFAYGJHAJUGQZYGJUHAHUGJLMUIZJUJUKZMULUMUNJEUBZPU
        OZUPUQZUEUFZHQZYHEPRIYLPUHZYOYGHYQYNUDJUEYQYNPPUOZUPUQUDYQYMYRUPYLPPURU
        SVEUTVAVBKVCVFFPIVGUKAYOJYEPUOUPUQUEUFZSTZEIVHZFRVIZYLYEVJTZEIVHZFRVIAC
        UBZBUBZSTZCHVHZBUGVIUUBNAUUHUUBBUGAUUHUUBAUUHVDZYSUUFUHZFRVIZUUBAUUHJUU
        FSTZUUKAJHQUUHUULMUUGUULCJHUUEJUUFSVKVLVMAUULVDJVNUEUFZUUFSTZUUKAUUMJST
        UULUUNAUUMYGJSAVNUDSTZUUMYGSTZVNUDUEUFZVNVOUEUFZVNUDSUDVOSTZUUQUURSTZVP
        UDUGQZVOUGQVNUGQZUUSUUTVQVRVSWSUDVOVNVTWAWBUVBUUQVNUHWSVNUJWCWDWEUVBUVA
        AYIUUOUUPVQWSVRYJVNUDJVTWFWGYKWHUUMJUUFSUGWIWJWKWLAUUNUUKVQZUULAYIUVCYJ
        FUUFJWMUKWNWOWPUUIUUJUUAFRUUIYERQZVDZUUJUUAUVEUUJVDZYTEIUVFYLIQZVDUUEYS
        STZYTCHYOUUEYOYSSVKUVFUVHCHVHZUVGUVFUVIUUHAUUHUVDUUJWQUUJUVIUUHVQUVEUUJ
        UVHUUGCHYSUUFUUESWRWTXAXBWNUVGYPUVFUVGYLRQZYPYPEIRKXCXDXAXEXFXGXHXIXGXJ
        XIAUUAUUDFRAUVDVDZYTUUCEIUVKUVGVDUVJUVDYIYTUUCVQUVGUVJUVKUVJYLJUAUBZPUO
        ZUPUQZUEUFZHQZUARXKZIUVPUAYLRXLIYPERXKUVQKYPUVPEUARYLUVLUHZYOUVOHUVRYNU
        VNJUEUVRYMUVMUPYLUVLPURUSVAVBXMXNXOXAAUVDUVGXPAYIUVDUVGYJXQYLYEJXRXSXTY
        AWOABCDEEFHIJGKLMNOYBYC $.
    $}

    $d A b u v w x y z $.  $d B a v w y $.  $d B b u v w y $.  $d B z $.
    $d B v w x y $.  $d C b u v w y z $.  $d C v w x y z $.  $d b ph u v w y $.
    $d ph v w x y $.  $d ph z $.

    $( Lemma for ~ suplocsr .  The set ` A ` has a least upper bound.
       (Contributed by Jim Kingdon, 16-Jan-2024.) $)
    suplocsrlem $p |- ( ph -> E. x e. R. ( A. y e. A -. x <R y /\
        A. y e. R. ( y <R x -> E. z e. A y <R z ) ) ) $=
      ( cv wbr cnp wa cltr cnr wcel vv vu va vb cltp wn wral wrex suplocsrlempr
      wi c1p cop cer cplr co cm1r simpr sseldd adantr mappsrprg syl2anc ltrelsr
      cec brel syl simprd weq breq2 notbid cbvralv wceq wfal ltsosr sotri sylan
      wb map2psrprg mpbid adantlr simplr simprr breqtrrd ad4antr simprl syl3anc
      ltpsrprg bicom 3imtr3i simp-4r simpllr eqeltrd rabeq2i sylanbrc pm2.21fal
      equcom rspcdva rexlimddv inegd ralrimiva ex biimtrid nfan simp-6r eqbrtrd
      nfv nfra1 simp-7r ad2antrr syl3c cbvrexv sylib opeq1 eceq1d oveq2d eleq1d
      rsp elrab2 simpld mpbird eqbrtrrd rspcev r19.29af ad5antr sylancom ltm1sr
      ad3antrrr wo m1r addclsr sylancl wor w3a sowlin mpan mpd mpjaodan anim12d
      breq1 ralbidv imbi1d anbi12d syl6an rexlimdva ) AUANZENZUEOZUFZEGUGZUUEUU
      DUEOZUUEUBNZUEOZUBGUHZUJZEPUGZQZUAPUHBNZCNZROZUFZCFUGZUUQUUPROZUUQDNZROZD
      FUHZUJZCSUGZQZBSUHZABCDEUAUBFGHIJKLMUIAUUOUVHUAPAUUDPTZQZHUUDUKULUMVCUNUO
      ZSTZUUOUVKUUQROZUFZCFUGZUUQUVKROZUVDUJZCSUGZQZUVHUVJHUPUNUOZSTZUVLUVJUVTU
      VKROZUWAUVLQUVJUVIHSTZUWBAUVIUQZAUWCUVIAFSHJKURZUSZUUDHUTVAZUVTUVKSSRVBVD
      VEVFUVJUUHUVOUUNUVRUUHUUDUCNZUEOZUFZUCGUGZUVJUVOUUGUWJEUCGEUCVGZUUFUWIUUE
      UWHUUDUEVHVIZVJUVJUWKUVOUVJUWKQZUVNCFUWNUUQFTZQZUVMUWPUVMQZHUUEUKULZUMVCZ
      UNUOZUUQVKZVLEPUWNUVMUXAEPUHZUWOUVJUVMUXBUWKUVJUVMQZUVTUUQROZUXBUVJUWBUVM
      UXDUWGUVTUVKUUQRSVMVBVNVOUXCUWCUXDUXBVPZUVJUWCUVMUWFUSEUUQHVQZVEVRVSVSUWQ
      UUEPTZUXAQZQZUUFUXIUVKUWTROZUUFUXIUVKUUQUWTRUWPUVMUXHVTUWQUXGUXAWAZWBUXIU
      VIUXGUWCUXJUUFVPUVJUVIUWKUWOUVMUXHUWDWCUWQUXGUXAWDZUVJUWCUWKUWOUVMUXHUWFW
      CUUDUUEHWFWEVRUXIUWJUUGUCGUUEUWLUUGUWJVPUCEVGUWJUUGVPUWMEUCWOUUGUWJWGWHUV
      JUWKUWOUVMUXHWIUXIUXGUWTFTZUUEGTUXLUXIUWTUUQFUXKUWNUWOUVMUXHWJWKUXMEGPIWL
      WMWPWNWQWRWSWTXAUVJUUNUVRUVJUUNQZUVQCSUXNUUQSTZQZUVPUVDUXPUVPQZUXDUVDUUQH
      ROZUXQUXDQZUXAUVDEPUXQUXDEUXPUVPEUXNUXOEUVJUUNEUVJEXEUUMEPXFXBUXOEXEXBUVP
      EXEXBUXDEXEXBUXSUXGQZUXAQZUUEUDNZUEOZUVDUDGUYAUULUYCUDGUHUYAUUNUXGUUIUULU
      VJUUNUXOUVPUXDUXGUXAXCUXSUXGUXAVTZUYAUWTUVKROZUUIUYAUWTUUQUVKRUXTUXAUQUXP
      UVPUXDUXGUXAWIXDUYAUXGUVIUWCUYEUUIVPUYDAUVIUUNUXOUVPUXDUXGUXAXGUXSUWCUXGU
      XAUVJUWCUUNUXOUVPUXDUWFWCZXHUUEUUDHWFWEVRUUMEPXPXIUUKUYCUBUDGUUJUYBUUEUEV
      HXJXKUYAUYBGTZUYCQZQZHUYBUKULZUMVCZUNUOZFTZUUQUYLROZUVDUYIUYBPTZUYMUYIUYG
      UYOUYMQUYAUYGUYCWDUXMUYMEUYBPGEUDVGZUWTUYLFUYPUWSUYKHUNUYPUWRUYJUMUUEUYBU
      KXLXMXNXOIXQXKZVFUYIUWTUUQUYLRUXTUXAUYHVTUYIUWTUYLROZUYCUYAUYGUYCWAUYIUXG
      UYOUWCUYRUYCVPUXSUXGUXAUYHWJUYIUYOUYMUYQXRUXSUWCUXGUXAUYHUYFYFUUEUYBHWFWE
      XSXTUVCUYNDUYLFUVBUYLUUQRVHYAVAWQUXSUXDUXBUXQUXDUQUXSUWCUXEUYFUXFVEVRYBUX
      QUXRHFTZUVDAUYSUVIUUNUXOUVPUXRKYCUVCUXRDHFUVBHUUQRVHYAYDUXQUVTHROZUXDUXRY
      GZAUYTUVIUUNUXOUVPAUWCUYTUWEHYEVEWCUXQUWAUWCUXOUYTVUAUJZUXQUWCUPSTUWAUVJU
      WCUUNUXOUVPUWFYFZYHHUPYIYJVUCUXNUXOUVPVTSRYKUWAUWCUXOYLVUBVMSUVTHUUQRYMYN
      WEYOYPWTWSWTYQUVGUVSBUVKSUUPUVKVKZUUTUVOUVFUVRVUDUUSUVNCFVUDUURUVMUUPUVKU
      UQRYRVIYSVUDUVEUVQCSVUDUVAUVPUVDUUPUVKUUQRVHYTYSUUAYAUUBUUCYO $.
  $}

  ${
    $d A a b c x y z $.  $d a c ph x y z $.
    suplocsr.m $e |- ( ph -> E. x x e. A ) $.
    suplocsr.ub $e |- ( ph -> E. x e. R. A. y e. A y <R x ) $.
    suplocsr.loc $e |- ( ph -> A. x e. R. A. y e. R. ( x <R y ->
      ( E. z e. A x <R z \/ A. z e. A z <R y ) ) ) $.
    $( An inhabited, bounded, located set of signed reals has a supremum.
       (Contributed by Jim Kingdon, 22-Jan-2024.) $)
    suplocsr $p |- ( ph ->
       E. x e. R. ( A. y e. A -. x <R y /\
            A. y e. R. ( y <R x -> E. z e. A y <R z ) ) ) $=
      ( va vc vb cv wcel cltr wbr wral wrex cnr c1p cer wn wi wex eleq1w cbvexv
      wa sylib cop cec cplr cnp crab weq opeq1 eceq1d oveq2d eleq1d cbvrabv wss
      ltrelsr simpld ralimi dfss3 sylibr rexlimivw syl adantr simpr suplocsrlem
      co brel wo exlimddv ) AILZEMZBLZCLZNOZUACEPVQVPNOZVQDLZNODEQUBCRPUFBRQIAV
      PEMZBUCVOIUCFWAVOBIBIEUDUEUGAVOUFBCDJEVNKLZSUHZTUIZUJVJZEMZKUKULVNWFVNJLZ
      SUHZTUIZUJVJZEMKJUKKJUMZWEWJEWKWDWIVNUJWKWCWHTWBWGSUNUOUPUQURAERUSZVOAVSC
      EPZBRQZWLGWMWLBRWMVQRMZCEPWLVSWOCEVSWOVPRMVQVPRRNUTVKVAVBCERVCVDVEVFVGAVO
      VHAWNVOGVGAVRVPVTNODEQVTVQNODEPVLUBCRPBRPVOHVGVIVM $.
  $}

  $( Class of complex numbers. $)
  cc $a class CC $.
  $( Class of real numbers. $)
  cr $a class RR $.
  $( Extend class notation to include the complex number 0. $)
  cc0 $a class 0 $.
  $( Extend class notation to include the complex number 1. $)
  c1 $a class 1 $.
  $( Extend class notation to include the complex number i. $)
  ci $a class _i $.
  $( Addition on complex numbers. $)
  caddc $a class + $.
  $( 'Less than' predicate (defined over real subset of complex numbers). $)
  cltrr $a class <RR $.
  $( Multiplication on complex numbers.  The token ` x. ` is a center dot. $)
  cmul $a class x. $.

  $( Define the set of complex numbers.  (Contributed by NM, 22-Feb-1996.) $)
  df-c $a |- CC = ( R. X. R. ) $.

  $( Define the complex number 0.  (Contributed by NM, 22-Feb-1996.) $)
  df-0 $a |- 0 = <. 0R , 0R >. $.

  $( Define the complex number 1.  (Contributed by NM, 22-Feb-1996.) $)
  df-1 $a |- 1 = <. 1R , 0R >. $.

  $( Define the complex number ` _i ` (the imaginary unit).  (Contributed by
     NM, 22-Feb-1996.) $)
  df-i $a |- _i = <. 0R , 1R >. $.

  $( Define the set of real numbers.  (Contributed by NM, 22-Feb-1996.) $)
  df-r $a |- RR = ( R. X. { 0R } ) $.

  ${
    $d x y z w v u f $.
    $( Define addition over complex numbers.  (Contributed by NM,
       28-May-1995.) $)
    df-add $a |- + = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\
                  E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. )
                  /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) } $.

    $( Define multiplication over complex numbers.  (Contributed by NM,
       9-Aug-1995.) $)
    df-mul $a |- x. = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\
               E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. )
                  /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) ,
                  ( ( v .R u ) +R ( w .R f ) ) >. ) ) } $.

    $( Define 'less than' on the real subset of complex numbers.  (Contributed
       by NM, 22-Feb-1996.) $)
    df-lt $a |- <RR = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\
     E. z E. w ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) /\ z <R w ) ) } $.
  $}

  $( Ordered pair membership in the class of complex numbers.  (Contributed by
     NM, 14-May-1996.) $)
  opelcn $p |- ( <. A , B >. e. CC <-> ( A e. R. /\ B e. R. ) ) $=
    ( cop cc wcel cnr cxp wa df-c eleq2i opelxp bitri ) ABCZDEMFFGZEAFEBFEHDNMI
    JABFFKL $.

  $( Ordered pair membership in class of real subset of complex numbers.
     (Contributed by NM, 22-Feb-1996.) $)
  opelreal $p |- ( <. A , 0R >. e. RR <-> A e. R. ) $=
    ( c0r cop cr wcel cnr wceq eqid csn cxp wa df-r eleq2i opelxp 0r elexi elsn
    anbi2i 3bitri mpbiran2 ) ABCZDEZAFEZBBGZBHUBUAFBIZJZEUCBUEEZKUCUDKDUFUALMAB
    FUENUGUDUCBBBFOPQRST $.

  ${
    $d x y A $.
    $( Membership in class of real numbers.  (Contributed by NM,
       31-Mar-1996.) $)
    elreal $p |- ( A e. RR <-> E. x e. R. <. x , 0R >. = A ) $=
      ( vy cr wcel cnr c0r csn cxp cv cop wceq wrex eleq2i elxp2 0r elexi opeq2
      df-r bitri eqeq2d rexsn eqcom rexbii ) BDEBFGHZIZEZAJZGKZBLZAFMZDUFBSNUGB
      UHCJZKZLZCUEMZAFMUKACBFUEOUOUJAFUOBUILZUJUNUPCGGFPQULGLUMUIBULGUHRUAUBBUI
      UCTUDTT $.
  $}

  ${
    $d A x y $.
    $( The real number mapping in ~ elreal is unique.  (Contributed by Jim
       Kingdon, 11-Jul-2021.) $)
    elrealeu $p |- ( A e. RR <-> E! x e. R. <. x , 0R >. = A ) $=
      ( vy cr wcel cv c0r cop wceq wreu wrex wrmo elreal biimpi wa wi ralrimiva
      cnr wral sylibr eqtr3 wb opthg mpan2 ad2antlr imbitrid simpl opeq1 eqeq1d
      0r syl6 rmo4 reu5 sylanbrc reurex impbii ) BDEZAFZGHZBIZARJZUQUTARKZUTARL
      ZVAUQVBABMZNUQUTCFZGHZBIZOZURVEIZPZCRSZARSVCUQVKARUQURREZOZVJCRVMVEREZOZV
      HVIGGIZOZVIVHUSVFIZVOVQUSVFBUAVLVRVQUBZUQVNVLGREVSUJURGVEGRRUCUDUEUFVIVPU
      GUKQQUTVGACRVIUSVFBURVEGUHUIULTUTARUMUNVAVBUQUTARUOVDTUP $.
  $}

  $( Ordered pair membership in the class of complex numbers.  (Contributed by
     Mario Carneiro, 15-Jun-2013.) $)
  elreal2 $p |- ( A e. RR <->
      ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) ) $=
    ( cr wcel cnr c0r csn cxp c1st cfv cop wceq wa df-r eleq2i xp1st c2nd xp2nd
    1st2nd2 elsni syl opeq2d eqtrd eleq1 0r elexi snid opelxp mpbiran2 biimparc
    jca bitrdi impbii bitri ) ABCADEFZGZCZAHIZDCZAUQEJZKZLZBUOAMNUPVAUPURUTADUN
    OUPAUQAPIZJUSADUNRUPVBEUQUPVBUNCVBEKADUNQVBESTUAUBUJUTUPURUTUPUSUOCZURAUSUO
    UCVCUREUNCEEDUDUEUFUQEDUNUGUHUKUIULUM $.

  $( The empty set is not a complex number.  Note: do not use this after the
     real number axioms are developed, since it is a construction-dependent
     property.  See also ~ cnm which is a related property.  (Contributed by
     NM, 2-May-1996.) $)
  0ncn $p |- -. (/) e. CC $=
    ( c0 cc wcel cnr cxp 0nelxp df-c eleq2i mtbir ) ABCADDEZCDDFBJAGHI $.

  ${
    $d A u v x $.
    $( A complex number is an inhabited set.  Note: do not use this after the
       real number axioms are developed, since it is a construction-dependent
       property.  (Contributed by Jim Kingdon, 23-Oct-2023.)
       (New usage is discouraged.) $)
    cnm $p |- ( A e. CC -> E. x x e. A ) $=
      ( vu vv cc wcel cv cop wceq cnr wa wex cxp elxpi df-c eleq2s cvv mpbir2an
      vex opm simprl eleq2d exbidv mpbiri ex exlimdvv mpd ) BEFZBCGZDGZHZIZUIJF
      UJJFKZKZDLCLZAGZBFZALZUOBJJMECDBJJNOPUHUNURCDUHUNURUHUNKZURUPUKFZALZVAUIQ
      FUJQFCSDSAUIUJTRUSUQUTAUSBUKUPUHULUMUAUBUCUDUEUFUG $.
  $}

  ${
    $d x y z w $.
    $( 'Less than' is a relation on real numbers.  (Contributed by NM,
       22-Feb-1996.) $)
    ltrelre $p |- <RR C_ ( RR X. RR ) $=
      ( vx vy vz vw cltrr cv cr wcel wa c0r cop wceq wbr wex copab cxp opabssxp
      cltr df-lt eqsstri ) EAFZGHBFZGHIUACFZJKLUBDFZJKLIUCUDRMIDNCNZIABOGGPABCD
      SUEABGGQT $.
  $}

  ${
    $d x y z w v u f A $.  $d x y z w v u f B $.  $d x y z w v u f C $.
    $d x y z w v u f D $.
    $( Addition of complex numbers in terms of signed reals.  (Contributed by
       NM, 28-May-1995.) $)
    addcnsr $p |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) ->
           ( <. A , B >. + <. C , D >. ) = <. ( A +R C ) , ( B +R D ) >. ) $=
      ( vx vy vz vw vv vu vf cv cplr co cop cnr wcel wa wceq cc addclsr opelxpi
      cxp ad2ant2r ad2ant2l syl2anc simpll simprl oveq12d simplr simprr opeq12d
      caddc wex coprab df-add df-c eleq2i anbi12i anbi1i oprabbii eqtri ovi3 )
      EFGHIJABCDHLZJLZMNZILZKLZMNZOZACMNZBDMNZOZKUMPAPQZBPQZRCPQZDPQZRRVKPQZVLP
      QZVMPPUCZQVNVPVRVOVQACUAUDVOVQVSVNVPBDUAUEVKVLPPUBUFVDASZVGBSZRZVECSZVHDS
      ZRZRZVFVKVIVLWGVDAVECMWAWBWFUGWCWDWEUHUIWGVGBVHDMWAWBWFUJWCWDWEUKUIULUMEL
      ZTQZFLZTQZRZWHVDVGOSWJVEVHOSRGLVJSRKUNJUNIUNHUNZRZEFGUOWHVTQZWJVTQZRZWMRZ
      EFGUOEFGHIJKUPWNWREFGWLWQWMWIWOWKWPTVTWHUQURTVTWJUQURUSUTVAVBVC $.

    $( Multiplication of complex numbers in terms of signed reals.
       (Contributed by NM, 9-Aug-1995.) $)
    mulcnsr $p |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) ->
               ( <. A , B >. x. <. C , D >. ) =
                   <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) ,
                          ( ( B .R C ) +R ( A .R D ) ) >. ) $=
      ( vx vy vz vw vv cv cmr co cm1r cplr cnr wcel wa mulclsr wceq oveq12d cop
      vu cmul cxp ad2ant2r ad2ant2l sylancr addclsr syl2anc ad2ant2lr ad2ant2rl
      vf m1r opelxpi simpll simprl simplr simprr oveq2d opeq12d wex coprab df-c
      cc df-mul eleq2i anbi12i anbi1i oprabbii eqtri ovi3 ) EFGHIUBABCDHJZUBJZK
      LZMIJZULJZKLZKLZNLZVOVMKLZVLVPKLZNLZUAZACKLZMBDKLZKLZNLZBCKLZADKLZNLZUAZU
      LUCOAOPZBOPZQCOPZDOPZQQZWGOPZWJOPZWKOOUDZPWPWDOPZWFOPZWQWLWNWTWMWOACRUEWP
      MOPWEOPZXAUMWMWOXBWLWNBDRUFMWERUGWDWFUHUIWPWHOPZWIOPZWRWMWNXCWLWOBCRUJWLW
      OXDWMWNADRUKWHWIUHUIWGWJOOUNUIVLASZVOBSZQZVMCSZVPDSZQZQZVSWGWBWJXKVNWDVRW
      FNXKVLAVMCKXEXFXJUOZXGXHXIUPZTXKVQWEMKXKVOBVPDKXEXFXJUQZXGXHXIURZTUSTXKVT
      WHWAWINXKVOBVMCKXNXMTXKVLAVPDKXLXOTTUTUCEJZVDPZFJZVDPZQZXPVLVOUASXRVMVPUA
      SQGJWCSQULVAUBVAIVAHVAZQZEFGVBXPWSPZXRWSPZQZYAQZEFGVBEFGHIUBULVEYBYFEFGXT
      YEYAXQYCXSYDVDWSXPVCVFVDWSXRVCVFVGVHVIVJVK $.
  $}

  ${
    eqresr.1 $e |- A e. _V $.
    $( Equality of real numbers in terms of intermediate signed reals.
       (Contributed by NM, 10-May-1996.) $)
    eqresr $p |- ( <. A , 0R >. = <. B , 0R >. <-> A = B ) $=
      ( c0r cop wceq eqid cnr 0r elexi opth mpbiran2 ) ADEBDEFABFDDFDGADBDCDHIJ
      KL $.
  $}

  $( Addition of real numbers in terms of intermediate signed reals.
     (Contributed by NM, 10-May-1996.) $)
  addresr $p |- ( ( A e. R. /\ B e. R. ) ->
           ( <. A , 0R >. + <. B , 0R >. ) = <. ( A +R B ) , 0R >. ) $=
    ( cnr wcel wa c0r cop caddc co cplr wceq 0r addcnsr an4s 0idsr ax-mp opeq2i
    mpanr12 eqtrdi ) ACDZBCDZEZAFGBFGHIZABJIZFFJIZGZUDFGUBFCDZUGUCUFKZLLTUGUAUG
    UHAFBFMNRUEFUDUGUEFKLFOPQS $.

  $( Multiplication of real numbers in terms of intermediate signed reals.
     (Contributed by NM, 10-May-1996.) $)
  mulresr $p |- ( ( A e. R. /\ B e. R. ) ->
         ( <. A , 0R >. x. <. B , 0R >. ) = <. ( A .R B ) , 0R >. ) $=
    ( cnr wcel wa c0r cop cmul co cmr cm1r cplr wceq mulcnsr ax-mp oveq2i 0idsr
    0r 00sr eqtrd an4s mpanr12 m1r mulclsr syl eqtrid mulcomsrg mpan oveqan12rd
    eqtri eqtrdi opeq12d ) ACDZBCDZEZAFGBFGHIZABJIZKFFJIZJIZLIZFBJIZAFJIZLIZGZU
    QFGUOFCDZVEUPVDMZRRUMVEUNVEVFAFBFNUAUBUOUTUQVCFUOUTUQFLIZUQUSFUQLUSKFJIZFUR
    FKJVEURFMRFSOPKCDVHFMUCKSOUJPUOUQCDVGUQMABUDUQQUEUFUOVCFFLIZFUNUMVAFVBFLUNV
    ABFJIZFVEUNVAVJMRFBUGUHBSTASUIVEVIFMRFQOUKULT $.

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Ordering of real subset of complex numbers in terms of signed reals.
       (Contributed by NM, 22-Feb-1996.) $)
    ltresr $p |- ( <. A , 0R >. <RR <. B , 0R >. <-> A <R B ) $=
      ( vz vw vx vy c0r cop cltrr wbr cnr wcel wa cltr cr wceq wex anbi1d eqcom
      cv ltrelre brel opelreal anbi12i sylib ltrelsr eleq1 eqeq1 2exbidv anbi2d
      anbi12d df-lt brabg bianabs vex eqresr 3bitr4i opth2 bitr4i anbi1i 2exbii
      wb bitrdi syl2anbr breq12 copsex2g bitrd pm5.21nii ) AGHZBGHZIJZAKLZBKLZM
      ZABNJZVKVIOLZVJOLZMZVNVIVJOOIUAUBVPVLVQVMAUCZBUCZUDUEABKKNUFUBVNVKABHCTZD
      TZHPZWAWBNJZMZDQCQZVOVLVPVQVKWFVBVMVSVTVRVKVIWAGHZPZVJWBGHZPZMZWDMZDQCQZW
      FVRVKWMETZOLZFTZOLZMZWNWGPZWPWIPZMZWDMZDQCQZMVPWQMZWHWTMZWDMZDQCQZMVRWMME
      FVIVJOOIWNVIPZWRXDXCXGXHWOVPWQWNVIOUGRXHXBXFCDXHXAXEWDXHWSWHWTWNVIWGUHRRU
      IUKWPVJPZXDVRXGWMXIWQVQVPWPVJOUGUJXIXFWLCDXIXEWKWDXIWTWJWHWPVJWIUHUJRUIUK
      EFCDULUMUNWLWECDWKWCWDWKAWAPZBWBPZMWCWHXJWJXKWGVIPWAAPWHXJWAACUOZUPVIWGSA
      WASUQWIVJPWBBPWJXKWBBDUOZUPVJWISBWBSUQUDABWAWBXLXMURUSUTVAVCVDWDVOCDABKKW
      AAWBBNVEVFVGVH $.
  $}

  $( Ordering of real subset of complex numbers in terms of signed reals.
     (Contributed by NM, 22-Feb-1996.) $)
  ltresr2 $p |- ( ( A e. RR /\ B e. RR ) ->
      ( A <RR B <-> ( 1st ` A ) <R ( 1st ` B ) ) ) $=
    ( cr wcel wa cltrr wbr c1st cfv c0r cop cltr wceq elreal2 simprbi breqan12d
    cnr ltresr bitrdi ) ACDZBCDZEABFGAHIZJKZBHIZJKZFGUBUDLGTUAAUCBUEFTUBQDAUCMA
    NOUAUDQDBUEMBNOPUBUDRS $.

  $( Technical trick to permit reuse of previous lemmas to prove arithmetic
     operation laws in ` CC ` from those in ` R. ` .  The trick involves
     ~ qsid , which shows that the coset of the converse epsilon relation
     (which is not an equivalence relation) acts as an identity divisor for the
     quotient set operation.  This lets us "pretend" that ` CC ` is a quotient
     set, even though it is not (compare ~ df-c ), and allows us to reuse some
     of the equivalence class lemmas we developed for the transition from
     positive reals to signed reals, etc.  (Contributed by NM, 13-Aug-1995.) $)
  dfcnqs $p |- CC = ( ( R. X. R. ) /. `' _E ) $=
    ( cc cnr cxp cep ccnv cqs df-c qsid eqtr4i ) ABBCZJDEFGJHI $.

  $( Technical trick to permit re-use of some equivalence class lemmas for
     operation laws.  See ~ dfcnqs and ~ mulcnsrec .  (Contributed by NM,
     13-Aug-1995.) $)
  addcnsrec $p |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) ->
                 ( [ <. A , B >. ] `' _E + [ <. C , D >. ] `' _E ) =
                   [ <. ( A +R C ) , ( B +R D ) >. ] `' _E ) $=
    ( cnr wcel wa cop caddc co cplr cep ccnv cec addcnsr wceq opelxpi ecidg syl
    addclsr cxp oveqan12d ad2ant2r ad2ant2l syl2anc 3eqtr4d ) AEFZBEFZGZCEFZDEF
    ZGZGZABHZCDHZIJACKJZBDKJZHZUNLMZNZUOUSNZIJURUSNZABCDOUIULUTUNVAUOIUIUNEEUAZ
    FUTUNPABEEQUNVCRSULUOVCFVAUOPCDEEQUOVCRSUBUMURVCFZVBURPUMUPEFZUQEFZVDUGUJVE
    UHUKACTUCUHUKVFUGUJBDTUDUPUQEEQUEURVCRSUF $.

  $( Technical trick to permit re-use of some equivalence class lemmas for
     operation laws.  The trick involves ~ ecidg , which shows that the coset
     of the converse epsilon relation (which is not an equivalence relation)
     leaves a set unchanged.  See also ~ dfcnqs .  (Contributed by NM,
     13-Aug-1995.) $)
  mulcnsrec $p |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) ->
               ( [ <. A , B >. ] `' _E x. [ <. C , D >. ] `' _E ) =
                    [ <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) ,
                          ( ( B .R C ) +R ( A .R D ) ) >. ] `' _E ) $=
    ( cnr wcel wa cop cmul cmr cm1r cplr cec wceq opelxpi ecidg mulclsr syl2anc
    co syl cep ccnv mulcnsr cxp simpll simprl m1r simplr simprr sylancr addclsr
    oveqan12d 3eqtr4d ) AEFZBEFZGZCEFZDEFZGZGZABHZCDHZISACJSZKBDJSZJSZLSZBCJSZA
    DJSZLSZHZVAUAUBZMZVBVKMZISVJVKMZABCDUCUPUSVLVAVMVBIUPVAEEUDZFVLVANABEEOVAVO
    PTUSVBVOFVMVBNCDEEOVBVOPTULUTVJVOFZVNVJNUTVFEFZVIEFZVPUTVCEFZVEEFZVQUTUNUQV
    SUNUOUSUEZUPUQURUFZACQRUTKEFVDEFZVTUGUTUOURWCUNUOUSUHZUPUQURUIZBDQRKVDQUJVC
    VEUKRUTVGEFZVHEFZVRUTUOUQWFWDWBBCQRUTUNURWGWAWEADQRVGVHUKRVFVIEEORVJVOPTUM
    $.

  ${
    $d f u v w x y z $.
    $( Existence of a sum.  This is dependent on how we define ` + ` so once we
       proceed to real number axioms we will replace it with theorems such as
       ~ addcl .  (Contributed by Jim Kingdon, 14-Jul-2021.) $)
    addvalex $p |- ( ( A e. V /\ B e. W ) -> ( A + B ) e. _V ) $=
      ( vx vy vz vw vv vu vf wcel wa caddc cop cvv cv cnr cnp cc cfv df-ov cplr
      co cxp cer cqs df-nr npex xpex qsex eqeltri wceq wex coprab df-add eleq2i
      df-c anbi12i anbi1i oprabbii eqtri oprabex3 opexg fvexg sylancr eqeltrid
      ) ACLBDLMZABNUDABOZNUAZPABNUBVHNPLVIPLVJPLEFGHIJHQZJQZUCUDIQZKQZUCUDOZKNR
      RSSUEZUFUGPUHVPUFSSUIUIUJUKULNEQZTLZFQZTLZMZVQVKVMOUMVSVLVNOUMMGQVOUMMKUN
      JUNIUNHUNZMZEFGUOVQRRUEZLZVSWDLZMZWBMZEFGUOEFGHIJKUPWCWHEFGWAWGWBVRWEVTWF
      TWDVQURUQTWDVSURUQUSUTVAVBVCABCDVDVINPPVEVFVG $.
  $}

  ${
    $d K l u $.
    $( Lemma for ~ pitonn .  Two ways to write the number one.  (Contributed by
       Jim Kingdon, 24-Apr-2020.) $)
    pitonnlem1 $p |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } ,
          { u | [ <. 1o , 1o >. ] ~Q <Q u } >.
          +P. 1P ) , 1P >. ] ~R
          , 0R >. = 1 $=
      ( c1r c0r cop cv c1o cec cltq wbr cab c1p cpp cer wceq c1q df-1nqqs abbii
      co eqtri c1 ceq df-1 df-1r df-i1p breq2i breq1i oveq1i opeq1i eceq1 ax-mp
      opeq12i eqtr2i ) UACDEBFZGGEUBHZIJZBKZUOAFZIJZAKZEZLMSZLEZNHZDEUCCVDDCLLM
      SZLEZNHZVDUDVFVCOVGVDOVEVBLLVALMLUNPIJZBKZPURIJZAKZEVAABUEVIUQVKUTVHUPBPU
      OUNIQUFRVJUSAPUOURIQUGRULTUHUIVFVCNUJUKTUIUM $.

    $( Lemma for ~ pitonn .  Simplifying an expression involving signed reals.
       (Contributed by Jim Kingdon, 26-Apr-2020.) $)
    pitonnlem1p1 $p |- ( A e. P. ->
        [ <. ( A +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R =
        [ <. ( A +P. 1P ) , 1P >. ] ~R ) $=
      ( cnp wcel c1p cpp co cop cer cec wceq 1pr addclpr mp2an addcomprg oveq1d
      mpan2 addassprg mp3an13 eqtrd a1i wb enreceq syl22anc mpbird ) ABCZADDEFZ
      EFZUFGHIADEFZDGHIJZUGDEFZUFUHEFZJZUEUJUFAEFZDEFZUKUEUGUMDEUEUFBCZUGUMJDBC
      ZUPUOKKDDLMZAUFNPOUOUEUPUNUKJUQKUFADQRSUEUGBCZUOUHBCZUPUIULUAUEUOURUQAUFL
      PUOUEUQTUEUPUSKADLPUPUEKTUGUFUHDUBUCUD $.

    $( Lemma for ~ pitonn .  Two ways to add one to a number.  (Contributed by
       Jim Kingdon, 24-Apr-2020.) $)
    pitonnlem2 $p |- ( K e. N. ->
        ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } ,
          { u | [ <. K , 1o >. ] ~Q <Q u } >.
          +P. 1P ) , 1P >. ] ~R
          , 0R >. + 1 ) =
        <. [ <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } ,
          { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >.
          +P. 1P ) , 1P >. ] ~R
          , 0R >. ) $=
      ( wcel c1o cop cec cltq wbr cab c1p cpp co cer c0r c1r wceq cnp 1pr syl
      cnpi cv ceq c1 caddc cplr cpli df-1 oveq2i cnr cxp nnprlu addclpr sylancl
      cqs opelxpi enrex ecelqsi df-nr eleqtrrdi 1sr addresr eqtrid pitonnlem1p1
      df-1r mp2an addsrpr mpanl2 mpanr12 c1q cplq addpinq1 breq2d abbidv breq1d
      wa opeq12d cnq nnnq addnqpr1 eqtrd oveq1d opeq1d eceq1d 3eqtr4d ) BUADZCU
      BZBEFUCGZHICJWHAUBZHIAJFZKLMZKFZNGZOFZUDUEMZWMPUFMZOFZWGBEUGMEFUCGZHIZCJZ
      WRWIHIZAJZFZKLMZKFZNGZOFWFWOWNPOFZUEMZWQUDXGWNUEUHUIWFWMUJDPUJDXHWQQWFWMR
      RUKZNUOZUJWFWLXIDZWMXJDWFWKRDZKRDZXKWFWJRDXMXLABCULSWJKUMUNZSWKKRRUPUNXIW
      LNUQURTUSUTVAWMPVBUNVCWFWPXFOWFWKKKLMZLMXOFNGZWKKLMZKFZNGZWPXFWFXLXPXSQXN
      WKVDTWFWPWMXOKFNGZUFMZXPPXTWMUFVEUIWFXLYAXPQZXNXLXORDZXMYBXMXMYCSSKKUMVFS
      XLXMYCXMVPYBSWKKXOKVGVHVITVCWFXEXRNWFXDXQKWFXCWKKLWFXCWGWHVJVKMZHIZCJZYDW
      IHIZAJZFZWKWFWTYFXBYHWFWSYECWFWRYDWGHBVLZVMVNWFXAYGAWFWRYDWIHYJVOVNVQWFWH
      VRDYIWKQBVSAWHCVTTWAWBWCWDWEWCWA $.
  $}

  ${
    $d N l u w z $.  $d k l u w y z $.  $d x y z $.
    $( Mapping from ` N. ` to ` NN ` .  (Contributed by Jim Kingdon,
       22-Apr-2020.) $)
    pitonn $p |- ( N e. N. ->
        <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
          { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e.
        |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } ) $=
      ( wcel c1o cop ceq cec cltq wbr cab c1p cpp co cer c0r eceq1d abbidv cnpi
      vz vw vk cv c1 caddc wral wa cint wi wal cpli opeq1 breq2d breq1d opeq12d
      oveq1d opeq1d eleq1d imbi2d pitonnlem1 eleq1i biimpri adantr oveq1 rspccv
      wceq ad2antll wb pitonnlem2 sylibd indpi alrimiv eleq2 raleqbi1dv anbi12d
      a2d ralab sylibr cnp cxp cqs cnr nnprlu 1pr addclpr sylancl opelxpi enrex
      ex ecelqsi syl 0r elintg mpbird ) DUAFZEUEZDGHZIJZKLZEMZWTCUEZKLZCMZHZNOP
      ZNHZQJZRHZUFAUEZFZBUEZUFUGPZXKFZBXKUHZUIZAMZUJFZXJUBUEZFZUBXRUHZWQUFXTFZX
      NXTFZBXTUHZUIZYAUKZUBULYBWQYGUBYFWRUCUEZGHZIJZKLZEMZYJXCKLZCMZHZNOPZNHZQJ
      ZRHZXTFZUKYFWRGGHZIJZKLZEMZUUBXCKLZCMZHZNOPZNHZQJZRHZXTFZUKYFWRUDUEZGHZIJ
      ZKLZEMZUUOXCKLZCMZHZNOPZNHZQJZRHZXTFZUKYFWRUUMGUMPZGHZIJZKLZEMZUVHXCKLZCM
      ZHZNOPZNHZQJZRHZXTFZUKYGUCUDDYHGVHZYTUULYFUVSYSUUKXTUVSYRUUJRUVSYQUUIQUVS
      YPUUHNUVSYOUUGNOUVSYLUUDYNUUFUVSYKUUCEUVSYJUUBWRKUVSYIUUAIYHGGUNSZUOTUVSY
      MUUECUVSYJUUBXCKUVTUPTUQURUSSUSUTVAYHUUMVHZYTUVEYFUWAYSUVDXTUWAYRUVCRUWAY
      QUVBQUWAYPUVANUWAYOUUTNOUWAYLUUQYNUUSUWAYKUUPEUWAYJUUOWRKUWAYIUUNIYHUUMGU
      NSZUOTUWAYMUURCUWAYJUUOXCKUWBUPTUQURUSSUSUTVAYHUVFVHZYTUVRYFUWCYSUVQXTUWC
      YRUVPRUWCYQUVOQUWCYPUVNNUWCYOUVMNOUWCYLUVJYNUVLUWCYKUVIEUWCYJUVHWRKUWCYIU
      VGIYHUVFGUNSZUOTUWCYMUVKCUWCYJUVHXCKUWDUPTUQURUSSUSUTVAYHDVHZYTYAYFUWEYSX
      JXTUWEYRXIRUWEYQXHQUWEYPXGNUWEYOXFNOUWEYLXBYNXEUWEYKXAEUWEYJWTWRKUWEYIWSI
      YHDGUNSZUOTUWEYMXDCUWEYJWTXCKUWFUPTUQURUSSUSUTVAYCUULYEUULYCUUKUFXTCEVBVC
      VDVEUUMUAFZYFUVEUVRUWGYFUVEUVRUKUWGYFUIUVEUVDUFUGPZXTFZUVRYEUVEUWIUKUWGYC
      YDUWIBUVDXTXMUVDVHXNUWHXTXMUVDUFUGVFUTVGVIUWGUWIUVRVJYFUWGUWHUVQXTCUUMEVK
      UTVEVLWKVRVMVNXQYFYAUBAXKXTVHXLYCXPYEXKXTUFVOXOYDBXKXTXKXTXNVOVPVQVSVTWQX
      JWAWAWBZQWCZWDWBZFZXSYBVJWQXIUWKFZRWDFUWMWQXHUWJFZUWNWQXGWAFZNWAFZUWOWQXF
      WAFUWQUWPCDEWEWFXFNWGWHWFXGNWAWAWIWHUWJXHQWJWLWMWNXIRUWKWDWIWHUBXJXRUWLWO
      WMWP $.
  $}

  ${
    $d N l u $.
    $( Embedding from ` N. ` to ` RR ` yields a number greater than zero.
       (Contributed by Jim Kingdon, 15-Jul-2021.) $)
    pitoregt0 $p |- ( N e. N. ->
        0 <RR <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
          { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
        ) $=
      ( wcel c1p cop cer cec cv cltq wbr cab cpp co cltr c0r cltrr cltp cnp 1pr
      cnpi c1o ceq cc0 addclpr mp2an ltaddpr sylancr wceq a1i addassprg syl3anc
      nnprlu addcomprg oveq2d eqtrd breqtrd sylancl ltsrprg mpanl12 mpbird df-0
      wb wa breq1i ltresr df-0r 3bitri sylibr ) BUADZEEFGHZCIBUBFUCHZJKCLVLAIJK
      ALFZEMNZEFGHZOKZUDVOPFZQKZVJVPEEMNZEVNMNZRKZVJVSVSVMMNZVTRVJVSSDZVMSDZVSW
      BRKESDZWEWCTTEEUEUFABCUMZVSVMUGUHVJWBEEVMMNZMNZVTVJWEWEWDWBWHUIWEVJTUJZWI
      WFEEVMUKULVJWGVNEMVJWEWDWGVNUITWFEVMUNUHUOUPUQVJVNSDZWEVPWAVCZVJWDWEWJWFT
      VMEUEURTWEWEWJWEVDWKTTEEVNEUSUTURVAVRPPFZVQQKPVOOKVPUDWLVQQVBVEPVOVFPVKVO
      OVGVEVHVI $.
  $}

  ${
    $d N l u $.
    $( Embedding from ` N. ` to ` RR ` .  Similar to ~ pitonn but separate in
       the sense that we have not proved ~ nnssre yet.  (Contributed by Jim
       Kingdon, 15-Jul-2021.) $)
    pitore $p |- ( N e. N. ->
        <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
          { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e.
        RR ) $=
      ( cnpi wcel c1o cop ceq cec cltq wbr cab c1p cpp cer cnr cnp 1pr sylancl
      cv co c0r cr cxp cqs nnprlu addclpr opelxpi enrex ecelqsi df-nr eleqtrrdi
      syl opelreal sylibr ) BDEZCTBFGHIZJKCLUQATJKALGZMNUAZMGZOIZPEVAUBGUCEUPVA
      QQUDZOUEZPUPUTVBEZVAVCEUPUSQEZMQEZVDUPURQEVFVEABCUFRURMUGSRUSMQQUHSVBUTOU
      IUJUMUKULVAUNUO $.
  $}

  ${
    $d N l u $.
    $( Embedding the reciprocal of a natural number into ` RR ` .  (Contributed
       by Jim Kingdon, 15-Jul-2021.) $)
    recnnre $p |- ( N e. N. ->
        <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } ,
          { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ,
        0R >. e. RR ) $=
      ( cnpi wcel c1o cop ceq cec crq cltq wbr cab c1p cer cnr cnp 1pr sylancl
      cv cfv cpp co c0r cxp cqs recnnpr addclpr opelxpi enrex ecelqsi syl df-nr
      cr eleqtrrdi opelreal sylibr ) BDEZCTBFGHIJUAZKLCMUSATKLAMGZNUBUCZNGZOIZP
      EVCUDGUNEURVCQQUEZOUFZPURVBVDEZVCVEEURVAQEZNQEZVFURUTQEVHVGABCUGRUTNUHSRV
      ANQQUISVDVBOUJUKULUMUOVCUPUQ $.
  $}

  ${
    $d x y $.
    peano1nnnn.n $e |- N =
      |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.

    ${
      $d x y z $.
      $( One is an element of ` NN ` .  This is a counterpart to ~ 1nn designed
         for real number axioms which involve natural numbers (notably,
         ~ axcaucvg ).  (Contributed by Jim Kingdon, 14-Jul-2021.)
         (New usage is discouraged.) $)
      peano1nnnn $p |- 1 e. N $=
        ( vz c1 wcel cv caddc co wral wa cab cint eleq2i cr wb c1r c0r eleq2
        cop df-1 cnr 1sr opelreal mpbir eqeltri elintg ax-mp vex weq raleqbi1dv
        bitri anbi12d elab simplbi mprgbir ) FCGZFEHZGZEFAHZGZBHFIJZVAGZBVAKZLZ
        AMZURFVGNZGZUTEVGKZCVHFDOFPGVIVJQFRSUAZPUBVKPGRUCGUDRUEUFUGEFVGPUHUIUMU
        SVGGUTVCUSGZBUSKZVFUTVMLAUSEUJAEUKVBUTVEVMVAUSFTVDVLBVAUSVAUSVCTULUNUOU
        PUQ $.
    $}

    ${
      $d A y z $.  $d x y z $.
      $( A successor of a positive integer is a positive integer.  This is a
         counterpart to ~ peano2nn designed for real number axioms which
         involve to natural numbers (notably, ~ axcaucvg ).  (Contributed by
         Jim Kingdon, 14-Jul-2021.)  (New usage is discouraged.) $)
      peano2nnnn $p |- ( A e. N -> ( A + 1 ) e. N ) $=
        ( vz wcel c1 caddc co cv wral wa eleq2i elintg bitrid cvv c1r c0r cnr
        cab cint ibi vex wceq eleq2 raleqbi1dv anbi12d elab oveq1 eleq1d rspcva
        simprbi sylan2 expcom ralimia syl wb cop df-1 0r opexg eqeltri addvalex
        1sr mp2an mpan2 mpbird ) CDGZCHIJZDGZVJFKZGZFHAKZGZBKZHIJZVNGZBVNLZMZAU
        AZLZVICVLGZFWALZWBVIWDVICWAUBZGVIWDDWECENFCWADOPUCWCVMFWAWCVLWAGZVMWFWC
        VQVLGZBVLLZVMWFHVLGZWHVTWIWHMAVLFUDVNVLUEVOWIVSWHVNVLHUFVRWGBVNVLVNVLVQ
        UFUGUHUIUMWGVMBCVLVPCUEVQVJVLVPCHIUJUKULUNUOUPUQVIVJQGZVKWBURVIHQGWJHRS
        USZQUTRTGSTGWKQGVEVARSTTVBVFVCCHDQVDVGVKVJWEGWJWBDWEVJENFVJWAQOPUQVH $.
    $}
  $}

  ${
    $d J l $.  $d J u $.  $d K l $.  $d K u $.

    $( Ordering of natural numbers with ` <N ` or ` <RR ` .  (Contributed by
       Jim Kingdon, 13-Jul-2021.) $)
    ltrennb $p |- ( ( J e. N. /\ K e. N. ) ->
        ( J <N K <-> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } ,
          { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
        <RR <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } ,
          { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
        ) ) $=
      ( cnpi wcel wbr cv c1o cop ceq cec cltq cab c1p cpp co cer c0r cnq ltnnnq
      wa clti cltr cltrr cltp wb adantr adantl ltnqpr syl2anc cnp nqprlu prsrlt
      nnnq syl 3bitrd ltresr bitr4di ) BEFZCEFZUBZBCUCGZDHZBIJKLZMGDNVEAHZMGANJ
      ZOPQOJRLZVDCIJKLZMGDNVIVFMGANJZOPQOJRLZUDGZVHSJVKSJUEGVBVCVEVIMGZVGVJUFGZ
      VLBCUAVBVETFZVITFZVMVNUGUTVOVABUOUHZVAVPUTCUOUIZAVEVIDUJUKVBVGULFZVJULFZV
      NVLUGVBVOVSVQAVEDUMUPVBVPVTVRAVIDUMUPVGVJUNUKUQVHVKURUS $.
  $}

  ${
    $d J l $.  $d J u $.  $d K l $.  $d K u $.

    $( Ordering of natural numbers with ` <N ` or ` <RR ` .  (Contributed by
       Jim Kingdon, 12-Jul-2021.) $)
    ltrenn $p |- ( J <N K -> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } ,
          { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
        <RR <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } ,
          { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
        ) $=
      ( cnpi wcel clti wbr cv c1o cop ceq cec cltq cab c1p cpp co cer c0r cltrr
      wa ltrelpi brel ltrennb biimpd mpcom ) BEFCEFUBZBCGHZDIZBJKLMZNHDOUKAIZNH
      AOKPQRPKSMTKUJCJKLMZNHDOUMULNHAOKPQRPKSMTKUAHZBCEEGUCUDUHUIUNABCDUEUFUG
      $.
  $}

  ${
    $d N l u $.
    $( Another way of saying that a number times its reciprocal is one.
       (Contributed by Jim Kingdon, 17-Jul-2021.) $)
    recidpipr $p |- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
          { u | [ <. N , 1o >. ] ~Q <Q u } >. .P.
        <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } ,
          { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. ) = 1P ) $=
      ( cnpi wcel cv c1o cop ceq cec cltq wbr cab c1p cnq wceq syl c1q abbidv
      co crq cfv cmq nnnq recclnq mulnqpr syl2anc recidnq breq2d breq1d opeq12d
      cmp df-i1p eqtr4di eqtr3d ) BDEZCFZBGHIJZURUAUBZUCTZKLZCMZUTAFZKLZAMZHZUQ
      URKLCMURVCKLAMHUQUSKLCMUSVCKLAMHULTZNUPUROEZUSOEZVFVGPBUDZUPVHVIVJURUEQAU
      RUSCUFUGUPVFUQRKLZCMZRVCKLZAMZHNUPVBVLVEVNUPVAVKCUPUTRUQKUPVHUTRPVJURUHQZ
      UISUPVDVMAUPUTRVCKVOUJSUKACUMUNUO $.
  $}

  ${
    recidpirqlemcalc.a $e |- ( ph -> A e. P. ) $.
    recidpirqlemcalc.b $e |- ( ph -> B e. P. ) $.
    recidpirqlemcalc.rec $e |- ( ph -> ( A .P. B ) = 1P ) $.
    $( Lemma for ~ recidpirq .  Rearranging some of the expressions.
       (Contributed by Jim Kingdon, 17-Jul-2021.) $)
    recidpirqlemcalc $p |- ( ph ->
        ( ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) +P. 1P ) =
        ( ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) +P.
          ( 1P +P. 1P ) ) ) $=
      ( c1p cpp co cmp cnp wcel wceq addclpr syl2anc syl3anc 1idpr eqtrd oveq1d
      syl 1pr addassprg distrprg oveq2d mulcomprg oveq12d 3eqtrd mp1i addcomprg
      a1i 3eqtr4d ) ABGHIZCGHIZHIZGHIZGHIZUNGGHIZHIZULUMJIZGGJIZHIZGHIULGJIZGUM
      JIZHIZUQHIAUNKLZGKLZVFUPURMAULKLZUMKLZVEABKLZVFVGDVFAUAUJZBGNOZACKLZVFVHE
      VJCGNOZULUMNOVJVJUNGGUBPAVAUOGHAVAGCHIZULHIZGHIUOAUSVOUTGHAUSULCJIZVBHIZV
      PULHIVOAVGVLVFUSVQMVKEVJULCGUCPAVBULVPHAVGVBULMVKULQTZUDAVPVNULHAVPCULJIZ
      CBJIZCGJIZHIZVNAVGVLVPVSMVKEULCUEOAVLVIVFVSWBMEDVJCBGUCPAVTGWACHAVTBCJIZG
      AVLVIVTWCMEDCBUEOFRAVLWACMECQTUFUGSUGVFUTGMAUAGQUHUFAVOUNGHAVOUMULHIZUNAV
      NUMULHAVFVLVNUMMVJEGCUIOSAVHVGWDUNMVMVKUMULUIORSRSAVDUNUQHAVBULVCUMHVRAVC
      UMGJIZUMAVFVHVCWEMVJVMGUMUEOAVHWEUMMVMUMQTRUFSUK $.
  $}

  ${
    $d N l u $.
    $( A real number times its reciprocal is one, where reciprocal is expressed
       with ` *Q ` .  (Contributed by Jim Kingdon, 15-Jul-2021.) $)
    recidpirq $p |- ( N e. N. -> (
        <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
          { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
        x. <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } ,
          { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >.
        ] ~R , 0R >. ) = 1 ) $=
      ( wcel cop cec cltq wbr cab c1p cpp cer c0r wceq cnp syl2anc c1r addclpr
      co cmp cnpi cv c1o ceq crq cfv cmul cmr cnr nnprlu prsrcl recnnpr mulresr
      c1 syl 1pr a1i mulsrpr syl22anc recidpipr recidpirqlemcalc eqeq2i mulclpr
      df-1r wb wa pm3.2i mp1i enreceq bitrid mpbird eqtrd opeq1d df-1 eqtr4di )
      BUADZCUBZBUCEUDFZGHCIVRAUBZGHAIEZJKSZJELFZMEVQVRUEUFZGHCIWCVSGHAIEZJKSZJE
      LFZMEUGSZWBWFUHSZMEZUNVPWBUIDZWFUIDZWGWINVPVTODZWJABCUJZVTUKUOVPWDODZWKAB
      CULZWDUKUOWBWFUMPVPWIQMEUNVPWHQMVPWHWAWETSZJJTSZKSZWAJTSZJWETSZKSZELFZQVP
      WAODZJODZWEODZXDWHXBNVPWLXDXCWMXDVPUPUQZVTJRPZXFVPWNXDXEWOXFWDJRPZXFWAJWE
      JURUSVPXBQNZWRJKSXAJJKSZKSNZVPVTWDWMWOABCUTVAXIXBXJJELFZNZVPXKQXLXBVDVBVP
      WRODZXAODZXJODZXDXMXKVEVPWPODZWQODZXNVPXCXEXQXGXHWAWEVCPXDXDVFZXRVPXDXDUP
      UPVGZJJVCVHWPWQRPVPWSODZWTODZXOVPXCXDYAXGXFWAJVCPVPXDXEYBXFXHJWEVCPWSWTRP
      XSXPVPXTJJRVHXFWRXAXJJVIUSVJVKVLVMVNVOVL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Final derivation of real and complex number postulates
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( The complex numbers form a set.  Use ~ cnex instead.  (Contributed by
     Mario Carneiro, 17-Nov-2014.)  (New usage is discouraged.) $)
  axcnex $p |- CC e. _V $=
    ( cc cnr cxp cvv df-c cnp cer cqs df-nr cpw npex xpex pwex wss wtru wer a1i
    enrer qsss eqeltri mptru ssexi ) ABBCDEBBBFFCZGHZDIUDUCJZUCFFKKLMUDUENOUCGU
    CGPORQSUAUBTZUFLT $.

  $( The real numbers are a subset of the complex numbers.  Axiom for real and
     complex numbers, derived from set theory.  This construction-dependent
     theorem should not be referenced directly; instead, use ~ ax-resscn .
     (Contributed by NM, 1-Mar-1995.)  (Proof shortened by Andrew Salmon,
     12-Aug-2011.)  (New usage is discouraged.) $)
  axresscn $p |- RR C_ CC $=
    ( cnr c0r csn cxp cr cc wcel wss 0r snssi xpss2 mp2b df-r df-c 3sstr4i ) AB
    CZDZAADZEFBAGPAHQRHIBAJPAAKLMNO $.

  $( 1 is a complex number.  Axiom for real and complex numbers, derived from
     set theory.  This construction-dependent theorem should not be referenced
     directly; instead, use ~ ax-1cn .  (Contributed by NM, 12-Apr-2007.)
     (New usage is discouraged.) $)
  ax1cn $p |- 1 e. CC $=
    ( cr cc c1 axresscn c1r c0r cop df-1 wcel cnr opelreal mpbir eqeltri sselii
    1sr ) ABCDCEFGZAHPAIEJIOEKLMN $.

  $( 1 is a real number.  Axiom for real and complex numbers, derived from set
     theory.  This construction-dependent theorem should not be referenced
     directly; instead, use ~ ax-1re .

     In the Metamath Proof Explorer, this is not a complex number axiom but is
     proved from ~ ax-1cn and the other axioms.  It is not known whether we can
     do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020)
     uses excluded middle.  (Contributed by Jim Kingdon, 13-Jan-2020.)
     (New usage is discouraged.) $)
  ax1re $p |- 1 e. RR $=
    ( c1 c1r c0r cop cr df-1 wcel cnr 1sr opelreal mpbir eqeltri ) ABCDZEFMEGBH
    GIBJKL $.

  $( ` _i ` is a complex number.  Axiom for real and complex numbers, derived
     from set theory.  This construction-dependent theorem should not be
     referenced directly; instead, use ~ ax-icn .  (Contributed by NM,
     23-Feb-1996.)  (New usage is discouraged.) $)
  axicn $p |- _i e. CC $=
    ( ci cc wcel c0r cnr c1r 0r 1sr cop wa df-i eleq1i opelcn bitri mpbir2an )
    ABCZDECZFECZGHPDFIZBCQRJASBKLDFMNO $.

  ${
    $d A w x y z $.  $d B w x y z $.
    $( Closure law for addition of complex numbers.  Axiom for real and complex
       numbers, derived from set theory.  This construction-dependent theorem
       should not be referenced directly, nor should the proven axiom
       ~ ax-addcl be used later.  Instead, in most cases use ~ addcl .
       (Contributed by NM, 14-Jun-1995.)  (New usage is discouraged.) $)
    axaddcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) $=
      ( vx vy vz vw cc wcel wa cv cop wceq cnr wex caddc co elxpi df-c ad2ant2l
      eleq2s cxp anim12i ee4anv sylibr cplr simpll simprl oveq12d addcnsr eqtrd
      addclsr ad2ant2r opelxpi syl2anc eleqtrrdi eqeltrd exlimivv syl ) AGHZBGH
      ZIZACJZDJZKZLZVBMHZVCMHZIZIZBEJZFJZKZLZVJMHZVKMHZIZIZIZFNENZDNCNZABOPZGHZ
      VAVIDNCNZVQFNENZIVTUSWCUTWDWCAMMUAZGCDAMMQRTWDBWEGEFBMMQRTUBVIVQCDEFUCUDV
      SWBCDVRWBEFVRWAVBVJUEPZVCVKUEPZKZGVRWAVDVLOPZWHVRAVDBVLOVEVHVQUFVIVMVPUGU
      HVHVPWIWHLVEVMVBVCVJVKUISUJVHVPWHGHVEVMVHVPIZWHWEGWJWFMHZWGMHZWHWEHVFVNWK
      VGVOVBVJUKULVGVOWLVFVNVCVKUKSWFWGMMUMUNRUOSUPUQUQUR $.

    $( Closure law for addition in the real subfield of complex numbers.  Axiom
       for real and complex numbers, derived from set theory.  This
       construction-dependent theorem should not be referenced directly, nor
       should the proven axiom ~ ax-addrcl be used later.  Instead, in most
       cases use ~ readdcl .  (Contributed by NM, 31-Mar-1996.)
       (New usage is discouraged.) $)
    axaddrcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) $=
      ( vx vy cv c0r cop caddc co cr wcel cnr elreal wceq oveq1 eleq1d oveq2 wa
      cplr addresr addclsr opelreal sylibr eqeltrd 2gencl ) CEZFGZDEZFGZHIZJKAU
      IHIZJKABHIZJKCDUGUIABLJCAMDBMUGANUJUKJUGAUIHOPUIBNUKULJUIBAHQPUFLKUHLKRZU
      JUFUHSIZFGZJUFUHTUMUNLKUOJKUFUHUAUNUBUCUDUE $.

    $( Closure law for multiplication of complex numbers.  Axiom for real and
       complex numbers, derived from set theory.  This construction-dependent
       theorem should not be referenced directly, nor should the proven axiom
       ~ ax-mulcl be used later.  Instead, in most cases use ~ mulcl .
       (Contributed by NM, 10-Aug-1995.)  (New usage is discouraged.) $)
    axmulcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC ) $=
      ( vx vy vz vw cc wcel wa cv cop wceq cnr wex cmul co df-c mulclsr syl2anc
      cmr elxpi eleq2s anim12i ee4anv sylibr cm1r simpll simprl oveq12d mulcnsr
      cxp cplr ad2ant2l simplrl simprrl m1r a1i simplrr simprrr addclsr opelxpi
      eqtrd eleqtrrdi eqeltrd exlimivv syl ) AGHZBGHZIZACJZDJZKZLZVJMHZVKMHZIZI
      ZBEJZFJZKZLZVRMHZVSMHZIZIZIZFNENZDNCNZABOPZGHZVIVQDNCNZWEFNENZIWHVGWKVHWL
      WKAMMUKZGCDAMMUAQUBWLBWMGEFBMMUAQUBUCVQWECDEFUDUEWGWJCDWFWJEFWFWIVJVRTPZU
      FVKVSTPZTPZULPZVKVRTPZVJVSTPZULPZKZGWFWIVLVTOPZXAWFAVLBVTOVMVPWEUGVQWAWDU
      HUIVPWDXBXALVMWAVJVKVRVSUJUMVBWFXAWMGWFWQMHZWTMHZXAWMHWFWNMHZWPMHZXCWFVNW
      BXEVMVNVOWEUNZVQWAWBWCUOZVJVRRSWFUFMHZWOMHZXFXIWFUPUQWFVOWCXJVMVNVOWEURZV
      QWAWBWCUSZVKVSRSUFWORSWNWPUTSWFWRMHZWSMHZXDWFVOWBXMXKXHVKVRRSWFVNWCXNXGXL
      VJVSRSWRWSUTSWQWTMMVASQVCVDVEVEVF $.

    $( Closure law for multiplication in the real subfield of complex numbers.
       Axiom for real and complex numbers, derived from set theory.  This
       construction-dependent theorem should not be referenced directly, nor
       should the proven axiom ~ ax-mulrcl be used later.  Instead, in most
       cases use ~ remulcl .  (New usage is discouraged.)  (Contributed by NM,
       31-Mar-1996.) $)
    axmulrcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) $=
      ( vx vy cv c0r cop cmul co cr wcel cnr elreal wceq oveq1 eleq1d oveq2 cmr
      wa mulresr mulclsr opelreal sylibr eqeltrd 2gencl ) CEZFGZDEZFGZHIZJKAUIH
      IZJKABHIZJKCDUGUIABLJCAMDBMUGANUJUKJUGAUIHOPUIBNUKULJUIBAHQPUFLKUHLKSZUJU
      FUHRIZFGZJUFUHTUMUNLKUOJKUFUHUAUNUBUCUDUE $.
  $}

  ${
    $d a b x y z w v u f $.
    $( Addition is an operation on the complex numbers.  This theorem can be
       used as an alternate axiom for complex numbers in place of the less
       specific ~ axaddcl .  This construction-dependent theorem should not be
       referenced directly; instead, use ~ ax-addf .  (Contributed by NM,
       8-Feb-2005.)  (New usage is discouraged.) $)
    axaddf $p |- + : ( CC X. CC ) --> CC $=
      ( vx vy vw vv vu vf vz vb va cc caddc cv co wcel wral wceq cop wex wtru
      wa cxp wf wfn wfun cdm cplr coprab wmo mosubop anass 2exbii 19.42vv bitri
      moeq mobii mpbir moani funoprab df-add funeqi dmeqi dmoprabss eqsstri wss
      cnm adantl axaddcl wrel funrel oprssdmm mptru eqssi df-fn mpbir2an rgen2a
      mp1i ffnov ) JJUAZJKUBKVRUCZALZBLZKMJNZBJOAJOVSKUDZKUEZVRPWCVTJNWAJNTZVTC
      LZDLZQPZWAELZFLZQPZTGLWFWIUFMWGWJUFMQZPZTZFRERZDRCRZTZABGUGZUDWQABGWPWEGW
      PGUHWHWKWMTZFRERZTZDRCRZGUHWTGCDVTWMGEFWAGWLUNUIUIWPXBGWOXACDWOWHWSTZFRER
      XAWNXCEFWHWKWMUJUKWHWSEFULUMUKUOUPUQURKWRABGCDEFUSZUTUPZWDVRWDWRUEVRKWRXD
      VAWPABGJJVBVCVRWDVDSABHIJKILZJNHLXFNHRSHXFVEVFWEWBSVTWAVGZVFWCKVHSXEKVIVP
      VJVKVLKVRVMVNWBABJXGVOABJJJKVQVN $.

    $( Multiplication is an operation on the complex numbers.  This is the
       construction-dependent version of ~ ax-mulf and it should not be
       referenced outside the construction.  We generally prefer to develop our
       theory using the less specific ~ mulcl .  (Contributed by NM,
       8-Feb-2005.)  (New usage is discouraged.) $)
    axmulf $p |- x. : ( CC X. CC ) --> CC $=
      ( vx vy vw vv vu vf vz vb va cc cmul cv co wcel wceq wa cop cmr wex wtru
      cxp wf wfn wral wfun cdm cm1r cplr coprab wmo moeq mosubop 2exbii 19.42vv
      anass bitri mobii mpbir moani funoprab df-mul dmeqi dmoprabss eqsstri wss
      funeqi cnm adantl axmulcl wrel funrel oprssdmm mptru eqssi df-fn mpbir2an
      mp1i rgen2a ffnov ) JJUAZJKUBKVTUCZALZBLZKMJNZBJUDAJUDWAKUEZKUFZVTOWEWBJN
      WCJNPZWBCLZDLZQOZWCELZFLZQOZPGLWHWKRMUGWIWLRMRMUHMWIWKRMWHWLRMUHMQZOZPZFS
      ESZDSCSZPZABGUIZUEWSABGWRWGGWRGUJWJWMWOPZFSESZPZDSCSZGUJXBGCDWBWOGEFWCGWN
      UKULULWRXDGWQXCCDWQWJXAPZFSESXCWPXEEFWJWMWOUOUMWJXAEFUNUPUMUQURUSUTKWTABG
      CDEFVAZVFURZWFVTWFWTUFVTKWTXFVBWRABGJJVCVDVTWFVETABHIJKILZJNHLXHNHSTHXHVG
      VHWGWDTWBWCVIZVHWEKVJTXGKVKVQVLVMVNKVTVOVPWDABJXIVRABJJJKVSVP $.
  $}

  ${
    $d A w x y z $.  $d B w z $.
    $( Addition is commutative.  Axiom for real and complex numbers, derived
       from set theory.  This construction-dependent theorem should not be
       referenced directly, nor should the proven axiom ~ ax-addcom be used
       later.  Instead, use ~ addcom .

       In the Metamath Proof Explorer this is not a complex number axiom but is
       instead proved from other axioms.  That proof relies on real number
       trichotomy and it is not known whether it is possible to prove this from
       the other axioms without it.  (Contributed by Jim Kingdon, 17-Jan-2020.)
       (New usage is discouraged.) $)
    axaddcom $p |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) $=
      ( vx vy vz vw cv cop caddc co wceq cnr oveq1 oveq2 eqeq12d wcel addcomsrg
      wa cplr addcnsr cc df-c ad2ant2r ad2ant2l opeq12d ancoms 3eqtr4d 2optocl
      ) CGZDGZHZEGZFGZHZIJZUNUKIJZKAUNIJZUNAIJZKABIJZBAIJZKCDEFABLLUAUBUKAKUOUQ
      UPURUKAUNIMUKAUNINOUNBKUQUSURUTUNBAINUNBAIMOUILPZUJLPZRZULLPZUMLPZRZRZUIU
      LSJZUJUMSJZHULUISJZUMUJSJZHZUOUPVGVHVJVIVKVAVDVHVJKVBVEUIULQUCVBVEVIVKKVA
      VDUJUMQUDUEUIUJULUMTVFVCUPVLKULUMUIUJTUFUGUH $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.
    $( Multiplication of complex numbers is commutative.  Axiom for real and
       complex numbers, derived from set theory.  This construction-dependent
       theorem should not be referenced directly, nor should the proven axiom
       ~ ax-mulcom be used later.  Instead, use ~ mulcom .  (Contributed by NM,
       31-Aug-1995.)  (New usage is discouraged.) $)
    axmulcom $p |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) $=
      ( vx vy vz vw cv cmr co cm1r cplr cnr mulcnsrec wcel wa mulcomsrg syl2anc
      wceq oveq12d mulclsr cc cmul dfcnqs simpll simprl simplr simprr addcomsrg
      cep ccnv oveq2d eqtrd ecovicom ) CDEFABUACGZEGZHIZJDGZFGZHIZHIZKIUBUIUJLU
      QUOHIZUNURHIZKIZUOUNHIZJURUQHIZHIZKIURUNHIZUOUQHIZKIZUCUNUQUOURMUOURUNUQM
      UNLNZUQLNZOZUOLNZURLNZOZOZUPVDUTVFKVPVJVMUPVDRVJVKVOUDZVLVMVNUEZUNUOPQVPU
      SVEJHVPVKVNUSVERVJVKVOUFZVLVMVNUGZUQURPQUKSVPVCVHVGKIZVIVPVAVHVBVGKVPVKVM
      VAVHRVSVRUQUOPQVPVJVNVBVGRVQVTUNURPQSVPVHLNZVGLNZWAVIRVPVMVKWBVRVSUOUQTQV
      PVNVJWCVTVQURUNTQVHVGUHQULUM $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $( Addition of complex numbers is associative.  This theorem transfers the
       associative laws for the real and imaginary signed real components of
       complex number pairs, to complex number addition itself.  Axiom for real
       and complex numbers, derived from set theory.  This
       construction-dependent theorem should not be referenced directly, nor
       should the proven axiom ~ ax-addass be used later.  Instead, use
       ~ addass .  (Contributed by NM, 2-Sep-1995.)
       (New usage is discouraged.) $)
    axaddass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                 ( ( A + B ) + C ) = ( A + ( B + C ) ) ) $=
      ( vx vy vz vw vv vu cv cplr co cnr addcnsrec wcel wa addclsr anim12i an4s
      wceq cc caddc ccnv addasssrg 3adant3r 3adant2r 3adant1r 3adant3l 3adant2l
      cep dfcnqs 3adant1l ecoviass ) DEFGHIABCUAUBGJZIJZKLZUJUCMDJZFJZKLZEJZUNK
      LZUSHJZKLZVAUOKLZUQURVBKLZKLZUTUPKLZVEUKUQUTURUNNURUNVBUONUSVAVBUONUQUTVE
      UPNUQMOZURMOZUTMOZUNMOZUSMOZVAMOZPVHVIPVLVJVKPVMUQURQUTUNQRSVIVBMOZVKUOMO
      ZVEMOZUPMOZPVIVNPVPVKVOPVQURVBQUNUOQRSVHVIVKPZVNVOPZVCVFTZVJVHVIVSVTVKVHV
      IVNVTVOUQURVBUDUEUFUGVJVRVSVDVGTZVHVJVKVSWAVIVJVKVOWAVNUTUNUOUDUHUIULUM
      $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u f g h $.
    $( Multiplication of complex numbers is associative.  Axiom for real and
       complex numbers, derived from set theory.  This construction-dependent
       theorem should not be referenced directly; instead, use ~ ax-mulass .
       (Contributed by NM, 3-Sep-1995.)  (New usage is discouraged.) $)
    axmulass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                 ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) $=
      ( vf vg vh cv cmr cplr cnr cm1r wcel mulclsr syl2anc wceq adantl distrsrg
      co wa syl3anc vx vy vz vw vv vu cc cmul cep ccnv dfcnqs mulcnsrec sylancr
      m1r addclsr syl2an an4s syl2anr an42s jca w3a simp1l simp2l simp3l simp2r
      simp3r simp1r addcomsrg addasssrg caov42d a1i eqtrd mulcomsrg 3coml simp3
      oveq2d oveq12d 3adant3 simp1 3eqtr3d mulasssrg caovdilemd caovcld caov12d
      simp2 3eqtrd 3eqtr4rd ecoviass ) UAUBUCUDUEUFABCUGUHUDGZUEGZHRZUCGZUFGZHR
      ZIRZUIUJJUAGZWLHRZKUBGZWIHRZHRZIRZWRWLHRZWPWIHRZIRZXAWJHRZKXDWMHRZHRZIRZX
      DWJHRZXAWMHRZIRZWPWLWJHRZKWIWMHRZHRZIRZHRZKWRWOHRZHRZIRZWRXOHRZWPWOHRZIRZ
      XOUKWPWRWLWIULWLWIWJWMULXAXDWJWMULWPWRXOWOULWPJLZWRJLZSZWLJLZWIJLZSZSXAJL
      ZXDJLZYCYFYDYGYIYCYFSWQJLWTJLZYIYDYGSZWPWLMYLKJLZWSJLZYKUNWRWIMZKWSMUMWQW
      TUOUPUQYCYGYDYFYJYDYFSXBJLXCJLYJYCYGSWRWLMWPWIMXBXCUOURUSUTYHWJJLZWMJLZSZ
      SXOJLZWOJLZYFYPYGYQYSYFYPSXLJLZXNJLZYSYGYQSZWLWJMZUUCYMXMJLZUUBUNWIWMMZKX
      MMZUMXLXNUOUPUQYFYQYGYPYTYGYPSWKJLZWNJLZYTYFYQSWIWJMZWLWMMZWKWNUOURUSUTYE
      YHYRVAZWPXLHRZWPXNHRZIRZKWRWKHRZHRZKWRWNHRZHRZIRZIRUUMUUQIRZUUSUUNIRZIRXS
      XHUULDEFUUMUUNUUQUUSJIUULYCUUAUUMJLYCYDYHYRVBZUULYFYPUUAYEYFYGYRVCZYEYHYP
      YQVDZUUDNZWPXLMNUULYCUUBUUNJLUVCUULYMUUEUUBUNUULYGYQUUEYEYFYGYRVEZYEYHYPY
      QVFZUUFNZUUGUMZWPXNMNUULYMUUPJLZUUQJLUNUULYDUUHUVKYCYDYHYRVGZUULYGYPUUHUV
      GUVEUUJNZWRWKMNZKUUPMUMDGZJLZEGZJLZSZUVOUVQIRZUVQUVOIROUULUVOUVQVHPZUVPUV
      RFGZJLZVAZUVTUWBIRUVOUVQUWBIRIROUULUVOUVQUWBVIPZUULYMUURJLZUUSJLUNUULYDUU
      IUWFUVLUULYFYQUUIUVDUVHUUKNZWRWNMNZKUURMUMUVSUVTJLZUULUVOUVQUOZPZVJUULXPU
      UOXRUUTIUULYCUUAUUBXPUUOOUVCUVFUVJWPXLXNQTUULXRKUUPUURIRZHRZUUTUULXQUWLKH
      UULYDUUHUUIXQUWLOUVLUVMUWGWRWKWNQTVPUULYMUVKUWFUWMUUTOYMUULUNVKZUVNUWHKUU
      PUURQTVLVQUULXEUVAXGUVBIUULXEUUMKWSWJHRZHRZIRUVAUULDEFWPKWLWSJIHWJUVSUVOU
      VQHRZUVQUVOHROUULUVOUVQVMPZUWDUVTUWBHRZUVOUWBHRZUVQUWBHRZIRZOUULUWDUWBUVT
      HRZUWBUVOHRZUWBUVQHRZIRZUWSUXBUWCUVPUVRUXCUXFOUWBUVOUVQQVNUWDUWCUWIUXCUWS
      OUVPUVRUWCVOZUVPUVRUWIUWCUWJVRUWBUVTVMNUWDUXDUWTUXEUXAIUWDUWCUVPUXDUWTOUX
      GUVPUVRUWCVSUWBUVOVMNUWDUWCUVRUXEUXAOUXGUVPUVRUWCWEUWBUVQVMNVQVTPZUWDUWQU
      WBHRUVOUXAHROUULUVOUVQUWBWAPZUVSUWQJLUULUVOUVQMPZUVCUWNUVDUULYDYGYNUVLUVG
      YONZUVEWBUULUWPUUQUUMIUULUWOUUPKHUULYDYGYPUWOUUPOUVLUVGUVEWRWIWJWATVPVPVL
      UULXGKUURWPXMHRZIRZHRZUUSKUXLHRZIRZUVBUULXFUXMKHUULDEFWRWPWLWIJIHWMUWRUXH
      UXIUXJUVLUVCUVDUVGUVHWBVPUULYMUWFUXLJLUXNUXPOUWNUWHUULDEWPXMJJJHUXJUVCUVI
      WCKUURUXLQTUULUXOUUNUUSIUULDEFKWPXMJHUWNUVCUVIUWRUXIWDVPWFVQWGUULWRXLHRZW
      RXNHRZIRZWPWKHRZWPWNHRZIRZIRUXQUXTIRZUYAUXRIRZIRYBXKUULDEFUXQUXRUXTUYAJIU
      ULDEWRXLJJJHUXJUVLUVFWCUULDEWRXNJJJHUXJUVLUVJWCUULDEWPWKJJJHUXJUVCUVMWCUW
      AUWEUULDEWPWNJJJHUXJUVCUWGWCUWKVJUULXTUXSYAUYBIUULYDUUAUUBXTUXSOUVLUVFUVJ
      WRXLXNQTUULYCUUHUUIYAUYBOUVCUVMUWGWPWKWNQTVQUULXIUYCXJUYDIUULDEFWRWPWLWIJ
      IHWJUWRUXHUXIUXJUVLUVCUVDUVGUVEWBUULXJUYAKWSWMHRZHRZIRUYDUULDEFWPKWLWSJIH
      WMUWRUXHUXIUXJUVCUWNUVDUXKUVHWBUULUYFUXRUYAIUULUYFKWRXMHRZHRUXRUULUYEUYGK
      HUULYDYGYQUYEUYGOUVLUVGUVHWRWIWMWATVPUULDEFKWRXMJHUWNUVLUVIUWRUXIWDVLVPVL
      VQWGWH $.
  $}

  ${
    $d x y z w v u A $.  $d x y z w v u B $.  $d x y z w v u C $.
    $d x y z w v u f g h $.
    $( Distributive law for complex numbers (left-distributivity).  Axiom for
       real and complex numbers, derived from set theory.  This
       construction-dependent theorem should not be referenced directly, nor
       should the proven axiom ~ ax-distr be used later.  Instead, use
       ~ adddi .  (Contributed by NM, 2-Sep-1995.)
       (New usage is discouraged.) $)
    axdistr $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) ) $=
      ( vf cnr cv cplr co cmr cm1r wcel wa addclsr mulclsr m1r sylancr distrsrg
      wceq syl3anc syl2anc vx vy vz vw vv vu vg vh cc caddc ccnv cmul addcnsrec
      cep dfcnqs mulcnsrec anim12i an4s syl2an syl2anr jca simp1l simp2l simp3l
      an42s w3a simp1r simp2r simp3r oveq2d a1i oveq12d addcomsrg adantl caov4d
      eqtrd addasssrg ecovidi ) UAUBUCUDUEUFABCUIUJUNUKEULUAFZUCFZUEFZGHZIHZJUB
      FZUDFZUFFZGHZIHZIHZGHZWDWBIHZVSWGIHZGHZVSVTIHZJWDWEIHZIHZGHZVSWAIHZJWDWFI
      HZIHZGHZGHZWDVTIHZVSWEIHZGHZWDWAIHZVSWFIHZGHZGHZWBWGWQXEXAXHUOVTWEWAWFUMV
      SWDWBWGUPVSWDVTWEUPVSWDWAWFUPWQXEXAXHUMVTEKZWAEKZWEEKZWFEKZWBEKZWGEKZLXJX
      KLXNXLXMLXOVTWAMWEWFMUQURVSEKZWDEKZLZXJXLLZLWQEKZXEEKZXPXJXQXLXTXPXJLWNEK
      ZWPEKZXTXQXLLZVSVTNZYDJEKZWOEKZYCOWDWENZJWONZPWNWPMUSURXPXLXQXJYAXQXJLXCE
      KZXDEKZYAXPXLLWDVTNZVSWENZXCXDMUTVEVAXRXKXMLZLXAEKZXHEKZXPXKXQXMYOXPXKLWR
      EKZWTEKZYOXQXMLZVSWANZYSYFWSEKZYROWDWFNZJWSNZPWRWTMUSURXPXMXQXKYPXQXKLXFE
      KZXGEKZYPXPXMLWDWANZVSWFNZXFXGMUTVEVAXRXSYNVFZWJWNWRGHZWPWTGHZGHXBUUHWCUU
      IWIUUJGUUHXPXJXKWCUUIRXPXQXSYNVBZXRXJXLYNVCZXRXSXKXMVDZVSVTWAQSUUHWIJWOWS
      GHZIHZUUJUUHWHUUNJIUUHXQXLXMWHUUNRXPXQXSYNVGZXRXJXLYNVHZXRXSXKXMVIZWDWEWF
      QSVJUUHYFYGUUAUUOUUJRYFUUHOVKUUHXQXLYGUUPUUQYHTZUUHXQXMUUAUUPUURUUBTZJWOW
      SQSVPVLUUHDUGUHWNWRWPWTEGUUHXPXJYBUUKUULYETUUHXPXKYQUUKUUMYTTUUHYFYGYCOUU
      SYIPDFZEKZUGFZEKZLZUVAUVCGHZUVCUVAGHRUUHUVAUVCVMVNZUVBUVDUHFZEKVFUVFUVHGH
      UVAUVCUVHGHGHRUUHUVAUVCUVHVQVNZUUHYFUUAYROUUTUUCPUVEUVFEKUUHUVAUVCMVNZVOV
      PUUHWMXCXFGHZXDXGGHZGHXIUUHWKUVKWLUVLGUUHXQXJXKWKUVKRUUPUULUUMWDVTWAQSUUH
      XPXLXMWLUVLRUUKUUQUURVSWEWFQSVLUUHDUGUHXCXFXDXGEGUUHXQXJYJUUPUULYLTUUHXQX
      KUUDUUPUUMUUFTUUHXPXLYKUUKUUQYMTUVGUVIUUHXPXMUUEUUKUURUUGTUVJVOVPVR $.
  $}

  $( i-squared equals -1 (expressed as i-squared plus 1 is 0).  Axiom for real
     and complex numbers, derived from set theory.  This construction-dependent
     theorem should not be referenced directly; instead, use ~ ax-i2m1 .
     (Contributed by NM, 5-May-1996.)  (New usage is discouraged.) $)
  axi2m1 $p |- ( ( _i x. _i ) + 1 ) = 0 $=
    ( c0r c1r cop cmul co caddc ci cm1r cplr cmr cnr wcel wceq 0r 1sr ax-mp m1r
    1idsr eqtri oveq12i c1 cc0 mulcnsr mp4an 00sr oveq2i addcomsrg mp2an 3eqtri
    0idsr opeq12i oveq1i addresr m1p1sr opeq1i df-i df-1 df-0 3eqtr4i ) ABCZUTD
    EZBACZFEZAACZGGDEZUAFEUBVCHACZVBFEZHBIEZACZVDVAVFVBFVAAAJEZHBBJEZJEZIEZBAJE
    ZABJEZIEZCZVFAKLZBKLZVRVSVAVQMNONOABABUCUDVMHVPAVMAHIEZHAIEZHVJAVLHIVRVJAMN
    AUEPVLHBJEZHVKBHJVSVKBMOBRPUFHKLZWBHMQHRPSTVRWCVTWAMNQAHUGUHWCWAHMQHUJPUIVP
    AAIEZAVNAVOAIVSVNAMOBUEPVRVOAMNARPTVRWDAMNAUJPSUKSULWCVSVGVIMQOHBUMUHVHAAUN
    UOUIVEVAUAVBFGUTGUTDUPUPTUQTURUS $.

  $( 0 is less than 1.  Axiom for real and complex numbers, derived from set
     theory.  This construction-dependent theorem should not be referenced
     directly; instead, use ~ ax-0lt1 .

     The version of this axiom in the Metamath Proof Explorer reads
     ` 1 =/= 0 ` ; here we change it to ` 0 <RR 1 ` .  The proof of ` 0 <RR 1 `
     from ` 1 =/= 0 ` in the Metamath Proof Explorer (accessed 12-Jan-2020)
     relies on real number trichotomy.  (Contributed by Jim Kingdon,
     12-Jan-2020.)  (New usage is discouraged.) $)
  ax0lt1 $p |- 0 <RR 1 $=
    ( c0r cop c1r cc0 c1 cltrr wbr cltr 0lt1sr ltresr mpbir df-0 df-1 3brtr4i )
    AABZCABZDEFOPFGACHGIACJKLMN $.

  ${
    $d A x y $.
    $( ` 1 ` is an identity element for real multiplication.  Axiom for real
       and complex numbers, derived from set theory.  This
       construction-dependent theorem should not be referenced directly;
       instead, use ~ ax-1rid .  (Contributed by Scott Fenton, 3-Jan-2013.)
       (New usage is discouraged.) $)
    ax1rid $p |- ( A e. RR -> ( A x. 1 ) = A ) $=
      ( vx vy cv cop c1 cmul co wceq cnr c0r csn cr df-r oveq1 id eqeq12d elsni
      wcel c1r df-1 oveq2i cmr 1sr mpan2 1idsr opeq1d eqtrd eqtrid opeq2 oveq1d
      mulresr imbitrrid impcom sylan2 optocl ) BDZCDZEZFGHZUSIZAFGHZAIBCAJKLZMN
      USAIZUTVBUSAUSAFGOVDPQURVCSUQJSZURKIZVAURKRVFVEVAVEVAVFUQKEZFGHZVGIVEVHVG
      TKEZGHZVGFVIVGGUAUBVEVJUQTUCHZKEZVGVETJSVJVLIUDUQTULUEVEVKUQKUQUFUGUHUIVF
      UTVHUSVGVFUSVGFGURKUQUJZUKVMQUMUNUOUP $.
  $}

  ${
    $d A x y $.
    $( ` 0 ` is an identity element for real addition.  Axiom for real and
       complex numbers, derived from set theory.  This construction-dependent
       theorem should not be referenced directly; instead, use ~ ax-0id .

       In the Metamath Proof Explorer this is not a complex number axiom but is
       instead proved from other axioms.  That proof relies on excluded middle
       and it is not known whether it is possible to prove this from the other
       axioms without excluded middle.  (Contributed by Jim Kingdon,
       16-Jan-2020.)  (New usage is discouraged.) $)
    ax0id $p |- ( A e. CC -> ( A + 0 ) = A ) $=
      ( vx vy cv cop cc0 caddc co wceq cnr cc df-c oveq1 id eqeq12d wcel c0r 0r
      cplr 0idsr wa addcnsr mpanr12 eqcomi oveq2d adantr adantl opeq12d 3eqtr3d
      df-0 a1i optocl ) BDZCDZEZFGHZUOIAFGHZAIBCAJJKLUOAIZUPUQUOAUOAFGMURNOUMJP
      ZUNJPZUAZUOQQEZGHZUMQSHZUNQSHZEZUPUOVAQJPZVGVCVFIRRUMUNQQUBUCVAVBFUOGVBFI
      VAFVBUJUDUKUEVAVDUMVEUNUSVDUMIUTUMTUFUTVEUNIUSUNTUGUHUIUL $.
  $}

  ${
    $d x A $.
    $( Existence of negative of real number.  Axiom for real and complex
       numbers, derived from set theory.  This construction-dependent theorem
       should not be referenced directly; instead, use ~ ax-rnegex .
       (Contributed by NM, 15-May-1996.)  (New usage is discouraged.) $)
    axrnegex $p |- ( A e. RR -> E. x e. RR ( A + x ) = 0 ) $=
      ( cr wcel c1st cfv cm1r cmr co c0r cop caddc cc0 wceq cv wrex cnr elreal2
      simplbi syl2anc mulclsr sylancl opelreal sylibr cplr simprbi oveq1d pn0sr
      m1r addresr opeq1d df-0 eqtr4di syl 3eqtrd oveq2 eqeq1d rspcev ) BCDZBEFZ
      GHIZJKZCDZBVBLIZMNZBAOZLIZMNZACPUSVAQDZVCUSUTQDZGQDVIUSVJBUTJKZNZBRZSZUIU
      TGUAUBZVAUCUDUSVDVKVBLIZUTVAUEIZJKZMUSBVKVBLUSVJVLVMUFUGUSVJVIVPVRNVNVOUT
      VAUJTUSVJVRMNVNVJVRJJKMVJVQJJUTUHUKULUMUNUOVHVEAVBCVFVBNVGVDMVFVBBLUPUQUR
      T $.
  $}

  ${
    $d x y A $.  $d x y z $.
    $( Existence of positive reciprocal of positive real number.  Axiom for
       real and complex numbers, derived from set theory.  This
       construction-dependent theorem should not be referenced directly;
       instead, use ~ ax-precex .

       In treatments which assume excluded middle, the ` 0 <RR A ` condition is
       generally replaced by ` A =/= 0 ` , and it may not be necessary to state
       that the reciproacal is positive.  (Contributed by Jim Kingdon,
       6-Feb-2020.)  (New usage is discouraged.) $)
    axprecex $p |- ( ( A e. RR /\ 0 <RR A ) ->
        E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) ) $=
      ( vy vz cr wcel cc0 cltrr wbr cmul co c1 wceq wrex c0r cop cnr bitri c1r
      wa cv wi wex elreal df-rex breq2 oveq1 eqeq1d anbi2d rexbidv imbi12d cltr
      df-0 breq1i ltresr cmr recexgt0sr opelreal anbi1i a1i mulresr df-1 eqeq2i
      wb eqid 1sr 0r opthg2 mp2an mpbiran2 bitrdi anbi12d pm5.32da bitrid oveq2
      rspcev biimtrrdi expd rexlimdv syl5 biimtrid gencl imp ) BEFZGBHIZGAUAZHI
      ZBWFJKZLMZTZAENZGCUAZOPZHIZWGWMWFJKZLMZTZAENZUBWEWKUBWLQFZWDCWMBWDWMBMZCQ
      NWSWTTCUCCBUDWTCQUERWTWNWEWRWKWMBGHUFWTWQWJAEWTWPWIWGWTWOWHLWMBWFJUGUHUIU
      JUKWNOWLULIZWSWRWNOOPZWMHIXAGXBWMHUMUNOWLUORXAODUAZULIZWLXCUPKZSMZTZDQNWS
      WRDWLUQWSXGWRDQWSXCQFZXGWRWSXHXGTZXCOPZEFZGXJHIZWMXJJKZLMZTZTZWRXPXHXOTWS
      XIXKXHXOXCURUSWSXHXOXGWSXHTZXLXDXNXFXLXDVDXQXLXBXJHIXDGXBXJHUMUNOXCUORUTX
      QXNXEOPZLMZXFXQXMXRLWLXCVAUHXSXRSOPZMZXFLXTXRVBVCYAXFOOMZOVESQFOQFYAXFYBT
      VDVFVGXEOSOQQVHVIVJRVKVLVMVNWQXOAXJEWFXJMZWGXLWPXNWFXJGHUFYCWOXMLWFXJWMJV
      OUHVLVPVQVRVSVTWAWBWC $.
  $}

  ${
    $d x y z w A $.
    $( A complex number can be expressed in terms of two reals.  Definition
       10-1.1(v) of [Gleason] p. 130.  Axiom for real and complex numbers,
       derived from set theory.  This construction-dependent theorem should not
       be referenced directly; instead, use ~ ax-cnre .  (Contributed by NM,
       13-May-1996.)  (New usage is discouraged.) $)
    axcnre $p |- ( A e. CC ->
                      E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) $=
      ( cop cmul co caddc wceq cr cnr wcel wa c0r cplr c1r cmr cm1r eqtrd cvv
      0r vz vw cv ci wrex df-c eqeq1 2rexbidv wex opelreal anbi12i biimpri df-i
      cc oveq1i 1sr mulcnsr mpanl12 mpan2 mulcomsrg mpan 00sr oveq1d oveq2i m1r
      ax-mp eqtri 0idsr eqtrdi 1idsr eqtrid opeq12d oveq2d adantl mpanl2 mpanr1
      addcnsr addcomsrg opeq12 syl2an 3eqtrrd vex opexg mp2an eleq1 oveq1 oveq2
      bi2anan9 sylan9eq eqeq2d anbi12d spc2ev syl2anc r2ex sylibr optocl ) UAUC
      ZUBUCZDZAUCZUDBUCZEFZGFZHZBIUEAIUEZCXCHZBIUEAIUEUAUBCJJUNUFWSCHXDXFABIIWS
      CXCUGUHWQJKZWRJKZLZWTIKZXAIKZLZXDLZBUIAUIZXEXIWQMDZIKZWRMDZIKZLZWSXOUDXQE
      FZGFZHZXNXSXIXPXGXRXHWQUJWRUJUKULXIYAXOMWRDZGFZWQMNFZMWRNFZDZWSXHYAYDHXGX
      HXTYCXOGXHXTMODZXQEFZYCUDYHXQEUMUOXHYIMWRPFZQOMPFZPFZNFZOWRPFZMMPFZNFZDZY
      CXHMJKZYIYQHZTYROJKZXHYRLYSTUPMOWRMUQURUSXHYMMYPWRXHYMMYLNFZMXHYJMYLNXHYJ
      WRMPFZMYRXHYJUUBHTMWRUTVAWRVBRVCUUAMMNFZMYLMMNYLQMPFZMYKMQPYTYKMHUPOVBVFV
      DQJKUUDMHVEQVBVFVGVDYRUUCMHTMVHVFVGVIXHYPWRYONFZWRXHYNWRYONXHYNWROPFZWRYT
      XHYNUUFHUPOWRUTVAWRVJRVCXHUUEWRMNFZWRYOMWRNYRYOMHTMVBVFVDWRVHZVKRVLRVKVMV
      NXGYRXHYDYGHZTXGYRYRXHLUUITWQMMWRVQVOVPXGYEWQHYFWRHYGWSHXHWQVHXHYFUUGWRYR
      XHYFUUGHTMWRVRVAUUHRYEYFWQWRVSVTWAXMXSYBLABXOXQWQSKYRXOSKUAWBTWQMSJWCWDWR
      SKYRXQSKUBWBTWRMSJWCWDWTXOHZXAXQHZLZXLXSXDYBUUJXJXPUUKXKXRWTXOIWEXAXQIWEW
      HUULXCYAWSUUJUUKXCXOXBGFYAWTXOXBGWFUUKXBXTXOGXAXQUDEWGVMWIWJWKWLWMXDABIIW
      NWOWP $.
  $}

  ${
    $d x A $.
    $( Real number less-than is irreflexive.  Axiom for real and complex
       numbers, derived from set theory.  This construction-dependent theorem
       should not be referenced directly; instead, use ~ ax-pre-ltirr .
       (Contributed by Jim Kingdon, 12-Jan-2020.)
       (New usage is discouraged.) $)
    axpre-ltirr $p |- ( A e. RR -> -. A <RR A ) $=
      ( vx cv c0r cop cltrr wbr wn cnr wcel cr wceq wrex wa elreal df-rex bitri
      wex id cltr breq12d notbid ltsosr ltrelsr soirri ltresr mtbir a1i gencl )
      BCZDEZUKFGZHZAAFGZHUJIJZAKJZBUKAUPUKALZBIMUOUQNBRBAOUQBIPQUQULUNUQUKAUKAF
      UQSZURUAUBUMUOULUJUJTGUJTIUCUDUEUJUJUFUGUHUI $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.
    $( Real number less-than is weakly linear.  Axiom for real and complex
       numbers, derived from set theory.  This construction-dependent theorem
       should not be referenced directly; instead, use ~ ax-pre-ltwlin .
       (Contributed by Jim Kingdon, 12-Jan-2020.)
       (New usage is discouraged.) $)
    axpre-ltwlin $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
        ( A <RR B -> ( A <RR C \/ C <RR B ) ) ) $=
      ( vx vy vz cv c0r cop cltrr wbr wo cnr elreal wceq breq1 breq2 wcel cltr
      wi orbi1d imbi12d orbi2d orbi12d imbi2d w3a wor ltsosr sowlin mpan ltresr
      cr orbi12i 3imtr4g 3gencl ) DGZHIZEGZHIZJKZUQFGZHIZJKZVBUSJKZLZTAUSJKZAVB
      JKZVDLZTABJKZVGVBBJKZLZTVIACJKZCBJKZLZTDEFUQUSVBAMULBCDANEBNFCNUQAOZUTVFV
      EVHUQAUSJPVOVCVGVDUQAVBJPUAUBUSBOZVFVIVHVKUSBAJQVPVDVJVGUSBVBJQUCUBVBCOZV
      KVNVIVQVGVLVJVMVBCAJQVBCBJPUDUEUPMRURMRVAMRUFZUPURSKZUPVASKZVAURSKZLZUTVE
      MSUGVRVSWBTUHMUPURVASUIUJUPURUKVCVTVDWAUPVAUKVAURUKUMUNUO $.
  $}

  ${
    $d x y z A $.  $d y z B $.  $d z C $.
    $( Ordering on reals is transitive.  Axiom for real and complex numbers,
       derived from set theory.  This construction-dependent theorem should not
       be referenced directly; instead, use ~ ax-pre-lttrn .  (Contributed by
       NM, 19-May-1996.)  (Revised by Mario Carneiro, 16-Jun-2013.)
       (New usage is discouraged.) $)
    axpre-lttrn $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
               ( ( A <RR B /\ B <RR C ) -> A <RR C ) ) $=
      ( vx vy vz cv c0r cop cltrr wbr wa cnr elreal wceq breq1 breq2 wcel cltr
      wi anbi1d imbi12d anbi12d imbi1d anbi2d w3a ltresr ltsosr ltrelsr syl2anb
      cr sotri sylibr a1i 3gencl ) DGZHIZEGZHIZJKZUSFGZHIZJKZLZUQVBJKZTZAUSJKZV
      CLZAVBJKZTABJKZBVBJKZLZVITVJBCJKZLZACJKZTDEFUQUSVBAMUKBCDANEBNFCNUQAOZVDV
      HVEVIVPUTVGVCUQAUSJPUAUQAVBJPUBUSBOZVHVLVIVQVGVJVCVKUSBAJQUSBVBJPUCUDVBCO
      ZVLVNVIVOVRVKVMVJVBCBJQUEVBCAJQUBVFUPMRURMRVAMRUFVDUPVASKZVEUTUPURSKURVAS
      KVSVCUPURUGURVAUGUPURVASMUHUIULUJUPVAUGUMUNUO $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Apartness of reals is tight.  Axiom for real and complex numbers,
       derived from set theory.  This construction-dependent theorem should not
       be referenced directly; instead, use ~ ax-pre-apti .

       (Contributed by Jim Kingdon, 29-Jan-2020.)
       (New usage is discouraged.) $)
    axpre-apti $p |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) ->
        A = B ) $=
      ( vx vy cr wcel cltrr wbr wo wn wceq cv c0r cop wi cnr elreal breq1 breq2
      orbi12d notbid eqeq1 imbi12d eqeq2 wa aptisr 3expia ltresr orbi12i notbii
      cltr vex eqresr 3imtr4g 2gencl 3impia ) AEFBEFABGHZBAGHZIZJZABKZCLZMNZDLZ
      MNZGHZVEVCGHZIZJZVCVEKZOAVEGHZVEAGHZIZJZAVEKZOUTVAOCDVCVEABPECAQDBQVCAKZV
      IVNVJVOVPVHVMVPVFVKVGVLVCAVEGRVCAVEGSTUAVCAVEUBUCVEBKZVNUTVOVAVQVMUSVQVKU
      QVLURVEBAGSVEBAGRTUAVEBAUDUCVBPFZVDPFZUEVBVDUKHZVDVBUKHZIZJZVBVDKZVIVJVRV
      SWCWDVBVDUFUGVHWBVFVTVGWAVBVDUHVDVBUHUIUJVBVDCULUMUNUOUP $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.
    $( Ordering property of addition on reals.  Axiom for real and complex
       numbers, derived from set theory.  This construction-dependent theorem
       should not be referenced directly; instead, use ~ ax-pre-ltadd .
       (Contributed by NM, 11-May-1996.)  (New usage is discouraged.) $)
    axpre-ltadd $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                  ( A <RR B -> ( C + A ) <RR ( C + B ) ) ) $=
      ( vx vy vz cr wcel w3a cltrr wbr caddc co cv c0r cop wb cnr elreal wceq
      breq1 oveq2 breq1d bibi12d breq2 breq2d breq12d bibi2d cltr ltasrg ltresr
      oveq1 cplr simp3 simp1 simp2 wa addresr breqan12d syl22anc bitrdi 3bitr4d
      a1i 3gencl biimpd ) AGHBGHCGHIABJKZCALMZCBLMZJKZDNZOPZENZOPZJKZFNZOPZVKLM
      ZVPVMLMZJKZQAVMJKZVPALMZVRJKZQVFWAVPBLMZJKZQVFVIQDEFVKVMVPARGBCDASEBSFCSV
      KATZVNVTVSWBVKAVMJUAWEVQWAVRJVKAVPLUBUCUDVMBTZVTVFWBWDVMBAJUEWFVRWCWAJVMB
      VPLUBUFUDVPCTZWDVIVFWGWAVGWCVHJVPCALULVPCBLULUGUHVJRHZVLRHZVORHZIZVJVLUIK
      ZVOVJUMMZVOVLUMMZUIKZVNVSVJVLVOUJVNWLQWKVJVLUKVCWKVSWMOPZWNOPZJKZWOWKWJWH
      WJWIVSWRQWHWIWJUNZWHWIWJUOWSWHWIWJUPWJWHUQWJWIUQVQWPVRWQJVOVJURVOVLURUSUT
      WMWNUKVAVBVDVE $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The product of two positive reals is positive.  Axiom for real and
       complex numbers, derived from set theory.  This construction-dependent
       theorem should not be referenced directly; instead, use
       ~ ax-pre-mulgt0 .  (Contributed by NM, 13-May-1996.)
       (New usage is discouraged.) $)
    axpre-mulgt0 $p |- ( ( A e. RR /\ B e. RR ) ->
                  ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) ) $=
      ( vx vy cc0 cv c0r cop cltrr wbr wa cmul cnr elreal wceq cltr df-0 ltresr
      co wi cr breq2 anbi1d oveq1 breq2d imbi12d anbi2d oveq2 wcel breq1i bitri
      cmr mulgt0sr syl2anb a1i mulresr breq12d bitrdi imbitrrid 2gencl ) ECFZGH
      ZIJZEDFZGHZIJZKZEVBVELSZIJZTEAIJZVFKZEAVELSZIJZTVJEBIJZKZEABLSZIJZTCDVBVE
      ABMUACANDBNVBAOZVGVKVIVMVRVCVJVFVBAEIUBUCVRVHVLEIVBAVELUDUEUFVEBOZVKVOVMV
      QVSVFVNVJVEBEIUBUGVSVLVPEIVEBALUHUEUFVGVIVAMUIVDMUIKZGVAVDULSZPJZVCGVAPJZ
      GVDPJZWBVFVCGGHZVBIJWCEWEVBIQUJGVARUKVFWEVEIJWDEWEVEIQUJGVDRUKVAVDUMUNVTV
      IWEWAGHZIJWBVTEWEVHWFIEWEOVTQUOVAVDUPUQGWARURUSUT $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d z C $.
    $( Strong extensionality of multiplication (expressed in terms of
       ` <RR ` ).  Axiom for real and complex numbers, derived from set theory.
       This construction-dependent theorem should not be referenced directly;
       instead, use ~ ax-pre-mulext .

       (Contributed by Jim Kingdon, 18-Feb-2020.)
       (New usage is discouraged.) $)
    axpre-mulext $p |- ( ( A e. RR /\ B e. RR /\ C e. RR )
        -> ( ( A x. C ) <RR ( B x. C ) -> ( A <RR B \/ B <RR A ) ) ) $=
      ( vx vz vy cv c0r cop cmul co cltrr wbr wo wi cnr elreal wceq wcel cltr
      cr oveq1 breq1d breq1 breq2 orbi12d imbi12d breq2d breq12d imbi1d w3a cmr
      oveq2 mulextsr1 mulresr 3adant2 3adant1 ltresr bitrdi orbi12i a1i 3imtr4d
      wb 3gencl ) DGZHIZEGZHIZJKZFGZHIZVHJKZLMZVFVKLMZVKVFLMZNZOAVHJKZVLLMZAVKL
      MZVKALMZNZOVQBVHJKZLMZABLMZBALMZNZOACJKZBCJKZLMZWFODFEVFVKVHAPUABCDAQFBQE
      CQVFARZVMVRVPWAWJVIVQVLLVFAVHJUBUCWJVNVSVOVTVFAVKLUDVFAVKLUEUFUGVKBRZVRWC
      WAWFWKVLWBVQLVKBVHJUBUHWKVSWDVTWEVKBALUEVKBALUDUFUGVHCRZWCWIWFWLVQWGWBWHL
      VHCAJUMVHCBJUMUIUJVEPSZVJPSZVGPSZUKZVEVGULKZVJVGULKZTMZVEVJTMZVJVETMZNZVM
      VPVEVJVGUNWPVMWQHIZWRHIZLMWSWPVIXCVLXDLWMWOVIXCRWNVEVGUOUPWNWOVLXDRWMVJVG
      UOUQUIWQWRURUSVPXBVCWPVNWTVOXAVEVJURVJVEURUTVAVBVD $.
  $}

  ${
    $d A x y z $.
    $( The reciprocal from ~ axprecex is unique.  (Contributed by Jim Kingdon,
       15-Jul-2021.) $)
    rereceu $p |- ( ( A e. RR /\ 0 <RR A ) ->
        E! x e. RR ( A x. x ) = 1 ) $=
      ( vy vz cr wcel cc0 cltrr wbr wa cv cmul co c1 wceq wrex adantr cc sselid
      axresscn wrmo wreu axprecex simpr reximi wi wral eqtr3 wb simpll axmulcom
      syl simprl syl2anc simprr eqeq12d oveq1 biimtrdi axmulass sylibd ad2antll
      syl3anc oveq2 ax1rid sylan9eqr rexlimddv syl5 ralrimivva eqeq1d rmo4 reu5
      sylibr sylanbrc ) BEFZGBHIZJZBAKZLMZNOZAEPZVSAEUAZVSAEUBVPGVQHIZVSJZAEPVT
      ABUCWCVSAEWBVSUDUEULVPVSBCKZLMZNOZJZVQWDOZUFZCEUGAEUGWAVPWIACEEWGVRWEOZVP
      VQEFZWDEFZJZJZWHVRWENUHWNGDKZHIZBWOLMZNOZJZWJWHUFDEVPWSDEPWMDBUCQWNWOEFZW
      SJZJZWJVQWQLMZWDWQLMZOZWHXBWJVQBLMZWOLMZWDBLMZWOLMZOZXEXBWJXFXHOZXJWNWJXK
      UIXAWNVRXFWEXHWNBRFZVQRFZVRXFOWNERBTVNVOWMUJSZWNERVQTVPWKWLUMZSZBVQUKUNWN
      XLWDRFZWEXHOXNWNERWDTVPWKWLUOSZBWDUKUNUPQXFXHWOLUQURXBXGXCXIXDXBXMXLWORFZ
      XGXCOWNXMXAXPQWNXLXAXNQZXBERWOTWNWTWSUMSZVQBWOUSVBXBXQXLXSXIXDOWNXQXAXRQX
      TYAWDBWOUSVBUPUTXBXCVQXDWDXAWNXCVQNLMZVQWRXCYBOWTWPWQNVQLVCVAWNWKYBVQOXOV
      QVDULVEXAWNXDWDNLMZWDWRXDYCOWTWPWQNWDLVCVAWLYCWDOVPWKWDVDVAVEUPUTVFVGVHVS
      WFACEWHVRWENVQWDBLVCVIVJVLVSAEVKVM $.
  $}

  ${
    $d N l r u y $.
    $( Two ways to express the reciprocal of a natural number.  (Contributed by
       Jim Kingdon, 11-Jul-2021.) $)
    recriota $p |- ( N e. N. -> ( iota_ r e. RR ( <. [ <. ( <.
        { l | l <Q [ <. N , 1o >. ] ~Q } ,
        { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R
        >. x. r ) = 1 ) =
        <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } ,
        { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >.
        ] ~R , 0R >. ) $=
      ( vy wcel cv wbr cop cltq cab c1p co cmul c1 wceq cr syl2anc adantr cc wa
      cnpi cc0 cltrr c1o ceq cec cpp cer c0r crio crq cfv wrex pitore pitoregt0
      axprecex simprrr wreu wb simprl oveq2 eqeq1d riota2 mpbid oveq2d axresscn
      rereceu sselid recnnre axmulcom recidpirq eqtr3d axmulass syl3anc sylancr
      oveq1d ax1cn 3eqtr3d ax1rid syl eqtrd rexlimddv ) BUBFZUCEGZUDHZDGZBUEIUF
      UGZJHDKWHAGZJHAKILUHMLIUIUGUJIZWENMZOPZUAZWJCGZNMZOPZCQUKZWGWHULUMZJHDKWR
      WIJHAKILUHMLIUIUGUJIZPEQWDWJQFZUCWJUDHZWMEQUNABDUOZABDUPZEWJUQRWDWEQFZWMU
      AZUAZWQWEWSXFWLWQWEPZWDXDWFWLURZXFXDWPCQUSZWLXGUTWDXDWMVAZXFWTXAXIWDWTXEX
      BSZWDXAXEXCSCWJVHRWPWLCQWEWNWEPWOWKOWNWEWJNVBVCVDRVEXFWSWKNMZWSONMZWEWSXF
      WKOWSNXHVFXFXLWEONMZWEXFWSWJNMZWENMZOWENMZXLXNXFXOOWENXFWJWSNMZXOOXFWJTFZ
      WSTFZXRXOPXFQTWJVGXKVIZXFQTWSVGWDWSQFZXEABDVJSZVIZWJWSVKRWDXROPXEABDVLSVM
      VQXFXTXSWETFZXPXLPYDYAXFQTWEVGXJVIZWSWJWEVNVOXFOTFYEXQXNPVRYFOWEVKVPVSXFX
      DXNWEPXJWEVTWAWBXFYBXMWSPYCWSVTWAVSWBWC $.
  $}

  ${
    $d A l n u w x y z $.

    $( Archimedean axiom.  The Archimedean property is more naturally stated
       once we have defined ` NN ` .  Unless we find another way to state it,
       we'll just use the right hand side of ~ dfnn2 in stating what we mean by
       "natural number" in the context of this axiom.

       This construction-dependent theorem should not be referenced directly;
       instead, use ~ ax-arch .  (Contributed by Jim Kingdon, 22-Apr-2020.)
       (New usage is discouraged.) $)
    axarch $p |- ( A e. RR -> E. n e.
        |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n ) $=
      ( vz vl vw vu wcel cv c0r cop wceq cltrr wbr wa cab wrex cnr cnpi cr wral
      c1 caddc co cint elreal biimpi c1o ceq cec cltq c1p cpp cer cltr ad2antrl
      archsr simplrr simprr ltresr sylibr pitonn simpr breq2d rspcedv rexlimddv
      eqbrtrrd mpd ) CUAIZEJZKLZCMZCDJZNOZDUCAJZIBJUCUDUEVPIBVPUBPAQUFZRZESVJVM
      ESRECUGUHVJVKSIZVMPPZVKFJGJZUILUJUKZULOFQWBHJULOHQLUMUNUEUMLUOUKZUPOZVRGT
      VSWDGTRVJVMGHVKFURUQVTWATIZWDPZPZCWCKLZNOZVRWGVLCWHNVJVSVMWFUSWGWDVLWHNOV
      TWEWDUTVKWCVAVBVHWGVOWIDWHVQWEWHVQIVTWDABHWAFVCUQWGVNWHMZPVNWHCNWGWJVDVEV
      FVIVGVG $.
  $}

  ${
    $d x y $.
    nntopi.n $e |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.

    ${
      $d A x y $.  $d A y z $.
      $( Peano's inductive postulate.  This is a counterpart to ~ peano5nni
         designed for real number axioms which involve natural numbers
         (notably, ~ axcaucvg ).  (Contributed by Jim Kingdon, 14-Jul-2021.)
         (New usage is discouraged.) $)
      peano5nnnn $p |- ( ( 1 e. A /\ A. z e. A ( z + 1 ) e. A ) -> N C_ A ) $=
        ( cv c1 caddc co wcel wral wss wceq wa cr ax1re elin mpan2 wi oveq1 cin
        eleq1d cbvralv inss1 ssralv ax-mp inss2 sseli axaddrcl simplbi2com 3syl
        biimpri ralimia syl cvv cc axcnex axresscn ssexi inex2 eleq2 raleqbi1dv
        cab anbi12d elabg cint intss1 eqsstrid biimtrrdi syl2an sstrdi sylan2br
        ) CGZHIJZDKZCDLHDKZBGZHIJZDKZBDLZEDMVTVPBCDVRVNNVSVODVRVNHIUAUCUDVQWAOE
        DPUBZDVQHWBKZVSWBKZBWBLZEWBMZWAVQHPKZWCQWCVQWGOHDPRUMSWAVTBWBLZWEWBDMWA
        WHTDPUEZVTBWBDUFUGVTWDBWBVRWBKVRPKZVSPKZVTWDTWBPVRDPUHUIWJWGWKQVRHUJSWD
        VTWKVSDPRUKULUNUOWBUPKZWCWEOZWFTPDPUQURUSUTVAWLWMWBHAGZKZVSWNKZBWNLZOZA
        VDZKZWFWRWMAWBUPWNWBNWOWCWQWEWNWBHVBWPWDBWNWBWNWBVSVBVCVEVFWTEWSVGWBFWB
        WSVHVIVJUGVKWIVLVM $.
    $}

    ${
      $d z k $.  $d z A $.  $d z ps $.  $d z ch $.  $d z th $.  $d z ta $.
      $d k ph $.  $d N k y z $.  $d N x y z $.  $d ph x y $.
      nnindnn.1 $e |- ( z = 1 -> ( ph <-> ps ) ) $.
      nnindnn.y $e |- ( z = k -> ( ph <-> ch ) ) $.
      nnindnn.y1 $e |- ( z = ( k + 1 ) -> ( ph <-> th ) ) $.
      nnindnn.a $e |- ( z = A -> ( ph <-> ta ) ) $.
      nnindnn.basis $e |- ps $.
      nnindnn.step $e |- ( k e. N -> ( ch -> th ) ) $.
      $( Principle of Mathematical Induction (inference schema).  This is a
         counterpart to ~ nnind designed for real number axioms which involve
         natural numbers (notably, ~ axcaucvg ).  (Contributed by Jim Kingdon,
         14-Jul-2021.)  (New usage is discouraged.) $)
      nnindnn $p |- ( A e. N -> ta ) $=
        ( wcel c1 crab wa cv caddc co wral wss peano1nnnn elrab mpbir2an elrabi
        peano2nnnn a1d anim12d 3imtr4g mpcom rgen peano5nnnn mp2an sseli simprd
        sylib ) IKSZVCEVCIAHKUAZSVCEUBKVDITVDSZJUCZTUDUEZVDSZJVDUFKVDUGVETKSBFG
        KLUHQABHTKMUIUJVHJVDVFKSZVFVDSZVHAHVFKUKVIVICUBVGKSZDUBVJVHVIVIVKCDVIVK
        VIFGVFKLULUMRUNACHVFKNUIADHVGKOUIUOUPUQFGJVDKLURUSUTAEHIKPUIVBVA $.
    $}

    ${
      $d A w z $.  $d N z $.  $d N k y $.  $d N w y $.  $d N x $.
      $d k l u v z $.  $d k l u w z $.  $d l u y $.  $d x y z $.  $d w x $.
      $d l x $.  $d u x $.
      $( Mapping from ` NN ` to ` N. ` .  (Contributed by Jim Kingdon,
         13-Jul-2021.) $)
      nntopi $p |- ( A e. N -> E. z e. N.
          <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } ,
            { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
          = A ) $=
        ( c1o cop ceq cec cltq wbr cab c1p cpp wceq cnpi c1 vw vk vv cv cer c0r
        co wrex caddc eqeq2 rexbidv wcel 1pi eqid c1r c1q opeq1 eceq1d df-1nqqs
        eqtr4di breq2d abbidv breq1d opeq12d df-i1p oveq1d opeq1d eqeq1d rspcev
        df-1r df-1 mp2an wa cpli simplr addclpi sylancl pitonnlem2 simpr eqtr3d
        syl syl2anc ex rexlimdva cbvrexv imbitrdi nnindnn ) GUDZCUDZIJZKLZMNZGO
        ZWKDUDZMNZDOZJZPQUGZPJZUELZUFJZUAUDZRZCSUHXATRZCSUHZXAUBUDZRZCSUHZXAXFT
        UIUGZRZCSUHZXAERZCSUHABUAEUBFHXBTRXCXDCSXBTXAUJUKXBXFRXCXGCSXBXFXAUJUKX
        BXIRXCXJCSXBXIXAUJUKXBERXCXLCSXBEXAUJUKISULZTTRZXEUMTUNXDXNCISWIIRZXATT
        XOXAUOUFJTXOWTUOUFXOWTPPQUGZPJZUELUOXOWSXQUEXOWRXPPXOWQPPQXOWQWHUPMNZGO
        ZUPWNMNZDOZJPXOWMXSWPYAXOWLXRGXOWKUPWHMXOWKIIJZKLUPXOWJYBKWIIIUQURUSUTZ
        VAVBXOWOXTDXOWKUPWNMYCVCVBVDDGVEUTVFVGURVJUTVGVKUTVHVIVLXFFULZXHWHUCUDZ
        IJZKLZMNZGOZYGWNMNZDOZJZPQUGZPJZUELZUFJZXIRZUCSUHZXKYDXGYRCSYDWISULZVMZ
        XGYRYTXGVMZWIIVNUGZSULZWHUUBIJZKLZMNZGOZUUEWNMNZDOZJZPQUGZPJZUELZUFJZXI
        RZYRUUAYSXMUUCYDYSXGVOZUMWIIVPVQUUAXATUIUGZUUNXIUUAYSUUQUUNRUUPDWIGVRWA
        UUAXAXFTUIYTXGVSVFVTYQUUOUCUUBSYEUUBRZYPUUNXIUURYOUUMUFUURYNUULUEUURYMU
        UKPUURYLUUJPQUURYIUUGYKUUIUURYHUUFGUURYGUUEWHMUURYFUUDKYEUUBIUQURZVAVBU
        URYJUUHDUURYGUUEWNMUUSVCVBVDVFVGURVGVHVIWBWCWDYQXJUCCSYEWIRZYPXAXIUUTYO
        WTUFUUTYNWSUEUUTYMWRPUUTYLWQPQUUTYIWMYKWPUUTYHWLGUUTYGWKWHMUUTYFWJKYEWI
        IUQURZVAVBUUTYJWODUUTYGWKWNMUVAVCVBVDVFVGURVGVHWEWFWG $.
    $}
  $}

  ${
    $d F z $.  $d J l u z $.  $d l u y $.  $d x y $.
    axcaucvglemcl.n $e |- N =
      |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
    axcaucvglemcl.f $e |- ( ph -> F : N --> RR ) $.
    $( Lemma for ~ axcaucvg .  Mapping to ` N. ` and ` R. ` .  (Contributed by
       Jim Kingdon, 10-Jul-2021.) $)
    axcaucvglemcl $p |- ( ( ph /\ J e. N. ) -> ( iota_ z e. R.
        ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } ,
        { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) =
        <. z , 0R >. ) e. R. ) $=
      ( wcel wa cv cop cec cltq wbr cab cnr cnpi c1o ceq c1p cpp co cer c0r cfv
      wceq wreu crio cr c1 caddc wral cint pitonn eleqtrrdi ffvelcdmda elrealeu
      sylan2 sylib eqcom reubii riotacl syl ) AGUALZMZINGUBOUCPZQRISVJENQRESOUD
      UEUFUDOUGPUHOZFUIZDNUHOZUJZDTUKZVNDTULTLVIVMVLUJZDTUKZVOVIVLUMLZVQVHAVKHL
      VRVHVKUNBNZLCNUNUOUFVSLCVSUPMBSUQHBCEGIURJUSAHUMVKFKUTVBDVLVAVCVPVNDTVMVL
      VDVEVCVNDTVFVG $.
  $}

  ${
    axcaucvg.n $e |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
    axcaucvg.f $e |- ( ph -> F : N --> RR ) $.
    axcaucvg.cau $e |- ( ph ->
      A. n e. N A. k e. N ( n <RR k ->
        ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\
        ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) ) $.

    ${
      axcaucvg.g $e |- G = ( j e. N. |-> ( iota_ z e. R.
        ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } ,
        { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) =
        <. z , 0R >. ) ) $.

      ${
        $d j ph $.  $d F z $.  $d j l u z $.  $d l u y $.  $d x y $.
        $( Lemma for ~ axcaucvg .  Mapping to ` N. ` and ` R. ` yields a
           sequence.  (Contributed by Jim Kingdon, 9-Jul-2021.) $)
        axcaucvglemf $p |- ( ph -> G : N. --> R. ) $=
          ( cv cop cec cnpi c1o ceq cltq wbr cab c1p cpp cer wceq axcaucvglemcl
          co c0r cfv cnr crio fmptd ) AFUAMRFRZUBSUCTZUDUEMUFUSERUDUEEUFSUGUHUL
          UGSUITUMSIUNDRUMSUJDUOUPUOJABCDEIURKMNOUKQUQ $.
      $}

      ${
        $d F j z $.  $d G z $.  $d J j l u z $.  $d j ph $.  $d l u y $.
        $d x y $.
        $( Lemma for ~ axcaucvg .  Value of sequence when mapping to ` N. ` and
           ` R. ` .  (Contributed by Jim Kingdon, 10-Jul-2021.) $)
        axcaucvglemval $p |- ( ( ph /\ J e. N. ) ->
          ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } ,
          { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >.
          ] ~R , 0R >. ) =
          <. ( G ` J ) , 0R >. ) $=
          ( cop cnr cnpi wcel wa cv c1o ceq cec cltq wbr cab c1p cpp co cer c0r
          cfv wceq crio opeq1 eceq1d breq2d abbidv breq1d opeq12d oveq1d opeq1d
          cmpt fveq2d eqeq1d riotabidv adantl simpr axcaucvglemcl fvmptd eqcomd
          wreu wb eqeltrd cr wf adantr c1 caddc wral pitonn eleqtrrdi ffvelcdmd
          a1i cint elrealeu sylib eqcom reubii eqeq2d riota2 syl2anc mpbird ) A
          KUAUBZUCZNUDZKUESZUFUGZUHUIZNUJZXBEUDZUHUIZEUJZSZUKULUMZUKSZUNUGZUOSZ
          IUPZKJUPZUOSZUQZXMDUDZUOSZUQZDTURZXNUQZWSXNXTWSFKWTFUDZUESZUFUGZUHUIZ
          NUJZYDXEUHUIZEUJZSZUKULUMZUKSZUNUGZUOSZIUPZXRUQZDTURZXTUAJTJFUAYPVGUQ
          WSRWHYBKUQZYPXTUQWSYQYOXSDTYQYNXMXRYQYMXLIYQYLXKUOYQYKXJUNYQYJXIUKYQY
          IXHUKULYQYFXDYHXGYQYEXCNYQYDXBWTUHYQYCXAUFYBKUEUSUTZVAVBYQYGXFEYQYDXB
          XEUHYRVCVBVDVEVFUTVFVHVIVJVKAWRVLABCDEIKLNOPVMZVNZVOWSXNTUBXSDTVPZXPY
          AVQWSXNXTTYTYSVRWSXRXMUQZDTVPZUUAWSXMVSUBUUCWSLVSXLIALVSIVTWRPWAWRXLL
          UBAWRXLWBBUDZUBCUDWBWCUMUUDUBCUUDWDUCBUJWILBCEKNWEOWFVKWGDXMWJWKUUBXS
          DTXRXMWLWMWKXSXPDTXNXQXNUQXRXOXMXQXNUOUSWNWOWPWQ $.
      $}

      ${
        $d F a b k n $.  $d F z $.  $d F j $.  $d N a b k n $.  $d G z $.
        $d a b k l n r u $.  $d j l u z $.  $d j ph $.  $d k n ph $.
        $d l u y $.  $d x y $.  $d j n $.  $d z n $.  $d j k $.  $d k z $.
        $( Lemma for ~ axcaucvg .  The result of mapping to ` N. ` and ` R. `
           satisfies the Cauchy condition.  (Contributed by Jim Kingdon,
           9-Jul-2021.) $)
        axcaucvglemcau $p |- ( ph -> A. n e. N. A. k e. N. ( n <N k ->
          ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <.
            { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
            +P. 1P ) , 1P >. ] ~R )
          /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <.
            { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
            { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
            +P. 1P ) , 1P >. ] ~R )
          ) ) ) $=
          ( wbr co cltrr vb va cv clti cfv c1o cop ceq cec crq cltq cab c1p cpp
          cer cplr cltr wa wi cnpi wral wcel c0r cmul c1 wceq crio caddc ltrenn
          cr adantl breq2 fveq2 oveq1d breq2d breq1d anbi12d breq1 oveq1 eqeq1d
          imbi12d riotabidv oveq2d breq12d oveq12d ralbidv sylib ad3antrrr cint
          cbvral2v pitonn eleqtrrdi ad3antlr ad2antlr mpd simpld axcaucvglemval
          rspcdva ad2antrr adantr recriota cnr wf axcaucvglemf simplr ffvelcdmd
          adantlr cnp recnnpr prsrcl syl addresr syl2anc 3brtr3d ltresr simpllr
          eqtrd simprd jca ex ralrimiva ) AHUCZGUCZUDRZYBJUEZYCJUEZMUCZYBUFUGUH
          UIZUJUEZUKRMULYIEUCZUKREULUGZUMUNSUMUGUOUIZUPSZUQRZYFYEYLUPSZUQRZURZU
          SZGUTVAHUTAYBUTVBZURZYRGUTYTYCUTVBZURZYDYQUUBYDURZYNYPUUCYEVCUGZYMVCU
          GZTRYNUUCYGYHUKRMULYHYJUKREULUGUMUNSUMUGUOUIVCUGZIUEZYGYCUFUGUHUIZUKR
          MULUUHYJUKREULUGUMUNSUMUGUOUIVCUGZIUEZUUFLUCZVDSZVEVFZLVJVGZVHSZUUDUU
          ETUUCUUGUUOTRZUUJUUGUUNVHSZTRZUUCUUFUUITRZUUPUURURZYDUUSUUBEYBYCMVIVK
          UUCUUFUAUCZTRZUUGUVAIUEZUUNVHSZTRZUVCUUQTRZURZUSZUUSUUTUSUAKUUIUVAUUI
          VFZUVBUUSUVGUUTUVAUUIUUFTVLUVIUVEUUPUVFUURUVIUVDUUOUUGTUVIUVCUUJUUNVH
          UVAUUIIVMZVNVOUVIUVCUUJUUQTUVJVPVQWAUUCUBUCZUVATRZUVKIUEZUVCUVKUUKVDS
          ZVEVFZLVJVGZVHSZTRZUVCUVMUVPVHSZTRZURZUSZUAKVAZUVHUAKVAUBKUUFUVKUUFVF
          ZUWBUVHUAKUWDUVLUVBUWAUVGUVKUUFUVATVRUWDUVRUVEUVTUVFUWDUVMUUGUVQUVDTU
          VKUUFIVMZUWDUVPUUNUVCVHUWDUVOUUMLVJUWDUVNUULVEUVKUUFUUKVDVSVTWBZWCWDU
          WDUVSUUQUVCTUWDUVMUUGUVPUUNVHUWEUWFWEVOVQWAWFAUWCUBKVAZYSUUAYDAYBYCTR
          ZYBIUEZYCIUEZYBUUKVDSZVEVFZLVJVGZVHSZTRZUWJUWIUWMVHSZTRZURZUSZGKVAHKV
          AUWGPUWSUWBUVKYCTRZUVMUWJUVPVHSZTRZUWJUVSTRZURZUSHGUBUAKKYBUVKVFZUWHU
          WTUWRUXDYBUVKYCTVRUXEUWOUXBUWQUXCUXEUWIUVMUWNUXATYBUVKIVMZUXEUWMUVPUW
          JVHUXEUWLUVOLVJUXEUWKUVNVEYBUVKUUKVDVSVTWBZWCWDUXEUWPUVSUWJTUXEUWIUVM
          UWMUVPVHUXFUXGWEVOVQWAYCUVAVFZUWTUVLUXDUWAYCUVAUVKTVLUXHUXBUVRUXCUVTU
          XHUXAUVQUVMTUXHUWJUVCUVPVHYCUVAIVMZVNVOUXHUWJUVCUVSTUXIVPVQWAWJWGWHYS
          UUFKVBAUUAYDYSUUFVEBUCZVBCUCVEVHSUXJVBCUXJVAURBULWIZKBCEYBMWKNWLWMWRU
          UAUUIKVBYTYDUUAUUIUXKKBCEYCMWKNWLWNWRWOZWPYTUUGUUDVFUUAYDABCDEFGHIJYB
          KLMNOPQWQWSZUUCUUOYFVCUGZYLVCUGZVHSZUUEUUCUUJUXNUUNUXOVHUUBUUJUXNVFZY
          DAUUAUXQYSABCDEFGHIJYCKLMNOPQWQXGWTZYSUUNUXOVFAUUAYDEYBLMXAWMZWEUUCYF
          XBVBYLXBVBZUXPUUEVFUUCUTXBYCJAUTXBJXCYSUUAYDABCDEFGHIJKLMNOPQXDWHZYTU
          UAYDXEXFYSUXTAUUAYDYSYKXHVBUXTEYBMXIYKXJXKWMZYFYLXLXMXQXNYEYMXOWGUUCU
          XNYOVCUGZTRYPUUCUUJUUQUXNUYCTUUCUUPUURUXLXRUXRUUCUUQUUDUXOVHSZUYCUUCU
          UGUUDUUNUXOVHUXMUXSWEUUCYEXBVBUXTUYDUYCVFUUCUTXBYBJUYAAYSUUAYDXPXFUYB
          YEYLXLXMXQXNYFYOXOWGXSXTYAYA $.
      $}

      ${
        $d F b e f g k $.  $d F b e f j k $.  $d F k n $.  $d F b j k y $.
        $d F j z $.  $d G a b c d e f g k x $.  $d G c f g k l u $.
        $d G k l n u $.  $d G z $.  $d N b e f g k $.  $d N b e f j k $.
        $d N k n $.  $d N b j k y $.  $d N x $.  $d a b c e f g k ph x $.
        $d j k l u y $.  $d j k ph x $.  $d k l n r u $.  $d l u z $.
        $d n ph $.  $d x y $.  $d j n $.  $d n z $.  $d k z $.  $d g j $.
        $d g z $.  $d g y $.  $d l x $.  $d u x $.
        $( Lemma for ~ axcaucvg .  Mapping the limit from ` N. ` and ` R. ` .
           (Contributed by Jim Kingdon, 10-Jul-2021.) $)
        axcaucvglemres $p |- ( ph ->
          E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k ->
          ( ( F ` k ) <RR ( y + x ) /\
            y <RR ( ( F ` k ) + x ) ) ) ) ) $=
          ( wbr wa cnpi va vc vb vd ve vf vg c0r cv cltr clti cplr co wral wrex
          cfv cnr cc0 cltrr caddc axcaucvglemf axcaucvglemcau caucvgsr wcel cop
          wi cr opelreal biimpri ad2antrl weq breq2 fveq2 breq1d oveq1d anbi12d
          breq2d imbi12d cbvralv rexbii imbi2i ralbii anbi2i wceq elreal biimpi
          ad2antlr simplrr ad2antrr simprr simplr breq1i ltresr bitr3di syl2anc
          df-0 bitri biimpa oveq2 imbi2d rexralbidv rspcv breq1 ralbidv cbvrexv
          mp2d imbi1d c1o ceq cec cltq cab c1p cpp cer c1 cint pitonn eleqtrrdi
          sylib nntopi adantl adantr rspcdva wb simplrl ltrennb simplll ad4antr
          simprl bitrd axcaucvglemval fveq2d eqtr3d addresr ad3antrrr rexlimddv
          bitr3id ralrimiva rspcev ad5antr oveq2d breq12d syl ffvelcdmd oveq12d
          wf 3imtr3d ex sylan2br oveq1 ) AUHUAUIZUJRZUBUIZGUIZUKRZUUOJUPZUCUIZU
          ULULUMZUJRZUURUUQUULULUMZUJRZSZVFZGTUNZUBTUOZVFZUAUQUNZURBUIZUSRZFUIZ
          UUOUSRZUUOIUPZCUIZUVIUTUMZUSRZUVNUVMUVIUTUMZUSRZSZVFZGKUNFKUOZVFZBVGU
          NZCVGUOZUCUQAUAUCEUBGHJMABCDEFGHIJKLMNOPQVAZABCDEFGHIJKLMNOPQVBVCAUUR
          UQVDZUVHSZSUURUHVEZVGVDZUVJUVLUVMUWHUVIUTUMZUSRZUWHUVQUSRZSZVFZGKUNZF
          KUOZVFZBVGUNZUWDUWFUWIAUVHUWIUWFUURVHVIVJUWGAUWFUUMUUNUDUIZUKRZUWSJUP
          ZUUSUJRZUURUXAUULULUMZUJRZSZVFZUDTUNZUBTUOZVFZUAUQUNZSZUWRUXJUVHUWFUX
          IUVGUAUQUXHUVFUUMUXGUVEUBTUXFUVDUDGTUDGVKZUWTUUPUXEUVCUWSUUOUUNUKVLUX
          LUXBUUTUXDUVBUXLUXAUUQUUSUJUWSUUOJVMZVNUXLUXCUVAUURUJUXLUXAUUQUULULUX
          MVOVQVPVRVSVTWAWBWCAUXKSZUWQBVGUXNUVIVGVDZSZUVJUWPUXPUVJSZUEUIZUHVEZU
          VIWDZUWPUEUQUXOUXTUEUQUOZUXNUVJUXOUYAUEUVIWEWFWGUXQUXRUQVDZUXTSZSZUFU
          IZUWSUKRZUXAUURUXRULUMZUJRZUURUXAUXRULUMZUJRZSZVFZUDTUNZUWPUFTUYDUWTU
          YKVFZUDTUNZUBTUOZUYMUFTUOUYDUXJUHUXRUJRZUYPUXPUXJUVJUYCAUWFUXJUXOWHWI
          UYDUXTUVJUYQUXQUYBUXTWJZUXPUVJUYCWKUXTUVJUYQUXTURUXSUSRZUVJUYQUXSUVIU
          RUSVLUYSUHUHVEZUXSUSRUYQURUYTUXSUSWPWLUHUXRWMWQWNWRWOUYBUXJUYQUYPVFZV
          FUXQUXTUXIVUAUAUXRUQUAUEVKZUUMUYQUXHUYPUULUXRUHUJVLVUBUXFUYNUBUDTTVUB
          UXEUYKUWTVUBUXBUYHUXDUYJVUBUUSUYGUXAUJUULUXRUURULWSVQVUBUXCUYIUURUJUU
          LUXRUXAULWSVQVPWTXAVRXBVJXFUYOUYMUBUFTUBUFVKZUYNUYLUDTVUCUWTUYFUYKUUN
          UYEUWSUKXCXGXDXEXTUYDUYETVDZUYMSZSZMUIZUYEXHVEXIXJZXKRMXLVUHEUIZXKREX
          LVEXMXNUMXMVEXOXJUHVEZKVDZVUJUUOUSRZUWMVFZGKUNZUWPVUDVUKUYDUYMVUDVUJX
          PUVIVDUVNXPUTUMUVIVDCUVIUNSBXLXQKBCEUYEMXRNXSVJVUFVUMGKVUFUUOKVDZSZVU
          GUGUIZXHVEXIXJZXKRMXLVURVUIXKREXLVEXMXNUMXMVEXOXJUHVEZUUOWDZVUMUGTVUO
          VUTUGTUOVUFBCUGEUUOKMNYAYBVUPVUQTVDZVUTSZSZUYEVUQUKRZVUQJUPZUYGUJRZUU
          RVVEUXRULUMZUJRZSZVULUWMVVCUYLVVDVVIVFUDTVUQUDUGVKZUYFVVDUYKVVIUWSVUQ
          UYEUKVLVVJUYHVVFUYJVVHVVJUXAVVEUYGUJUWSVUQJVMZVNVVJUYIVVGUURUJVVJUXAV
          VEUXRULVVKVOVQVPVRVUPUYMVVBUYDVUDUYMVUOWHYCVUPVVAVUTYJZYDVVCVVDVUJVUS
          USRZVULVVCVUDVVAVVDVVMYEVUPVUDVVBUYDVUDUYMVUOYFYCVVLEUYEVUQMYGWOVVCVU
          SUUOVUJUSVUPVVAVUTWJZVQYKVVCVVFUWKVVHUWLVVFVVEUHVEZUYGUHVEZUSRVVCUWKV
          VEUYGWMVVCVVOUVMVVPUWJUSVVCVUSIUPZVVOUVMVVCAVVAVVQVVOWDUXQAUYCVUEVUOV
          VBAUXKUXOUVJYHYIZVVLABCDEFGHIJVUQKLMNOPQYLWOVVCVUSUUOIVVNYMYNZVVCUWHU
          XSUTUMZVVPUWJVVCUWFUYBVVTVVPWDUXPUWFUVJUYCVUEVUOVVBAUWFUXJUXOYFUUAVUF
          UYBVUOVVBUXQUYBUXTVUEYFWIZUURUXRYOWOUYDVVTUWJWDVUEVUOVVBUYDUXSUVIUWHU
          TUYRUUBYPYNUUCYRVVHUWHVVGUHVEZUSRVVCUWLUURVVGWMVVCVWBUVQUWHUSVVCVVOUX
          SUTUMZVWBUVQVVCVVEUQVDUYBVWCVWBWDVVCTUQVUQJVVCATUQJUUGVVRUWEUUDVVLUUE
          VWAVVEUXRYOWOVVCVVOUVMUXSUVIUTVVSUYDUXTVUEVUOVVBUYRYPUUFYNVQYRVPUUHYQ
          YSUWOVUNFVUJKUVKVUJWDZUWNVUMGKVWDUVLVULUWMUVKVUJUUOUSXCXGXDYTWOYQYQUU
          IYSUUJUWCUWRCUWHVGUVNUWHWDZUWBUWQBVGVWEUWAUWPUVJVWEUVTUWNFGKKVWEUVSUW
          MUVLVWEUVPUWKUVRUWLVWEUVOUWJUVMUSUVNUWHUVIUTUUKVQUVNUWHUVQUSXCVPWTXAW
          TXDYTWOYQ $.
      $}
    $}

    ${
      $d F a j k l n u z $.  $d F a b c j k l u x y $.  $d N j k n $.
      $d N x y $.  $d b c j k l n u z $.  $d j k n ph $.  $d k l n r u $.
      $d ph x $.  $d x y $.  $d l x $.  $d u x $.
      $( Real number completeness axiom.  A Cauchy sequence with a modulus of
         convergence converges.  This is basically Corollary 11.2.13 of [HoTT],
         p.  (varies).  The HoTT book theorem has a modulus of convergence
         (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas
         this theorem fixes the rate of convergence to say that all terms after
         the nth term must be within ` 1 / n ` of the nth term (it should later
         be able to prove versions of this theorem with a different fixed rate
         or a modulus of convergence supplied as a hypothesis).

         Because we are stating this axiom before we have introduced notations
         for ` NN ` or division, we use ` N ` for the natural numbers and
         express a reciprocal in terms of ` iota_ ` .

         This construction-dependent theorem should not be referenced directly;
         instead, use ~ ax-caucvg .  (Contributed by Jim Kingdon, 8-Jul-2021.)
         (New usage is discouraged.) $)
      axcaucvg $p |- ( ph ->
          E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k ->
          ( ( F ` k ) <RR ( y + x ) /\
            y <RR ( ( F ` k ) + x ) ) ) ) ) $=
        ( vz vu cv cop cltq c1p c0r wceq vb vc va vl c1o ceq cec wbr cab cpp co
        cnpi cer cfv cnr crio cmpt breq1 cbvabv breq2 oveq1i opeq1i eceq1 ax-mp
        opeq12i fveq2i a1i opeq1 eqeq12d cbvriotav mpteq2i axcaucvglemres ) ABC
        MNDEFGDULUAOZDOUEPUFUGZQUHZUAUIZVNUBOZQUHZUBUIZPZRUJUKZRPZUMUGZSPZGUNZU
        COZSPZTZUCUOUPZUQHIUDJKLDULWIUDOZVNQUHZUDUIZVNNOZQUHZNUIZPZRUJUKZRPZUMU
        GZSPZGUNZMOZSPZTZMUOUPWHXDUCMUOWFXBTZWEXAWGXCWEXATXEWDWTGWCWSSWBWRTWCWS
        TWAWQRVTWPRUJVPWLVSWOVOWKUAUDVMWJVNQURUSVRWNUBNVQWMVNQUTUSVEVAVBWBWRUMV
        CVDVBVFVGWFXBSVHVIVJVKVL $.
    $}
  $}

  ${
    $d A a b c y z $.  $d A b c w y z $.  $d A a b x y z $.  $d B a b c y z $.
    $d B a b x y z $.  $d C a $.  $d C w $.  $d a b c ph y z $.  $d ph x y z $.
    axpre-suploclem.ss $e |- ( ph -> A C_ RR ) $.
    axpre-suploclem.m $e |- ( ph -> C e. A ) $.
    axpre-suploclem.ub $e |- ( ph -> E. x e. RR A. y e. A y <RR x ) $.
    axpre-suploclem.loc $e |- ( ph -> A. x e. RR A. y e. RR ( x <RR y
      -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) ) $.
    axpre-suploclem.b $e |- B = { w e. R. | <. w , 0R >. e. A } $.
    $( Lemma for ~ axpre-suploc .  The result.  The proof just needs to define
       ` B ` as basically the same set as ` A ` (but expressed as a subset of
       ` R. ` rather than a subset of ` RR ` ), and apply ~ suplocsr .
       (Contributed by Jim Kingdon, 24-Jan-2024.) $)
    axpre-suploclemres $p |- ( ph -> E. x e. RR ( A. y e. A -. x <RR y /\
        A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) ) $=
      ( vc wbr cnr wa cltrr cr wcel va vb cv cltr wn wral wrex c1st cfv wex c0r
      wi wceq sseldd elreal2 sylib simpld simprd eqeltrrd opeq1 eleq1d sylanbrc
      cop elrab2 eleq1 spcegv sylc simprl breq1 simplrr rspcdva simplrl breqtrd
      simpr ltresr ralrimiva brralrspcev syl2anc rexlimddv sylibr breq2 ralbidv
      orbi2d imbi12d rexbidv orbi1d ad2antrr opelreal mpd simplll simprr rspcev
      sseld rexlimdvaa simplr orim12d ralrimivva suplocsr notbid sselda adantlr
      wo ex breq2d bitrdi mtbird adantrrr eqbrtrrd nfcv nfre1 nfim nfralya nfan
      nfv adantr eqbrtrd exp32 rexlimd adantrrl imbi1d anbi12d syl12anc ) AUAUC
      ZUBUCZUDOZUEZUBGUFZYDYCUDOZYDNUCZUDOZNGUGZULZUBPUFZQZBUCZCUCZROZUEZCFUFZY
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      TZUYTUKVCZFTZVUCVUDVUEYPVUFUMZVUDYPSTZVUEVUHQZAFSYPIWTYPUOZUPZUQVUDYPVUFF
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      SWBUXCUUDUYNCSUXCYTUYMUUCYOUWEYPRWAXTWBYAWLYBVS $.
  $}

  ${
    $d A a b c d w $.  $d A a b c d x y z $.
    $( An inhabited, bounded-above, located set of reals has a supremum.

       Locatedness here means that given ` x < y ` , either there is an element
       of the set greater than ` x ` , or ` y ` is an upper bound.

       This construction-dependent theorem should not be referenced directly;
       instead, use ~ ax-pre-suploc .  (Contributed by Jim Kingdon,
       23-Jan-2024.)  (New usage is discouraged.) $)
    axpre-suploc $p |- ( ( ( A C_ RR /\ E. x x e. A )
        /\ ( E. x e. RR A. y e. A y <RR x
        /\ A. x e. RR A. y e. RR ( x <RR y
          -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) ) )
        -> E. x e. RR ( A. y e. A -. x <RR y /\
          A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) ) $=
      ( vd va vb vc cr cv wa cltrr wbr wral wrex wo wi breq2 ralbidv breq1 wcel
      vw wss wex wn simplr eleq1w cbvexv sylib c0r cop cnr crab simplll simplrl
      simpr cbvrexv cbvralv rexbii sylibr simplrr rexbidv orbi1d imbi12d orbi2d
      weq cbvral2v orbi12i imbi2i 2ralbii eqid axpre-suploclemres notbid imbi1d
      bitri anbi12d anbi12i ralbii anbi2i exlimddv ) DIUCZAJZDUAZAUDZKZBJZWBLMZ
      BDNZAIOZWBWFLMZWBCJZLMZCDOZWKWFLMZCDNZPZQZBINAINZKZKZEJZDUAZWJUEZBDNZWGWF
      WKLMZCDOZQZBINZKZAIOZEWTWDXBEUDWAWDWSUFWCXBAEAEDUGUHUIWTXBKZFJZGJZLMZUEZG
      DNZXMXLLMZXMHJZLMZHDOZQZGINZKZFIOZXJXKFGHUBDUBJUJUKDUAUBULUMZXAWAWDWSXBUN
      WTXBUPXKWIXQGDNZFIOZWEWIWRXBUOYGXMWBLMZGDNZAIOWIYFYIFAIFAVFZXQYHGDXLWBXML
      RZSUQYIWHAIYHWGGBDXMWFWBLTZURUSVOUTXKWRXNXLXRLMZHDOZXRXMLMZHDNZPZQZGINFIN
      ZWEWIWRXBVAYSWJWBXRLMZHDOZXRWFLMZHDNZPZQZBINAINWRYRUUEWBXMLMZUUAYPPZQFGAB
      IIYJXNUUFYQUUGXLWBXMLTZYJYNUUAYPYJYMYTHDXLWBXRLTVBVCVDGBVFZUUFWJUUGUUDXMW
      FWBLRZUUIYPUUCUUAUUIYOUUBHDXMWFXRLRSVEVDVGUUEWQABIIUUDWPWJUUAWMUUCWOYTWLH
      CDXRWKWBLRUQUUBWNHCDXRWKWFLTURVHVIVJVOUTYEVKVLYDXDWGWFXRLMZHDOZQZBINZKZAI
      OZXJYDUUFUEZGDNZYHXTQZGINZKZAIOUUPYCUVAFAIYJXPUURYBUUTYJXOUUQGDYJXNUUFUUH
      VMSYJYAUUSGIYJXQYHXTYKVNSVPUQUVAUUOAIUURXDUUTUUNUUQXCGBDUUIUUFWJUUJVMURUU
      SUUMGBIUUIYHWGXTUULYLUUIXSUUKHDXMWFXRLTVBVDURVQUSVOUUOXIAIUUNXHXDUUMXGBIU
      ULXFWGUUKXEHCDXRWKWFLRUQVIVRVSUSVOUIVT $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Real and complex number postulates restated as axioms
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( The complex numbers form a set.  Proofs should normally use ~ cnex
     instead.  (New usage is discouraged.)  (Contributed by NM, 1-Mar-1995.) $)
  ax-cnex $a |- CC e. _V $.
  $( $j restatement 'ax-cnex' of 'axcnex'; $)

  $( The real numbers are a subset of the complex numbers.  Axiom for real and
     complex numbers, justified by Theorem ~ axresscn .  (Contributed by NM,
     1-Mar-1995.) $)
  ax-resscn $a |- RR C_ CC $.
  $( $j restatement 'ax-resscn' of 'axresscn'; $)

  $( 1 is a complex number.  Axiom for real and complex numbers, justified by
     Theorem ~ ax1cn .  (Contributed by NM, 1-Mar-1995.) $)
  ax-1cn $a |- 1 e. CC $.
  $( $j restatement 'ax-1cn' of 'ax1cn'; $)

  $( 1 is a real number.  Axiom for real and complex numbers, justified by
     Theorem ~ ax1re .  Proofs should use ~ 1re instead.  (Contributed by Jim
     Kingdon, 13-Jan-2020.)  (New usage is discouraged.) $)
  ax-1re $a |- 1 e. RR $.
  $( $j restatement 'ax-1re' of 'ax1re'; $)

  $( ` _i ` is a complex number.  Axiom for real and complex numbers, justified
     by Theorem ~ axicn .  (Contributed by NM, 1-Mar-1995.) $)
  ax-icn $a |- _i e. CC $.
  $( $j restatement 'ax-icn' of 'axicn'; $)

  $( Closure law for addition of complex numbers.  Axiom for real and complex
     numbers, justified by Theorem ~ axaddcl .  Proofs should normally use
     ~ addcl instead, which asserts the same thing but follows our naming
     conventions for closures.  (New usage is discouraged.)  (Contributed by
     NM, 22-Nov-1994.) $)
  ax-addcl $a |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) $.
  $( $j restatement 'ax-addcl' of 'axaddcl'; $)

  $( Closure law for addition in the real subfield of complex numbers.  Axiom
     for real and complex numbers, justified by Theorem ~ axaddrcl .  Proofs
     should normally use ~ readdcl instead.  (New usage is discouraged.)
     (Contributed by NM, 22-Nov-1994.) $)
  ax-addrcl $a |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) $.
  $( $j restatement 'ax-addrcl' of 'axaddrcl'; $)

  $( Closure law for multiplication of complex numbers.  Axiom for real and
     complex numbers, justified by Theorem ~ axmulcl .  Proofs should normally
     use ~ mulcl instead.  (New usage is discouraged.)  (Contributed by NM,
     22-Nov-1994.) $)
  ax-mulcl $a |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC ) $.
  $( $j restatement 'ax-mulcl' of 'axmulcl'; $)

  $( Closure law for multiplication in the real subfield of complex numbers.
     Axiom for real and complex numbers, justified by Theorem ~ axmulrcl .
     Proofs should normally use ~ remulcl instead.  (New usage is discouraged.)
     (Contributed by NM, 22-Nov-1994.) $)
  ax-mulrcl $a |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) $.
  $( $j restatement 'ax-mulrcl' of 'axmulrcl'; $)

  $( Addition is commutative.  Axiom for real and complex numbers, justified by
     Theorem ~ axaddcom .  Proofs should normally use ~ addcom instead.
     (New usage is discouraged.)  (Contributed by Jim Kingdon, 17-Jan-2020.) $)
  ax-addcom $a |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) $.
  $( $j restatement 'ax-addcom' of 'axaddcom'; $)

  $( Multiplication of complex numbers is commutative.  Axiom for real and
     complex numbers, justified by Theorem ~ axmulcom .  Proofs should normally
     use ~ mulcom instead.  (New usage is discouraged.)  (Contributed by NM,
     22-Nov-1994.) $)
  ax-mulcom $a |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) $.
  $( $j restatement 'ax-mulcom' of 'axmulcom'; $)

  $( Addition of complex numbers is associative.  Axiom for real and complex
     numbers, justified by Theorem ~ axaddass .  Proofs should normally use
     ~ addass instead.  (New usage is discouraged.)  (Contributed by NM,
     22-Nov-1994.) $)
  ax-addass $a |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( ( A + B ) + C ) = ( A + ( B + C ) ) ) $.
  $( $j restatement 'ax-addass' of 'axaddass'; $)

  $( Multiplication of complex numbers is associative.  Axiom for real and
     complex numbers, justified by Theorem ~ axmulass .  Proofs should normally
     use ~ mulass instead.  (New usage is discouraged.)  (Contributed by NM,
     22-Nov-1994.) $)
  ax-mulass $a |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) $.
  $( $j restatement 'ax-mulass' of 'axmulass'; $)

  $( Distributive law for complex numbers (left-distributivity).  Axiom for
     real and complex numbers, justified by Theorem ~ axdistr .  Proofs should
     normally use ~ adddi instead.  (New usage is discouraged.)  (Contributed
     by NM, 22-Nov-1994.) $)
  ax-distr $a |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
              ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) ) $.
  $( $j restatement 'ax-distr' of 'axdistr'; $)

  $( i-squared equals -1 (expressed as i-squared plus 1 is 0).  Axiom for real
     and complex numbers, justified by Theorem ~ axi2m1 .  (Contributed by NM,
     29-Jan-1995.) $)
  ax-i2m1 $a |- ( ( _i x. _i ) + 1 ) = 0 $.
  $( $j restatement 'ax-i2m1' of 'axi2m1'; $)

  $( 0 is less than 1.  Axiom for real and complex numbers, justified by
     Theorem ~ ax0lt1 .  Proofs should normally use ~ 0lt1 instead.
     (New usage is discouraged.)  (Contributed by Jim Kingdon, 12-Jan-2020.) $)
  ax-0lt1 $a |- 0 <RR 1 $.
  $( $j restatement 'ax-0lt1' of 'ax0lt1'; $)

  $( ` 1 ` is an identity element for real multiplication.  Axiom for real and
     complex numbers, justified by Theorem ~ ax1rid .  (Contributed by NM,
     29-Jan-1995.) $)
  ax-1rid $a |- ( A e. RR -> ( A x. 1 ) = A ) $.
  $( $j restatement 'ax-1rid' of 'ax1rid'; $)

  $( ` 0 ` is an identity element for real addition.  Axiom for real and
     complex numbers, justified by Theorem ~ ax0id .

     Proofs should normally use ~ addrid instead.  (New usage is discouraged.)
     (Contributed by Jim Kingdon, 16-Jan-2020.) $)
  ax-0id $a |- ( A e. CC -> ( A + 0 ) = A ) $.
  $( $j restatement 'ax-0id' of 'ax0id'; $)

  ${
    $d x A $.
    $( Existence of negative of real number.  Axiom for real and complex
       numbers, justified by Theorem ~ axrnegex .  (Contributed by Eric
       Schmidt, 21-May-2007.) $)
    ax-rnegex $a |- ( A e. RR -> E. x e. RR ( A + x ) = 0 ) $.
    $( $j restatement 'ax-rnegex' of 'axrnegex'; $)
  $}

  ${
    $d x A $.
    $( Existence of reciprocal of positive real number.  Axiom for real and
       complex numbers, justified by Theorem ~ axprecex .  (Contributed by Jim
       Kingdon, 6-Feb-2020.) $)
    ax-precex $a |- ( ( A e. RR /\ 0 <RR A ) ->
        E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) ) $.
    $( $j restatement 'ax-precex' of 'axprecex'; $)
  $}

  ${
    $d x y A $.
    $( A complex number can be expressed in terms of two reals.  Definition
       10-1.1(v) of [Gleason] p. 130.  Axiom for real and complex numbers,
       justified by Theorem ~ axcnre .  For naming consistency, use ~ cnre for
       new proofs.  (New usage is discouraged.)  (Contributed by NM,
       9-May-1999.) $)
    ax-cnre $a |- ( A e. CC ->
                      E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) $.
    $( $j restatement 'ax-cnre' of 'axcnre'; $)
  $}

  $( Real number less-than is irreflexive.  Axiom for real and complex numbers,
     justified by Theorem ~ ax-pre-ltirr .  (Contributed by Jim Kingdon,
     12-Jan-2020.) $)
  ax-pre-ltirr $a |- ( A e. RR -> -. A <RR A ) $.
  $( $j restatement 'ax-pre-ltirr' of 'axpre-ltirr'; $)

  $( Real number less-than is weakly linear.  Axiom for real and complex
     numbers, justified by Theorem ~ axpre-ltwlin .  (Contributed by Jim
     Kingdon, 12-Jan-2020.) $)
  ax-pre-ltwlin $a |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
    ( A <RR B -> ( A <RR C \/ C <RR B ) ) ) $.
  $( $j restatement 'ax-pre-ltwlin' of 'axpre-ltwlin'; $)

  $( Ordering on reals is transitive.  Axiom for real and complex numbers,
     justified by Theorem ~ axpre-lttrn .  (Contributed by NM, 13-Oct-2005.) $)
  ax-pre-lttrn $a |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
             ( ( A <RR B /\ B <RR C ) -> A <RR C ) ) $.
  $( $j restatement 'ax-pre-lttrn' of 'axpre-lttrn'; $)

  $( Apartness of reals is tight.  Axiom for real and complex numbers,
     justified by Theorem ~ axpre-apti .  (Contributed by Jim Kingdon,
     29-Jan-2020.) $)
  ax-pre-apti $a |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) ->
    A = B ) $.
  $( $j restatement 'ax-pre-apti' of 'axpre-apti'; $)

  $( Ordering property of addition on reals.  Axiom for real and complex
     numbers, justified by Theorem ~ axpre-ltadd .  (Contributed by NM,
     13-Oct-2005.) $)
  ax-pre-ltadd $a |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A <RR B -> ( C + A ) <RR ( C + B ) ) ) $.
  $( $j restatement 'ax-pre-ltadd' of 'axpre-ltadd'; $)

  $( The product of two positive reals is positive.  Axiom for real and complex
     numbers, justified by Theorem ~ axpre-mulgt0 .  (Contributed by NM,
     13-Oct-2005.) $)
  ax-pre-mulgt0 $a |- ( ( A e. RR /\ B e. RR ) ->
                ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) ) $.
  $( $j restatement 'ax-pre-mulgt0' of 'axpre-mulgt0'; $)

  $( Strong extensionality of multiplication (expressed in terms of ` <RR ` ).
     Axiom for real and complex numbers, justified by Theorem ~ axpre-mulext

     (Contributed by Jim Kingdon, 18-Feb-2020.) $)
  ax-pre-mulext $a |- ( ( A e. RR /\ B e. RR /\ C e. RR )
      -> ( ( A x. C ) <RR ( B x. C ) -> ( A <RR B \/ B <RR A ) ) ) $.
  $( $j restatement 'ax-pre-mulext' of 'axpre-mulext'; $)

  ${
    $d A n x y $.

    $( Archimedean axiom.  Definition 3.1(2) of [Geuvers], p. 9.  Axiom for
       real and complex numbers, justified by Theorem ~ axarch .

       This axiom should not be used directly; instead use ~ arch (which is the
       same, but stated in terms of ` NN ` and ` < ` ).  (Contributed by Jim
       Kingdon, 2-May-2020.)  (New usage is discouraged.) $)
    ax-arch $a |- ( A e. RR -> E. n e.
        |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n ) $.
    $( $j restatement 'ax-arch' of 'axarch'; $)
  $}

  ${
    ax-caucvg.n $e |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
    ax-caucvg.f $e |- ( ph -> F : N --> RR ) $.
    ax-caucvg.cau $e |- ( ph ->
      A. n e. N A. k e. N ( n <RR k ->
        ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\
        ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) ) $.

    $d F j k n $.  $d F j k x y $.  $d N j k n $.  $d N x y $.  $d j k n $.
    $d j k n ph $.  $d k n r $.  $d ph x $.  $d x y $.
    $( Completeness.  Axiom for real and complex numbers, justified by Theorem
       ~ axcaucvg .

       A Cauchy sequence (as defined here, which has a rate convergence built
       in) of real numbers converges to a real number.  Specifically on rate of
       convergence, all terms after the nth term must be within ` 1 / n ` of
       the nth term.

       This axiom should not be used directly; instead use ~ caucvgre (which is
       the same, but stated in terms of the ` NN ` and ` 1 / n ` notations).
       (Contributed by Jim Kingdon, 19-Jul-2021.)
       (New usage is discouraged.) $)
    ax-caucvg $a |- ( ph ->
      E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k ->
      ( ( F ` k ) <RR ( y + x ) /\
        y <RR ( ( F ` k ) + x ) ) ) ) ) $.

    $( $j restatement 'ax-caucvg' of 'axcaucvg'; $)
  $}

  ${
    $d A x y z $.
    $( An inhabited, bounded-above, located set of reals has a supremum.

       Locatedness here means that given ` x < y ` , either there is an element
       of the set greater than ` x ` , or ` y ` is an upper bound.

       Although this and ~ ax-caucvg are both completeness properties,
       countable choice would probably be needed to derive this from
       ~ ax-caucvg .

       (Contributed by Jim Kingdon, 23-Jan-2024.) $)
    ax-pre-suploc $a |- ( ( ( A C_ RR /\ E. x x e. A )
        /\ ( E. x e. RR A. y e. A y <RR x
        /\ A. x e. RR A. y e. RR ( x <RR y
          -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) ) )
        -> E. x e. RR ( A. y e. A -. x <RR y /\
          A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) ) $.

    $( $j restatement 'ax-pre-suploc' of 'axpre-suploc'; $)
  $}

  $( Addition is an operation on the complex numbers.  This deprecated axiom is
     provided for historical compatibility but is not a bona fide axiom for
     complex numbers (independent of set theory) since it cannot be interpreted
     as a first- or second-order statement (see
     ~ https://us.metamath.org/downloads/schmidt-cnaxioms.pdf ).  It may be
     deleted in the future and should be avoided for new theorems.  Instead,
     the less specific ~ addcl should be used.  Note that uses of ~ ax-addf can
     be eliminated by using the defined operation
     ` ( x e. CC , y e. CC |-> ( x + y ) ) ` in place of ` + ` , from which
     this axiom (with the defined operation in place of ` + ` ) follows as a
     theorem.

     This axiom is justified by Theorem ~ axaddf .  (New usage is discouraged.)
     (Contributed by NM, 19-Oct-2004.) $)
  ax-addf $a |- + : ( CC X. CC ) --> CC $.
  $( $j restatement 'ax-addf' of 'axaddf'; $)

  $( Multiplication is an operation on the complex numbers.  This axiom tells
     us that ` x. ` is defined only on complex numbers which is analogous to
     the way that other operations are defined, for example see ~ subf or
     ~ eff .  However, while Metamath can handle this axiom, if we wish to work
     with weaker complex number axioms, we can avoid it by using the less
     specific ~ mulcl .  Note that uses of ~ ax-mulf can be eliminated by using
     the defined operation ` ( x e. CC , y e. CC |-> ( x x. y ) ) ` in place of
     ` x. ` , as seen in ~ mpomulf .

     This axiom is justified by Theorem ~ axmulf .  (New usage is discouraged.)
     (Contributed by NM, 19-Oct-2004.) $)
  ax-mulf $a |- x. : ( CC X. CC ) --> CC $.
  $( $j restatement 'ax-mulf' of 'axmulf'; $)


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Derive the basic properties from the field axioms
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Some deductions from the field axioms for complex numbers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Alias for ~ ax-cnex .  (Contributed by Mario Carneiro, 17-Nov-2014.) $)
  cnex $p |- CC e. _V $=
    ( ax-cnex ) A $.

  $( Alias for ~ ax-addcl , for naming consistency with ~ addcli .  Use this
     theorem instead of ~ ax-addcl or ~ axaddcl .  (Contributed by NM,
     10-Mar-2008.) $)
  addcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) $=
    ( ax-addcl ) ABC $.

  $( Alias for ~ ax-addrcl , for naming consistency with ~ readdcli .
     (Contributed by NM, 10-Mar-2008.) $)
  readdcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) $=
    ( ax-addrcl ) ABC $.

  $( Alias for ~ ax-mulcl , for naming consistency with ~ mulcli .
     (Contributed by NM, 10-Mar-2008.) $)
  mulcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC ) $=
    ( ax-mulcl ) ABC $.

  $( Alias for ~ ax-mulrcl , for naming consistency with ~ remulcli .
     (Contributed by NM, 10-Mar-2008.) $)
  remulcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) $=
    ( ax-mulrcl ) ABC $.

  $( Alias for ~ ax-mulcom , for naming consistency with ~ mulcomi .
     (Contributed by NM, 10-Mar-2008.) $)
  mulcom $p |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) $=
    ( ax-mulcom ) ABC $.

  $( Alias for ~ ax-addass , for naming consistency with ~ addassi .
     (Contributed by NM, 10-Mar-2008.) $)
  addass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                 ( ( A + B ) + C ) = ( A + ( B + C ) ) ) $=
    ( ax-addass ) ABCD $.

  $( Alias for ~ ax-mulass , for naming consistency with ~ mulassi .
     (Contributed by NM, 10-Mar-2008.) $)
  mulass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                 ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) $=
    ( ax-mulass ) ABCD $.

  $( Alias for ~ ax-distr , for naming consistency with ~ adddii .
     (Contributed by NM, 10-Mar-2008.) $)
  adddi $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) ) $=
    ( ax-distr ) ABCD $.

  $( A real number is a complex number.  (Contributed by NM, 10-Aug-1999.) $)
  recn $p |- ( A e. RR -> A e. CC ) $=
    ( cr cc ax-resscn sseli ) BCADE $.

  $( The real numbers form a set.  (Contributed by Mario Carneiro,
     17-Nov-2014.) $)
  reex $p |- RR e. _V $=
    ( cr cc cnex ax-resscn ssexi ) ABCDE $.

  $( Reals are a subset of the pair of real and complex numbers (common case).
     (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  reelprrecn $p |- RR e. { RR , CC } $=
    ( cr cc reex prid1 ) ABCD $.

  $( Complex numbers are a subset of the pair of real and complex numbers
     (common case).  (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  cnelprrecn $p |- CC e. { RR , CC } $=
    ( cr cc cnex prid2 ) ABCD $.

  ${
    $d x y z $.
    $( Multiplication is an operation on complex numbers.  Version of ~ ax-mulf
       using maps-to notation, proved from the axioms of set theory and
       ~ ax-mulcl .  (Contributed by GG, 16-Mar-2025.) $)
    mpomulf $p |- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC $=
      ( vz cc cxp cv cmul co cmpo wf wfn wss wcel wral mulcl rgen2 fnmpo coprab
      eqid wa ax-mp wceq w3a simpll simplr eleq1a syl imp 3jca ssoprab2i df-mpo
      wi dfxp3 3sstr4i dff2 mpbir2an ) DDEZDABDDAFZBFZGHZIZJVAUQKZVAUQDEZLUTDMZ
      BDNADNVBVDABDDURUSOZPABDDUTVADVASQUAURDMZUSDMZTZCFZUTUBZTZABCRVFVGVIDMZUC
      ZABCRVAVCVKVMABCVKVFVGVLVFVGVJUDVFVGVJUEVHVJVLVHVDVJVLULVEUTDVIUFUGUHUIUJ
      ABCDDUTUKABCDDDUMUNUQDVAUOUP $.
    $( $j usage 'mpomulf' avoids 'ax-mulf'; $)
  $}

  $( Distributive law for complex numbers (right-distributivity).  (Contributed
     by NM, 10-Oct-2004.) $)
  adddir $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
              ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) ) $=
    ( cc wcel w3a caddc co cmul adddi 3coml wa addcl mulcom sylan 3impa 3adant2
    wceq 3adant1 oveq12d 3eqtr4d ) ADEZBDEZCDEZFZCABGHZIHZCAIHZCBIHZGHZUFCIHZAC
    IHZBCIHZGHUDUBUCUGUJRCABJKUBUCUDUKUGRZUBUCLUFDEUDUNABMUFCNOPUEULUHUMUIGUBUD
    ULUHRUCACNQUCUDUMUIRUBBCNSTUA $.

  $( 0 is a complex number.  (Contributed by NM, 19-Feb-2005.) $)
  0cn $p |- 0 e. CC $=
    ( ci cmul co c1 caddc cc0 cc ax-i2m1 wcel ax-icn mulcl mp2an addcl eqeltrri
    ax-1cn ) AABCZDECZFGHPGIZDGIQGIAGIZSRJJAAKLOPDMLN $.

  $( 0 is a complex number, deductive form.  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  0cnd $p |- ( ph -> 0 e. CC ) $=
    ( cc0 cc wcel 0cn a1i ) BCDAEF $.

  $( 0 is a set (common case).  (Contributed by David A. Wheeler,
     7-Jul-2016.) $)
  c0ex $p |- 0 e. _V $=
    ( cc0 cc 0cn elexi ) ABCD $.

  $( 1 is a set.  Common special case.  (Contributed by David A. Wheeler,
     7-Jul-2016.) $)
  1ex $p |- 1 e. _V $=
    ( c1 cc ax-1cn elexi ) ABCD $.

  ${
    $d A x y $.
    $( Alias for ~ ax-cnre , for naming consistency.  (Contributed by NM,
       3-Jan-2013.) $)
    cnre $p |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) $=
      ( ax-cnre ) ABCD $.
  $}

  ${
    $d A x y $.
    $( ` 1 ` is an identity element for multiplication.  Based on ideas by Eric
       Schmidt.  (Contributed by Scott Fenton, 3-Jan-2013.) $)
    mulrid $p |- ( A e. CC -> ( A x. 1 ) = A ) $=
      ( vx vy cc wcel cv ci cmul co caddc wceq cr wrex c1 ax-icn ax-1cn ax-1rid
      recn syl eqtrd cnre wa sylancr adddir mp3an3 syl2an mulass mp3an13 oveq2d
      mulcl oveqan12d oveq1 id eqeq12d syl5ibrcom rexlimivv ) ADEABFZGCFZHIZJIZ
      KZCLMBLMANHIZAKZBCAUAVAVCBCLLUQLEZURLEZUBZVCVAUTNHIZUTKVFVGUQNHIZUSNHIZJI
      ZUTVDUQDEZUSDEZVGVJKZVEUQRVEGDEZURDEZVLOURRZGURUJUCVKVLNDEZVMPUQUSNUDUEUF
      VDVEVHUQVIUSJUQQVEVIGURNHIZHIZUSVEVOVIVSKZVPVNVOVQVTOPGURNUGUHSVEVRURGHUR
      QUITUKTVAVBVGAUTAUTNHULVAUMUNUOUPS $.
  $}

  $( Identity law for multiplication.  Note: see ~ mulrid for commuted version.
     (Contributed by NM, 8-Oct-1999.) $)
  mullid $p |- ( A e. CC -> ( 1 x. A ) = A ) $=
    ( cc wcel c1 cmul co wceq ax-1cn mulcom mpan mulrid eqtrd ) ABCZDAEFZADEFZA
    DBCMNOGHDAIJAKL $.

  $( ` 1 ` is a real number.  (Contributed by Jim Kingdon, 13-Jan-2020.) $)
  1re $p |- 1 e. RR $=
    ( ax-1re ) A $.

  $( ` 0 ` is a real number.  (Contributed by Eric Schmidt, 21-May-2007.)
     (Revised by Scott Fenton, 3-Jan-2013.) $)
  0re $p |- 0 e. RR $=
    ( vx c1 cr wcel cv caddc cc0 wceq wrex 1re ax-rnegex readdcl mpan syl5ibcom
    co eleq1 rexlimiv mp2b ) BCDZBAEZFOZGHZACIGCDZJABKUBUCACTCDZUACDZUBUCSUDUEJ
    BTLMUAGCPNQR $.

  $( ` 0 ` is a real number, deductive form.  (Contributed by David A. Wheeler,
     6-Dec-2018.) $)
  0red $p |- ( ph -> 0 e. RR ) $=
    ( cc0 cr wcel 0re a1i ) BCDAEF $.

  ${
    axi.1 $e |- A e. CC $.
    $( Identity law for multiplication.  (Contributed by NM, 14-Feb-1995.) $)
    mulridi $p |- ( A x. 1 ) = A $=
      ( cc wcel c1 cmul co wceq mulrid ax-mp ) ACDAEFGAHBAIJ $.

    $( Identity law for multiplication.  (Contributed by NM, 14-Feb-1995.) $)
    mullidi $p |- ( 1 x. A ) = A $=
      ( cc wcel c1 cmul co wceq mullid ax-mp ) ACDEAFGAHBAIJ $.

    axi.2 $e |- B e. CC $.
    $( Closure law for addition.  (Contributed by NM, 23-Nov-1994.) $)
    addcli $p |- ( A + B ) e. CC $=
      ( cc wcel caddc co addcl mp2an ) AEFBEFABGHEFCDABIJ $.

    $( Closure law for multiplication.  (Contributed by NM, 23-Nov-1994.) $)
    mulcli $p |- ( A x. B ) e. CC $=
      ( cc wcel cmul co mulcl mp2an ) AEFBEFABGHEFCDABIJ $.

    $( Commutative law for multiplication.  (Contributed by NM,
       23-Nov-1994.) $)
    mulcomi $p |- ( A x. B ) = ( B x. A ) $=
      ( cc wcel cmul co wceq mulcom mp2an ) AEFBEFABGHBAGHICDABJK $.

    ${
      mulcomli.3 $e |- ( A x. B ) = C $.
      $( Commutative law for multiplication.  (Contributed by NM,
         23-Nov-1994.) $)
      mulcomli $p |- ( B x. A ) = C $=
        ( cmul co mulcomi eqtri ) BAGHABGHCBAEDIFJ $.
    $}

    axi.3 $e |- C e. CC $.
    $( Associative law for addition.  (Contributed by NM, 23-Nov-1994.) $)
    addassi $p |- ( ( A + B ) + C ) = ( A + ( B + C ) ) $=
      ( cc wcel caddc co wceq addass mp3an ) AGHBGHCGHABIJCIJABCIJIJKDEFABCLM
      $.

    $( Associative law for multiplication.  (Contributed by NM,
       23-Nov-1994.) $)
    mulassi $p |- ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) $=
      ( cc wcel cmul co wceq mulass mp3an ) AGHBGHCGHABIJCIJABCIJIJKDEFABCLM $.

    $( Distributive law (left-distributivity).  (Contributed by NM,
       23-Nov-1994.) $)
    adddii $p |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) $=
      ( cc wcel caddc co cmul wceq adddi mp3an ) AGHBGHCGHABCIJKJABKJACKJIJLDEF
      ABCMN $.

    $( Distributive law (right-distributivity).  (Contributed by NM,
       16-Feb-1995.) $)
    adddiri $p |- ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) $=
      ( cc wcel caddc co cmul wceq adddir mp3an ) AGHBGHCGHABIJCKJACKJBCKJIJLDE
      FABCMN $.
  $}

  ${
    recni.1 $e |- A e. RR $.
    $( A real number is a complex number.  (Contributed by NM, 1-Mar-1995.) $)
    recni $p |- A e. CC $=
      ( cr cc ax-resscn sselii ) CDAEBF $.

    axri.2 $e |- B e. RR $.
    $( Closure law for addition of reals.  (Contributed by NM, 17-Jan-1997.) $)
    readdcli $p |- ( A + B ) e. RR $=
      ( cr wcel caddc co readdcl mp2an ) AEFBEFABGHEFCDABIJ $.

    $( Closure law for multiplication of reals.  (Contributed by NM,
       17-Jan-1997.) $)
    remulcli $p |- ( A x. B ) e. RR $=
      ( cr wcel cmul co remulcl mp2an ) AEFBEFABGHEFCDABIJ $.
  $}

  $( 1 is an real number, deductive form (common case).  (Contributed by David
     A. Wheeler, 6-Dec-2018.) $)
  1red $p |- ( ph -> 1 e. RR ) $=
    ( c1 cr wcel 1re a1i ) BCDAEF $.

  $( 1 is a complex number, deductive form (common case).  (Contributed by
     David A. Wheeler, 6-Dec-2018.) $)
  1cnd $p |- ( ph -> 1 e. CC ) $=
    ( c1 cc wcel ax-1cn a1i ) BCDAEF $.

  ${
    addcld.1 $e |- ( ph -> A e. CC ) $.
    $( Identity law for multiplication.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mulridd $p |- ( ph -> ( A x. 1 ) = A ) $=
      ( cc wcel c1 cmul co wceq mulrid syl ) ABDEBFGHBICBJK $.

    $( Identity law for multiplication.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mullidd $p |- ( ph -> ( 1 x. A ) = A ) $=
      ( cc wcel c1 cmul co wceq mullid syl ) ABDEFBGHBICBJK $.

    addcld.2 $e |- ( ph -> B e. CC ) $.
    $( Closure law for addition.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    addcld $p |- ( ph -> ( A + B ) e. CC ) $=
      ( cc wcel caddc co addcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Closure law for multiplication.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mulcld $p |- ( ph -> ( A x. B ) e. CC ) $=
      ( cc wcel cmul co mulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Commutative law for multiplication.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mulcomd $p |- ( ph -> ( A x. B ) = ( B x. A ) ) $=
      ( cc wcel cmul co wceq mulcom syl2anc ) ABFGCFGBCHICBHIJDEBCKL $.

    addassd.3 $e |- ( ph -> C e. CC ) $.
    $( Associative law for addition.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    addassd $p |- ( ph -> ( ( A + B ) + C ) = ( A + ( B + C ) ) ) $=
      ( cc wcel caddc co wceq addass syl3anc ) ABHICHIDHIBCJKDJKBCDJKJKLEFGBCDM
      N $.

    $( Associative law for multiplication.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mulassd $p |- ( ph -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) $=
      ( cc wcel cmul co wceq mulass syl3anc ) ABHICHIDHIBCJKDJKBCDJKJKLEFGBCDMN
      $.

    $( Distributive law (left-distributivity).  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    adddid $p |- ( ph -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) ) $=
      ( cc wcel caddc co cmul wceq adddi syl3anc ) ABHICHIDHIBCDJKLKBCLKBDLKJKM
      EFGBCDNO $.

    $( Distributive law (right-distributivity).  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    adddird $p |- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) ) $=
      ( cc wcel caddc co cmul wceq adddir syl3anc ) ABHICHIDHIBCJKDLKBDLKCDLKJK
      MEFGBCDNO $.
  $}

  ${
    adddirp1d.a $e |- ( ph -> A e. CC ) $.
    adddirp1d.b $e |- ( ph -> B e. CC ) $.
    $( Distributive law, plus 1 version.  (Contributed by Glauco Siliprandi,
       11-Dec-2019.) $)
    adddirp1d $p |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) ) $=
      ( c1 caddc co cmul 1cnd adddird mullidd oveq2d eqtrd ) ABFGHCIHBCIHZFCIHZ
      GHOCGHABFCDAJEKAPCOGACELMN $.
  $}

  ${
    joinlmuladdmuld.1 $e |- ( ph -> A e. CC ) $.
    joinlmuladdmuld.2 $e |- ( ph -> B e. CC ) $.
    joinlmuladdmuld.3 $e |- ( ph -> C e. CC ) $.
    joinlmuladdmuld.4 $e |- ( ph -> ( ( A x. B ) + ( C x. B ) ) = D ) $.
    $( Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler,
       26-Oct-2019.) $)
    joinlmuladdmuld $p |- ( ph -> ( ( A + C ) x. B ) = D ) $=
      ( caddc co cmul adddird eqtrd ) ABDJKCLKBCLKDCLKJKEABDCFHGMIN $.
  $}

  ${
    recnd.1 $e |- ( ph -> A e. RR ) $.
    $( Deduction from real number to complex number.  (Contributed by NM,
       26-Oct-1999.) $)
    recnd $p |- ( ph -> A e. CC ) $=
      ( cr wcel cc recn syl ) ABDEBFECBGH $.

    readdcld.2 $e |- ( ph -> B e. RR ) $.
    $( Closure law for addition of reals.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    readdcld $p |- ( ph -> ( A + B ) e. RR ) $=
      ( cr wcel caddc co readdcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Closure law for multiplication of reals.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    remulcld $p |- ( ph -> ( A x. B ) e. RR ) $=
      ( cr wcel cmul co remulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Infinity and the extended real number system
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c <_ $. $( 'Less than or equal to' relation. $)
  $c +oo $. $( Plus infinity $)
  $c -oo $. $( Minus infinity $)
  $c RR* $. $( The set of extended reals $)
  $c < $. $( 'Less than' relation (over extended reals) $)

  $( Plus infinity. $)
  cpnf $a class +oo $.
  $( Minus infinity. $)
  cmnf $a class -oo $.
  $( The set of extended reals (includes plus and minus infinity). $)
  cxr $a class RR* $.
  $( 'Less than' predicate (extended to include the extended reals). $)
  clt $a class < $.
  $( Extend wff notation to include the 'less than or equal to' relation. $)
  cle $a class <_ $.

  $( Define plus infinity.  Note that the definition is arbitrary, requiring
     only that ` +oo ` be a set not in ` RR ` and different from ` -oo `
     ( ~ df-mnf ).  We use ` ~P U. CC ` to make it independent of the
     construction of ` CC ` , and Cantor's Theorem will show that it is
     different from any member of ` CC ` and therefore ` RR ` .  See ~ pnfnre
     and ~ mnfnre , and we'll also be able to prove ` +oo =/= -oo ` .

     A simpler possibility is to define ` +oo ` as ` CC ` and ` -oo ` as
     ` { CC } ` , but that approach requires the Axiom of Regularity to show
     that ` +oo ` and ` -oo ` are different from each other and from all
     members of ` RR ` .  (Contributed by NM, 13-Oct-2005.)
     (New usage is discouraged.) $)
  df-pnf $a |- +oo = ~P U. CC $.

  $( Define minus infinity as the power set of plus infinity.  Note that the
     definition is arbitrary, requiring only that ` -oo ` be a set not in
     ` RR ` and different from ` +oo ` (see ~ mnfnre ).  (Contributed by NM,
     13-Oct-2005.)  (New usage is discouraged.) $)
  df-mnf $a |- -oo = ~P +oo $.

  $( Define the set of extended reals that includes plus and minus infinity.
     Definition 12-3.1 of [Gleason] p. 173.  (Contributed by NM,
     13-Oct-2005.) $)
  df-xr $a |- RR* = ( RR u. { +oo , -oo } ) $.

  ${
    $d x y $.
    $( Define 'less than' on the set of extended reals.  Definition 12-3.1 of
       [Gleason] p. 173.  Note that in our postulates for complex numbers,
       ` <RR ` is primitive and not necessarily a relation on ` RR ` .
       (Contributed by NM, 13-Oct-2005.) $)
    df-ltxr $a |- < = ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) }
      u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) ) $.
  $}

  $( Define 'less than or equal to' on the extended real subset of complex
     numbers.  (Contributed by NM, 13-Oct-2005.) $)
  df-le $a |- <_ = ( ( RR* X. RR* ) \ `' < ) $.

  $( Plus infinity is not a real number.  (Contributed by NM, 13-Oct-2005.) $)
  pnfnre $p |- +oo e/ RR $=
    ( cpnf cr wcel cc cuni cpw cvv wn uniex pwuninel2 ax-mp df-pnf eleq1i mtbir
    cnex recn mto nelir ) ABABCADCZSDEZFZDCZTGCUBHDOIDGJKAUADLMNAPQR $.

  $( Minus infinity is not a real number.  (Contributed by NM, 13-Oct-2005.) $)
  mnfnre $p |- -oo e/ RR $=
    ( cmnf cr wcel cc cuni cpw cvv wn cnex 2pwuninelg ax-mp df-mnf df-pnf pweqi
    cpnf eqtri eleq1i mtbir recn mto nelir ) ABABCADCZUBDEFZFZDCZDGCUEHIDGJKAUD
    DAOFUDLOUCMNPQRASTUA $.

  $( The standard reals are a subset of the extended reals.  (Contributed by
     NM, 14-Oct-2005.) $)
  ressxr $p |- RR C_ RR* $=
    ( cr cpnf cmnf cpr cun cxr ssun1 df-xr sseqtrri ) AABCDZEFAJGHI $.

  $( The Cartesian product of standard reals are a subset of the Cartesian
     product of extended reals (common case).  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  rexpssxrxp $p |- ( RR X. RR ) C_ ( RR* X. RR* ) $=
    ( cr cxr wss cxp ressxr xpss12 mp2an ) ABCZHAADBBDCEEABABFG $.

  $( A standard real is an extended real.  (Contributed by NM, 14-Oct-2005.) $)
  rexr $p |- ( A e. RR -> A e. RR* ) $=
    ( cr cxr ressxr sseli ) BCADE $.

  $( Zero is an extended real.  (Contributed by Mario Carneiro,
     15-Jun-2014.) $)
  0xr $p |- 0 e. RR* $=
    ( cr cxr cc0 ressxr 0re sselii ) ABCDEF $.

  $( No (finite) real equals plus infinity.  (Contributed by NM, 14-Oct-2005.)
     (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  renepnf $p |- ( A e. RR -> A =/= +oo ) $=
    ( cr wcel cpnf wceq pnfnre neli eleq1 mtbiri necon2ai ) ABCZADADEKDBCDBFGAD
    BHIJ $.

  $( No real equals minus infinity.  (Contributed by NM, 14-Oct-2005.)  (Proof
     shortened by Andrew Salmon, 19-Nov-2011.) $)
  renemnf $p |- ( A e. RR -> A =/= -oo ) $=
    ( cr wcel cmnf wceq mnfnre neli eleq1 mtbiri necon2ai ) ABCZADADEKDBCDBFGAD
    BHIJ $.

  ${
    rexrd.1 $e |- ( ph -> A e. RR ) $.
    $( A standard real is an extended real.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    rexrd $p |- ( ph -> A e. RR* ) $=
      ( cr cxr ressxr sselid ) ADEBFCG $.

    $( No (finite) real equals plus infinity.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    renepnfd $p |- ( ph -> A =/= +oo ) $=
      ( cr wcel cpnf wne renepnf syl ) ABDEBFGCBHI $.

    $( No real equals minus infinity.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    renemnfd $p |- ( ph -> A =/= -oo ) $=
      ( cr wcel cmnf wne renemnf syl ) ABDEBFGCBHI $.
  $}

  $( Plus infinity belongs to the set of extended reals.  (Contributed by NM,
     13-Oct-2005.)  (Proof shortened by Anthony Hart, 29-Aug-2011.) $)
  pnfxr $p |- +oo e. RR* $=
    ( cpnf cr cmnf cpr cun cxr ssun2 cc cuni cpw df-pnf cnex uniex pwex eqeltri
    cvv prid1 sselii df-xr eleqtrri ) ABACDZEZFUAUBAUABGACAHIZJPKUCHLMNOQRST $.

  $( Plus infinity exists (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  pnfex $p |- +oo e. _V $=
    ( cpnf cxr pnfxr elexi ) ABCD $.

  $( Plus and minus infinity are different elements of ` RR* ` .  (Contributed
     by NM, 14-Oct-2005.) $)
  pnfnemnf $p |- +oo =/= -oo $=
    ( cpnf cpw cmnf cxr wcel wne pnfxr pwne ax-mp necomi df-mnf neeqtrri ) AABZ
    CMAADEMAFGADHIJKL $.

  $( Minus and plus infinity are different (common case).  (Contributed by
     David A. Wheeler, 8-Dec-2018.) $)
  mnfnepnf $p |- -oo =/= +oo $=
    ( cpnf cmnf pnfnemnf necomi ) ABCD $.

  $( Minus infinity belongs to the set of extended reals.  (Contributed by NM,
     13-Oct-2005.)  (Proof shortened by Anthony Hart, 29-Aug-2011.)  (Proof
     shortened by Andrew Salmon, 19-Nov-2011.) $)
  mnfxr $p |- -oo e. RR* $=
    ( cmnf cr cpnf cpr cun cxr wcel cpw cvv df-mnf pnfex pwex prid2 elun2 ax-mp
    eqeltri df-xr eleqtrri ) ABCADZEZFASGATGCAACHIJCKLPMASBNOQR $.

  ${
    rexri.1 $e |- A e. RR $.
    $( A standard real is an extended real (inference form.)  (Contributed by
       David Moews, 28-Feb-2017.) $)
    rexri $p |- A e. RR* $=
      ( cr wcel cxr rexr ax-mp ) ACDAEDBAFG $.
  $}

  $( ` 1 ` is an extended real number.  (Contributed by Glauco Siliprandi,
     2-Jan-2022.) $)
  1xr $p |- 1 e. RR* $=
    ( c1 1re rexri ) ABC $.

  $( The reals and the infinities are disjoint.  (Contributed by NM,
     25-Oct-2005.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  renfdisj $p |- ( RR i^i { +oo , -oo } ) = (/) $=
    ( vx cr cpnf cmnf cpr cin c0 wceq cv wcel wn disj vex elpr renepnf necon2bi
    wo renemnf jaoi sylbi con2i mprgbir ) BCDEZFGHAIZUCJZKABABUCLUEUDBJZUEUDCHZ
    UDDHZQUFKZUDCDAMNUGUIUHUFUDCUDOPUFUDDUDRPSTUAUB $.

  ${
    $d x y $.
    $( 'Less than' is a relation on extended reals.  (Contributed by Mario
       Carneiro, 28-Apr-2015.) $)
    ltrelxr $p |- < C_ ( RR* X. RR* ) $=
      ( vx vy cv cr wcel copab cmnf csn cun cxp cxr wa eqsstri sstri wss ressxr
      cpnf unssi xpss12 mp2an clt cltrr wbr w3a df-ltxr df-3an opabbii opabssxp
      rexpssxrxp cpr snsspr2 ssun2 df-xr sseqtrri snsspr1 ) UAACZDEZBCZDEZUPURU
      BUCZUDZABFZDGHZIZQHZJZVCDJZIZIKKJZABUEVBVHVIVBDDJZVIVBUQUSLUTLZABFVJVAVKA
      BUQUSUTUFUGUTABDDUHMUINVFVGVIVDKOVEKOVFVIODVCKPVCQGUJZKQGUKVLDVLIKVLDULUM
      UNZNZRVEVLKQGUOVMNVDKVEKSTVCKODKOVGVIOVNPVCKDKSTRRM $.
  $}

  $( 'Less than' is a relation.  (Contributed by NM, 14-Oct-2005.) $)
  ltrel $p |- Rel < $=
    ( clt cxr cxp wss wrel ltrelxr relxp relss mp2 ) ABBCZDJEAEFBBGAJHI $.

  $( 'Less than or equal' is a relation on extended reals.  (Contributed by
     Mario Carneiro, 28-Apr-2015.) $)
  lerelxr $p |- <_ C_ ( RR* X. RR* ) $=
    ( cle cxr cxp clt ccnv cdif df-le difss eqsstri ) ABBCZDEZFJGJKHI $.

  $( 'Less or equal to' is a relation.  (Contributed by FL, 2-Aug-2009.)
     (Revised by Mario Carneiro, 28-Apr-2015.) $)
  lerel $p |- Rel <_ $=
    ( cle cxr cxp wss wrel lerelxr relxp relss mp2 ) ABBCZDJEAEFBBGAJHI $.

  $( 'Less than or equal to' expressed in terms of 'less than', for extended
     reals.  (Contributed by NM, 14-Oct-2005.) $)
  xrlenlt $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) ) $=
    ( cxr wcel wa cle wbr cop clt ccnv wn df-br cxp wb opelxpi cdif df-le eldif
    eleq2i bitri baib syl bitrid opelcnvg bitr4id notbid bitr4d ) ACDBCDEZABFGZ
    ABHZIJZDZKZBAIGZKUIUJFDZUHUMABFLUHUJCCMZDZUOUMNABCCOUOUQUMUOUJUPUKPZDUQUMEF
    URUJQSUJUPUKRTUAUBUCUHUNULUHUNBAHIDULBAILABCCIUDUEUFUG $.

  ${
    $d x y A $.  $d x y B $.
    $( The standard less-than ` <RR ` and the extended real less-than ` < ` are
       identical when restricted to the non-extended reals ` RR ` .
       (Contributed by NM, 13-Oct-2005.)  (Revised by Mario Carneiro,
       28-Apr-2015.) $)
    ltxrlt $p |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) ) $=
      ( vx vy cr wcel clt wbr cltrr cv w3a cmnf csn cun cpnf cxp wo brun wceq
      wi wa copab df-ltxr breqi bitri eleq1 breq1 3anbi13d breq2 3anbi23d brabg
      eqid simp3 biimtrdi wn simprbi elsni syl a1i renepnf neneqd pm2.24 syl6ci
      brxp adantl simplbi renemnf adantr jaod biimtrid 3adant3 ibir orcd sylibr
      wb 3expia impbid ) AEFZBEFZUAZABGHZABIHZWAABCJZEFZDJZEFZWCWEIHZKZCDUBZHZA
      BELMZNZOMZPZWKEPZNZHZQZVTWBWAABWIWPNZHWRABGWSCDUCUDABWIWPRUEZVTWJWBWQVTWJ
      VRVSWBKZWBWHVRWFAWEIHZKXACDABEEWIWCASWDVRWGXBWFWCAEUFWCAWEIUGUHWEBSWFVSXB
      WBVRWEBEUFWEBAIUIUJWIULUKZVRVSWBUMUNWQABWNHZABWOHZQVTWBABWNWORVTXDWBXEVSX
      DWBTVRVSXDBOSZXFUOWBXDXFTVSXDBWMFZXFXDAWLFXGABWLWMVDUPBOUQURUSVSBOBUTVAXF
      WBVBVCVEVRXEWBTVSVRXEALSZXHUOWBXEXHTVRXEAWKFZXHXEXIVSABWKEVDVFALUQURUSVRA
      LAVGVAXHWBVBVCVHVIVJVIVJVRVSWBWAXAWRWAXAWJWQXAWJVRVSWJXAVOWBXCVKVLVMWTVNV
      PVQ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Restate the ordering postulates with extended real "less than"
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Real number less-than is irreflexive.  Axiom for real and complex numbers,
     derived from set theory.  This restates ~ ax-pre-ltirr with ordering on
     the extended reals.  New proofs should use ~ ltnr instead for naming
     consistency.  (New usage is discouraged.)  (Contributed by Jim Kingdon,
     15-Jan-2020.) $)
  axltirr $p |- ( A e. RR -> -. A < A ) $=
    ( cr wcel clt wbr cltrr ax-pre-ltirr wb ltxrlt anidms mtbird ) ABCZAADEZAAF
    EZAGLMNHAAIJK $.

  $( Real number less-than is weakly linear.  Axiom for real and complex
     numbers, derived from set theory.  This restates ~ ax-pre-ltwlin with
     ordering on the extended reals.  (Contributed by Jim Kingdon,
     15-Jan-2020.) $)
  axltwlin $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
      ( A < B -> ( A < C \/ C < B ) ) ) $=
    ( cr wcel w3a cltrr wbr wo clt ax-pre-ltwlin ltxrlt 3adant3 3adant2 3adant1
    wb ancoms orbi12d 3imtr4d ) ADEZBDEZCDEZFZABGHZACGHZCBGHZIABJHZACJHZCBJHZIA
    BCKTUAUGUDPUBABLMUCUHUEUIUFTUBUHUEPUAACLNUAUBUIUFPZTUBUAUJCBLQORS $.

  $( Ordering on reals is transitive.  Axiom for real and complex numbers,
     derived from set theory.  This restates ~ ax-pre-lttrn with ordering on
     the extended reals.  New proofs should use ~ lttr instead for naming
     consistency.  (New usage is discouraged.)  (Contributed by NM,
     13-Oct-2005.) $)
  axlttrn $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
             ( ( A < B /\ B < C ) -> A < C ) ) $=
    ( cr wcel w3a cltrr wbr wa clt ax-pre-lttrn 3adant3 3adant1 anbi12d 3adant2
    wb ltxrlt 3imtr4d ) ADEZBDEZCDEZFZABGHZBCGHZIACGHZABJHZBCJHZIACJHZABCKUBUFU
    CUGUDSTUFUCPUAABQLTUAUGUDPSBCQMNSUAUHUEPTACQOR $.

  $( Ordering property of addition on reals.  Axiom for real and complex
     numbers, derived from set theory.  (This restates ~ ax-pre-ltadd with
     ordering on the extended reals.)  (Contributed by NM, 13-Oct-2005.) $)
  axltadd $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A < B -> ( C + A ) < ( C + B ) ) ) $=
    ( cr wcel w3a cltrr wbr caddc co clt ax-pre-ltadd wb ltxrlt 3adant3 readdcl
    wa syl2an 3impdi 3coml 3imtr4d ) ADEZBDEZCDEZFABGHZCAIJZCBIJZGHZABKHZUFUGKH
    ZABCLUBUCUIUEMUDABNOUDUBUCUJUHMZUDUBUCUKUDUBQUFDEUGDEUKUDUCQCAPCBPUFUGNRSTU
    A $.

  $( Apartness of reals is tight.  Axiom for real and complex numbers, derived
     from set theory.  (This restates ~ ax-pre-apti with ordering on the
     extended reals.)  (Contributed by Jim Kingdon, 29-Jan-2020.) $)
  axapti $p |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B ) $=
    ( cr wcel clt wbr wo wn wceq cltrr ltxrlt ancoms orbi12d notbid ax-pre-apti
    wa wb 3expia sylbid 3impia ) ACDZBCDZABEFZBAEFZGZHZABIZUAUBPZUFABJFZBAJFZGZ
    HZUGUHUEUKUHUCUIUDUJABKUBUAUDUJQBAKLMNUAUBULUGABORST $.

  $( The product of two positive reals is positive.  Axiom for real and complex
     numbers, derived from set theory.  (This restates ~ ax-pre-mulgt0 with
     ordering on the extended reals.)  (Contributed by NM, 13-Oct-2005.) $)
  axmulgt0 $p |- ( ( A e. RR /\ B e. RR ) ->
                ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) ) $=
    ( cr wcel wa cc0 cltrr wbr cmul co clt ax-pre-mulgt0 wb 0re ltxrlt bi2anan9
    mpan remulcl sylancr 3imtr4d ) ACDZBCDZEZFAGHZFBGHZEFABIJZGHZFAKHZFBKHZEFUF
    KHZABLUAUHUDUBUIUEFCDZUAUHUDMNFAOQUKUBUIUEMNFBOQPUCUKUFCDUJUGMNABRFUFOST $.

  ${
    $d x y z A $.
    $( An inhabited, bounded-above, located set of reals has a supremum.  Axiom
       for real and complex numbers, derived from ZF set theory.  (This
       restates ~ ax-pre-suploc with ordering on the extended reals.)
       (Contributed by Jim Kingdon, 30-Jan-2024.) $)
    axsuploc $p |- ( ( ( A C_ RR /\ E. x x e. A )
        /\ ( E. x e. RR A. y e. A y < x
        /\ A. x e. RR A. y e. RR ( x < y
          -> ( E. z e. A x < z \/ A. z e. A z < y ) ) ) )
        -> E. x e. RR ( A. y e. A -. x < y /\
          A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $=
      ( cr cv wcel wa clt wral wrex wi cltrr wb ltxrlt ralbidva rexbidva simplr
      wbr syl2anc wss wex wo wn ssel2 sylan an32s simpr simpllr adantlr orbi12d
      imbi12d anbi12d adantr pm5.32i ax-pre-suploc sylbi bicomd notbid ad2antrr
      mpbid ) DEUAZAFZDGAUBZHZBFZVCISZBDJZAEKZVCVFISZVCCFZISZCDKZVKVFISZCDJZUCZ
      LZBEJZAEJZHZHZVCVFMSZUDZBDJZVFVCMSZVFVKMSZCDKZLZBEJZHZAEKZVJUDZBDJZVGVFVK
      ISZCDKZLZBEJZHZAEKZWAVEWEBDJZAEKZWBVCVKMSZCDKZVKVFMSZCDJZUCZLZBEJZAEJZHZH
      WKVEVTXJVBVTXJNVDVBVIXAVSXIVBVHWTAEVBVCEGZHZVGWEBDVBVFDGZXKVGWENZVBXMHVFE
      GZXKXNDEVFUEZVFVCOZUFUGPQVBVRXHAEXLVQXGBEXLXOHZVJWBVPXFXRXKXOVJWBNZVBXKXO
      RZXLXOUHZVCVFOZTXRVMXCVOXEXRVLXBCDXRVKDGZHZXKVKEGZVLXBNVBXKXOYCUIXLYCYEXO
      VBYCYEXKDEVKUEUJUJZVCVKOTQXRVNXDCDYDYEXOVNXDNYFXLXOYCRZVKVFOTPUKULPPUMUNU
      OABCDUPUQVBWKWSNVDVTVBWJWRAEXLWDWMWIWQXLWCWLBDXLXMHZWBVJYHVJWBYHXKXOXSVBX
      KXMRVBXMXOXKXPUJYBTURUSPXLWHWPBEXRWEVGWGWOXRVGWEXRXOXKXNYAXTXQTURXRWFWNCD
      YDWNWFYDXOYEWNWFNYGYFVFVKOTURQULPUMQUTVA $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordering on reals
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Alias for ~ axlttrn , for naming consistency with ~ lttri .  New proofs
     should generally use this instead of ~ ax-pre-lttrn .  (Contributed by NM,
     10-Mar-2008.) $)
  lttr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
             ( ( A < B /\ B < C ) -> A < C ) ) $=
    ( axlttrn ) ABCD $.

  $( The product of two positive numbers is positive.  (Contributed by NM,
     10-Mar-2008.) $)
  mulgt0 $p |- ( ( ( A e. RR /\ 0 < A ) /\
                ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) ) $=
    ( cr wcel cc0 clt wbr cmul co wa axmulgt0 imp an4s ) ACDZBCDZEAFGZEBFGZEABH
    IFGZNOJPQJRABKLM $.

  $( 'Less than or equal to' expressed in terms of 'less than'.  Part of
     definition 11.2.7(vi) of [HoTT], p.  (varies).  (Contributed by NM,
     13-May-1999.) $)
  lenlt $p |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) ) $=
    ( cr wcel cxr cle wbr clt wn wb rexr xrlenlt syl2an ) ACDAEDBEDABFGBAHGIJBC
    DAKBKABLM $.

  $( 'Less than' is irreflexive.  (Contributed by NM, 18-Aug-1999.) $)
  ltnr $p |- ( A e. RR -> -. A < A ) $=
    ( axltirr ) AB $.

  ${
    $d x y z $.
    $( 'Less than' is a strict ordering.  (Contributed by NM, 19-Jan-1997.) $)
    ltso $p |- < Or RR $=
      ( vx vy vz cr clt wor wpo cv wbr wo wi wral wtru wcel wn ltnr adantl lttr
      w3a wa ispod mptru axltwlin rgen3 df-iso mpbir2an ) DEFDEGZAHZBHZEIZUHCHZ
      EIZUKUIEIJKZCDLBDLADLUGMABCDEUHDNZUHUHEIOMUHPQUNUIDNUKDNSUJUIUKEITULKMUHU
      IUKRQUAUBUMABCDDDUHUIUKUCUDABCDEUEUF $.
  $}

  $( 'Greater than' is a strict ordering.  (Contributed by JJ, 11-Oct-2018.) $)
  gtso $p |- `' < Or RR $=
    ( vx cr clt wor ccnv ltso cc0 wcel cv wex wb 0re elex2 cnvsom mp2b mpbi ) B
    CDZBCEDZFGBHAIBHAJQRKLAGBMABCNOP $.

  $( Tightness of real apartness.  (Contributed by NM, 5-May-1999.) $)
  lttri3 $p |- ( ( A e. RR /\ B e. RR ) ->
                ( A = B <-> ( -. A < B /\ -. B < A ) ) ) $=
    ( cr wcel wa wceq clt wbr wn wi ltnr breq2 notbid syl5ibcom breq1 adantr wo
    jcad ioran axapti 3expia biimtrrid impbid ) ACDZBCDZEZABFZABGHZIZBAGHZIZEZU
    DUGULJUEUDUGUIUKUDAAGHZIZUGUIAKZUGUMUHABAGLMNUDUNUGUKUOUGUMUJABAGOMNRPULUHU
    JQIZUFUGUHUJSUDUEUPUGABTUAUBUC $.

  $( Tightness of real apartness.  (Contributed by NM, 14-May-1999.) $)
  letri3 $p |- ( ( A e. RR /\ B e. RR ) ->
                ( A = B <-> ( A <_ B /\ B <_ A ) ) ) $=
    ( cr wcel wa wceq clt wbr wn cle lttri3 ancom bitr4di ancoms anbi12d bitr4d
    lenlt wb ) ACDZBCDZEZABFZBAGHIZABGHIZEZABJHZBAJHZEUAUBUDUCEUEABKUCUDLMUAUFU
    CUGUDABQTSUGUDRBAQNOP $.

  $( Transitive law, weaker form of ` ( A < B /\ B <_ C ) -> A < C ` .
     (Contributed by AV, 14-Oct-2018.) $)
  ltleletr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A < B /\ B <_ C ) -> A <_ C ) ) $=
    ( cr wcel w3a clt wbr cle wn wi wa lttr 3coml expcomd con3 wb lenlt 3adant1
    syl6 3adant2 imbi12d sylibrd impd ) ADEZBDEZCDEZFZABGHZBCIHZACIHZUHUICBGHZJ
    ZCAGHZJZKZUJUKKUHUIUNULKUPUHUNUIULUGUEUFUNUILULKCABMNOUNULPTUHUJUMUKUOUFUGU
    JUMQUEBCRSUEUGUKUOQUFACRUAUBUCUD $.

  $( Transitive law.  (Contributed by NM, 12-Nov-1999.) $)
  letr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
             ( ( A <_ B /\ B <_ C ) -> A <_ C ) ) $=
    ( cr wcel w3a clt wbr wo wn wa wi axltwlin 3coml orcom imbitrdi con3d lenlt
    cle wb 3adant3 3adant1 anbi12d ioran bitr4di 3adant2 3imtr4d ) ADEZBDEZCDEZ
    FZBAGHZCBGHZIZJZCAGHZJZABSHZBCSHZKZACSHZUKUPUNUKUPUMULIZUNUJUHUIUPVBLCABMNU
    MULOPQUKUTULJZUMJZKUOUKURVCUSVDUHUIURVCTUJABRUAUIUJUSVDTUHBCRUBUCULUMUDUEUH
    UJVAUQTUIACRUFUG $.

  $( 'Less than or equal to' is reflexive.  (Contributed by NM,
     18-Aug-1999.) $)
  leid $p |- ( A e. RR -> A <_ A ) $=
    ( cr wcel cle wbr clt wn ltnr wb lenlt anidms mpbird ) ABCZAADEZAAFEGZAHMNO
    IAAJKL $.

  $( 'Less than' implies not equal.  See also ~ ltap which is the same but for
     apartness.  (Contributed by NM, 9-Oct-1999.)  (Revised by Mario Carneiro,
     16-Sep-2015.) $)
  ltne $p |- ( ( A e. RR /\ A < B ) -> B =/= A ) $=
    ( cr wcel clt wbr wne wn wceq ltnr breq2 notbid syl5ibrcom necon2ad imp ) A
    CDZABEFZBAGPQBAPQHBAIZAAEFZHAJRQSBAAEKLMNO $.

  $( 'Less than' is not symmetric.  (Contributed by NM, 8-Jan-2002.) $)
  ltnsym $p |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) ) $=
    ( cr wcel wa clt wbr wi wn lttr 3anidm13 expd ltnr adantr con3 syl6ci ) ACD
    ZBCDZEZABFGZBAFGZAAFGZHUBIZUAISTUAUBQRTUAEUBHABAJKLQUCRAMNUAUBOP $.

  $( Equality in terms of 'less than or equal to', 'less than'.  (Contributed
     by NM, 7-Apr-2001.) $)
  eqlelt $p |- ( ( A e. RR /\ B e. RR ) ->
               ( A = B <-> ( A <_ B /\ -. A < B ) ) ) $=
    ( cr wcel wa wceq cle wbr clt wn letri3 wb lenlt ancoms anbi2d bitrd ) ACDZ
    BCDZEZABFABGHZBAGHZETABIHJZEABKSUAUBTRQUAUBLBAMNOP $.

  $( 'Less than' implies 'less than or equal to'.  (Contributed by NM,
     25-Aug-1999.) $)
  ltle $p |- ( ( A e. RR /\ B e. RR ) ->
               ( A < B -> A <_ B ) ) $=
    ( cr wcel wa clt wbr wn cle ltnsym lenlt sylibrd ) ACDBCDEABFGBAFGHABIGABJA
    BKL $.

  $( Transitive law.  Part of Definition 11.2.7(vi) of [HoTT], p.  (varies).
     (Contributed by NM, 23-May-1999.) $)
  lelttr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
             ( ( A <_ B /\ B < C ) -> A < C ) ) $=
    ( cr wcel w3a cle wbr clt wa wn simprl wb simpl1 simpl2 lenlt syl2anc mpbid
    pm2.21d idd wo simprr wi simpl3 axltwlin syl3anc mpd mpjaod ex ) ADEZBDEZCD
    EZFZABGHZBCIHZJZACIHZUMUPJZBAIHZUQUQURUSUQURUNUSKZUMUNUOLURUJUKUNUTMUJUKULU
    PNZUJUKULUPOZABPQRSURUQTURUOUSUQUAZUMUNUOUBURUKULUJUOVCUCVBUJUKULUPUDVABCAU
    EUFUGUHUI $.

  $( Transitive law.  Part of Definition 11.2.7(vi) of [HoTT], p.  (varies).
     (Contributed by NM, 25-Aug-1999.) $)
  ltletr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
             ( ( A < B /\ B <_ C ) -> A < C ) ) $=
    ( cr wcel w3a clt wbr cle wa wn simprr wb simpl2 simpl3 lenlt syl2anc mpbid
    wo simprl wi axltwlin adantr mpd ecased ex ) ADEZBDEZCDEZFZABGHZBCIHZJZACGH
    ZUJUMJZUNCBGHZUOULUPKZUJUKULLUOUHUIULUQMUGUHUIUMNUGUHUIUMOBCPQRUOUKUNUPSZUJ
    UKULTUJUKURUAUMABCUBUCUDUEUF $.

  $( 'Less than' is antisymmetric and irreflexive.  (Contributed by NM,
     13-Aug-2005.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  ltnsym2 $p |- ( ( A e. RR /\ B e. RR ) -> -. ( A < B /\ B < A ) ) $=
    ( cr clt wor wcel wa wbr wn ltso so2nr mpan ) CDEACFBCFGABDHBADHGIJCABDKL
    $.

  $( Equality implies 'less than or equal to'.  (Contributed by NM,
     4-Apr-2005.) $)
  eqle $p |- ( ( A e. RR /\ A = B ) -> A <_ B ) $=
    ( cr wcel cle wbr wceq leid breq2 biimpac sylan ) ACDAAEFZABGZABEFZAHMLNABA
    EIJK $.

  ${
    lt.1 $e |- A e. RR $.
    $( 'Less than' is irreflexive.  (Contributed by NM, 18-Aug-1999.) $)
    ltnri $p |- -. A < A $=
      ( cr wcel clt wbr wn ltnr ax-mp ) ACDAAEFGBAHI $.

    $( Equality implies 'less than or equal to'.  (Contributed by NM,
       23-May-1999.)  (Revised by Alexander van der Vekens, 20-Mar-2018.) $)
    eqlei $p |- ( A = B -> A <_ B ) $=
      ( cr wcel wceq cle wbr eqle mpan ) ADEABFABGHCABIJ $.

    $( Equality implies 'less than or equal to'.  (Contributed by Alexander van
       der Vekens, 20-Mar-2018.) $)
    eqlei2 $p |- ( B = A -> B <_ A ) $=
      ( cr wcel wceq cle wbr eleq1 mpbiri eqle mpancom ) BDEZBAFZBAGHNMADECBADI
      JBAKL $.

    ${
      ltneii.2 $e |- A < B $.
      $( 'Less than' implies not equal.  See also ~ gtapii which is the same
         for apartness.  (Contributed by Mario Carneiro, 30-Sep-2013.) $)
      gtneii $p |- B =/= A $=
        ( cr wcel clt wbr wne ltne mp2an ) AEFABGHBAICDABJK $.

      $( 'Greater than' implies not equal.  (Contributed by Mario Carneiro,
         16-Sep-2015.) $)
      ltneii $p |- A =/= B $=
        ( gtneii necomi ) BAABCDEF $.
    $}

    lt.2 $e |- B e. RR $.
    $( Tightness of real apartness.  (Contributed by NM, 14-May-1999.) $)
    lttri3i $p |- ( A = B <-> ( -. A < B /\ -. B < A ) ) $=
      ( cr wcel wceq clt wbr wn wa wb lttri3 mp2an ) AEFBEFABGABHIJBAHIJKLCDABM
      N $.

    $( Tightness of real apartness.  (Contributed by NM, 14-May-1999.) $)
    letri3i $p |- ( A = B <-> ( A <_ B /\ B <_ A ) ) $=
      ( cr wcel wceq cle wbr wa wb letri3 mp2an ) AEFBEFABGABHIBAHIJKCDABLM $.

    $( 'Less than' is not symmetric.  (Contributed by NM, 6-May-1999.) $)
    ltnsymi $p |- ( A < B -> -. B < A ) $=
      ( cr wcel clt wbr wn wi ltnsym mp2an ) AEFBEFABGHBAGHIJCDABKL $.

    $( 'Less than or equal to' in terms of 'less than'.  (Contributed by NM,
       24-May-1999.) $)
    lenlti $p |- ( A <_ B <-> -. B < A ) $=
      ( cr wcel cle wbr clt wn wb lenlt mp2an ) AEFBEFABGHBAIHJKCDABLM $.

    $( 'Less than' implies 'less than or equal to'.  (Contributed by NM,
       14-May-1999.) $)
    ltlei $p |- ( A < B -> A <_ B ) $=
      ( cr wcel clt wbr cle wi ltle mp2an ) AEFBEFABGHABIHJCDABKL $.

    ${
      ltlei.1 $e |- A < B $.
      $( 'Less than' implies 'less than or equal to' (inference).  (Contributed
         by NM, 22-Aug-1999.) $)
      ltleii $p |- A <_ B $=
        ( clt wbr cle ltlei ax-mp ) ABFGABHGEABCDIJ $.
    $}

    $( 'Less than' implies not equal.  (Contributed by NM, 28-Jul-1999.) $)
    ltnei $p |- ( A < B -> B =/= A ) $=
      ( cr wcel clt wbr wne ltne mpan ) AEFABGHBAICABJK $.

    ${
      lt.3 $e |- C e. RR $.
      $( 'Less than' is transitive.  Theorem I.17 of [Apostol] p. 20.
         (Contributed by NM, 14-May-1999.) $)
      lttri $p |- ( ( A < B /\ B < C ) -> A < C ) $=
        ( cr wcel clt wbr wa wi lttr mp3an ) AGHBGHCGHABIJBCIJKACIJLDEFABCMN $.

      $( 'Less than or equal to', 'less than' transitive law.  (Contributed by
         NM, 14-May-1999.) $)
      lelttri $p |- ( ( A <_ B /\ B < C ) -> A < C ) $=
        ( cr wcel cle wbr clt wa wi lelttr mp3an ) AGHBGHCGHABIJBCKJLACKJMDEFAB
        CNO $.

      $( 'Less than', 'less than or equal to' transitive law.  (Contributed by
         NM, 14-May-1999.) $)
      ltletri $p |- ( ( A < B /\ B <_ C ) -> A < C ) $=
        ( cr wcel clt wbr cle wa wi ltletr mp3an ) AGHBGHCGHABIJBCKJLACIJMDEFAB
        CNO $.

      $( 'Less than or equal to' is transitive.  (Contributed by NM,
         14-May-1999.) $)
      letri $p |- ( ( A <_ B /\ B <_ C ) -> A <_ C ) $=
        ( cr wcel cle wbr wa wi letr mp3an ) AGHBGHCGHABIJBCIJKACIJLDEFABCMN $.

      $( Extended trichotomy law for 'less than or equal to'.  (Contributed by
         NM, 14-Aug-2000.) $)
      le2tri3i $p |- ( ( A <_ B /\ B <_ C /\ C <_ A ) <->
                      ( A = B /\ B = C /\ C = A ) ) $=
        ( cle wbr wceq wa letri letri3i biimpri sylan2 3impb 3comr eqcomd sylan
        w3a eqlei 3impa 3jca 3anim123i impbii ) ABGHZBCGHZCAGHZSZABIZBCIZCAIZSU
        HUIUJUKUEUFUGUIUFUGJUEBAGHZUIBCAEFDKUIUEULJABDELMNOUFUGUEUJUFUGUEUJUGUE
        JUFCBGHZUJCABFDEKUJUFUMJBCEFLMNOPUEUFUGUKUEUFJACGHZUGUKABCDEFKUNUGJZACA
        CIUOACDFLMQRUAUBUIUEUJUFUKUGABDTBCETCAFTUCUD $.
    $}

    $( The product of two positive numbers is positive.  (Contributed by NM,
       16-May-1999.) $)
    mulgt0i $p |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) $=
      ( cr wcel cc0 clt wbr wa cmul co wi axmulgt0 mp2an ) AEFBEFGAHIGBHIJGABKL
      HIMCDABNO $.

    mulgt0i.3 $e |- 0 < A $.
    mulgt0i.4 $e |- 0 < B $.
    $( The product of two positive numbers is positive.  (Contributed by NM,
       18-May-1999.) $)
    mulgt0ii $p |- 0 < ( A x. B ) $=
      ( cc0 clt wbr cmul co mulgt0i mp2an ) GAHIGBHIGABJKHIEFABCDLM $.
  $}

  ${
    ltd.1 $e |- ( ph -> A e. RR ) $.
    $( 'Less than' is irreflexive.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    ltnrd $p |- ( ph -> -. A < A ) $=
      ( cr wcel clt wbr wn ltnr syl ) ABDEBBFGHCBIJ $.

    ${
      ltned.2 $e |- ( ph -> A < B ) $.
      $( 'Less than' implies not equal.  See also ~ gtapd which is the same but
         for apartness.  (Contributed by Mario Carneiro, 27-May-2016.) $)
      gtned $p |- ( ph -> B =/= A ) $=
        ( cr wcel clt wbr wne ltne syl2anc ) ABFGBCHICBJDEBCKL $.

      $( 'Greater than' implies not equal.  (Contributed by Mario Carneiro,
         27-May-2016.) $)
      ltned $p |- ( ph -> A =/= B ) $=
        ( gtned necomd ) ACBABCDEFG $.
    $}

    ltd.2 $e |- ( ph -> B e. RR ) $.
    $( Tightness of real apartness.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    lttri3d $p |- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) ) $=
      ( cr wcel wceq clt wbr wn wa wb lttri3 syl2anc ) ABFGCFGBCHBCIJKCBIJKLMDE
      BCNO $.

    $( Tightness of real apartness.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    letri3d $p |- ( ph -> ( A = B <-> ( A <_ B /\ B <_ A ) ) ) $=
      ( cr wcel wceq cle wbr wa wb letri3 syl2anc ) ABFGCFGBCHBCIJCBIJKLDEBCMN
      $.

    ${
      letrid.3 $e |- ( ph -> A <_ B ) $.
      letrid.4 $e |- ( ph -> B <_ A ) $.
      $( Tightness of real apartness.  (Contributed by Matthew House,
         28-Jun-2026.) $)
      letrid $p |- ( ph -> A = B ) $=
        ( wceq cle wbr letri3d mpbir2and ) ABCHBCIZJCBMJFGABCDEKL $.
    $}

    $( Equality in terms of 'less than or equal to', 'less than'.  (Contributed
       by NM, 7-Apr-2001.) $)
    eqleltd $p |- ( ph -> ( A = B <-> ( A <_ B /\ -. A < B ) ) ) $=
      ( cr wcel wceq cle wbr clt wn wa wb eqlelt syl2anc ) ABFGCFGBCHBCIJBCKJLM
      NDEBCOP $.

    $( 'Less than or equal to' in terms of 'less than'.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    lenltd $p |- ( ph -> ( A <_ B <-> -. B < A ) ) $=
      ( cr wcel cle wbr clt wn wb lenlt syl2anc ) ABFGCFGBCHICBJIKLDEBCMN $.

    ${
      ltled.1 $e |- ( ph -> A < B ) $.
      $( 'Less than' implies 'less than or equal to'.  (Contributed by Mario
         Carneiro, 27-May-2016.) $)
      ltled $p |- ( ph -> A <_ B ) $=
        ( clt wbr cle cr wcel wi ltle syl2anc mpd ) ABCGHZBCIHZFABJKCJKPQLDEBCM
        NO $.

      $( 'Less than' implies 'less than or equal to'.  (Contributed by Mario
         Carneiro, 27-May-2016.) $)
      ltnsymd $p |- ( ph -> -. B < A ) $=
        ( cle wbr clt wn ltled lenltd mpbid ) ABCGHCBIHJABCDEFKABCDELM $.
    $}

    ${
      nltled.1 $e |- ( ph -> -. B < A ) $.
      $( 'Not less than ' implies 'less than or equal to'.  (Contributed by
         Glauco Siliprandi, 11-Dec-2019.) $)
      nltled $p |- ( ph -> A <_ B ) $=
        ( cle wbr clt wn lenltd mpbird ) ABCGHCBIHJFABCDEKL $.
    $}

    ${
      lensymd.3 $e |- ( ph -> A <_ B ) $.
      $( 'Less than or equal to' implies 'not less than'.  (Contributed by
         Glauco Siliprandi, 11-Dec-2019.) $)
      lensymd $p |- ( ph -> -. B < A ) $=
        ( cle wbr clt wn lenltd mpbid ) ABCGHCBIHJFABCDEKL $.
    $}

    ${
      mulgt0d.3 $e |- ( ph -> 0 < A ) $.
      mulgt0d.4 $e |- ( ph -> 0 < B ) $.
      $( The product of two positive numbers is positive.  (Contributed by
         Mario Carneiro, 27-May-2016.) $)
      mulgt0d $p |- ( ph -> 0 < ( A x. B ) ) $=
        ( cr wcel cc0 clt wbr cmul co mulgt0 syl22anc ) ABHIJBKLCHIJCKLJBCMNKLD
        FEGBCOP $.
    $}

    letrd.3 $e |- ( ph -> C e. RR ) $.
    ${
      letrd.4 $e |- ( ph -> A <_ B ) $.
      letrd.5 $e |- ( ph -> B <_ C ) $.
      $( Transitive law deduction for 'less than or equal to'.  (Contributed by
         NM, 20-May-2005.) $)
      letrd $p |- ( ph -> A <_ C ) $=
        ( cle wbr cr wcel wa wi letr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABLMCLM
        DLMSTNUAOEFGBCDPQR $.
    $}

    ${
      lelttrd.4 $e |- ( ph -> A <_ B ) $.
      lelttrd.5 $e |- ( ph -> B < C ) $.
      $( Transitive law deduction for 'less than or equal to', 'less than'.
         (Contributed by NM, 8-Jan-2006.) $)
      lelttrd $p |- ( ph -> A < C ) $=
        ( cle wbr clt cr wcel wa wi lelttr syl3anc mp2and ) ABCJKZCDLKZBDLKZHIA
        BMNCMNDMNTUAOUBPEFGBCDQRS $.
    $}

    ${
      lttrd.4 $e |- ( ph -> A < B ) $.
      lttrd.5 $e |- ( ph -> B < C ) $.
      $( Transitive law deduction for 'less than'.  (Contributed by NM,
         9-Jan-2006.) $)
      lttrd $p |- ( ph -> A < C ) $=
        ( clt wbr cr wcel wa wi lttr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABLMCLM
        DLMSTNUAOEFGBCDPQR $.
    $}
  $}

  $( 0 is less than 1.  Theorem I.21 of [Apostol] p. 20.  Part of definition
     11.2.7(vi) of [HoTT], p.  (varies).  (Contributed by NM, 17-Jan-1997.) $)
  0lt1 $p |- 0 < 1 $=
    ( cc0 c1 clt wbr cltrr ax-0lt1 cr wcel wb 0re 1re ltxrlt mp2an mpbir ) ABCD
    ZABEDZFAGHBGHOPIJKABLMN $.

  $( Negative trichotomy property for real numbers.  It is well known that we
     cannot prove real number trichotomy, ` A < B \/ A = B \/ B < A ` .  Does
     that mean there is a pair of real numbers where none of those hold (that
     is, where we can refute each of those three relationships)?  Actually, no,
     as shown here.  This is another example of distinguishing between being
     unable to prove something, or being able to refute it.  (Contributed by
     Jim Kingdon, 13-Aug-2023.) $)
  ltntri $p |- ( ( A e. RR /\ B e. RR )
      -> -. ( -. A < B /\ -. A = B /\ -. B < A ) ) $=
    ( cr wcel wa clt wbr wn wceq w3a simpll simplr simpr3 nltled simpr1 letri3d
    cle mpbir2and simpr2 pm2.65da ) ACDZBCDZEZABFGHZABIZHZBAFGHZJZUEUCUHEZUEABQ
    GBAQGUIABUAUBUHKZUAUBUHLZUCUDUFUGMNUIBAUKUJUCUDUFUGONUIABUJUKPRUCUDUFUGST
    $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Initial properties of the complex numbers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Commutative/associative law for multiplication.  (Contributed by NM,
     30-Apr-2005.) $)
  mul12 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) ) $=
    ( cc wcel w3a cmul co wceq wa mulcom oveq1d 3adant3 mulass 3com12 3eqtr3d )
    ADEZBDEZCDEZFABGHZCGHZBAGHZCGHZABCGHGHBACGHGHZQRUAUCISQRJTUBCGABKLMABCNRQSU
    CUDIBACNOP $.

  $( Commutative/associative law.  (Contributed by NM, 8-Oct-1999.) $)
  mul32 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) ) $=
    ( cc wcel w3a cmul co wceq wa mulcom oveq2d 3adant1 mulass 3com23 3eqtr4d )
    ADEZBDEZCDEZFABCGHZGHZACBGHZGHZABGHCGHACGHBGHZRSUAUCIQRSJTUBAGBCKLMABCNQSRU
    DUCIACBNOP $.

  $( Commutative/associative law.  (Contributed by Scott Fenton,
     3-Jan-2013.) $)
  mul31 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) ) $=
    ( cc wcel cmul co wceq wa mulcom oveq2d 3adant1 mulass mulcl ancoms mulcomd
    w3a simp1 3eqtr4d ) ADEZBDEZCDEZQZABCFGZFGZACBFGZFGZABFGCFGUFAFGUAUBUEUGHTU
    AUBIUDUFAFBCJKLABCMUCUFAUAUBUFDEZTUBUAUHCBNOLTUAUBRPS $.

  $( Rearrangement of 4 factors.  (Contributed by NM, 8-Oct-1999.) $)
  mul4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
              ( ( A x. B ) x. ( C x. D ) ) =
                 ( ( A x. C ) x. ( B x. D ) ) ) $=
    ( cc wcel wa cmul co wceq w3a mul32 oveq1d 3expa adantrr mulcl mulass 3expb
    sylan an4s 3eqtr3d ) AEFZBEFZGZCEFZDEFZGZGABHIZCHIZDHIZACHIZBHIZDHIZUHCDHIH
    IZUKBDHIHIZUDUEUJUMJZUFUBUCUEUPUBUCUEKUIULDHABCLMNOUDUHEFZUGUJUNJZABPUQUEUF
    URUHCDQRSUBUEUCUFUMUOJZUBUEGUKEFZUCUFGUSACPUTUCUFUSUKBDQRSTUA $.

  $( A simple product of sums expansion.  (Contributed by NM, 21-Feb-2005.) $)
  muladd11 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) =
                 ( ( 1 + A ) + ( B + ( A x. B ) ) ) ) $=
    ( cc wcel wa c1 caddc cmul wceq ax-1cn addcl mpan adddi mp3an2 sylan adantr
    co mulridd adddir eqtrd mp3an1 mullid adantl oveq1d oveq12d ) ACDZBCDZEZFAG
    QZFBGQHQZUIFHQZUIBHQZGQZUIBABHQZGQZGQUFUICDZUGUJUMIZFCDZUFUPJFAKLZUPURUGUQJ
    UIFBMNOUHUKUIULUOGUFUKUIIUGUFUIUSRPUHULFBHQZUNGQZUOURUFUGULVAIJFABSUAUHUTBU
    NGUGUTBIUFBUBUCUDTUET $.

  $( Two times a number.  (Contributed by NM, 18-May-1999.)  (Revised by Mario
     Carneiro, 27-May-2016.) $)
  1p1times $p |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) ) $=
    ( cc wcel c1 caddc co cmul ax-1cn a1i id adddird mullid oveq12d eqtrd ) ABC
    ZDDEFAGFDAGFZPEFAAEFODDADBCOHIZQOJKOPAPAEALZRMN $.

  $( A theorem for complex numbers analogous the second Peano postulate
     ~ peano2 .  (Contributed by NM, 17-Aug-2005.) $)
  peano2cn $p |- ( A e. CC -> ( A + 1 ) e. CC ) $=
    ( cc wcel c1 caddc co ax-1cn addcl mpan2 ) ABCDBCADEFBCGADHI $.

  $( A theorem for reals analogous the second Peano postulate ~ peano2 .
     (Contributed by NM, 5-Jul-2005.) $)
  peano2re $p |- ( A e. RR -> ( A + 1 ) e. RR ) $=
    ( cr wcel c1 caddc co 1re readdcl mpan2 ) ABCDBCADEFBCGADHI $.

  $( Addition is commutative.  (Contributed by Jim Kingdon, 17-Jan-2020.) $)
  addcom $p |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) $=
    ( ax-addcom ) ABC $.

  $( ` 0 ` is an additive identity.  (Contributed by Jim Kingdon,
     16-Jan-2020.) $)
  addrid $p |- ( A e. CC -> ( A + 0 ) = A ) $=
    ( ax-0id ) AB $.

  $( ` 0 ` is a left identity for addition.  (Contributed by Scott Fenton,
     3-Jan-2013.) $)
  addlid $p |- ( A e. CC -> ( 0 + A ) = A ) $=
    ( cc wcel cc0 caddc co wceq 0cn addcom mpan2 addrid eqtr3d ) ABCZADEFZDAEFZ
    AMDBCNOGHADIJAKL $.

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Cancellation law for addition over the reals.  (Contributed by Scott
       Fenton, 3-Jan-2013.) $)
    readdcan $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
        ( ( C + A ) = ( C + B ) <-> A = B ) ) $=
      ( vx cr wcel caddc co wceq wa oveq2 recnd addassd adantr cc oveq1d addlid
      cc0 syl eqtrd w3a cv wi ax-rnegex 3ad2ant3 adantl wb simprl simpl3 simpl1
      simpl2 eqeq12d mpbird addcom syl2anc simplrr 3eqtr3d ex rexlimddv impbid1
      wrex eqtr3d ) AEFZBEFZCEFZUAZCAGHZCBGHZIZABIZVFCDUBZGHZRIZVIVJUCDEVEVCVMD
      EVAVDDCUDUEVFVKEFZVMJZJZVIVJVPVIJZVKCGHZAGHZVRBGHZABVQVSVTIZVKVGGHZVKVHGH
      ZIZVIWDVPVGVHVKGKUFVPWAWDUGVIVPVSWBVTWCVPVKCAVPVKVFVNVMUHLZVPCVCVDVEVOUIL
      ZVPAVCVDVEVOUJLZMVPVKCBWEWFVPBVCVDVEVOUKLZMULNUMVQVSRAGHZAVQVRRAGVQVLVRRV
      QCOFZVKOFZVLVRIVPWJVIWFNVPWKVIWENCVKUNUOVFVNVMVIUPVBZPVQAOFZWIAIVPWMVIWGN
      AQSTVQVTRBGHZBVQVRRBGWLPVQBOFZWNBIVPWOVIWHNBQSTUQURUSABCGKUT $.
  $}

  $( ` 0 ` is its own additive identity.  (Contributed by Scott Fenton,
     3-Jan-2013.) $)
  00id $p |- ( 0 + 0 ) = 0 $=
    ( cc0 cc wcel caddc co wceq 0cn addrid ax-mp ) ABCAADEAFGAHI $.

  ${
    mul.1 $e |- A e. CC $.
    $( ` 0 ` is an additive identity.  (Contributed by NM, 23-Nov-1994.)
       (Revised by Scott Fenton, 3-Jan-2013.) $)
    addridi $p |- ( A + 0 ) = A $=
      ( cc wcel cc0 caddc co wceq addrid ax-mp ) ACDAEFGAHBAIJ $.

    $( ` 0 ` is a left identity for addition.  (Contributed by NM,
       3-Jan-2013.) $)
    addlidi $p |- ( 0 + A ) = A $=
      ( cc wcel cc0 caddc co wceq addlid ax-mp ) ACDEAFGAHBAIJ $.

    mul.2 $e |- B e. CC $.
    $( Addition is commutative.  Based on ideas by Eric Schmidt.  (Contributed
       by Scott Fenton, 3-Jan-2013.) $)
    addcomi $p |- ( A + B ) = ( B + A ) $=
      ( cc wcel caddc co wceq addcom mp2an ) AEFBEFABGHBAGHICDABJK $.

    ${
      addcomli.2 $e |- ( A + B ) = C $.
      $( Addition is commutative.  (Contributed by Mario Carneiro,
         19-Apr-2015.) $)
      addcomli $p |- ( B + A ) = C $=
        ( caddc co addcomi eqtri ) BAGHABGHCBAEDIFJ $.
    $}

    mul.3 $e |- C e. CC $.
    $( Commutative/associative law that swaps the first two factors in a triple
       product.  (Contributed by NM, 11-May-1999.)  (Proof shortened by Andrew
       Salmon, 19-Nov-2011.) $)
    mul12i $p |- ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) $=
      ( cc wcel cmul co wceq mul12 mp3an ) AGHBGHCGHABCIJIJBACIJIJKDEFABCLM $.

    $( Commutative/associative law that swaps the last two factors in a triple
       product.  (Contributed by NM, 11-May-1999.) $)
    mul32i $p |- ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) $=
      ( cc wcel cmul co wceq mul32 mp3an ) AGHBGHCGHABIJCIJACIJBIJKDEFABCLM $.

    mul4.4 $e |- D e. CC $.
    $( Rearrangement of 4 factors.  (Contributed by NM, 16-Feb-1995.) $)
    mul4i $p |- ( ( A x. B ) x. ( C x. D ) ) =
               ( ( A x. C ) x. ( B x. D ) ) $=
      ( cc wcel cmul co wceq mul4 mp4an ) AIJBIJCIJDIJABKLCDKLKLACKLBDKLKLMEFGH
      ABCDNO $.
  $}

  ${
    muld.1 $e |- ( ph -> A e. CC ) $.
    $( ` 0 ` is an additive identity.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    addridd $p |- ( ph -> ( A + 0 ) = A ) $=
      ( cc wcel cc0 caddc co wceq addrid syl ) ABDEBFGHBICBJK $.

    $( ` 0 ` is a left identity for addition.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    addlidd $p |- ( ph -> ( 0 + A ) = A ) $=
      ( cc wcel cc0 caddc co wceq addlid syl ) ABDEFBGHBICBJK $.

    addcomd.2 $e |- ( ph -> B e. CC ) $.
    $( Addition is commutative.  Based on ideas by Eric Schmidt.  (Contributed
       by Scott Fenton, 3-Jan-2013.)  (Revised by Mario Carneiro,
       27-May-2016.) $)
    addcomd $p |- ( ph -> ( A + B ) = ( B + A ) ) $=
      ( cc wcel caddc co wceq addcom syl2anc ) ABFGCFGBCHICBHIJDEBCKL $.

    mul12d.3 $e |- ( ph -> C e. CC ) $.
    $( Commutative/associative law that swaps the first two factors in a triple
       product.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    mul12d $p |- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) ) $=
      ( cc wcel cmul co wceq mul12 syl3anc ) ABHICHIDHIBCDJKJKCBDJKJKLEFGBCDMN
      $.

    $( Commutative/associative law that swaps the last two factors in a triple
       product.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    mul32d $p |- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) ) $=
      ( cc wcel cmul co wceq mul32 syl3anc ) ABHICHIDHIBCJKDJKBDJKCJKLEFGBCDMN
      $.

    $( Commutative/associative law.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mul31d $p |- ( ph -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) ) $=
      ( cc wcel cmul co wceq mul31 syl3anc ) ABHICHIDHIBCJKDJKDCJKBJKLEFGBCDMN
      $.

    mul4d.4 $e |- ( ph -> D e. CC ) $.
    $( Rearrangement of 4 factors.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mul4d $p |- ( ph -> ( ( A x. B ) x. ( C x. D ) ) =
                        ( ( A x. C ) x. ( B x. D ) ) ) $=
      ( cc wcel cmul co wceq mul4 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBDLMCELMLMN
      FGHIBCDEOP $.
  $}

  $( A simple product of sums expansion.  (Contributed by AV, 30-Jul-2021.) $)
  muladd11r $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) =
                 ( ( ( A x. B ) + ( A + B ) ) + 1 ) ) $=
    ( cc wcel wa c1 caddc cmul simpl 1cnd addcomd simpr oveq12d muladd11 addcld
    co mulcl addassd addcl 3eqtrd eqtr3d oveq1d ) ACDZBCDZEZAFGPZBFGPZHPFAGPZFB
    GPZHPUHBABHPZGPZGPZUJABGPZGPZFGPZUEUFUHUGUIHUEAFUCUDIZUEJZKUEBFUCUDLZUQKMAB
    NUEULFAUKGPZGPUSFGPUOUEFAUKUQUPUEBUJURABQZOZRUEFUSUQUEAUKUPVAOKUEUSUNFGUEUM
    UJGPUSUNUEABUJUPURUTRUEUMUJABSUTKUAUBTT $.

  ${
    comraddd.1 $e |- ( ph -> B e. CC ) $.
    comraddd.2 $e |- ( ph -> C e. CC ) $.
    comraddd.3 $e |- ( ph -> A = ( B + C ) ) $.
    $( Commute RHS addition, in deduction form.  (Contributed by David A.
       Wheeler, 11-Oct-2018.) $)
    comraddd $p |- ( ph -> A = ( C + B ) ) $=
      ( caddc co addcomd eqtrd ) ABCDHIDCHIGACDEFJK $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Real and complex numbers - basic operations
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Addition
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Commutative/associative law that swaps the first two terms in a triple
     sum.  (Contributed by NM, 11-May-2004.) $)
  add12 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( A + ( B + C ) ) = ( B + ( A + C ) ) ) $=
    ( cc wcel w3a caddc co wceq wa addcom oveq1d 3adant3 addass 3com12 3eqtr3d
    ) ADEZBDEZCDEZFABGHZCGHZBAGHZCGHZABCGHGHBACGHGHZQRUAUCISQRJTUBCGABKLMABCNRQ
    SUCUDIBACNOP $.

  $( Commutative/associative law that swaps the last two terms in a triple sum.
     (Contributed by NM, 13-Nov-1999.) $)
  add32 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( ( A + B ) + C ) = ( ( A + C ) + B ) ) $=
    ( cc wcel w3a caddc co wceq wa addcom oveq2d 3adant1 addass 3com23 3eqtr4d
    ) ADEZBDEZCDEZFABCGHZGHZACBGHZGHZABGHCGHACGHBGHZRSUAUCIQRSJTUBAGBCKLMABCNQS
    RUDUCIACBNOP $.

  $( Commutative/associative law that swaps the last two terms in a triple sum,
     rearranging the parentheses.  (Contributed by Paul Chapman,
     18-May-2007.) $)
  add32r $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                          ( A + ( B + C ) ) = ( ( A + C ) + B ) ) $=
    ( cc wcel w3a caddc co addass add32 eqtr3d ) ADEBDECDEFABGHCGHABCGHGHACGHBG
    HABCIABCJK $.

  $( Rearrangement of 4 terms in a sum.  (Contributed by NM, 13-Nov-1999.)
     (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
  add4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
              ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) ) $=
    ( cc wcel wa caddc wceq add12 3expb oveq2d adantll addcl addass sylan2 an4s
    co 3expa 3eqtr4d ) AEFZBEFZGZCEFZDEFZGZGABCDHRZHRZHRZACBDHRZHRZHRZABHRUGHRZ
    ACHRUJHRZUBUFUIULIUAUBUFGUHUKAHUBUDUEUHUKIBCDJKLMUFUCUGEFZUMUIIZCDNUAUBUOUP
    ABUGOSPUAUDUBUEUNULIZUBUEGUAUDGUJEFZUQBDNUAUDURUQACUJOSPQT $.

  $( Rearrangement of 4 terms in a sum.  (Contributed by NM, 12-May-2005.) $)
  add42 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
              ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) ) $=
    ( cc wcel wa caddc co add4 wceq addcom ad2ant2l oveq2d eqtrd ) AEFZBEFZGCEF
    ZDEFZGGZABHICDHIHIACHIZBDHIZHIUADBHIZHIABCDJTUBUCUAHQSUBUCKPRBDLMNO $.

  ${
    add.1 $e |- A e. CC $.
    add.2 $e |- B e. CC $.
    add.3 $e |- C e. CC $.
    $( Commutative/associative law that swaps the first two terms in a triple
       sum.  (Contributed by NM, 21-Jan-1997.) $)
    add12i $p |- ( A + ( B + C ) ) = ( B + ( A + C ) ) $=
      ( cc wcel caddc co wceq add12 mp3an ) AGHBGHCGHABCIJIJBACIJIJKDEFABCLM $.

    $( Commutative/associative law that swaps the last two terms in a triple
       sum.  (Contributed by NM, 21-Jan-1997.) $)
    add32i $p |- ( ( A + B ) + C ) = ( ( A + C ) + B ) $=
      ( cc wcel caddc co wceq add32 mp3an ) AGHBGHCGHABIJCIJACIJBIJKDEFABCLM $.

    add4.4 $e |- D e. CC $.
    $( Rearrangement of 4 terms in a sum.  (Contributed by NM, 9-May-1999.) $)
    add4i $p |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) $=
      ( cc wcel caddc co wceq add4 mp4an ) AIJBIJCIJDIJABKLCDKLKLACKLBDKLKLMEFG
      HABCDNO $.

    $( Rearrangement of 4 terms in a sum.  (Contributed by NM, 22-Aug-1999.) $)
    add42i $p |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) $=
      ( caddc co add4i addcomi oveq2i eqtri ) ABIJCDIJIJACIJZBDIJZIJODBIJZIJABC
      DEFGHKPQOIBDFHLMN $.
  $}

  ${
    addd.1 $e |- ( ph -> A e. CC ) $.
    addd.2 $e |- ( ph -> B e. CC ) $.
    addd.3 $e |- ( ph -> C e. CC ) $.
    $( Commutative/associative law that swaps the first two terms in a triple
       sum.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    add12d $p |- ( ph -> ( A + ( B + C ) ) = ( B + ( A + C ) ) ) $=
      ( cc wcel caddc co wceq add12 syl3anc ) ABHICHIDHIBCDJKJKCBDJKJKLEFGBCDMN
      $.

    $( Commutative/associative law that swaps the last two terms in a triple
       sum.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    add32d $p |- ( ph -> ( ( A + B ) + C ) = ( ( A + C ) + B ) ) $=
      ( cc wcel caddc co wceq add32 syl3anc ) ABHICHIDHIBCJKDJKBDJKCJKLEFGBCDMN
      $.

    add4d.4 $e |- ( ph -> D e. CC ) $.
    $( Rearrangement of 4 terms in a sum.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    add4d $p |- ( ph ->
      ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) ) $=
      ( cc wcel caddc co wceq add4 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBDLMCELMLM
      NFGHIBCDEOP $.

    $( Rearrangement of 4 terms in a sum.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    add42d $p |- ( ph ->
      ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) ) $=
      ( cc wcel caddc co wceq add42 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBDLMECLML
      MNFGHIBCDEOP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Subtraction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare symbols needed for some defined terms. $)
  $c - $. $( Subtraction (binary minus). $)
  $c -u $. $( Unary minus sign. $)

  $( Extend class notation to include subtraction. $)
  cmin $a class - $.

  $( Extend class notation to include unary minus.  The symbol ` -u ` is not a
     class by itself but part of a compound class definition.  We do this
     rather than making it a formal function since it is so commonly used.
     Note:  We use different symbols for unary minus ( ` -u ` ) and subtraction
     ~ cmin ( ` - ` ) to prevent syntax ambiguity.  For example, looking at the
     syntax definition ~ co , if we used the same symbol
     then " ` ( - A - B ) ` " could mean either " ` - A ` " minus " ` B ` ", or
     it could represent the (meaningless) operation of
     classes " ` - ` " and " ` - B ` " connected with "operation" " ` A ` ".
     On the other hand, " ` ( -u A - B ) ` " is unambiguous. $)
  cneg $a class -u A $.

  ${
    $d x y z $.
    $( Define subtraction.  Theorem ~ subval shows its value (and describes how
       this definition works), Theorem ~ subaddi relates it to addition, and
       Theorems ~ subcli and ~ resubcli prove its closure laws.  (Contributed
       by NM, 26-Nov-1994.) $)
    df-sub $a |- - = ( x e. CC , y e. CC |->
                   ( iota_ z e. CC ( y + z ) = x ) ) $.
  $}

  $( Define the negative of a number (unary minus).  We use different symbols
     for unary minus ( ` -u ` ) and subtraction ( ` - ` ) to prevent syntax
     ambiguity.  See ~ cneg for a discussion of this.  (Contributed by NM,
     10-Feb-1995.) $)
  df-neg $a |- -u A = ( 0 - A ) $.

  ${
    $d a b c d x y z A $.  $d x B $.  $d x C $.
    $( Addition cancellation of a real number from two complex numbers.  Lemma
       for ~ cnegex .  (Contributed by Eric Schmidt, 22-May-2007.) $)
    cnegexlem1 $p |- ( ( A e. RR /\ B e. CC /\ C e. CC ) ->
                  ( ( A + B ) = ( A + C ) <-> B = C ) ) $=
      ( vx cr wcel cc caddc co wceq cc0 wi recn wa oveq2 addassd eqeq12d adantl
      wb oveq1 w3a cv ax-rnegex 3ad2ant1 simpr simpll simplrl simplrr imbitrrid
      wrex adantr addcom eqeq1d adantlr addlid eqeqan12d ad2antrr ex sylbid imp
      bitrd sylibd sylan2 rexlimdva 3impb syl3an1 mpd impbid1 ) AEFZBGFZCGFZUAZ
      ABHIZACHIZJZBCJZVLADUBZHIZKJZDEUJZVOVPLZVIVJVTVKDAUCUDVIAGFZVJVKVTWALZAMW
      BVJVKWCWBVJVKNZNZVSWADEVQEFWEVQGFZVSWALVQMWEWFNZVSWAWGVSNVOVQAHIZBHIZWHCH
      IZJZVPWGVOWKLVSVOWKWGVQVMHIZVQVNHIZJVMVNVQHOWGWIWLWJWMWGVQABWEWFUEZWBWDWF
      UFZWBVJVKWFUGPWGVQACWNWOWBVJVKWFUHPQUIUKWGVSWKVPSZWGVSWHKJZWPWBWFVSWQSWDW
      BWFNVRWHKAVQULUMUNWGWQWPWGWQNWKKBHIZKCHIZJZVPWQWKWTSWGWQWIWRWJWSWHKBHTWHK
      CHTQRWEWTVPSZWFWQWDXAWBVJVKWRBWSCBUOCUOUPRUQVAURUSUTVBURVCVDVEVFVGBCAHOVH
      $.

    $( Existence of a real number which produces a real number when multiplied
       by ` _i ` .  (Hint: zero is such a number, although we don't need to
       prove that yet).  Lemma for ~ cnegex .  (Contributed by Eric Schmidt,
       22-May-2007.) $)
    cnegexlem2 $p |- E. y e. RR ( _i x. y ) e. RR $=
      ( vx vz cc wcel cv ci cmul co caddc wceq cr wrex wa wi adantr recn addlid
      cc0 oveq1 0cn cnre ax-rnegex ax-icn mulcl sylancr 3ad2ant3 ad2antrl add32
      w3a 3com23 eqcomd sylan9eq adantrl 3ad2ant2 eqtr3d ex syl3an 3expa simplr
      3eqtr3d imp eqeltrrd exp32 rexlimdva mpd reximdva rexlimiv mp2b ) SDESBFZ
      GAFZHIZJIZKZALMZBLMVLLEZALMZUABASUBVOVQBLVJLEZVNVPALVRVKLEZNZVJCFZJIZSKZC
      LMZVNVPOZVRWDVSCVJUCPVTWCWECLVTWALEZNZWCVNVPWGWCVNNZNWAVLLWGWHWAVLKZVRVSW
      FWHWIOZVRVJDEZVSVLDEZWFWADEZWJVJQVSGDEVKDEWLUDVKQGVKUEUFWAQWKWLWMUJZWHWIW
      NWHNZSWAJIZWAVLWNWPWAKZWHWMWKWQWLWARUGPWOWBVLJIZSVLJIZWPVLWCWRWSKWNVNWBSV
      LJTUHWNVNWRWPKWCWNVNWRVMWAJIZWPWKWMWLWRWTKVJWAVLUIUKVNWPWTSVMWAJTULUMUNWN
      WSVLKZWHWLWKXAWMVLRUOPVAUPUQURUSVBVTWFWHUTVCVDVEVFVGVHVI $.

    $( Existence of real number difference.  Lemma for ~ cnegex .  (Contributed
       by Eric Schmidt, 22-May-2007.) $)
    cnegexlem3 $p |- ( ( b e. RR /\ y e. RR ) -> E. c e. RR ( b + c ) = y ) $=
      ( vx cv cr wcel wa caddc co wceq wrex ax-rnegex adantlr adantr cc wb recn
      cc0 anim1i readdcl anim12i add32 3expa addcl addcom sylan eqtr2d adantllr
      syl sylanl2 ancoms sylan2 id sylan9eq adantlll eqeq12d simpllr cnegexlem1
      an32s simplr syl3anc bitr3d syl2an rexbidva mpbid adantl r19.29a ) BEZFGZ
      AEZFGZHZVKDEZIJZSKZVICEZIJZVKKZCFLZDFVMVNFGZHZVPHZVIVNIJZVQIJZSKZCFLZVTWB
      WGVPVJWAWGVLVJWAHWDFGWGVIVNUACWDMUJNOWCWFVSCFWCVIPGZVKPGZHZWAHZVPHZVQPGZW
      FVSQVQFGWBWKVPVMWJWAVJWHVLWIVIRVKRUBTTVQRWLWMHZVNVRIJZVNVKIJZKZWFVSWNWOWE
      WPSWKWMWOWEKZVPWHWAWMWRWIWAWHVNPGZWMWRVNRZWHWSHWMHWEVRVNIJZWOWHWSWMWEXAKV
      IVNVQUCUDWHWMWSXAWOKZWHWMHVRPGZWSXBVIVQUEZVRVNUFUGUTUHUKUINWLWPSKZWMWIWAV
      PXEWHWIWAHVPWPVOSWAWIWSWPVOKZWTWSWIXFVNVKUFULUMVPUNUOUPOUQWKWMWQVSQZVPWKW
      MHWAXCWIXGWJWAWMVAWJWMXCWAWHWMXCWIXDNNWHWIWAWMURVNVRVKUSVBNVCVDVEVFVLVPDF
      LVJDVKMVGVH $.

    $( Existence of the negative of a complex number.  (Contributed by Eric
       Schmidt, 21-May-2007.) $)
    cnegex $p |- ( A e. CC -> E. x e. CC ( A + x ) = 0 ) $=
      ( va vb vy vc vd cc wcel cv ci cmul co caddc wceq cr wrex cc0 wa ax-icn
      cnre cnegexlem2 cnegexlem3 ad2ant2lr mulcl sylancr syl2an adantlr adantll
      addcl adantr anim12i sylan2 ad2antrr ad2antlr simpr addassd simpll adantl
      recn mpan adddi mp3an1 oveq2d eqtr4d oveq1d eqtr3d adantlrr eqtr2d eqeq1d
      oveq2 biimpa rspcev syl2anc readdcl ax-rnegex ad2ant2rl r19.29a rexlimddv
      adantllr syl rexlimdvaa mpi oveq1 rexbidv syl5ibrcom rexlimivv ) BHIBCJZK
      DJZLMZNMZOZDPQCPQBAJZNMZROZAHQZCDBUAWLWPCDPPWHPIZWIPIZSZWPWLWKWMNMZROZAHQ
      ZWSKEJZLMZPIZEPQXBEUBWSXEXBEPWSXCPIZXESZSZWIFJZNMZXCOZXBFPWRXFXKFPQWQXEED
      FUCUDXHXIPIZXKSZSZWHXDNMZGJZNMZROZXBGPXNXPPIZSZXRSKXILMZXPNMZHIZWKYBNMZRO
      ZXBXTYCXRXMXSYCXHXLXSYCXKXLYAHIZXPHIZYCXSXLKHIZXIHIZYFTXIUTZKXIUEZUFXPUTZ
      YAXPUJUGUHUIUKXTXRYEXTXQYDRWSXMXSXQYDOXGWSXMSXSSYDWHKXJLMZNMZXPNMZXQWSXLX
      SYDYOOZXKWSXLSWHHIZWIHIZSZYISZYGYPXSWSYSXLYIWQYQWRYRWHUTWIUTULYJULYLYTYGS
      ZWKYANMZXPNMYDYOUUAWKYAXPYSWKHIZYIYGYRYQWJHIZUUCYHYRUUDTKWIUEVAZWHWJUJUMU
      NYIYFYSYGYHYIYFTYKVAZUOYTYGUPUQUUAUUBYNXPNYTUUBYNOYGYTUUBWHWJYANMZNMYNYTW
      HWJYAYQYRYIURYRUUDYQYIUUEUOYIYFYSUUFUSUQYTYMUUGWHNYRYIYMUUGOZYQYHYRYIUUHT
      KWIXIVBVCUIVDVEUKVFVGUGVHXMYOXQOZWSXSXKUUIXLXKYNXOXPNXKYMXDWHNXJXCKLVKVDV
      FUSUOVIVTVJVLXAYEAYBHWMYBOWTYDRWMYBWKNVKVJVMVNXHXRGPQZXMWQXEUUJWRXFWQXESX
      OPIUUJWHXDVOGXOVPWAVQUKVRVSWBWCWLWOXAAHWLWNWTRBWKWMNWDVJWEWFWGWA $.
  $}

  ${
    $d A x $.
    $( Existence of a left inverse for addition.  (Contributed by Scott Fenton,
       3-Jan-2013.) $)
    cnegex2 $p |- ( A e. CC -> E. x e. CC ( x + A ) = 0 ) $=
      ( cc wcel cv caddc co cc0 wceq wrex cnegex addcom eqeq1d rexbidva mpbid
      wa ) BCDZBAEZFGZHIZACJRBFGZHIZACJABKQTUBACQRCDPSUAHBRLMNO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Cancellation law for addition.  Theorem I.1 of [Apostol] p. 18.
       (Contributed by NM, 22-Nov-1994.)  (Proof shortened by Mario Carneiro,
       27-May-2016.) $)
    addcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                   ( ( A + B ) = ( A + C ) <-> B = C ) ) $=
      ( vx cc wcel w3a cv caddc co cc0 wceq wb wa oveq1d addassd addlid 3eqtr3d
      oveq2 syl wrex cnegex2 simprr simprl simpl1 simpl2 simpl3 eqeq12d impbid1
      3ad2ant1 imbitrid rexlimddv ) AEFZBEFZCEFZGZDHZAIJZKLZABIJZACIJZLZBCLZMDE
      UMUNUSDEUAUODAUBUJUPUQEFZUSNZNZVBVCVBUQUTIJZUQVAIJZLVFVCUTVAUQISVFVGBVHCV
      FURBIJKBIJZVGBVFURKBIUPVDUSUCZOVFUQABUPVDUSUDZUMUNUOVEUEZUMUNUOVEUFZPVFUN
      VIBLVMBQTRVFURCIJKCIJZVHCVFURKCIVJOVFUQACVKVLUMUNUOVEUGZPVFUOVNCLVOCQTRUH
      UKBCAISUIUL $.

    $( Cancellation law for addition.  (Contributed by NM, 30-Jul-2004.)
       (Revised by Scott Fenton, 3-Jan-2013.) $)
    addcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                    ( ( A + C ) = ( B + C ) <-> A = B ) ) $=
      ( vx cc wcel w3a cv caddc co cc0 wb wa oveq1 addassd oveq2d addrid 3eqtrd
      wceq syl wrex cnegex 3ad2ant3 simpl1 simpl3 simprl simprr simpl2 imbitrid
      eqeq12d impbid1 rexlimddv ) AEFZBEFZCEFZGZCDHZIJZKSZACIJZBCIJZSZABSZLDEUO
      UMUSDEUAUNDCUBUCUPUQEFZUSMZMZVBVCVBUTUQIJZVAUQIJZSVFVCUTVAUQINVFVGAVHBVFV
      GAURIJAKIJZAVFACUQUMUNUOVEUDZUMUNUOVEUEZUPVDUSUFZOVFURKAIUPVDUSUGZPVFUMVI
      ASVJAQTRVFVHBURIJBKIJZBVFBCUQUMUNUOVEUHZVKVLOVFURKBIVMPVFUNVNBSVOBQTRUJUI
      ABCINUKUL $.
  $}

  ${
    addcani.1 $e |- A e. CC $.
    addcani.2 $e |- B e. CC $.
    addcani.3 $e |- C e. CC $.
    $( Cancellation law for addition.  Theorem I.1 of [Apostol] p. 18.
       (Contributed by NM, 27-Oct-1999.)  (Revised by Scott Fenton,
       3-Jan-2013.) $)
    addcani $p |- ( ( A + B ) = ( A + C ) <-> B = C ) $=
      ( cc wcel caddc co wceq wb addcan mp3an ) AGHBGHCGHABIJACIJKBCKLDEFABCMN
      $.

    $( Cancellation law for addition.  Theorem I.1 of [Apostol] p. 18.
       (Contributed by NM, 14-May-2003.)  (Revised by Scott Fenton,
       3-Jan-2013.) $)
    addcan2i $p |- ( ( A + C ) = ( B + C ) <-> A = B ) $=
      ( cc wcel caddc co wceq wb addcan2 mp3an ) AGHBGHCGHACIJBCIJKABKLDEFABCMN
      $.
  $}

  ${
    addcand.1 $e |- ( ph -> A e. CC ) $.
    addcand.2 $e |- ( ph -> B e. CC ) $.
    addcand.3 $e |- ( ph -> C e. CC ) $.
    $( Cancellation law for addition.  Theorem I.1 of [Apostol] p. 18.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    addcand $p |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) ) $=
      ( cc wcel caddc co wceq wb addcan syl3anc ) ABHICHIDHIBCJKBDJKLCDLMEFGBCD
      NO $.

    $( Cancellation law for addition.  Theorem I.1 of [Apostol] p. 18.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    addcan2d $p |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) ) $=
      ( cc wcel caddc co wceq wb addcan2 syl3anc ) ABHICHIDHIBDJKCDJKLBCLMEFGBC
      DNO $.

    ${
      addcanad.4 $e |- ( ph -> ( A + B ) = ( A + C ) ) $.
      $( Cancelling a term on the left-hand side of a sum in an equality.
         Consequence of ~ addcand .  (Contributed by David Moews,
         28-Feb-2017.) $)
      addcanad $p |- ( ph -> B = C ) $=
        ( caddc co wceq addcand mpbid ) ABCIJBDIJKCDKHABCDEFGLM $.
    $}

    ${
      addcan2ad.4 $e |- ( ph -> ( A + C ) = ( B + C ) ) $.
      $( Cancelling a term on the right-hand side of a sum in an equality.
         Consequence of ~ addcan2d .  (Contributed by David Moews,
         28-Feb-2017.) $)
      addcan2ad $p |- ( ph -> A = B ) $=
        ( caddc co wceq addcan2d mpbid ) ABDIJCDIJKBCKHABCDEFGLM $.
    $}

    ${
      addneintrd.4 $e |- ( ph -> B =/= C ) $.
      $( Introducing a term on the left-hand side of a sum in a negated
         equality.  Contrapositive of ~ addcanad .  Consequence of ~ addcand .
         (Contributed by David Moews, 28-Feb-2017.) $)
      addneintrd $p |- ( ph -> ( A + B ) =/= ( A + C ) ) $=
        ( caddc co wne addcand necon3bid mpbird ) ABCIJZBDIJZKCDKHAOPCDABCDEFGL
        MN $.
    $}

    ${
      addneintr2d.4 $e |- ( ph -> A =/= B ) $.
      $( Introducing a term on the right-hand side of a sum in a negated
         equality.  Contrapositive of ~ addcan2ad .  Consequence of
         ~ addcan2d .  (Contributed by David Moews, 28-Feb-2017.) $)
      addneintr2d $p |- ( ph -> ( A + C ) =/= ( B + C ) ) $=
        ( caddc co wne addcan2d necon3bid mpbird ) ABDIJZCDIJZKBCKHAOPBCABCDEFG
        LMN $.
    $}
  $}

  $( Alternate proof of ~ 0cn .  (Contributed by NM, 19-Feb-2005.)  (Revised by
     Mario Carneiro, 27-May-2016.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  0cnALT $p |- 0 e. CC $=
    ( vx ci cv caddc co cc0 wceq wrex wcel ax-icn cnegex ax-mp addcl mpan eleq1
    cc syl5ibcom rexlimiv ) BACZDEZFGZAPHZFPIZBPIZUBJABKLUAUCAPSPIZTPIZUAUCUDUE
    UFJBSMNTFPOQRL $.

  ${
    $d x y A $.  $d x y B $.
    $( Existential uniqueness of negatives.  Theorem I.2 of [Apostol] p. 18.
       (Contributed by NM, 22-Nov-1994.)  (Proof shortened by Mario Carneiro,
       27-May-2016.) $)
    negeu $p |- ( ( A e. CC /\ B e. CC ) -> E! x e. CC ( A + x ) = B ) $=
      ( vy cc wcel wa cv caddc co wceq wreu wrex cnegex adantr wral simpl simpr
      cc0 wb addcl syl2anr simplrr oveq1d simplll simplrl simpllr eqeq2d addcld
      addassd addlidd 3eqtr3rd addcand bitrd ralrimiva reu6i syl2anc rexlimddv
      ) BEFZCEFZGZBDHZIJZSKZBAHZIJZCKZAELZDEUSVDDEMUTDBNOVAVBEFZVDGZGZVBCIJZEFZ
      VGVEVLKZTZAEPVHVJVIUTVMVAVIVDQUSUTRVBCUAUBVKVOAEVKVEEFZGZVGVFBVLIJZKVNVQC
      VRVFVQVCCIJSCIJVRCVQVCSCIVAVIVDVPUCUDVQBVBCUSUTVJVPUEZVAVIVDVPUFZUSUTVJVP
      UGZUJVQCWAUKULUHVQBVEVLVSVKVPRVQVBCVTWAUIUMUNUOVGAEVLUPUQUR $.
  $}

  ${
    $d x y z A $.  $d x y z B $.
    $( Value of subtraction, which is the (unique) element ` x ` such that
       ` B + x = A ` .  (Contributed by NM, 4-Aug-2007.)  (Revised by Mario
       Carneiro, 2-Nov-2013.) $)
    subval $p |- ( ( A e. CC /\ B e. CC ) ->
                  ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) ) $=
      ( vy vz cc wcel cv caddc co wceq crio cmin wa negeu riotacl syl riotabidv
      wreu ancoms eqeq2 oveq1 eqeq1d df-sub ovmpog mpd3an3 ) BFGZCFGZCAHZIJZBKZ
      AFLZFGZBCMJULKUHUGUMUHUGNUKAFSUMACBOUKAFPQTDEBCFFEHZUIIJZDHZKZAFLULMUOBKZ
      AFLFUPBKUQURAFUPBUOUARUNCKZURUKAFUSUOUJBUNCUIIUBUCRDEAUDUEUF $.
  $}

  $( Equality theorem for negatives.  (Contributed by NM, 10-Feb-1995.) $)
  negeq $p |- ( A = B -> -u A = -u B ) $=
    ( wceq cc0 cmin co cneg oveq2 df-neg 3eqtr4g ) ABCDAEFDBEFAGBGABDEHAIBIJ $.

  ${
    negeqi.1 $e |- A = B $.
    $( Equality inference for negatives.  (Contributed by NM, 14-Feb-1995.) $)
    negeqi $p |- -u A = -u B $=
      ( wceq cneg negeq ax-mp ) ABDAEBEDCABFG $.
  $}

  ${
    negeqd.1 $e |- ( ph -> A = B ) $.
    $( Equality deduction for negatives.  (Contributed by NM, 14-May-1999.) $)
    negeqd $p |- ( ph -> -u A = -u B ) $=
      ( wceq cneg negeq syl ) ABCEBFCFEDBCGH $.
  $}

  ${
    nfnegd.1 $e |- ( ph -> F/_ x A ) $.
    $( Deduction version of ~ nfneg .  (Contributed by NM, 29-Feb-2008.)
       (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nfnegd $p |- ( ph -> F/_ x -u A ) $=
      ( cneg cc0 cmin co df-neg nfcvd nfovd nfcxfrd ) ABCEFCGHCIABFCGABFJABGJDK
      L $.
  $}

  ${
    nfneg.1 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for the negative of a complex number.
       (Contributed by NM, 12-Jun-2005.)  (Revised by Mario Carneiro,
       15-Oct-2016.) $)
    nfneg $p |- F/_ x -u A $=
      ( cneg wnfc wtru a1i nfnegd mptru ) ABDEFABABEFCGHI $.
  $}

  $( Move class substitution in and out of the negative of a number.
     (Contributed by NM, 1-Mar-2008.)  (Proof shortened by Andrew Salmon,
     22-Oct-2011.) $)
  csbnegg $p |- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B ) $=
    ( wcel cc0 cmin co csb cneg csbov2g df-neg csbeq2i 3eqtr4g ) BDEABFCGHZIFAB
    CIZGHABCJZIPJABFCGDKABQOCLMPLN $.

  ${
    $d x A $.  $d x B $.
    $( Closure law for subtraction.  (Contributed by NM, 10-May-1999.)
       (Revised by Mario Carneiro, 21-Dec-2013.) $)
    subcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC ) $=
      ( vx cc wcel wa cmin co cv caddc wceq crio subval wreu ancoms riotacl syl
      negeu eqeltrd ) ADEZBDEZFZABGHBCIJHAKZCDLZDCABMUBUCCDNZUDDEUATUECBAROUCCD
      PQS $.
  $}

  $( Closure law for negative.  (Contributed by NM, 6-Aug-2003.) $)
  negcl $p |- ( A e. CC -> -u A e. CC ) $=
    ( cc wcel cneg cc0 cmin co df-neg 0cn subcl mpan eqeltrid ) ABCZADEAFGZBAHE
    BCMNBCIEAJKL $.

  $( ` -u _i ` is a complex number (common case).  (Contributed by David A.
     Wheeler, 7-Dec-2018.) $)
  negicn $p |- -u _i e. CC $=
    ( ci cc wcel cneg ax-icn negcl ax-mp ) ABCADBCEAFG $.

  ${
    $d x y z $.
    $( Subtraction is an operation on the complex numbers.  (Contributed by NM,
       4-Aug-2007.)  (Revised by Mario Carneiro, 16-Nov-2013.) $)
    subf $p |- - : ( CC X. CC ) --> CC $=
      ( vy vz vx cv caddc co wceq cc crio wcel wral cxp cmin wf wa subval subcl
      eqeltrrd rgen2a df-sub fmpo mpbi ) ADZBDEFCDZGBHIZHJZAHKCHKHHLHMNUFCAHUDH
      JUCHJOUDUCMFUEHBUDUCPUDUCQRSCAHHUEHMCABTUAUB $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Relationship between subtraction and addition.  (Contributed by NM,
       20-Jan-1997.)  (Revised by Mario Carneiro, 21-Dec-2013.) $)
    subadd $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                  ( ( A - B ) = C <-> ( B + C ) = A ) ) $=
      ( vx cc wcel w3a cmin co wceq cv caddc crio wb subval eqeq1d 3adant3 wreu
      wa negeu oveq2 riota2 sylan2 3impb 3com13 bitr4d ) AEFZBEFZCEFZGABHIZCJZB
      DKZLIZAJZDEMZCJZBCLIZAJZUGUHUKUPNUIUGUHSUJUOCDABOPQUIUHUGURUPNZUIUHUGUSUH
      UGSUIUNDERUSDBATUNURDECULCJUMUQAULCBLUAPUBUCUDUEUF $.
  $}

  $( Relationship between subtraction and addition.  (Contributed by Scott
     Fenton, 5-Jul-2013.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  subadd2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
            ( ( A - B ) = C <-> ( C + B ) = A ) ) $=
    ( cc wcel w3a cmin co wceq caddc subadd simp2 simp3 addcomd eqeq1d bitrd )
    ADEZBDEZCDEZFZABGHCIBCJHZAICBJHZAIABCKTUAUBATBCQRSLQRSMNOP $.

  $( Swap subtrahend and result of subtraction.  (Contributed by NM,
     14-Dec-2007.) $)
  subsub23 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - B ) = C <-> ( A - C ) = B ) ) $=
    ( cc wcel caddc co wceq cmin addcom 3adant1 eqeq1d subadd wb 3com23 3bitr4d
    w3a ) ADEZBDEZCDEZQZBCFGZAHCBFGZAHZABIGCHACIGBHZUAUBUCASTUBUCHRBCJKLABCMRTS
    UEUDNACBMOP $.

  $( Cancellation law for subtraction.  (Contributed by NM, 10-May-2004.)
     (Revised by Mario Carneiro, 27-May-2016.) $)
  pncan $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A ) $=
    ( cc wcel wa caddc co cmin wceq simpr simpl addcomd wb addcl subadd syl3anc
    mpbird ) ACDZBCDZEZABFGZBHGAIZBAFGUAIZTBARSJZRSKZLTUACDSRUBUCMABNUDUEUABAOP
    Q $.

  $( Cancellation law for subtraction.  (Contributed by NM, 17-Apr-2005.) $)
  pncan2 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B ) $=
    ( cc wcel caddc co cmin wceq wa addcom oveq1d pncan eqtr3d ancoms ) BCDZACD
    ZABEFZAGFZBHOPIZBAEFZAGFRBSTQAGBAJKBALMN $.

  $( Subtraction and addition of equals.  (Contributed by NM, 14-Mar-2005.) $)
  pncan3 $p |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B - A ) ) = B ) $=
    ( cc wcel wa cmin co wceq caddc eqid simpr simpl subcl ancoms syl3anc mpbii
    wb subadd ) ACDZBCDZEZBAFGZUBHZAUBIGBHZUBJUATSUBCDZUCUDQSTKSTLTSUEBAMNBAUBR
    OP $.

  $( Cancellation law for subtraction.  (Contributed by NM, 10-May-2004.)
     (Revised by Mario Carneiro, 27-May-2016.) $)
  npcan $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = A ) $=
    ( cc wcel wa cmin co caddc subcl simpr addcomd wceq pncan3 ancoms eqtrd ) A
    CDZBCDZEZABFGZBHGBSHGZARSBABIPQJKQPTALBAMNO $.

  $( Associative-type law for addition and subtraction.  (Contributed by NM,
     6-Aug-2003.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  addsubass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                   ( ( A + B ) - C ) = ( A + ( B - C ) ) ) $=
    ( cc wcel cmin co caddc simp1 subcl 3adant1 simp3 addassd wceq npcan oveq2d
    w3a eqtrd oveq1d addcld pncan syl2anc eqtr3d ) ADEZBDEZCDEZQZABCFGZHGZCHGZC
    FGZABHGZCFGUIUGUJULCFUGUJAUHCHGZHGULUGAUHCUDUEUFIZUEUFUHDEUDBCJKZUDUEUFLZMU
    GUMBAHUEUFUMBNUDBCOKPRSUGUIDEUFUKUINUGAUHUNUOTUPUICUAUBUC $.

  $( Law for addition and subtraction.  (Contributed by NM, 19-Aug-2001.)
     (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
  addsub $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                  ( ( A + B ) - C ) = ( ( A - C ) + B ) ) $=
    ( cc wcel w3a caddc co cmin wa addcom oveq1d 3adant3 addsubass 3com12 subcl
    wceq sylan2 3impb 3eqtrd ) ADEZBDEZCDEZFABGHZCIHZBAGHZCIHZBACIHZGHZUHBGHZUA
    UBUEUGQUCUAUBJUDUFCIABKLMUBUAUCUGUIQBACNOUBUAUCUIUJQZUBUAUCUKUAUCJUBUHDEUKA
    CPBUHKRSOT $.

  $( Commutative/associative law for addition and subtraction.  (Contributed by
     NM, 1-Feb-2007.) $)
  subadd23 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( ( A - B ) + C ) = ( A + ( C - B ) ) ) $=
    ( cc wcel cmin co caddc wceq w3a addsub addsubass eqtr3d 3com23 ) ADEZCDEZB
    DEZABFGCHGZACBFGHGZIOPQJACHGBFGRSACBKACBLMN $.

  $( Commutative/associative law for addition and subtraction.  (Contributed by
     NM, 8-Feb-2005.) $)
  addsub12 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( A + ( B - C ) ) = ( B + ( A - C ) ) ) $=
    ( cc wcel cmin co caddc w3a subadd23 subcl addcom sylan 3impa eqtr3d 3com23
    wceq wa ) ADEZCDEZBDEZABCFGHGZBACFGZHGZQSTUAIUCBHGZUBUDACBJSTUAUEUDQZSTRUCD
    EUAUFACKUCBLMNOP $.

  $( Law for subtraction and addition.  (Contributed by NM, 20-Nov-2005.) $)
  2addsub $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
                 ( ( ( A + B ) + C ) - D ) = ( ( ( A + C ) - D ) + B ) ) $=
    ( cc wcel wa caddc co cmin wceq add32 3expa oveq1d addcl addsub 3expb sylan
    adantrr an4s eqtrd ) AEFZBEFZGZCEFZDEFZGGZABHICHIZDJIACHIZBHIZDJIZUIDJIBHIZ
    UGUHUJDJUDUEUHUJKZUFUBUCUEUMABCLMSNUBUEUCUFUKULKZUBUEGUIEFZUCUFGUNACOUOUCUF
    UNUIBDPQRTUA $.

  $( Relation between sums and differences.  (Contributed by Jeff Madsen,
     17-Jun-2010.) $)
  addsubeq4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
                    ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) ) $=
    ( cc wcel wa cmin co wceq caddc eqcom subcl ancoms subadd 3expa an4s bitrid
    wb sylan addcom adantl oveq1d addsubass 3com12 eqtrd adantlr addcl 3bitr2rd
    eqeq1d 3expb sylan2 ) AEFZBEFZGZCEFZDEFZGZGZCAHIZBDHIZJZDUTKIZBJZCDKIZAHIZB
    JZABKIVEJZVBVAUTJZUSVDUTVALUMUPUNUQVIVDSZUMUPGUTEFZUNUQGZVJUPUMVKCAMNVLVKVJ
    UNUQVKVJBDUTOPNTQRUSVFVCBUMURVFVCJUNUMURGZVFDCKIZAHIZVCVMVEVNAHURVEVNJUMCDU
    AUBUCURUMVOVCJZUPUQUMVPUQUPUMVPDCAUDUEPNUFUGUJURUOVEEFZVGVHSZCDUHVQUOVRVQUM
    UNVRVEABOUKNULUI $.

  ${
    pncan3oi.1 $e |- A e. CC $.
    pncan3oi.2 $e |- B e. CC $.
    $( Subtraction and addition of equals.  Almost but not exactly the same as
       ~ pncan3i and ~ pncan , this order happens often when applying
       "operations to both sides" so create a theorem specifically for it.  A
       deduction version of this is available as ~ pncand .  (Contributed by
       David A. Wheeler, 11-Oct-2018.) $)
    pncan3oi $p |- ( ( A + B ) - B ) = A $=
      ( cc wcel caddc co cmin wceq pncan mp2an ) AEFBEFABGHBIHAJCDABKL $.
  $}

  ${
    mvrraddi.1 $e |- B e. CC $.
    mvrraddi.2 $e |- C e. CC $.
    mvrraddi.3 $e |- A = ( B + C ) $.
    $( Move RHS right addition to LHS. (Contributed by David A. Wheeler,
       11-Oct-2018.) $)
    mvrraddi $p |- ( A - C ) = B $=
      ( cmin co caddc oveq1i pncan3oi eqtri ) ACGHBCIHZCGHBAMCGFJBCDEKL $.
  $}

  ${
    mvlladdi.1 $e |- A e. CC $.
    mvlladdi.2 $e |- B e. CC $.
    mvlladdi.3 $e |- ( A + B ) = C $.
    $( Move LHS left addition to RHS. (Contributed by David A. Wheeler,
       11-Oct-2018.) $)
    mvlladdi $p |- B = ( C - A ) $=
      ( caddc co cmin pncan3oi addcomi eqtr3i oveq1i ) BAGHZAIHBCAIHBAEDJNCAIAB
      GHNCABDEKFLML $.
  $}

  $( Subtraction of a number from itself.  (Contributed by NM, 8-Oct-1999.)
     (Revised by Mario Carneiro, 27-May-2016.) $)
  subid $p |- ( A e. CC -> ( A - A ) = 0 ) $=
    ( cc wcel cc0 caddc co cmin addrid oveq1d wceq 0cn pncan2 mpan2 eqtr3d ) AB
    CZADEFZAGFZAAGFDOPAAGAHIODBCQDJKADLMN $.

  $( Identity law for subtraction.  (Contributed by NM, 9-May-2004.)  (Revised
     by Mario Carneiro, 27-May-2016.) $)
  subid1 $p |- ( A e. CC -> ( A - 0 ) = A ) $=
    ( cc wcel cc0 caddc co cmin addrid oveq1d wceq 0cn pncan mpan2 eqtr3d ) ABC
    ZADEFZDGFZADGFAOPADGAHIODBCQAJKADLMN $.

  $( Cancellation law for subtraction.  (Contributed by NM, 8-Feb-2005.) $)
  npncan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - B ) + ( B - C ) ) = ( A - C ) ) $=
    ( cc wcel w3a cmin co caddc wceq subcl 3adant3 addsubass syld3an1 wa oveq1d
    npcan eqtr3d ) ADEZBDEZCDEZFABGHZBIHZCGHZUBBCGHIHZACGHZUBDEZTSUAUDUEJSTUGUA
    ABKLUBBCMNSTUDUFJUASTOUCACGABQPLR $.

  $( Cancellation law for subtraction.  (Contributed by NM, 1-Sep-2005.) $)
  nppcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( ( A - B ) + C ) + B ) = ( A + C ) ) $=
    ( cc wcel w3a cmin co caddc subcl 3adant3 simp3 simp2 add32d wceq wa oveq1d
    npcan eqtrd ) ADEZBDEZCDEZFZABGHZCIHBIHUDBIHZCIHZACIHZUCUDCBTUAUDDEUBABJKTU
    AUBLTUAUBMNTUAUFUGOUBTUAPUEACIABRQKS $.

  $( Cancellation law for subtraction:  ((a-b)-c)+b = a-c holds for complex
     numbers a,b,c.  (Contributed by Alexander van der Vekens, 24-Mar-2018.) $)
  nnpcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC )
                    -> ( ( ( A - B ) - C ) + B ) = ( A - C ) ) $=
    ( cc wcel w3a cmin co caddc wceq subcl 3adant3 addsub eqcomd syld3an1 npcan
    oveq1d eqtrd ) ADEZBDEZCDEZFZABGHZCGHBIHZUCBIHZCGHZACGHUCDEZTSUAUDUFJSTUGUA
    ABKLUGTUAFUFUDUCBCMNOUBUEACGSTUEAJUAABPLQR $.

  $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
     14-Sep-2015.) $)
  nppcan3 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - B ) + ( C + B ) ) = ( A + C ) ) $=
    ( cc wcel w3a cmin co caddc subcl 3adant3 simp3 simp2 addassd nppcan eqtr3d
    ) ADEZBDEZCDEZFZABGHZCIHBIHUACBIHIHACIHTUACBQRUADESABJKQRSLQRSMNABCOP $.

  $( Cancellation law for subtraction.  (Contributed by NM, 8-Feb-2005.) $)
  subcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - C ) = ( B - C ) <-> A = B ) ) $=
    ( cc wcel w3a cmin co wceq caddc wb simp1 simp3 subcl 3adant1 subadd2 npcan
    syl3anc eqeq1d eqcom bitrdi bitrd ) ADEZBDEZCDEZFZACGHBCGHZIZUGCJHZAIZABIZU
    FUCUEUGDEZUHUJKUCUDUELUCUDUEMUDUEULUCBCNOACUGPRUFUJBAIUKUFUIBAUDUEUIBIUCBCQ
    OSBATUAUB $.

  $( If the difference between two numbers is zero, they are equal.
     (Contributed by NM, 16-Nov-1999.) $)
  subeq0 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = 0 <-> A = B ) ) $=
    ( cc wcel wa cmin co wceq cc0 subid adantl eqeq2d subcan2 3anidm23 bitr3d
    wb ) ACDZBCDZEZABFGZBBFGZHZTIHABHZSUAITRUAIHQBJKLQRUBUCPABBMNO $.

  $( Cancellation law for subtraction.  (Contributed by Scott Fenton,
     21-Jun-2013.) $)
  npncan2 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + ( B - A ) ) = 0 ) $=
    ( cc wcel wa cmin co caddc cc0 wceq npncan 3anidm13 subid adantr eqtrd ) AC
    DZBCDZEABFGBAFGHGZAAFGZIPQRSJABAKLPSIJQAMNO $.

  $( Law for double subtraction.  (Contributed by NM, 30-Jun-2005.)  (Revised
     by Mario Carneiro, 27-May-2016.) $)
  subsub2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                  ( A - ( B - C ) ) = ( A + ( C - B ) ) ) $=
    ( cc wcel w3a cmin co caddc wceq cc0 subcl 3adant1 simp1 simp3 simp2 add12d
    syl2anc npncan2 oveq2d addridd 3eqtrd wb addcld subadd syl3anc mpbird ) ADE
    ZBDEZCDEZFZABCGHZGHACBGHZIHZJZULUNIHZAJZUKUPAULUMIHZIHAKIHAUKULAUMUIUJULDEZ
    UHBCLMZUHUIUJNZUKUJUIUMDEUHUIUJOUHUIUJPCBLRZQUKURKAIUIUJURKJUHBCSMTUKAVAUAU
    BUKUHUSUNDEUOUQUCVAUTUKAUMVAVBUDAULUNUEUFUG $.

  $( Cancellation law for subtraction.  (Contributed by NM, 21-Jun-2005.)
     (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  nncan $p |- ( ( A e. CC /\ B e. CC ) -> ( A - ( A - B ) ) = B ) $=
    ( cc wcel wa cmin co caddc wceq subsub2 3anidm12 pncan3 eqtrd ) ACDZBCDZEAA
    BFGFGZABAFGHGZBNOPQIAABJKABLM $.

  $( Law for double subtraction.  (Contributed by NM, 13-May-2004.) $)
  subsub $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                  ( A - ( B - C ) ) = ( ( A - B ) + C ) ) $=
    ( cc wcel w3a cmin caddc subsub2 wceq addsubass addsub eqtr3d 3com23 eqtrd
    co ) ADEZBDEZCDEZFABCGPGPACBGPHPZABGPCHPZABCIQSRTUAJQSRFACHPBGPTUAACBKACBLM
    NO $.

  $( Cancellation law for subtraction.  (Contributed by NM, 29-Sep-2005.) $)
  nppcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - ( B + C ) ) + C ) = ( A - B ) ) $=
    ( cc wcel w3a caddc co cmin wceq addcl 3adant1 subsub syld3an2 pncan oveq2d
    eqtr3d ) ADEZBDEZCDEZFZABCGHZCIHZIHZAUBIHCGHZABIHRUBDEZSTUDUEJSTUFRBCKLAUBC
    MNUAUCBAISTUCBJRBCOLPQ $.

  $( Law for double subtraction.  (Contributed by NM, 27-Jul-2005.) $)
  subsub3 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                  ( A - ( B - C ) ) = ( ( A + C ) - B ) ) $=
    ( cc wcel w3a cmin co caddc subsub2 wceq addsubass 3com23 eqtr4d ) ADEZBDEZ
    CDEZFABCGHGHACBGHIHZACIHBGHZABCJOQPSRKACBLMN $.

  $( Law for double subtraction.  (Contributed by NM, 19-Aug-2005.)  (Revised
     by Mario Carneiro, 27-May-2016.) $)
  subsub4 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                  ( ( A - B ) - C ) = ( A - ( B + C ) ) ) $=
    ( cc wcel w3a cmin co caddc wceq nppcan2 wb simp1 simp2 subcl syl2anc simp3
    addcld subadd2 syl3anc mpbird ) ADEZBDEZCDEZFZABGHZCGHABCIHZGHZJZUHCIHUFJZA
    BCKUEUFDEZUDUHDEZUIUJLUEUBUCUKUBUCUDMZUBUCUDNZABOPUBUCUDQZUEUBUGDEULUMUEBCU
    NUORAUGOPUFCUHSTUA $.

  $( Swap the second and third terms in a double subtraction.  (Contributed by
     NM, 19-Aug-2005.) $)
  sub32 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                  ( ( A - B ) - C ) = ( ( A - C ) - B ) ) $=
    ( cc wcel w3a caddc cmin wceq addcom 3adant1 oveq2d subsub4 3com23 3eqtr4d
    co ) ADEZBDEZCDEZFZABCGPZHPACBGPZHPZABHPCHPACHPBHPZTUAUBAHRSUAUBIQBCJKLABCM
    QSRUDUCIACBMNO $.

  $( Cancellation law for subtraction.  (Contributed by NM, 4-Sep-2005.) $)
  nnncan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - ( B - C ) ) - C ) = ( A - B ) ) $=
    ( cc wcel cmin co caddc wceq subcl 3adant1 subsub4 syld3an2 wa npcan oveq2d
    w3a eqtrd ) ADEZBDEZCDEZQABCFGZFGCFGZAUBCHGZFGZABFGZSUBDEZTUAUCUEITUAUGSBCJ
    KAUBCLMTUAUEUFISTUANUDBAFBCOPKR $.

  $( Cancellation law for subtraction.  (Contributed by NM, 8-Feb-2005.)
     (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  nnncan1 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - B ) - ( A - C ) ) = ( C - B ) ) $=
    ( cc wcel w3a cmin co wceq subcl 3adant2 sub32 syld3an3 nncan oveq1d eqtrd
    ) ADEZBDEZCDEZFZABGHACGHZGHZAUAGHZBGHZCBGHQRSUADEZUBUDIQSUERACJKABUALMTUCCB
    GQSUCCIRACNKOP $.

  $( Cancellation law for subtraction.  (Contributed by NM, 1-Oct-2005.) $)
  nnncan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - C ) - ( B - C ) ) = ( A - B ) ) $=
    ( cc wcel w3a cmin co wceq subcl 3adant1 sub32 syld3an2 nnncan eqtr3d ) ADE
    ZBDEZCDEZFABCGHZGHCGHZACGHSGHZABGHPSDEZQRTUAIQRUBPBCJKASCLMABCNO $.

  $( Cancellation law for subtraction.  (Contributed by Scott Fenton,
     23-Jun-2013.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  npncan3 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
    ( ( A - B ) + ( C - A ) ) = ( C - B ) ) $=
    ( cc wcel w3a cmin co caddc simp1 subcl ancoms 3adant2 simp2 addsub syl3anc
    wceq pncan3 oveq1d eqtr3d ) ADEZBDEZCDEZFZACAGHZIHZBGHZABGHUEIHZCBGHUDUAUED
    EZUBUGUHQUAUBUCJUAUCUIUBUCUAUICAKLMUAUBUCNAUEBOPUDUFCBGUAUCUFCQUBACRMST $.

  $( Cancellation law for mixed addition and subtraction.  (Contributed by NM,
     4-Mar-2005.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  pnpcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A + B ) - ( A + C ) ) = ( B - C ) ) $=
    ( cc wcel w3a caddc co cmin simp1 simp2 addcld simp3 subsub4 syl3anc pncan2
    wceq 3adant3 oveq1d eqtr3d ) ADEZBDEZCDEZFZABGHZAIHZCIHZUEACGHIHZBCIHUDUEDE
    UAUCUGUHQUDABUAUBUCJZUAUBUCKLUIUAUBUCMUEACNOUDUFBCIUAUBUFBQUCABPRST $.

  $( Cancellation law for mixed addition and subtraction.  (Contributed by
     Scott Fenton, 9-Jun-2006.) $)
  pnpcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( ( A + C ) - ( B + C ) ) = ( A - B ) ) $=
    ( cc wcel w3a caddc co cmin wceq addcom 3adant2 3adant1 oveq12d 3coml eqtrd
    pnpcan ) ADEZBDEZCDEZFZACGHZBCGHZIHCAGHZCBGHZIHZABIHZUAUBUDUCUEIRTUBUDJSACK
    LSTUCUEJRBCKMNTRSUFUGJCABQOP $.

  $( Cancellation law for mixed addition and subtraction.  (Contributed by NM,
     30-Jun-2005.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  pnncan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A + B ) - ( A - C ) ) = ( B + C ) ) $=
    ( cc wcel caddc co cmin wceq simp1 simp2 addcld simp3 subsub syl3anc pncan2
    w3a 3adant3 oveq1d eqtrd ) ADEZBDEZCDEZQZABFGZACHGHGZUEAHGZCFGZBCFGUDUEDEUA
    UCUFUHIUDABUAUBUCJZUAUBUCKLUIUAUBUCMUEACNOUDUGBCFUAUBUGBIUCABPRST $.

  $( Cancellation law for mixed addition and subtraction.  (Contributed by NM,
     30-Jun-2005.) $)
  ppncan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A + B ) + ( C - B ) ) = ( A + C ) ) $=
    ( cc wcel w3a caddc co cmin wceq addcom 3adant3 oveq1d addcl subsub2 pnncan
    syld3an1 3com12 3eqtr3d ) ADEZBDEZCDEZFZABGHZBCIHZIHZBAGHZUEIHZUDCBIHGHZACG
    HZUCUDUGUEITUAUDUGJUBABKLMUDDEZUATUBUFUIJTUAUKUBABNLUDBCOQUATUBUHUJJBACPRS
    $.

  $( Rearrangement of 4 terms in a mixed addition and subtraction.
     (Contributed by NM, 4-Mar-2005.) $)
  addsub4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
              ( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) ) ) $=
    ( cc wcel caddc cmin wceq simpll simplr simprl addsub syl3anc oveq1d addcld
    wa co simprr subsub4 subcl ad2ant2r addsubass 3eqtr3d ) AEFZBEFZQZCEFZDEFZQ
    ZQZABGRZCHRZDHRZACHRZBGRZDHRZULCDGRHRZUOBDHRGRZUKUMUPDHUKUEUFUHUMUPIUEUFUJJ
    ZUEUFUJKZUGUHUILZABCMNOUKULEFUHUIUNURIUKABUTVAPVBUGUHUISZULCDTNUKUOEFZUFUIU
    QUSIUEUHVDUFUIACUAUBVAVCUOBDUCNUD $.

  $( Rearrangement of 4 terms in a mixed addition and subtraction.
     (Contributed by NM, 24-Aug-2006.) $)
  subadd4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
              ( ( A - B ) - ( C - D ) ) = ( ( A + D ) - ( B + C ) ) ) $=
    ( cc wcel wa cmin caddc wceq subcl subsub2 3expb sylan addsub4 an42s eqtr4d
    co ) AEFZBEFZGZCEFZDEFZGZGABHRZCDHRHRZUEDCHRIRZADIRBCIRHRZUAUEEFZUDUFUGJZAB
    KUIUBUCUJUECDLMNSUCTUBUHUGJADBCOPQ $.

  $( Rearrangement of 4 terms in a subtraction.  (Contributed by NM,
     23-Nov-2007.) $)
  sub4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
              ( ( A - B ) - ( C - D ) ) = ( ( A - C ) - ( B - D ) ) ) $=
    ( cc wcel wa caddc cmin wceq addcom ad2ant2lr oveq2d subadd4 an4s 3eqtr4d
    co ) AEFZBEFZGCEFZDEFZGGZADHQZBCHQZIQUCCBHQZIQZABIQCDIQIQACIQBDIQIQZUBUDUEU
    CISTUDUEJRUABCKLMABCDNRTSUAUGUFJACBDNOP $.

  $( Minus 0 equals 0.  (Contributed by NM, 17-Jan-1997.) $)
  neg0 $p |- -u 0 = 0 $=
    ( cc0 cneg cmin co df-neg cc wcel wceq 0cn subid ax-mp eqtri ) ABAACDZAAEAF
    GMAHIAJKL $.

  $( Addition of a number and its negative.  (Contributed by NM,
     14-Mar-2005.) $)
  negid $p |- ( A e. CC -> ( A + -u A ) = 0 ) $=
    ( cc wcel cneg caddc co cc0 cmin df-neg oveq2i wceq 0cn pncan3 mpan2 eqtrid
    ) ABCZAADZEFAGAHFZEFZGQRAEAIJPGBCSGKLAGMNO $.

  $( Relationship between subtraction and negative.  Theorem I.3 of [Apostol]
     p. 18.  (Contributed by NM, 21-Jan-1997.)  (Proof shortened by Mario
     Carneiro, 27-May-2016.) $)
  negsub $p |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) ) $=
    ( cc wcel wa cneg caddc co cc0 cmin wceq df-neg oveq2i a1i addsubass mp3an2
    0cn simpl addridd oveq1d 3eqtr2d ) ACDZBCDZEZABFZGHZAIBJHZGHZAIGHZBJHZABJHU
    FUHKUDUEUGAGBLMNUBICDUCUJUHKQAIBOPUDUIABJUDAUBUCRSTUA $.

  $( Relationship between subtraction and negative.  (Contributed by NM,
     10-May-2004.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  subneg $p |- ( ( A e. CC /\ B e. CC ) -> ( A - -u B ) = ( A + B ) ) $=
    ( cc wcel wa cneg cmin co cc0 caddc df-neg oveq2i wceq subsub mp3an2 eqtrid
    0cn subid1 adantr oveq1d eqtrd ) ACDZBCDZEZABFZGHZAIGHZBJHZABJHUDUFAIBGHZGH
    ZUHUEUIAGBKLUBICDUCUJUHMQAIBNOPUDUGABJUBUGAMUCARSTUA $.

  $( A number is equal to the negative of its negative.  Theorem I.4 of
     [Apostol] p. 18.  (Contributed by NM, 12-Jan-2002.)  (Revised by Mario
     Carneiro, 27-May-2016.) $)
  negneg $p |- ( A e. CC -> -u -u A = A ) $=
    ( cc wcel cneg cc0 caddc co cmin df-neg wceq 0cn subneg eqtrid addlid eqtrd
    mpan ) ABCZADZDZEAFGZAQSERHGZTRIEBCQUATJKEALPMANO $.

  $( Negative is one-to-one.  (Contributed by NM, 8-Feb-2005.)  (Revised by
     Mario Carneiro, 27-May-2016.) $)
  neg11 $p |- ( ( A e. CC /\ B e. CC ) -> ( -u A = -u B <-> A = B ) ) $=
    ( cc wcel wa cneg wceq negeq negneg eqeqan12d imbitrid impbid1 ) ACDZBCDZEZ
    AFZBFZGZABGZRPFZQFZGOSPQHMNTAUABAIBIJKABHL $.

  $( Negative contraposition law.  (Contributed by NM, 9-May-2004.) $)
  negcon1 $p |- ( ( A e. CC /\ B e. CC ) -> ( -u A = B <-> -u B = A ) ) $=
    ( cc wcel wa cneg wceq negcl neg11 sylan negneg adantr eqeq1d bitr3d bitrdi
    wb eqcom ) ACDZBCDZEZAFZBGZABFZGZUCAGTUAFZUCGZUBUDRUACDSUFUBPAHUABIJTUEAUCR
    UEAGSAKLMNAUCQO $.

  $( Negative contraposition law.  (Contributed by NM, 14-Nov-2004.) $)
  negcon2 $p |- ( ( A e. CC /\ B e. CC ) -> ( A = -u B <-> B = -u A ) ) $=
    ( cc wcel wa cneg wceq eqcom negcon1 bitr4id bitrdi ) ACDBCDEZABFZGZAFZBGZB
    OGLNMAGPAMHABIJOBHK $.

  $( A number is zero iff its negative is zero.  (Contributed by NM,
     12-Jul-2005.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  negeq0 $p |- ( A e. CC -> ( A = 0 <-> -u A = 0 ) ) $=
    ( cc wcel cneg cc0 wceq wb 0cn neg11 mpan2 neg0 eqeq2i bitr3di ) ABCZADZEDZ
    FZAEFZOEFNEBCQRGHAEIJPEOKLM $.

  $( Cancellation law for subtraction.  (Contributed by NM, 8-Feb-2005.)
     (Revised by Mario Carneiro, 27-May-2016.) $)
  subcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                ( ( A - B ) = ( A - C ) <-> B = C ) ) $=
    ( cc wcel w3a caddc co wceq cmin simp2 simp1 addcomd eqeq1d simp3 addsubeq4
    wb syl22anc addcan 3bitr3d ) ADEZBDEZCDEZFZBAGHZACGHZIZABGHZUFIABJHACJHIZBC
    IUDUEUHUFUDBAUAUBUCKZUAUBUCLZMNUDUBUAUAUCUGUIQUJUKUKUAUBUCOBAACPRABCST $.

  $( Distribution of negative over subtraction.  (Contributed by NM,
     15-Nov-2004.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  negsubdi $p |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( -u A + B ) ) $=
    ( cc wcel wa cc0 cmin co caddc cneg 0cn subsub mp3an1 df-neg oveq1i 3eqtr4g
    wceq ) ACDZBCDZEFABGHZGHZFAGHZBIHZTJAJZBIHFCDRSUAUCQKFABLMTNUDUBBIANOP $.

  $( Distribution of negative over addition.  (Contributed by NM, 10-May-2004.)
     (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  negdi $p |- ( ( A e. CC /\ B e. CC ) ->
                 -u ( A + B ) = ( -u A + -u B ) ) $=
    ( cc wcel wa cneg cmin co caddc subneg negeqd negcl negsubdi sylan2 eqtr3d
    wceq ) ACDZBCDZEZABFZGHZFZABIHZFAFTIHZSUAUCABJKRQTCDUBUDPBLATMNO $.

  $( Distribution of negative over addition.  (Contributed by NM,
     1-Jan-2006.) $)
  negdi2 $p |- ( ( A e. CC /\ B e. CC ) ->
                 -u ( A + B ) = ( -u A - B ) ) $=
    ( cc wcel wa caddc co cneg cmin negdi wceq negcl negsub sylan eqtrd ) ACDZB
    CDZEABFGHAHZBHFGZRBIGZABJPRCDQSTKALRBMNO $.

  $( Distribution of negative over subtraction.  (Contributed by NM,
     4-Oct-1999.) $)
  negsubdi2 $p |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) ) $=
    ( cc wcel wa cmin cneg caddc negsubdi wceq negcl addcom sylan negsub ancoms
    co 3eqtrd ) ACDZBCDZEABFPGAGZBHPZBTHPZBAFPZABIRTCDSUAUBJAKTBLMSRUBUCJBANOQ
    $.

  $( Relationship between subtraction and negative.  (Contributed by Paul
     Chapman, 8-Oct-2007.) $)
  neg2sub $p |- ( ( A e. CC /\ B e. CC ) -> ( -u A - -u B ) = ( B - A ) ) $=
    ( cc wcel wa cneg cmin co caddc wceq negcl sylan negsubdi negsubdi2 3eqtr2d
    subneg ) ACDZBCDZEAFZBFGHZSBIHZABGHFBAGHQSCDRTUAJAKSBPLABMABNO $.

  ${
    $d A x $.
    $( Closure law for negative of reals.  (Contributed by NM, 20-Jan-1997.) $)
    renegcl $p |- ( A e. RR -> -u A e. RR ) $=
      ( vx cr wcel cv caddc co cc0 wceq wrex cneg ax-rnegex wa recn cmin df-neg
      cc wb eqeq1i 0cn subadd mp3an1 bitrid sylan2 wi eleq1a adantl sylbird mpd
      sylan rexlimdva ) ACDZABEZFGHIZBCJAKZCDZBALULUNUPBCULUMCDZMUNUOUMIZUPUQUL
      UMQDZURUNRUMNURHAOGZUMIZULUSMUNUOUTUMAPSULAQDZUSVAUNRZANHQDVBUSVCTHAUMUAU
      BUJUCUDUQURUPUEULUMCUOUFUGUHUKUI $.
  $}

  ${
    renegcl.1 $e |- A e. RR $.
    $( Closure law for negative of reals.  (Note: this inference proof style
       and the deduction theorem usage in ~ renegcl is deprecated, but is
       retained for its demonstration value.)  (Contributed by NM,
       17-Jan-1997.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
    renegcli $p |- -u A e. RR $=
      ( cr wcel cneg renegcl ax-mp ) ACDAECDBAFG $.

    resubcl.2 $e |- B e. RR $.
    $( Closure law for subtraction of reals.  (Contributed by NM, 17-Jan-1997.)
       (Revised by Mario Carneiro, 27-May-2016.) $)
    resubcli $p |- ( A - B ) e. RR $=
      ( cneg caddc co cmin cr cc wcel wceq recni negsub mp2an renegcli readdcli
      eqeltrri ) ABEZFGZABHGZIAJKBJKTUALACMBDMABNOASCBDPQR $.
  $}

  $( Closure law for subtraction of reals.  (Contributed by NM,
     20-Jan-1997.) $)
  resubcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR ) $=
    ( cr wcel wa cneg caddc cmin wceq recn negsub syl2an renegcl readdcl sylan2
    co cc eqeltrrd ) ACDZBCDZEABFZGPZABHPZCSAQDBQDUBUCITAJBJABKLTSUACDUBCDBMAUA
    NOR $.

  $( The negative of a real is real.  (Contributed by NM, 11-Aug-1999.)
     (Revised by Mario Carneiro, 14-Jul-2014.) $)
  negreb $p |- ( A e. CC -> ( -u A e. RR <-> A e. RR ) ) $=
    ( cc wcel cneg cr renegcl negneg eleq1d imbitrid impbid1 ) ABCZADZECZAECZML
    DZECKNLFKOAEAGHIAFJ $.

  $( "Reverse" second Peano postulate analog for complex numbers:  A complex
     number minus 1 is a complex number.  (Contributed by Alexander van der
     Vekens, 18-Mar-2018.) $)
  peano2cnm $p |- ( N e. CC -> ( N - 1 ) e. CC ) $=
    ( cc wcel c1 cmin co ax-1cn subcl mpan2 ) ABCDBCADEFBCGADHI $.

  $( "Reverse" second Peano postulate analog for reals.  (Contributed by NM,
     6-Feb-2007.) $)
  peano2rem $p |- ( N e. RR -> ( N - 1 ) e. RR ) $=
    ( cr wcel c1 cmin co 1re resubcl mpan2 ) ABCDBCADEFBCGADHI $.

  ${
    negidi.1 $e |- A e. CC $.
    $( Closure law for negative.  (Contributed by NM, 26-Nov-1994.) $)
    negcli $p |- -u A e. CC $=
      ( cc wcel cneg negcl ax-mp ) ACDAECDBAFG $.

    $( Addition of a number and its negative.  (Contributed by NM,
       26-Nov-1994.) $)
    negidi $p |- ( A + -u A ) = 0 $=
      ( cc wcel cneg caddc co cc0 wceq negid ax-mp ) ACDAAEFGHIBAJK $.

    $( A number is equal to the negative of its negative.  Theorem I.4 of
       [Apostol] p. 18.  (Contributed by NM, 8-Feb-1995.)  (Proof shortened by
       Andrew Salmon, 22-Oct-2011.) $)
    negnegi $p |- -u -u A = A $=
      ( cc wcel cneg wceq negneg ax-mp ) ACDAEEAFBAGH $.

    $( Subtraction of a number from itself.  (Contributed by NM,
       26-Nov-1994.) $)
    subidi $p |- ( A - A ) = 0 $=
      ( cc wcel cmin co cc0 wceq subid ax-mp ) ACDAAEFGHBAIJ $.

    $( Identity law for subtraction.  (Contributed by NM, 29-May-1999.) $)
    subid1i $p |- ( A - 0 ) = A $=
      ( cc wcel cc0 cmin co wceq subid1 ax-mp ) ACDAEFGAHBAIJ $.

    $( A number is nonzero iff its negative is nonzero.  (Contributed by NM,
       10-Aug-1999.) $)
    negne0bi $p |- ( A =/= 0 <-> -u A =/= 0 ) $=
      ( cc0 cneg cc wcel wceq wb negeq0 ax-mp necon3bii ) ACADZCAEFACGLCGHBAIJK
      $.

    $( The negative of a real is real.  (Contributed by NM, 11-Aug-1999.) $)
    negrebi $p |- ( -u A e. RR <-> A e. RR ) $=
      ( cc wcel cneg cr wb negreb ax-mp ) ACDAEFDAFDGBAHI $.

    ${
      negne0i.2 $e |- A =/= 0 $.
      $( The negative of a nonzero number is nonzero.  (Contributed by NM,
         30-Jul-2004.) $)
      negne0i $p |- -u A =/= 0 $=
        ( cc0 wne cneg negne0bi mpbi ) ADEAFDECABGH $.
    $}

    pncan3i.2 $e |- B e. CC $.
    $( Closure law for subtraction.  (Contributed by NM, 26-Nov-1994.)
       (Revised by Mario Carneiro, 21-Dec-2013.) $)
    subcli $p |- ( A - B ) e. CC $=
      ( cc wcel cmin co subcl mp2an ) AEFBEFABGHEFCDABIJ $.

    $( Subtraction and addition of equals.  (Contributed by NM,
       26-Nov-1994.) $)
    pncan3i $p |- ( A + ( B - A ) ) = B $=
      ( cc wcel cmin co caddc wceq pncan3 mp2an ) AEFBEFABAGHIHBJCDABKL $.

    $( Relationship between subtraction and negative.  Theorem I.3 of [Apostol]
       p. 18.  (Contributed by NM, 26-Nov-1994.)  (Proof shortened by Andrew
       Salmon, 22-Oct-2011.) $)
    negsubi $p |- ( A + -u B ) = ( A - B ) $=
      ( cc wcel cneg caddc co cmin wceq negsub mp2an ) AEFBEFABGHIABJIKCDABLM
      $.

    $( Relationship between subtraction and negative.  (Contributed by NM,
       1-Dec-2005.) $)
    subnegi $p |- ( A - -u B ) = ( A + B ) $=
      ( cc wcel cneg cmin co caddc wceq subneg mp2an ) AEFBEFABGHIABJIKCDABLM
      $.

    $( If the difference between two numbers is zero, they are equal.
       (Contributed by NM, 8-May-1999.) $)
    subeq0i $p |- ( ( A - B ) = 0 <-> A = B ) $=
      ( cc wcel cmin co cc0 wceq wb subeq0 mp2an ) AEFBEFABGHIJABJKCDABLM $.

    $( Negative is one-to-one.  (Contributed by NM, 1-Aug-1999.) $)
    neg11i $p |- ( -u A = -u B <-> A = B ) $=
      ( cc wcel cneg wceq wb neg11 mp2an ) AEFBEFAGBGHABHICDABJK $.

    $( Negative contraposition law.  (Contributed by NM, 25-Aug-1999.) $)
    negcon1i $p |- ( -u A = B <-> -u B = A ) $=
      ( cc wcel cneg wceq wb negcon1 mp2an ) AEFBEFAGBHBGAHICDABJK $.

    $( Negative contraposition law.  (Contributed by NM, 25-Aug-1999.) $)
    negcon2i $p |- ( A = -u B <-> B = -u A ) $=
      ( cc wcel cneg wceq wb negcon2 mp2an ) AEFBEFABGHBAGHICDABJK $.

    $( Distribution of negative over addition.  (Contributed by NM,
       28-Jul-1999.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
    negdii $p |- -u ( A + B ) = ( -u A + -u B ) $=
      ( caddc co cneg wceq cc0 addcli negidi oveq12i eqtri negcli add4i 3eqtr2i
      00id addcani mpbi ) ABEFZTGZEFZTAGZBGZEFZEFZHUAUEHUBIAUCEFZBUDEFZEFZUFTAB
      CDJZKUIIIEFIUGIUHIEACKBDKLQMAUCBUDCACNZDBDNZOPTUAUEUJTUJNUCUDUKULJRS $.

    $( Distribution of negative over subtraction.  (Contributed by NM,
       6-Aug-1999.) $)
    negsubdii $p |- -u ( A - B ) = ( -u A + B ) $=
      ( cneg caddc co cmin negcli negdii negsubi negeqi negnegi oveq2i 3eqtr3i
      ) ABEZFGZEAEZPEZFGABHGZERBFGAPCBDIJQTABCDKLSBRFBDMNO $.

    $( Distribution of negative over subtraction.  (Contributed by NM,
       1-Oct-1999.) $)
    negsubdi2i $p |- -u ( A - B ) = ( B - A ) $=
      ( cmin co cneg caddc negsubdii negcli negsubi addcomli eqtri ) ABEFGAGZBH
      FBAEFZABCDIBNODACJBADCKLM $.

    subadd.3 $e |- C e. CC $.
    $( Relationship between subtraction and addition.  (Contributed by NM,
       26-Nov-1994.)  (Revised by Mario Carneiro, 21-Dec-2013.) $)
    subaddi $p |- ( ( A - B ) = C <-> ( B + C ) = A ) $=
      ( cc wcel cmin co wceq caddc wb subadd mp3an ) AGHBGHCGHABIJCKBCLJAKMDEFA
      BCNO $.

    $( Relationship between subtraction and addition.  (Contributed by NM,
       15-Dec-2006.) $)
    subadd2i $p |- ( ( A - B ) = C <-> ( C + B ) = A ) $=
      ( cc wcel cmin co wceq caddc wb subadd2 mp3an ) AGHBGHCGHABIJCKCBLJAKMDEF
      ABCNO $.

    ${
      subaddri.4 $e |- ( B + C ) = A $.
      $( Relationship between subtraction and addition.  (Contributed by NM,
         16-Dec-2006.) $)
      subaddrii $p |- ( A - B ) = C $=
        ( cmin co wceq caddc subaddi mpbir ) ABHICJBCKIAJGABCDEFLM $.
    $}

    $( Swap subtrahend and result of subtraction.  (Contributed by NM,
       7-Oct-1999.) $)
    subsub23i $p |- ( ( A - B ) = C <-> ( A - C ) = B ) $=
      ( cc wcel cmin co wceq wb subsub23 mp3an ) AGHBGHCGHABIJCKACIJBKLDEFABCMN
      $.

    $( Associative-type law for subtraction and addition.  (Contributed by NM,
       16-Sep-1999.) $)
    addsubassi $p |- ( ( A + B ) - C ) = ( A + ( B - C ) ) $=
      ( cc wcel caddc co cmin wceq addsubass mp3an ) AGHBGHCGHABIJCKJABCKJIJLDE
      FABCMN $.

    $( Law for subtraction and addition.  (Contributed by NM, 6-Aug-2003.) $)
    addsubi $p |- ( ( A + B ) - C ) = ( ( A - C ) + B ) $=
      ( cc wcel caddc co cmin wceq addsub mp3an ) AGHBGHCGHABIJCKJACKJBIJLDEFAB
      CMN $.

    $( Cancellation law for subtraction.  (Contributed by NM, 8-Feb-2005.) $)
    subcani $p |- ( ( A - B ) = ( A - C ) <-> B = C ) $=
      ( cc wcel cmin co wceq wb subcan mp3an ) AGHBGHCGHABIJACIJKBCKLDEFABCMN
      $.

    $( Cancellation law for subtraction.  (Contributed by NM, 8-Feb-2005.) $)
    subcan2i $p |- ( ( A - C ) = ( B - C ) <-> A = B ) $=
      ( cc wcel cmin co wceq wb subcan2 mp3an ) AGHBGHCGHACIJBCIJKABKLDEFABCMN
      $.

    $( Cancellation law for mixed addition and subtraction.  (Contributed by
       NM, 14-Jan-2006.) $)
    pnncani $p |- ( ( A + B ) - ( A - C ) ) = ( B + C ) $=
      ( cc wcel caddc co cmin wceq pnncan mp3an ) AGHBGHCGHABIJACKJKJBCIJLDEFAB
      CMN $.

    addsub4i.4 $e |- D e. CC $.
    $( Rearrangement of 4 terms in a mixed addition and subtraction.
       (Contributed by NM, 17-Oct-1999.) $)
    addsub4i $p |- ( ( A + B ) - ( C + D ) ) =
                 ( ( A - C ) + ( B - D ) ) $=
      ( cc wcel caddc co cmin wceq addsub4 mp4an ) AIJBIJCIJDIJABKLCDKLMLACMLBD
      MLKLNEFGHABCDOP $.
  $}

  $( Alternate proof of ~ 0re .  (Contributed by NM, 19-Feb-2005.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  0reALT $p |- 0 e. RR $=
    ( c1 cmin co cc0 cr ax-1cn subidi 1re resubcli eqeltrri ) AABCDEAFGAAHHIJ
    $.

  ${
    negidd.1 $e |- ( ph -> A e. CC ) $.
    $( Closure law for negative.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    negcld $p |- ( ph -> -u A e. CC ) $=
      ( cc wcel cneg negcl syl ) ABDEBFDECBGH $.

    $( Subtraction of a number from itself.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subidd $p |- ( ph -> ( A - A ) = 0 ) $=
      ( cc wcel cmin co cc0 wceq subid syl ) ABDEBBFGHICBJK $.

    $( Identity law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subid1d $p |- ( ph -> ( A - 0 ) = A ) $=
      ( cc wcel cc0 cmin co wceq subid1 syl ) ABDEBFGHBICBJK $.

    $( Addition of a number and its negative.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    negidd $p |- ( ph -> ( A + -u A ) = 0 ) $=
      ( cc wcel cneg caddc co cc0 wceq negid syl ) ABDEBBFGHIJCBKL $.

    $( A number is equal to the negative of its negative.  Theorem I.4 of
       [Apostol] p. 18.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    negnegd $p |- ( ph -> -u -u A = A ) $=
      ( cc wcel cneg wceq negneg syl ) ABDEBFFBGCBHI $.

    $( A number is zero iff its negative is zero.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    negeq0d $p |- ( ph -> ( A = 0 <-> -u A = 0 ) ) $=
      ( cc wcel cc0 wceq cneg wb negeq0 syl ) ABDEBFGBHFGICBJK $.

    $( A number is nonzero iff its negative is nonzero.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    negne0bd $p |- ( ph -> ( A =/= 0 <-> -u A =/= 0 ) ) $=
      ( cc0 cneg negeq0d necon3bid ) ABDBEDABCFG $.

    ${
      negcon1d.2 $e |- ( ph -> B e. CC ) $.
      $( Contraposition law for unary minus.  Deduction form of ~ negcon1 .
         (Contributed by David Moews, 28-Feb-2017.) $)
      negcon1d $p |- ( ph -> ( -u A = B <-> -u B = A ) ) $=
        ( cc wcel cneg wceq wb negcon1 syl2anc ) ABFGCFGBHCICHBIJDEBCKL $.
    $}

    ${
      negcon1ad.2 $e |- ( ph -> -u A = B ) $.
      $( Contraposition law for unary minus.  One-way deduction form of
         ~ negcon1 .  (Contributed by David Moews, 28-Feb-2017.) $)
      negcon1ad $p |- ( ph -> -u B = A ) $=
        ( cneg wceq cc negcld eqeltrrd negcon1d mpbid ) ABFZCGCFBGEABCDAMCHEABD
        IJKL $.
    $}

    ${
      neg11ad.2 $e |- ( ph -> B e. CC ) $.
      $( The negatives of two complex numbers are equal iff they are equal.
         Deduction form of ~ neg11 .  Generalization of ~ neg11d .
         (Contributed by David Moews, 28-Feb-2017.) $)
      neg11ad $p |- ( ph -> ( -u A = -u B <-> A = B ) ) $=
        ( cc wcel cneg wceq wb neg11 syl2anc ) ABFGCFGBHCHIBCIJDEBCKL $.
    $}

    ${
      negned.2 $e |- ( ph -> B e. CC ) $.
      negned.3 $e |- ( ph -> A =/= B ) $.
      $( If two complex numbers are unequal, so are their negatives.
         Contrapositive of ~ neg11d .  (Contributed by David Moews,
         28-Feb-2017.) $)
      negned $p |- ( ph -> -u A =/= -u B ) $=
        ( cneg wne neg11ad necon3bid mpbird ) ABGZCGZHBCHFALMBCABCDEIJK $.
    $}

    ${
      negne0d.2 $e |- ( ph -> A =/= 0 ) $.
      $( The negative of a nonzero number is nonzero.  See also ~ negap0d which
         is similar but for apart from zero rather than not equal to zero.
         (Contributed by Mario Carneiro, 27-May-2016.) $)
      negne0d $p |- ( ph -> -u A =/= 0 ) $=
        ( cc0 wne cneg negne0bd mpbid ) ABEFBGEFDABCHI $.
    $}

    ${
      negrebd.2 $e |- ( ph -> -u A e. RR ) $.
      $( The negative of a real is real.  (Contributed by Mario Carneiro,
         28-May-2016.) $)
      negrebd $p |- ( ph -> A e. RR ) $=
        ( cneg cr wcel cc wb negreb syl mpbid ) ABEFGZBFGZDABHGMNICBJKL $.
    $}

    pncand.2 $e |- ( ph -> B e. CC ) $.
    $( Closure law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subcld $p |- ( ph -> ( A - B ) e. CC ) $=
      ( cc wcel cmin co subcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    pncand $p |- ( ph -> ( ( A + B ) - B ) = A ) $=
      ( cc wcel caddc co cmin wceq pncan syl2anc ) ABFGCFGBCHICJIBKDEBCLM $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    pncan2d $p |- ( ph -> ( ( A + B ) - A ) = B ) $=
      ( cc wcel caddc co cmin wceq pncan2 syl2anc ) ABFGCFGBCHIBJICKDEBCLM $.

    $( Subtraction and addition of equals.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    pncan3d $p |- ( ph -> ( A + ( B - A ) ) = B ) $=
      ( cc wcel cmin co caddc wceq pncan3 syl2anc ) ABFGCFGBCBHIJICKDEBCLM $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    npcand $p |- ( ph -> ( ( A - B ) + B ) = A ) $=
      ( cc wcel cmin co caddc wceq npcan syl2anc ) ABFGCFGBCHICJIBKDEBCLM $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nncand $p |- ( ph -> ( A - ( A - B ) ) = B ) $=
      ( cc wcel cmin co wceq nncan syl2anc ) ABFGCFGBBCHIHICJDEBCKL $.

    $( Relationship between subtraction and negative.  Theorem I.3 of [Apostol]
       p. 18.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    negsubd $p |- ( ph -> ( A + -u B ) = ( A - B ) ) $=
      ( cc wcel cneg caddc co cmin wceq negsub syl2anc ) ABFGCFGBCHIJBCKJLDEBCM
      N $.

    $( Relationship between subtraction and negative.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    subnegd $p |- ( ph -> ( A - -u B ) = ( A + B ) ) $=
      ( cc wcel cneg cmin co caddc wceq subneg syl2anc ) ABFGCFGBCHIJBCKJLDEBCM
      N $.

    ${
      subeq0d.3 $e |- ( ph -> ( A - B ) = 0 ) $.
      $( If the difference between two numbers is zero, they are equal.
         (Contributed by Mario Carneiro, 27-May-2016.) $)
      subeq0d $p |- ( ph -> A = B ) $=
        ( cmin co cc0 wceq cc wcel wb subeq0 syl2anc mpbid ) ABCGHIJZBCJZFABKLC
        KLQRMDEBCNOP $.
    $}

    ${
      subne0d.3 $e |- ( ph -> A =/= B ) $.
      $( Two unequal numbers have nonzero difference.  See also ~ subap0d which
         is the same thing for apartness rather than negated equality.
         (Contributed by Mario Carneiro, 1-Jan-2017.) $)
      subne0d $p |- ( ph -> ( A - B ) =/= 0 ) $=
        ( cmin co cc0 wne cc wcel wceq wb subeq0 syl2anc necon3bid mpbird ) ABC
        GHZIJBCJFASIBCABKLCKLSIMBCMNDEBCOPQR $.
    $}

    $( The difference of two complex numbers is zero iff they are equal.
       Deduction form of ~ subeq0 .  Generalization of ~ subeq0d .
       (Contributed by David Moews, 28-Feb-2017.) $)
    subeq0ad $p |- ( ph -> ( ( A - B ) = 0 <-> A = B ) ) $=
      ( cc wcel cmin co cc0 wceq wb subeq0 syl2anc ) ABFGCFGBCHIJKBCKLDEBCMN $.

    ${
      subne0ad.3 $e |- ( ph -> ( A - B ) =/= 0 ) $.
      $( If the difference of two complex numbers is nonzero, they are unequal.
         Converse of ~ subne0d .  Contrapositive of ~ subeq0bd .  (Contributed
         by David Moews, 28-Feb-2017.) $)
      subne0ad $p |- ( ph -> A =/= B ) $=
        ( cmin co cc0 wne subeq0ad necon3bid mpbid ) ABCGHZIJBCJFANIBCABCDEKLM
        $.
    $}

    ${
      neg11d.3 $e |- ( ph -> -u A = -u B ) $.
      $( If the difference between two numbers is zero, they are equal.
         (Contributed by Mario Carneiro, 27-May-2016.) $)
      neg11d $p |- ( ph -> A = B ) $=
        ( cneg wceq neg11ad mpbid ) ABGCGHBCHFABCDEIJ $.
    $}

    $( Distribution of negative over addition.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    negdid $p |- ( ph -> -u ( A + B ) = ( -u A + -u B ) ) $=
      ( cc wcel caddc co cneg wceq negdi syl2anc ) ABFGCFGBCHIJBJCJHIKDEBCLM $.

    $( Distribution of negative over addition.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    negdi2d $p |- ( ph -> -u ( A + B ) = ( -u A - B ) ) $=
      ( cc wcel caddc co cneg cmin wceq negdi2 syl2anc ) ABFGCFGBCHIJBJCKILDEBC
      MN $.

    $( Distribution of negative over subtraction.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    negsubdid $p |- ( ph -> -u ( A - B ) = ( -u A + B ) ) $=
      ( cc wcel cmin co cneg caddc wceq negsubdi syl2anc ) ABFGCFGBCHIJBJCKILDE
      BCMN $.

    $( Distribution of negative over subtraction.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    negsubdi2d $p |- ( ph -> -u ( A - B ) = ( B - A ) ) $=
      ( cc wcel cmin co cneg wceq negsubdi2 syl2anc ) ABFGCFGBCHIJCBHIKDEBCLM
      $.

    $( Relationship between subtraction and negative.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    neg2subd $p |- ( ph -> ( -u A - -u B ) = ( B - A ) ) $=
      ( cc wcel cneg cmin co wceq neg2sub syl2anc ) ABFGCFGBHCHIJCBIJKDEBCLM $.

    subaddd.3 $e |- ( ph -> C e. CC ) $.
    $( Relationship between subtraction and addition.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    subaddd $p |- ( ph -> ( ( A - B ) = C <-> ( B + C ) = A ) ) $=
      ( cc wcel cmin co wceq caddc wb subadd syl3anc ) ABHICHIDHIBCJKDLCDMKBLNE
      FGBCDOP $.

    $( Relationship between subtraction and addition.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    subadd2d $p |- ( ph -> ( ( A - B ) = C <-> ( C + B ) = A ) ) $=
      ( cc wcel cmin co wceq caddc wb subadd2 syl3anc ) ABHICHIDHIBCJKDLDCMKBLN
      EFGBCDOP $.

    $( Associative-type law for subtraction and addition.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    addsubassd $p |- ( ph -> ( ( A + B ) - C ) = ( A + ( B - C ) ) ) $=
      ( cc wcel caddc co cmin wceq addsubass syl3anc ) ABHICHIDHIBCJKDLKBCDLKJK
      MEFGBCDNO $.

    $( Law for subtraction and addition.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    addsubd $p |- ( ph -> ( ( A + B ) - C ) = ( ( A - C ) + B ) ) $=
      ( cc wcel caddc co cmin wceq addsub syl3anc ) ABHICHIDHIBCJKDLKBDLKCJKMEF
      GBCDNO $.

    $( Commutative/associative law for addition and subtraction.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    subadd23d $p |- ( ph -> ( ( A - B ) + C ) = ( A + ( C - B ) ) ) $=
      ( cc wcel cmin co caddc wceq subadd23 syl3anc ) ABHICHIDHIBCJKDLKBDCJKLKM
      EFGBCDNO $.

    $( Commutative/associative law for addition and subtraction.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    addsub12d $p |- ( ph -> ( A + ( B - C ) ) = ( B + ( A - C ) ) ) $=
      ( cc wcel cmin co caddc wceq addsub12 syl3anc ) ABHICHIDHIBCDJKLKCBDJKLKM
      EFGBCDNO $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    npncand $p |- ( ph -> ( ( A - B ) + ( B - C ) ) = ( A - C ) ) $=
      ( cc wcel cmin co caddc wceq npncan syl3anc ) ABHICHIDHIBCJKCDJKLKBDJKMEF
      GBCDNO $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nppcand $p |- ( ph -> ( ( ( A - B ) + C ) + B ) = ( A + C ) ) $=
      ( cc wcel cmin co caddc wceq nppcan syl3anc ) ABHICHIDHIBCJKDLKCLKBDLKMEF
      GBCDNO $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nppcan2d $p |- ( ph -> ( ( A - ( B + C ) ) + C ) = ( A - B ) ) $=
      ( cc wcel caddc co cmin wceq nppcan2 syl3anc ) ABHICHIDHIBCDJKLKDJKBCLKME
      FGBCDNO $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nppcan3d $p |- ( ph -> ( ( A - B ) + ( C + B ) ) = ( A + C ) ) $=
      ( cc wcel cmin co caddc wceq nppcan3 syl3anc ) ABHICHIDHIBCJKDCLKLKBDLKME
      FGBCDNO $.

    $( Law for double subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subsubd $p |- ( ph -> ( A - ( B - C ) ) = ( ( A - B ) + C ) ) $=
      ( cc wcel cmin co caddc wceq subsub syl3anc ) ABHICHIDHIBCDJKJKBCJKDLKMEF
      GBCDNO $.

    $( Law for double subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subsub2d $p |- ( ph -> ( A - ( B - C ) ) = ( A + ( C - B ) ) ) $=
      ( cc wcel cmin co caddc wceq subsub2 syl3anc ) ABHICHIDHIBCDJKJKBDCJKLKME
      FGBCDNO $.

    $( Law for double subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subsub3d $p |- ( ph -> ( A - ( B - C ) ) = ( ( A + C ) - B ) ) $=
      ( cc wcel cmin co caddc wceq subsub3 syl3anc ) ABHICHIDHIBCDJKJKBDLKCJKME
      FGBCDNO $.

    $( Law for double subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subsub4d $p |- ( ph -> ( ( A - B ) - C ) = ( A - ( B + C ) ) ) $=
      ( cc wcel cmin co caddc wceq subsub4 syl3anc ) ABHICHIDHIBCJKDJKBCDLKJKME
      FGBCDNO $.

    $( Swap the second and third terms in a double subtraction.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    sub32d $p |- ( ph -> ( ( A - B ) - C ) = ( ( A - C ) - B ) ) $=
      ( cc wcel cmin co wceq sub32 syl3anc ) ABHICHIDHIBCJKDJKBDJKCJKLEFGBCDMN
      $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nnncand $p |- ( ph -> ( ( A - ( B - C ) ) - C ) = ( A - B ) ) $=
      ( cc wcel cmin co wceq nnncan syl3anc ) ABHICHIDHIBCDJKJKDJKBCJKLEFGBCDMN
      $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nnncan1d $p |- ( ph -> ( ( A - B ) - ( A - C ) ) = ( C - B ) ) $=
      ( cc wcel cmin co wceq nnncan1 syl3anc ) ABHICHIDHIBCJKBDJKJKDCJKLEFGBCDM
      N $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nnncan2d $p |- ( ph -> ( ( A - C ) - ( B - C ) ) = ( A - B ) ) $=
      ( cc wcel cmin co wceq nnncan2 syl3anc ) ABHICHIDHIBDJKCDJKJKBCJKLEFGBCDM
      N $.

    $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    npncan3d $p |- ( ph -> ( ( A - B ) + ( C - A ) ) = ( C - B ) ) $=
      ( cc wcel cmin co caddc wceq npncan3 syl3anc ) ABHICHIDHIBCJKDBJKLKDCJKME
      FGBCDNO $.

    $( Cancellation law for mixed addition and subtraction.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    pnpcand $p |- ( ph -> ( ( A + B ) - ( A + C ) ) = ( B - C ) ) $=
      ( cc wcel caddc co cmin wceq pnpcan syl3anc ) ABHICHIDHIBCJKBDJKLKCDLKMEF
      GBCDNO $.

    $( Cancellation law for mixed addition and subtraction.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    pnpcan2d $p |- ( ph -> ( ( A + C ) - ( B + C ) ) = ( A - B ) ) $=
      ( cc wcel caddc co cmin wceq pnpcan2 syl3anc ) ABHICHIDHIBDJKCDJKLKBCLKME
      FGBCDNO $.

    $( Cancellation law for mixed addition and subtraction.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    pnncand $p |- ( ph -> ( ( A + B ) - ( A - C ) ) = ( B + C ) ) $=
      ( cc wcel caddc co cmin wceq pnncan syl3anc ) ABHICHIDHIBCJKBDLKLKCDJKMEF
      GBCDNO $.

    $( Cancellation law for mixed addition and subtraction.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    ppncand $p |- ( ph -> ( ( A + B ) + ( C - B ) ) = ( A + C ) ) $=
      ( cc wcel caddc co cmin wceq ppncan syl3anc ) ABHICHIDHIBCJKDCLKJKBDJKMEF
      GBCDNO $.

    ${
      subcand.4 $e |- ( ph -> ( A - B ) = ( A - C ) ) $.
      $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
         27-May-2016.) $)
      subcand $p |- ( ph -> B = C ) $=
        ( cmin co wceq cc wcel wb subcan syl3anc mpbid ) ABCIJBDIJKZCDKZHABLMCL
        MDLMRSNEFGBCDOPQ $.
    $}

    ${
      subcan2d.4 $e |- ( ph -> ( A - C ) = ( B - C ) ) $.
      $( Cancellation law for subtraction.  (Contributed by Mario Carneiro,
         22-Sep-2016.) $)
      subcan2d $p |- ( ph -> A = B ) $=
        ( cmin co wceq cc wcel wb subcan2 syl3anc mpbid ) ABDIJCDIJKZBCKZHABLMC
        LMDLMRSNEFGBCDOPQ $.
    $}

    $( Cancellation law for subtraction.  Deduction form of ~ subcan .
       Generalization of ~ subcand .  (Contributed by David Moews,
       28-Feb-2017.) $)
    subcanad $p |- ( ph -> ( ( A - B ) = ( A - C ) <-> B = C ) ) $=
      ( cc wcel cmin co wceq wb subcan syl3anc ) ABHICHIDHIBCJKBDJKLCDLMEFGBCDN
      O $.

    ${
      subneintrd.4 $e |- ( ph -> B =/= C ) $.
      $( Introducing subtraction on both sides of a statement of inequality.
         Contrapositive of ~ subcand .  (Contributed by David Moews,
         28-Feb-2017.) $)
      subneintrd $p |- ( ph -> ( A - B ) =/= ( A - C ) ) $=
        ( cmin co wne subcanad necon3bid mpbird ) ABCIJZBDIJZKCDKHAOPCDABCDEFGL
        MN $.
    $}

    $( Cancellation law for subtraction.  Deduction form of ~ subcan2 .
       Generalization of ~ subcan2d .  (Contributed by David Moews,
       28-Feb-2017.) $)
    subcan2ad $p |- ( ph -> ( ( A - C ) = ( B - C ) <-> A = B ) ) $=
      ( cc wcel cmin co wceq wb subcan2 syl3anc ) ABHICHIDHIBDJKCDJKLBCLMEFGBCD
      NO $.

    ${
      subneintr2d.4 $e |- ( ph -> A =/= B ) $.
      $( Introducing subtraction on both sides of a statement of inequality.
         Contrapositive of ~ subcan2d .  (Contributed by David Moews,
         28-Feb-2017.) $)
      subneintr2d $p |- ( ph -> ( A - C ) =/= ( B - C ) ) $=
        ( cmin co wne subcan2ad necon3bid mpbird ) ABDIJZCDIJZKBCKHAOPBCABCDEFG
        LMN $.
    $}

    addsub4d.4 $e |- ( ph -> D e. CC ) $.
    $( Rearrangement of 4 terms in a mixed addition and subtraction.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    addsub4d $p |- ( ph ->
                ( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) ) ) $=
      ( cc wcel caddc co cmin wceq addsub4 syl22anc ) ABJKCJKDJKEJKBCLMDELMNMBD
      NMCENMLMOFGHIBCDEPQ $.

    $( Rearrangement of 4 terms in a mixed addition and subtraction.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    subadd4d $p |- ( ph ->
                ( ( A - B ) - ( C - D ) ) = ( ( A + D ) - ( B + C ) ) ) $=
      ( cc wcel cmin co caddc wceq subadd4 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBE
      NMCDNMLMOFGHIBCDEPQ $.

    $( Rearrangement of 4 terms in a subtraction.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    sub4d $p |- ( ph ->
                ( ( A - B ) - ( C - D ) ) = ( ( A - C ) - ( B - D ) ) ) $=
      ( cc wcel cmin co wceq sub4 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBDLMCELMLMN
      FGHIBCDEOP $.

    $( Law for subtraction and addition.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    2addsubd $p |- ( ph ->
      ( ( ( A + B ) + C ) - D ) = ( ( ( A + C ) - D ) + B ) ) $=
      ( cc wcel caddc co cmin wceq 2addsub syl22anc ) ABJKCJKDJKEJKBCLMDLMENMBD
      LMENMCLMOFGHIBCDEPQ $.

    $( Relation between sums and differences.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    addsubeq4d $p |- ( ph ->
        ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) ) $=
      ( cc wcel caddc co wceq cmin wb addsubeq4 syl22anc ) ABJKCJKDJKEJKBCLMDEL
      MNDBOMCEOMNPFGHIBCDEQR $.
  $}

  ${
    subeqxfrd.a $e |- ( ph -> A e. CC ) $.
    subeqxfrd.b $e |- ( ph -> B e. CC ) $.
    subeqxfrd.c $e |- ( ph -> C e. CC ) $.
    subeqxfrd.d $e |- ( ph -> D e. CC ) $.
    subeqxfrd.1 $e |- ( ph -> ( A - B ) = ( C - D ) ) $.
    $( Transfer two terms of a subtraction in an equality.  (Contributed by
       Thierry Arnoux, 2-Feb-2020.) $)
    subeqxfrd $p |- ( ph -> ( A - C ) = ( B - D ) ) $=
      ( cmin co caddc oveq1d npncand npncan3d 3eqtr3d ) ABCKLZCDKLZMLDEKLZSMLBD
      KLCEKLARTSMJNABCDFGHOADECHIGPQ $.
  $}

  ${
    mvlraddd.1 $e |- ( ph -> A e. CC ) $.
    mvlraddd.2 $e |- ( ph -> B e. CC ) $.
    mvlraddd.3 $e |- ( ph -> ( A + B ) = C ) $.
    $( Move LHS right addition to RHS. (Contributed by David A. Wheeler,
       15-Oct-2018.) $)
    mvlraddd $p |- ( ph -> A = ( C - B ) ) $=
      ( caddc co cmin pncand oveq1d eqtr3d ) ABCHIZCJIBDCJIABCEFKANDCJGLM $.

    $( Move LHS left addition to RHS. (Contributed by David A. Wheeler,
       15-Oct-2018.) $)
    mvlladdd $p |- ( ph -> B = ( C - A ) ) $=
      ( caddc co cmin pncand addcomd eqtr3d oveq1d ) ACBHIZBJICDBJIACBFEKAODBJA
      BCHIODABCEFLGMNM $.
  $}

  ${
    mvrraddd.1 $e |- ( ph -> B e. CC ) $.
    mvrraddd.2 $e |- ( ph -> C e. CC ) $.
    mvrraddd.3 $e |- ( ph -> A = ( B + C ) ) $.
    $( Move RHS right addition to LHS. (Contributed by David A. Wheeler,
       15-Oct-2018.) $)
    mvrraddd $p |- ( ph -> ( A - C ) = B ) $=
      ( cmin co caddc oveq1d pncand eqtrd ) ABDHICDJIZDHICABNDHGKACDEFLM $.

    $( Move RHS left addition to LHS. (Contributed by David A. Wheeler,
       11-Oct-2018.) $)
    mvrladdd $p |- ( ph -> ( A - B ) = C ) $=
      ( comraddd mvrraddd ) ABDCFEABCDEFGHI $.
  $}

  ${
    assraddsubd.1 $e |- ( ph -> B e. CC ) $.
    assraddsubd.2 $e |- ( ph -> C e. CC ) $.
    assraddsubd.3 $e |- ( ph -> D e. CC ) $.
    assraddsubd.4 $e |- ( ph -> A = ( ( B + C ) - D ) ) $.
    $( Associate RHS addition-subtraction.  (Contributed by David A. Wheeler,
       15-Oct-2018.) $)
    assraddsubd $p |- ( ph -> A = ( B + ( C - D ) ) ) $=
      ( caddc co cmin addsubassd eqtrd ) ABCDJKELKCDELKJKIACDEFGHMN $.
  $}

  ${
    subaddeqd.a $e |- ( ph -> A e. CC ) $.
    subaddeqd.b $e |- ( ph -> B e. CC ) $.
    subaddeqd.c $e |- ( ph -> C e. CC ) $.
    subaddeqd.d $e |- ( ph -> D e. CC ) $.
    subaddeqd.1 $e |- ( ph -> ( A + B ) = ( C + D ) ) $.
    $( Transfer two terms of a subtraction to an addition in an equality.
       (Contributed by Thierry Arnoux, 2-Feb-2020.) $)
    subaddeqd $p |- ( ph -> ( A - D ) = ( C - B ) ) $=
      ( caddc co cmin oveq1d addcomd eqtrd pnpcan2d pnpcand 3eqtr3d ) ABCKLZECK
      LZMLZEDKLZUAMLZBEMLDCMLAUBDEKLZUAMLUDATUEUAMJNAUEUCUAMADEHIONPABECFIGQAED
      CIHGRS $.
  $}

  ${
    addlsub.a $e |- ( ph -> A e. CC ) $.
    addlsub.b $e |- ( ph -> B e. CC ) $.
    addlsub.c $e |- ( ph -> C e. CC ) $.
    $( Left-subtraction:  Subtraction of the left summand from the result of an
       addition.  (Contributed by BJ, 6-Jun-2019.) $)
    addlsub $p |- ( ph -> ( ( A + B ) = C <-> A = ( C - B ) ) ) $=
      ( caddc co wceq cmin oveq1 pncand wa wi eqtr2 eqcomd a1i mpan2d syl5 eqtr
      npcand impbid ) ABCHIZDJZBDCKIZJZUEUDCKIZUFJZAUGUDDCKLAUIUHBJZUGABCEFMUIU
      JNZUGOAUKUFBUHUFBPQRSTUGUDUFCHIZJZAUEBUFCHLAUMULDJZUEADCGFUBUMUNNUEOAUDUL
      DUARSTUC $.

    $( Right-subtraction:  Subtraction of the right summand from the result of
       an addition.  (Contributed by BJ, 6-Jun-2019.) $)
    addrsub $p |- ( ph -> ( ( A + B ) = C <-> B = ( C - A ) ) ) $=
      ( caddc co wceq cmin addcomd eqeq1d addlsub bitrd ) ABCHIZDJCBHIZDJCDBKIJ
      APQDABCEFLMACBDFEGNO $.

    $( A subtraction law:  Exchanging the subtrahend and the result of the
       subtraction.  (Contributed by BJ, 6-Jun-2019.) $)
    subexsub $p |- ( ph -> ( A = ( C - B ) <-> B = ( C - A ) ) ) $=
      ( caddc co wceq cmin addlsub addrsub bitr3d ) ABCHIDJBDCKIJCDBKIJABCDEFGL
      ABCDEFGMN $.
  $}

  $( If adding a number to a another number yields the other number, the added
     number must be ` 0 ` .  This shows that ` 0 ` is the unique (right)
     identity of the complex numbers.  (Contributed by AV, 17-Jan-2021.) $)
  addid0 $p |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) ) $=
    ( cc wcel wa caddc co wceq cc0 simpl simpr subaddd eqcom subid adantr eqtrd
    cmin wi ex biimtrid sylbird oveq2 addrid sylan9eqr impbid ) ACDZBCDZEZABFGZ
    AHZBIHZUHUJAAQGZBHZUKUHAABUFUGJZUNUFUGKLUFUMUKRUGUMBULHZUFUKULBMUFUOUKUFUOE
    BULIUFUOKUFULIHUOANOPSTOUAUFUKUJRUGUFUKUJUKUFUIAIFGABIAFUBAUCUDSOUE $.

  $( Adding a nonzero number to a complex number does not yield the complex
     number.  (Contributed by AV, 17-Jan-2021.) $)
  addn0nid $p |- ( ( X e. CC /\ Y e. CC /\ Y =/= 0 ) -> ( X + Y ) =/= X ) $=
    ( cc wcel cc0 wne caddc co wa wceq addid0 biimpd necon3d 3impia ) ACDZBCDZB
    EFABGHZAFOPIZQABERQAJBEJABKLMN $.

  ${
    pnpncand.1 $e |- ( ph -> A e. CC ) $.
    pnpncand.2 $e |- ( ph -> B e. CC ) $.
    pnpncand.3 $e |- ( ph -> C e. CC ) $.
    $( Addition/subtraction cancellation law.  (Contributed by Scott Fenton,
       14-Dec-2017.) $)
    pnpncand $p |- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A ) $=
      ( cmin co caddc subcld addcld subsub2d pncand eqtr3d ) ABCDHIZJIZPHIQDCHI
      JIBAQCDABPEACDFGKZLFGMABPERNO $.
  $}

  $( Reverse the order of subtraction in an equality.  (Contributed by Scott
     Fenton, 8-Jul-2013.) $)
  subeqrev $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
    ( ( A - B ) = ( C - D ) <-> ( B - A ) = ( D - C ) ) ) $=
    ( cc wcel wa cmin co cneg wb subcl neg11 syl2an negsubdi2 eqeqan12d bitr3d
    wceq ) AEFBEFGZCEFDEFGZGABHIZJZCDHIZJZRZUAUCRZBAHIZDCHIZRSUAEFUCEFUEUFKTABL
    CDLUAUCMNSTUBUGUDUHABOCDOPQ $.

  $( Two complex numbers add up to zero iff they are each other's opposites.
     (Contributed by Thierry Arnoux, 2-May-2017.) $)
  addeq0 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) = 0 <-> A = -u B ) ) $=
    ( cc wcel wa cc0 cmin co wceq caddc cneg simpr simpl subadd2d df-neg eqeq1i
    0cnd eqcom bitr3i bitr3di ) ACDZBCDZEZFBGHZAIZABJHFIABKZIZUCFBAUCQUAUBLUAUB
    MNUEUFAIUGUFUDABOPUFARST $.

  $( Cancellation law for addition and subtraction with 1.  (Contributed by
     Alexander van der Vekens, 3-Oct-2018.) $)
  pncan1 $p |- ( A e. CC -> ( ( A + 1 ) - 1 ) = A ) $=
    ( cc wcel c1 id ax-1cn a1i pncand ) ABCZADIEDBCIFGH $.

  $( Cancellation law for subtraction and addition with 1.  (Contributed by
     Alexander van der Vekens, 5-Oct-2018.) $)
  npcan1 $p |- ( A e. CC -> ( ( A - 1 ) + 1 ) = A ) $=
    ( cc wcel c1 id ax-1cn a1i npcand ) ABCZADIEDBCIFGH $.

  ${
    subeq0bd.1 $e |- ( ph -> A e. CC ) $.
    subeq0bd.2 $e |- ( ph -> A = B ) $.
    $( If two complex numbers are equal, their difference is zero.  Consequence
       of ~ subeq0ad .  Converse of ~ subeq0d .  Contrapositive of ~ subne0ad .
       (Contributed by David Moews, 28-Feb-2017.) $)
    subeq0bd $p |- ( ph -> ( A - B ) = 0 ) $=
      ( cmin co cc0 wceq cc eqeltrrd subeq0ad mpbird ) ABCFGHIBCIEABCDABCJEDKLM
      $.
  $}

  ${
    renegcld.1 $e |- ( ph -> A e. RR ) $.
    $( Closure law for negative of reals.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    renegcld $p |- ( ph -> -u A e. RR ) $=
      ( cr wcel cneg renegcl syl ) ABDEBFDECBGH $.

    resubcld.2 $e |- ( ph -> B e. RR ) $.
    $( Closure law for subtraction of reals.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    resubcld $p |- ( ph -> ( A - B ) e. RR ) $=
      ( cr wcel cmin co resubcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
  $}

  ${
    $d A n x y $.
    negf1o.1 $e |- F = ( x e. A |-> -u x ) $.
    $( Negation is an isomorphism of a subset of the real numbers to the
       negated elements of the subset.  (Contributed by AV, 9-Aug-2020.) $)
    negf1o $p |- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } ) $=
      ( vy cr cv cneg wcel wceq wa imp wi cc recn eleq1d adantl negeq elrab wss
      crab wf1o ccnv cmpt ssel renegcl syl6 negneg syl biimpcd mpd sylanbrc weq
      eqcomd simpr a1i biimtrid wb syl6com ad3antrrr negcon2 exp31 sylbi impcom
      syl2anc f1ocnv2d simpld ) BGUAZBCHZIZBJZCGUBZDUCDUDFVMFHZIZUEKVIAFBVMAHZI
      ZVODEVIVPBJZLZVQGJZVQIZBJZVQVMJVIVRVTVIVRVPGJZVTBGVPUFZVPUGUHMVSWCWBVIVRW
      CWDMVRWCWBNVIWCVRWBWCVPWABWCVPOJZVPWAKVPPZWEWAVPVPUIUOUJQUKRULVLWBCVQGVJV
      QKVKWABVJVQSQTUMVIVNVMJZVOBJZWGVNGJZWHLZVIWHVLWHCVNGCFUNVKVOBVJVNSQTZWJWH
      NVIWIWHUPUQURMVRWGLVIVPVOKVNVQKUSZWGVRVIWLNZWGWJVRWMNWKWJVRVIWLWJVRLZVILW
      EVNOJZWLWNVIWEVRVIWENWJVIVRWCWEWDWFUTRMWIWOWHVRVIVNPVAVPVNVBVFVCVDVEVEVGV
      H $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Multiplication
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( k times k minus 1 is a complex number if k is a complex number.
     (Contributed by Alexander van der Vekens, 11-Mar-2018.) $)
  kcnktkm1cn $p |- ( K e. CC -> ( K x. ( K - 1 ) ) e. CC ) $=
    ( cc wcel c1 cmin co id ax-1cn a1i subcld mulcld ) ABCZAADEFLGZLADMDBCLHIJK
    $.

  $( Product of two sums.  (Contributed by NM, 14-Jan-2006.)  (Proof shortened
     by Andrew Salmon, 19-Nov-2011.) $)
  muladd $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
               ( ( A + B ) x. ( C + D ) ) =
       ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) ) $=
    ( cc wcel wa caddc co cmul wceq addcl 3expa adantrl oveq12d mulcl ad2ant2lr
    adddir syl2an ad2ant2l adddi adantrr ad2ant2r anandirs add32d mulcom oveq2d
    3expb sylan ad2ant2rl addassd ancoms 3eqtr3d an4s 3eqtrd ) AEFZBEFZGZCEFZDE
    FZGZGZABHIZCDHIJIZVCCJIZVCDJIZHIZACJIZBCJIZHIZADJIZBDJIZHIZHIZVHDBJIZHIZVKC
    BJIZHIHIZURVCEFZVAVDVGKZABLVSUSUTVTVCCDUAUHUIVBVEVJVFVMHURUSVEVJKZUTUPUQUSW
    AABCRMUBURUTVFVMKZUSUPUQUTWBABDRMNOVBVNVHVMHIZVIHIVPVKHIZVQHIVRVBVHVIVMUPUS
    VHEFZUQUTACPZUCZUQUSVIEFUPUTBCPQURUTVMEFZUSUPUQUTWHUPUTGVKEFZVLEFZWHUQUTGZA
    DPZBDPZVKVLLSUDNUEVBWCWDVIVQHVBVHVKHIZVLHIWNVOHIWCWDVBVLVOWNHUQUTVLVOKUPUSB
    DUFTUGVBVHVKVLWGUPUTWIUQUSWLUJZUQUTWJUPUSWMTUKVBVHVKVOWGWOUQUTVOEFZUPUSUTUQ
    WPDBPULZTUEUMUQUSVIVQKUPUTBCUFQOVBVPVKVQUPUSUQUTVPEFZUPUSGWEWPWRWKWFWQVHVOL
    SUNWOUQUSVQEFZUPUTUSUQWSCBPULQUKUOUO $.

  $( Distribution of multiplication over subtraction.  Theorem I.5 of [Apostol]
     p. 18.  (Contributed by NM, 18-Nov-2004.) $)
  subdi $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) ) $=
    ( cc wcel w3a cmul cmin wceq caddc simp1 simp3 3adant1 adddid pncan3 ancoms
    co subcl oveq2d mulcl eqtr3d 3adant3 3adant2 wa sylan2 3impb subaddd mpbird
    eqcomd ) ADEZBDEZCDEZFZABGQZACGQZHQZABCHQZGQZUMUPURIUOURJQZUNIUMACUQJQZGQUS
    UNUMACUQUJUKULKUJUKULLUKULUQDEZUJBCRZMNUMUTBAGUKULUTBIZUJULUKVCCBOPMSUAUMUN
    UOURUJUKUNDEULABTUBUJULUODEUKACTUCUJUKULURDEZUKULUDUJVAVDVBAUQTUEUFUGUHUI
    $.

  $( Distribution of multiplication over subtraction.  Theorem I.5 of [Apostol]
     p. 18.  (Contributed by NM, 30-Dec-2005.) $)
  subdir $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
               ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) ) $=
    ( cc wcel cmin co cmul wceq subdi 3coml wa subcl mulcom sylan 3impa 3adant2
    w3a 3adant1 oveq12d 3eqtr4d ) ADEZBDEZCDEZRZCABFGZHGZCAHGZCBHGZFGZUFCHGZACH
    GZBCHGZFGUDUBUCUGUJICABJKUBUCUDUKUGIZUBUCLUFDEUDUNABMUFCNOPUEULUHUMUIFUBUDU
    LUHIUCACNQUCUDUMUIIUBBCNSTUA $.

  $( Multiplication by ` 0 ` .  Theorem I.6 of [Apostol] p. 18.  (Contributed
     by NM, 10-Aug-1999.) $)
  mul02 $p |- ( A e. CC -> ( 0 x. A ) = 0 ) $=
    ( cc wcel cc0 cmul co cmin 0cn subidi oveq1i subdir mp3an12 wa mulcl subidd
    wceq mpan eqtrd eqtr3id ) ABCZDAEFZDDGFZAEFZDUBDAEDHIJTUCUAUAGFZDDBCZUETUCU
    DPHHDDAKLUETUDDPHUETMUADANOQRS $.

  $( Zero times a real is zero.  Although we prove it as a corollary of
     ~ mul02 , the name is for consistency with the Metamath Proof Explorer
     which proves it before ~ mul02 .  (Contributed by Scott Fenton,
     3-Jan-2013.) $)
  mul02lem2 $p |- ( A e. RR -> ( 0 x. A ) = 0 ) $=
    ( cr wcel cc cc0 cmul co wceq recn mul02 syl ) ABCADCEAFGEHAIAJK $.

  $( Multiplication by ` 0 ` .  Theorem I.6 of [Apostol] p. 18.  (Contributed
     by NM, 15-May-1999.)  (Revised by Scott Fenton, 3-Jan-2013.) $)
  mul01 $p |- ( A e. CC -> ( A x. 0 ) = 0 ) $=
    ( cc wcel cc0 cmul co wceq 0cn mulcom mpan2 mul02 eqtrd ) ABCZADEFZDAEFZDMD
    BCNOGHADIJAKL $.

  ${
    mul01i.1 $e |- A e. CC $.
    $( Multiplication by 0.  Theorem I.6 of [Apostol] p. 18.  (Contributed by
       NM, 23-Nov-1994.) $)
    mul02i $p |- ( 0 x. A ) = 0 $=
      ( cc wcel cc0 cmul co wceq mul02 ax-mp ) ACDEAFGEHBAIJ $.

    $( Multiplication by ` 0 ` .  Theorem I.6 of [Apostol] p. 18.  (Contributed
       by NM, 23-Nov-1994.)  (Revised by Scott Fenton, 3-Jan-2013.) $)
    mul01i $p |- ( A x. 0 ) = 0 $=
      ( cc wcel cc0 cmul co wceq mul01 ax-mp ) ACDAEFGEHBAIJ $.
  $}

  ${
    mul01d.1 $e |- ( ph -> A e. CC ) $.
    $( Multiplication by 0.  Theorem I.6 of [Apostol] p. 18.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    mul02d $p |- ( ph -> ( 0 x. A ) = 0 ) $=
      ( cc wcel cc0 cmul co wceq mul02 syl ) ABDEFBGHFICBJK $.

    $( Multiplication by ` 0 ` .  Theorem I.6 of [Apostol] p. 18.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    mul01d $p |- ( ph -> ( A x. 0 ) = 0 ) $=
      ( cc wcel cc0 cmul co wceq mul01 syl ) ABDEBFGHFICBJK $.
  $}

  $( The imaginary unit ` _i ` is not zero.  (Contributed by NM,
     6-May-1999.) $)
  ine0 $p |- _i =/= 0 $=
    ( ci cc0 wceq c1 0lt1 gtneii neii caddc co cmul oveq2 ax-icn mul01i eqtr2di
    0re oveq1d ax-1cn addlidi ax-i2m1 3eqtr3g mto neir ) ABABCZDBCDBBDOEFGUCBDH
    IAAJIZDHIDBUCBUDDHUCUDABJIBABAJKALMNPDQRSTUAUB $.

  $( Product with negative is negative of product.  Theorem I.12 of [Apostol]
     p. 18.  (Contributed by NM, 14-May-1999.)  (Proof shortened by Mario
     Carneiro, 27-May-2016.) $)
  mulneg1 $p |- ( ( A e. CC /\ B e. CC ) ->
                 ( -u A x. B ) = -u ( A x. B ) ) $=
    ( cc wcel wa cmin co cmul cneg wceq subdir mp3an1 simpr mul02d oveq1d eqtrd
    cc0 0cn df-neg oveq1i 3eqtr4g ) ACDZBCDZEZQAFGZBHGZQABHGZFGZAIZBHGUGIUDUFQB
    HGZUGFGZUHQCDUBUCUFUKJRQABKLUDUJQUGFUDBUBUCMNOPUIUEBHASTUGSUA $.

  $( The product with a negative is the negative of the product.  (Contributed
     by NM, 30-Jul-2004.) $)
  mulneg2 $p |- ( ( A e. CC /\ B e. CC ) ->
                 ( A x. -u B ) = -u ( A x. B ) ) $=
    ( cc wcel wa cneg cmul co mulneg1 ancoms negcl mulcom sylan2 negeqd 3eqtr4d
    wceq ) ACDZBCDZEZBFZAGHZBAGHZFZATGHZABGHZFRQUAUCPBAIJRQTCDUDUAPBKATLMSUEUBA
    BLNO $.

  $( Swap the negative sign in a product.  (Contributed by NM, 30-Jul-2004.) $)
  mulneg12 $p |- ( ( A e. CC /\ B e. CC ) ->
                  ( -u A x. B ) = ( A x. -u B ) ) $=
    ( cc wcel wa cneg cmul co mulneg1 mulneg2 eqtr4d ) ACDBCDEAFBGHABGHFABFGHAB
    IABJK $.

  $( Product of two negatives.  Theorem I.12 of [Apostol] p. 18.  (Contributed
     by NM, 30-Jul-2004.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  mul2neg $p |- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) ) $=
    ( cc wcel wa cneg cmul co negcl mulneg12 sylan2 negneg adantl oveq2d eqtrd
    wceq ) ACDZBCDZEZAFBFZGHZATFZGHZABGHRQTCDUAUCPBIATJKSUBBAGRUBBPQBLMNO $.

  $( Convert a subtraction to addition using multiplication by a negative.
     (Contributed by NM, 2-Feb-2007.) $)
  submul2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
                 ( A - ( B x. C ) ) = ( A + ( B x. -u C ) ) ) $=
    ( cc wcel cmul co cmin cneg caddc wceq wa adantl oveq2d mulcl negsub sylan2
    mulneg2 eqtr2d 3impb ) ADEZBDEZCDEZABCFGZHGZABCIFGZJGZKUAUBUCLZLZUGAUDIZJGZ
    UEUIUFUJAJUHUFUJKUABCRMNUHUAUDDEUKUEKBCOAUDPQST $.

  $( Product with minus one is negative.  (Contributed by NM, 16-Nov-1999.) $)
  mulm1 $p |- ( A e. CC -> ( -u 1 x. A ) = -u A ) $=
    ( cc wcel c1 cneg cmul co wceq ax-1cn mulneg1 mpan mullid negeqd eqtrd ) AB
    CZDEAFGZDAFGZEZAEDBCOPRHIDAJKOQAALMN $.

  $( Product of two differences.  (Contributed by NM, 14-Jan-2006.) $)
  mulsub $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
               ( ( A - B ) x. ( C - D ) ) =
       ( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) ) $=
    ( cc wcel wa cneg caddc cmul cmin negsub oveqan12d wceq negcl mulneg2 mulcl
    co ancoms syl2an muladd sylanr2 sylanl2 mul2neg oveq2d negdi eqtr4d ancom2s
    ad2ant2l an42s oveq12d addcl an4s negsubd 3eqtrd eqtr3d ) AEFZBEFZGZCEFZDEF
    ZGZGZABHZIRZCDHZIRZJRZABKRZCDKRZJRACJRZDBJRZIRZADJRZCBJRZIRZKRZUSVBVEVIVGVJ
    JABLCDLMVCVHVKVFVDJRZIRZAVFJRZCVDJRZIRZIRZVMVPHZIRVQURUQVDEFZVBVHWCNZBOVAUQ
    WEGUTVFEFWFDOAVDCVFUAUBUCVCVSVMWBWDIURVAVSVMNUQUTURVAGZVRVLVKIVAURVRVLNDBUD
    SUEUIUQVAURUTWBWDNZUQVAGZUTURWHWIUTURGZGWBVNHZVOHZIRZWDWIWJVTWKWAWLIADPCBPM
    WIVNEFZVOEFZWDWMNWJADQZCBQZVNVOUFTUGUHUJUKVCVMVPUQUTURVAVMEFZUQUTGVKEFVLEFZ
    WRWGACQVAURWSDBQSVKVLULTUMUQVAURUTVPEFZWIWNWOWTURUTGWPUTURWOWQSVNVOULTUJUNU
    OUP $.

  $( Swap the order of subtraction in a multiplication.  (Contributed by Scott
     Fenton, 24-Jun-2013.) $)
  mulsub2 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
    ( ( A - B ) x. ( C - D ) ) = ( ( B - A ) x. ( D - C ) ) ) $=
    ( cc wcel wa cmin co cneg cmul wceq subcl syl2an negsubdi2 oveqan12d eqtr3d
    mul2neg ) AEFBEFGZCEFDEFGZGABHIZJZCDHIZJZKIZUAUCKIZBAHIZDCHIZKISUAEFUCEFUEU
    FLTABMCDMUAUCRNSTUBUGUDUHKABOCDOPQ $.

  ${
    mulm1.1 $e |- A e. CC $.
    $( Product with minus one is negative.  (Contributed by NM,
       31-Jul-1999.) $)
    mulm1i $p |- ( -u 1 x. A ) = -u A $=
      ( cc wcel c1 cneg cmul co wceq mulm1 ax-mp ) ACDEFAGHAFIBAJK $.

    mulneg.2 $e |- B e. CC $.
    $( Product with negative is negative of product.  Theorem I.12 of [Apostol]
       p. 18.  (Contributed by NM, 10-Feb-1995.)  (Revised by Mario Carneiro,
       27-May-2016.) $)
    mulneg1i $p |- ( -u A x. B ) = -u ( A x. B ) $=
      ( cc wcel cneg cmul co wceq mulneg1 mp2an ) AEFBEFAGBHIABHIGJCDABKL $.

    $( Product with negative is negative of product.  (Contributed by NM,
       31-Jul-1999.)  (Revised by Mario Carneiro, 27-May-2016.) $)
    mulneg2i $p |- ( A x. -u B ) = -u ( A x. B ) $=
      ( cc wcel cneg cmul co wceq mulneg2 mp2an ) AEFBEFABGHIABHIGJCDABKL $.

    $( Product of two negatives.  Theorem I.12 of [Apostol] p. 18.
       (Contributed by NM, 14-Feb-1995.)  (Revised by Mario Carneiro,
       27-May-2016.) $)
    mul2negi $p |- ( -u A x. -u B ) = ( A x. B ) $=
      ( cc wcel cneg cmul co wceq mul2neg mp2an ) AEFBEFAGBGHIABHIJCDABKL $.

    subdi.3 $e |- C e. CC $.
    $( Distribution of multiplication over subtraction.  Theorem I.5 of
       [Apostol] p. 18.  (Contributed by NM, 26-Nov-1994.) $)
    subdii $p |- ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) $=
      ( cc wcel cmin co cmul wceq subdi mp3an ) AGHBGHCGHABCIJKJABKJACKJIJLDEFA
      BCMN $.

    $( Distribution of multiplication over subtraction.  Theorem I.5 of
       [Apostol] p. 18.  (Contributed by NM, 8-May-1999.) $)
    subdiri $p |- ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) $=
      ( cc wcel cmin co cmul wceq subdir mp3an ) AGHBGHCGHABIJCKJACKJBCKJIJLDEF
      ABCMN $.

    muladdi.4 $e |- D e. CC $.
    $( Product of two sums.  (Contributed by NM, 17-May-1999.) $)
    muladdi $p |- ( ( A + B ) x. ( C + D ) ) =
        ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) $=
      ( cc wcel caddc co cmul wceq muladd mp4an ) AIJBIJCIJDIJABKLCDKLMLACMLDBM
      LKLADMLCBMLKLKLNEFGHABCDOP $.
  $}

  ${
    mulm1d.1 $e |- ( ph -> A e. CC ) $.
    $( Product with minus one is negative.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mulm1d $p |- ( ph -> ( -u 1 x. A ) = -u A ) $=
      ( cc wcel c1 cneg cmul co wceq mulm1 syl ) ABDEFGBHIBGJCBKL $.

    mulnegd.2 $e |- ( ph -> B e. CC ) $.
    $( Product with negative is negative of product.  Theorem I.12 of [Apostol]
       p. 18.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    mulneg1d $p |- ( ph -> ( -u A x. B ) = -u ( A x. B ) ) $=
      ( cc wcel cneg cmul co wceq mulneg1 syl2anc ) ABFGCFGBHCIJBCIJHKDEBCLM $.

    $( Product with negative is negative of product.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    mulneg2d $p |- ( ph -> ( A x. -u B ) = -u ( A x. B ) ) $=
      ( cc wcel cneg cmul co wceq mulneg2 syl2anc ) ABFGCFGBCHIJBCIJHKDEBCLM $.

    $( Product of two negatives.  Theorem I.12 of [Apostol] p. 18.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    mul2negd $p |- ( ph -> ( -u A x. -u B ) = ( A x. B ) ) $=
      ( cc wcel cneg cmul co wceq mul2neg syl2anc ) ABFGCFGBHCHIJBCIJKDEBCLM $.

    subdid.3 $e |- ( ph -> C e. CC ) $.
    $( Distribution of multiplication over subtraction.  Theorem I.5 of
       [Apostol] p. 18.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    subdid $p |- ( ph -> ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) ) $=
      ( cc wcel cmin co cmul wceq subdi syl3anc ) ABHICHIDHIBCDJKLKBCLKBDLKJKME
      FGBCDNO $.

    $( Distribution of multiplication over subtraction.  Theorem I.5 of
       [Apostol] p. 18.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    subdird $p |- ( ph -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) ) $=
      ( cc wcel cmin co cmul wceq subdir syl3anc ) ABHICHIDHIBCJKDLKBDLKCDLKJKM
      EFGBCDNO $.

    muladdd.4 $e |- ( ph -> D e. CC ) $.
    $( Product of two sums.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    muladdd $p |- ( ph -> ( ( A + B ) x. ( C + D ) ) =
        ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) ) $=
      ( cc wcel caddc co cmul wceq muladd syl22anc ) ABJKCJKDJKEJKBCLMDELMNMBDN
      MECNMLMBENMDCNMLMLMOFGHIBCDEPQ $.

    $( Product of two differences.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    mulsubd $p |- ( ph -> ( ( A - B ) x. ( C - D ) ) =
         ( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) ) $=
      ( cc wcel cmin co cmul caddc wceq mulsub syl22anc ) ABJKCJKDJKEJKBCLMDELM
      NMBDNMECNMOMBENMDCNMOMLMPFGHIBCDEQR $.
  $}

  ${
    muls1d.1 $e |- ( ph -> A e. CC ) $.
    muls1d.2 $e |- ( ph -> B e. CC ) $.
    $( Multiplication by one minus a number.  (Contributed by Scott Fenton,
       23-Dec-2017.) $)
    muls1d $p |- ( ph -> ( A x. ( B - 1 ) ) = ( ( A x. B ) - A ) ) $=
      ( c1 cmin co cmul 1cnd subdid mulridd oveq2d eqtrd ) ABCFGHIHBCIHZBFIHZGH
      OBGHABCFDEAJKAPBOGABDLMN $.

    $( Multiplication followed by the subtraction of a factor.  (Contributed by
       Alexander van der Vekens, 28-Aug-2018.) $)
    mulsubfacd $p |- ( ph -> ( ( A x. B ) - B ) = ( ( A - 1 ) x. B ) ) $=
      ( c1 cmin co cmul cc wcel ax-1cn a1i subdird mullidd oveq2d eqtr2d ) ABFG
      HCIHBCIHZFCIHZGHRCGHABFCDFJKALMENASCRGACEOPQ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordering on reals (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Addition to both sides of 'less than'.  (Contributed by NM,
       12-Nov-1999.)  (Revised by Mario Carneiro, 27-May-2016.) $)
    ltadd2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                  ( A < B <-> ( C + A ) < ( C + B ) ) ) $=
      ( vx cr wcel clt wbr caddc co axltadd wceq wi wa readdcld addassd breq12d
      cc0 recnd cc w3a cv wrex ax-rnegex 3ad2ant3 simpl3 simpl1 syl3anc sylibrd
      simpl2 simprl simprr wb addcom eqeq1d syl2anc mpbid oveq1d addlidd sylibd
      eqtrd rexlimddv impbid ) AEFZBEFZCEFZUAZABGHZCAIJZCBIJZGHZABCKVGCDUBZIJZR
      LZVKVHMDEVFVDVNDEUCVEDCUDUEVGVLEFZVNNZNZVKVLCIJZAIJZVRBIJZGHZVHVQVKVLVIIJ
      ZVLVJIJZGHZWAVQVIEFVJEFVOVKWDMVQCAVDVEVFVPUFZVDVEVFVPUGZOVQCBWEVDVEVFVPUJ
      ZOVGVOVNUKZVIVJVLKUHVQVSWBVTWCGVQVLCAVQVLWHSZVQCWESZVQAWFSZPVQVLCBWIWJVQB
      WGSZPQUIVQVSAVTBGVQVSRAIJAVQVRRAIVQVNVRRLZVGVOVNULVQCTFZVLTFZVNWMUMWJWIWN
      WONVMVRRCVLUNUOUPUQZURVQAWKUSVAVQVTRBIJBVQVRRBIWPURVQBWLUSVAQUTVBVC $.
  $}

  ${
    ltadd2i.1 $e |- A e. RR $.
    ltadd2i.2 $e |- B e. RR $.
    ltadd2i.3 $e |- C e. RR $.
    $( Addition to both sides of 'less than'.  (Contributed by NM,
       21-Jan-1997.) $)
    ltadd2i $p |- ( A < B <-> ( C + A ) < ( C + B ) ) $=
      ( cr wcel clt wbr caddc co wb ltadd2 mp3an ) AGHBGHCGHABIJCAKLCBKLIJMDEFA
      BCNO $.
  $}

  ${
    ltadd2d.1 $e |- ( ph -> A e. RR ) $.
    ltadd2d.2 $e |- ( ph -> B e. RR ) $.
    ltadd2d.3 $e |- ( ph -> C e. RR ) $.
    $( Addition to both sides of 'less than'.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    ltadd2d $p |- ( ph -> ( A < B <-> ( C + A ) < ( C + B ) ) ) $=
      ( cr wcel clt wbr caddc co wb ltadd2 syl3anc ) ABHICHIDHIBCJKDBLMDCLMJKNE
      FGBCDOP $.

    ${
      ltletrd.4 $e |- ( ph -> A < B ) $.
      $( Addition to both sides of 'less than'.  (Contributed by Mario
         Carneiro, 30-May-2016.) $)
      ltadd2dd $p |- ( ph -> ( C + A ) < ( C + B ) ) $=
        ( clt wbr caddc co ltadd2d mpbid ) ABCIJDBKLDCKLIJHABCDEFGMN $.

      ltletrd.5 $e |- ( ph -> B <_ C ) $.
      $( Transitive law deduction for 'less than', 'less than or equal to'.
         (Contributed by NM, 9-Jan-2006.) $)
      ltletrd $p |- ( ph -> A < C ) $=
        ( clt wbr cle cr wcel wa wi ltletr syl3anc mp2and ) ABCJKZCDLKZBDJKZHIA
        BMNCMNDMNTUAOUBPEFGBCDQRS $.
    $}
  $}

  $( Adding a negative number to another number decreases it.  (Contributed by
     Glauco Siliprandi, 11-Dec-2019.) $)
  ltaddneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) ) $=
    ( cr wcel wa cc0 clt wbr caddc co wb ltadd2 mp3an2 wceq recn addridd adantl
    0re breq2d bitrd ) ACDZBCDZEZAFGHZBAIJZBFIJZGHZUEBGHUAFCDUBUDUGKRAFBLMUCUFB
    UEGUBUFBNUAUBBBOPQST $.

  $( Adding a negative number to another number decreases it.  (Contributed by
     AV, 19-Mar-2021.) $)
  ltaddnegr $p |- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( A + B ) < B ) ) $=
    ( cr wcel wa cc0 clt wbr caddc co ltaddneg cc wceq recn addcom breq1d bitrd
    syl2anr ) ACDZBCDZEZAFGHBAIJZBGHABIJZBGHABKUAUBUCBGTBLDALDUBUCMSBNANBAORPQ
    $.

  ${
    lelttrdi.r $e |- ( ph -> ( A e. RR /\ B e. RR /\ C e. RR ) ) $.
    lelttrdi.l $e |- ( ph -> B <_ C ) $.
    $( If a number is less than another number, and the other number is less
       than or equal to a third number, the first number is less than the third
       number.  (Contributed by Alexander van der Vekens, 24-Mar-2018.) $)
    lelttrdi $p |- ( ph -> ( A < B -> A < C ) ) $=
      ( clt wbr wa cr wcel simp1d adantr simp2d simp3d simpr cle ltletrd ex ) A
      BCGHZBDGHATIBCDABJKZTAUACJKZDJKZELMAUBTAUAUBUCENMAUCTAUAUBUCEOMATPACDQHTF
      MRS $.
  $}

  $( Positive implies nonzero.  (Contributed by NM, 3-Oct-1999.)  (Proof
     shortened by Mario Carneiro, 27-May-2016.) $)
  gt0ne0 $p |- ( ( A e. RR /\ 0 < A ) -> A =/= 0 ) $=
    ( cr wcel cc0 clt wbr wne 0red ltne sylan ) ABCZDBCDAEFADGKHDAIJ $.

  $( A number which is less than zero is not zero.  See also ~ lt0ap0 which is
     similar but for apartness.  (Contributed by Stefan O'Rear,
     13-Sep-2014.) $)
  lt0ne0 $p |- ( ( A e. RR /\ A < 0 ) -> A =/= 0 ) $=
    ( cr wcel cc0 clt wbr wa ltne necomd ) ABCADEFGDAADHI $.

  $( Addition to both sides of 'less than'.  Part of definition 11.2.7(vi) of
     [HoTT], p.  (varies).  (Contributed by NM, 12-Nov-1999.)  (Proof shortened
     by Mario Carneiro, 27-May-2016.) $)
  ltadd1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A < B <-> ( A + C ) < ( B + C ) ) ) $=
    ( cr wcel w3a clt wbr caddc co ltadd2 simp3 recnd simp1 addcomd simp2 bitrd
    breq12d ) ADEZBDEZCDEZFZABGHCAIJZCBIJZGHACIJZBCIJZGHABCKUBUCUEUDUFGUBCAUBCS
    TUALMZUBASTUANMOUBCBUGUBBSTUAPMORQ $.

  $( Addition to both sides of 'less than or equal to'.  Part of definition
     11.2.7(vi) of [HoTT], p.  (varies).  (Contributed by NM, 18-Oct-1999.)
     (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  leadd1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A <_ B <-> ( A + C ) <_ ( B + C ) ) ) $=
    ( cr w3a clt wbr wn caddc co cle wb ltadd1 3com12 notbid simp1 simp2 lenltd
    wcel readdcld simp3 3bitr4d ) ADSZBDSZCDSZEZBAFGZHBCIJZACIJZFGZHABKGUIUHKGU
    FUGUJUDUCUEUGUJLBACMNOUFABUCUDUEPZUCUDUEQZRUFUIUHUFACUKUCUDUEUAZTUFBCULUMTR
    UB $.

  $( Addition to both sides of 'less than or equal to'.  (Contributed by NM,
     26-Oct-1999.) $)
  leadd2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A <_ B <-> ( C + A ) <_ ( C + B ) ) ) $=
    ( cr wcel w3a cle wbr caddc co leadd1 simp1 recnd simp3 addcomd simp2 bitrd
    breq12d ) ADEZBDEZCDEZFZABGHACIJZBCIJZGHCAIJZCBIJZGHABCKUBUCUEUDUFGUBACUBAS
    TUALMUBCSTUANMZOUBBCUBBSTUAPMUGORQ $.

  $( 'Less than' relationship between subtraction and addition.  (Contributed
     by NM, 21-Jan-1997.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  ltsubadd $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                  ( ( A - B ) < C <-> A < ( C + B ) ) ) $=
    ( cr wcel w3a co clt wbr caddc wb simp1 simp2 resubcld simp3 ltadd1 syl3anc
    cmin recnd npcand breq1d bitrd ) ADEZBDEZCDEZFZABRGZCHIZUGBJGZCBJGZHIZAUJHI
    UFUGDEUEUDUHUKKUFABUCUDUELZUCUDUEMZNUCUDUEOUMUGCBPQUFUIAUJHUFABUFAULSUFBUMS
    TUAUB $.

  $( 'Less than' relationship between subtraction and addition.  (Contributed
     by NM, 21-Jan-1997.) $)
  ltsubadd2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A - B ) < C <-> A < ( B + C ) ) ) $=
    ( cr wcel w3a cmin clt wbr caddc ltsubadd simp2 recnd addcomd breq2d bitr4d
    co simp3 ) ADEZBDEZCDEZFZABGQCHIACBJQZHIABCJQZHIABCKUBUDUCAHUBBCUBBSTUALMUB
    CSTUARMNOP $.

  $( 'Less than or equal to' relationship between subtraction and addition.
     (Contributed by NM, 17-Nov-2004.)  (Proof shortened by Mario Carneiro,
     27-May-2016.) $)
  lesubadd $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A - B ) <_ C <-> A <_ ( C + B ) ) ) $=
    ( cr wcel w3a co cle wbr caddc wb simp1 simp2 resubcld simp3 leadd1 syl3anc
    cmin recnd npcand breq1d bitrd ) ADEZBDEZCDEZFZABRGZCHIZUGBJGZCBJGZHIZAUJHI
    UFUGDEUEUDUHUKKUFABUCUDUELZUCUDUEMZNUCUDUEOUMUGCBPQUFUIAUJHUFABUFAULSUFBUMS
    TUAUB $.

  $( 'Less than or equal to' relationship between subtraction and addition.
     (Contributed by NM, 10-Aug-1999.) $)
  lesubadd2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A - B ) <_ C <-> A <_ ( B + C ) ) ) $=
    ( cr wcel w3a cmin cle wbr caddc lesubadd simp2 recnd addcomd breq2d bitr4d
    co simp3 ) ADEZBDEZCDEZFZABGQCHIACBJQZHIABCJQZHIABCKUBUDUCAHUBBCUBBSTUALMUB
    CSTUARMNOP $.

  $( 'Less than' relationship between addition and subtraction.  (Contributed
     by NM, 17-Nov-2004.) $)
  ltaddsub $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A + B ) < C <-> A < ( C - B ) ) ) $=
    ( cr wcel w3a co clt wbr caddc wb simp1 simp3 simp2 resubcld ltadd1 syl3anc
    cmin recnd npcand breq2d bitr2d ) ADEZBDEZCDEZFZACBRGZHIZABJGZUGBJGZHIZUICH
    IUFUCUGDEUDUHUKKUCUDUELUFCBUCUDUEMZUCUDUENZOUMAUGBPQUFUJCUIHUFCBUFCULSUFBUM
    STUAUB $.

  $( 'Less than' relationship between addition and subtraction.  (Contributed
     by NM, 17-Nov-2004.) $)
  ltaddsub2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A + B ) < C <-> B < ( C - A ) ) ) $=
    ( cr wcel w3a caddc co clt wbr cmin wceq cc addcom syl2an 3adant3 breq1d wb
    recn ltaddsub 3com12 bitrd ) ADEZBDEZCDEZFZABGHZCIJBAGHZCIJZBCAKHIJZUFUGUHC
    IUCUDUGUHLZUEUCAMEBMEUKUDASBSABNOPQUDUCUEUIUJRBACTUAUB $.

  $( 'Less than or equal to' relationship between addition and subtraction.
     (Contributed by NM, 6-Apr-2005.) $)
  leaddsub $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A + B ) <_ C <-> A <_ ( C - B ) ) ) $=
    ( cr wcel w3a co cle wbr caddc wb simp1 simp3 simp2 resubcld leadd1 syl3anc
    cmin recnd npcand breq2d bitr2d ) ADEZBDEZCDEZFZACBRGZHIZABJGZUGBJGZHIZUICH
    IUFUCUGDEUDUHUKKUCUDUELUFCBUCUDUEMZUCUDUENZOUMAUGBPQUFUJCUIHUFCBUFCULSUFBUM
    STUAUB $.

  $( 'Less than or equal to' relationship between and addition and subtraction.
     (Contributed by NM, 6-Apr-2005.) $)
  leaddsub2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A + B ) <_ C <-> B <_ ( C - A ) ) ) $=
    ( cr wcel w3a caddc co cle wbr cmin wceq cc addcom syl2an 3adant3 breq1d wb
    recn leaddsub 3com12 bitrd ) ADEZBDEZCDEZFZABGHZCIJBAGHZCIJZBCAKHIJZUFUGUHC
    IUCUDUGUHLZUEUCAMEBMEUKUDASBSABNOPQUDUCUEUIUJRBACTUAUB $.

  $( Swap subtrahends in an inequality.  (Contributed by NM, 29-Sep-2005.) $)
  suble $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A - B ) <_ C <-> ( A - C ) <_ B ) ) $=
    ( cr wcel w3a cmin co cle wbr caddc lesubadd wb lesubadd2 3com23 bitr4d ) A
    DEZBDEZCDEZFABGHCIJACBKHIJZACGHBIJZABCLQSRUATMACBNOP $.

  $( Swap subtrahends in an inequality.  (Contributed by NM, 29-Sep-2005.)
     (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  lesub $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( A <_ ( B - C ) <-> C <_ ( B - A ) ) ) $=
    ( cr wcel cmin co cle wbr wb w3a caddc leaddsub leaddsub2 bitr3d 3com23 ) A
    DEZCDEZBDEZABCFGHIZCBAFGHIZJQRSKACLGBHITUAACBMACBNOP $.

  $( 'Less than' relationship between subtraction and addition.  (Contributed
     by NM, 4-Oct-1999.) $)
  ltsub23 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( ( A - B ) < C <-> ( A - C ) < B ) ) $=
    ( cr wcel w3a cmin co clt wbr caddc ltsubadd wb ltsubadd2 3com23 bitr4d ) A
    DEZBDEZCDEZFABGHCIJACBKHIJZACGHBIJZABCLQSRUATMACBNOP $.

  $( 'Less than' relationship between subtraction and addition.  (Contributed
     by NM, 17-Nov-2004.) $)
  ltsub13 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                   ( A < ( B - C ) <-> C < ( B - A ) ) ) $=
    ( cr wcel cmin co clt wbr wb w3a caddc ltaddsub ltaddsub2 bitr3d 3com23 ) A
    DEZCDEZBDEZABCFGHIZCBAFGHIZJQRSKACLGBHITUAACBMACBNOP $.

  $( Adding both sides of two 'less than or equal to' relations.  (Contributed
     by NM, 17-Apr-2005.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  le2add $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
                ( ( A <_ C /\ B <_ D ) -> ( A + B ) <_ ( C + D ) ) ) $=
    ( cr wcel wa cle wbr caddc co wb simpll simprl simplr leadd1 syl3anc simprr
    leadd2 readdcld anbi12d wi letr sylbid ) AEFZBEFZGZCEFZDEFZGZGZACHIZBDHIZGA
    BJKZCBJKZHIZUOCDJKZHIZGZUNUQHIZUKULUPUMURUKUEUHUFULUPLUEUFUJMZUGUHUINZUEUFU
    JOZACBPQUKUFUIUHUMURLVCUGUHUIRZVBBDCSQUAUKUNEFUOEFUQEFUSUTUBUKABVAVCTUKCBVB
    VCTUKCDVBVDTUNUOUQUCQUD $.

  $( Adding both sides of two 'less than' relations.  Theorem I.25 of [Apostol]
     p. 20.  (Contributed by NM, 15-Aug-1999.)  (Proof shortened by Mario
     Carneiro, 27-May-2016.) $)
  lt2add $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
                ( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) ) ) $=
    ( cr wcel wa clt wbr caddc co wb simpll simprl simplr ltadd1 syl3anc simprr
    ltadd2d readdcld anbi12d wi lttr sylbid ) AEFZBEFZGZCEFZDEFZGZGZACHIZBDHIZG
    ABJKZCBJKZHIZUOCDJKZHIZGZUNUQHIZUKULUPUMURUKUEUHUFULUPLUEUFUJMZUGUHUINZUEUF
    UJOZACBPQUKBDCVCUGUHUIRZVBSUAUKUNEFUOEFUQEFUSUTUBUKABVAVCTUKCBVBVCTUKCDVBVD
    TUNUOUQUCQUD $.

  $( Adding both sides of two orderings.  (Contributed by NM, 23-Dec-2007.) $)
  ltleadd $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
                ( ( A < C /\ B <_ D ) -> ( A + B ) < ( C + D ) ) ) $=
    ( cr wcel wa clt wbr caddc co wb ltadd1 3com23 3expa adantrr leadd2 readdcl
    cle 3expb adantll anbi12d wi adantr ancoms ad2ant2lr adantl ltletr syl3anc
    sylbid ) AEFZBEFZGZCEFZDEFZGZGZACHIZBDSIZGABJKZCBJKZHIZVACDJKZSIZGZUTVCHIZU
    QURVBUSVDUMUNURVBLZUOUKULUNVGUKUNULVGACBMNOPULUPUSVDLZUKULUNUOVHULUOUNVHBDC
    QNTUAUBUQUTEFZVAEFZVCEFZVEVFUCUMVIUPABRUDULUNVJUKUOUNULVJCBRUEUFUPVKUMCDRUG
    UTVAVCUHUIUJ $.

  $( Adding both sides of two orderings.  (Contributed by NM, 15-Aug-2008.) $)
  leltadd $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
                ( ( A <_ C /\ B < D ) -> ( A + B ) < ( C + D ) ) ) $=
    ( cr wcel wa cle wbr caddc co wi ltleadd ancomsd cc wceq recn addcom syl2an
    clt ancom2s ancom1s breqan12d sylibrd ) AEFZBEFZGZCEFZDEFZGZGACHIZBDTIZGZBA
    JKZDCJKZTIZABJKZCDJKZTIUFUEUJUMUPLZUFUEGZUIUHUSUTUIUHGGULUKUPBADCMNUAUBUGUJ
    UQUNURUOTUEAOFBOFUQUNPUFAQBQABRSUHCOFDOFURUOPUICQDQCDRSUCUD $.

  $( The sum of 2 positive numbers is positive.  (Contributed by NM,
     1-Jun-2005.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  addgt0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) ->
                0 < ( A + B ) ) $=
    ( cr wcel wa cc0 clt wbr caddc co 00id wi 0re lt2add mpanl12 imp eqbrtrrid
    ) ACDBCDEZFAGHFBGHEZEFFFIJZABIJZGKRSTUAGHZFCDZUCRSUBLMMFFABNOPQ $.

  $( The sum of nonnegative and positive numbers is positive.  (Contributed by
     NM, 28-Dec-2005.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  addgegt0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < B ) ) ->
                  0 < ( A + B ) ) $=
    ( cr wcel wa cc0 cle wbr clt caddc co 00id wi 0re leltadd mpanl12 eqbrtrrid
    imp ) ACDBCDEZFAGHFBIHEZEFFFJKZABJKZILSTUAUBIHZFCDZUDSTUCMNNFFABOPRQ $.

  $( The sum of nonnegative and positive numbers is positive.  (Contributed by
     NM, 28-Dec-2005.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  addgtge0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ B ) ) ->
                  0 < ( A + B ) ) $=
    ( cr wcel wa cc0 clt wbr cle caddc co 00id wi 0re ltleadd mpanl12 eqbrtrrid
    imp ) ACDBCDEZFAGHFBIHEZEFFFJKZABJKZGLSTUAUBGHZFCDZUDSTUCMNNFFABOPRQ $.

  $( The sum of 2 nonnegative numbers is nonnegative.  (Contributed by NM,
     17-Mar-2005.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
  addge0 $p |- ( ( ( A e. RR /\ B e. RR ) /\
               ( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A + B ) ) $=
    ( cr wcel wa cc0 cle wbr caddc co 00id wi 0re le2add mpanl12 imp eqbrtrrid
    ) ACDBCDEZFAGHFBGHEZEFFFIJZABIJZGKRSTUAGHZFCDZUCRSUBLMMFFABNOPQ $.

  $( Adding a positive number to another number increases it.  (Contributed by
     NM, 17-Nov-2004.) $)
  ltaddpos $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( B + A ) ) ) $=
    ( cr wcel wa cc0 clt wbr caddc co wb ltadd2 mp3an1 wceq recn addridd adantl
    0re breq1d bitrd ) ACDZBCDZEZFAGHZBFIJZBAIJZGHZBUFGHFCDUAUBUDUGKRFABLMUCUEB
    UFGUBUEBNUAUBBBOPQST $.

  $( Adding a positive number to another number increases it.  (Contributed by
     NM, 8-Apr-2005.) $)
  ltaddpos2 $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( A + B ) ) ) $=
    ( cr wcel wa cc0 clt wbr caddc co ltaddpos wceq addcom syl2an breq2d bitr4d
    cc recn ) ACDZBCDZEZFAGHBBAIJZGHBABIJZGHABKUAUCUBBGSAQDBQDUCUBLTARBRABMNOP
    $.

  $( Subtracting a positive number from another number decreases it.
     (Contributed by NM, 17-Nov-2004.)  (Proof shortened by Andrew Salmon,
     19-Nov-2011.) $)
  ltsubpos $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> ( B - A ) < B ) ) $=
    ( cr wcel wa cc0 clt wbr caddc co cmin ltaddpos wb ltsubadd 3anidm13 ancoms
    bitr4d ) ACDZBCDZEFAGHBBAIJGHZBAKJBGHZABLSRUATMZSRUBBABNOPQ $.

  $( Comparison of two numbers whose difference is positive.  (Contributed by
     NM, 17-Nov-2004.) $)
  posdif $p |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) ) $=
    ( cr wcel wa cc0 cmin co clt caddc wb resubcl ancoms simpl ltaddpos syl2anc
    wbr cc wceq recn pncan3 syl2an breq2d bitr2d ) ACDZBCDZEZFBAGHZIQZAAUHJHZIQ
    ZABIQUGUHCDZUEUIUKKUFUEULBALMUEUFNUHAOPUGUJBAIUEARDBRDUJBSUFATBTABUAUBUCUD
    $.

  $( Subtraction from both sides of 'less than or equal to'.  (Contributed by
     NM, 13-May-2004.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  lesub1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A <_ B <-> ( A - C ) <_ ( B - C ) ) ) $=
    ( cr wcel w3a cmin co cle caddc simp1 simp3 simp2 resubcld lesubadd syl3anc
    wbr wb recnd npcand breq2d bitr2d ) ADEZBDEZCDEZFZACGHBCGHZIQZAUGCJHZIQZABI
    QUFUCUEUGDEUHUJRUCUDUEKUCUDUELZUFBCUCUDUEMZUKNACUGOPUFUIBAIUFBCUFBULSUFCUKS
    TUAUB $.

  $( Subtraction of both sides of 'less than or equal to'.  (Contributed by NM,
     29-Sep-2005.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  lesub2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A <_ B <-> ( C - B ) <_ ( C - A ) ) ) $=
    ( cr wcel w3a cle wbr cmin co caddc leadd2 wb simp3 simp1 readdcld lesubadd
    simp2 syl3anc recnd addsubd breq1d 3bitr2d resubcld leaddsub bitrd ) ADEZBD
    EZCDEZFZABGHZCBIJZAKJZCGHZULCAIJGHZUJUKCAKJZCBKJGHZUPBIJZCGHZUNABCLUJUPDEUH
    UIUSUQMUJCAUGUHUINZUGUHUIOZPUGUHUIRZUTUPBCQSUJURUMCGUJCABUJCUTTUJAVATUJBVBT
    UAUBUCUJULDEUGUIUNUOMUJCBUTVBUDVAUTULACUESUF $.

  $( Subtraction from both sides of 'less than'.  (Contributed by FL,
     3-Jan-2008.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  ltsub1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A < B <-> ( A - C ) < ( B - C ) ) ) $=
    ( cr wcel w3a cmin co clt caddc simp1 simp3 simp2 resubcld ltsubadd syl3anc
    wbr wb recnd npcand breq2d bitr2d ) ADEZBDEZCDEZFZACGHBCGHZIQZAUGCJHZIQZABI
    QUFUCUEUGDEUHUJRUCUDUEKUCUDUELZUFBCUCUDUEMZUKNACUGOPUFUIBAIUFBCUFBULSUFCUKS
    TUAUB $.

  $( Subtraction of both sides of 'less than'.  (Contributed by NM,
     29-Sep-2005.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  ltsub2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( A < B <-> ( C - B ) < ( C - A ) ) ) $=
    ( cr wcel w3a clt wbr cmin co caddc ltadd2 wb simp3 simp1 readdcld ltsubadd
    simp2 syl3anc recnd addsubd breq1d 3bitr2d resubcld ltaddsub bitrd ) ADEZBD
    EZCDEZFZABGHZCBIJZAKJZCGHZULCAIJGHZUJUKCAKJZCBKJGHZUPBIJZCGHZUNABCLUJUPDEUH
    UIUSUQMUJCAUGUHUINZUGUHUIOZPUGUHUIRZUTUPBCQSUJURUMCGUJCABUJCUTTUJAVATUJBVBT
    UAUBUCUJULDEUGUIUNUOMUJCBUTVBUDVAUTULACUESUF $.

  $( Subtracting both sides of two 'less than' relations.  (Contributed by
     Mario Carneiro, 14-Apr-2016.) $)
  lt2sub $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
                ( ( A < C /\ D < B ) -> ( A - B ) < ( C - D ) ) ) $=
    ( cr wcel wa clt wbr cmin simpll simprl simplr ltsub1 syl3anc simprr ltsub2
    co wb resubcl anbi12d wi adantr resubcld adantl lttr sylbid ) AEFZBEFZGZCEF
    ZDEFZGZGZACHIZDBHIZGABJRZCBJRZHIZURCDJRZHIZGZUQUTHIZUNUOUSUPVAUNUHUKUIUOUSS
    UHUIUMKUJUKULLZUHUIUMMZACBNOUNULUIUKUPVASUJUKULPVEVDDBCQOUAUNUQEFZUREFUTEFZ
    VBVCUBUJVFUMABTUCUNCBVDVEUDUMVGUJCDTUEUQURUTUFOUG $.

  $( Subtracting both sides of two 'less than or equal to' relations.
     (Contributed by Mario Carneiro, 14-Apr-2016.) $)
  le2sub $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
                ( ( A <_ C /\ D <_ B ) -> ( A - B ) <_ ( C - D ) ) ) $=
    ( cr wcel wa cle wbr cmin simpll simprl simplr lesub1 syl3anc simprr lesub2
    co wb resubcl anbi12d wi adantr resubcld adantl letr sylbid ) AEFZBEFZGZCEF
    ZDEFZGZGZACHIZDBHIZGABJRZCBJRZHIZURCDJRZHIZGZUQUTHIZUNUOUSUPVAUNUHUKUIUOUSS
    UHUIUMKUJUKULLZUHUIUMMZACBNOUNULUIUKUPVASUJUKULPVEVDDBCQOUAUNUQEFZUREFUTEFZ
    VBVCUBUJVFUMABTUCUNCBVDVEUDUMVGUJCDTUEUQURUTUFOUG $.

  $( Negative of both sides of 'less than'.  Theorem I.23 of [Apostol] p. 20.
     (Contributed by NM, 27-Aug-1999.)  (Proof shortened by Mario Carneiro,
     27-May-2016.) $)
  ltneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) ) $=
    ( cr wcel wa clt wbr cc0 cmin cneg 0re ltsub2 mp3an3 df-neg breq12i bitr4di
    co wb ) ACDZBCDZEABFGZHBIQZHAIQZFGZBJZAJZFGSTHCDUAUDRKABHLMUEUBUFUCFBNANOP
    $.

  $( Contraposition of negative in 'less than'.  (Contributed by NM,
     8-Nov-2004.) $)
  ltnegcon1 $p |- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) ) $=
    ( cr wcel wa cneg clt wbr wb renegcl ltneg sylan simpl recnd negnegd breq2d
    bitrd ) ACDZBCDZEZAFZBGHZBFZUAFZGHZUCAGHRUACDSUBUEIAJUABKLTUDAUCGTATARSMNOP
    Q $.

  $( Contraposition of negative in 'less than'.  (Contributed by Mario
     Carneiro, 25-Feb-2015.) $)
  ltnegcon2 $p |- ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> B < -u A ) ) $=
    ( cr wcel wa cneg clt wbr wb renegcl ltneg sylan2 simpr recnd negnegd bitrd
    breq1d ) ACDZBCDZEZABFZGHZUAFZAFZGHZBUDGHSRUACDUBUEIBJAUAKLTUCBUDGTBTBRSMNO
    QP $.

  $( Negative of both sides of 'less than or equal to'.  (Contributed by NM,
     12-Sep-1999.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  leneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) ) $=
    ( cr wcel wa cle wbr cc0 cmin cneg 0re lesub2 mp3an3 df-neg breq12i bitr4di
    co wb ) ACDZBCDZEABFGZHBIQZHAIQZFGZBJZAJZFGSTHCDUAUDRKABHLMUEUBUFUCFBNANOP
    $.

  $( Contraposition of negative in 'less than or equal to'.  (Contributed by
     NM, 10-May-2004.) $)
  lenegcon1 $p |- ( ( A e. RR /\ B e. RR ) -> ( -u A <_ B <-> -u B <_ A ) ) $=
    ( cr wcel wa cneg cle wbr wb renegcl leneg sylan recn negnegd breq2d adantr
    bitrd ) ACDZBCDZEAFZBGHZBFZTFZGHZUBAGHZRTCDSUAUDIAJTBKLRUDUEISRUCAUBGRAAMNO
    PQ $.

  $( Contraposition of negative in 'less than or equal to'.  (Contributed by
     NM, 8-Oct-2005.) $)
  lenegcon2 $p |- ( ( A e. RR /\ B e. RR ) -> ( A <_ -u B <-> B <_ -u A ) ) $=
    ( cr wcel wa cneg cle wbr wb renegcl sylan2 wceq recn negnegd adantl breq1d
    leneg bitrd ) ACDZBCDZEZABFZGHZUBFZAFZGHZBUEGHTSUBCDUCUFIBJAUBQKUAUDBUEGTUD
    BLSTBBMNOPR $.

  $( Comparison of a number and its negative to zero.  Theorem I.23 of
     [Apostol] p. 20.  (Contributed by NM, 14-May-1999.) $)
  lt0neg1 $p |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) ) $=
    ( cr wcel cc0 clt wbr cneg wb 0re ltneg mpan2 neg0 breq1i bitrdi ) ABCZADEF
    ZDGZAGZEFZDREFODBCPSHIADJKQDRELMN $.

  $( Comparison of a number and its negative to zero.  (Contributed by NM,
     10-May-2004.) $)
  lt0neg2 $p |- ( A e. RR -> ( 0 < A <-> -u A < 0 ) ) $=
    ( cr wcel cc0 clt wbr cneg wb 0re ltneg mpan neg0 breq2i bitrdi ) ABCZDAEFZ
    AGZDGZEFZQDEFDBCOPSHIDAJKRDQELMN $.

  $( Comparison of a number and its negative to zero.  (Contributed by NM,
     10-May-2004.) $)
  le0neg1 $p |- ( A e. RR -> ( A <_ 0 <-> 0 <_ -u A ) ) $=
    ( cr wcel cc0 cle wbr cneg wb 0re leneg mpan2 neg0 breq1i bitrdi ) ABCZADEF
    ZDGZAGZEFZDREFODBCPSHIADJKQDRELMN $.

  $( Comparison of a number and its negative to zero.  (Contributed by NM,
     24-Aug-1999.) $)
  le0neg2 $p |- ( A e. RR -> ( 0 <_ A <-> -u A <_ 0 ) ) $=
    ( cr wcel cc0 cle wbr cneg wb 0re leneg mpan neg0 breq2i bitrdi ) ABCZDAEFZ
    AGZDGZEFZQDEFDBCOPSHIDAJKRDQELMN $.

  $( A number is less than or equal to itself plus a nonnegative number.
     (Contributed by NM, 21-Feb-2005.) $)
  addge01 $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> A <_ ( A + B ) ) ) $=
    ( cr wcel wa cc0 cle wbr caddc co wb leadd2 mp3an1 ancoms wceq recn addridd
    0re adantr breq1d bitrd ) ACDZBCDZEZFBGHZAFIJZABIJZGHZAUGGHUCUBUEUHKZFCDUCU
    BUIRFBALMNUDUFAUGGUBUFAOUCUBAAPQSTUA $.

  $( A number is less than or equal to itself plus a nonnegative number.
     (Contributed by NM, 27-Jul-2005.) $)
  addge02 $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> A <_ ( B + A ) ) ) $=
    ( cr wcel wa cc0 cle wbr caddc co addge01 cc wceq recn addcom syl2an breq2d
    bitrd ) ACDZBCDZEZFBGHAABIJZGHABAIJZGHABKUAUBUCAGSALDBLDUBUCMTANBNABOPQR $.

  $( Two nonnegative numbers are zero iff their sum is zero.  (Contributed by
     Jeff Madsen, 2-Sep-2009.)  (Proof shortened by Mario Carneiro,
     27-May-2016.) $)
  add20 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) )
                        -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) ) $=
    ( cr wcel cc0 cle wbr wa caddc wceq simpllr simplrl simplll addge02 syl2anc
    co wb mpbid simpr breqtrd simplrr letri3d mpbir2and oveq2d addridd 3eqtr3rd
    0red recnd jca ex oveq12 00id eqtrdi impbid1 ) ACDZEAFGZHZBCDZEBFGZHZHZABIP
    ZEJZAEJZBEJZHZVAVCVFVAVCHZVDVEVGVBAEIPEAVGBEAIVGVEBEFGUSVGBVBEFVGUPBVBFGZUO
    UPUTVCKVGURUOUPVHQUQURUSVCLZUOUPUTVCMZBANORVAVCSZTUQURUSVCUAVGBEVIVGUGUBUCZ
    UDVKVGAVGAVJUHUEUFVLUIUJVFVBEEIPEAEBEIUKULUMUN $.

  $( Nonnegative subtraction.  (Contributed by NM, 14-Mar-2005.)  (Proof
     shortened by Mario Carneiro, 27-May-2016.) $)
  subge0 $p |- ( ( A e. RR /\ B e. RR ) ->
                ( 0 <_ ( A - B ) <-> B <_ A ) ) $=
    ( cr wcel wa cc0 caddc co cle wbr cmin wb 0red simpr simpl leaddsub syl3anc
    recnd addlidd breq1d bitr3d ) ACDZBCDZEZFBGHZAIJZFABKHIJZBAIJUDFCDUCUBUFUGL
    UDMUBUCNZUBUCOFBAPQUDUEBAIUDBUDBUHRSTUA $.

  $( Nonpositive subtraction.  (Contributed by NM, 20-Mar-2008.)  (Proof
     shortened by Mario Carneiro, 27-May-2016.) $)
  suble0 $p |- ( ( A e. RR /\ B e. RR ) ->
                ( ( A - B ) <_ 0 <-> A <_ B ) ) $=
    ( cr wcel wa cmin co cc0 cle wbr wb suble mp3an3 simpl recnd subid1d breq1d
    0re bitrd ) ACDZBCDZEZABFGHIJZAHFGZBIJZABIJTUAHCDUCUEKRABHLMUBUDABIUBAUBATU
    ANOPQS $.

  $( The sum of a real number and a second real number is less then the real
     number iff the second real number is negative.  (Contributed by Alexander
     van der Vekens, 30-May-2018.) $)
  leaddle0 $p |- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) <_ A <-> B <_ 0 ) ) $=
    ( cr wcel wa caddc co cle wbr cmin wb leaddsub2 3anidm13 wceq subidd adantr
    cc0 recn breq2d bitrd ) ACDZBCDZEZABFGAHIZBAAJGZHIZBQHIUAUBUDUFKABALMUCUEQB
    HUAUEQNUBUAAAROPST $.

  $( Nonnegative subtraction.  (Contributed by NM, 27-Jul-2005.) $)
  subge02 $p |- ( ( A e. RR /\ B e. RR ) ->
                ( 0 <_ B <-> ( A - B ) <_ A ) ) $=
    ( cr wcel wa cc0 cle wbr caddc co cmin addge01 wb lesubadd 3anidm13 bitr4d
    ) ACDZBCDZEFBGHAABIJGHZABKJAGHZABLQRTSMABANOP $.

  $( Lemma to show a nonnegative number is zero.  (Contributed by NM,
     8-Oct-1999.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  lesub0 $p |- ( ( A e. RR /\ B e. RR ) ->
                ( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 ) ) $=
    ( cr wcel wa cc0 wceq cle wbr cmin co 0red letri3 sylan2 ancom simpr lesub2
    wb simpl syl3anc recnd subid1d breq1d bitrd ancoms anbi2d bitrid bitr2d ) A
    CDZBCDZEZAFGZAFHIZFAHIZEZUNBBAJKZHIZEZUJUIFCDZULUORUJLAFMNUOUNUMEUKURUMUNOU
    KUMUQUNUJUIUMUQRUJUIEZUMBFJKZUPHIZUQUTUIUSUJUMVBRUJUIPUTLUJUISZAFBQTUTVABUP
    HUTBUTBVCUAUBUCUDUEUFUGUH $.

  $( The product of two negative numbers is positive.  (Contributed by Jeff
     Hankins, 8-Jun-2009.) $)
  mullt0 $p |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) )
               -> 0 < ( A x. B ) ) $=
    ( cr wcel cc0 clt wbr wa cneg cmul renegcl adantr lt0neg1 biimpa jca mulgt0
    co syl2an cc recn wceq mul2neg ad2ant2r breqtrd ) ACDZAEFGZHZBCDZBEFGZHZHEA
    IZBIZJQZABJQZFUGUKCDZEUKFGZHULCDZEULFGZHEUMFGUJUGUOUPUEUOUFAKLUEUFUPAMNOUJU
    QURUHUQUIBKLUHUIURBMNOUKULPRUEUHUMUNUAZUFUIUEASDBSDUSUHATBTABUBRUCUD $.

  $( 0 is less than or equal to 1.  (Contributed by Mario Carneiro,
     29-Apr-2015.) $)
  0le1 $p |- 0 <_ 1 $=
    ( cc0 c1 0re 1re 0lt1 ltleii ) ABCDEF $.

  ${
    $d x B $.  $d x y C $.  $d x y D $.  $d x y M $.  $d x y N $.  $d x y ph $.
    $d x y S $.
    ltord.1 $e |- ( x = y -> A = B ) $.
    ltord.2 $e |- ( x = C -> A = M ) $.
    ltord.3 $e |- ( x = D -> A = N ) $.
    ltord.4 $e |- S C_ RR $.
    ltord.5 $e |- ( ( ph /\ x e. S ) -> A e. RR ) $.
    ${
      ltord.6 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) ) $.
      $( Lemma for ~ eqord1 .  (Contributed by Mario Carneiro, 14-Jun-2014.) $)
      ltordlem $p |- ( ( ph /\ ( C e. S /\ D e. S ) ) ->
                        ( C < D -> M < N ) ) $=
        ( clt wbr wi wceq cv wral wcel wa ralrimivva breq1 breq1d imbi12d breq2
        eqeq1 eqeq1d chvarv breq2d rspc2v mpan9 ) ABUAZCUAZQRZDEQRZSZCHUBBHUBFH
        UCGHUCUDFGQRZIJQRZSZAUTBCHHPUEUTVCFUQQRZIEQRZSBCFGHHUPFTZURVDUSVEUPFUQQ
        UFVFDIEQLUGUHUQGTZVDVAVEVBUQGFQUIVGEJIQUPGTZDJTZSVGEJTZSBCUPUQTZVHVGVIV
        JUPUQGUJVKDEJKUKUHMULUMUHUNUO $.

      $( A strictly increasing real function on a subset of ` RR ` is
         one-to-one.  (Contributed by Mario Carneiro, 14-Jun-2014.)  (Revised
         by Jim Kingdon, 20-Dec-2022.) $)
      eqord1 $p |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) ) $=
        ( wcel wa wceq cr cv wex simprl elisset syl adantr adantl biimpa eqtr3d
        wb eqeq2 exlimddv ex clt wn ltordlem con3d wi ancom2s anim12d ralrimiva
        wral eleq1d rspccva sylan adantrr adantrl lttri3d sselid simprr 3imtr4d
        wbr impbid ) AFHQZGHQZRRZFGSZIJSZVPVQVRVPVQRZBUAZFSZVRBVPWABUBZVQVPVNWB
        AVNVOUCZBFHUDUEUFVSWARZDIJWADISVSLUGWDVTGSZDJSVSWAWEVQWAWEUJVPFGVTUKUGU
        HMUEUIULUMVPIJUNVLZUOZJIUNVLZUOZRFGUNVLZUOZGFUNVLZUOZRVRVQVPWGWKWIWMVPW
        JWFABCDEFGHIJKLMNOPUPUQAVOVNWIWMURAVOVNRRWLWHABCDEGFHJIKMLNOPUPUQUSUTVP
        IJAVNITQZVOADTQZBHVBZVNWNAWOBHOVAZWOWNBFHWADITLVCVDVEVFAVOJTQZVNAWPVOWR
        WQWOWRBGHWEDJTMVCVDVEVGVHVPFGVPHTFNWCVIVPHTGNAVNVOVJVIVHVKVM $.
    $}

    ltord2.6 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> B < A ) ) $.
    $( A strictly decreasing real function on a subset of ` RR ` is one-to-one.
       (Contributed by Mario Carneiro, 14-Jun-2014.) $)
    eqord2 $p |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) ) $=
      ( wcel wa wceq cr cneg cv negeqd renegcld clt wb ralrimiva eleq1d rspccva
      wbr sylan adantrl adantrr ltneg syl2anc sylibd eqord1 recnd neg11ad bitrd
      wral ) AFHQZGHQZRRZFGSIUAZJUAZSIJSABCDUAZEUAZFGHVEVFBUBZCUBZSZDEKUCVIFSZD
      ILUCVIGSZDJMUCNAVIHQZRDOUDAVNVJHQZRRZVIVJUEUJEDUEUJZVGVHUEUJZPVPETQZDTQZV
      QVRUFAVOVSVNAVTBHVAZVOVSAVTBHOUGZVTVSBVJHVKDETKUHUIUKULAVNVTVOOUMEDUNUOUP
      UQVDIJVDIAVBITQZVCAWAVBWCWBVTWCBFHVLDITLUHUIUKUMURVDJAVCJTQZVBAWAVCWDWBVT
      WDBGHVMDJTMUHUIUKULURUSUT $.
  $}

  ${
    lt2.1 $e |- A e. RR $.
    $( 'Less than or equal to' is reflexive.  (Contributed by NM,
       18-Aug-1999.) $)
    leidi $p |- A <_ A $=
      ( cr wcel cle wbr leid ax-mp ) ACDAAEFBAGH $.

    $( Positive means nonzero (useful for ordering theorems involving
       division).  (Contributed by NM, 16-Sep-1999.) $)
    gt0ne0i $p |- ( 0 < A -> A =/= 0 ) $=
      ( cc0 0re ltnei ) CADBE $.

    ${
      gt0ne0i.2 $e |- 0 < A $.
      $( Positive implies nonzero.  (Contributed by NM, 15-May-1999.) $)
      gt0ne0ii $p |- A =/= 0 $=
        ( cc0 clt wbr wne gt0ne0i ax-mp ) DAEFADGCABHI $.
    $}

    lt2.2 $e |- B e. RR $.
    $( Addition of 2 positive numbers is positive.  (Contributed by NM,
       16-May-1999.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
    addgt0i $p |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A + B ) ) $=
      ( cr wcel cc0 clt wbr wa caddc co addgt0 mpanl12 ) AEFBEFGAHIGBHIJGABKLHI
      CDABMN $.

    $( Addition of 2 nonnegative numbers is nonnegative.  (Contributed by NM,
       28-May-1999.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
    addge0i $p |- ( ( 0 <_ A /\ 0 <_ B ) -> 0 <_ ( A + B ) ) $=
      ( cr wcel cc0 cle wbr wa caddc co addge0 mpanl12 ) AEFBEFGAHIGBHIJGABKLHI
      CDABMN $.

    $( Addition of nonnegative and positive numbers is positive.  (Contributed
       by NM, 25-Sep-1999.)  (Revised by Mario Carneiro, 27-May-2016.) $)
    addgegt0i $p |- ( ( 0 <_ A /\ 0 < B ) -> 0 < ( A + B ) ) $=
      ( cr wcel cc0 cle wbr clt wa caddc co addgegt0 mpanl12 ) AEFBEFGAHIGBJIKG
      ABLMJICDABNO $.

    ${
      addgt0i.3 $e |- 0 < A $.
      addgt0i.4 $e |- 0 < B $.
      $( Addition of 2 positive numbers is positive.  (Contributed by NM,
         18-May-1999.) $)
      addgt0ii $p |- 0 < ( A + B ) $=
        ( cc0 clt wbr caddc co addgt0i mp2an ) GAHIGBHIGABJKHIEFABCDLM $.
    $}

    $( Two nonnegative numbers are zero iff their sum is zero.  (Contributed by
       NM, 28-Jul-1999.) $)
    add20i $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
      ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) ) $=
      ( cr wcel cc0 cle wbr wa caddc co wceq wb add20 an4s mpanl12 ) AEFZBEFZGA
      HIZGBHIZJABKLGMAGMBGMJNZCDRTSUAUBABOPQ $.

    $( Negative of both sides of 'less than'.  Theorem I.23 of [Apostol] p. 20.
       (Contributed by NM, 21-Jan-1997.) $)
    ltnegi $p |- ( A < B <-> -u B < -u A ) $=
      ( cr wcel clt wbr cneg wb ltneg mp2an ) AEFBEFABGHBIAIGHJCDABKL $.

    $( Negative of both sides of 'less than or equal to'.  (Contributed by NM,
       1-Aug-1999.) $)
    lenegi $p |- ( A <_ B <-> -u B <_ -u A ) $=
      ( cr wcel cle wbr cneg wb leneg mp2an ) AEFBEFABGHBIAIGHJCDABKL $.

    $( Contraposition of negative in 'less than'.  (Contributed by NM,
       14-May-1999.) $)
    ltnegcon2i $p |- ( A < -u B <-> B < -u A ) $=
      ( cr wcel cneg clt wbr wb ltnegcon2 mp2an ) AEFBEFABGHIBAGHIJCDABKL $.

    $( Lemma to show a nonnegative number is zero.  (Contributed by NM,
       8-Oct-1999.)  (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
    lesub0i $p |- ( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 ) $=
      ( cr wcel cc0 cle wbr cmin co wa wceq wb lesub0 mp2an ) AEFBEFGAHIBBAJKHI
      LAGMNCDABOP $.

    $( Adding a positive number to another number increases it.  (Contributed
       by NM, 25-Aug-1999.) $)
    ltaddposi $p |- ( 0 < A <-> B < ( B + A ) ) $=
      ( cr wcel cc0 clt wbr caddc co wb ltaddpos mp2an ) AEFBEFGAHIBBAJKHILCDAB
      MN $.

    $( Comparison of two numbers whose difference is positive.  (Contributed by
       NM, 19-Aug-2001.) $)
    posdifi $p |- ( A < B <-> 0 < ( B - A ) ) $=
      ( cr wcel clt wbr cc0 cmin co wb posdif mp2an ) AEFBEFABGHIBAJKGHLCDABMN
      $.

    $( Contraposition of negative in 'less than'.  (Contributed by NM,
       14-May-1999.) $)
    ltnegcon1i $p |- ( -u A < B <-> -u B < A ) $=
      ( cr wcel cneg clt wbr wb ltnegcon1 mp2an ) AEFBEFAGBHIBGAHIJCDABKL $.

    $( Contraposition of negative in 'less than or equal to'.  (Contributed by
       NM, 6-Apr-2005.) $)
    lenegcon1i $p |- ( -u A <_ B <-> -u B <_ A ) $=
      ( cr wcel cneg cle wbr wb lenegcon1 mp2an ) AEFBEFAGBHIBGAHIJCDABKL $.

    $( Nonnegative subtraction.  (Contributed by NM, 13-Aug-2000.) $)
    subge0i $p |- ( 0 <_ ( A - B ) <-> B <_ A ) $=
      ( cr wcel cc0 cmin co cle wbr wb subge0 mp2an ) AEFBEFGABHIJKBAJKLCDABMN
      $.

    lt2.3 $e |- C e. RR $.
    $( Addition to both sides of 'less than'.  Theorem I.18 of [Apostol] p. 20.
       (Contributed by NM, 21-Jan-1997.) $)
    ltadd1i $p |- ( A < B <-> ( A + C ) < ( B + C ) ) $=
      ( cr wcel clt wbr caddc co wb ltadd1 mp3an ) AGHBGHCGHABIJACKLBCKLIJMDEFA
      BCNO $.

    $( Addition to both sides of 'less than or equal to'.  (Contributed by NM,
       11-Aug-1999.) $)
    leadd1i $p |- ( A <_ B <-> ( A + C ) <_ ( B + C ) ) $=
      ( cr wcel cle wbr caddc co wb leadd1 mp3an ) AGHBGHCGHABIJACKLBCKLIJMDEFA
      BCNO $.

    $( Addition to both sides of 'less than or equal to'.  (Contributed by NM,
       11-Aug-1999.) $)
    leadd2i $p |- ( A <_ B <-> ( C + A ) <_ ( C + B ) ) $=
      ( cr wcel cle wbr caddc co wb leadd2 mp3an ) AGHBGHCGHABIJCAKLCBKLIJMDEFA
      BCNO $.

    $( 'Less than' relationship between subtraction and addition.  (Contributed
       by NM, 21-Jan-1997.)  (Proof shortened by Andrew Salmon,
       19-Nov-2011.) $)
    ltsubaddi $p |- ( ( A - B ) < C <-> A < ( C + B ) ) $=
      ( cr wcel cmin co clt wbr caddc wb ltsubadd mp3an ) AGHBGHCGHABIJCKLACBMJ
      KLNDEFABCOP $.

    $( 'Less than or equal to' relationship between subtraction and addition.
       (Contributed by NM, 30-Sep-1999.)  (Proof shortened by Andrew Salmon,
       19-Nov-2011.) $)
    lesubaddi $p |- ( ( A - B ) <_ C <-> A <_ ( C + B ) ) $=
      ( cr wcel cmin co cle wbr caddc wb lesubadd mp3an ) AGHBGHCGHABIJCKLACBMJ
      KLNDEFABCOP $.

    $( 'Less than' relationship between subtraction and addition.  (Contributed
       by NM, 21-Jan-1997.) $)
    ltsubadd2i $p |- ( ( A - B ) < C <-> A < ( B + C ) ) $=
      ( cr wcel cmin co clt wbr caddc wb ltsubadd2 mp3an ) AGHBGHCGHABIJCKLABCM
      JKLNDEFABCOP $.

    $( 'Less than or equal to' relationship between subtraction and addition.
       (Contributed by NM, 3-Aug-1999.) $)
    lesubadd2i $p |- ( ( A - B ) <_ C <-> A <_ ( B + C ) ) $=
      ( cr wcel cmin co cle wbr caddc wb lesubadd2 mp3an ) AGHBGHCGHABIJCKLABCM
      JKLNDEFABCOP $.

    $( 'Less than' relationship between subtraction and addition.  (Contributed
       by NM, 14-May-1999.) $)
    ltaddsubi $p |- ( ( A + B ) < C <-> A < ( C - B ) ) $=
      ( cr wcel caddc co clt wbr cmin wb ltaddsub mp3an ) AGHBGHCGHABIJCKLACBMJ
      KLNDEFABCOP $.

    lt.4 $e |- D e. RR $.
    $( Adding both side of two inequalities.  Theorem I.25 of [Apostol] p. 20.
       (Contributed by NM, 14-May-1999.) $)
    lt2addi $p |- ( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) ) $=
      ( cr wcel clt wbr wa caddc co wi lt2add mp4an ) AIJBIJCIJDIJACKLBDKLMABNO
      CDNOKLPEFGHABCDQR $.

    $( Adding both side of two inequalities.  (Contributed by NM,
       16-Sep-1999.) $)
    le2addi $p |- ( ( A <_ C /\ B <_ D ) -> ( A + B ) <_ ( C + D ) ) $=
      ( cr wcel cle wbr wa caddc co wi le2add mp4an ) AIJBIJCIJDIJACKLBDKLMABNO
      CDNOKLPEFGHABCDQR $.
  $}

  ${
    gt0ne0d.1 $e |- ( ph -> 0 < A ) $.
    $( Positive implies nonzero.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    gt0ne0d $p |- ( ph -> A =/= 0 ) $=
      ( cc0 cr wcel clt wbr wne 0re ltne sylancr ) ADEFDBGHBDIJCDBKL $.
  $}

  ${
    lt0ne0d.1 $e |- ( ph -> A < 0 ) $.
    $( Something less than zero is not zero.  Deduction form.  See also
       ~ lt0ap0d which is similar but for apartness.  (Contributed by David
       Moews, 28-Feb-2017.) $)
    lt0ne0d $p |- ( ph -> A =/= 0 ) $=
      ( cc0 clt wbr wne wceq 0re ltnri breq1 mtbiri necon2ai syl ) ABDEFZBDGCOB
      DBDHODDEFDIJBDDEKLMN $.
  $}

  ${
    leidd.1 $e |- ( ph -> A e. RR ) $.
    $( 'Less than or equal to' is reflexive.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    leidd $p |- ( ph -> A <_ A ) $=
      ( cr wcel cle wbr leid syl ) ABDEBBFGCBHI $.

    $( Comparison of a number and its negative to zero.  Theorem I.23 of
       [Apostol] p. 20.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    lt0neg1d $p |- ( ph -> ( A < 0 <-> 0 < -u A ) ) $=
      ( cr wcel cc0 clt wbr cneg wb lt0neg1 syl ) ABDEBFGHFBIGHJCBKL $.

    $( Comparison of a number and its negative to zero.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    lt0neg2d $p |- ( ph -> ( 0 < A <-> -u A < 0 ) ) $=
      ( cr wcel cc0 clt wbr cneg wb lt0neg2 syl ) ABDEFBGHBIFGHJCBKL $.

    $( Comparison of a number and its negative to zero.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    le0neg1d $p |- ( ph -> ( A <_ 0 <-> 0 <_ -u A ) ) $=
      ( cr wcel cc0 cle wbr cneg wb le0neg1 syl ) ABDEBFGHFBIGHJCBKL $.

    $( Comparison of a number and its negative to zero.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    le0neg2d $p |- ( ph -> ( 0 <_ A <-> -u A <_ 0 ) ) $=
      ( cr wcel cc0 cle wbr cneg wb le0neg2 syl ) ABDEFBGHBIFGHJCBKL $.

    ltnegd.2 $e |- ( ph -> B e. RR ) $.
    ${
      addgegt0d.3 $e |- ( ph -> 0 <_ A ) $.
      addgegt0d.4 $e |- ( ph -> 0 < B ) $.
      $( Addition of nonnegative and positive numbers is positive.
         (Contributed by Mario Carneiro, 27-May-2016.) $)
      addgegt0d $p |- ( ph -> 0 < ( A + B ) ) $=
        ( cr wcel cc0 cle wbr clt caddc co addgegt0 syl22anc ) ABHICHIJBKLJCMLJ
        BCNOMLDEFGBCPQ $.
    $}

    ${
      addgtge0d.3 $e |- ( ph -> 0 < A ) $.
      addgtge0d.4 $e |- ( ph -> 0 <_ B ) $.
      $( Addition of positive and nonnegative numbers is positive.
         (Contributed by Asger C. Ipsen, 12-May-2021.) $)
      addgtge0d $p |- ( ph -> 0 < ( A + B ) ) $=
        ( cr wcel cc0 clt wbr cle caddc co addgtge0 syl22anc ) ABHICHIJBKLJCMLJ
        BCNOKLDEFGBCPQ $.
    $}

    ${
      addgt0d.3 $e |- ( ph -> 0 < A ) $.
      addgt0d.4 $e |- ( ph -> 0 < B ) $.
      $( Addition of 2 positive numbers is positive.  (Contributed by Mario
         Carneiro, 27-May-2016.) $)
      addgt0d $p |- ( ph -> 0 < ( A + B ) ) $=
        ( cc0 0red ltled addgegt0d ) ABCDEAHBAIDFJGK $.
    $}

    ${
      addge0d.3 $e |- ( ph -> 0 <_ A ) $.
      addge0d.4 $e |- ( ph -> 0 <_ B ) $.
      $( Addition of 2 nonnegative numbers is nonnegative.  (Contributed by
         Mario Carneiro, 27-May-2016.) $)
      addge0d $p |- ( ph -> 0 <_ ( A + B ) ) $=
        ( cr wcel cc0 cle wbr caddc co addge0 syl22anc ) ABHICHIJBKLJCKLJBCMNKL
        DEFGBCOP $.
    $}

    $( Negative of both sides of 'less than'.  Theorem I.23 of [Apostol] p. 20.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    ltnegd $p |- ( ph -> ( A < B <-> -u B < -u A ) ) $=
      ( cr wcel clt wbr cneg wb ltneg syl2anc ) ABFGCFGBCHICJBJHIKDEBCLM $.

    $( Negative of both sides of 'less than or equal to'.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    lenegd $p |- ( ph -> ( A <_ B <-> -u B <_ -u A ) ) $=
      ( cr wcel cle wbr cneg wb leneg syl2anc ) ABFGCFGBCHICJBJHIKDEBCLM $.

    ${
      ltnegcon1d.3 $e |- ( ph -> -u A < B ) $.
      $( Contraposition of negative in 'less than'.  (Contributed by Mario
         Carneiro, 27-May-2016.) $)
      ltnegcon1d $p |- ( ph -> -u B < A ) $=
        ( cneg clt wbr cr wcel wb ltnegcon1 syl2anc mpbid ) ABGCHIZCGBHIZFABJKC
        JKPQLDEBCMNO $.
    $}

    ${
      ltnegcon2d.3 $e |- ( ph -> A < -u B ) $.
      $( Contraposition of negative in 'less than'.  (Contributed by Mario
         Carneiro, 27-May-2016.) $)
      ltnegcon2d $p |- ( ph -> B < -u A ) $=
        ( cneg clt wbr cr wcel wb ltnegcon2 syl2anc mpbid ) ABCGHIZCBGHIZFABJKC
        JKPQLDEBCMNO $.
    $}

    ${
      lenegcon1d.3 $e |- ( ph -> -u A <_ B ) $.
      $( Contraposition of negative in 'less than or equal to'.  (Contributed
         by Mario Carneiro, 27-May-2016.) $)
      lenegcon1d $p |- ( ph -> -u B <_ A ) $=
        ( cneg cle wbr cr wcel wb lenegcon1 syl2anc mpbid ) ABGCHIZCGBHIZFABJKC
        JKPQLDEBCMNO $.
    $}

    ${
      lenegcon2d.3 $e |- ( ph -> A <_ -u B ) $.
      $( Contraposition of negative in 'less than or equal to'.  (Contributed
         by Mario Carneiro, 27-May-2016.) $)
      lenegcon2d $p |- ( ph -> B <_ -u A ) $=
        ( cneg cle wbr cr wcel wb lenegcon2 syl2anc mpbid ) ABCGHIZCBGHIZFABJKC
        JKPQLDEBCMNO $.
    $}

    $( Adding a positive number to another number increases it.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    ltaddposd $p |- ( ph -> ( 0 < A <-> B < ( B + A ) ) ) $=
      ( cr wcel cc0 clt wbr caddc co wb ltaddpos syl2anc ) ABFGCFGHBIJCCBKLIJMD
      EBCNO $.

    $( Adding a positive number to another number increases it.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    ltaddpos2d $p |- ( ph -> ( 0 < A <-> B < ( A + B ) ) ) $=
      ( cr wcel cc0 clt wbr caddc co wb ltaddpos2 syl2anc ) ABFGCFGHBIJCBCKLIJM
      DEBCNO $.

    $( Subtracting a positive number from another number decreases it.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    ltsubposd $p |- ( ph -> ( 0 < A <-> ( B - A ) < B ) ) $=
      ( cr wcel cc0 clt wbr cmin co wb ltsubpos syl2anc ) ABFGCFGHBIJCBKLCIJMDE
      BCNO $.

    $( Comparison of two numbers whose difference is positive.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    posdifd $p |- ( ph -> ( A < B <-> 0 < ( B - A ) ) ) $=
      ( cr wcel clt wbr cc0 cmin co wb posdif syl2anc ) ABFGCFGBCHIJCBKLHIMDEBC
      NO $.

    $( A number is less than or equal to itself plus a nonnegative number.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    addge01d $p |- ( ph -> ( 0 <_ B <-> A <_ ( A + B ) ) ) $=
      ( cr wcel cc0 cle wbr caddc co wb addge01 syl2anc ) ABFGCFGHCIJBBCKLIJMDE
      BCNO $.

    $( A number is less than or equal to itself plus a nonnegative number.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    addge02d $p |- ( ph -> ( 0 <_ B <-> A <_ ( B + A ) ) ) $=
      ( cr wcel cc0 cle wbr caddc co wb addge02 syl2anc ) ABFGCFGHCIJBCBKLIJMDE
      BCNO $.

    $( Nonnegative subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subge0d $p |- ( ph -> ( 0 <_ ( A - B ) <-> B <_ A ) ) $=
      ( cr wcel cc0 cmin co cle wbr wb subge0 syl2anc ) ABFGCFGHBCIJKLCBKLMDEBC
      NO $.

    $( Nonpositive subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    suble0d $p |- ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) ) $=
      ( cr wcel cmin co cc0 cle wbr wb suble0 syl2anc ) ABFGCFGBCHIJKLBCKLMDEBC
      NO $.

    $( Nonnegative subtraction.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    subge02d $p |- ( ph -> ( 0 <_ B <-> ( A - B ) <_ A ) ) $=
      ( cr wcel cc0 cle wbr cmin co wb subge02 syl2anc ) ABFGCFGHCIJBCKLBIJMDEB
      CNO $.

    ltadd1d.3 $e |- ( ph -> C e. RR ) $.
    $( Addition to both sides of 'less than'.  Theorem I.18 of [Apostol] p. 20.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    ltadd1d $p |- ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) ) $=
      ( cr wcel clt wbr caddc co wb ltadd1 syl3anc ) ABHICHIDHIBCJKBDLMCDLMJKNE
      FGBCDOP $.

    $( Addition to both sides of 'less than or equal to'.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    leadd1d $p |- ( ph -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) ) $=
      ( cr wcel cle wbr caddc co wb leadd1 syl3anc ) ABHICHIDHIBCJKBDLMCDLMJKNE
      FGBCDOP $.

    $( Addition to both sides of 'less than or equal to'.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    leadd2d $p |- ( ph -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) ) $=
      ( cr wcel cle wbr caddc co wb leadd2 syl3anc ) ABHICHIDHIBCJKDBLMDCLMJKNE
      FGBCDOP $.

    $( 'Less than' relationship between subtraction and addition.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    ltsubaddd $p |- ( ph -> ( ( A - B ) < C <-> A < ( C + B ) ) ) $=
      ( cr wcel cmin co clt wbr caddc wb ltsubadd syl3anc ) ABHICHIDHIBCJKDLMBD
      CNKLMOEFGBCDPQ $.

    $( 'Less than or equal to' relationship between subtraction and addition.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    lesubaddd $p |- ( ph -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) ) $=
      ( cr wcel cmin co cle wbr caddc wb lesubadd syl3anc ) ABHICHIDHIBCJKDLMBD
      CNKLMOEFGBCDPQ $.

    $( 'Less than' relationship between subtraction and addition.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    ltsubadd2d $p |- ( ph -> ( ( A - B ) < C <-> A < ( B + C ) ) ) $=
      ( cr wcel cmin co clt wbr caddc wb ltsubadd2 syl3anc ) ABHICHIDHIBCJKDLMB
      CDNKLMOEFGBCDPQ $.

    $( 'Less than or equal to' relationship between subtraction and addition.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    lesubadd2d $p |- ( ph -> ( ( A - B ) <_ C <-> A <_ ( B + C ) ) ) $=
      ( cr wcel cmin co cle wbr caddc wb lesubadd2 syl3anc ) ABHICHIDHIBCJKDLMB
      CDNKLMOEFGBCDPQ $.

    $( 'Less than' relationship between subtraction and addition.  (Contributed
       by Mario Carneiro, 27-May-2016.) $)
    ltaddsubd $p |- ( ph -> ( ( A + B ) < C <-> A < ( C - B ) ) ) $=
      ( cr wcel caddc co clt wbr cmin wb ltaddsub syl3anc ) ABHICHIDHIBCJKDLMBD
      CNKLMOEFGBCDPQ $.

    $( 'Less than' relationship between subtraction and addition.  (Contributed
       by Mario Carneiro, 29-Dec-2016.) $)
    ltaddsub2d $p |- ( ph -> ( ( A + B ) < C <-> B < ( C - A ) ) ) $=
      ( cr wcel caddc co clt wbr cmin wb ltaddsub2 syl3anc ) ABHICHIDHIBCJKDLMC
      DBNKLMOEFGBCDPQ $.

    $( 'Less than or equal to' relationship between and addition and
       subtraction.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    leaddsub2d $p |- ( ph -> ( ( A + B ) <_ C <-> B <_ ( C - A ) ) ) $=
      ( cr wcel caddc co cle wbr cmin wb leaddsub2 syl3anc ) ABHICHIDHIBCJKDLMC
      DBNKLMOEFGBCDPQ $.

    ${
      subled.4 $e |- ( ph -> ( A - B ) <_ C ) $.
      $( Swap subtrahends in an inequality.  (Contributed by Mario Carneiro,
         27-May-2016.) $)
      subled $p |- ( ph -> ( A - C ) <_ B ) $=
        ( cmin co cle wbr cr wcel wb suble syl3anc mpbid ) ABCIJDKLZBDIJCKLZHAB
        MNCMNDMNSTOEFGBCDPQR $.
    $}

    ${
      lesubd.4 $e |- ( ph -> A <_ ( B - C ) ) $.
      $( Swap subtrahends in an inequality.  (Contributed by Mario Carneiro,
         27-May-2016.) $)
      lesubd $p |- ( ph -> C <_ ( B - A ) ) $=
        ( cmin co cle wbr cr wcel wb lesub syl3anc mpbid ) ABCDIJKLZDCBIJKLZHAB
        MNCMNDMNSTOEFGBCDPQR $.
    $}

    ${
      ltsub23d.4 $e |- ( ph -> ( A - B ) < C ) $.
      $( 'Less than' relationship between subtraction and addition.
         (Contributed by Mario Carneiro, 27-May-2016.) $)
      ltsub23d $p |- ( ph -> ( A - C ) < B ) $=
        ( cmin co clt wbr cr wcel wb ltsub23 syl3anc mpbid ) ABCIJDKLZBDIJCKLZH
        ABMNCMNDMNSTOEFGBCDPQR $.
    $}

    ${
      ltsub13d.4 $e |- ( ph -> A < ( B - C ) ) $.
      $( 'Less than' relationship between subtraction and addition.
         (Contributed by Mario Carneiro, 27-May-2016.) $)
      ltsub13d $p |- ( ph -> C < ( B - A ) ) $=
        ( cmin co clt wbr cr wcel wb ltsub13 syl3anc mpbid ) ABCDIJKLZDCBIJKLZH
        ABMNCMNDMNSTOEFGBCDPQR $.
    $}

    $( Subtraction from both sides of 'less than or equal to'.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    lesub1d $p |- ( ph -> ( A <_ B <-> ( A - C ) <_ ( B - C ) ) ) $=
      ( cr wcel cle wbr cmin co wb lesub1 syl3anc ) ABHICHIDHIBCJKBDLMCDLMJKNEF
      GBCDOP $.

    $( Subtraction of both sides of 'less than or equal to'.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    lesub2d $p |- ( ph -> ( A <_ B <-> ( C - B ) <_ ( C - A ) ) ) $=
      ( cr wcel cle wbr cmin co wb lesub2 syl3anc ) ABHICHIDHIBCJKDCLMDBLMJKNEF
      GBCDOP $.

    $( Subtraction from both sides of 'less than'.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    ltsub1d $p |- ( ph -> ( A < B <-> ( A - C ) < ( B - C ) ) ) $=
      ( cr wcel clt wbr cmin co wb ltsub1 syl3anc ) ABHICHIDHIBCJKBDLMCDLMJKNEF
      GBCDOP $.

    $( Subtraction of both sides of 'less than'.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    ltsub2d $p |- ( ph -> ( A < B <-> ( C - B ) < ( C - A ) ) ) $=
      ( cr wcel clt wbr cmin co wb ltsub2 syl3anc ) ABHICHIDHIBCJKDCLMDBLMJKNEF
      GBCDOP $.

    ${
      ltadd1dd.4 $e |- ( ph -> A < B ) $.
      $( Addition to both sides of 'less than'.  Theorem I.18 of [Apostol]
         p. 20.  (Contributed by Mario Carneiro, 30-May-2016.) $)
      ltadd1dd $p |- ( ph -> ( A + C ) < ( B + C ) ) $=
        ( clt wbr caddc co ltadd1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.

      $( Subtraction from both sides of 'less than'.  (Contributed by Mario
         Carneiro, 30-May-2016.) $)
      ltsub1dd $p |- ( ph -> ( A - C ) < ( B - C ) ) $=
        ( clt wbr cmin co ltsub1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.

      $( Subtraction of both sides of 'less than'.  (Contributed by Mario
         Carneiro, 30-May-2016.) $)
      ltsub2dd $p |- ( ph -> ( C - B ) < ( C - A ) ) $=
        ( clt wbr cmin co ltsub2d mpbid ) ABCIJDCKLDBKLIJHABCDEFGMN $.
    $}

    ${
      leadd1dd.4 $e |- ( ph -> A <_ B ) $.
      $( Addition to both sides of 'less than or equal to'.  (Contributed by
         Mario Carneiro, 30-May-2016.) $)
      leadd1dd $p |- ( ph -> ( A + C ) <_ ( B + C ) ) $=
        ( cle wbr caddc co leadd1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.

      $( Addition to both sides of 'less than or equal to'.  (Contributed by
         Mario Carneiro, 30-May-2016.) $)
      leadd2dd $p |- ( ph -> ( C + A ) <_ ( C + B ) ) $=
        ( cle wbr caddc co leadd2d mpbid ) ABCIJDBKLDCKLIJHABCDEFGMN $.

      $( Subtraction from both sides of 'less than or equal to'.  (Contributed
         by Mario Carneiro, 30-May-2016.) $)
      lesub1dd $p |- ( ph -> ( A - C ) <_ ( B - C ) ) $=
        ( cle wbr cmin co lesub1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.

      $( Subtraction of both sides of 'less than or equal to'.  (Contributed by
         Mario Carneiro, 30-May-2016.) $)
      lesub2dd $p |- ( ph -> ( C - B ) <_ ( C - A ) ) $=
        ( cle wbr cmin co lesub2d mpbid ) ABCIJDCKLDBKLIJHABCDEFGMN $.
    $}

    lt2addd.4 $e |- ( ph -> D e. RR ) $.
    ${
      le2addd.5 $e |- ( ph -> A <_ C ) $.
      le2addd.6 $e |- ( ph -> B <_ D ) $.
      $( Adding both side of two inequalities.  (Contributed by Mario Carneiro,
         27-May-2016.) $)
      le2addd $p |- ( ph -> ( A + B ) <_ ( C + D ) ) $=
        ( cle wbr caddc co cr wcel wa wi le2add syl22anc mp2and ) ABDLMZCELMZBC
        NODENOLMZJKABPQCPQDPQEPQUCUDRUESFGHIBCDETUAUB $.

      $( Subtracting both sides of two 'less than or equal to' relations.
         (Contributed by Mario Carneiro, 27-May-2016.) $)
      le2subd $p |- ( ph -> ( A - D ) <_ ( C - B ) ) $=
        ( cle wbr cmin co cr wcel wa wi le2sub syl22anc mp2and ) ABDLMZCELMZBEN
        ODCNOLMZJKABPQEPQDPQCPQUCUDRUESFIHGBEDCTUAUB $.
    $}

    ${
      ltleaddd.5 $e |- ( ph -> A < C ) $.
      ltleaddd.6 $e |- ( ph -> B <_ D ) $.
      $( Adding both sides of two orderings.  (Contributed by Mario Carneiro,
         27-May-2016.) $)
      ltleaddd $p |- ( ph -> ( A + B ) < ( C + D ) ) $=
        ( clt wbr cle caddc co cr wcel wa wi ltleadd syl22anc mp2and ) ABDLMZCE
        NMZBCOPDEOPLMZJKABQRCQRDQREQRUDUESUFTFGHIBCDEUAUBUC $.
    $}

    ${
      leltaddd.5 $e |- ( ph -> A <_ C ) $.
      leltaddd.6 $e |- ( ph -> B < D ) $.
      $( Adding both sides of two orderings.  (Contributed by Mario Carneiro,
         27-May-2016.) $)
      leltaddd $p |- ( ph -> ( A + B ) < ( C + D ) ) $=
        ( cle wbr clt caddc co cr wcel wa wi leltadd syl22anc mp2and ) ABDLMZCE
        NMZBCOPDEOPNMZJKABQRCQRDQREQRUDUESUFTFGHIBCDEUAUBUC $.
    $}

    ${
      lt2addd.5 $e |- ( ph -> A < C ) $.
      lt2addd.6 $e |- ( ph -> B < D ) $.
      $( Adding both side of two inequalities.  Theorem I.25 of [Apostol]
         p. 20.  (Contributed by Mario Carneiro, 27-May-2016.) $)
      lt2addd $p |- ( ph -> ( A + B ) < ( C + D ) ) $=
        ( ltled ltleaddd ) ABCDEFGHIJACEGIKLM $.

      $( Subtracting both sides of two 'less than' relations.  (Contributed by
         Mario Carneiro, 27-May-2016.) $)
      lt2subd $p |- ( ph -> ( A - D ) < ( C - B ) ) $=
        ( clt wbr cmin co cr wcel wa wi lt2sub syl22anc mp2and ) ABDLMZCELMZBEN
        ODCNOLMZJKABPQEPQDPQCPQUCUDRUESFIHGBEDCTUAUB $.
    $}
  $}

  ${
    possumd.1 $e |- ( ph -> A e. RR ) $.
    possumd.2 $e |- ( ph -> B e. RR ) $.
    $( Condition for a positive sum.  (Contributed by Scott Fenton,
       16-Dec-2017.) $)
    possumd $p |- ( ph -> ( 0 < ( A + B ) <-> -u B < A ) ) $=
      ( cneg clt wbr cmin co caddc renegcld posdifd recnd subnegd breq2d bitr2d
      cc0 ) ACFZBGHRBSIJZGHRBCKJZGHASBACELDMATUARGABCABDNACENOPQ $.
  $}

  ${
    sublt0d.1 $e |- ( ph -> A e. RR ) $.
    sublt0d.2 $e |- ( ph -> B e. RR ) $.
    $( When a subtraction gives a negative result.  (Contributed by Glauco
       Siliprandi, 11-Dec-2019.) $)
    sublt0d $p |- ( ph -> ( ( A - B ) < 0 <-> A < B ) ) $=
      ( cmin co cc0 clt wbr caddc 0red ltsubaddd recnd addlidd breq2d bitrd ) A
      BCFGHIJBHCKGZIJBCIJABCHDEALMARCBIACACENOPQ $.
  $}

  $( Addition and subtraction on one side of 'less than'.  (Contributed by AV,
     24-Nov-2018.) $)
  ltaddsublt $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                    ( B < C <-> ( ( A + B ) - C ) < A ) ) $=
    ( cr wcel w3a clt caddc co cmin wb ltadd2 3comr readdcl 3adant3 simp3 simp1
    wbr ltsubaddd bitr4d ) ADEZBDEZCDEZFZBCGRZABHIZACHIGRZUFCJIAGRUBUCUAUEUGKBC
    ALMUDUFCAUAUBUFDEUCABNOUAUBUCPUAUBUCQST $.

  $( ` 1 <_ 1 ` .  Common special case.  (Contributed by David A. Wheeler,
     16-Jul-2016.) $)
  1le1 $p |- 1 <_ 1 $=
    ( c1 1re leidi ) ABC $.

  $( A positive sum must have a positive addend.  Part of Definition 11.2.7(vi)
     of [HoTT], p.  (varies).  (Contributed by Jim Kingdon, 26-Jan-2020.) $)
  gt0add $p |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) ->
      ( 0 < A \/ 0 < B ) ) $=
    ( cr wcel cc0 caddc co clt wbr w3a wo simp3 wi 0red simp1 readdcld axltwlin
    simp2 syl3anc mpd ltaddposd orbi2d mpbird ) ACDZBCDZEABFGZHIZJZEAHIZEBHIZKU
    IAUFHIZKZUHUGULUDUEUGLUHECDUFCDUDUGULMUHNUHABUDUEUGOZUDUEUGRZPUMEUFAQSTUHUJ
    UKUIUHBAUNUMUAUBUC $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Real Apartness
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c #RR $. $( 'Apart' relation (over real numbers) $)

  $( Class of real apartness relation. $)
  creap $a class #RR $.

  ${
    $d x y $.
    $( Define real apartness.  Definition in Section 11.2.1 of [HoTT], p.
       (varies).  Although ` #RR ` is an apartness relation on the reals (see
       ~ df-ap for more discussion of apartness relations), for our purposes it
       is just a stepping stone to defining ` =//= ` which is an apartness
       relation on complex numbers.  On the reals, ` #RR ` and ` =//= ` agree
       ( ~ apreap ).  (Contributed by Jim Kingdon, 26-Jan-2020.) $)
    df-reap $a |- #RR = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\
      ( x < y \/ y < x ) ) } $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( Real apartness in terms of classes.  Beyond the development of ` =//= `
       itself, proofs should use ~ reaplt instead.  (New usage is discouraged.)
       (Contributed by Jim Kingdon, 29-Jan-2020.) $)
    reapval $p |- ( ( A e. RR /\ B e. RR ) ->
        ( A #RR B <-> ( A < B \/ B < A ) ) ) $=
      ( vx vy creap wbr cr wcel wa clt wo cv breq12 simpr simpl breq12d orbi12d
      wceq df-reap brab2ga baib ) ABEFAGHBGHIABJFZBAJFZKZCLZDLZJFZUFUEJFZKUDCDA
      BGGEUEARZUFBRZIZUGUBUHUCUEAUFBJMUKUFBUEAJUIUJNUIUJOPQCDSTUA $.
  $}

  $( Real apartness is irreflexive.  Part of Definition 11.2.7(v) of [HoTT],
     p.  (varies).  Beyond the development of ` =//= ` itself, proofs should
     use ~ apirr instead.  (Contributed by Jim Kingdon, 26-Jan-2020.) $)
  reapirr $p |- ( A e. RR -> -. A #RR A ) $=
    ( cr wcel creap wbr clt ltnr wo wb reapval anidms oridm bitrdi mtbird ) ABC
    ZAADEZAAFEZAGOPQQHZQOPRIAAJKQLMN $.

  ${
    $d x y A $.
    $( Existence of reciprocal of real number.  (Contributed by Jim Kingdon,
       29-Jan-2020.) $)
    recexre $p |- ( ( A e. RR /\ A #RR 0 ) -> E. x e. RR ( A x. x ) = 1 ) $=
      ( vy cr wcel cc0 wbr cmul co c1 wceq wrex clt wb 0re wa cneg pm5.32i recn
      cltrr creap cv reapval mpan2 lt0neg1 renegcl ltxrlt bitrd ax-precex simpr
      sylancr reximi syl sylan sylbi negnegd oveq2d eqeq1d rspcev sylbir adantl
      wo negeq rexlimddv cc mul2neg syl2an rexbidva adantr mpbid ex mpan sylbid
      jaod imp ) BDEZBFUAGZBAUBZHIZJKZADLZVPVQBFMGZFBMGZVBZWAVPFDEZVQWDNOBFUCUD
      VPWBWAWCVPWBWAVPWBPZBQZVRQZHIZJKZADLZWAWFWGCUBZHIZJKZWKCDWFVPFWGTGZPWNCDL
      ZVPWBWOVPWBFWGMGZWOBUEVPWEWGDEZWQWONOBUFZFWGUGUKUHRVPWRWOWPWSWRWOPFWLTGZW
      NPZCDLWPCWGUIXAWNCDWTWNUJULUMUNUOWLDEZWNPZWKWFXCXBWGWLQZQZHIZJKZPWKXBXGWN
      XBXFWMJXBXEWLWGHXBWLWLSUPUQURRXBXDDEXGWKWLUFWJXGAXDDVRXDKZWIXFJXHWHXEWGHV
      RXDVCUQURUSUNUTVAVDVPWKWANWBVPWJVTADVPVRDEZPWIVSJVPBVEEVRVEEWIVSKXIBSVRSB
      VRVFVGURVHVIVJVKVPWCWAVPWCPVPFBTGZPZWAVPWCXJWEVPWCXJNOFBUGVLRXKFVRTGZVTPZ
      ADLWAABUIXMVTADXLVTUJULUMUOVKVNVMVO $.
  $}

  $( Real apartness is tight.  Beyond the development of apartness itself,
     proofs should use ~ apti .  (Contributed by Jim Kingdon, 30-Jan-2020.)
     (New usage is discouraged.) $)
  reapti $p |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #RR B ) ) $=
    ( cr wcel wa wceq creap wbr wn clt wo ltnr adantr oridm breq2 breq1 orbi12d
    bitr3id notbid syl5ibcom reapval sylibrd axapti 3expia sylbid impbid ) ACDZ
    BCDZEZABFZABGHZIZUIUJABJHZBAJHZKZIZULUIAAJHZIZUJUPUGURUHALMUJUQUOUQUQUQKUJU
    OUQNUJUQUMUQUNABAJOABAJPQRSTUIUKUOABUASZUBUIULUPUJUSUGUHUPUJABUCUDUEUF $.

  ${
    $d x A $.
    $( Existence of reciprocal of positive real number.  (Contributed by Jim
       Kingdon, 6-Feb-2020.) $)
    recexgt0 $p |- ( ( A e. RR /\ 0 < A ) ->
        E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) ) $=
      ( cr wcel cc0 cltrr wbr wa cv cmul co c1 wceq clt ax-precex wb 0re ltxrlt
      wrex mpan pm5.32i anbi1d rexbiia 3imtr4i ) BCDZEBFGZHEAIZFGZBUGJKLMZHZACS
      UEEBNGZHEUGNGZUIHZACSABOUEUKUFECDZUEUKUFPQEBRTUAUMUJACUGCDZULUHUIUNUOULUH
      PQEUGRTUBUCUD $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Complex Apartness
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c =//= $. $( 'Apart' relation (over complex numbers) $)

  $( Class of complex apartness relation. $)
  cap $a class =//= $.

  ${
    $d r s t u x y $.
    $( Define complex apartness.  Definition 6.1 of [Geuvers], p. 17.

       Two numbers are considered apart if it is possible to separate them.
       One common usage is that we can divide by a number if it is apart from
       zero (see for example ~ recclap which says that a number apart from zero
       has a reciprocal).

       The defining characteristics of an apartness are irreflexivity
       ( ~ apirr ), symmetry ( ~ apsym ), and cotransitivity ( ~ apcotr ).
       Apartness implies negated equality, as seen at ~ apne , and the converse
       would also follow if we assumed excluded middle.

       In addition, apartness of complex numbers is tight, which means that two
       numbers which are not apart are equal ( ~ apti ).

       (Contributed by Jim Kingdon, 26-Jan-2020.) $)
    df-ap $a |- =//= = { <. x , y >. |
      E. r e. RR E. s e. RR E. t e. RR E. u e. RR
      ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\
      ( r #RR t \/ s #RR u ) ) } $.
  $}

  $( ` _i ` times itself is minus 1.  (Contributed by NM, 6-May-1999.)  (Proof
     shortened by Andrew Salmon, 19-Nov-2011.) $)
  ixi $p |- ( _i x. _i ) = -u 1 $=
    ( c1 cneg cc0 cmin co ci cmul df-neg caddc ax-i2m1 0cn ax-1cn ax-icn mulcli
    wceq subadd2i mpbir eqtr2i ) ABCADEZFFGEZAHSTOTAIECOJCATKLFFMMNPQR $.

  $( The imaginary unit ` _i ` is not a real number.  (Contributed by NM,
     6-May-1999.) $)
  inelr $p |- -. _i e. RR $=
    ( ci cr wcel cc0 wceq ine0 neii clt wbr wn co c1 0re 1re caddc wa anidms ex
    cmul mtoi 0lt1 ltnsymi cneg renegcli eqeltri ltadd1i ax-1cn addlidi ax-i2m1
    ax-mp ixi breq12i bitri mtbir mullt0 mulgt0 wb lttri3 mpan2 mpbir2and mto )
    ABCZADEZADFGVBVCADHIZJZDAHIZJZVBVDDAASKZHIZVILDHIZDLHIVJJUADLMNUBUJVIDLOKZV
    HLOKZHIVJDVHLMVHLUCBUKLNUDUENUFVKLVLDHLUGUHUIULUMUNZVBVDVIVBVDPVIAAUOQRTVBV
    FVIVMVBVFVIVBVFPVIAAUPQRTVBDBCVCVEVGPUQMADURUSUTVA $.

  ${
    $d A x $.
    $( A real number times the imaginary unit is real only if the number is 0.
       (Contributed by NM, 28-May-1999.)  (Revised by Mario Carneiro,
       27-May-2016.) $)
    rimul $p |- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> A = 0 ) $=
      ( vx cr wcel ci cmul co wa cc0 wceq creap wbr wn inelr cv c1 recnd ax-icn
      wrex cc recexre adantlr simplll simprl mulass mp3an1 syl2anc oveq2 eqtrdi
      mulridi ad2antll eqtrd simpllr remulcld eqeltrrd rexlimddv ex mtoi wb 0re
      reapti mpan2 adantr mpbird ) ACDZEAFGZCDZHZAIJZAIKLZMZVHVJECDZNVHVJVLVHVJ
      HZABOZFGZPJZVLBCVEVJVPBCSVGBAUAUBVMVNCDZVPHZHZVFVNFGZECVSVTEVOFGZEVSATDZV
      NTDZVTWAJZVSAVEVGVJVRUCQVSVNVMVQVPUDZQETDWBWCWDREAVNUEUFUGVPWAEJVMVQVPWAE
      PFGEVOPEFUHERUJUIUKULVSVFVNVEVGVJVRUMWEUNUOUPUQURVEVIVKUSZVGVEICDWFUTAIVA
      VBVCVD $.
  $}

  $( Decomposition of a real number into real part (itself) and imaginary part
     (zero).  (Contributed by Jim Kingdon, 30-Jan-2020.) $)
  rereim $p |- ( ( ( A e. RR /\ B e. RR ) /\
      ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B = A /\ C = 0 ) ) $=
    ( cr wcel wa ci cmul co caddc wceq cc0 simpll recnd simplr simprr eqcomd cc
    cmin ax-icn a1i simprl mulcld subaddd mpbird resubcld eqeltrrd rimul oveq2d
    syl2anc mul01i eqtrdi eqtrd subeq0d jca ) ADEZBDEZFZCDEZABGCHIZJIZKZFZFZBAK
    CLKZVDABVDABVDAUPUQVCMZNZVDBUPUQVCOZNZVDABSIZUTLVDVJUTKVAAKVDAVAURUSVBPQVDA
    BUTVGVIVDGCGREVDTUAVDCURUSVBUBZNUCUDUEZVDUTGLHILVDCLGHVDUSUTDEVEVKVDVJUTDVL
    VDABVFVHUFUGCUHUJZUIGTUKULUMUNQVMUO $.

  ${
    $d A r s t u x y $.  $d B r s t u x y $.

    $( Complex apartness and real apartness agree on the real numbers.
       (Contributed by Jim Kingdon, 31-Jan-2020.) $)
    apreap $p |- ( ( A e. RR /\ B e. RR ) -> ( A =//= B <-> A #RR B ) ) $=
      ( vr vs vt vu cr wcel wa wbr creap cv ci co caddc wceq wrex 2rexbidv cc0
      wo vx vy cap cmul eqeq1 anbi1d anbi2d df-ap brabg simplll simplrl simplrr
      wn adantr simprll rereim syl22anc simprd simpllr simprlr eqtr4d wb reapti
      syl2anc mpbid simprr ecased simpld ex rexlimdvva sylbid w3a ax-icn mul01i
      3brtr3d oveq2i simp1 recnd addridd eqtr2id simp2 olc 3ad2ant3 orcomd 0red
      jca31 simpr oveq2d eqeq2d breq2d orbi2d anbi12d oveq1d orbi1d syld breq1d
      rspcedv rexbidv 3syld 3adant3 sylibrd mpd 3expia impbid ) AGHZBGHZIZABUCJ
      ZABKJZXGXHACLZMDLZUDNZONZPZBELZMFLZUDNZONZPZIZXJXOKJZXKXPKJZTZIZFGQZEGQZD
      GQZCGQZXIUALZXMPZUBLZXRPZIZYCIZFGQEGQZDGQCGQXNYLIZYCIZFGQEGQZDGQCGQYHUAUB
      ABGGUCYIAPZYOYRCDGGYSYNYQEFGGYSYMYPYCYSYJXNYLYIAXMUEUFUFRRYKBPZYRYFCDGGYT
      YQYDEFGGYTYPXTYCYTYLXSXNYKBXRUEUGUFRRUAUBFEDCUHUIZXGYFXICDGGXGXJGHZXKGHZI
      ZIZYDXIEFGGUUEXOGHZXPGHZIZIZYDXIUUIYDIZXJXOABKUUJYAYBUUJXKXPPZYBUMZUUJXKS
      XPUUJXJAPZXKSPZUUJXEUUBUUCXNUUMUUNIUUIXEYDXEXFUUDUUHUJUNUUIUUBYDXGUUBUUCU
      UHUKUNUUIUUCYDXGUUBUUCUUHULUNZUUIXNXSYCUOAXJXKUPUQZURUUJXOBPZXPSPZUUJXFUU
      FUUGXSUUQUURIUUIXFYDXEXFUUDUUHUSUNUUEUUFUUGYDUKUUEUUFUUGYDULZUUIXNXSYCUTB
      XOXPUPUQZURVAUUJUUCUUGUUKUULVBUUOUUSXKXPVCVDVEUUIXTYCVFVGUUJUUMUUNUUPVHUU
      JUUQUURUUTVHVOVIVJVJVKXEXFXIXHXEXFXIVLZAAMSUDNZONZPZBBUVBONZPZIZXISSKJZTZ
      IZXHUVAUVDUVFUVIUVAUVCASONAUVBSAOMVMVNZVPUVAAUVAAXEXFXIVQZVRVSVTUVAUVEBSO
      NBUVBSBOUVKVPUVABUVABXEXFXIWAZVRVSVTUVAUVHXIXIXEUVHXITXFXIUVHWBWCWDWFUVAU
      VJYHXHUVAUVJUVDXSIZAXOKJZSXPKJZTZIZFGQZEGQZAAXLONZPZXSIZUVOYBTZIZFGQZEGQZ
      DGQZYHUVAUVJUVDBBXQONZPZIZXIUVPTZIZFGQZUVTUVAUWMUVJFSGUVAWEZUVAUURIZUWKUV
      GUWLUVIUWPUWJUVFUVDUWPUWIUVEBUWPXQUVBBOUWPXPSMUDUVAUURWGZWHWHWIUGUWPUVPUV
      HXIUWPXPSSKUWQWJWKWLWQUVAUVSUWNEBGUVMUVAUUQIZUVRUWMFGUWRUVNUWKUVQUWLUWRXS
      UWJUVDUWRXRUWIBUWRXOBXQOUVAUUQWGZWMWIUGUWRUVOXIUVPUWRXOBAKUWSWJWNWLWRWQWO
      UVAUWGUVTDSGUWOUVAUUNIZUWEUVREFGGUWTUWCUVNUWDUVQUWTUWBUVDXSUWTUWAUVCAUWTX
      LUVBAOUWTXKSMUDUVAUUNWGZWHWHWIUFUWTYBUVPUVOUWTXKSXPKUXAWPWKWLRWQUVAYGUWHC
      AGUVLUVAUUMIZYEUWFDEGGUXBYDUWEFGUXBXTUWCYCUWDUXBXNUWBXSUXBXMUWAAUXBXJAXLO
      UVAUUMWGZWMWIUFUXBYAUVOYBUXBXJAXOKUXCWPWNWLWRRWQWSXEXFXHYHVBXIUUAWTXAXBXC
      XD $.
  $}

  $( Real apartness in terms of less than.  Part of Definition 11.2.7(vi) of
     [HoTT], p.  (varies).  (Contributed by Jim Kingdon, 1-Feb-2020.) $)
  reaplt $p |- ( ( A e. RR /\ B e. RR ) ->
      ( A =//= B <-> ( A < B \/ B < A ) ) ) $=
    ( cr wcel wa cap wbr creap clt wo apreap reapval bitrd ) ACDBCDEABFGABHGABI
    GBAIGJABKABLM $.

  $( Real apartness in terms of less than (exclusive-or version).  (Contributed
     by Jim Kingdon, 23-Mar-2020.) $)
  reapltxor $p |- ( ( A e. RR /\ B e. RR ) ->
      ( A =//= B <-> ( A < B \/_ B < A ) ) ) $=
    ( cr wcel wa cap wbr clt wo reaplt wn ltnsym2 biantrud df-xor bitr4di bitrd
    wxo ) ACDBCDEZABFGABHGZBAHGZIZSTQZABJRUAUASTEKZEUBRUCUAABLMSTNOP $.

  $( One is apart from zero.  (Contributed by Jim Kingdon, 24-Feb-2020.) $)
  1ap0 $p |- 1 =//= 0 $=
    ( c1 cc0 cap wbr clt wo 0lt1 olci cr wcel wb 1re 0re reaplt mp2an mpbir ) A
    BCDZABEDZBAEDZFZSRGHAIJBIJQTKLMABNOP $.

  $( Multiplication of both sides of 'less than' by a positive number.  Theorem
     I.19 of [Apostol] p. 20.  (Contributed by NM, 15-May-1999.)  (Revised by
     Mario Carneiro, 27-May-2016.) $)
  ltmul1a $p |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ A < B )
      -> ( A x. C ) < ( B x. C ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cmul co simpl2 simpl1 resubcld simpl3l posdifd
    cmin recnd remulcld simpr mpbid simpl3r mulgt0d subdird breqtrd mpbird ) AD
    EZBDEZCDEZFCGHZIZJZABGHZIZACKLZBCKLZGHFUQUPRLZGHUOFBARLZCKLURGUOUSCUOBAUHUI
    ULUNMZUHUIULUNNZOUJUKUHUIUNPZUOUNFUSGHUMUNUAUOABVAUTQUBUJUKUHUIUNUCUDUOBACU
    OBUTSUOAVASUOCVBSUEUFUOUPUQUOACVAVBTUOBCUTVBTQUG $.

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Multiplication of both sides of 'less than' by a positive number.
       Theorem I.19 of [Apostol] p. 20.  Part of Definition 11.2.7(vi) of
       [HoTT], p.  (varies).  (Contributed by NM, 13-Feb-2005.)  (Revised by
       Mario Carneiro, 27-May-2016.) $)
    ltmul1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
                 -> ( A < B <-> ( A x. C ) < ( B x. C ) ) ) $=
      ( vx cr wcel cc0 clt wbr wa cmul co ltmul1a ex c1 cc recnd adantr 3brtr3d
      w3a cv wceq wrex recexgt0 3ad2ant3 simpl1 simpl3l remulcld simpl2 simprrl
      wi simprl jca 3jca sylan mulassd simprrr oveq2d mulridd rexlimddv impbid
      ) AEFZBEFZCEFZGCHIZJZTZABHIZACKLZBCKLZHIZVGVHVKABCMNVGGDUAZHIZCVLKLZOUBZJ
      ZVKVHUKDEVFVBVPDEUCVCDCUDUEVGVLEFZVPJZJZVKVHVSVKJZAOKLZBOKLZABHVTAVNKLZBV
      NKLZWAWBHVTVIVLKLZVJVLKLZWCWDHVSVIEFZVJEFZVQVMJZTVKWEWFHIVSWGWHWIVSACVBVC
      VFVRUFZVDVEVBVCVRUGZUHVSBCVBVCVFVRUIZWKUHVSVQVMVGVQVPULZVGVQVMVOUJUMUNVIV
      JVLMUOVTACVLVSAPFVKVSAWJQRZVSCPFVKVSCWKQRZVSVLPFVKVSVLWMQRZUPVTBCVLVSBPFV
      KVSBWLQRZWOWPUPSVTVNOAKVSVOVKVGVQVMVOUQRZURVTVNOBKWRURSVTAWNUSVTBWQUSSNUT
      VA $.
  $}

  $( Multiplication of both sides of 'less than or equal to' by a positive
     number.  (Contributed by NM, 21-Feb-2005.) $)
  lemul1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
                -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a wn cmul co cle wb ltmul1 notbid lenlt remulcld
    3com12 3adant3 simp1 simp3l simp2 lenltd 3bitr4d ) ADEZBDEZCDEZFCGHZIZJZBAG
    HZKZBCLMZACLMZGHZKZABNHZUPUONHUHUGUKUNUROUHUGUKJUMUQBACPQTUGUHUSUNOUKABRUAU
    LUPUOULACUGUHUKUBUGUHUIUJUCZSULBCUGUHUKUDUTSUEUF $.

  $( Lemma for ~ reapmul1 .  (Contributed by Jim Kingdon, 8-Feb-2020.) $)
  reapmul1lem $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
      ( A =//= B <-> ( A x. C ) =//= ( B x. C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a wo cmul co cap ltmul1 wb 3com12 orbi12d reaplt
    remulcld 3adant3 simp1 simp3l simp2 syl2anc 3bitr4d ) ADEZBDEZCDEZFCGHZIZJZ
    ABGHZBAGHZKZACLMZBCLMZGHZUQUPGHZKZABNHZUPUQNHZULUMURUNUSABCOUHUGUKUNUSPBACO
    QRUGUHVAUOPUKABSUAULUPDEUQDEVBUTPULACUGUHUKUBUGUHUIUJUCZTULBCUGUHUKUDVCTUPU
    QSUEUF $.

  $( Multiplication of both sides of real apartness by a real number apart from
     zero.  Special case of ~ apmul1 .  (Contributed by Jim Kingdon,
     8-Feb-2020.) $)
  reapmul1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C =//= 0 ) ) ->
      ( A =//= B <-> ( A x. C ) =//= ( B x. C ) ) ) $=
    ( cr wcel cc0 cap wbr wa cmul co wb clt reaplt cneg recnd mulneg2d renegcld
    wo reapmul1lem 0re mpan2 pm5.32i simp1 simp3l simp2 breq12d simp3r lt0neg1d
    mpbid syl112anc remulcld ltnegd orbi12d syl2anc orcom 3bitr4d 3expa anassrs
    w3a bitrdi jaodan anasss sylan2b 3impa ) ADEZBDEZCDEZCFGHZIZABGHZACJKZBCJKZ
    GHZLZVJVFVGIZVHCFMHZFCMHZSZIVOVHVIVSVHFDEVIVSLUACFNUBUCVPVHVSVOVPVHIVQVOVRV
    PVHVQVOVFVGVHVQIZVOVFVGVTUTZACOZJKZBWBJKZGHZVLOZVMOZGHZVKVNWAWCWFWDWGGWAACW
    AAVFVGVTUDZPWACVFVGVHVQUEZPZQWABCWABVFVGVTUFZPWKQUGWAVFVGWBDEFWBMHZVKWELWIW
    LWACWJRWAVQWMVFVGVHVQUHWACWJUIUJABWBTUKWAVLVMMHZVMVLMHZSZWGWFMHZWFWGMHZSZVN
    WHWAWNWQWOWRWAVLVMWAACWIWJULZWABCWLWJULZUMWAVMVLXAWTUMUNWAVLDEVMDEVNWPLWTXA
    VLVMNUOWAWHWRWQSZWSWAWFDEWGDEWHXBLWAVLWTRWAVMXARWFWGNUOWRWQUPVAUQUQURUSVPVH
    VRVOVFVGVHVRIVOABCTURUSVBVCVDVE $.

  $( Real addition respects apartness.  (Contributed by Jim Kingdon,
     13-Feb-2020.) $)
  reapadd1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
      ( A =//= B <-> ( A + C ) =//= ( B + C ) ) ) $=
    ( cr wcel w3a clt wbr wo caddc co cap ltadd1 wb 3com12 orbi12d reaplt simp1
    3adant3 readdcld simp3 simp2 syl2anc 3bitr4d ) ADEZBDEZCDEZFZABGHZBAGHZIZAC
    JKZBCJKZGHZUMULGHZIZABLHZULUMLHZUHUIUNUJUOABCMUFUEUGUJUONBACMOPUEUFUQUKNUGA
    BQSUHULDEUMDEURUPNUHACUEUFUGRUEUFUGUAZTUHBCUEUFUGUBUSTULUMQUCUD $.

  $( Real negation respects apartness.  (Contributed by Jim Kingdon,
     13-Feb-2020.) $)
  reapneg $p |- ( ( A e. RR /\ B e. RR )
      -> ( A =//= B <-> -u A =//= -u B ) ) $=
    ( cr wcel wa cap wbr cneg clt wo reaplt ltneg wb ancoms orbi12d bitrd orcom
    bitrdi simpl renegcld simpr syl2anc bitr4d ) ACDZBCDZEZABFGZAHZBHZIGZUIUHIG
    ZJZUHUIFGZUFUGUKUJJZULUFUGABIGZBAIGZJUNABKUFUOUKUPUJABLUEUDUPUJMBALNOPUKUJQ
    RUFUHCDUICDUMULMUFAUDUESTUFBUDUEUATUHUIKUBUC $.

  $( Real apartness is cotransitive.  Part of Definition 11.2.7(v) of [HoTT],
     p.  (varies).  (Contributed by Jim Kingdon, 16-Feb-2020.) $)
  reapcotr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
      ( A =//= B -> ( A =//= C \/ B =//= C ) ) ) $=
    ( cr w3a cap wbr clt wo wb reaplt 3adant3 axltwlin wi 3com12 orim12d sylbid
    wcel orcom orbi2i or42 bitri imbitrdi 3adant2 3adant1 orbi12d sylibrd ) ADR
    ZBDRZCDRZEZABFGZACHGZCAHGZIZBCHGZCBHGZIZIZACFGZBCFGZIUKULUMUQIZUPUNIZIZUSUK
    ULABHGZBAHGZIZVDUHUIULVGJUJABKLUKVEVBVFVCABCMUIUHUJVFVCNBACMOPQVDVBUNUPIZIU
    SVCVHVBUPUNSTUMUQUNUPUAUBUCUKUTUOVAURUHUJUTUOJUIACKUDUIUJVAURJUHBCKUEUFUG
    $.

  $( Left extensionality for multiplication.  (Contributed by Jim Kingdon,
     19-Feb-2020.) $)
  remulext1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> (
      ( A x. C ) =//= ( B x. C ) -> A =//= B ) ) $=
    ( cr wcel w3a cmul co cap wbr clt wo wb remulcld reaplt cltrr ax-pre-mulext
    syl2anc ltxrlt 3imtr4d simp1 simp3 simp2 orbi12d bitrd wi 3com12 orcom jaod
    bitrdi sylbid ) ADEZBDEZCDEZFZACGHZBCGHZIJZUPUQKJZUQUPKJZLZABIJZUOUPDEZUQDE
    ZURVAMUOACULUMUNUAZULUMUNUBZNZUOBCULUMUNUCZVFNZUPUQORUOUSVBUTUOUPUQPJZABPJZ
    BAPJZLZUSVBABCQUOVCVDUSVJMVGVIUPUQSRUOVBABKJZBAKJZLZVMUOULUMVBVPMVEVHABORUO
    VNVKVOVLUOULUMVNVKMVEVHABSRUOUMULVOVLMVHVEBASRUDUEZTUOUQUPPJZVLVKLZUTVBUMUL
    UNVRVSUFBACQUGUOVDVCUTVRMVIVGUQUPSRUOVBVMVSVQVKVLUHUJTUIUK $.

  $( Right extensionality for real multiplication.  (Contributed by Jim
     Kingdon, 22-Feb-2020.) $)
  remulext2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> (
      ( C x. A ) =//= ( C x. B ) -> A =//= B ) ) $=
    ( cr wcel w3a cmul co cap simp1 recnd simp3 mulcomd simp2 breq12d remulext1
    wbr sylbird ) ADEZBDEZCDEZFZCAGHZCBGHZIQACGHZBCGHZIQABIQUBUEUCUFUDIUBACUBAS
    TUAJKUBCSTUALKZMUBBCUBBSTUANKUGMOABCPR $.

  $( The square of a real number apart from zero is positive.  (Contributed by
     Jim Kingdon, 7-Feb-2020.) $)
  apsqgt0 $p |- ( ( A e. RR /\ A =//= 0 ) -> 0 < ( A x. A ) ) $=
    ( cr wcel cc0 cap wbr wa clt wo cmul co wb 0re reaplt pm5.32i mullt0 anidms
    mpan2 mulgt0 jaodan sylbi ) ABCZADEFZGUBADHFZDAHFZIZGDAAJKHFZUBUCUFUBDBCUCU
    FLMADNROUBUDUGUEUBUDGUGAAPQUBUEGUGAASQTUA $.

  $( The representation of complex numbers in terms of real and imaginary parts
     is unique.  Proposition 10-1.3 of [Gleason] p. 130.  (Contributed by NM,
     9-May-1999.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  cru $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
      ( ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) <-> ( A = C /\ B = D ) ) ) $=
    ( cr wcel wa ci cmul co caddc wceq simplrl simplll cmin cc0 mulcld resubcld
    recnd subeq0d simpr cc ax-icn a1i simpllr simplrr addsubeq4d subdid eqeltrd
    mpbid eqtr4d rimul syl2anc oveq2d oveq1d subidd 3eqtrd eqcomd jca ex oveq12
    oveq2 sylan2 impbid1 ) AEFZBEFZGZCEFZDEFZGZGZAHBIJZKJCHDIJZKJLZACLZBDLZGZVK
    VNVQVKVNGZVOVPVRCAVRCAVRCVGVHVIVNMZSZVRAVEVFVJVNNZSZVRCAOJZVLVMOJZVMVMOJPVR
    VNWCWDLVKVNUAVRAVLCVMWBVRHBHUBFVRUCUDZVRBVEVFVJVNUEZSZQVTVRHDWEVRDVGVHVIVNU
    FZSZQZUGUJZVRVLVMVMOVRBDHIVRBDWGWIVRBDOJZEFHWLIJZEFWLPLVRBDWFWHRVRWMWCEVRWM
    WDWCVRHBDWEWGWIUHWKUKVRCAVSWARUIWLULUMTZUNUOVRVMWJUPUQTURWNUSUTVPVOVLVMLVNB
    DHIVBACVLVMKVAVCVD $.

  ${
    $d A r s t u x y $.  $d B r s t u x y $.  $d C r s t u x y $.
    $d D r s t u x y $.
    $( Complex apartness in terms of real and imaginary parts.  (Contributed by
       Jim Kingdon, 12-Feb-2020.) $)
    apreim $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
        ( ( A + ( _i x. B ) ) =//= ( C + ( _i x. D ) )
        <-> ( A =//= C \/ B =//= D ) ) ) $=
      ( vr vs vt vu cr wcel wa ci cmul co caddc wbr wo wceq creap wrex vx vy cv
      cc wb simpll recnd ax-icn simplr mulcld addcld simprl simprr eqeq1 anbi1d
      cap 2rexbidv anbi2d df-ap brabg syl2anc ad3antrrr orbi12d simprll simpllr
      a1i apreap syl21anc mpbid simpld simprlr breq12d simprd mpbird rexlimdvva
      cru bitrd ex sylbid ad2ant2r ad2ant2l pm5.32i eqid pm3.2i biantrur eqeq2d
      oveq1 breq2 orbi1d anbi12d oveq2d orbi2d rspc2ev syl3an3b syl3an3 anassrs
      oveq2 3expa breq1 adantr sylbi impbid ) AIJZBIJZKZCIJZDIJZKZKZALBMNZONZCL
      DMNZONZUPPZACUPPZBDUPPZQZXIXNXKEUCZLFUCZMNZONZRZXMGUCZLHUCZMNZONZRZKZXRYC
      SPZXSYDSPZQZKZHITGITZFITEITZXQXIXKUDJXMUDJXNYNUEZXIAXJXIAXCXDXHUFZUGXILBL
      UDJXIUHVFZXIBXCXDXHUIZUGUJUKXICXLXICXEXFXGULZUGXILDYQXIDXEXFXGUMZUGUJUKUA
      UCZYARZUBUCZYFRZKZYKKZHITGITZFITEITYBUUDKZYKKZHITGITZFITEITYNUAUBXKXMUDUD
      UPUUAXKRZUUGUUJEFIIUUKUUFUUIGHIIUUKUUEUUHYKUUKUUBYBUUDUUAXKYAUNUOUOUQUQUU
      CXMRZUUJYMEFIIUULUUIYLGHIIUULUUHYHYKUULUUDYGYBUUCXMYFUNURUOUQUQUAUBHGFEUS
      UTVAZXIYMXQEFIIXIXRIJXSIJKZKZYLXQGHIIUUOYCIJYDIJKZKZYLXQUUQYLKZXQYKUUQYHY
      KUMUURXQACSPZBDSPZQZYKUURXOUUSXPUUTUURXCXFXOUUSUEZXIXCUUNUUPYLYPVBZXIXFUU
      NUUPYLYSVBZACVGZVAUURXDXGXPUUTUEZXIXDUUNUUPYLYRVBZXIXGUUNUUPYLYTVBZBDVGZV
      AVCUURUUSYIUUTYJUURAXRCYCSUURAXRRZBXSRZUURYBUVJUVKKZUUQYBYGYKVDUURXCXDUUN
      YBUVLUEUVCUVGXIUUNUUPYLVEABXRXSVPVHVIZVJUURCYCRZDYDRZUURYGUVNUVOKZUUQYBYG
      YKVKUURXFXGUUPYGUVPUEUVDUVHUUOUUPYLUICDYCYDVPVHVIZVJVLUURBXSDYDSUURUVJUVK
      UVMVMUURUVNUVOUVQVMVLVCVQVNVRVOVOVSXIXQXNXIXQKXIUVAKZXNXIXQUVAXIXOUUSXPUU
      TXCXFUVBXDXGUVEVTXDXGUVFXCXFUVIWAVCWBUVRXNYNXEXHUVAYNXCXDXHUVAKZYNUVSXCXD
      XKXKRZYGKZAYCSPZBYDSPZQZKZHITGITZYNXFXGUVAUWFUVAXFXGUVTXMXMRZKZUVAKZUWFUW
      HUVAUVTUWGXKWCXMWCWDWEUWEUWIUVTXMCYEONZRZKZUUSUWCQZKGHCDIIYCCRZUWAUWLUWDU
      WMUWNYGUWKUVTUWNYFUWJXMYCCYEOWGWFURUWNUWBUUSUWCYCCASWHWIWJYDDRZUWLUWHUWMU
      VAUWOUWKUWGUVTUWOUWJXMXMUWOYEXLCOYDDLMWQWKWFURUWOUWCUUTUUSYDDBSWHWLWJWMWN
      WRYMUWFXKAXTONZRZYGKZUWBYJQZKZHITGITEFABIIXRARZYLUWTGHIIUXAYHUWRYKUWSUXAY
      BUWQYGUXAYAUWPXKXRAXTOWGWFUOUXAYIUWBYJXRAYCSWSWIWJUQXSBRZUWTUWEGHIIUXBUWR
      UWAUWSUWDUXBUWQUVTYGUXBUWPXKXKUXBXTXJAOXSBLMWQWKWFUOUXBYJUWCUWBXSBYDSWSWL
      WJUQWMWOWRWPXIYOUVAUUMWTVNXAVRXB $.
  $}

  $( Complex multiplication in terms of real and imaginary parts.  (Contributed
     by Jim Kingdon, 23-Feb-2020.) $)
  mulreim $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
      ( ( A + ( _i x. B ) ) x. ( C + ( _i x. D ) ) ) =
      ( ( ( A x. C ) + -u ( B x. D ) ) +
        ( _i x. ( ( C x. B ) + ( D x. A ) ) ) ) ) $=
    ( cr wcel wa ci cmul co caddc cneg simpll recnd mulcld mulcomd oveq2d eqtrd
    mul12d oveq12d cc ax-icn a1i simplr simprl simprr muladdd ixi oveq1i eqtrdi
    c1 mul4d mulm1d negeqd 3eqtrd addcomd adddid 3eqtr4d ) AEFZBEFZGZCEFZDEFZGZ
    GZAHBIJZKJCHDIJZKJIJACIJZVGVFIJZKJZAVGIJZCVFIJZKJZKJVHBDIJZLZKJZHCBIJZDAIJZ
    KJIJZKJVEAVFCVGVEAUSUTVDMNZVEHBHUAFVEUBUCZVEBUSUTVDUDNZOVECVAVBVCUENZVEHDWA
    VEDVAVBVCUFNZOUGVEVJVPVMVSKVEVIVOVHKVEVIUKLZDBIJZIJZWFLVOVEVIHHIJZWFIJWGVEH
    DHBWAWDWAWBULWHWEWFIUHUIUJVEWFVEDBWDWBOUMVEWFVNVEDBWDWBPUNUOQVEHVRIJZHVQIJZ
    KJWJWIKJVMVSVEWIWJVEHVRWAVEDAWDVTOZOVEHVQWAVECBWCWBOZOUPVEVKWIVLWJKVEVKHADI
    JZIJWIVEAHDVTWAWDSVEWMVRHIVEADVTWDPQRVECHBWCWAWBSTVEHVQVRWAWLWKUQURTR $.

  ${
    $d A x y $.
    $( Apartness is irreflexive.  (Contributed by Jim Kingdon, 16-Feb-2020.) $)
    apirr $p |- ( A e. CC -> -. A =//= A ) $=
      ( vx vy cc wcel cv ci co cr wrex cap wn wa creap reapirr wb apreap anidms
      wbr mtbird cmul caddc wceq cnre wo anim12i sylibr apreim ad2antlr breq12d
      ioran id notbid adantl mpbird ex rexlimdvva mpd ) ADEZABFZGCFZUAHUBHZUCZC
      IJBIJAAKSZLZBCAUDUSVCVEBCIIUSUTIEZVAIEZMZMZVCVEVIVCMVEVBVBKSZLZVHVKUSVCVH
      VJUTUTKSZVAVAKSZUEZVHVLLZVMLZMVNLVFVOVGVPVFVLUTUTNSZUTOVFVLVQPUTUTQRTVGVM
      VAVANSZVAOVGVMVRPVAVAQRTUFVLVMUKUGVHVJVNPUTVAUTVAUHRTUIVCVEVKPVIVCVDVJVCA
      VBAVBKVCULZVSUJUMUNUOUPUQUR $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.
    $( Apartness is symmetric.  This theorem for real numbers is part of
       Definition 11.2.7(v) of [HoTT], p.  (varies).  (Contributed by Jim
       Kingdon, 16-Feb-2020.) $)
    apsym $p |- ( ( A e. CC /\ B e. CC ) -> ( A =//= B <-> B =//= A ) ) $=
      ( vz vw vx vy cc wcel wa cv co cr wrex cap wbr wb wo clt reaplt syl2anc
      ci cmul caddc wceq adantl ad3antrrr simplrl ad2antrr orcom bitr4di bitr4d
      cnre simplrr orbi12d syl22anc 3bitr4d simpr simpllr breq12d ex rexlimdvva
      apreim mpd ) AGHZBGHZIZBCJZUADJZUBKUCKZUDZDLMCLMZABNOZBANOZPZVEVKVDCDBULU
      EVFVJVNCDLLVFVGLHZVHLHZIZIZVJVNVRVJIZAEJZUAFJZUBKUCKZUDZFLMELMZVNVDWDVEVQ
      VJEFAULUFVSWCVNEFLLVSVTLHZWALHZIZIZWCVNWHWCIZWBVINOZVIWBNOZVLVMWIVTVGNOZW
      AVHNOZQZVGVTNOZVHWANOZQZWJWKWIWLWOWMWPWIWLVTVGROZVGVTROZQZWOWIWEVOWLWTPVS
      WEWFWCUGZVSVOWGWCVFVOVPVJUGUHZVTVGSTWIWOWSWRQZWTWIVOWEWOXCPXBXAVGVTSTWRWS
      UIUJUKWIWMWAVHROZVHWAROZQZWPWIWFVPWMXFPVSWEWFWCUMZVSVPWGWCVFVOVPVJUMUHZWA
      VHSTWIWPXEXDQZXFWIVPWFWPXIPXHXGVHWASTXDXEUIUJUKUNWIWEWFVOVPWJWNPXAXGXBXHV
      TWAVGVHVBUOWIVOVPWEWFWKWQPXBXHXAXGVGVHVTWAVBUOUPWIAWBBVINWHWCUQZVRVJWGWCU
      RZUSWIBVIAWBNXKXJUSUPUTVAVCUTVAVC $.
  $}

  ${
    $d A u v w x y z $.  $d B u v w x y z $.  $d C u v w x y z $.
    $( Apartness is cotransitive.  (Contributed by Jim Kingdon,
       16-Feb-2020.) $)
    apcotr $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
        ( A =//= B -> ( A =//= C \/ B =//= C ) ) ) $=
      ( vu vv vz vw vx vy cc wcel cv co cr wrex cap wbr wo wa ad3antrrr ci cmul
      w3a caddc wceq wi cnre 3ad2ant3 3ad2ant2 ad2antrr 3ad2ant1 adantr simpllr
      simpr breq12d simplrl simplrr simprl simprr apreim syl22anc bitrd syl3anc
      wb reapcotr orim12d sylbid or4 imbitrdi simplr orbi12d sylibrd rexlimdvva
      ex mpd ) AJKZBJKZCJKZUCZCDLZUAELZUBMUDMZUEZENODNOZABPQZACPQZBCPQZRZUFZVRV
      PWDVQDECUGUHVSWCWIDENNVSVTNKZWANKZSZSZWCWIWMWCSZBFLZUAGLZUBMUDMZUEZGNOFNO
      ZWIVSWSWLWCVQVPWSVRFGBUGUIUJWNWRWIFGNNWNWONKZWPNKZSZSZWRWIXCWRSZAHLZUAILZ
      UBMUDMZUEZINOHNOZWIWMXIWCXBWRVSXIWLVPVQXIVRHIAUGUKULTXDXHWIHINNXDXENKZXFN
      KZSZSZXHWIXMXHSZWEXEVTPQZXFWAPQZRZWOVTPQZWPWAPQZRZRZWHXNWEXOXRRZXPXSRZRZY
      AXNWEXEWOPQZXFWPPQZRZYDXNWEXGWQPQZYGXNAXGBWQPXMXHUNZXCWRXLXHUMZUOXNXJXKWT
      XAYHYGVDXDXJXKXHUPZXDXJXKXHUQZXCWTWRXLXHWNWTXAURTZXCXAWRXLXHWNWTXAUSTZXEX
      FWOWPUTVAVBXNYEYBYFYCXNXJWTWJYEYBUFYKYMXCWJWRXLXHWMWJWCXBVSWJWKURUJTZXEWO
      VTVEVCXNXKXAWKYFYCUFYLYNXCWKWRXLXHWMWKWCXBVSWJWKUSUJTZXFWPWAVEVCVFVGXOXRX
      PXSVHVIXNWFXQWGXTXNWFXGWBPQZXQXNAXGCWBPYIXCWCWRXLXHWMWCXBVJTZUOXNXJXKWJWK
      YQXQVDYKYLYOYPXEXFVTWAUTVAVBXNWGWQWBPQZXTXNBWQCWBPYJYRUOXNWTXAWJWKYSXTVDY
      MYNYOYPWOWPVTWAUTVAVBVKVLVNVMVOVNVMVOVNVMVO $.
  $}

  ${
    $d A u v w x y z $.  $d B u v w x y z $.  $d C u v w x y z $.
    $( Addition respects apartness.  Analogue of ~ addcan for apartness.
       (Contributed by Jim Kingdon, 13-Feb-2020.) $)
    apadd1 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
        ( A =//= B <-> ( A + C ) =//= ( B + C ) ) ) $=
      ( vu vv wcel cv ci cmul co caddc cr wrex cap wbr wb wa ad2antrr ad3antrrr
      recnd vz vw vx vy cc wceq cnre 3ad2ant3 3ad2ant2 3ad2ant1 simplrl simplrr
      w3a wo simprl simprr apreim syl22anc simpr simpllr readdcld ax-icn mulcld
      breq12d a1i simplr oveq12d adddid oveq2d 3eqtr4d reapadd1 syl3anc orbi12d
      add4d 3bitr4d ex rexlimdvva mpd ) AUEFZBUEFZCUEFZUMZCDGZHEGZIJZKJZUFZELMD
      LMZABNOZACKJZBCKJZNOZPZWAVSWHVTDECUGUHWBWGWMDELLWBWCLFZWDLFZQZQZWGWMWQWGQ
      ZBUAGZHUBGZIJZKJZUFZUBLMUALMZWMWBXDWPWGVTVSXDWAUAUBBUGUIRWRXCWMUAUBLLWRWS
      LFZWTLFZQZQZXCWMXHXCQZAUCGZHUDGZIJZKJZUFZUDLMUCLMZWMWRXOXGXCWBXOWPWGVSVTX
      OWAUCUDAUGUJRRXIXNWMUCUDLLXIXJLFZXKLFZQZQZXNWMXSXNQZXMXBNOZXJWSNOZXKWTNOZ
      UNZWIWLXTXPXQXEXFYAYDPXIXPXQXNUKZXIXPXQXNULZXHXEXCXRXNWRXEXFUOSZXHXFXCXRX
      NWRXEXFUPSZXJXKWSWTUQURXTAXMBXBNXSXNUSZXHXCXRXNUTZVDXTXJWCKJZHXKWDKJZIJZK
      JZWSWCKJZHWTWDKJZIJZKJZNOZYKYONOZYLYPNOZUNZWLYDXTYKLFYLLFYOLFYPLFYSUUBPXT
      XJWCYEXHWNXCXRXNWQWNWGXGWBWNWOUORSZVAXTXKWDYFXHWOXCXRXNWQWOWGXGWBWNWOUPRS
      ZVAXTWSWCYGUUCVAXTWTWDYHUUDVAYKYLYOYPUQURXTWJYNWKYRNXTXMWFKJYKXLWEKJZKJWJ
      YNXTXJXLWCWEXTXJYETXTHXKHUEFXTVBVEZXTXKYFTZVCXTWCUUCTZXTHWDUUFXTWDUUDTZVC
      ZVNXTAXMCWFKYIXHWGXCXRXNWQWGXGVFSZVGXTYMUUEYKKXTHXKWDUUFUUGUUIVHVIVJXTXBW
      FKJYOXAWEKJZKJWKYRXTWSXAWCWEXTWSYGTXTHWTUUFXTWTYHTZVCUUHUUJVNXTBXBCWFKYJU
      UKVGXTYQUULYOKXTHWTWDUUFUUMUUIVHVIVJVDXTYBYTYCUUAXTXPXEWNYBYTPYEYGUUCXJWS
      WCVKVLXTXQXFWOYCUUAPYFYHUUDXKWTWDVKVLVMVOVOVPVQVRVPVQVRVPVQVR $.
  $}

  $( Addition respects apartness.  (Contributed by Jim Kingdon,
     16-Feb-2020.) $)
  apadd2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
      ( A =//= B <-> ( C + A ) =//= ( C + B ) ) ) $=
    ( cc wcel w3a cap wbr caddc apadd1 simp1 simp3 addcomd simp2 breq12d bitrd
    co ) ADEZBDEZCDEZFZABGHACIQZBCIQZGHCAIQZCBIQZGHABCJUAUBUDUCUEGUAACRSTKRSTLZ
    MUABCRSTNUFMOP $.

  $( Strong extensionality for addition.  Given excluded middle, apartness
     would be equivalent to negated equality and this would follow readily (for
     all operations) from ~ oveq12 .  For us, it is proved a different way.
     (Contributed by Jim Kingdon, 15-Feb-2020.) $)
  addext $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> (
      ( A + B ) =//= ( C + D ) -> ( A =//= C \/ B =//= D ) ) ) $=
    ( cc wcel wa caddc co cap wbr wo simpll simplr addcld simprl simprr syl3anc
    wi wb apcotr apadd1 apadd2 apsym syl2anc bitrd orbi12d sylibrd ) AEFZBEFZGZ
    CEFZDEFZGZGZABHIZCDHIZJKZUPCBHIZJKZUQUSJKZLZACJKZBDJKZLUOUPEFUQEFZUSEFZURVB
    SUOABUIUJUNMZUIUJUNNZOUOCDUKULUMPZUKULUMQZOZUOCBVIVHOZUPUQUSUARUOVCUTVDVAUO
    UIULUJVCUTTVGVIVHACBUBRUOVDUSUQJKZVAUOUJUMULVDVMTVHVJVIBDCUCRUOVFVEVMVATVLV
    KUSUQUDUEUFUGUH $.

  ${
    $d A w x y z $.  $d B w x y z $.
    $( Negation respects apartness.  (Contributed by Jim Kingdon,
       14-Feb-2020.) $)
    apneg $p |- ( ( A e. CC /\ B e. CC )
        -> ( A =//= B <-> -u A =//= -u B ) ) $=
      ( vz vw vx vy wcel wa cv ci cmul co caddc cr wrex cap wbr cneg renegcld
      wb cc wceq cnre adantl ad3antrrr wo simpr simpllr breq12d simplrl simplrr
      simprl simprr apreim syl22anc ax-icn mulcld negdid negeqd mulneg2d oveq2d
      recnd a1i 3eqtr4d reapneg syl2anc orbi12d 3bitr4rd 3bitrd rexlimdvva mpd
      ex ) AUAGZBUAGZHZBCIZJDIZKLZMLZUBZDNOCNOZABPQZARZBRZPQZTZVNWAVMCDBUCUDVOV
      TWFCDNNVOVPNGZVQNGZHZHZVTWFWJVTHZAEIZJFIZKLZMLZUBZFNOENOZWFVMWQVNWIVTEFAU
      CUEWKWPWFEFNNWKWLNGZWMNGZHZHZWPWFXAWPHZWBWOVSPQZWLVPPQZWMVQPQZUFZWEXBAWOB
      VSPXAWPUGZWJVTWTWPUHZUIXBWRWSWGWHXCXFTWKWRWSWPUJZWKWRWSWPUKZWJWGVTWTWPVOW
      GWHULUEZWJWHVTWTWPVOWGWHUMUEZWLWMVPVQUNUOXBWLRZJWMRZKLZMLZVPRZJVQRZKLZMLZ
      PQZXMXQPQZXNXRPQZUFZWEXFXBXMNGXNNGXQNGXRNGYAYDTXBWLXISXBWMXJSXBVPXKSXBVQX
      LSXMXNXQXRUNUOXBWCXPWDXTPXBWORXMWNRZMLWCXPXBWLWNXBWLXIVBXBJWMJUAGXBUPVCZX
      BWMXJVBZUQURXBAWOXGUSXBXOYEXMMXBJWMYFYGUTVAVDXBVSRXQVRRZMLWDXTXBVPVRXBVPX
      KVBXBJVQYFXBVQXLVBZUQURXBBVSXHUSXBXSYHXQMXBJVQYFYIUTVAVDUIXBXDYBXEYCXBWRW
      GXDYBTXIXKWLVPVEVFXBWSWHXEYCTXJXLWMVQVEVFVGVHVIVLVJVKVLVJVK $.
  $}

  ${
    $d A u v w x y z $.  $d B u v w x y z $.  $d C u v w x y z $.
    $( Left extensionality for complex multiplication.  (Contributed by Jim
       Kingdon, 22-Feb-2020.) $)
    mulext1 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> (
        ( A x. C ) =//= ( B x. C ) -> A =//= B ) ) $=
      ( cc wcel cv cmul co caddc cr wrex cap wbr wi wa ad3antrrr recnd syl22anc
      mulcld remulcld vu vv vz vw vx vy ci wceq cnre 3ad2ant3 3ad2ant2 ad2antrr
      w3a 3ad2ant1 adantr cneg wo simplrl simprl simprr negcld addext remulext1
      simplrr syl3anc wb apneg syl2anc sylbird orim12d remulext2 orcom imbitrdi
      syld jaod simplr oveq12d simpllr breq12d mulreim renegcld readdcld apreim
      simpr 3bitrd 3imtr4d sylibrd ex rexlimdvva mpd ) ADEZBDEZCDEZUMZCUAFZUGUB
      FZGHIHZUHZUBJKUAJKZACGHZBCGHZLMZABLMZNZWMWKWSWLUAUBCUIUJWNWRXDUAUBJJWNWOJ
      EZWPJEZOZOZWRXDXHWROZBUCFZUGUDFZGHIHZUHZUDJKUCJKZXDWNXNXGWRWLWKXNWMUCUDBU
      IUKULXIXMXDUCUDJJXIXJJEZXKJEZOZOZXMXDXRXMOZAUEFZUGUFFZGHIHZUHZUFJKUEJKZXD
      XHYDWRXQXMWNYDXGWKWLYDWMUEUFAUIUNUOPXSYCXDUEUFJJXSXTJEZYAJEZOZOZYCXDYHYCO
      ZXBYBXLLMZXCYIXTWOGHZYAWPGHZUPZIHZXJWOGHZXKWPGHZUPZIHZLMZWOYAGHZWPXTGHZIH
      ZWOXKGHZWPXJGHZIHZLMZUQZXTXJLMZYAXKLMZUQZXBYJYIYSUUJUUFYIYSYKYOLMZYMYQLMZ
      UQZUUJYIYKDEYMDEYODEYQDEYSUUMNYIXTWOYIXTXSYEYFYCURZQZYIWOXRXEXMYGYCXHXEWR
      XQWNXEXFUSULPZQZSYIYLYIYAWPYIYAXSYEYFYCVDZQZYIWPXRXFXMYGYCXHXFWRXQWNXEXFU
      TULPZQZSZVAYIXJWOYIXJXRXOXMYGYCXIXOXPUSPZQZUUQSYIYPYIXKWPYIXKXRXPXMYGYCXI
      XOXPUTPZQZUVASZVAYKYMYOYQVBRYIUUKUUHUULUUIYIYEXOXEUUKUUHNUUNUVCUUPXTXJWOV
      CVEYIUULYLYPLMZUUIYIYLDEYPDEUVHUULVFUVBUVGYLYPVGVHYIYFXPXFUVHUUINUURUVEUU
      TYAXKWPVCVEVIVJVNYIUUFUUIUUHUQZUUJYIUUFYTUUCLMZUUAUUDLMZUQZUVIYIYTDEUUADE
      UUCDEUUDDEUUFUVLNYIWOYAUUQUUSSYIWPXTUVAUUOSYIWOXKUUQUVFSYIWPXJUVAUVDSYTUU
      AUUCUUDVBRYIUVJUUIUVKUUHYIYFXPXEUVJUUINUURUVEUUPYAXKWOVKVEYIYEXOXFUVKUUHN
      UUNUVCUUTXTXJWPVKVEVJVNUUIUUHVLVMVOYIXBYBWQGHZXLWQGHZLMYNUGUUBGHIHZYRUGUU
      EGHIHZLMZUUGYIWTUVMXAUVNLYIAYBCWQGYHYCWDZXRWRXMYGYCXHWRXQVPPZVQYIBXLCWQGX
      RXMYGYCVRZUVSVQVSYIUVMUVOUVNUVPLYIYEYFXEXFUVMUVOUHUUNUURUUPUUTXTYAWOWPVTR
      YIXOXPXEXFUVNUVPUHUVCUVEUUPUUTXJXKWOWPVTRVSYIYNJEUUBJEYRJEUUEJEUVQUUGVFYI
      YKYMYIXTWOUUNUUPTYIYLYIYAWPUURUUTTWAWBYIYTUUAYIWOYAUUPUURTYIWPXTUUTUUNTWB
      YIYOYQYIXJWOUVCUUPTYIYPYIXKWPUVEUUTTWAWBYIUUCUUDYIWOXKUUPUVETYIWPXJUUTUVC
      TWBYNUUBYRUUEWCRWEYIYEYFXOXPYJUUJVFUUNUURUVCUVEXTYAXJXKWCRWFYIAYBBXLLUVRU
      VTVSWGWHWIWJWHWIWJWHWIWJ $.
  $}

  $( Right extensionality for complex multiplication.  (Contributed by Jim
     Kingdon, 22-Feb-2020.) $)
  mulext2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> (
      ( C x. A ) =//= ( C x. B ) -> A =//= B ) ) $=
    ( cc wcel w3a cmul cap wbr simp3 simp1 mulcomd simp2 breq12d mulext1 sylbid
    co ) ADEZBDEZCDEZFZCAGQZCBGQZHIACGQZBCGQZHIABHIUAUBUDUCUEHUACARSTJZRSTKLUAC
    BUFRSTMLNABCOP $.

  $( Strong extensionality for multiplication.  Given excluded middle,
     apartness would be equivalent to negated equality and this would follow
     readily (for all operations) from ~ oveq12 .  For us, it is proved a
     different way.  (Contributed by Jim Kingdon, 23-Feb-2020.) $)
  mulext $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> (
      ( A x. B ) =//= ( C x. D ) -> ( A =//= C \/ B =//= D ) ) ) $=
    ( cc wcel wa cmul cap wbr simpll simplr mulcld simprl simprr apcotr syl3anc
    co wo wi mulext1 mulext2 wb apsym syl2anc sylibd orim12d syld ) AEFZBEFZGZC
    EFZDEFZGZGZABHRZCDHRZIJZUPCBHRZIJZUQUSIJZSZACIJZBDIJZSUOUPEFUQEFUSEFURVBTUO
    ABUIUJUNKZUIUJUNLZMUOCDUKULUMNZUKULUMOZMUOCBVGVFMUPUQUSPQUOUTVCVAVDUOUIULUJ
    UTVCTVEVGVFACBUAQUOVADBIJZVDUOUMUJULVAVITVHVFVGDBCUBQUOUMUJVIVDUCVHVFDBUDUE
    UFUGUH $.

  $( A product apart from zero.  Lemma 2.13 of [Geuvers], p. 6.  (Contributed
     by Jim Kingdon, 24-Feb-2020.) $)
  mulap0r $p |- ( ( A e. CC /\ B e. CC /\ ( A x. B ) =//= 0 ) ->
      ( A =//= 0 /\ B =//= 0 ) ) $=
    ( cc wcel cmul co cc0 cap wbr wo breqtrrd wi mulext syl22anc wn apirr biorf
    mpd wb 3syl w3a simp3 simp2 mul02d simp1 0cnd orcomd mpbird mul01d jca ) AC
    DZBCDZABEFZGHIZUAZAGHIZBGHIZUOUPBBHIZUPJZUOUPURUOUMGBEFZHIZUPURJZUOUMGUTHUK
    ULUNUBZUOBUKULUNUCZUDKUOUKULGCDZULVAVBLUKULUNUEZVDUOUFZVDABGBMNRUGUOULUROUP
    USSVDBPURUPQTUHUOUQAAHIZUQJZUOUMAGEFZHIZVIUOUMGVJHVCUOAVFUIKUOUKULUKVEVKVIL
    VFVDVFVGABAGMNRUOUKVHOUQVISVFAPVHUQQTUHUJ $.

  $( A square is nonnegative.  Lemma 2.35 of [Geuvers], p. 9.  (Contributed by
     NM, 23-May-2007.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  msqge0 $p |- ( A e. RR -> 0 <_ ( A x. A ) ) $=
    ( cr wcel cc0 cmul co cle wbr clt wn remulcl anidms 0re ltnsym2 sylancl cap
    wa wo orc wb reaplt imbitrrid wi w3a cc recn mulap0r syl3an1 syl3an2 simpld
    3expia apsqgt0 ex 3syld ancld mtod lenlt sylancr mpbird ) ABCZDAAEFZGHZVADI
    HZJZUTVCVCDVAIHZQZUTVABCZDBCZVFJUTVGAAKLZMVADNOUTVCVEUTVCVADPHZADPHZVEVCVJU
    TVCVERZVCVESUTVGVHVJVLTVIMVADUAOUBUTVJVKUCUTUTVJVKUTUTVJUDVKVKUTUTAUECZVJVK
    VKQZAUFZUTVMVMVJVNVOAAUGUHUIUJUKLUTVKVEAULUMUNUOUPUTVHVGVBVDTMVIDVAUQURUS
    $.

  ${
    msqge0i.1 $e |- A e. RR $.
    $( A square is nonnegative.  (Contributed by NM, 14-May-1999.)  (Proof
       shortened by Andrew Salmon, 19-Nov-2011.) $)
    msqge0i $p |- 0 <_ ( A x. A ) $=
      ( cr wcel cc0 cmul co cle wbr msqge0 ax-mp ) ACDEAAFGHIBAJK $.
  $}

  ${
    msqge0d.1 $e |- ( ph -> A e. RR ) $.
    $( A square is nonnegative.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    msqge0d $p |- ( ph -> 0 <_ ( A x. A ) ) $=
      ( cr wcel cc0 cmul co cle wbr msqge0 syl ) ABDEFBBGHIJCBKL $.
  $}

  $( The product of two nonnegative numbers is nonnegative.  (Contributed by
     NM, 8-Oct-1999.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  mulge0 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\
                ( B e. RR /\ 0 <_ B ) ) -> 0 <_ ( A x. B ) ) $=
    ( cr wcel cc0 cle wbr wa clt wn ad2ant2r 0re sylancl cap wo wb reaplt lenlt
    cc mpbird cmul co remulcl ltnsym2 imbitrrid simplll simplrl wi recn mulap0r
    orc syl3an1 syl3an2 3expia imp simpld mpan2 ad3antrrr mpbid biimpa ad2antrr
    mpan biorf syl simprd ad2antrl adantr ad2antlr mulgt0d ex syld mtod sylancr
    ancld ) ACDZEAFGZHZBCDZEBFGZHZHZEABUAUBZFGZWBEIGZJZWAWDWDEWBIGZHZWAWBCDZECD
    ZWGJVOVRWHVPVSABUCKZLWBEUDMWAWDWFWAWDWBENGZWFWDWKWAWDWFOZWDWFUKWAWHWIWKWLPW
    JLWBEQMUEWAWKWFWAWKHZABVOVPVTWKUFVQVRVSWKUGWMEAIGZAEIGZWNOZWMAENGZWPWMWQBEN
    GZWAWKWQWRHZVOVRWKWSUHVPVSVOVRWKWSVRVOBSDZWKWSBUIVOASDWTWKWSAUIABUJULUMUNKU
    OZUPVOWQWPPZVPVTWKVOWIXBLAEQUQURUSWMWOJZWNWPPVQXCVTWKVOVPXCWIVOVPXCPLEARVBU
    TVAWOWNVCVDTWMEBIGZBEIGZXDOZWMWRXFWMWQWRXAVEWAWRXFPZWKVRXGVQVSVRWIXGLBEQUQV
    FVGUSWMXEJZXDXFPVTXHVQWKVRVSXHWIVRVSXHPLEBRVBUTVHXEXDVCVDTVIVJVKVNVLWAWIWHW
    CWEPLWJEWBRVMT $.

  ${
    mulge0i.1 $e |- A e. RR $.
    mulge0i.2 $e |- B e. RR $.
    $( The product of two nonnegative numbers is nonnegative.  (Contributed by
       NM, 30-Jul-1999.) $)
    mulge0i $p |- ( ( 0 <_ A /\ 0 <_ B ) -> 0 <_ ( A x. B ) ) $=
      ( cr wcel cc0 cle wbr wa cmul co mulge0 an4s mpanl12 ) AEFZBEFZGAHIZGBHIZ
      JGABKLHIZCDPRQSTABMNO $.
  $}

  ${
    mulge0d.1 $e |- ( ph -> A e. RR ) $.
    mulge0d.2 $e |- ( ph -> B e. RR ) $.
    mulge0d.3 $e |- ( ph -> 0 <_ A ) $.
    mulge0d.4 $e |- ( ph -> 0 <_ B ) $.
    $( The product of two nonnegative numbers is nonnegative.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    mulge0d $p |- ( ph -> 0 <_ ( A x. B ) ) $=
      ( cr wcel cc0 cle wbr cmul co mulge0 syl22anc ) ABHIJBKLCHIJCKLJBCMNKLDFE
      GBCOP $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.
    $( Complex apartness is tight.  (Contributed by Jim Kingdon,
       21-Feb-2020.) $)
    apti $p |- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A =//= B ) ) $=
      ( vx vy vz vw cc wcel wa cv co wceq cr wrex cap wbr wn wb syl2anc notbid
      cmul caddc cnre adantr adantl ad2antrr simpr ad3antrrr simplr cru simpllr
      ci eqeq12d apreim ioran bitrdi breq12d simpld reapti apreap bitr4d simprd
      wo creap anbi12d 3bitr4d ex rexlimdvva mpd ) AGHZBGHZIZACJZULDJZUAKUBKZLZ
      DMNCMNZABLZABOPZQZRZVJVQVKCDAUCUDVLVPWACDMMVLVMMHZVNMHZIZIZVPWAWEVPIZBEJZ
      ULFJZUAKUBKZLZFMNEMNZWAVLWKWDVPVKWKVJEFBUCUEUFWFWJWAEFMMWFWGMHZWHMHZIZIZW
      JWAWOWJIZVOWILZVMWGLZVNWHLZIZVRVTWPWDWNWQWTRWEWDVPWNWJVLWDUGUHZWFWNWJUIZV
      MVNWGWHUJSWPAVOBWIWEVPWNWJUKZWOWJUGZUMWPVOWIOPZQZVMWGOPZQZVNWHOPZQZIZVTWT
      WPWDWNXFXKRXAXBWDWNIZXFXGXIVCZQXKXLXEXMVMVNWGWHUNTXGXIUOUPSWPVSXEWPAVOBWI
      OXCXDUQTWPWRXHWSXJWPWBWLWRXHRWPWBWCXAURWPWLWMXBURWBWLIZWRVMWGVDPZQXHVMWGU
      SXNXGXOVMWGUTTVASWPWCWMWSXJRWPWBWCXAVBWPWLWMXBVBWCWMIZWSVNWHVDPZQXJVNWHUS
      XPXIXQVNWHUTTVASVEVFVFVGVHVIVGVHVI $.
  $}

  $( Apartness implies negated equality.  We cannot in general prove the
     converse (as shown at ~ neapmkv ), which is the whole point of having
     separate notations for apartness and negated equality.  (Contributed by
     Jim Kingdon, 21-Feb-2020.) $)
  apne $p |- ( ( A e. CC /\ B e. CC ) -> ( A =//= B -> A =/= B ) ) $=
    ( cc wcel wa cap wbr wceq wn apti biimpd necon2ad ) ACDBCDEZABFGZABMABHNIAB
    JKL $.

  ${
    apcon4bid.a $e |- ( ph -> A e. CC ) $.
    apcon4bid.b $e |- ( ph -> B e. CC ) $.
    apcon4bid.c $e |- ( ph -> C e. CC ) $.
    apcon4bid.d $e |- ( ph -> D e. CC ) $.
    apcon4bid.1 $e |- ( ph -> ( A =//= B <-> C =//= D ) ) $.
    $( Contrapositive law deduction for apartness.  (Contributed by Jim
       Kingdon, 31-Jul-2023.) $)
    apcon4bid $p |- ( ph -> ( A = B <-> C = D ) ) $=
      ( cap wbr wn wceq notbid cc wcel wb apti syl2anc 3bitr4d ) ABCKLZMZDEKLZM
      ZBCNZDENZAUBUDJOABPQCPQUFUCRFGBCSTADPQEPQUGUERHIDESTUA $.
  $}

  $( ` <_ ` implies 'less than' is 'apart'.  (Contributed by Jim Kingdon,
     13-Aug-2021.) $)
  leltap $p |- ( ( A e. RR /\ B e. RR /\ A <_ B ) ->
      ( A < B <-> B =//= A ) ) $=
    ( cr wcel cle wbr w3a clt wo cap wn wb simp3 simp1 simp2 lenltd mpbid biorf
    syl reaplt syl2anc bitr4d ) ACDZBCDZABEFZGZABHFZBAHFZUGIZBAJFZUFUHKZUGUILUF
    UEUKUCUDUEMUFABUCUDUENZUCUDUEOZPQUHUGRSUFUDUCUJUILUMULBATUAUB $.

  $( Positive implies apart from zero.  (Contributed by Jim Kingdon,
     27-Feb-2020.) $)
  gt0ap0 $p |- ( ( A e. RR /\ 0 < A ) -> A =//= 0 ) $=
    ( cr wcel cc0 clt wbr wa cap wo simpr olcd wb 0red reaplt syldan mpbird ) A
    BCZDAEFZGZADHFZADEFZRIZSRUAQRJKQRDBCTUBLSMADNOP $.

  ${
    gt0ap0i.1 $e |- A e. RR $.
    $( Positive means apart from zero (useful for ordering theorems involving
       division).  (Contributed by Jim Kingdon, 27-Feb-2020.) $)
    gt0ap0i $p |- ( 0 < A -> A =//= 0 ) $=
      ( cr wcel cc0 clt wbr cap gt0ap0 mpan ) ACDEAFGAEHGBAIJ $.

    ${
      gt0ap0i.2 $e |- 0 < A $.
      $( Positive implies apart from zero.  (Contributed by Jim Kingdon,
         27-Feb-2020.) $)
      gt0ap0ii $p |- A =//= 0 $=
        ( cc0 clt wbr cap gt0ap0i ax-mp ) DAEFADGFCABHI $.
    $}
  $}

  ${
    gt0ap0d.1 $e |- ( ph -> A e. RR ) $.
    gt0ap0d.2 $e |- ( ph -> 0 < A ) $.
    $( Positive implies apart from zero.  Because of the way we define
       ` =//= ` , ` A ` must be an element of ` RR ` , not just ` RR* ` .
       (Contributed by Jim Kingdon, 27-Feb-2020.) $)
    gt0ap0d $p |- ( ph -> A =//= 0 ) $=
      ( cr wcel cc0 clt wbr cap gt0ap0 syl2anc ) ABEFGBHIBGJICDBKL $.
  $}

  $( A number is apart from zero iff its negative is apart from zero.
     (Contributed by Jim Kingdon, 27-Feb-2020.) $)
  negap0 $p |- ( A e. CC -> ( A =//= 0 <-> -u A =//= 0 ) ) $=
    ( cc wcel cc0 cap wbr cneg wb 0cn apneg mpan2 neg0 breq2i bitrdi ) ABCZADEF
    ZAGZDGZEFZQDEFODBCPSHIADJKRDQELMN $.

  ${
    negap0d.1 $e |- ( ph -> A e. CC ) $.
    negap0d.2 $e |- ( ph -> A =//= 0 ) $.
    $( The negative of a number apart from zero is apart from zero.
       (Contributed by Jim Kingdon, 25-Feb-2024.) $)
    negap0d $p |- ( ph -> -u A =//= 0 ) $=
      ( cc0 cap wbr cneg cc wcel wb negap0 syl mpbid ) ABEFGZBHEFGZDABIJOPKCBLM
      N $.
  $}

  $( Less than in terms of non-strict order and apartness.  (Contributed by Jim
     Kingdon, 28-Feb-2020.) $)
  ltleap $p |- ( ( A e. RR /\ B e. RR ) ->
      ( A < B <-> ( A <_ B /\ A =//= B ) ) ) $=
    ( cr wcel wa clt wbr cle cap ltle wo orc reaplt imbitrrid jcad wn simprl wb
    adantr mpbid lenlt simprr ecased ex impbid ) ACDBCDEZABFGZABHGZABIGZEZUFUGU
    HUIABJUGUIUFUGBAFGZKZUGUKLABMZNOUFUJUGUFUJEZUGUKUNUHUKPZUFUHUIQUFUHUORUJABU
    ASTUNUIULUFUHUIUBUFUIULRUJUMSTUCUDUE $.

  $( 'Less than' implies apart.  (Contributed by Jim Kingdon, 12-Aug-2021.) $)
  ltap $p |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B =//= A ) $=
    ( cr wcel clt wbr w3a cap wo simp3 olcd simp2 simp1 reaplt syl2anc mpbird
    wb ) ACDZBCDZABEFZGZBAHFZBAEFZTIZUATUCRSTJKUASRUBUDQRSTLRSTMBANOP $.

  ${
    ltapii.a $e |- A e. RR $.
    ltapii.b $e |- B e. RR $.
    ${
      ltapii.lt $e |- A < B $.
      $( 'Greater than' implies apart.  (Contributed by Jim Kingdon,
         12-Aug-2021.) $)
      gtapii $p |- B =//= A $=
        ( cr wcel clt wbr cap ltap mp3an ) AFGBFGABHIBAJICDEABKL $.

      $( 'Less than' implies apart.  (Contributed by Jim Kingdon,
         12-Aug-2021.) $)
      ltapii $p |- A =//= B $=
        ( cap wbr gtapii cc wcel wb recni apsym mp2an mpbi ) BAFGZABFGZABCDEHBI
        JAIJPQKBDLACLBAMNO $.
    $}

    $( 'Less than' implies apart.  (Contributed by Jim Kingdon,
       12-Aug-2021.) $)
    ltapi $p |- ( A < B -> B =//= A ) $=
      ( cr wcel clt wbr cap ltap mp3an12 ) AEFBEFABGHBAIHCDABJK $.
  $}

  ${
    ltapd.a $e |- ( ph -> A e. RR ) $.
    ltapd.b $e |- ( ph -> B e. RR ) $.

    ${
      ltapd.lt $e |- ( ph -> A < B ) $.
      $( 'Greater than' implies apart.  (Contributed by Jim Kingdon,
         12-Aug-2021.) $)
      gtapd $p |- ( ph -> B =//= A ) $=
        ( cr wcel clt wbr cap ltap syl3anc ) ABGHCGHBCIJCBKJDEFBCLM $.

      $( 'Less than' implies apart.  (Contributed by Jim Kingdon,
         12-Aug-2021.) $)
      ltapd $p |- ( ph -> A =//= B ) $=
        ( cap wbr gtapd cc wcel wb recnd apsym syl2anc mpbid ) ACBGHZBCGHZABCDE
        FIACJKBJKQRLACEMABDMCBNOP $.
    $}

    ${
      leltapd.3 $e |- ( ph -> A <_ B ) $.
      $( ` <_ ` implies 'less than' is 'apart'.  (Contributed by Jim Kingdon,
         13-Aug-2021.) $)
      leltapd $p |- ( ph -> ( A < B <-> B =//= A ) ) $=
        ( cr wcel cle wbr clt cap wb leltap syl3anc ) ABGHCGHBCIJBCKJCBLJMDEFBC
        NO $.
    $}
  $}

  $( A nonnegative number is apart from zero if and only if it is positive.
     (Contributed by Jim Kingdon, 11-Aug-2021.) $)
  ap0gt0 $p |- ( ( A e. RR /\ 0 <_ A ) -> ( A =//= 0 <-> 0 < A ) ) $=
    ( cr wcel cc0 cle wbr wa cap wo wb 0red reaplt syldan wn simpr simpl lenltd
    clt mpbid biorf syl bitr4d ) ABCZDAEFZGZADHFZADRFZDARFZIZUHUCUDDBCUFUIJUEKZ
    ADLMUEUGNZUHUIJUEUDUKUCUDOUEDAUJUCUDPQSUGUHTUAUB $.

  ${
    ap0gt0d.1 $e |- ( ph -> A e. RR ) $.
    ${
      ap0gt0d.2 $e |- ( ph -> 0 <_ A ) $.
      ap0gt0d.3 $e |- ( ph -> A =//= 0 ) $.
      $( A nonzero nonnegative number is positive.  (Contributed by Jim
         Kingdon, 11-Aug-2021.) $)
      ap0gt0d $p |- ( ph -> 0 < A ) $=
        ( cc0 cap wbr clt cr wcel cle wb ap0gt0 syl2anc mpbid ) ABFGHZFBIHZEABJ
        KFBLHQRMCDBNOP $.
    $}
  $}

  ${
    $d A u v w x y z $.  $d B u v w x y z $.  $d C u v w x y z $.
    $( Subtraction respects apartness.  Analogue of ~ subcan2 for apartness.
       (Contributed by Jim Kingdon, 6-Jan-2022.) $)
    apsub1 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
        ( A =//= B <-> ( A - C ) =//= ( B - C ) ) ) $=
      ( cc wcel w3a cap wbr cneg caddc co cmin negcl apadd1 syl3an3 simp1 simp3
      wb negsubd simp2 breq12d bitrd ) ADEZBDEZCDEZFZABGHZACIZJKZBUHJKZGHZACLKZ
      BCLKZGHUEUCUDUHDEUGUKRCMABUHNOUFUIULUJUMGUFACUCUDUEPUCUDUEQZSUFBCUCUDUETU
      NSUAUB $.
  $}

  $( Two numbers being apart is equivalent to their difference being apart from
     zero.  (Contributed by Jim Kingdon, 25-Dec-2022.) $)
  subap0 $p |- ( ( A e. CC /\ B e. CC )
      -> ( ( A - B ) =//= 0 <-> A =//= B ) ) $=
    ( cc wcel wa cap wbr cneg caddc co cmin cc0 wb adantl apadd1 mpd3an3 negsub
    negcl simpr negidd breq12d bitr2d ) ACDZBCDZEZABFGZABHZIJZBUGIJZFGZABKJZLFG
    UCUDUGCDZUFUJMUDULUCBRNABUGOPUEUHUKUILFABQUEBUCUDSTUAUB $.

  ${
    subap0d.a $e |- ( ph -> A e. CC ) $.
    subap0d.b $e |- ( ph -> B e. CC ) $.
    ${
      subap0d.ap $e |- ( ph -> A =//= B ) $.
      $( Two numbers apart from each other have difference apart from zero.
         (Contributed by Jim Kingdon, 12-Aug-2021.)  (Proof shortened by BJ,
         15-Aug-2024.) $)
      subap0d $p |- ( ph -> ( A - B ) =//= 0 ) $=
        ( cmin co cc0 cap wbr cc wcel wb subap0 syl2anc mpbird ) ABCGHIJKZBCJKZ
        FABLMCLMRSNDEBCOPQ $.
    $}
  $}

  $( Equality of complex numbers is stable.  Stability here means
     ` -. -. A = B -> A = B ` as defined at ~ df-stab .  This theorem for real
     numbers is Proposition 5.2 of [BauerHanson], p. 27.  (Contributed by Jim
     Kingdon, 1-Aug-2023.)  (Proof shortened by BJ, 15-Aug-2024.) $)
  cnstab $p |- ( ( A e. CC /\ B e. CC ) -> STAB A = B ) $=
    ( cc wcel wa wceq wstab cap wbr wn stabnot apti stbid mpbiri ) ACDBCDEZABFZ
    GABHIZJZGQKOPRABLMN $.

  ${
    $d A r s t u x y $.  $d B r s t u x y $.
    $( Reverse closure for apartness.  (Contributed by Jim Kingdon,
       19-Dec-2023.) $)
    aprcl $p |- ( A =//= B -> ( A e. CC /\ B e. CC ) ) $=
      ( vr vs vt vu vx vy cap cv ci co wceq wa cr wrex cc wcel 2rexbidv cvv wbr
      cop c1st cmul caddc c2nd creap wo df-br copab eqeq1 anbi1d anbi2d elopabi
      cfv df-ap eleq2s reximi syl relopabi brrelex1i ad3antrrr brrelex2i op1stg
      sylbi simpl syl2anc simprl ad2antrr recnd ax-icn a1i simprr mulcld addcld
      eqeltrd eqeltrrd op2ndg recn adantr adantl jca ex rexlimdvva mpd ) ABIUAZ
      ABUBZUCUOZCJZKDJZUDLZUELZMZWGUFUOZEJZKFJZUDLZUELZMZNZFOPZEOPZDOPZCOPZAQRZ
      BQRZNZWFWTWIWOUGUAWJWPUGUAUHZNZFOPZEOPZDOPZCOPZXDWFWGIRXMABIUIXMWGGJZWLMZ
      HJZWRMZNZXHNZFOPEOPZDOPCOPZGHUJIYAWMXQNZXHNZFOPEOPZDOPCOPXMGHWGXNWHMZXTYD
      CDOOYEXSYCEFOOYEXRYBXHYEXOWMXQXNWHWLUKULULSSXPWNMZYDXKCDOOYFYCXIEFOOYFYBW
      TXHYFXQWSWMXPWNWRUKUMULSSUNGHFEDCUPZUQVEXLXCCOXKXBDOXJXAEOXIWTFOWTXHVFURU
      RURURUSWFXBXGCDOOWFWIORZWJORZNZNZWTXGEFOOYKWOORZWPORZNZNZWTXGYOWTNZXEXFYP
      WHAQYPATRZBTRZWHAMWFYQYJYNWTABIYAGHIYGUTZVAVBZWFYRYJYNWTABIYSVCVBZABTTVDV
      GYPWHWLQYOWMWSVHYPWIWKYPWIYKYHYNWTWFYHYIVHVIVJYPKWJKQRZYPVKVLYPWJYKYIYNWT
      WFYHYIVMVIVJVNVOVPVQYPWNBQYPYQYRWNBMYTUUAABTTVRVGYPWNWRQYOWMWSVMYOWRQRZWT
      YNUUCYKYNWOWQYLWOQRYMWOVSVTYNKWPUUBYNVKVLYMWPQRYLWPVSWAVNVOWAVTVPVQWBWCWD
      WDWE $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( The points apart from a given point are complex numbers.  (Contributed
       by Jim Kingdon, 19-Dec-2023.) $)
    apsscn $p |- { x e. A | x =//= B } C_ CC $=
      ( vy cv cap wbr crab cc wcel wa breq1 elrab aprcl simplbiim simpld ssriv
      ) DAEZCFGZABHZIDEZTJZUAIJZCIJZUBUABJUACFGZUCUDKSUEAUABRUACFLMUACNOPQ $.
  $}

  $( A number which is less than zero is apart from zero.  (Contributed by Jim
     Kingdon, 25-Feb-2024.) $)
  lt0ap0 $p |- ( ( A e. RR /\ A < 0 ) -> A =//= 0 ) $=
    ( cr wcel cc0 clt wbr wa simpl 0red simpr ltapd ) ABCZADEFZGZADLMHNILMJK $.

  ${
    lt0ap0d.a $e |- ( ph -> A e. RR ) $.
    lt0ap0d.lt $e |- ( ph -> A < 0 ) $.
    $( A real number less than zero is apart from zero.  Deduction form.
       (Contributed by Jim Kingdon, 24-Feb-2024.) $)
    lt0ap0d $p |- ( ph -> A =//= 0 ) $=
      ( cr wcel cc0 clt wbr cap lt0ap0 syl2anc ) ABEFBGHIBGJICDBKL $.
  $}

  ${
    $d p q r s t u v $.  $d x y z $.
    $( Complex apartness (as defined at ~ df-ap ) is a tight apartness (as
       defined at ~ df-tap ).  (Contributed by Jim Kingdon, 16-Feb-2025.) $)
    aptap $p |- =//= TAp CC $=
      ( vx vy vz vp vq vr vs vu vv cc cap cv wbr wral wa wceq wcel ci cr wrex
      vt wtap cxp wss wn wi wo c1st cmul co caddc c2nd creap copab eqeq1 anbi1d
      cfv 2rexbidv anbi2d elopabi df-ap eleq2s cop wrel relopabi simp-5l 1st2nd
      sylancr simprll simp-5r ax-icn a1i simp-4r mulcld eqeltrd simprlr simpllr
      recnd addcld jca elxp6 sylanbrc rexlimdva2 rexlimdva mpd ssriv apirr rgen
      simplr apsym biimpd rgen2 pm3.2i apcotr rgen3 biimprd dftap2 mpbir3an
      apti ) JKUBKJJUCZUDALZXAKMUEZAJNZXABLZKMZXDXAKMZUFZBJNAJNZOXEXACLZKMXDXIK
      MUGUFZCJNBJNAJNZXEUEZXAXDPZUFZBJNAJNZOUAKWTUALZKQZXPUHUQZDLZRELZUIUJZUKUJ
      ZPZXPULUQZFLZRGLZUIUJZUKUJZPZOZXSYEUMMXTYFUMMUGZOZGSTZFSTZESTZDSTZXPWTQZY
      PXPHLZYBPZILZYHPZOZYKOZGSTFSTZESTDSTZHIUNKUUEYCUUAOZYKOZGSTFSTZESTDSTYPHI
      XPYRXRPZUUDUUHDESSUUIUUCUUGFGSSUUIUUBUUFYKUUIYSYCUUAYRXRYBUOUPUPURURYTYDP
      ZUUHYNDESSUUJUUGYLFGSSUUJUUFYJYKUUJUUAYIYCYTYDYHUOUSUPURURUTHIGFEDVAZVBXQ
      YOYQDSXQXSSQZOZYNYQESUUMXTSQZOZYMYQFSUUOYESQZOZYLYQGSUUQYFSQZOZYLOZXPXRYD
      VCPZXRJQZYDJQZOYQUUTKVDXQUVAUUEHIKUUKVEXQUULUUNUUPUURYLVFXPKVGVHUUTUVBUVC
      UUTXRYBJUUSYCYIYKVIUUTXSYAUUTXSXQUULUUNUUPUURYLVJVRUUTRXTRJQUUTVKVLZUUTXT
      UUMUUNUUPUURYLVMVRVNVSVOUUTYDYHJUUSYCYIYKVPUUTYEYGUUTYEUUOUUPUURYLVQVRUUT
      RYFUVDUUTYFUUQUURYLWIVRVNVSVOVTXPJJWAWBWCWDWDWDWEWFXCXHXBAJXAWGWHXGABJJXA
      JQXDJQOZXEXFXAXDWJWKWLWMXKXOXJABCJJJXAXDXIWNWOXNABJJUVEXMXLXAXDWSWPWLWMAB
      CJKWQWR $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Reciprocals
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Lemma for ~ recexap .  (Contributed by Eric Schmidt, 23-May-2007.) $)
  recextlem1 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) x.
      ( A - ( _i x. B ) ) ) = ( ( A x. A ) + ( B x. B ) ) ) $=
    ( cc wcel wa ci cmul caddc cmin cneg ax-icn mulcl adantl sylan2 subdid wceq
    co eqtrd anidms oveq12d simpl subcl adddird mulcom c1 oveq1i mulm1d eqtr2id
    mpan ixi mul4 mpanl12 eqtr4d adantr negcld npncand subneg syl2an 3eqtrd ) A
    CDZBCDZEZAFBGQZHQAVCIQZGQAVDGQZVCVDGQZHQAAGQZAVCGQZIQZVHBBGQZJZIQZHQZVGVJHQ
    ZVBAVCVDUTVAUAZVAVCCDZUTFCDZVAVPKFBLUIZMZVAUTVPVDCDVRAVCUBNUCVBVEVIVFVLHVBA
    AVCVOVOVSOVBVFVCAGQZVCVCGQZIQVLVBVCAVCVSVOVSOVBVHVTVKWAIVAUTVPVHVTPVRAVCUDN
    VAVKWAPZUTVAWBVAVAEZVKFFGQZVJGQZWAWCWEUEJZVJGQVKWDWFVJGUJUFWCVJBBLZUGUHVQVQ
    WCWEWAPKKFFBBUKULRSMTUMTVBVMVGVKIQZVNVBVGVHVKUTVGCDZVAUTWIAALSZUNVAUTVPVHCD
    VRAVCLNVAVKCDZUTVAWKWCVJWGUOSMUPUTWIVJCDZWHVNPVAWJVAWLWGSVGVJUQURRUS $.

  $( Lemma for ~ recexap .  (Contributed by Jim Kingdon, 20-Feb-2020.) $)
  recexaplem2 $p |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) =//= 0 ) ->
      ( ( A x. A ) + ( B x. B ) ) =//= 0 ) $=
    ( cr wcel ci cmul co caddc cc0 cap wbr clt wo wa wb 0re cle remulcl anim12i
    anidms w3a ax-icn mul01i oveq2i eqtr2i breq2i apreim mpanr12 bitrid pm5.32i
    00id adantr apsqgt0 msqge0 an32s addgtge0 syl2anc anassrs jaodan sylbi olcd
    addgegt0 3impa simp1 remulcld simp2 readdcld reaplt sylancl mpbird ) ACDZBC
    DZAEBFGHGZIJKZUAZAAFGZBBFGZHGZIJKZVRILKZIVRLKZMZVOWAVTVKVLVNWAVKVLNZVNNWCAI
    JKZBIJKZMZNWAWCVNWFVNVMIEIFGZHGZJKZWCWFIWHVMJWHIIHGIWGIIHEUBUCUDUKUEUFWCICD
    ZWJWIWFOPPABIIUGUHUIUJWCWDWAWEWCWDNVPCDZVQCDZNZIVPLKZIVQQKZNZWAWCWMWDVKWKVL
    WLVKWKAARTVLWLBBRTSZULVKWDVLWPVKWDNWNVLWOAUMBUNSUOVPVQUPUQWCWENWMIVPQKZIVQL
    KZNZWAWCWMWEWQULVKVLWEWTVKWRVLWENWSAUNBUMSURVPVQVBUQUSUTVCVAVOVRCDWJVSWBOVO
    VPVQVOAAVKVLVNVDZXAVEVOBBVKVLVNVFZXBVEVGPVRIVHVIVJ $.

  ${
    $d x y a b A $.
    $( Existence of reciprocal of nonzero complex number.  (Contributed by Jim
       Kingdon, 20-Feb-2020.) $)
    recexap $p |- ( ( A e. CC /\ A =//= 0 ) -> E. x e. CC ( A x. x ) = 1 ) $=
      ( va vb vy cc wcel cc0 cap wbr cv cmul co c1 wceq wrex cr wi wa adantr ci
      caddc cnre recexaplem2 3expia wb remulcl anidms readdcl syl2an 0re apreap
      creap sylancl recexre sylan recn cmin ax-icn mulcl mpan subcl addcl simpr
      sylan2 mulassd recextlem1 oveq1d eqtr3d id sylan9eq eqeq1d rspcev syl2anc
      oveq2 exp31 syl5 rexlimdv mpd ex sylbid syld breq1 adantl rexbidv 3imtr4d
      oveq1 rexlimivv syl imp ) BFGZBHIJZBAKZLMZNOZAFPZWKBCKZUADKZLMZUBMZOZDQPC
      QPWLWPRZCDBUCXAXBCDQQWQQGZWRQGZSZXAXBXEXASWTHIJZWTWMLMZNOZAFPZWLWPXEXFXIR
      XAXEXFWQWQLMZWRWRLMZUBMZHIJZXIXCXDXFXMWQWRUDUEXEXMXLHUMJZXIXEXLQGZHQGXMXN
      UFXCXJQGZXKQGZXOXDXCXPWQWQUGUHXDXQWRWRUGUHXJXKUIUJZUKXLHULUNXEXNXIXEXNSXL
      EKZLMZNOZEQPZXIXEXOXNYBXREXLUOUPXEYBXIRZXNXCWQFGZWRFGZYCXDWQUQWRUQYDYESZY
      AXIEQXSQGXSFGZYFYAXIRXSUQYFYGYAXIYFYGSZYASWQWSURMZXSLMZFGZWTYJLMZNOZXIYHY
      KYAYFYIFGZYGYKYEYDWSFGZYNUAFGYEYOUSUAWRUTVAZWQWSVBVEZYIXSUTUPTYHYAYLXTNYH
      WTYILMZXSLMYLXTYHWTYIXSYFWTFGZYGYEYDYOYSYPWQWSVCVETYFYNYGYQTYFYGVDVFYHYRX
      LXSLYFYRXLOYGWQWRVGTVHVIYAVJVKXHYMAYJFWMYJOXGYLNWMYJWTLVOVLVMVNVPVQVRUJTV
      SVTWAWBTXAWLXFUFXEBWTHIWCWDXAWPXIUFXEXAWOXHAFXAWNXGNBWTWMLWGVLWEWDWFVTWHW
      IWJ $.
  $}

  ${
    $d A x $.  $d B x $.
    $( The product of two numbers apart from zero is apart from zero.  Lemma
       2.15 of [Geuvers], p. 6.  (Contributed by Jim Kingdon, 22-Feb-2020.) $)
    mulap0 $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) ) ->
        ( A x. B ) =//= 0 ) $=
      ( vx cc wcel cc0 cap wa cv cmul co c1 wceq recexap adantl simpllr simplll
      wbr wrex simplrl simprl mulassd simprr oveq2d mulridd 3eqtrd 3brtr4d 0cnd
      mul02d wi mulcld mulext1 syl3anc mpd rexlimddv ) ADEZAFGRZHZBDEZBFGRZHZHZ
      BCIZJKZLMZABJKZFGRZCDVAVECDSURCBNOVBVCDEZVEHZHZVFVCJKZFVCJKZGRZVGVJAFVKVL
      GUPUQVAVIPVJVKAVDJKALJKAVJABVCUPUQVAVIQZURUSUTVITZVBVHVEUAZUBVJVDLAJVBVHV
      EUCUDVJAVNUEUFVJVCVPUIUGVJVFDEFDEVHVMVGUJVJABVNVOUKVJUHVPVFFVCULUMUNUO $.
  $}

  $( The product of two numbers apart from zero is apart from zero.
     (Contributed by Jim Kingdon, 24-Feb-2020.) $)
  mulap0b $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A =//= 0 /\ B =//= 0 ) <->
      ( A x. B ) =//= 0 ) ) $=
    ( cc wcel wa cc0 cap wbr cmul co mulap0 an4s ex mulap0r 3expia impbid ) ACD
    ZBCDZEZAFGHZBFGHZEZABIJFGHZSUBUCQTRUAUCABKLMQRUCUBABNOP $.

  ${
    mulap0i.1 $e |- A e. CC $.
    mulap0i.2 $e |- B e. CC $.
    mulap0i.3 $e |- A =//= 0 $.
    mulap0i.4 $e |- B =//= 0 $.
    $( The product of two numbers apart from zero is apart from zero.
       (Contributed by Jim Kingdon, 23-Feb-2020.) $)
    mulap0i $p |- ( A x. B ) =//= 0 $=
      ( cc wcel cc0 cap wbr cmul co mulap0 mp4an ) AGHAIJKBGHBIJKABLMIJKCEDFABN
      O $.
  $}

  ${
    mulap0d.1 $e |- ( ph -> A e. CC ) $.
    mulap0d.2 $e |- ( ph -> B e. CC ) $.
    $( The product of two numbers apart from zero is apart from zero.  Exercise
       11.11 of [HoTT], p.  (varies).  (Contributed by Jim Kingdon,
       24-Feb-2020.) $)
    mulap0bd $p |- ( ph
        -> ( ( A =//= 0 /\ B =//= 0 ) <-> ( A x. B ) =//= 0 ) ) $=
      ( cc wcel cc0 cap wbr wa cmul co wb mulap0b syl2anc ) ABFGCFGBHIJCHIJKBCL
      MHIJNDEBCOP $.

    mulap0d.3 $e |- ( ph -> A =//= 0 ) $.
    mulap0d.4 $e |- ( ph -> B =//= 0 ) $.
    $( The product of two numbers apart from zero is apart from zero.
       (Contributed by Jim Kingdon, 23-Feb-2020.) $)
    mulap0d $p |- ( ph -> ( A x. B ) =//= 0 ) $=
      ( cc wcel cc0 cap wbr cmul co mulap0 syl22anc ) ABHIBJKLCHICJKLBCMNJKLDFE
      GBCOP $.
  $}

  ${
    mulap0bad.1 $e |- ( ph -> A e. CC ) $.
    mulap0bad.2 $e |- ( ph -> B e. CC ) $.
    mulap0bad.3 $e |- ( ph -> ( A x. B ) =//= 0 ) $.
    $( A factor of a complex number apart from zero is apart from zero.
       Partial converse of ~ mulap0d and consequence of ~ mulap0bd .
       (Contributed by Jim Kingdon, 24-Feb-2020.) $)
    mulap0bad $p |- ( ph -> A =//= 0 ) $=
      ( cc0 cap wbr wa cmul co mulap0bd mpbird simpld ) ABGHIZCGHIZAPQJBCKLGHIF
      ABCDEMNO $.

    $( A factor of a complex number apart from zero is apart from zero.
       Partial converse of ~ mulap0d and consequence of ~ mulap0bd .
       (Contributed by Jim Kingdon, 24-Feb-2020.) $)
    mulap0bbd $p |- ( ph -> B =//= 0 ) $=
      ( cc0 cap wbr wa cmul co mulap0bd mpbird simprd ) ABGHIZCGHIZAPQJBCKLGHIF
      ABCDEMNO $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ph $.
    mulcand.1 $e |- ( ph -> A e. CC ) $.
    mulcand.2 $e |- ( ph -> B e. CC ) $.
    mulcand.3 $e |- ( ph -> C e. CC ) $.
    mulcand.4 $e |- ( ph -> C =//= 0 ) $.
    $( Cancellation law for multiplication.  (Contributed by Jim Kingdon,
       21-Feb-2020.) $)
    mulcanapd $p |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) ) $=
      ( vx cmul co wceq c1 cc wcel wa oveq2 adantr oveq1d mulassd cv wi cc0 cap
      wrex recexap syl2anc simprl mulcomd simprr eqtrd mullidd 3eqtr3d imbitrid
      wbr eqeq12d rexlimddv impbid1 ) ADBJKZDCJKZLZBCLZADIUAZJKZMLZVAVBUBINADNO
      ZDUCUDUOVEINUEGHIDUFUGVAVCUSJKZVCUTJKZLAVCNOZVEPZPZVBUSUTVCJQVKVGBVHCVKVC
      DJKZBJKMBJKVGBVKVLMBJVKVLVDMVKVCDAVIVEUHZAVFVJGRZUIAVIVEUJUKZSVKVCDBVMVNA
      BNOVJERZTVKBVPULUMVKVLCJKMCJKVHCVKVLMCJVOSVKVCDCVMVNACNOVJFRZTVKCVQULUMUP
      UNUQBCDJQUR $.

    $( Cancellation law for multiplication.  (Contributed by Jim Kingdon,
       21-Feb-2020.) $)
    mulcanap2d $p |- ( ph -> ( ( A x. C ) = ( B x. C ) <-> A = B ) ) $=
      ( cmul co wceq mulcomd eqeq12d mulcanapd bitrd ) ABDIJZCDIJZKDBIJZDCIJZKB
      CKAPRQSABDEGLACDFGLMABCDEFGHNO $.
  $}

  ${
    mulcanapad.1 $e |- ( ph -> A e. CC ) $.
    mulcanapad.2 $e |- ( ph -> B e. CC ) $.
    mulcanapad.3 $e |- ( ph -> C e. CC ) $.
    mulcanapad.4 $e |- ( ph -> C =//= 0 ) $.
    ${
      mulcanapad.5 $e |- ( ph -> ( C x. A ) = ( C x. B ) ) $.
      $( Cancellation of a nonzero factor on the left in an equation.  One-way
         deduction form of ~ mulcanapd .  (Contributed by Jim Kingdon,
         21-Feb-2020.) $)
      mulcanapad $p |- ( ph -> A = B ) $=
        ( cmul co wceq mulcanapd mpbid ) ADBJKDCJKLBCLIABCDEFGHMN $.
    $}

    ${
      mulcanap2ad.5 $e |- ( ph -> ( A x. C ) = ( B x. C ) ) $.
      $( Cancellation of a nonzero factor on the right in an equation.  One-way
         deduction form of ~ mulcanap2d .  (Contributed by Jim Kingdon,
         21-Feb-2020.) $)
      mulcanap2ad $p |- ( ph -> A = B ) $=
        ( cmul co wceq mulcanap2d mpbid ) ABDJKCDJKLBCLIABCDEFGHMN $.
    $}
  $}

  $( Cancellation law for multiplication (full theorem form).  (Contributed by
     Jim Kingdon, 21-Feb-2020.) $)
  mulcanap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) ) ->
                 ( ( C x. A ) = ( C x. B ) <-> A = B ) ) $=
    ( cc wcel cc0 cap wbr wa w3a simp1 simp2 simp3l simp3r mulcanapd ) ADEZBDEZ
    CDEZCFGHZIZJABCPQTKPQTLPQRSMPQRSNO $.

  $( Cancellation law for multiplication.  (Contributed by Jim Kingdon,
     21-Feb-2020.) $)
  mulcanap2 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) ) ->
                 ( ( A x. C ) = ( B x. C ) <-> A = B ) ) $=
    ( cc wcel cc0 cap wbr wa w3a simp1 simp2 simp3l simp3r mulcanap2d ) ADEZBDE
    ZCDEZCFGHZIZJABCPQTKPQTLPQRSMPQRSNO $.

  ${
    mulcanapi.1 $e |- A e. CC $.
    mulcanapi.2 $e |- B e. CC $.
    mulcanapi.3 $e |- C e. CC $.
    mulcanapi.4 $e |- C =//= 0 $.
    $( Cancellation law for multiplication.  (Contributed by Jim Kingdon,
       21-Feb-2020.) $)
    mulcanapi $p |- ( ( C x. A ) = ( C x. B ) <-> A = B ) $=
      ( cc wcel cc0 cap wbr wa cmul co wceq wb pm3.2i mulcanap mp3an ) AHIBHICH
      IZCJKLZMCANOCBNOPABPQDEUAUBFGRABCST $.
  $}

  $( A number is apart from zero iff its square is apart from zero.
     (Contributed by Matthew House, 28-Jun-2026.) $)
  msqap0 $p |- ( A e. CC -> ( ( A x. A ) =//= 0 <-> A =//= 0 ) ) $=
    ( cc wcel cc0 cap wbr wa cmul co wb mulap0b anidms anidm bitr3di ) ABCZADZE
    ZFZRGZAAHIPQFZROSTJAAKLRMN $.

  $( A number is zero iff its square is zero.  (Contributed by Matthew House,
     28-Jun-2026.) $)
  msq0 $p |- ( A e. CC -> ( ( A x. A ) = 0 <-> A = 0 ) ) $=
    ( cc wcel cmul co cc0 id mulcld 0cnd msqap0 apcon4bid ) ABCZAADEFZAMLAALGZN
    HLIZNOAJK $.

  $( Property of numbers whose product equals their sum.  Equation 5 of
     [Kreyszig] p. 12.  (Contributed by NM, 13-Nov-2006.) $)
  muleqadd $p |- ( ( A e. CC /\ B e. CC ) ->
             ( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) ) $=
    ( cc wcel wa c1 cmin co cmul wceq caddc ax-1cn mulsub mpanr2 mpanl2 mulridi
    cc0 oveq2i a1i mulrid oveqan12d oveq12d addsub mp3an2 syl2anc 3eqtrd eqeq1d
    mulcl addcl subcld 0cn addcan2 mp3an23 syl addlidi eqeq2i subeq0ad 3bitr2rd
    wb bitr3di ) ACDZBCDZEZAFGHBFGHIHZFJABIHZABKHZGHZFKHZFJZVGQJZVEVFJVCVDVHFVC
    VDVEFFIHZKHZAFIHZBFIHZKHZGHZVEFKHZVFGHZVHVAFCDZVBVDVPJZLVAVSEVBVSVTLAFBFMNO
    VCVLVQVOVFGVLVQJVCVKFVEKFLPRSVAVBVMAVNBKATBTUAUBVCVECDZVFCDZVRVHJZABUHZABUI
    ZWAVSWBWCLVEFVFUCUDUEUFUGVCVHQFKHZJZVJVIVCVGCDZWGVJUSZVCVEVFWDWEUJWHQCDVSWI
    UKLVGQFULUMUNWFFVHFLUOUPUTVCVEVFWDWEUQUR $.

  ${
    $d x y A $.  $d x y B $.
    $( Existential uniqueness of reciprocals.  (Contributed by Jim Kingdon,
       21-Feb-2020.) $)
    receuap $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
        E! x e. CC ( B x. x ) = A ) $=
      ( vy cc wcel cc0 w3a cv cmul co wceq wrex wa wi wral 3adant1 oveq2 eqeq1d
      c1 cap wbr wreu recexap simprl simpll mulcld oveq1 simplr mulassd mullidd
      ad2antll 3eqtr3d syl2anc rexlimdvaa 3adant3 eqtr3 mulcanap imbitrid 3expa
      rspcev mpd expcom ralrimivv reu4 sylanbrc ) BEFZCEFZCGUAUBZHZCAIZJKZBLZAE
      MZVMCDIZJKZBLZNZVKVOLZOZDEPAEPVMAEUCVJVPTLZDEMZVNVHVIWBVGDCUDQVGVHWBVNOVI
      VGVHNZWAVNDEWCVOEFZWANZNZVOBJKZEFCWGJKZBLZVNWFVOBWCWDWAUEZVGVHWEUFZUGWFVP
      BJKZTBJKZWHBWAWLWMLWCWDVPTBJUHULWFCVOBVGVHWEUIWJWKUJWFBWKUKUMVMWIAWGEVKWG
      LVLWHBVKWGCJRSVAUNUOUPVBVJVTADEEVHVIVKEFZWDNZVTOVGWOVHVINZVTWNWDWPVTVRVLV
      PLWNWDWPHVSVLVPBUQVKVOCURUSUTVCQVDVMVQADEVSVLVPBVKVOCJRSVEVF $.
  $}

  ${
    mul0eqap.a $e |- ( ph -> A e. CC ) $.
    mul0eqap.b $e |- ( ph -> B e. CC ) $.
    mul0eqap.ab $e |- ( ph -> A =//= B ) $.
    mul0eqap.0 $e |- ( ph -> ( A x. B ) = 0 ) $.
    $( If two numbers are apart from each other and their product is zero, one
       of them must be zero.  (Contributed by Jim Kingdon, 31-Jul-2023.) $)
    mul0eqap $p |- ( ph -> ( A = 0 \/ B = 0 ) ) $=
      ( cc0 wceq cap wbr wo cc wcel 0cnd mpd wa adantr sylan9bbr apcon4bid cmul
      wi apcotr syl3anc co mulcld ibar mulap0bd mpbird ex iba orim12d orcomd )
      ACHIZBHIZABHJKZCHJKZLZUNUOLABCJKZURFABMNZCMNZHMNUSURUBDEAOBCHUCUDPAUPUNUQ
      UOAUPUNAUPQZUNBCUAUEZHIZAVDUPGRVBCHVCHAVAUPERVBOZAVCMNZUPABCDEUFZRVEUPUQU
      PUQQZAVCHJKZUPUQUGABCDEUHZSTUIUJAUQUOAUQQZUOVDAVDUQGRVKBHVCHAUTUQDRVKOZAV
      FUQVGRVLUQUPVHAVIUQUPUKVJSTUIUJULPUM $.
  $}

  ${
    $d A x $.
    $( A complex number has a multiplicative inverse if and only if it is apart
       from zero.  Theorem 11.2.4 of [HoTT], p.  (varies), generalized from
       real to complex numbers.  (Contributed by Jim Kingdon, 18-Jan-2025.) $)
    recapb $p |- ( A e. CC -> ( A =//= 0 <-> E. x e. CC ( A x. x ) = 1 ) ) $=
      ( cc wcel cc0 cap wbr cv cmul co c1 wceq wrex recexap ex wa simprl simprr
      simpl 1ap0 eqbrtrdi mulap0bad rexlimdvaa impbid ) BCDZBEFGZBAHZIJZKLZACMZ
      UEUFUJABNOUEUIUFACUEUGCDZUIPZPZBUGUEULSUEUKUIQUMUHKEFUEUKUIRTUAUBUCUD $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Division
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Extend class notation to include division. $)
  cdiv $a class / $.

  ${
    $d x y z $.
    $( Define division.  Theorem ~ divmulap relates it to multiplication, and
       ~ divclap and ~ redivclap prove its closure laws.  (Contributed by NM,
       2-Feb-1995.)  Use ~ divvalap instead.  (Revised by Mario Carneiro,
       1-Apr-2014.)  (New usage is discouraged.) $)
    df-div $a |- / = ( x e. CC , y e. ( CC \ { 0 } ) |->
                   ( iota_ z e. CC ( y x. z ) = x ) ) $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y C $.
    $( Value of division: the (unique) element ` x ` such that
       ` ( B x. x ) = A ` .  This is meaningful only when ` B ` is apart from
       zero.  (Contributed by Jim Kingdon, 21-Feb-2020.) $)
    divvalap $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
        ( A / B ) = ( iota_ x e. CC ( B x. x ) = A ) ) $=
      ( vz vy cc wcel cc0 cap wbr w3a csn cdif cv cmul wceq crio cdiv riotabidv
      co simp1 wne simp2 wi 0cn apne mpan2 adantl eldifsn sylanbrc wreu receuap
      3impia riotacl syl eqeq2 oveq1 eqeq1d df-div ovmpog syl3anc ) BFGZCFGZCHI
      JZKZVBCFHLMZGZCANZOTZBPZAFQZFGZBCRTVKPVBVCVDUAVEVCCHUBZVGVBVCVDUCVBVCVDVM
      VCVDVMUDZVBVCHFGVNUECHUFUGUHUMCFHUIUJVEVJAFUKVLABCULVJAFUNUODEBCFVFENZVHO
      TZDNZPZAFQVKRVPBPZAFQFVQBPVRVSAFVQBVPUPSVOCPZVSVJAFVTVPVIBVOCVHOUQURSDEAU
      SUTVA $.

    $( Relationship between division and multiplication.  (Contributed by Jim
       Kingdon, 22-Feb-2020.) $)
    divmulap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
        -> ( ( A / C ) = B <-> ( C x. B ) = A ) ) $=
      ( vx cc wcel cc0 cap wbr wa w3a cdiv co wceq cv cmul 3expb 3adant2 eqeq1d
      crio divvalap wreu wb simp2 receuap oveq2 riota2 syl2anc bitr4d ) AEFZBEF
      ZCEFZCGHIZJZKZACLMZBNCDOZPMZANZDETZBNZCBPMZANZUOUPUTBUJUNUPUTNZUKUJULUMVD
      DACUAQRSUOUKUSDEUBZVCVAUCUJUKUNUDUJUNVEUKUJULUMVEDACUEQRUSVCDEBUQBNURVBAU
      QBCPUFSUGUHUI $.
  $}

  $( Relationship between division and multiplication.  (Contributed by Jim
     Kingdon, 22-Feb-2020.) $)
  divmulap2 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
      -> ( ( A / C ) = B <-> A = ( C x. B ) ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cdiv co wceq cmul divmulap eqcom bitrdi ) ADEB
    DECDECFGHIJACKLBMCBNLZAMARMABCORAPQ $.

  $( Relationship between division and multiplication.  (Contributed by Jim
     Kingdon, 22-Feb-2020.) $)
  divmulap3 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
      -> ( ( A / C ) = B <-> A = ( B x. C ) ) ) $=
    ( cc wcel cc0 cap wbr wa cdiv co wceq cmul divmulap2 mulcom adantrr 3adant1
    w3a eqeq2d bitr4d ) ADEZBDEZCDEZCFGHZIZRZACJKBLACBMKZLABCMKZLABCNUFUHUGAUBU
    EUHUGLZUAUBUCUIUDBCOPQST $.

  ${
    $d x A $.  $d x B $.
    $( Closure law for division.  (Contributed by Jim Kingdon, 22-Feb-2020.) $)
    divclap $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) -> ( A / B ) e. CC ) $=
      ( vx cc wcel cc0 cap wbr w3a cdiv co cmul wceq crio divvalap wreu receuap
      cv riotacl syl eqeltrd ) ADEBDEBFGHIZABJKBCRLKAMZCDNZDCABOUBUCCDPUDDECABQ
      UCCDSTUA $.
  $}

  $( Closure law for reciprocal.  (Contributed by Jim Kingdon, 22-Feb-2020.) $)
  recclap $p |- ( ( A e. CC /\ A =//= 0 ) -> ( 1 / A ) e. CC ) $=
    ( c1 cc wcel cc0 cap wbr cdiv co ax-1cn divclap mp3an1 ) BCDACDAEFGBAHICDJB
    AKL $.

  $( A cancellation law for division.  (Contributed by Jim Kingdon,
     22-Feb-2020.) $)
  divcanap2 $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
      ( B x. ( A / B ) ) = A ) $=
    ( cc wcel cc0 cap wbr w3a cdiv co wceq cmul eqid wa wb simp1 divclap 3simpc
    divmulap syl3anc mpbii ) ACDZBCDZBEFGZHZABIJZUFKZBUFLJAKZUFMUEUBUFCDUCUDNUG
    UHOUBUCUDPABQUBUCUDRAUFBSTUA $.

  $( A cancellation law for division.  (Contributed by Jim Kingdon,
     22-Feb-2020.) $)
  divcanap1 $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 )
      -> ( ( A / B ) x. B ) = A ) $=
    ( cc wcel cc0 cap wbr w3a cdiv cmul divclap simp2 mulcomd divcanap2 eqtrd
    co ) ACDZBCDZBEFGZHZABIPZBJPBUAJPATUABABKQRSLMABNO $.

  $( A ratio is zero iff the numerator is zero.  (Contributed by Jim Kingdon,
     22-Feb-2020.) $)
  diveqap0 $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
      ( ( A / B ) = 0 <-> A = 0 ) ) $=
    ( cc wcel cc0 cap wbr w3a cdiv co wceq cmul wa wb simp1 0cnd 3simpc syl3anc
    divmulap2 simp2 mul01d eqeq2d bitrd ) ACDZBCDZBEFGZHZABIJEKZABELJZKZAEKUGUD
    ECDUEUFMUHUJNUDUEUFOUGPUDUEUFQAEBSRUGUIEAUGBUDUEUFTUAUBUC $.

  $( The ratio of numbers apart from zero is apart from zero.  (Contributed by
     Jim Kingdon, 22-Feb-2020.) $)
  divap0b $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
      ( A =//= 0 <-> ( A / B ) =//= 0 ) ) $=
    ( cc wcel cc0 cap wbr cdiv co wa cmul simp3 biantrud divclap simp2 mulap0bd
    w3a divcanap1 breq1d 3bitrrd ) ACDZBCDZBEFGZQZABHIZEFGZUFUCJUEBKIZEFGAEFGUD
    UCUFUAUBUCLMUDUEBABNUAUBUCOPUDUGAEFABRST $.

  $( The ratio of numbers apart from zero is apart from zero.  (Contributed by
     Jim Kingdon, 22-Feb-2020.) $)
  divap0 $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) ) ->
      ( A / B ) =//= 0 ) $=
    ( cc wcel cc0 cap wbr wa cdiv co wb divap0b 3expb biimpa an32s ) ACDZBCDZBE
    FGZHZAEFGZABIJEFGZPSHTUAPQRTUAKABLMNO $.

  $( The reciprocal of a number apart from zero is apart from zero.
     (Contributed by Jim Kingdon, 24-Feb-2020.) $)
  recap0 $p |- ( ( A e. CC /\ A =//= 0 ) -> ( 1 / A ) =//= 0 ) $=
    ( c1 cc wcel cc0 cap wbr wa cdiv co ax-1cn 1ap0 divap0 mpanl12 ) BCDBEFGACD
    AEFGHBAIJEFGKLBAMN $.

  $( Multiplication of a number and its reciprocal.  (Contributed by Jim
     Kingdon, 24-Feb-2020.) $)
  recidap $p |- ( ( A e. CC /\ A =//= 0 ) -> ( A x. ( 1 / A ) ) = 1 ) $=
    ( c1 cc wcel cc0 cap wbr cdiv co cmul wceq ax-1cn divcanap2 mp3an1 ) BCDACD
    AEFGABAHIJIBKLBAMN $.

  $( Multiplication of a number and its reciprocal.  (Contributed by Jim
     Kingdon, 24-Feb-2020.) $)
  recidap2 $p |- ( ( A e. CC /\ A =//= 0 ) -> ( ( 1 / A ) x. A ) = 1 ) $=
    ( c1 cc wcel cc0 cap wbr cdiv co cmul wceq ax-1cn divcanap1 mp3an1 ) BCDACD
    AEFGBAHIAJIBKLBAMN $.

  $( Relationship between division and reciprocal.  (Contributed by Jim
     Kingdon, 24-Feb-2020.) $)
  divrecap $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
      ( A / B ) = ( A x. ( 1 / B ) ) ) $=
    ( cc wcel cc0 cap wbr w3a cdiv co c1 cmul wceq simp2 recclap 3adant1 mul12d
    simp1 recidap oveq2d mulridd 3eqtrd mulcld 3simpc divmulap syl3anc mpbird
    wa wb ) ACDZBCDZBEFGZHZABIJAKBIJZLJZMZBUOLJZAMZUMUQABUNLJZLJAKLJAUMBAUNUJUK
    ULNUJUKULRZUKULUNCDUJBOPZQUMUSKALUKULUSKMUJBSPTUMAUTUAUBUMUJUOCDUKULUHUPURU
    IUTUMAUNUTVAUCUJUKULUDAUOBUEUFUG $.

  $( Relationship between division and reciprocal.  (Contributed by Jim
     Kingdon, 25-Feb-2020.) $)
  divrecap2 $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
      ( A / B ) = ( ( 1 / B ) x. A ) ) $=
    ( cc wcel cc0 cap wbr w3a cdiv co c1 divrecap simp1 recclap 3adant1 mulcomd
    cmul eqtrd ) ACDZBCDZBEFGZHZABIJAKBIJZQJUCAQJABLUBAUCSTUAMTUAUCCDSBNOPR $.

  $( An associative law for division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  divassap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) ) ->
      ( ( A x. B ) / C ) = ( A x. ( B / C ) ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cmul cdiv wceq recclap mulass syl3an3 divrecap
    co c1 syl3anc mulcl 3adant3 simp3l simp3r simp2 oveq2d 3eqtr4d ) ADEZBDEZCD
    EZCFGHZIZJZABKRZSCLRZKRZABUOKRZKRZUNCLRZABCLRZKRULUHUIUODEUPURMCNABUOOPUMUN
    DEZUJUKUSUPMUHUIVAULABUAUBUHUIUJUKUCZUHUIUJUKUDZUNCQTUMUTUQAKUMUIUJUKUTUQMU
    HUIULUEVBVCBCQTUFUG $.

  $( A commutative/associative law for division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  div23ap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
      -> ( ( A x. B ) / C ) = ( ( A / C ) x. B ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cmul co cdiv wceq mulcom oveq1d 3adant3 3com12
    divassap simp2 divclap 3expb 3adant2 mulcomd 3eqtrd ) ADEZBDEZCDEZCFGHZIZJZ
    ABKLZCMLZBAKLZCMLZBACMLZKLZUPBKLUFUGUMUONUJUFUGIULUNCMABOPQUGUFUJUOUQNBACSR
    UKBUPUFUGUJTUFUJUPDEZUGUFUHUIURACUAUBUCUDUE $.

  $( A commutative/associative law for division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  div32ap $p |- ( ( A e. CC /\ ( B e. CC /\ B =//= 0 ) /\ C e. CC )
      -> ( ( A / B ) x. C ) = ( A x. ( C / B ) ) ) $=
    ( cc wcel cc0 cap wbr wa cdiv cmul wceq w3a div23ap divassap eqtr3d 3com23
    co ) ADEZCDEZBDEBFGHIZABJRCKRZACBJRKRZLSTUAMACKRBJRUBUCACBNACBOPQ $.

  $( A commutative/associative law for division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  div13ap $p |- ( ( A e. CC /\ ( B e. CC /\ B =//= 0 ) /\ C e. CC )
      -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cmul co cdiv wceq mulcom oveq1d 3adant2 3com23
    div23ap 3coml 3eqtr3d ) ADEZBDEBFGHIZCDEZJACKLZBMLZCAKLZBMLZABMLCKLZCBMLAKL
    ZUBUDUFUHNUCUBUDIUEUGBMACOPQUBUDUCUFUINACBSRUDUBUCUHUJNCABSTUA $.

  $( A commutative/associative law for division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  div12ap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
      -> ( A x. ( B / C ) ) = ( B x. ( A / C ) ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cdiv co cmul divclap 3expb mulcom sylan2 3impb
    wceq div13ap 3comr sylan 3impa 3com23 3eqtrd ) ADEZBDEZCDEZCFGHZIZJABCKLZML
    ZUKAMLZACKLZBMLZBUNMLZUFUGUJULUMSZUGUJIUFUKDEZUQUGUHUIURBCNOAUKPQRUGUJUFUMU
    OSBCATUAUFUJUGUOUPSZUFUJUGUSUFUJIUNDEZUGUSUFUHUIUTACNOUNBPUBUCUDUE $.

  $( An associative law for division and multiplication.  (Contributed by Jim
     Kingdon, 24-Jan-2022.) $)
  divmulassap $p |- ( ( ( A e. CC /\ B e. CC /\ C e. CC )
        /\ ( D e. CC /\ D =//= 0 ) )
      -> ( ( A x. ( B / D ) ) x. C ) = ( ( A x. B ) x. ( C / D ) ) ) $=
    ( cc wcel w3a cc0 cap wbr wa cdiv co cmul wceq simpl1 simpl2 simpr divassap
    syl3anc eqcomd oveq1d mulcl 3adant3 adantr simpl3 div32ap eqtrd ) AEFZBEFZC
    EFZGZDEFDHIJKZKZABDLMNMZCNMABNMZDLMZCNMZUPCDLMNMZUNUOUQCNUNUQUOUNUIUJUMUQUO
    OUIUJUKUMPUIUJUKUMQULUMRZABDSTUAUBUNUPEFZUMUKURUSOULVAUMUIUJVAUKABUCUDUEUTU
    IUJUKUMUFUPDCUGTUH $.

  $( An associative/commutative law for division and multiplication.
     (Contributed by Jim Kingdon, 24-Jan-2022.) $)
  divmulasscomap $p |- ( ( ( A e. CC /\ B e. CC /\ C e. CC )
        /\ ( D e. CC /\ D =//= 0 ) )
      -> ( ( A x. ( B / D ) ) x. C ) = ( B x. ( ( A x. C ) / D ) ) ) $=
    ( cc wcel w3a cc0 cap wbr wa cdiv co cmul divmulassap mulcom 3adant3 adantr
    wceq oveq1d simpl2 simpl1 anim1i 3anass sylibr divclap syl mulassd divassap
    simp3 simpr syl3anc eqcomd oveq2d eqtrd 3eqtrd ) AEFZBEFZCEFZGZDEFZDHIJZKZK
    ZABDLMNMCNMABNMZCDLMZNMBANMZVFNMZBACNMDLMZNMZABCDOVDVEVGVFNUTVEVGSZVCUQURVK
    USABPQRTVDVHBAVFNMZNMVJVDBAVFUQURUSVCUAUQURUSVCUBZVDUSVAVBGZVFEFVDUSVCKVNUT
    USVCUQURUSUJZUCUSVAVBUDUECDUFUGUHVDVLVIBNVDVIVLVDUQUSVCVIVLSVMUTUSVCVORUTVC
    UKACDUIULUMUNUOUP $.

  $( Distribution of division over addition.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  divdirap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
               -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) ) $=
    ( cc wcel cc0 cap wbr wa w3a caddc co c1 cdiv cmul simp1 simp2 wceq syl3anc
    divrecap recclap 3ad2ant3 adddird addcld simp3l simp3r oveq12d 3eqtr4d ) AD
    EZBDEZCDEZCFGHZIZJZABKLZMCNLZOLZAUPOLZBUPOLZKLUOCNLZACNLZBCNLZKLUNABUPUIUJU
    MPZUIUJUMQZUMUIUPDEUJCUAUBUCUNUODEUKULUTUQRUNABVCVDUDUIUJUKULUEZUIUJUKULUFZ
    UOCTSUNVAURVBUSKUNUIUKULVAURRVCVEVFACTSUNUJUKULVBUSRVDVEVFBCTSUGUH $.

  $( A cancellation law for division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  divcanap3 $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
                 ( ( B x. A ) / B ) = A ) $=
    ( cc wcel cc0 cap wbr w3a cmul co cdiv wceq wa wb simp2 simp1 mulcld 3simpc
    eqid divmulap syl3anc mpbiri ) ACDZBCDZBEFGZHZBAIJZBKJALZUGUGLZUGSUFUGCDUCU
    DUEMUHUINUFBAUCUDUEOUCUDUEPZQUJUCUDUERUGABTUAUB $.

  $( A cancellation law for division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  divcanap4 $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
                 ( ( A x. B ) / B ) = A ) $=
    ( cc wcel cc0 cap wbr w3a cmul co cdiv wceq mulcom 3adant3 oveq1d divcanap3
    eqtrd ) ACDZBCDZBEFGZHZABIJZBKJBAIJZBKJAUAUBUCBKRSUBUCLTABMNOABPQ $.

  $( One-to-one relationship for division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  div11ap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) ) ->
               ( ( A / C ) = ( B / C ) <-> A = B ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cdiv co cmul wceq simp1 simp3l divclap syl3anc
    simp3r divcanap2 simp2 mulcanapd eqeq12d bitr3d ) ADEZBDEZCDEZCFGHZIZJZCACK
    LZMLZCBCKLZMLZNUKUMNABNUJUKUMCUJUEUGUHUKDEUEUFUIOZUEUFUGUHPZUEUFUGUHSZACQRU
    JUFUGUHUMDEUEUFUIUAZUPUQBCQRUPUQUBUJULAUNBUJUEUGUHULANUOUPUQACTRUJUFUGUHUNB
    NURUPUQBCTRUCUD $.

  $( A number divided by itself is one.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  dividap $p |- ( ( A e. CC /\ A =//= 0 ) -> ( A / A ) = 1 ) $=
    ( cc wcel cc0 cap wbr wa cdiv co cmul wceq divrecap 3anidm12 recidap eqtrd
    c1 ) ABCZADEFZGAAHIZAPAHIJIZPQRSTKAALMANO $.

  $( Division into zero is zero.  (Contributed by Jim Kingdon, 25-Feb-2020.) $)
  div0ap $p |- ( ( A e. CC /\ A =//= 0 ) -> ( 0 / A ) = 0 ) $=
    ( cc wcel cc0 cap wbr wa cdiv co wceq cmul simpl mul01d wb divmulap mp3an12
    0cn mpbird ) ABCZADEFZGZDAHIDJZADKIDJZUAASTLMDBCZUDUAUBUCNQQDDAOPR $.

  $( A number divided by 1 is itself.  (Contributed by NM, 9-Jan-2002.)  (Proof
     shortened by Mario Carneiro, 27-May-2016.) $)
  div1 $p |- ( A e. CC -> ( A / 1 ) = A ) $=
    ( cc wcel c1 cdiv co wceq cmul mullid wb cc0 cap wbr ax-1cn pm3.2i divmulap
    wa 1ap0 mp3an3 anidms mpbird ) ABCZADEFAGZDAHFAGZAIUBUCUDJZUBUBDBCZDKLMZQUE
    UFUGNROAADPSTUA $.

  $( 1 divided by 1 is 1 (common case).  (Contributed by David A. Wheeler,
     7-Dec-2018.) $)
  1div1e1 $p |- ( 1 / 1 ) = 1 $=
    ( c1 cc wcel cdiv co wceq ax-1cn div1 ax-mp ) ABCAADEAFGAHI $.

  $( Equality in terms of unit ratio.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  diveqap1 $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
      ( ( A / B ) = 1 <-> A = B ) ) $=
    ( cc wcel cc0 cap wbr w3a cdiv co c1 wceq cmul wb wa ax-1cn divmulap2 3impb
    mp3an2 simp2 mulridd eqeq2d bitrd ) ACDZBCDZBEFGZHZABIJKLZABKMJZLZABLUDUEUF
    UHUJNZUDKCDUEUFOUKPAKBQSRUGUIBAUGBUDUEUFTUAUBUC $.

  $( Move negative sign inside of a division.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  divnegap $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
                -u ( A / B ) = ( -u A / B ) ) $=
    ( cc wcel cc0 cap wbr w3a cneg c1 cdiv co cmul wceq wa recclap sylan2 3impb
    mulneg1 divrecap negcl syl3an1 negeqd 3eqtr4rd ) ACDZBCDZBEFGZHZAIZJBKLZMLZ
    AUJMLZIZUIBKLZABKLZIUEUFUGUKUMNZUFUGOUEUJCDUPBPAUJSQRUEUICDUFUGUNUKNAUAUIBT
    UBUHUOULABTUCUD $.

  $( Distribution of division over addition with a multiplication.
     (Contributed by Jim Kingdon, 11-Nov-2021.) $)
  muldivdirap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
               -> ( ( ( C x. A ) + B ) / C ) = ( A + ( B / C ) ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cmul co caddc cdiv wceq simp3l mulcld divdirap
    simp1 syld3an1 3anass biimpri 3adant2 divcanap3 syl oveq1d eqtrd ) ADEZBDEZ
    CDEZCFGHZIZJZCAKLZBMLCNLZUNCNLZBCNLZMLZAUQMLUNDEUIUHULUOUROUMCAUHUIUJUKPUHU
    IULSQUNBCRTUMUPAUQMUMUHUJUKJZUPAOUHULUSUIUSUHULIUHUJUKUAUBUCACUDUEUFUG $.

  $( Distribution of division over subtraction.  (Contributed by NM,
     4-Mar-2005.) $)
  divsubdirap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
               -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cdiv cneg caddc cmin wceq eqtr3d 3expb 3adant1
    co divclap divdirap syl3an2 negsub oveq1d 3adant3 divnegap 3adant2 negsubd
    negcl oveq2d ) ADEZBDEZCDEZCFGHZIZJZACKSZBLZCKSZMSZABNSZCKSZUQBCKSZNSZUPAUR
    MSZCKSZUTVBULUKURDEUOVFUTOBUIAURCUAUBUKULVFVBOUOUKULIVEVACKABUCUDUEPUPUQVCL
    ZMSUTVDUPVGUSUQMULUOVGUSOZUKULUMUNVHBCUFQRUJUPUQVCUKUOUQDEZULUKUMUNVIACTQUG
    ULUOVCDEZUKULUMUNVJBCTQRUHPP $.

  $( A number is equal to the reciprocal of its reciprocal.  (Contributed by
     Jim Kingdon, 25-Feb-2020.) $)
  recrecap $p |- ( ( A e. CC /\ A =//= 0 ) -> ( 1 / ( 1 / A ) ) = A ) $=
    ( cc wcel cc0 cap wbr wa c1 cdiv co wceq cmul recidap2 wb 1cnd simpl recap0
    recclap divmulap syl112anc mpbird ) ABCZADEFZGZHHAIJZIJAKZUEALJHKZAMUDHBCUB
    UEBCUEDEFUFUGNUDOUBUCPARAQHAUESTUA $.

  $( Reciprocal is one-to-one.  (Contributed by Jim Kingdon, 25-Feb-2020.) $)
  rec11ap $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) )
                  -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) $=
    ( cc wcel cc0 cap wbr wa c1 cdiv co wceq cmul wb 1cnd adantl simpl divmulap
    recclap syl3anc simpll simprl simprr divrecap eqeq1d diveqap1 3bitr2d ) ACD
    ZAEFGZHZBCDZBEFGZHZHZIAJKIBJKZLZAUOMKZILZABJKZILZABLZUNICDUOCDZUJUPURNUNOUM
    VBUJBSPUJUMQIUOARTUNUSUQIUNUHUKULUSUQLUHUIUMUAZUJUKULUBZUJUKULUCZABUDTUEUNU
    HUKULUTVANVCVDVEABUFTUG $.

  $( Mutual reciprocals.  (Contributed by Jim Kingdon, 25-Feb-2020.) $)
  rec11rap $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) )
                  -> ( ( 1 / A ) = B <-> ( 1 / B ) = A ) ) $=
    ( cc wcel cc0 cap wa c1 cdiv co wceq cmul wb simprl simpll simplr divmulap2
    wbr 1cnd syl112anc simprr divmulap3 bitr4d ) ACDZAEFRZGZBCDZBEFRZGZGZHAIJBK
    ZHABLJKZHBIJAKZUJHCDZUGUDUEUKULMUJSZUFUGUHNZUDUEUIOZUDUEUIPHBAQTUJUNUDUGUHU
    MULMUOUQUPUFUGUHUAHABUBTUC $.

  $( Multiplication of two ratios.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  divmuldivap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C =//= 0 )
                    /\ ( D e. CC /\ D =//= 0 ) ) )
               -> ( ( A / C ) x. ( B / D ) ) = ( ( A x. B ) / ( C x. D ) ) ) $=
    ( cc wcel cc0 cap wbr wa cdiv co cmul wceq w3a 3anass mulcl syl2an 3adantr1
    divclap ad2ant2r 3adantl1 mulap0 divcanap3 syl3anc simp2 jca mul4 divcanap2
    oveqan12d eqtr3d oveq1d syl2anbr an4s ) AEFZCEFZCGHIZJZBEFZDEFZDGHIZJZACKLZ
    BDKLZMLZABMLZCDMLZKLZNZUOURJUOUPUQOZUSUTVAOZVIUSVBJUOUPUQPUSUTVAPVJVKJZVGVE
    MLZVGKLZVEVHVLVEEFZVGEFZVGGHIZVNVENVJVCEFZVDEFZVOVKACTZBDTZVCVDQRUPUQVKVPUO
    URUTVAVPUSUPUTVPUQVACDQUASUBUPUQVKVQUOURUTVAVQUSCDUCSUBVEVGUDUEVLVMVFVGKVLC
    VCMLZDVDMLZMLZVMVFVJUPVRJUTVSJWDVMNVKVJUPVRUOUPUQUFVTUGVKUTVSUSUTVAUFWAUGCV
    CDVDUHRVJVKWBAWCBMACUIBDUIUJUKULUKUMUN $.

  $( Division of two ratios.  Theorem I.15 of [Apostol] p. 18.  (Contributed by
     Jim Kingdon, 25-Feb-2020.) $)
  divdivdivap $p |- ( ( ( A e. CC /\ ( B e. CC /\ B =//= 0 ) ) /\
                  ( ( C e. CC /\ C =//= 0 ) /\ ( D e. CC /\ D =//= 0 ) ) )
                -> ( ( A / B ) / ( C / D ) ) = ( ( A x. D ) / ( B x. C ) ) ) $=
    ( cc wcel cc0 cap wbr wa cdiv co cmul wceq divclap syl3anc mulcomd eqtrd c1
    mulcld simprrl simprll simprlr simpll simplrl simplr simprl syl22anc oveq2d
    simplrr divmuldivap simprrr mulap0d dividap syl2anc mulassd mullidd 3eqtr3d
    simprr oveq1d eqtr3d wb mulap0 ad2ant2lr divap0 adantl divmulap syl112anc
    mpbird ) AEFZBEFZBGHIZJZJZCEFZCGHIZJZDEFZDGHIZJZJZJZABKLZCDKLZKLADMLZBCMLZK
    LZNZWDWGMLZWCNZWBWDDCKLZWCMLZMLZWIWCWBWLWGWDMWBWLWCWKMLZWGWBWKWCWBVRVOVPWKE
    FVNVQVRVSUAZVNVOVPVTUBZVNVOVPVTUCZDCOPZWBVJVKVLWCEFZVJVMWAUDZVJVKVLWAUEZVJV
    KVLWAUJABOPZQWBVJVRVMVQWNWGNWTWOVJVMWAUFVNVQVTUGZADBCUKUHRUIWBWDWKMLZWCMLSW
    CMLWMWCWBXDSWCMWBXDCDMLZDCMLZKLZSWBVOVRVTVQXDXGNWPWOVNVQVTUSXCCDDCUKUHWBXGX
    FXFKLZSWBXEXFXFKWBCDWPWOQUTWBXFEFXFGHIXHSNWBDCWOWPTWBDCWOWPVNVQVRVSULZWQUMX
    FUNUORRUTWBWDWKWCWBVOVRVSWDEFZWPWOXICDOPZWRXBUPWBWCXBUQURVAWBWSWGEFZXJWDGHI
    ZWHWJVBXBWBWEEFWFEFWFGHIZXLWBADWTWOTWBBCXAWPTVMVQXNVJVTBCVCVDWEWFOPXKWAXMVN
    CDVEVFWCWGWDVGVHVI $.

  $( Cancellation of common factor in a ratio.  (Contributed by Jim Kingdon,
     25-Feb-2020.) $)
  divcanap5 $p |- ( ( A e. CC /\ ( B e. CC /\ B =//= 0 ) /\
      ( C e. CC /\ C =//= 0 ) ) -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cdiv co cmul c1 dividap oveq1d 3ad2ant3 simp3l
    wceq simp1 simp3 simp2 divmuldivap syl22anc divclap mullidd 3adant3 3eqtr3d
    3expb ) ADEZBDEZBFGHZIZCDEZCFGHZIZJZCCKLZABKLZMLZNUSMLZCAMLCBMLKLZUSUPUJUTV
    ASUMUPURNUSMCOPQUQUNUJUPUMUTVBSUJUMUNUORUJUMUPTUJUMUPUAUJUMUPUBCACBUCUDUJUM
    VAUSSUPUJUMIUSUJUKULUSDEABUEUIUFUGUH $.

  $( Swap the denominators in the product of two ratios.  (Contributed by Jim
     Kingdon, 26-Feb-2020.) $)
  divmul13ap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C =//= 0 )
                    /\ ( D e. CC /\ D =//= 0 ) ) )
                -> ( ( A / C ) x. ( B / D ) ) = ( ( B / C ) x. ( A / D ) ) ) $=
    ( cc wcel wa cc0 cap wbr cmul co cdiv wceq mulcom adantr oveq1d divmuldivap
    ancom1s 3eqtr4d ) AEFZBEFZGZCEFCHIJGDEFDHIJGGZGZABKLZCDKLZMLBAKLZUGMLZACMLB
    DMLKLBCMLADMLKLZUEUFUHUGMUCUFUHNUDABOPQABCDRUBUAUDUJUINBACDRST $.

  $( Swap the numerators in the product of two ratios.  (Contributed by Jim
     Kingdon, 26-Feb-2020.) $)
  divmul24ap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C =//= 0 )
                    /\ ( D e. CC /\ D =//= 0 ) ) )
                -> ( ( A / C ) x. ( B / D ) ) = ( ( A / D ) x. ( B / C ) ) ) $=
    ( cc wcel wa cc0 cap wbr cmul co cdiv wceq mulcom adantl oveq2d divmuldivap
    ad2ant2r ancom2s 3eqtr4d ) AEFBEFGZCEFZCHIJZGZDEFZDHIJZGZGZGZABKLZCDKLZMLUK
    DCKLZMLZACMLBDMLKLADMLBCMLKLZUJULUMUKMUIULUMNZUBUCUFUPUDUGCDOSPQABCDRUBUHUE
    UOUNNABDCRTUA $.

  $( Cross-multiply in an equality of ratios.  (Contributed by Jim Kingdon,
     26-Feb-2020.) $)
  divmuleqap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C =//= 0 ) /\
                     ( D e. CC /\ D =//= 0 ) ) ) ->
                   ( ( A / C ) = ( B / D ) <-> ( A x. D ) = ( B x. C ) ) ) $=
    ( cc wcel wa cc0 cap cdiv cmul wceq divclap 3expb ad2ant2r ad2ant2l mulassd
    wbr co divcanap1 mulap0 jca adantl mulcanap2 syl3anc simprll simprrl oveq1d
    wb mulcl eqtr3d mulcomd oveq2d 3eqtr2d eqeq12d bitr3d ) AEFZBEFZGZCEFZCHIRZ
    GZDEFZDHIRZGZGZGZACJSZCDKSZKSZBDJSZVIKSZLZVHVKLZADKSZBCKSZLVGVHEFZVKEFZVIEF
    ZVIHIRZGZVMVNUIUQVBVQURVEUQUTVAVQACMNOZURVEVRUQVBURVCVDVRBDMNPZVFWAUSVFVSVT
    UTVCVSVAVDCDUJOCDUAUBUCVHVKVIUDUEVGVJVOVLVPVGVHCKSZDKSVJVOVGVHCDWBUSUTVAVEU
    FZUSVBVCVDUGZQVGWDADKUQVBWDALZURVEUQUTVAWGACTNOUHUKVGVLVKDCKSZKSVKDKSZCKSVP
    VGVIWHVKKVGCDWEWFULUMVGVKDCWCWFWEQVGWIBCKURVEWIBLZUQVBURVCVDWJBDTNPUHUNUOUP
    $.

  $( The reciprocal of a ratio.  (Contributed by Jim Kingdon, 26-Feb-2020.) $)
  recdivap $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) ) ->
                ( 1 / ( A / B ) ) = ( B / A ) ) $=
    ( cc wcel cc0 cap wa c1 cdiv co cmul 1div1e1 oveq1i wceq ax-1cn 1ap0 pm3.2i
    wbr divdivdivap mullid mpanl12 eqtr3id oveqan12rd ad2ant2r eqtrd ) ACDZAEFR
    ZGBCDZBEFRZGGZHABIJZIJZHBKJZHAKJZIJZBAIJZUJULHHIJZUKIJZUOUQHUKILMHCDZUSHEFR
    ZGUJURUONOUSUTOPQHHABSUAUBUFUHUOUPNUGUIUHUFUMBUNAIBTATUCUDUE $.

  $( Cancellation of inverted fractions.  (Contributed by Jim Kingdon,
     26-Feb-2020.) $)
  divcanap6 $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) ) ->
         ( ( A / B ) x. ( B / A ) ) = 1 ) $=
    ( cc wcel cc0 cap wbr wa cdiv co c1 cmul recdivap oveq2d wceq divclap 3expb
    adantlr divap0 recidap syl2anc eqtr3d ) ACDZAEFGZHBCDZBEFGZHZHZABIJZKUIIJZL
    JZUIBAIJZLJKUHUJULUILABMNUHUICDZUIEFGUKKOUCUGUMUDUCUEUFUMABPQRABSUITUAUB $.

  $( Swap denominators in a division.  (Contributed by Jim Kingdon,
     26-Feb-2020.) $)
  divdiv32ap $p |- ( ( A e. CC /\ ( B e. CC /\ B =//= 0 )
      /\ ( C e. CC /\ C =//= 0 ) )
      -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) $=
    ( cc wcel cc0 cap wbr wa c1 cdiv cmul wceq recclap div23ap syl3an2 divrecap
    w3a co 3expb 3adant3 oveq1d divclap syl3an1 3impa 3com23 3eqtr4d ) ADEZBDEZ
    BFGHZIZCDEZCFGHZIZRZAJBKSZLSZCKSZACKSZUPLSZABKSZCKSUSBKSZUKUHUPDEUNURUTMBNA
    UPCOPUOVAUQCKUHUKVAUQMZUNUHUIUJVCABQTUAUBUHUNUKVBUTMZUHUNUKVDUHUNIZUIUJVDVE
    USDEZUIUJVDUHULUMVFACUCTUSBQUDTUEUFUG $.

  $( Cancel equal divisors in a division.  (Contributed by Jim Kingdon,
     26-Feb-2020.) $)
  divcanap7 $p |- ( ( A e. CC /\ ( B e. CC /\ B =//= 0 ) /\
    ( C e. CC /\ C =//= 0 ) ) -> ( ( A / C ) / ( B / C ) ) = ( A / B ) ) $=
    ( cc wcel cc0 cap wbr w3a cdiv cmul wceq divdivdivap 3impdir mulcom adantrr
    wa co 3adant2 oveq1d divcanap5 3eqtrd ) ADEZBDEBFGHQZCDEZCFGHZQZIZACJRBCJRJ
    RZACKRZCBKRZJRZCAKRZUKJRABJRUCUGUDUIULLACBCMNUHUJUMUKJUCUGUJUMLZUDUCUEUNUFA
    COPSTABCUAUB $.

  $( Cancellation law for division and multiplication.  (Contributed by Jim
     Kingdon, 26-Feb-2020.) $)
  dmdcanap $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) /\
         C e. CC ) ->
         ( ( A / B ) x. ( C / A ) ) = ( C / B ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cdiv co cmul wceq simp1l simp3 divclap syl3anc
    simp1r simp2l simp2r div23ap syl112anc divcanap2 oveq1d eqtr3d ) ADEZAFGHZI
    ZBDEZBFGHZIZCDEZJZACAKLZMLZBKLZABKLUOMLZCBKLUNUGUODEZUJUKUQURNUGUHULUMOZUNU
    MUGUHUSUIULUMPZUTUGUHULUMSZCAQRUIUJUKUMTUIUJUKUMUAAUOBUBUCUNUPCBKUNUMUGUHUP
    CNVAUTVBCAUDRUEUF $.

  $( Division into a fraction.  (Contributed by Jim Kingdon, 26-Feb-2020.) $)
  divdivap1 $p |- ( ( A e. CC /\ ( B e. CC /\ B =//= 0 ) /\
       ( C e. CC /\ C =//= 0 ) ) -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cdiv co c1 cmul wceq ax-1cn pm3.2i divdivdivap
    1ap0 mpanr2 3impa oveq2d ad2antrl 3adant1 mulrid oveq1d 3ad2ant1 3eqtr3d
    div1 ) ADEZBDEBFGHIZCDEZCFGHZIZJABKLZCMKLZKLZAMNLZBCNLZKLZUOCKLZAUSKLZUJUKU
    NUQUTOZUJUKIUNMDEZMFGHZIVCVDVEPSQABCMRTUAUKUNUQVAOZUJULVFUKUMULUPCUOKCUIUBU
    CUDUJUKUTVBOUNUJURAUSKAUEUFUGUH $.

  $( Division by a fraction.  (Contributed by Jim Kingdon, 26-Feb-2020.) $)
  divdivap2 $p |- ( ( A e. CC /\ ( B e. CC /\ B =//= 0 )
      /\ ( C e. CC /\ C =//= 0 ) )
      -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) ) $=
    ( cc wcel cc0 cap wbr wa w3a c1 cdiv co cmul wceq ax-1cn pm3.2i divdivdivap
    1ap0 mpanl2 3impb 3ad2ant1 oveq1d mullid ad2antrl 3adant3 oveq2d 3eqtr3d
    div1 ) ADEZBDEZBFGHZIZCDECFGHIZJZAKLMZBCLMZLMZACNMZKBNMZLMZAUQLMUSBLMUJUMUN
    URVAOZUJKDEZKFGHZIUMUNIVBVCVDPSQAKBCRTUAUOUPAUQLUJUMUPAOUNAUIUBUCUOUTBUSLUJ
    UMUTBOZUNUKVEUJULBUDUEUFUGUH $.

  $( Division into a reciprocal.  (Contributed by Jim Kingdon, 26-Feb-2020.) $)
  recdivap2 $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) ) ->
                ( ( 1 / A ) / B ) = ( 1 / ( A x. B ) ) ) $=
    ( c1 cc wcel cc0 cap wbr wa cdiv co cmul wceq ax-1cn divdivap1 mp3an1 ) CDE
    ADEAFGHIBDEBFGHICAJKBJKCABLKJKMNCABOP $.

  $( Cancellation in a double division.  (Contributed by Jim Kingdon,
     26-Feb-2020.) $)
  ddcanap $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) ) ->
                ( A / ( A / B ) ) = B ) $=
    ( cc wcel cc0 cap wbr cdiv wceq cmul simpll simprl simprr divcanap1 syl3anc
    wa co wb divclap divap0 divmulap syl112anc mpbird ) ACDZAEFGZPZBCDZBEFGZPZP
    ZAABHQZHQBIZUKBJQAIZUJUDUGUHUMUDUEUIKZUFUGUHLZUFUGUHMZABNOUJUDUGUKCDZUKEFGU
    LUMRUNUOUJUDUGUHUQUNUOUPABSOABTABUKUAUBUC $.

  $( Addition of two ratios.  (Contributed by Jim Kingdon, 26-Feb-2020.) $)
  divadddivap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C =//= 0 )
                    /\ ( D e. CC /\ D =//= 0 ) ) ) ->
  ( ( A / C ) + ( B / D ) ) = ( ( ( A x. D ) + ( B x. C ) ) / ( C x. D ) ) ) $=
    ( cc wcel wa cc0 cap cmul co caddc cdiv wceq mulcl ad2ant2r syl3anc mulcomd
    wbr oveq12d adantrl adantrr mulap0 jca adantl divdirap simpll simprr simpld
    ad2ant2lr simprll simprl divcanap5 eqtrd simplr oveq1d eqtr2d ) AEFZBEFZGZC
    EFZCHISZGZDEFZDHISZGZGZGZADJKZBCJKZLKCDJKZMKZVIVKMKZVJVKMKZLKZACMKZBDMKZLKV
    HVIEFZVJEFZVKEFZVKHISZGZVLVONUTVFVRVCURVDVRUSVEADOPUAUSVCVSURVFUSVAVSVBBCOU
    BUJVGWBUTVGVTWAVAVDVTVBVECDOPCDUCUDUEVIVJVKUFQVHVMVPVNVQLVHVMDAJKZDCJKZMKZV
    PVHVIWCVKWDMVHADURUSVGUGZVHVDVEUTVCVFUHZUIZRVHCDUTVAVBVFUKZWHRTVHURVCVFWEVP
    NWFUTVCVFULZWGACDUMQUNVHVNCBJKZVKMKZVQVHVJWKVKMVHBCURUSVGUOZWIRUPVHUSVFVCWL
    VQNWMWGWJBDCUMQUNTUQ $.

  $( Subtraction of two ratios.  (Contributed by Jim Kingdon, 26-Feb-2020.) $)
  divsubdivap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C =//= 0 )
                    /\ ( D e. CC /\ D =//= 0 ) ) ) ->
  ( ( A / C ) - ( B / D ) ) = ( ( ( A x. D ) - ( B x. C ) ) / ( C x. D ) ) ) $=
    ( cc wcel wa cc0 cap wbr cmul co cneg caddc cdiv cmin syl3anc oveq2d eqtr3d
    wceq sylanl2 simplr simprrl simprrr divnegap simpll simprll simprlr divclap
    negcl divadddivap negsubd mulneg1d mulcld eqtrd oveq1d ) AEFZBEFZGZCEFZCHIJ
    ZGZDEFZDHIJZGZGZGZADKLZBMZCKLZNLZCDKLZOLZACOLZBDOLZPLZVHBCKLZPLZVLOLVGVNVID
    OLZNLZVMVPURUQVIEFVFVTVMTBUJAVICDUKUAVGVNVOMZNLVTVPVGWAVSVNNVGURVCVDWAVSTUQ
    URVFUBZUSVBVCVDUCZUSVBVCVDUDZBDUEQRVGVNVOVGUQUTVAVNEFUQURVFUFZUSUTVAVEUGZUS
    UTVAVEUHACUIQVGURVCVDVOEFWBWCWDBDUIQULSSVGVKVRVLOVGVKVHVQMZNLVRVGVJWGVHNVGB
    CWBWFUMRVGVHVQVGADWEWCUNVGBCWBWFUNULUOUPS $.

  $( Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy
     this relationship.  (Contributed by Jim Kingdon, 26-Feb-2020.) $)
  conjmulap $p |- ( ( ( P e. CC /\ P =//= 0 ) /\ ( Q e. CC /\ Q =//= 0 ) ) ->
      ( ( ( 1 / P ) + ( 1 / Q ) ) = 1 <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) ) $=
    ( cc wcel cc0 cap wa cmul co c1 cdiv caddc wceq cmin recclap adantr recidap
    wbr 3eqtrd ad2ant2r simpll simprl mul32d oveq1d mullid adantl oveq2d mulrid
    ad2antrl mulassd ad2antrr oveq12d mulcl adddid addcom 3eqtr4d mulridd addcl
    eqeq12d syl2an mulap0 ax-1cn mulcanap mp3an2 syl12anc eqcom muleqadd bitrid
    wb 3bitr3d ) ACDZAEFRZGZBCDZBEFRZGZGZABHIZJAKIZJBKIZLIZHIZVRJHIZMZABLIZVRMZ
    WAJMZAJNIBJNIHIJMZVQWBWEWCVRVQVRVSHIZVRVTHIZLIBALIZWBWEVQWIBWJALVQWIAVSHIZB
    HIZJBHIZBVQABVSVKVLVPUAZVMVNVOUBZVMVSCDZVPAOZPZUCVMWMWNMVPVMWLJBHAQUDPVNWNB
    MVMVOBUEUISVQWJABVTHIZHIZAJHIZAVQABVTWOWPVPVTCDZVMBOZUFZUJVPXAXBMVMVPWTJAHB
    QUGUFVKXBAMVLVPAUHUKSULVQVRVSVTVKVNVRCDZVLVOABUMZTZWSXEUNVKVNWEWKMVLVOABUOT
    UPVKVNWCVRMVLVOVKVNGZVRXGUQTUSVQWACDZXFVREFRZWDWGVIZVMWQXCXJVPWRXDVSVTURUTX
    HABVAXJJCDXFXKGXLVBWAJVRVCVDVEVKVNWFWHVIVLVOWFVRWEMXIWHWEVRVFABVGVHTVJ $.

  ${
    $d x A $.
    $( Closure law for reciprocal.  (Contributed by Jim Kingdon,
       26-Feb-2020.) $)
    rerecclap $p |- ( ( A e. RR /\ A =//= 0 ) -> ( 1 / A ) e. RR ) $=
      ( vx cr wcel cc0 cap wbr wa cv c1 cdiv co wceq wrex creap wb 0re cc recnd
      cmul apreap mpan2 pm5.32i recexre sylbi eqcom 1cnd simpll simplr divmulap
      simpr syl112anc bitrid rexbidva mpbird risset sylibr ) ACDZAEFGZHZBIZJAKL
      ZMZBCNZVBCDUTVDAVATLJMZBCNZUTURAEOGZHVFURUSVGURECDUSVGPQAEUAUBUCBAUDUEUTV
      CVEBCVCVBVAMZUTVACDZHZVEVAVBUFVJJRDVARDARDUSVHVEPVJUGVJVAUTVIUKSVJAURUSVI
      UHSURUSVIUIJVAAUJULUMUNUOBVBCUPUQ $.
  $}

  $( Closure law for division of reals.  (Contributed by Jim Kingdon,
     26-Feb-2020.) $)
  redivclap $p |- ( ( A e. RR /\ B e. RR /\ B =//= 0 ) -> ( A / B ) e. RR ) $=
    ( cr wcel cc0 cap wbr cdiv co c1 cmul wceq simp1 recnd simp2 simp3 divrecap
    w3a cc syl3anc rerecclap 3adant1 remulcld eqeltrd ) ACDZBCDZBEFGZRZABHIZAJB
    HIZKIZCUHASDBSDUGUIUKLUHAUEUFUGMZNUHBUEUFUGONUEUFUGPABQTUHAUJULUFUGUJCDUEBU
    AUBUCUD $.

  $( A number equal to its negative is zero.  (Contributed by NM, 12-Jul-2005.)
     (Revised by Mario Carneiro, 27-May-2016.) $)
  eqneg $p |- ( A e. CC -> ( A = -u A <-> A = 0 ) ) $=
    ( cc wcel c1 caddc cmul cc0 wceq cneg 1p1times ax-1cn addcli mul01i eqtr4id
    co negid eqeq12d a1i 1re 0lt1 0cnd cap readdcli addgt0ii gt0ap0ii mulcanapd
    id wbr negcl addcand 3bitr3rd ) ABCZDDEOZAFOZUMGFOZHAAEOZAAIZEOZHAGHAUQHULU
    NUPUOURAJULUOGURUMDDKKLZMAPNQULAGUMULUGZULUAUMBCULUSRUMGUBUHULUMDDSSUCDDSST
    TUDUERUFULAAUQUTUTAUIUJUK $.

  ${
    eqnegd.1 $e |- ( ph -> A e. CC ) $.
    $( A complex number equals its negative iff it is zero.  Deduction form of
       ~ eqneg .  (Contributed by David Moews, 28-Feb-2017.) $)
    eqnegd $p |- ( ph -> ( A = -u A <-> A = 0 ) ) $=
      ( cc wcel cneg wceq cc0 wb eqneg syl ) ABDEBBFGBHGICBJK $.
  $}

  ${
    eqnegad.1 $e |- ( ph -> A e. CC ) $.
    eqnegad.2 $e |- ( ph -> A = -u A ) $.
    $( If a complex number equals its own negative, it is zero.  One-way
       deduction form of ~ eqneg .  (Contributed by David Moews,
       28-Feb-2017.) $)
    eqnegad $p |- ( ph -> A = 0 ) $=
      ( cneg wceq cc0 eqnegd mpbid ) ABBEFBGFDABCHI $.
  $}

  $( Quotient of two negatives.  (Contributed by Jim Kingdon, 27-Feb-2020.) $)
  div2negap $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
                  ( -u A / -u B ) = ( A / B ) ) $=
    ( cc wcel cc0 cap wbr w3a cneg cdiv co wceq cmul 3ad2ant2 syl112anc 3adant1
    negcl c1 eqtrd 3ad2ant1 simp1 simp2 simp3 div12ap divnegap syld3an1 dividap
    negeqd eqtr3d oveq2d ax-1cn negcli mulcom mpan2 mulm1 divclap negap0 biimpa
    wb divmulap mpbird ) ACDZBCDZBEFGZHZAIZBIZJKABJKZLZVGVHMKZVFLZVEVJAVGBJKZMK
    ZVFVEVGCDZVBVCVDVJVMLVCVBVNVDBQNZVBVCVDUAVBVCVDUBZVBVCVDUCVGABUDOVEVMARIZMK
    ZVFVEVLVQAMVEBBJKZIZVLVQVCVCVBVDVTVLLVPBBUEUFVEVSRVCVDVSRLVBBUGPUHUIUJVBVCV
    RVFLVDVBVRVQAMKZVFVBVQCDVRWALRUKULAVQUMUNAUOSTSSVEVFCDZVHCDVNVGEFGZVIVKUSVB
    VCWBVDAQTABUPVOVCVDWCVBVCVDWCBUQURPVFVHVGUTOVA $.

  $( Move negative sign inside of a division.  (Contributed by Jim Kingdon,
     27-Feb-2020.) $)
  divneg2ap $p |- ( ( A e. CC /\ B e. CC /\ B =//= 0 ) ->
                   -u ( A / B ) = ( A / -u B ) ) $=
    ( cc wcel cc0 cap wbr w3a cdiv co cneg divnegap wceq negcl div2negap negneg
    syl3an1 3ad2ant1 oveq1d 3eqtr2d ) ACDZBCDZBEFGZHZABIJKAKZBIJZUEKZBKZIJZAUHI
    JABLUAUECDUBUCUIUFMANUEBOQUDUGAUHIUAUBUGAMUCAPRST $.

  ${
    divclz.1 $e |- A e. CC $.
    $( Closure law for reciprocal.  (Contributed by Jim Kingdon,
       27-Feb-2020.) $)
    recclapzi $p |- ( A =//= 0 -> ( 1 / A ) e. CC ) $=
      ( cc wcel cc0 cap wbr c1 cdiv co recclap mpan ) ACDAEFGHAIJCDBAKL $.

    $( The reciprocal of a number apart from zero is apart from zero.
       (Contributed by Jim Kingdon, 27-Feb-2020.) $)
    recap0apzi $p |- ( A =//= 0 -> ( 1 / A ) =//= 0 ) $=
      ( cc wcel cc0 cap wbr c1 cdiv co recap0 mpan ) ACDAEFGHAIJEFGBAKL $.

    $( Multiplication of a number and its reciprocal.  (Contributed by Jim
       Kingdon, 27-Feb-2020.) $)
    recidapzi $p |- ( A =//= 0 -> ( A x. ( 1 / A ) ) = 1 ) $=
      ( cc wcel cc0 cap wbr c1 cdiv co cmul wceq recidap mpan ) ACDAEFGAHAIJKJH
      LBAMN $.

    $( A number divided by 1 is itself.  (Contributed by NM, 9-Jan-2002.) $)
    div1i $p |- ( A / 1 ) = A $=
      ( cc wcel c1 cdiv co wceq div1 ax-mp ) ACDAEFGAHBAIJ $.

    $( A number equal to its negative is zero.  (Contributed by NM,
       29-May-1999.) $)
    eqnegi $p |- ( A = -u A <-> A = 0 ) $=
      ( cc wcel cneg wceq cc0 wb eqneg ax-mp ) ACDAAEFAGFHBAIJ $.

    ${
      recclap.2 $e |- A =//= 0 $.
      $( Closure law for reciprocal.  (Contributed by NM, 30-Apr-2005.) $)
      recclapi $p |- ( 1 / A ) e. CC $=
        ( cc0 cap wbr c1 cdiv co cc wcel recclapzi ax-mp ) ADEFGAHIJKCABLM $.

      $( Multiplication of a number and its reciprocal.  (Contributed by NM,
         9-Feb-1995.) $)
      recidapi $p |- ( A x. ( 1 / A ) ) = 1 $=
        ( cc0 cap wbr c1 cdiv co cmul wceq recidapzi ax-mp ) ADEFAGAHIJIGKCABLM
        $.

      $( A number is equal to the reciprocal of its reciprocal.  Theorem I.10
         of [Apostol] p. 18.  (Contributed by NM, 9-Feb-1995.) $)
      recrecapi $p |- ( 1 / ( 1 / A ) ) = A $=
        ( cc wcel cc0 cap wbr c1 cdiv co wceq recrecap mp2an ) ADEAFGHIIAJKJKAL
        BCAMN $.

      $( A number divided by itself is one.  (Contributed by NM,
         9-Feb-1995.) $)
      dividapi $p |- ( A / A ) = 1 $=
        ( cc wcel cc0 cap wbr cdiv co c1 wceq dividap mp2an ) ADEAFGHAAIJKLBCAM
        N $.

      $( Division into zero is zero.  (Contributed by NM, 12-Aug-1999.) $)
      div0api $p |- ( 0 / A ) = 0 $=
        ( cc wcel cc0 cap wbr cdiv co wceq div0ap mp2an ) ADEAFGHFAIJFKBCALM $.
    $}

    divclz.2 $e |- B e. CC $.
    $( Closure law for division.  (Contributed by Jim Kingdon, 27-Feb-2020.) $)
    divclapzi $p |- ( B =//= 0 -> ( A / B ) e. CC ) $=
      ( cc wcel cc0 cap wbr cdiv co divclap mp3an12 ) AEFBEFBGHIABJKEFCDABLM $.

    $( A cancellation law for division.  (Contributed by Jim Kingdon,
       27-Feb-2020.) $)
    divcanap1zi $p |- ( B =//= 0 -> ( ( A / B ) x. B ) = A ) $=
      ( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap1 mp3an12 ) AEFBEFBGHIABJ
      KBLKAMCDABNO $.

    $( A cancellation law for division.  (Contributed by Jim Kingdon,
       27-Feb-2020.) $)
    divcanap2zi $p |- ( B =//= 0 -> ( B x. ( A / B ) ) = A ) $=
      ( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap2 mp3an12 ) AEFBEFBGHIBAB
      JKLKAMCDABNO $.

    $( Relationship between division and reciprocal.  (Contributed by Jim
       Kingdon, 27-Feb-2020.) $)
    divrecapzi $p |- ( B =//= 0 -> ( A / B ) = ( A x. ( 1 / B ) ) ) $=
      ( cc wcel cc0 cap wbr cdiv co c1 cmul wceq divrecap mp3an12 ) AEFBEFBGHIA
      BJKALBJKMKNCDABOP $.

    $( A cancellation law for division.  (Contributed by Jim Kingdon,
       27-Feb-2020.) $)
    divcanap3zi $p |- ( B =//= 0 -> ( ( B x. A ) / B ) = A ) $=
      ( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap3 mp3an12 ) AEFBEFBGHIBAJ
      KBLKAMCDABNO $.

    $( A cancellation law for division.  (Contributed by Jim Kingdon,
       27-Feb-2020.) $)
    divcanap4zi $p |- ( B =//= 0 -> ( ( A x. B ) / B ) = A ) $=
      ( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap4 mp3an12 ) AEFBEFBGHIABJ
      KBLKAMCDABNO $.

    $( Reciprocal is one-to-one.  (Contributed by Jim Kingdon, 28-Feb-2020.) $)
    rec11api $p |- ( ( A =//= 0 /\ B =//= 0 ) ->
               ( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) $=
      ( cc0 cap wbr cc wcel c1 cdiv co wceq wb wa rec11ap mpanl1 mpanr1 ) AEFGZ
      BHIZBEFGZJAKLJBKLMABMNZDAHISTUAOUBCABPQR $.

    ${
      divclap.3 $e |- B =//= 0 $.
      $( Closure law for division.  (Contributed by Jim Kingdon,
         28-Feb-2020.) $)
      divclapi $p |- ( A / B ) e. CC $=
        ( cc wcel cc0 cap wbr cdiv co divclap mp3an ) AFGBFGBHIJABKLFGCDEABMN
        $.

      $( A cancellation law for division.  (Contributed by Jim Kingdon,
         28-Feb-2020.) $)
      divcanap2i $p |- ( B x. ( A / B ) ) = A $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap2 mp3an ) AFGBFGBHIJBAB
        KLMLANCDEABOP $.

      $( A cancellation law for division.  (Contributed by Jim Kingdon,
         28-Feb-2020.) $)
      divcanap1i $p |- ( ( A / B ) x. B ) = A $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap1 mp3an ) AFGBFGBHIJABK
        LBMLANCDEABOP $.

      $( Relationship between division and reciprocal.  (Contributed by Jim
         Kingdon, 28-Feb-2020.) $)
      divrecapi $p |- ( A / B ) = ( A x. ( 1 / B ) ) $=
        ( cc wcel cc0 cap wbr cdiv co c1 cmul wceq divrecap mp3an ) AFGBFGBHIJA
        BKLAMBKLNLOCDEABPQ $.

      $( A cancellation law for division.  (Contributed by Jim Kingdon,
         28-Feb-2020.) $)
      divcanap3i $p |- ( ( B x. A ) / B ) = A $=
        ( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap3 mp3an ) AFGBFGBHIJBAK
        LBMLANCDEABOP $.

      $( A cancellation law for division.  (Contributed by Jim Kingdon,
         28-Feb-2020.) $)
      divcanap4i $p |- ( ( A x. B ) / B ) = A $=
        ( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap4 mp3an ) AFGBFGBHIJABK
        LBMLANCDEABOP $.
    $}

    ${
      divap0i.3 $e |- A =//= 0 $.
      divap0.i4 $e |- B =//= 0 $.
      $( The ratio of numbers apart from zero is apart from zero.  (Contributed
         by Jim Kingdon, 28-Feb-2020.) $)
      divap0i $p |- ( A / B ) =//= 0 $=
        ( cc wcel cc0 cap wbr cdiv co divap0 mp4an ) AGHAIJKBGHBIJKABLMIJKCEDFA
        BNO $.

      $( Reciprocal is one-to-one.  (Contributed by Jim Kingdon,
         28-Feb-2020.) $)
      rec11apii $p |- ( ( 1 / A ) = ( 1 / B ) <-> A = B ) $=
        ( cc0 cap wbr c1 cdiv co wceq wb rec11api mp2an ) AGHIBGHIJAKLJBKLMABMN
        EFABCDOP $.
    $}

    divmulz.3 $e |- C e. CC $.
    $( An associative law for division.  (Contributed by Jim Kingdon,
       28-Feb-2020.) $)
    divassapzi $p |- ( C =//= 0 -> ( ( A x. B ) / C ) = ( A x. ( B / C ) ) ) $=
      ( cc wcel cc0 cap wbr cmul co cdiv wceq wa divassap mp3an12 mpan ) CGHZCI
      JKZABLMCNMABCNMLMOZFAGHBGHTUAPUBDEABCQRS $.

    $( Relationship between division and multiplication.  (Contributed by Jim
       Kingdon, 28-Feb-2020.) $)
    divmulapzi $p |- ( B =//= 0 -> ( ( A / B ) = C <-> ( B x. C ) = A ) ) $=
      ( cc wcel cc0 cap wbr cdiv co wceq cmul wb wa divmulap mp3an12 mpan ) BGH
      ZBIJKZABLMCNBCOMANPZEAGHCGHUAUBQUCDFACBRST $.

    $( Distribution of division over addition.  (Contributed by Jim Kingdon,
       28-Feb-2020.) $)
    divdirapzi $p |- ( C =//= 0 ->
                    ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) ) $=
      ( cc wcel cc0 cap wbr caddc co cdiv wceq wa divdirap mp3an12 mpan ) CGHZC
      IJKZABLMCNMACNMBCNMLMOZFAGHBGHTUAPUBDEABCQRS $.

    $( Swap denominators in a division.  (Contributed by Jim Kingdon,
       28-Feb-2020.) $)
    divdiv23apzi $p |- ( ( B =//= 0 /\ C =//= 0 ) ->
                ( ( A / B ) / C ) = ( ( A / C ) / B ) ) $=
      ( cc0 cap wbr cc wcel cdiv co wceq wa divdiv32ap mp3an1 mpanl1 mpanr1 ) B
      GHIZCJKZCGHIZABLMCLMACLMBLMNZFBJKZTUAUBOZUCEAJKUDTOUEUCDABCPQRS $.

    ${
      divmulapi.4 $e |- B =//= 0 $.
      $( Relationship between division and multiplication.  (Contributed by Jim
         Kingdon, 29-Feb-2020.) $)
      divmulapi $p |- ( ( A / B ) = C <-> ( B x. C ) = A ) $=
        ( cc0 cap wbr cdiv co wceq cmul wb divmulapzi ax-mp ) BHIJABKLCMBCNLAMO
        GABCDEFPQ $.

      divdiv32api.5 $e |- C =//= 0 $.
      $( Swap denominators in a division.  (Contributed by Jim Kingdon,
         29-Feb-2020.) $)
      divdiv32api $p |- ( ( A / B ) / C ) = ( ( A / C ) / B ) $=
        ( cc0 cap wbr cdiv co wceq divdiv23apzi mp2an ) BIJKCIJKABLMCLMACLMBLMN
        GHABCDEFOP $.
    $}

    ${
      divassap.4 $e |- C =//= 0 $.
      $( An associative law for division.  (Contributed by Jim Kingdon,
         9-Mar-2020.) $)
      divassapi $p |- ( ( A x. B ) / C ) = ( A x. ( B / C ) ) $=
        ( cc wcel cc0 cap wbr wa cmul co cdiv wceq pm3.2i divassap mp3an ) AHIB
        HICHIZCJKLZMABNOCPOABCPONOQDEUAUBFGRABCST $.

      $( Distribution of division over addition.  (Contributed by Jim Kingdon,
         9-Mar-2020.) $)
      divdirapi $p |- ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) $=
        ( cc wcel cc0 cap wbr wa caddc co cdiv wceq pm3.2i divdirap mp3an ) AHI
        BHICHIZCJKLZMABNOCPOACPOBCPONOQDEUAUBFGRABCST $.

      $( A commutative/associative law for division.  (Contributed by Jim
         Kingdon, 9-Mar-2020.) $)
      div23api $p |- ( ( A x. B ) / C ) = ( ( A / C ) x. B ) $=
        ( cc wcel cc0 cap wbr wa cmul co cdiv wceq pm3.2i div23ap mp3an ) AHIBH
        ICHIZCJKLZMABNOCPOACPOBNOQDEUAUBFGRABCST $.

      $( One-to-one relationship for division.  (Contributed by Jim Kingdon,
         9-Mar-2020.) $)
      div11api $p |- ( ( A / C ) = ( B / C ) <-> A = B ) $=
        ( cc wcel cc0 cap wbr wa cdiv co wceq wb pm3.2i div11ap mp3an ) AHIBHIC
        HIZCJKLZMACNOBCNOPABPQDEUAUBFGRABCST $.
    $}

    divmuldivap.4 $e |- D e. CC $.
    divmuldivap.5 $e |- B =//= 0 $.
    divmuldivap.6 $e |- D =//= 0 $.
    $( Multiplication of two ratios.  (Contributed by Jim Kingdon,
       9-Mar-2020.) $)
    divmuldivapi $p |- ( ( A / B ) x. ( C / D ) ) =
                      ( ( A x. C ) / ( B x. D ) ) $=
      ( cc wcel cc0 cap wbr wa cdiv co cmul pm3.2i wceq divmuldivap mp4an ) AKL
      CKLBKLZBMNOZPDKLZDMNOZPABQRCDQRSRACSRBDSRQRUAEGUDUEFITUFUGHJTACBDUBUC $.

    $( Swap denominators of two ratios.  (Contributed by Jim Kingdon,
       9-Mar-2020.) $)
    divmul13api $p |- ( ( A / B ) x. ( C / D ) ) =
                   ( ( C / B ) x. ( A / D ) ) $=
      ( cc wcel cc0 cap wbr wa cdiv co cmul pm3.2i wceq divmul13ap mp4an ) AKLC
      KLBKLZBMNOZPDKLZDMNOZPABQRCDQRSRCBQRADQRSRUAEGUDUEFITUFUGHJTACBDUBUC $.

    $( Addition of two ratios.  (Contributed by Jim Kingdon, 9-Mar-2020.) $)
    divadddivapi $p |- ( ( A / B ) + ( C / D ) ) =
                      ( ( ( A x. D ) + ( C x. B ) ) / ( B x. D ) ) $=
      ( cc wcel cc0 cap wbr wa cdiv co caddc cmul wceq pm3.2i divadddivap mp4an
      ) AKLCKLBKLZBMNOZPDKLZDMNOZPABQRCDQRSRADTRCBTRSRBDTRQRUAEGUEUFFIUBUGUHHJU
      BACBDUCUD $.

    divdivdivap.7 $e |- C =//= 0 $.
    $( Division of two ratios.  (Contributed by Jim Kingdon, 9-Mar-2020.) $)
    divdivdivapi $p |- ( ( A / B ) / ( C / D ) )
                        = ( ( A x. D ) / ( B x. C ) ) $=
      ( cc wcel cc0 cap wbr wa cdiv co pm3.2i cmul wceq divdivdivap mp4an ) ALM
      BLMZBNOPZQCLMZCNOPZQDLMZDNOPZQABRSCDRSRSADUASBCUASRSUBEUEUFFITUGUHGKTUIUJ
      HJTABCDUCUD $.
  $}

  ${
    redivclap.1 $e |- A e. RR $.
    $( Closure law for reciprocal.  (Contributed by Jim Kingdon,
       9-Mar-2020.) $)
    rerecclapzi $p |- ( A =//= 0 -> ( 1 / A ) e. RR ) $=
      ( cr wcel cc0 cap wbr c1 cdiv co rerecclap mpan ) ACDAEFGHAIJCDBAKL $.

    ${
      rerecclap.2 $e |- A =//= 0 $.
      $( Closure law for reciprocal.  (Contributed by Jim Kingdon,
         9-Mar-2020.) $)
      rerecclapi $p |- ( 1 / A ) e. RR $=
        ( cc0 cap wbr c1 cdiv co cr wcel rerecclapzi ax-mp ) ADEFGAHIJKCABLM $.
    $}

    redivclap.2 $e |- B e. RR $.
    $( Closure law for division of reals.  (Contributed by Jim Kingdon,
       9-Mar-2020.) $)
    redivclapzi $p |- ( B =//= 0 -> ( A / B ) e. RR ) $=
      ( cr wcel cc0 cap wbr cdiv co redivclap mp3an12 ) AEFBEFBGHIABJKEFCDABLM
      $.

    redivclap.3 $e |- B =//= 0 $.
    $( Closure law for division of reals.  (Contributed by Jim Kingdon,
       9-Mar-2020.) $)
    redivclapi $p |- ( A / B ) e. RR $=
      ( cc0 cap wbr cdiv co cr wcel redivclapzi ax-mp ) BFGHABIJKLEABCDMN $.
  $}

  ${
    div1d.1 $e |- ( ph -> A e. CC ) $.
    $( A number divided by 1 is itself.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    div1d $p |- ( ph -> ( A / 1 ) = A ) $=
      ( cc wcel c1 cdiv co wceq div1 syl ) ABDEBFGHBICBJK $.

    ${
      reccld.2 $e |- ( ph -> A =//= 0 ) $.
      $( Closure law for reciprocal.  (Contributed by Jim Kingdon,
         3-Mar-2020.) $)
      recclapd $p |- ( ph -> ( 1 / A ) e. CC ) $=
        ( cc wcel cc0 cap wbr c1 cdiv co recclap syl2anc ) ABEFBGHIJBKLEFCDBMN
        $.

      $( The reciprocal of a number apart from zero is apart from zero.
         (Contributed by Jim Kingdon, 3-Mar-2020.) $)
      recap0d $p |- ( ph -> ( 1 / A ) =//= 0 ) $=
        ( cc wcel cc0 cap wbr c1 cdiv co recap0 syl2anc ) ABEFBGHIJBKLGHICDBMN
        $.

      $( Multiplication of a number and its reciprocal.  (Contributed by Jim
         Kingdon, 3-Mar-2020.) $)
      recidapd $p |- ( ph -> ( A x. ( 1 / A ) ) = 1 ) $=
        ( cc wcel cc0 cap wbr c1 cdiv co cmul wceq recidap syl2anc ) ABEFBGHIBJ
        BKLMLJNCDBOP $.

      $( Multiplication of a number and its reciprocal.  (Contributed by Jim
         Kingdon, 3-Mar-2020.) $)
      recidap2d $p |- ( ph -> ( ( 1 / A ) x. A ) = 1 ) $=
        ( cc wcel cc0 cap wbr c1 cdiv co cmul wceq recidap2 syl2anc ) ABEFBGHIJ
        BKLBMLJNCDBOP $.

      $( A number is equal to the reciprocal of its reciprocal.  (Contributed
         by Jim Kingdon, 3-Mar-2020.) $)
      recrecapd $p |- ( ph -> ( 1 / ( 1 / A ) ) = A ) $=
        ( cc wcel cc0 cap wbr c1 cdiv co wceq recrecap syl2anc ) ABEFBGHIJJBKLK
        LBMCDBNO $.

      $( A number divided by itself is one.  (Contributed by Jim Kingdon,
         3-Mar-2020.) $)
      dividapd $p |- ( ph -> ( A / A ) = 1 ) $=
        ( cc wcel cc0 cap wbr cdiv co c1 wceq dividap syl2anc ) ABEFBGHIBBJKLMC
        DBNO $.

      $( Division into zero is zero.  (Contributed by Jim Kingdon,
         3-Mar-2020.) $)
      div0apd $p |- ( ph -> ( 0 / A ) = 0 ) $=
        ( cc wcel cc0 cap wbr cdiv co wceq div0ap syl2anc ) ABEFBGHIGBJKGLCDBMN
        $.
    $}
  $}

  $( Multiplication of both sides of complex apartness by a complex number
     apart from zero.  (Contributed by Jim Kingdon, 20-Mar-2020.) $)
  apmul1 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
      -> ( A =//= B <-> ( A x. C ) =//= ( B x. C ) ) ) $=
    ( cc wcel cc0 cap wbr wa cmul co c1 mulassd oveq2d mulridd 3eqtrd wi mulcld
    w3a mulext1 cdiv simp1 simp3l simp3r recclapd simp2 breq12d syl3anc sylbird
    recidapd 3adant3r impbid ) ADEZBDEZCDEZCFGHZIZSZABGHZACJKZBCJKZGHZURUSUTLCU
    AKZJKZVAVCJKZGHZVBURVDAVEBGURVDACVCJKZJKALJKAURACVCUMUNUQUBZUMUNUOUPUCZURCV
    IUMUNUOUPUDZUEZMURVGLAJURCVIVJUJZNURAVHOPURVEBVGJKBLJKBURBCVCUMUNUQUFZVIVKM
    URVGLBJVLNURBVMOPUGURUTDEVADEVCDEVFVBQURACVHVIRURBCVMVIRVKUTVAVCTUHUIUMUNUO
    VBUSQUPABCTUKUL $.

  $( Multiplication of both sides of complex apartness by a complex number
     apart from zero.  (Contributed by Jim Kingdon, 6-Jan-2023.) $)
  apmul2 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =//= 0 ) )
      -> ( A =//= B <-> ( C x. A ) =//= ( C x. B ) ) ) $=
    ( cc wcel cc0 cap wbr wa w3a cmul apmul1 simp1 simp3l mulcomd simp2 breq12d
    co bitrd ) ADEZBDEZCDEZCFGHZIZJZABGHACKRZBCKRZGHCAKRZCBKRZGHABCLUEUFUHUGUIG
    UEACTUAUDMTUAUBUCNZOUEBCTUAUDPUJOQS $.

  ${
    divcld.1 $e |- ( ph -> A e. CC ) $.
    divcld.2 $e |- ( ph -> B e. CC ) $.
    ${
      divclapd.3 $e |- ( ph -> B =//= 0 ) $.
      $( Closure law for division.  (Contributed by Jim Kingdon,
         29-Feb-2020.) $)
      divclapd $p |- ( ph -> ( A / B ) e. CC ) $=
        ( cc wcel cc0 cap wbr cdiv co divclap syl3anc ) ABGHCGHCIJKBCLMGHDEFBCN
        O $.

      $( A cancellation law for division.  (Contributed by Jim Kingdon,
         29-Feb-2020.) $)
      divcanap1d $p |- ( ph -> ( ( A / B ) x. B ) = A ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap1 syl3anc ) ABGHCGHCIJK
        BCLMCNMBODEFBCPQ $.

      $( A cancellation law for division.  (Contributed by Jim Kingdon,
         29-Feb-2020.) $)
      divcanap2d $p |- ( ph -> ( B x. ( A / B ) ) = A ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap2 syl3anc ) ABGHCGHCIJK
        CBCLMNMBODEFBCPQ $.

      $( Relationship between division and reciprocal.  Theorem I.9 of
         [Apostol] p. 18.  (Contributed by Jim Kingdon, 29-Feb-2020.) $)
      divrecapd $p |- ( ph -> ( A / B ) = ( A x. ( 1 / B ) ) ) $=
        ( cc wcel cc0 cap wbr cdiv co c1 cmul wceq divrecap syl3anc ) ABGHCGHCI
        JKBCLMBNCLMOMPDEFBCQR $.

      $( Relationship between division and reciprocal.  (Contributed by Jim
         Kingdon, 29-Feb-2020.) $)
      divrecap2d $p |- ( ph -> ( A / B ) = ( ( 1 / B ) x. A ) ) $=
        ( cc wcel cc0 cap wbr cdiv co c1 cmul wceq divrecap2 syl3anc ) ABGHCGHC
        IJKBCLMNCLMBOMPDEFBCQR $.

      $( A cancellation law for division.  (Contributed by Jim Kingdon,
         29-Feb-2020.) $)
      divcanap3d $p |- ( ph -> ( ( B x. A ) / B ) = A ) $=
        ( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap3 syl3anc ) ABGHCGHCIJK
        CBLMCNMBODEFBCPQ $.

      $( A cancellation law for division.  (Contributed by Jim Kingdon,
         29-Feb-2020.) $)
      divcanap4d $p |- ( ph -> ( ( A x. B ) / B ) = A ) $=
        ( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap4 syl3anc ) ABGHCGHCIJK
        BCLMCNMBODEFBCPQ $.

      ${
        diveqap0d.4 $e |- ( ph -> ( A / B ) = 0 ) $.
        $( If a ratio is zero, the numerator is zero.  (Contributed by Jim
           Kingdon, 19-Mar-2020.) $)
        diveqap0d $p |- ( ph -> A = 0 ) $=
          ( cdiv co cc0 wceq cc wcel cap wbr wb diveqap0 syl3anc mpbid ) ABCHIJ
          KZBJKZGABLMCLMCJNOTUAPDEFBCQRS $.
      $}

      ${
        diveqap1d.4 $e |- ( ph -> ( A / B ) = 1 ) $.
        $( Equality in terms of unit ratio.  (Contributed by Jim Kingdon,
           19-Mar-2020.) $)
        diveqap1d $p |- ( ph -> A = B ) $=
          ( cdiv co c1 wceq cc wcel cc0 cap wbr wb diveqap1 syl3anc mpbid ) ABC
          HIJKZBCKZGABLMCLMCNOPUAUBQDEFBCRST $.
      $}

      $( The quotient of two complex numbers is one iff they are equal.
         Deduction form of ~ diveqap1 .  Generalization of ~ diveqap1d .
         (Contributed by Jim Kingdon, 19-Mar-2020.) $)
      diveqap1ad $p |- ( ph -> ( ( A / B ) = 1 <-> A = B ) ) $=
        ( cc wcel cc0 cap wbr cdiv co c1 wceq wb diveqap1 syl3anc ) ABGHCGHCIJK
        BCLMNOBCOPDEFBCQR $.

      $( A fraction of complex numbers is zero iff its numerator is.  Deduction
         form of ~ diveqap0 .  (Contributed by Jim Kingdon, 19-Mar-2020.) $)
      diveqap0ad $p |- ( ph -> ( ( A / B ) = 0 <-> A = 0 ) ) $=
        ( cc wcel cc0 cap wbr cdiv co wceq wb diveqap0 syl3anc ) ABGHCGHCIJKBCL
        MINBINODEFBCPQ $.

      ${
        divap1d.4 $e |- ( ph -> A =//= B ) $.
        $( If two complex numbers are apart, their quotient is apart from one.
           (Contributed by Jim Kingdon, 20-Mar-2020.) $)
        divap1d $p |- ( ph -> ( A / B ) =//= 1 ) $=
          ( c1 cdiv co cmul cap wbr cc wcel cc0 wb recclapd recap0d apmul1
          syl112anc mpbid divrecapd eqcomd recidapd 3brtr3d ) ABHCIJZKJZCUGKJZB
          CIJZHLABCLMZUHUILMZGABNOCNOUGNOUGPLMUKULQDEACEFRACEFSBCUGTUAUBAUJUHAB
          CDEFUCUDACEFUEUF $.
      $}

      $( A ratio is zero iff the numerator is zero.  (Contributed by Jim
         Kingdon, 19-Mar-2020.) $)
      divap0bd $p |- ( ph -> ( A =//= 0 <-> ( A / B ) =//= 0 ) ) $=
        ( cc wcel cc0 cap wbr cdiv co wb divap0b syl3anc ) ABGHCGHCIJKBIJKBCLMI
        JKNDEFBCOP $.

      $( Move negative sign inside of a division.  (Contributed by Jim Kingdon,
         19-Mar-2020.) $)
      divnegapd $p |- ( ph -> -u ( A / B ) = ( -u A / B ) ) $=
        ( cc wcel cc0 cap wbr cdiv co cneg wceq divnegap syl3anc ) ABGHCGHCIJKB
        CLMNBNCLMODEFBCPQ $.

      $( Move negative sign inside of a division.  (Contributed by Jim Kingdon,
         19-Mar-2020.) $)
      divneg2apd $p |- ( ph -> -u ( A / B ) = ( A / -u B ) ) $=
        ( cc wcel cc0 cap wbr cdiv co cneg wceq divneg2ap syl3anc ) ABGHCGHCIJK
        BCLMNBCNLMODEFBCPQ $.

      $( Quotient of two negatives.  (Contributed by Jim Kingdon,
         19-Mar-2020.) $)
      div2negapd $p |- ( ph -> ( -u A / -u B ) = ( A / B ) ) $=
        ( cc wcel cc0 cap wbr cneg cdiv co wceq div2negap syl3anc ) ABGHCGHCIJK
        BLCLMNBCMNODEFBCPQ $.
    $}

    ${
      divap0d.3 $e |- ( ph -> A =//= 0 ) $.
      divap0d.4 $e |- ( ph -> B =//= 0 ) $.
      $( The ratio of numbers apart from zero is apart from zero.  (Contributed
         by Jim Kingdon, 3-Mar-2020.) $)
      divap0d $p |- ( ph -> ( A / B ) =//= 0 ) $=
        ( cc wcel cc0 cap wbr cdiv co divap0 syl22anc ) ABHIBJKLCHICJKLBCMNJKLD
        FEGBCOP $.

      $( The reciprocal of a ratio.  (Contributed by Jim Kingdon,
         3-Mar-2020.) $)
      recdivapd $p |- ( ph -> ( 1 / ( A / B ) ) = ( B / A ) ) $=
        ( cc wcel cc0 cap wbr c1 cdiv co wceq recdivap syl22anc ) ABHIBJKLCHICJ
        KLMBCNONOCBNOPDFEGBCQR $.

      $( Division into a reciprocal.  (Contributed by Jim Kingdon,
         3-Mar-2020.) $)
      recdivap2d $p |- ( ph -> ( ( 1 / A ) / B ) = ( 1 / ( A x. B ) ) ) $=
        ( cc wcel cc0 cap wbr c1 cdiv co cmul wceq recdivap2 syl22anc ) ABHIBJK
        LCHICJKLMBNOCNOMBCPONOQDFEGBCRS $.

      $( Cancellation of inverted fractions.  (Contributed by Jim Kingdon,
         3-Mar-2020.) $)
      divcanap6d $p |- ( ph -> ( ( A / B ) x. ( B / A ) ) = 1 ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul c1 wceq divcanap6 syl22anc ) ABHIBJK
        LCHICJKLBCMNCBMNONPQDFEGBCRS $.

      $( Cancellation in a double division.  (Contributed by Jim Kingdon,
         3-Mar-2020.) $)
      ddcanapd $p |- ( ph -> ( A / ( A / B ) ) = B ) $=
        ( cc wcel cc0 cap wbr cdiv co wceq ddcanap syl22anc ) ABHIBJKLCHICJKLBB
        CMNMNCODFEGBCPQ $.

      rec11d.5 $e |- ( ph -> ( 1 / A ) = ( 1 / B ) ) $.
      $( Reciprocal is one-to-one.  (Contributed by Jim Kingdon,
         3-Mar-2020.) $)
      rec11apd $p |- ( ph -> A = B ) $=
        ( c1 cdiv co wceq cc wcel cc0 cap wbr wb rec11ap syl22anc mpbid ) AIBJK
        ICJKLZBCLZHABMNBOPQCMNCOPQUBUCRDFEGBCSTUA $.
    $}

    divmuld.3 $e |- ( ph -> C e. CC ) $.
    ${
      divmulapd.4 $e |- ( ph -> B =//= 0 ) $.
      $( Relationship between division and multiplication.  (Contributed by Jim
         Kingdon, 8-Mar-2020.) $)
      divmulapd $p |- ( ph -> ( ( A / B ) = C <-> ( B x. C ) = A ) ) $=
        ( cc wcel cc0 cap wbr cdiv co wceq cmul wb divmulap syl112anc ) ABIJDIJ
        CIJCKLMBCNODPCDQOBPREGFHBDCST $.

      $( Relationship between division and multiplication.  (Contributed by Jim
         Kingdon, 26-Dec-2022.) $)
      apdivmuld $p |- ( ph -> ( ( A / B ) =//= C <-> ( B x. C ) =//= A ) ) $=
        ( cdiv co cap wbr cmul cc wcel cc0 wb divclapd apmul1 syl112anc mulcomd
        divcanap1d breq12d mulcld apsym syl2anc 3bitrd ) ABCIJZDKLZUHCMJZDCMJZK
        LZBCDMJZKLZUMBKLZAUHNODNOCNOCPKLUIULQABCEFHRGFHUHDCSTAUJBUKUMKABCEFHUBA
        DCGFUAUCABNOUMNOUNUOQEACDFGUDBUMUEUFUG $.

      $( A commutative/associative law for division.  (Contributed by Jim
         Kingdon, 8-Mar-2020.) $)
      div32apd $p |- ( ph -> ( ( A / B ) x. C ) = ( A x. ( C / B ) ) ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq div32ap syl121anc ) ABIJCIJCKLM
        DIJBCNODPOBDCNOPOQEFHGBCDRS $.

      $( A commutative/associative law for division.  (Contributed by Jim
         Kingdon, 8-Mar-2020.) $)
      div13apd $p |- ( ph -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq div13ap syl121anc ) ABIJCIJCKLM
        DIJBCNODPODCNOBPOQEFHGBCDRS $.

      divdiv23apd.5 $e |- ( ph -> C =//= 0 ) $.
      $( Swap denominators in a division.  (Contributed by Jim Kingdon,
         8-Mar-2020.) $)
      divdiv32apd $p |- ( ph -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) $=
        ( cc wcel cc0 cap wbr cdiv co wceq divdiv32ap syl122anc ) ABJKCJKCLMNDJ
        KDLMNBCOPDOPBDOPCOPQEFHGIBCDRS $.

      $( Cancellation of common factor in a ratio.  (Contributed by Jim
         Kingdon, 8-Mar-2020.) $)
      divcanap5d $p |- ( ph -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) ) $=
        ( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap5 syl122anc ) ABJKCJKCL
        MNDJKDLMNDBOPDCOPQPBCQPREFHGIBCDST $.

      $( Cancellation of common factor in a ratio.  (Contributed by Jim
         Kingdon, 8-Mar-2020.) $)
      divcanap5rd $p |- ( ph -> ( ( A x. C ) / ( B x. C ) ) = ( A / B ) ) $=
        ( cmul co cdiv mulcomd oveq12d divcanap5d eqtrd ) ABDJKZCDJKZLKDBJKZDCJ
        KZLKBCLKAQSRTLABDEGMACDFGMNABCDEFGHIOP $.

      $( Cancel equal divisors in a division.  (Contributed by Jim Kingdon,
         8-Mar-2020.) $)
      divcanap7d $p |- ( ph -> ( ( A / C ) / ( B / C ) ) = ( A / B ) ) $=
        ( cc wcel cc0 cap wbr cdiv co wceq divcanap7 syl122anc ) ABJKCJKCLMNDJK
        DLMNBDOPCDOPOPBCOPQEFHGIBCDRS $.

      $( Cancellation law for division and multiplication.  (Contributed by Jim
         Kingdon, 8-Mar-2020.) $)
      dmdcanapd $p |- ( ph -> ( ( B / C ) x. ( A / B ) ) = ( A / C ) ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq dmdcanap syl221anc ) ACJKCLMNDJ
        KDLMNBJKCDOPBCOPQPBDOPRFHGIECDBST $.

      $( Cancellation law for division and multiplication.  (Contributed by Jim
         Kingdon, 8-Mar-2020.) $)
      dmdcanap2d $p |- ( ph -> ( ( A / B ) x. ( B / C ) ) = ( A / C ) ) $=
        ( cdiv co cmul divclapd mulcomd dmdcanapd eqtrd ) ABCJKZCDJKZLKRQLKBDJK
        AQRABCEFHMACDFGIMNABCDEFGHIOP $.

      $( Division into a fraction.  (Contributed by Jim Kingdon,
         8-Mar-2020.) $)
      divdivap1d $p |- ( ph -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq divdivap1 syl122anc ) ABJKCJKCL
        MNDJKDLMNBCOPDOPBCDQPOPREFHGIBCDST $.

      $( Division by a fraction.  (Contributed by Jim Kingdon, 8-Mar-2020.) $)
      divdivap2d $p |- ( ph -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq divdivap2 syl122anc ) ABJKCJKCL
        MNDJKDLMNBCDOPOPBDQPCOPREFHGIBCDST $.
    $}

    ${
      divassapd.4 $e |- ( ph -> C =//= 0 ) $.
      $( Relationship between division and multiplication.  (Contributed by Jim
         Kingdon, 2-Mar-2020.) $)
      divmulap2d $p |- ( ph -> ( ( A / C ) = B <-> A = ( C x. B ) ) ) $=
        ( cc wcel cc0 cap wbr cdiv co wceq cmul wb divmulap2 syl112anc ) ABIJCI
        JDIJDKLMBDNOCPBDCQOPREFGHBCDST $.

      $( Relationship between division and multiplication.  (Contributed by Jim
         Kingdon, 2-Mar-2020.) $)
      divmulap3d $p |- ( ph -> ( ( A / C ) = B <-> A = ( B x. C ) ) ) $=
        ( cc wcel cc0 cap wbr cdiv co wceq cmul wb divmulap3 syl112anc ) ABIJCI
        JDIJDKLMBDNOCPBCDQOPREFGHBCDST $.

      $( An associative law for division.  (Contributed by Jim Kingdon,
         2-Mar-2020.) $)
      divassapd $p |- ( ph -> ( ( A x. B ) / C ) = ( A x. ( B / C ) ) ) $=
        ( cc wcel cc0 cap wbr cmul co cdiv wceq divassap syl112anc ) ABIJCIJDIJ
        DKLMBCNODPOBCDPONOQEFGHBCDRS $.

      $( A commutative/associative law for division.  (Contributed by Jim
         Kingdon, 2-Mar-2020.) $)
      div12apd $p |- ( ph -> ( A x. ( B / C ) ) = ( B x. ( A / C ) ) ) $=
        ( cc wcel cc0 cap wbr cdiv co cmul wceq div12ap syl112anc ) ABIJCIJDIJD
        KLMBCDNOPOCBDNOPOQEFGHBCDRS $.

      $( A commutative/associative law for division.  (Contributed by Jim
         Kingdon, 2-Mar-2020.) $)
      div23apd $p |- ( ph -> ( ( A x. B ) / C ) = ( ( A / C ) x. B ) ) $=
        ( cc wcel cc0 cap wbr cmul co cdiv wceq div23ap syl112anc ) ABIJCIJDIJD
        KLMBCNODPOBDPOCNOQEFGHBCDRS $.

      $( Distribution of division over addition.  (Contributed by Jim Kingdon,
         2-Mar-2020.) $)
      divdirapd $p |- ( ph ->
          ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) ) $=
        ( cc wcel cc0 cap wbr caddc co cdiv wceq divdirap syl112anc ) ABIJCIJDI
        JDKLMBCNODPOBDPOCDPONOQEFGHBCDRS $.

      $( Distribution of division over subtraction.  (Contributed by Jim
         Kingdon, 2-Mar-2020.) $)
      divsubdirapd $p |- ( ph ->
          ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) ) $=
        ( cc wcel cc0 cap wbr cmin co cdiv wceq divsubdirap syl112anc ) ABIJCIJ
        DIJDKLMBCNODPOBDPOCDPONOQEFGHBCDRS $.

      div11apd.5 $e |- ( ph -> ( A / C ) = ( B / C ) ) $.
      $( One-to-one relationship for division.  (Contributed by Jim Kingdon,
         2-Mar-2020.) $)
      div11apd $p |- ( ph -> A = B ) $=
        ( cdiv co wceq cc wcel cc0 cap wbr wb div11ap syl112anc mpbid ) ABDJKCD
        JKLZBCLZIABMNCMNDMNDOPQUBUCREFGHBCDSTUA $.
    $}

    divmuldivapd.4 $e |- ( ph -> D e. CC ) $.
    divmuldivapd.5 $e |- ( ph -> B =//= 0 ) $.
    divmuldivapd.6 $e |- ( ph -> D =//= 0 ) $.
    $( Multiplication of two ratios.  (Contributed by Jim Kingdon,
       30-Jul-2021.) $)
    divmuldivapd $p |- ( ph -> ( ( A / B ) x. ( C / D ) ) =
        ( ( A x. C ) / ( B x. D ) ) ) $=
      ( cc wcel cc0 cap wbr wa cdiv co cmul wceq jca divmuldivap syl22anc ) ABL
      MDLMCLMZCNOPZQELMZENOPZQBCRSDERSTSBDTSCETSRSUAFHAUEUFGJUBAUGUHIKUBBDCEUCU
      D $.

    $( Cross-multiply in an equality of ratios.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    divmuleqapd $p |- ( ph ->
        ( ( A / B ) = ( C / D ) <-> ( A x. D ) = ( C x. B ) ) ) $=
      ( cc wcel cc0 cap wbr wa cdiv co wceq cmul wb jca divmuleqap syl22anc ) A
      BLMDLMCLMZCNOPZQELMZENOPZQBCRSDERSTBEUASDCUASTUBFHAUFUGGJUCAUHUIIKUCBDCEU
      DUE $.
  $}

  ${
    redivclapd.1 $e |- ( ph -> A e. RR ) $.
    ${
      rerecclapd.2 $e |- ( ph -> A =//= 0 ) $.
      $( Closure law for reciprocal.  (Contributed by Jim Kingdon,
         29-Feb-2020.) $)
      rerecclapd $p |- ( ph -> ( 1 / A ) e. RR ) $=
        ( cr wcel cc0 cap wbr c1 cdiv co rerecclap syl2anc ) ABEFBGHIJBKLEFCDBM
        N $.
    $}

    redivclapd.2 $e |- ( ph -> B e. RR ) $.
    redivclapd.3 $e |- ( ph -> B =//= 0 ) $.
    $( Closure law for division of reals.  (Contributed by Jim Kingdon,
       29-Feb-2020.) $)
    redivclapd $p |- ( ph -> ( A / B ) e. RR ) $=
      ( cr wcel cc0 cap wbr cdiv co redivclap syl3anc ) ABGHCGHCIJKBCLMGHDEFBCN
      O $.
  $}

  ${
    diveq1bd.1 $e |- ( ph -> B e. CC ) $.
    diveqap1bd.2 $e |- ( ph -> B =//= 0 ) $.
    diveq1bd.3 $e |- ( ph -> A = B ) $.
    $( If two complex numbers are equal, their quotient is one.  One-way
       deduction form of ~ diveqap1 .  Converse of ~ diveqap1d .  (Contributed
       by David Moews, 28-Feb-2017.)  (Revised by Jim Kingdon, 2-Aug-2023.) $)
    diveqap1bd $p |- ( ph -> ( A / B ) = 1 ) $=
      ( cdiv co c1 wceq cc eqeltrd diveqap1ad mpbird ) ABCGHIJBCJFABCABCKFDLDEM
      N $.
  $}

  $( Swap the order of subtraction in a division.  (Contributed by Scott
     Fenton, 24-Jun-2013.) $)
  div2subap $p |- ( ( ( A e. CC /\ B e. CC ) /\
    ( C e. CC /\ D e. CC /\ C =//= D ) ) ->
    ( ( A - B ) / ( C - D ) ) = ( ( B - A ) / ( D - C ) ) ) $=
    ( cc wcel wa cap wbr cmin co cneg cdiv cc0 wceq subcl 3adant3 caddc negcld
    wb w3a apneg biimp3a simp1 simp2 apadd2 syl3anc negidd negsubd 3brtr3d 0cnd
    mpbid apsym syl2anc jca div2negap 3expb syl2an negsubdi2 oveqan12d eqtr3d )
    AEFBEFGZCEFZDEFZCDHIZUAZGABJKZLZCDJKZLZMKZVGVIMKZBAJKZDCJKZMKVBVGEFZVIEFZVI
    NHIZGVKVLOZVFABPVFVPVQVCVDVPVECDPQZVFNVIHIZVQVFCCLZRKZCDLZRKZNVIHVFWAWCHIZW
    BWDHIZVCVDVEWECDUBUCVFWAEFWCEFVCWEWFTVFCVCVDVEUDZSVFDVCVDVEUEZSWGWAWCCUFUGU
    LVFCWGUHVFCDWGWHUIUJVFNEFVPVTVQTVFUKVSNVIUMUNULUOVOVPVQVRVGVIUPUQURVBVFVHVM
    VJVNMABUSVCVDVJVNOVECDUSQUTVA $.

  ${
    div2subd.1 $e |- ( ph -> A e. CC ) $.
    div2subd.2 $e |- ( ph -> B e. CC ) $.
    div2subd.3 $e |- ( ph -> C e. CC ) $.
    div2subd.4 $e |- ( ph -> D e. CC ) $.
    div2subd.5 $e |- ( ph -> C =//= D ) $.
    $( Swap subtrahend and minuend inside the numerator and denominator of a
       fraction.  Deduction form of ~ div2subap .  (Contributed by David Moews,
       28-Feb-2017.) $)
    div2subapd $p |- ( ph -> ( ( A - B ) / ( C - D ) ) =
                           ( ( B - A ) / ( D - C ) ) ) $=
      ( cc wcel cap wbr cmin co cdiv wceq div2subap syl23anc ) ABKLCKLDKLEKLDEM
      NBCOPDEOPQPCBOPEDOPQPRFGHIJBCDEST $.
  $}

  $( Subtraction of reciprocals.  (Contributed by Scott Fenton, 9-Jul-2015.) $)
  subrecap $p |- ( ( ( A e. CC /\ A =//= 0 ) /\ ( B e. CC /\ B =//= 0 ) ) ->
      ( ( 1 / A ) - ( 1 / B ) ) = ( ( B - A ) / ( A x. B ) ) ) $=
    ( cc wcel cc0 cap wa c1 cdiv co cmin cmul wceq 1cnd id divsubdivap syl21anc
    wbr simprl mullidd simpll oveq12d oveq1d eqtrd ) ACDZAEFRZGZBCDZBEFRZGZGZHA
    IJHBIJKJZHBLJZHALJZKJZABLJZIJZBAKJZUPIJUKHCDZUSUKULUQMUKNZUTUKOHHABPQUKUOUR
    UPIUKUMBUNAKUKBUGUHUISTUKAUEUFUJUATUBUCUD $.

  ${
    subreci.1 $e |- A e. CC $.
    subreci.2 $e |- B e. CC $.
    subrecapi.3 $e |- A =//= 0 $.
    subrecapi.4 $e |- B =//= 0 $.
    $( Subtraction of reciprocals.  (Contributed by Scott Fenton,
       9-Jan-2017.) $)
    subrecapi $p |- ( ( 1 / A ) - ( 1 / B ) ) = ( ( B - A ) / ( A x. B ) ) $=
      ( cc wcel cc0 cap wbr c1 cdiv co cmin cmul wceq subrecap mp4an ) AGHAIJKB
      GHBIJKLAMNLBMNONBAONABPNMNQCEDFABRS $.
  $}

  ${
    subrecd.1 $e |- ( ph -> A e. CC ) $.
    subrecd.2 $e |- ( ph -> B e. CC ) $.
    subrecapd.3 $e |- ( ph -> A =//= 0 ) $.
    subrecapd.4 $e |- ( ph -> B =//= 0 ) $.
    $( Subtraction of reciprocals.  (Contributed by Scott Fenton,
       9-Jan-2017.) $)
    subrecapd $p |- ( ph ->
      ( ( 1 / A ) - ( 1 / B ) ) = ( ( B - A ) / ( A x. B ) ) ) $=
      ( cc wcel cc0 cap wbr c1 cdiv co cmin cmul wceq subrecap syl22anc ) ABHIB
      JKLCHICJKLMBNOMCNOPOCBPOBCQONORDFEGBCST $.
  $}

  ${
    mvllmulapd.1 $e |- ( ph -> A e. CC ) $.
    mvllmulapd.2 $e |- ( ph -> B e. CC ) $.
    mvllmulapd.3 $e |- ( ph -> A =//= 0 ) $.
    mvllmulapd.4 $e |- ( ph -> ( A x. B ) = C ) $.
    $( Move LHS left multiplication to RHS. (Contributed by Jim Kingdon,
       10-Jun-2020.) $)
    mvllmulapd $p |- ( ph -> B = ( C / A ) ) $=
      ( cmul co cdiv divcanap4d mulcomd eqtr3d oveq1d ) ACBIJZBKJCDBKJACBFEGLAP
      DBKABCIJPDABCEFMHNON $.
  $}

  ${
    $d A x $.
    $( A real number has a multiplicative inverse if and only if it is apart
       from zero.  Theorem 11.2.4 of [HoTT], p.  (varies).  (Contributed by Jim
       Kingdon, 18-Jan-2025.) $)
    rerecapb $p |- ( A e. RR -> ( A =//= 0 <-> E. x e. RR ( A x. x ) = 1 ) ) $=
      ( cr wcel cc0 cap wbr cv cmul co c1 wceq wrex cdiv rerecclap recn recidap
      wa cc sylan oveq2 eqeq1d rspcev syl2anc ex wss wi ax-resscn ssrexv recapb
      ax-mp biimprd syl2im impbid ) BCDZBEFGZBAHZIJZKLZACMZUOUPUTUOUPRKBNJZCDBV
      AIJZKLZUTBOUOBSDZUPVCBPZBQTUSVCAVACUQVALURVBKUQVABIUAUBUCUDUEUOVDUTUSASMZ
      UPVECSUFUTVFUGUHUSACSUIUKVDUPVFABUJULUMUN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordering on reals (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( A number is less than itself plus 1.  (Contributed by NM, 20-Aug-2001.) $)
  ltp1 $p |- ( A e. RR -> A < ( A + 1 ) ) $=
    ( c1 cr wcel caddc co clt wbr 1re wa cc0 0lt1 ltaddpos mpbii mpan ) BCDZACD
    ZAABEFGHZIPQJKBGHRLBAMNO $.

  $( A number is less than or equal to itself plus 1.  (Contributed by NM,
     5-Jan-2006.) $)
  lep1 $p |- ( A e. RR -> A <_ ( A + 1 ) ) $=
    ( cr wcel c1 caddc co clt wbr cle ltp1 wi peano2re ltle mpdan mpd ) ABCZAAD
    EFZGHZAQIHZAJPQBCRSKALAQMNO $.

  $( A number minus 1 is less than itself.  (Contributed by NM, 9-Apr-2006.) $)
  ltm1 $p |- ( A e. RR -> ( A - 1 ) < A ) $=
    ( cr wcel c1 cmin co cc0 clt wbr 0lt1 wb 0re 1re mp3an12 mpbii recn subid1d
    ltsub2 breqtrd ) ABCZADEFZAGEFZAHTGDHIZUAUBHIZJGBCDBCTUCUDKLMGDARNOTAAPQS
    $.

  $( A number minus 1 is less than or equal to itself.  (Contributed by Mario
     Carneiro, 2-Oct-2015.) $)
  lem1 $p |- ( A e. RR -> ( A - 1 ) <_ A ) $=
    ( cr wcel c1 cmin co clt wbr cle ltm1 wi peano2rem ltle mpancom mpd ) ABCZA
    DEFZAGHZQAIHZAJQBCPRSKALQAMNO $.

  $( A transitive property of 'less than or equal' and plus 1.  (Contributed by
     NM, 5-Aug-2005.) $)
  letrp1 $p |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> A <_ ( B + 1 ) ) $=
    ( cr wcel cle wbr w3a c1 caddc co clt wa adantl peano2re ancli lelttr 3expb
    ltp1 wi sylan2 mpan2d 3impia ltle 3adant3 mpd ) ACDZBCDZABEFZGABHIJZKFZAUIE
    FZUFUGUHUJUFUGLUHBUIKFZUJUGULUFBRMUGUFUGUICDZLUHULLUJSZUGUMBNZOUFUGUMUNABUI
    PQTUAUBUFUGUJUKSZUHUGUFUMUPUOAUIUCTUDUE $.

  $( A transitive property of plus 1 and 'less than or equal'.  (Contributed by
     NM, 16-Aug-2005.) $)
  p1le $p |- ( ( A e. RR /\ B e. RR /\ ( A + 1 ) <_ B ) -> A <_ B ) $=
    ( cr wcel c1 caddc co cle wa lep1 adantr wi peano2re ancli letr 3expa sylan
    wbr mpand 3impia ) ACDZBCDZAEFGZBHRZABHRZUAUBIAUCHRZUDUEUAUFUBAJKUAUAUCCDZI
    UBUFUDIUELZUAUGAMNUAUGUBUHAUCBOPQST $.

  $( The reciprocal of a positive number is positive.  Exercise 4 of [Apostol]
     p. 21.  (Contributed by NM, 25-Aug-1999.)  (Revised by Mario Carneiro,
     27-May-2016.) $)
  recgt0 $p |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) ) $=
    ( cr wcel cc0 clt wbr wa c1 co cap 0re cneg cmul adantr rerecclapd lt0neg1d
    wn cc syl2anc wb cdiv cle 0lt1 1re ax-mp simpll gt0ap0 renegcld simpr simpl
    ltnsymi mpbid simplr mulgt0d recnd recclap mulneg1d recidap2 negeqd breqtrd
    wceq eqtrd 1red mpbird ex mtoi lenlt sylancr recap0 apsym sylancl mpbir2and
    0cn ltleap ) ABCZDAEFZGZDHAUAIZEFZDVRUBFZDVRJFZVQVTVRDEFZQZVQWBHDEFZDHEFWDQ
    UCDHKUDUKUEVQWBWDVQWBGZWDDHLZEFWEDVRLZAMIZWFEWEWGAWEVRWEAVOVPWBUFZVQADJFZWB
    AUGZNZOUHWIWEWBDWGEFVQWBUIWEVRVQVRBCZWBVQAVOVPUJZWKOZNPULVOVPWBUMUNWEWHVRAM
    IZLWFWEVRAWEARCZWJVRRCZVQWQWBVQAWNUOZNZWLAUPZSWTUQWEWPHWEWQWJWPHVAWTWLAURSU
    SVBUTWEHWEVCPVDVEVFVQDBCZWMVTWCTKWODVRVGVHVDVQVRDJFZWAVQWQWJXCWSWKAVISVQWRD
    RCXCWATVQWQWJWRWSWKXASVMVRDVJVKULVQXBWMVSVTWAGTKWODVRVNVHVL $.

  $( Infer that a multiplicand is positive from a positive multiplier and
     positive product.  See ~ prodgt0 for the same theorem with ` 0 < A `
     replaced by the weaker condition ` 0 <_ A ` .  (Contributed by Jim
     Kingdon, 29-Feb-2020.) $)
  prodgt0gt0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) )
                 -> 0 < B ) $=
    ( cr wcel wa cc0 clt wbr cmul co cdiv simpll simplr remulcld simprl gt0ap0d
    c1 rerecclapd simprr recnd recgt0 mulgt0d divrecapd simpr adantr divcanap3d
    ad2ant2r cc eqtr3d breqtrd ) ACDZBCDZEZFAGHZFABIJZGHZEZEZFUOQAKJZIJZBGURUOU
    SURABUKULUQLZUKULUQMNZURAVAURAVAUMUNUPOPZRUMUNUPSUKUNFUSGHULUPAUAUGUBURUOAK
    JUTBURUOAURUOVBTURAVATZVCUCURBAUMBUHDUQUMBUKULUDTUEVDVCUFUIUJ $.

  $( Infer that a multiplicand is positive from a nonnegative multiplier and
     positive product.  (Contributed by NM, 24-Apr-2005.)  (Revised by Mario
     Carneiro, 27-May-2016.) $)
  prodgt0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) )
                 -> 0 < B ) $=
    ( cr wcel wa cc0 cle wbr cmul co clt wn cneg renegcld lt0neg1d recnd negcld
    cap mpbird 0red simpllr simplll simplr biimpa simprr simpll mulcomd breqtrd
    mul2negd breqtrrd adantr prodgt0gt0 syl22anc lenltd mpbid pm2.65da remulcld
    simplrl gt0ap0d mulap0bbd cc wb 0cnd apsym syl2anc ltleap mpbir2and ) ACDZB
    CDZEZFAGHZFABIJZKHZEZEZFBKHZFBGHZFBRHZVOVQBFKHZLVOVSAFKHZVOVSEZVTFAMZKHZWAB
    MZCDWBCDFWDKHZFWDWBIJZKHZWCWABVHVIVNVSUANWAAVHVIVNVSUBZNVOVSWEVOBVHVIVNUCZO
    UDVOWGVSVOFWBWDIJZWFKVOFVLWJKVJVKVMUEZVOABVOAVHVIVNUFZPZVOBWIPZUIUJVOWBWDVO
    AWMQVOBWNQUGUHUKWDWBULUMWAAWHOSWAVKVTLVJVKVMVSURWAFAWATWHUNUOUPVOFBVOTZWIUN
    SVOBFRHZVRVOABWMWNVOVLVOABWLWIUQWKUSUTVOBVADFVADWPVRVBWNVOVCBFVDVEUOVOFCDVI
    VPVQVREVBWOWIFBVFVEVG $.

  $( Infer that a multiplier is positive from a nonnegative multiplicand and
     positive product.  (Contributed by NM, 24-Apr-2005.) $)
  prodgt02 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ B /\ 0 < ( A x. B ) ) )
                 -> 0 < A ) $=
    ( cr wcel wa cc0 cle wbr cmul co clt cc wceq mulcom syl2an breq2d biimpd wi
    recn prodgt0 ex ancoms sylan2d imp ) ACDZBCDZEZFBGHZFABIJZKHZEFAKHZUGUJFBAI
    JZKHZUHUKUGUJUMUGUIULFKUEALDBLDUIULMUFASBSABNOPQUFUEUHUMEZUKRUFUEEUNUKBATUA
    UBUCUD $.

  $( Infer that a multiplicand is nonnegative from a positive multiplier and
     nonnegative product.  (Contributed by NM, 2-Jul-2005.)  (Revised by Mario
     Carneiro, 27-May-2016.) $)
  prodge0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ ( A x. B ) ) )
                 -> 0 <_ B ) $=
    ( cr wcel wa cc0 clt wbr cmul co cle wn cneg simpll renegcld recnd lt0neg1d
    simplr 3imtr4d lenltd simprl simprr mulgt0d mulneg2d breqtrd remulcld con3d
    expr 0red impr ) ACDZBCDZEZFAGHZFABIJZKHZFBKHZUMUNEZUOFGHZLBFGHZLUPUQURUTUS
    URFBMZGHZFUOMZGHZUTUSUMUNVBVDUMUNVBEZEZFAVAIJVCGVFAVAUKULVENZVFBUKULVERZOUM
    UNVBUAUMUNVBUBUCVFABVFAVGPVFBVHPUDUEUHURBUKULUNRZQURUOURABUKULUNNVIUFZQSUGU
    RFUOURUIZVJTURFBVKVITSUJ $.

  $( Infer that a multiplier is nonnegative from a positive multiplicand and
     nonnegative product.  (Contributed by NM, 2-Jul-2005.) $)
  prodge02 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < B /\ 0 <_ ( A x. B ) ) )
                 -> 0 <_ A ) $=
    ( cr wcel wa cc0 clt wbr cmul co cle cc wceq mulcom syl2an breq2d biimpd wi
    recn prodge0 ex ancoms sylan2d imp ) ACDZBCDZEZFBGHZFABIJZKHZEFAKHZUGUJFBAI
    JZKHZUHUKUGUJUMUGUIULFKUEALDBLDUIULMUFASBSABNOPQUFUEUHUMEZUKRUFUEEUNUKBATUA
    UBUCUD $.

  $( Multiplication of both sides of 'less than' by a positive number.  Theorem
     I.19 of [Apostol] p. 20.  (Contributed by NM, 13-Feb-2005.) $)
  ltmul2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
                -> ( A < B <-> ( C x. A ) < ( C x. B ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cmul co ltmul1 wb cc recn mulcom sylan 3adant2
    wceq 3adant1 breq12d syl3an3 3adant3r bitrd ) ADEZBDEZCDEZFCGHZIJABGHACKLZB
    CKLZGHZCAKLZCBKLZGHZABCMUFUGUHULUONZUIUHUFUGCOEZUPCPUFUGUQJUJUMUKUNGUFUQUJU
    MTZUGUFAOEUQURAPACQRSUGUQUKUNTZUFUGBOEUQUSBPBCQRUAUBUCUDUE $.

  $( Multiplication of both sides of 'less than or equal to' by a positive
     number.  (Contributed by NM, 16-Mar-2005.) $)
  lemul2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
               -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cle cmul co lemul1 wb wceq recn mulcom syl2an
    cc 3adant2 3adant1 breq12d 3adant3r bitrd ) ADEZBDEZCDEZFCGHZIJABKHACLMZBCL
    MZKHZCALMZCBLMZKHZABCNUFUGUHULUOOUIUFUGUHJUJUMUKUNKUFUHUJUMPZUGUFATECTEZUPU
    HAQCQZACRSUAUGUHUKUNPZUFUGBTEUQUSUHBQURBCRSUBUCUDUE $.

  $( Multiplication of both sides of 'less than or equal to' by a nonnegative
     number.  Part of Definition 11.2.7(vi) of [HoTT], p.  (varies).
     (Contributed by NM, 21-Feb-2005.) $)
  lemul1a $p |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 <_ C ) ) /\
                 A <_ B ) -> ( A x. C ) <_ ( B x. C ) ) $=
    ( cr wcel cc0 cle wbr wa w3a cmul co simpl2 simpl1 resubcld simpl3l subge0d
    cmin recnd remulcld simpr mpbird simpl3r mulge0d subdird breqtrd mpbid ) AD
    EZBDEZCDEZFCGHZIZJZABGHZIZFBCKLZACKLZRLZGHUQUPGHUOFBARLZCKLURGUOUSCUOBAUHUI
    ULUNMZUHUIULUNNZOUJUKUHUIUNPZUOFUSGHUNUMUNUAUOBAUTVAQUBUJUKUHUIUNUCUDUOBACU
    OBUTSUOAVASUOCVBSUEUFUOUPUQUOBCUTVBTUOACVAVBTQUG $.

  $( Multiplication of both sides of 'less than or equal to' by a nonnegative
     number.  (Contributed by Paul Chapman, 7-Sep-2007.) $)
  lemul2a $p |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 <_ C ) ) /\
                      A <_ B ) -> ( C x. A ) <_ ( C x. B ) ) $=
    ( cr wcel cc0 cle wbr wa cmul co lemul1a wceq cc recn mulcom syl2an adantrr
    w3a adantr 3adant2 3adant1 3brtr3d ) ADEZBDEZCDEZFCGHZIZSZABGHZIACJKZBCJKZC
    AJKZCBJKZGABCLUIUKUMMZUJUDUHUOUEUDUFUOUGUDANECNEZUOUFAOCOZACPQRUATUIULUNMZU
    JUEUHURUDUEUFURUGUEBNEUPURUFBOUQBCPQRUBTUC $.

  $( Comparison of product of two positive numbers.  (Contributed by NM,
     30-Dec-2005.) $)
  ltmul12a $p |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\
               ( ( C e. RR /\ D e. RR ) /\ ( 0 <_ C /\ C < D ) ) ) ->
                 ( A x. C ) < ( B x. D ) ) $=
    ( cr wcel cc0 cle wbr clt cmul simpllr ad2ant2l imp ad2ant2r lelttr remulcl
    wa co wi simplll simpll simprl jca ltle adantrl lemul1a syl31anc wb simplrl
    simplrr 0re mp3an1 adantlr ltmul2 syl112anc biimpa anasss ad2ant2lr syl3anc
    adantrrl adantr mp2and an4s ) AEFZBEFZRZCEFZDEFZRZGAHIZABJIZRZGCHIZCDJIZRZA
    CKSZBDKSZJIZVGVJRZVMVPRZRZVQBCKSZHIZWCVRJIZVSWBVEVFVHVNRZABHIZWDVEVFVJWAUAV
    EVFVJWALVJVPWFVGVMVJVPRVHVNVHVIVPUBVJVNVOUCUDMVGVMWGVJVPVGVLWGVKVGVLWGABUEN
    UFOABCUGUHVTVMVOWEVNVTVMVOWEVTVMRZVOWEWHVHVIVFGBJIZVOWEUIVGVHVIVMUJVGVHVIVM
    UKVEVFVJVMLVGVMWIVJVGVMWIGEFVEVFVMWITULGABPUMNUNCDBUOUPUQURVAVTWDWERVSTZWAV
    TVQEFZWCEFZVREFZWJVEVHWKVFVIACQOVFVHWLVEVIBCQUSVFVIWMVEVHBDQMVQWCVRPUTVBVCV
    D $.

  $( Comparison of product of two nonnegative numbers.  (Contributed by NM,
     22-Feb-2008.) $)
  lemul12b $p |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR ) /\
       ( C e. RR /\ ( D e. RR /\ 0 <_ D ) ) ) -> ( ( A <_ B /\ C <_ D ) ->
                 ( A x. C ) <_ ( B x. D ) ) ) $=
    ( cr wcel cc0 cle wbr wa cmul co wi w3a lemul2a ex adantlr remulcl ad2ant2r
    3comr 3expb adantrrr lemul1a 3expa adantllr adantrl anim12d ancomsd adantrr
    ad2ant2rl ad2ant2l letr syl3anc syld ) AEFZGAHIZJZBEFZJZCEFZDEFZGDHIZJZJZJZ
    ABHIZCDHIZJACKLZADKLZHIZVIBDKLZHIZJZVHVKHIZVEVGVFVMVEVGVJVFVLUQVDVGVJMZURUQ
    UTVAVOVBUQUTVAVOUTVAUQVOUTVAUQNVGVJCDAOPTUAUBQUSVCVFVLMZUTUOURVCVPUPUOURVCV
    PUOURVCNVFVLABDUCPUDUEUFUGUHVEVHEFZVIEFZVKEFZVMVNMUQUTVQURVCUOUTVQUPACRQSUQ
    VCVRURUTUOVAVRUPVBADRSUJURVCVSUQUTURVAVSVBBDRUIUKVHVIVKULUMUN $.

  $( Comparison of product of two nonnegative numbers.  (Contributed by NM,
     22-Feb-2008.) $)
  lemul12a $p |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR ) /\
       ( ( C e. RR /\ 0 <_ C ) /\ D e. RR ) ) -> ( ( A <_ B /\ C <_ D ) ->
                 ( A x. C ) <_ ( B x. D ) ) ) $=
    ( cr wcel cc0 cle wbr wa cmul co simpll ad2antlr simplrr wi 0re letr mp3an1
    exp4b com23 imp41 ad2ant2l jca jca32 simpr lemul12b sylc ex ) AEFGAHIJBEFJZ
    CEFZGCHIZJZDEFZJZJZABHIZCDHIZJZACKLBDKLHIZUPUSJZUJUKUNGDHIZJZJJUSUTVAUJUKVC
    UJUOUSMUOUKUJUSUKULUNMNVAUNVBUJUMUNUSOUOURVBUJUQUKULUNURVBUKUNULURVBPUKUNUL
    URVBGEFUKUNULURJVBPQGCDRSTUAUBUCUDUEUPUSUFABCDUGUHUI $.

  $( The product of two numbers greater than 1 is greater than 1.  (Contributed
     by NM, 13-Feb-2005.) $)
  mulgt1 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) ->
                1 < ( A x. B ) ) $=
    ( cr wcel wa c1 clt wbr cmul co wi a1i cc0 0lt1 1re lttr adantr mp3an1 syld
    simpl 0re mp3an12 mpani ltmul2 biimpd exp32 impcom impd wceq ax-1rid breq1d
    w3a sylibd jcad remulcl syldan imp ) ACDZBCDZEZFAGHZFBGHZEZFABIJZGHZUTVCVAA
    VDGHZEZVEUTVCVAVFVCVAKUTVAVBTLUTVCAFIJZVDGHZVFUTVAVBVIUTVAMAGHZVBVIKZURVAVJ
    KUSURMFGHZVAVJNMCDFCDZURVLVAEVJKUAOMFAPUBUCQUSURVJVKKUSURVJVKVMUSURVJEZVKOV
    MUSVNULVBVIFBAUDUERUFUGSUHUTVHAVDGURVHAUIUSAUJQUKUMUNURUSVDCDZVGVEKZABUOVMU
    RVOVPOFAVDPRUPSUQ $.

  $( Multiplication by a number greater than 1.  (Contributed by NM,
     24-Dec-2005.) $)
  ltmulgt11 $p |- ( ( A e. RR /\ B e. RR /\ 0 < A ) ->
                ( 1 < B <-> A < ( A x. B ) ) ) $=
    ( cr wcel cc0 clt wbr w3a c1 cmul co wb 1re ltmul2 mp3an1 3impb 3com12 wceq
    wa ax-1rid 3ad2ant1 breq1d bitrd ) ACDZBCDZEAFGZHZIBFGZAIJKZABJKZFGZAUJFGUE
    UDUFUHUKLZUEUDUFULICDUEUDUFSULMIBANOPQUGUIAUJFUDUEUIARUFATUAUBUC $.

  $( Multiplication by a number greater than 1.  (Contributed by NM,
     24-Dec-2005.) $)
  ltmulgt12 $p |- ( ( A e. RR /\ B e. RR /\ 0 < A ) ->
                ( 1 < B <-> A < ( B x. A ) ) ) $=
    ( cr wcel cc0 clt wbr w3a c1 cmul ltmulgt11 wceq recn mulcom syl2an 3adant3
    co cc breq2d bitrd ) ACDZBCDZEAFGZHZIBFGAABJQZFGABAJQZFGABKUDUEUFAFUAUBUEUF
    LZUCUAARDBRDUGUBAMBMABNOPST $.

  $( Multiplication by a number greater than or equal to 1.  (Contributed by
     NM, 17-Dec-2005.) $)
  lemulge11 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 1 <_ B ) ) ->
                A <_ ( A x. B ) ) $=
    ( cr wcel wa cc0 cle wbr c1 cmul co wceq ax-1rid ad2antrr simpll simprl jca
    simplr 1re 0le1 pm3.2i jctil jca31 leid simprr lemul12a sylc eqbrtrrd ) ACD
    ZBCDZEZFAGHZIBGHZEZEZAIJKZAABJKZGUIUPALUJUNAMNUOUIULEZUIEICDZFIGHZEZUJEZEAA
    GHZUMEUPUQGHUOURUIVBUOUIULUIUJUNOZUKULUMPQVDUOUJVAUIUJUNRUSUTSTUAUBUCUOVCUM
    UIVCUJUNAUDNUKULUMUEQAAIBUFUGUH $.

  $( Multiplication by a number greater than or equal to 1.  (Contributed by
     Paul Chapman, 21-Mar-2011.) $)
  lemulge12 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 1 <_ B ) ) ->
                    A <_ ( B x. A ) ) $=
    ( cr wcel wa cc0 cle wbr c1 cmul co lemulge11 wb cc wceq recn mulcom syl2an
    breq2d adantr mpbid ) ACDZBCDZEZFAGHIBGHEZEAABJKZGHZABAJKZGHZABLUDUGUIMUEUD
    UFUHAGUBANDBNDUFUHOUCAPBPABQRSTUA $.

  $( Division of both sides of 'less than' by a positive number.  (Contributed
     by NM, 10-Oct-2004.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  ltdiv1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
               ( A < B <-> ( A / C ) < ( B / C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a c1 cdiv co cmul wb simp1 simp2 recnd divrecapd
    simp3l simp3r gt0ap0d rerecclapd recgt0 3ad2ant3 ltmul1 syl112anc breq12d
    bitr4d ) ADEZBDEZCDEZFCGHZIZJZABGHZAKCLMZNMZBUQNMZGHZACLMZBCLMZGHUOUJUKUQDE
    FUQGHZUPUTOUJUKUNPZUJUKUNQZUOCUJUKULUMTZUOCVFUJUKULUMUAUBZUCUNUJVCUKCUDUEAB
    UQUFUGUOVAURVBUSGUOACUOAVDRUOCVFRZVGSUOBCUOBVERVHVGSUHUI $.

  $( Division of both sides of a less than or equal to relation by a positive
     number.  (Contributed by NM, 18-Nov-2004.) $)
  lediv1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
               ( A <_ B <-> ( A / C ) <_ ( B / C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a wn co cle wb ltdiv1 3adant1 redivclap syld3an3
    cdiv 3expb 3com12 notbid lenlt 3adant3 cap gt0ap0 3adant2 lenltd 3bitr4d )
    ADEZBDEZCDEZFCGHZIZJZBAGHZKZBCSLZACSLZGHZKABMHZUSURMHUOUPUTUKUJUNUPUTNBACOU
    AUBUJUKVAUQNUNABUCUDUOUSURUJUNUSDEZUKUJULUMVBUJULUMCFUEHZVBULUMVCUJCUFZPACQ
    RTUGUKUNURDEZUJUKULUMVEUKULUMVCVEULUMVCUKVDPBCQRTPUHUI $.

  $( Division of a positive number by a positive number.  (Contributed by NM,
     28-Sep-2005.) $)
  gt0div $p |- ( ( A e. RR /\ B e. RR /\ 0 < B ) ->
               ( 0 < A <-> 0 < ( A / B ) ) ) $=
    ( cr wcel cc0 clt wbr w3a cdiv co wb wa 0re ltdiv1 mp3an1 3impb wceq gt0ap0
    cap cc recn div0ap sylan syldan breq1d 3adant1 bitrd ) ACDZBCDZEBFGZHEAFGZE
    BIJZABIJZFGZEUMFGZUHUIUJUKUNKZECDUHUIUJLZUPMEABNOPUIUJUNUOKUHUQULEUMFUIUJBE
    SGZULEQZBRUIBTDURUSBUABUBUCUDUEUFUG $.

  $( Division of a nonnegative number by a positive number.  (Contributed by
     NM, 28-Sep-2005.) $)
  ge0div $p |- ( ( A e. RR /\ B e. RR /\ 0 < B ) ->
               ( 0 <_ A <-> 0 <_ ( A / B ) ) ) $=
    ( cr wcel cc0 clt wbr w3a cle cdiv co wb 0re lediv1 mp3an1 3impb cap gt0ap0
    wa wceq cc recn div0ap sylan syldan breq1d 3adant1 bitrd ) ACDZBCDZEBFGZHEA
    IGZEBJKZABJKZIGZEUNIGZUIUJUKULUOLZECDUIUJUKSZUQMEABNOPUJUKUOUPLUIURUMEUNIUJ
    UKBEQGZUMETZBRUJBUADUSUTBUBBUCUDUEUFUGUH $.

  $( The ratio of two positive numbers is positive.  (Contributed by NM,
     12-Oct-1999.) $)
  divgt0 $p |- ( ( ( A e. RR /\ 0 < A ) /\
             ( B e. RR /\ 0 < B ) ) -> 0 < ( A / B ) ) $=
    ( cr wcel cc0 clt wbr cdiv co wi w3a gt0div biimpd 3exp com34 com23 imp43 )
    ACDZEAFGZBCDZEBFGZEABHIFGZRTSUAUBJRTUASUBRTUASUBJRTUAKSUBABLMNOPQ $.

  $( The ratio of nonnegative and positive numbers is nonnegative.
     (Contributed by NM, 27-Sep-1999.) $)
  divge0 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\
             ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) ) $=
    ( cr wcel cc0 cle wbr clt cdiv co w3a ge0div biimpd 3exp com34 com23 imp43
    wi ) ACDZEAFGZBCDZEBHGZEABIJFGZSUATUBUCRSUAUBTUCSUAUBTUCRSUAUBKTUCABLMNOPQ
    $.

  $( 'Less than' relationship between division and multiplication.
     (Contributed by NM, 12-Oct-1999.)  (Proof shortened by Mario Carneiro,
     27-May-2016.) $)
  ltmuldiv $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
                -> ( ( A x. C ) < B <-> A < ( B / C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cmul co cdiv wb simp1 simp3l remulcld syld3an1
    ltdiv1 recnd simp3r gt0ap0d divcanap4d breq1d bitrd ) ADEZBDEZCDEZFCGHZIZJZ
    ACKLZBGHZULCMLZBCMLZGHZAUOGHULDEUGUFUJUMUPNUKACUFUGUJOZUFUGUHUIPZQULBCSRUKU
    NAUOGUKACUKAUQTUKCURTUKCURUFUGUHUIUAUBUCUDUE $.

  $( 'Less than' relationship between division and multiplication.
     (Contributed by NM, 18-Nov-2004.) $)
  ltmuldiv2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
                   -> ( ( C x. A ) < B <-> A < ( B / C ) ) ) $=
    ( cr wcel cc0 clt wbr wa cmul co cdiv wceq cc mulcom syl2an adantrr 3adant2
    w3a recn breq1d ltmuldiv bitr3d ) ADEZBDEZCDEZFCGHZIZSZACJKZBGHCAJKZBGHABCL
    KGHUIUJUKBGUDUHUJUKMZUEUDUFULUGUDANECNEULUFATCTACOPQRUAABCUBUC $.

  $( 'Less than' relationship between division and multiplication.
     (Contributed by NM, 18-Nov-2004.) $)
  ltdivmul $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
              -> ( ( A / C ) < B <-> A < ( C x. B ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cmul co cdiv wb remulcl ancoms adantrr 3adant1
    cc recn ltdiv1 syld3an2 adantr ad2antrl cap gt0ap0 adantl divcanap3d breq2d
    wceq bitr2d ) ADEZBDEZCDEZFCGHZIZJZACBKLZGHZACMLZURCMLZGHZUTBGHULURDEZUMUPU
    SVBNUMUPVCULUMUNVCUOUNUMVCCBOPQRAURCUAUBUQVABUTGUMUPVABUJULUMUPIBCUMBSEUPBT
    UCUNCSEUMUOCTUDUPCFUEHUMCUFUGUHRUIUK $.

  $( 'Less than or equal to' relationship between division and multiplication.
     (Contributed by NM, 9-Dec-2005.) $)
  ledivmul $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
                  ( ( A / C ) <_ B <-> A <_ ( C x. B ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cmul co cle simp3l simp2 remulcld lediv1 recnd
    cdiv wb syld3an2 simp3r gt0ap0d divcanap3d breq2d bitr2d ) ADEZBDEZCDEZFCGH
    ZIZJZACBKLZMHZACSLZUMCSLZMHZUOBMHUGUMDEUHUKUNUQTULCBUGUHUIUJNZUGUHUKOZPAUMC
    QUAULUPBUOMULBCULBUSRULCURRULCURUGUHUIUJUBUCUDUEUF $.

  $( 'Less than' relationship between division and multiplication.
     (Contributed by NM, 24-Feb-2005.) $)
  ltdivmul2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
                 -> ( ( A / C ) < B <-> A < ( B x. C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cdiv co cmul ltdivmul wceq recn mulcom adantrr
    cc syl2an 3adant1 breq2d bitr4d ) ADEZBDEZCDEZFCGHZIZJZACKLBGHACBMLZGHABCML
    ZGHABCNUIUKUJAGUEUHUKUJOZUDUEUFULUGUEBSECSEULUFBPCPBCQTRUAUBUC $.

  $( 'Less than' relationship between division and multiplication.
     (Contributed by NM, 8-Jan-2006.) $)
  lt2mul2div $p |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) )
                   /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) ->
                    ( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) ) $=
    ( cr wcel cc0 clt wbr wa cmul co cdiv simprl simprrl mulcomd oveq1d syl3anc
    recnd wb simplrl simplrr gt0ap0d divassapd eqtrd breq2d remulcld redivclapd
    simpll simplr ltmuldiv simprr ltdivmul 3bitr4d ) AEFZBEFZGBHIZJZJZCEFZDEFZG
    DHIZJZJZJZACDKLZBMLZHIZADCBMLZKLZHIZABKLVFHIZADMLVIHIZVEVGVJAHVEVGDCKLZBMLV
    JVEVFVNBMVECDVECUSUTVCNZSZVEDUSUTVAVBOZSZPQVEDCBVRVPVEBUOUPUQVDUAZSVEBVSUOU
    PUQVDUBUCZUDUEUFVEUOVFEFURVLVHTUOURVDUIZVECDVOVQUGUOURVDUJAVFBUKRVEUOVIEFVC
    VMVKTWAVECBVOVSVTUHUSUTVCULAVIDUMRUN $.

  $( 'Less than or equal to' relationship between division and multiplication.
     (Contributed by NM, 9-Dec-2005.) $)
  ledivmul2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
                  ( ( A / C ) <_ B <-> A <_ ( B x. C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cdiv cle cmul ledivmul wceq recn mulcom syl2an
    co cc adantrr 3adant1 breq2d bitr4d ) ADEZBDEZCDEZFCGHZIZJZACKSBLHACBMSZLHA
    BCMSZLHABCNUJULUKALUFUIULUKOZUEUFUGUMUHUFBTECTEUMUGBPCPBCQRUAUBUCUD $.

  $( 'Less than or equal' relationship between division and multiplication.
     (Contributed by NM, 10-Mar-2006.) $)
  lemuldiv $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
                  ( ( A x. C ) <_ B <-> A <_ ( B / C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cdiv co wn cmul cle wb ltdivmul2 3com12 notbid
    lenltd simp1 simp2 simp3l simp3r gt0ap0d redivclapd remulcld 3bitr4rd ) ADE
    ZBDEZCDEZFCGHZIZJZBCKLZAGHZMBACNLZGHZMAUOOHUQBOHUNUPURUJUIUMUPURPBACQRSUNAU
    OUIUJUMUAZUNBCUIUJUMUBZUIUJUKULUCZUNCVAUIUJUKULUDUEUFTUNUQBUNACUSVAUGUTTUH
    $.

  $( 'Less than or equal' relationship between division and multiplication.
     (Contributed by NM, 10-Mar-2006.) $)
  lemuldiv2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
                      ( ( C x. A ) <_ B <-> A <_ ( B / C ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cmul co cle cdiv wceq cc mulcom syl2an adantrr
    recn 3adant2 breq1d lemuldiv bitr3d ) ADEZBDEZCDEZFCGHZIZJZACKLZBMHCAKLZBMH
    ABCNLMHUJUKULBMUEUIUKULOZUFUEUGUMUHUEAPECPEUMUGATCTACQRSUAUBABCUCUD $.

  $( The reciprocal of both sides of 'less than'.  (Contributed by NM,
     26-Sep-1999.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  ltrec $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) ->
              ( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) $=
    ( cr wcel cc0 clt wbr wa c1 cdiv co cmul 1red simprl simpll simplr ltmuldiv
    wb syl112anc recnd mullidd breq1d gt0ap0d divrecapd breq2d 3bitr3d ltdivmul
    rerecclapd simprr bitr4d ) ACDZEAFGZHZBCDZEBFGZHZHZABFGZIBIAJKZLKZFGZIBJKUS
    FGZUQIALKZBFGZIBAJKZFGZURVAUQICDZUNUKULVDVFRUQMZUMUNUONZUKULUPOZUKULUPPZIBA
    QSUQVCABFUQAUQAVJTZUAUBUQVEUTIFUQBAUQBVITVLUQAVJVKUCZUDUEUFUQVGUSCDUNUOVBVA
    RVHUQAVJVMUHVIUMUNUOUIIUSBUGSUJ $.

  $( The reciprocal of both sides of 'less than or equal to'.  (Contributed by
     NM, 3-Oct-1999.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  lerec $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) ->
              ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) ) $=
    ( cr wcel cc0 clt wbr wa wn c1 co cle wb ltrec ancoms notbid lenltd gt0ap0d
    cdiv rerecclapd simpll simprl simprr simplr 3bitr4d ) ACDZEAFGZHZBCDZEBFGZH
    ZHZBAFGZIJASKZJBSKZFGZIABLGUOUNLGULUMUPUKUHUMUPMBANOPULABUFUGUKUAZUHUIUJUBZ
    QULUOUNULBURULBURUHUIUJUCRTULAUQULAUQUFUGUKUDRTQUE $.

  $( Lemma for ~ lt2msq .  (Contributed by Mario Carneiro, 27-May-2016.) $)
  lt2msq1 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) ->
                  ( A x. A ) < ( B x. B ) ) $=
    ( cr wcel cc0 cle wbr wa clt w3a co simp1l remulcld simp2 simp1 simp3 ltled
    cmul lemul1a lelttrd syl31anc wb 0red simp1r ltmul2 syl112anc mpbid ) ACDZE
    AFGZHZBCDZABIGZJZAARKZBARKZBBRKZUMAAUHUIUKULLZUQMUMBAUJUKULNZUQMUMBBURURMUM
    UHUKUJABFGUNUOFGUQURUJUKULOUMABUQURUJUKULPZQABASUAUMULUOUPIGZUSUMUHUKUKEBIG
    ULUTUBUQURURUMEABUMUCUQURUHUIUKULUDUSTABBUEUFUGT $.

  $( Two nonnegative numbers compare the same as their squares.  (Contributed
     by Roy F. Longton, 8-Aug-2005.)  (Revised by Mario Carneiro,
     27-May-2016.) $)
  lt2msq $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
                  ( A < B <-> ( A x. A ) < ( B x. B ) ) ) $=
    ( cr wcel cc0 cle wbr wa clt cmul co wi lt2msq1 3expia cap syl2anc remulcld
    wn lenltd 3imtr4d adantrr simpr simpll con3d simprl wo recnd syl22anc oridm
    cc mulext imbitrdi anim12d wb ltleap impbid ) ACDZEAFGZHZBCDZEBFGZHZHZABIGZ
    AAJKZBBJKZIGZUSUTVDVGLVAUSUTVDVGABMNUAVCVEVFFGZVEVFOGZHZABFGZABOGZHZVGVDVCV
    HVKVIVLVCVFVEIGZRBAIGZRVHVKVCVOVNVCVBUQVOVNLUSVBUBUQURVBUCZVBUQVOVNBAMNPUDV
    CVEVFVCAAVPVPQZVCBBUSUTVAUEZVRQZSVCABVPVRSTVCVIVLVLUFZVLVCAUJDZWABUJDZWBVIV
    TLVCAVPUGZWCVCBVRUGZWDAABBUKUHVLUIULUMVCVECDVFCDVGVJUNVQVSVEVFUOPVCUQUTVDVM
    UNVPVRABUOPTUP $.

  $( Division of a positive number by both sides of 'less than'.  (Contributed
     by NM, 27-Apr-2005.) $)
  ltdiv2 $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\
      ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C / B ) < ( C / A ) ) ) $=
    ( cr wcel cc0 clt wbr wa c1 cdiv co cmul wb gt0ap0 3ad2ant1 recnd divrecapd
    w3a cap simp2l 3ad2ant2 rerecclap syldan simp3 ltmul2 syl3anc ltrec 3adant3
    rerecclapd simp3l simp1l breq12d 3bitr4d ) ADEZFAGHZIZBDEZFBGHZIZCDEZFCGHZI
    ZSZJBKLZJAKLZGHZCVEMLZCVFMLZGHZABGHZCBKLZCAKLZGHVDVEDEVFDEZVCVGVJNVDBUQURUS
    VCUAZUTUQBFTHVCBOUBZUJUQUTVNVCUOUPAFTHZVNAOZAUCUDPUQUTVCUEVEVFCUFUGUQUTVKVG
    NVCABUHUIVDVLVHVMVIGVDCBVDCUQUTVAVBUKQZVDBVOQVPRVDCAVSVDAUOUPUTVCULQUQUTVQV
    CVRPRUMUN $.

  $( Reciprocal swap in a 'less than' relation.  (Contributed by NM,
     24-Feb-2005.) $)
  ltrec1 $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) ->
              ( ( 1 / A ) < B <-> ( 1 / B ) < A ) ) $=
    ( cr wcel cc0 clt wbr wa c1 cdiv co simpll simplr gt0ap0d rerecclapd recgt0
    wb adantr simpr ltrec syl21anc recnd recrecapd breq2d bitrd ) ACDZEAFGZHZBC
    DEBFGHZHZIAJKZBFGZIBJKZIUKJKZFGZUMAFGUJUKCDEUKFGZUIULUOQUJAUFUGUILZUJAUQUFU
    GUIMNZOUHUPUIAPRUHUISUKBTUAUJUNAUMFUJAUJAUQUBURUCUDUE $.

  $( Reciprocal swap in a 'less than or equal to' relation.  (Contributed by
     NM, 24-Feb-2005.) $)
  lerec2 $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) ->
              ( A <_ ( 1 / B ) <-> B <_ ( 1 / A ) ) ) $=
    ( cr wcel cc0 clt wbr wa c1 cdiv co cle wb simprl simprr gt0ap0d rerecclapd
    simpl recgt0 adantl lerec syl12anc recnd recrecapd breq1d bitrd ) ACDEAFGHZ
    BCDZEBFGZHZHZAIBJKZLGZIULJKZIAJKZLGZBUOLGUKUGULCDEULFGZUMUPMUGUJRUKBUGUHUIN
    ZUKBURUGUHUIOPZQUJUQUGBSTAULUAUBUKUNBUOLUKBUKBURUCUSUDUEUF $.

  $( Invert ratios of positive numbers and swap their ordering.  (Contributed
     by NM, 9-Jan-2006.) $)
  ledivdiv $p |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\
                      ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) ) ->
                  ( ( A / B ) <_ ( C / D ) <-> ( D / C ) <_ ( B / A ) ) ) $=
    ( cr wcel cc0 clt wbr wa cdiv co cle c1 gt0ap0d redivclapd divgt0 recdivapd
    wb recnd simplll simplrl simplrr adantr simprll simprrl simprrr lerec bitrd
    adantl syl22anc simprlr simpllr breq12d ) AEFZGAHIZJZBEFZGBHIZJZJZCEFZGCHIZ
    JZDEFZGDHIZJZJZJZABKLZCDKLZMIZNVKKLZNVJKLZMIZDCKLZBAKLZMIVIVJEFGVJHIZVKEFGV
    KHIZVLVOSVIABUOUPUTVHUAZUQURUSVHUBZVIBWAUQURUSVHUCOZPVAVRVHABQUDVICDVAVBVCV
    GUEZVAVDVEVFUFZVIDWDVAVDVEVFUGOZPVHVSVACDQUJVJVKUHUKVIVMVPVNVQMVICDVICWCTVI
    DWDTVICWCVAVBVCVGULOWERVIABVIAVTTVIBWATVIAVTUOUPUTVHUMOWBRUNUI $.

  $( Division of a positive number by both sides of 'less than or equal to'.
     (Contributed by NM, 10-Jan-2006.) $)
  lediv2 $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\
       ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) ) $=
    ( cr wcel cc0 clt wbr wa w3a c1 cdiv co cle wb gt0ap0d rerecclapd divrecapd
    cmul recnd simp2l simp2r simp1l simp1r simp3l simp3r lemul2 syl112anc lerec
    3adant3 breq12d 3bitr4d ) ADEZFAGHZIZBDEZFBGHZIZCDEZFCGHZIZJZKBLMZKALMZNHZC
    VCSMZCVDSMZNHZABNHZCBLMZCALMZNHVBVCDEVDDEUSUTVEVHOVBBUOUPUQVAUAZVBBVLUOUPUQ
    VAUBPZQVBAUMUNURVAUCZVBAVNUMUNURVAUDPZQUOURUSUTUEZUOURUSUTUFVCVDCUGUHUOURVI
    VEOVAABUIUJVBVJVFVKVGNVBCBVBCVPTZVBBVLTVMRVBCAVQVBAVNTVORUKUL $.

  $( Swap denominator with other side of 'less than'.  (Contributed by NM,
     3-Oct-1999.) $)
  ltdiv23 $p |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) )
              -> ( ( A / B ) < C <-> ( A / C ) < B ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cdiv co cmul wb simp1 simp2l simp2r redivclapd
    gt0ap0d recnd simp3l simp2 ltmul1 syl3anc divcanap1d breq1d remulcld ltdiv1
    syld3an2 simp3r divcanap3d breq2d bitrd 3bitrd ) ADEZBDEZFBGHZIZCDEZFCGHZIZ
    JZABKLZCGHZVCBMLZCBMLZGHZAVFGHZACKLZBGHZVBVCDEUSURVDVGNVBABUOURVAOZUOUPUQVA
    PZVBBVLUOUPUQVAQSZRUOURUSUTUAZUOURVAUBVCCBUCUDVBVEAVFGVBABVBAVKTVBBVLTZVMUE
    UFVBVHVIVFCKLZGHZVJUOVFDEURVAVHVQNVBCBVNVLUGAVFCUHUIVBVPBVIGVBBCVOVBCVNTVBC
    VNUOURUSUTUJSUKULUMUN $.

  $( Swap denominator with other side of 'less than or equal to'.  (Contributed
     by NM, 30-May-2005.) $)
  lediv23 $p |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) )
              -> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) ) $=
    ( cr wcel cc0 clt wbr wa w3a cdiv co cle cmul wb simp1 simp2l gt0ap0d recnd
    simp2r redivclapd simp3l lemul1 syl112anc divcanap1d breq1d remulcld lediv1
    syld3an2 simp3r divcanap3d breq2d bitrd 3bitrd ) ADEZBDEZFBGHZIZCDEZFCGHZIZ
    JZABKLZCMHZVCBNLZCBNLZMHZAVFMHZACKLZBMHZVBVCDEUSUPUQVDVGOVBABUOURVAPZUOUPUQ
    VAQZVBBVLUOUPUQVATZRZUAUOURUSUTUBZVLVMVCCBUCUDVBVEAVFMVBABVBAVKSVBBVLSZVNUE
    UFVBVHVIVFCKLZMHZVJUOVFDEURVAVHVROVBCBVOVLUGAVFCUHUIVBVQBVIMVBBCVPVBCVOSVBC
    VOUOURUSUTUJRUKULUMUN $.

  $( Comparison of ratio of two nonnegative numbers.  (Contributed by NM,
     31-Dec-2005.) $)
  lediv12a $p |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A <_ B ) ) /\
               ( ( C e. RR /\ D e. RR ) /\ ( 0 < C /\ C <_ D ) ) ) ->
                 ( A / D ) <_ ( B / C ) ) $=
    ( cr wcel wa cc0 cle wbr clt c1 cdiv cmul syl22anc gt0ap0d rerecclapd recnd
    co jca simplrr simprrr simprll simprrl simprlr 0red ltletrd simplll simplrl
    wb lerec mpbid wi simpllr recgt0 syl2anc lemul12a mp2and divrecapd 3brtr4d
    ltled ) AEFZBEFZGZHAIJZABIJZGZGZCEFZDEFZGZHCKJZCDIJZGZGZGZALDMSZNSZBLCMSZNS
    ZADMSBCMSIVPVFVQVSIJZVRVTIJZVDVEVFVOUAVPVMWAVHVKVLVMUBZVPVIVLVJHDKJZVMWAUJV
    HVIVJVNUCZVHVKVLVMUDZVHVIVJVNUEZVPHCDVPUFZWEWGWFWCUGZCDUKOULVPVBVEGVCVQEFZH
    VQIJZGVSEFVFWAGWBUMVPVBVEVBVCVGVOUHZVDVEVFVOUITVBVCVGVOUNZVPWJWKVPDWGVPDWGW
    IPZQZVPHVQWHWOVPVJWDHVQKJWGWIDUOUPVATVPCWEVPCWEWFPZQABVQVSUQOURVPADVPAWLRVP
    DWGRWNUSVPBCVPBWMRVPCWERWPUSUT $.

  $( Division of both sides of 'less than or equal to' into a nonnegative
     number.  (Contributed by Paul Chapman, 7-Sep-2007.) $)
  lediv2a $p |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\
                     ( C e. RR /\ 0 <_ C ) ) /\
                   A <_ B ) -> ( C / B ) <_ ( C / A ) ) $=
    ( cr wcel cc0 clt wbr wa cle w3a cdiv co pm3.2 pm2.43i adantr anim2i ancoms
    leid jca ad2antlr 3adantl2 ad2ant2r simplr anim1i 3adantl3 lediv12a syl2anc
    id ) ADEZFAGHZIZBDEZFBGHZIZCDEZFCJHZIZKABJHZIUPUPIZUQCCJHZIZIZUJUMIZUKUSIZI
    ZCBLMCALMJHULURUSVCUOURVCULUSURUTVBUPUTUQUPUTUPUPNOPUQUPVBUPVAUQCSQRTUAUBUL
    UOUSVFURULUOIZUSIVDVEVGVDUSUJUMVDUKUNVDUIUCPVGUKUSUJUKUOUDUETUFCCABUGUH $.

  $( The reciprocal of a positive number less than 1 is greater than 1.
     (Contributed by NM, 23-Feb-2005.) $)
  reclt1 $p |- ( ( A e. RR /\ 0 < A ) -> ( A < 1 <-> 1 < ( 1 / A ) ) ) $=
    ( cr wcel cc0 clt wbr wa c1 cdiv co wb 1re 0lt1 ltrec mpanr12 breq1i bitrdi
    1div1e1 ) ABCDAEFGZAHEFZHHIJZHAIJZEFZHUBEFSHBCDHEFTUCKLMAHNOUAHUBERPQ $.

  $( The reciprocal of a positive number greater than 1 is less than 1.
     (Contributed by NM, 28-Dec-2005.) $)
  recgt1 $p |- ( ( A e. RR /\ 0 < A ) -> ( 1 < A <-> ( 1 / A ) < 1 ) ) $=
    ( cr wcel cc0 clt wbr wa c1 cdiv co wb 1re 0lt1 ltrec mpanl12 breq2i bitrdi
    1div1e1 ) ABCDAEFGZHAEFZHAIJZHHIJZEFZUAHEFHBCDHEFSTUCKLMHANOUBHUAERPQ $.

  $( The reciprocal of a number greater than 1 is positive and less than 1.
     (Contributed by NM, 23-Feb-2005.) $)
  recgt1i $p |- ( ( A e. RR /\ 1 < A ) ->
               ( 0 < ( 1 / A ) /\ ( 1 / A ) < 1 ) ) $=
    ( cr wcel c1 clt wbr wa cc0 cdiv co 0lt1 wi 0re 1re mp3an12 mpani imdistani
    lttr recgt0 syl recgt1 biimpa sylancom jca ) ABCZDAEFZGZHDAIJZEFZUHDEFZUGUE
    HAEFZGZUIUEUFUKUEHDEFZUFUKKHBCDBCUEUMUFGUKLMNHDAROPQZASTUEUFULUJUNULUFUJAUA
    UBUCUD $.

  $( Construct a number less than 1 from any nonnegative number.  (Contributed
     by NM, 30-Dec-2005.) $)
  recp1lt1 $p |- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) < 1 ) $=
    ( cr wcel cc0 cle wbr wa c1 caddc co cdiv clt cmul simpl ltp1 recnd addcomd
    syl 1cnd breqtrd addcld 1red readdcld 1re addgtge0 mpanr1 mpanl1 divcanap1d
    0lt1 gt0ap0d mullidd 3brtr4d wb redivclapd ltmul1 syl112anc mpbird ) ABCZDA
    EFZGZAHAIJZKJZHLFZVBVAMJZHVAMJZLFZUTAVAVDVELUTAAHIJZVALUTURAVGLFURUSNZAORUT
    AHUTAVHPZUTSZQTUTAVAVIUTHAVJVIUAZUTVAUTHAUTUBZVHUCZHBCZURUSDVALFZUDVNURGDHL
    FUSVOUIHAUEUFUGZUJZUHUTVAVKUKULUTVBBCVNVABCVOVCVFUMUTAVAVHVMVQUNVLVMVPVBHVA
    UOUPUQ $.

  $( Given a positive number ` A ` , construct a new positive number less than
     both ` A ` and 1.  (Contributed by NM, 28-Dec-2005.) $)
  recreclt $p |- ( ( A e. RR /\ 0 < A ) ->
        ( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) ) $=
    ( cr wcel cc0 clt wbr wa c1 cdiv co caddc recgt0 1re ltaddpos sylancl mpbid
    wb sylancr 0lt1 cc simpl gt0ap0 rerecclapd readdcl wi 0re mp3an12 syl mpani
    lttr mpd recgt1 syl2anc mpbii wceq recnd ax-1cn addcom breqtrd simpr ltrec1
    syl22anc jca ) ABCZDAEFZGZHHHAIJZKJZIJZHEFZVIAEFZVFHVHEFZVJVFDVGEFZVLALVFVG
    BCZHBCZVMVLQVFAVDVEUAZAUBUCZMVGHNOPZVFVHBCZDVHEFZVLVJQVFVOVNVSMVQHVGUDRZVFV
    LVTVRVFDHEFZVLVTSVFVSWBVLGVTUEZWADBCVOVSWCUFMDHVHUJUGUHUIUKZVHULUMPVFVGVHEF
    ZVKVFVGVGHKJZVHEVFWBVGWFEFZSVFVOVNWBWGQMVQHVGNRUNVFVGTCHTCWFVHUOVFVGVQUPUQV
    GHUROUSVFVDVEVSVTWEVKQVPVDVEUTWAWDAVHVAVBPVC $.

  $( The square function on nonnegative reals is monotonic.  (Contributed by
     NM, 3-Aug-1999.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  le2msq $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
              ( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) ) $=
    ( cr wcel cc0 cle wbr wa clt cmul lt2msq ancoms notbid simpll simprl lenltd
    wn co wb remulcld 3bitr4d ) ACDZEAFGZHZBCDZEBFGZHZHZBAIGZQBBJRZAAJRZIGZQABF
    GUKUJFGUHUIULUGUDUIULSBAKLMUHABUBUCUGNZUDUEUFOZPUHUKUJUHAAUMUMTUHBBUNUNTPUA
    $.

  $( The square of a nonnegative number is a one-to-one function.  (Contributed
     by NM, 29-Jul-1999.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  msq11 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
              ( ( A x. A ) = ( B x. B ) <-> A = B ) ) $=
    ( cr wcel cc0 cle wbr cmul wceq le2msq ancoms anbi12d simpll simprl letri3d
    wa co wb remulcld 3bitr4rd ) ACDZEAFGZPZBCDZEBFGZPZPZABFGZBAFGZPAAHQZBBHQZF
    GZUKUJFGZPABIUJUKIUGUHULUIUMABJUFUCUIUMRBAJKLUGABUAUBUFMZUCUDUENZOUGUJUKUGA
    AUNUNSUGBBUOUOSOT $.

  $( Less-than-or-equal-to and division relation.  (Lemma for computing upper
     bounds of products.  The "+ 1" prevents division by zero.)  (Contributed
     by NM, 28-Sep-2005.) $)
  ledivp1 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
                  ( ( A / ( B + 1 ) ) x. B ) <_ A ) $=
    ( cr wcel cc0 cle wbr wa c1 caddc cdiv cmul simprl peano2re syl simpll 0red
    co clt recnd simprr ltp1 lelttrd redivclapd simpl divge0 syl12anc jca ltled
    gt0ap0d lemul2a syl31anc divcanap1d breqtrd ) ACDZEAFGZHZBCDZEBFGZHZHZABIJR
    ZKRZBLRZVCVBLRZAFVAURVBCDZVCCDZEVCFGZHBVBFGVDVEFGUQURUSMZVAURVFVIBNOZVAVGVH
    VAAVBUOUPUTPZVJVAVBVJVAEBVBVAQVIVJUQURUSUAVAURBVBSGVIBUBOZUCZUJZUDVAUQVFEVB
    SGVHUQUTUEVJVMAVBUFUGUHVABVBVIVJVLUIBVBVCUKULVAAVBVAAVKTVAVBVJTVNUMUN $.

  ${
    $d x A $.
    $( If a nonnegative number is less than any positive number, it is zero.
       (Contributed by NM, 11-Feb-2006.) $)
    squeeze0 $p |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) ->
                   A = 0 ) $=
      ( cr wcel cc0 cle wbr cv clt wi wral w3a wceq ltnr 3ad2ant1 breq2 imbi12d
      wn rspcva 3adant2 mtod simp1 0red lenltd mpbird simp2 letri3d mpbir2and )
      BCDZEBFGZEAHZIGZBUKIGZJZACKZLZBEMBEFGZUJUPUQEBIGZRUPURBBIGZUIUJUSRUOBNOUI
      UOURUSJZUJUNUTABCUKBMULURUMUSUKBEIPUKBBIPQSTUAUPBEUIUJUOUBZUPUCZUDUEUIUJU
      OUFUPBEVAVBUGUH $.
  $}

  ${
    ltplus1.1 $e |- A e. RR $.
    $( A number is less than itself plus 1.  (Contributed by NM,
       20-Aug-2001.) $)
    ltp1i $p |- A < ( A + 1 ) $=
      ( cr wcel c1 caddc co clt wbr ltp1 ax-mp ) ACDAAEFGHIBAJK $.

    $( The reciprocal of a positive number is positive.  Exercise 4 of
       [Apostol] p. 21.  (Contributed by NM, 15-May-1999.) $)
    recgt0i $p |- ( 0 < A -> 0 < ( 1 / A ) ) $=
      ( cr wcel cc0 clt wbr c1 cdiv co recgt0 mpan ) ACDEAFGEHAIJFGBAKL $.

    ${
      recgt0i.2 $e |- 0 < A $.
      $( The reciprocal of a positive number is positive.  Exercise 4 of
         [Apostol] p. 21.  (Contributed by NM, 15-May-1999.) $)
      recgt0ii $p |- 0 < ( 1 / A ) $=
        ( cr wcel cc0 clt wbr c1 cdiv co recgt0 mp2an ) ADEFAGHFIAJKGHBCALM $.
    $}

    prodgt0.2 $e |- B e. RR $.
    $( Infer that a multiplicand is positive from a nonnegative multiplier and
       positive product.  (Contributed by NM, 15-May-1999.) $)
    prodgt0i $p |- ( ( 0 <_ A /\ 0 < ( A x. B ) ) -> 0 < B ) $=
      ( cr wcel cc0 cle wbr cmul co clt wa prodgt0 mpanl12 ) AEFBEFGAHIGABJKLIM
      GBLICDABNO $.

    $( Infer that a multiplicand is nonnegative from a positive multiplier and
       nonnegative product.  (Contributed by NM, 2-Jul-2005.) $)
    prodge0i $p |- ( ( 0 < A /\ 0 <_ ( A x. B ) ) -> 0 <_ B ) $=
      ( cr wcel cc0 clt wbr cmul co cle wa prodge0 mpanl12 ) AEFBEFGAHIGABJKLIM
      GBLICDABNO $.

    $( The ratio of two positive numbers is positive.  (Contributed by NM,
       16-May-1999.) $)
    divgt0i $p |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A / B ) ) $=
      ( cr wcel cc0 clt wbr cdiv co wa divgt0 mpanr1 mpanl1 ) AEFZGAHIZGBHIZGAB
      JKHIZCPQLBEFRSDABMNO $.

    $( The ratio of nonnegative and positive numbers is nonnegative.
       (Contributed by NM, 12-Aug-1999.) $)
    divge0i $p |- ( ( 0 <_ A /\ 0 < B ) -> 0 <_ ( A / B ) ) $=
      ( cr wcel cc0 cle wbr clt cdiv co wa divge0 mpanr1 mpanl1 ) AEFZGAHIZGBJI
      ZGABKLHIZCQRMBEFSTDABNOP $.

    $( The reciprocal of both sides of 'less than'.  (Contributed by NM,
       15-Sep-1999.) $)
    ltreci $p |- ( ( 0 < A /\ 0 < B ) ->
                ( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) $=
      ( cr wcel cc0 clt wbr c1 cdiv co wb wa ltrec mpanr1 mpanl1 ) AEFZGAHIZGBH
      IZABHIJBKLJAKLHIMZCRSNBEFTUADABOPQ $.

    $( The reciprocal of both sides of 'less than or equal to'.  (Contributed
       by NM, 16-Sep-1999.) $)
    lereci $p |- ( ( 0 < A /\ 0 < B ) ->
                ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) ) $=
      ( cr wcel cc0 clt wbr cle c1 cdiv co wb wa lerec mpanr1 mpanl1 ) AEFZGAHI
      ZGBHIZABJIKBLMKALMJINZCSTOBEFUAUBDABPQR $.

    $( The square function on nonnegative reals is strictly monotonic.
       (Contributed by NM, 3-Aug-1999.) $)
    lt2msqi $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
                ( A < B <-> ( A x. A ) < ( B x. B ) ) ) $=
      ( cr wcel cc0 cle wbr clt cmul co wb wa lt2msq mpanr1 mpanl1 ) AEFZGAHIZG
      BHIZABJIAAKLBBKLJIMZCRSNBEFTUADABOPQ $.

    $( The square function on nonnegative reals is monotonic.  (Contributed by
       NM, 2-Aug-1999.) $)
    le2msqi $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
                ( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) ) $=
      ( cr wcel cc0 cle wbr cmul co wb wa le2msq mpanr1 mpanl1 ) AEFZGAHIZGBHIZ
      ABHIAAJKBBJKHILZCQRMBEFSTDABNOP $.

    $( The square of a nonnegative number is a one-to-one function.
       (Contributed by NM, 29-Jul-1999.) $)
    msq11i $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
                ( ( A x. A ) = ( B x. B ) <-> A = B ) ) $=
      ( cr wcel cc0 cle wbr cmul co wceq wb wa msq11 mpanr1 mpanl1 ) AEFZGAHIZG
      BHIZAAJKBBJKLABLMZCRSNBEFTUADABOPQ $.

    ${
      divgt0i2.3 $e |- 0 < B $.
      $( The ratio of two positive numbers is positive.  (Contributed by NM,
         16-May-1999.) $)
      divgt0i2i $p |- ( 0 < A -> 0 < ( A / B ) ) $=
        ( cc0 clt wbr cdiv co divgt0i mpan2 ) FAGHFBGHFABIJGHEABCDKL $.
    $}

    ${
      ltreci.3 $e |- 0 < A $.
      ltreci.4 $e |- 0 < B $.
      $( The reciprocal of both sides of 'less than'.  (Contributed by NM,
         15-Sep-1999.) $)
      ltrecii $p |- ( A < B <-> ( 1 / B ) < ( 1 / A ) ) $=
        ( cc0 clt wbr c1 cdiv co wb ltreci mp2an ) GAHIGBHIABHIJBKLJAKLHIMEFABC
        DNO $.

      $( The ratio of two positive numbers is positive.  (Contributed by NM,
         18-May-1999.) $)
      divgt0ii $p |- 0 < ( A / B ) $=
        ( cc0 clt wbr cdiv co divgt0i2i ax-mp ) GAHIGABJKHIEABCDFLM $.
    $}

    ltmul1.3 $e |- C e. RR $.
    $( Multiplication of both sides of 'less than' by a positive number.
       Theorem I.19 of [Apostol] p. 20.  (Contributed by NM, 16-May-1999.) $)
    ltmul1i $p |- ( 0 < C -> ( A < B <-> ( A x. C ) < ( B x. C ) ) ) $=
      ( cr wcel cc0 clt wbr cmul co wb wa ltmul1 mp3an12 mpan ) CGHZICJKZABJKAC
      LMBCLMJKNZFAGHBGHSTOUADEABCPQR $.

    $( Division of both sides of 'less than' by a positive number.
       (Contributed by NM, 16-May-1999.) $)
    ltdiv1i $p |- ( 0 < C -> ( A < B <-> ( A / C ) < ( B / C ) ) ) $=
      ( cr wcel cc0 clt wbr cdiv co wb wa ltdiv1 mp3an12 mpan ) CGHZICJKZABJKAC
      LMBCLMJKNZFAGHBGHSTOUADEABCPQR $.

    $( 'Less than' relationship between division and multiplication.
       (Contributed by NM, 12-Oct-1999.) $)
    ltmuldivi $p |- ( 0 < C -> ( ( A x. C ) < B <-> A < ( B / C ) ) ) $=
      ( cr wcel cc0 clt wbr cmul co cdiv wb wa ltmuldiv mp3an12 mpan ) CGHZICJK
      ZACLMBJKABCNMJKOZFAGHBGHTUAPUBDEABCQRS $.

    $( Multiplication of both sides of 'less than' by a positive number.
       Theorem I.19 of [Apostol] p. 20.  (Contributed by NM, 16-May-1999.) $)
    ltmul2i $p |- ( 0 < C -> ( A < B <-> ( C x. A ) < ( C x. B ) ) ) $=
      ( cr wcel cc0 clt wbr cmul co wb wa ltmul2 mp3an12 mpan ) CGHZICJKZABJKCA
      LMCBLMJKNZFAGHBGHSTOUADEABCPQR $.

    $( Multiplication of both sides of 'less than or equal to' by a positive
       number.  (Contributed by NM, 2-Aug-1999.) $)
    lemul1i $p |- ( 0 < C -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) $=
      ( cr wcel cc0 clt wbr cle cmul co wb wa lemul1 mp3an12 mpan ) CGHZICJKZAB
      LKACMNBCMNLKOZFAGHBGHTUAPUBDEABCQRS $.

    $( Multiplication of both sides of 'less than or equal to' by a positive
       number.  (Contributed by NM, 1-Aug-1999.) $)
    lemul2i $p |- ( 0 < C -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) ) $=
      ( cr wcel cc0 clt wbr cle cmul co wb wa lemul2 mp3an12 mpan ) CGHZICJKZAB
      LKCAMNCBMNLKOZFAGHBGHTUAPUBDEABCQRS $.

    $( Swap denominator with other side of 'less than'.  (Contributed by NM,
       26-Sep-1999.) $)
    ltdiv23i $p |- ( ( 0 < B /\ 0 < C ) ->
                ( ( A / B ) < C <-> ( A / C ) < B ) ) $=
      ( cc0 clt wbr cr wcel cdiv co wb wa ltdiv23 mp3an1 mpanl1 mpanr1 ) GBHIZC
      JKZGCHIZABLMCHIACLMBHINZFBJKZTUAUBOZUCEAJKUDTOUEUCDABCPQRS $.

    ${
      ltdiv23i.4 $e |- 0 < B $.
      ltdiv23i.5 $e |- 0 < C $.
      $( Swap denominator with other side of 'less than'.  (Contributed by NM,
         26-Sep-1999.) $)
      ltdiv23ii $p |- ( ( A / B ) < C <-> ( A / C ) < B ) $=
        ( cc0 clt wbr cdiv co wb ltdiv23i mp2an ) IBJKICJKABLMCJKACLMBJKNGHABCD
        EFOP $.
    $}

    ${
      ltmul1i.4 $e |- 0 < C $.
      $( Multiplication of both sides of 'less than' by a positive number.
         Theorem I.19 of [Apostol] p. 20.  (Contributed by NM, 16-May-1999.)
         (Proof shortened by Paul Chapman, 25-Jan-2008.) $)
      ltmul1ii $p |- ( A < B <-> ( A x. C ) < ( B x. C ) ) $=
        ( cc0 clt wbr cmul co wb ltmul1i ax-mp ) HCIJABIJACKLBCKLIJMGABCDEFNO
        $.

      $( Division of both sides of 'less than' by a positive number.
         (Contributed by NM, 16-May-1999.) $)
      ltdiv1ii $p |- ( A < B <-> ( A / C ) < ( B / C ) ) $=
        ( cc0 clt wbr cdiv co wb ltdiv1i ax-mp ) HCIJABIJACKLBCKLIJMGABCDEFNO
        $.
    $}
  $}

  ${
    ltp1d.1 $e |- ( ph -> A e. RR ) $.
    $( A number is less than itself plus 1.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    ltp1d $p |- ( ph -> A < ( A + 1 ) ) $=
      ( cr wcel c1 caddc co clt wbr ltp1 syl ) ABDEBBFGHIJCBKL $.

    $( A number is less than or equal to itself plus 1.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    lep1d $p |- ( ph -> A <_ ( A + 1 ) ) $=
      ( cr wcel c1 caddc co cle wbr lep1 syl ) ABDEBBFGHIJCBKL $.

    $( A number minus 1 is less than itself.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    ltm1d $p |- ( ph -> ( A - 1 ) < A ) $=
      ( cr wcel c1 cmin co clt wbr ltm1 syl ) ABDEBFGHBIJCBKL $.

    $( A number minus 1 is less than or equal to itself.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    lem1d $p |- ( ph -> ( A - 1 ) <_ A ) $=
      ( cr wcel c1 cmin co cle wbr lem1 syl ) ABDEBFGHBIJCBKL $.

    ${
      recgt0d.2 $e |- ( ph -> 0 < A ) $.
      $( The reciprocal of a positive number is positive.  Exercise 4 of
         [Apostol] p. 21.  (Contributed by Mario Carneiro, 28-May-2016.) $)
      recgt0d $p |- ( ph -> 0 < ( 1 / A ) ) $=
        ( cr wcel cc0 clt wbr c1 cdiv co recgt0 syl2anc ) ABEFGBHIGJBKLHICDBMN
        $.
    $}

    divgt0d.2 $e |- ( ph -> B e. RR ) $.
    ${
      divgt0d.3 $e |- ( ph -> 0 < A ) $.
      divgt0d.4 $e |- ( ph -> 0 < B ) $.
      $( The ratio of two positive numbers is positive.  (Contributed by Mario
         Carneiro, 28-May-2016.) $)
      divgt0d $p |- ( ph -> 0 < ( A / B ) ) $=
        ( cr wcel cc0 clt wbr cdiv co divgt0 syl22anc ) ABHIJBKLCHIJCKLJBCMNKLD
        FEGBCOP $.
    $}

    ${
      mulgt1d.3 $e |- ( ph -> 1 < A ) $.
      mulgt1d.4 $e |- ( ph -> 1 < B ) $.
      $( The product of two numbers greater than 1 is greater than 1.
         (Contributed by Mario Carneiro, 28-May-2016.) $)
      mulgt1d $p |- ( ph -> 1 < ( A x. B ) ) $=
        ( cr wcel c1 clt wbr cmul co mulgt1 syl22anc ) ABHICHIJBKLJCKLJBCMNKLDE
        FGBCOP $.
    $}

    ${
      lemulge11d.3 $e |- ( ph -> 0 <_ A ) $.
      lemulge11d.4 $e |- ( ph -> 1 <_ B ) $.
      $( Multiplication by a number greater than or equal to 1.  (Contributed
         by Mario Carneiro, 28-May-2016.) $)
      lemulge11d $p |- ( ph -> A <_ ( A x. B ) ) $=
        ( cr wcel cc0 cle wbr c1 cmul co lemulge11 syl22anc ) ABHICHIJBKLMCKLBB
        CNOKLDEFGBCPQ $.

      $( Multiplication by a number greater than or equal to 1.  (Contributed
         by Mario Carneiro, 28-May-2016.) $)
      lemulge12d $p |- ( ph -> A <_ ( B x. A ) ) $=
        ( cr wcel cc0 cle wbr c1 cmul co lemulge12 syl22anc ) ABHICHIJBKLMCKLBC
        BNOKLDEFGBCPQ $.
    $}

    lemul1ad.3 $e |- ( ph -> C e. RR ) $.
    ${
      lemul1ad.4 $e |- ( ph -> 0 <_ C ) $.
      lemul1ad.5 $e |- ( ph -> A <_ B ) $.
      $( Multiplication of both sides of 'less than or equal to' by a
         nonnegative number.  (Contributed by Mario Carneiro, 28-May-2016.) $)
      lemul1ad $p |- ( ph -> ( A x. C ) <_ ( B x. C ) ) $=
        ( cr wcel cc0 cle wbr wa cmul co jca lemul1a syl31anc ) ABJKCJKDJKZLDMN
        ZOBCMNBDPQCDPQMNEFAUAUBGHRIBCDST $.

      $( Multiplication of both sides of 'less than or equal to' by a
         nonnegative number.  (Contributed by Mario Carneiro, 28-May-2016.) $)
      lemul2ad $p |- ( ph -> ( C x. A ) <_ ( C x. B ) ) $=
        ( cr wcel cc0 cle wbr wa cmul co jca lemul2a syl31anc ) ABJKCJKDJKZLDMN
        ZOBCMNDBPQDCPQMNEFAUAUBGHRIBCDST $.
    $}

    ltmul12ad.3 $e |- ( ph -> D e. RR ) $.
    ${
      ltmul12ad.4 $e |- ( ph -> 0 <_ A ) $.
      ltmul12ad.5 $e |- ( ph -> A < B ) $.
      ltmul12ad.6 $e |- ( ph -> 0 <_ C ) $.
      ltmul12ad.7 $e |- ( ph -> C < D ) $.
      $( Comparison of product of two positive numbers.  (Contributed by Mario
         Carneiro, 28-May-2016.) $)
      ltmul12ad $p |- ( ph -> ( A x. C ) < ( B x. D ) ) $=
        ( cr wcel wa cc0 wbr clt jca cle cmul co ltmul12a syl22anc ) ABNOZCNOZP
        QBUARZBCSRZPDNOZENOZPQDUARZDESRZPBDUBUCCEUBUCSRAUFUGFGTAUHUIJKTAUJUKHIT
        AULUMLMTBCDEUDUE $.
    $}

    ${
      lemul12ad.4 $e |- ( ph -> 0 <_ A ) $.
      lemul12ad.5 $e |- ( ph -> 0 <_ C ) $.
      lemul12ad.6 $e |- ( ph -> A <_ B ) $.
      lemul12ad.7 $e |- ( ph -> C <_ D ) $.
      $( Comparison of product of two nonnegative numbers.  (Contributed by
         Mario Carneiro, 28-May-2016.) $)
      lemul12ad $p |- ( ph -> ( A x. C ) <_ ( B x. D ) ) $=
        ( cle wbr cmul co cr wcel wa cc0 wi jca lemul12a syl22anc mp2and ) ABCN
        OZDENOZBDPQCEPQNOZLMABRSZUABNOZTCRSDRSZUADNOZTERSUGUHTUIUBAUJUKFJUCGAUL
        UMHKUCIBCDEUDUEUF $.
    $}

    ${
      lemul12bd.4 $e |- ( ph -> 0 <_ A ) $.
      lemul12bd.5 $e |- ( ph -> 0 <_ D ) $.
      lemul12bd.6 $e |- ( ph -> A <_ B ) $.
      lemul12bd.7 $e |- ( ph -> C <_ D ) $.
      $( Comparison of product of two nonnegative numbers.  (Contributed by
         Mario Carneiro, 28-May-2016.) $)
      lemul12bd $p |- ( ph -> ( A x. C ) <_ ( B x. D ) ) $=
        ( cle wbr cmul co cr wcel wa cc0 wi jca lemul12b syl22anc mp2and ) ABCN
        OZDENOZBDPQCEPQNOZLMABRSZUABNOZTCRSDRSERSZUAENOZTUGUHTUIUBAUJUKFJUCGHAU
        LUMIKUCBCDEUDUEUF $.
    $}
  $}

  $( Multiplying a nonnegative number by a nonpositive number yields a
     nonpositive number.  (Contributed by Jim Kingdon, 28-Oct-2021.) $)
  mulle0r $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ 0 /\ 0 <_ B ) ) ->
      ( A x. B ) <_ 0 ) $=
    ( cr wcel wa cc0 cle cmul co simpll recnd simplr mulcomd 0red simprr simprl
    wbr lemul2ad mul01d breqtrd eqbrtrd ) ACDZBCDZEZAFGQZFBGQZEZEZABHIBAHIZFGUH
    ABUHAUBUCUGJZKUHBUBUCUGLZKZMUHUIBFHIFGUHAFBUJUHNUKUDUEUFOUDUEUFPRUHBULSTUA
    $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Suprema
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d w x y S $.  $d w y A $.
    $( If a set of reals contains a lower bound, it contains a unique lower
       bound.  (Contributed by NM, 9-Oct-2005.) $)
    lbreu $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) ->
                E! x e. S A. y e. S x <_ y ) $=
      ( vw cr wss cv cle wbr wral wrex wa weq wi wreu wcel breq2 im2anan9r ssel
      rspcv anim12d impcom letri3 syl exbiri com23 syld ralrimivv anim2i ancoms
      wb com3r breq1 ralbidv reu4 sylibr ) CEFZAGZBGZHIZBCJZACKZLVBVADGZUSHIZBC
      JZLZADMZNZDCJACJZLZVAACOVBUQVJUQVIVBUQVHADCCURCPZVCCPZLZVFUQVGVMVFURVCHIZ
      VCURHIZLZUQVGNVLVAVNVKVEVOUTVNBVCCUSVCURHQTVDVOBURCUSURVCHQTRVMUQVPVGVMUQ
      VGVPVMUQLUREPZVCEPZLZVGVPUKUQVMVSUQVKVQVLVRCEURSCEVCSUAUBURVCUCUDUEUFUGUL
      UHUIUJVAVEADCVGUTVDBCURVCUSHUMUNUOUP $.

    $( If a set of reals contains a lower bound, it contains a unique lower
       bound that belongs to the set.  (Contributed by NM, 9-Oct-2005.)
       (Revised by Mario Carneiro, 24-Dec-2016.) $)
    lbcl $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) ->
                ( iota_ x e. S A. y e. S x <_ y ) e. S ) $=
      ( cr wss cv cle wbr wral wrex wa wreu crio wcel lbreu riotacl syl ) CDEAF
      BFGHBCIZACJKRACLRACMCNABCORACPQ $.

    $( If a set of reals contains a lower bound, the lower bound is less than
       or equal to all members of the set.  (Contributed by NM, 9-Oct-2005.)
       (Proof shortened by Mario Carneiro, 24-Dec-2016.) $)
    lble $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y /\ A e. S ) ->
                ( iota_ x e. S A. y e. S x <_ y ) <_ A ) $=
      ( cr wss cv cle wral wrex wcel crio wreu lbreu nfcv nfriota1 nfbr nfralxy
      wbr wa eqid wceq nfra1 nfeq2 breq1 ralbid riotaprop syl simprd breq2 rspc
      nfriota mpan9 3impa ) DEFZAGZBGZHSZBDIZADJZCDKZUSADLZCHSZUOUTTZVBUQHSZBDI
      ZVAVCVDVBDKZVFVDUSADMVGVFTABDNUSVFADVBVEABDADOAVBUQHUSADPAHOAUQOQRVBUAUPV
      BUBURVEBDBUPVBUSBADURBDUCBDOULZUDUPVBUQHUEUFUGUHUIVEVCBCDBVBCHVHBHOBCOQUQ
      CVBHUJUKUMUN $.
  $}

  ${
    $d S f g x y z $.
    $( If a set of reals contains a lower bound, the lower bound is its
       infimum.  (Contributed by NM, 9-Oct-2005.)  (Revised by AV,
       4-Sep-2020.) $)
    lbinf $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) ->
                    inf ( S , RR , < ) = ( iota_ x e. S A. y e. S x <_ y ) ) $=
      ( vz vg vf cr wss cv cle wbr wral wrex wa crio clt wcel weq wn adantr mpd
      wb lttri3 adantl lbcl wi ssel ssel2 adantlr lble 3expa lensymd infminti )
      CGHZAIBIJKBCLZACMZNZDEFGCUOACOZPFIZGQEIZGQNFERUSUTPKSUTUSPKSNUBUQUSUTUCUD
      UQURCQZURGQZABCUEZUNVAVBUFUPCGURUGTUAZVCUQDIZCQZNURVEUQVBVFVDTUNVFVEGQUPC
      GVEUHUIUNUPVFURVEJKABVECUJUKULUM $.

    $( If a set of reals contains a lower bound, it contains its infimum.
       (Contributed by NM, 11-Oct-2005.)  (Revised by AV, 4-Sep-2020.) $)
    lbinfcl $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y )
                         -> inf ( S , RR , < ) e. S ) $=
      ( cr wss cv cle wbr wral wrex wa clt cinf crio lbinf lbcl eqeltrd ) CDEAF
      BFGHBCIZACJKCDLMRACNCABCOABCPQ $.

    $d A y $.
    $( If a set of reals contains a lower bound, its infimum is less than or
       equal to all members of the set.  (Contributed by NM, 11-Oct-2005.)
       (Revised by AV, 4-Sep-2020.) $)
    lbinfle $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y /\ A e. S )
                         -> inf ( S , RR , < ) <_ A ) $=
      ( cr wss cle wbr wral wrex wcel w3a clt cinf crio wceq lbinf 3adant3 lble
      cv eqbrtrd ) DEFZATBTGHBDIZADJZCDKZLDEMNZUCADOZCGUBUDUFUGPUEABDQRABCDSUA
      $.
  $}

  ${
    $d A x y z $.  $d f g ph x $.
    suprubex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. x < y
      /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $.
    suprubex.ss $e |- ( ph -> A C_ RR ) $.

    ${
      suprubex.b $e |- ( ph -> B e. A ) $.
      $( A member of a nonempty bounded set of reals is less than or equal to
         the set's upper bound.  (Contributed by Jim Kingdon, 18-Jan-2022.) $)
      suprubex $p |- ( ph -> B <_ sup ( A , RR , < ) ) $=
        ( vg vf cr clt csup sseldd cv wcel wa wbr wn weq lttri3 supclti supubti
        wb adantl mpd nltled ) AFELMNZAELFHIOABCDJKLEMKPZLQJPZLQRKJUAUJUKMSTUKU
        JMSTRUEAUJUKUBUFZGUCAFEQUIFMSTIABCDJKLEFMULGUDUGUH $.
    $}

    ${
      $d A w x y $.  $d A w z $.  $d B w z $.
      suprlubex.b $e |- ( ph -> B e. RR ) $.
      $( The supremum of a nonempty bounded set of reals is the least upper
         bound.  (Contributed by Jim Kingdon, 19-Jan-2022.) $)
      suprlubex $p |- ( ph
          -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) ) $=
        ( vg vf cr wcel clt csup wbr cv wb wa wn wrex weq lttri3 adantl wor a1i
        ltso suplub2ti mpdan ) AFLMFELNONPFDQNPDEUARIABCDJKLEFNKQZLMJQZLMSKJUBU
        JUKNPTUKUJNPTSRAUJUKUCUDGLNUEAUGUFHUHUI $.

      $( An upper bound is not less than the supremum of a nonempty bounded set
         of reals.  (Contributed by Jim Kingdon, 19-Jan-2022.) $)
      suprnubex $p |- ( ph
          -> ( -. B < sup ( A , RR , < ) <-> A. z e. A -. B < z ) ) $=
        ( cr clt csup wbr wn cv wrex wral suprlubex notbid ralnex bitr4di ) AFE
        JKLKMZNFDOKMZDEPZNUCNDEQAUBUDABCDEFGHIRSUCDETUA $.

      $d A w x y z $.  $d f g ph x $.  $d ph w x $.
      $( The supremum of a nonempty bounded set of reals is less than or equal
         to an upper bound.  (Contributed by NM, 18-Mar-2005.)  (Revised by
         Mario Carneiro, 6-Sep-2014.) $)
      suprleubex $p |- ( ph
          -> ( sup ( A , RR , < ) <_ B <-> A. z e. A z <_ B ) ) $=
        ( vw vg vf cr clt cle wbr cv wral wn wcel csup wa weq wb lttri3 supclti
        adantl lenltd bitrd breq2 notbid cbvralv bitr4di sselda adantr ralbidva
        suprnubex bitr4d breq1 bitrdi ) AEMNUAZFOPZJQZFOPZJERZDQZFOPZDERAVBFVCN
        PZSZJERZVEAVBFVFNPZSZDERZVJAVBFVANPSVMAVAFABCDKLMENLQZMTKQZMTUBLKUCVNVO
        NPSVOVNNPSUBUDAVNVOUEUGGUFIUHABCDEFGHIUQUIVIVLJDEJDUCVHVKVCVFFNUJUKULUM
        AVDVIJEAVCETZUBVCFAEMVCHUNAFMTVPIUOUHUPURVDVGJDEVCVFFOUSULUT $.
    $}
  $}

  ${
    $d x y z $.  $d y z F $.
    negiso.1 $e |- F = ( x e. RR |-> -u x ) $.
    $( Negation is an order anti-isomorphism of the real numbers, which is its
       own inverse.  (Contributed by Mario Carneiro, 24-Dec-2016.) $)
    negiso $p |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F ) $=
      ( vz vy cr clt ccnv wceq cv wbr wb cneg wa wtru wcel simpr adantl negeq
      cc wiso wf1o wral cmpt renegcld recn negcon2 syl2an f1ocnv2d mptru simpli
      cfv simpl recnd negcld brcnvg syl2anc a1i fvmptd breq12d 3bitr4rd df-isom
      ltneg rgen2a mpbir2an cbvmptv simpri 3eqtr4i pm3.2i ) FFGGHZBUAZBHZBIVKFF
      BUBZDJZEJZGKZVNBULZVOBULZVJKZLZEFUCDFUCVMVLEFVOMZUDZIZVMWCNOAEFFAJZMZWABC
      OWDFPZNWDOWFQUEOVOFPZNVOOWGQUEWFWGNWDWAIVOWEILZOWFWDTPVOTPZWHWGWDUFVOUFZW
      DVOUGUHRUIUJZUKVTDEFVNFPZWGNZVNMZWAVJKZWAWNGKZVSVPWMWNTPWATPWOWPLWMVNWMVN
      WLWGUMZUNUOZWMVOWGWIWLWJRUOZWNWATTGUPUQWMVQWNVRWAVJWMAVNWEWNFBTBAFWEUDZIW
      MCURZWDVNIWEWNIWMWDVNSRWQWRUSWMAVOWEWAFBTXAWDVOIWEWAIWMWDVOSRWLWGQWSUSUTV
      NVOVCVAVDDEFFGVJBVBVEWBWTVLBEAFWAWEVOWDSVFVMWCWKVGCVHVI $.
  $}

  ${
    $d A x y z $.
    $( The infimum of a set of reals ` A ` .  (Contributed by NM, 9-Oct-2005.)
       (Revised by AV, 4-Sep-2020.) $)
    dfinfre $p |- ( A C_ RR -> inf ( A , RR , < ) = U. { x e. RR |
        ( A. y e. A x <_ y /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) } ) $=
      ( cr clt cv wbr wral wrex wi wa crab cuni wn wcel wb vex brcnv eqtrid wss
      cinf ccnv csup cle df-inf df-sup ssel2 notbii lenlt bitr4id sylan2 ancoms
      an32s ralbidva rexbii imbi12i ralbii a1i anbi12d rabbidva unieqd ) DEUAZD
      EFUBDEFUCZUDZAGZBGZUEHZBDIZVFVGFHZCGZVGFHZCDJZKZBEIZLZAEMZNZDEFUFVCVEVFVG
      VDHZOZBDIZVGVFVDHZVGVKVDHZCDJZKZBEIZLZAEMZNVRABCDEVDUGVCWHVQVCWGVPAEVCVFE
      PZLZWAVIWFVOWJVTVHBDVCVGDPZWIVTVHQZWIVCWKLZWLWMWIVGEPZWLDEVGUHWIWNLVTVGVF
      FHZOVHVSWOVFVGFARZBRZSUIVFVGUJUKULUMUNUOWFVOQWJWEVNBEWBVJWDVMVGVFFWQWPSWC
      VLCDVGVKFWQCRSUPUQURUSUTVAVBTT $.
  $}

  ${
    $d a b x z $.  $d j k ph $.  $d j ph u w $.  $d j ph u x y z $.
    sup3exmid.ex $e |- (
      ( u C_ RR /\ E. w w e. u /\ E. x e. RR A. y e. u y <_ x ) ->
      E. x e. RR ( A. y e. u -. x < y /\
      A. y e. RR ( y < x -> E. z e. u y < z ) ) ) $.
    $( If any inhabited set of real numbers bounded from above has a supremum,
       excluded middle follows.  (Contributed by Jim Kingdon, 2-Apr-2023.) $)
    sup3exmid $p |- DECID ph $=
      ( vj wo cc0 cv wceq c1 cr clt wbr wcel 1re wtru wa vb va vk wdc crab csup
      wn cpr 0lt1 wi 0re lttri3 adantl wral wrex wss wex cle elrabi elpri eleq1
      wb syl mpbiri jaoi ssriv eqid elexi prid1 eqeq1 orbi1d elrab3 ax-mp mpbir
      orci elex2 0le1 breq1 eqlei2 rgen breq2 ralbidv mp2an w3a cvv prexg rabex
      rspcev sseq1 eleq2 exbidv raleq rexbidv rexeq anbi12d imbi12d vtocl mp3an
      3anbi123d imbi2d supclti mptru axltwlin suplubti df-rex sylib elrab simpr
      a1i mpan gt0ne0d neneqd simplr orel1 sylc sylanb exlimiv iba olcs bitr4id
      ltnri eqrdv supeq1d prid2 lenlti mpbi mtbiri supmaxti eqtrdi breq1d con2i
      orim12i df-dc ) AUDAAUGZIZJHKZJLZAIZHJMUHZUEZNOUFZOPZUUAMOPZIZYOJMOPZUUDU
      IJNQZMNQZUUANQZUUEUUDUJUKRUUHSBCDUAUBNYTOUBKZNQUAKZNQTUUIUUJLUUIUUJOPUGUU
      JUUIOPUGTVBSUUIUUJULUMZBKZCKZOPUGZCYTUNZUUMUULOPZUUMDKZOPZDYTUOZUJZCNUNZT
      ZBNUOZSYTNUPZEKZYTQZEUQZUUMUULURPZCYTUNZBNUOZUVCUCYTNUCKZYTQZUVKJLZUVKMLZ
      IZUVKNQZUVLUVKYSQUVOYRHUVKYSUSUVKJMUTVCUVMUVPUVNUVMUVPUUFUKUVKJNVAVDUVNUV
      PUUGRUVKMNVAVDVEVCVFJYTQZUVGUVQJJLZAIZUVRAJVGVOJYSQUVQUVSVBJMJNUKVHVIYRUV
      SHJYSYQYQUVRAYPJJVJVKVLVMVNEJYTVPVMUUGUUMMURPZCYTUNZUVJRUVTCYTUUMYTQUUMYS
      QZUVTYRHUUMYSUSUWBUUMJLZUUMMLZIUVTUUMJMUTUWCUVTUWDUWCUVTJMURPZVQUUMJMURVR
      VDMUUMRVSVEVCVCVTUVIUWABMNUULMLUVHUVTCYTUULMUUMURWAWBWHWCFKZNUPZUVEUWFQZE
      UQZUVHCUWFUNZBNUOZWDZUUNCUWFUNZUUPUURDUWFUOZUJZCNUNZTZBNUOZUJUVDUVGUVJWDZ
      UVCUJFYTYRHYSUUFUUGYSWEQUKRJMNNWFWCWGUWFYTLZUWLUWSUWRUVCUWTUWGUVDUWIUVGUW
      KUVJUWFYTNWIUWTUWHUVFEUWFYTUVEWJWKUWTUWJUVIBNUVHCUWFYTWLWMWSUWTUWQUVBBNUW
      TUWMUUOUWPUVAUUNCUWFYTWLUWTUWOUUTCNUWTUWNUUSUUPUURDUWFYTWNWTWBWOWMWPGWQWR
      XIZXAXBJMUUAXCWRVMUUBAUUCYNUUBUUQYTQZJUUQOPZTZDUQZAUUBUXCDYTUOZUXEUUFUUBU
      XFUKUUFUUBTUXFUJSBCDUAUBNYTJOUUKUXAXDXBXJUXCDYTXEXFUXDADUXBUUQYSQZUUQJLZA
      IZTZUXCAYRUXIHUUQYSYPUUQLYQUXHAYPUUQJVJVKXGZUXJUXCTZUXHUGUXIAUXLUUQJUXLUU
      QUXJUXCXHXKXLUXGUXIUXCXMUXHAXNXOXPXQVCAUUCAUUCMMOPZMRYAZAUUAMMOAUUAYSNOUF
      ZMANYTYSOADYTYSAUXBUXJUXGUXKUXHAUXGUXJVBUXIUXGXRXSXTYBYCUXOMLSDUAUBNYSMOU
      UKUUGSRXIMYSQSJMMNRVHYDXIUXGMUUQOPZUGZSUXGUXHUUQMLZIUXQUUQJMUTUXHUXQUXRUX
      HUXPMJOPZUWEUXSUGVQJMUKRYEYFUUQJMOWAYGUXRUXPUXMUXNUUQMMOWAYGVEVCUMYHXBYIY
      JYGYKYLVMAYMVN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Imaginary and complex number properties
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( The real representation of complex numbers is apart from zero iff one of
     its terms is apart from zero.  (Contributed by Jim Kingdon,
     5-Mar-2020.) $)
  crap0 $p |- ( ( A e. RR /\ B e. RR ) ->
      ( ( A =//= 0 \/ B =//= 0 ) <-> ( A + ( _i x. B ) ) =//= 0 ) ) $=
    ( cr wcel wa ci co caddc cc0 cap wbr wo wb 0re apreim mpanr12 ax-icn mul01i
    cmul oveq2i 00id eqtri breq2i bitr3di ) ACDBCDEZAFBSGHGZIFISGZHGZJKZAIJKBIJ
    KLZUFIJKUEICDZUKUIUJMNNABIIOPUHIUFJUHIIHGIUGIIHFQRTUAUBUCUD $.

  ${
    $d x y z w A $.
    $( The real part of a complex number is unique.  Proposition 10-1.3 of
       [Gleason] p. 130.  (Contributed by NM, 9-May-1999.)  (Proof shortened by
       Mario Carneiro, 27-May-2016.) $)
    creur $p |- ( A e. CC -> E! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) $=
      ( vz vw cc wcel cv ci cmul co caddc wceq cr wrex wreu cnre wa wb wral cru
      ancoms eqcom ancom 3bitr4g anassrs rexbidva biidd ceqsrexv ad2antlr bitrd
      ralrimiva reu6i syldan eqeq1 rexbidv reubidv syl5ibrcom rexlimivv syl ) C
      FGCDHZIEHZJKLKZMZENODNOCAHZIBHZJKLKZMZBNOZANPZDECQVDVJDENNVANGZVBNGZRZVJV
      DVCVGMZBNOZANPZVKVLVOVEVAMZSZANTVPVMVRANVMVENGZRZVOVFVBMZVQRZBNOZVQVTVNWB
      BNVMVSVFNGZVNWBSVMVSWDRZRVGVCMZVQWARZVNWBWEVMWFWGSVEVFVAVBUAUBVCVGUCWAVQU
      DUEUFUGVLWCVQSVKVSVQVQBVBNWAVQUHUIUJUKULVOANVAUMUNVDVIVOANVDVHVNBNCVCVGUO
      UPUQURUSUT $.

    $( The imaginary part of a complex number is unique.  Proposition 10-1.3 of
       [Gleason] p. 130.  (Contributed by NM, 9-May-1999.)  (Proof shortened by
       Mario Carneiro, 27-May-2016.) $)
    creui $p |- ( A e. CC -> E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) ) $=
      ( vz vw cc wcel cv ci cmul co caddc wceq cr wrex wreu cnre wa wb wral cru
      simpr eqcom ancoms bitrid anass1rs rexbidva biidd ceqsrexv ad2antrr bitrd
      ralrimiva reu6i syl2anc eqeq1 rexbidv reubidv syl5ibrcom rexlimivv syl )
      CFGCDHZIEHZJKLKZMZENODNOCAHZIBHZJKLKZMZANOZBNPZDECQVDVJDENNVANGZVBNGZRZVJ
      VDVCVGMZANOZBNPZVMVLVOVFVBMZSZBNTVPVKVLUBVMVRBNVMVFNGZRZVOVEVAMZVQRZANOZV
      QVTVNWBANVMVENGZVSVNWBSVNVGVCMZVMWDVSRZRWBVCVGUCWFVMWEWBSVEVFVAVBUAUDUEUF
      UGVKWCVQSVLVSVQVQAVANWAVQUHUIUJUKULVOBNVBUMUNVDVIVOBNVDVHVNANCVCVGUOUPUQU
      RUSUT $.
  $}

  ${
    $d x y z A $.
    $( The complex conjugate of a complex number is unique.  (Contributed by
       Mario Carneiro, 6-Nov-2013.) $)
    cju $p |- ( A e. CC -> E! x e. CC
               ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) $=
      ( vy vz cc wcel cv caddc co cr ci cmin cmul wrex wceq ax-icn oveq2d oveq2
      wa eleq1d wi wral wreu cnre recn mulcl sylancr subcl syl2an adantr adantl
      ppncand readdcl anidms eqeltrd pnncand a1i adddid eqtr4d addcld mulass c1
      mp3an12 syl cneg ixi 1re renegcli eqeltri readdcld remulcl anbi12d rspcev
      simpr syl12anc oveq1 rexbidv syl5ibrcom rexlimivv an4 cc0 pnpcan imbitrid
      resubcl 3expb ancoms adantrl adantrr subdid nnncan1 3com23 eqtr3d anim12d
      rimul subeq0 biimpd 3syld biimtrid ralrimivva reu4 sylanbrc ) BEFZBAGZHIZ
      JFZKBXCLIZMIZJFZSZAENZXIBCGZHIZJFZKBXKLIZMIZJFZSZSZXCXKOZUAZCEUBAEUBXIAEU
      CXBBXKKDGZMIZHIZOZDJNCJNXJCDBUDYDXJCDJJXKJFZYAJFZSZXJYDYCXCHIZJFZKYCXCLIZ
      MIZJFZSZAENZYGXKYBLIZEFZYCYOHIZJFZKYCYOLIZMIZJFZYNYEXKEFZYBEFZYPYFXKUEZYF
      KEFZYAEFZUUCPYAUEZKYAUFUGZXKYBUHUIYGYQXKXKHIZJYGXKYBXKYEUUBYFUUDUJZYFUUCY
      EUUHUKZUUJULYEUUIJFZYFYEUULXKXKUMUNUJUOYGYTKKMIZYAYAHIZMIZJYGYTKKUUNMIZMI
      ZUUOYGYSUUPKMYGYSYBYBHIUUPYGXKYBYBUUJUUKUUKUPYGKYAYAUUEYGPUQYFUUFYEUUGUKZ
      UURURUSQYGUUNEFZUUOUUQOZYGYAYAUURUURUTUUEUUEUUSUUTPPKKUUNVAVCVDUSYGUUMJFU
      UNJFUUOJFUUMVBVEJVFVBVGVHVIYGYAYAYEYFVNZUVAVJUUMUUNVKUGUOYMYRUUASAYOEXCYO
      OZYIYRYLUUAUVBYHYQJXCYOYCHRTUVBYKYTJUVBYJYSKMXCYOYCLRQTVLVMVOYDXIYMAEYDXE
      YIXHYLYDXDYHJBYCXCHVPTYDXGYKJYDXFYJKMBYCXCLVPQTVLVQVRVSVDXBXTACEEXRXEXMSZ
      XHXPSZSZXBXCEFZUUBSZSZXSXEXHXMXPVTUVHUVEXCXKLIZJFZKUVIMIZJFZSZUVIWAOZXSUV
      HUVCUVJUVDUVLUVCXDXLLIZJFUVHUVJXDXLWDUVHUVOUVIJXBUVFUUBUVOUVIOBXCXKWBWETW
      CUVDXOXGLIZJFZUVHUVLXPXHUVQXOXGWDWFUVHUVPUVKJUVHKXNXFLIZMIUVPUVKUVHKXNXFU
      UEUVHPUQXBUUBXNEFUVFBXKUHWGXBUVFXFEFUUBBXCUHWHWIUVHUVRUVIKMXBUVFUUBUVRUVI
      OZXBUUBUVFUVSBXKXCWJWKWEQWLTWCWMUVMUVNUAUVHUVIWNUQUVGUVNXSUAXBUVGUVNXSXCX
      KWOWPUKWQWRWSXIXQACEXSXEXMXHXPXSXDXLJXCXKBHRTXSXGXOJXSXFXNKMXCXKBLRQTVLWT
      XA $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Function operation analogue theorems
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A x y $.  $d F x y $.  $d G x y $.  $d V x y $.
    $( Function analogue of ~ negsub .  (Contributed by Mario Carneiro,
       24-Jul-2014.) $)
    ofnegsub $p |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) ->
      ( F oF + ( ( A X. { -u 1 } ) oF x. G ) ) = ( F oF - G ) ) $=
      ( vx vy wcel cc wf cv cfv caddc cneg c1 cmul cof co cmin adantl a1i addcl
      w3a csn cxp simp2 mulcl ax-1cn negcli fconst6 simp3 simp1 inidm off subcl
      wa eqidd ffnd ffvelcdmda mulcld ofc1g mulm1d negsubd subcld ofvalg eqtr4d
      eqtrd offeq ) ADGZAHBIZAHCIZUBZEFAAAEJZBKZLHHHVLCKZMZBANMZUCUDZCOPQZBCRPQ
      ZDDVLHGFJZHGUOZVLVTLQHGVKVLVTUASVHVIVJUEZVKEFAAAOHHHVQCDDWAVLVTOQHGVKVLVT
      UFSAHVQIVKAVPHNUGUHZUITVHVIVJUJZVHVIVJUKZWEAULZUMWEWEWFVKEFAAARHHHBCDDWAV
      LVTRQHGVKVLVTUNSWBWDWEWEWFUMVKVLAGUOZVMUPZWGVLVRKVPVNOQVOVKAVPVNOHCDHVLWE
      VPHGZVKWCTVKAHCWDUQZWGVNUPZWGVPVNWIWGWCTVKAHVLCWDURZUSUTWGVNWLVAVFWGVMVOL
      QVMVNRQVLVSKWGVMVNVKAHVLBWBURZWLVBVKAAVMVNRAHBCDDVLVKAHBWBUQWJWEWEWFWHWKW
      GVMVNWMWLVCVDVEVG $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Indicator Functions
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  According to Wikipedia ( ~ https://en.wikipedia.org/wiki/Indicator_function ,
  "Indicator function", 11-Apr-2026): "In mathematics, an _indicator function_
  or a _characteristic function_ of a subset of a set is a function that maps
  elements of the subset to one, and all other elements to zero.  That is, if
  ` A ` is a subset of some set ` X `, then the indicator function of ` A ` is
  the function ` _Ind `_A defined by ` _Ind `_A ( x ) = 1 if ` x e. A `, and
  ` _Ind `_A ( x ) = 0 otherwise."

  See also definition in [Lang2] p. 3: "The _characteristic function_ of a
  subset S' of S is the function ` ch ` such that ` ch `(x) = 1 if ` x e. S `'
  and ` ch `(x) = 0 if ` x e/ S `'".
$)

  $c _Ind $.

  $( Extend class notation with the indicator function generator. $)
  cind $a class _Ind $.

  ${
    $d a o x $.
    $( Define the indicator function generator.  It generates an indicator
       function ` ( ( _Ind `` O ) `` A ) ` for a given domain ` O ` and a given
       subset ` A ` of the domain, see ~ indval .  In contrast to the
       definitions and notations in Wikipedia and [Lang2] p. 3, the domain and
       the subset are always mentioned explicitly.  (Contributed by Thierry
       Arnoux, 20-Jan-2017.) $)
    df-ind $a |- _Ind = ( o e. _V |-> ( a e. ~P o |->
                                    ( x e. o |-> if ( x e. a , 1 , 0 ) ) ) ) $.
  $}

  ${
    $d a o x O $.  $d a o V $.
    $( Value of the indicator function generator with domain ` O ` .
       (Contributed by Thierry Arnoux, 23-Aug-2017.) $)
    indv $p |- ( O e. V ->
     ( _Ind ` O ) = ( a e. ~P O |-> ( x e. O |-> if ( x e. a , 1 , 0 ) ) ) ) $=
      ( vo wcel cv cpw c1 cc0 cif cmpt cvv cind df-ind wceq pweq mpteq12dv elex
      mpteq1 pwexg mptexg 3syl fvmptd3 ) BCFZEBDEGZHZAUFAGDGFIJKZLZLDBHZABUHLZL
      ZMNMAEDOUFBPDUGUIUJUKUFBQAUFBUHTRBCSZUEBMFUJMFULMFUMBMUADUJUKMUBUCUD $.

    $d a x A $.
    $( Value of the indicator function generator for a set ` A ` and a domain
       ` O ` , i.e., an indicator function for a given domain ` O ` and a given
       subset ` A ` of the domain.  (Contributed by Thierry Arnoux,
       2-Feb-2017.) $)
    indval $p |- ( ( O e. V /\ A C_ O ) ->
               ( ( _Ind ` O ) ` A ) = ( x e. O |-> if ( x e. A , 1 , 0 ) ) ) $=
      ( va wcel wss wa cv c1 cc0 cif cmpt cpw cind cfv cvv wceq indv adantr
      eleq2 ifbid mpteq2dv adantl ssexg ancoms simpr elpwd mptexg fvmptd ) CDFZ
      BCGZHZEBACAIZEIZFZJKLZMZACUNBFZJKLZMZCNZCOPZQUKVCEVBURMRULACDESTUOBRZURVA
      RUMVDACUQUTVDUPUSJKUOBUNUAUBUCUDUMBCQULUKBQFBCDUEUFUKULUGUHUKVAQFULACUTDU
      ITUJ $.

    $d O j z $.  $d A z $.
    $( The indicator function generator does not generate a (meaningful)
       indicator function for a class which is not a subset of the domain.
       (Contributed by AV, 11-Apr-2026.) $)
    indval0 $p |- ( -. A C_ O -> ( ( _Ind ` O ) ` A ) = (/) ) $=
      ( vz vj vo va vx wn cind cfv c0 cvv wcel cv wi cpw cmpt syl a1i wceq noel
      wss wex elfvm c1 cc0 cif df-ind mptrcl exlimiv pm2.21i indv fveq1d adantr
      wb elpwi con3i adantl eqid fvmptndm eqtrd eleq2d expcom pm5.21ndd eqrdv
      wa ) ABUBZHZCABIJZJZKVHBLMZCNZVJMZVLKMZVMVKOVHVMDNZVIMZDUCVKVLADVIUDVPVKD
      ELFENZPGVQGNFNMUEUFUGZQQIVOBGEFUHUIUJRSVNVKOVHVNVKVLUAUKSVKVHVMVNUOVKVHVF
      ZVJKVLVSVJAFBPZGBVRQZQZJZKVKVJWCTVHVKAVIWBGBLFULUMUNVSAVTMZHZWCKTVHWEVKWD
      VGABUPUQURFVTWAWBAWBUSUTRVAVBVCVDVE $.
  $}

  ${
    $d x y A $.  $d x y O $.  $d y V $.
    $( An indicator function as a function with domain and codomain.
       (Contributed by Thierry Arnoux, 13-Aug-2017.) $)
    indfdc $p |- ( ( O e. V /\ A C_ O /\ A. x e. O DECID x e. A ) ->
                                    ( ( _Ind ` O ) ` A ) : O --> { 0 , 1 } ) $=
      ( vy wcel wss cv wdc wral w3a c1 cc0 cif cpr cind cfv cmpt wceq cr indval
      3adant3 wa 1red eleq1w dcbid simpl3 simpr rspcdva ifprdc syl3anc eleqtrdi
      0red prcom fmpt3d ) CDFZBCGZAHZBFZIZACJZKZECEHZBFZLMNZMLOZBCPQQZUPUQVGECV
      ERSVAEBCDUAUBVBVCCFZUCZVELMOZVFVILTFMTFVDIZVEVJFVIUDVIUMVIUTVKACVCURVCSUS
      VDAEBUEUFUPUQVAVHUGVBVHUHUIVDLMTTUJUKLMUNULUO $.
  $}

  ${
    $d x A $.  $d x O $.  $d x V $.  $d x X $.
    $( Value of the indicator function.  (Contributed by Thierry Arnoux,
       13-Aug-2017.) $)
    indfval $p |- ( ( O e. V /\ A C_ O /\ X e. O ) ->
      ( ( ( _Ind ` O ) ` A ) ` X ) = if ( X e. A , 1 , 0 ) ) $=
      ( vx wcel wss w3a cv c1 cc0 cif cind cfv cvv cmpt wceq indval cr elexi wa
      3adant3 simpr eleq1d ifbid simp3 1re 0re ifex a1i fvmptd ) BCFZABGZDBFZHZ
      EDEIZAFZJKLZDAFZJKLZBABMNNZOULUMVAEBURPQUNEABCRUBUOUPDQZUAZUQUSJKVCUPDAUO
      VBUCUDUEULUMUNUFUTOFUOUSJKJSUGTKSUHTUIUJUK $.
  $}

  $( Value of the indicator function where it is ` 1 ` .  (Contributed by
     Thierry Arnoux, 14-Aug-2017.) $)
  ind1 $p |- ( ( O e. V /\ A C_ O /\ X e. A ) ->
                                          ( ( ( _Ind ` O ) ` A ) ` X ) = 1 ) $=
    ( wcel wss w3a cind cfv c1 cc0 cif wceq simp2 simp3 sseldd indfval syld3an3
    iftrue 3ad2ant3 eqtrd ) BCEZABFZDAEZGZDABHIIIZUDJKLZJUBUCUDDBEUFUGMUEABDUBU
    CUDNUBUCUDOPABCDQRUDUBUGJMUCUDJKSTUA $.

  $( Value of the indicator function where it is ` 0 ` .  (Contributed by
     Thierry Arnoux, 14-Aug-2017.) $)
  ind0 $p |- ( ( O e. V /\ A C_ O /\ X e. ( O \ A ) ) ->
                                          ( ( ( _Ind ` O ) ` A ) ` X ) = 0 ) $=
    ( wcel wss cdif w3a cind cfv c1 cc0 wceq eldifi indfval syl3an3 wn 3ad2ant3
    cif eldifn iffalsed eqtrd ) BCEZABFZDBAGEZHZDABIJJJZDAEZKLSZLUEUCUDDBEUGUIM
    DBANABCDOPUFUHKLUEUCUHQUDDBATRUAUB $.

  ${
    $d O y $.  $d V y $.
    $( Indicator of the empty set.  (Contributed by Thierry Arnoux,
       25-Jan-2026.) $)
    indconst0 $p |- ( O e. V -> ( ( _Ind ` O ) ` (/) ) = ( O X. { 0 } ) ) $=
      ( vy wcel cc0 csn c0 cind cfv wf cxp wceq cv c1 cif wss cmpt 0ss cr 0re
      indval mpan2 wa noel iffalsei elexi elsn2 mpbir a1i fmpt3d fconst2g ax-mp
      wb sylib ) ABDZAEFZGAHIIZJZUQAUPKLZUOCACMZGDZNEOZUPUQUOGAPUQCAVBQLARCGABU
      AUBVBUPDZUOUTADUCVCVBELVANEUTUDUEVBEESTUFUGUHUIUJESDURUSUMTAESUQUKULUN $.
  $}

  ${
    $d O x $.  $d V x $.
    $( Indicator of the whole set.  (Contributed by Thierry Arnoux,
       25-Jan-2026.) $)
    indconst1 $p |- ( O e. V -> ( ( _Ind ` O ) ` O ) = ( O X. { 1 } ) ) $=
      ( vx wcel c1 csn cind cfv wf cxp wceq cv cc0 cif wss cmpt ssid indval 1re
      cr mpan2 wa simpr iftrued elexi elsn2 sylibr fmpt3d fconst2g ax-mp sylib
      wb ) ABDZAEFZAAGHHZIZUOAUNJKZUMCACLADZEMNZUNUOUMAAOUOCAUSPKAQCAABRUAUMURU
      BZUSEKUSUNDUTUREMUMURUCUDUSEETSUEUFUGUHETDUPUQULSAETUOUIUJUK $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Integer sets
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Positive integers (as a subset of complex numbers)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c NN $. $( The set of positive integers (blackboard bold N). $)

  $( Extend class notation to include the class of positive integers. $)
  cn $a class NN $.

  ${
    $d x y $.
    $( Definition of the set of positive integers.  For naming consistency with
       the Metamath Proof Explorer usages should refer to ~ dfnn2 instead.
       (Contributed by Jeff Hankins, 12-Sep-2013.)  (Revised by Mario Carneiro,
       3-May-2014.)  (New usage is discouraged.) $)
    df-inn $a |- NN = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.

    $( Definition of the set of positive integers.  Another name for ~ df-inn .
       (Contributed by Jeff Hankins, 12-Sep-2013.)  (Revised by Mario Carneiro,
       3-May-2014.) $)
    dfnn2 $p |- NN = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $=
      ( df-inn ) ABC $.
  $}

  ${
    $d x y A $.
    $( Peano's inductive postulate.  Theorem I.36 (principle of mathematical
       induction) of [Apostol] p. 34.  (Contributed by NM, 10-Jan-1997.)
       (Revised by Mario Carneiro, 17-Nov-2014.) $)
    peano5nni $p |- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ A ) $=
      ( vy c1 wcel cv caddc co wral wa cn cr cin wss elin ax-mp syl cvv eleq2
      wi 1re biimpri mpan2 inss1 ssralv inss2 1red readdcld simplbi2com ralimia
      sseli reex inex2 wceq raleqbi1dv anbi12d elabg cint dfnn2 intss1 eqsstrid
      cab biimtrrdi syl2an sstrdi ) DBEZAFZDGHZBEZABIZJKBLMZBVFDVKEZVHVKEZAVKIZ
      KVKNZVJVFDLEZVLUAVLVFVPJDBLOUBUCVJVIAVKIZVNVKBNVJVQTBLUDZVIAVKBUEPVIVMAVK
      VGVKEZVHLEZVIVMTVSVGDVKLVGBLUFUKVSUGUHVMVIVTVHBLOUIQUJQVKREZVLVNJZVOTLBUL
      UMWAWBVKDCFZEZVHWCEZAWCIZJZCVBZEZVOWGWBCVKRWCVKUNWDVLWFVNWCVKDSWEVMAWCVKW
      CVKVHSUOUPUQWIKWHURVKCAUSVKWHUTVAVCPVDVRVE $.
  $}

  $( The positive integers are a subset of the reals.  (Contributed by NM,
     10-Jan-1997.)  (Revised by Mario Carneiro, 16-Jun-2013.) $)
  nnssre $p |- NN C_ RR $=
    ( vx c1 cr wcel cv caddc co wral cn wss 1re peano2re rgen peano5nni mp2an )
    BCDAEZBFGCDZACHICJKQACPLMACNO $.

  $( The positive integers are a subset of the complex numbers.  (Contributed
     by NM, 2-Aug-2004.) $)
  nnsscn $p |- NN C_ CC $=
    ( cn cr cc nnssre ax-resscn sstri ) ABCDEF $.

  $( The set of positive integers exists.  (Contributed by NM, 3-Oct-1999.)
     (Revised by Mario Carneiro, 17-Nov-2014.) $)
  nnex $p |- NN e. _V $=
    ( cn cc cnex nnsscn ssexi ) ABCDE $.

  $( A positive integer is a real number.  (Contributed by NM, 18-Aug-1999.) $)
  nnre $p |- ( A e. NN -> A e. RR ) $=
    ( cn cr nnssre sseli ) BCADE $.

  $( A positive integer is a complex number.  (Contributed by NM,
     18-Aug-1999.) $)
  nncn $p |- ( A e. NN -> A e. CC ) $=
    ( cn cc nnsscn sseli ) BCADE $.

  ${
    nnre.1 $e |- A e. NN $.
    $( A positive integer is a real number.  (Contributed by NM,
       18-Aug-1999.) $)
    nnrei $p |- A e. RR $=
      ( cn wcel cr nnre ax-mp ) ACDAEDBAFG $.

    $( A positive integer is a complex number.  (Contributed by NM,
       18-Aug-1999.) $)
    nncni $p |- A e. CC $=
      ( nnrei recni ) AABCD $.
  $}

  ${
    $d x y z A $.
    $( Peano postulate: 1 is a positive integer.  (Contributed by NM,
       11-Jan-1997.) $)
    1nn $p |- 1 e. NN $=
      ( vz vx vy c1 cn wcel cv caddc co wral wa cab cint dfnn2 eleq2i cr wb 1re
      elintg eleq2 ax-mp bitri vex weq raleqbi1dv anbi12d elab simplbi mprgbir
      ) DEFZDAGZFZADBGZFZCGDHIZUMFZCUMJZKZBLZUJDUSMZFZULAUSJZEUTDBCNODPFVAVBQRA
      DUSPSUAUBUKUSFULUOUKFZCUKJZURULVDKBUKAUCBAUDUNULUQVDUMUKDTUPVCCUMUKUMUKUO
      TUEUFUGUHUI $.

    $( Peano postulate: a successor of a positive integer is a positive
       integer.  (Contributed by NM, 11-Jan-1997.)  (Revised by Mario Carneiro,
       17-Nov-2014.) $)
    peano2nn $p |- ( A e. NN -> ( A + 1 ) e. NN ) $=
      ( vz vx vy cn wcel c1 caddc co cv wral wa eleq2i elintg bitrid wceq eleq2
      cab syl cr cint dfnn2 ibi vex raleqbi1dv anbi12d elab oveq1 eleq1d rspcva
      simprbi sylan2 expcom ralimia wb nnre 1red readdcld mpbird ) AEFZAGHIZEFZ
      VABJZFZBGCJZFZDJZGHIZVEFZDVEKZLZCRZKZUTAVCFZBVLKZVMUTVOUTAVLUAZFUTVOEVPAC
      DUBZMBAVLENOUCVNVDBVLVNVCVLFZVDVRVNVHVCFZDVCKZVDVRGVCFZVTVKWAVTLCVCBUDVEV
      CPVFWAVJVTVEVCGQVIVSDVEVCVEVCVHQUEUFUGUKVSVDDAVCVGAPVHVAVCVGAGHUHUIUJULUM
      UNSUTVATFZVBVMUOUTAGAUPUTUQURVBVAVPFWBVMEVPVAVQMBVAVLTNOSUS $.
  $}

  ${
    nnred.1 $e |- ( ph -> A e. NN ) $.
    $( A positive integer is a real number.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nnred $p |- ( ph -> A e. RR ) $=
      ( cn cr nnssre sselid ) ADEBFCG $.

    $( A positive integer is a complex number.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nncnd $p |- ( ph -> A e. CC ) $=
      ( cn cc nnsscn sselid ) ADEBFCG $.

    $( Peano postulate: a successor of a positive integer is a positive
       integer.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    peano2nnd $p |- ( ph -> ( A + 1 ) e. NN ) $=
      ( cn wcel c1 caddc co peano2nn syl ) ABDEBFGHDECBIJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Principle of mathematical induction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y $.  $d x A $.  $d x ps $.  $d x ch $.  $d x th $.  $d x ta $.
    $d y ph $.
    $( Substitutions. $)
    nnind.1 $e |- ( x = 1 -> ( ph <-> ps ) ) $.
    nnind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    nnind.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
    nnind.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Basis. $)
    nnind.5 $e |- ps $.
    $( Induction step. $)
    nnind.6 $e |- ( y e. NN -> ( ch -> th ) ) $.
    $( Principle of Mathematical Induction (inference schema).  The first four
       hypotheses give us the substitution instances we need; the last two are
       the basis and the induction step.  See ~ nnaddcl for an example of its
       use.  This is an alternative for Metamath 100 proof #74.  (Contributed
       by NM, 10-Jan-1997.)  (Revised by Mario Carneiro, 16-Jun-2013.) $)
    nnind $p |- ( A e. NN -> ta ) $=
      ( cn wcel crab wa c1 elrab cv caddc wral wss 1nn mpbir2an elrabi peano2nn
      co a1d anim12d 3imtr4g mpcom rgen peano5nni mp2an sseli sylib simprd ) HO
      PZUTEUTHAFOQZPUTEROVAHSVAPZGUAZSUBUIZVAPZGVAUCOVAUDVBSOPBUEMABFSOITUFVEGV
      AVCOPZVCVAPZVEAFVCOUGVFVFCRVDOPZDRVGVEVFVFVHCDVFVHVFVCUHUJNUKACFVCOJTADFV
      DOKTULUMUNGVAUOUPUQAEFHOLTURUS $.
  $}

  ${
    $d x y $.  $d x A $.  $d x ps $.  $d x ch $.  $d x th $.  $d x ta $.
    $d y ph $.
    $( Induction step. $)
    nnindALT.6 $e |- ( y e. NN -> ( ch -> th ) ) $.
    $( Basis. $)
    nnindALT.5 $e |- ps $.
    $( Substitutions. $)
    nnindALT.1 $e |- ( x = 1 -> ( ph <-> ps ) ) $.
    nnindALT.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    nnindALT.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
    nnindALT.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Principle of Mathematical Induction (inference schema).  The last four
       hypotheses give us the substitution instances we need; the first two are
       the induction step and the basis.

       This ALT version of ~ nnind has a different hypothesis order.  It may be
       easier to use with the metamath program's Proof Assistant, because
       "MM-PA> assign last" will be applied to the substitution instances
       first.  We may eventually use this one as the official version.  You may
       use either version.  After the proof is complete, the ALT version can be
       changed to the non-ALT version with "MM-PA> minimize nnind /allow".
       (Contributed by NM, 7-Dec-2005.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    nnindALT $p |- ( A e. NN -> ta ) $=
      ( nnind ) ABCDEFGHKLMNJIO $.
  $}

  ${
    $d x y A $.
    $( Every positive integer is one or a successor.  (Contributed by Mario
       Carneiro, 16-May-2014.) $)
    nn1m1nn $p |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) ) $=
      ( vx vy cv c1 wceq cmin co cn wcel wo caddc orc 1cnd eqeq1 eleq1d orbi12d
      cc oveq1 ax-1cn 2thd nncn pncan sylancl id eqeltrd olcd a1d nnind ) BDZEF
      ZUJEGHZIJZKZERJZCDZEFZUPEGHZIJZKZUPELHZEFZVAEGHZIJZKZAEFZAEGHZIJZKBCAUKUN
      UOUKUMMUKNUAUJUPFZUKUQUMUSUJUPEOVIULURIUJUPEGSPQUJVAFZUKVBUMVDUJVAEOVJULV
      CIUJVAEGSPQUJAFZUKVFUMVHUJAEOVKULVGIUJAEGSPQTUPIJZVEUTVLVDVBVLVCUPIVLUPRJ
      UOVCUPFUPUBTUPEUCUDVLUEUFUGUHUI $.
  $}

  ${
    $d x y A $.  $d x ps $.  $d x ch $.  $d x th $.  $d y ph $.
    $( Substitutions. $)
    nn1suc.1 $e |- ( x = 1 -> ( ph <-> ps ) ) $.
    nn1suc.3 $e |- ( x = ( y + 1 ) -> ( ph <-> ch ) ) $.
    nn1suc.4 $e |- ( x = A -> ( ph <-> th ) ) $.
    $( Hypothesis showing it holds for 1. $)
    nn1suc.5 $e |- ps $.
    $( Hypothesis showing it holds for a successor. $)
    nn1suc.6 $e |- ( y e. NN -> ch ) $.
    $( If a statement holds for 1 and also holds for a successor, it holds for
       all positive integers.  The first three hypotheses give us the
       substitution instances we need; the last two show that it holds for 1
       and for a successor.  (Contributed by NM, 11-Oct-2004.)  (Revised by
       Mario Carneiro, 16-May-2014.) $)
    nn1suc $p |- ( A e. NN -> th ) $=
      ( cn wcel c1 wceq co wsbc wb caddc cmin 1ex sbcie mpbir 1nn mpbiri sbcieg
      wi eleq1 syl dfsbcq bitr3d cv elisset wa pm5.32ri adantr cr nnre peano2re
      a1i 3syl oveq1 sbceq1d adantl mpbid sylbir exlimddv cc nncn npcan sylancl
      ax-1cn bitrd imbitrid nn1m1nn mpjaod ) GMNZGOPZDGOUAQZMNZVSDUHVRVSDAEORZW
      BBKABEOUBHUCUDVSAEGRZDWBVSVRWCDSVSVROMNUEGOMUIUFADEGMJUGZUJAEGOUKULUFVAWA
      AEVTOTQZRZVRDWAFUMZVTPZWFFFVTMUNWAWHUOWGMNZWHUOZWFWHWIWAWGVTMUIUPWJCWFWIC
      WHLUQWJAEWGOTQZRZCWFWIWLCSZWHWIWGURNWKURNWMWGUSWGUTACEWKURIUGVBUQWHWLWFSW
      IWHAEWKWEWGVTOTVCVDVEULVFVGVHVRWFWCDVRAEWEGVRGVINOVINWEGPGVJVMGOVKVLVDWDV
      NVOGVPVQ $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Closure of addition of positive integers, proved by induction on the
       second addend.  (Contributed by NM, 12-Jan-1997.) $)
    nnaddcl $p |- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN ) $=
      ( vx vy cn wcel caddc co cv wi c1 wceq oveq2 eleq1d imbi2d peano2nn wa cc
      nncn ax-1cn addass mp3an3 syl2an imbitrid expcom a2d nnind impcom ) BEFAE
      FZABGHZEFZUIACIZGHZEFZJUIAKGHZEFZJUIADIZGHZEFZJUIAUQKGHZGHZEFZJUIUKJCDBUL
      KLZUNUPUIVCUMUOEULKAGMNOULUQLZUNUSUIVDUMUREULUQAGMNOULUTLZUNVBUIVEUMVAEUL
      UTAGMNOULBLZUNUKUIVFUMUJEULBAGMNOAPUQEFZUIUSVBUIVGUSVBJUSURKGHZEFUIVGQZVB
      URPVIVHVAEUIARFZUQRFZVHVALZVGASUQSVJVKKRFVLTAUQKUAUBUCNUDUEUFUGUH $.

    $( Closure of multiplication of positive integers.  (Contributed by NM,
       12-Jan-1997.) $)
    nnmulcl $p |- ( ( A e. NN /\ B e. NN ) -> ( A x. B ) e. NN ) $=
      ( vx vy cn wcel cmul co cv wi c1 caddc wceq oveq2 eleq1d imbi2d cc mulrid
      nncn wa biimprd mpcom nnaddcl ancoms ax-1cn adddi mp3an3 oveq2d imbitrrid
      adantr eqtrd syl2an exp4b pm2.43b a2d nnind impcom ) BEFAEFZABGHZEFZURACI
      ZGHZEFZJURAKGHZEFZJURADIZGHZEFZJURAVFKLHZGHZEFZJURUTJCDBVAKMZVCVEURVLVBVD
      EVAKAGNOPVAVFMZVCVHURVMVBVGEVAVFAGNOPVAVIMZVCVKURVNVBVJEVAVIAGNOPVABMZVCU
      TURVOVBUSEVABAGNOPAQFZURVEASZVPVEURVPVDAEARZOUAUBVFEFZURVHVKVSURVHVKJURVS
      URVHVKURVHTVKURVSTZVGALHZEFZVHURWBVGAUCUDVTVJWAEURVPVFQFZVJWAMVSVQVFSVPWC
      TVJVGVDLHZWAVPWCKQFVJWDMUEAVFKUFUGVPWDWAMWCVPVDAVGLVRUHUJUKULOUIUMUNUOUPU
      Q $.
  $}

  ${
    nnmulcli.1 $e |- A e. NN $.
    nnmulcli.2 $e |- B e. NN $.
    $( Closure of multiplication of positive integers.  (Contributed by Mario
       Carneiro, 18-Feb-2014.) $)
    nnmulcli $p |- ( A x. B ) e. NN $=
      ( cn wcel cmul co nnmulcl mp2an ) AEFBEFABGHEFCDABIJ $.
  $}

  ${
    $d x y A $.
    $( A positive integer is one or greater.  (Contributed by NM,
       25-Aug-1999.) $)
    nnge1 $p |- ( A e. NN -> 1 <_ A ) $=
      ( vx vy c1 cv cle wbr caddc co breq2 wcel cr wi cc0 clt wn 0re 1re lenlt
      wb 1le1 cn nnre recn addridd breq2d 0lt1 axltadd mp3an12 wa readdcl mpan2
      mpi peano2re mp3an3 syl2anc mpand con3d sylancr 3imtr4d sylbird syl nnind
      lttr ) DBEZFGDDFGDCEZFGZDVFDHIZFGZDAFGBCAVEDDFJVEVFDFJVEVHDFJVEADFJUAVFUB
      KVFLKZVGVIMVFUCVJVGDVFNHIZFGZVIVJVKVFDFVJVFVFUDUEUFVJVKDOGZPZVHDOGZPZVLVI
      VJVOVMVJVKVHOGZVOVMVJNDOGZVQUGNLKZDLKZVJVRVQMQRNDVFUHUIUMVJVKLKZVHLKZVQVO
      UJVMMZVJVSWAQVFNUKULZVFUNZWAWBVTWCRVKVHDVDUOUPUQURVJVTWAVLVNTRWDDVKSUSVJV
      TWBVIVPTRWEDVHSUSUTVAVBVC $.
  $}

  $( A positive integer is less than or equal to one iff it is equal to one.
     (Contributed by NM, 3-Apr-2005.) $)
  nnle1eq1 $p |- ( A e. NN -> ( A <_ 1 <-> A = 1 ) ) $=
    ( cn wcel c1 cle wbr wa wceq nnge1 biantrud cr wb 1re letri3 sylancl bitr4d
    nnre ) ABCZADEFZSDAEFZGZADHZRTSAIJRAKCDKCUBUALAQMADNOP $.

  $( A positive integer is positive.  (Contributed by NM, 26-Sep-1999.) $)
  nngt0 $p |- ( A e. NN -> 0 < A ) $=
    ( cn wcel cr c1 cle wbr cc0 clt nnre nnge1 0lt1 wa wi 0re 1re mp3an12 mpani
    ltletr sylc ) ABCADCZEAFGZHAIGZAJAKUAHEIGZUBUCLHDCEDCUAUDUBMUCNOPHEASQRT $.

  $( A positive integer is not less than one.  (Contributed by NM,
     18-Jan-2004.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  nnnlt1 $p |- ( A e. NN -> -. A < 1 ) $=
    ( cn wcel c1 cle wbr clt wn nnge1 cr wb 1re nnre lenlt sylancr mpbid ) ABCZ
    DAEFZADGFHZAIQDJCAJCRSKLAMDANOP $.

  $( Zero is not a positive integer.  (Contributed by NM, 25-Aug-1999.) $)
  0nnn $p |- -. 0 e. NN $=
    ( cc0 cn wcel c1 clt wbr 0lt1 nnnlt1 mt2 ) ABCADEFGAHI $.

  $( A positive integer is nonzero.  (Contributed by NM, 27-Sep-1999.) $)
  nnne0 $p |- ( A e. NN -> A =/= 0 ) $=
    ( cn wcel cc0 wceq 0nnn eleq1 mtbiri necon2ai ) ABCZADADEJDBCFADBGHI $.

  $( A positive integer is apart from zero.  (Contributed by Jim Kingdon,
     8-Mar-2020.) $)
  nnap0 $p |- ( A e. NN -> A =//= 0 ) $=
    ( cn wcel nnre nngt0 gt0ap0d ) ABCAADAEF $.

  ${
    nngt0.1 $e |- A e. NN $.
    $( A positive integer is positive (inference version).  (Contributed by NM,
       17-Sep-1999.) $)
    nngt0i $p |- 0 < A $=
      ( cn wcel cc0 clt wbr nngt0 ax-mp ) ACDEAFGBAHI $.

    $( A positive integer is apart from zero (inference version).  (Contributed
       by Jim Kingdon, 1-Jan-2023.) $)
    nnap0i $p |- A =//= 0 $=
      ( nnrei nngt0i gt0ap0ii ) AABCABDE $.

    $( A positive integer is nonzero (inference version).  (Contributed by NM,
       25-Aug-1999.) $)
    nnne0i $p |- A =/= 0 $=
      ( nnrei nngt0i gt0ne0ii ) AABCABDE $.
  $}

  ${
    $d x A $.  $d x B $.
    $( There exists a positive integer greater than or equal to any two others.
       (Contributed by NM, 18-Aug-1999.) $)
    nn2ge $p |- ( ( A e. NN /\ B e. NN ) ->
                E. x e. NN ( A <_ x /\ B <_ x ) ) $=
      ( cn wcel wa caddc co cle wbr cc0 cr nnre adantl nngt0 ltled adantr mpbid
      clt breq2 cv wrex nnaddcl 0red addge01d addge02d anbi12d rspcev syl12anc
      wceq ) BDEZCDEZFZBCGHZDEBUNIJZCUNIJZBAUAZIJZCUQIJZFZADUBBCUCUMKCIJUOUMKCU
      MUDZULCLEUKCMNZULKCSJUKCONPUMBCUKBLEULBMQZVBUERUMKBIJUPUMKBVAVCUKKBSJULBO
      QPUMCBVBVCUFRUTUOUPFAUNDUQUNUJURUOUSUPUQUNBITUQUNCITUGUHUI $.
  $}

  ${
    $d x y A $.
    $( A positive integer is either one or greater than one.  This is for
       ` NN ` ; ~ 0elnn is a similar theorem for ` _om ` (the natural numbers
       as ordinals).  (Contributed by Jim Kingdon, 7-Mar-2020.) $)
    nn1gt1 $p |- ( A e. NN -> ( A = 1 \/ 1 < A ) ) $=
      ( vx vy cv c1 wceq clt wbr wo caddc co eqeq1 breq2 orbi12d eqid orci wcel
      cn cc0 cr nngt0 wb nnre 1re ltaddpos2 sylancl mpbid olcd a1d nnind ) BDZE
      FZEUKGHZIEEFZEEGHZICDZEFZEUPGHZIZUPEJKZEFZEUTGHZIZAEFZEAGHZIBCAULULUNUMUO
      UKEELUKEEGMNUKUPFULUQUMURUKUPELUKUPEGMNUKUTFULVAUMVBUKUTELUKUTEGMNUKAFULV
      DUMVEUKAELUKAEGMNUNUOEOPUPRQZVCUSVFVBVAVFSUPGHZVBUPUAVFUPTQETQVGVBUBUPUCU
      DUPEUEUFUGUHUIUJ $.
  $}

  $( A positive integer is greater than one iff it is not equal to one.
     (Contributed by NM, 7-Oct-2004.) $)
  nngt1ne1 $p |- ( A e. NN -> ( 1 < A <-> A =/= 1 ) ) $=
    ( cn wcel c1 clt wbr wne cr 1re ltne mpan wceq wn df-ne nn1gt1 ord biimtrid
    impbid2 ) ABCZDAEFZADGZDHCTUAIDAJKUAADLZMSTADNSUBTAOPQR $.

  $( The quotient of a real and a positive integer is real.  (Contributed by
     NM, 28-Nov-2008.) $)
  nndivre $p |- ( ( A e. RR /\ N e. NN ) -> ( A / N ) e. RR ) $=
    ( cn wcel cr cc0 cap wbr wa cdiv co nnre nnap0 jca redivclap 3expb sylan2 )
    BCDZAEDZBEDZBFGHZIABJKEDZRTUABLBMNSTUAUBABOPQ $.

  $( The reciprocal of a positive integer is real.  (Contributed by NM,
     8-Feb-2008.) $)
  nnrecre $p |- ( N e. NN -> ( 1 / N ) e. RR ) $=
    ( c1 cr wcel cn cdiv co 1re nndivre mpan ) BCDAEDBAFGCDHBAIJ $.

  $( The reciprocal of a positive integer is positive.  (Contributed by NM,
     25-Aug-1999.) $)
  nnrecgt0 $p |- ( A e. NN -> 0 < ( 1 / A ) ) $=
    ( cn wcel c1 cle wbr cc0 cdiv co clt nnge1 0lt1 cr wa wi 0re ltletr mp3an12
    nnre 1re recgt0 ex syld syl mpani mpd ) ABCZDAEFZGDAHIJFZAKUGGDJFZUHUILUGAM
    CZUJUHNZUIOASUKULGAJFZUIGMCDMCUKULUMOPTGDAQRUKUMUIAUAUBUCUDUEUF $.

  ${
    $d z A $.  $d x y z B $.
    $( Subtraction of positive integers.  (Contributed by NM, 20-Aug-2001.)
       (Revised by Mario Carneiro, 16-May-2014.) $)
    nnsub $p |- ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( B - A ) e. NN ) ) $=
      ( vz vx cn wcel clt wbr cmin co wi wral wceq breq2 eleq1d imbi12d ralbidv
      c1 oveq1 cr vy wa cv caddc nnnlt1 pm2.21d rgen breq1 oveq2 cbvralv adantr
      cc nncn ax-1cn pncan sylancl simpl eqeltrd syl5ibrcom a1dd rspcv nnre 1re
      ltsubadd mp3an2 syl2anr subsub3 mp3an3 syl2an biimpd syl9r nn1m1nn adantl
      wb wo mpjaod ralrimdva biimtrid rspcva sylan2 cc0 posdif imbitrrid impbid
      nnind nngt0 ) AEFZBEFZUBZABGHZBAIJZEFZWHWGCUCZBGHZBWMIJZEFZKZCELZWJWLKZWM
      DUCZGHZWTWMIJZEFZKZCELWMRGHZRWMIJZEFZKZCELWMUAUCZGHZXIWMIJZEFZKZCELZWMXIR
      UDJZGHZXOWMIJZEFZKZCELZWRDUABWTRMZXDXHCEYAXAXEXCXGWTRWMGNYAXBXFEWTRWMISOP
      QWTXIMZXDXMCEYBXAXJXCXLWTXIWMGNYBXBXKEWTXIWMISOPQWTXOMZXDXSCEYCXAXPXCXRWT
      XOWMGNYCXBXQEWTXOWMISOPQWTBMZXDWQCEYDXAWNXCWPWTBWMGNYDXBWOEWTBWMISOPQXHCE
      WMEFZXEXGWMUEUFUGXNWTXIGHZXIWTIJZEFZKZDELZXIEFZXTXMYICDEWMWTMZXJYFXLYHWMW
      TXIGUHYLXKYGEWMWTXIIUIOPUJYKYJXSCEYKYEUBZWMRMZYJXSKWMRIJZEFZYMYNXSYJYMYNX
      RXPYMXRYNXORIJZEFYMYQXIEYMXIULFZRULFZYQXIMYKYRYEXIUMZUKUNXIRUOUPYKYEUQURY
      NXQYQEWMRXOIUIOUSUTUTYPYJYOXIGHZXIYOIJZEFZKZYMXSYIUUDDYOEWTYOMZYFUUAYHUUC
      WTYOXIGUHUUEYGUUBEWTYOXIIUIOPVAYMUUDXSYMUUAXPUUCXRYEWMTFZXITFZUUAXPVNZYKW
      MVBXIVBUUFRTFUUGUUHVCWMRXIVDVEVFYMUUBXQEYKYRWMULFZUUBXQMZYEYTWMUMYRUUIYSU
      UJUNXIWMRVGVHVIOPVJVKYEYNYPVOYKWMVLVMVPVQVRWEWQWSCAEWMAMZWNWJWPWLWMABGUHU
      UKWOWKEWMABIUIOPVSVTWLWJWIWAWKGHZWKWFWGATFBTFWJUULVNWHAVBBVBABWBVIWCWD $.

    nnsub.1 $e |- A e. NN $.
    nnsub.2 $e |- B e. NN $.
    $( Subtraction of positive integers.  (Contributed by NM, 19-Aug-2001.) $)
    nnsubi $p |- ( A < B <-> ( B - A ) e. NN ) $=
      ( cn wcel clt wbr cmin co wb nnsub mp2an ) AEFBEFABGHBAIJEFKCDABLM $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Two ways to express " ` A ` divides ` B ` " for positive integers.
       (Contributed by NM, 3-Feb-2004.)  (Proof shortened by Mario Carneiro,
       16-May-2014.) $)
    nndiv $p |- ( ( A e. NN /\ B e. NN ) ->
                ( E. x e. NN ( A x. x ) = B <-> ( B / A ) e. NN ) ) $=
      ( cdiv co cn wcel cv wceq wrex cmul risset eqcom ad2antlr ad2antrr adantl
      wa cc nncn cc0 cap wbr nnap0 divmulapd bitrid rexbidva bitr2id ) CBDEZFGA
      HZUHIZAFJBFGZCFGZQZBUIKECIZAFJAUHFLUMUJUNAFUJUHUIIUMUIFGZQZUNUIUHMUPCBUIU
      LCRGUKUOCSNUKBRGULUOBSOUOUIRGUMUISPUKBTUAUBULUOBUCOUDUEUFUG $.
  $}

  $( Transitive property of divisibility: if ` A ` divides ` B ` and ` B `
     divides ` C ` , then ` A ` divides ` C ` .  Typically, ` C ` would be an
     integer, although the theorem holds for complex ` C ` .  (Contributed by
     NM, 3-May-2005.) $)
  nndivtr $p |- ( ( ( A e. NN /\ B e. NN /\ C e. CC ) /\
               ( ( B / A ) e. NN /\ ( C / B ) e. NN ) ) -> ( C / A ) e. NN ) $=
    ( cn wcel cc w3a cdiv co wa cmul cc0 cap wbr wceq nncn 3ad2ant2 nnap0 jca
    c1 nnmulcl simp3 3ad2ant1 divmul24ap syl22anc dividapd oveq1d divclap 3expb
    sylan2 ancoms mullidd 3adant2 3eqtrd eleq1d imbitrid imp ) ADEZBDEZCFEZGZBA
    HIZDECBHIZDEJZCAHIZDEZVDVBVCKIZDEVAVFVBVCUAVAVGVEDVAVGBBHIZVEKIZTVEKIZVEVAB
    FEZUTAFEZALMNZJZVKBLMNZJZVGVIOUSURVKUTBPZQURUSUTUBURUSVNUTURVLVMAPARSZUCUSU
    RVPUTUSVKVOVQBRZSQBCABUDUEUSURVIVJOUTUSVHTVEKUSBVQVSUFUGQURUTVJVEOUSURUTJVE
    UTURVEFEZURUTVNVTVRUTVLVMVTCAUHUIUJUKULUMUNUOUPUQ $.

  ${
    nnge1d.1 $e |- ( ph -> A e. NN ) $.
    $( A positive integer is one or greater.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nnge1d $p |- ( ph -> 1 <_ A ) $=
      ( cn wcel c1 cle wbr nnge1 syl ) ABDEFBGHCBIJ $.

    $( A positive integer is positive.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nngt0d $p |- ( ph -> 0 < A ) $=
      ( cn wcel cc0 clt wbr nngt0 syl ) ABDEFBGHCBIJ $.

    $( A positive integer is nonzero.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nnne0d $p |- ( ph -> A =/= 0 ) $=
      ( cn wcel cc0 wne nnne0 syl ) ABDEBFGCBHI $.

    $( A positive integer is apart from zero.  (Contributed by Jim Kingdon,
       25-Aug-2021.) $)
    nnap0d $p |- ( ph -> A =//= 0 ) $=
      ( cn wcel cc0 cap wbr nnap0 syl ) ABDEBFGHCBIJ $.

    $( The reciprocal of a positive integer is real.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    nnrecred $p |- ( ph -> ( 1 / A ) e. RR ) $=
      ( cn wcel c1 cdiv co cr nnrecre syl ) ABDEFBGHIECBJK $.

    nnmulcld.2 $e |- ( ph -> B e. NN ) $.
    $( Closure of addition of positive integers.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    nnaddcld $p |- ( ph -> ( A + B ) e. NN ) $=
      ( cn wcel caddc co nnaddcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Closure of multiplication of positive integers.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    nnmulcld $p |- ( ph -> ( A x. B ) e. NN ) $=
      ( cn wcel cmul co nnmulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
  $}

  ${
    nndivred.1 $e |- ( ph -> A e. RR ) $.
    nndivred.2 $e |- ( ph -> B e. NN ) $.
    $( A positive integer is one or greater.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nndivred $p |- ( ph -> ( A / B ) e. RR ) $=
      ( cr wcel cn cdiv co nndivre syl2anc ) ABFGCHGBCIJFGDEBCKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Decimal representation of numbers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The decimal representation of numbers/integers is based on the decimal digits
  0 through 9 ( ~ df-0 through ~ df-9 ), which are explicitly defined in the
  following.  Note that the numbers 0 and 1 are constants defined as primitives
  of the complex number axiom system (see ~ df-0 and ~ df-1 ).

  Integers can also be exhibited as sums of powers of 10 (e.g., the number 103
  can be expressed as ` ( ( ; 1 0 ^ 2 ) + 3 ) `) or as some other expression
  built from operations on the numbers 0 through 9.  For example, the prime
  number 823541 can be expressed as ` ( 7 ^ 7 ) - 2 ` .

  Most abstract math rarely requires numbers larger than 4.  Even in Wiles'
  proof of Fermat's Last Theorem, the largest number used appears to be 12.

$)

  $c 2 $. $( The decimal number 2 $)
  $c 3 $. $( The decimal number 3 $)
  $c 4 $. $( The decimal number 4 $)
  $c 5 $. $( The decimal number 5 $)
  $c 6 $. $( The decimal number 6 $)
  $c 7 $. $( The decimal number 7 $)
  $c 8 $. $( The decimal number 8 $)
  $c 9 $. $( The decimal number 9 $)

  $( Extend class notation to include the number 2. $)
  c2 $a class 2 $.
  $( Extend class notation to include the number 3. $)
  c3 $a class 3 $.
  $( Extend class notation to include the number 4. $)
  c4 $a class 4 $.
  $( Extend class notation to include the number 5. $)
  c5 $a class 5 $.
  $( Extend class notation to include the number 6. $)
  c6 $a class 6 $.
  $( Extend class notation to include the number 7. $)
  c7 $a class 7 $.
  $( Extend class notation to include the number 8. $)
  c8 $a class 8 $.
  $( Extend class notation to include the number 9. $)
  c9 $a class 9 $.

  $( Define the number 2.  (Contributed by NM, 27-May-1999.) $)
  df-2 $a |- 2 = ( 1 + 1 ) $.

  $( Define the number 3.  (Contributed by NM, 27-May-1999.) $)
  df-3 $a |- 3 = ( 2 + 1 ) $.
  $( Define the number 4.  (Contributed by NM, 27-May-1999.) $)
  df-4 $a |- 4 = ( 3 + 1 ) $.
  $( Define the number 5.  (Contributed by NM, 27-May-1999.) $)
  df-5 $a |- 5 = ( 4 + 1 ) $.
  $( Define the number 6.  (Contributed by NM, 27-May-1999.) $)
  df-6 $a |- 6 = ( 5 + 1 ) $.
  $( Define the number 7.  (Contributed by NM, 27-May-1999.) $)
  df-7 $a |- 7 = ( 6 + 1 ) $.
  $( Define the number 8.  (Contributed by NM, 27-May-1999.) $)
  df-8 $a |- 8 = ( 7 + 1 ) $.
  $( Define the number 9.  (Contributed by NM, 27-May-1999.) $)
  df-9 $a |- 9 = ( 8 + 1 ) $.

  $( ` 0 =/= 1 ` (common case).  See aso ~ 1ap0 .  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  0ne1 $p |- 0 =/= 1 $=
    ( cc0 c1 0re 0lt1 ltneii ) ABCDE $.

  $( ` 1 =/= 0 ` .  See aso ~ 1ap0 .  (Contributed by Jim Kingdon,
     9-Mar-2020.) $)
  1ne0 $p |- 1 =/= 0 $=
    ( cc0 c1 0ne1 necomi ) ABCD $.

  $( ` ( 1 - 1 ) = 0 ` (common case).  (Contributed by David A. Wheeler,
     7-Jul-2016.) $)
  1m1e0 $p |- ( 1 - 1 ) = 0 $=
    ( c1 ax-1cn subidi ) ABC $.

  $( The number 2 is real.  (Contributed by NM, 27-May-1999.) $)
  2re $p |- 2 e. RR $=
    ( c2 c1 caddc co cr df-2 1re readdcli eqeltri ) ABBCDEFBBGGHI $.

  $( The number 2 is a complex number.  (Contributed by NM, 30-Jul-2004.) $)
  2cn $p |- 2 e. CC $=
    ( c2 2re recni ) ABC $.

  $( 2 is a set (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  2ex $p |- 2 e. _V $=
    ( c2 cc 2cn elexi ) ABCD $.

  $( 2 is a complex number, deductive form (common case).  (Contributed by
     David A. Wheeler, 8-Dec-2018.) $)
  2cnd $p |- ( ph -> 2 e. CC ) $=
    ( c2 cc wcel 2cn a1i ) BCDAEF $.

  $( The number 3 is real.  (Contributed by NM, 27-May-1999.) $)
  3re $p |- 3 e. RR $=
    ( c3 c2 c1 caddc co cr df-3 2re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.

  $( The number 3 is a complex number.  (Contributed by FL, 17-Oct-2010.) $)
  3cn $p |- 3 e. CC $=
    ( c3 3re recni ) ABC $.

  $( 3 is a set (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  3ex $p |- 3 e. _V $=
    ( c3 cc 3cn elexi ) ABCD $.

  $( The number 4 is real.  (Contributed by NM, 27-May-1999.) $)
  4re $p |- 4 e. RR $=
    ( c4 c3 c1 caddc co cr df-4 3re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.

  $( The number 4 is a complex number.  (Contributed by David A. Wheeler,
     7-Jul-2016.) $)
  4cn $p |- 4 e. CC $=
    ( c4 4re recni ) ABC $.

  $( The number 5 is real.  (Contributed by NM, 27-May-1999.) $)
  5re $p |- 5 e. RR $=
    ( c5 c4 c1 caddc co cr df-5 4re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.

  $( The number 5 is complex.  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  5cn $p |- 5 e. CC $=
    ( c5 5re recni ) ABC $.

  $( The number 6 is real.  (Contributed by NM, 27-May-1999.) $)
  6re $p |- 6 e. RR $=
    ( c6 c5 c1 caddc co cr df-6 5re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.

  $( The number 6 is complex.  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  6cn $p |- 6 e. CC $=
    ( c6 6re recni ) ABC $.

  $( The number 7 is real.  (Contributed by NM, 27-May-1999.) $)
  7re $p |- 7 e. RR $=
    ( c7 c6 c1 caddc co cr df-7 6re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.

  $( The number 7 is complex.  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  7cn $p |- 7 e. CC $=
    ( c7 7re recni ) ABC $.

  $( The number 8 is real.  (Contributed by NM, 27-May-1999.) $)
  8re $p |- 8 e. RR $=
    ( c8 c7 c1 caddc co cr df-8 7re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.

  $( The number 8 is complex.  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  8cn $p |- 8 e. CC $=
    ( c8 8re recni ) ABC $.

  $( The number 9 is real.  (Contributed by NM, 27-May-1999.) $)
  9re $p |- 9 e. RR $=
    ( c9 c8 c1 caddc co cr df-9 8re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.

  $( The number 9 is complex.  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  9cn $p |- 9 e. CC $=
    ( c9 9re recni ) ABC $.

  $( Zero is nonnegative.  (Contributed by David A. Wheeler, 7-Jul-2016.) $)
  0le0 $p |- 0 <_ 0 $=
    ( cc0 0re leidi ) ABC $.

  $( 0 is less than or equal to 2.  (Contributed by David A. Wheeler,
     7-Dec-2018.) $)
  0le2 $p |- 0 <_ 2 $=
    ( cc0 c1 caddc co c2 cle wbr 0le1 1re addge0i mp2an df-2 breqtrri ) ABBCDZE
    FABFGZOANFGHHBBIIJKLM $.

  $( The number 2 is positive.  (Contributed by NM, 27-May-1999.) $)
  2pos $p |- 0 < 2 $=
    ( cc0 c1 caddc co c2 clt 1re 0lt1 addgt0ii df-2 breqtrri ) ABBCDEFBBGGHHIJK
    $.

  $( The number 2 is nonzero.  (Contributed by NM, 9-Nov-2007.) $)
  2ne0 $p |- 2 =/= 0 $=
    ( c2 2re 2pos gt0ne0ii ) ABCD $.

  $( The number 2 is apart from zero.  (Contributed by Jim Kingdon,
     9-Mar-2020.) $)
  2ap0 $p |- 2 =//= 0 $=
    ( c2 2re 2pos gt0ap0ii ) ABCD $.

  $( The number 3 is positive.  (Contributed by NM, 27-May-1999.) $)
  3pos $p |- 0 < 3 $=
    ( cc0 c2 c1 caddc co c3 clt 2re 1re 2pos 0lt1 addgt0ii df-3 breqtrri ) ABCD
    EFGBCHIJKLMN $.

  $( The number 3 is nonzero.  (Contributed by FL, 17-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 7-May-2011.) $)
  3ne0 $p |- 3 =/= 0 $=
    ( c3 3re 3pos gt0ne0ii ) ABCD $.

  $( The number 3 is apart from zero.  (Contributed by Jim Kingdon,
     10-Oct-2021.) $)
  3ap0 $p |- 3 =//= 0 $=
    ( c3 3re 3pos gt0ap0ii ) ABCD $.

  $( The number 4 is positive.  (Contributed by NM, 27-May-1999.) $)
  4pos $p |- 0 < 4 $=
    ( cc0 c3 c1 caddc co c4 clt 3re 1re 3pos 0lt1 addgt0ii df-4 breqtrri ) ABCD
    EFGBCHIJKLMN $.

  $( The number 4 is nonzero.  (Contributed by David A. Wheeler,
     5-Dec-2018.) $)
  4ne0 $p |- 4 =/= 0 $=
    ( c4 4re 4pos gt0ne0ii ) ABCD $.

  $( The number 4 is apart from zero.  (Contributed by Jim Kingdon,
     10-Oct-2021.) $)
  4ap0 $p |- 4 =//= 0 $=
    ( c4 4re 4pos gt0ap0ii ) ABCD $.

  $( The number 5 is positive.  (Contributed by NM, 27-May-1999.) $)
  5pos $p |- 0 < 5 $=
    ( cc0 c4 c1 caddc co c5 clt 4re 1re 4pos 0lt1 addgt0ii df-5 breqtrri ) ABCD
    EFGBCHIJKLMN $.

  $( The number 6 is positive.  (Contributed by NM, 27-May-1999.) $)
  6pos $p |- 0 < 6 $=
    ( cc0 c5 c1 caddc co c6 clt 5re 1re 5pos 0lt1 addgt0ii df-6 breqtrri ) ABCD
    EFGBCHIJKLMN $.

  $( The number 7 is positive.  (Contributed by NM, 27-May-1999.) $)
  7pos $p |- 0 < 7 $=
    ( cc0 c6 c1 caddc co c7 clt 6re 1re 6pos 0lt1 addgt0ii df-7 breqtrri ) ABCD
    EFGBCHIJKLMN $.

  $( The number 8 is positive.  (Contributed by NM, 27-May-1999.) $)
  8pos $p |- 0 < 8 $=
    ( cc0 c7 c1 caddc co c8 clt 7re 1re 7pos 0lt1 addgt0ii df-8 breqtrri ) ABCD
    EFGBCHIJKLMN $.

  $( The number 9 is positive.  (Contributed by NM, 27-May-1999.) $)
  9pos $p |- 0 < 9 $=
    ( cc0 c8 c1 caddc co c9 clt 8re 1re 8pos 0lt1 addgt0ii df-9 breqtrri ) ABCD
    EFGBCHIJKLMN $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Some properties of specific numbers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This includes adding two pairs of values 1..10
  (where the right is less than the left)
  and where the left is less than the right for the values 1..10.

$)

  $( -1 is a complex number (common case).  (Contributed by David A. Wheeler,
     7-Jul-2016.) $)
  neg1cn $p |- -u 1 e. CC $=
    ( c1 ax-1cn negcli ) ABC $.

  $( -1 is a real number (common case).  (Contributed by David A. Wheeler,
     5-Dec-2018.) $)
  neg1rr $p |- -u 1 e. RR $=
    ( c1 1re renegcli ) ABC $.

  $( -1 is nonzero (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  neg1ne0 $p |- -u 1 =/= 0 $=
    ( c1 ax-1cn 1ne0 negne0i ) ABCD $.

  $( -1 is less than 0 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  neg1lt0 $p |- -u 1 < 0 $=
    ( c1 cneg cc0 clt wbr neg0 0lt1 eqbrtri 1re 0re ltnegcon1i mpbir ) ABCDECBZ
    ADEMCADFGHACIJKL $.

  $( -1 is apart from zero.  (Contributed by Jim Kingdon, 9-Jun-2020.) $)
  neg1ap0 $p |- -u 1 =//= 0 $=
    ( c1 cc0 cap wbr cneg 1ap0 cc wcel wb ax-1cn negap0 ax-mp mpbi ) ABCDZAEBCD
    ZFAGHNOIJAKLM $.

  $( ` -u -u 1 ` is 1 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  negneg1e1 $p |- -u -u 1 = 1 $=
    ( c1 ax-1cn negnegi ) ABC $.

  $( ` 1 + -u 1 ` is 0 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  1pneg1e0 $p |- ( 1 + -u 1 ) = 0 $=
    ( c1 ax-1cn negidi ) ABC $.

  $( 0 minus 0 equals 0 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  0m0e0 $p |- ( 0 - 0 ) = 0 $=
    ( cc0 0cn subidi ) ABC $.

  $( 1 - 0 = 1 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  1m0e1 $p |- ( 1 - 0 ) = 1 $=
    ( c1 ax-1cn subid1i ) ABC $.

  $( 0 + 1 = 1.  (Contributed by David A. Wheeler, 7-Jul-2016.) $)
  0p1e1 $p |- ( 0 + 1 ) = 1 $=
    ( c1 ax-1cn addlidi ) ABC $.

  $( Function value at ` N + 1 ` with ` N ` replaced by ` 0 ` .  Technical
     theorem to be used to reduce the size of a significant number of proofs.
     (Contributed by AV, 13-Aug-2022.) $)
  fv0p1e1 $p |- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) ) $=
    ( cc0 wceq c1 caddc co oveq1 0p1e1 eqtrdi fveq2d ) BCDZBEFGZEALMCEFGEBCEFHI
    JK $.

  $( 1 + 0 = 1.  (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  1p0e1 $p |- ( 1 + 0 ) = 1 $=
    ( c1 ax-1cn addridi ) ABC $.

  $( 1 + 1 = 2.  (Contributed by NM, 1-Apr-2008.) $)
  1p1e2 $p |- ( 1 + 1 ) = 2 $=
    ( c2 c1 caddc co df-2 eqcomi ) ABBCDEF $.

  $( 2 - 1 = 1.  The result is on the right-hand-side to be consistent with
     similar proofs like ~ 4p4e8 .  (Contributed by David A. Wheeler,
     4-Jan-2017.) $)
  2m1e1 $p |- ( 2 - 1 ) = 1 $=
    ( c2 c1 2cn ax-1cn 1p1e2 subaddrii ) ABBCDDEF $.

  $( 1 = 2 - 1 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  1e2m1 $p |- 1 = ( 2 - 1 ) $=
    ( c2 c1 cmin co 2m1e1 eqcomi ) ABCDBEF $.

  $( 3 - 1 = 2.  (Contributed by FL, 17-Oct-2010.)  (Revised by NM,
     10-Dec-2017.) $)
  3m1e2 $p |- ( 3 - 1 ) = 2 $=
    ( c3 c1 c2 3cn ax-1cn 2cn caddc co addcomi df-3 eqtr4i subaddrii ) ABCDEFBC
    GHCBGHABCEFIJKL $.

  $( 4 - 1 = 3.  (Contributed by AV, 8-Feb-2021.)  (Proof shortened by AV,
     6-Sep-2021.) $)
  4m1e3 $p |- ( 4 - 1 ) = 3 $=
    ( c4 c3 c1 3cn ax-1cn df-4 mvrraddi ) ABCDEFG $.

  $( 5 - 1 = 4.  (Contributed by AV, 6-Sep-2021.) $)
  5m1e4 $p |- ( 5 - 1 ) = 4 $=
    ( c5 c4 c1 4cn ax-1cn df-5 mvrraddi ) ABCDEFG $.

  $( 6 - 1 = 5.  (Contributed by AV, 6-Sep-2021.) $)
  6m1e5 $p |- ( 6 - 1 ) = 5 $=
    ( c6 c5 c1 5cn ax-1cn df-6 mvrraddi ) ABCDEFG $.

  $( 7 - 1 = 6.  (Contributed by AV, 6-Sep-2021.) $)
  7m1e6 $p |- ( 7 - 1 ) = 6 $=
    ( c7 c6 c1 6cn ax-1cn df-7 mvrraddi ) ABCDEFG $.

  $( 8 - 1 = 7.  (Contributed by AV, 6-Sep-2021.) $)
  8m1e7 $p |- ( 8 - 1 ) = 7 $=
    ( c8 c7 c1 7cn ax-1cn df-8 mvrraddi ) ABCDEFG $.

  $( 9 - 1 = 8.  (Contributed by AV, 6-Sep-2021.) $)
  9m1e8 $p |- ( 9 - 1 ) = 8 $=
    ( c9 c8 c1 8cn ax-1cn df-9 mvrraddi ) ABCDEFG $.

  $( Two plus two equals four.  For more information, see "2+2=4 Trivia" on the
     Metamath Proof Explorer Home Page:
     ~ https://us.metamath.org/mpeuni/mmset.html#trivia .  (Contributed by NM,
     27-May-1999.) $)
  2p2e4 $p |- ( 2 + 2 ) = 4 $=
    ( c2 caddc co c1 c4 df-2 oveq2i c3 df-4 oveq1i ax-1cn addassi 3eqtri eqtr4i
    df-3 2cn ) AABCADDBCZBCZEAQABFGEHDBCADBCZDBCRIHSDBOJADDPKKLMN $.

  $( Two times a number.  (Contributed by NM, 10-Oct-2004.)  (Revised by Mario
     Carneiro, 27-May-2016.)  (Proof shortened by AV, 26-Feb-2020.) $)
  2times $p |- ( A e. CC -> ( 2 x. A ) = ( A + A ) ) $=
    ( cc wcel c2 cmul co c1 caddc df-2 oveq1i 1p1times eqtrid ) ABCDAEFGGHFZAEF
    AAHFDMAEIJAKL $.

  $( A number times 2.  (Contributed by NM, 16-Oct-2007.) $)
  times2 $p |- ( A e. CC -> ( A x. 2 ) = ( A + A ) ) $=
    ( cc wcel c2 cmul co caddc wceq 2cn mulcom mpan2 2times eqtrd ) ABCZADEFZDA
    EFZAAGFNDBCOPHIADJKALM $.

  ${
    2times.1 $e |- A e. CC $.
    $( Two times a number.  (Contributed by NM, 1-Aug-1999.) $)
    2timesi $p |- ( 2 x. A ) = ( A + A ) $=
      ( cc wcel c2 cmul co caddc wceq 2times ax-mp ) ACDEAFGAAHGIBAJK $.

    $( A number times 2.  (Contributed by NM, 11-May-2004.) $)
    times2i $p |- ( A x. 2 ) = ( A + A ) $=
      ( cc wcel c2 cmul co caddc wceq times2 ax-mp ) ACDAEFGAAHGIBAJK $.
  $}

  $( Two times a complex number minus the number itself results in the number
     itself.  (Contributed by Alexander van der Vekens, 8-Jun-2018.) $)
  2txmxeqx $p |- ( X e. CC -> ( ( 2 x. X ) - X ) = X ) $=
    ( cc wcel c2 cmul co id 2times mvrladdd ) ABCZDAEFAAJGZKAHI $.

  $( 2 divided by 2 is 1 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  2div2e1 $p |- ( 2 / 2 ) = 1 $=
    ( c2 2cn 2ap0 dividapi ) ABCD $.

  $( 2 + 1 = 3.  (Contributed by Mario Carneiro, 18-Apr-2015.) $)
  2p1e3 $p |- ( 2 + 1 ) = 3 $=
    ( c3 c2 c1 caddc co df-3 eqcomi ) ABCDEFG $.

  $( 1 + 2 = 3 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  1p2e3 $p |- ( 1 + 2 ) = 3 $=
    ( c2 c1 c3 2cn ax-1cn 2p1e3 addcomli ) ABCDEFG $.

  $( 3 + 1 = 4.  (Contributed by Mario Carneiro, 18-Apr-2015.) $)
  3p1e4 $p |- ( 3 + 1 ) = 4 $=
    ( c4 c3 c1 caddc co df-4 eqcomi ) ABCDEFG $.

  $( 4 + 1 = 5.  (Contributed by Mario Carneiro, 18-Apr-2015.) $)
  4p1e5 $p |- ( 4 + 1 ) = 5 $=
    ( c5 c4 c1 caddc co df-5 eqcomi ) ABCDEFG $.

  $( 5 + 1 = 6.  (Contributed by Mario Carneiro, 18-Apr-2015.) $)
  5p1e6 $p |- ( 5 + 1 ) = 6 $=
    ( c6 c5 c1 caddc co df-6 eqcomi ) ABCDEFG $.

  $( 6 + 1 = 7.  (Contributed by Mario Carneiro, 18-Apr-2015.) $)
  6p1e7 $p |- ( 6 + 1 ) = 7 $=
    ( c7 c6 c1 caddc co df-7 eqcomi ) ABCDEFG $.

  $( 7 + 1 = 8.  (Contributed by Mario Carneiro, 18-Apr-2015.) $)
  7p1e8 $p |- ( 7 + 1 ) = 8 $=
    ( c8 c7 c1 caddc co df-8 eqcomi ) ABCDEFG $.

  $( 8 + 1 = 9.  (Contributed by Mario Carneiro, 18-Apr-2015.) $)
  8p1e9 $p |- ( 8 + 1 ) = 9 $=
    ( c9 c8 c1 caddc co df-9 eqcomi ) ABCDEFG $.

  $( 3 + 2 = 5.  (Contributed by NM, 11-May-2004.) $)
  3p2e5 $p |- ( 3 + 2 ) = 5 $=
    ( c3 c2 caddc co c4 c1 c5 df-2 oveq2i 3cn ax-1cn addassi eqtr4i df-4 oveq1i
    df-5 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.

  $( 3 + 3 = 6.  (Contributed by NM, 11-May-2004.) $)
  3p3e6 $p |- ( 3 + 3 ) = 6 $=
    ( c3 caddc co c2 c1 c6 df-3 oveq2i 3cn 2cn ax-1cn addassi eqtr4i df-6 3p2e5
    c5 oveq1i ) AABCZADBCZEBCZFRADEBCZBCTAUAABGHADEIJKLMFPEBCTNSPEBOQMM $.

  $( 4 + 2 = 6.  (Contributed by NM, 11-May-2004.) $)
  4p2e6 $p |- ( 4 + 2 ) = 6 $=
    ( c4 c2 caddc co c5 c1 c6 df-2 oveq2i 4cn ax-1cn addassi eqtr4i df-5 oveq1i
    df-6 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.

  $( 4 + 3 = 7.  (Contributed by NM, 11-May-2004.) $)
  4p3e7 $p |- ( 4 + 3 ) = 7 $=
    ( c4 c3 caddc co c2 c1 c7 df-3 oveq2i 4cn 2cn ax-1cn addassi eqtr4i c6 df-7
    4p2e6 oveq1i ) ABCDZAECDZFCDZGSAEFCDZCDUABUBACHIAEFJKLMNGOFCDUAPTOFCQRNN $.

  $( 4 + 4 = 8.  (Contributed by NM, 11-May-2004.) $)
  4p4e8 $p |- ( 4 + 4 ) = 8 $=
    ( c4 caddc co c3 c1 c8 df-4 oveq2i 4cn 3cn ax-1cn addassi eqtr4i df-8 4p3e7
    c7 oveq1i ) AABCZADBCZEBCZFRADEBCZBCTAUAABGHADEIJKLMFPEBCTNSPEBOQMM $.

  $( 5 + 2 = 7.  (Contributed by NM, 11-May-2004.) $)
  5p2e7 $p |- ( 5 + 2 ) = 7 $=
    ( c5 c2 caddc co c6 c1 c7 df-2 oveq2i 5cn ax-1cn addassi eqtr4i df-6 oveq1i
    df-7 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.

  $( 5 + 3 = 8.  (Contributed by NM, 11-May-2004.) $)
  5p3e8 $p |- ( 5 + 3 ) = 8 $=
    ( c5 c3 caddc co c2 c1 c8 df-3 oveq2i 5cn 2cn ax-1cn addassi eqtr4i c7 df-8
    5p2e7 oveq1i ) ABCDZAECDZFCDZGSAEFCDZCDUABUBACHIAEFJKLMNGOFCDUAPTOFCQRNN $.

  $( 5 + 4 = 9.  (Contributed by NM, 11-May-2004.) $)
  5p4e9 $p |- ( 5 + 4 ) = 9 $=
    ( c5 c4 caddc co c3 c1 c9 df-4 oveq2i 5cn 3cn ax-1cn addassi eqtr4i c8 df-9
    5p3e8 oveq1i ) ABCDZAECDZFCDZGSAEFCDZCDUABUBACHIAEFJKLMNGOFCDUAPTOFCQRNN $.

  $( 6 + 2 = 8.  (Contributed by NM, 11-May-2004.) $)
  6p2e8 $p |- ( 6 + 2 ) = 8 $=
    ( c6 c2 caddc co c7 c1 c8 df-2 oveq2i 6cn ax-1cn addassi eqtr4i df-7 oveq1i
    df-8 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.

  $( 6 + 3 = 9.  (Contributed by NM, 11-May-2004.) $)
  6p3e9 $p |- ( 6 + 3 ) = 9 $=
    ( c6 c3 caddc co c2 c1 c9 df-3 oveq2i 6cn 2cn ax-1cn addassi eqtr4i c8 df-9
    6p2e8 oveq1i ) ABCDZAECDZFCDZGSAEFCDZCDUABUBACHIAEFJKLMNGOFCDUAPTOFCQRNN $.

  $( 7 + 2 = 9.  (Contributed by NM, 11-May-2004.) $)
  7p2e9 $p |- ( 7 + 2 ) = 9 $=
    ( c7 c2 caddc co c8 c1 c9 df-2 oveq2i 7cn ax-1cn addassi eqtr4i df-8 oveq1i
    df-9 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.

  $( 1 times 1 equals 1.  (Contributed by David A. Wheeler, 7-Jul-2016.) $)
  1t1e1 $p |- ( 1 x. 1 ) = 1 $=
    ( c1 ax-1cn mulridi ) ABC $.

  $( 2 times 1 equals 2.  (Contributed by David A. Wheeler, 6-Dec-2018.) $)
  2t1e2 $p |- ( 2 x. 1 ) = 2 $=
    ( c2 2cn mulridi ) ABC $.

  $( 2 times 2 equals 4.  (Contributed by NM, 1-Aug-1999.) $)
  2t2e4 $p |- ( 2 x. 2 ) = 4 $=
    ( c2 cmul co caddc c4 2cn 2timesi 2p2e4 eqtri ) AABCAADCEAFGHI $.

  $( 3 times 1 equals 3.  (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  3t1e3 $p |- ( 3 x. 1 ) = 3 $=
    ( c3 3cn mulridi ) ABC $.

  $( 3 times 2 equals 6.  (Contributed by NM, 2-Aug-2004.) $)
  3t2e6 $p |- ( 3 x. 2 ) = 6 $=
    ( c3 c2 cmul co caddc c6 3cn times2i 3p3e6 eqtri ) ABCDAAEDFAGHIJ $.

  $( 3 times 3 equals 9.  (Contributed by NM, 11-May-2004.) $)
  3t3e9 $p |- ( 3 x. 3 ) = 9 $=
    ( c3 cmul co c2 c1 caddc c9 df-3 oveq2i 3cn 2cn ax-1cn adddii 3t2e6 oveq12i
    c6 3t1e3 eqtri 6p3e9 ) AABCADEFCZBCZGATABHIUAPAFCZGUAADBCZAEBCZFCUBADEJKLMU
    CPUDAFNQORSRR $.

  $( 4 times 2 equals 8.  (Contributed by NM, 2-Aug-2004.) $)
  4t2e8 $p |- ( 4 x. 2 ) = 8 $=
    ( c4 c2 cmul co caddc c8 4cn times2i 4p4e8 eqtri ) ABCDAAEDFAGHIJ $.

  $( 2 times 0 equals 0.  (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  2t0e0 $p |- ( 2 x. 0 ) = 0 $=
    ( c2 2cn mul01i ) ABC $.

  $( One half of four is two.  (Contributed by NM, 3-Sep-1999.) $)
  4d2e2 $p |- ( 4 / 2 ) = 2 $=
    ( c4 c2 cdiv co wceq cmul 2t2e4 4cn 2cn 2ap0 divmulapi mpbir ) ABCDBEBBFDAE
    GABBHIIJKL $.

  $( 2 is a positive integer.  (Contributed by NM, 20-Aug-2001.) $)
  2nn $p |- 2 e. NN $=
    ( c2 c1 caddc co cn df-2 wcel 1nn peano2nn ax-mp eqeltri ) ABBCDZEFBEGLEGHB
    IJK $.

  $( 3 is a positive integer.  (Contributed by NM, 8-Jan-2006.) $)
  3nn $p |- 3 e. NN $=
    ( c3 c2 c1 caddc co cn df-3 wcel 2nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
    HIBJKL $.

  $( 4 is a positive integer.  (Contributed by NM, 8-Jan-2006.) $)
  4nn $p |- 4 e. NN $=
    ( c4 c3 c1 caddc co cn df-4 wcel 3nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
    HIBJKL $.

  $( 5 is a positive integer.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  5nn $p |- 5 e. NN $=
    ( c5 c4 c1 caddc co cn df-5 wcel 4nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
    HIBJKL $.

  $( 6 is a positive integer.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  6nn $p |- 6 e. NN $=
    ( c6 c5 c1 caddc co cn df-6 wcel 5nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
    HIBJKL $.

  $( 7 is a positive integer.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  7nn $p |- 7 e. NN $=
    ( c7 c6 c1 caddc co cn df-7 wcel 6nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
    HIBJKL $.

  $( 8 is a positive integer.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  8nn $p |- 8 e. NN $=
    ( c8 c7 c1 caddc co cn df-8 wcel 7nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
    HIBJKL $.

  $( 9 is a positive integer.  (Contributed by NM, 21-Oct-2012.) $)
  9nn $p |- 9 e. NN $=
    ( c9 c8 c1 caddc co cn df-9 wcel 8nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
    HIBJKL $.

  $( 1 is less than 2.  (Contributed by NM, 24-Feb-2005.) $)
  1lt2 $p |- 1 < 2 $=
    ( c1 caddc co c2 clt 1re ltp1i df-2 breqtrri ) AAABCDEAFGHI $.

  $( 2 is less than 3.  (Contributed by NM, 26-Sep-2010.) $)
  2lt3 $p |- 2 < 3 $=
    ( c2 c1 caddc co c3 clt 2re ltp1i df-3 breqtrri ) AABCDEFAGHIJ $.

  $( 1 is less than 3.  (Contributed by NM, 26-Sep-2010.) $)
  1lt3 $p |- 1 < 3 $=
    ( c1 c2 clt wbr c3 1lt2 2lt3 1re 2re 3re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 3 is less than 4.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  3lt4 $p |- 3 < 4 $=
    ( c3 c1 caddc co c4 clt 3re ltp1i df-4 breqtrri ) AABCDEFAGHIJ $.

  $( 2 is less than 4.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  2lt4 $p |- 2 < 4 $=
    ( c2 c3 clt wbr c4 2lt3 3lt4 2re 3re 4re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 1 is less than 4.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  1lt4 $p |- 1 < 4 $=
    ( c1 c2 clt wbr c4 1lt2 2lt4 1re 2re 4re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 4 is less than 5.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  4lt5 $p |- 4 < 5 $=
    ( c4 c1 caddc co c5 clt 4re ltp1i df-5 breqtrri ) AABCDEFAGHIJ $.

  $( 3 is less than 5.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  3lt5 $p |- 3 < 5 $=
    ( c3 c4 clt wbr c5 3lt4 4lt5 3re 4re 5re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 2 is less than 5.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  2lt5 $p |- 2 < 5 $=
    ( c2 c4 clt wbr c5 2lt4 4lt5 2re 4re 5re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 1 is less than 5.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  1lt5 $p |- 1 < 5 $=
    ( c1 c4 clt wbr c5 1lt4 4lt5 1re 4re 5re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 5 is less than 6.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  5lt6 $p |- 5 < 6 $=
    ( c5 c1 caddc co c6 clt 5re ltp1i df-6 breqtrri ) AABCDEFAGHIJ $.

  $( 4 is less than 6.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  4lt6 $p |- 4 < 6 $=
    ( c4 c5 clt wbr c6 4lt5 5lt6 4re 5re 6re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 3 is less than 6.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  3lt6 $p |- 3 < 6 $=
    ( c3 c4 clt wbr c6 3lt4 4lt6 3re 4re 6re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 2 is less than 6.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  2lt6 $p |- 2 < 6 $=
    ( c2 c3 clt wbr c6 2lt3 3lt6 2re 3re 6re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 1 is less than 6.  (Contributed by NM, 19-Oct-2012.) $)
  1lt6 $p |- 1 < 6 $=
    ( c1 c2 clt wbr c6 1lt2 2lt6 1re 2re 6re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 6 is less than 7.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  6lt7 $p |- 6 < 7 $=
    ( c6 c1 caddc co c7 clt 6re ltp1i df-7 breqtrri ) AABCDEFAGHIJ $.

  $( 5 is less than 7.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  5lt7 $p |- 5 < 7 $=
    ( c5 c6 clt wbr c7 5lt6 6lt7 5re 6re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 4 is less than 7.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  4lt7 $p |- 4 < 7 $=
    ( c4 c5 clt wbr c7 4lt5 5lt7 4re 5re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 3 is less than 7.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  3lt7 $p |- 3 < 7 $=
    ( c3 c4 clt wbr c7 3lt4 4lt7 3re 4re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 2 is less than 7.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  2lt7 $p |- 2 < 7 $=
    ( c2 c3 clt wbr c7 2lt3 3lt7 2re 3re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 1 is less than 7.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  1lt7 $p |- 1 < 7 $=
    ( c1 c2 clt wbr c7 1lt2 2lt7 1re 2re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 7 is less than 8.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  7lt8 $p |- 7 < 8 $=
    ( c7 c1 caddc co c8 clt 7re ltp1i df-8 breqtrri ) AABCDEFAGHIJ $.

  $( 6 is less than 8.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  6lt8 $p |- 6 < 8 $=
    ( c6 c7 clt wbr c8 6lt7 7lt8 6re 7re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 5 is less than 8.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  5lt8 $p |- 5 < 8 $=
    ( c5 c6 clt wbr c8 5lt6 6lt8 5re 6re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 4 is less than 8.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  4lt8 $p |- 4 < 8 $=
    ( c4 c5 clt wbr c8 4lt5 5lt8 4re 5re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 3 is less than 8.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  3lt8 $p |- 3 < 8 $=
    ( c3 c4 clt wbr c8 3lt4 4lt8 3re 4re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 2 is less than 8.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  2lt8 $p |- 2 < 8 $=
    ( c2 c3 clt wbr c8 2lt3 3lt8 2re 3re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 1 is less than 8.  (Contributed by Mario Carneiro, 15-Sep-2013.) $)
  1lt8 $p |- 1 < 8 $=
    ( c1 c2 clt wbr c8 1lt2 2lt8 1re 2re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 8 is less than 9.  (Contributed by Mario Carneiro, 19-Feb-2014.) $)
  8lt9 $p |- 8 < 9 $=
    ( c8 c1 caddc co c9 clt 8re ltp1i df-9 breqtrri ) AABCDEFAGHIJ $.

  $( 7 is less than 9.  (Contributed by Mario Carneiro, 9-Mar-2015.) $)
  7lt9 $p |- 7 < 9 $=
    ( c7 c8 clt wbr c9 7lt8 8lt9 7re 8re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 6 is less than 9.  (Contributed by Mario Carneiro, 9-Mar-2015.) $)
  6lt9 $p |- 6 < 9 $=
    ( c6 c7 clt wbr c9 6lt7 7lt9 6re 7re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 5 is less than 9.  (Contributed by Mario Carneiro, 9-Mar-2015.) $)
  5lt9 $p |- 5 < 9 $=
    ( c5 c6 clt wbr c9 5lt6 6lt9 5re 6re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 4 is less than 9.  (Contributed by Mario Carneiro, 9-Mar-2015.) $)
  4lt9 $p |- 4 < 9 $=
    ( c4 c5 clt wbr c9 4lt5 5lt9 4re 5re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 3 is less than 9.  (Contributed by Mario Carneiro, 9-Mar-2015.) $)
  3lt9 $p |- 3 < 9 $=
    ( c3 c4 clt wbr c9 3lt4 4lt9 3re 4re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 2 is less than 9.  (Contributed by Mario Carneiro, 9-Mar-2015.) $)
  2lt9 $p |- 2 < 9 $=
    ( c2 c3 clt wbr c9 2lt3 3lt9 2re 3re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 1 is less than 9.  (Contributed by NM, 19-Oct-2012.)  (Revised by Mario
     Carneiro, 9-Mar-2015.) $)
  1lt9 $p |- 1 < 9 $=
    ( c1 c2 clt wbr c9 1lt2 2lt9 1re 2re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
    KL $.

  $( 0 is not equal to 2.  (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  0ne2 $p |- 0 =/= 2 $=
    ( c2 cc0 2ne0 necomi ) ABCD $.

  $( 1 is not equal to 2.  (Contributed by NM, 19-Oct-2012.) $)
  1ne2 $p |- 1 =/= 2 $=
    ( c1 c2 1re 1lt2 ltneii ) ABCDE $.

  $( 1 is apart from 2.  (Contributed by Jim Kingdon, 29-Oct-2022.) $)
  1ap2 $p |- 1 =//= 2 $=
    ( c1 c2 1re 2re 1lt2 ltapii ) ABCDEF $.

  $( 1 is less than or equal to 2 (common case).  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  1le2 $p |- 1 <_ 2 $=
    ( c1 c2 1re 2re 1lt2 ltleii ) ABCDEF $.

  $( 2 is a nonzero complex number (common case).  (Contributed by David A.
     Wheeler, 7-Dec-2018.) $)
  2cnne0 $p |- ( 2 e. CC /\ 2 =/= 0 ) $=
    ( c2 cc wcel cc0 wne 2cn 2ne0 pm3.2i ) ABCADEFGH $.

  $( 2 is a nonzero real number (common case).  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  2rene0 $p |- ( 2 e. RR /\ 2 =/= 0 ) $=
    ( c2 cr wcel cc0 wne 2re 2ne0 pm3.2i ) ABCADEFGH $.

  $( 1 is less than or equal to 3.  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  1le3 $p |- 1 <_ 3 $=
    ( c1 c3 1re 3re 1lt3 ltleii ) ABCDEF $.

  $( ` -u 1 x. -u 1 ` is 1 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  neg1mulneg1e1 $p |- ( -u 1 x. -u 1 ) = 1 $=
    ( c1 cneg cmul co ax-1cn mul2negi 1t1e1 eqtri ) ABZICDAACDAAAEEFGH $.

  $( One-half is real.  (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  halfre $p |- ( 1 / 2 ) e. RR $=
    ( c2 2re 2ap0 rerecclapi ) ABCD $.

  $( One-half is complex.  (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  halfcn $p |- ( 1 / 2 ) e. CC $=
    ( c1 c2 cdiv co halfre recni ) ABCDEF $.

  $( One-half is greater than zero.  (Contributed by NM, 24-Feb-2005.) $)
  halfgt0 $p |- 0 < ( 1 / 2 ) $=
    ( c2 2re 2pos recgt0ii ) ABCD $.

  $( One-half is not negative.  (Contributed by AV, 7-Jun-2020.) $)
  halfge0 $p |- 0 <_ ( 1 / 2 ) $=
    ( cc0 c1 c2 cdiv co 0re halfre halfgt0 ltleii ) ABCDEFGHI $.

  $( One-half is less than one.  (Contributed by NM, 24-Feb-2005.) $)
  halflt1 $p |- ( 1 / 2 ) < 1 $=
    ( c1 cdiv co clt wbr 1div1e1 1lt2 eqbrtri 1re 2re 0lt1 2pos ltdiv23ii mpbi
    c2 ) AABCZODEAOBCADEPAODFGHAAOIIJKLMN $.

  $( Prove that 1 - 1/2 = 1/2.  (Contributed by David A. Wheeler,
     4-Jan-2017.) $)
  1mhlfehlf $p |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 ) $=
    ( c2 c1 cmin co cdiv cc wcel cc0 cap wbr wceq 2cn ax-1cn pm3.2i divsubdirap
    wa 2ap0 mp3an 2m1e1 oveq1i 2div2e1 3eqtr3ri ) ABCDZAEDZAAEDZBAEDZCDZUFBUFCD
    AFGZBFGUHAHIJZPUDUGKLMUHUILQNABAORUCBAESTUEBUFCUATUB $.

  $( An eighth of four thirds is a sixth.  (Contributed by Paul Chapman,
     24-Nov-2007.) $)
  8th4div3 $p |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 ) $=
    ( c1 c8 cdiv co c4 c3 cmul c6 ax-1cn 8re recni 4cn gt0ap0ii c2 eqtr3i eqtri
    3cn 2cn cc wcel 8pos divmuldivapi mulcomi mul32i 4t2e8 oveq1i mulassi 3t2e6
    3re 3pos oveq2i oveq12i cc0 cap wbr wceq 6re 6pos 4re 4pos divcanap5 mp3an1
    wa mp4an ) ABCDEFCDGDZEAGDZEHGDZCDZAHCDZVEAEGDZBFGDZCDVHABEFIBJKLQBJUAMFUIU
    JMUBVJVFVKVGCAEILUCVKEFNGDZGDZVGEFGDNGDZVKVMENGDZFGDVNVKENFLRQUDVOBFGUEUFOE
    FNLQRUGOVLHEGUHUKPULPHSTZHUMUNUOZESTZEUMUNUOZVHVIUPZHUQKHUQURMLEUSUTMASTVPV
    QVCVRVSVCVTIAHEVAVBVDP $.

  $( One half plus or minus one sixth.  (Contributed by Paul Chapman,
     17-Jan-2008.) $)
  halfpm6th $p |- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\
                       ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) ) $=
    ( c1 c2 cdiv co c6 cmin c3 wceq caddc cmul 3cn ax-1cn 2cn 2ap0 divmuldivapi
    oveq1i mulridi 3t2e6 oveq12i 3eqtr3i 3re 3pos gt0ap0ii halfcn mullidi eqtri
    dividapi cc wcel cc0 cap wbr wa 6cn 6re 6pos pm3.2i divsubdirap mp3an 3m1e2
    oveq2i recclapi 3eqtr2i c4 divdirapi df-4 3eqtr4ri 2t2e4 divclapi ) ABCDZAE
    CDZFDZAGCDZHVJVKIDZBGCDZHVLGECDZVKFDZGAFDZECDZVMVJVPVKFGGCDZVJJDZGAJDZGBJDZ
    CDVJVPGGABKKLMGUAUBUCZNOWAAVJJDVJVTAVJJGKWDUGPVJUDUEUFWBGWCECGKQRSTZPGUHUIA
    UHUIEUHUIZEUJUKULZUMVSVQHKLWFWGUNEUOUPUCZUQGAEURUSVSBECDABJDZWCCDZVMVRBECUT
    PWIBWCECBMUERSVMBBCDZJDVMAJDWJVMWKAVMJBMNUGZVAAGBBLKMMWDNOVMGKWDVBQTVCVCVNV
    DECDZBBJDZWCCDZVOGAIDZECDVPVKIDWMVNGAEKLUNWHVEVDWPECVFPVJVPVKIWEPVGWNVDWCEC
    VHRSVOWKJDVOAJDWOVOWKAVOJWLVABGBBMKMMWDNOVOBGMKWDVIQTVCUQ $.

  $( i times 0 equals 0 (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  it0e0 $p |- ( _i x. 0 ) = 0 $=
    ( ci ax-icn mul01i ) ABC $.

  $( ` ( 2 x. _i ) e. CC ` (common case).  (Contributed by David A. Wheeler,
     8-Dec-2018.) $)
  2mulicn $p |- ( 2 x. _i ) e. CC $=
    ( c2 ci 2cn ax-icn mulcli ) ABCDE $.

  $( The imaginary unit ` _i ` is apart from zero.  (Contributed by Jim
     Kingdon, 9-Mar-2020.) $)
  iap0 $p |- _i =//= 0 $=
    ( cc0 ci c1 cmul co caddc cap wbr wo 1ap0 olci cr wcel wb 0re apreim ax-icn
    1re oveq2i eqtri mp4an mpbir mulridi addlidi it0e0 00id 3brtr3i ) ABCDEZFEZ
    ABADEZFEZBAGUIUKGHZAAGHZCAGHZIZUNUMJKALMZCLMUPUPULUONOROOACAAPUAUBUIABFEBUH
    BAFBQUCSBQUDTUKAAFEAUJAAFUESUFTUG $.

  $( ` 2 x. _i ` is apart from zero.  (Contributed by Jim Kingdon,
     9-Mar-2020.) $)
  2muliap0 $p |- ( 2 x. _i ) =//= 0 $=
    ( c2 ci 2cn ax-icn 2ap0 iap0 mulap0i ) ABCDEFG $.

  $( ` ( 2 x. _i ) =/= 0 ` .  See also ~ 2muliap0 .  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  2muline0 $p |- ( 2 x. _i ) =/= 0 $=
    ( c2 ci cmul co cc0 cap wbr wne 2muliap0 cc wcel wi 2mulicn 0cn mp2an ax-mp
    apne ) ABCDZEFGZREHZIRJKEJKSTLMNREQOP $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Simple number properties
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Closure of half of a number (common case).  (Contributed by NM,
     1-Jan-2006.) $)
  halfcl $p |- ( A e. CC -> ( A / 2 ) e. CC ) $=
    ( cc wcel c2 cc0 cap wbr cdiv co 2cn 2ap0 divclap mp3an23 ) ABCDBCDEFGADHIB
    CJKADLM $.

  $( Real closure of half.  (Contributed by NM, 1-Jan-2006.) $)
  rehalfcl $p |- ( A e. RR -> ( A / 2 ) e. RR ) $=
    ( cr wcel c2 cc0 cap wbr cdiv co 2re 2ap0 redivclap mp3an23 ) ABCDBCDEFGADH
    IBCJKADLM $.

  $( Half of a number is zero iff the number is zero.  (Contributed by NM,
     20-Apr-2006.) $)
  half0 $p |- ( A e. CC -> ( ( A / 2 ) = 0 <-> A = 0 ) ) $=
    ( cc wcel c2 cc0 cap wbr cdiv co wceq wb 2cn 2ap0 diveqap0 mp3an23 ) ABCDBC
    DEFGADHIEJAEJKLMADNO $.

  $( Two halves make a whole.  (Contributed by NM, 11-Apr-2005.) $)
  2halves $p |- ( A e. CC -> ( ( A / 2 ) + ( A / 2 ) ) = A ) $=
    ( cc wcel c2 cmul co cdiv 2times oveq1d cc0 cap wbr wceq 2cn 2ap0 divcanap3
    caddc mp3an23 wa pm3.2i divdirap mp3an3 anidms 3eqtr3rd ) ABCZDAEFZDGFZAAQF
    ZDGFZAADGFZUJQFZUEUFUHDGAHIUEDBCZDJKLZUGAMNOADPRUEUIUKMZUEUEULUMSUNULUMNOTA
    ADUAUBUCUD $.

  $( A number is positive iff its half is positive.  (Contributed by NM,
     10-Apr-2005.) $)
  halfpos2 $p |- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) ) $=
    ( cr wcel c2 cc0 clt wbr cdiv co wb 2re 2pos gt0div mp3an23 ) ABCDBCEDFGEAF
    GEADHIFGJKLADMN $.

  $( A positive number is greater than its half.  (Contributed by NM,
     28-Oct-2004.)  (Proof shortened by Mario Carneiro, 27-May-2016.) $)
  halfpos $p |- ( A e. RR -> ( 0 < A <-> ( A / 2 ) < A ) ) $=
    ( cr wcel cc0 clt c2 cdiv co caddc halfpos2 rehalfcl ltaddposd cc wceq recn
    wbr 2halves syl breq2d 3bitrd ) ABCZDAEPDAFGHZEPUBUBUBIHZEPUBAEPAJUAUBUBAKZ
    UDLUAUCAUBEUAAMCUCANAOAQRST $.

  $( A number is nonnegative iff its half is nonnegative.  (Contributed by NM,
     9-Dec-2005.) $)
  halfnneg2 $p |- ( A e. RR -> ( 0 <_ A <-> 0 <_ ( A / 2 ) ) ) $=
    ( cr wcel c2 cc0 clt wbr cle cdiv co wb 2re 2pos ge0div mp3an23 ) ABCDBCEDF
    GEAHGEADIJHGKLMADNO $.

  $( Closure of half-sum and half-difference.  (Contributed by Paul Chapman,
     12-Oct-2007.) $)
  halfaddsubcl $p |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) / 2 ) e. CC /\
                         ( ( A - B ) / 2 ) e. CC ) ) $=
    ( cc wcel wa caddc co c2 cdiv cmin addcl halfcl syl subcl jca ) ACDBCDEZABF
    GZHIGCDZABJGZHIGCDZPQCDRABKQLMPSCDTABNSLMO $.

  $( Sum and difference of half-sum and half-difference.  (Contributed by Paul
     Chapman, 12-Oct-2007.) $)
  halfaddsub $p |- ( ( A e. CC /\ B e. CC ) ->
                      ( ( ( ( A + B ) / 2 ) + ( ( A - B ) / 2 ) ) = A /\
                        ( ( ( A + B ) / 2 ) - ( ( A - B ) / 2 ) ) = B ) ) $=
    ( cc wcel wa caddc cdiv cmin wceq cmul 2times adantr eqtr4d oveq1d 2cn 2ap0
    co c2 mp3an3 syl2anc ppncan 3anidm13 addcl subcl cc0 cap divdirap divcanap3
    wbr pm3.2i mp3an23 3eqtr3d pnncan 3anidm23 adantl divsubdirap jca ) ACDZBCD
    ZEZABFQZRGQZABHQZRGQZFQZAIVBVDHQZBIUTVAVCFQZRGQZRAJQZRGQZVEAUTVGVIRGUTVGAAF
    QZVIURUSVGVKIABAUAUBURVIVKIUSAKLMNUTVACDZVCCDZVHVEIZABUCZABUDZVLVMRCDZRUEUF
    UIZEZVNVQVROPUJZVAVCRUGSTURVJAIZUSURVQVRWAOPARUHUKLULUTVAVCHQZRGQZRBJQZRGQZ
    VFBUTWBWDRGUTWBBBFQZWDURUSWBWFIABBUMUNUSWDWFIURBKUOMNUTVLVMWCVFIZVOVPVLVMVS
    WGVTVAVCRUPSTUSWEBIZURUSVQVRWHOPBRUHUKUOULUQ $.

  $( Subtracting the half of a number from the number yields the half of the
     number.  (Contributed by AV, 28-Jun-2021.) $)
  subhalfhalf $p |- ( A e. CC -> ( A - ( A / 2 ) ) = ( A / 2 ) ) $=
    ( cc wcel c2 cdiv co cmin cmul id 2cnd cc0 cap wbr divcanap1d eqcomd oveq1d
    2ap0 a1i c1 3eqtrd halfcl mulcomd mulsubfacd wceq 2m1e1 mullidd ) ABCZAADEF
    ZGFUHDHFZUHGFDUHHFZUHGFZUHUGAUIUHGUGUIAUGADUGIUGJZDKLMUGQRNOPUGUIUJUHGUGUHD
    AUAZULUBPUGUKDSGFZUHHFSUHHFUHUGDUHULUMUCUGUNSUHHUNSUDUGUERPUGUHUMUFTT $.

  $( TODO - use this to reduce:
  Statement "lt2add" is directly referenced in the proofs of 9 statements:
    2climnn 2climnn0 climaddlem3 climmullem5 climcaui ser1f0i
  MM> sh us 2halves
  Statement "2halves" is directly referenced in the proofs of 10 statements:
    2climnn 2climnn0 climmullem5 climcaui caucvgi ser1f0i
  MM>
  $)
  $( A sum is less than the whole if each term is less than half.  (Contributed
     by NM, 13-Dec-2006.) $)
  lt2halves $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                ( ( A < ( C / 2 ) /\ B < ( C / 2 ) ) -> ( A + B ) < C ) ) $=
    ( cr wcel w3a c2 co clt wbr wa caddc wi 3simpa rehalfcl jca 3ad2ant3 lt2add
    cdiv syl2anc wb cc wceq recn 2halves syl breq2d sylibd ) ADEZBDEZCDEZFZACGS
    HZIJBUMIJKZABLHZUMUMLHZIJZUOCIJZULUIUJKUMDEZUSKZUNUQMUIUJUKNUKUIUTUJUKUSUSC
    OZVAPQABUMUMRTUKUIUQURUAUJUKUPCUOIUKCUBEUPCUCCUDCUEUFUGQUH $.

  $( Sum is less than product for numbers greater than 2.  (Contributed by
     Stefan Allan, 24-Sep-2010.) $)
  addltmul $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) )
              -> ( A + B ) < ( A x. B ) ) $=
    ( cr wcel wa c2 clt wbr caddc co cmul c1 cmin 2re 1re ltsub1 syl2an remulcl
    wb cc mp3an13 2m1e1 breq1i bitrdi bi2anan9 wi peano2rem mulgt1 ex wceq recn
    sylbid ax-1cn mulsub mpanl2 mpanr2 mpan2 readdcl remulcli sylancl ltaddsub2
    breq2d mp3an2 syl2anc oveq2i breq2i bitr3di ltadd1 mp3an3 ax-1rid oveqan12d
    1t1e1 breq1d bitr3d 3bitrd sylibd imp ) ACDZBCDZEZFAGHZFBGHZEZABIJZABKJZGHZ
    VTWCLALMJZBLMJZKJZGHZWFVTWCLWGGHZLWHGHZEZWJVRWAWKVSWBWLVRWAFLMJZWGGHZWKFCDZ
    VRLCDZWAWOSNOFALPUAWNLWGGUBUCUDVSWBWNWHGHZWLWPVSWQWBWRSNOFBLPUAWNLWHGUBUCUD
    UEVRWGCDZWHCDZWMWJUFVSAUGBUGWSWTEWMWJWGWHUHUIQULVTWJLWELLKJZIJZALKJZBLKJZIJ
    ZMJZGHZXELIJZWELIJZGHZWFVTWIXFLGVRATDZBTDZWIXFUJZVSAUKBUKXKXLLTDZXMUMXKXNXL
    XNEXMUMALBLUNUOUPQVBVTXHXBGHZXGXJVTXECDZXBCDZXOXGSZVRXCCDZXDCDZXPVSVRWQXSOA
    LRUQVSWQXTOBLRUQXCXDURQZVTWECDZXACDXQABRZLLOOUSWEXAURUTXPWQXQXROXELXBVAVCVD
    XBXIXHGXALWEIVLVEVFVGVTXEWEGHZXJWFVTXPYBYDXJSZYAYCXPYBWQYEOXEWELVHVIVDVTXEW
    DWEGVRVSXCAXDBIAVJBVJVKVMVNVOVPVQ $.

  ${
    $d x y $.
    $( There is no smallest positive real number.  (Contributed by NM,
       28-Oct-2004.) $)
    nominpos $p |- -. E. x e. RR ( 0 < x /\
                   -. E. y e. RR ( 0 < y /\ y < x ) ) $=
      ( cc0 cv clt wbr wa cr wrex wn wcel wi c2 cdiv co rehalfcl divgt0 mpanr12
      2re 2pos ex halfpos biimpd jcad wceq breq2 breq1 rspcev syl6an imanim syl
      anbi12d nrex ) CADZEFZCBDZEFZUPUNEFZGZBHIZJGZAHUNHKZUOUTLVAJVBUNMNOZHKUOC
      VCEFZVCUNEFZGZUTUNPVBUOVDVEVBUOVDVBUOGMHKCMEFVDSTUNMQRUAVBUOVEUNUBUCUDUSV
      FBVCHUPVCUEUQVDURVEUPVCCEUFUPVCUNEUGULUHUIUOUTUJUKUM $.
  $}

  $( Ordering property for average.  (Contributed by Mario Carneiro,
     28-May-2014.) $)
  avglt1 $p |- ( ( A e. RR /\ B e. RR ) ->
                 ( A < B <-> A < ( ( A + B ) / 2 ) ) ) $=
    ( cr wcel wa clt wbr caddc co c2 cmul cdiv wb ltadd2 3anidm13 cc wceq simpl
    recnd times2 syl breq1d cc0 readdcl 2re pm3.2i a1i ltmuldiv syl3anc 3bitr2d
    2pos ) ACDZBCDZEZABFGZAAHIZABHIZFGZAJKIZUQFGZAUQJLIFGZULUMUOURMABANOUNUSUPU
    QFUNAPDUSUPQUNAULUMRZSATUAUBUNULUQCDJCDZUCJFGZEZUTVAMVBABUDVEUNVCVDUEUKUFUG
    AUQJUHUIUJ $.

  $( Ordering property for average.  (Contributed by Mario Carneiro,
     28-May-2014.) $)
  avglt2 $p |- ( ( A e. RR /\ B e. RR ) ->
                 ( A < B <-> ( ( A + B ) / 2 ) < B ) ) $=
    ( cr wcel wa caddc co c2 cmul clt wbr cdiv cc simpr recnd 2times syl breq2d
    wceq wb cc0 readdcl 2re 2pos pm3.2i a1i ltdivmul syl3anc 3anidm23 3bitr4rd
    ltadd1 ) ACDZBCDZEZABFGZHBIGZJKZUOBBFGZJKZUOHLGBJKZABJKZUNUPURUOJUNBMDUPURS
    UNBULUMNZOBPQRUNUOCDUMHCDZUAHJKZEZUTUQTABUBVBVEUNVCVDUCUDUEUFUOBHUGUHULUMVA
    USTABBUKUIUJ $.

  $( Ordering property for average.  (Contributed by Mario Carneiro,
     28-May-2014.) $)
  avgle1 $p |- ( ( A e. RR /\ B e. RR ) ->
                 ( A <_ B <-> A <_ ( ( A + B ) / 2 ) ) ) $=
    ( cr wcel wa clt wbr wn caddc co c2 cdiv cle wb avglt2 ancoms cc wceq lenlt
    recn addcom syl2an oveq1d breq1d bitr4d notbid readdcl rehalfcl syl 3bitr4d
    syldan ) ACDZBCDZEZBAFGZHABIJZKLJZAFGZHZABMGAUQMGZUNUOURUNUOBAIJZKLJZAFGZUR
    UMULUOVCNBAOPUNUQVBAFUNUPVAKLULAQDBQDUPVARUMATBTABUAUBUCUDUEUFABSULUMUQCDZU
    TUSNUNUPCDVDABUGUPUHUIAUQSUKUJ $.

  $( Ordering property for average.  (Contributed by Jeff Hankins,
     15-Sep-2013.)  (Revised by Mario Carneiro, 28-May-2014.) $)
  avgle2 $p |- ( ( A e. RR /\ B e. RR ) ->
                 ( A <_ B <-> ( ( A + B ) / 2 ) <_ B ) ) $=
    ( cr wcel wa clt wbr wn caddc co c2 cdiv cle wb avglt1 ancoms cc wceq lenlt
    recn addcom syl2an oveq1d breq2d bitr4d notbid readdcl rehalfcl syl 3bitr4d
    sylancom ) ACDZBCDZEZBAFGZHBABIJZKLJZFGZHZABMGUQBMGZUNUOURUNUOBBAIJZKLJZFGZ
    URUMULUOVCNBAOPUNUQVBBFUNUPVAKLULAQDBQDUPVARUMATBTABUAUBUCUDUEUFABSULUMUQCD
    ZUTUSNUNUPCDVDABUGUPUHUIUQBSUKUJ $.

  ${
    2timesd.1 $e |- ( ph -> A e. CC ) $.
    $( Two times a number.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    2timesd $p |- ( ph -> ( 2 x. A ) = ( A + A ) ) $=
      ( cc wcel c2 cmul co caddc wceq 2times syl ) ABDEFBGHBBIHJCBKL $.

    $( A number times 2.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    times2d $p |- ( ph -> ( A x. 2 ) = ( A + A ) ) $=
      ( cc wcel c2 cmul co caddc wceq times2 syl ) ABDEBFGHBBIHJCBKL $.

    $( Closure of half of a number (frequently used special case).
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    halfcld $p |- ( ph -> ( A / 2 ) e. CC ) $=
      ( cc wcel c2 cdiv co halfcl syl ) ABDEBFGHDECBIJ $.

    $( Two halves make a whole.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    2halvesd $p |- ( ph -> ( ( A / 2 ) + ( A / 2 ) ) = A ) $=
      ( cc wcel c2 cdiv co caddc wceq 2halves syl ) ABDEBFGHZMIHBJCBKL $.
  $}

  ${
    rehalfcld.1 $e |- ( ph -> A e. RR ) $.
    $( Real closure of half.  (Contributed by Mario Carneiro, 27-May-2016.) $)
    rehalfcld $p |- ( ph -> ( A / 2 ) e. RR ) $=
      ( cr wcel c2 cdiv co rehalfcl syl ) ABDEBFGHDECBIJ $.

    lt2halvesd.2 $e |- ( ph -> B e. RR ) $.
    lt2halvesd.3 $e |- ( ph -> C e. RR ) $.
    lt2halvesd.4 $e |- ( ph -> A < ( C / 2 ) ) $.
    lt2halvesd.5 $e |- ( ph -> B < ( C / 2 ) ) $.
    $( A sum is less than the whole if each term is less than half.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    lt2halvesd $p |- ( ph -> ( A + B ) < C ) $=
      ( c2 cdiv co clt wbr caddc cr wcel wa wi lt2halves syl3anc mp2and ) ABDJK
      LZMNZCUCMNZBCOLDMNZHIABPQCPQDPQUDUERUFSEFGBCDTUAUB $.
  $}

  ${
    rehalfcli.1 $e |- A e. RR $.
    $( Half a real number is real.  Inference form.  (Contributed by David
       Moews, 28-Feb-2017.) $)
    rehalfcli $p |- ( A / 2 ) e. RR $=
      ( cr wcel c2 cdiv co rehalfcl ax-mp ) ACDAEFGCDBAHI $.
  $}

  $( Adding two times 1 to a number.  (Contributed by AV, 22-Sep-2018.) $)
  add1p1 $p |- ( N e. CC -> ( ( N + 1 ) + 1 ) = ( N + 2 ) ) $=
    ( cc wcel c1 caddc co c2 id 1cnd addassd wceq 1p1e2 a1i oveq2d eqtrd ) ABCZ
    ADEFDEFADDEFZEFAGEFPADDPHPIZRJPQGAEQGKPLMNO $.

  $( Subtracting two times 1 from a number.  (Contributed by AV,
     23-Oct-2018.) $)
  sub1m1 $p |- ( N e. CC -> ( ( N - 1 ) - 1 ) = ( N - 2 ) ) $=
    ( cc wcel c1 cmin co caddc c2 id 1cnd subsub4d wceq 1p1e2 a1i oveq2d eqtrd
    ) ABCZADEFDEFADDGFZEFAHEFQADDQIQJZSKQRHAERHLQMNOP $.

  $( Subtracting 2 and afterwards 1 from a number results in the difference
     between the number and 3.  (Contributed by Alexander van der Vekens,
     16-Sep-2018.) $)
  cnm2m1cnm3 $p |- ( A e. CC -> ( ( A - 2 ) - 1 ) = ( A - 3 ) ) $=
    ( cc wcel c2 cmin co c1 caddc c3 id 2cnd 1cnd subsub4d wceq 2p1e3 a1i eqtrd
    oveq2d ) ABCZADEFGEFADGHFZEFAIEFSADGSJSKSLMSTIAETINSOPRQ $.

  $( A complex number increased by 1, then divided by 2, then decreased by 1
     equals the complex number decreased by 1 and then divided by 2.
     (Contributed by AV, 24-May-2020.) $)
  xp1d2m1eqxm1d2 $p |- ( X e. CC
                         -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) ) $=
    ( cc wcel c1 caddc co cdiv cmin peano2cn halfcld peano2cnm syl 2cnd cc0 cap
    c2 wbr a1i cmul divcanap1d 2ap0 1cnd subdird mullidd oveq12d wceq oveq2d id
    2m1e1 subsub3d 3eqtr2rd 3eqtrd mulcanap2ad ) ABCZADEFZPGFZDHFZADHFZPGFZPUNU
    PBCUQBCUNUOAIZJZUPKLUNURAKZJUNMZPNOQUNUARZUNUQPSFUPPSFZDPSFZHFUOPHFZUSPSFZU
    NUPDPVAUNUBZVCUCUNVEUOVFPHUNUOPUTVCVDTUNPVCUDUEUNVHURAPDHFZHFVGUNURPVBVCVDT
    UNVJDAHVJDUFUNUIRUGUNAPDUNUHVCVIUJUKULUM $.

  $( An integer greater than 5, divided by 4 and increased by 1, is less than
     or equal to the half of the integer minus 1.  (Contributed by AV,
     8-Jul-2021.) $)
  div4p1lem1div2 $p |- ( ( N e. RR /\ 6 <_ N )
                         -> ( ( N / 4 ) + 1 ) <_ ( ( N - 1 ) / 2 ) ) $=
    ( cr wcel c6 cle wbr c4 co c1 caddc cmin c2 cmul a1i wceq adantr wb syl3anc
    cc cc0 wa cdiv 6re id leadd2d biimpa times2d breqtrrd 4cn 2cn addassd 4p2e6
    recn oveq2i eqtrdi breq1d mpbird clt 4re 4ap0 redivclapd peano2re peano2rem
    cap syl rehalfcld 4pos pm3.2i recnd 1cnd divcanap1d mullidi joinlmuladdmuld
    lemul1 oveq12d 2t2e4 eqcomi oveq2d mulass eqcomd 2ap0 oveq1d subdird 3eqtrd
    w3a breq12d readdcld 2re remulcld leaddsub bicomd 3bitrd ) ABCZDAEFZUAZAGUB
    HZIJHZAIKHZLUBHZEFZAGJHZLJHZALMHZEFZWOXDADJHZXCEFZWOXEAAJHZXCEWMWNXEXGEFWMD
    AADBCWMUCNWMUDZXHUEUFWMXCXGOWNWMAAUMZUGPUHWMXDXFQWNWMXBXEXCEWMXBAGLJHZJHXEW
    MAGLXIGSCWMUINZLSCZWMUJNZUKXJDAJULUNUOUPPUQWMWTXDQWNWMWTWQGMHZWSGMHZEFZXAXC
    LKHZEFZXDWMWQBCZWSBCGBCZTGURFZUAZWTXPQWMWPBCXSWMAGXHXTWMUSNZGTVDFWMUTNZVAZW
    PVBVEWMWRAVCZVFZYBWMXTYAUSVGVHNWQWSGVNRWMXNXAXOXQEWMWPGIXAWMWPYEVIXKWMVJZWM
    WPGMHAIGMHZGJWMAGXIXKYDVKYIGOWMGUIVLNVOVMWMXOWSLLMHZMHZWSLMHZLMHZXQWMGYJWSM
    GYJOWMYJGVPVQNVRWMWSSCZXLXLYKYMOWMWSYGVIXMXMYNXLXLWEYMYKWSLLVSVTRWMYMWRLMHX
    CILMHZKHXQWMYLWRLMWMWRLWMWRYFVIXMLTVDFWMWANVKWBWMAILXIYHXMWCWMYOLXCKYOLOWML
    UJVLNVRWDWDWFWMXABCZLBCZXCBCZXRXDQWMAGXHYCWGYQWMWHNZWMALXHYSWIYPYQYRWEXDXRX
    ALXCWJWKRWLPUQ $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The Archimedean property
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d n x y A $.
    $( Archimedean property of real numbers.  For any real number, there is an
       integer greater than it.  Theorem I.29 of [Apostol] p. 26.  (Contributed
       by NM, 21-Jan-1997.) $)
    arch $p |- ( A e. RR -> E. n e. NN A < n ) $=
      ( vx vy cr wcel cv clt wbr cn wrex cltrr c1 caddc co wral wa cint ax-arch
      cab dfnn2 rexeqi sylibr wb nnre ltxrlt sylan2 rexbidva mpbird ) AEFZABGZH
      IZBJKAUKLIZBJKZUJUMBMCGZFDGMNOUOFDUOPQCTRZKUNCDABSUMBJUPCDUAUBUCUJULUMBJU
      KJFUJUKEFULUMUDUKUEAUKUFUGUHUI $.
  $}

  ${
    $d n A $.
    $( There exists a positive integer whose reciprocal is less than a given
       positive real.  Exercise 3 of [Apostol] p. 28.  (Contributed by NM,
       8-Nov-2004.) $)
    nnrecl $p |- ( ( A e. RR /\ 0 < A ) -> E. n e. NN ( 1 / n ) < A ) $=
      ( cr wcel cc0 clt wbr wa c1 cdiv co cv cn wrex simpl gt0ap0 rerecclapd wb
      jca adantr arch syl recgt0 nnre nngt0 ltrec syl2an recrecapd breq2d bitrd
      cc recn rexbidva mpbid ) ACDZEAFGZHZIAJKZBLZFGZBMNZIUSJKZAFGZBMNUQURCDZVA
      UQAUOUPOAPZQZURBUAUBUQUTVCBMUQUSMDZHUTVBIURJKZFGZVCUQVDEURFGZHUSCDZEUSFGZ
      HUTVIRVGUQVDVJVFAUCSVGVKVLUSUDUSUESURUSUFUGUQVIVCRVGUQVHAVBFUQAUOAUKDUPAU
      LTVEUHUITUJUMUN $.
  $}

  ${
    $d x A $.  $d x k $.
    $( A bounded real sequence ` A ( k ) ` is less than or equal to at least
       one of its indices.  (Contributed by NM, 18-Jan-2008.) $)
    bndndx $p |- ( E. x e. RR A. k e. NN ( A e. RR /\ A <_ x ) ->
                  E. k e. NN A <_ k ) $=
      ( cr wcel cv cle wbr wa cn wral wrex wi clt arch nnre lelttr ltle 3adant2
      w3a syld exp5o com3l imp4b com23 sylan2 reximdva mpd r19.35-1 rexlimiv
      syl ) BDEZBAFZGHZIZCJKZBCFZGHZCJLZADUMDEZUOURMZCJLZUPUSMUTUMUQNHZCJLVBUMC
      OUTVCVACJUQJEUTUQDEZVCVAMUQPUTVDIUOVCURUTVDULUNVCURMZULUTVDUNVEMULUTVDUNV
      CURULUTVDTUNVCIBUQNHZURBUMUQQULVDVFURMUTBUQRSUAUBUCUDUEUFUGUHUOURCJUIUKUJ
      $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Nonnegative integers (as a subset of complex numbers)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c NN0 $. $( The set of nonnegative integers (blackboard bold N_0). $)

  $( Extend class notation to include the class of nonnegative integers. $)
  cn0 $a class NN0 $.

  $( Define the set of nonnegative integers.  (Contributed by Raph Levien,
     10-Dec-2002.) $)
  df-n0 $a |- NN0 = ( NN u. { 0 } ) $.

  $( Nonnegative integers expressed in terms of naturals and zero.
     (Contributed by Raph Levien, 10-Dec-2002.) $)
  elnn0 $p |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) ) $=
    ( cn0 wcel cn cc0 csn cun wceq df-n0 eleq2i elun c0ex elsn2 orbi2i 3bitri
    wo ) ABCADEFZGZCADCZAQCZPSAEHZPBRAIJADQKTUASAELMNO $.

  $( Positive naturals are a subset of nonnegative integers.  (Contributed by
     Raph Levien, 10-Dec-2002.) $)
  nnssnn0 $p |- NN C_ NN0 $=
    ( cn cc0 csn cun cn0 ssun1 df-n0 sseqtrri ) AABCZDEAIFGH $.

  $( Nonnegative integers are a subset of the reals.  (Contributed by Raph
     Levien, 10-Dec-2002.) $)
  nn0ssre $p |- NN0 C_ RR $=
    ( cn0 cn cc0 csn cun cr df-n0 nnssre wcel wss 0re snssi ax-mp unssi eqsstri
    ) ABCDZEFGBPFHCFIPFJKCFLMNO $.

  $( Nonnegative integers are a subset of the complex numbers.)  (Contributed
     by NM, 9-May-2004.) $)
  nn0sscn $p |- NN0 C_ CC $=
    ( cn0 cr cc nn0ssre ax-resscn sstri ) ABCDEF $.

  $( The set of nonnegative integers exists.  (Contributed by NM,
     18-Jul-2004.) $)
  nn0ex $p |- NN0 e. _V $=
    ( cn0 cn cc0 csn cun cvv df-n0 nnex c0ex snex unex eqeltri ) ABCDZEFGBMHCIJ
    KL $.

  $( A positive integer is a nonnegative integer.  (Contributed by NM,
     9-May-2004.) $)
  nnnn0 $p |- ( A e. NN -> A e. NN0 ) $=
    ( cn cn0 nnssnn0 sseli ) BCADE $.

  ${
    nnnn0.1 $e |- N e. NN $.
    $( A positive integer is a nonnegative integer.  (Contributed by NM,
       20-Jun-2005.) $)
    nnnn0i $p |- N e. NN0 $=
      ( cn wcel cn0 nnnn0 ax-mp ) ACDAEDBAFG $.
  $}

  $( A nonnegative integer is a real number.  (Contributed by NM,
     9-May-2004.) $)
  nn0re $p |- ( A e. NN0 -> A e. RR ) $=
    ( cn0 cr nn0ssre sseli ) BCADE $.

  $( A nonnegative integer is a complex number.  (Contributed by NM,
     9-May-2004.) $)
  nn0cn $p |- ( A e. NN0 -> A e. CC ) $=
    ( cn0 cc nn0sscn sseli ) BCADE $.

  ${
    nn0re.1 $e |- A e. NN0 $.
    $( A nonnegative integer is a real number.  (Contributed by NM,
       14-May-2003.) $)
    nn0rei $p |- A e. RR $=
      ( cn0 cr nn0ssre sselii ) CDAEBF $.

    $( A nonnegative integer is a complex number.  (Contributed by NM,
       14-May-2003.) $)
    nn0cni $p |- A e. CC $=
      ( nn0rei recni ) AABCD $.
  $}

  $( The set of positive integers defined in terms of nonnegative integers.
     (Contributed by NM, 23-Sep-2007.)  (Proof shortened by Mario Carneiro,
     13-Feb-2013.) $)
  dfn2 $p |- NN = ( NN0 \ { 0 } ) $=
    ( cn0 cc0 csn cdif cn cun df-n0 difeq1i difun2 wcel wceq 0nnn difsn 3eqtrri
    wn ax-mp ) ABCZDEQFZQDEQDZEARQGHEQIBEJOSEKLBEMPN $.

  $( The positive integer property expressed in terms of difference from zero.
     (Contributed by Stefan O'Rear, 12-Sep-2015.) $)
  elnnne0 $p |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) ) $=
    ( cn wcel cn0 cc0 csn cdif wne wa dfn2 eleq2i eldifsn bitri ) ABCADEFGZCADC
    AEHIBNAJKADELM $.

  $( 0 is a nonnegative integer.  (Contributed by Raph Levien, 10-Dec-2002.) $)
  0nn0 $p |- 0 e. NN0 $=
    ( cc0 wceq cn0 wcel eqid cn wo elnn0 biimpri olcs ax-mp ) AABZACDZAEAFDZLMM
    NLGAHIJK $.

  $( 1 is a nonnegative integer.  (Contributed by Raph Levien, 10-Dec-2002.) $)
  1nn0 $p |- 1 e. NN0 $=
    ( c1 1nn nnnn0i ) ABC $.

  $( 2 is a nonnegative integer.  (Contributed by Raph Levien, 10-Dec-2002.) $)
  2nn0 $p |- 2 e. NN0 $=
    ( c2 2nn nnnn0i ) ABC $.

  $( 3 is a nonnegative integer.  (Contributed by Mario Carneiro,
     18-Feb-2014.) $)
  3nn0 $p |- 3 e. NN0 $=
    ( c3 3nn nnnn0i ) ABC $.

  $( 4 is a nonnegative integer.  (Contributed by Mario Carneiro,
     18-Feb-2014.) $)
  4nn0 $p |- 4 e. NN0 $=
    ( c4 4nn nnnn0i ) ABC $.

  $( 5 is a nonnegative integer.  (Contributed by Mario Carneiro,
     19-Apr-2015.) $)
  5nn0 $p |- 5 e. NN0 $=
    ( c5 5nn nnnn0i ) ABC $.

  $( 6 is a nonnegative integer.  (Contributed by Mario Carneiro,
     19-Apr-2015.) $)
  6nn0 $p |- 6 e. NN0 $=
    ( c6 6nn nnnn0i ) ABC $.

  $( 7 is a nonnegative integer.  (Contributed by Mario Carneiro,
     19-Apr-2015.) $)
  7nn0 $p |- 7 e. NN0 $=
    ( c7 7nn nnnn0i ) ABC $.

  $( 8 is a nonnegative integer.  (Contributed by Mario Carneiro,
     19-Apr-2015.) $)
  8nn0 $p |- 8 e. NN0 $=
    ( c8 8nn nnnn0i ) ABC $.

  $( 9 is a nonnegative integer.  (Contributed by Mario Carneiro,
     19-Apr-2015.) $)
  9nn0 $p |- 9 e. NN0 $=
    ( c9 9nn nnnn0i ) ABC $.

  $( A nonnegative integer is greater than or equal to zero.  (Contributed by
     NM, 9-May-2004.)  (Revised by Mario Carneiro, 16-May-2014.) $)
  nn0ge0 $p |- ( N e. NN0 -> 0 <_ N ) $=
    ( cn0 wcel cn cc0 wceq wo cle wbr elnn0 cr clt nnre nngt0 wi ltle mpan sylc
    0re 0le0 breq2 mpbiri jaoi sylbi ) ABCADCZAEFZGEAHIZAJUEUGUFUEAKCZEALIZUGAM
    ANEKCUHUIUGOSEAPQRUFUGEEHITAEEHUAUBUCUD $.

  $( A nonnegative integer is not less than zero.  (Contributed by NM,
     9-May-2004.)  (Revised by Mario Carneiro, 27-May-2016.) $)
  nn0nlt0 $p |- ( A e. NN0 -> -. A < 0 ) $=
    ( cn0 wcel cc0 cle wbr clt wn nn0ge0 cr wb 0re nn0re lenlt sylancr mpbid )
    ABCZDAEFZADGFHZAIQDJCAJCRSKLAMDANOP $.

  ${
    nn0ge0.1 $e |- N e. NN0 $.
    $( Nonnegative integers are nonnegative.  (Contributed by Raph Levien,
       10-Dec-2002.) $)
    nn0ge0i $p |- 0 <_ N $=
      ( cn0 wcel cc0 cle wbr nn0ge0 ax-mp ) ACDEAFGBAHI $.
  $}

  $( A nonnegative integer is less than or equal to zero iff it is equal to
     zero.  (Contributed by NM, 9-Dec-2005.) $)
  nn0le0eq0 $p |- ( N e. NN0 -> ( N <_ 0 <-> N = 0 ) ) $=
    ( cn0 wcel cc0 cle wbr wa nn0ge0 biantrud cr wb nn0re letri3 sylancl bitr4d
    wceq 0re ) ABCZADEFZSDAEFZGZADPZRTSAHIRAJCDJCUBUAKALQADMNO $.

  $( A nonnegative integer increased by 1 is greater than 0.  (Contributed by
     Alexander van der Vekens, 3-Oct-2018.) $)
  nn0p1gt0 $p |- ( N e. NN0 -> 0 < ( N + 1 ) ) $=
    ( cn0 wcel c1 nn0re 1red nn0ge0 cc0 clt wbr 0lt1 a1i addgegt0d ) ABCZADAENF
    AGHDIJNKLM $.

  $( A positive integer plus a nonnegative integer is a positive integer.
     (Contributed by NM, 20-Apr-2005.)  (Proof shortened by Mario Carneiro,
     16-May-2014.) $)
  nnnn0addcl $p |- ( ( M e. NN /\ N e. NN0 ) -> ( M + N ) e. NN ) $=
    ( cn0 wcel cn cc0 wceq wo caddc co elnn0 nnaddcl wa oveq2 addridd sylan9eqr
    nncn simpl eqeltrd jaodan sylan2b ) BCDAEDZBEDZBFGZHABIJZEDZBKUBUCUFUDABLUB
    UDMUEAEUDUBUEAFIJABFAINUBAAQOPUBUDRSTUA $.

  $( A nonnegative integer plus a positive integer is a positive integer.
     (Contributed by NM, 22-Dec-2005.) $)
  nn0nnaddcl $p |- ( ( M e. NN0 /\ N e. NN ) -> ( M + N ) e. NN ) $=
    ( cn wcel cn0 caddc co wa wceq nncn nn0cn addcom syl2an nnnn0addcl eqeltrrd
    cc ancoms ) BCDZAEDZABFGZCDRSHBAFGZTCRBPDAPDUATISBJAKBALMBANOQ $.

  $( The result of subtracting a positive integer from 0 is not a nonnegative
     integer.  (Contributed by Alexander van der Vekens, 19-Mar-2018.) $)
  0mnnnnn0 $p |- ( N e. NN -> ( 0 - N ) e/ NN0 ) $=
    ( cn wcel cc0 cmin co cn0 wnel 0re cneg df-neg eqcomi eleq1i cle wbr nn0ge0
    wn cr nnre le0neg1d clt nngt0 0red lenltd pm2.21 biimtrdi mpid sylbird syl5
    wi biimtrid mt2i df-nel sylibr ) ABCZDAEFZGCZQUPGHUOUQDRCZIUQAJZGCZUOURQZUP
    USGUSUPAKLMUTDUSNOZUOVAUSPUOVBADNOZVAUOAASZTUOVCDAUAOZVAAUBUOVCVEQVEVAUJUOA
    DVDUOUCUDVEVAUEUFUGUHUIUKULUPGUMUN $.

  ${
    un0addcl.1 $e |- ( ph -> S C_ CC ) $.
    un0addcl.2 $e |- T = ( S u. { 0 } ) $.
    ${
      un0addcl.3 $e |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M + N ) e. S ) $.
      $( If ` S ` is closed under addition, then so is ` S u. { 0 } ` .
         (Contributed by Mario Carneiro, 17-Jul-2014.) $)
      un0addcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M + N ) e. T ) $=
        ( wcel caddc co cc0 wo wa eleq2i elun bitri cc sselda eqeltrd csn ssun1
        cun sseqtrri sselid expr addlidd wss a1i elsni oveq1d eleq1d syl5ibrcom
        wi impancom jaodan sylan2b 0cnd snssd unssd eqsstrid addridd simpr jaod
        oveq2d biimtrid impr ) ADCIZECIZDEJKZCIZVIEBIZELUAZIZMZAVHNZVKVIEBVMUCZ
        IVOCVQEGOEBVMPQVPVLVKVNVHADBIZDVMIZMZVLVKUNZVHDVQIVTCVQDGODBVMPQAVRWAVS
        AVRVLVKAVRVLNNBCVJBVQCBVMUBGUDZHUEUFAVLVSVKAVLNZVKVSLEJKZCIWCWDECWCEABR
        EFSUGABCEBCUHAWBUISTVSVJWDCVSDLEJDLUJUKULUMUOUPUQVPVKVNDLJKZCIVPWEDCVPD
        ACRDACVQRGABVMRFALRAURUSUTVASVBAVHVCTVNVJWECVNELDJELUJVEULUMVDVFVG $.
    $}

    un0mulcl.3 $e |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M x. N ) e. S ) $.
    $( If ` S ` is closed under multiplication, then so is ` S u. { 0 } ` .
       (Contributed by Mario Carneiro, 17-Jul-2014.) $)
    un0mulcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M x. N ) e. T ) $=
      ( wcel cmul co cc0 wo wa eleq2i elun bitri sseqtrri cc sselda csn wi expr
      cun ssun1 sselid mul02d wss ssun2 c0ex mpbir eqeltrdi elsni oveq1d eleq1d
      snss syl5ibrcom impancom jaodan sylan2b 0cnd snssd eqsstrid mul01d oveq2d
      unssd jaod biimtrid impr ) ADCIZECIZDEJKZCIZVKEBIZELUAZIZMZAVJNZVMVKEBVOU
      DZIVQCVSEGOEBVOPQVRVNVMVPVJADBIZDVOIZMZVNVMUBZVJDVSIWBCVSDGODBVOPQAVTWCWA
      AVTVNVMAVTVNNNBCVLBVSCBVOUEGRHUFUCAVNWAVMAVNNZVMWALEJKZCIWDWELCWDEABSEFTU
      GLCIVOCUHVOVSCVOBUIGRLCUJUPUKZULWAVLWECWADLEJDLUMUNUOUQURUSUTVRVMVPDLJKZC
      IVRWGLCVRDACSDACVSSGABVOSFALSAVAVBVFVCTVDWFULVPVLWGCVPELDJELUMVEUOUQVGVHV
      I $.
  $}

  $( Closure of addition of nonnegative integers.  (Contributed by Raph Levien,
     10-Dec-2002.)  (Proof shortened by Mario Carneiro, 17-Jul-2014.) $)
  nn0addcl $p |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M + N ) e. NN0 ) $=
    ( cn cc wss cn0 wcel wa caddc co nnsscn df-n0 nnaddcl adantl un0addcl mpan
    id ) CDEZAFGBFGHABIJZFGKRCFABRQLACGBCGHSCGRABMNOP $.

  $( Closure of multiplication of nonnegative integers.  (Contributed by NM,
     22-Jul-2004.)  (Proof shortened by Mario Carneiro, 17-Jul-2014.) $)
  nn0mulcl $p |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M x. N ) e. NN0 ) $=
    ( cn cc wss cn0 wcel wa cmul co nnsscn df-n0 nnmulcl adantl un0mulcl mpan
    id ) CDEZAFGBFGHABIJZFGKRCFABRQLACGBCGHSCGRABMNOP $.

  ${
    nn0addcl.1 $e |- M e. NN0 $.
    nn0addcl.2 $e |- N e. NN0 $.
    $( Closure of addition of nonnegative integers, inference form.
       (Contributed by Raph Levien, 10-Dec-2002.) $)
    nn0addcli $p |- ( M + N ) e. NN0 $=
      ( cn0 wcel caddc co nn0addcl mp2an ) AEFBEFABGHEFCDABIJ $.

    $( Closure of multiplication of nonnegative integers, inference form.
       (Contributed by Raph Levien, 10-Dec-2002.) $)
    nn0mulcli $p |- ( M x. N ) e. NN0 $=
      ( cn0 wcel cmul co nn0mulcl mp2an ) AEFBEFABGHEFCDABIJ $.
  $}

  $( A nonnegative integer plus 1 is a positive integer.  (Contributed by Raph
     Levien, 30-Jun-2006.)  (Revised by Mario Carneiro, 16-May-2014.) $)
  nn0p1nn $p |- ( N e. NN0 -> ( N + 1 ) e. NN ) $=
    ( cn0 wcel c1 cn caddc co 1nn nn0nnaddcl mpan2 ) ABCDECADFGECHADIJ $.

  $( Second Peano postulate for nonnegative integers.  (Contributed by NM,
     9-May-2004.) $)
  peano2nn0 $p |- ( N e. NN0 -> ( N + 1 ) e. NN0 ) $=
    ( cn0 wcel c1 caddc co 1nn0 nn0addcl mpan2 ) ABCDBCADEFBCGADHI $.

  $( A positive integer minus 1 is a nonnegative integer.  (Contributed by
     Jason Orendorff, 24-Jan-2007.)  (Revised by Mario Carneiro,
     16-May-2014.) $)
  nnm1nn0 $p |- ( N e. NN -> ( N - 1 ) e. NN0 ) $=
    ( cn wcel c1 cmin co cc0 wceq wo cn0 nn1m1nn oveq1 eqtrdi orim1i syl orcomd
    1m1e0 elnn0 sylibr ) ABCZADEFZBCZUAGHZIUAJCTUCUBTADHZUBIUCUBIAKUDUCUBUDUADD
    EFGADDELQMNOPUARS $.

  $( The nonnegative integer property expressed in terms of positive integers.
     (Contributed by NM, 10-May-2004.)  (Proof shortened by Mario Carneiro,
     16-May-2014.) $)
  elnn0nn $p |- ( N e. NN0 <-> ( N e. CC /\ ( N + 1 ) e. NN ) ) $=
    ( cn0 wcel cc c1 caddc co cn nn0cn nn0p1nn jca cmin wceq simpl ax-1cn pncan
    wa sylancl nnm1nn0 adantl eqeltrrd impbii ) ABCZADCZAEFGZHCZQZUCUDUFAIAJKUG
    UEELGZABUGUDEDCUHAMUDUFNOAEPRUFUHBCUDUESTUAUB $.

  $( The positive integer property expressed in terms of nonnegative integers.
     (Contributed by NM, 10-May-2004.) $)
  elnnnn0 $p |- ( N e. NN <-> ( N e. CC /\ ( N - 1 ) e. NN0 ) ) $=
    ( cn wcel cc c1 cmin co cn0 nncn caddc wa npcan1 eleq1d peano2cnm biantrurd
    bitr3d elnn0nn bitr4di biadan2 ) ABCZADCZAEFGZHCZAIUATUBDCZUBEJGZBCZKZUCUAU
    FTUGUAUEABALMUAUDUFANOPUBQRS $.

  $( The positive integer property expressed in terms of nonnegative integers.
     (Contributed by NM, 1-Sep-2005.) $)
  elnnnn0b $p |- ( N e. NN <-> ( N e. NN0 /\ 0 < N ) ) $=
    ( cn wcel cn0 cc0 clt wbr wa nnnn0 nngt0 jca wceq wo elnn0 ax-1 breq2 ltnri
    wi 0re pm2.21i biimtrdi jaoi sylbi imp impbii ) ABCZADCZEAFGZHUFUGUHAIAJKUG
    UHUFUGUFAELZMUHUFRZANUFUJUIUFUHOUIUHEEFGZUFAEEFPUKUFESQTUAUBUCUDUE $.

  $( The positive integer property expressed in terms of nonnegative integers.
     (Contributed by NM, 10-Jan-2006.) $)
  elnnnn0c $p |- ( N e. NN <-> ( N e. NN0 /\ 1 <_ N ) ) $=
    ( cn wcel cn0 c1 cle wbr wa nnnn0 nnge1 jca cc0 clt 0lt1 cr wi nn0re ltletr
    0re 1re mp3an12 syl mpani imdistani elnnnn0b sylibr impbii ) ABCZADCZEAFGZH
    ZUHUIUJAIAJKUKUILAMGZHUHUIUJULUILEMGZUJULNUIAOCZUMUJHULPZAQLOCEOCUNUOSTLEAR
    UAUBUCUDAUEUFUG $.

  $( A number is less than or equal to itself plus a nonnegative integer.
     (Contributed by NM, 10-Mar-2005.) $)
  nn0addge1 $p |- ( ( A e. RR /\ N e. NN0 ) -> A <_ ( A + N ) ) $=
    ( cn0 wcel cr cc0 cle wbr wa caddc co nn0re nn0ge0 jca addge01 3expb sylan2
    biimp3a ) BCDZAEDZBEDZFBGHZIAABJKGHZSUAUBBLBMNTUAUBUCTUAUBUCABORPQ $.

  $( A number is less than or equal to itself plus a nonnegative integer.
     (Contributed by NM, 10-Mar-2005.) $)
  nn0addge2 $p |- ( ( A e. RR /\ N e. NN0 ) -> A <_ ( N + A ) ) $=
    ( cn0 wcel cr cc0 cle wbr wa caddc co nn0re nn0ge0 jca addge02 3expb sylan2
    biimp3a ) BCDZAEDZBEDZFBGHZIABAJKGHZSUAUBBLBMNTUAUBUCTUAUBUCABORPQ $.

  ${
    nn0addge1.1 $e |- A e. RR $.
    nn0addge1.2 $e |- N e. NN0 $.
    $( A number is less than or equal to itself plus a nonnegative integer.
       (Contributed by NM, 10-Mar-2005.) $)
    nn0addge1i $p |- A <_ ( A + N ) $=
      ( cr wcel cn0 caddc co cle wbr nn0addge1 mp2an ) AEFBGFAABHIJKCDABLM $.

    $( A number is less than or equal to itself plus a nonnegative integer.
       (Contributed by NM, 10-Mar-2005.) $)
    nn0addge2i $p |- A <_ ( N + A ) $=
      ( cr wcel cn0 caddc co cle wbr nn0addge2 mp2an ) AEFBGFABAHIJKCDABLM $.
  $}

  ${
    nn0le2x.1 $e |- N e. NN0 $.
    $( A nonnegative integer is less than or equal to twice itself.
       (Contributed by Raph Levien, 10-Dec-2002.) $)
    nn0le2xi $p |- N <_ ( 2 x. N ) $=
      ( caddc co c2 cmul cle nn0rei nn0addge1i nn0cni 2timesi breqtrri ) AAACDE
      AFDGAAABHBIAABJKL $.
  $}

  ${
    nn0lele2x.1 $e |- M e. NN0 $.
    nn0lele2x.2 $e |- N e. NN0 $.
    $( 'Less than or equal to' implies 'less than or equal to twice' for
       nonnegative integers.  (Contributed by Raph Levien, 10-Dec-2002.) $)
    nn0lele2xi $p |- ( N <_ M -> N <_ ( 2 x. M ) ) $=
      ( cle wbr c2 cmul co nn0le2xi nn0rei 2re remulcli letri mpan2 ) BAEFAGAHI
      ZEFBPEFACJBAPBDKACKZGALQMNO $.
  $}

  $( Two ways to write the support of a function into ` NN0 ` .  (Contributed
     by Mario Carneiro, 29-Dec-2014.)  (Revised by AV, 7-Jul-2019.) $)
  fcdmnn0supp $p |- ( ( I e. V /\ F : I --> NN0 )
                      -> ( F supp 0 ) = ( `' F " NN ) ) $=
    ( wcel cn0 wf wa cc0 csupp co ccnv csn cdif cima cn wceq cvv c0ex fsuppeq
    wi mpan2 imp dfn2 imaeq2i eqtr4di ) BCDZBEAFZGAHIJZAKZEHLMZNZUIONUFUGUHUKPZ
    UFHQDUGULTREABCQHSUAUBOUJUIUCUDUE $.

  $( A function into ` NN0 ` is finitely supported iff its support is finite.
     (Contributed by AV, 8-Jul-2019.) $)
  fcdmnn0fsupp $p |- ( ( I e. V /\ F : I --> NN0 )
                       -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) $=
    ( wcel cn0 wf wa cc0 cfsupp wbr ccnv csn cdif cima cfn cn wb cvv wi c0ex
    ffsuppbi mpan2 imp dfn2 imaeq2i eleq1i bitr4di ) BCDZBEAFZGAHIJZAKZEHLMZNZO
    DZUKPNZODUHUIUJUNQZUHHRDUIUPSTEABCRHUAUBUCUOUMOPULUKUDUEUFUG $.

  $( Version of ~ fcdmnn0supp avoiding ~ ax-coll by assuming ` F ` is a set
     rather than its domain ` I ` .  (Contributed by SN, 5-Aug-2024.) $)
  fcdmnn0suppg $p |- ( ( F e. V /\ F : I --> NN0 )
                       -> ( F supp 0 ) = ( `' F " NN ) ) $=
    ( wcel cn0 wf wa cc0 csupp co ccnv csn cdif cima cn wceq cvv c0ex fsuppeqg
    wi mpan2 imp dfn2 imaeq2i eqtr4di ) ACDZBEAFZGAHIJZAKZEHLMZNZUIONUFUGUHUKPZ
    UFHQDUGULTREABCQHSUAUBOUJUIUCUDUE $.
  $( $j usage 'fcdmnn0suppg' avoids 'ax-coll'; $)

  $( Version of ~ fcdmnn0fsupp avoiding ~ ax-coll by assuming ` F ` is a set
     rather than its domain ` I ` .  (Contributed by SN, 5-Aug-2024.) $)
  fcdmnn0fsuppg $p |- ( ( F e. V /\ F : I --> NN0 )
                        -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) $=
    ( wcel cn0 wf wa cc0 cfsupp wbr csupp co cfn ccnv cn cima wfun wb ffun cvv
    simpl c0ex funisfsupp mp3an3 syl2an2 fcdmnn0suppg eleq1d bitrd ) ACDZBEAFZG
    ZAHIJZAHKLZMDZANOPZMDUJAQZUIUIULUNRZBEASUIUJUAUPUIHTDUQUBACTHUCUDUEUKUMUOMA
    BCUFUGUH $.
  $( $j usage 'fcdmnn0fsuppg' avoids 'ax-coll'; $)

  $( Two ways to write the support of a function on ` NN0 ` .  (Contributed by
     Mario Carneiro, 29-Dec-2014.) $)
  nn0supp $p |- ( F : I --> NN0 ->
    ( `' F " ( _V \ { 0 } ) ) = ( `' F " NN ) ) $=
    ( cn0 wf ccnv cn cima cvv cc0 csn cdif dfn2 invdif eqtr4i imaeq2i wfun wceq
    cin ffun inpreima syl wss cdm cnvimass fimacnv sseqtrid sseqin2 sylib eqtrd
    fdm eqtr4d eqtr2id ) BCADZAEZFGUNCHIJZKZRZGZUNUPGZFUQUNFCUOKUQLCUOMNOUMURUN
    CGZUSRZUSUMAPURVAQBCASCUPATUAUMUSUTUBVAUSQUMAUCZUSUTAUPUDUMVBBUTBCAUJBCAUEU
    KUFUSUTUGUHUIUL $.

  ${
    nnnn0d.1 $e |- ( ph -> A e. NN ) $.
    $( A positive integer is a nonnegative integer.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    nnnn0d $p |- ( ph -> A e. NN0 ) $=
      ( cn cn0 nnssnn0 sselid ) ADEBFCG $.
  $}

  ${
    nn0red.1 $e |- ( ph -> A e. NN0 ) $.
    $( A nonnegative integer is a real number.  (Contributed by Mario Carneiro,
       27-May-2016.) $)
    nn0red $p |- ( ph -> A e. RR ) $=
      ( cn0 cr nn0ssre sselid ) ADEBFCG $.

    $( A nonnegative integer is a complex number.  (Contributed by Mario
       Carneiro, 27-May-2016.) $)
    nn0cnd $p |- ( ph -> A e. CC ) $=
      ( nn0red recnd ) ABABCDE $.

    $( A nonnegative integer is greater than or equal to zero.  (Contributed by
       Mario Carneiro, 27-May-2016.) $)
    nn0ge0d $p |- ( ph -> 0 <_ A ) $=
      ( cn0 wcel cc0 cle wbr nn0ge0 syl ) ABDEFBGHCBIJ $.

    nn0addcld.2 $e |- ( ph -> B e. NN0 ) $.
    $( Closure of addition of nonnegative integers, inference form.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    nn0addcld $p |- ( ph -> ( A + B ) e. NN0 ) $=
      ( cn0 wcel caddc co nn0addcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Closure of multiplication of nonnegative integers, inference form.
       (Contributed by Mario Carneiro, 27-May-2016.) $)
    nn0mulcld $p |- ( ph -> ( A x. B ) e. NN0 ) $=
      ( cn0 wcel cmul co nn0mulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
  $}

  $( Closure law for addition of reals, restricted to nonnegative integers.
     (Contributed by Alexander van der Vekens, 6-Apr-2018.) $)
  nn0readdcl $p |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A + B ) e. RR ) $=
    ( cn0 wcel wa caddc co nn0addcl nn0red ) ACDBCDEABFGABHI $.

  $( If a nonnegative integer is greater than or equal to two, the integer
     decreased by 1 is a positive integer.  (Contributed by Alexander van der
     Vekens, 1-Aug-2018.)  (Revised by AV, 4-Jan-2020.) $)
  nn0ge2m1nn $p |- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN ) $=
    ( cn0 wcel c2 cle wbr wa c1 cmin co cn wo wceq simpl w3a clt 1red 2re 1lt2
    cr a1i nn0re 3jca adantr simpr jctil ltleletr elnnnn0c sylanbrc nn1m1nn syl
    sylc wi wn 1re lenlti biimpi breq2 mtbiri pm2.21d com12 adantl orim1d oridm
    mt2 mpd sylib ) ABCZDAEFZGZAHIJKCZVKLZVKVJAHMZVKLZVLVJAKCZVNVJVHHAEFZVOVHVI
    NVJHTCZDTCZATCZOZHDPFZVIGVPVHVTVIVHVQVRVSVHQVRVHRUAAUBUCUDVJVIWAVHVIUESUFHD
    AUGULAUHUIAUJUKVJVMVKVKVIVMVKUMVHVMVIVKVMVIVKVMVIDHEFZWBWASWBWAUNDHRUOUPUQV
    EAHDEURUSUTVAVBVCVFVKVDVG $.

  $( If a nonnegative integer is greater than or equal to two, the integer
     decreased by 1 is also a nonnegative integer.  (Contributed by Alexander
     van der Vekens, 1-Aug-2018.) $)
  nn0ge2m1nn0 $p |- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN0 ) $=
    ( cn0 wcel c2 cle wbr wa c1 cmin co nn0ge2m1nn nnnn0d ) ABCDAEFGAHIJAKL $.

  $( Closure law for dividing of a nonnegative integer by a positive integer.
     (Contributed by Alexander van der Vekens, 14-Apr-2018.) $)
  nn0nndivcl $p |- ( ( K e. NN0 /\ L e. NN ) -> ( K / L ) e. RR ) $=
    ( cn0 wcel cn wa cr nn0re adantr simpr nndivred ) ACDZBEDZFABLAGDMAHILMJK
    $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Extended nonnegative integers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The function values of the hash (set size) function are either nonnegative
  integers or positive infinity.  To avoid the need to distinguish between
  finite and infinite sets (and therefore if the set size is a nonnegative
  integer or positive infinity), it is useful to provide a definition of the
  set of nonnegative integers extended by positive infinity, analogously to
  the extension of the real numbers ` RR* ` , see ~ df-xr .

$)

  $c NN0* $. $( The set of extended nonnegative integers $)

  $( The set of extended nonnegative integers. $)
  cxnn0 $a class NN0* $.

  $( Define the set of extended nonnegative integers that includes positive
     infinity.  Analogue of the extension of the real numbers ` RR* ` , see
     ~ df-xr .  If we assumed excluded middle, this would be essentially the
     same as ` NN+oo ` as defined at ~ df-nninf but in its absence the
     relationship between the two is more complicated.  (Contributed by AV,
     10-Dec-2020.) $)
  df-xnn0 $a |- NN0* = ( NN0 u. { +oo } ) $.

  $( An extended nonnegative integer is either a standard nonnegative integer
     or positive infinity.  (Contributed by AV, 10-Dec-2020.) $)
  elxnn0 $p |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) ) $=
    ( cxnn0 wcel cn0 cpnf csn cun wceq df-xnn0 eleq2i pnfex elsn2 orbi2i 3bitri
    wo elun ) ABCADEFZGZCADCZAQCZOSAEHZOBRAIJADQPTUASAEKLMN $.

  $( The standard nonnegative integers are a subset of the extended nonnegative
     integers.  (Contributed by AV, 10-Dec-2020.) $)
  nn0ssxnn0 $p |- NN0 C_ NN0* $=
    ( cn0 cpnf csn cun cxnn0 ssun1 df-xnn0 sseqtrri ) AABCZDEAIFGH $.

  $( A standard nonnegative integer is an extended nonnegative integer.
     (Contributed by AV, 10-Dec-2020.) $)
  nn0xnn0 $p |- ( A e. NN0 -> A e. NN0* ) $=
    ( cn0 cxnn0 nn0ssxnn0 sseli ) BCADE $.

  $( An extended nonnegative integer is an extended real.  (Contributed by AV,
     10-Dec-2020.) $)
  xnn0xr $p |- ( A e. NN0* -> A e. RR* ) $=
    ( cxnn0 wcel cn0 cpnf wceq wo cxr elxnn0 nn0re rexrd pnfxr eleq1 jaoi sylbi
    mpbiri ) ABCADCZAEFZGAHCZAIQSRQAAJKRSEHCLAEHMPNO $.

  $( Zero is an extended nonnegative integer.  (Contributed by AV,
     10-Dec-2020.) $)
  0xnn0 $p |- 0 e. NN0* $=
    ( cn0 cxnn0 cc0 nn0ssxnn0 0nn0 sselii ) ABCDEF $.

  $( Positive infinity is an extended nonnegative integer.  (Contributed by AV,
     10-Dec-2020.) $)
  pnf0xnn0 $p |- +oo e. NN0* $=
    ( cpnf cxnn0 wcel cn0 wceq wo eqid olci elxnn0 mpbir ) ABCADCZAAEZFLKAGHAIJ
    $.

  $( No standard nonnegative integer equals positive infinity.  (Contributed by
     AV, 10-Dec-2020.) $)
  nn0nepnf $p |- ( A e. NN0 -> A =/= +oo ) $=
    ( cn0 wcel cpnf wceq cr pnfnre neli nn0re mto eleq1 mtbiri necon2ai ) ABCZA
    DADENDBCZODFCDFGHDIJADBKLM $.

  ${
    nn0xnn0d.1 $e |- ( ph -> A e. NN0 ) $.
    $( A standard nonnegative integer is an extended nonnegative integer,
       deduction form.  (Contributed by AV, 10-Dec-2020.) $)
    nn0xnn0d $p |- ( ph -> A e. NN0* ) $=
      ( cn0 cxnn0 nn0ssxnn0 sselid ) ADEBFCG $.

    $( No standard nonnegative integer equals positive infinity, deduction
       form.  (Contributed by AV, 10-Dec-2020.) $)
    nn0nepnfd $p |- ( ph -> A =/= +oo ) $=
      ( cn0 wcel cpnf wne nn0nepnf syl ) ABDEBFGCBHI $.
  $}

  $( No extended nonnegative integer equals negative infinity.  (Contributed by
     AV, 10-Dec-2020.) $)
  xnn0nemnf $p |- ( A e. NN0* -> A =/= -oo ) $=
    ( cxnn0 wcel cn0 cpnf wceq wo cmnf wne elxnn0 nn0re renemnfd pnfnemnf neeq1
    mpbiri jaoi sylbi ) ABCADCZAEFZGAHIZAJRTSRAAKLSTEHIMAEHNOPQ $.

  $( The extended nonnegative integers are extended reals without negative
     infinity.  (Contributed by AV, 10-Dec-2020.) $)
  xnn0xrnemnf $p |- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) ) $=
    ( cxnn0 wcel cxr cmnf wne xnn0xr xnn0nemnf jca ) ABCADCAEFAGAHI $.

  $( An extended nonnegative integer which is not a standard nonnegative
     integer is positive infinity.  (Contributed by AV, 10-Dec-2020.) $)
  xnn0nnn0pnf $p |- ( ( N e. NN0* /\ -. N e. NN0 ) -> N = +oo ) $=
    ( cxnn0 wcel cn0 wn cpnf wceq wo wi elxnn0 pm2.53 sylbi imp ) ABCZADCZEZAFG
    ZNOQHPQIAJOQKLM $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Integers (as a subset of complex numbers)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c ZZ $. $( The set of integers (blackboard bold Z). $)

  $( Extend class notation to include the class of integers. $)
  cz $a class ZZ $.

  $( Define the set of integers, which are the positive and negative integers
     together with zero.  Definition of integers in [Apostol] p. 22.  The
     letter Z abbreviates the German word Zahlen meaning "numbers."
     (Contributed by NM, 8-Jan-2002.) $)
  df-z $a |- ZZ = { n e. RR | ( n = 0 \/ n e. NN \/ -u n e. NN ) } $.

  ${
    $d x N $.
    $( Membership in the set of integers.  (Contributed by NM, 8-Jan-2002.) $)
    elz $p |- ( N e. ZZ <->
              ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) ) $=
      ( vx cv cc0 wceq cn wcel cneg w3o cr cz eqeq1 eleq1 eleq1d 3orbi123d df-z
      negeq elrab2 ) BCZDEZSFGZSHZFGZIADEZAFGZAHZFGZIBAJKSAEZTUDUAUEUCUGSADLSAF
      MUHUBUFFSAQNOBPR $.
  $}

  $( The negative of a positive integer is an integer.  (Contributed by NM,
     12-Jan-2002.) $)
  nnnegz $p |- ( N e. NN -> -u N e. ZZ ) $=
    ( cn wcel cneg cr cc0 wceq w3o cz nnre renegcld cc nncn negneg eleq1d mpcom
    biimprd 3mix3d elz sylanbrc ) ABCZADZECUBFGZUBBCZUBDZBCZHUBICUAAAJKUAUFUCUD
    ALCZUAUFAMUGUFUAUGUEABANOQPRUBST $.

  $( An integer is a real.  (Contributed by NM, 8-Jan-2002.) $)
  zre $p |- ( N e. ZZ -> N e. RR ) $=
    ( cz wcel cr cc0 wceq cn cneg w3o elz simplbi ) ABCADCAEFAGCAHGCIAJK $.

  $( An integer is a complex number.  (Contributed by NM, 9-May-2004.) $)
  zcn $p |- ( N e. ZZ -> N e. CC ) $=
    ( cz wcel zre recnd ) ABCAADE $.

  ${
    zre.1 $e |- A e. ZZ $.
    $( An integer is a real number.  (Contributed by NM, 14-Jul-2005.) $)
    zrei $p |- A e. RR $=
      ( cz wcel cr zre ax-mp ) ACDAEDBAFG $.
  $}

  $( The integers are a subset of the reals.  (Contributed by NM,
     2-Aug-2004.) $)
  zssre $p |- ZZ C_ RR $=
    ( vx cz cr cv zre ssriv ) ABCADEF $.

  $( The integers are a subset of the complex numbers.  (Contributed by NM,
     2-Aug-2004.) $)
  zsscn $p |- ZZ C_ CC $=
    ( vx cz cc cv zcn ssriv ) ABCADEF $.

  $( The set of integers exists.  (Contributed by NM, 30-Jul-2004.)  (Revised
     by Mario Carneiro, 17-Nov-2014.) $)
  zex $p |- ZZ e. _V $=
    ( cz cc cnex zsscn ssexi ) ABCDE $.

  $( Positive integer property expressed in terms of integers.  (Contributed by
     NM, 8-Jan-2002.) $)
  elnnz $p |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) ) $=
    ( cn wcel cr cneg cc0 wceq wo wa clt wbr cz orc nngt0 jca31 wi wn w3o bitri
    nnre idd lt0neg2 renegcl 0re ltnsym sylancl sylbid nsyl gt0ne0 neneqd ioran
    imp sylanbrc pm2.21d jaod ex com23 imp31 impbii 3orrot 3orass anbi2i anbi1i
    elz bitr4i ) ABCZADCZVFAEZBCZAFGZHZHZIZFAJKZIZALCZVNIVFVOVFVGVLVNATVFVKMANO
    VGVLVNVFVGVNVLVFVGVNVLVFPVGVNIZVFVFVKVQVFUAVQVKVFVQVIQVJQVKQVQFVHJKZVIVGVNV
    RQZVGVNVHFJKZVSAUBVGVHDCFDCVTVSPAUCUDVHFUEUFUGULVHNUHVQAFAUIUJVIVJUKUMUNUOU
    PUQURUSVPVMVNVPVGVJVFVIRZIVMAVDWAVLVGWAVFVIVJRVLVJVFVIUTVFVIVJVASVBSVCVE $.

  $( Zero is an integer.  (Contributed by NM, 12-Jan-2002.) $)
  0z $p |- 0 e. ZZ $=
    ( cc0 cz wcel cr wceq cn cneg w3o 0re eqid 3mix1i elz mpbir2an ) ABCADCAAEZ
    AFCZAGFCZHINOPAJKALM $.

  $( Zero is an integer, deductive form (common case).  (Contributed by David
     A. Wheeler, 8-Dec-2018.) $)
  0zd $p |- ( ph -> 0 e. ZZ ) $=
    ( cc0 cz wcel 0z a1i ) BCDAEF $.

  $( Nonnegative integer property expressed in terms of integers.  (Contributed
     by NM, 9-May-2004.) $)
  elnn0z $p |- ( N e. NN0 <-> ( N e. ZZ /\ 0 <_ N ) ) $=
    ( cn0 wcel cz cc0 cle wbr wa cr wceq cn cneg w3o nn0re elnn0 adantr a1i clt
    wo wi biimpi orcomd 3mix1 3mix2 jaoi syl elz sylanbrc nn0ge0 jca 0nn0 eleq1
    simprbi mpbiri nnnn0 wn simpr 0red zre lenltd mpbid lt0neg1d imbitrrid mtod
    nngt0 pm2.21d 3jaod mpd impbii ) ABCZADCZEAFGZHZVJVKVLVJAICZAEJZAKCZALZKCZM
    ZVKANVJVOVPSVSVJVPVOVJVPVOSAOUAUBVOVSVPVOVPVRUCVPVOVRUDUEUFAUGZUHAUIUJVMVSV
    JVKVSVLVKVNVSVTUMPVMVOVJVPVRVOVJTVMVOVJEBCUKAEBULUNQVPVJTVMAUOQVMVRVJVMVRAE
    RGZVMVLWAUPVKVLUQVMEAVMURVKVNVLAUSPZUTVAVRWAVMEVQRGVQVEVMAWBVBVCVDVFVGVHVI
    $.

  $( Integer property expressed in terms nonnegative integers and positive
     integers.  (Contributed by NM, 10-May-2004.) $)
  elznn0nn $p |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) ) $=
    ( cz wcel cr cc0 wceq cn cneg w3o wa cn0 wo elz andi df-3or anbi2i pm4.71ri
    nn0re elnn0 bitri orcom orbi1i 3bitr4i ) ABCADCZAEFZAGCZAHGCZIZJZAKCZUDUGJZ
    LZAMUDUEUFLZUGLZJUDUMJZUKLUIULUDUMUGNUHUNUDUEUFUGOPUJUOUKUJUDUJJUOUJUDARQUJ
    UMUDUJUFUELUMASUFUEUATPTUBUCT $.

  $( Integer property expressed in terms of nonnegative integers.  (Contributed
     by NM, 9-May-2004.) $)
  elznn0 $p |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) ) $=
    ( cz wcel cr cc0 wceq cn cneg w3o wa cn0 wo elz wb elnn0 a1i recn 0cn bitri
    cc negcon1 sylancl eqeq1i eqcom bitrdi orbi2d bitrid orbi12d 3orass orordir
    neg0 orcom 3bitrri bitr2di pm5.32i ) ABCADCZAEFZAGCZAHZGCZIZJUPAKCZUSKCZLZJ
    AMUPVAVDUPVDURUQLZUTUQLZLZVAUPVBVEVCVFVBVENUPAOPVCUTUSEFZLUPVFUSOUPVHUQUTUP
    VHEHZAFZUQUPATCETCVHVJNAQRAEUAUBVJEAFUQVIEAUKUCEAUDSUEUFUGUHVAUQURUTLZLVKUQ
    LVGUQURUTUIUQVKULURUTUQUJUMUNUOS $.

  $( Integer property expressed in terms of positive integers and nonnegative
     integers.  (Contributed by NM, 12-Jul-2005.) $)
  elznn $p |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) ) $=
    ( cz wcel cr cc0 wceq cn cneg w3o wa cn0 elz 3orrot 3orass bitri elnn0 recn
    wo orbi2d bitr4id negeq0d pm5.32i ) ABCADCZAEFZAGCZAHZGCZIZJUCUEUFKCZRZJALU
    CUHUJUCUHUEUGUDRZRZUJUHUEUGUDIULUDUEUGMUEUGUDNOUCUIUKUEUCUIUGUFEFZRUKUFPUCU
    DUMUGUCAAQUASTSTUBO $.

  $( Positive integers are a subset of integers.  (Contributed by NM,
     9-Jan-2002.) $)
  nnssz $p |- NN C_ ZZ $=
    ( vx cn cz cv wcel cc0 clt wbr elnnz simplbi ssriv ) ABCADZBELCEFLGHLIJK $.

  $( Nonnegative integers are a subset of the integers.  (Contributed by NM,
     9-May-2004.) $)
  nn0ssz $p |- NN0 C_ ZZ $=
    ( cn0 cn cc0 csn cun cz df-n0 nnssz wcel wss c0ex snss mpbi unssi eqsstri
    0z ) ABCDZEFGBQFHCFIQFJPCFKLMNO $.

  $( A positive integer is an integer.  (Contributed by NM, 9-May-2004.) $)
  nnz $p |- ( N e. NN -> N e. ZZ ) $=
    ( cn cz nnssz sseli ) BCADE $.

  $( A nonnegative integer is an integer.  (Contributed by NM, 9-May-2004.) $)
  nn0z $p |- ( N e. NN0 -> N e. ZZ ) $=
    ( cn0 cz nn0ssz sseli ) BCADE $.

  ${
    nnzi.1 $e |- N e. NN $.
    $( A positive integer is an integer.  (Contributed by Mario Carneiro,
       18-Feb-2014.) $)
    nnzi $p |- N e. ZZ $=
      ( cn cz nnssz sselii ) CDAEBF $.
  $}

  ${
    nn0zi.1 $e |- N e. NN0 $.
    $( A nonnegative integer is an integer.  (Contributed by Mario Carneiro,
       18-Feb-2014.) $)
    nn0zi $p |- N e. ZZ $=
      ( cn0 cz nn0ssz sselii ) CDAEBF $.
  $}

  $( Positive integer property expressed in terms of integers.  (Contributed by
     NM, 10-May-2004.)  (Proof shortened by Mario Carneiro, 16-May-2014.) $)
  elnnz1 $p |- ( N e. NN <-> ( N e. ZZ /\ 1 <_ N ) ) $=
    ( cn wcel cz c1 cle wbr wa nnz nnge1 jca cc0 clt 0lt1 cr zre 0re 1re ltletr
    wi mp3an12 syl mpani imdistani elnnz sylibr impbii ) ABCZADCZEAFGZHZUHUIUJA
    IAJKUKUILAMGZHUHUIUJULUILEMGZUJULNUIAOCZUMUJHULTZAPLOCEOCUNUOQRLEASUAUBUCUD
    AUEUFUG $.

  $( Positive integers expressed as a subset of integers.  (Contributed by NM,
     3-Oct-2004.) $)
  nnzrab $p |- NN = { x e. ZZ | 1 <_ x } $=
    ( cn cv cz wcel c1 cle wbr wa cab crab elnnz1 abbi2i df-rab eqtr4i ) BACZDE
    FPGHZIZAJQADKRABPLMQADNO $.

  $( Nonnegative integers expressed as a subset of integers.  (Contributed by
     NM, 3-Oct-2004.) $)
  nn0zrab $p |- NN0 = { x e. ZZ | 0 <_ x } $=
    ( cn0 cv cz wcel cc0 cle wbr wa cab crab elnn0z abbi2i df-rab eqtr4i ) BACZ
    DEFPGHZIZAJQADKRABPLMQADNO $.

  $( One is an integer.  (Contributed by NM, 10-May-2004.) $)
  1z $p |- 1 e. ZZ $=
    ( c1 1nn nnzi ) ABC $.

  $( 1 is an integer, deductive form (common case).  (Contributed by David A.
     Wheeler, 6-Dec-2018.) $)
  1zzd $p |- ( ph -> 1 e. ZZ ) $=
    ( c1 cz wcel 1z a1i ) BCDAEF $.

  $( Two is an integer.  (Contributed by NM, 10-May-2004.) $)
  2z $p |- 2 e. ZZ $=
    ( c2 2nn nnzi ) ABC $.

  $( 3 is an integer.  (Contributed by David A. Wheeler, 8-Dec-2018.) $)
  3z $p |- 3 e. ZZ $=
    ( c3 3nn nnzi ) ABC $.

  $( 4 is an integer.  (Contributed by BJ, 26-Mar-2020.) $)
  4z $p |- 4 e. ZZ $=
    ( c4 4nn nnzi ) ABC $.

  $( Closure law for negative integers.  (Contributed by NM, 9-May-2004.) $)
  znegcl $p |- ( N e. ZZ -> -u N e. ZZ ) $=
    ( cz wcel cr cc0 wceq cn cneg w3o elz negeq neg0 eqtrdi eqeltrdi nnnegz nnz
    wa 0z 3jaoi adantl sylbi ) ABCADCZAEFZAGCZAHZGCZIZQUEBCZAJUGUHUBUCUHUDUFUCU
    EEBUCUEEHEAEKLMRNAOUEPSTUA $.

  $( -1 is an integer (common case).  (Contributed by David A. Wheeler,
     5-Dec-2018.) $)
  neg1z $p |- -u 1 e. ZZ $=
    ( c1 cn wcel cneg cz 1nn nnnegz ax-mp ) ABCADECFAGH $.

  $( A number is an integer iff its negative is.  (Contributed by Stefan
     O'Rear, 13-Sep-2014.) $)
  znegclb $p |- ( A e. CC -> ( A e. ZZ <-> -u A e. ZZ ) ) $=
    ( cc wcel cz cneg znegcl negneg eleq1d imbitrid impbid2 ) ABCZADCZAEZDCZAFN
    MEZDCKLMFKOADAGHIJ $.

  $( The negative of a nonnegative integer is an integer.  (Contributed by NM,
     9-May-2004.) $)
  nn0negz $p |- ( N e. NN0 -> -u N e. ZZ ) $=
    ( cn0 wcel cz cneg nn0z znegcl syl ) ABCADCAEDCAFAGH $.

  ${
    nn0negzi.1 $e |- N e. NN0 $.
    $( The negative of a nonnegative integer is an integer.  (Contributed by
       Mario Carneiro, 18-Feb-2014.) $)
    nn0negzi $p |- -u N e. ZZ $=
      ( cn0 wcel cneg cz nn0negz ax-mp ) ACDAEFDBAGH $.
  $}

  $( Second Peano postulate generalized to integers.  (Contributed by NM,
     13-Feb-2005.) $)
  peano2z $p |- ( N e. ZZ -> ( N + 1 ) e. ZZ ) $=
    ( cz wcel c1 caddc co cr cn0 cneg wo readdcld cn wa a1i cc recnd cc0 neg1cn
    1red eqtrd elznn0nn biimpi biantrurd orbi2d mpbird wi peano2nn0 adantr 1cnd
    renegcld negdid oveq1d negcld addassd ax-1cn 1pneg1e0 oveq2i eqtrdi addridd
    zre addcomli simpr eqeltrd elnn0nn sylanbrc ex orim12d mpd elznn0 ) ABCZADE
    FZGCVKHCZVKIZHCZJZVKBCVJADAUTZVJSKVJAHCZAIZLCZJZVOVJVTVQAGCZVSMZJZVJWCAUAUB
    VJVSWBVQVJWAVSVPUCUDUEVJVQVLVSVNVQVLUFVJAUGNVJVSVNVJVSMZVMOCVMDEFZLCVNWDVMW
    DVKWDADVJWAVSVPUHZWDSKUJPWDWEVRLWDWEVRQEFZVRWDWEVRDIZDEFZEFZWGWDWEVRWHEFZDE
    FWJWDVMWKDEWDADWDAWFPZWDUIZUKULWDVRWHDWDAWLUMZWHOCWDRNWMUNTWIQVREDWHQUORUPV
    AUQURWDVRWNUSTVJVSVBVCVMVDVEVFVGVHVKVIVE $.

  ${
    $d M x y $.  $d N x $.
    $( Lemma for ~ zaddcl .  Special case in which ` N ` is a positive integer.
       (Contributed by Jim Kingdon, 14-Mar-2020.) $)
    zaddcllempos $p |- ( ( M e. ZZ /\ N e. NN ) -> ( M + N ) e. ZZ ) $=
      ( vx vy cn wcel cz caddc co cv wi c1 wceq oveq2 eleq1d imbi2d peano2z zcn
      wa cc adantl nncn adantr 1cnd addassd imbitrid ex a2d nnind impcom ) BEFA
      GFZABHIZGFZUKACJZHIZGFZKUKALHIZGFZKUKADJZHIZGFZKUKAUSLHIZHIZGFZKUKUMKCDBU
      NLMZUPURUKVEUOUQGUNLAHNOPUNUSMZUPVAUKVFUOUTGUNUSAHNOPUNVBMZUPVDUKVGUOVCGU
      NVBAHNOPUNBMZUPUMUKVHUOULGUNBAHNOPAQUSEFZUKVAVDVIUKVAVDKVAUTLHIZGFVIUKSZV
      DUTQVKVJVCGVKAUSLUKATFVIARUAVIUSTFUKUSUBUCVKUDUEOUFUGUHUIUJ $.
  $}

  $( "Reverse" second Peano postulate for integers.  (Contributed by NM,
     12-Sep-2005.) $)
  peano2zm $p |- ( N e. ZZ -> ( N - 1 ) e. ZZ ) $=
    ( cz wcel c1 cmin co cneg caddc zcn negsubdid znegcl peano2z syl eqeltrd cc
    1cnd wb subcld znegclb mpbird ) ABCZADEFZBCZUBGZBCZUAUDAGZDHFZBUAADAIZUAPZJ
    UAUFBCUGBCAKUFLMNUAUBOCUCUEQUAADUHUIRUBSMT $.

  ${
    $d M x y $.  $d N x y $.
    $( Lemma for ~ zaddcl .  Special case in which ` -u N ` is a positive
       integer.  (Contributed by Jim Kingdon, 14-Mar-2020.) $)
    zaddcllemneg $p |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) ->
        ( M + N ) e. ZZ ) $=
      ( vx vy cz wcel cneg cn caddc co oveq2d wa cv wi wceq negeq eleq1d imbi2d
      c1 cmin cr w3a simp2 recnd negnegd cc zcn adantr negsubd peano2zm eqeltrd
      1cnd nncn ad2antrr negdi2d ad2antlr negcld addsubassd adantl eqeltrrd a2d
      exp31 nnind impcom 3impa ) AEFZBUAFZBGZHFZUBZAVHGZIJZABIJEVJVKBAIVJBVJBVF
      VGVIUCUDUEKVFVGVIVLEFZVIVFVGLZVMVNACMZGZIJZEFZNVNASGZIJZEFZNVNADMZGZIJZEF
      ZNVNAWBSIJZGZIJZEFZNVNVMNCDVHVOSOZVRWAVNWJVQVTEWJVPVSAIVOSPKQRVOWBOZVRWEV
      NWKVQWDEWKVPWCAIVOWBPKQRVOWFOZVRWIVNWLVQWHEWLVPWGAIVOWFPKQRVOVHOZVRVMVNWM
      VQVLEWMVPVKAIVOVHPKQRVNVTASTJZEVNASVFAUFFZVGAUGUHZVNULUIVFWNEFVGAUJUHUKWB
      HFZVNWEWIWQVNWEWIWQVNLZWELZWHAWCSTJZIJZEWSWGWTAIWSWBSWQWBUFFVNWEWBUMUNZWS
      ULZUOKWSWDSTJZXAEWSAWCSVNWOWQWEWPUPWSWBXBUQXCURWEXDEFWRWDUJUSUTUKVBVAVCVD
      VEUT $.
  $}

  $( Closure of addition of integers.  (Contributed by NM, 9-May-2004.)  (Proof
     shortened by Mario Carneiro, 16-May-2014.) $)
  zaddcl $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ ) $=
    ( cz wcel wa cc0 wceq cn cneg w3o caddc co cr elz simprbi adantl zcn adantr
    cc wi addridd simpl eqeltrd eleq1d syl5ibrcom zaddcllempos zre zaddcllemneg
    oveq2 ex 3expia sylan2 3jaod mpd ) ACDZBCDZEZBFGZBHDZBIHDZJZABKLZCDZUPVAUOU
    PBMDZVABNOPUQURVCUSUTUQVCURAFKLZCDUQVEACUQAUOASDUPAQRUAUOUPUBUCURVBVECBFAKU
    IUDUEUOUSVCTUPUOUSVCABUFUJRUPUOVDUTVCTBUGUOVDUTVCABUHUKULUMUN $.

  $( Closure of subtraction of integers.  (Contributed by NM, 11-May-2004.) $)
  zsubcl $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ ) $=
    ( cz wcel wa cneg caddc co cmin wceq zcn negsub syl2an znegcl zaddcl sylan2
    cc eqeltrrd ) ACDZBCDZEABFZGHZABIHZCSAQDBQDUBUCJTAKBKABLMTSUACDUBCDBNAUAOPR
    $.

  $( Integer trichotomy (with zero).  (Contributed by Jim Kingdon,
     14-Mar-2020.) $)
  ztri3or0 $p |- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) ) $=
    ( cz wcel cc0 wceq clt wbr w3o cn cneg cr elz simprbi idd wi nngt0 lt0neg1d
    a1i zre imbitrrid 3orim123d mpd 3orrot sylibr ) ABCZADEZDAFGZADFGZHZUHUFUGH
    UEUFAICZAJZICZHZUIUEAKCUMALMUEUFUFUJUGULUHUEUFNUJUGOUEAPRULUHUEDUKFGUKPUEAA
    SQTUAUBUHUFUGUCUD $.

  $( Integer trichotomy.  (Contributed by Jim Kingdon, 14-Mar-2020.) $)
  ztri3or $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) ) $=
    ( cz wcel wa clt wbr wceq w3o cmin cc0 zsubcl ztri3or0 syl cneg zre posdifd
    co cr recnd adantr adantl resubcld lt0neg2d negsubdi2d breq1d 3bitrd bicomd
    subeq0ad 3orbi123d mpbird ) ACDZBCDZEZABFGZABHZBAFGZIABJRZKFGZURKHZKURFGZIZ
    UNURCDVBABLURMNUNUOUSUPUTUQVAUNUOKBAJRZFGVCOZKFGUSUNABULASDUMAPUAZUMBSDULBP
    UBZQUNVCUNBAVFVEUCUDUNVDURKFUNBAUNBVFTZUNAVETZUEUFUGUNUTUPUNABVHVGUIUHUNBAV
    FVEQUJUK $.

  $( Trichotomy law.  (Contributed by Jim Kingdon, 27-Mar-2020.) $)
  zletric $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) ) $=
    ( cz wcel wa cr clt wbr wceq w3o cle zre anim12i ztri3or ltle orc syl6 eqle
    wo wi ex adantr ancoms olc 3jaod sylc ) ACDZBCDZEAFDZBFDZEZABGHZABIZBAGHZJA
    BKHZBAKHZSZUGUIUHUJALBLMABNUKULUQUMUNUKULUOUQABOUOUPPZQUKUMUOUQUIUMUOTUJUIU
    MUOABRUAUBURQUKUNUPUQUJUIUNUPTBAOUCUPUOUDQUEUF $.

  $( Trichotomy law.  (Contributed by Jim Kingdon, 17-Apr-2020.) $)
  zlelttric $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B < A ) ) $=
    ( cz wcel wa cr clt wbr wceq w3o cle zre anim12i ztri3or ltle orc syl6 eqle
    wo wi ex adantr olc a1i 3jaod sylc ) ACDZBCDZEAFDZBFDZEZABGHZABIZBAGHZJABKH
    ZUNSZUGUIUHUJALBLMABNUKULUPUMUNUKULUOUPABOUOUNPZQUKUMUOUPUIUMUOTUJUIUMUOABR
    UAUBUQQUNUPTUKUNUOUCUDUEUF $.

  $( 'Less than' expressed in terms of 'less than or equal to'.  (Contributed
     by Jim Kingdon, 14-Mar-2020.) $)
  zltnle $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> -. B <_ A ) ) $=
    ( cz wcel wa clt wbr cle wn cr zre lenlt syl2anr biimpd con2d wceq w3o syl6
    wb wi ztri3or ax-1 a1i eqcom eqle sylan2b ex adantl sylan2 pm2.24 3jaod mpd
    ltle impbid ) ACDZBCDZEZABFGZBAHGZIZUQUSURUQUSURIZUPBJDZAJDZUSVASUOBKZAKZBA
    LMNOUQURABPZBAFGZQUTURTZABUAUQURVHVFVGURVHTUQURUTUBUCUQVFUSVHUPUOVBVFUSTZVD
    VBVIUOVBVFUSVFVBBAPUSABUDBAUEUFUGUHUIUSURUJZRUQVGUSVHUPVBVCVGUSTUOVDVEBAUMM
    VJRUKULUN $.

  $( Integer 'Less than or equal to' expressed in terms of 'less than' or
     'equals'.  (Contributed by Jim Kingdon, 8-Apr-2020.) $)
  zleloe $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
      ( A <_ B <-> ( A < B \/ A = B ) ) ) $=
    ( cz wcel wa cle wbr clt wo wn cr wb zre lenlt syl2an w3o ztri3or df-3or wi
    wceq sylib orcomd ord sylbid ltle eqle ex adantr jaod impbid ) ACDZBCDZEZAB
    FGZABHGZABTZIZUMUNBAHGZJZUQUKAKDZBKDZUNUSLULAMZBMZABNOUMURUQUMUQURUMUOUPURP
    UQURIABQUOUPURRUAUBUCUDUKUTVAUQUNSULVBVCUTVAEUOUNUPABUEUTUPUNSVAUTUPUNABUFU
    GUHUIOUJ $.

  $( An integer is not a positive integer iff it is less than one.
     (Contributed by NM, 13-Jul-2005.) $)
  znnnlt1 $p |- ( N e. ZZ -> ( -. N e. NN <-> N < 1 ) ) $=
    ( cz wcel cn wn c1 cle wbr clt elnnz1 baib notbid wb 1z zltnle mpan2 bitr4d
    ) ABCZADCZEFAGHZEZAFIHZRSTSRTAJKLRFBCUBUAMNAFOPQ $.

  $( A positive integer is not less than or equal to zero.  (Contributed by AV,
     13-May-2020.) $)
  nnnle0 $p |- ( A e. NN -> -. A <_ 0 ) $=
    ( cn wcel cc0 clt wbr cle wn nngt0 cz wb 0z nnz zltnle sylancr mpbid ) ABCZ
    DAEFZADGFHZAIQDJCAJCRSKLAMDANOP $.

  $( Transitive law of ordering for integers.  (Contributed by Alexander van
     der Vekens, 3-Apr-2018.) $)
  zletr $p |- ( ( J e. ZZ /\ K e. ZZ /\ L e. ZZ )
                -> ( ( J <_ K /\ K <_ L ) -> J <_ L ) ) $=
    ( cz wcel cr cle wbr wa wi zre letr syl3an ) ADEAFEBDEBFECDECFEABGHBCGHIACG
    HJAKBKCKABCLM $.

  $( Reverse closure law for addition of integers.  (Contributed by NM,
     11-May-2004.) $)
  zrevaddcl $p |- ( N e. ZZ ->
                   ( ( M e. CC /\ ( M + N ) e. ZZ ) <-> M e. ZZ ) ) $=
    ( cz wcel cc caddc co wa cmin zcn pncan sylan2 ancoms adantr zsubcl adantlr
    wceq eqeltrrd ex wi zaddcl expcom impbid pm5.32da pm4.71ri bitr4di ) BCDZAE
    DZABFGZCDZHUHACDZHUKUGUHUJUKUGUHHZUJUKULUJUKULUJHUIBIGZACULUMAQZUJUHUGUNUGU
    HBEDUNBJABKLMNUGUJUMCDZUHUJUGUOUIBOMPRSUGUKUJTUHUKUGUJABUAUBNUCUDUKUHAJUEUF
    $.

  $( The positive difference of unequal integers is a positive integer.
     (Generalization of ~ nnsub .)  (Contributed by NM, 11-May-2004.) $)
  znnsub $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) ) $=
    ( cz wcel wa clt wbr cmin co cn cr wb posdif syl2an zsubcl ancoms biantrurd
    cc0 zre bitrd elnnz bitr4di ) ACDZBCDZEZABFGZBAHIZCDZRUGFGZEZUGJDUEUFUIUJUC
    AKDBKDUFUILUDASBSABMNUEUHUIUDUCUHBAOPQTUGUAUB $.

  $( The sum of a real number not being an integer and an integer is not an
     integer.  Note that "not being an integer" in this case means "the
     negation of is an integer" rather than "is apart from any integer" (given
     excluded middle, those two would be equivalent).  (Contributed by AV,
     19-Jul-2021.) $)
  nzadd $p |- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ )
                -> ( A + B ) e. ( RR \ ZZ ) ) $=
    ( cr cz cdif wcel wa caddc co wn eldif zre readdcl sylan2 adantlr wi zsubcl
    cmin expcom cc adantl wceq zcn pncan syl2an eleq1d sylibd con3d com23 imp31
    recn ex jca sylanb sylibr ) ACDEZFZBDFZGABHIZCFZUSDFZJZGZUSUPFUQACFZADFZJZG
    ZURVCACDKVGURGUTVBVDURUTVFURVDBCFUTBLABMNOVDVFURVBVDURVFVBVDURVFVBPVDURGZVA
    VEVHVAUSBRIZDFZVEURVAVJPVDVAURVJUSBQSUAVHVIADVDATFBTFVIAUBURAUKBUCABUDUEUFU
    GUHULUIUJUMUNUSCDKUO $.

  $( Closure of multiplication of integers.  (Contributed by NM,
     30-Jul-2004.) $)
  zmulcl $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ ) $=
    ( cz wcel cr cn0 cneg wo wa cmul co elznn0 nn0mulcl jctild cc syl2an eleq1d
    wceq imbitrid syl6 wi orcd a1i remulcl recn mulneg1 olc mulneg2 mul2neg orc
    ccased imbitrrdi imp an4s syl2anb ) ACDAEDZAFDZAGZFDZHZIBEDZBFDZBGZFDZHZIAB
    JKZCDZBCDALBLUPVAUTVEVGUPVAIZUTVEIZVGVHVIVFEDZVFFDZVFGZFDZHZIZVGVHUQVBUSVDV
    OVHUQVBIZVNVJVPVNUAVHVPVKVMABMUBUCABUDZNVHUSVBIZVNVJVHVRVMVNVRURBJKZFDVHVMU
    RBMVHVSVLFUPAODZBODZVSVLRVAAUEZBUEZABUFPQSVMVKUGZTVQNVHUQVDIZVNVJVHWEVMVNWE
    AVCJKZFDVHVMAVCMVHWFVLFUPVTWAWFVLRVAWBWCABUHPQSWDTVQNVHUSVDIZVNVJVHWGVKVNWG
    URVCJKZFDVHVKURVCMVHWHVFFUPVTWAWHVFRVAWBWCABUIPQSVKVMUJTVQNUKVFLULUMUNUO $.

  $( Integer ordering relation.  (Contributed by NM, 10-May-2004.)  (Proof
     shortened by Mario Carneiro, 16-May-2014.) $)
  zltp1le $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) ) $=
    ( cz wcel wa clt wbr c1 caddc co cle cmin cn wi nnge1 a1i znnsub cr wb zre
    1re leaddsub2 mp3an2 syl2an 3imtr4d adantr ltp1d peano2re syl adantl ltletr
    syl3anc mpand impbid ) ACDZBCDZEZABFGZAHIJZBKGZUQBALJZMDZHVAKGZURUTVBVCNUQV
    AOPABQUOARDZBRDZUTVCSZUPATZBTZVDHRDVEVFUAAHBUBUCUDUEUQAUSFGZUTURUQAUOVDUPVG
    UFZUGUQVDUSRDZVEVIUTEURNVJUQVDVKVJAUHUIUPVEUOVHUJAUSBUKULUMUN $.

  $( Integer ordering relation.  (Contributed by NM, 10-May-2004.) $)
  zleltp1 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> M < ( N + 1 ) ) ) $=
    ( cz wcel wa cle wbr c1 caddc co clt cr wb zre leadd1 mp3an3 syl2an peano2z
    1re zltp1le sylan2 bitr4d ) ACDZBCDZEABFGZAHIJBHIJZFGZAUFKGZUCALDZBLDZUEUGM
    ZUDANBNUIUJHLDUKSABHOPQUDUCUFCDUHUGMBRAUFTUAUB $.

  $( Integer ordering relation.  (Contributed by NM, 13-Nov-2004.) $)
  zlem1lt $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) ) $=
    ( cz wcel wa c1 cmin co clt wbr caddc cle wb peano2zm zltp1le sylan wceq cc
    zcn ax-1cn npcan sylancl adantr breq1d bitr2d ) ACDZBCDZEZAFGHZBIJZUIFKHZBL
    JZABLJUFUICDUGUJULMANUIBOPUHUKABLUFUKAQZUGUFARDFRDUMASTAFUAUBUCUDUE $.

  $( Integer ordering relation.  (Contributed by NM, 13-Nov-2004.) $)
  zltlem1 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) ) $=
    ( cz wcel wa c1 cmin co cle wbr caddc clt wb peano2zm zleltp1 sylan2 cc zcn
    wceq ax-1cn npcan sylancl adantl breq2d bitr2d ) ACDZBCDZEZABFGHZIJZAUIFKHZ
    LJZABLJUGUFUICDUJULMBNAUIOPUHUKBALUGUKBSZUFUGBQDFQDUMBRTBFUAUBUCUDUE $.

  $( An integer greater than ` 0 ` is greater than or equal to ` 1 ` .
     (Contributed by AV, 14-Oct-2018.) $)
  zgt0ge1 $p |- ( Z e. ZZ -> ( 0 < Z <-> 1 <_ Z ) ) $=
    ( cz wcel cc0 clt wbr c1 caddc co cle wb zltp1le mpan wceq 0p1e1 a1i breq1d
    0z bitrd ) ABCZDAEFZDGHIZAJFZGAJFDBCTUAUCKRDALMTUBGAJUBGNTOPQS $.

$( TODO: The following 14 theorems do not contain ` ZZ ` - these theorems are
   about positive or nonnegative integers which are defined earlier.  But the
   proofs of these theorems are based on theorems for ` ZZ `.  It should be
   clarified if these theorems could be proven without theorems for ` ZZ ` and
   then could be moved up into the corresponding subsections.  Alternately,
   these theorems could be moved into a new separate subsection "Positive and
   nonnegative integers (cont.)". $)

  $( Positive integer ordering relation.  (Contributed by NM, 13-Aug-2001.)
     (Proof shortened by Mario Carneiro, 16-May-2014.) $)
  nnleltp1 $p |- ( ( A e. NN /\ B e. NN ) ->
                ( A <_ B <-> A < ( B + 1 ) ) ) $=
    ( cn wcel cz cle wbr c1 caddc co clt wb nnz zleltp1 syl2an ) ACDAEDBEDABFGA
    BHIJKGLBCDAMBMABNO $.

  $( Positive integer ordering relation.  (Contributed by NM, 19-Aug-2001.) $)
  nnltp1le $p |- ( ( A e. NN /\ B e. NN ) ->
                ( A < B <-> ( A + 1 ) <_ B ) ) $=
    ( cn wcel cz clt wbr c1 caddc co cle wb nnz zltp1le syl2an ) ACDAEDBEDABFGA
    HIJBKGLBCDAMBMABNO $.

  $( Closure of addition of positive integers minus one.  (Contributed by NM,
     6-Aug-2003.)  (Proof shortened by Mario Carneiro, 16-May-2014.) $)
  nnaddm1cl $p |- ( ( A e. NN /\ B e. NN ) -> ( ( A + B ) - 1 ) e. NN ) $=
    ( cn wcel wa caddc co c1 cmin wceq nncn ax-1cn addsub mp3an3 syl2an nnm1nn0
    cc cn0 nn0nnaddcl sylan eqeltrd ) ACDZBCDZEABFGHIGZAHIGZBFGZCUBAQDZBQDZUDUF
    JZUCAKBKUGUHHQDUILABHMNOUBUERDUCUFCDAPUEBSTUA $.

  $( Nonnegative integer ordering relation.  (Contributed by Raph Levien,
     10-Dec-2002.)  (Proof shortened by Mario Carneiro, 16-May-2014.) $)
  nn0ltp1le $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
                ( M < N <-> ( M + 1 ) <_ N ) ) $=
    ( cn0 wcel cz clt wbr c1 caddc co cle wb nn0z zltp1le syl2an ) ACDAEDBEDABF
    GAHIJBKGLBCDAMBMABNO $.

  $( Nonnegative integer ordering relation.  (Contributed by Raph Levien,
     10-Apr-2004.) $)
  nn0leltp1 $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
                ( M <_ N <-> M < ( N + 1 ) ) ) $=
    ( cn0 wcel cz cle wbr c1 caddc co clt wb nn0z zleltp1 syl2an ) ACDAEDBEDABF
    GABHIJKGLBCDAMBMABNO $.

  $( Nonnegative integer ordering relation.  (Contributed by NM, 10-May-2004.)
     (Proof shortened by Mario Carneiro, 16-May-2014.) $)
  nn0ltlem1 $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
                ( M < N <-> M <_ ( N - 1 ) ) ) $=
    ( cn0 wcel cz clt wbr c1 cmin co cle wb nn0z zltlem1 syl2an ) ACDAEDBEDABFG
    ABHIJKGLBCDAMBMABNO $.

  $( The nonnegative difference of integers is a nonnegative integer.
     (Generalization of ~ nn0sub .)  (Contributed by NM, 14-Jul-2005.) $)
  znn0sub $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
                ( M <_ N <-> ( N - M ) e. NN0 ) ) $=
    ( cz wcel wa cle wbr co cc0 cn0 wb cr subge0 syl2an zsubcl biantrurd bitr3d
    cmin zre ancoms elnn0z bitr4di ) ACDZBCDZEABFGZBARHZCDZIUFFGZEZUFJDUDUCUEUI
    KUDUCEZUHUEUIUDBLDALDUHUEKUCBSASBAMNUJUGUHBAOPQTUFUAUB $.

  $( Subtraction of nonnegative integers.  (Contributed by NM, 9-May-2004.) $)
  nn0sub $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
                ( M <_ N <-> ( N - M ) e. NN0 ) ) $=
    ( cn0 wcel cz cle wbr cmin co wb nn0z znn0sub syl2an ) ACDAEDBEDABFGBAHICDJ
    BCDAKBKABLM $.

  $( Subtracting a nonnegative integer from a nonnegative integer which is
     greater than the first one results in a nonnegative integer.  (Contributed
     by Alexander van der Vekens, 6-Apr-2018.) $)
  ltsubnn0 $p |- ( ( A e. NN0 /\ B e. NN0 )
                   -> ( B < A -> ( A - B ) e. NN0 ) ) $=
    ( cn0 wcel wa clt wbr cle cmin co cr wi nn0re ltle syl2anr wb nn0sub ancoms
    sylibd ) ACDZBCDZEBAFGZBAHGZABIJCDZUABKDAKDUBUCLTBMAMBANOUATUCUDPBAQRS $.

  $( A nonnegative integer is greater than or equal to its negative.
     (Contributed by AV, 13-Aug-2021.) $)
  nn0negleid $p |- ( A e. NN0 -> -u A <_ A ) $=
    ( cn0 wcel cneg cc0 nn0re renegcld 0red cle wbr nn0ge0 le0neg2d mpbid letrd
    ) ABCZADZEAOAAFZGOHQOEAIJPEIJAKZOAQLMRN $.

  $( If the difference of a real number and a nonnegative integer is greater
     than another real number, the sum of the real number and the nonnegative
     integer is also greater than the other real number.  (Contributed by AV,
     13-Aug-2021.) $)
  difgtsumgt $p |- ( ( A e. RR /\ B e. NN0 /\ C e. RR )
                     -> ( C < ( A - B ) -> C < ( A + B ) ) ) $=
    ( cr wcel cn0 w3a cmin co clt wbr cneg caddc cc wa wceq recn nn0cn 3ad2ant2
    readdcld anim12i 3adant3 negsub syl eqcomd breq2d simp3 simp1 renegcld 3jca
    nn0re cle nn0negleid leadd2dd lelttrdi sylbid ) ADEZBFEZCDEZGZCABHIZJKCABLZ
    MIZJKCABMIZJKUTVAVCCJUTVCVAUTANEZBNEZOZVCVAPUQURVGUSUQVEURVFAQBRUAUBABUCUDU
    EUFUTCVCVDUTUSVCDEVDDEUQURUSUGUTAVBUQURUSUHZURUQVBDEUSURBBUKZUISZTUTABVHURU
    QBDEUSVISZTUJUTVBBAVJVKVHURUQVBBULKUSBUMSUNUOUP $.

  $( A nonnegative integer which is neither 0 nor 1 is greater than or equal to
     2.  (Contributed by Alexander van der Vekens, 6-Dec-2017.) $)
  nn0n0n1ge2 $p |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N ) $=
    ( cn0 wcel cc0 wne c1 w3a c2 cle wbr cmin co wceq caddc 3ad2ant1 cn elnnne0
    wa nnm1nn0 syl nn0cn subsub4d oveq2i eqtr2di 3simpa sylibr subeq0ad necon3d
    1cnd 1p1e2 biimpd imp 3adant2 sylanbrc eqeltrd wb 2nn0 jctl nn0sub mpbird )
    ABCZADEZAFEZGZHAIJZAHKLZBCZVDVFAFKLZFKLZBVAVBVFVIMVCVAVIAFFNLZKLVFVAAFFAUAZ
    VAUIZVLUBVJHAKUJUCUDOVDVHPCZVIBCVDVHBCZVHDEZVMVDAPCZVNVDVAVBRVPVAVBVCUEAQUF
    ASTVAVCVOVBVAVCVOVAVHDAFVAVHDMAFMVAAFVKVLUGUKUHULUMVHQUNVHSTUOVDHBCZVARZVEV
    GUPVAVBVRVCVAVQUQUROHAUSTUT $.

  ${
    $d x y z N $.
    $( Membership in the set of integers.  Commonly used in constructions of
       the integers as equivalence classes under subtraction of the positive
       integers.  (Contributed by Mario Carneiro, 16-May-2014.) $)
    elz2 $p |- ( N e. ZZ <-> E. x e. NN E. y e. NN N = ( x - y ) ) $=
      ( cz wcel cr cn0 cneg wo wa cmin co wceq cn c1 1nn ax-1cn sylancr syl2an
      cc wrex elznn0 caddc nn0p1nn adantl a1i recn adantr pncan sylancl rspceov
      eqcomd syl3anc negsub simpr nnnn0addcl eqeltrrd nncan jaodan nnre resubcl
      cv cle wbr nnz zletric syl2anr nnnn0 nn0sub nncn negsubdi2 eleq1d orbi12d
      wb bitr4d mpbid jca eleq1 negeq anbi12d syl5ibrcom rexlimivv impbii bitri
      ) CDECFEZCGEZCHZGEZIZJZCAVBZBVBZKLZMZBNUAANUAZCUBWJWOWEWFWOWHWEWFJZCOUCLZ
      NEZONEZCWQOKLZMWOWFWRWECUDUEWSWPPUFWPWTCWPCTEZOTEZWTCMWEXAWFCUGZUHQCOUIUJ
      ULABNNWQOCKUKUMWEWHJZWSOCKLZNECOXEKLZMWOWSXDPUFXDOWGUCLZXENXDXBXAXGXEMQWE
      XAWHXCUHZOCUNRXDWSWHXGNEPWEWHUOOWGUPRUQXDXFCXDXBXAXFCMQXHOCURRULABNNOXECK
      UKUMUSWNWJABNNWKNEZWLNEZJZWJWNWMFEZWMGEZWMHZGEZIZJXKXLXPXIWKFEWLFEXLXJWKU
      TWLUTWKWLVASXKWLWKVCVDZWKWLVCVDZIZXPXJWLDEWKDEXSXIWLVEWKVEWLWKVFVGXKXQXMX
      RXOXJWLGEZWKGEZXQXMVNXIWLVHZWKVHZWLWKVIVGXKXRWLWKKLZGEZXOXIYAXTXRYEVNXJYC
      YBWKWLVISXKXNYDGXIWKTEWLTEXNYDMXJWKVJWLVJWKWLVKSVLVOVMVPVQWNWEXLWIXPCWMFV
      RWNWFXMWHXOCWMGVRWNWGXNGCWMVSVLVMVTWAWBWCWD $.

    $( Alternate definition of the integers, based on ~ elz2 .  (Contributed by
       Mario Carneiro, 16-May-2014.) $)
    dfz2 $p |- ZZ = ( - " ( NN X. NN ) ) $=
      ( vx vy vz cz cmin cn cxp cima cv wcel co wceq wrex elz2 cc wfn wb nnsscn
      wss mp2an wf subf ffn ax-mp xpss12 ovelimab bitr4i eqriv ) ADEFFGZHZAIZDJ
      UKBICIEKLCFMBFMZUKUJJZBCUKNEOOGZPZUIUNSZUMULQUNOEUAUOUBUNOEUCUDFOSZUQUPRR
      FOFOUETBCUNFFUKEUFTUGUH $.
  $}

  $( Subtraction of nonnegative integers.  (Contributed by NM, 4-Sep-2005.) $)
  nn0sub2 $p |- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) -> ( N - M ) e. NN0 ) $=
    ( cn0 wcel cle wbr cmin co nn0sub biimp3a ) ACDBCDABEFBAGHCDABIJ $.

  $( Apartness is equivalent to not equal for integers.  (Contributed by Jim
     Kingdon, 14-Mar-2020.) $)
  zapne $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M =//= N <-> M =/= N ) ) $=
    ( cz wcel wa cap wbr wne cc wi zcn apne syl2an wceq wn clt wo w3o cr zre wb
    df-ne ztri3or 3orrot 3orass bitri sylib reaplt orcom bitrdi biimtrid impbid
    ord sylibrd ) ACDZBCDZEZABFGZABHZUOAIDBIDURUSJUPAKBKABLMUSABNZOZUQURABUBUQV
    ABAPGZABPGZQZURUQUTVDUQVCUTVBRZUTVDQZABUCVEUTVBVCRVFVCUTVBUDUTVBVCUEUFUGUMU
    OASDZBSDZURVDUAUPATBTVGVHEURVCVBQVDABUHVCVBUIUJMUNUKUL $.

  $( Equality of integers is decidable.  (Contributed by Jim Kingdon,
     14-Mar-2020.) $)
  zdceq $p |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A = B ) $=
    ( cz wcel wa clt wbr wceq w3o wdc ztri3or cr zre wne ltne necomd sylibr syl
    wi ex wo olc dcne adantr sylan orc a1i adantl 3jaod mpd ) ACDZBCDZEZABFGZAB
    HZBAFGZIUOJZABKUMUNUQUOUPUKALDZULUNUQSZAMURUSULURUNUQURUNEZABNZUQUTBAABOPVA
    UOVAUAZUQVAUOUBABUCZQZRTUDUEUOUQSUMUOVBUQUOVAUFVCQUGULUPUQSZUKULBLDZVEBMVFU
    PUQVFUPEVAUQBAOVDRTRUHUIUJ $.

  $( Integer ` <_ ` is decidable.  (Contributed by Jim Kingdon, 7-Apr-2020.) $)
  zdcle $p |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A <_ B ) $=
    ( cz wcel wa clt wbr wceq w3o cle wdc ztri3or cr wi zre ltle wn sylibr syl6
    wo orc df-dc eqle syl ex adantr lenlt biimpd con2d olc 3jaod syl2an mpd ) A
    CDZBCDZEABFGZABHZBAFGZIZABJGZKZABLUNAMDZBMDZUSVANUOAOBOVBVCEZUPVAUQURVDUPUT
    VAABPUTUTUTQZTZVAUTVEUAUTUBZRZSVBUQVANVCVBUQVAVBUQEUTVAABUCVHUDUEUFVDURVEVA
    VDUTURVDUTURQABUGUHUIVEVFVAVEUTUJVGRSUKULUM $.

  $( Integer ` < ` is decidable.  (Contributed by Jim Kingdon, 1-Jun-2020.) $)
  zdclt $p |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B ) $=
    ( cz wcel wa clt wbr wceq w3o wdc ztri3or cr wi zre wn wo orc sylibr adantr
    df-dc a1i ltnr wb breq2 adantl mtbid olc ex ltnsym ancoms syl6 3jaod syl2an
    syl mpd ) ACDZBCDZEABFGZABHZBAFGZIZURJZABKUPALDZBLDZVAVBMUQANBNVCVDEZURVBUS
    UTURVBMVEURURUROZPZVBURVFQURTZRUAVCUSVBMVDVCUSVBVCUSEZVFVBVIAAFGZURVCVJOUSA
    UBSUSVJURUCVCABAFUDUEUFVFVGVBVFURUGVHRZUNUHSVEUTVFVBVDVCUTVFMBAUIUJVKUKULUM
    UO $.

  ${
    $d A x y $.
    $( Whether an integer is an element of a finite set of integers is
       decidable.  (Contributed by Jim Kingdon, 8-Jun-2026.) $)
    zfidc $p |- ( ( S C_ ZZ /\ A e. ZZ /\ S e. Fin ) -> DECID A e. S ) $=
      ( vx vy cz wss wcel cfn w3a weq wdc wral cv zdceq rgen2 simp2 simp1 simp3
      a1i elssdc ) BEFZAEGZBHGZIZCDBEACDJKZDELCELUDUECDEECMDMNOSUAUBUCPUAUBUCQU
      AUBUCRT $.
  $}

  $( Integer 'Less than' expressed in terms of 'less than or equal to'.  Also
     see ~ ltleap which is a similar result for real numbers.  (Contributed by
     Jim Kingdon, 14-Mar-2020.) $)
  zltlen $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
               ( A < B <-> ( A <_ B /\ B =/= A ) ) ) $=
    ( cz wcel wa clt wbr cle wne cap cr wb zre ltleap syl2an zapne anbi2d bitrd
    necom anbi2i bitrdi ) ACDZBCDZEZABFGZABHGZABIZEZUFBAIZEUDUEUFABJGZEZUHUBAKD
    BKDUEUKLUCAMBMABNOUDUJUGUFABPQRUGUIUFABSTUA $.

  $( A nonnegative integer is neither 0 nor 1 if and only if it is greater than
     or equal to 2.  (Contributed by Alexander van der Vekens, 17-Jan-2018.) $)
  nn0n0n1ge2b $p |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) <-> 2 <_ N ) ) $=
    ( wcel cc0 wne c1 wa c2 wbr wdc wn wi cz wceq zdceq sylancl dcned wo wb syl
    clt cn0 nn0n0n1ge2 3expib nn0z 0z 1z dcand ianordc nnedc orbi12d bitrd 2pos
    cle breq1 mpbiri a1d 1lt2 jaoi impcom 2z zltnle adantr mpbid ex sylbid sylc
    condc impbid ) AUABZACDZAEDZFZGAUMHZVIVJVKVMAUBUCVIVLIVLJZVMJZKVMVLKVIVJVKV
    IACVIALBZCLBACMZIZAUDZUEACNOZPZVIAEVIVPELBAEMZIZVSUFAENOZPUGVIVNVQWBQZVOVIV
    NVJJZVKJZQZWEVIVJIVNWHRWAVJVKUHSVIWFVQWGWBVIVRWFVQRVTACUISVIWCWGWBRWDAEUISU
    JUKVIWEVOVIWEFAGTHZVOWEVIWIVQVIWIKWBVQWIVIVQWICGTHULACGTUNUOUPWBWIVIWBWIEGT
    HUQAEGTUNUOUPURUSVIWIVORZWEVIVPGLBWJVSUTAGVAOVBVCVDVEVLVMVGVFVH $.

  $( A nonnegative integer less than ` 1 ` is ` 0 ` .  (Contributed by Paul
     Chapman, 22-Jun-2011.) $)
  nn0lt10b $p |- ( N e. NN0 -> ( N < 1 <-> N = 0 ) ) $=
    ( cn0 wcel cc0 wceq cle wbr wa c1 clt cr wb nn0re 0re letri3 mpan2 biantrud
    syl nn0ge0 cz nn0z caddc co 0z zleltp1 0p1e1 breq2i bitrdi 3bitr2rd ) ABCZA
    DEZADFGZDAFGZHZULAIJGZUJAKCZUKUNLZAMUPDKCUQNADOPRUJUMULASQUJATCZULUOLAUAURU
    LADIUBUCZJGZUOURDTCULUTLUDADUEPUSIAJUFUGUHRUI $.

  $( A nonnegative integer less than 2 must be 0 or 1.  (Contributed by
     Alexander van der Vekens, 16-Sep-2018.) $)
  nn0lt2 $p |- ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) ) $=
    ( cn0 wcel c2 clt wbr wa c1 wceq cc0 wo wne wi olc a1i cle cz wb sylancl 1z
    cmin co nn0z 2z zltlem1 2m1e1 breq2i bitrdi necom zltlen nn0lt10b biimpa ex
    orcd sylbird expd syl7bi sylbid imp wdc zdceq adantr dcne sylib mpjaod ) AB
    CZADEFZGZAHIZAJIZVIKZAHLZVIVKMVHVIVJNOVFVGVLVKMZVFVGAHPFZVMVFVGADHUAUBZPFZV
    NVFAQCZDQCVGVPRAUCZUDADUESVOHAPUFUGUHVLHALZVFVNVKAHUIVFVNVSVKVFVNVSGZAHEFZV
    KVFVQHQCZWAVTRVRTAHUJSVFWAVKVFWAGVJVIVFWAVJAUKULUNUMUOUPUQURUSVHVIUTZVIVLKV
    FWCVGVFVQWBWCVRTAHVASVBAHVCVDVE $.

  $( A nonnegative integer which is less than or equal to 2 is either 0 or 1 or
     2.  (Contributed by AV, 16-Mar-2019.) $)
  nn0le2is012 $p |- ( ( N e. NN0 /\ N <_ 2 )
                      -> ( N = 0 \/ N = 1 \/ N = 2 ) ) $=
    ( cn0 wcel c2 cle wbr cc0 wceq c1 clt wo cz wb 2z zleloe sylancl com12 jaoi
    a1d sylbid w3o nn0z wi cmin co zltlem1 2m1e1 breq2d bitrd 1z nn0lt10b 3mix1
    a1i biimtrdi 3mix2 3mix3 imp ) ABCZADEFZAGHZAIHZADHZUAZURUSADJFZVBKZVCURALC
    ZDLCZUSVEMAUBZNADOPVEURVCVDURVCUCZVBURVDVCURVDAIEFZVCURVDADIUDUEZEFZVJURVFV
    GVDVLMVHNADUFPURVKIAEVKIHURUGUMUHUIURVJAIJFZVAKZVCURVFILCVJVNMVHUJAIOPVNURV
    CVMVIVAURVMVCURVMUTVCAUKUTVAVBULUNQVAVCURVAUTVBUOSRQTTQVBVCURVBUTVAUPSRQTUQ
    $.

  $( Nonnegative integer ordering relation.  (Contributed by NM,
     21-Jun-2005.) $)
  nn0lem1lt $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
                   ( M <_ N <-> ( M - 1 ) < N ) ) $=
    ( cn0 wcel cz cle wbr c1 cmin co clt wb nn0z zlem1lt syl2an ) ACDAEDBEDABFG
    AHIJBKGLBCDAMBMABNO $.

  $( Positive integer ordering relation.  (Contributed by NM, 21-Jun-2005.) $)
  nnlem1lt $p |- ( ( M e. NN /\ N e. NN ) -> ( M <_ N <-> ( M - 1 ) < N ) ) $=
    ( cn wcel cz cle wbr c1 cmin co clt wb nnz zlem1lt syl2an ) ACDAEDBEDABFGAH
    IJBKGLBCDAMBMABNO $.

  $( Positive integer ordering relation.  (Contributed by NM, 21-Jun-2005.) $)
  nnltlem1 $p |- ( ( M e. NN /\ N e. NN ) -> ( M < N <-> M <_ ( N - 1 ) ) ) $=
    ( cn wcel cz clt wbr c1 cmin co cle wb nnz zltlem1 syl2an ) ACDAEDBEDABFGAB
    HIJKGLBCDAMBMABNO $.

  $( A positive integer decreased by 1 is greater than or equal to 0.
     (Contributed by AV, 30-Oct-2018.) $)
  nnm1ge0 $p |- ( N e. NN -> 0 <_ ( N - 1 ) ) $=
    ( cn wcel cc0 clt wbr c1 cmin co cle nngt0 cz wb nnz zltlem1 sylancr mpbid
    0z ) ABCZDAEFZDAGHIJFZAKSDLCALCTUAMRANDAOPQ $.

  $( Division of a nonnegative integer by a positive number is not negative.
     (Contributed by Alexander van der Vekens, 14-Apr-2018.) $)
  nn0ge0div $p |- ( ( K e. NN0 /\ L e. NN ) -> 0 <_ ( K / L ) ) $=
    ( cn0 wcel cn wa cc0 cle wbr cdiv co nn0ge0 adantr cr clt wb cz elnnz nn0re
    w3a zre ad2antrl simprr 3jca sylan2b ge0div syl mpbid ) ACDZBEDZFZGAHIZGABJ
    KHIZUIULUJALMUKANDZBNDZGBOIZTZULUMPUJUIBQDZUPFZUQBRUIUSFUNUOUPUIUNUSASMURUO
    UIUPBUAUBUIURUPUCUDUEABUFUGUH $.

  ${
    $d k M $.  $d k N $.
    $( Two ways to express " ` M ` divides ` N ` .  (Contributed by NM,
       3-Oct-2008.) $)
    zdiv $p |- ( ( M e. NN /\ N e. ZZ )
                -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) ) $=
      ( cn wcel cz wa cc0 cap wbr cv cmul co wceq wrex cdiv wb cc wi zcn adantr
      nnap0 nncn divcanap3 3coml 3expa sylan2 3adantl2 oveq1 sylan9req eqeltrrd
      w3a simplr exp31 rexlimdv divcanap2 3com12 oveq2 eqeq1d rspcev expcom syl
      impbid 3expia syl2an mpd ) BDEZCFEZGBHIJZBAKZLMZCNZAFOZCBPMZFEZQZVGVIVHBU
      BUAVGBREZCREZVIVPSVHBUCCTVQVRVIVPVQVRVIULZVMVOVSVLVOAFVSVJFEZVLVOVSVTGZVL
      GVJVNFWAVLVJVKBPMZVNVQVIVTWBVJNZVRVTVQVIGVJREZWCVJTVQVIWDWCWDVQVIWCVJBUDU
      EUFUGUHVKCBPUIUJVSVTVLUMUKUNUOVSBVNLMZCNZVOVMSVRVQVIWFCBUPUQVOWFVMVLWFAVN
      FVJVNNVKWECVJVNBLURUSUTVAVBVCVDVEVF $.
  $}

  $( Property of divisibility: if ` D ` divides ` A ` and ` B ` then it divides
     ` A + B ` .  (Contributed by NM, 3-Oct-2008.) $)
  zdivadd $p |- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\
       ( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) ) -> ( ( A + B ) / D ) e. ZZ ) $=
    ( cn wcel cz w3a cdiv co wa caddc wceq cc cc0 cap wbr zcn nncn nnap0 jca
    divdirap syl3an 3comr adantr zaddcl adantl eqeltrd ) CDEZAFEZBFEZGZACHIZFEB
    CHIZFEJZJABKICHIZULUMKIZFUKUOUPLZUNUIUJUHUQUIAMEUJBMEUHCMEZCNOPZJUQAQBQUHUR
    USCRCSTABCUAUBUCUDUNUPFEUKULUMUEUFUG $.

  $( Property of divisibility: if ` D ` divides ` A ` then it divides
     ` B x. A ` .  (Contributed by NM, 3-Oct-2008.) $)
  zdivmul $p |- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\
       ( A / D ) e. ZZ ) -> ( ( B x. A ) / D ) e. ZZ ) $=
    ( cn wcel cz w3a cdiv co wa cmul wceq cc cc0 cap wbr 3ad2ant2 3ad2ant1 nncn
    zcn nnap0 3ad2ant3 divassap syl3anc 3comr adantr zmulcl 3ad2antl3 eqeltrd
    jca ) CDEZAFEZBFEZGZACHIZFEZJBAKICHIZBUOKIZFUNUQURLZUPULUMUKUSULUMUKGBMEZAM
    EZCMEZCNOPZJZUSUMULUTUKBTQULUMVAUKATRUKULVDUMUKVBVCCSCUAUJUBBACUCUDUEUFUMUK
    UPURFEULBUOUGUHUI $.

  ${
    $d k M $.  $d k N $.
    $( An extensionality-like property for integer ordering.  (Contributed by
       NM, 29-Oct-2005.) $)
    zextle $p |- ( ( M e. ZZ /\ N e. ZZ /\ A. k e. ZZ ( k <_ M <-> k <_ N ) )
                  -> M = N ) $=
      ( cz wcel cv cle wbr wb wral wceq wa zre leidd adantr breq1 bibi12d mpbid
      rspcva cr adantlr mpbird adantll jca ex letri3 syl2an sylibrd 3impia ) BD
      EZCDEZAFZBGHZULCGHZIZADJZBCKZUJUKLZUPBCGHZCBGHZLZUQURUPVAURUPLUSUTUJUPUSU
      KUJUPLBBGHZUSUJVBUPUJBBMZNOUOVBUSIABDULBKUMVBUNUSULBBGPULBCGPQSRUAUKUPUTU
      JUKUPLUTCCGHZUKVDUPUKCCMZNOUOUTVDIACDULCKUMUTUNVDULCBGPULCCGPQSUBUCUDUEUJ
      BTECTEUQVAIUKVCVEBCUFUGUHUI $.

    $( An extensionality-like property for integer ordering.  (Contributed by
       NM, 29-Oct-2005.) $)
    zextlt $p |- ( ( M e. ZZ /\ N e. ZZ /\ A. k e. ZZ ( k < M <-> k < N ) )
                  -> M = N ) $=
      ( cz wcel clt wbr wb wral wceq wa c1 cmin cle zltlem1 peano2zm syl2an zcn
      co cc cv adantrr adantrl bibi12d ancoms ralbidva wi zextle 3expia subcan2
      ax-1cn mp3an3 sylibd sylbid 3impia ) BDEZCDEZAUAZBFGZURCFGZHZADIZBCJZUPUQ
      KZVBURBLMSZNGZURCLMSZNGZHZADIZVCVDVAVIADURDEZVDVAVIHVKVDKUSVFUTVHVKUPUSVF
      HUQURBOUBVKUQUTVHHUPURCOUCUDUEUFVDVJVEVGJZVCUPVEDEZVGDEZVJVLUGUQBPCPVMVNV
      JVLAVEVGUHUIQUPBTEZCTEZVLVCHZUQBRCRVOVPLTEVQUKBCLUJULQUMUNUO $.
  $}

  $( The reciprocal of a number greater than 1 is not an integer.  (Contributed
     by NM, 3-May-2005.) $)
  recnz $p |- ( ( A e. RR /\ 1 < A ) -> -. ( 1 / A ) e. ZZ ) $=
    ( cr wcel c1 clt wbr wa cdiv co cz cc0 recgt1i simprd cle wn simpld zgt0ge1
    syl5ibcom wb 1re cap 0lt1 0re lttr mp3an12 mpani imdistani gt0ap0 rerecclap
    wi syl syldan lenlt sylancr sylibd mt2d ) ABCZDAEFZGZDAHIZJCZUTDEFZUSKUTEFZ
    VBALZMUSVADUTNFZVBOZUSVCVAVEUSVCVBVDPUTQRUSDBCZUTBCZVEVFSTUQURAKUAFZVHUSUQK
    AEFZGVIUQURVJUQKDEFZURVJUBKBCVGUQVKURGVJUJUCTKDAUDUEUFUGAUHUKAUIULDUTUMUNUO
    UP $.

  $( A number between an integer and its successor is not an integer.
     (Contributed by NM, 3-May-2005.) $)
  btwnnz $p |- ( ( A e. ZZ /\ A < B /\ B < ( A + 1 ) ) -> -. B e. ZZ ) $=
    ( cz wcel clt wbr c1 caddc co wn wa cle zltp1le cr wb peano2z zre syl lenlt
    syl2an bitrd biimpd impancom con2d 3impia ) ACDZABEFZBAGHIZEFZBCDZJUFUGKUJU
    IUFUJUGUIJZUFUJKZUGUKULUGUHBLFZUKABMUFUHNDZBNDUMUKOUJUFUHCDUNAPUHQRBQUHBSTU
    AUBUCUDUE $.

  $( A larger number does not divide a smaller positive integer.  (Contributed
     by NM, 3-May-2005.) $)
  gtndiv $p |- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ ) $=
    ( cr wcel cn clt wbr w3a cc0 cdiv co c1 caddc cz wn nnre 3ad2ant2 wa mp3an1
    wb simp1 nngt0 adantl 0re lttr sylan ancoms mpand 3impia divgt0d simp3 cmul
    wi 1re ltdivmul2 mp3an2 syl12anc mullidd breq2d 3ad2ant1 bitrd mpbird 0p1e1
    recn breqtrrdi 0z btwnnz syl2anc ) ACDZBEDZBAFGZHZIBAJKZFGZVMILMKZFGZVMNDOZ
    VLBAVJVIBCDZVKBPZQZVIVJVKUAZVJVIIBFGZVKBUBZQVIVJVKIAFGZVIVJRWBVKWDVJWBVIWCU
    CVJVIWBVKRWDUMZVJVRVIWEVSICDVRVIWEUDIBAUESUFUGUHUIZUJVLVMLVOFVLVMLFGZVKVIVJ
    VKUKVLWGBLAULKZFGZVKVLVRVIWDWGWITZVTWAWFVRLCDVIWDRWJUNBLAUOUPUQVIVJWIVKTVKV
    IWHABFVIAAVDURUSUTVAVBVCVEINDVNVPVQVFIVMVGSVH $.

  $( One-half is not an integer.  (Contributed by NM, 31-Jul-2004.) $)
  halfnz $p |- -. ( 1 / 2 ) e. ZZ $=
    ( c2 cr wcel c1 clt wbr cdiv co cz wn 2re 1lt2 recnz mp2an ) ABCDAEFDAGHICJ
    KLAMN $.

  $( Three halves is not an integer.  (Contributed by AV, 2-Jun-2020.) $)
  3halfnz $p |- -. ( 3 / 2 ) e. ZZ $=
    ( c1 cz wcel c3 c2 cdiv co clt wbr caddc wn cmul cr wb 3re 2re mp3an breq2i
    c4 mpbir 1z 2cn mullidi 2lt3 eqbrtri cc0 1re 2pos pm3.2i ltmuldiv mpbi 3lt4
    wa 2t2e4 1p1e2 ltdivmul bitri btwnnz ) ABCADEFGZHIZUSAAJGZHIZUSBCKUAAELGZDH
    IZUTVCEDHEUBUCUDUEAMCDMCZEMCZUFEHIZUMZVDUTNUGOVFVGPUHUIZADEUJQUKVBDEELGZHIZ
    VKDSHIULVJSDHUNRTVBUSEHIZVKVAEUSHUORVEVFVHVLVKNOPVIDEEUPQUQTAUSURQ $.

  ${
    $d A w x y z $.  $d f g ph x $.  $d ph w x z $.
    suprzclex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\
        A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $.
    suprzclex.ss $e |- ( ph -> A C_ ZZ ) $.
    $( The supremum of a set of integers is an element of the set.
       (Contributed by Jim Kingdon, 20-Dec-2021.) $)
    suprzclex $p |- ( ph -> sup ( A , RR , < ) e. A ) $=
      ( vw cr clt c1 cv wbr wcel wrex wa wn cz wral adantr vg vf csup cmin wceq
      co lttri3 adantl supclti ltm1d zssre sstrdi peano2rem syl suprlubex mpbid
      wb cle caddc sselda sselid simprl sseldd peano2re ad2antrr simpr suprubex
      wss zre wi simprr 1red ltsubaddd lelttrd zleltp1 syl2anc mpbird ralrimiva
      breq2 cbvrexv imbi2i ralbii anbi2i rexbii sylib letri3d mpbir2and eqeltrd
      suprleubex rexlimddv ) AEIJUCZKUDUFZDLZJMZWKENDEAWLWKJMWNDEOAWKABCDUAUBIE
      JUBLZINUALZINPWOWPUEWOWPJMQWPWOJMQPUQAWOWPUGUHFUIZUJABCDEWLFAERIGUKULZAWK
      INZWLINWQWKUMUNUOUPAWMENZWNPZPZWKWMEXBWKWMUEWKWMURMZWMWKURMXBXCHLZWMURMZH
      ESXBXEHEXBXDENZPZXEXDWMKUSUFZJMZXGXDWKXHXGRIXDUKXBERXDAERVHXAGTZUTZVAXBWS
      XFAWSXAWQTZTXBXHINZXFXBWMINZXMXBWMRNZXNXBERWMXJAWTWNVBZVCZWMVIUNZWMVDUNTX
      GBCDEXDABLZCLZJMQCESZXTXSJMZXTWMJMZDEOZVJZCISZPZBIOZXAXFFVEAEIVHXAXFWRVEX
      BXFVFVGXBWKXHJMZXFXBWNYIAWTWNVKXBWKKWMXLXBVLXRVMUPTVNXGXDRNXOXEXIUQXKXBXO
      XFXQTXDWMVOVPVQVRXBBCHEWMAYAYBXTXDJMZHEOZVJZCISZPZBIOZXAAYHYOFYGYNBIYFYMY
      AYEYLCIYDYKYBYCYJDHEWMXDXTJVSVTWAWBWCWDWETZXBERIXJUKULZXRWIVQXBBCHEWMYPYQ
      XPVGXBWKWMXLXRWFWGXPWHWJ $.
  $}

  ${
    $d x A $.
    $( Two ways to express " ` A ` is a prime number (or 1)".  (Contributed by
       NM, 4-May-2005.) $)
    prime $p |- ( A e. NN -> (
                 A. x e. NN ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <->
          A. x e. NN ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) -> x = A ) ) ) $=
      ( cn wcel cv cdiv c1 wceq wi clt wbr wa wb cz nnz wn bitr4di adantl nnre
      cr co wo cle w3a wne wdc 1z zdceq mpan2 dfordc df-ne imbi1i imbi2d impexp
      3syl bi2.04 bitri nngt1ne1 anbi1d gtndiv 3expia sylan con2d lenlt sylibrd
      syl2an ancoms syl5 pm4.71rd anbi2d 3anass bitr3d imbi1d bitrd ralbidva )
      BCDZBAEZFUAZCDZVQGHZVQBHZUBZIZGVQJKZVQBUCKZVSUDZWAIZACVPVQCDZLZWCVQGUEZVS
      LZWAIZWGWHWCWLMVPWHWCVSWJWAIZIZWLWHWBWMVSWHVQNDZVTUFZWBWMMVQOWOGNDWPUGVQG
      UHUIWPWBVTPZWAIWMVTWAUJWJWQWAVQGUKULQUOUMWLWJVSWAIIWNWJVSWAUNWJVSWAUPUQQR
      WIWKWFWAWIWDVSLZWKWFWIWDWJVSWHWDWJMVPVQURRUSWIWRWDWEVSLZLWFWIVSWSWDWIVSWE
      VSVRNDZWIWEVROWHVPWTWEIWHVPLZWTBVQJKZPZWEXAXBWTWHVQTDZVPXBWTPZIVQSZXDVPXB
      XEVQBUTVAVBVCWHXDBTDWEXCMVPXFBSVQBVDVFVEVGVHVIVJWDWEVSVKQVLVMVNVO $.
  $}

  $( The square of a nonzero integer is a positive integer.  (Contributed by
     NM, 2-Aug-2004.) $)
  msqznn $p |- ( ( A e. ZZ /\ A =/= 0 ) -> ( A x. A ) e. NN ) $=
    ( cz wcel cc0 wne wa cmul co clt wbr cn zmulcl anidms adantr wb zapne mpan2
    cap 0z pm5.32i cr zre apsqgt0 sylan sylbir elnnz sylanbrc ) ABCZADEZFZAAGHZ
    BCZDUKIJZUKKCUHULUIUHULAALMNUJUHADRJZFUMUHUNUIUHDBCUNUIOSADPQTUHAUACUNUMAUB
    AUCUDUEUKUFUG $.

  $( No even integer equals an odd integer (i.e. no integer can be both even
     and odd).  Exercise 10(a) of [Apostol] p. 28.  (Contributed by NM,
     31-Jul-2004.)  (Proof shortened by Mario Carneiro, 18-May-2014.) $)
  zneo $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
             ( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) ) $=
    ( cz wcel wa c2 cmul co c1 caddc cdiv wn wceq cmin cc 2cn zcn mulcl sylancr
    a1i wne halfnz adantr adantl 1cnd subaddd subdid oveq1d zsubcl syl cc0 2ap0
    cap wbr divcanap3d eqtr3d eqeltrd oveq1 eleq1d syl5ibcom sylbird mpi necomd
    necon3bd ) ACDZBCDZEZFBGHZIJHZFAGHZVGIFKHZCDZLVIVJUAUBVGVLVIVJVGVIVJMVJVHNH
    ZIMZVLVGVJVHIVGFODZAODZVJODPVEVPVFAQUCZFARSVGVOBODZVHODPVFVRVEBQUDZFBRSVGUE
    UFVGVMFKHZCDVNVLVGVTABNHZCVGFWAGHZFKHVTWAVGWBVMFKVGFABVOVGPTZVQVSUGUHVGWAFV
    GWACDWAODABUIZWAQUJWCFUKUMUNVGULTUOUPWDUQVNVTVKCVMIFKURUSUTVAVDVBVC $.

  ${
    $d j k N $.
    $( A positive integer is even or odd.  (Contributed by Jim Kingdon,
       15-Mar-2020.) $)
    nneoor $p |- ( N e. NN ->
                ( ( N / 2 ) e. NN \/ ( ( N + 1 ) / 2 ) e. NN ) ) $=
      ( vj vk cn wcel c1 caddc co c2 cdiv cv wceq oveq1d eleq1d orbi12d 2div2e1
      wo oveq1 cc 2cn df-2 oveq1i eqtr3i 1nn eqeltri orci peano2nn nncn cc0 cap
      add1p1 wbr 2ap0 pm3.2i divdirap mp3an23 oveq2i eqtrdi eqtrd syl imbitrrid
      wa orim2d orcom imbitrdi nnind orcomd ) ADEAFGHZIJHZDEZAIJHZDEZBKZFGHZIJH
      ZDEZVMIJHZDEZQFFGHZIJHZDEZFIJHZDEZQCKZFGHZIJHZDEZWDIJHZDEZQZWEFGHZIJHZDEZ
      WGQZVJVLQBCAVMFLZVPWAVRWCWOVOVTDWOVNVSIJVMFFGRMNWOVQWBDVMFIJRNOVMWDLZVPWG
      VRWIWPVOWFDWPVNWEIJVMWDFGRMNWPVQWHDVMWDIJRNOVMWELZVPWMVRWGWQVOWLDWQVNWKIJ
      VMWEFGRMNWQVQWFDVMWEIJRNOVMALZVPVJVRVLWRVOVIDWRVNVHIJVMAFGRMNWRVQVKDVMAIJ
      RNOWAWCVTFDIIJHZVTFIVSIJUAUBPUCUDUEUFWDDEZWJWGWMQWNWTWIWMWGWIWMWTWHFGHZDE
      WHUGWTWLXADWTWDSEZWLXALWDUHXBWLWDIGHZIJHZXAXBWKXCIJWDUKMXBXDWHWSGHZXAXBIS
      EZXFIUIUJULZVBXDXELTXFXGTUMUNWDIIUOUPWSFWHGPUQURUSUTNVAVCWGWMVDVEVFVG $.

    $( A positive integer is even or odd but not both.  (Contributed by NM,
       1-Jan-2006.)  (Proof shortened by Mario Carneiro, 18-May-2014.) $)
    nneo $p |- ( N e. NN ->
                ( ( N / 2 ) e. NN <-> -. ( ( N + 1 ) / 2 ) e. NN ) ) $=
      ( cn wcel c2 cdiv co c1 caddc cmul wceq nncn peano2cn syl 2cnd divcanap2d
      wn cc cc0 cz nnz cap wbr 2ap0 a1i oveq1d eqtr4d wne syl2an expcom syl5com
      zneo necon2bd nneoor orcomd ord impbid ) ABCZADEFZBCZAGHFZDEFZBCZPZUQDVAI
      FZDURIFZGHFZJUSVCUQVDUTVFUQUTDUQAQCUTQCAKZALMUQNZDRUAUBUQUCUDZOUQVEAGHUQA
      DVGVHVIOUEUFUSVBVDVFVBUSVDVFUGZVBVASCURSCVJUSVATURTVAURUKUHUIULUJUQVBUSUQ
      USVBAUMUNUOUP $.

    nneo.1 $e |- N e. NN $.
    $( A positive integer is even or odd but not both.  (Contributed by NM,
       20-Aug-2001.) $)
    nneoi $p |- ( ( N / 2 ) e. NN <-> -. ( ( N + 1 ) / 2 ) e. NN ) $=
      ( cn wcel c2 cdiv co c1 caddc wn wb nneo ax-mp ) ACDAEFGCDAHIGEFGCDJKBALM
      $.
  $}

  $( An integer is even or odd.  (Contributed by NM, 1-Jan-2006.) $)
  zeo $p |- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) ) $=
    ( cz wcel cc0 wceq cn cneg c2 cdiv co c1 caddc 2cn 2ap0 syl cc mp3an23 cmin
    eleq1d ax-1cn cr w3o wa wo elz oveq1 div0api eqeltri eqeltrdi adantl nneoor
    0z orcd nnz orim12i wi recn cap wbr nnnegz biimtrrdi halfcld negnegd sylibd
    divnegap peano2zm negsubdi2i 2m1e1 eqtr2i subcli negcon2i mpbi oveq2i negcl
    addsubass negdi mpan2 3eqtr4a oveq1d 2div2e1 eqcomi peano2cn pm3.2i eqtr4id
    divsubdirap 3eqtr4d imbitrid znegcl syl6 peano2re recnd syl5 orim12d adantr
    mpd 3jaodan sylbi ) ABCAUACZADEZAFCZAGZFCZUBUCAHIJZBCZAKLJZHIJZBCZUDZAUEWRW
    SXHWTXBWSXHWRWSXDXGWSXCDHIJZBADHIUFXIDBHMNUGULUHUIUMUJWTXHWRWTXCFCZXFFCZUDX
    HAUKXJXDXKXGXCUNXFUNUOOUJWRXBUCXAHIJZFCZXAKLJZHIJZFCZUDZXHXBXQWRXAUKUJWRXQX
    HUPXBWRXMXDXPXGWRXMXCGZGZBCZXDWRXMXRFCXTWRXRXLFWRAPCZXRXLEZAUQZYAHPCZHDURUS
    ZYBMNAHVEQOSXRUTVAWRXSXCBWRXCWRAYCVBVCSVDXPXOBCZWRXGXOUNWRYFXFGZGZBCZXGWRYF
    YGBCZYIYFXOKRJZBCWRYJXOVFWRYKYGBWRYAYKYGEYCYAXNHRJZHIJZXEGZHIJZYKYGYAYLYNHI
    YAXAKHRJZLJZXAKGZLJZYLYNYPYRXALKYPGZEYPYREYTHKRJKKHTMVGVHVIKYPTKHTMVJVKVLVM
    YAXAPCZYLYQEZAVNZUUAKPCZYDUUBTMXAKHVOQOYAUUDYNYSETAKVPVQVRVSYAYKXOHHIJZRJZY
    MKUUEXORUUEKVTWAVMYAXNPCZYMUUFEZYAUUAUUGUUCXAWBOUUGYDYDYEUCUUHMYDYEMNWCXNHH
    WEQOWDYAXEPCZYGYOEZAWBUUIYDYEUUJMNXEHVEQOWFOSWGYGWHWIWRYHXFBWRXFWRXEWRXEAWJ
    WKVBVCSVDWLWMWNWOWPWQ $.

  $( An integer is even or odd but not both.  (Contributed by Mario Carneiro,
     12-Sep-2015.) $)
  zeo2 $p |- ( N e. ZZ ->
              ( ( N / 2 ) e. ZZ <-> -. ( ( N + 1 ) / 2 ) e. ZZ ) ) $=
    ( cz wcel c2 cdiv co c1 caddc wn cmul wceq cc zcn peano2cn syl 2cnd cc0 cap
    wbr divcanap2d a1i oveq1d eqtr4d wne expcom necon2bd syl5com zeo orcomd ord
    2ap0 zneo impbid ) ABCZADEFZBCZAGHFZDEFZBCZIZUNDURJFZDUOJFZGHFZKUPUTUNVAUQV
    CUNUQDUNALCUQLCAMZANOUNPZDQRSUNUKUAZTUNVBAGHUNADVDVEVFTUBUCUPUSVAVCUSUPVAVC
    UDURUOULUEUFUGUNUSUPUNUPUSAUHUIUJUM $.

  ${
    $d x A $.  $d x B $.
    $( Second Peano postulate for upper integers.  (Contributed by NM,
       3-Oct-2004.) $)
    peano2uz2 $p |- ( ( A e. ZZ /\ B e. { x e. ZZ | A <_ x } ) ->
                             ( B + 1 ) e. { x e. ZZ | A <_ x } ) $=
      ( cz wcel cle wbr wa c1 caddc co cv crab peano2z ad2antrl cr wi zre breq2
      elrab lep1 adantl peano2re ancli letr 3expb sylan2 mpan2d syl2an impr jca
      anbi2i 3imtr4i ) BDEZCDEZBCFGZHZHZCIJKZDEZBUSFGZHUNCBALZFGZADMZEZHUSVDEUR
      UTVAUOUTUNUPCNOUNUOUPVAUNBPEZCPEZUPVAQUOBRCRVFVGHUPCUSFGZVAVGVHVFCUAUBVGV
      FVGUSPEZHUPVHHVAQZVGVICUCUDVFVGVIVJBCUSUEUFUGUHUIUJUKVEUQUNVCUPACDVBCBFST
      ULVCVAAUSDVBUSBFSTUM $.
  $}

  ${
    $d k n x A $.  $d k n x N $.
    $( Peano's inductive postulate for upper integers.  (Contributed by NM,
       6-Jul-2005.)  (Revised by Mario Carneiro, 25-Jul-2013.) $)
    peano5uzti $p |- ( N e. ZZ -> ( ( N e. A /\ A. x e. A ( x + 1 ) e. A )
                          -> { k e. ZZ | N <_ k } C_ A ) ) $=
      ( vn cz wcel cv c1 caddc co wa cc wceq cn wi oveq1 eleq1d imbi2d ex breq2
      wral cle wbr crab wss elrab anbi2i cmin ad2antrl 1cnd subcld npcan syl2an
      zcn ax-1cn subsub mp3an3 cn0 znn0sub biimpa anasss ancoms adantll nn0p1nn
      eqeltrd simpr simpll adantr pncan3d simprl rspccv ad2antll nncnd ad2antlr
      syl add32d sylibd a2d nnind syl3c eqeltrrd sylanb expcom expdimp ssrdv )
      DFGZDBGZAHZIJKZBGZABUBZLZDCHZUCUDZCFUEZBUFWGWMLZEWPBWGWMEHZWPGZWRBGZWMWSL
      ZWGWTXAWMWRFGZDWRUCUDZLZLZWGWTWSXDWMWOXCCWRFWNWRDUCUAUGUHXEWGLZWRDIUIKZUI
      KZXGJKZWRBXEWRMGZXGMGZXIWRNWGXBXJWMXCWRUOUJZWGDIDUOZWGUKULZWRXGUMUNXFXHOG
      WGWMXIBGZXFXHWRDUIKZIJKZOXEXJDMGZXHXQNZWGXLXMXJXRIMGXSUPWRDIUQURUNXFXPUSG
      ZXQOGXDWGXTWMWGXDXTWGXBXCXTWGXBLXCXTDWRUTVAVBVCVDXPVEVPVFXEWGVGWMXDWGVHWG
      WMWNXGJKZBGZPZPWGWMIXGJKZBGZPZPWGWMWRXGJKZBGZPZPWGWMWRIJKZXGJKZBGZPZPWGWM
      XOPZPCEXHWNINZYCYFWGYOYBYEWMYOYAYDBWNIXGJQRSSWNWRNZYCYIWGYPYBYHWMYPYAYGBW
      NWRXGJQRSSWNYJNZYCYMWGYQYBYLWMYQYAYKBWNYJXGJQRSSWNXHNZYCYNWGYRYBXOWMYRYAX
      IBWNXHXGJQRSSWGWMYEWQYDDBWQIDWQUKWGXRWMXMVIVJWGWHWLVKVFTWROGZWGYIYMYSWGYI
      YMPYSWGLZWMYHYLYTWMYHYLPYTWMLZYHYGIJKZBGZYLWLYHUUCPYTWHWKUUCAYGBWIYGNWJUU
      BBWIYGIJQRVLVMUUAUUBYKBUUAWRXGIUUAWRYSWGWMVHVNWGXKYSWMXNVOUUAUKVQRVRTVSTV
      SVTWAWBWCWDWEWFT $.

    ${
      peano5uz.1 $e |- N e. ZZ $.
      $( Peano's inductive postulate for upper integers.  (Contributed by NM,
         6-Jul-2005.)  (Revised by Mario Carneiro, 3-May-2014.) $)
      peano5uzi $p |- ( ( N e. A /\ A. x e. A ( x + 1 ) e. A ) ->
                    { k e. ZZ | N <_ k } C_ A ) $=
        ( cz wcel cv c1 caddc co wral wa cle wbr crab wss wi peano5uzti ax-mp )
        DFGDBGAHIJKBGABLMDCHNOCFPBQREABCDST $.
    $}
  $}

  ${
    $d x y z N $.
    dfuz.1 $e |- N e. ZZ $.
    $( An expression for the upper integers that start at ` N ` that is
       analogous to ~ dfnn2 for positive integers.  (Contributed by NM,
       6-Jul-2005.)  (Proof shortened by Mario Carneiro, 3-May-2014.) $)
    dfuzi $p |- { z e. ZZ | N <_ z } =
               |^| { x | ( N e. x /\ A. y e. x ( y + 1 ) e. x ) } $=
      ( cv cle wbr cz crab wcel c1 caddc co wral wa cab wss mpbir2an eleq2 cint
      ssintab peano5uzi mpgbir zrei leidi breq2 elrab peano2uz2 mpan rgen rabex
      wi zex wceq raleqbi1dv anbi12d elab intss1 ax-mp eqssi ) DCFZGHZCIJZDAFZK
      ZBFZLMNZVEKZBVEOZPZAQZUAZVDVMRVKVDVERUMAVKAVDUBBVECDEUCUDVDVLKZVMVDRVNDVD
      KZVHVDKZBVDOZVODIKZDDGHZEDDEUEUFVCVSCDIVBDDGUGUHSVPBVDVRVGVDKVPECDVGUIUJU
      KVKVOVQPAVDVCCIUNULVEVDUOVFVOVJVQVEVDDTVIVPBVEVDVEVDVHTUPUQURSVDVLUSUTVA
      $.
  $}

  ${
    $d j w N $.  $d j ps $.  $d j ch $.  $d j th $.  $d j ta $.  $d k w ph $.
    $d j k w M $.
    $( Substitutions. $)
    uzind.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
    uzind.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
    uzind.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
    uzind.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
    $( Basis. $)
    uzind.5 $e |- ( M e. ZZ -> ps ) $.
    $( Induction step. $)
    uzind.6 $e |- ( ( M e. ZZ /\ k e. ZZ /\ M <_ k ) -> ( ch -> th ) ) $.
    $( Induction on the upper integers that start at ` M ` .  The first four
       hypotheses give us the substitution instances we need; the last two are
       the basis and the induction step.  (Contributed by NM, 5-Jul-2005.) $)
    uzind $p |- ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> ta ) $=
      ( cz wcel cle wbr wa vw w3a cv crab c1 caddc wral wss zre leidd jca ancli
      co wceq breq2 anbi12d elrab sylibr wi peano2z a1i adantrd clt ltp1 adantl
      cr peano2re lelttr 3expb sylan2 mpan2d ltle syld syl2an expimpd 3exp jcad
      imp4d 3imtr4g ralrimiv peano5uzti mp2and sseld 3imtr3g 3impib simprd ) HP
      QZIPQZHIRSZUBZWIEWJWHWIETZWGWHWIWHWKTZWGIHUAUCZRSZUAPUDZQIHFUCZRSZATZFPUD
      ZQWHWITWLWGWOWSIWGHWSQZGUCZUEUFUMZWSQZGWSUGWOWSUHWGWGHHRSZBTZTWTWGXEWGXDB
      WGHHUIZUJNUKULWRXEFHPWPHUNWQXDABWPHHRUOJUPUQURWGXCGWSWGXAPQZHXARSZCTZTZXB
      PQZHXBRSZDTZTXAWSQXCWGXJXKXMWGXGXKXIXGXKUSWGXAUTVAVBWGXJXLDWGXGXIXLWGXGTX
      HXLCWGHVFQZXAVFQZXHXLUSXGXFXAUIXNXOTZXHHXBVCSZXLXPXHXAXBVCSZXQXOXRXNXAVDV
      EXOXNXOXBVFQZTXHXRTXQUSZXOXSXAVGZULXNXOXSXTHXAXBVHVIVJVKXOXNXSXQXLUSYAHXB
      VLVJVMVNVBVOWGXGXHCDWGXGXHCDUSOVPVRVQVQWRXIFXAPWPXAUNWQXHACWPXAHRUOKUPUQW
      RXMFXBPWPXBUNWQXLADWPXBHRUOLUPUQVSVTGWSUAHWAWBWCWNWIUAIPWMIHRUOUQWRWKFIPW
      PIUNWQWIAEWPIHRUOMUPUQWDWEWFWF $.
  $}

  ${
    $d j N $.  $d j ps $.  $d j ch $.  $d j th $.  $d j ta $.  $d k ph $.
    $d j k M $.
    $( Substitutions. $)
    uzind2.1 $e |- ( j = ( M + 1 ) -> ( ph <-> ps ) ) $.
    uzind2.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
    uzind2.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
    uzind2.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
    $( Basis. $)
    uzind2.5 $e |- ( M e. ZZ -> ps ) $.
    $( Induction step. $)
    uzind2.6 $e |- ( ( M e. ZZ /\ k e. ZZ /\ M < k ) -> ( ch -> th ) ) $.
    $( Induction on the upper integers that start _after_ an integer ` M ` .
       The first four hypotheses give us the substitution instances we need;
       the last two are the basis and the induction step.  (Contributed by NM,
       25-Jul-2005.) $)
    uzind2 $p |- ( ( M e. ZZ /\ N e. ZZ /\ M < N ) -> ta ) $=
      ( cz wcel wbr wi wceq clt wa c1 caddc co zltp1le peano2z cv imbi2d 3expia
      cle a1i w3a sylbird ex com3l imp 3adant1 a2d uzind 3exp syl com34 pm2.43a
      sylbid 3impia ) HPQZIPQZHIUARZEVGVHUBVIHUCUDUEZIUKRZEHIUFVGVHVKESZVHVGVLV
      GVHVKVGEVGVJPQZVHVKVGESZSSHUGVMVHVKVNVGASVGBSZVGCSVGDSVNFGVJIFUHZVJTABVGJ
      UIVPGUHZTACVGKUIVPVQUCUDUETADVGLUIVPITAEVGMUIVOVMNULVMVQPQZVJVQUKRZUMVGCD
      VRVSVGCDSZSZVMVRVSWAVGVRVSVTVGVRVSVTSVGVRUBVSHVQUARZVTHVQUFVGVRWBVTOUJUNU
      OUPUQURUSUTVAVBVCVDUQVEVF $.
  $}

  ${
    $d j k N $.  $d j ps $.  $d j ch $.  $d j th $.  $d j ta $.  $d m ph $.
    $d j m k M $.
    $( Substitutions. $)
    uzind3.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
    uzind3.2 $e |- ( j = m -> ( ph <-> ch ) ) $.
    uzind3.3 $e |- ( j = ( m + 1 ) -> ( ph <-> th ) ) $.
    uzind3.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
    $( Basis. $)
    uzind3.5 $e |- ( M e. ZZ -> ps ) $.
    $( Induction step. $)
    uzind3.6 $e |- ( ( M e. ZZ /\ m e. { k e. ZZ | M <_ k } ) ->
                   ( ch -> th ) ) $.
    $( Induction on the upper integers that start at an integer ` M ` .  The
       first four hypotheses give us the substitution instances we need, and
       the last two are the basis and the induction step.  (Contributed by NM,
       26-Jul-2005.) $)
    uzind3 $p |- ( ( M e. ZZ /\ N e. { k e. ZZ | M <_ k } ) -> ta ) $=
      ( cle wbr cz wcel cv wa breq2 elrab wi sylan2br 3impb uzind 3expb sylan2b
      crab ) JIGUAZQRZGSUKZTISTZJSTZIJQRZUBEUMUQGJSULJIQUCUDUOUPUQEABCDEFHIJKLM
      NOUOHUAZSTZIURQRZCDUEZUSUTUBUOURUNTVAUMUTGURSULURIQUCUDPUFUGUHUIUJ $.
  $}

  ${
    $d x y $.  $d A x $.  $d ps x $.  $d ch x $.  $d th x $.  $d ta x $.
    $d ph y $.
    nn0ind.1 $e |- ( x = 0 -> ( ph <-> ps ) ) $.
    nn0ind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    nn0ind.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
    nn0ind.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    nn0ind.5 $e |- ps $.
    nn0ind.6 $e |- ( y e. NN0 -> ( ch -> th ) ) $.
    $( Principle of Mathematical Induction (inference schema) on nonnegative
       integers.  The first four hypotheses give us the substitution instances
       we need; the last two are the basis and the induction step.
       (Contributed by NM, 13-May-2004.) $)
    nn0ind $p |- ( A e. NN0 -> ta ) $=
      ( cn0 wcel cz cc0 cle wbr wa elnn0z 0z a1i cv sylbir 3adant1 uzind mp3an1
      wi sylbi ) HOPHQPZRHSTZUAEHUBRQPZULUMEUCABCDEFGRHIJKLBUNMUDGUEZQPZRUOSTZC
      DUJZUNUPUQUAUOOPURUOUBNUFUGUHUIUK $.
  $}

  ${
    $d K x $.  $d M x y $.  $d N x y $.  $d ch x $.  $d ph y $.  $d ps x $.
    $d ta x $.  $d th x $.
    fzind.1 $e |- ( x = M -> ( ph <-> ps ) ) $.
    fzind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    fzind.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
    fzind.4 $e |- ( x = K -> ( ph <-> ta ) ) $.
    fzind.5 $e |- ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> ps ) $.
    fzind.6 $e |- ( ( ( M e. ZZ /\ N e. ZZ ) /\
                      ( y e. ZZ /\ M <_ y /\ y < N ) ) -> ( ch -> th ) ) $.
    $( Induction on the integers from ` M ` to ` N ` inclusive .  The first
       four hypotheses give us the substitution instances we need; the last two
       are the basis and the induction step.  (Contributed by Paul Chapman,
       31-Mar-2011.) $)
    fzind $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\
                    ( K e. ZZ /\ M <_ K /\ K <_ N ) ) -> ta ) $=
      ( cle wbr wa wi cz wcel w3a cv c1 caddc co breq1 anbi2d imbi12d 3expib cr
      wceq zre p1le 3expia syl2an imdistanda imim1d 3ad2ant2 wb zltp1le adantlr
      clt expcom pm5.32d adantl 3expa com12 sylbird com23 expd 3impib impd syld
      ex a2d uzind expcomd 3expb 3impia impcom ) HUAUBZIHQRZHJQRZUCZIUAUBZJUAUB
      ZSZEWFWGWHEWCWDWEWGWHETZTWCWDSZWGWEWJWGWKWEWJTZWGWCWDWLWGWCWDUCWHWEEWHFUD
      ZJQRZSZATWHIJQRZSZBTWHGUDZJQRZSZCTZWHWRUEUFUGZJQRZSZDTZWHWESZETFGIHWMIUMZ
      WOWQABXGWNWPWHWMIJQUHUIKUJWMWRUMZWOWTACXHWNWSWHWMWRJQUHUILUJWMXBUMZWOXDAD
      XIWNXCWHWMXBJQUHUIMUJWMHUMZWOXFAEXJWNWEWHWMHJQUHUINUJWGWHWPBOUKWGWRUAUBZI
      WRQRZUCZXAXDCTZXEXKWGXAXNTXLXKXDWTCXKWHXCWSXKWRULUBZJULUBZXCWSTWHWRUNJUNX
      OXPXCWSWRJUOUPUQURUSUTXMXDCDXMWHXCCDTZXMXCWHXQWGXKXLXCWHXQTZTWGXKXLSZXCXR
      WGWHXSXCSZXQWGWHXTXQTWIXTXSWRJVDRZSZXQWHYBXTVAWGWHXSYAXCXSWHYAXCVAZXKWHYC
      XLWRJVBVCVEVFVGYBWIXQXKXLYAWIXQTWIXKXLYAUCXQPVEVHVIVJVPVKVLVMVKVNVQVOVRVS
      VTVEVKWAVNWB $.
  $}

  ${
    $d K x $.  $d N x y $.  $d ch x $.  $d ph y $.  $d ps x $.  $d ta x $.
    $d th x $.
    fnn0ind.1 $e |- ( x = 0 -> ( ph <-> ps ) ) $.
    fnn0ind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    fnn0ind.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
    fnn0ind.4 $e |- ( x = K -> ( ph <-> ta ) ) $.
    fnn0ind.5 $e |- ( N e. NN0 -> ps ) $.
    fnn0ind.6 $e |- ( ( N e. NN0 /\ y e. NN0 /\ y < N ) -> ( ch -> th ) ) $.
    $( Induction on the integers from ` 0 ` to ` N ` inclusive .  The first
       four hypotheses give us the substitution instances we need; the last two
       are the basis and the induction step.  (Contributed by Paul Chapman,
       31-Mar-2011.) $)
    fnn0ind $p |- ( ( N e. NN0 /\ K e. NN0 /\ K <_ N ) -> ta ) $=
      ( wcel wbr wa cc0 wi cn0 cle cz elnn0z w3a nn0z 0z sylbir 3adant1 clt zre
      cv cr 0re lelttr ltle 3adant2 mp3an1 syl2an ex com23 3impib impcom anbi1i
      syld 3expb syl2anbr expcom 3impa expd mpd adantll fzind mpanl1 syl5 3expa
      sylanb 3impb ) IUAPZHUAPZHIUBQZEVTWARVSEVTHUCPZSHUBQZRWAVSETZHUDWBWCWAWDV
      SIUCPZWBWCWAUEZEIUFWEWFESUCPZWEWFEUGABCDEFGHSIJKLMWESIUBQZBWGWEWHRZVSBIUD
      ZNUHUIWEGULZUCPZSWKUBQZWKIUJQZUEZCDTZWGWEWORWHWPWOWEWHWLWMWNWEWHTWLWEWMWN
      RZWHWLWEWQWHTZWLWKUMPZIUMPZWRWEWKUKIUKSUMPZWSWTWRUNXAWSWTUEWQSIUJQZWHSWKI
      UOXAWTXBWHTWSSIUPUQVEURUSUTVAVBVCWOWEWHWPTWOWEWHWPWLWMWNWIWPTWIWLWMRZWNRZ
      WPWIVSWKUAPZWNRWPXDWJXEXCWNWKUDVDVSXEWNWPOVFVGVHVIVJVCVKVLVMVNVHVOVPVQVCV
      R $.
  $}

  ${
    $d x y z $.  $d A x z $.  $d ps x z $.  $d ch x z $.  $d th x z $.
    $d ta x z $.  $d ph y z $.
    nn0ind-raph.1 $e |- ( x = 0 -> ( ph <-> ps ) ) $.
    nn0ind-raph.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    nn0ind-raph.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
    nn0ind-raph.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    nn0ind-raph.5 $e |- ps $.
    nn0ind-raph.6 $e |- ( y e. NN0 -> ( ch -> th ) ) $.
    $( Principle of Mathematical Induction (inference schema) on nonnegative
       integers.  The first four hypotheses give us the substitution instances
       we need; the last two are the basis and the induction step.  Raph Levien
       remarks:  "This seems a bit painful.  I wonder if an explicit
       substitution version would be easier."  (Contributed by Raph Levien,
       10-Apr-2004.) $)
    nn0ind-raph $p |- ( A e. NN0 -> ta ) $=
      ( vz wcel cc0 wceq c1 wi cn0 cn wo elnn0 wsb wsbc dfsbcq2 cv sbhypf caddc
      nfv co nfsbc1v 1ex c0ex wa eleq1a ax-mp mpbiri wb eqeq2 biimtrrdi pm5.74d
      0nn0 mpbii com12 vtocle sylc adantr oveq1 0p1e1 eqtrdi eqeq2d imp sbceq1a
      mpbird mpbid vtoclef nnnn0 syl nnind eqeq1 bicomd sylan9bb sylbird eqcoms
      ex jaoi sylbi ) HUAPHUBPZHQRZUCEHUDWJEWKAFOUEAFSUFZCDEOGHAFOSUGACFOGUHZCF
      UKJUIADFOWMSUJULZDFUKKUIAEFOHEFUKLUIWLFSAFSUMUNFUHZSRZAWLWPATGQUOWMQRZWPA
      WQWPUPADWQDWPWQWMUAPZCDQUAPWQWRTVDQUAWMUQURWQCTFQUOWQWOQRZCWQWSATWSCTWSAB
      MIUSWQWSACWQWSWOWMRACUTWMQWOVAJVBVCVEVFVGNVHVIWQWPADUTZWQWPWOWNRWTWQWNSWO
      WQWNQSUJULSWMQSUJVJVKVLVMKVBVNVPWGVGAFSVOVQVRWMUBPWRCDTWMVSNVTWAEQHQHRZET
      ZFQXBFUKUOWSXAWOHRZEWOQHWBWSXCEWSXCUPBEMWSBAXCEWSABIWCLWDVEWGWEVRWFWHWI
      $.
  $}

  ${
    $d A x $.  $d ch x $.  $d et x $.  $d ph y $.  $d ps x $.  $d ta x $.
    $d th x $.  $d x y ze $.
    zindd.1 $e |- ( x = 0 -> ( ph <-> ps ) ) $.
    zindd.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    zindd.3 $e |- ( x = ( y + 1 ) -> ( ph <-> ta ) ) $.
    zindd.4 $e |- ( x = -u y -> ( ph <-> th ) ) $.
    zindd.5 $e |- ( x = A -> ( ph <-> et ) ) $.
    zindd.6 $e |- ( ze -> ps ) $.
    zindd.7 $e |- ( ze -> ( y e. NN0 -> ( ch -> ta ) ) ) $.
    zindd.8 $e |- ( ze -> ( y e. NN -> ( ch -> th ) ) ) $.
    $( Principle of Mathematical Induction on all integers, deduction version.
       The first five hypotheses give the substitutions; the last three are the
       basis, the induction, and the extension to negative numbers.
       (Contributed by Paul Chapman, 17-Apr-2009.)  (Proof shortened by Mario
       Carneiro, 4-Jan-2017.) $)
    zindd $p |- ( ze -> ( A e. ZZ -> et ) ) $=
      ( cz wcel wral wi cv cneg cn0 cn wo cr znegcl elznn0nn sylib simpr orim2i
      wa syl zcn negnegd eleq1d orbi2d mpbid cc0 wceq imbi2d c1 caddc com12 a2d
      co nn0ind nnnn0 mpdd jaod syl5 ralrimiv wb sylan9eqr eqcomd bicomd rspcdv
      negeq rspccv 3syl ) GDISUAZAHSUAJSTFUBGDISIUCZSTZWDUDZUETZWDUFTZUGZGDWEWG
      WFUDZUFTZUGZWIWEWGWFUHTZWKUNZUGZWLWEWFSTWOWDUIWFUJUKWNWKWGWMWKULUMUOWEWKW
      HWGWEWJWDUFWEWDWDUPUQURUSUTGWGDWHWGGDGAUBZGBUBZGCUBZGEUBZGDUBHIWFHUCZVAVB
      ABGKVCZWTWDVBACGLVCZWTWDVDVEVHVBAEGMVCZWTWFVBZADGNVCPWDUETZGCEGXECEUBQVFV
      GZVIVFGWHCDWHGCWHXEWRWDVJWPWQWRWSWRHIWDXAXBXCXBPXFVIUOVFRVKVLVMVNWCAHSWTS
      TZWCAXGDAIWTUDZSWTUIXGWDXHVBZUNZADXJXDADVOXJWFWTXIXGWFXHUDWTWDXHVTXGWTWTU
      PUQVPVQNUOVRVSVFVNAFHJSOWAWB $.
  $}

  ${
    $d x z A $.  $d y A $.
    $( Any real number can be sandwiched between two integers.  Exercise 2 of
       [Apostol] p. 28.  (Contributed by NM, 10-Nov-2004.) $)
    btwnz $p |- ( A e. RR -> ( E. x e. ZZ x < A /\ E. y e. ZZ A < y ) ) $=
      ( vz cr wcel cv clt wbr cz wrex cneg cn renegcl arch wa wb nnre ltnegcon1
      syl syl5 pm5.32d nnnegz breq1 rspcev sylan biimtrdi expd rexlimdv mpd nnz
      ex anim1i reximi2 jca ) CEFZAGZCHIZAJKZCBGZHIZBJKZUPCLZDGZHIZDMKZUSUPVCEF
      VFCNVCDOTUPVEUSDMUPVDMFZVEUSUPVGVEPVGVDLZCHIZPUSUPVGVEVIVGVDEFZUPVEVIQZVD
      RUPVJVKCVDSULUAUBVGVHJFVIUSVDUCURVIAVHJUQVHCHUDUEUFUGUHUIUJUPVABMKVBCBOVA
      VABMJUTMFUTJFVAUTUKUMUNTUO $.
  $}

  ${
    nn0zd.1 $e |- ( ph -> A e. NN0 ) $.
    $( A positive integer is an integer.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    nn0zd $p |- ( ph -> A e. ZZ ) $=
      ( cn0 cz nn0ssz sselid ) ADEBFCG $.
  $}

  ${
    nnzd.1 $e |- ( ph -> A e. NN ) $.
    $( A nonnegative integer is an integer.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    nnzd $p |- ( ph -> A e. ZZ ) $=
      ( nnnn0d nn0zd ) ABABCDE $.
  $}

  ${
    zred.1 $e |- ( ph -> A e. ZZ ) $.
    $( An integer is a real number.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    zred $p |- ( ph -> A e. RR ) $=
      ( cz cr zssre sselid ) ADEBFCG $.

    $( An integer is a complex number.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    zcnd $p |- ( ph -> A e. CC ) $=
      ( zred recnd ) ABABCDE $.

    $( Closure law for negative integers.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    znegcld $p |- ( ph -> -u A e. ZZ ) $=
      ( cz wcel cneg znegcl syl ) ABDEBFDECBGH $.

    $( Deduction from second Peano postulate generalized to integers.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    peano2zd $p |- ( ph -> ( A + 1 ) e. ZZ ) $=
      ( cz wcel c1 caddc co peano2z syl ) ABDEBFGHDECBIJ $.

    zaddcld.1 $e |- ( ph -> B e. ZZ ) $.
    $( Closure of addition of integers.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    zaddcld $p |- ( ph -> ( A + B ) e. ZZ ) $=
      ( cz wcel caddc co zaddcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Closure of subtraction of integers.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    zsubcld $p |- ( ph -> ( A - B ) e. ZZ ) $=
      ( cz wcel cmin co zsubcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Closure of multiplication of integers.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    zmulcld $p |- ( ph -> ( A x. B ) e. ZZ ) $=
      ( cz wcel cmul co zmulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
  $}

  $( Increasing an integer by 2 results in an integer.  (Contributed by
     Alexander van der Vekens, 16-Sep-2018.) $)
  zadd2cl $p |- ( N e. ZZ -> ( N + 2 ) e. ZZ ) $=
    ( cz wcel c2 id 2z a1i zaddcld ) ABCZADIEDBCIFGH $.

  ${
    btwnapz.a $e |- ( ph -> A e. ZZ ) $.
    btwnapz.b $e |- ( ph -> B e. RR ) $.
    btwnapz.c $e |- ( ph -> C e. ZZ ) $.
    btwnapz.ab $e |- ( ph -> A < B ) $.
    btwnapz.ba $e |- ( ph -> B < ( A + 1 ) ) $.
    $( A number between an integer and its successor is apart from any integer.
       (Contributed by Jim Kingdon, 6-Jan-2023.) $)
    btwnapz $p |- ( ph -> B =//= C ) $=
      ( cle wbr cap clt wa cr wcel zred adantr cz syl2anc simpr gtapd caddc syl
      lelttrd c1 co peano2re wb zltp1le biimpa ltletrd ltapd zlelttric mpjaodan
      wo ) ADBJKZCDLKBDMKZAUQNZDCADOPZUQADGQZRZACOPZUQFRZUSDBCVBABOPZUQABEQZRVD
      AUQUAABCMKUQHRUEUBAURNZCDAVCURFRZAUTURVARZVGCBUFUCUGZDVHAVJOPZURAVEVKVFBU
      HUDRVIACVJMKURIRAURVJDJKZABSPZDSPZURVLUIEGBDUJTUKULUMAVNVMUQURUPGEDBUNTUO
      $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Decimal arithmetic
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c ; $.
  $( Constant used for decimal constructor. $)
  cdc $a class ; A B $.

  $( Define the "decimal constructor", which is used to build up "decimal
     integers" or "numeric terms" in base ` 1 0 ` .  For example,
     ` ( ; ; ; 1 0 0 0 + ; ; ; 2 0 0 0 ) = ; ; ; 3 0 0 0 ` ~ 1kp2ke3k .
     (Contributed by Mario Carneiro, 17-Apr-2015.)  (Revised by AV,
     1-Aug-2021.) $)
  df-dec $a |- ; A B = ( ( ( 9 + 1 ) x. A ) + B ) $.

  $( 9 + 1 = 10.  (Contributed by Mario Carneiro, 18-Apr-2015.)  (Revised by
     Stanislas Polu, 7-Apr-2020.)  (Revised by AV, 1-Aug-2021.) $)
  9p1e10 $p |- ( 9 + 1 ) = ; 1 0 $=
    ( c1 cc0 cdc c9 caddc co cmul df-dec cn 9nn 1nn nnaddcl mp2an nncni mulridi
    wcel oveq1i addridi 3eqtrri ) ABCDAEFZAGFZBEFTBEFTABHUATBETTDIPAIPTIPJKDALM
    NZOQTUBRS $.

  $( Version of the definition of the "decimal constructor" using ` ; 1 0 `
     instead of the symbol 10.  Of course, this statement cannot be used as
     definition, because it uses the "decimal constructor".  (Contributed by
     AV, 1-Aug-2021.) $)
  dfdec10 $p |- ; A B = ( ( ; 1 0 x. A ) + B ) $=
    ( cdc c9 c1 caddc co cmul cc0 df-dec 9p1e10 oveq1i eqtri ) ABCDEFGZAHGZBFGE
    ICZAHGZBFGABJOQBFNPAHKLLM $.

  $( Equality theorem for the decimal constructor.  (Contributed by Mario
     Carneiro, 17-Apr-2015.)  (Revised by AV, 6-Sep-2021.) $)
  deceq1 $p |- ( A = B -> ; A C = ; B C ) $=
    ( wceq c9 c1 caddc co cmul cdc oveq2 oveq1d df-dec 3eqtr4g ) ABDZEFGHZAIHZC
    GHPBIHZCGHACJBCJOQRCGABPIKLACMBCMN $.

  $( Equality theorem for the decimal constructor.  (Contributed by Mario
     Carneiro, 17-Apr-2015.)  (Revised by AV, 6-Sep-2021.) $)
  deceq2 $p |- ( A = B -> ; C A = ; C B ) $=
    ( wceq c9 c1 caddc co cmul cdc oveq2 df-dec 3eqtr4g ) ABDEFGHCIHZAGHNBGHCAJ
    CBJABNGKCALCBLM $.

  ${
    deceq1i.1 $e |- A = B $.
    $( Equality theorem for the decimal constructor.  (Contributed by Mario
       Carneiro, 17-Apr-2015.) $)
    deceq1i $p |- ; A C = ; B C $=
      ( wceq cdc deceq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality theorem for the decimal constructor.  (Contributed by Mario
       Carneiro, 17-Apr-2015.) $)
    deceq2i $p |- ; C A = ; C B $=
      ( wceq cdc deceq2 ax-mp ) ABECAFCBFEDABCGH $.

    deceq12i.2 $e |- C = D $.
    $( Equality theorem for the decimal constructor.  (Contributed by Mario
       Carneiro, 17-Apr-2015.) $)
    deceq12i $p |- ; A C = ; B D $=
      ( cdc deceq1i deceq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
  $}

  ${
    numnncl.1 $e |- T e. NN0 $.
    numnncl.2 $e |- A e. NN0 $.
    ${
      numnncl.3 $e |- B e. NN $.
      $( Closure for a numeral (with units place).  (Contributed by Mario
         Carneiro, 18-Feb-2014.) $)
      numnncl $p |- ( ( T x. A ) + B ) e. NN $=
        ( cmul co cn0 wcel cn caddc nn0mulcli nn0nnaddcl mp2an ) CAGHZIJBKJPBLH
        KJCADEMFPBNO $.
    $}

    $( Add a zero in the units place.  (Contributed by Mario Carneiro,
       18-Feb-2014.) $)
    num0u $p |- ( T x. A ) = ( ( T x. A ) + 0 ) $=
      ( cmul co cc0 caddc nn0mulcli nn0cni addridi eqcomi ) BAEFZGHFMMMBACDIJKL
      $.

    $( Add a zero in the higher places.  (Contributed by Mario Carneiro,
       18-Feb-2014.) $)
    num0h $p |- A = ( ( T x. 0 ) + A ) $=
      ( cc0 cmul co caddc nn0cni mul01i oveq1i addlidi eqtr2i ) BEFGZAHGEAHGANE
      AHBBCIJKAADILM $.

    numcl.2 $e |- B e. NN0 $.
    $( Closure for a decimal integer (with units place).  (Contributed by Mario
       Carneiro, 18-Feb-2014.) $)
    numcl $p |- ( ( T x. A ) + B ) e. NN0 $=
      ( cmul co nn0mulcli nn0addcli ) CAGHBCADEIFJ $.

    numsuc.4 $e |- ( B + 1 ) = C $.
    numsuc.5 $e |- N = ( ( T x. A ) + B ) $.
    $( The successor of a decimal integer (no carry).  (Contributed by Mario
       Carneiro, 18-Feb-2014.) $)
    numsuc $p |- ( N + 1 ) = ( ( T x. A ) + C ) $=
      ( c1 caddc co cmul oveq1i nn0mulcli nn0cni ax-1cn addassi oveq2i 3eqtri )
      EKLMDANMZBLMZKLMUBBKLMZLMUBCLMEUCKLJOUBBKUBDAFGPQBHQRSUDCUBLITUA $.
  $}

  ${
    deccl.1 $e |- A e. NN0 $.
    deccl.2 $e |- B e. NN0 $.
    $( Closure for a numeral.  (Contributed by Mario Carneiro, 17-Apr-2015.)
       (Revised by AV, 6-Sep-2021.) $)
    deccl $p |- ; A B e. NN0 $=
      ( cdc c9 c1 caddc co cmul cn0 df-dec 9nn0 1nn0 nn0addcli numcl eqeltri )
      ABEFGHIZAJIBHIKABLABRFGMNOCDPQ $.
  $}

  $( 10 is a positive integer.  (Contributed by NM, 8-Nov-2012.)  (Revised by
     AV, 6-Sep-2021.) $)
  10nn $p |- ; 1 0 e. NN $=
    ( c9 c1 caddc co cc0 cdc cn 9p1e10 wcel 9nn peano2nn ax-mp eqeltrri ) ABCDZ
    BEFGHAGINGIJAKLM $.

  $( The number 10 is positive.  (Contributed by NM, 5-Feb-2007.)  (Revised by
     AV, 8-Sep-2021.) $)
  10pos $p |- 0 < ; 1 0 $=
    ( c1 cc0 cdc 10nn nngt0i ) ABCDE $.

  $( 10 is a nonnegative integer.  (Contributed by Mario Carneiro,
     19-Apr-2015.)  (Revised by AV, 6-Sep-2021.) $)
  10nn0 $p |- ; 1 0 e. NN0 $=
    ( c1 cc0 1nn0 0nn0 deccl ) ABCDE $.

  $( The number 10 is real.  (Contributed by NM, 5-Feb-2007.)  (Revised by AV,
     8-Sep-2021.) $)
  10re $p |- ; 1 0 e. RR $=
    ( c1 cc0 cdc 10nn nnrei ) ABCDE $.

  ${
    decnncl.1 $e |- A e. NN0 $.
    decnncl.2 $e |- B e. NN $.
    $( Closure for a numeral.  (Contributed by Mario Carneiro, 17-Apr-2015.)
       (Revised by AV, 6-Sep-2021.) $)
    decnncl $p |- ; A B e. NN $=
      ( cdc c1 cc0 cmul co caddc cn dfdec10 10nn0 numnncl eqeltri ) ABEFGEZAHIB
      JIKABLABPMCDNO $.
  $}

  ${
    dec0u.1 $e |- A e. NN0 $.
    $( Add a zero in the units place.  (Contributed by Mario Carneiro,
       17-Apr-2015.)  (Revised by AV, 6-Sep-2021.) $)
    dec0u $p |- ( ; 1 0 x. A ) = ; A 0 $=
      ( c1 cc0 cdc cmul co caddc 10nn0 num0u dfdec10 eqtr4i ) CDEZAFGZNDHGADEAM
      IBJADKL $.

    $( Add a zero in the higher places.  (Contributed by Mario Carneiro,
       17-Apr-2015.)  (Revised by AV, 6-Sep-2021.) $)
    dec0h $p |- A = ; 0 A $=
      ( c1 cc0 cdc cmul co caddc 10nn0 num0h dfdec10 eqtr4i ) ACDEZDFGAHGDAEAMI
      BJDAKL $.
  $}

  ${
    numnncl2.1 $e |- T e. NN $.
    numnncl2.2 $e |- A e. NN $.
    $( Closure for a decimal integer (zero units place).  (Contributed by Mario
       Carneiro, 9-Mar-2015.) $)
    numnncl2 $p |- ( ( T x. A ) + 0 ) e. NN $=
      ( cmul co cc0 caddc cn nnmulcli nncni addridi eqeltri ) BAEFZGHFNINNBACDJ
      ZKLOM $.
  $}

  ${
    decnncl2.1 $e |- A e. NN $.
    $( Closure for a decimal integer (zero units place).  (Contributed by Mario
       Carneiro, 17-Apr-2015.)  (Revised by AV, 6-Sep-2021.) $)
    decnncl2 $p |- ; A 0 e. NN $=
      ( cc0 cdc c1 cmul co caddc cn dfdec10 10nn numnncl2 eqeltri ) ACDECDZAFGC
      HGIACJANKBLM $.
  $}

  ${
    numlt.1 $e |- T e. NN $.
    numlt.2 $e |- A e. NN0 $.
    numlt.3 $e |- B e. NN0 $.
    ${
      numlt.4 $e |- C e. NN $.
      numlt.5 $e |- B < C $.
      $( Comparing two decimal integers (equal higher places).  (Contributed by
         Mario Carneiro, 18-Feb-2014.) $)
      numlt $p |- ( ( T x. A ) + B ) < ( ( T x. A ) + C ) $=
        ( clt wbr cmul co caddc nn0rei nnrei nnnn0i nn0mulcli ltadd2i mpbi ) BC
        JKDALMZBNMUACNMJKIBCUABGOCHPUADADEQFROST $.
    $}

    numltc.3 $e |- C e. NN0 $.
    numltc.4 $e |- D e. NN0 $.
    numltc.5 $e |- C < T $.
    numltc.6 $e |- A < B $.
    $( Comparing two decimal integers (unequal higher places).  (Contributed by
       Mario Carneiro, 18-Feb-2014.) $)
    numltc $p |- ( ( T x. A ) + C ) < ( ( T x. B ) + D ) $=
      ( cmul co caddc clt wbr cle nn0rei wcel numlt nnrei ax-1cn adddii mulridi
      c1 recni oveq2i eqtri breqtrri cn0 wb nn0ltp1le mp2an cc0 nngt0i peano2re
      mpbi cr ax-mp lemul2i remulcli readdcli ltletri nn0addge1i ) EAMNZCONZEBM
      NZPQZVHVHDONZRQVGVJPQVGEAUFONZMNZPQVLVHRQZVIVGVFEONZVLPACEEFGIFKUAVLVFEUF
      MNZONVNEAUFEEFUBZUGZAAGSZUGUCUDVOEVFOEVQUEUHUIUJVKBRQZVMABPQZVSLAUKTBUKTV
      TVSULGHABUMUNURUOEPQVSVMULEFUPVKBEAUSTVKUSTVRAUQUTZBHSZVPVAUTURVGVLVHVFCE
      AVPVRVBCISVCZEVKVPWAVBEBVPWBVBZVDUNVHDWDJVEVGVHVJWCWDVHDWDDJSVCVDUN $.
  $}

  ${
    le9lt10.c $e |- A e. NN0 $.
    le9lt10.e $e |- A <_ 9 $.
    $( A "decimal digit" (i.e. a nonnegative integer less than or equal to 9)
       is less then 10.  (Contributed by AV, 8-Sep-2021.) $)
    le9lt10 $p |- A < ; 1 0 $=
      ( c9 c1 caddc co cc0 cdc clt cle cz wcel wb nn0zi 9nn0 zleltp1 mp2an mpbi
      wbr 9p1e10 breqtri ) ADEFGZEHIJADKTZAUCJTZCALMDLMUDUENABODPOADQRSUAUB $.
  $}

  ${
    declt.a $e |- A e. NN0 $.
    declt.b $e |- B e. NN0 $.
    ${
      declt.c $e |- C e. NN $.
      declt.l $e |- B < C $.
      $( Comparing two decimal integers (equal higher places).  (Contributed by
         Mario Carneiro, 17-Apr-2015.)  (Revised by AV, 6-Sep-2021.) $)
      declt $p |- ; A B < ; A C $=
        ( c1 cc0 cdc cmul co caddc clt 10nn numlt dfdec10 3brtr4i ) HIJZAKLZBML
        TCMLABJACJNABCSODEFGPABQACQR $.
    $}

    ${
      decltc.c $e |- C e. NN0 $.
      decltc.d $e |- D e. NN0 $.
      decltc.s $e |- C < ; 1 0 $.
      decltc.l $e |- A < B $.
      $( Comparing two decimal integers (unequal higher places).  (Contributed
         by Mario Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      decltc $p |- ; A C < ; B D $=
        ( c1 cc0 cdc cmul co caddc clt 10nn numltc dfdec10 3brtr4i ) KLMZANOCPO
        UBBNODPOACMBDMQABCDUBREFGHIJSACTBDTUA $.
    $}

    ${
      declth.c $e |- C e. NN0 $.
      declth.d $e |- D e. NN0 $.
      declth.e $e |- C <_ 9 $.
      declth.l $e |- A < B $.
      $( Comparing two decimal integers (unequal higher places).  (Contributed
         by AV, 8-Sep-2021.) $)
      declth $p |- ; A C < ; B D $=
        ( le9lt10 decltc ) ABCDEFGHCGIKJL $.
    $}

    decsuc.c $e |- ( B + 1 ) = C $.
    decsuc.n $e |- N = ; A B $.
    $( The successor of a decimal integer (no carry).  (Contributed by Mario
       Carneiro, 17-Apr-2015.)  (Revised by AV, 6-Sep-2021.) $)
    decsuc $p |- ( N + 1 ) = ; A C $=
      ( c1 caddc co cc0 cdc cmul 10nn0 dfdec10 eqtri numsuc eqtr4i ) DIJKILMZAN
      KZCJKACMABCTDOEFGDABMUABJKHABPQRACPS $.
  $}

  ${
    3decltc.a $e |- A e. NN0 $.
    3decltc.b $e |- B e. NN0 $.
    3decltc.c $e |- C e. NN0 $.
    3decltc.d $e |- D e. NN0 $.
    3decltc.e $e |- E e. NN0 $.
    3decltc.f $e |- F e. NN0 $.
    3decltc.3 $e |- A < B $.
    ${
      3declth.1 $e |- C <_ 9 $.
      3declth.2 $e |- E <_ 9 $.
      $( Comparing two decimal integers with three "digits" (unequal higher
         places).  (Contributed by AV, 8-Sep-2021.) $)
      3declth $p |- ; ; A C E < ; ; B D F $=
        ( cdc deccl declth ) ACPBDPEFACGIQBDHJQKLOABCDGHIJNMRR $.
    $}

    3decltc.1 $e |- C < ; 1 0 $.
    3decltc.2 $e |- E < ; 1 0 $.
    $( Comparing two decimal integers with three "digits" (unequal higher
       places).  (Contributed by AV, 15-Jun-2021.)  (Revised by AV,
       6-Sep-2021.) $)
    3decltc $p |- ; ; A C E < ; ; B D F $=
      ( cdc deccl decltc ) ACPBDPEFACGIQBDHJQKLOABCDGHIJNMRR $.
  $}

  ${
    decle.1 $e |- A e. NN0 $.
    decle.2 $e |- B e. NN0 $.
    decle.3 $e |- C e. NN0 $.
    ${
      decle.4 $e |- B <_ C $.
      $( Comparing two decimal integers (equal higher places).  (Contributed by
         AV, 17-Aug-2021.)  (Revised by AV, 8-Sep-2021.) $)
      decle $p |- ; A B <_ ; A C $=
        ( c1 cc0 cdc cmul caddc cle wbr nn0rei 10nn0 nn0mulcli leadd2i dfdec10
        co mpbi 3brtr4i ) HIJZAKTZBLTZUDCLTZABJACJMBCMNUEUFMNGBCUDBEOCFOUDUCAPD
        QORUAABSACSUB $.
    $}

    decleh.4 $e |- D e. NN0 $.
    decleh.5 $e |- C <_ 9 $.
    decleh.6 $e |- A < B $.
    $( Comparing two decimal integers (unequal higher places).  (Contributed by
       AV, 17-Aug-2021.)  (Revised by AV, 8-Sep-2021.) $)
    decleh $p |- ; A C <_ ; B D $=
      ( cdc deccl nn0rei declth ltleii ) ACKZBDKZPACEGLMQBDFHLMABCDEFGHIJNO $.
  $}

  ${
    declei.1 $e |- A e. NN $.
    declei.2 $e |- B e. NN0 $.
    declei.3 $e |- C e. NN0 $.
    declei.4 $e |- C <_ 9 $.
    $( Comparing a digit to a decimal integer.  (Contributed by AV,
       17-Aug-2021.) $)
    declei $p |- C <_ ; A B $=
      ( cc0 cdc cle dec0h 0nn0 nnnn0i nngt0i decleh eqbrtri ) CHCIABIJCFKHACBLA
      DMFEGADNOP $.
  $}

  ${
    numlti.1 $e |- T e. NN $.
    numlti.2 $e |- A e. NN $.
    numlti.3 $e |- B e. NN0 $.
    numlti.4 $e |- C e. NN0 $.
    numlti.5 $e |- C < T $.
    $( Comparing a digit to a decimal integer.  (Contributed by Mario Carneiro,
       18-Feb-2014.) $)
    numlti $p |- C < ( ( T x. A ) + B ) $=
      ( cc0 cmul co caddc clt nnnn0i num0h 0nn0 nngt0i numltc eqbrtri ) CDJKLCM
      LDAKLBMLNCDDEOHPJACBDEQAFOHGIAFRST $.
  $}

  ${
    declti.a $e |- A e. NN $.
    declti.b $e |- B e. NN0 $.
    declti.c $e |- C e. NN0 $.
    ${
      declti.l $e |- C < ; 1 0 $.
      $( Comparing a digit to a decimal integer.  (Contributed by Mario
         Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      declti $p |- C < ; A B $=
        ( c1 cc0 cdc cmul co caddc clt 10nn numlti dfdec10 breqtrri ) CHIJZAKLB
        MLABJNABCSODEFGPABQR $.
    $}

    decltdi.4 $e |- C <_ 9 $.
    $( Comparing a digit to a decimal integer.  (Contributed by AV,
       8-Sep-2021.) $)
    decltdi $p |- C < ; A B $=
      ( le9lt10 declti ) ABCDEFCFGHI $.
  $}

  ${
    numsucc.1 $e |- Y e. NN0 $.
    numsucc.2 $e |- T = ( Y + 1 ) $.
    numsucc.3 $e |- A e. NN0 $.
    numsucc.4 $e |- ( A + 1 ) = B $.
    numsucc.5 $e |- N = ( ( T x. A ) + Y ) $.
    $( The successor of a decimal integer (with carry).  (Contributed by Mario
       Carneiro, 18-Feb-2014.) $)
    numsucc $p |- ( N + 1 ) = ( ( T x. B ) + 0 ) $=
      ( c1 caddc co cmul cc0 cn0 1nn0 nn0addcli nn0cni oveq2i 3eqtr4ri eqeltrri
      eqeltri mulridi ax-1cn adddii eqcomi numsuc num0u 3eqtri ) DKLMZCAKLMZNMZ
      CBNMZUNOLMCANMZCKNMZLMUOCLMUMUKUPCUOLCCCEKLMZPGEKFQRUCZSZUDTCAKUSAHSUEUFA
      ECCDURHFCUQGUGJUHUAULBCNITBCURULBPIAKHQRUBUIUJ $.
  $}

  ${
    decsucc.1 $e |- A e. NN0 $.
    decsucc.2 $e |- ( A + 1 ) = B $.
    decsucc.3 $e |- N = ; A 9 $.
    $( The successor of a decimal integer (with carry).  (Contributed by Mario
       Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
    decsucc $p |- ( N + 1 ) = ; B 0 $=
      ( c1 caddc co cc0 cmul c9 9nn0 9p1e10 eqcomi dfdec10 eqtri numsucc eqtr4i
      cdc ) CGHIGJTZBKIJHIBJTABUACLMLGHIUANODECALTUAAKILHIFALPQRBJPS $.
  $}

  $( The successor of zero.  (Contributed by Mario Carneiro, 18-Feb-2014.) $)
  1e0p1 $p |- 1 = ( 0 + 1 ) $=
    ( cc0 c1 caddc co 0p1e1 eqcomi ) ABCDBEF $.

  $( Ten plus an integer.  (Contributed by Mario Carneiro, 19-Apr-2015.)
     (Revised by AV, 6-Sep-2021.) $)
  dec10p $p |- ( ; 1 0 + A ) = ; 1 A $=
    ( c1 cdc cc0 cmul co caddc dfdec10 10nn nncni mulridi oveq1i eqtr2i ) BACBD
    CZBEFZAGFNAGFBAHONAGNNIJKLM $.

  ${
    numma.1 $e |- T e. NN0 $.
    numma.2 $e |- A e. NN0 $.
    numma.3 $e |- B e. NN0 $.
    numma.4 $e |- C e. NN0 $.
    numma.5 $e |- D e. NN0 $.
    numma.6 $e |- M = ( ( T x. A ) + B ) $.
    numma.7 $e |- N = ( ( T x. C ) + D ) $.
    ${
      numma.8 $e |- P e. NN0 $.
      numma.9 $e |- ( ( A x. P ) + C ) = E $.
      numma.10 $e |- ( ( B x. P ) + D ) = F $.
      $( Perform a multiply-add of two decimal integers ` M ` and ` N ` against
         a fixed multiplicand ` P ` (no carry).  (Contributed by Mario
         Carneiro, 18-Feb-2014.) $)
      numma $p |- ( ( M x. P ) + N ) = ( ( T x. E ) + F ) $=
        ( cmul caddc oveq1i oveq12i nn0cni mulcli adddii mulassi eqtr4i adddiri
        co add4i oveq2i 3eqtr2i ) IEUAUKZJUBUKFAUAUKZBUBUKZEUAUKZFCUAUKZDUBUKZU
        BUKZFAEUAUKZCUBUKZUAUKZBEUAUKZDUBUKZUBUKZFGUAUKZHUBUKUOURJUTUBIUQEUAPUC
        QUDVGUPEUAUKZUSUBUKZVFUBUKZVAVDVJVFUBVDFVBUAUKZUSUBUKVJFVBCFKUEZAEALUEZ
        ERUEZUFCNUEZUGVIVLUSUBFAEVMVNVOUHUCUIUCVAVIVEUBUKZUTUBUKVKURVQUTUBUPBEF
        AVMVNUFZBMUEZVOUJUCVIUSVEDUPEVRVOUFFCVMVPUFBEVSVOUFDOUEULUIUIVDVHVFHUBV
        CGFUASUMTUDUN $.
    $}

    ${
      nummac.8 $e |- P e. NN0 $.
      nummac.9 $e |- F e. NN0 $.
      nummac.10 $e |- G e. NN0 $.
      nummac.11 $e |- ( ( A x. P ) + ( C + G ) ) = E $.
      nummac.12 $e |- ( ( B x. P ) + D ) = ( ( T x. G ) + F ) $.
      $( Perform a multiply-add of two decimal integers ` M ` and ` N ` against
         a fixed multiplicand ` P ` (with carry).  (Contributed by Mario
         Carneiro, 18-Feb-2014.) $)
      nummac $p |- ( ( M x. P ) + N ) = ( ( T x. E ) + F ) $=
        ( cmin co cmul caddc nn0cni mulcli addassi eqtri addcli eqeltrri subdii
        oveq1i wceq subadd2i mpbir eqcomi numma wcel npcan mp2an subcli 3eqtr4i
        cc eqtr3i ) FGIUDUEZUFUEZFIUFUEZHUGUEZUGUEFGUFUEZVJUDUEZVKUGUEZJEUFUEKU
        GUEVLHUGUEZVIVMVKUGFGIFLUHZAEUFUEZCUGUEZIUGUEZGVFVSVQCIUGUEUGUEGVQCIAEA
        MUHESUHUIZCOUHZIUAUHZUJUBUKZVRIVQCVTWAULZWBULUMZWBUNUOABCDEFVHVKJKLMNOP
        QRSVHVRVHVRUPVSGUPWCGIVRWEWBWDUQURUSUCUTVMVJUGUEZHUGUEVOVNWFVLHUGVLVFVA
        VJVFVAWFVLUPFGVPWEUIZFIVPWBUIZVLVJVBVCUOVMVJHVLVJWGWHVDWHHTUHUJVGVE $.
    $}

    ${
      numma2c.8 $e |- P e. NN0 $.
      numma2c.9 $e |- F e. NN0 $.
      numma2c.10 $e |- G e. NN0 $.
      numma2c.11 $e |- ( ( P x. A ) + ( C + G ) ) = E $.
      numma2c.12 $e |- ( ( P x. B ) + D ) = ( ( T x. G ) + F ) $.
      $( Perform a multiply-add of two decimal integers ` M ` and ` N ` against
         a fixed multiplicand ` P ` (with carry).  (Contributed by Mario
         Carneiro, 18-Feb-2014.) $)
      numma2c $p |- ( ( P x. M ) + N ) = ( ( T x. E ) + F ) $=
        ( cmul co caddc nn0cni cn0 numcl eqeltri mulcomi oveq1i eqtri nummac )
        EJUDUEZKUFUEJEUDUEZKUFUEFGUDUEHUFUEUOUPKUFEJESUGZJJFAUDUEBUFUEUHQABFLMN
        UIUJUGUKULABCDEFGHIJKLMNOPQRSTUAAEUDUEZCIUFUEZUFUEEAUDUEZUSUFUEGURUTUSU
        FAEAMUGUQUKULUBUMBEUDUEZDUFUEEBUDUEZDUFUEFIUDUEHUFUEVAVBDUFBEBNUGUQUKUL
        UCUMUNUM $.
    $}

    ${
      numadd.8 $e |- ( A + C ) = E $.
      numadd.9 $e |- ( B + D ) = F $.
      $( Add two decimal integers ` M ` and ` N ` (no carry).  (Contributed by
         Mario Carneiro, 18-Feb-2014.) $)
      numadd $p |- ( M + N ) = ( ( T x. E ) + F ) $=
        ( co caddc c1 cmul numcl eqeltri nn0cni mulridi oveq1i 1nn0 eqtri numma
        cn0 eqtr3i ) HUAUBSZITSHITSEFUBSGTSUMHITHHHEAUBSBTSUKOABEJKLUCUDUEUFUGA
        BCDUAEFGHIJKLMNOPUHAUAUBSZCTSACTSFUNACTAAKUEUFUGQUIBUAUBSZDTSBDTSGUOBDT
        BBLUEUFUGRUIUJUL $.
    $}

    ${
      numaddc.8 $e |- F e. NN0 $.
      numaddc.9 $e |- ( ( A + C ) + 1 ) = E $.
      numaddc.10 $e |- ( B + D ) = ( ( T x. 1 ) + F ) $.
      $( Add two decimal integers ` M ` and ` N ` (with carry).  (Contributed
         by Mario Carneiro, 18-Feb-2014.) $)
      numaddc $p |- ( M + N ) = ( ( T x. E ) + F ) $=
        ( co c1 cmul caddc cn0 numcl eqeltri nn0cni mulridi oveq1i 1nn0 addassi
        ax-1cn 3eqtr2i eqtri nummac eqtr3i ) HUAUBTZIUCTHIUCTEFUBTGUCTUQHIUCHHH
        EAUBTBUCTUDOABEJKLUEUFUGUHUIABCDUAEFGUAHIJKLMNOPUJQUJAUAUBTZCUAUCTZUCTA
        USUCTACUCTUAUCTFURAUSUCAAKUGZUHUIACUAUTCMUGULUKRUMBUAUBTZDUCTBDUCTEUAUB
        TGUCTVABDUCBBLUGUHUISUNUOUP $.
    $}
  $}

  ${
    nummul1c.1 $e |- T e. NN0 $.
    nummul1c.2 $e |- P e. NN0 $.
    nummul1c.3 $e |- A e. NN0 $.
    nummul1c.4 $e |- B e. NN0 $.
    nummul1c.5 $e |- N = ( ( T x. A ) + B ) $.
    nummul1c.6 $e |- D e. NN0 $.
    nummul1c.7 $e |- E e. NN0 $.
    ${
      nummul1c.8 $e |- ( ( A x. P ) + E ) = C $.
      nummul1c.9 $e |- ( B x. P ) = ( ( T x. E ) + D ) $.
      $( The product of a decimal integer with a number.  (Contributed by Mario
         Carneiro, 18-Feb-2014.) $)
      nummul1c $p |- ( N x. P ) = ( ( T x. C ) + D ) $=
        ( co cc0 caddc cmul cn0 numcl eqeltri num0u num0h nn0cni addlidi oveq2i
        0nn0 eqtri eqtr3i nummac ) HEUARZUNSTRFCUARDTREHHFAUARBTRUBMABFIKLUCUDJ
        UEABSSEFCDGHSIKLUJUJMSFIUJUFJNOAEUARZSGTRZTRUOGTRCUPGUOTGGOUGUHUIPUKBEU
        ARZUQSTRFGUARDTREBLJUEQULUMUK $.
    $}

    ${
      nummul2c.7 $e |- ( ( P x. A ) + E ) = C $.
      nummul2c.8 $e |- ( P x. B ) = ( ( T x. E ) + D ) $.
      $( The product of a decimal integer with a number (with carry).
         (Contributed by Mario Carneiro, 18-Feb-2014.) $)
      nummul2c $p |- ( P x. N ) = ( ( T x. C ) + D ) $=
        ( cmul co caddc cn0 numcl eqeltri nn0cni oveq1i eqtri mulcomli nummul1c
        mulcomi ) HEFCRSDTSHHFARSBTSUAMABFIKLUBUCUDEJUDZABCDEFGHIJKLMNOAERSZGTS
        EARSZGTSCUKULGTAEAKUDUJUIUEPUFEBFGRSDTSUJBLUDQUGUHUG $.
    $}
  $}

  ${
    decma.a $e |- A e. NN0 $.
    decma.b $e |- B e. NN0 $.
    decma.c $e |- C e. NN0 $.
    decma.d $e |- D e. NN0 $.
    decma.m $e |- M = ; A B $.
    decma.n $e |- N = ; C D $.
    ${
      decma.p $e |- P e. NN0 $.
      decma.e $e |- ( ( A x. P ) + C ) = E $.
      decma.f $e |- ( ( B x. P ) + D ) = F $.
      $( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
         multiplicand ` P ` (no carry).  (Contributed by Mario Carneiro,
         18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      decma $p |- ( ( M x. P ) + N ) = ; E F $=
        ( cmul co caddc c1 cc0 cdc 10nn0 dfdec10 eqtri numma eqtr4i ) HESTIUATU
        BUCUDZFSTGUATFGUDABCDEUJFGHIUEJKLMHABUDUJASTBUATNABUFUGICDUDUJCSTDUATOC
        DUFUGPQRUHFGUFUI $.
    $}

    ${
      decmac.p $e |- P e. NN0 $.
      decmac.f $e |- F e. NN0 $.
      decmac.g $e |- G e. NN0 $.
      decmac.e $e |- ( ( A x. P ) + ( C + G ) ) = E $.
      decmac.2 $e |- ( ( B x. P ) + D ) = ; G F $.
      $( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
         multiplicand ` P ` (with carry).  (Contributed by Mario Carneiro,
         18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      decmac $p |- ( ( M x. P ) + N ) = ; E F $=
        ( cmul co caddc c1 cc0 cdc 10nn0 dfdec10 eqtri nummac eqtr4i ) IEUBUCJU
        DUCUEUFUGZFUBUCGUDUCFGUGABCDEUMFGHIJUHKLMNIABUGUMAUBUCBUDUCOABUIUJJCDUG
        UMCUBUCDUDUCPCDUIUJQRSTBEUBUCDUDUCHGUGUMHUBUCGUDUCUAHGUIUJUKFGUIUL $.
    $}

    ${
      decma2c.p $e |- P e. NN0 $.
      decma2c.f $e |- F e. NN0 $.
      decma2c.g $e |- G e. NN0 $.
      decma2c.e $e |- ( ( P x. A ) + ( C + G ) ) = E $.
      decma2c.2 $e |- ( ( P x. B ) + D ) = ; G F $.
      $( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
         multiplier ` P ` (with carry).  (Contributed by Mario Carneiro,
         18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      decma2c $p |- ( ( P x. M ) + N ) = ; E F $=
        ( cmul co caddc c1 cc0 cdc 10nn0 dfdec10 eqtri numma2c eqtr4i ) EIUBUCJ
        UDUCUEUFUGZFUBUCGUDUCFGUGABCDEUMFGHIJUHKLMNIABUGUMAUBUCBUDUCOABUIUJJCDU
        GUMCUBUCDUDUCPCDUIUJQRSTEBUBUCDUDUCHGUGUMHUBUCGUDUCUAHGUIUJUKFGUIUL $.
    $}

    ${
      decadd.e $e |- ( A + C ) = E $.
      decadd.f $e |- ( B + D ) = F $.
      $( Add two numerals ` M ` and ` N ` (no carry).  (Contributed by Mario
         Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      decadd $p |- ( M + N ) = ; E F $=
        ( caddc co cdc cmul c1 cc0 10nn0 dfdec10 eqtri numadd eqtr4i ) GHQRUAUB
        SZETRFQREFSABCDUHEFGHUCIJKLGABSUHATRBQRMABUDUEHCDSUHCTRDQRNCDUDUEOPUFEF
        UDUG $.
    $}

    decaddc.e $e |- ( ( A + C ) + 1 ) = E $.
    ${
      decaddc.f $e |- F e. NN0 $.
      decaddc.2 $e |- ( B + D ) = ; 1 F $.
      $( Add two numerals ` M ` and ` N ` (with carry).  (Contributed by Mario
         Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      decaddc $p |- ( M + N ) = ; E F $=
        ( caddc co cdc c1 cc0 cmul 10nn0 dfdec10 eqtri numaddc eqtr4i ) GHRSUAU
        BTZEUCSFRSEFTABCDUIEFGHUDIJKLGABTUIAUCSBRSMABUEUFHCDTUICUCSDRSNCDUEUFPO
        BDRSUAFTUIUAUCSFRSQUAFUEUFUGEFUEUH $.
    $}

    decaddc2.t $e |- ( B + D ) = ; 1 0 $.
    $( Add two numerals ` M ` and ` N ` (with carry).  (Contributed by Mario
       Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
    decaddc2 $p |- ( M + N ) = ; E 0 $=
      ( cc0 0nn0 decaddc ) ABCDEPFGHIJKLMNQOR $.
  $}

  ${
    decrmanc.a $e |- A e. NN0 $.
    decrmanc.b $e |- B e. NN0 $.
    decrmanc.n $e |- N e. NN0 $.
    decrmanc.m $e |- M = ; A B $.
    decrmanc.p $e |- P e. NN0 $.
    ${
      decrmanc.e $e |- ( A x. P ) = E $.
      decrmanc.f $e |- ( ( B x. P ) + N ) = F $.
      $( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
         multiplicand ` P ` (no carry).  (Contributed by AV, 16-Sep-2021.) $)
      decrmanc $p |- ( ( M x. P ) + N ) = ; E F $=
        ( cc0 0nn0 dec0h cmul co caddc nn0mulcli nn0cni addridi eqtri decma ) A
        BOGCDEFGHIPJKGJQLACRSZOTSUFDUFUFACHLUAUBUCMUDNUE $.
    $}

    ${
      decrmac.f $e |- F e. NN0 $.
      decrmac.g $e |- G e. NN0 $.
      decrmac.e $e |- ( ( A x. P ) + G ) = E $.
      decrmac.2 $e |- ( ( B x. P ) + N ) = ; G F $.
      $( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
         multiplicand ` P ` (with carry).  (Contributed by AV, 16-Sep-2021.) $)
      decrmac $p |- ( ( M x. P ) + N ) = ; E F $=
        ( cc0 co caddc 0nn0 dec0h cmul nn0cni addlidi oveq2i eqtri decmac ) ABR
        HCDEFGHIJUAKLHKUBMNOACUCSZRFTSZTSUIFTSDUJFUITFFOUDUEUFPUGQUH $.
    $}
  $}

  ${
    decaddm10.a $e |- A e. NN0 $.
    decaddm10.b $e |- B e. NN0 $.
    $( The sum of two multiples of 10 is a multiple of 10.  (Contributed by AV,
       30-Jul-2021.) $)
    decaddm10 $p |- ( ; A 0 + ; B 0 ) = ; ( A + B ) 0 $=
      ( cc0 caddc co cdc 0nn0 eqid 00id decadd ) AEBEABFGZEAEHZBEHZCIDINJOJMJKL
      $.
  $}

  ${
    decaddi.1 $e |- A e. NN0 $.
    decaddi.2 $e |- B e. NN0 $.
    decaddi.3 $e |- N e. NN0 $.
    decaddi.4 $e |- M = ; A B $.
    ${
      decaddi.5 $e |- ( B + N ) = C $.
      $( Add two numerals ` M ` and ` N ` (no carry).  (Contributed by Mario
         Carneiro, 18-Feb-2014.) $)
      decaddi $p |- ( M + N ) = ; A C $=
        ( cc0 0nn0 dec0h nn0cni addridi decadd ) ABKEACDEFGLHIEHMAAFNOJP $.
    $}

    decaddci.5 $e |- ( A + 1 ) = D $.
    ${
      decaddci.6 $e |- C e. NN0 $.
      decaddci.7 $e |- ( B + N ) = ; 1 C $.
      $( Add two numerals ` M ` and ` N ` (no carry).  (Contributed by Mario
         Carneiro, 18-Feb-2014.) $)
      decaddci $p |- ( M + N ) = ; D C $=
        ( cc0 0nn0 dec0h caddc co c1 nn0cni addridi oveq1i eqtri decaddc ) ABNF
        DCEFGHOIJFIPANQRZSQRASQRDUEASQAAGTUAUBKUCLMUD $.
    $}

    ${
      decaddci2.6 $e |- ( B + N ) = ; 1 0 $.
      $( Add two numerals ` M ` and ` N ` (no carry).  (Contributed by Mario
         Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      decaddci2 $p |- ( M + N ) = ; D 0 $=
        ( cc0 0nn0 decaddci ) ABLCDEFGHIJMKN $.
    $}

    ${
      decsubi.5 $e |- ( B - N ) = C $.
      $( Difference between a numeral ` M ` and a nonnegative integer ` N ` (no
         underflow).  (Contributed by AV, 22-Jul-2021.)  (Revised by AV,
         6-Sep-2021.) $)
      decsubi $p |- ( M - N ) = ; A C $=
        ( c1 cdc co caddc cmin nn0cni dfdec10 eqtri cmul 10nn0 nn0mulcli oveq1i
        cc0 addsubassi eqcomi oveq2i 3eqtr4i ) MUENZAUAOZBPOZFQOUKBFQOZPOZEFQOA
        CNZUKBFUKUJAUBGUCRBHRFIRUFEULFQEABNULJABSTUDUOUKCPOUNACSCUMUKPUMCLUGUHT
        UI $.
    $}
  $}

  ${
    decmul1.p $e |- P e. NN0 $.
    decmul1.a $e |- A e. NN0 $.
    decmul1.b $e |- B e. NN0 $.
    decmul1.n $e |- N = ; A B $.
    decmul1.0 $e |- D e. NN0 $.
    ${
      decmul1.c $e |- ( A x. P ) = C $.
      decmul1.d $e |- ( B x. P ) = D $.
      $( The product of a numeral with a number (no carry).  (Contributed by
         AV, 22-Jul-2021.)  (Revised by AV, 6-Sep-2021.) $)
      decmul1 $p |- ( N x. P ) = ; C D $=
        ( cmul co cc0 cdc caddc 10nn0 nn0cni c1 dfdec10 eqtri nn0mulcli addridi
        0nn0 oveq2i addlidi mul01i eqcomi oveq1i 3eqtr3i nummul1c eqtr4i ) FENO
        UAPQZCNODROCDQABCDEUOPFSGHIFABQUOANOBROJABUBUCKUFAENOZPROUPCUPUPAEHGUDT
        UELUCPBENOZROPDROUQUOPNOZDROUQDPRMUGUQUQBEIGUDTUHPURDRURPUOUOSTUIUJUKUL
        UMCDUBUN $.
    $}

    decmul1c.e $e |- E e. NN0 $.
    ${
      decmul1c.c $e |- ( ( A x. P ) + E ) = C $.
      decmul1c.2 $e |- ( B x. P ) = ; E D $.
      $( The product of a numeral with a number (with carry).  (Contributed by
         Mario Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
      decmul1c $p |- ( N x. P ) = ; C D $=
        ( cmul co cdc caddc dfdec10 c1 cc0 10nn0 eqtri nummul1c eqtr4i ) GEPQUA
        UBRZCPQDSQCDRABCDEUGFGUCHIJGABRUGAPQBSQKABTUDLMNBEPQFDRUGFPQDSQOFDTUDUE
        CDTUF $.
    $}

    decmul2c.c $e |- ( ( P x. A ) + E ) = C $.
    decmul2c.2 $e |- ( P x. B ) = ; E D $.
    $( The product of a numeral with a number (with carry).  (Contributed by
       Mario Carneiro, 18-Feb-2014.)  (Revised by AV, 6-Sep-2021.) $)
    decmul2c $p |- ( P x. N ) = ; C D $=
      ( cmul co cdc caddc dfdec10 c1 cc0 10nn0 eqtri nummul2c eqtr4i ) EGPQUAUB
      RZCPQDSQCDRABCDEUGFGUCHIJGABRUGAPQBSQKABTUDLMNEBPQFDRUGFPQDSQOFDTUDUECDTU
      F $.
  $}

  ${
    decmulnc.n $e |- N e. NN0 $.
    decmulnc.a $e |- A e. NN0 $.
    decmulnc.b $e |- B e. NN0 $.
    $( The product of a numeral with a number (no carry).  (Contributed by AV,
       15-Jun-2021.) $)
    decmulnc $p |- ( N x. ; A B ) = ; ( N x. A ) ( N x. B ) $=
      ( cmul co cc0 cdc eqid nn0mulcli 0nn0 nn0cni addridi dec0h decmul2c ) ABC
      AGHZCBGHZCIABJZDEFTKCBDFLZMRRCADELNOSUAPQ $.
  $}

  ${
    11multnc.n $e |- N e. NN0 $.
    $( The product of 11 (as numeral) with a number (no carry).  (Contributed
       by AV, 15-Jun-2021.) $)
    11multnc $p |- ( N x. ; 1 1 ) = ; N N $=
      ( c1 cdc cmul co 1nn0 decmulnc nn0cni mulridi deceq12i eqtri ) ACCDEFACEF
      ZMDAADCCABGGHMAMAAABIJZNKL $.
  $}

  ${
    decmul10add.1 $e |- A e. NN0 $.
    decmul10add.2 $e |- B e. NN0 $.
    decmul10add.3 $e |- M e. NN0 $.
    decmul10add.4 $e |- E = ( M x. A ) $.
    decmul10add.5 $e |- F = ( M x. B ) $.
    $( A multiplication of a number and a numeral expressed as addition with
       first summand as multiple of 10.  (Contributed by AV, 22-Jul-2021.)
       (Revised by AV, 6-Sep-2021.) $)
    decmul10add $p |- ( M x. ; A B ) = ( ; E 0 + F ) $=
      ( cdc cmul co cc0 caddc nn0cni 10nn0 nn0mulcli eqcomi 3eqtri oveq2i dec0u
      c1 dfdec10 adddii mul12i deceq1i oveq12i ) EABKZLMEUCNKZALMZBOMZLMEUKLMZE
      BLMZOMCNKZDOMUIULELABUDUAEUKBEHPZUKUJAQFRPBGPUEUMUOUNDOUMUJEALMZLMUQNKUOE
      UJAUPUJQPAFPUFUQEAHFRUBUQCNCUQISUGTDUNJSUHT $.
  $}

  ${
    6p5lem.1 $e |- A e. NN0 $.
    6p5lem.2 $e |- D e. NN0 $.
    6p5lem.3 $e |- E e. NN0 $.
    6p5lem.4 $e |- B = ( D + 1 ) $.
    6p5lem.5 $e |- C = ( E + 1 ) $.
    6p5lem.6 $e |- ( A + D ) = ; 1 E $.
    $( Lemma for ~ 6p5e11 and related theorems.  (Contributed by Mario
       Carneiro, 19-Apr-2015.) $)
    6p5lem $p |- ( A + B ) = ; 1 C $=
      ( caddc co c1 cdc oveq2i nn0cni ax-1cn addassi 1nn0 eqcomi decsuc 3eqtr2i
      ) ABLMADNLMZLMADLMZNLMNCOBUDALIPADNAFQDGQRSNECUETHCENLMJUAKUBUC $.
  $}

  $( 5 + 5 = 10.  (Contributed by NM, 5-Feb-2007.)  (Revised by Stanislas Polu,
     7-Apr-2020.)  (Revised by AV, 6-Sep-2021.) $)
  5p5e10 $p |- ( 5 + 5 ) = ; 1 0 $=
    ( c5 caddc co c4 c1 cc0 cdc df-5 oveq2i 5cn 4cn ax-1cn addassi eqtr4i 5p4e9
    c9 oveq1i 9p1e10 3eqtri ) AABCZADBCZEBCZPEBCEFGTADEBCZBCUBAUCABHIADEJKLMNUA
    PEBOQRS $.

  $( 6 + 4 = 10.  (Contributed by NM, 5-Feb-2007.)  (Revised by Stanislas Polu,
     7-Apr-2020.)  (Revised by AV, 6-Sep-2021.) $)
  6p4e10 $p |- ( 6 + 4 ) = ; 1 0 $=
    ( c6 c4 caddc co c3 c1 c9 cc0 cdc df-4 oveq2i 6cn 3cn ax-1cn addassi eqtr4i
    6p3e9 oveq1i 9p1e10 3eqtri ) ABCDZAECDZFCDZGFCDFHIUAAEFCDZCDUCBUDACJKAEFLMN
    OPUBGFCQRST $.

  $( 6 + 5 = 11.  (Contributed by Mario Carneiro, 19-Apr-2015.)  (Revised by
     AV, 6-Sep-2021.) $)
  6p5e11 $p |- ( 6 + 5 ) = ; 1 1 $=
    ( c6 c5 c1 c4 cc0 6nn0 4nn0 0nn0 df-5 1e0p1 6p4e10 6p5lem ) ABCDEFGHIJKL $.

  $( 6 + 6 = 12.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  6p6e12 $p |- ( 6 + 6 ) = ; 1 2 $=
    ( c6 c2 c5 c1 6nn0 5nn0 1nn0 df-6 df-2 6p5e11 6p5lem ) AABCDEFGHIJK $.

  $( 7 + 3 = 10.  (Contributed by NM, 5-Feb-2007.)  (Revised by Stanislas Polu,
     7-Apr-2020.)  (Revised by AV, 6-Sep-2021.) $)
  7p3e10 $p |- ( 7 + 3 ) = ; 1 0 $=
    ( c7 c3 caddc co c2 c1 c9 cc0 cdc df-3 oveq2i 7cn 2cn ax-1cn addassi eqtr4i
    7p2e9 oveq1i 9p1e10 3eqtri ) ABCDZAECDZFCDZGFCDFHIUAAEFCDZCDUCBUDACJKAEFLMN
    OPUBGFCQRST $.

  $( 7 + 4 = 11.  (Contributed by Mario Carneiro, 19-Apr-2015.)  (Revised by
     AV, 6-Sep-2021.) $)
  7p4e11 $p |- ( 7 + 4 ) = ; 1 1 $=
    ( c7 c4 c1 c3 cc0 7nn0 3nn0 0nn0 df-4 1e0p1 7p3e10 6p5lem ) ABCDEFGHIJKL $.

  $( 7 + 5 = 12.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7p5e12 $p |- ( 7 + 5 ) = ; 1 2 $=
    ( c7 c5 c2 c4 c1 7nn0 4nn0 1nn0 df-5 df-2 7p4e11 6p5lem ) ABCDEFGHIJKL $.

  $( 7 + 6 = 13.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7p6e13 $p |- ( 7 + 6 ) = ; 1 3 $=
    ( c7 c6 c3 c5 c2 7nn0 5nn0 2nn0 df-6 df-3 7p5e12 6p5lem ) ABCDEFGHIJKL $.

  $( 7 + 7 = 14.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7p7e14 $p |- ( 7 + 7 ) = ; 1 4 $=
    ( c7 c4 c6 c3 7nn0 6nn0 3nn0 df-7 df-4 7p6e13 6p5lem ) AABCDEFGHIJK $.

  $( 8 + 2 = 10.  (Contributed by NM, 5-Feb-2007.)  (Revised by Stanislas Polu,
     7-Apr-2020.)  (Revised by AV, 6-Sep-2021.) $)
  8p2e10 $p |- ( 8 + 2 ) = ; 1 0 $=
    ( c8 c2 caddc co c1 c9 cc0 cdc df-2 oveq2i 8cn ax-1cn addassi eqtr4i oveq1i
    df-9 9p1e10 3eqtr2i ) ABCDZAECDZECDZFECDEGHSAEECDZCDUABUBACIJAEEKLLMNFTECPO
    QR $.

  $( 8 + 3 = 11.  (Contributed by Mario Carneiro, 19-Apr-2015.)  (Revised by
     AV, 6-Sep-2021.) $)
  8p3e11 $p |- ( 8 + 3 ) = ; 1 1 $=
    ( c8 c3 c1 c2 cc0 8nn0 2nn0 0nn0 df-3 1e0p1 8p2e10 6p5lem ) ABCDEFGHIJKL $.

  $( 8 + 4 = 12.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8p4e12 $p |- ( 8 + 4 ) = ; 1 2 $=
    ( c8 c4 c2 c3 c1 8nn0 3nn0 1nn0 df-4 df-2 8p3e11 6p5lem ) ABCDEFGHIJKL $.

  $( 8 + 5 = 13.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8p5e13 $p |- ( 8 + 5 ) = ; 1 3 $=
    ( c8 c5 c3 c4 c2 8nn0 4nn0 2nn0 df-5 df-3 8p4e12 6p5lem ) ABCDEFGHIJKL $.

  $( 8 + 6 = 14.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8p6e14 $p |- ( 8 + 6 ) = ; 1 4 $=
    ( c8 c6 c4 c5 c3 8nn0 5nn0 3nn0 df-6 df-4 8p5e13 6p5lem ) ABCDEFGHIJKL $.

  $( 8 + 7 = 15.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8p7e15 $p |- ( 8 + 7 ) = ; 1 5 $=
    ( c8 c7 c5 c6 c4 8nn0 6nn0 4nn0 df-7 df-5 8p6e14 6p5lem ) ABCDEFGHIJKL $.

  $( 8 + 8 = 16.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8p8e16 $p |- ( 8 + 8 ) = ; 1 6 $=
    ( c8 c6 c7 c5 8nn0 7nn0 5nn0 df-8 df-6 8p7e15 6p5lem ) AABCDEFGHIJK $.

  $( 9 + 2 = 11.  (Contributed by Mario Carneiro, 19-Apr-2015.)  (Revised by
     AV, 6-Sep-2021.) $)
  9p2e11 $p |- ( 9 + 2 ) = ; 1 1 $=
    ( c9 c2 c1 cc0 9nn0 1nn0 0nn0 df-2 1e0p1 9p1e10 6p5lem ) ABCCDEFGHIJK $.

  $( 9 + 3 = 12.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9p3e12 $p |- ( 9 + 3 ) = ; 1 2 $=
    ( c9 c3 c2 c1 9nn0 2nn0 1nn0 df-3 df-2 9p2e11 6p5lem ) ABCCDEFGHIJK $.

  $( 9 + 4 = 13.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9p4e13 $p |- ( 9 + 4 ) = ; 1 3 $=
    ( c9 c4 c3 c2 9nn0 3nn0 2nn0 df-4 df-3 9p3e12 6p5lem ) ABCCDEFGHIJK $.

  $( 9 + 5 = 14.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9p5e14 $p |- ( 9 + 5 ) = ; 1 4 $=
    ( c9 c5 c4 c3 9nn0 4nn0 3nn0 df-5 df-4 9p4e13 6p5lem ) ABCCDEFGHIJK $.

  $( 9 + 6 = 15.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9p6e15 $p |- ( 9 + 6 ) = ; 1 5 $=
    ( c9 c6 c5 c4 9nn0 5nn0 4nn0 df-6 df-5 9p5e14 6p5lem ) ABCCDEFGHIJK $.

  $( 9 + 7 = 16.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9p7e16 $p |- ( 9 + 7 ) = ; 1 6 $=
    ( c9 c7 c6 c5 9nn0 6nn0 5nn0 df-7 df-6 9p6e15 6p5lem ) ABCCDEFGHIJK $.

  $( 9 + 8 = 17.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9p8e17 $p |- ( 9 + 8 ) = ; 1 7 $=
    ( c9 c8 c7 c6 9nn0 7nn0 6nn0 df-8 df-7 9p7e16 6p5lem ) ABCCDEFGHIJK $.

  $( 9 + 9 = 18.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9p9e18 $p |- ( 9 + 9 ) = ; 1 8 $=
    ( c9 c8 c7 9nn0 8nn0 7nn0 df-9 df-8 9p8e17 6p5lem ) AABBCDEFGHIJ $.

  $( 10 + 10 = 20.  (Contributed by Mario Carneiro, 19-Apr-2015.)  (Revised by
     AV, 6-Sep-2021.) $)
  10p10e20 $p |- ( ; 1 0 + ; 1 0 ) = ; 2 0 $=
    ( c1 cc0 c2 cdc 1nn0 0nn0 eqid 1p1e2 00id decadd ) ABABCBABDZKEFEFKGZLHIJ
    $.

  $( 10 - 1 = 9.  (Contributed by AV, 6-Sep-2021.) $)
  10m1e9 $p |- ( ; 1 0 - 1 ) = 9 $=
    ( c1 cc0 cdc c9 9cn ax-1cn caddc co 9p1e10 eqcomi mvrraddi ) ABCZDAEFDAGHLI
    JK $.

  ${
    4t3lem.1 $e |- A e. NN0 $.
    4t3lem.2 $e |- B e. NN0 $.
    4t3lem.3 $e |- C = ( B + 1 ) $.
    4t3lem.4 $e |- ( A x. B ) = D $.
    4t3lem.5 $e |- ( D + A ) = E $.
    $( Lemma for ~ 4t3e12 and related theorems.  (Contributed by Mario
       Carneiro, 19-Apr-2015.) $)
    4t3lem $p |- ( A x. C ) = E $=
      ( cmul co c1 caddc oveq2i nn0cni ax-1cn adddii mulridi eqtri oveq12i ) AC
      KLABMNLZKLZECUBAKHOUCDANLZEUCABKLZAMKLZNLUDABMAFPZBGPQRUEDUFANIAUGSUATJTT
      $.
  $}

  $( 4 times 3 equals 12.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  4t3e12 $p |- ( 4 x. 3 ) = ; 1 2 $=
    ( c4 c2 c3 c8 c1 cdc 4nn0 2nn0 df-3 4t2e8 8p4e12 4t3lem ) ABCDEBFGHIJKL $.

  $( 4 times 4 equals 16.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  4t4e16 $p |- ( 4 x. 4 ) = ; 1 6 $=
    ( c4 c3 c1 c2 cdc c6 4nn0 3nn0 df-4 4t3e12 1nn0 2nn0 4cn 2cn 4p2e6 addcomli
    eqid decaddi 4t3lem ) ABACDEZCFEGHIJCDFTAKLGTQADFMNOPRS $.

  $( 5 times 2 equals 10.  (Contributed by NM, 5-Feb-2007.)  (Revised by AV,
     4-Sep-2021.) $)
  5t2e10 $p |- ( 5 x. 2 ) = ; 1 0 $=
    ( c5 c1 c2 cc0 cdc 5nn0 1nn0 df-2 5cn mulridi 5p5e10 4t3lem ) ABCABDEFGHAIJ
    KL $.

  $( 5 times 3 equals 15.  (Contributed by Mario Carneiro, 19-Apr-2015.)
     (Revised by AV, 6-Sep-2021.) $)
  5t3e15 $p |- ( 5 x. 3 ) = ; 1 5 $=
    ( c5 c2 c3 c1 cc0 cdc 5nn0 2nn0 df-3 5t2e10 dec10p 4t3lem ) ABCDEFDAFGHIJAK
    L $.

  $( 5 times 4 equals 20.  (Contributed by Mario Carneiro, 19-Apr-2015.)
     (Revised by AV, 6-Sep-2021.) $)
  5t4e20 $p |- ( 5 x. 4 ) = ; 2 0 $=
    ( c5 c3 c4 c1 cdc c2 5nn0 3nn0 df-4 5t3e15 1nn0 eqid 1p1e2 5p5e10 decaddci2
    cc0 4t3lem ) ABCDAEZFPEGHIJDAFRAKGGRLMNOQ $.

  $( 5 times 5 equals 25.  (Contributed by Mario Carneiro, 19-Apr-2015.)
     (Revised by AV, 6-Sep-2021.) $)
  5t5e25 $p |- ( 5 x. 5 ) = ; 2 5 $=
    ( c5 c4 c1 cc0 cdc c2 cmul co 5nn0 4nn0 df-5 5t4e20 2nn0 dec0u eqtr4i caddc
    dfdec10 eqcomi 4t3lem ) ABACDEFGHZFAEZIJKABGHFDETLFMNOUATAPHFAQRS $.

  $( 6 times 2 equals 12.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  6t2e12 $p |- ( 6 x. 2 ) = ; 1 2 $=
    ( c6 c2 cmul co caddc c1 cdc 6cn times2i 6p6e12 eqtri ) ABCDAAEDFBGAHIJK $.

  $( 6 times 3 equals 18.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  6t3e18 $p |- ( 6 x. 3 ) = ; 1 8 $=
    ( c6 c2 c3 c1 cdc c8 6nn0 2nn0 df-3 6t2e12 1nn0 eqid 6cn 2cn 6p2e8 addcomli
    decaddi 4t3lem ) ABCDBEZDFEGHIJDBFSAKHGSLABFMNOPQR $.

  $( 6 times 4 equals 24.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  6t4e24 $p |- ( 6 x. 4 ) = ; 2 4 $=
    ( c6 c3 c4 c1 c8 cdc 6nn0 3nn0 df-4 6t3e18 1nn0 8nn0 eqid 1p1e2 4nn0 8p6e14
    c2 decaddci 4t3lem ) ABCDEFZQCFGHIJDECQTAKLGTMNOPRS $.

  $( 6 times 5 equals 30.  (Contributed by Mario Carneiro, 19-Apr-2015.)
     (Revised by AV, 6-Sep-2021.) $)
  6t5e30 $p |- ( 6 x. 5 ) = ; 3 0 $=
    ( c6 c4 c5 c2 cdc c3 cc0 6nn0 4nn0 df-5 6t4e24 2nn0 2p1e3 c1 6cn 4cn 6p4e10
    eqid addcomli decaddci2 4t3lem ) ABCDBEZFGEHIJKDBFUBALIHUBRMABNGEOPQSTUA $.

  $( 6 times 6 equals 36.  (Contributed by Mario Carneiro, 19-Apr-2015.)
     (Revised by AV, 6-Sep-2021.) $)
  6t6e36 $p |- ( 6 x. 6 ) = ; 3 6 $=
    ( c6 c5 c1 cc0 cdc c3 cmul co 6nn0 5nn0 df-6 6t5e30 3nn0 dec0u eqtr4i caddc
    dfdec10 eqcomi 4t3lem ) ABACDEFGHZFAEZIJKABGHFDETLFMNOUATAPHFAQRS $.

  $( 7 times 2 equals 14.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7t2e14 $p |- ( 7 x. 2 ) = ; 1 4 $=
    ( c7 c2 cmul co caddc c1 c4 cdc 7cn times2i 7p7e14 eqtri ) ABCDAAEDFGHAIJKL
    $.

  $( 7 times 3 equals 21.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7t3e21 $p |- ( 7 x. 3 ) = ; 2 1 $=
    ( c7 c2 c3 c1 c4 7nn0 2nn0 df-3 7t2e14 1nn0 4nn0 eqid 1p1e2 nn0cni addcomli
    cdc 7p4e11 decaddci 4t3lem ) ABCDEPZBDPFGHIDEDBTAJKFTLMJAEDDPAFNEKNQORS $.

  $( 7 times 4 equals 28.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7t4e28 $p |- ( 7 x. 4 ) = ; 2 8 $=
    ( c7 c3 c4 c2 c1 cdc c8 7nn0 3nn0 df-4 7t3e21 2nn0 1nn0 7cn ax-1cn addcomli
    eqid 7p1e8 decaddi 4t3lem ) ABCDEFZDGFHIJKDEGUAALMHUAQAEGNORPST $.

  $( 7 times 5 equals 35.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7t5e35 $p |- ( 7 x. 5 ) = ; 3 5 $=
    ( c7 c4 c5 c2 c8 cdc 7nn0 4nn0 df-5 7t4e28 2nn0 8nn0 eqid 2p1e3 5nn0 8p7e15
    c3 decaddci 4t3lem ) ABCDEFZQCFGHIJDECQTAKLGTMNOPRS $.

  $( 7 times 6 equals 42.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7t6e42 $p |- ( 7 x. 6 ) = ; 4 2 $=
    ( c7 c5 c6 c3 cdc c4 c2 7nn0 5nn0 df-6 7t5e35 3nn0 eqid 3p1e4 nn0cni 7p5e12
    2nn0 c1 addcomli decaddci 4t3lem ) ABCDBEZFGEHIJKDBGFUBALIHUBMNQABRGEAHOBIO
    PSTUA $.

  $( 7 times 7 equals 49.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  7t7e49 $p |- ( 7 x. 7 ) = ; 4 9 $=
    ( c7 c6 c4 c2 cdc c9 7nn0 6nn0 df-7 7t6e42 4nn0 2nn0 7cn 2cn 7p2e9 addcomli
    eqid decaddi 4t3lem ) ABACDEZCFEGHIJCDFTAKLGTQADFMNOPRS $.

  $( 8 times 2 equals 16.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8t2e16 $p |- ( 8 x. 2 ) = ; 1 6 $=
    ( c8 c2 cmul co caddc c1 c6 cdc 8cn times2i 8p8e16 eqtri ) ABCDAAEDFGHAIJKL
    $.

  $( 8 times 3 equals 24.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8t3e24 $p |- ( 8 x. 3 ) = ; 2 4 $=
    ( c8 c2 c3 c1 c6 cdc 8nn0 2nn0 df-3 8t2e16 1nn0 6nn0 eqid 1p1e2 4nn0 nn0cni
    c4 8p6e14 addcomli decaddci 4t3lem ) ABCDEFZBQFGHIJDEQBUBAKLGUBMNOAEDQFAGPE
    LPRSTUA $.

  $( 8 times 4 equals 32.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8t4e32 $p |- ( 8 x. 4 ) = ; 3 2 $=
    ( c8 c3 c4 c2 8nn0 3nn0 df-4 8t3e24 2nn0 4nn0 eqid 2p1e3 c1 nn0cni addcomli
    cdc 8p4e12 decaddci 4t3lem ) ABCDCPZBDPEFGHDCDBTAIJETKLIACMDPAENCJNQORS $.

  $( 8 times 5 equals 40.  (Contributed by Mario Carneiro, 19-Apr-2015.)
     (Revised by AV, 6-Sep-2021.) $)
  8t5e40 $p |- ( 8 x. 5 ) = ; 4 0 $=
    ( c8 c4 c5 c3 c2 cdc cc0 8nn0 4nn0 df-5 8t4e32 3nn0 2nn0 eqid 3p1e4 8cn 2cn
    c1 8p2e10 addcomli decaddci2 4t3lem ) ABCDEFZBGFHIJKDEBUCALMHUCNOAERGFPQSTU
    AUB $.

  $( 8 times 6 equals 48.  (Contributed by Mario Carneiro, 19-Apr-2015.)
     (Revised by AV, 6-Sep-2021.) $)
  8t6e48 $p |- ( 8 x. 6 ) = ; 4 8 $=
    ( c8 c5 c6 c1 cc0 cdc c4 cmul 8nn0 5nn0 df-6 8t5e40 4nn0 dec0u eqtr4i caddc
    co dfdec10 eqcomi 4t3lem ) ABCDEFGHQZGAFZIJKABHQGEFUALGMNOUBUAAPQGARST $.

  $( 8 times 7 equals 56.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8t7e56 $p |- ( 8 x. 7 ) = ; 5 6 $=
    ( c8 c6 c7 c4 cdc c5 8nn0 6nn0 df-7 8t6e48 4nn0 eqid 8p8e16 decaddci 4t3lem
    4p1e5 ) ABCDAEZFBEGHIJDABFQAKGGQLPHMNO $.

  $( 8 times 8 equals 64.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  8t8e64 $p |- ( 8 x. 8 ) = ; 6 4 $=
    ( c8 c7 c5 c6 cdc c4 8nn0 7nn0 df-8 8t7e56 5nn0 6nn0 eqid 5p1e6 4nn0 nn0cni
    c1 8p6e14 addcomli decaddci 4t3lem ) ABACDEZDFEGHIJCDFDUBAKLGUBMNOADQFEAGPD
    LPRSTUA $.

  $( 9 times 2 equals 18.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9t2e18 $p |- ( 9 x. 2 ) = ; 1 8 $=
    ( c9 c2 cmul co caddc c1 c8 cdc 9cn times2i 9p9e18 eqtri ) ABCDAAEDFGHAIJKL
    $.

  $( 9 times 3 equals 27.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9t3e27 $p |- ( 9 x. 3 ) = ; 2 7 $=
    ( c9 c2 c3 c1 c8 cdc 9nn0 2nn0 df-3 9t2e18 1nn0 8nn0 eqid 1p1e2 7nn0 nn0cni
    c7 9p8e17 addcomli decaddci 4t3lem ) ABCDEFZBQFGHIJDEQBUBAKLGUBMNOAEDQFAGPE
    LPRSTUA $.

  $( 9 times 4 equals 36.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9t4e36 $p |- ( 9 x. 4 ) = ; 3 6 $=
    ( c9 c3 c4 c2 c7 cdc 9nn0 3nn0 df-4 9t3e27 2nn0 7nn0 eqid 2p1e3 6nn0 nn0cni
    c6 c1 9p7e16 addcomli decaddci 4t3lem ) ABCDEFZBQFGHIJDEQBUCAKLGUCMNOAERQFA
    GPELPSTUAUB $.

  $( 9 times 5 equals 45.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9t5e45 $p |- ( 9 x. 5 ) = ; 4 5 $=
    ( c9 c4 c5 c3 c6 cdc 9nn0 4nn0 df-5 9t4e36 3nn0 6nn0 eqid 3p1e4 5nn0 nn0cni
    c1 9p6e15 addcomli decaddci 4t3lem ) ABCDEFZBCFGHIJDECBUBAKLGUBMNOAEQCFAGPE
    LPRSTUA $.

  $( 9 times 6 equals 54.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9t6e54 $p |- ( 9 x. 6 ) = ; 5 4 $=
    ( c9 c5 c6 cdc 9nn0 5nn0 df-6 9t5e45 4nn0 eqid 4p1e5 nn0cni 9p5e14 addcomli
    c4 c1 decaddci 4t3lem ) ABCOBDZBODEFGHOBOBSAIFESJKIABPODAELBFLMNQR $.

  $( 9 times 7 equals 63.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9t7e63 $p |- ( 9 x. 7 ) = ; 6 3 $=
    ( c9 c6 c7 c5 c4 cdc 9nn0 6nn0 df-7 9t6e54 5nn0 4nn0 eqid 5p1e6 3nn0 nn0cni
    c3 c1 9p4e13 addcomli decaddci 4t3lem ) ABCDEFZBQFGHIJDEQBUCAKLGUCMNOAERQFA
    GPELPSTUAUB $.

  $( 9 times 8 equals 72.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9t8e72 $p |- ( 9 x. 8 ) = ; 7 2 $=
    ( c9 c7 c8 c6 c3 cdc c2 9nn0 7nn0 df-8 9t7e63 6nn0 3nn0 eqid 6p1e7 2nn0 9cn
    c1 3cn 9p3e12 addcomli decaddci 4t3lem ) ABCDEFZBGFHIJKDEGBUDALMHUDNOPAERGF
    QSTUAUBUC $.

  $( 9 times 9 equals 81.  (Contributed by Mario Carneiro, 19-Apr-2015.) $)
  9t9e81 $p |- ( 9 x. 9 ) = ; 8 1 $=
    ( c9 c8 c7 c2 cdc c1 9nn0 8nn0 df-9 9t8e72 7nn0 2nn0 eqid 7p1e8 1nn0 9p2e11
    9cn 2cn addcomli decaddci 4t3lem ) ABACDEZBFEGHIJCDFBUBAKLGUBMNOADFFEQRPSTU
    A $.

  $( 9 times 11 equals 99.  (Contributed by AV, 14-Jun-2021.)  (Revised by AV,
     6-Sep-2021.) $)
  9t11e99 $p |- ( 9 x. ; 1 1 ) = ; 9 9 $=
    ( c9 c1 cc0 cdc cmul co caddc 9cn 10nn0 nn0cni ax-1cn mulcli adddii mulridi
    oveq2i mulcomi eqtri oveq12i dfdec10 3eqtr4i ) ABCDZBEFZBGFZEFZUAAEFZAGFZAB
    BDZEFAADUDAUBEFZABEFZGFUFAUBBHUABUAIJZKLKMUHUEUIAGUHAUAEFUEUBUAAEUAUJNOAUAH
    UJPQAHNRQUGUCAEBBSOAAST $.

  $( 9 is less than 10.  (Contributed by Mario Carneiro, 8-Feb-2015.)  (Revised
     by AV, 8-Sep-2021.) $)
  9lt10 $p |- 9 < ; 1 0 $=
    ( c9 c1 caddc co cc0 cdc clt 9re ltp1i 9p1e10 breqtri ) AABCDBEFGAHIJK $.

  $( 8 is less than 10.  (Contributed by Mario Carneiro, 8-Feb-2015.)  (Revised
     by AV, 8-Sep-2021.) $)
  8lt10 $p |- 8 < ; 1 0 $=
    ( c8 c9 clt wbr c1 cc0 cdc 8lt9 9lt10 8re 9re 10re lttri mp2an ) ABCDBEFGZC
    DAOCDHIABOJKLMN $.

  $( 7 is less than 10.  (Contributed by Mario Carneiro, 10-Mar-2015.)
     (Revised by AV, 8-Sep-2021.) $)
  7lt10 $p |- 7 < ; 1 0 $=
    ( c7 c8 clt wbr c1 cc0 cdc 7lt8 8lt10 7re 8re 10re lttri mp2an ) ABCDBEFGZC
    DAOCDHIABOJKLMN $.

  $( 6 is less than 10.  (Contributed by Mario Carneiro, 10-Mar-2015.)
     (Revised by AV, 8-Sep-2021.) $)
  6lt10 $p |- 6 < ; 1 0 $=
    ( c6 c7 clt wbr c1 cc0 cdc 6lt7 7lt10 6re 7re 10re lttri mp2an ) ABCDBEFGZC
    DAOCDHIABOJKLMN $.

  $( 5 is less than 10.  (Contributed by Mario Carneiro, 10-Mar-2015.)
     (Revised by AV, 8-Sep-2021.) $)
  5lt10 $p |- 5 < ; 1 0 $=
    ( c5 c6 clt wbr c1 cc0 cdc 5lt6 6lt10 5re 6re 10re lttri mp2an ) ABCDBEFGZC
    DAOCDHIABOJKLMN $.

  $( 4 is less than 10.  (Contributed by Mario Carneiro, 10-Mar-2015.)
     (Revised by AV, 8-Sep-2021.) $)
  4lt10 $p |- 4 < ; 1 0 $=
    ( c4 c5 clt wbr c1 cc0 cdc 4lt5 5lt10 4re 5re 10re lttri mp2an ) ABCDBEFGZC
    DAOCDHIABOJKLMN $.

  $( 3 is less than 10.  (Contributed by Mario Carneiro, 10-Mar-2015.)
     (Revised by AV, 8-Sep-2021.) $)
  3lt10 $p |- 3 < ; 1 0 $=
    ( c3 c4 clt wbr c1 cc0 cdc 3lt4 4lt10 3re 4re 10re lttri mp2an ) ABCDBEFGZC
    DAOCDHIABOJKLMN $.

  $( 2 is less than 10.  (Contributed by Mario Carneiro, 10-Mar-2015.)
     (Revised by AV, 8-Sep-2021.) $)
  2lt10 $p |- 2 < ; 1 0 $=
    ( c2 c3 clt wbr c1 cc0 cdc 2lt3 3lt10 2re 3re 10re lttri mp2an ) ABCDBEFGZC
    DAOCDHIABOJKLMN $.

  $( 1 is less than 10.  (Contributed by NM, 7-Nov-2012.)  (Revised by Mario
     Carneiro, 9-Mar-2015.)  (Revised by AV, 8-Sep-2021.) $)
  1lt10 $p |- 1 < ; 1 0 $=
    ( c1 c2 clt wbr cc0 cdc 1lt2 2lt10 1re 2re 10re lttri mp2an ) ABCDBAEFZCDAN
    CDGHABNIJKLM $.

  ${
    decbin.1 $e |- A e. NN0 $.
    $( Decompose base 4 into base 2.  (Contributed by Mario Carneiro,
       18-Feb-2014.) $)
    decbin0 $p |- ( 4 x. A ) = ( 2 x. ( 2 x. A ) ) $=
      ( c2 cmul co c4 2t2e4 oveq1i 2cn nn0cni mulassi eqtr3i ) CCDEZADEFADECCAD
      EDEMFADGHCCAIIABJKL $.

    $( Decompose base 4 into base 2.  (Contributed by Mario Carneiro,
       18-Feb-2014.) $)
    decbin2 $p |- ( ( 4 x. A ) + 2 ) = ( 2 x. ( ( 2 x. A ) + 1 ) ) $=
      ( c2 cmul co c1 caddc c4 2t1e2 oveq2i nn0cni mulcli ax-1cn adddii decbin0
      2cn oveq1i 3eqtr4ri ) CCADEZDEZCFDEZGETCGECSFGEDEHADEZCGEUACTGIJCSFPCAPAB
      KLMNUBTCGABOQR $.

    $( Decompose base 4 into base 2.  (Contributed by Mario Carneiro,
       18-Feb-2014.) $)
    decbin3 $p |- ( ( 4 x. A ) + 3 ) = ( ( 2 x. ( ( 2 x. A ) + 1 ) ) + 1 ) $=
      ( c2 cmul co c1 caddc c4 c3 4nn0 2nn0 2p1e3 decbin2 eqcomi numsuc ) CCADE
      FGEDEZFGEHADEZIGEACIHPJBKLQCGEPABMNON $.
  $}

  $( Half minus a third.  (Contributed by Scott Fenton, 8-Jul-2015.) $)
  halfthird $p |- ( ( 1 / 2 ) - ( 1 / 3 ) ) = ( 1 / 6 ) $=
    ( c1 c2 cdiv co c3 cmin cmul c6 2cn 3cn 2ap0 3ap0 subrecapi 2p1e3 subaddrii
    ax-1cn 3t2e6 mulcomli oveq12i eqtri ) ABCDAECDFDEBFDZBEGDZCDAHCDBEIJKLMUAAU
    BHCEBAJIPNOEBHJIQRST $.

  $( One fifth minus one sixth.  (Contributed by Scott Fenton, 9-Jan-2017.) $)
  5recm6rec $p |- ( ( 1 / 5 ) - ( 1 / 6 ) ) = ( 1 / ; 3 0 ) $=
    ( c1 c5 cdiv co c6 cmin cmul c3 cc0 cdc 5cn 6cn 5re 5pos gt0ap0ii subrecapi
    6re 6pos ax-1cn 5p1e6 subaddrii 6t5e30 mulcomli oveq12i eqtri ) ABCDAECDFDE
    BFDZBEGDZCDAHIJZCDBEKLBMNOEQROPUFAUGUHCEBALKSTUAEBUHLKUBUCUDUE $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Upper sets of integers
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c ZZ>= $. $( ` ZZ ` with greater-than-or-equal-to subscript. $)

  $( Extend class notation with the upper integer function.
     Read " ` ZZ>= `` M ` " as "the set of integers greater than or equal to
     ` M ` ". $)
  cuz $a class ZZ>= $.

  ${
    $d j k N $.  $d k M $.
    $( Define a function whose value at ` j ` is the semi-infinite set of
       contiguous integers starting at ` j ` , which we will also call the
       upper integers starting at ` j ` .  Read " ` ZZ>= `` M ` " as "the set
       of integers greater than or equal to ` M ` ".  See ~ uzval for its
       value, ~ uzssz for its relationship to ` ZZ ` , ~ nnuz and ~ nn0uz for
       its relationships to ` NN ` and ` NN0 ` , and ~ eluz1 and ~ eluz2 for
       its membership relations.  (Contributed by NM, 5-Sep-2005.) $)
    df-uz $a |- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } ) $.

    $( The value of the upper integers function.  (Contributed by NM,
       5-Sep-2005.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    uzval $p |- ( N e. ZZ -> ( ZZ>= ` N ) = { k e. ZZ | N <_ k } ) $=
      ( vj cv cle wbr cz crab cuz wceq breq1 rabbidv df-uz zex rabex fvmpt ) CB
      CDZADZEFZAGHBREFZAGHGIQBJSTAGQBREKLCAMTAGNOP $.

    $( The domain and codomain of the upper integers function.  (Contributed by
       Scott Fenton, 8-Aug-2013.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    uzf $p |- ZZ>= : ZZ --> ~P ZZ $=
      ( vj vk cv cle wbr cz crab cpw wcel wral cuz wss ssrab2 elpw2 mpbir rgenw
      wf zex df-uz fmpt mpbi ) ACBCDEZBFGZFHZIZAFJFUDKQUEAFUEUCFLUBBFMUCFRNOPAF
      UDUCKABSTUA $.

    $( Membership in the upper set of integers starting at ` M ` .
       (Contributed by NM, 5-Sep-2005.) $)
    eluz1 $p |- ( M e. ZZ ->
           ( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) ) ) $=
      ( vk cz wcel cuz cfv cv cle wbr crab wa uzval eleq2d breq2 elrab bitrdi )
      ADEZBAFGZEBACHZIJZCDKZEBDEABIJZLRSUBBCAMNUAUCCBDTBAIOPQ $.

    $( Implication of membership in an upper set of integers.  (Contributed by
       NM, 6-Sep-2005.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    eluzel2 $p |- ( N e. ( ZZ>= ` M ) -> M e. ZZ ) $=
      ( cuz cfv wcel cdm cz wrel cpw wf frel ax-mp relelfvdm mpan fdmi eleqtrdi
      uzf ) BACDEZACFZGCHZRASEGGIZCJTQGUACKLBACMNGUACQOP $.

    $( Membership in an upper set of integers.  We use the fact that a
       function's value (under our function value definition) is empty outside
       of its domain to show ` M e. ZZ ` .  (Contributed by NM, 5-Sep-2005.)
       (Revised by Mario Carneiro, 3-Nov-2013.) $)
    eluz2 $p |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) ) $=
      ( cuz cfv wcel cz cle wbr w3a eluzel2 simp1 wa eluz1 bitrd 3anass bitr4di
      ibar pm5.21nii ) BACDEZAFEZTBFEZABGHZIZABJTUAUBKTSTUAUBLZLZUCTSUDUEABMTUD
      QNTUAUBOPR $.
  $}

  $( Membership in an earlier upper set of integers.  (Contributed by Thierry
     Arnoux, 8-Oct-2018.) $)
  eluzmn $p |- ( ( M e. ZZ /\ N e. NN0 ) -> M e. ( ZZ>= ` ( M - N ) ) ) $=
    ( cz wcel cn0 wa cmin co cle wbr cuz simpl simpr nn0zd zsubcld caddc nn0red
    cfv zred recnd readdcld cr nn0addge1 sylancom lesub1dd pncand breqtrd eluz2
    syl3anbrc ) ACDZBEDZFZABGHZCDUJUMAIJAUMKRDULABUJUKLZULBUJUKMZNOUNULUMABPHZB
    GHAIULAUPBULAUNSZULABUQULBUOQZUAURUJUKAUBDAUPIJUQABUCUDUEULABULAUQTULBURTUF
    UGUMAUHUI $.

  ${
    eluz.1 $e |- M e. ZZ $.
    $( Membership in an upper set of integers.  (Contributed by NM,
       5-Sep-2005.) $)
    eluz1i $p |- ( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) ) $=
      ( cz wcel cuz cfv cle wbr wa wb eluz1 ax-mp ) ADEBAFGEBDEABHIJKCABLM $.
  $}

  $( An integer in an upper set of integers is an element of an upper set of
     integers with a smaller bound.  (Contributed by Alexander van der Vekens,
     17-Jun-2018.) $)
  eluzuzle $p |- ( ( B e. ZZ /\ B <_ A )
                   -> ( C e. ( ZZ>= ` A ) -> C e. ( ZZ>= ` B ) ) ) $=
    ( cuz cfv wcel cz cle wbr w3a wa eluz2 simpll simpr2 cr zre ad2antrr adantl
    3ad2ant1 3ad2ant2 simplr simpr3 letrd syl3anbrc ex biimtrid ) CADEFAGFZCGFZ
    ACHIZJZBGFZBAHIZKZCBDEFZACLUMUJUNUMUJKZUKUHBCHIUNUKULUJMUMUGUHUINUOBACUKBOF
    ULUJBPQUJAOFZUMUGUHUPUIAPSRUJCOFZUMUHUGUQUICPTRUKULUJUAUMUGUHUIUBUCBCLUDUEU
    F $.

  $( A member of an upper set of integers is an integer.  (Contributed by NM,
     6-Sep-2005.) $)
  eluzelz $p |- ( N e. ( ZZ>= ` M ) -> N e. ZZ ) $=
    ( cuz cfv wcel cz cle wbr eluz2 simp2bi ) BACDEAFEBFEABGHABIJ $.

  $( A member of an upper set of integers is a real.  (Contributed by Mario
     Carneiro, 31-Aug-2013.) $)
  eluzelre $p |- ( N e. ( ZZ>= ` M ) -> N e. RR ) $=
    ( cuz cfv wcel eluzelz zred ) BACDEBABFG $.

  $( A member of an upper set of integers is a complex number.  (Contributed by
     Glauco Siliprandi, 29-Jun-2017.) $)
  eluzelcn $p |- ( N e. ( ZZ>= ` M ) -> N e. CC ) $=
    ( cuz cfv wcel eluzelre recnd ) BACDEBABFG $.

  $( Implication of membership in an upper set of integers.  (Contributed by
     NM, 6-Sep-2005.) $)
  eluzle $p |- ( N e. ( ZZ>= ` M ) -> M <_ N ) $=
    ( cuz cfv wcel cz cle wbr eluz2 simp3bi ) BACDEAFEBFEABGHABIJ $.

  $( Membership in an upper set of integers.  (Contributed by NM,
     2-Oct-2005.) $)
  eluz $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
         ( N e. ( ZZ>= ` M ) <-> M <_ N ) ) $=
    ( cz wcel cuz cfv cle wbr eluz1 baibd ) ACDBAEFDBCDABGHABIJ $.

  $( Membership of the least member in an upper set of integers.  (Contributed
     by NM, 2-Sep-2005.) $)
  uzid $p |- ( M e. ZZ -> M e. ( ZZ>= ` M ) ) $=
    ( cz wcel cuz cfv cle wbr wa zre leidd ancli eluz1 mpbird ) ABCZAADECNAAFGZ
    HNONAAIJKAALM $.

  ${
    uzidd.1 $e |- ( ph -> M e. ZZ ) $.
    $( Membership of the least member in an upper set of integers.
       (Contributed by Glauco Siliprandi, 23-Oct-2021.) $)
    uzidd $p |- ( ph -> M e. ( ZZ>= ` M ) ) $=
      ( cz wcel cuz cfv uzid syl ) ABDEBBFGECBHI $.
  $}

  ${
    $d k M $.
    $( The upper integers are all nonempty.  (Contributed by Mario Carneiro,
       16-Jan-2014.) $)
    uzn0 $p |- ( M e. ran ZZ>= -> M =/= (/) ) $=
      ( vk cuz crn wcel cv cfv wceq cz wrex c0 wne cpw wfn uzf ffn fvelrnb mp2b
      wf wb uzid ne0i syl neeq1 syl5ibcom rexlimiv sylbi ) ACDEZBFZCGZAHZBIJZAK
      LZIIMZCSCINUHULTOIUNCPBIACQRUKUMBIUIIEZUJKLZUKUMUOUIUJEUPUIUAUJUIUBUCUJAK
      UDUEUFUG $.
  $}

  $( Transitive law for sets of upper integers.  (Contributed by NM,
     20-Sep-2005.) $)
  uztrn $p |- ( ( M e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` N ) ) ->
              M e. ( ZZ>= ` N ) ) $=
    ( cuz cfv wcel wa cz cle eluzel2 adantl eluzelz adantr eluzle zletr syl3anc
    wbr wi mp2and eluz2 syl3anbrc ) BADEFZACDEZFZGZCHFZBHFZCBIQZBUCFUDUFUBCAJKZ
    UBUGUDABLMZUECAIQZABIQZUHUDUKUBCANKUBULUDABNMUEUFAHFZUGUKULGUHRUIUDUMUBCALK
    UJCABOPSCBTUA $.

  ${
    uztrn2.1 $e |- Z = ( ZZ>= ` K ) $.
    $( Transitive law for sets of upper integers.  (Contributed by Mario
       Carneiro, 26-Dec-2013.) $)
    uztrn2 $p |- ( ( N e. Z /\ M e. ( ZZ>= ` N ) ) -> M e. Z ) $=
      ( wcel cuz cfv wa eleq2i uztrn ancoms sylanb eleqtrrdi ) CDFZBCGHFZIBAGHZ
      DOCQFZPBQFZDQCEJPRSCBAKLMEN $.
  $}

  $( Contraposition law for upper integers.  (Contributed by NM,
     28-Nov-2005.) $)
  uzneg $p |- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) ) $=
    ( cuz cfv wcel cneg cle wbr eluzle cz wb eluzel2 eluzelz zre syl2an syl2anc
    cr leneg mpbid znegcl eluz mpbird ) BACDEZAFZBFZCDEZUEUDGHZUCABGHZUGABIUCAJ
    EZBJEZUHUGKZABLZABMZUIAQEBQEUKUJANBNABROPSUCUJUIUFUGKZUMULUJUEJEUDJEUNUIBTA
    TUEUDUAOPUB $.

  ${
    $d k y M $.  $d k N $.
    $( An upper set of integers is a subset of all integers.  (Contributed by
       NM, 2-Sep-2005.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    uzssz $p |- ( ZZ>= ` M ) C_ ZZ $=
      ( vy cuz cfv cz cv eluzelz ssriv ) BACDEABFGH $.

    $( Subset relationship for two sets of upper integers.  (Contributed by NM,
       5-Sep-2005.) $)
    uzss $p |- ( N e. ( ZZ>= ` M ) ->
               ( ZZ>= ` N ) C_ ( ZZ>= ` M ) ) $=
      ( vk cuz cfv wcel cv cz cle wbr wa eluzle adantr wi eluzel2 eluzelz eluz1
      jca wb syl zletr 3expa sylan mpand imdistanda 3imtr4d ssrdv ) BADEZFZCBDE
      ZUHUICGZHFZBUKIJZKZULAUKIJZKZUKUJFZUKUHFZUIULUMUOUIULKABIJZUMUOUIUSULABLM
      UIAHFZBHFZKULUSUMKUONZUIUTVAABOZABPZRUTVAULVBABUKUAUBUCUDUEUIVAUQUNSVDBUK
      QTUIUTURUPSVCAUKQTUFUG $.
  $}

  $( Trichotomy of the ordering relation on integers, stated in terms of upper
     integers.  (Contributed by NM, 6-Jul-2005.)  (Revised by Mario Carneiro,
     25-Jun-2013.) $)
  uztric $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
      ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) ) $=
    ( cz wcel wa cuz cfv wo cle wbr zletric eluz wb ancoms orbi12d mpbird ) ACD
    ZBCDZEZBAFGDZABFGDZHABIJZBAIJZHABKSTUBUAUCABLRQUAUCMBALNOP $.

  $( The upper integers function is one-to-one.  (Contributed by NM,
     12-Dec-2005.) $)
  uz11 $p |- ( M e. ZZ ->
                ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) ) $=
    ( cz wcel cuz cfv wceq wa eleq2 eluzel2 biimtrdi mpan9 cle imbitrrid eluzle
    uzid wbr syl6 cr zre imbitrid anim12d impl ancoms anassrs wb letri3 adantlr
    syl2an mpbird mpdan ex fveq2 impbid1 ) ACDZAEFZBEFZGZABGZUOURUSUOURHZBCDZUS
    UOAUPDZURVAAPZURVBAUQDZVAUPUQAIZBAJKLUTVAHUSABMQZBAMQZHZUOURVAVHURVAHUOVHUR
    VAUOVHURVAVFUOVGURVABUPDZVFVAVIURBUQDBPUPUQBINABORURUOVDVGUOVBURVDVCVEUABAO
    RUBUCUDUEUOVAUSVHUFZURUOASDBSDVJVAATBTABUGUIUHUJUKULABEUMUN $.

  $( Membership in the next upper set of integers.  (Contributed by NM,
     12-Sep-2005.) $)
  eluzp1m1 $p |- ( ( M e. ZZ /\
             N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) ) $=
    ( cz wcel c1 caddc co cuz cfv cmin cle wbr wa peano2zm ad2antrl cr wb eluz1
    zre 1re leaddsub mp3an2 syl2an biimpa anasss jca ex peano2z syl 3imtr4d imp
    ) ACDZBAEFGZHIDZBEJGZAHIDZULBCDZUMBKLZMZUOCDZAUOKLZMZUNUPULUSVBULUSMUTVAUQU
    TULURBNOULUQURVAULUQMURVAULAPDZBPDZURVAQZUQASBSVCEPDVDVETAEBUAUBUCUDUEUFUGU
    LUMCDUNUSQAUHUMBRUIAUORUJUK $.

  $( Strict ordering implied by membership in the next upper set of integers.
     (Contributed by NM, 12-Sep-2005.) $)
  eluzp1l $p |- ( ( M e. ZZ /\
             N e. ( ZZ>= ` ( M + 1 ) ) ) -> M < N ) $=
    ( cz wcel c1 caddc cuz cfv clt wbr cle eluzle adantl eluzelz zltp1le sylan2
    co wa wb mpbird ) ACDZBAEFQZGHDZRABIJZUBBKJZUCUEUAUBBLMUCUABCDUDUESUBBNABOP
    T $.

  $( Membership in the next upper set of integers.  (Contributed by NM,
     5-Oct-2005.) $)
  eluzp1p1 $p |- ( N e. ( ZZ>= ` M ) ->
                       ( N + 1 ) e. ( ZZ>= ` ( M + 1 ) ) ) $=
    ( cz wcel cle wbr w3a c1 caddc co cuz cfv peano2z 3ad2ant1 3ad2ant2 zre 1re
    cr wb eluz2 leadd1 mp3an3 syl2an biimp3a 3jca 3imtr4i ) ACDZBCDZABEFZGZAHIJ
    ZCDZBHIJZCDZUKUMEFZGBAKLDUMUKKLDUJULUNUOUGUHULUIAMNUHUGUNUIBMOUGUHUIUOUGARD
    ZBRDZUIUOSZUHAPBPUPUQHRDURQABHUAUBUCUDUEABTUKUMTUF $.

  ${
    eluzaddi.1 $e |- M e. ZZ $.
    eluzaddi.2 $e |- K e. ZZ $.
    $( Membership in a later upper set of integers.  (Contributed by Paul
       Chapman, 22-Nov-2007.) $)
    eluzaddi $p |- ( N e. ( ZZ>= ` M )
          -> ( N + K ) e. ( ZZ>= ` ( M + K ) ) ) $=
      ( cuz cfv wcel caddc co cz cle wbr eluzelz zaddcl sylancl wa eluz1i zrei
      cr wb zre leadd1 mp3an13 syl biimpa sylbi mp2an sylanbrc ) CBFGHZCAIJZKHZ
      BAIJZUKLMZUKUMFGHUJCKHZAKHZULBCNECAOPUJUOBCLMZQUNBCDRUOUQUNUOCTHZUQUNUAZC
      UBBTHURATHUSBDSAESBCAUCUDUEUFUGUMUKBKHUPUMKHDEBAOUHRUI $.

    $( Membership in an earlier upper set of integers.  (Contributed by Paul
       Chapman, 22-Nov-2007.) $)
    eluzsubi $p |- ( N e. ( ZZ>= ` ( M + K ) )
         -> ( N - K ) e. ( ZZ>= ` M ) ) $=
      ( caddc co cuz cfv wcel cmin cz cle eluzelz zsubcl sylancl eluz1i cr zrei
      wbr wa zaddcl mp2an wb zre leaddsub mp3an12 syl biimpa sylbi sylanbrc ) C
      BAFGZHIJZCAKGZLJZBUNMTZUNBHIJUMCLJZALJZUOULCNECAOPUMUQULCMTZUAUPULCBLJURU
      LLJDEBAUBUCQUQUSUPUQCRJZUSUPUDZCUEBRJARJUTVABDSAESBACUFUGUHUIUJBUNDQUK $.
  $}

  $( Membership in a later upper set of integers.  (Contributed by Jeff Madsen,
     2-Sep-2009.) $)
  eluzadd $p |- ( ( N e. ( ZZ>= ` M ) /\ K e. ZZ ) ->
                                      ( N + K ) e. ( ZZ>= ` ( M + K ) ) ) $=
    ( cuz cfv wcel cz wa caddc co cle eluzelz zaddcl sylan adantr zred wb eluz1
    wbr syl eluzel2 simpr simpl mpbid simprd leadd1dd zaddcld mpbir2and ) CBDEF
    ZAGFZHZCAIJZBAIJZDEFZULGFZUMULKSZUICGFZUJUOBCLZCAMNUKBCAUKBUIBGFZUJBCUAOZPU
    KCUIUQUJUROPUKAUIUJUBZPUKUQBCKSZUKUIUQVBHZUIUJUCUKUSUIVCQUTBCRTUDUEUFUKUMGF
    UNUOUPHQUKBAUTVAUGUMULRTUH $.

  $( Membership in an earlier upper set of integers.  (Contributed by Jeff
     Madsen, 2-Sep-2009.) $)
  eluzsub $p |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) )
                                          -> ( N - K ) e. ( ZZ>= ` M ) ) $=
    ( cz wcel caddc co cuz cfv w3a cmin cle wbr wa wb eluz1 syl mpbid cr zred
    eluzelz 3ad2ant3 simp2 zsubcld simp3 simp1 zaddcld simprd syl3anc mpbir2and
    leaddsub ) BDEZADEZCBAFGZHIEZJZCAKGZBHIEZUQDEZBUQLMZUPCAUOULCDEZUMUNCUAUBZU
    LUMUOUCZUDUPUNCLMZUTUPVAVDUPUOVAVDNZULUMUOUEUPUNDEUOVEOUPBAULUMUOUFZVCUGUNC
    PQRUHUPBSEASECSEVDUTOUPBVFTUPAVCTUPCVBTBACUKUIRUPULURUSUTNOVFBUQPQUJ $.

  $( Choices for an element of an upper interval of integers.  (Contributed by
     Jeff Madsen, 2-Sep-2009.) $)
  uzm1 $p |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) ) $=
    ( cuz cfv wcel wceq clt wbr wo c1 cmin co wn eluzle eluzel2 zred cz syl2anc
    cle wb eluzelz lenltd mpbid w3o ztri3or df-3or sylib ecased orcomd wi eqcom
    biimpi a1i zltlem1 1zzd zsubcld eluz bitr4d biimpd orim12d mpd ) BACDZEZABF
    ZABGHZIBAFZBJKLZVBEZIVCVEVDVCVEVDIZBAGHZVCABSHVJMABNVCABVCAABOZPVCBABUAZPUB
    UCVCVEVDVJUDZVIVJIVCAQEZBQEZVMVKVLABUERVEVDVJUFUGUHUIVCVDVFVEVHVDVFUJVCVDVF
    ABUKULUMVCVEVHVCVEAVGSHZVHVCVNVOVEVPTVKVLABUNRVCVNVGQEVHVPTVKVCBJVLVCUOUPAV
    GUQRURUSUTVA $.

  $( The nonnegative difference of integers is a nonnegative integer.
     (Contributed by NM, 4-Sep-2005.) $)
  uznn0sub $p |- ( N e. ( ZZ>= ` M ) -> ( N - M ) e. NN0 ) $=
    ( cuz cfv wcel cz cle wbr w3a cmin co cn0 eluz2 znn0sub biimp3a sylbi ) BAC
    DEAFEZBFEZABGHZIBAJKLEZABMQRSTABNOP $.

  $( Intersection of two upper intervals of integers.  (Contributed by Mario
     Carneiro, 24-Dec-2013.) $)
  uzin $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
    ( ( ZZ>= ` M ) i^i ( ZZ>= ` N ) ) = ( ZZ>= ` if ( M <_ N , N , M ) ) ) $=
    ( cz wcel wa cuz cfv wo cle wbr wceq wss uzss sylib eluzle fveq2d eqtr4d cr
    syl zre cin uztric sseqin2 iftrue df-ss wn wb eluzel2 eluzelz letri3 syl2an
    cif syl2anc biantrurd bitr4d biimprcd eqeq1d sylibrd com12 wi iffalse zdcle
    a1i wdc df-dc mpjaod jaoi ) ACDZBCDZEBAFGZDZABFGZDZHVJVLUAZABIJZBAULZFGZKZA
    BUBVKVRVMVKVNVLVQVKVLVJLVNVLKABMVLVJUCNVKVPBFVKVOVPBKABOVOBAUDZSPQVMVNVJVQV
    MVJVLLVNVJKBAMVJVLUENVMVPAFVMVOVPAKZVOUFZVOVMVTVOVMBAKZVTVMWBVOVMWBBAIJZVOE
    ZVOVMVIVHWBWDUGZBAUHZBAUIZVIBRDARDWEVHBTATBAUJUKUMVMWCVOBAOUNUOUPVOVPBAVSUQ
    URUSWAVTUTVMVOBAVAVCVMVOVDZVOWAHVMVHVIWHWGWFABVBUMVOVENVFPQVGS $.

  $( Choices for an element of an upper interval of integers.  (Contributed by
     Jeff Madsen, 2-Sep-2009.) $)
  uzp1 $p |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) ) $=
    ( cuz cfv wcel wceq c1 cmin co wo caddc uzm1 eluzp1p1 eluzelcn ax-1cn npcan
    cc sylancl eleq1d imbitrid orim2d mpd ) BACDZEZBAFZBGHIZUCEZJUEBAGKICDZEZJA
    BLUDUGUIUEUGUFGKIZUHEUDUIAUFMUDUJBUHUDBQEGQEUJBFABNOBGPRSTUAUB $.

  $( Nonnegative integers expressed as an upper set of integers.  (Contributed
     by NM, 2-Sep-2005.) $)
  nn0uz $p |- NN0 = ( ZZ>= ` 0 ) $=
    ( vk cn0 cc0 cv cle wbr cz crab cuz nn0zrab wcel wceq 0z uzval ax-mp eqtr4i
    cfv ) BCADEFAGHZCIQZAJCGKSRLMACNOP $.

  $( Positive integers expressed as an upper set of integers.  (Contributed by
     NM, 2-Sep-2005.) $)
  nnuz $p |- NN = ( ZZ>= ` 1 ) $=
    ( vk cn c1 cv cle wbr cz crab cuz cfv nnzrab wcel wceq uzval ax-mp eqtr4i
    1z ) BCADEFAGHZCIJZAKCGLSRMQACNOP $.

  $( A positive integer expressed as a member of an upper set of integers.
     (Contributed by NM, 6-Jun-2006.) $)
  elnnuz $p |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) ) $=
    ( cn c1 cuz cfv nnuz eleq2i ) BCDEAFG $.

  $( A nonnegative integer expressed as a member an upper set of integers.
     (Contributed by NM, 6-Jun-2006.) $)
  elnn0uz $p |- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) ) $=
    ( cn0 cc0 cuz cfv nn0uz eleq2i ) BCDEAFG $.

  $( 5 is an integer greater than or equal to 3.  (Contributed by AV,
     7-Sep-2025.) $)
  5eluz3 $p |- 5 e. ( ZZ>= ` 3 ) $=
    ( c5 c3 cuz cfv wcel cz cle wbr 5nn nnzi 3re 5re 3lt5 ltleii eluz2 mpbir3an
    3z ) ABCDEBFEAFEBAGHQAIJBAKLMNBAOP $.

  $( An integer in the upper set of integers starting at 3 is element of the
     upper set of integers starting at 2.  (Contributed by Alexander van der
     Vekens, 17-Sep-2018.) $)
  uzuzle23 $p |- ( A e. ( ZZ>= ` 3 ) -> A e. ( ZZ>= ` 2 ) ) $=
    ( c2 cz wcel c3 cle wbr cuz cfv wi 2z 2re 3re 2lt3 ltleii eluzuzle mp2an )
    BCDBEFGAEHIDABHIDJKBELMNOEBAPQ $.

  $( An integer greater than or equal to 4 is an integer greater than or equal
     to 2.  (Contributed by AV, 30-May-2023.) $)
  uzuzle24 $p |- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 2 ) ) $=
    ( c2 cz wcel c4 cle wbr cuz cfv wi 2z 2re 4re 2lt4 ltleii eluzuzle mp2an )
    BCDBEFGAEHIDABHIDJKBELMNOEBAPQ $.

  $( An integer greater than or equal to 4 is an integer greater than or equal
     to 3.  (Contributed by AV, 5-Sep-2025.) $)
  uzuzle34 $p |- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 3 ) ) $=
    ( c3 cz wcel c4 cle wbr cuz cfv wi 3z 3re 4re 3lt4 ltleii eluzuzle mp2an )
    BCDBEFGAEHIDABHIDJKBELMNOEBAPQ $.

  $( An integer greater than or equal to 5 is an integer greater than or equal
     to 3.  (Contributed by AV, 15-Nov-2025.) $)
  uzuzle35 $p |- ( A e. ( ZZ>= ` 5 ) -> A e. ( ZZ>= ` 3 ) ) $=
    ( c5 cuz cfv c3 wcel wss 5eluz3 uzss ax-mp sseli ) BCDZECDZABMFLMGHEBIJK $.

  $( An integer is greater than or equal to 2 is a positive integer.
     (Contributed by AV, 3-Nov-2018.) $)
  eluz2nn $p |- ( A e. ( ZZ>= ` 2 ) -> A e. NN ) $=
    ( c2 cuz cfv wcel c1 cn cz cle wbr wi 1z 1le2 eluzuzle mp2an nnuz eleqtrrdi
    ) ABCDEZAFCDZGFHEFBIJRASEKLMBFANOPQ $.

  $( An integer greater than or equal to 3 is a positive integer.  (Contributed
     by Alexander van der Vekens, 17-Sep-2018.)  (Proof shortened by AV,
     30-Nov-2025.) $)
  eluz3nn $p |- ( N e. ( ZZ>= ` 3 ) -> N e. NN ) $=
    ( c3 cuz cfv wcel c2 cn uzuzle23 eluz2nn syl ) ABCDEAFCDEAGEAHAIJ $.

  $( An integer greater than or equal to 4 is an integer greater than or equal
     to 2.  (Contributed by AV, 30-May-2023.) $)
  eluz4eluz2 $p |- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 2 ) ) $=
    ( c2 cz wcel c4 cle wbr cuz cfv wi 2z 2re 4re 2lt4 ltleii eluzuzle mp2an )
    BCDBEFGAEHIDABHIDJKBELMNOEBAPQ $.

  $( An integer greater than or equal to 4 is a positive integer.  (Contributed
     by AV, 30-May-2023.) $)
  eluz4nn $p |- ( X e. ( ZZ>= ` 4 ) -> X e. NN ) $=
    ( c4 cuz cfv wcel c2 cn eluz4eluz2 eluz2nn syl ) ABCDEAFCDEAGEAHAIJ $.

  $( If an integer is greater than or equal to 2, then it is a nonnegative
     integer.  (Contributed by AV, 27-Aug-2018.)  (Proof shortened by AV,
     3-Nov-2018.) $)
  eluzge2nn0 $p |- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 ) $=
    ( c2 cuz cfv wcel eluz2nn nnnn0d ) ABCDEAAFG $.

  $( An integer greater than or equal to 2 is not 0.  (Contributed by AV,
     25-May-2020.) $)
  eluz2n0 $p |- ( N e. ( ZZ>= ` 2 ) -> N =/= 0 ) $=
    ( c2 cuz cfv wcel eluz2nn nnne0d ) ABCDEAAFG $.

  $( If an integer is greater than 3, then it is a positive integer.
     (Contributed by Alexander van der Vekens, 17-Sep-2018.) $)
  eluzge3nn $p |- ( N e. ( ZZ>= ` 3 ) -> N e. NN ) $=
    ( c3 cuz cfv wcel c1 cn cz cle wbr wi 1z 1le3 eluzuzle mp2an elnnuz sylibr
    ) ABCDEZAFCDEZAGEFHEFBIJRSKLMBFANOAPQ $.

  $( An integer greater than or equal to 3 decreased by 2 is a positive
     integer.  (Contributed by Alexander van der Vekens, 17-Sep-2018.) $)
  uz3m2nn $p |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN ) $=
    ( c3 cuz cfv wcel c2 clt wbr cmin co cn cz cle w3a eluz2 2lt3 cr wa wi a1i
    2re 3re zre ltletr syl3anc mpani imp 3adant1 sylbi wb 2nn eluzge3nn sylancr
    nnsub mpbid ) ABCDEZFAGHZAFIJKEZUPBLEZALEZBAMHZNUQBAOUTVAUQUSUTVAUQUTFBGHZV
    AUQPUTFQEZBQEZAQEVBVARUQSVCUTUATVDUTUBTAUCFBAUDUEUFUGUHUIUPFKEAKEUQURUJUKAU
    LFAUNUMUO $.

  $( 1 is an integer greater than or equal to 0.  (Contributed by Alexander van
     der Vekens, 8-Jun-2018.) $)
  1eluzge0 $p |- 1 e. ( ZZ>= ` 0 ) $=
    ( c1 cc0 cuz cfv wcel cz cle wbr 0z 1z 0le1 eluz2 mpbir3an ) ABCDEBFEAFEBAG
    HIJKBALM $.

  $( 2 is an integer greater than or equal to 0.  (Contributed by Alexander van
     der Vekens, 8-Jun-2018.)  (Proof shortened by OpenAI, 25-Mar-2020.) $)
  2eluzge0 $p |- 2 e. ( ZZ>= ` 0 ) $=
    ( c2 cn0 cc0 cuz cfv 2nn0 nn0uz eleqtri ) ABCDEFGH $.

  $( 2 is an integer greater than or equal to 1.  (Contributed by Alexander van
     der Vekens, 8-Jun-2018.) $)
  2eluzge1 $p |- 2 e. ( ZZ>= ` 1 ) $=
    ( c2 c1 cuz cfv wcel cz cle wbr 1z 2z 1le2 eluz2 mpbir3an ) ABCDEBFEAFEBAGH
    IJKBALM $.

  $( The upper integers starting from a natural are a subset of the naturals.
     (Contributed by Scott Fenton, 29-Jun-2013.) $)
  uznnssnn $p |- ( N e. NN -> ( ZZ>= ` N ) C_ NN ) $=
    ( cn wcel cuz cfv c1 wss elnnuz uzss sylbi nnuz sseqtrrdi ) ABCZADEZFDEZBMA
    OCNOGAHFAIJKL $.

  ${
    $d m n M $.
    $( Restricted universal quantification in an upper set of integers.
       (Contributed by NM, 9-Sep-2005.) $)
    raluz $p |- ( M e. ZZ -> ( A. n e. ( ZZ>= ` M ) ph <->
               A. n e. ZZ ( M <_ n -> ph ) ) ) $=
      ( cz wcel cv cle wbr wi cuz cfv wa eluz1 imbi1d impexp bitrdi ralbidv2 )
      CDEZACBFZGHZAIZBCJKZDRSUBEZAISDEZTLZAIUDUAIRUCUEACSMNUDTAOPQ $.

    $( Restricted universal quantification in an upper set of integers.
       (Contributed by NM, 9-Sep-2005.) $)
    raluz2 $p |- ( A. n e. ( ZZ>= ` M ) ph <->
               ( M e. ZZ -> A. n e. ZZ ( M <_ n -> ph ) ) ) $=
      ( cuz cfv wral cz wcel cv cle wbr wi w3a eluz2 3anass bitri imbi1i impexp
      wa imbi2i bi2.04 ralbii2 r19.21v ) ABCDEZFCGHZCBIZJKZALZLZBGFUEUHBGFLAUIB
      UDGUFUDHZALUEUFGHZUGSZSZALZUKUILZUJUMAUJUEUKUGMUMCUFNUEUKUGOPQUNUEUKUHLZL
      ZUOUNUEULALZLUQUEULARURUPUEUKUGARTPUEUKUHUAPPUBUEUHBGUCP $.

    $( Restricted existential quantification in an upper set of integers.
       (Contributed by NM, 9-Sep-2005.) $)
    rexuz $p |- ( M e. ZZ -> ( E. n e. ( ZZ>= ` M ) ph <->
               E. n e. ZZ ( M <_ n /\ ph ) ) ) $=
      ( cz wcel cv cle wbr wa cuz cfv eluz1 anbi1d anass bitrdi rexbidv2 ) CDEZ
      ACBFZGHZAIZBCJKZDQRUAEZAIRDEZSIZAIUCTIQUBUDACRLMUCSANOP $.

    $( Restricted existential quantification in an upper set of integers.
       (Contributed by NM, 9-Sep-2005.) $)
    rexuz2 $p |- ( E. n e. ( ZZ>= ` M ) ph <->
               ( M e. ZZ /\ E. n e. ZZ ( M <_ n /\ ph ) ) ) $=
      ( cuz cfv wrex cz wcel cv cle wbr wa eluz2 df-3an bitri anbi1i anass an12
      w3a rexbii2 r19.42v ) ABCDEZFCGHZCBIZJKZALZLZBGFUCUFBGFLAUGBUBGUDUBHZALUC
      UDGHZLZUELZALZUIUGLZUHUKAUHUCUIUESUKCUDMUCUIUENOPULUJUFLZUMUJUEAQUNUCUIUF
      LLUMUCUIUFQUCUIUFROOOTUCUFBGUAO $.

    $( Double existential quantification in an upper set of integers.
       (Contributed by NM, 3-Nov-2005.) $)
    2rexuz $p |- ( E. m E. n e. ( ZZ>= ` m ) ph <->
                  E. m e. ZZ E. n e. ZZ ( m <_ n /\ ph ) ) $=
      ( cv cuz cfv wrex wex cz wcel cle wbr wa rexuz2 exbii df-rex bitr4i ) ACB
      DZEFGZBHRIJRCDKLAMCIGZMZBHTBIGSUABACRNOTBIPQ $.
  $}

  $( Second Peano postulate for an upper set of integers.  (Contributed by NM,
     7-Sep-2005.) $)
  peano2uz $p |- ( N e. ( ZZ>= ` M ) ->
                 ( N + 1 ) e. ( ZZ>= ` M ) ) $=
    ( cz wcel cle wbr w3a c1 caddc co cuz cfv simp1 peano2z 3ad2ant2 zre letrp1
    cr syl3an2 eluz2 syl3an1 3jca 3imtr4i ) ACDZBCDZABEFZGZUDBHIJZCDZAUHEFZGBAK
    LZDUHUKDUGUDUIUJUDUEUFMUEUDUIUFBNOUDARDZUEUFUJAPUEULBRDUFUJBPABQSUAUBABTAUH
    TUC $.

  ${
    peano2uzs.1 $e |- Z = ( ZZ>= ` M ) $.
    $( Second Peano postulate for an upper set of integers.  (Contributed by
       Mario Carneiro, 26-Dec-2013.) $)
    peano2uzs $p |- ( N e. Z -> ( N + 1 ) e. Z ) $=
      ( c1 caddc co wcel cuz cfv peano2uz eleqtrrdi eleq2s ) BEFGZCHBAIJZCBOHNO
      CABKDLDM $.
  $}

  $( Reversed second Peano axiom for upper integers.  (Contributed by NM,
     2-Jan-2006.) $)
  peano2uzr $p |- ( ( M e. ZZ /\
             N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` M ) ) $=
    ( cz wcel c1 caddc co cuz cfv wa cmin wceq cc eluzelcn ax-1cn npcan sylancl
    adantl eluzp1m1 peano2uz syl eqeltrrd ) ACDZBAEFGZHIDZJZBEKGZEFGZBAHIZUEUHB
    LZUCUEBMDEMDUJUDBNOBEPQRUFUGUIDUHUIDABSAUGTUAUB $.

  ${
    $d j K $.  $d j k M $.  $d j k N $.
    $( Addition closure law for an upper set of integers.  (Contributed by NM,
       4-Jun-2006.) $)
    uzaddcl $p |- ( ( N e. ( ZZ>= ` M ) /\ K e. NN0 ) ->
                   ( N + K ) e. ( ZZ>= ` M ) ) $=
      ( vj vk cn0 wcel cuz caddc co cv wi cc0 c1 wceq oveq2 eleq1d imbi2d wa cc
      eluzelcn addridd ibir ax-1cn addass mp3an3 syl2anr adantr peano2uz adantl
      cfv nn0cn eqeltrrd exp31 a2d nn0ind impcom ) AFGCBHUKZGZCAIJZURGZUSCDKZIJ
      ZURGZLUSCMIJZURGZLUSCEKZIJZURGZLUSCVGNIJZIJZURGZLUSVALDEAVBMOZVDVFUSVMVCV
      EURVBMCIPQRVBVGOZVDVIUSVNVCVHURVBVGCIPQRVBVJOZVDVLUSVOVCVKURVBVJCIPQRVBAO
      ZVDVAUSVPVCUTURVBACIPQRUSVFUSVECURUSCBCUAZUBQUCVGFGZUSVIVLVRUSVIVLVRUSSZV
      ISVHNIJZVKURVSVTVKOZVIUSCTGZVGTGZWAVRVQVGULWBWCNTGWAUDCVGNUEUFUGUHVIVTURG
      VSBVHUIUJUMUNUOUPUQ $.
  $}

  $( The sum of a nonnegative integer and an integer is an integer greater than
     or equal to that integer.  (Contributed by Alexander van der Vekens,
     3-Oct-2018.) $)
  nn0pzuz $p |- ( ( N e. NN0 /\ Z e. ZZ ) -> ( N + Z ) e. ( ZZ>= ` Z ) ) $=
    ( cn0 wcel cz wa caddc co cle wbr cuz cfv simpr nn0z zaddcl sylan nn0addge2
    cr zre ancoms eluz2 syl3anbrc ) ACDZBEDZFUDABGHZEDZBUEIJZUEBKLDUCUDMUCAEDUD
    UFANABOPUDUCUGUDBRDUCUGBSBAQPTBUEUAUB $.

  ${
    $d j m N $.  $d j ps $.  $d j ch $.  $d j th $.  $d j ta $.  $d k ph $.
    $d j k m M $.
    $( Substitutions. $)
    uzind4.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
    uzind4.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
    uzind4.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
    uzind4.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
    $( Basis. $)
    uzind4.5 $e |- ( M e. ZZ -> ps ) $.
    $( Induction step. $)
    uzind4.6 $e |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) ) $.
    $( Induction on the upper set of integers that starts at an integer ` M ` .
       The first four hypotheses give us the substitution instances we need,
       and the last two are the basis and the induction step.  (Contributed by
       NM, 7-Sep-2005.) $)
    uzind4 $p |- ( N e. ( ZZ>= ` M ) -> ta ) $=
      ( vm wcel cz cle wbr cuz cfv cv crab eluzel2 eluzelz breq2 elrab sylanbrc
      eluzle wa wi w3a eluz2 biimpri 3expb sylan2b syl uzind3 syl2anc ) IHUAUBZ
      QZHRQZIHPUCZSTZPRUDZQZEHIUEVBIRQHISTZVGHIUFHIUJVEVHPIRVDIHSUGUHUIABCDEFPG
      HIJKLMNVCGUCZVFQZUKVIVAQZCDULVJVCVIRQZHVISTZUKVKVEVMPVIRVDVIHSUGUHVCVLVMV
      KVKVCVLVMUMHVIUNUOUPUQOURUSUT $.
  $}

  ${
    $d j N $.  $d j ps $.  $d j ch $.  $d j th $.  $d j ta $.  $d k ph $.
    $d j k M $.
    $( Basis. $)
    uzind4ALT.5 $e |- ( M e. ZZ -> ps ) $.
    $( Induction step. $)
    uzind4ALT.6 $e |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) ) $.
    $( Substitutions. $)
    uzind4ALT.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
    uzind4ALT.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
    uzind4ALT.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
    uzind4ALT.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
    $( Induction on the upper set of integers that starts at an integer ` M ` .
       The last four hypotheses give us the substitution instances we need; the
       first two are the basis and the induction step.  Either ~ uzind4 or
       ~ uzind4ALT may be used; see comment for ~ nnind .  (Contributed by NM,
       7-Sep-2005.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    uzind4ALT $p |- ( N e. ( ZZ>= ` M ) -> ta ) $=
      ( uzind4 ) ABCDEFGHILMNOJKP $.
  $}

  ${
    $d m k j M $.  $d j N $.  $d j m ph $.
    $( Basis. $)
    uzind4s.1 $e |- ( M e. ZZ -> [. M / k ]. ph ) $.
    $( Induction step. $)
    uzind4s.2 $e |- ( k e. ( ZZ>= ` M )
         -> ( ph -> [. ( k + 1 ) / k ]. ph ) ) $.
    $( Induction on the upper set of integers that starts at an integer ` M ` ,
       using explicit substitution.  The hypotheses are the basis and the
       induction step.  (Contributed by NM, 4-Nov-2005.) $)
    uzind4s $p |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph ) $=
      ( vj vm wsb wsbc cv c1 caddc co dfsbcq2 sbequ wcel wi nfim imbi12d uzind4
      cuz cfv nfv nfs1v nfsbc1v weq eleq1 sbequ12 oveq1 sbceq1d chvar ) ABGIABC
      JABHIZABHKZLMNZJZABDJGHCDABGCOAGHBPABGUOOABGDOEBKZCUBUCZQZAABUQLMNZJZRZRU
      NURQZUMUPRZRBHVCVDBVCBUDUMUPBABHUEABUOUFSSBHUGZUSVCVBVDUQUNURUHVEAUMVAUPA
      BHUIVEABUTUOUQUNLMUJUKTTFULUA $.
  $}

  ${
    $d k m n M $.  $d m N $.  $d k m n ph $.  $d j k m n $.
    $( Basis. $)
    uzind4s2.1 $e |- ( M e. ZZ -> [. M / j ]. ph ) $.
    $( Induction step. $)
    uzind4s2.2 $e |- ( k e. ( ZZ>= ` M ) ->
                     ( [. k / j ]. ph -> [. ( k + 1 ) / j ]. ph ) ) $.
    $( Induction on the upper set of integers that starts at an integer ` M ` ,
       using explicit substitution.  The hypotheses are the basis and the
       induction step.  Use this instead of ~ uzind4s when ` j ` and ` k ` must
       be distinct in ` [. ( k + 1 ) / j ]. ph ` .  (Contributed by NM,
       16-Nov-2005.) $)
    uzind4s2 $p |- ( N e. ( ZZ>= ` M ) -> [. N / j ]. ph ) $=
      ( vm vn cv wsbc c1 caddc co dfsbcq wi cuz cfv weq oveq1 sbceq1d imbi12d
      vtoclga uzind4 ) ABHJZKABDKABIJZKZABUFLMNZKZABEKHIDEABUEDOABUEUFOABUEUHOA
      BUEEOFABCJZKZABUJLMNZKZPUGUIPCUFDQRCISZUKUGUMUIABUJUFOUNABULUHUJUFLMTUAUB
      GUCUD $.
  $}

  ${
    $d j N $.  $d j ps $.  $d j ch $.  $d j th $.  $d j ta $.  $d k ph $.
    $d j k M $.
    $( Substitutions. $)
    uzind4i.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
    uzind4i.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
    uzind4i.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
    uzind4i.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
    $( Basis. $)
    uzind4i.5 $e |- ps $.
    $( Induction step. $)
    uzind4i.6 $e |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) ) $.
    $( Induction on the upper integers that start at ` M ` .  The first four
       give us the substitution instances we need, and the last two are the
       basis and the induction step.  This is a stronger version of ~ uzind4
       assuming that ` ps ` holds unconditionally.  Notice that
       ` N e. ( ZZ>= `` M ) ` implies that the lower bound ` M ` is an integer
       ( ` M e. ZZ ` , see ~ eluzel2 ).  (Contributed by NM, 4-Sep-2005.)
       (Revised by AV, 13-Jul-2022.) $)
    uzind4i $p |- ( N e. ( ZZ>= ` M ) -> ta ) $=
      ( cz wcel a1i uzind4 ) ABCDEFGHIJKLMBHPQNROS $.
  $}

  ${
    $d x y z $.  $d y w ph $.  $d x w z ps $.
    $( Substitutions. $)
    indstr.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Induction step. $)
    indstr.2 $e |- ( x e. NN -> ( A. y e. NN ( y < x -> ps ) -> ph ) ) $.
    $( Strong Mathematical Induction for positive integers (inference schema).
       (Contributed by NM, 17-Aug-2001.) $)
    indstr $p |- ( x e. NN -> ph ) $=
      ( vz vw vu cv cn wcel clt wbr wi wral c1 breq2 imbi1d ralbidv co wceq weq
      caddc nnnlt1 pm2.21d rgen wex wb 1nn elex2 nfra1 r19.3rm mp2b wa wo com12
      rsp adantl wsbc nfv nfsbc1v nfim sbceq1a imbi12d cbvral mpbi rspec dfsbcq
      vex sbcie bitr3id biimprcd syl6 adantr jcad imbitrrdi cle nnleltp1 cz nnz
      jaob zleloe syl2an bitr3d ancoms sylibrd ralimdva biimtrid nnind mpd ) CJ
      ZKLDJZWLMNZBOZDKPZAWMGJZMNZBOZDKPWMQMNZBOZDKPWMHJZMNZBOZDKPZWMXBQUDUAZMNZ
      BOZDKPZWPGHWLWQQUBZWSXADKXJWRWTBWQQWMMRSTGHUCZWSXDDKXKWRXCBWQXBWMMRSTWQXF
      UBZWSXHDKXLWRXGBWQXFWMMRSTGCUCZWSWODKXMWRWNBWQWLWMMRSTXADKWMKLZWTBWMUEUFU
      GXEXEDKPZXBKLZXIQKLIJKLIUHXEXOUIUJIQKUKXEDIKXDDKULUMUNXPXEXHDKXPXNUOZXEXC
      DHUCZUPZBOZXHXQXEXDXRBOZUOXTXQXEXDYAXNXEXDOXPXEXNXDXDDKURUQUSXPXEYAOXNXPX
      EACXBUTZYAXEYBOZHKWPAOZCKPYCHKPYDCKFUGYDYCCHKYDHVAXEYBCXECVAACXBVBVCCHUCZ
      WPXEAYBYEWOXDDKYEWNXCBWLXBWMMRSTACXBVDVEVFVGVHXRBYBBACWMUTXRYBABCWMDVJEVK
      ACWMXBVIVLVMVNVOVPXCBXRWBVQXQXGXSBXNXPXGXSUIXNXPUOWMXBVRNZXGXSWMXBVSXNWMV
      TLXBVTLYFXSUIXPWMWAXBWAWMXBWCWDWEWFSWGWHWIWJFWK $.
  $}

  ${
    $d A f g w x y z $.  $d f g ph w x y z $.
    infrenegsupex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. y < x
        /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) ) $.
    infrenegsupex.ss $e |- ( ph -> A C_ RR ) $.
    $( The infimum of a set of reals ` A ` is the negative of the supremum of
       the negatives of its elements.  (Contributed by Jim Kingdon,
       14-Jan-2022.) $)
    infrenegsupex $p |- ( ph -> inf ( A , RR , < )
        = -u sup ( { z e. RR | -u z e. A } , RR , < ) ) $=
      ( vg vf vw cr clt cneg cv wcel csup wa wceq wbr wiso cinf crab wn infclti
      wb lttri3 adantl recnd negnegd ccnv cmpt cfv cima negeq cbvmptv mptpreima
      eqid negiso simpri imaeq1i eqtr3i supeq1i simpli isocnv ax-mp isoeq1 mpbi
      a1i cnvinfex supisoti eqtrid df-inf eqcomi fveq2i cc negcld fvmptd eqtr2d
      cnvti eqidd negeqd eqtr3d ) AEKLUAZMZMWCDNZMZEODKUBZKLPZMAWCAWCABCDHIKELI
      NZKOHNZKOQWIWJRWIWJLSUCWJWILSUCQUEAWIWJUFUGZFUDZUHZUIAWDWHAWHEKLUJZPZJKJN
      ZMZUKZULZWDAWHWREUMZKLPWSKWGWTLWRUJZEUMWGWTDKWFEWRJDKWQWFWPWEUNUOUPXAWREK
      KLWNWRTZXAWRRZJWRWRUQURZUSZUTVAVBABCDHIKKEWNLWRKKWNLWRTZAKKWNLXATZXFXBXGX
      BXCXDVCKKLWNWRVDVEXCXGXFUEXEKKWNLWRXAVFVEVGVHGABCDKELFVIAHIKLWKVSVJVKAWSW
      CWRULWDWOWCWRWCWOEKLVLVMVNAJWCWQWDKWRVOAWRVTWPWCRWQWDRAWPWCUNUGWLAWCWMVPV
      QVKVRWAWB $.
  $}

  ${
    $d A a b c y z $.  $d A a c w y z $.  $d A a w x y z $.  $d a c ph y $.
    supinfneg.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. x < y
        /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $.
    supinfneg.ss $e |- ( ph -> A C_ RR ) $.
    $( If a set of real numbers has a least upper bound, the set of the
       negation of those numbers has a greatest lower bound.  For a theorem
       which is similar but only for the boundedness part, see ~ ublbneg .
       (Contributed by Jim Kingdon, 15-Jan-2022.) $)
    supinfneg $p |- ( ph -> E. x e. RR (
        A. y e. { w e. RR | -u w e. A } -. y < x /\
        A. y e. RR ( x < y -> E. z e. { w e. RR | -u w e. A } z < y ) ) ) $=
      ( va vb vc cv clt wbr wral wrex wi cr wa wcel cneg crab weq breq1 ralbidv
      notbid breq2 imbi1d anbi12d cbvrexv sylibr cbvralv imbi2i rexbidv imbi12d
      wn ralbii bitri anbi12i rexbii ad2antlr simplr simprl elrabi negeq eleq1d
      renegcl elrab3 biimpd mpcom wceq rspcv syl adantr ltnegcon2 sylan sylibrd
      ancoms ralrimdva sylc nfcv nfre1 nfim nfralya nfan sseld ad6antr renegcld
      wb nfv mpd recnd negnegd eqeltrd elrab sylanbrc simp-4r ltnegcon1d rspcev
      simpr syl2anc simpllr jca31 ltnegcon1 adantl sylbid r19.29af ex ralrimiva
      imp adantrl syl12anc rexlimdva ) AILZJLZMNZUPZJFOZXOXNMNZXOKLZMNZKFPZQZJR
      OZSZIRPZCLZBLZMNZUPZCELZUAZFTZERUBZOZYHYGMNZDLZYGMNZDYNPZQZCROZSZBRPZAXNY
      GMNZUPZCFOZYGXNMNZYGYQMNZDFPZQZCROZSZIRPZYFAYPUPZCFOZYIUUIQZCROZSZBRPUUMG
      UULUURIBRIBUCZUUFUUOUUKUUQUUSUUEUUNCFUUSUUDYPXNYHYGMUDUFUEUUSUUJUUPCRUUSU
      UGYIUUIXNYHYGMUGUHUEUIUJUKYEUULIRXRUUFYDUUKXQUUEJCFJCUCZXPUUDXOYGXNMUGUFU
      LYDXSXOYQMNZDFPZQZJROUUKYCUVCJRYBUVBXSYAUVAKDFXTYQXOMUGUJUMUQUVCUUJJCRUUT
      XSUUGUVBUUIXOYGXNMUDUUTUVAUUHDFXOYGYQMUDUNUOULURUSUTUKAYEUUCIRAXNRTZSZYEU
      UCUVEYESZXNUAZRTZYGUVGMNZUPZCYNOZUVGYGMNZYSQZCROZUUCUVDUVHAYEXNVGVAUVFUVD
      XRUVKAUVDYEVBUVEXRYDVCUVDXRUVJCYNYGYNTZUVDXRUVJQUVOUVDSXRXNYGUAZMNZUPZUVJ
      UVOXRUVRQZUVDUVOUVPFTZUVSYGRTZUVOUVTYMEYGRVDZUWAUVOUVTYMUVTEYGRECUCYLUVPF
      YKYGVEVFVHVIVJXQUVRJUVPFXOUVPVKZXPUVQXOUVPXNMUGUFVLVMVNUVOUWAUVDUVJUVRWIU
      WBUWAUVDSUVIUVQYGXNVOUFVPVQVRVSVTUVEYDUVNXRUVEYDSZUVMCRUWDUWASZUVLYSUWEUV
      LSZUVPXTMNZYSKFUWEUVLKUWDUWAKUVEYDKUVEKWJYCKJRKRWAXSYBKXSKWJYAKFWBWCWDWEU
      WAKWJWEUVLKWJWEUWFXTFTZSZUWGSZXTUAZYNTZUWKYGMNZYSUWJUWKRTUWKUAZFTZUWLUWJX
      TUWJUWHXTRTZUWFUWHUWGVBZAUWHUWPQUVDYDUWAUVLUWHUWGAFRXTHWFWGWKZWHUWJUWNXTF
      UWJXTUWJXTUWRWLWMUWQWNYMUWOEUWKRYKUWKVKYLUWNFYKUWKVEVFWOWPUWJYGXTUWDUWAUV
      LUWHUWGWQUWRUWIUWGWTWRYRUWMDUWKYNYQUWKYGMUDWSXAUWEUVDUWASZYDSZUVLUWGKFPZU
      WEUVDUWAYDAUVDYDUWAXBUWDUWAWTUVEYDUWAVBXCUWTUVLUXAUWTUVLUVPXNMNZUXAUWSUVL
      UXBWIYDXNYGXDVNUWSYDUXBUXAQZUWAYDUXCQZUVDUWAUVPRTUXDYGVGYCUXCJUVPRUWCXSUX
      BYBUXAXOUVPXNMUDUWCYAUWGKFXOUVPXTMUDUNUOVLVMXEXJXFXJVPXGXHXIXKUUBUVKUVNSB
      UVGRYHUVGVKZYOUVKUUAUVNUXEYJUVJCYNUXEYIUVIYHUVGYGMUGUFUEUXEYTUVMCRUXEYPUV
      LYSYHUVGYGMUDUHUEUIWSXLXHXMWK $.
  $}

  ${
    $d A a b c y z $.  $d A a c w y z $.  $d A a w x y z $.  $d a c ph y $.
    infsupneg.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. y < x
        /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) ) $.
    infsupneg.ss $e |- ( ph -> A C_ RR ) $.
    $( If a set of real numbers has a greatest lower bound, the set of the
       negation of those numbers has a least upper bound.  To go in the other
       direction see ~ supinfneg .  (Contributed by Jim Kingdon,
       15-Jan-2022.) $)
    infsupneg $p |- ( ph -> E. x e. RR (
        A. y e. { w e. RR | -u w e. A } -. x < y /\
        A. y e. RR ( y < x -> E. z e. { w e. RR | -u w e. A } y < z ) ) ) $=
      ( vb va vc cv clt wbr wral wrex wi cr wa wcel cneg crab weq breq2 ralbidv
      notbid breq1 imbi1d anbi12d cbvrexv sylibr cbvralv imbi2i rexbidv imbi12d
      wn ralbii bitri anbi12i rexbii ad2antlr simplr simprl elrabi negeq eleq1d
      renegcl elrab3 biimpd mpcom wceq rspcv adantr wb ltnegcon1 ancoms sylibrd
      syl sylan ralrimdva sylc nfv nfcv nfre1 nfim nfralya ad6antr mpd renegcld
      nfan sseld recnd negnegd eqeltrd elrab sylanbrc simp-4r ltnegcon2d rspcev
      simpr syl2anc simpllr jca31 ltnegcon2 adantl sylbid r19.29af ex ralrimiva
      imp adantrl syl12anc rexlimdva ) AILZJLZMNZUPZIFOZXOXNMNZKLZXNMNZKFPZQZIR
      OZSZJRPZBLZCLZMNZUPZCELZUAZFTZERUBZOZYHYGMNZYHDLZMNZDYNPZQZCROZSZBRPZAYHX
      OMNZUPZCFOZXOYHMNZYQYHMNZDFPZQZCROZSZJRPZYFAYPUPZCFOZYIUUIQZCROZSZBRPUUMG
      UULUURJBRJBUCZUUFUUOUUKUUQUUSUUEUUNCFUUSUUDYPXOYGYHMUDUFUEUUSUUJUUPCRUUSU
      UGYIUUIXOYGYHMUGUHUEUIUJUKYEUULJRXRUUFYDUUKXQUUEICFICUCZXPUUDXNYHXOMUGUFU
      LYDXSYQXNMNZDFPZQZIROUUKYCUVCIRYBUVBXSYAUVAKDFXTYQXNMUGUJUMUQUVCUUJICRUUT
      XSUUGUVBUUIXNYHXOMUDUUTUVAUUHDFXNYHYQMUDUNUOULURUSUTUKAYEUUCJRAXORTZSZYEU
      UCUVEYESZXOUAZRTZUVGYHMNZUPZCYNOZYHUVGMNZYSQZCROZUUCUVDUVHAYEXOVGVAUVFUVD
      XRUVKAUVDYEVBUVEXRYDVCUVDXRUVJCYNYHYNTZUVDXRUVJQUVOUVDSXRYHUAZXOMNZUPZUVJ
      UVOXRUVRQZUVDUVOUVPFTZUVSYHRTZUVOUVTYMEYHRVDZUWAUVOUVTYMUVTEYHRECUCYLUVPF
      YKYHVEVFVHVIVJXQUVRIUVPFXNUVPVKZXPUVQXNUVPXOMUGUFVLVRVMUVOUWAUVDUVJUVRVNU
      WBUWAUVDSUVIUVQUVDUWAUVIUVQVNXOYHVOVPUFVSVQVPVTWAUVEYDUVNXRUVEYDSZUVMCRUW
      DUWASZUVLYSUWEUVLSZXTUVPMNZYSKFUWEUVLKUWDUWAKUVEYDKUVEKWBYCKIRKRWCXSYBKXS
      KWBYAKFWDWEWFWJUWAKWBWJUVLKWBWJUWFXTFTZSZUWGSZXTUAZYNTZYHUWKMNZYSUWJUWKRT
      UWKUAZFTZUWLUWJXTUWJUWHXTRTZUWFUWHUWGVBZAUWHUWPQUVDYDUWAUVLUWHUWGAFRXTHWK
      WGWHZWIUWJUWNXTFUWJXTUWJXTUWRWLWMUWQWNYMUWOEUWKRYKUWKVKYLUWNFYKUWKVEVFWOW
      PUWJXTYHUWRUWDUWAUVLUWHUWGWQUWIUWGWTWRYRUWMDUWKYNYQUWKYHMUDWSXAUWEUVDUWAS
      ZYDSZUVLUWGKFPZUWEUVDUWAYDAUVDYDUWAXBUWDUWAWTUVEYDUWAVBXCUWTUVLUXAUWTUVLX
      OUVPMNZUXAUWSUVLUXBVNZYDUWAUVDUXCYHXOXDVPVMUWSYDUXBUXAQZUWAYDUXDQZUVDUWAU
      VPRTUXEYHVGYCUXDIUVPRUWCXSUXBYBUXAXNUVPXOMUDUWCYAUWGKFXNUVPXTMUDUNUOVLVRX
      EXJXFXJVSXGXHXIXKUUBUVKUVNSBUVGRYGUVGVKZYOUVKUUAUVNUXFYJUVJCYNUXFYIUVIYGU
      VGYHMUGUFUEUXFYTUVMCRUXFYPUVLYSYGUVGYHMUDUHUEUIWSXLXHXMWH $.
  $}

  ${
    $d A f g w x y z $.  $d f g ph x y z $.
    supminfex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. x < y
        /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $.
    supminfex.ss $e |- ( ph -> A C_ RR ) $.
    $( A supremum is the negation of the infimum of that set's image under
       negation.  (Contributed by Jim Kingdon, 14-Jan-2022.) $)
    supminfex $p |- ( ph ->
        sup ( A , RR , < ) = -u inf ( { w e. RR | -u w e. A } , RR , < ) ) $=
      ( vg vf cv cneg wcel cr crab clt wceq adantl wb eleq1d cinf supinfneg wss
      csup ssrab2 a1i infrenegsupex elrabi sselda negeq elrab3 renegcl syl recn
      negnegd 3bitrd eqrdav supeq1d negeqd eqtrd cc wa wbr lttri3 infclti recnd
      wn supclti negcon2 syl2anc mpbid ) AEKZLZFMZENOZNPUAZFNPUDZLZQZVQVPLQZAVP
      DKZLZVOMZDNOZNPUDZLVRABCDVOABCDEFGHUBZVONUCAVNENUEUFUGAWEVQANWDFPABWDFNBK
      ZWDMZWGNMZAWCDWGNUHRAFNWGHUIWIWHWGFMZSAWIWHWGLZVOMZWKLZFMZWJWCWLDWGNWAWGQ
      WBWKVOWAWGUJTUKWIWKNMWLWNSWGULVNWNEWKNVLWKQVMWMFVLWKUJTUKUMWIWMWGFWIWGWGU
      NUOTUPRUQURUSUTAVPVAMVQVAMVSVTSAVPABCDIJNVOPJKZNMIKZNMVBWOWPQWOWPPVCVGWPW
      OPVCVGVBSAWOWPVDRZWFVEVFAVQABCDIJNFPWQGVHVFVPVQVIVJVK $.
  $}

  ${
    $d A x y z a b c d $.  $d B x y z a b c d $.  $d ph x y z a b c d $.
    infregelbex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. y < x
      /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) ) $.
    infregelbex.ss $e |- ( ph -> A C_ RR ) $.
    infregelbex.b $e |- ( ph -> B e. RR ) $.
    $( Any lower bound of a set of real numbers with an infimum is less than or
       equal to the infimum.  (Contributed by Jim Kingdon, 27-Sep-2024.) $)
    infregelbex $p |- ( ph
        -> ( B <_ inf ( A , RR , < ) <-> A. z e. A B <_ z ) ) $=
      ( vc vb vd cr clt cle wbr cv wa wcel wn va cinf ad2antrr wb lttri3 adantl
      wral weq infclti sselda adantlr simplr adantr inflbti imp letrd ralrimiva
      nltled breq2 cbvralv sylib simpl simpr sseldd lenltd ralbidva mpbid breq1
      wss notbid jca infnlbti sylc impbida ) AFEMNUBZOPZFDQZOPZDEUGZAVPRZFUAQZO
      PZUAEUGVSVTWBUAEVTWAESZRFVOWAAFMSZVPWCIUCAVOMSZVPWCABCDJKMENKQZMSJQZMSRKJ
      UHWFWGNPTWGWFNPTRUDAWFWGUEUFZGUIZUCAWCWAMSVPAEMWAHUJZUKAVPWCULAWCVOWAOPVP
      AWCRVOWAAWEWCWIUMWJAWCWAVONPTABCDJKMEWANWHGUNUOURUKUPUQWBVRUADEWAVQFOUSUT
      VAAVSRZFVOAWDVSIUMZAWEVSWIUMWKAWDVQFNPZTZDEUGZRVOFNPTAVSVBWKWDWOWLWKLQZFN
      PZTZLEUGZWOWKFWPOPZLEUGZWSWKVSXAAVSVCVRWTDLEVQWPFOUSUTVAWKWTWRLEWKWPESZRZ
      FWPAWDVSXBIUCXCEMWPAEMVIVSXBHUCWKXBVCVDVEVFVGWRWNLDELDUHWQWMWPVQFNVHVJUTV
      AVKABCDJKMEFNWHGVLVMURVN $.
  $}

  $( Membership in a nonnegative upper set of integers implies membership in
     ` NN0 ` .  (Contributed by Paul Chapman, 22-Jun-2011.) $)
  eluznn0 $p |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 ) $=
    ( cc0 cn0 nn0uz uztrn2 ) CABDEF $.

  $( Membership in a positive upper set of integers implies membership in
     ` NN ` .  (Contributed by JJ, 1-Oct-2018.) $)
  eluznn $p |- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN ) $=
    ( c1 cn nnuz uztrn2 ) CABDEF $.

  $( Two ways to say "an integer greater than or equal to 2".  (Contributed by
     Paul Chapman, 23-Nov-2012.) $)
  eluz2b1 $p |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) ) $=
    ( c2 cuz cfv wcel cz cle wbr wa c1 clt 2z eluz1i caddc co zltp1le mpan df-2
    wb 1z breq1i bitr4di pm5.32i bitr4i ) ABCDEAFEZBAGHZIUEJAKHZIBALMUEUGUFUEUG
    JJNOZAGHZUFJFEUEUGUISTJAPQBUHAGRUAUBUCUD $.

  $( An integer greater than or equal to 2 is greater than 1.  (Contributed by
     AV, 24-May-2020.) $)
  eluz2gt1 $p |- ( N e. ( ZZ>= ` 2 ) -> 1 < N ) $=
    ( c2 cuz cfv wcel cz c1 clt wbr eluz2b1 simprbi ) ABCDEAFEGAHIAJK $.

  $( Two ways to say "an integer greater than or equal to 2".  (Contributed by
     Paul Chapman, 23-Nov-2012.) $)
  eluz2b2 $p |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ 1 < N ) ) $=
    ( c2 cuz cfv wcel cz c1 clt wbr wa cn eluz2b1 cle cr 1re zre ltle imdistani
    wi sylancr elnnz1 sylibr simpr jca nnz anim1i impbii bitri ) ABCDEAFEZGAHIZ
    JZAKEZUJJZALUKUMUKULUJUKUIGAMIZJULUIUJUNUIGNEANEUJUNSOAPGAQTRAUAUBUIUJUCUDU
    LUIUJAUEUFUGUH $.

  $( Two ways to say "an integer greater than or equal to 2".  (Contributed by
     Paul Chapman, 23-Nov-2012.) $)
  eluz2b3 $p |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) ) $=
    ( c2 cuz cfv wcel cn c1 clt wbr wa wne eluz2b2 nngt1ne1 pm5.32i bitri ) ABC
    DEAFEZGAHIZJPAGKZJALPQRAMNO $.

  $( One less than an integer greater than or equal to 2 is a positive integer.
     (Contributed by Paul Chapman, 17-Nov-2012.) $)
  uz2m1nn $p |- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN ) $=
    ( c2 cuz cfv wcel cz c1 clt wbr wa cmin co cn eluz2b1 wb znnsub mpan biimpa
    1z sylbi ) ABCDEAFEZGAHIZJAGKLMEZANUAUBUCGFEUAUBUCOSGAPQRT $.

  $( 1 is not in ` ( ZZ>= `` 2 ) ` .  (Contributed by Paul Chapman,
     21-Nov-2012.) $)
  1nuz2 $p |- -. 1 e. ( ZZ>= ` 2 ) $=
    ( c1 c2 cuz cfv wcel wne neirr cn eluz2b3 simprbi mto ) ABCDEZAAFZAGLAHEMAI
    JK $.

  $( A positive integer is either 1 or greater than or equal to 2.
     (Contributed by Paul Chapman, 17-Nov-2012.) $)
  elnn1uz2 $p |- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) ) $=
    ( cn wcel c1 wceq c2 cuz cfv wo wne wa olc cz nnz wn 1z zdceq orbi2i sylibr
    wdc mpan2 df-dc sylib df-ne syl ordi sylanbrc eluz2b3 1nn eleq1 mpbiri jaoi
    eluz2nn impbii ) ABCZADEZAFGHCZIZUOUPUOADJZKZIZURUOUPUOIUPUSIZVAUOUPLUOAMCZ
    VBANVCUPUPOZIZVBVCUPTZVEVCDMCVFPADQUAUPUBUCUSVDUPADUDRSUEUPUOUSUFUGUQUTUPAU
    HRSUPUOUQUPUODBCUIADBUJUKAUMULUN $.

  $( Closure of multiplication of integers greater than or equal to 2.
     (Contributed by Paul Chapman, 26-Oct-2012.) $)
  uz2mulcl $p |- ( ( M e. ( ZZ>= ` 2 ) /\ N e. ( ZZ>= ` 2 ) ) ->
                   ( M x. N ) e. ( ZZ>= ` 2 ) ) $=
    ( c2 cuz cfv wcel wa cmul co cz c1 clt wbr eluzelz syl2an cr eluz2b1 anim1i
    zre sylbi zmulcl mulgt1 an4s sylanbrc ) ACDEZFZBUEFZGABHIZJFZKUHLMZUHUEFUFA
    JFZBJFZUIUGCANCBNABUAOUFAPFZKALMZGZBPFZKBLMZGZUJUGUFUKUNGUOAQUKUMUNASRTUGUL
    UQGURBQULUPUQBSRTUMUPUNUQUJABUBUCOUHQUD $.

  ${
    $d ph y $.  $d ps x $.  $d x y $.
    indstr2.1 $e |- ( x = 1 -> ( ph <-> ch ) ) $.
    indstr2.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    indstr2.3 $e |- ch $.
    indstr2.4 $e |- ( x e. ( ZZ>= ` 2 ) ->
                      ( A. y e. NN ( y < x -> ps ) -> ph ) ) $.
    $( Strong Mathematical Induction for positive integers (inference schema).
       The first two hypotheses give us the substitution instances we need; the
       last two are the basis and the induction step.  (Contributed by Paul
       Chapman, 21-Nov-2012.) $)
    indstr2 $p |- ( x e. NN -> ph ) $=
      ( cv cn wcel c1 wceq c2 cuz clt wbr wi wb cfv wral elnn1uz2 nnnlt1 adantl
      wo wa wn breq2 adantr mtbird pm2.21d ralrimiva pm5.5 syl bitrd jaoi sylbi
      mpbiri indstr ) ABDEGDJZKLVAMNZVAOPUALZUFEJZVAQRZBSZEKUBZASZVAUCVBVHVCVBV
      HCHVBVHACVBVGVHATVBVFEKVBVDKLZUGZVEBVJVEVDMQRZVIVKUHVBVDUDUEVBVEVKTVIVAMV
      DQUIUJUKULUMVGAUNUOFUPUSIUQURUT $.
  $}

  $( Membership of an integer in an upper set of integers is decidable.
     (Contributed by Jim Kingdon, 18-Apr-2020.) $)
  eluzdc $p |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID N e. ( ZZ>= ` M ) ) $=
    ( cz wcel wa cle wbr clt wo cuz cfv wdc zlelttric wn eluz biimprd wb zltnle
    ancoms notbid sylbid orim12d df-dc imbitrrdi mpd ) ACDZBCDZEZABFGZBAHGZIZBA
    JKDZLZABMUHUKULULNZIUMUHUIULUJUNUHULUIABOZPUHUJUINZUNUGUFUJUPQBARSUHUNUPUHU
    LUIUOTPUAUBULUCUDUE $.

  $( Membership of an integer in ` NN0 ` is decidable.  (Contributed by Jim
     Kingdon, 8-Oct-2024.) $)
  elnn0dc $p |- ( N e. ZZ -> DECID N e. NN0 ) $=
    ( cz wcel cc0 cuz cfv wdc cn0 0z eluzdc mpan elnn0uz dcbii sylibr ) ABCZADE
    FCZGZAHCZGDBCOQIDAJKRPALMN $.

  $( Membership of an integer in ` NN ` is decidable.  (Contributed by Jim
     Kingdon, 17-Oct-2024.) $)
  elnndc $p |- ( N e. ZZ -> DECID N e. NN ) $=
    ( cz wcel c1 cuz cfv wdc cn 1z eluzdc mpan elnnuz dcbii sylibr ) ABCZADEFCZ
    GZAHCZGDBCOQIDAJKRPALMN $.

  ${
    $d A a b x y z $.
    $( The image under negation of a bounded-above set of reals is bounded
       below.  For a theorem which is similar but also adds that the bounds
       need to be the tightest possible, see ~ supinfneg .  (Contributed by
       Paul Chapman, 21-Mar-2011.) $)
    ublbneg $p |- ( E. x e. RR A. y e. A y <_ x ->
                    E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y ) $=
      ( vb va cv cle wbr wral cr wrex cneg wcel crab breq1 cbvralv wceq ralbidv
      rexbii breq2 cbvrexv bitri renegcl wa wi elrabi negeq eleq1d elrab3 mpcom
      biimpd rspcv adantl wb lenegcon1 sylan2 sylibrd ralrimdva rspcev rexlimiv
      syl syl6an sylbir ) BGZAGZHIZBDJZAKLZEGZFGZHIZEDJZFKLZVFVEHIZBCGZMZDNZCKO
      ZJZAKLZVNVEVKHIZBDJZFKLVIVMWCFKVLWBEBDVJVEVKHPQTWCVHFAKVKVFRWBVGBDVKVFVEH
      UASUBUCVMWAFKVKKNZVKMZKNVMWEVEHIZBVSJZWAVKUDWDVMWFBVSWDVEVSNZUEVMVEMZVKHI
      ZWFWHVMWJUFZWDWHWIDNZWKVEKNZWHWLVRCVEKUGZWMWHWLVRWLCVEKVPVERVQWIDVPVEUHUI
      UJULUKVLWJEWIDVJWIVKHPUMVBUNWHWDWMWFWJUOWNVKVEUPUQURUSVTWGAWEKVFWERVOWFBV
      SVFWEVEHPSUTVCVAVD $.
  $}

  ${
    $d A w z $.
    $( Two ways to express the image under negation of a set of integers.
       (Contributed by Paul Chapman, 21-Mar-2011.) $)
    eqreznegel $p |- ( A C_ ZZ ->
                       { z e. RR | -u z e. A } = { z e. ZZ | -u z e. A } ) $=
      ( vw cz wss cv cneg wcel cr crab wa wi ssel cc recn caddc cc0 negid elrab
      co 0z eqeltrdi pm4.71i zrevaddcl bitrid imbitrid syl6 com23 impd a1i jcad
      simpr zre anim1i impbid1 weq negeq eleq1d 3bitr4g eqrdv ) BDEZCAFZGZBHZAI
      JZVDADJZVACFZIHZVGGZBHZKZVGDHZVJKZVGVEHVGVFHVAVKVMVAVKVLVJVAVHVJVLVAVJVHV
      LVAVJVIDHZVHVLLBDVIMVHVGNHZVNVLVGOVOVOVGVIPTZDHZKVNVLVOVQVOVPQDVGRUAUBUCV
      GVIUDUEUFUGUHUIVKVJLVAVHVJULUJUKVLVHVJVGUMUNUOVDVJAVGIACUPVCVIBVBVGUQURZS
      VDVJAVGDVRSUSUT $.
  $}

  ${
    $d A x y z $.
    $( The image under negation of an inhabited set of reals is inhabited.
       (Contributed by Jim Kingdon, 10-Apr-2020.) $)
    negm $p |- ( ( A C_ RR /\ E. x x e. A ) ->
        E. y y e. { z e. RR | -u z e. A } ) $=
      ( cr wss cv wcel wex cneg crab ssel renegcl wceq negeq eleq1d elrab3 recn
      wb syl negnegd bitrd biimprd syli elex2 syl6 exlimdv imp ) DEFZAGZDHZAIBG
      CGZJZDHZCEKZHBIZUIUKUPAUIUKUJJZUOHZUPUKUIUJEHZURDEUJLUSURUKUSURUQJZDHZUKU
      SUQEHURVASUJMUNVACUQEULUQNUMUTDULUQOPQTUSUTUJDUSUJUJRUAPUBUCUDBUQUOUEUFUG
      UH $.

    $( If a set of reals is bounded below, it is bounded below by an integer.
       (Contributed by Paul Chapman, 21-Mar-2011.) $)
    lbzbi $p |- ( A C_ RR -> ( E. x e. RR A. y e. A x <_ y <->
                               E. x e. ZZ A. y e. A x <_ y ) ) $=
      ( vz cr wss cv cle wbr wral wrex cz wcel wi wa clt expdimp com23 imp ex
      nfv nfre1 btwnz simpld w3a zre ltleletr syl3an1 expd 3expia syl5 ralrimiv
      ssel2 ralim syl anasss expcom imdistand breq1 ralbidv rspcev syl6 ancomsd
      weq rexlimdv mpdi rexlimd zssre ssrexv ax-mp impbid1 ) CEFZAGZBGZHIZBCJZA
      EKZVPALKZVLVPVRAEVLAUAVPALUBVLVPVMEMZVRVLVPVSVRNVLVPOZVSDGZVMPIZDLKZVRVSW
      CVMWAPIDLKDDVMUCUDVSVTWCVRNZVSVLVPWDVSVLOZVPOWBVRDLWEVPWALMZWBVRNZWEWFVPW
      GWEWBWFVPOZVRWEWBWHVRNWEWBOZWHWFWAVNHIZBCJZOVRWIWFVPWKWEWBWFVPWKNZNWEWFWB
      WLWFWEWBWLNZWFVSVLWMWFVSOZVLOZWBWLWOWBOZVOWJNZBCJWLWPWQBCWOWBVNCMZWQNWOWR
      WBWQWNVLWRWBWQNZVLWROVNEMZWNWSCEVNUMWFVSWTWSWFVSWTUEWBVOWJWFWAEMVSWTWBVOO
      WJNWAUFWAVMVNUGUHUIUJUKQRSULVOWJBCUNUOTUPUQRSURVPWKAWALADVDVOWJBCVMWAVNHU
      SUTVAVBTRVCQVEUPUQVFTRVGLEFVRVQNVHVPALEVIVJVK $.
  $}

  $( A (nonnegative) integer between 1 and 3 must be 1, 2 or 3.  (Contributed
     by Alexander van der Vekens, 13-Sep-2018.) $)
  nn01to3 $p |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 )
      -> ( N = 1 \/ N = 2 \/ N = 3 ) ) $=
    ( cn0 wcel c1 cle wbr c3 c2 clt w3o wo wb cz zleloe sylancr syl mpbid caddc
    wceq co w3a simp2 simp1 1z nn0z 1nn0 nn0ltp1le mpan breq1i bitr4di 2z bitrd
    orbi1d orcomd orcom orbi2i sylib 3orass sylibr wi 3mix1 eqcoms a1i 3mix2 wa
    df-2 simp3 biantrurd 2nn0 df-3 cr nn0red 3re sylancl 3bitr4d 3mix3 biimtrdi
    letri3 3jaod mpd ) ABCZDAEFZAGEFZUAZDASZHASZHAIFZJZADSZAHSZAGSZJZWDWEWFWGKZ
    KZWHWDWEWGWFKZKWNWDWOWEWDDAIFZWEKZWOWEKZWDWBWQWAWBWCUBWDWAWBWQLZWAWBWCUCZWA
    DMCAMCZWSUDAUEZDANOPQWDWAWQWRLWTWAWPWOWEWAWPHAEFZWOWAWPDDRTZAEFZXCDBCWAWPXE
    LUFDAUGUHHXDAEVFUIUJWAHMCXAXCWOLUKXBHANOULUMPQUNWOWMWEWGWFUOUPUQWEWFWGURUSW
    DWEWLWFWGWEWLUTWDWLADWIWJWKVAVBVCWFWLUTWDWLAHWJWIWKVDVBVCWDWGWKWLWDGAEFZWCX
    FVEZWGWKWDWCXFWAWBWCVGVHWDWAWGXFLWTWAWGHDRTZAEFZXFHBCWAWGXILVIHAUGUHGXHAEVJ
    UIUJPWDAVKCGVKCWKXGLWDAWTVLVMAGVRVNVOWKWIWJVPVQVSVT $.

  $( Alternate proof of ~ nn0ge2m1nn :  If a nonnegative integer is greater
     than or equal to two, the integer decreased by 1 is a positive integer.
     This version is proved using ~ eluz2 , a theorem for upper sets of
     integers, which are defined later than the positive and nonnegative
     integers.  This proof is, however, much shorter than the proof of
     ~ nn0ge2m1nn .  (Contributed by Alexander van der Vekens, 1-Aug-2018.)
     (New usage is discouraged.)  (Proof modification is discouraged.) $)
  nn0ge2m1nnALT $p |- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN ) $=
    ( cn0 wcel c2 cle wbr wa cuz cfv c1 cmin co cn cz 2z a1i adantr simpr eluz2
    nn0z syl3anbrc uz2m1nn syl ) ABCZDAEFZGZADHICZAJKLMCUFDNCZANCZUEUGUHUFOPUDU
    IUEATQUDUERDASUAAUBUC $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Rational numbers (as a subset of complex numbers)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c QQ $. $( The set of rational numbers (blackboard bold Q). $)

  $( Extend class notation to include the class of rationals. $)
  cq $a class QQ $.

  $( Define the set of rational numbers.  Based on definition of rationals in
     [Apostol] p. 22.  See ~ elq for the relation "is rational".  (Contributed
     by NM, 8-Jan-2002.) $)
  df-q $a |- QQ = ( / " ( ZZ X. NN ) ) $.

  ${
    $d x y z $.

    $( Division restricted to ` ZZ X. NN ` is a function.  Given excluded
       middle, it would be easy to prove this for ` CC X. ( CC \ { 0 } ) ` .
       The key difference is that an element of ` NN ` is apart from zero,
       whereas being an element of ` CC \ { 0 } ` implies being not equal to
       zero.  (Contributed by Jim Kingdon, 19-Mar-2020.) $)
    divfnzn $p |- ( / |` ( ZZ X. NN ) ) Fn ( ZZ X. NN ) $=
      ( vy vz vx cv co wceq cc crio wcel cn wral cz cdiv cres nncn ad2antlr cc0
      wa adantl eqeltrrd cmul cxp wfn zcn ad2antrr cap wbr divmulapd riotabidva
      simpr nnap0 wb eqcom a1i riotabidv simpl divclapd reueq sylib riotacl syl
      wreu sylan rgen2 csn cdif cmpo df-div reseq1i wss zsscn wne nnne0 eldifsn
      sylanbrc ssriv resmpo mp2an eqtri fnmpo ax-mp ) ADZBDZUAECDZFZBGHZGIZAJKC
      LKMLJUBZNZWHUCWGCALJWDLIZWBJIZRZWDWBMEZWCFZBGHZWFGWLWNWEBGWLWCGIZRWDWBWCW
      JWDGIZWKWPWDUDZUEWKWBGIZWJWPWBOZPWLWPUJWKWBQUFUGZWJWPWBUKZPUHUIWLWCWMFZBG
      HZWOGWLXCWNBGXCWNULWLWCWMUMUNUOWJWQWKXDGIZWRWQWKRZXCBGVBZXEXFWMGIXGXFWDWB
      WQWKUPWKWSWQWTSWKXAWQXBSUQBGWMURUSXCBGUTVAVCTTVDCALJWFWIGWICAGGQVEVFZWFVG
      ZWHNZCALJWFVGZMXIWHCABVHVILGVJJXHVJXJXKFVKCJXHWDJIWQWDQVLWDXHIWDOWDVMWDGQ
      VNVOVPCAGXHLJWFVQVRVSVTWA $.
  $}

  ${
    $d x y A $.
    $( Membership in the set of rationals.  (Contributed by NM, 8-Jan-2002.)
       (Revised by Mario Carneiro, 28-Jan-2014.) $)
    elq $p |- ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) ) $=
      ( cq wcel cv cdiv cz cn cxp cres co wceq wrex cima df-q eleq2i resima wfn
      wss wb divfnzn ssid ovelimab mp2an 3bitr2i wa ovres eqeq2d 2rexbiia bitri
      ) CDEZCAFZBFZGHIJZKZLZMZBINAHNZCUMUNGLZMZBINAHNULCGUOOZECUPUOOZEZUSDVBCPQ
      VCVBCGUORQUPUOSUOUOTVDUSUAUBUOUCABUOHICUPUDUEUFURVAABHIUMHEUNIEUGUQUTCUMU
      NHIGUHUIUJUK $.

    $( If ` A ` is rational, then some integer multiple of it is an integer.
       (Contributed by NM, 7-Nov-2008.)  (Revised by Mario Carneiro,
       22-Jul-2014.) $)
    qmulz $p |- ( A e. QQ -> E. x e. NN ( A x. x ) e. ZZ ) $=
      ( vy cq wcel cv cdiv co wceq cn wrex cz elq rexcom wa cc zcn adantr sylbi
      cmul adantl nncn cc0 cap nnap0 divcanap1d simpr eqeltrd eleq1d syl5ibrcom
      wbr oveq1 rexlimdva reximia ) BDEBCFZAFZGHZIZAJKCLKZBUPTHZLEZAJKZCABMUSUR
      CLKZAJKVBURCALJNVCVAAJUPJEZURVACLVDUOLEZOZVAURUQUPTHZLEVFVGUOLVFUOUPVEUOP
      EVDUOQUAVDUPPEVEUPUBRVDUPUCUDUKVEUPUERUFVDVEUGUHURUTVGLBUQUPTULUIUJUMUNSS
      $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The ratio of an integer and a positive integer is a rational number.
       (Contributed by NM, 12-Jan-2002.) $)
    znq $p |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. QQ ) $=
      ( vx vy cz wcel cn wa cdiv co cv wceq wrex eqid rspceov mp3an3 elq sylibr
      cq ) AEFZBGFZHABIJZCKDKIJLDGMCEMZUBSFTUAUBUBLUCUBNCDEGABUBIOPCDUBQR $.
  $}

  ${
    $d x y A $.
    $( A rational number is a real number.  (Contributed by NM,
       14-Nov-2002.) $)
    qre $p |- ( A e. QQ -> A e. RR ) $=
      ( vx vy cq wcel cv cdiv co wceq cn wrex cz cr elq wa cc0 cap wbr zre nnre
      nnap0 jca redivclap 3expb syl2an eleq1 syl5ibrcom rexlimivv sylbi ) ADEAB
      FZCFZGHZIZCJKBLKAMEZBCANUMUNBCLJUJLEZUKJEZOUNUMULMEZUOUJMEZUKMEZUKPQRZOUQ
      UPUJSUPUSUTUKTUKUAUBURUSUTUQUJUKUCUDUEAULMUFUGUHUI $.

    $( An integer is a rational number.  (Contributed by NM, 9-Jan-2002.) $)
    zq $p |- ( A e. ZZ -> A e. QQ ) $=
      ( vx vy cv wceq cz wrex cdiv co cn wcel cq eqcom zcn div1d eqeq2d bitr4id
      c1 1nn oveq2 rspcev mpan biimtrdi reximia risset elq 3imtr4i ) BDZAEZBFGA
      UHCDZHIZEZCJGZBFGAFKALKUIUMBFUHFKZUIAUHRHIZEZUMUNUIAUHEUPUHAMUNUOUHAUNUHU
      HNOPQRJKUPUMSULUPCRJUJREUKUOAUJRUHHTPUAUBUCUDBAFUEBCAUFUG $.
  $}

  $( The integers are a subset of the rationals.  (Contributed by NM,
     9-Jan-2002.) $)
  zssq $p |- ZZ C_ QQ $=
    ( vx cz cq cv zq ssriv ) ABCADEF $.

  $( The nonnegative integers are a subset of the rationals.  (Contributed by
     NM, 31-Jul-2004.) $)
  nn0ssq $p |- NN0 C_ QQ $=
    ( cn0 cz cq nn0ssz zssq sstri ) ABCDEF $.

  $( The positive integers are a subset of the rationals.  (Contributed by NM,
     31-Jul-2004.) $)
  nnssq $p |- NN C_ QQ $=
    ( cn cz cq nnssz zssq sstri ) ABCDEF $.

  $( The rationals are a subset of the reals.  (Contributed by NM,
     9-Jan-2002.) $)
  qssre $p |- QQ C_ RR $=
    ( vx cq cr cv qre ssriv ) ABCADEF $.

  $( The rationals are a subset of the complex numbers.  (Contributed by NM,
     2-Aug-2004.) $)
  qsscn $p |- QQ C_ CC $=
    ( cq cr cc qssre ax-resscn sstri ) ABCDEF $.

  $( The set of rational numbers exists.  (Contributed by NM, 30-Jul-2004.)
     (Revised by Mario Carneiro, 17-Nov-2014.) $)
  qex $p |- QQ e. _V $=
    ( cq cc cnex qsscn ssexi ) ABCDE $.

  $( A positive integer is rational.  (Contributed by NM, 17-Nov-2004.) $)
  nnq $p |- ( A e. NN -> A e. QQ ) $=
    ( cn cq nnssq sseli ) BCADE $.

  $( A rational number is a complex number.  (Contributed by NM,
     2-Aug-2004.) $)
  qcn $p |- ( A e. QQ -> A e. CC ) $=
    ( cq cc qsscn sseli ) BCADE $.

  ${
    $d x y z w v u A $.  $d x y z w v u B $.
    $( Closure of addition of rationals.  (Contributed by NM, 1-Aug-2004.) $)
    qaddcl $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A + B ) e. QQ ) $=
      ( vx vy vz vw vv vu cq wcel cv cdiv co wceq cn wrex cz caddc wa cc elq wi
      cmul zmulcl sylan2 ad2ant2rl simpl adantl syl2anr zaddcld adantr ad2ant2l
      nnz nnmulcl oveq12 cc0 cap wbr anim12i nncn nnap0 divadddivap syl2an an4s
      zcn jca sylan9eqr w3a rspceov sylibr syl3anc rexlimivv rexlimdvv syl2anb
      exp43 imp ) AIJACKZDKZLMZNZDOPCQPZBEKZFKZLMZNZFOPEQPZABRMZIJZBIJCDAUAEFBU
      AWAWFWHWAWEWHEFQOVTWBQJZWCOJZSZWEWHUBUBCDQOVQQJZVROJZSZVTWKWEWHWNWKVTWEWH
      WNWKSZVTWESZSVQWCUCMZWBVRUCMZRMZQJZVRWCUCMZOJZWGWSXALMZNZWHWOWTWPWOWQWRWL
      WJWQQJZWMWIWJWLWCQJXEWCUMVQWCUDUEUFWKWIVRQJZWRQJWNWIWJUGWMXFWLVRUMUHWBVRU
      DUIUJUKWOXBWPWMWJXBWLWIVRWCUNULUKWPWOWGVSWDRMZXCAVSBWDRUOWLWIWMWJXGXCNZWL
      WISVQTJZWBTJZSVRTJZVRUPUQURZSZWCTJZWCUPUQURZSZSXHWMWJSWLXIWIXJVQVEWBVEUSW
      MXMWJXPWMXKXLVRUTVRVAVFWJXNXOWCUTWCVAVFUSVQWBVRWCVBVCVDVGWTXBXDVHWGGKHKLM
      NHOPGQPWHGHQOWSXAWGLVIGHWGUAVJVKVDVOVLVMVPVN $.

    $( Closure law for the negative of a rational.  (Contributed by NM,
       2-Aug-2004.)  (Revised by Mario Carneiro, 15-Sep-2014.) $)
    qnegcl $p |- ( A e. QQ -> -u A e. QQ ) $=
      ( vx vy cq wcel cv cdiv co wceq cn wrex cz cneg elq wa cc zcn adantr nncn
      adantl cc0 cap wbr nnap0 divnegapd znegcl sylan eqeltrd eleq1d syl5ibrcom
      znq negeq rexlimivv sylbi ) ADEABFZCFZGHZIZCJKBLKAMZDEZBCANURUTBCLJUOLEZU
      PJEZOZUTURUQMZDEVCVDUOMZUPGHZDVCUOUPVAUOPEVBUOQRVBUPPEVAUPSTVBUPUAUBUCVAU
      PUDTUEVAVELEVBVFDEUOUFVEUPUKUGUHURUSVDDAUQULUIUJUMUN $.

    $( Closure of multiplication of rationals.  (Contributed by NM,
       1-Aug-2004.) $)
    qmulcl $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ ) $=
      ( vx vy vz vw vv vu cq wcel cv cdiv co wceq cn wrex cz cmul wa cc nnmulcl
      elq zmulcl anim12i an4s adantr oveq12 cc0 cap wbr zcn ad2ant2r nncn nnap0
      jca ad2ant2l divmuldivap syl2anc sylan9eqr rspceov 3expa sylibr rexlimivv
      wi exp43 rexlimdvv imp syl2anb ) AIJACKZDKZLMZNZDOPCQPZBEKZFKZLMZNZFOPEQP
      ZABRMZIJZBIJCDAUBEFBUBVMVRVTVMVQVTEFQOVLVNQJZVOOJZSZVQVTVDVDCDQOVIQJZVJOJ
      ZSZVLWCVQVTWFWCVLVQVTWFWCSZVLVQSZSVIVNRMZQJZVJVORMZOJZSZVSWIWKLMZNZVTWGWM
      WHWDWAWEWBWMWDWASWJWEWBSWLVIVNUCVJVOUAUDUEUFWHWGVSVKVPRMZWNAVKBVPRUGWGVIT
      JZVNTJZSZVJTJZVJUHUIUJZSZVOTJZVOUHUIUJZSZSZWPWNNWDWAWSWEWBWDWQWAWRVIUKVNU
      KUDULWEWBXFWDWAWEXBWBXEWEWTXAVJUMVJUNUOWBXCXDVOUMVOUNUOUDUPVIVNVJVOUQURUS
      WMWOSVSGKHKLMNHOPGQPZVTWJWLWOXGGHQOWIWKVSLUTVAGHVSUBVBURUEVEVCVFVGVH $.
  $}

  $( Closure of subtraction of rationals.  (Contributed by NM, 2-Aug-2004.) $)
  qsubcl $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A - B ) e. QQ ) $=
    ( cq wcel wa cneg caddc co cmin wceq qcn negsub syl2an qnegcl qaddcl sylan2
    cc eqeltrrd ) ACDZBCDZEABFZGHZABIHZCSAQDBQDUBUCJTAKBKABLMTSUACDUBCDBNAUAOPR
    $.

  ${
    $d A w x y z $.  $d B w x y z $.
    $( Apartness is equivalent to not equal for rationals.  (Contributed by Jim
       Kingdon, 20-Mar-2020.) $)
    qapne $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A =//= B <-> A =/= B ) ) $=
      ( vz vw vx vy wcel wa cv cdiv co wceq cn cz cap wbr wb cmul c1 cc cq wrex
      wne elq biimpi adantl simplll sylib simplrl simprl ad3antrrr simprr nncnd
      cc0 zcnd syl divclapd simplrr mulcld recclapd recap0d syl112anc divrecapd
      nnap0 apmul1 eqcomd mulassd recidapd oveq2d mulridd breq12d bitrd zmulcld
      3eqtrd breq1d mulcomd 3eqtr2d breq2d simpr simpllr 3bitr4d syl2anc bitr3d
      nnzd zapne wn notbid apti qcn ad2antrr eqeltrd necon3bid rexlimdvva mpd
      ex ) AUAGZBUAGZHZBCIZDIZJKZLZDMUBCNUBZABOPZABUCZQZWQXCWPWQXCCDBUDUEUFWRXB
      XFCDNMWRWSNGZWTMGZHZHZXBXFXJXBHZAEIZFIZJKZLZFMUBENUBZXFXKWPXPWPWQXIXBUGEF
      AUDUHXKXOXFEFNMXKXLNGZXMMGZHZHZXOXFXTXOHZXDXLWTRKZXMWSRKZUCZXEYAYBYCOPZXD
      YDYAXLXAXMRKZOPZXNXAOPZYEXDYAYGXLSXMJKZRKZYFYIRKZOPZYHYAXLTGYFTGYITGYIUNO
      PYGYLQYAXLXKXQXRXOUIZUOZYAXAXMYAWSWTYAWSXJXGXBXSXOWRXGXHUJUKZUOZYAWTXJXHX
      BXSXOWRXGXHULUKZUMZYAXHWTUNOPYQWTVDUPZUQZYAXMXKXQXRXOURZUMZUSYAXMUUBYAXRX
      MUNOPUUAXMVDUPZUTZYAXMUUBUUCVAXLYFYIVEVBYAYJXNYKXAOYAXNYJYAXLXMYNUUBUUCVC
      VFYAYKXAXMYIRKZRKXASRKXAYAXAXMYIYTUUBUUDVGYAUUESXARYAXMUUBUUCVHVIYAXAYTVJ
      VNVKVLYAYEXLYCSWTJKZRKZOPZYGYAYEYBUUFRKZUUGOPZUUHYAYBTGZYCTGZUUFTGUUFUNOP
      YEUUJQYAYBYAXLWTYMYAWTYQWDVMZUOZYAYCYAXMWSYAXMUUAWDYOVMZUOZYAWTYRYSUTZYAW
      TYRYSVAYBYCUUFVEVBYAUUIXLUUGOYAUUIXLWTUUFRKZRKXLSRKXLYAXLWTUUFYNYRUUQVGYA
      UURSXLRYAWTYRYSVHVIYAXLYNVJVNVOVLYAUUGYFXLOYAUUGXMWSUUFRKZRKXMXARKYFYAXMW
      SUUFUUBYPUUQVGYAXAUUSXMRYAWSWTYPYRYSVCVIYAXMXAUUBYTVPVQVRVLYAAXNBXAOXTXOV
      SXJXBXSXOVTZVKWAZYAYBNGYCNGYEYDQUUMUUOYBYCWEWBWCYAYBYCABYAYEWFZXDWFZYBYCL
      ZABLZYAYEXDUVAWGYAUUKUULUVDUVBQUUNUUPYBYCWHWBYAATGZBTGUVEUVCQXJUVFXBXSXOW
      PUVFWQXIAWIWJUKYABXATUUTYTWKABWHWBWAWLVLWOWMWNWOWMWN $.
  $}

  $( Rational 'Less than' expressed in terms of 'less than or equal to'.  Also
     see ~ ltleap which is a similar result for real numbers.  (Contributed by
     Jim Kingdon, 11-Oct-2021.) $)
  qltlen $p |- ( ( A e. QQ /\ B e. QQ ) ->
      ( A < B <-> ( A <_ B /\ B =/= A ) ) ) $=
    ( cq wcel wa clt wbr cle wne cap cr wb qre ltleap syl2an qapne anbi2d bitrd
    necom anbi2i bitrdi ) ACDZBCDZEZABFGZABHGZABIZEZUFBAIZEUDUEUFABJGZEZUHUBAKD
    BKDUEUKLUCAMBMABNOUDUJUGUFABPQRUGUIUFABSTUA $.

  $( Apartness is equivalent to not equal for rationals.  (Contributed by Jim
     Kingdon, 9-Nov-2021.) $)
  qlttri2 $p |- ( ( A e. QQ /\ B e. QQ ) ->
      ( A =/= B <-> ( A < B \/ B < A ) ) ) $=
    ( cq wcel wa cap wbr wne clt wo qapne cr wb qre reaplt syl2an bitr3d ) ACDZ
    BCDZEABFGZABHABIGBAIGJZABKRALDBLDTUAMSANBNABOPQ $.

  ${
    $d x y z w A $.
    $( Closure of reciprocal of rationals.  (Contributed by NM, 3-Aug-2004.) $)
    qreccl $p |- ( ( A e. QQ /\ A =/= 0 ) -> ( 1 / A ) e. QQ ) $=
      ( vx vy vz vw cq wcel cc0 cap wbr c1 cdiv co wb cz cn cv wceq wrex wa wne
      ax-1cn 1ap0 div0api 0z 1nn znq mp2an eqeltrri qapne mpan2 biimpar wi cmul
      elq nnne0 ancli nnz zapne sylancl adantl pm5.32i anbi1i breq1 cc zcn nncn
      anim12i divap0b 3expa sylan bicomd sylan9bbr sylbir simplll zmulcl sylan2
      bitrd adantr msqznn adantlr bitri oveq2 dividap oveq1d simpll simpl simpr
      jca divdivdivap syl22anc eqtr3d anass1rs sylan9eqr an32s ex sylbid anasss
      an4s rspceov sylibr syl8 rexlimivv sylbi imp syldan ) AFGZAHUAZAHIJZKALMZ
      FGZXGXIXHXGHFGXIXHNHKLMZHFKUBUCUDHOGZKPGXLFGUEUFHKUGUHUIAHUJUKULXGXIXKXGA
      BQZCQZLMZRZCPSBOSXIXKUMZBCAUOXQXRBCOPXNOGZXOPGZTZXQXIXNXOUNMZOGZXNXNUNMZP
      GZTZXJYBYDLMZRZTZXKXTXSXTXOHUAZTXQXIYIUMZUMZXTYJXOUPUQXSXTYJYLYAYJTZXQYKY
      MXQTZXIXNHUAZYIYNXIXNHIJZYOYNYAXOHIJZTZXQTZXIYPNYRYMXQYAYQYJXTYQYJNZXSXTX
      OOGZXMYTXOURZUEXOHUSUTVAVBVCZXQXIXPHIJZYRYPAXPHIVDYRYPUUDYAXNVEGZXOVEGZTZ
      YQYPUUDNZXSUUEXTUUFXNVFXOVGVHZUUEUUFYQUUHXNXOVIVJVKVLVMVNYNXSXMYPYONXSXTY
      JXQVOUEXNHUSUTZVRYNYOYIYNYOTZYFYHYMYOYFXQYAYOYFYJYAYOTYCYEYAYCYOXTXSUUAYC
      UUBXNXOVPVQVSXSYOYEXTXNVTWAWIWAWAUUKYSYPTZYHUULYNYPTUUKYSYNYPUUCVCYNYPYOU
      UJVBWBYRYPXQYHXQYRYPTXJKXPLMZYGAXPKLWCYAYPYQUUMYGRZYAUUGYPYQTUUNUUIUUEYPU
      UFYQUUNUUEYPTZUUFYQTZTZXNXNLMZXPLMZUUMYGUUQUURKXPLUUOUURKRUUPXNWDVSWEUUQU
      UEUUOUUOUUPUUSYGRUUEYPUUPWFUUOUUPWGZUUTUUOUUPWHXNXNXNXOWJWKWLWSVKWMWNWOVN
      WIWPWQWPWRVQYIXJDQEQLMREPSDOSZXKYCYEYHUVADEOPYBYDXJLWTVJDEXJUOXAXBXCXDXEX
      F $.
  $}

  $( Closure of division of rationals.  (Contributed by NM, 3-Aug-2004.) $)
  qdivcl $p |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ ) $=
    ( cq wcel cc0 wne w3a cdiv co c1 cmul cc qcn 3ad2ant1 3ad2ant2 cap simp3 wb
    wbr cz 0z zq ax-mp qapne mpan2 mpbird divrecapd qreccl qmulcl 3impb eqeltrd
    wa sylan2 ) ACDZBCDZBEFZGZABHIAJBHIZKIZCUQABUNUOALDUPAMNUOUNBLDUPBMOUQBEPSZ
    UPUNUOUPQUOUNUTUPRZUPUOECDZVAETDVBUAEUBUCBEUDUEOUFUGUNUOUPUSCDZUOUPULUNURCD
    VCBUHAURUIUMUJUK $.

  $( Reverse closure law for addition of rationals.  (Contributed by NM,
     2-Aug-2004.) $)
  qrevaddcl $p |- ( B e. QQ ->
                   ( ( A e. CC /\ ( A + B ) e. QQ ) <-> A e. QQ ) ) $=
    ( cq wcel cc caddc co wa cmin qcn pncan sylan2 ancoms adantr qsubcl adantlr
    wceq eqeltrrd ex wi qaddcl expcom impbid pm5.32da pm4.71ri bitr4di ) BCDZAE
    DZABFGZCDZHUHACDZHUKUGUHUJUKUGUHHZUJUKULUJUKULUJHUIBIGZACULUMAQZUJUHUGUNUGU
    HBEDUNBJABKLMNUGUJUMCDZUHUJUGUOUIBOMPRSUGUKUJTUHUKUGUJABUAUBNUCUDUKUHAJUEUF
    $.

  $( The reciprocal of a positive integer is rational.  (Contributed by NM,
     17-Nov-2004.) $)
  nnrecq $p |- ( A e. NN -> ( 1 / A ) e. QQ ) $=
    ( c1 cz wcel cn cdiv co cq 1z znq mpan ) BCDAEDBAFGHDIBAJK $.

  $( The sum of an irrational number and a rational number is irrational.
     (Contributed by NM, 7-Nov-2008.) $)
  irradd $p |- ( ( A e. ( RR \ QQ ) /\ B e. QQ )
                -> ( A + B ) e. ( RR \ QQ ) ) $=
    ( cr cq cdif wcel wa caddc co wn eldif qre readdcl sylan2 adantlr wi qsubcl
    cmin expcom cc adantl wceq qcn pncan syl2an eleq1d sylibd con3d com23 imp31
    recn ex jca sylanb sylibr ) ACDEZFZBDFZGABHIZCFZUSDFZJZGZUSUPFUQACFZADFZJZG
    ZURVCACDKVGURGUTVBVDURUTVFURVDBCFUTBLABMNOVDVFURVBVDURVFVBVDURVFVBPVDURGZVA
    VEVHVAUSBRIZDFZVEURVAVJPVDVAURVJUSBQSUAVHVIADVDATFBTFVIAUBURAUKBUCABUDUEUFU
    GUHULUIUJUMUNUSCDKUO $.

  $( The product of a real which is not rational with a nonzero rational is not
     rational.  Note that by "not rational" we mean the negation of "is
     rational" (whereas "irrational" is often defined to mean apart from any
     rational number - given excluded middle these two definitions would be
     equivalent).  For a similar theorem with irrational in place of not
     rational, see ~ irrmulap .  (Contributed by NM, 7-Nov-2008.) $)
  irrmul $p |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 )
                -> ( A x. B ) e. ( RR \ QQ ) ) $=
    ( cr cq cdif wcel cc0 wne w3a cmul co wn wa eldif remulcl wi 3expb 3ad2ant2
    qre cc sylan2 ad2ant2r cdiv qdivcl expcom adantl wceq recn 3ad2ant1 qcn cap
    simp3 wb cz 0z zq ax-mp qapne mpan2 mpbird divcanap4d eleq1d con3d ex com23
    wbr sylibd imp31 jca 3impb syl3an1b sylibr ) ACDEZFZBDFZBGHZIABJKZCFZVQDFZL
    ZMZVQVMFVNACFZADFZLZMZVOVPWAACDNWEVOVPWAWEVOVPMZMVRVTWBVOVRWDVPVOWBBCFVRBSA
    BOUAUBWBWDWFVTWBWFWDVTWBWFWDVTPWBWFMZVSWCWGVSVQBUCKZDFZWCWFVSWIPWBVSWFWIVSV
    OVPWIVQBUDQUEUFWGWHADWBVOVPWHAUGWBVOVPIZABWBVOATFVPAUHUIVOWBBTFVPBUJRWJBGUK
    VFZVPWBVOVPULVOWBWKVPUMZVPVOGDFZWLGUNFWMUOGUPUQBGURUSRUTVAQVBVGVCVDVEVHVIVJ
    VKVQCDNVL $.

  ${
    $d A q $.  $d B q $.  $d Q q $.
    irrmulap.a $e |- ( ph -> A e. RR ) $.
    irrmulap.aq $e |- ( ph -> A. q e. QQ A =//= q ) $.
    irrmulap.b $e |- ( ph -> B e. QQ ) $.
    irrmulap.b0 $e |- ( ph -> B =/= 0 ) $.
    irrmulap.q $e |- ( ph -> Q e. QQ ) $.
    $( The product of an irrational with a nonzero rational is irrational.  By
       irrational we mean apart from any rational number.  For a similar
       theorem with not rational in place of irrational, see ~ irrmul .
       (Contributed by Jim Kingdon, 25-Aug-2025.) $)
    irrmulap $p |- ( ph -> ( A x. B ) =//= Q ) $=
      ( co cap wbr cmul cq wcel cc0 cc qcn syl cdiv cv breq2 wne qdivcl syl3anc
      rspcdva wb recnd apsym syl2anc mpbird cz 0z ax-mp qapne sylancl apdivmuld
      zq mulcomd breq1d bitrd mpbid ) ADCUAKZBLMZBCNKZDLMZAVEBVDLMZABEUBZLMVHEO
      VDVIVDBLUCGADOPZCOPZCQUDZVDOPZJHIDCUEUFZUGAVDRPZBRPVEVHUHAVMVOVNVDSTABFUI
      ZVDBUJUKULAVECBNKZDLMVGADCBAVJDRPJDSTAVKCRPHCSTZVPACQLMZVLIAVKQOPZVSVLUHH
      QUMPVTUNQUSUOCQUPUQULURAVQVFDLACBVRVPUTVAVBVC $.
  $}

  ${
    $d A x y z $.
    $( A positive rational is the quotient of two positive integers.
       (Contributed by AV, 29-Dec-2022.) $)
    elpq $p |- ( ( A e. QQ /\ 0 < A )
                 -> E. x e. NN E. y e. NN A = ( x / y ) ) $=
      ( vz wcel cc0 clt wbr wa cv cdiv co wceq cn wrex cz rexcom wi cr adantl
      cq elq bitri breq2 wb zre nnre adantr nngt0 gt0div syl3anc sylan9bb elnnz
      bicomd simplbi2 imp weq oveq1 eqeq2d simpll rspcedvd ex sylbid com13 impl
      rexlimdva reximdva biimtrid impcom sylibr ) CUAEZFCGHZICAJZBJZKLZMZANOZBN
      OZVPBNOANOVLVKVRVKCDJZVNKLZMZDPOZBNOZVLVRVKWABNODPOWCDBCUBWADBPNQUCVLWBVQ
      BNVLVNNEZIWAVQDPVLWDVSPEZWAVQRWAWDWEIZVLVQWAWFVLVQRWAWFIZVLFVSGHZVQWAVLFV
      TGHZWFWHCVTFGUDWFWHWIWFVSSEZVNSEZFVNGHZWHWIUEWEWJWDVSUFTWDWKWEVNUGUHWDWLW
      EVNUIUHVSVNUJUKUNULWGWHVQWGWHIZVPWAAVSNWGWHVSNEZWFWHWNRZWAWEWOWDWNWEWHVSU
      MUOTTUPADUQZVPWAUEWMWPVOVTCVMVSVNKURUSTWAWFWHUTVAVBVCVBVDVEVFVGVHVIVPABNN
      QVJ $.

    $( A class is a positive rational iff it is the quotient of two positive
       integers.  (Contributed by AV, 30-Dec-2022.) $)
    elpqb $p |- ( ( A e. QQ /\ 0 < A )
                  <-> E. x e. NN E. y e. NN A = ( x / y ) ) $=
      ( cq wcel cc0 clt wbr wa cv cdiv co wceq cn wrex elpq cr nnre nngt0 jca
      cz nnz znq sylan divgt0 syl2an eleq1 anbi12d syl5ibrcom rexlimivv impbii
      breq2 ) CDEZFCGHZIZCAJZBJZKLZMZBNOANOABCPUSUOABNNUPNEZUQNEZIZUOUSURDEZFUR
      GHZIVBVCVDUTUPUAEVAVCUPUBUPUQUCUDUTUPQEZFUPGHZIUQQEZFUQGHZIVDVAUTVEVFUPRU
      PSTVAVGVHUQRUQSTUPUQUEUFTUSUMVCUNVDCURDUGCURFGULUHUIUJUK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Complex numbers as pairs of reals
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d F u v w z $.  $d x y u v w z $.
    cnref1o.1 $e |- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) ) $.
    $( There is a natural one-to-one mapping from ` ( RR X. RR ) ` to ` CC ` ,
       where we map ` <. x , y >. ` to ` ( x + ( _i x. y ) ) ` .  In our
       construction of the complex numbers, this is in fact our _definition_ of
       ` CC ` (see ~ df-c ), but in the axiomatic treatment we can only show
       that there is the expected mapping between these two sets.  (Contributed
       by Mario Carneiro, 16-Jun-2013.)  (Revised by Mario Carneiro,
       17-Feb-2014.) $)
    cnref1o $p |- F : ( RR X. RR ) -1-1-onto-> CC $=
      ( vz vw vu vv cr cc cv cfv wceq wral wcel ci cmul co caddc wa cxp wf1 wfo
      wf1o wf wfn simpl recnd ax-icn a1i simpr mulcld addcld rgen2a fnmpo ax-mp
      c1st c2nd cop 1st2nd2 fveq2d df-ov eqtr4di xp1st xp2nd oveq1 oveq2 oveq2d
      wi ovmpog syl3anc eqtrd eqeltrd rgen ffnfv mpbir2an wb jca syl2an oveq12d
      cru fveq2 eqeq12d vtoclga eqeqan12d cvv 1stexg 2ndexg opth bitrdi 3bitr4d
      vex biimpd dff13 wrex cnre eqeq2d 2rexbiia sylibr rexxp dffo3 df-f1o ) II
      UAZJCUDXCJCUBZXCJCUCZXDXCJCUEZEKZCLZFKZCLZMZXGXIMZVIZFXCNEXCNXFCXCUFZXHJO
      ZEXCNAKZPBKZQRZSRZJOZBINAINXNXTABIXPIOZXQIOZTZXPXRYCXPYAYBUGUHYCPXQPJOZYC
      UIUJYCXQYAYBUKUHULUMUNABIIXSCJDUOUPXOEXCXGXCOZXHXGUQLZPXGURLZQRZSRZJYEXHY
      FYGCRZYIYEXHYFYGUSZCLYJYEXGYKCXGIIUTZVAYFYGCVBVCYEYFIOZYGIOZYIJOYJYIMXGII
      VDZXGIIVEZYEYFYHYEYFYOUHYEPYGYDYEUIUJYEYGYPUHULUMZABYFYGIIXSYICYFXRSRJXPY
      FXRSVFXQYGMXRYHYFSXQYGPQVGVHDVJVKVLZYQVMVNEXCJCVOVPZXMEFXCYEXIXCOZTZXKXLU
      UAYIXIUQLZPXIURLZQRZSRZMZYFUUBMYGUUCMTZXKXLYEYMYNTUUBIOZUUCIOZTUUFUUGVQYT
      YEYMYNYOYPVRYTUUHUUIXIIIVDXIIIVEVRYFYGUUBUUCWAVSYEYTXHYIXJUUEYRXHYIMXJUUE
      MEXIXCXLXHXJYIUUEXGXICWBXLYFUUBYHUUDSXGXIUQWBXLYGUUCPQXGXIURWBVHVTWCYRWDW
      EUUAXLYKUUBUUCUSZMUUGYEYTXGYKXIUUJYLXIIIUTWEYFYGUUBUUCXGWFOZYFWFOEWLZXGWF
      WGUPUUKYGWFOUULXGWFWHUPWIWJWKWMUNEFXCJCWNVPXEXFXIXHMZEXCWOZFJNYSUUNFJXIJO
      ZXIGKZHKZCRZMZHIWOGIWOZUUNUUOXIUUPPUUQQRZSRZMZHIWOGIWOUUTGHXIWPUUSUVCGHII
      UUPIOZUUQIOZTZUURUVBXIUVFUVDUVEUVBJOUURUVBMUVDUVEUGZUVDUVEUKZUVFUUPUVAUVF
      UUPUVGUHUVFPUUQYDUVFUIUJUVFUUQUVHUHULUMABUUPUUQIIXSUVBCUUPXRSRJXPUUPXRSVF
      XQUUQMXRUVAUUPSXQUUQPQVGVHDVJVKWQWRWSUUMUUSEGHIIXGUUPUUQUSZMZXHUURXIUVJXH
      UVICLUURXGUVICWBUUPUUQCVBVCWQWTWSVNEFXCJCXAVPXCJCXBVP $.
  $}

  $( The addition operation is a set.  (Contributed by NM, 19-Oct-2004.)
     (Revised by Mario Carneiro, 17-Nov-2014.) $)
  addex $p |- + e. _V $=
    ( cc cxp caddc wf cvv wcel ax-addf cnex xpex fex2 mp3an ) AABZACDLEFAEFCEFG
    AAHHIHLACEEJK $.

  $( The multiplication operation is a set.  (Contributed by NM, 19-Oct-2004.)
     (Revised by Mario Carneiro, 17-Nov-2014.) $)
  mulex $p |- x. e. _V $=
    ( cc cxp cmul wf cvv wcel ax-mulf cnex xpex fex2 mp3an ) AABZACDLEFAEFCEFGA
    AHHIHLACEEJK $.


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Order sets
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Positive reals (as a subset of complex numbers)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c RR+ $. $( The set of positive reals (blackboard bold R^+). $)

  $( Extend class notation to include the class of positive reals. $)
  crp $a class RR+ $.

  $( Define the set of positive reals.  Definition of positive numbers in
     [Apostol] p. 20.  (Contributed by NM, 27-Oct-2007.) $)
  df-rp $a |- RR+ = { x e. RR | 0 < x } $.

  ${
    $d x A $.
    $( Membership in the set of positive reals.  (Contributed by NM,
       27-Oct-2007.) $)
    elrp $p |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) ) $=
      ( vx cc0 cv clt wbr cr crp breq2 df-rp elrab2 ) CBDZEFCAEFBAGHLACEIBJK $.
  $}

  ${
    elrpi.1 $e |- A e. RR $.
    elrpi.2 $e |- 0 < A $.
    $( Membership in the set of positive reals.  (Contributed by NM,
       23-Feb-2008.) $)
    elrpii $p |- A e. RR+ $=
      ( crp wcel cr cc0 clt wbr elrp mpbir2an ) ADEAFEGAHIBCAJK $.
  $}

  $( 1 is a positive real.  (Contributed by Jeff Hankins, 23-Nov-2008.) $)
  1rp $p |- 1 e. RR+ $=
    ( c1 1re 0lt1 elrpii ) ABCD $.

  $( 2 is a positive real.  (Contributed by Mario Carneiro, 28-May-2016.) $)
  2rp $p |- 2 e. RR+ $=
    ( c2 2re 2pos elrpii ) ABCD $.

  $( 3 is a positive real.  (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
  3rp $p |- 3 e. RR+ $=
    ( c3 3re 3pos elrpii ) ABCD $.

  $( A positive real is a real.  (Contributed by NM, 27-Oct-2007.) $)
  rpre $p |- ( A e. RR+ -> A e. RR ) $=
    ( vx crp cr cc0 cv clt wbr crab df-rp ssrab2 eqsstri sseli ) CDACEBFGHZBDID
    BJNBDKLM $.

  $( A positive real is an extended real.  (Contributed by Mario Carneiro,
     21-Aug-2015.) $)
  rpxr $p |- ( A e. RR+ -> A e. RR* ) $=
    ( crp wcel rpre rexrd ) ABCAADE $.

  $( A positive real is a complex number.  (Contributed by NM, 11-Nov-2008.) $)
  rpcn $p |- ( A e. RR+ -> A e. CC ) $=
    ( crp wcel rpre recnd ) ABCAADE $.

  $( A positive integer is a positive real.  (Contributed by NM,
     28-Nov-2008.) $)
  nnrp $p |- ( A e. NN -> A e. RR+ ) $=
    ( cn wcel cr cc0 clt wbr crp nnre nngt0 elrp sylanbrc ) ABCADCEAFGAHCAIAJAK
    L $.

  $( The positive reals are a subset of the reals.  (Contributed by NM,
     24-Feb-2008.) $)
  rpssre $p |- RR+ C_ RR $=
    ( vx crp cr cv rpre ssriv ) ABCADEF $.

  $( A positive real is greater than zero.  (Contributed by FL,
     27-Dec-2007.) $)
  rpgt0 $p |- ( A e. RR+ -> 0 < A ) $=
    ( crp wcel cr cc0 clt wbr elrp simprbi ) ABCADCEAFGAHI $.

  $( A positive real is greater than or equal to zero.  (Contributed by NM,
     22-Feb-2008.) $)
  rpge0 $p |- ( A e. RR+ -> 0 <_ A ) $=
    ( crp wcel cr cc0 clt wbr cle rpre rpgt0 wi 0re ltle mpan sylc ) ABCADCZEAF
    GZEAHGZAIAJEDCPQRKLEAMNO $.

  $( A positive real is a positive real number.  (Contributed by NM,
     11-Nov-2008.)  (Revised by Mario Carneiro, 31-Jan-2014.) $)
  rpregt0 $p |- ( A e. RR+ -> ( A e. RR /\ 0 < A ) ) $=
    ( crp wcel cr cc0 clt wbr wa elrp biimpi ) ABCADCEAFGHAIJ $.

  $( A positive real is a nonnegative real number.  (Contributed by Mario
     Carneiro, 31-Jan-2014.) $)
  rprege0 $p |- ( A e. RR+ -> ( A e. RR /\ 0 <_ A ) ) $=
    ( crp wcel cr cc0 cle wbr rpre rpge0 jca ) ABCADCEAFGAHAIJ $.

  $( A positive real is nonzero.  (Contributed by NM, 18-Jul-2008.) $)
  rpne0 $p |- ( A e. RR+ -> A =/= 0 ) $=
    ( crp wcel cr cc0 clt wbr wa wne rpregt0 gt0ne0 syl ) ABCADCEAFGHAEIAJAKL
    $.

  $( A positive real is apart from zero.  (Contributed by Jim Kingdon,
     22-Mar-2020.) $)
  rpap0 $p |- ( A e. RR+ -> A =//= 0 ) $=
    ( crp wcel rpre rpgt0 gt0ap0d ) ABCAADAEF $.

  $( A positive real is a nonzero real number.  (Contributed by NM,
     11-Nov-2008.) $)
  rprene0 $p |- ( A e. RR+ -> ( A e. RR /\ A =/= 0 ) ) $=
    ( crp wcel cr cc0 wne rpre rpne0 jca ) ABCADCAEFAGAHI $.

  $( A positive real is a real number apart from zero.  (Contributed by Jim
     Kingdon, 22-Mar-2020.) $)
  rpreap0 $p |- ( A e. RR+ -> ( A e. RR /\ A =//= 0 ) ) $=
    ( crp wcel cr cc0 cap wbr rpre rpap0 jca ) ABCADCAEFGAHAIJ $.

  $( A positive real is a nonzero complex number.  (Contributed by NM,
     11-Nov-2008.) $)
  rpcnne0 $p |- ( A e. RR+ -> ( A e. CC /\ A =/= 0 ) ) $=
    ( crp wcel cc cc0 wne rpcn rpne0 jca ) ABCADCAEFAGAHI $.

  $( A positive real is a complex number apart from zero.  (Contributed by Jim
     Kingdon, 22-Mar-2020.) $)
  rpcnap0 $p |- ( A e. RR+ -> ( A e. CC /\ A =//= 0 ) ) $=
    ( crp wcel cc cc0 cap wbr rpcn rpap0 jca ) ABCADCAEFGAHAIJ $.

  $( Quantification over positive reals.  (Contributed by NM, 12-Feb-2008.) $)
  ralrp $p |- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) ) $=
    ( cc0 cv clt wbr wi crp cr wcel wa elrp imbi1i impexp bitri ralbii2 ) ACBDZ
    EFZAGZBHIQHJZAGQIJZRKZAGUASGTUBAQLMUARANOP $.

  $( Quantification over positive reals.  (Contributed by Mario Carneiro,
     21-May-2014.) $)
  rexrp $p |- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) ) $=
    ( cc0 cv clt wbr wa crp cr wcel elrp anbi1i anass bitri rexbii2 ) ACBDZEFZA
    GZBHIPHJZAGPIJZQGZAGTRGSUAAPKLTQAMNO $.

  $( Closure law for addition of positive reals.  Part of Axiom 7 of [Apostol]
     p. 20.  (Contributed by NM, 27-Oct-2007.) $)
  rpaddcl $p |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A + B ) e. RR+ ) $=
    ( crp wcel wa caddc co cr cc0 clt wbr rpre readdcl syl2an elrp an4s syl2anb
    addgt0 sylanbrc ) ACDZBCDZEABFGZHDZIUBJKZUBCDTAHDZBHDZUCUAALBLABMNTUEIAJKZE
    UFIBJKZEUDUAAOBOUEUFUGUHUDABRPQUBOS $.

  $( Closure law for multiplication of positive reals.  Part of Axiom 7 of
     [Apostol] p. 20.  (Contributed by NM, 27-Oct-2007.) $)
  rpmulcl $p |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR+ ) $=
    ( crp wcel wa cmul co cc0 clt wbr rpre remulcl syl2an elrp syl2anb sylanbrc
    cr mulgt0 ) ACDZBCDZEABFGZQDZHUAIJZUACDSAQDZBQDZUBTAKBKABLMSUDHAIJEUEHBIJEU
    CTANBNABROUANP $.

  $( Closure law for division of positive reals.  (Contributed by FL,
     27-Dec-2007.) $)
  rpdivcl $p |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A / B ) e. RR+ ) $=
    ( crp wcel wa cdiv co cr cc0 clt wbr cap rpre rpreap0 redivclap syl2an elrp
    3expb divgt0 syl2anb sylanbrc ) ACDZBCDZEABFGZHDZIUDJKZUDCDUBAHDZBHDZBILKZE
    UEUCAMBNUGUHUIUEABORPUBUGIAJKEUHIBJKEUFUCAQBQABSTUDQUA $.

  $( Closure law for reciprocation of positive reals.  (Contributed by Jeff
     Hankins, 23-Nov-2008.) $)
  rpreccl $p |- ( A e. RR+ -> ( 1 / A ) e. RR+ ) $=
    ( c1 crp wcel cdiv co 1rp rpdivcl mpan ) BCDACDBAEFCDGBAHI $.

  $( Closure law for half of a positive real.  (Contributed by Mario Carneiro,
     31-Jan-2014.) $)
  rphalfcl $p |- ( A e. RR+ -> ( A / 2 ) e. RR+ ) $=
    ( crp wcel c2 cdiv co 2rp rpdivcl mpan2 ) ABCDBCADEFBCGADHI $.

  $( A number greater or equal to a positive real is positive real.
     (Contributed by Mario Carneiro, 28-May-2016.) $)
  rpgecl $p |- ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> B e. RR+ ) $=
    ( crp wcel cle wbr w3a cc0 clt simp2 0red rpre 3ad2ant1 rpgt0 simp3 ltletrd
    cr elrp sylanbrc ) ACDZBQDZABEFZGZUAHBIFBCDTUAUBJZUCHABUCKTUAAQDUBALMUDTUAH
    AIFUBANMTUAUBOPBRS $.

  $( Half of a positive real is less than the original number.  (Contributed by
     Mario Carneiro, 21-May-2014.) $)
  rphalflt $p |- ( A e. RR+ -> ( A / 2 ) < A ) $=
    ( crp wcel cr cc0 clt wbr wa c2 cdiv co elrp halfpos biimpa sylbi ) ABCADCZ
    EAFGZHAIJKAFGZALPQRAMNO $.

  $( Closure law for division of a real by a positive real.  (Contributed by
     NM, 10-Nov-2008.) $)
  rerpdivcl $p |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR ) $=
    ( crp wcel cr cc0 cap wbr wa cdiv co rpreap0 redivclap 3expb sylan2 ) BCDAE
    DZBEDZBFGHZIABJKEDZBLPQRSABMNO $.

  $( A nonnegative number plus one is a positive number.  (Contributed by Mario
     Carneiro, 5-Oct-2015.) $)
  ge0p1rp $p |- ( ( A e. RR /\ 0 <_ A ) -> ( A + 1 ) e. RR+ ) $=
    ( cr wcel cc0 cle wbr wa c1 caddc co clt crp peano2re 0red simpl simpr ltp1
    adantr lelttrd elrp sylanbrc ) ABCZDAEFZGZAHIJZBCZDUEKFUELCUBUFUCAMRZUDDAUE
    UDNUBUCOUGUBUCPUBAUEKFUCAQRSUETUA $.

  $( Either a real apart from zero or its negation is a positive real, but not
     both.  (Contributed by Jim Kingdon, 23-Mar-2020.) $)
  rpnegap $p |- ( ( A e. RR /\ A =//= 0 ) -> ( A e. RR+ \/_ -u A e. RR+ ) ) $=
    ( cr wcel cc0 cap wbr wa crp cneg wxo clt 0re reapltxor mpan2 xorcom bitrdi
    wb pm5.32i anxordi elrp bitri biimpi renegcl biantrurd bitr4id lt0neg1 ibar
    a1i 3bitr2d adantr xorbi12d mpbird ) ABCZADEFZGZAHCZAIZHCZJUMDAKFZGZUMADKFZ
    GZJZUOVCUOUMUSVAJZGVCUMUNVDUMUNVAUSJZVDUMDBCUNVEQLADMNVAUSOPRUMUSVASUAUBUOU
    PUTURVBUPUTQUOATUHUMURVBQUNUMURDUQKFZVAVBUMURUQBCZVFGVFUQTUMVGVFAUCUDUEAUFU
    MVAUGUIUJUKUL $.

  $( Elementhood of a negation in the positive real numbers.  (Contributed by
     Thierry Arnoux, 19-Sep-2018.) $)
  negelrp $p |- ( A e. RR -> ( -u A e. RR+ <-> A < 0 ) ) $=
    ( cneg crp wcel cr cc0 clt wbr elrp lt0neg1 renegcl biantrurd bitr2d bitrid
    wa ) ABZCDPEDZFPGHZOZAEDZAFGHZPITUARSAJTQRAKLMN $.

  ${
    negelrpd.1 $e |- ( ph -> A e. RR ) $.
    negelrpd.2 $e |- ( ph -> A < 0 ) $.
    $( The negation of a negative number is in the positive real numbers.
       (Contributed by Glauco Siliprandi, 26-Jun-2021.) $)
    negelrpd $p |- ( ph -> -u A e. RR+ ) $=
      ( cneg crp wcel cc0 clt wbr cr wb negelrp syl mpbird ) ABEFGZBHIJZDABKGPQ
      LCBMNO $.
  $}

  $( Zero is not a positive real.  Axiom 9 of [Apostol] p. 20.  (Contributed by
     NM, 27-Oct-2007.) $)
  0nrp $p |- -. 0 e. RR+ $=
    ( cc0 crp wcel clt wbr 0re ltnri rpgt0 mto ) ABCAADEAFGAHI $.

  $( Subtracting a positive real from another number decreases it.
     (Contributed by FL, 27-Dec-2007.) $)
  ltsubrp $p |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A ) $=
    ( crp wcel cr cc0 clt wbr wa cmin co elrp wi ltsubpos biimpd expcom sylan2b
    imp32 ) BCDAEDZBEDZFBGHZIABJKAGHZBLSTUAUBTSUAUBMTSIUAUBBANOPRQ $.

  $( Adding a positive number to another number increases it.  (Contributed by
     FL, 27-Dec-2007.) $)
  ltaddrp $p |- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) ) $=
    ( crp wcel cr cc0 clt wbr wa caddc co elrp wi ltaddpos biimpd imp32 sylan2b
    expcom ) BCDAEDZBEDZFBGHZIAABJKGHZBLSTUAUBTSUAUBMTSIUAUBBANORPQ $.

  $( Two ways to say one number is less than another.  (Contributed by Mario
     Carneiro, 21-May-2014.) $)
  difrp $p |- ( ( A e. RR /\ B e. RR ) ->
                ( A < B <-> ( B - A ) e. RR+ ) ) $=
    ( cr wcel wa clt wbr cc0 cmin co crp posdif wb resubcl ancoms elrp baib syl
    bitr4d ) ACDZBCDZEZABFGHBAIJZFGZUCKDZABLUBUCCDZUEUDMUATUFBANOUEUFUDUCPQRS
    $.

  ${
    elrpd.1 $e |- ( ph -> A e. RR ) $.
    elrpd.2 $e |- ( ph -> 0 < A ) $.
    $( Membership in the set of positive reals.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    elrpd $p |- ( ph -> A e. RR+ ) $=
      ( cr wcel cc0 clt wbr crp elrp sylanbrc ) ABEFGBHIBJFCDBKL $.
  $}

  ${
    nnrpd.1 $e |- ( ph -> A e. NN ) $.
    $( A positive integer is a positive real.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    nnrpd $p |- ( ph -> A e. RR+ ) $=
      ( cn wcel crp nnrp syl ) ABDEBFECBGH $.
  $}

  $( An integer greater than 1 is a positive real number not equal to 0 or 1.
     Useful for working with integer logarithm bases (which is a common case,
     e.g., base 2, base 3, or base 10).  (Contributed by Thierry Arnoux,
     26-Sep-2017.)  (Proof shortened by AV, 9-Jul-2022.) $)
  zgt1rpn0n1 $p |- ( B e. ( ZZ>= ` 2 ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) )
    $=
    ( c2 cuz cfv wcel crp cc0 wne c1 eluz2nn nnrpd eluz2n0 wn 1nuz2 nelne2 3jca
    mpan2 ) ABCDZEZAFEAGHAIHZSAAJKALSIREMTNAIROQP $.

  ${
    rpred.1 $e |- ( ph -> A e. RR+ ) $.
    $( A positive real is a real.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    rpred $p |- ( ph -> A e. RR ) $=
      ( crp cr rpssre sselid ) ADEBFCG $.

    $( A positive real is an extended real.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    rpxrd $p |- ( ph -> A e. RR* ) $=
      ( rpred rexrd ) ABABCDE $.

    $( A positive real is a complex number.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    rpcnd $p |- ( ph -> A e. CC ) $=
      ( rpred recnd ) ABABCDE $.

    $( A positive real is greater than zero.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    rpgt0d $p |- ( ph -> 0 < A ) $=
      ( crp wcel cc0 clt wbr rpgt0 syl ) ABDEFBGHCBIJ $.

    $( A positive real is greater than or equal to zero.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    rpge0d $p |- ( ph -> 0 <_ A ) $=
      ( crp wcel cc0 cle wbr rpge0 syl ) ABDEFBGHCBIJ $.

    $( A positive real is nonzero.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    rpne0d $p |- ( ph -> A =/= 0 ) $=
      ( crp wcel cc0 wne rpne0 syl ) ABDEBFGCBHI $.

    $( A positive real is apart from zero.  (Contributed by Jim Kingdon,
       28-Jul-2021.) $)
    rpap0d $p |- ( ph -> A =//= 0 ) $=
      ( crp wcel cc0 cap wbr rpap0 syl ) ABDEBFGHCBIJ $.

    $( A positive real is real and greater than zero.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    rpregt0d $p |- ( ph -> ( A e. RR /\ 0 < A ) ) $=
      ( cr wcel cc0 clt wbr rpred rpgt0d jca ) ABDEFBGHABCIABCJK $.

    $( A positive real is real and greater or equal to zero.  (Contributed by
       Mario Carneiro, 28-May-2016.) $)
    rprege0d $p |- ( ph -> ( A e. RR /\ 0 <_ A ) ) $=
      ( cr wcel cc0 cle wbr rpred rpge0d jca ) ABDEFBGHABCIABCJK $.

    $( A positive real is a nonzero real number.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    rprene0d $p |- ( ph -> ( A e. RR /\ A =/= 0 ) ) $=
      ( cr wcel cc0 wne rpred rpne0d jca ) ABDEBFGABCHABCIJ $.

    $( A positive real is a nonzero complex number.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    rpcnne0d $p |- ( ph -> ( A e. CC /\ A =/= 0 ) ) $=
      ( cc wcel cc0 wne rpcnd rpne0d jca ) ABDEBFGABCHABCIJ $.

    $( Closure law for reciprocation of positive reals.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    rpreccld $p |- ( ph -> ( 1 / A ) e. RR+ ) $=
      ( crp wcel c1 cdiv co rpreccl syl ) ABDEFBGHDECBIJ $.

    $( Closure law for reciprocation of positive reals.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    rprecred $p |- ( ph -> ( 1 / A ) e. RR ) $=
      ( c1 cdiv co rpreccld rpred ) ADBEFABCGH $.

    $( Closure law for half of a positive real.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    rphalfcld $p |- ( ph -> ( A / 2 ) e. RR+ ) $=
      ( crp wcel c2 cdiv co rphalfcl syl ) ABDEBFGHDECBIJ $.

    $( The reciprocal of a positive number less than 1 is greater than 1.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    reclt1d $p |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) ) $=
      ( cr wcel cc0 clt wbr wa c1 cdiv co wb rpregt0d reclt1 syl ) ABDEFBGHIBJG
      HJJBKLGHMABCNBOP $.

    $( The reciprocal of a positive number greater than 1 is less than 1.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    recgt1d $p |- ( ph -> ( 1 < A <-> ( 1 / A ) < 1 ) ) $=
      ( cr wcel cc0 clt wbr wa c1 cdiv co wb rpregt0d recgt1 syl ) ABDEFBGHIJBG
      HJBKLJGHMABCNBOP $.

    rpaddcld.1 $e |- ( ph -> B e. RR+ ) $.
    $( Closure law for addition of positive reals.  Part of Axiom 7 of
       [Apostol] p. 20.  (Contributed by Mario Carneiro, 28-May-2016.) $)
    rpaddcld $p |- ( ph -> ( A + B ) e. RR+ ) $=
      ( crp wcel caddc co rpaddcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Closure law for multiplication of positive reals.  Part of Axiom 7 of
       [Apostol] p. 20.  (Contributed by Mario Carneiro, 28-May-2016.) $)
    rpmulcld $p |- ( ph -> ( A x. B ) e. RR+ ) $=
      ( crp wcel cmul co rpmulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( Closure law for division of positive reals.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    rpdivcld $p |- ( ph -> ( A / B ) e. RR+ ) $=
      ( crp wcel cdiv co rpdivcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.

    $( The reciprocal of both sides of 'less than'.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    ltrecd $p |- ( ph -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) $=
      ( cr wcel cc0 clt wbr wa c1 cdiv co wb rpregt0d ltrec syl2anc ) ABFGHBIJK
      CFGHCIJKBCIJLCMNLBMNIJOABDPACEPBCQR $.

    $( The reciprocal of both sides of 'less than or equal to'.  (Contributed
       by Mario Carneiro, 28-May-2016.) $)
    lerecd $p |- ( ph -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) ) $=
      ( cr wcel cc0 clt wbr wa cle c1 cdiv co wb rpregt0d lerec syl2anc ) ABFGH
      BIJKCFGHCIJKBCLJMCNOMBNOLJPABDQACEQBCRS $.

    ${
      ltrec1d.2 $e |- ( ph -> ( 1 / A ) < B ) $.
      $( Reciprocal swap in a 'less than' relation.  (Contributed by Mario
         Carneiro, 28-May-2016.) $)
      ltrec1d $p |- ( ph -> ( 1 / B ) < A ) $=
        ( c1 cdiv co clt wbr cr wcel cc0 wa wb rpregt0d ltrec1 syl2anc mpbid )
        AGBHICJKZGCHIBJKZFABLMNBJKOCLMNCJKOUAUBPABDQACEQBCRST $.
    $}

    ${
      lerec2d.2 $e |- ( ph -> A <_ ( 1 / B ) ) $.
      $( Reciprocal swap in a 'less than or equal to' relation.  (Contributed
         by Mario Carneiro, 28-May-2016.) $)
      lerec2d $p |- ( ph -> B <_ ( 1 / A ) ) $=
        ( c1 cdiv co cle wbr cr wcel cc0 clt wa wb rpregt0d lerec2 syl2anc
        mpbid ) ABGCHIJKZCGBHIJKZFABLMNBOKPCLMNCOKPUBUCQABDRACERBCSTUA $.
    $}

    ${
      lediv2ad.3 $e |- ( ph -> C e. RR ) $.
      lediv2ad.4 $e |- ( ph -> 0 <_ C ) $.
      lediv2ad.5 $e |- ( ph -> A <_ B ) $.
      $( Division of both sides of 'less than or equal to' into a nonnegative
         number.  (Contributed by Mario Carneiro, 28-May-2016.) $)
      lediv2ad $p |- ( ph -> ( C / B ) <_ ( C / A ) ) $=
        ( cr wcel cc0 clt wbr wa cle cdiv co rpregt0d jca lediv2a syl31anc ) AB
        JKLBMNOCJKLCMNODJKZLDPNZOBCPNDCQRDBQRPNABESACFSAUCUDGHTIBCDUAUB $.
    $}

    ltdiv2d.3 $e |- ( ph -> C e. RR+ ) $.
    $( Division of a positive number by both sides of 'less than'.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    ltdiv2d $p |- ( ph -> ( A < B <-> ( C / B ) < ( C / A ) ) ) $=
      ( cr wcel cc0 clt wbr wa cdiv co wb rpregt0d ltdiv2 syl3anc ) ABHIJBKLMCH
      IJCKLMDHIJDKLMBCKLDCNODBNOKLPABEQACFQADGQBCDRS $.

    $( Division of a positive number by both sides of 'less than or equal to'.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    lediv2d $p |- ( ph -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) ) $=
      ( cr wcel cc0 clt wbr wa cle cdiv co wb rpregt0d lediv2 syl3anc ) ABHIJBK
      LMCHIJCKLMDHIJDKLMBCNLDCOPDBOPNLQABERACFRADGRBCDST $.

    ledivdivd.4 $e |- ( ph -> D e. RR+ ) $.
    ${
      ledivdivd.5 $e |- ( ph -> ( A / B ) <_ ( C / D ) ) $.
      $( Invert ratios of positive numbers and swap their ordering.
         (Contributed by Mario Carneiro, 28-May-2016.) $)
      ledivdivd $p |- ( ph -> ( D / C ) <_ ( B / A ) ) $=
        ( cdiv co cle wbr cr wcel cc0 clt wa rpregt0d ledivdiv syl22anc mpbid
        wb ) ABCKLDEKLMNZEDKLCBKLMNZJABOPQBRNSCOPQCRNSDOPQDRNSEOPQERNSUEUFUDABF
        TACGTADHTAEITBCDEUAUBUC $.
    $}
  $}

  $( The ratio of a number over a smaller positive number is larger than 1.
     (Contributed by Glauco Siliprandi, 5-Apr-2020.) $)
  divge1 $p |- ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> 1 <_ ( B / A ) ) $=
    ( crp wcel cr cle wbr w3a c1 cdiv co wceq rpgecl rpcn rpap0 dividapd eqcomd
    syl simp3 simp1 lediv2d mpbid eqbrtrd ) ACDZBEDZABFGZHZIBBJKZBAJKZFUGBCDZIU
    HLABMZUJUHIUJBBNBOPQRUGUFUHUIFGUDUEUFSUGABBUDUEUFTUKUKUAUBUC $.

  $( A real number divided by a positive real number is less than 1 iff the
     real number is less than the positive real number.  (Contributed by AV,
     25-May-2020.) $)
  divlt1lt $p |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) < 1 <-> A < B ) ) $=
    ( cr wcel crp wa cdiv co c1 clt wbr cc0 wb simpl rpregt0 adantl 0lt1 pm3.2i
    1re a1i ltdiv23 syl3anc wceq recn div1d adantr breq1d bitrd ) ACDZBEDZFZABG
    HIJKZAIGHZBJKZABJKUKUIBCDLBJKFZICDZLIJKZFZULUNMUIUJNUJUOUIBOPURUKUPUQSQRTAB
    IUAUBUKUMABJUIUMAUCUJUIAAUDUEUFUGUH $.

  $( A real number divided by a positive real number is less than or equal to 1
     iff the real number is less than or equal to the positive real number.
     (Contributed by AV, 29-Jun-2021.) $)
  divle1le $p |- ( ( A e. RR /\ B e. RR+ )
                   -> ( ( A / B ) <_ 1 <-> A <_ B ) ) $=
    ( cr wcel crp wa cdiv co c1 cle wbr cc0 clt simpl rpregt0 adantl 1re pm3.2i
    wb 0lt1 a1i lediv23 syl3anc wceq recn div1d adantr breq1d bitrd ) ACDZBEDZF
    ZABGHIJKZAIGHZBJKZABJKULUJBCDLBMKFZICDZLIMKZFZUMUOSUJUKNUKUPUJBOPUSULUQURQT
    RUAABIUBUCULUNABJUJUNAUDUKUJAAUEUFUGUHUI $.

  $( If a number is less than or equal to another number, the number divided by
     a positive number greater than or equal to one is less than or equal to
     the other number.  (Contributed by AV, 29-Jun-2021.) $)
  ledivge1le $p |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) )
                     -> ( A <_ B -> ( A / C ) <_ B ) ) $=
    ( cr wcel crp c1 cle wbr wa w3a cdiv co wi wb adantr 1red syl3anc cc0 clt
    divle1le rerpdivcl rpre letr expd sylbird com23 expimpd ex 3imp1 simp1 0lt1
    adantl 0red ltletr mpani imp jca 3ad2ant3 rpregt0 3ad2ant2 3jca lediv23 syl
    mpbird ) ADEZBFEZCFEZGCHIZJZKZABHIZACLMBHIZVKVLJZVMABLMZCHIZVFVGVJVLVPVFVGV
    JVLVPNZNVFVGJZVHVIVQVRVHJZVLVIVPVSVLVOGHIZVIVPNVRVTVLOVHABUAPVSVTVIVPVSVODE
    ZGDEZCDEZVTVIJVPNVRWAVHABUBPVSQVHWCVRCUCZUMVOGCUDRUEUFUGUHUIUJVNVFWCSCTIZJZ
    BDESBTIJZKZVMVPOVKWHVLVKVFWFWGVFVGVJUKVJVFWFVGVJWCWEVHWCVIWDPVHVIWEVHSGTIZV
    IWEULVHSDEWBWCWIVIJWENVHUNVHQWDSGCUORUPUQURUSVGVFWGVJBUTVAVBPACBVCVDVEUI $.

  ${
    rpgecld.1 $e |- ( ph -> A e. RR ) $.
    ${
      ge0p1rp.2 $e |- ( ph -> 0 <_ A ) $.
      $( A nonnegative number plus one is a positive number.  (Contributed by
         Mario Carneiro, 28-May-2016.) $)
      ge0p1rpd $p |- ( ph -> ( A + 1 ) e. RR+ ) $=
        ( cr wcel cc0 cle wbr c1 caddc co crp ge0p1rp syl2anc ) ABEFGBHIBJKLMFC
        DBNO $.
    $}

    rpgecld.2 $e |- ( ph -> B e. RR+ ) $.
    $( Closure law for division of a real by a positive real.  (Contributed by
       Mario Carneiro, 28-May-2016.) $)
    rerpdivcld $p |- ( ph -> ( A / B ) e. RR ) $=
      ( cr wcel crp cdiv co rerpdivcl syl2anc ) ABFGCHGBCIJFGDEBCKL $.

    $( Subtracting a positive real from another number decreases it.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    ltsubrpd $p |- ( ph -> ( A - B ) < A ) $=
      ( cr wcel crp cmin co clt wbr ltsubrp syl2anc ) ABFGCHGBCIJBKLDEBCMN $.

    $( Adding a positive number to another number increases it.  (Contributed
       by Mario Carneiro, 28-May-2016.) $)
    ltaddrpd $p |- ( ph -> A < ( A + B ) ) $=
      ( cr wcel crp caddc co clt wbr ltaddrp syl2anc ) ABFGCHGBBCIJKLDEBCMN $.

    $( Adding a positive number to another number increases it.  (Contributed
       by Mario Carneiro, 28-May-2016.) $)
    ltaddrp2d $p |- ( ph -> A < ( B + A ) ) $=
      ( caddc co clt ltaddrpd recnd rpcnd addcomd breqtrd ) ABBCFGCBFGHABCDEIAB
      CABDJACEKLM $.

    $( Multiplication by a number greater than 1.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    ltmulgt11d $p |- ( ph -> ( 1 < A <-> B < ( B x. A ) ) ) $=
      ( cr wcel cc0 clt wbr c1 cmul co wb rpred rpgt0d ltmulgt11 syl3anc ) ACFG
      BFGHCIJKBIJCCBLMIJNACEODACEPCBQR $.

    $( Multiplication by a number greater than 1.  (Contributed by Mario
       Carneiro, 28-May-2016.) $)
    ltmulgt12d $p |- ( ph -> ( 1 < A <-> B < ( A x. B ) ) ) $=
      ( cr wcel cc0 clt wbr c1 cmul co wb rpred rpgt0d ltmulgt12 syl3anc ) ACFG
      BFGHCIJKBIJCBCLMIJNACEODACEPCBQR $.

    $( Division of a positive number by a positive number.  (Contributed by
       Mario Carneiro, 28-May-2016.) $)
    gt0divd $p |- ( ph -> ( 0 < A <-> 0 < ( A / B ) ) ) $=
      ( cr wcel cc0 clt wbr cdiv co wb rpred rpgt0d gt0div syl3anc ) ABFGCFGHCI
      JHBIJHBCKLIJMDACENACEOBCPQ $.

    $( Division of a nonnegative number by a positive number.  (Contributed by
       Mario Carneiro, 28-May-2016.) $)
    ge0divd $p |- ( ph -> ( 0 <_ A <-> 0 <_ ( A / B ) ) ) $=
      ( cr wcel cc0 clt wbr cle cdiv co wb rpred rpgt0d ge0div syl3anc ) ABFGCF
      GHCIJHBKJHBCLMKJNDACEOACEPBCQR $.

    ${
      rpgecld.3 $e |- ( ph -> B <_ A ) $.
      $( A number greater or equal to a positive real is positive real.
         (Contributed by Mario Carneiro, 28-May-2016.) $)
      rpgecld $p |- ( ph -> A e. RR+ ) $=
        ( crp wcel cr cle wbr rpgecl syl3anc ) ACGHBIHCBJKBGHEDFCBLM $.
    $}

    divge0d.3 $e |- ( ph -> 0 <_ A ) $.
    $( The ratio of nonnegative and positive numbers is nonnegative.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    divge0d $p |- ( ph -> 0 <_ ( A / B ) ) $=
      ( cr wcel cc0 cle wbr clt wa cdiv co rpregt0d divge0 syl21anc ) ABGHIBJKC
      GHICLKMIBCNOJKDFACEPBCQR $.
  $}

  ${
    ltmul1d.1 $e |- ( ph -> A e. RR ) $.
    ltmul1d.2 $e |- ( ph -> B e. RR ) $.
    ltmul1d.3 $e |- ( ph -> C e. RR+ ) $.
    $( The ratio of nonnegative and positive numbers is nonnegative.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    ltmul1d $p |- ( ph -> ( A < B <-> ( A x. C ) < ( B x. C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cmul co wb rpregt0d ltmul1 syl3anc ) ABHICHIDHIJ
      DKLMBCKLBDNOCDNOKLPEFADGQBCDRS $.

    $( Multiplication of both sides of 'less than' by a positive number.
       Theorem I.19 of [Apostol] p. 20.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    ltmul2d $p |- ( ph -> ( A < B <-> ( C x. A ) < ( C x. B ) ) ) $=
      ( cr wcel cc0 clt wbr wa cmul co wb rpregt0d ltmul2 syl3anc ) ABHICHIDHIJ
      DKLMBCKLDBNODCNOKLPEFADGQBCDRS $.

    $( Multiplication of both sides of 'less than or equal to' by a positive
       number.  (Contributed by Mario Carneiro, 28-May-2016.) $)
    lemul1d $p |- ( ph -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cle cmul co wb rpregt0d lemul1 syl3anc ) ABHICHI
      DHIJDKLMBCNLBDOPCDOPNLQEFADGRBCDST $.

    $( Multiplication of both sides of 'less than or equal to' by a positive
       number.  (Contributed by Mario Carneiro, 28-May-2016.) $)
    lemul2d $p |- ( ph -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) ) $=
      ( cr wcel cc0 clt wbr wa cle cmul co wb rpregt0d lemul2 syl3anc ) ABHICHI
      DHIJDKLMBCNLDBOPDCOPNLQEFADGRBCDST $.

    $( Division of both sides of 'less than' by a positive number.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    ltdiv1d $p |- ( ph -> ( A < B <-> ( A / C ) < ( B / C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cdiv co wb rpregt0d ltdiv1 syl3anc ) ABHICHIDHIJ
      DKLMBCKLBDNOCDNOKLPEFADGQBCDRS $.

    $( Division of both sides of a less than or equal to relation by a positive
       number.  (Contributed by Mario Carneiro, 28-May-2016.) $)
    lediv1d $p |- ( ph -> ( A <_ B <-> ( A / C ) <_ ( B / C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cle cdiv co wb rpregt0d lediv1 syl3anc ) ABHICHI
      DHIJDKLMBCNLBDOPCDOPNLQEFADGRBCDST $.

    $( 'Less than' relationship between division and multiplication.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    ltmuldivd $p |- ( ph -> ( ( A x. C ) < B <-> A < ( B / C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cmul co cdiv wb rpregt0d ltmuldiv syl3anc ) ABHI
      CHIDHIJDKLMBDNOCKLBCDPOKLQEFADGRBCDST $.

    $( 'Less than' relationship between division and multiplication.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    ltmuldiv2d $p |- ( ph -> ( ( C x. A ) < B <-> A < ( B / C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cmul co cdiv wb rpregt0d ltmuldiv2 syl3anc ) ABH
      ICHIDHIJDKLMDBNOCKLBCDPOKLQEFADGRBCDST $.

    $( 'Less than or equal to' relationship between division and
       multiplication.  (Contributed by Mario Carneiro, 30-May-2016.) $)
    lemuldivd $p |- ( ph -> ( ( A x. C ) <_ B <-> A <_ ( B / C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cmul co cle cdiv wb rpregt0d lemuldiv syl3anc )
      ABHICHIDHIJDKLMBDNOCPLBCDQOPLREFADGSBCDTUA $.

    $( 'Less than or equal to' relationship between division and
       multiplication.  (Contributed by Mario Carneiro, 30-May-2016.) $)
    lemuldiv2d $p |- ( ph -> ( ( C x. A ) <_ B <-> A <_ ( B / C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cmul co cle cdiv wb rpregt0d lemuldiv2 syl3anc )
      ABHICHIDHIJDKLMDBNOCPLBCDQOPLREFADGSBCDTUA $.

    $( 'Less than' relationship between division and multiplication.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    ltdivmuld $p |- ( ph -> ( ( A / C ) < B <-> A < ( C x. B ) ) ) $=
      ( cr wcel cc0 clt wbr wa cdiv co cmul wb rpregt0d ltdivmul syl3anc ) ABHI
      CHIDHIJDKLMBDNOCKLBDCPOKLQEFADGRBCDST $.

    $( 'Less than' relationship between division and multiplication.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    ltdivmul2d $p |- ( ph -> ( ( A / C ) < B <-> A < ( B x. C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cdiv co cmul wb rpregt0d ltdivmul2 syl3anc ) ABH
      ICHIDHIJDKLMBDNOCKLBCDPOKLQEFADGRBCDST $.

    $( 'Less than or equal to' relationship between division and
       multiplication.  (Contributed by Mario Carneiro, 28-May-2016.) $)
    ledivmuld $p |- ( ph -> ( ( A / C ) <_ B <-> A <_ ( C x. B ) ) ) $=
      ( cr wcel cc0 clt wbr wa cdiv co cle cmul wb rpregt0d ledivmul syl3anc )
      ABHICHIDHIJDKLMBDNOCPLBDCQOPLREFADGSBCDTUA $.

    $( 'Less than or equal to' relationship between division and
       multiplication.  (Contributed by Mario Carneiro, 28-May-2016.) $)
    ledivmul2d $p |- ( ph -> ( ( A / C ) <_ B <-> A <_ ( B x. C ) ) ) $=
      ( cr wcel cc0 clt wbr wa cdiv co cle cmul wb rpregt0d ledivmul2 syl3anc )
      ABHICHIDHIJDKLMBDNOCPLBCDQOPLREFADGSBCDTUA $.

    ${
      ltdiv1dd.4 $e |- ( ph -> A < B ) $.
      $( The ratio of nonnegative and positive numbers is nonnegative.
         (Contributed by Mario Carneiro, 30-May-2016.) $)
      ltmul1dd $p |- ( ph -> ( A x. C ) < ( B x. C ) ) $=
        ( clt wbr cmul co ltmul1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.

      $( Multiplication of both sides of 'less than' by a positive number.
         Theorem I.19 of [Apostol] p. 20.  (Contributed by Mario Carneiro,
         30-May-2016.) $)
      ltmul2dd $p |- ( ph -> ( C x. A ) < ( C x. B ) ) $=
        ( clt wbr cmul co ltmul2d mpbid ) ABCIJDBKLDCKLIJHABCDEFGMN $.

      $( Division of both sides of 'less than' by a positive number.
         (Contributed by Mario Carneiro, 30-May-2016.) $)
      ltdiv1dd $p |- ( ph -> ( A / C ) < ( B / C ) ) $=
        ( clt wbr cdiv co ltdiv1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
    $}

    ${
      lediv1dd.4 $e |- ( ph -> A <_ B ) $.
      $( Division of both sides of a less than or equal to relation by a
         positive number.  (Contributed by Mario Carneiro, 30-May-2016.) $)
      lediv1dd $p |- ( ph -> ( A / C ) <_ ( B / C ) ) $=
        ( cle wbr cdiv co lediv1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
    $}

    ${
      lediv12ad.4 $e |- ( ph -> D e. RR ) $.
      lediv12ad.5 $e |- ( ph -> 0 <_ A ) $.
      lediv12ad.6 $e |- ( ph -> A <_ B ) $.
      lediv12ad.7 $e |- ( ph -> C <_ D ) $.
      $( Comparison of ratio of two nonnegative numbers.  (Contributed by Mario
         Carneiro, 28-May-2016.) $)
      lediv12ad $p |- ( ph -> ( A / D ) <_ ( B / C ) ) $=
        ( cr wcel wa cc0 cle wbr cdiv jca clt co rpred rpgt0d lediv12a syl22anc
        ) ABMNZCMNZOPBQRZBCQRZODMNZEMNZOPDUARZDEQRZOBESUBCDSUBQRAUGUHFGTAUIUJJK
        TAUKULADHUCITAUMUNADHUDLTBCDEUEUF $.
    $}
  $}

  ${
    ltdiv23d.1 $e |- ( ph -> A e. RR ) $.
    ltdiv23d.2 $e |- ( ph -> B e. RR+ ) $.
    ltdiv23d.3 $e |- ( ph -> C e. RR+ ) $.
    ${
      ltdiv23d.4 $e |- ( ph -> ( A / B ) < C ) $.
      $( Swap denominator with other side of 'less than'.  (Contributed by
         Mario Carneiro, 28-May-2016.) $)
      ltdiv23d $p |- ( ph -> ( A / C ) < B ) $=
        ( cdiv co clt wbr cr wcel cc0 wa wb rpregt0d ltdiv23 syl3anc mpbid ) AB
        CIJDKLZBDIJCKLZHABMNCMNOCKLPDMNODKLPUBUCQEACFRADGRBCDSTUA $.
    $}

    ${
      lediv23d.4 $e |- ( ph -> ( A / B ) <_ C ) $.
      $( Swap denominator with other side of 'less than or equal to'.
         (Contributed by Mario Carneiro, 28-May-2016.) $)
      lediv23d $p |- ( ph -> ( A / C ) <_ B ) $=
        ( cdiv co cle wbr cr wcel cc0 clt wa wb rpregt0d lediv23 syl3anc mpbid
        ) ABCIJDKLZBDIJCKLZHABMNCMNOCPLQDMNODPLQUCUDREACFSADGSBCDTUAUB $.
    $}
  $}

  ${
    mul2lt0.1 $e |- ( ph -> A e. RR ) $.
    mul2lt0.2 $e |- ( ph -> B e. RR ) $.
    ${
      mul2lt0.3 $e |- ( ph -> ( A x. B ) < 0 ) $.
      $( If the result of a multiplication is strictly negative, then
         multiplicands are of different signs.  (Contributed by Thierry Arnoux,
         19-Sep-2018.) $)
      mul2lt0rlt0 $p |- ( ( ph /\ B < 0 ) -> 0 < A ) $=
        ( cc0 clt wbr wa cneg cmul co cdiv cr wcel remulcld adantr cc recnd
        0red simpr negelrpd ltdiv1dd mulcld divneg2apd divcanap4d negeqd eqtr3d
        lt0ap0d negcld negap0d div0apd 3brtr3d lt0neg2d mpbird ) ACGHIZJZGBHIBK
        ZGHIURBCLMZCKZNMZGVANMUSGHURUTGVAAUTOPUQABCDEQRURUAURCACOPUQERZAUQUBZUC
        AUTGHIUQFRUDURUTCNMZKVBUSURUTCURBCABSPUQABDTRZACSPUQACETRZUEVGURCVCVDUJ
        ZUFURVEBURBCVFVGVHUGUHUIURVAURCVGUKURCVGVHULUMUNURBABOPUQDRUOUP $.

      $( If the result of a multiplication is strictly negative, then
         multiplicands are of different signs.  (Contributed by Thierry Arnoux,
         19-Sep-2018.) $)
      mul2lt0rgt0 $p |- ( ( ph /\ 0 < B ) -> A < 0 ) $=
        ( cc0 clt wbr wa cmul co adantr wcel recnd mul02d breqtrrd 0red simpr
        cr elrpd ltmul1d mpbird ) AGCHIZJZBGHIBCKLZGCKLZHIUEUFGUGHAUFGHIUDFMUEC
        UECACTNUDEMZOPQUEBGCABTNUDDMUERUECUHAUDSUAUBUC $.

      $( If the result of a multiplication is strictly negative, then
         multiplicands are of different signs.  (Contributed by Thierry Arnoux,
         19-Sep-2018.) $)
      mul2lt0llt0 $p |- ( ( ph /\ A < 0 ) -> 0 < B ) $=
        ( cmul co cc0 clt recnd mulcomd eqbrtrrd mul2lt0rlt0 ) ACBEDABCGHCBGHIJ
        ABCABDKACEKLFMN $.

      $( If the result of a multiplication is strictly negative, then
         multiplicands are of different signs.  (Contributed by Thierry Arnoux,
         2-Oct-2018.) $)
      mul2lt0lgt0 $p |- ( ( ph /\ 0 < A ) -> B < 0 ) $=
        ( cmul co cc0 clt recnd mulcomd eqbrtrrd mul2lt0rgt0 ) ACBEDABCGHCBGHIJ
        ABCABDKACEKLFMN $.
    $}

    ${
      mul2lt0.an $e |- ( ph -> A < 0 ) $.
      mul2lt0.bp $e |- ( ph -> 0 < B ) $.
      $( The product of multiplicands of different signs is negative.
         (Contributed by Jim Kingdon, 25-Feb-2024.) $)
      mul2lt0np $p |- ( ph -> ( A x. B ) < 0 ) $=
        ( cmul co cc0 clt 0red elrpd ltmul1dd recnd mul02d breqtrd ) ABCHIJCHIJ
        KABJCDALACEGMFNACACEOPQ $.

      $( The product of multiplicands of different signs is negative.
         (Contributed by Jim Kingdon, 25-Feb-2024.) $)
      mul2lt0pn $p |- ( ph -> ( B x. A ) < 0 ) $=
        ( cmul co cc0 clt recnd mulcomd mul2lt0np eqbrtrrd ) ABCHICBHIJKABCABDL
        ACELMABCDEFGNO $.
    $}
  $}

  ${
    lt2mul2divd.1 $e |- ( ph -> A e. RR ) $.
    lt2mul2divd.2 $e |- ( ph -> B e. RR+ ) $.
    lt2mul2divd.3 $e |- ( ph -> C e. RR ) $.
    lt2mul2divd.4 $e |- ( ph -> D e. RR+ ) $.
    $( The ratio of nonnegative and positive numbers is nonnegative.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    lt2mul2divd $p |- ( ph ->
      ( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) ) $=
      ( cr wcel cc0 clt wbr wa cmul co cdiv wb rpregt0d lt2mul2div syl22anc ) A
      BJKCJKLCMNODJKEJKLEMNOBCPQDEPQMNBERQDCRQMNSFACGTHAEITBCDEUAUB $.
  $}

  $( Division of a positive integer by a positive number is less than or equal
     to the integer iff the number is greater than or equal to 1.  (Contributed
     by AV, 19-Jun-2021.) $)
  nnledivrp $p |- ( ( A e. NN /\ B e. RR+ )
                     -> ( 1 <_ B <-> ( A / B ) <_ A ) ) $=
    ( cn wcel crp wa c1 cle wbr cdiv co cr cc0 clt wb 1re pm3.2i rpregt0 adantr
    0lt1 adantl nnre nngt0 jca lediv2 mp3an2i wceq nncn div1d breq2d bitrd ) AC
    DZBEDZFZGBHIZABJKZAGJKZHIZUPAHIGLDZMGNIZFUNBLDMBNIFZALDZMANIZFZUOUROUSUTPTQ
    UMVAULBRUAULVDUMULVBVCAUBAUCUDSGBAUEUFUNUQAUPHULUQAUGUMULAAUHUISUJUK $.

  $( Division of a nonnegative integer by a positive integer is less than or
     equal to the integer.  (Contributed by AV, 19-Jun-2021.) $)
  nn0ledivnn $p |- ( ( A e. NN0 /\ B e. NN ) -> ( A / B ) <_ A ) $=
    ( cn0 wcel cn cdiv co cle wbr cc0 wceq wo wi elnn0 wa c1 nnge1 adantl wb ex
    crp nnrp nnledivrp sylan2 mpbid cap nncn nnap0 jca div0ap syl 0le0 eqbrtrdi
    cc oveq1 id breq12d adantr mpbird jaoi sylbi imp ) ACDZBEDZABFGZAHIZVCAEDZA
    JKZLVDVFMZANVGVIVHVGVDVFVGVDOPBHIZVFVDVJVGBQRVDVGBUADVJVFSBUBABUCUDUETVHVDV
    FVHVDOZVFJBFGZJHIZVKVLJJHVKBUNDZBJUFIZOZVLJKVDVPVHVDVNVOBUGBUHUIRBUJUKULUMV
    HVFVMSVDVHVEVLAJHAJBFUOVHUPUQURUSTUTVAVB $.

  $( If the sum of a real number and a positive real number is less than or
     equal to a third real number, the first real number is less than the third
     real number.  (Contributed by AV, 1-Jul-2021.) $)
  addlelt $p |- ( ( M e. RR /\ N e. RR /\ A e. RR+ )
                      -> ( ( M + A ) <_ N -> M < N ) ) $=
    ( cr wcel crp w3a caddc clt wbr cle cc0 rpgt0 3ad2ant3 rpre simp1 ltaddposd
    co mpbid wa wi simpl adantl readdcld 3adant2 simp2 ltletr syl3anc mpand ) B
    DEZCDEZAFEZGZBBAHRZIJZUNCKJZBCIJZUMLAIJZUOULUJURUKAMNUMABULUJADEZUKAOZNUJUK
    ULPZQSUMUJUNDEZUKUOUPTUQUAVAUJULVBUKUJULTBAUJULUBULUSUJUTUCUDUEUJUKULUFBUNC
    UGUHUI $.

  ${
    ltesubnnd.1 $e |- ( ph -> M e. ZZ ) $.
    ltesubnnd.2 $e |- ( ph -> N e. NN ) $.
    $( Subtracting an integer number from another number decreases it.  See
       ~ ltsubrpd .  (Contributed by Thierry Arnoux, 18-Apr-2017.) $)
    ltesubnnd $p |- ( ph -> ( ( M + 1 ) - N ) <_ M ) $=
      ( c1 caddc co cmin cle zcnd 1cnd nncnd addsubd clt wbr zred nnrpd cz wcel
      ltsubrpd wb nnzd zsubcld zltp1le syl2anc mpbid eqbrtrd ) ABFGHCIHBCIHZFGH
      ZBJABFCABDKALACEMNAUIBOPZUJBJPZABCABDQACERUAAUISTBSTUKULUBABCDACEUCUDDUIB
      UEUFUGUH $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Infinity and the extended real number system (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
  $c -e $. $( the negative of an extended real number. $)
  $c +e $. $( Addition of extended real numbers. $)
  $c *e $. $( Multiplication of extended real numbers. $)

  $( Extend class notation to include the negative of an extended real. $)
  cxne $a class -e A $.

  $( Extend class notation to include addition of extended reals. $)
  cxad $a class +e $.

  $( Extend class notation to include multiplication of extended reals. $)
  cxmu $a class *e $.

  $( Define the negative of an extended real number.  (Contributed by FL,
     26-Dec-2011.) $)
  df-xneg $a |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) $.

  ${
    $d x y $.
    $( Define addition over extended real numbers.  (Contributed by Mario
       Carneiro, 20-Aug-2015.) $)
    df-xadd $a |- +e = ( x e. RR* , y e. RR* |-> if ( x = +oo ,
            if ( y = -oo , 0 , +oo ) , if ( x = -oo ,
            if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo ,
                 if ( y = -oo , -oo , ( x + y ) ) ) ) ) ) $.

    $( Define multiplication over extended real numbers.  (Contributed by Mario
       Carneiro, 20-Aug-2015.) $)
    df-xmul $a |- *e = ( x e. RR* , y e. RR* |-> if ( ( x = 0 \/ y = 0 ) , 0 ,
            if ( ( ( ( 0 < y /\ x = +oo ) \/ ( y < 0 /\ x = -oo ) ) \/
                   ( ( 0 < x /\ y = +oo ) \/ ( x < 0 /\ y = -oo ) ) ) , +oo ,
            if ( ( ( ( 0 < y /\ x = -oo ) \/ ( y < 0 /\ x = +oo ) ) \/
                   ( ( 0 < x /\ y = -oo ) \/ ( x < 0 /\ y = +oo ) ) ) , -oo ,
                 ( x x. y ) ) ) ) ) $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The 'less than' binary relation on the set of extended reals.
       Definition 12-3.1 of [Gleason] p. 173.  (Contributed by NM,
       14-Oct-2005.) $)
    ltxr $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <->
              ( ( ( ( A e. RR /\ B e. RR ) /\ A <RR B ) \/
                   ( A = -oo /\ B = +oo ) ) \/ ( ( A e. RR /\ B = +oo )
                  \/ ( A = -oo /\ B e. RR ) ) ) ) ) $=
      ( vx vy cxr wcel wa cv cr cltrr wbr cmnf csn cun cpnf cxp wceq clt bitrid
      wo w3a copab breq12 df-3an opabbii brab2ga a1i brun brxp elun orcom bitri
      wb elsng orbi1d bi2anan9 andir bitrdi anbi1d adantr orbi12d orass df-ltxr
      breqi 3bitr4g ) AEFZBEFZGZABCHZIFZDHZIFZVIVKJKZUAZCDUBZKZABILMZNZOMZPZVQI
      PZNZKZTZAIFZBIFZGABJKZGZALQZBOQZGZWEWJGZWIWFGZTZTZTABRKZWHWKTWNTVHVPWHWCW
      OVPWHUMVHVMWGCDABIIVOVIAVKBJUCVNVJVLGVMGCDVJVLVMUDUEUFUGWCABVTKZABWAKZTZV
      HWOABVTWAUHVHWSWKWLTZWMTWOVHWQWTWRWMWQAVRFZBVSFZGZVHWTABVRVSUIVHXCWIWETZW
      JGWTVFXAXDVGXBWJXAAVQFZWETZVFXDXAWEXETXFAIVQUJWEXEUKULVFXEWIWEALEUNZUOSBO
      EUNUPWIWEWJUQURSWRXEWFGZVHWMABVQIUIVFXHWMUMVGVFXEWIWFXGUSUTSVAWKWLWMVBURS
      VAWPABVOWBNZKWDABRXICDVCVDABVOWBUHULWHWKWNVBVE $.
  $}

  $( Membership in the set of extended reals.  (Contributed by NM,
     14-Oct-2005.) $)
  elxr $p |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) ) $=
    ( cxr wcel cr cpnf cmnf cpr cun wo wceq df-xr eleq2i elun pnfex mnfxr elexi
    w3o elpr2 orbi2i 3orass bitr4i 3bitri ) ABCADEFGZHZCADCZAUCCZIZUEAEJZAFJZQZ
    BUDAKLADUCMUGUEUHUIIZIUJUFUKUEAEFNFBOPRSUEUHUITUAUB $.

  $( An extended real other than minus infinity is real or positive infinite.
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xrnemnf $p |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) ) $=
    ( cr wcel cpnf wceq wo cmnf wn cxr wne pm5.61 w3o elxr df-3or bitri anbi12i
    wa df-ne renemnf pnfnemnf neeq1 mpbiri jaoi neneqd pm4.71i 3bitr4i ) ABCZAD
    EZFZAGEZFZUJHZQUIULQAICZAGJZQUIUIUJKUMUKUNULUMUGUHUJLUKAMUGUHUJNOAGRPUIULUI
    AGUGUNUHASUHUNDGJTADGUAUBUCUDUEUF $.

  $( An extended real other than plus infinity is real or negative infinite.
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xrnepnf $p |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) ) $=
    ( cr wcel cmnf wceq wo cpnf wn wa cxr wne pm5.61 w3o elxr df-3or or32 df-ne
    3bitri anbi12i renepnf mnfnepnf neeq1 mpbiri jaoi neneqd pm4.71i 3bitr4i )
    ABCZADEZFZAGEZFZUKHZIUJUMIAJCZAGKZIUJUJUKLUNULUOUMUNUHUKUIMUHUKFUIFULANUHUK
    UIOUHUKUIPRAGQSUJUMUJAGUHUOUIATUIUODGKUAADGUBUCUDUEUFUG $.

  $( The extended real 'less than' is irreflexive.  (Contributed by NM,
     14-Oct-2005.) $)
  xrltnr $p |- ( A e. RR* -> -. A < A ) $=
    ( cxr wcel cr cpnf wceq cmnf clt wbr wa cltrr neli intnanr pnfnemnf pm3.2ni
    wo intnan wb pnfxr ltxr w3o elxr ltnr pnfnre neii mp2an mtbir breq12 anidms
    wn mtbiri mnfnre nesymi mnfxr 3jaoi sylbi ) ABCADCZAEFZAGFZUAAAHIZUJZAUBUQV
    AURUSAUCURUTEEHIZVBEDCZVCJZEEKIZJZEGFZEEFZJZPZVCVHJZVGVCJZPZPZVJVMVFVIVDVEV
    CVCEDUDLZQMVGVHEGNUEMOVKVLVCVHVOMVCVGVOQOOEBCZVPVBVNRSSEETUFUGURUTVBRAEAEHU
    HUIUKUSUTGGHIZVQGDCZVRJZGGKIZJZGGFZGEFZJZPZVRWCJZWBVRJZPZPZWEWHWAWDVSVTVRVR
    GDULLZQMWCWBEGNUMQOWFWGVRWCWJMVRWBWJQOOGBCZWKVQWIRUNUNGGTUFUGUSUTVQRAGAGHUH
    UIUKUOUP $.

  $( Any (finite) real is less than plus infinity.  (Contributed by NM,
     14-Oct-2005.) $)
  ltpnf $p |- ( A e. RR -> A < +oo ) $=
    ( cr wcel cpnf clt wbr wa cltrr cmnf wceq wo eqid orc mpan2 olcd rexr pnfxr
    cxr wb ltxr sylancl mpbird ) ABCZADEFZUCDBCZGADHFGAIJZDDJZGKZUCUGGZUFUEGZKZ
    KZUCUKUHUCUGUKDLUIUJMNOUCARCDRCUDULSAPQADTUAUB $.

  ${
    ltpnfd.a $e |- ( ph -> A e. RR ) $.
    $( Any (finite) real is less than plus infinity.  (Contributed by Glauco
       Siliprandi, 11-Dec-2019.) $)
    ltpnfd $p |- ( ph -> A < +oo ) $=
      ( cr wcel cpnf clt wbr ltpnf syl ) ABDEBFGHCBIJ $.
  $}

  $( Zero is less than plus infinity (common case).  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  0ltpnf $p |- 0 < +oo $=
    ( cc0 cr wcel cpnf clt wbr 0re ltpnf ax-mp ) ABCADEFGAHI $.

  $( Minus infinity is less than any (finite) real.  (Contributed by NM,
     14-Oct-2005.) $)
  mnflt $p |- ( A e. RR -> -oo < A ) $=
    ( cr wcel cmnf clt wbr wa cltrr wceq cpnf wo eqid olc mpan olcd cxr wb rexr
    mnfxr ltxr sylancr mpbird ) ABCZDAEFZDBCZUCGDAHFGDDIZAJIZGKZUEUGGZUFUCGZKZK
    ZUCUKUHUFUCUKDLUJUIMNOUCDPCAPCUDULQSARDATUAUB $.

  $( Minus infinity is less than 0 (common case).  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  mnflt0 $p |- -oo < 0 $=
    ( cc0 cr wcel cmnf clt wbr 0re mnflt ax-mp ) ABCDAEFGAHI $.

  $( Minus infinity is less than plus infinity.  (Contributed by NM,
     14-Oct-2005.) $)
  mnfltpnf $p |- -oo < +oo $=
    ( cmnf cpnf clt wbr cr wcel wa cltrr wceq wo eqid olc mp2an cxr mnfxr pnfxr
    orci wb ltxr mpbir ) ABCDZAEFZBEFZGABHDGZAAIZBBIZGZJZUBUFGUEUCGJZJZUHUIUEUF
    UHAKBKUGUDLMQANFBNFUAUJROPABSMT $.

  $( Minus infinity is less than an extended real that is either real or plus
     infinity.  (Contributed by NM, 2-Feb-2006.) $)
  mnfltxr $p |- ( ( A e. RR \/ A = +oo ) -> -oo < A ) $=
    ( cr wcel cmnf clt wbr cpnf wceq mnflt mnfltpnf breq2 mpbiri jaoi ) ABCDAEF
    ZAGHZAIONDGEFJAGDEKLM $.

  $( No extended real is greater than plus infinity.  (Contributed by NM,
     15-Oct-2005.) $)
  pnfnlt $p |- ( A e. RR* -> -. +oo < A ) $=
    ( cxr wcel cpnf clt wbr cr wa cltrr cmnf wceq wo neli intnanr pnfnemnf neii
    pnfnre pm3.2ni wb pnfxr ltxr mpan mtbiri ) ABCZDAEFZDGCZAGCZHZDAIFZHZDJKZAD
    KZHZLZUFULHZUKUGHZLZLZUNUQUJUMUHUIUFUGDGQMZNNUKULDJOPZNRUOUPUFULUSNUKUGUTNR
    RDBCUDUEURSTDAUAUBUC $.

  $( No extended real is less than minus infinity.  (Contributed by NM,
     15-Oct-2005.) $)
  nltmnf $p |- ( A e. RR* -> -. A < -oo ) $=
    ( cxr wcel cmnf clt wbr cr wa cltrr wceq cpnf wo mnfnre neli intnan intnanr
    pnfnemnf nesymi pm3.2ni wb mnfxr ltxr mpan2 mtbiri ) ABCZADEFZAGCZDGCZHZADI
    FZHZADJZDKJZHZLZUGUMHZULUHHZLZLZUOURUKUNUIUJUHUGDGMNZOPUMULKDQRZOSUPUQUMUGV
    AOUHULUTOSSUEDBCUFUSTUAADUBUCUD $.

  $( Plus infinity is an upper bound for extended reals.  (Contributed by NM,
     30-Jan-2006.) $)
  pnfge $p |- ( A e. RR* -> A <_ +oo ) $=
    ( cxr wcel cpnf cle wbr clt wn pnfnlt wb pnfxr xrlenlt mpan2 mpbird ) ABCZA
    DEFZDAGFHZAIODBCPQJKADLMN $.

  $( 0 less than or equal to positive infinity.  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  0lepnf $p |- 0 <_ +oo $=
    ( cc0 cxr wcel cpnf cle wbr 0xr pnfge ax-mp ) ABCADEFGAHI $.

  $( If a number is a nonnegative integer or positive infinity, it is greater
     than or equal to 0.  (Contributed by Alexander van der Vekens,
     6-Jan-2018.) $)
  nn0pnfge0 $p |- ( ( N e. NN0 \/ N = +oo ) -> 0 <_ N ) $=
    ( cn0 wcel cc0 cle wbr cpnf wceq nn0ge0 0lepnf breq2 mpbiri jaoi ) ABCDAEFZ
    AGHZAIONDGEFJAGDEKLM $.

  $( Minus infinity is less than or equal to any extended real.  (Contributed
     by NM, 19-Jan-2006.) $)
  mnfle $p |- ( A e. RR* -> -oo <_ A ) $=
    ( cxr wcel cmnf cle wbr clt wn nltmnf wb mnfxr xrlenlt mpan mpbird ) ABCZDA
    EFZADGFHZAIDBCOPQJKDALMN $.

  $( Ordering on the extended reals is not symmetric.  (Contributed by NM,
     15-Oct-2005.) $)
  xrltnsym $p |- ( ( A e. RR* /\ B e. RR* ) ->
                ( A < B -> -. B < A ) ) $=
    ( cxr wcel cr cpnf wceq cmnf clt wbr wn pnfnlt syl adantr adantl mtbird a1d
    wa wb pm2.21d wi elxr ltnsym rexr breq1 nltmnf breq2 3jaodan sylan2br mnfxr
    w3o ax-mp breq12 mtbiri ancoms xrltnr 3jaoian syl2anb ) ACDZAEDZAFGZAHGZUKB
    EDZBFGZBHGZUKZABIJZBAIJZKZUAZBCDZAUBBUBZUTVFVJVAVBUTVCVJVDVEABUCUTVDRZVIVGV
    MVHFAIJZUTVNKZVDUTUSVOAUDZALMNVDVHVNSUTBFAIUEOPQUTVERZVGVIVQVGAHIJZUTVRKZVE
    UTUSVSVPAUFMNVEVGVRSUTBHAIUGOPTUHVFVAVKVJVLVAVKRZVGVIVTVGFBIJZVKWAKVABLOVAV
    GWASVKAFBIUENPTUIVBVCVJVDVEVBVCRZVIVGWBVHBHIJZVCWCKZVBVCVKWDBUDBUFMOVBVHWCS
    VCAHBIUGNPQVBVDRVIVGVDVBVIVDVBRVHFHIJZHCDZWEKUJHLULBFAHIUMUNUOQVBVERZVGVIWG
    VGHHIJZWFWHKUJHUPULAHBHIUMUNTUHUQUR $.

  $( 'Less than' is antisymmetric and irreflexive for extended reals.
     (Contributed by NM, 6-Feb-2007.) $)
  xrltnsym2 $p |- ( ( A e. RR* /\ B e. RR* ) -> -. ( A < B /\ B < A ) ) $=
    ( cxr wcel wa clt wbr wn wi xrltnsym imnan sylib ) ACDBCDEABFGZBAFGZHIMNEHA
    BJMNKL $.

  $( Ordering on the extended reals is transitive.  (Contributed by NM,
     15-Oct-2005.) $)
  xrlttr $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
             ( ( A < B /\ B < C ) -> A < C ) ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o clt wbr wa wn adantr adantl mtbird pm2.21d
    wb adantlr wi elxr lttr 3expa an32s pnfnlt syl breq1 adantll adantld nltmnf
    rexr breq2 adantrd 3jaodan sylan2b ltpnf mpbird a1d anasss adantrr mnfltpnf
    mnflt breq12 mpbiri 3jaoian 3impb syl3an3b syl3an1b ) ADEZAFEZAGHZAIHZJZBDE
    ZCDEZABKLZBCKLZMZACKLZUAZAUBVPVNVOCFEZCGHZCIHZJZWACUBVNVOWEWAVKVOWEMWAVLVMV
    KVOWEWAVKVOMZWBWAWCWDVKWBVOWAVOVKWBMZBFEZBGHZBIHZJWABUBWGWHWAWIWJVKWHWBWAVK
    WHWBWAABCUCUDUEWGWIMVRVTVQWBWIVRVTUAVKWBWIMZVRVTWKVRGCKLZWBWLNZWIWBVPWMCULC
    UFUGOWIVRWLSWBBGCKUHPQRUIUJWGWJMVQVTVRVKWJVQVTUAWBVKWJMZVQVTWNVQAIKLZVKWONZ
    WJVKVJWPAULAUKUGOWJVQWOSVKBIAKUMPQRTUNUOUPUEWFWCMVTVSVKWCVTVOVKWCMVTAGKLZVK
    WQWCAUQOWCVTWQSVKCGAKUMPURTUSVOWDWAVKVOWDMZVRVTVQWRVRVTWRVRBIKLZVOWSNWDBUKO
    WDVRWSSVOCIBKUMPQRUJZUIUOUTVLVOWAWEVLVOMZVQVTVRXAVQVTXAVQGBKLZVOXBNVLBUFPVL
    VQXBSVOAGBKUHOQRUNVAVMVOWEWAVMVOMWBWAWCWDVMWBWAVOVMWBMZVTVSXCVTICKLZWBXDVMC
    VCPVMVTXDSWBAICKUHOURUSTVMWCWAVOVMWCMZVTVSXEVTIGKLVBAICGKVDVEUSTVOWDWAVMWTU
    IUOUTVFVGVHVI $.

  ${
    $d x y z $.
    $( 'Less than' is a weakly linear ordering on the extended reals.
       (Contributed by NM, 15-Oct-2005.) $)
    xrltso $p |- < Or RR* $=
      ( vx vy vz cxr clt cv wbr wo wcel adantl wa cpnf wceq cmnf mpbird 3jaodan
      wb a1d sylan2b ancoms wor wpo wi wral wn xrltnr w3a xrlttr ispod mptru cr
      wtru elxr simplr simpll simpr axltwlin syl3anc ltpnf ad2antlr breq2 mnflt
      w3o orcd ad2antrr breq1 olcd anasss mnfltpnf breq12 mpbiri adantlr nltmnf
      rexr syl mtbird pm2.21d pnfnlt df-3or bitri mnfltxr adantr eqtr3 biimtrdi
      breq1d olc jaodan 3impa syl3an3b 3com13 rgen3 df-iso mpbir2an ) DEUADEUBZ
      AFZBFZEGZWOCFZEGZWRWPEGZHZUCZCDUDBDUDADUDWNULABCDEWODIZWOWOEGUEULWOUFJXCW
      PDIZWRDIZUGWQWPWREGKWSUCULWOWPWRUHJUIUJXBABCDDDXEXDXCXBXCXEXDWOUKIZWOLMZW
      ONMZVCZXBWOUMXEXDXIXBXEXDKZXFXBXGXHXFXJXBXFXEXDXBXDXFXEKZWPUKIZWPLMZWPNMZ
      VCXBWPUMXKXLXBXMXNXLXKXBXLXFXEXBXEXLXFKZWRUKIZWRLMZWRNMZVCZXBWRUMZXOXPXBX
      QXRXOXPKXFXLXPXBXLXFXPUNXLXFXPUOXOXPUPWOWPWRUQURXOXQKZXAWQYAWSWTYAWSWOLEG
      ZXFYBXLXQWOUSZUTXQWSYBQZXOWRLWOEVAZJOVDRXOXRKZXAWQYFWTWSYFWTNWPEGZXLYGXFX
      RWPVBVEXRWTYGQXOWRNWPEVFJOVGRPSVHTXMXKXBXMXFXEXBXEXMXFKZXSXBXTYHXPXBXQXRY
      HXPKZXAWQYIWTWSYIWTWRLEGZXPYJYHWRUSJXMWTYJQXFXPWPLWREVAVEOVGRYHXQKZXAWQYK
      WSWTYKWSYBXFYBXMXQYCUTXQYDYHYEJOVDRYHXRKZXAWQYLWTWSXMXRWTXFXMXRKWTNLEGZVI
      XRXMWTYMQWRNWPLEVJTVKVLVGRPSVHTXKXNKZWQXAYNWQWONEGZXFYOUEZXEXNXFXCYPWOVNW
      OVMVOVEXNWQYOQXKWPNWOEVAJVPVQPSVHTXJXGKZWQXAYQWQLWPEGZXDYRUEXEXGWPVRUTXGW
      QYRQXJWOLWPEVFJVPVQXEXHXBXDXHXEXBXEXHXPXQHZXRHZXBXEXSYTXTXPXQXRVSVTXHYSXB
      XRXHYSKZXAWQUUAWSWTUUAWSNWREGZYSUUBXHWRWAJXHWSUUBQYSWONWREVFWBOVDRXHXRKZW
      QWTXAUUCWOWRWPEWOWRNWCWEWTWSWFWDWGSTVLPWHWIWJWKABCDEWLWM $.
  $}

  $( Extended real version of ~ lttri3 .  (Contributed by NM, 9-Feb-2006.) $)
  xrlttri3 $p |- ( ( A e. RR* /\ B e. RR* ) ->
                ( A = B <-> ( -. A < B /\ -. B < A ) ) ) $=
    ( wcel cpnf wceq cmnf clt wbr wn wa ancoms wne adantr adantl mpbird 2falsed
    wb neneqd breq12 mtbiri cxr cr elxr lttri3 renepnf neeq2 necomd ltpnf breq2
    w3o wo notnot olcs ioran sylnib syl renemnf mnflt breq1 oranim 3syl 3jaodan
    orc eqtr3 eqcomd pnfxr xrltnr ax-mp 2thd mnfnepnf eqeq12 necon3bid mnfltpnf
    jca mpbiri neii neneqad mnfxr syl2anb ) AUACAUBCZADEZAFEZUJZBUBCZBDEZBFEZUJ
    ABEZABGHZIZBAGHZIZJZQZBUACAUCBUCWCWDWMWEWFWDWCWMWDVTWMWAWBVTWDWMABUDKWDWAJZ
    WGWLWNABWNBAWNBALZBDLZWDWPWABUEMWAWOWPQWDADBUFNOUGRWNWJWLIZWNWJBDGHZWDWRWAB
    UHMWAWJWRQWDADBGUINOWJWHWJUKZIZWLWHWJWTIWSULUMWHWJUNUOZUPPWDWBJZWGWLXBABXBB
    AXBWOBFLZWDXCWBBUQMWBWOXCQWDAFBUFNOUGRXBWHWSWQXBWHFBGHZWDXDWBBURMWBWHXDQWDA
    FBGUSNOWHWJVCZWHWJUTZVAPVBKWEWCWMWEVTWMWAWBWEVTJZWGWLXGABXGABLZADLZVTXIWEAU
    ENWEXHXIQVTBDAUFMORXGWHWSWQXGWHADGHZVTXJWEAUHNWEWHXJQVTBDAGUIMOXEXFVAPWEWAJ
    ZWGWLXKBABADVDVEXKWIWKXKWHDDGHZDUACXLIVFDVGVHZWAWEWHXLQADBDGSKTXKWJXLXMBDAD
    GSTVNVIWEWBJZWGWLXNABWBWEXHWBWEJZXHFDLVJXOABFDAFBDVKVLVOKRXNWHWSWQWBWEWHXOW
    HFDGHZVMAFBDGSVOKXEXFVAPVBKWFWCWMWFVTWMWAWBWFVTJZWGWLXQABXQXHAFLZVTXRWFAUQN
    WFXHXRQVTBFAUFMORXQWJWQXQWJFAGHZVTXSWFAURNWFWJXSQVTBFAGUSMOXAUPPWFWAJZWGWLX
    TABXTBAXTBAXTBAEFDEFDVJVPBFADVKTVQUGRXTWJWQXTWJXPVMBFADGSVOXAUPPWFWBJZWGWLW
    BWFWGABFVDKYAWIWKYAWHFFGHZFUACYBIVRFVGVHZWBWFWHYBQAFBFGSKTYAWJYBYCBFAFGSTVN
    VIVBKVBVS $.

  $( 'Less than' implies 'less than or equal' for extended reals.  (Contributed
     by NM, 19-Jan-2006.) $)
  xrltle $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> A <_ B ) ) $=
    ( cxr wcel wa clt wbr wn cle xrltnsym xrlenlt sylibrd ) ACDBCDEABFGBAFGHABI
    GABJABKL $.

  ${
    xrltled.a $e |- ( ph -> A e. RR* ) $.
    xrltled.b $e |- ( ph -> B e. RR* ) $.
    xrltled.altb $e |- ( ph -> A < B ) $.
    $( 'Less than' implies 'less than or equal to' for extended reals.
       Deduction form of ~ xrltle .  (Contributed by Glauco Siliprandi,
       11-Dec-2019.) $)
    xrltled $p |- ( ph -> A <_ B ) $=
      ( clt wbr cle cxr wcel wi xrltle syl2anc mpd ) ABCGHZBCIHZFABJKCJKPQLDEBC
      MNO $.
  $}

  $( 'Less than or equal to' is reflexive for extended reals.  (Contributed by
     NM, 7-Feb-2007.) $)
  xrleid $p |- ( A e. RR* -> A <_ A ) $=
    ( cxr wcel cle wbr clt wn xrltnr wb xrlenlt anidms mpbird ) ABCZAADEZAAFEGZ
    AHMNOIAAJKL $.

  ${
    xrleidd.1 $e |- ( ph -> A e. RR* ) $.
    $( 'Less than or equal to' is reflexive for extended reals.  Deduction form
       of ~ xrleid .  (Contributed by Glauco Siliprandi, 26-Jun-2021.) $)
    xrleidd $p |- ( ph -> A <_ A ) $=
      ( cxr wcel cle wbr xrleid syl ) ABDEBBFGCBHI $.
  $}

  $( Decidability of ` <_ ` for extended nonnegative integers.  (Contributed by
     Jim Kingdon, 13-Oct-2024.) $)
  xnn0dcle $p |- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B ) $=
    ( cxnn0 wcel wa cn0 cle cpnf wceq cz simpr nn0zd simplr wn wo sylibr elxnn0
    wbr cxr biimpi wdc zdcle syl2anc clt nn0red ltpnfd wb pnfxr xrlenlt sylancr
    rexrd biimpd mt2d eqnbrtrd olcd df-dc ad2antrr mpjaodan xnn0xr syl breqtrrd
    pnfge orcd adantl ) ACDZBCDZEZBFDZABGRZUAZBHIZVGVHEZAFDZVJAHIZVLVMEZAJDBJDV
    JVOAVLVMKLVOBVGVHVMMLABUBUCVLVNEZVIVINZOZVJVPVQVIVPAHBGVLVNKVPHBGRZBHUDRZVP
    BVPBVGVHVNMUEZUFVPVSVTNZVPHSDBSDVSWBUGUHVPBWAUKHBUIUJULUMUNUOVIUPZPVEVMVNOZ
    VFVHVEWDAQTUQURVGVKEZVRVJWEVIVQWEAHBGWEASDZAHGRVEWFVFVKAUSUQAVBUTVGVKKVAVCW
    CPVFVHVKOZVEVFWGBQTVDUR $.

  $( Dichotomy for extended nonnegative integers.  (Contributed by Jim Kingdon,
     13-Oct-2024.) $)
  xnn0letri $p |- ( ( A e. NN0* /\ B e. NN0* ) -> ( A <_ B \/ B <_ A ) ) $=
    ( cxnn0 wcel wa cn0 cle wbr wo cpnf wceq cz simpr nn0zd cxr xnn0xr breqtrrd
    pnfge syl elxnn0 simplr zletric ad3antlr olcd biimpi ad2antrr mpjaodan orcd
    syl2anc adantl ) ACDZBCDZEZBFDZABGHZBAGHZIZBJKZUMUNEZAFDZUQAJKZUSUTEZALDBLD
    UQVBAUSUTMNVBBUMUNUTUANABUBUIUSVAEZUPUOVCBJAGULBJGHZUKUNVAULBODVDBPBRSUCUSV
    AMQUDUKUTVAIZULUNUKVEATUEUFUGUMUREZUOUPVFAJBGVFAODZAJGHUKVGULURAPUFARSUMURM
    QUHULUNURIZUKULVHBTUEUJUG $.

  $( Trichotomy law for extended reals.  (Contributed by FL, 2-Aug-2009.) $)
  xrletri3 $p |- ( ( A e. RR* /\ B e. RR* ) ->
                   ( A = B <-> ( A <_ B /\ B <_ A ) ) ) $=
    ( cxr wcel wa wceq clt wbr wn cle xrlttri3 ancom bitr4di xrlenlt wb anbi12d
    ancoms bitr4d ) ACDZBCDZEZABFZBAGHIZABGHIZEZABJHZBAJHZEUAUBUDUCEUEABKUCUDLM
    UAUFUCUGUDABNTSUGUDOBANQPR $.

  ${
    xrletrid.1 $e |- ( ph -> A e. RR* ) $.
    xrletrid.2 $e |- ( ph -> B e. RR* ) $.
    xrletrid.3 $e |- ( ph -> A <_ B ) $.
    xrletrid.4 $e |- ( ph -> B <_ A ) $.
    $( Trichotomy law for extended reals.  (Contributed by Glauco Siliprandi,
       17-Aug-2020.) $)
    xrletrid $p |- ( ph -> A = B ) $=
      ( wceq cle wbr cxr wcel wa wb xrletri3 syl2anc mpbir2and ) ABCHZBCIJZCBIJ
      ZFGABKLCKLRSTMNDEBCOPQ $.
  $}

  $( Transitive law for ordering on extended reals.  (Contributed by NM,
     19-Jan-2006.) $)
  xrlelttr $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
             ( ( A <_ B /\ B < C ) -> A < C ) ) $=
    ( cxr wcel w3a cle wbr clt wa wn simprl simpl1 simpl2 xrlenlt syl2anc mpbid
    wb pm2.21d idd wo simprr wi simpl3 wor xrltso sowlin mpan syl3anc mpjaod ex
    mpd ) ADEZBDEZCDEZFZABGHZBCIHZJZACIHZUPUSJZBAIHZUTUTVAVBUTVAUQVBKZUPUQURLVA
    UMUNUQVCRUMUNUOUSMZUMUNUOUSNZABOPQSVAUTTVAURVBUTUAZUPUQURUBVAUNUOUMURVFUCZV
    EUMUNUOUSUDVDDIUEUNUOUMFVGUFDBCAIUGUHUIULUJUK $.

  $( Transitive law for ordering on extended reals.  (Contributed by NM,
     19-Jan-2006.) $)
  xrltletr $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
             ( ( A < B /\ B <_ C ) -> A < C ) ) $=
    ( cxr wcel w3a clt wbr cle wa wn simprr simpl2 simpl3 xrlenlt syl2anc mpbid
    wb wo simprl wi wor xrltso sowlin mpan adantr mpd ecased ex ) ADEZBDEZCDEZF
    ZABGHZBCIHZJZACGHZUMUPJZUQCBGHZURUOUSKZUMUNUOLURUKULUOUTRUJUKULUPMUJUKULUPN
    BCOPQURUNUQUSSZUMUNUOTUMUNVAUAZUPDGUBUMVBUCDABCGUDUEUFUGUHUI $.

  $( Transitive law for ordering on extended reals.  (Contributed by NM,
     9-Feb-2006.) $)
  xrletr $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
             ( ( A <_ B /\ B <_ C ) -> A <_ C ) ) $=
    ( cxr wcel w3a clt wbr wo wn cle wa wi wor xrltso sowlin mpan 3coml xrlenlt
    wb orcom imbitrdi con3d 3adant3 3adant1 anbi12d bitr4di 3adant2 3imtr4d
    ioran ) ADEZBDEZCDEZFZBAGHZCBGHZIZJZCAGHZJZABKHZBCKHZLZACKHZUNUSUQUNUSUPUOI
    ZUQUMUKULUSVEMZDGNUMUKULFVFODCABGPQRUPUOUAUBUCUNVCUOJZUPJZLURUNVAVGVBVHUKUL
    VAVGTUMABSUDULUMVBVHTUKBCSUEUFUOUPUJUGUKUMVDUTTULACSUHUI $.

  ${
    xrlttrd.1 $e |- ( ph -> A e. RR* ) $.
    xrlttrd.2 $e |- ( ph -> B e. RR* ) $.
    xrlttrd.3 $e |- ( ph -> C e. RR* ) $.
    ${
      xrlttrd.4 $e |- ( ph -> A < B ) $.
      xrlttrd.5 $e |- ( ph -> B < C ) $.
      $( Transitive law for ordering on extended reals.  (Contributed by Mario
         Carneiro, 23-Aug-2015.) $)
      xrlttrd $p |- ( ph -> A < C ) $=
        ( clt wbr cxr wcel wa wi xrlttr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABLM
        CLMDLMSTNUAOEFGBCDPQR $.
    $}

    ${
      xrlelttrd.4 $e |- ( ph -> A <_ B ) $.
      xrlelttrd.5 $e |- ( ph -> B < C ) $.
      $( Transitive law for ordering on extended reals.  (Contributed by Mario
         Carneiro, 23-Aug-2015.) $)
      xrlelttrd $p |- ( ph -> A < C ) $=
        ( cle wbr clt cxr wcel wa wi xrlelttr syl3anc mp2and ) ABCJKZCDLKZBDLKZ
        HIABMNCMNDMNTUAOUBPEFGBCDQRS $.
    $}

    ${
      xrltletrd.4 $e |- ( ph -> A < B ) $.
      xrltletrd.5 $e |- ( ph -> B <_ C ) $.
      $( Transitive law for ordering on extended reals.  (Contributed by Mario
         Carneiro, 23-Aug-2015.) $)
      xrltletrd $p |- ( ph -> A < C ) $=
        ( clt wbr cle cxr wcel wa wi xrltletr syl3anc mp2and ) ABCJKZCDLKZBDJKZ
        HIABMNCMNDMNTUAOUBPEFGBCDQRS $.
    $}

    ${
      xrletrd.4 $e |- ( ph -> A <_ B ) $.
      xrletrd.5 $e |- ( ph -> B <_ C ) $.
      $( Transitive law for ordering on extended reals.  (Contributed by Mario
         Carneiro, 23-Aug-2015.) $)
      xrletrd $p |- ( ph -> A <_ C ) $=
        ( cle wbr cxr wcel wa wi xrletr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABLM
        CLMDLMSTNUAOEFGBCDPQR $.
    $}
  $}

  $( 'Less than' implies not equal for extended reals.  (Contributed by NM,
     20-Jan-2006.) $)
  xrltne $p |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B =/= A ) $=
    ( cxr wcel clt wbr wne wceq xrltnr breq2 notbid syl5ibrcom necon2ad 3adant2
    wn imp ) ACDZABEFZBAGZBCDQRSQRBAQROBAHZAAEFZOAITRUABAAEJKLMPN $.

  $( An extended real is not less than plus infinity iff they are equal.
     (Contributed by NM, 30-Jan-2006.) $)
  nltpnft $p |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o clt wn wb elxr renepnf neneqd ltpnf notnot
    wbr 2falsed breq1 mtbiri syl id pnfxr xrltnr ax-mp 2thd mnfnepnf neii eqeq1
    mnfltpnf mpbiri necon3bi necon2bi 3jaoi sylbi ) ABCADCZAEFZAGFZHUQAEIQZJZKZ
    ALUPVAUQURUPUQUTUPAEAMNUPUSUTJAOUSPUARUQUQUTUQUBUQUSEEIQZEBCVBJUCEUDUEAEEIS
    TUFURUQUTURUQGEFGEUGUHAGEUITUTAGUSAGURUSGEIQUJAGEISUKULUMRUNUO $.

  $( An extended real is less than plus infinity iff they are not equal.
     (Contributed by Jim Kingdon, 17-Apr-2023.) $)
  npnflt $p |- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) ) $=
    ( cxr wcel cpnf clt wbr wne wceq wn nltpnft biimpd necon2ad wa ltpnf adantl
    cr cmnf simpr simplr pm2.21ddne mnfltpnf breq1 mpbiri elxr biimpi mpjao3dan
    w3o adantr ex impbid ) ABCZADEFZADGZUKULADUKADHZULIAJKLUKUMULUKUMMZAPCZULUN
    AQHZUPULUOANOUOUNMULADUOUNRUKUMUNSTUQULUOUQULQDEFUAAQDEUBUCOUKUPUNUQUGZUMUK
    URAUDUEUHUFUIUJ $.

  $( An extended real which is greater than plus infinity is plus infinity.
     (Contributed by Thierry Arnoux, 18-Dec-2016.) $)
  xgepnf $p |- ( A e. RR* -> ( +oo <_ A <-> A = +oo ) ) $=
    ( cxr wcel cpnf cle wbr clt wn wceq wb pnfxr xrlenlt mpan nltpnft bitr4d )
    ABCZDAEFZADGFHZADIDBCPQRJKDALMANO $.

  $( An extended real is not greater than minus infinity iff they are equal.
     (Contributed by NM, 2-Feb-2006.) $)
  ngtmnft $p |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o clt wbr wn wb renemnf neneqd mnflt 2falsed
    elxr wne mpbiri breq2 notnot syl pnfnemnf mnfltpnf necon3bi necon2bi xrltnr
    neeq1 id mnfxr ax-mp mtbiri 2thd 3jaoi sylbi ) ABCADCZAEFZAGFZHURGAIJZKZLZA
    QUPVAUQURUPURUTUPAGAMNUPUSUTKAOUSUAUBPUQURUTUQAGUQAGREGRUCAEGUHSNUTAEUSAEUQ
    USGEIJUDAEGITSUEUFPURURUTURUIURUSGGIJZGBCVBKUJGUGUKAGGITULUMUNUO $.

  $( An extended real is greater than minus infinite iff they are not equal.
     (Contributed by Jim Kingdon, 17-Apr-2023.) $)
  nmnfgt $p |- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) ) $=
    ( cxr wcel cmnf clt wbr wne wceq wn ngtmnft biimpd necon2ad wa mnflt adantl
    cr cpnf mnfltpnf breq2 mpbiri simpr simplr pm2.21ddne elxr biimpi mpjao3dan
    w3o adantr ex impbid ) ABCZDAEFZADGZUKULADUKADHZULIAJKLUKUMULUKUMMZAPCZULAQ
    HZUNUPULUOANOUQULUOUQULDQEFRAQDESTOUOUNMULADUOUNUAUKUMUNUBUCUKUPUQUNUGZUMUK
    URAUDUEUHUFUIUJ $.

  $( An extended real is real iff it is strictly bounded by infinities.
     (Contributed by NM, 2-Feb-2006.) $)
  xrrebnd $p |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o clt wa wb elxr id necon2bi wn xrltnr ax-mp
    wbr mtbiri 2falsed mnflt ltpnf jca 2thd renepnf pnfxr breq1 intnand renemnf
    mnfxr breq2 intnanrd 3jaoi sylbi ) ABCADCZAEFZAGFZHUOGAIRZAEIRZJZKZALUOVAUP
    UQUOUOUTUOMUOURUSAUAAUBUCUDUPUOUTUOAEAUENUPUSURUPUSEEIRZEBCVBOUFEPQAEEIUGSU
    HTUQUOUTUOAGAUINUQURUSUQURGGIRZGBCVCOUJGPQAGGIUKSULTUMUN $.

  $( A way of proving that an extended real is real.  (Contributed by NM,
     9-Mar-2006.) $)
  xrre $p |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) ->
              A e. RR ) $=
    ( cxr wcel cr wa cmnf clt wbr cle cpnf simprl ltpnf adantl wi rexr xrlelttr
    pnfxr mp3an3 sylan2 mpan2d imp adantrl wb xrrebnd ad2antrr mpbir2and ) ACDZ
    BEDZFZGAHIZABJIZFZFAEDZUKAKHIZUJUKULLUJULUOUKUJULUOUJULBKHIZUOUIUPUHBMNUIUH
    BCDZULUPFUOOZBPUHUQKCDURRABKQSTUAUBUCUHUNUKUOFUDUIUMAUEUFUG $.

  $( An extended real between two others is real.  (Contributed by NM,
     6-Feb-2007.) $)
  xrre2 $p |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\
              ( A < B /\ B < C ) ) -> B e. RR ) $=
    ( cxr wcel w3a clt wbr wa cr cmnf wi cle mnfle adantr mnfxr xrlelttr mp3an1
    cpnf mpand 3adant3 pnfge adantl pnfxr mp3an3 mpan2d 3adant1 anim12d xrrebnd
    xrltletr wb 3ad2ant2 sylibrd imp ) ADEZBDEZCDEZFZABGHZBCGHZIZBJEZURVAKBGHZB
    SGHZIZVBURUSVCUTVDUOUPUSVCLUQUOUPIKAMHZUSVCUOVFUPANOKDEUOUPVFUSIVCLPKABQRTU
    AUPUQUTVDLUOUPUQIUTCSMHZVDUQVGUPCUBUCUPUQSDEUTVGIVDLUDBCSUJUEUFUGUHUPUOVBVE
    UKUQBUIULUMUN $.

  $( A way of proving that an extended real is real.  (Contributed by FL,
     29-May-2014.) $)
  xrre3 $p |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) )
      -> A e. RR ) $=
    ( cxr wcel cr wa cle wbr cpnf clt cmnf mnflt adantl wi mnfxr a1i rexr simpl
    xrltletr syl3anc mpand imp adantrr simprr wb xrrebnd ad2antrr mpbir2and ) A
    CDZBEDZFZBAGHZAIJHZFZFAEDZKAJHZUMUKULUPUMUKULUPUKKBJHZULUPUJUQUIBLMUKKCDZBC
    DZUIUQULFUPNURUKOPUJUSUIBQMUIUJRKBASTUAUBUCUKULUMUDUIUOUPUMFUEUJUNAUFUGUH
    $.

  $( A nonnegative extended real is greater than negative infinity.
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  ge0gtmnf $p |- ( ( A e. RR* /\ 0 <_ A ) -> -oo < A ) $=
    ( cxr wcel cmnf cc0 clt wbr cle mnflt0 wa wi mnfxr 0xr xrltletr mp3an12 imp
    mpanr1 ) ABCZDEFGZEAHGZDAFGZIRSTJZUADBCEBCRUBUAKLMDEANOPQ $.

  $( A nonnegative extended real is greater than negative infinity.
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  ge0nemnf $p |- ( ( A e. RR* /\ 0 <_ A ) -> A =/= -oo ) $=
    ( cxr wcel cc0 cle wbr wa cmnf clt wne ge0gtmnf wn wb ngtmnft adantr biimpd
    wceq necon2ad mpd ) ABCZDAEFZGZHAIFZAHJAKUBUCAHUBAHQZUCLZTUDUEMUAANOPRS $.

  $( A nonnegative extended real that is less than a real bound is real.
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xrrege0 $p |- ( ( ( A e. RR* /\ B e. RR ) /\ ( 0 <_ A /\ A <_ B ) ) ->
              A e. RR ) $=
    ( cxr wcel cr cc0 cle wbr cmnf clt ge0gtmnf ad2ant2r simprr jca xrre syldan
    wa ) ACDZBEDZQZFAGHZABGHZQZIAJHZUBQAEDTUCQUDUBRUAUDSUBAKLTUAUBMNABOP $.

  ${
    $d k M $.  $d k N $.
    $( There exists an integer greater than or equal to any two others.
       (Contributed by NM, 28-Aug-2005.) $)
    z2ge $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
                E. k e. ZZ ( M <_ k /\ N <_ k ) ) $=
      ( cz wcel wa cle wbr cv wrex simplr simpr zred leidd breq2 anbi12d rspcev
      wceq syl12anc simpll zletric mpjaodan ) BDEZCDEZFZBCGHZBAIZGHZCUGGHZFZADJ
      ZCBGHZUEUFFZUDUFCCGHZUKUCUDUFKZUEUFLUMCUMCUOMNUJUFUNFACDUGCRUHUFUIUNUGCBG
      OUGCCGOPQSUEULFZUCBBGHZULUKUCUDULTZUPBUPBURMNUEULLUJUQULFABDUGBRUHUQUIULU
      GBBGOUGBCGOPQSBCUAUB $.
  $}

  $( Equality of two extended numbers with ` -e ` in front of them.
     (Contributed by FL, 26-Dec-2011.)  (Proof shortened by Mario Carneiro,
     20-Aug-2015.) $)
  xnegeq $p |- ( A = B -> -e A = -e B ) $=
    ( wceq cpnf cmnf cneg cif cxne eqeq1 negeq ifbieq2d df-xneg 3eqtr4g ) ABCZA
    DCZEAECZDAFZGZGBDCZEBECZDBFZGZGAHBHNOSRUBEABDINPTQUADABEIABJKKALBLM $.

  $( Minus ` +oo ` .  Remark of [BourbakiTop1] p.  IV.15.  (Contributed by FL,
     26-Dec-2011.) $)
  xnegpnf $p |- -e +oo = -oo $=
    ( cpnf cxne wceq cmnf cneg cif df-xneg eqid iftruei eqtri ) ABAACZDADCAAEFZ
    FDAGKDLAHIJ $.

  $( Minus ` -oo ` .  Remark of [BourbakiTop1] p.  IV.15.  (Contributed by FL,
     26-Dec-2011.)  (Revised by Mario Carneiro, 20-Aug-2015.) $)
  xnegmnf $p |- -e -oo = +oo $=
    ( cmnf cxne cpnf wceq cneg cif df-xneg wne mnfnepnf ifnefalse ax-mp iftruei
    eqid 3eqtri ) ABACDAAADZCAEZFZFZQCAGACHRQDIACAQJKOCPAMLN $.

  $( Minus a real number.  Remark [BourbakiTop1] p.  IV.15.  (Contributed by
     FL, 26-Dec-2011.)  (Proof shortened by Mario Carneiro, 20-Aug-2015.) $)
  rexneg $p |- ( A e. RR -> -e A = -u A ) $=
    ( cr wcel cxne cpnf wceq cmnf cif df-xneg wne renepnf ifnefalse syl renemnf
    cneg eqtrd eqtrid ) ABCZADAEFGAGFEAOZHZHZSAIRUATSRAEJUATFAKAEGTLMRAGJTSFANA
    GESLMPQ $.

  $( The negative of zero.  (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xneg0 $p |- -e 0 = 0 $=
    ( cc0 cxne cneg cr wcel wceq 0re rexneg ax-mp neg0 eqtri ) ABZACZAADELMFGAH
    IJK $.

  $( Closure of extended real negative.  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xnegcl $p |- ( A e. RR* -> -e A e. RR* ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o cxne elxr cneg rexneg renegcl rexrd xnegeq
    eqeltrd xnegpnf mnfxr eqeltri eqeltrdi xnegmnf pnfxr 3jaoi sylbi ) ABCADCZA
    EFZAGFZHAIZBCZAJUEUIUFUGUEUHUEUHAKDALAMPNUFUHEIZBAEOUJGBQRSTUGUHGIZBAGOUKEB
    UAUBSTUCUD $.

  $( Extended real version of ~ negneg .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xnegneg $p |- ( A e. RR* -> -e -e A = A ) $=
    ( cxr wcel cpnf wceq cmnf w3o cxne elxr cneg rexneg renegcl xnegmnf xnegpnf
    cr xnegeq syl eqtrdi id 3eqtr4a recn negnegd 3eqtrd 3jaoi sylbi ) ABCAOCZAD
    EZAFEZGAHZHZAEZAIUFUKUGUHUFUJAJZHZULJZAUFUIULEUJUMEAKUIULPQUFULOCUMUNEALULK
    QUFAAUAUBUCUGFHZDUJAMUGUIFEUJUOEUGUIDHZFADPNRUIFPQUGSTUHUPFUJANUHUIDEUJUPEU
    HUIUODAFPMRUIDPQUHSTUDUE $.

  $( Extended real version of ~ neg11 .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xneg11 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( -e A = -e B <-> A = B ) ) $=
    ( cxr wcel wa cxne wceq xnegeq xnegneg eqeqan12d imbitrid impbid1 ) ACDZBCD
    ZEZAFZBFZGZABGZRPFZQFZGOSPQHMNTAUABAIBIJKABHL $.

  $( Forward direction of ~ xltneg .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xltnegi $p |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A ) $=
    ( cxr wcel clt wbr cxne cr cpnf wceq cmnf w3o wa wi elxr cneg syl adantr wn
    sylbid rexneg breqan12rd bitr4d biimpd xnegeq xnegpnf eqtrdi adantl renegcl
    ltneg eqeltrd mnflt eqbrtrd a1d simpr breq2d nltmnf pm2.21d 3jaodan sylan2b
    rexr expimpd simpl breq1d pnfnlt breq1 anbi2d ltpnf mnfltpnf eqbrtrdi breq2
    mnfxr ax-mp pm2.21i imp 3jaoian sylanb xnegmnf imbitrrid 3jaoi sylbi 3impib
    biimtrdi ) ACDZBCDZABEFZBGZAGZEFZWDAHDZAIJZAKJZLWEWFMZWINZAOWJWNWKWLWJWEWFW
    IWEWJBHDZBIJZBKJZLZWFWINZBOZWJWOWSWPWQWJWOMZWFWIXAWFBPZAPZEFWIABUJWOWJWGXBW
    HXCEBUAZAUAZUBUCUDWJWPMZWIWFXFWGKWHEWPWGKJZWJWPWGIGKBIUEUFUGZUHWJKWHEFZWPWJ
    WHHDXIWJWHXCHXEAUIUKWHULQRUMUNWJWQMZWFAKEFZWIXJBKAEWJWQUOUPXJXKWIWJXKSZWQWJ
    WDXLAVAAUQQRURTUSUTVBWKWEWFWIWKWEMZWFIBEFZWIXMAIBEWKWEVCVDXMXNWIWEXNSWKBVEU
    HURTVBWLWMWEKBEFZMZWIWLWFXOWEAKBEVFVGXPWIWLWGIEFZWEWRXOXQWTWOXOXQWPWQWOXOMW
    GHDZXQWOXRXOWOWGXBHXDBUIUKRWGVHQWPXOMWGKIEWPXGXOXHRVIVJWQXOXQWQXOKKEFZXQBKK
    EVKXSXQKCDXSSVLKUQVMVNWCVOVPVQWLWHIWGEWLWHKGIAKUEVRUGUPVSTVTWAWB $.

  $( Extended real version of ~ ltneg .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xltneg $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) ) $=
    ( cxr wcel wa clt xltnegi 3expia wi xnegcl syl2anr xnegneg breqan12d sylibd
    wbr cxne impbid ) ACDZBCDZEZABFOZBPZAPZFOZRSUAUDABGHTUDUCPZUBPZFOZUASUBCDZU
    CCDZUDUGIRBJAJUHUIUDUGUBUCGHKRSUEAUFBFALBLMNQ $.

  $( Extended real version of ~ leneg .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xleneg $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -e B <_ -e A ) ) $=
    ( cxr wcel wa clt wbr wn cle wb xltneg ancoms notbid xrlenlt xnegcl syl2anr
    cxne 3bitr4d ) ACDZBCDZEZBAFGZHAQZBQZFGZHZABIGUDUCIGZUAUBUETSUBUEJBAKLMABNT
    UDCDUCCDUGUFJSBOAOUDUCNPR $.

  $( Extended real version of ~ lt0neg1 .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xlt0neg1 $p |- ( A e. RR* -> ( A < 0 <-> 0 < -e A ) ) $=
    ( cxr wcel cc0 clt wbr cxne wb 0xr xltneg mpan2 xneg0 breq1i bitrdi ) ABCZA
    DEFZDGZAGZEFZDREFODBCPSHIADJKQDRELMN $.

  $( Extended real version of ~ lt0neg2 .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xlt0neg2 $p |- ( A e. RR* -> ( 0 < A <-> -e A < 0 ) ) $=
    ( cxr wcel cc0 clt wbr cxne wb 0xr xltneg mpan xneg0 breq2i bitrdi ) ABCZDA
    EFZAGZDGZEFZQDEFDBCOPSHIDAJKRDQELMN $.

  $( Extended real version of ~ le0neg1 .  (Contributed by Mario Carneiro,
     9-Sep-2015.) $)
  xle0neg1 $p |- ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) ) $=
    ( cxr wcel cc0 cle wbr cxne wb 0xr xleneg mpan2 xneg0 breq1i bitrdi ) ABCZA
    DEFZDGZAGZEFZDREFODBCPSHIADJKQDRELMN $.

  $( Extended real version of ~ le0neg2 .  (Contributed by Mario Carneiro,
     9-Sep-2015.) $)
  xle0neg2 $p |- ( A e. RR* -> ( 0 <_ A <-> -e A <_ 0 ) ) $=
    ( cxr wcel cc0 cle wbr cxne wb 0xr xleneg mpan xneg0 breq2i bitrdi ) ABCZDA
    EFZAGZDGZEFZQDEFDBCOPSHIDAJKRDQELMN $.

  $( An extended real is or is not plus infinity.  (Contributed by Jim Kingdon,
     13-Apr-2023.) $)
  xrpnfdc $p |- ( A e. RR* -> DECID A = +oo ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o wdc elxr wn wo renepnf neneqd df-dc sylibr
    olcd orc mnfnepnf neii eqeq1 mtbiri 3jaoi sylbi ) ABCADCZAEFZAGFZHUFIZAJUEU
    HUFUGUEUFUFKZLZUHUEUIUFUEAEAMNQUFOZPUFUJUHUFUIRUKPUGUJUHUGUIUFUGUFGEFGESTAG
    EUAUBQUKPUCUD $.

  $( An extended real is or is not minus infinity.  (Contributed by Jim
     Kingdon, 13-Apr-2023.) $)
  xrmnfdc $p |- ( A e. RR* -> DECID A = -oo ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o wdc elxr wn wo renemnf neneqd df-dc sylibr
    olcd pnfnemnf neii eqeq1 mtbiri orc 3jaoi sylbi ) ABCADCZAEFZAGFZHUGIZAJUEU
    HUFUGUEUGUGKZLZUHUEUIUGUEAGAMNQUGOZPUFUJUHUFUIUGUFUGEGFEGRSAEGTUAQUKPUGUJUH
    UGUIUBUKPUCUD $.

  ${
    $d x y A $.  $d x y B $.
    $( The extended real addition operation is closed in extended reals.
       (Contributed by Mario Carneiro, 21-Aug-2015.) $)
    xaddf $p |- +e : ( RR* X. RR* ) --> RR* $=
      ( vx vy cv cpnf wceq cmnf cc0 cif cxr wcel wral cxad wa 0xr a1i pnfxr wdc
      xrmnfdc wn ifcldadc caddc co cxp wf adantl ifcldcd mnfxr xrpnfdc ad3antlr
      adantr cr simp-4r wo simp-5l simpllr neqned xrnemnf biimpi syl2anc ecased
      wne simplr simp-5r neqne readdcld rexrd ad3antrrr ad2antrr rgen2a df-xadd
      fmpo mpbi ) ACZDEZBCZFEZGDHZVMFEZVODEZGFHZVSDVPFVMVOUAUBZHZHZHZHZIJZBIKAI
      KIIUCILUDWFABIVMIJZVOIJZMZVNVQWDIWIVQIJVNWIVPGDIGIJZWINODIJZWIPOWHVPQZWGV
      ORUEZUFUJWIVNSZMZVRVTWCIWOVRMZVSGFIWJWPNOFIJZWPUGOWHVSQZWGWNVRVOUHZUIUFWO
      VRSZMZVSDWBIWKXAVSMPOXAVSSZMZVPFWAIWQXCVPMUGOXCVPSZMZWAXEVMVOXEVMUKJZVNWI
      WNWTXBXDULXEWGVMFVAZXFVNUMZWGWHWNWTXBXDUNXEVMFWOWTXBXDUOUPWGXGMXHVMUQURUS
      UTXEVOUKJZVSXAXBXDVBXEWHVOFVAZXIVSUMZWGWHWNWTXBXDVCXDXJXCVOFVDUEWHXJMXKVO
      UQURUSUTVEVFWIWLWNWTXBWMVGTWHWRWGWNWTWSUITWGVRQWHWNVMRVHTWGVNQWHVMUHUJTVI
      ABIIWEILABVJVKVL $.

    $( Value of the extended real addition operation.  (Contributed by Mario
       Carneiro, 20-Aug-2015.) $)
    xaddval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = if ( A = +oo ,
            if ( B = -oo , 0 , +oo ) , if ( A = -oo ,
            if ( B = +oo , 0 , -oo ) , if ( B = +oo , +oo ,
                 if ( B = -oo , -oo , ( A + B ) ) ) ) ) ) $=
      ( vx vy cxr wcel cpnf wceq cmnf cc0 cif caddc co wa a1i ad2antrr ifcldadc
      wdc wn eqeq1d cxad 0xr pnfxr xrmnfdc adantl ifcldcd mnfxr xrpnfdc simp-4r
      adantr cr wne ad4antr simpllr neqned xrnemnf biimpi syl2anc ecased simplr
      wo simpl simpr readdcld rexrd ad3antrrr cv ifbid oveq12 ifbieq12d df-xadd
      ifbieq2d ovmpoga mpd3an3 ) AEFZBEFZAGHZBIHZJGKZAIHZBGHZJIKZWAGVRIABLMZKZK
      ZKZKZEFABUAMWGHVOVPNZVQVSWFEWHVSEFVQWHVRJGEJEFWHUBOZGEFZWHUCOVPVRRZVOBUDU
      EZUFUJWHVQSZNZVTWBWEEWHWBEFWMVTWHWAJIEWIIEFZWHUGOVPWARZVOBUHUEZUFPWNVTSZN
      ZWAGWDEWJWSWANUCOWSWASZNZVRIWCEWOXAVRNUGOXAVRSZNZWCXCABXCAUKFZVQWHWMWRWTX
      BUIXCVOAIULZXDVQVAZWHVOWMWRWTXBVOVPVBUMXCAIWNWRWTXBUNUOVOXENXFAUPUQURUSXC
      BUKFZWAWSWTXBUTXCVPBIULZXGWAVAZWHVPWMWRWTXBVOVPVCUMXCBIXAXBVCUOVPXHNXIBUP
      UQURUSVDVEWHWKWMWRWTWLVFQWHWPWMWRWQPQVOVTRVPWMAUDPQVOVQRVPAUHUJQCDABEECVG
      ZGHZDVGZIHZJGKZXJIHZXLGHZJIKZXPGXMIXJXLLMZKZKZKZKWGUAEXJAHZXLBHZNZXKVQXNY
      AVSWFYDXJAGYBYCVBZTYDXMVRJGYDXLBIYBYCVCZTZVHYDXOVTXQXTWBWEYDXJAIYETYDXPWA
      JIYDXLBGYFTZVHYDXPWAXSWDGYHYDXMVRXRWCIYGXJAXLBLVIVLVLVJVJCDVKVMVN $.
  $}

  $( Addition of positive infinity on the right.  (Contributed by Mario
     Carneiro, 20-Aug-2015.) $)
  xaddpnf1 $p |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo ) $=
    ( cxr wcel cmnf wne wa cpnf cxad co wceq cc0 cif caddc pnfxr xaddval adantr
    mpan2 pnfnemnf ifnefalse mp1i iftruei eqtrdi ifeq12d wdc xrpnfdc ifiddc syl
    eqid sylan9eqr eqtrd ) ABCZADEZFAGHIZAGJZGDJZKGLZADJGGJZKDLZUQGUODAGMILZLZL
    ZLZGUKUMVBJZULUKGBCVCNAGOQPULUKVBUNGGLZGULUNUPGVAGGDEUPGJULRGDKGSTULVAUTGAD
    URUTSUQGUSGUHUAUBUCUKUNUDVDGJAUEUNGUFUGUIUJ $.

  $( Addition of positive infinity on the left.  (Contributed by Mario
     Carneiro, 20-Aug-2015.) $)
  xaddpnf2 $p |- ( ( A e. RR* /\ A =/= -oo ) -> ( +oo +e A ) = +oo ) $=
    ( cxr wcel cmnf wne cpnf cxad co wceq cc0 cif caddc pnfxr xaddval mpan eqid
    iftruei ifnefalse eqtrid sylan9eq ) ABCZADEZFAGHZFFIZADIZJFKZFDIAFIZJDKUGFU
    EDFALHKKKZKZFFBCUAUCUIIMFANOUBUIUFFUDUFUHFPQADJFRST $.

  $( Addition of negative infinity on the right.  (Contributed by Mario
     Carneiro, 20-Aug-2015.) $)
  xaddmnf1 $p |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo ) $=
    ( cxr wcel cpnf wne wa cmnf cxad co wceq cc0 cif caddc mnfxr xaddval adantr
    mpan2 ifnefalse mnfnepnf ax-mp eqid iftruei eqtri ifeq12 wdc xrmnfdc ifiddc
    mp2an syl eqtrid sylan9eqr eqtrd ) ABCZADEZFAGHIZADJGGJZKDLZAGJZGDJZKGLZUSD
    UPGAGMIZLZLZLZLZGUMUOVEJZUNUMGBCVFNAGOQPUNUMVEVDGADUQVDRUMVDURGGLZGUTGJZVCG
    JVDVGJGDEZVHSGDKGRTVCVBGVIVCVBJSGDDVBRTUPGVAGUAUBUCURUTGVCGUDUHUMURUEVGGJAU
    FURGUGUIUJUKUL $.

  $( Addition of negative infinity on the left.  (Contributed by Mario
     Carneiro, 20-Aug-2015.) $)
  xaddmnf2 $p |- ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo ) $=
    ( cxr wcel cpnf wne cmnf cxad co wceq cc0 caddc mnfxr xaddval mpan mnfnepnf
    cif ifnefalse ax-mp eqid iftruei eqtri eqtrid sylan9eq ) ABCZADEZFAGHZFDIAF
    IZJDPZFFIZADIZJFPZUJDUGFFAKHPPZPZPZFFBCUDUFUNILFAMNUEUNUKFUNUMUKFDEUNUMIOFD
    UHUMQRUIUKULFSTUAADJFQUBUC $.

  $( Addition of positive and negative infinity.  This is often taken to be a
     "null" value or out of the domain, but we define it (somewhat arbitrarily)
     to be zero so that the resulting function is total, which simplifies
     proofs.  (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  pnfaddmnf $p |- ( +oo +e -oo ) = 0 $=
    ( cpnf cmnf cxad wceq cc0 cif caddc cxr wcel pnfxr mnfxr xaddval mp2an eqid
    co iftruei 3eqtri ) ABCOZAADZBBDZEAFZABDBADZEBFUBATBABGOFFFZFZUAEAHIBHIRUDD
    JKABLMSUAUCANPTEABNPQ $.

  $( Addition of negative and positive infinity.  This is often taken to be a
     "null" value or out of the domain, but we define it (somewhat arbitrarily)
     to be zero so that the resulting function is total, which simplifies
     proofs.  (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  mnfaddpnf $p |- ( -oo +e +oo ) = 0 $=
    ( cmnf cpnf cxad wceq cc0 cif caddc wcel mnfxr pnfxr xaddval mp2an mnfnepnf
    co cxr wne ifnefalse eqid iftruei eqtri ax-mp ) ABCNZABDBADZEBFZAADZBBDZEAF
    ZUFBUCAABGNFFZFZFZEAOHBOHUBUJDIJABKLUJUIEABPUJUIDMABUDUIQUAUIUGEUEUGUHARSUF
    EABRSTTT $.

  $( The extended real addition operation when both arguments are real.
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  rexadd $p |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) ) $=
    ( cr wcel wa cxad cpnf wceq cmnf cc0 cif cxr rexr wne renepnf ifnefalse syl
    co renemnf eqtrd caddc xaddval syl2an sylan9eq ) ACDZBCDZEABFRZAGHBIHZJGKZA
    IHBGHZJIKZUJGUHIABUARZKZKZKZKZULUEALDBLDUGUPHUFAMBMABUBUCUEUFUPUNULUEUPUOUN
    UEAGNUPUOHAOAGUIUOPQUEAINUOUNHASAIUKUNPQTUFUNUMULUFBGNUNUMHBOBGGUMPQUFBINUM
    ULHBSBIIULPQTUDT $.

  $( Extended real subtraction when both arguments are real.  (Contributed by
     Mario Carneiro, 23-Aug-2015.) $)
  rexsub $p |- ( ( A e. RR /\ B e. RR ) -> ( A +e -e B ) = ( A - B ) ) $=
    ( cr wcel wa cxne cxad co cneg caddc cmin wceq rexneg adantl oveq2d renegcl
    rexadd sylan2 cc recn negsub syl2an 3eqtrd ) ACDZBCDZEZABFZGHABIZGHZAUHJHZA
    BKHZUFUGUHAGUEUGUHLUDBMNOUEUDUHCDUIUJLBPAUHQRUDASDBSDUJUKLUEATBTABUAUBUC $.

  ${
    rexaddd.1 $e |- ( ph -> A e. RR ) $.
    rexaddd.2 $e |- ( ph -> B e. RR ) $.
    $( The extended real addition operation when both arguments are real.
       Deduction version of ~ rexadd .  (Contributed by Glauco Siliprandi,
       24-Dec-2020.) $)
    rexaddd $p |- ( ph -> ( A +e B ) = ( A + B ) ) $=
      ( cr wcel cxad co caddc wceq rexadd syl2anc ) ABFGCFGBCHIBCJIKDEBCLM $.
  $}

  ${
    xnegcld.1 $e |- ( ph -> A e. RR* ) $.
    $( Closure of extended real negative.  (Contributed by Mario Carneiro,
       28-May-2016.) $)
    xnegcld $p |- ( ph -> -e A e. RR* ) $=
      ( cxr wcel cxne xnegcl syl ) ABDEBFDECBGH $.
  $}

  $( The set of extended reals exists.  (Contributed by NM, 24-Dec-2006.) $)
  xrex $p |- RR* e. _V $=
    ( cxr cr cpnf cmnf cpr cun cvv df-xr reex wcel pnfxr mnfxr prexg mp2an unex
    eqeltri ) ABCDEZFGHBQICAJDAJQGJKLCDAAMNOP $.

  $( Closure of extended real addition in the subset ` RR* / { -oo } ` .
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xaddnemnf $p |- ( ( ( A e. RR* /\ A =/= -oo ) /\
                      ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo ) $=
    ( cxr wcel cmnf wne wa cr cpnf wceq wo cxad co xrnemnf caddc rexadd readdcl
    pnfnemnf a1i eqnetrd eqeltrd renemnfd oveq2 rexr renemnf xaddpnf1 sylan9eqr
    syl2anc jaodan sylan2b oveq1 xaddpnf2 sylan9eq jaoian sylanb ) ACDZAEFZGAHD
    ZAIJZKBCDBEFGZABLMZEFZANURUTVBUSUTURBHDZBIJZKVBBNURVCVBVDURVCGZVAVEVAABOMHA
    BPABQUAUBURVDGZVAIEVDURVAAILMZIBIALUCURUPUQVGIJAUDAUEAUFUHUGIEFZVFRSTUIUJUS
    UTGZVAIEUSUTVAIBLMIAIBLUKBULUMVHVIRSTUNUO $.

  $( Closure of extended real addition in the subset ` RR* / { +oo } ` .
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xaddnepnf $p |- ( ( ( A e. RR* /\ A =/= +oo ) /\
                      ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo ) $=
    ( cxr wcel cpnf wne wa cr cmnf wceq wo cxad co xrnepnf caddc rexadd readdcl
    mnfnepnf a1i eqnetrd eqeltrd renepnfd oveq2 rexr renepnf xaddmnf1 sylan9eqr
    syl2anc jaodan sylan2b oveq1 xaddmnf2 sylan9eq jaoian sylanb ) ACDZAEFZGAHD
    ZAIJZKBCDBEFGZABLMZEFZANURUTVBUSUTURBHDZBIJZKVBBNURVCVBVDURVCGZVAVEVAABOMHA
    BPABQUAUBURVDGZVAIEVDURVAAILMZIBIALUCURUPUQVGIJAUDAUEAUFUHUGIEFZVFRSTUIUJUS
    UTGZVAIEUSUTVAIBLMIAIBLUKBULUMVHVIRSTUNUO $.

  $( Extended real version of ~ negid .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xnegid $p |- ( A e. RR* -> ( A +e -e A ) = 0 ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o cxne cxad co cc0 elxr cneg caddc rexneg id
    xnegeq eqtrdi oveq12d oveq2d renegcl rexadd negidd 3eqtrd xnegpnf pnfaddmnf
    mpdan recn xnegmnf mnfaddpnf 3jaoi sylbi ) ABCADCZAEFZAGFZHAAIZJKZLFZAMUNUS
    UOUPUNURAANZJKZAUTOKZLUNUQUTAJAPUAUNUTDCVAVBFAUBAUTUCUHUNAAUIUDUEUOUREGJKLU
    OAEUQGJUOQUOUQEIGAERUFSTUGSUPURGEJKLUPAGUQEJUPQUPUQGIEAGRUJSTUKSULUM $.

  $( The extended real addition operation is closed in extended reals.
     (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xaddcl $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* ) $=
    ( cxr cxad xaddf fovcl ) ABCCCDEF $.

  $( The extended real addition operation is commutative.  (Contributed by NM,
     26-Dec-2011.) $)
  xaddcom $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o co wa 3eqtr4d wne syl2anc sylan9eqr eqtr4d
    cxad oveq2d oveq1d wo elxr caddc cc recn addcom syl2an rexadd oveq2 renemnf
    ancoms rexr xaddpnf1 xaddpnf2 renepnf xaddmnf1 xaddmnf2 3jaodan sylan2b cc0
    oveq1 pnfaddmnf mnfaddpnf eqtr4i simpr 3eqtr4a wn wdc xrmnfdc exmiddc df-ne
    syl orbi2i sylibr mpjaodan adantl simpl xrpnfdc 3jaoian sylanb ) ACDZAEDZAF
    GZAHGZIBCDZABQJZBAQJZGZAUAWAWDWGWBWCWDWABEDZBFGZBHGZIWGBUAWAWHWGWIWJWAWHKAB
    UBJZBAUBJZWEWFWAAUCDBUCDWKWLGWHAUDBUDABUEUFABUGWHWAWFWLGBAUGUJLWAWIKWEFWFWI
    WAWEAFQJZFBFAQUHWAVTAHMZWMFGAUKZAUIZAULNOWIWAWFFAQJZFBFAQUTWAVTWNWQFGWOWPAU
    MNOPWAWJKWEHWFWJWAWEAHQJZHBHAQUHWAVTAFMZWRHGWOAUNZAUONOWJWAWFHAQJZHBHAQUTWA
    VTWSXAHGWOWTAUPNOPUQURWBWDKZFBQJZBFQJZWEWFWDXCXDGZWBWDWJXEBHMZWDWJKZFHQJZHF
    QJZXCXDXHUSXIVAVBVCXGBHFQWDWJVDZRXGBHFQXJSVEWDXFKXCFXDBUMBULPWDWJWJVFZTZWJX
    FTWDWJVGXLBVHWJVIVKXFXKWJBHVJVLVMVNVOXBAFBQWBWDVPZSXBAFBQXMRLWCWDKZHBQJZBHQ
    JZWEWFWDXOXPGZWCWDWIXQBFMZWDWIKZXIXHXOXPXIUSXHVBVAVCXSBFHQWDWIVDZRXSBFHQXTS
    VEWDXRKXOHXPBUPBUOPWDWIWIVFZTZWIXRTWDWIVGYBBVQWIVIVKXRYAWIBFVJVLVMVNVOXNAHB
    QWCWDVPZSXNAHBQYCRLVRVS $.

  $( Extended real version of ~ addrid .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xaddid1 $p |- ( A e. RR* -> ( A +e 0 ) = A ) $=
    ( cxr wcel cr cpnf wceq cmnf w3o cc0 cxad co elxr 0re wne ax-mp mp2an oveq1
    0xr id 3eqtr4a caddc rexadd mpan2 addridd renemnf xaddpnf2 renepnf xaddmnf2
    recn eqtrd 3jaoi sylbi ) ABCADCZAEFZAGFZHAIJKZAFZALUMUQUNUOUMUPAIUAKZAUMIDC
    ZUPURFMAIUBUCUMAAUIUDUJUNEIJKZEUPAIBCZIGNZUTEFRUSVBMIUEOIUFPAEIJQUNSTUOGIJK
    ZGUPAVAIENZVCGFRUSVDMIUGOIUHPAGIJQUOSTUKUL $.

  $( Extended real version of ~ addlid .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xaddid2 $p |- ( A e. RR* -> ( 0 +e A ) = A ) $=
    ( cxr wcel cc0 cxad co wceq 0xr xaddcom mpan xaddid1 eqtrd ) ABCZDAEFZADEFZ
    ADBCMNOGHDAIJAKL $.

  ${
    xaddid1d.1 $e |- ( ph -> A e. RR* ) $.
    $( ` 0 ` is a right identity for extended real addition.  (Contributed by
       Glauco Siliprandi, 17-Aug-2020.) $)
    xaddid1d $p |- ( ph -> ( A +e 0 ) = A ) $=
      ( cxr wcel cc0 cxad co wceq xaddid1 syl ) ABDEBFGHBICBJK $.
  $}

  $( An extended nonnegative integer which is less than or equal to a
     nonnegative integer is a nonnegative integer.  (Contributed by AV,
     24-Nov-2021.) $)
  xnn0lenn0nn0 $p |- ( ( M e. NN0* /\ N e. NN0 /\ M <_ N ) -> M e. NN0 ) $=
    ( cxnn0 wcel cn0 cle wbr cpnf wo wi elxnn0 2a1 wa wb breq1 nn0re syl sylbid
    wceq cr adantr rexrd xgepnf wnel pnfnre eleq1 elnelall biimtrdi com13 ax-mp
    cxr adantl ex jaoi sylbi 3imp ) ACDZBEDZABFGZAEDZUQUTAHSZIURUSUTJZJZAKUTVCV
    AUTURUSLVAURVBVAURMUSHBFGZUTVAUSVDNURAHBFOUAURVDUTJVAURVDBHSZUTURBUKDVDVENU
    RBBPUBBUCQHTUDZURVEUTJJUEVEURVFUTVEURHEDZVFUTJZBHEUFVGHTDVHHPUTHTUGQUHUIUJR
    ULRUMUNUOUP $.

  $( An extended nonnegative integer which is less than or equal to 2 is either
     0 or 1 or 2.  (Contributed by AV, 24-Nov-2021.) $)
  xnn0le2is012 $p |- ( ( N e. NN0* /\ N <_ 2 )
                       -> ( N = 0 \/ N = 1 \/ N = 2 ) ) $=
    ( cxnn0 wcel c2 cle wbr cn0 cc0 wceq c1 w3o xnn0lenn0nn0 mp3an2 nn0le2is012
    2nn0 sylancom ) ABCZADEFZAGCZAHIAJIADIKQDGCRSOADLMANP $.

  $( The sum of two extended nonnegative integers is ` 0 ` iff each of the two
     extended nonnegative integers is ` 0 ` .  (Contributed by AV,
     14-Dec-2020.) $)
  xnn0xadd0 $p |- ( ( A e. NN0* /\ B e. NN0* )
                   -> ( ( A +e B ) = 0 <-> ( A = 0 /\ B = 0 ) ) ) $=
    ( cxnn0 wcel wa cxad co cc0 wceq wi cn0 cpnf wo elxnn0 cr syl2an eqeq1d cxr
    nn0re wne caddc rexadd cle wb nn0ge0 add20 bitrd biimpd expcom oveq2 adantr
    wbr jca cmnf nn0xnn0 xnn0xrnemnf xaddpnf1 3syl adantl renepnf ax-mp pm2.21i
    0re nesymi biimtrdi jaoi sylbi com12 oveq1 xaddpnf2 syl sylan9bb imp oveq12
    ex 0xr xaddid1 eqtrdi impbid1 ) ACDZBCDZEABFGZHIZAHIBHIEZVTWAWCWDJZVTAKDZAL
    IZMWAWEJZANWFWHWGWAWFWEWABKDZBLIZMWFWEJZBNWIWKWJWFWIWEWFWIEZWCWDWLWCABUAGZH
    IZWDWLWBWMHWFAODZBODZWBWMIWIASZBSZABUBPQWFWOHAUCULZEWPHBUCULZEWNWDUDWIWFWOW
    SWQAUEUMWIWPWTWRBUEUMABUFPUGUHUIWJWFWEWJWFEZWCLHIZWDXAWCALFGZHIZXBWJWCXDUDW
    FWJWBXCHBLAFUJQUKXAXCLHWFXCLIZWJWFVTARDAUNTEXEAUOAUPAUQURUSQUGXBWDHLHODHLTV
    CHUTVAVDVBZVEVOVFVGVHWGWAWEWGWAEWCXBWDWGWCLBFGZHIWAXBWGWBXGHALBFVIQWAXGLHWA
    BRDBUNTEXGLIBUPBVJVKQVLXFVEVOVFVGVMWDWBHHFGZHAHBHFVNHRDXHHIVPHVQVAVRVS $.

  $( Extended real version of ~ negdi .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xnegdi $p |- ( ( A e. RR* /\ B e. RR* ) ->
    -e ( A +e B ) = ( -e A +e -e B ) ) $=
    ( wcel cr cpnf wceq cmnf cxad co cxne syl xnegeq xnegpnf wne eqtrdi 3eqtr4a
    wa oveq2d xnegmnf cc0 cxr w3o elxr caddc cneg cc recn syl2an readdcl rexneg
    negdi renegcl rexadd 3eqtr4d oveqan12d oveq2 rexr renemnf syl2anc sylan9eqr
    xaddpnf1 eqeltrd renepnf xaddmnf1 3jaodan sylan2b xneg0 pnfaddmnf mnfaddpnf
    simpr adantl xaddpnf2 xnegcl xnegneg imbitrid necon3d imp xaddmnf2 syl2an2r
    eqeq1d wn wo wdc xrmnfdc exmiddc df-ne orbi2i sylibr mpjaodan oveq1d adantr
    simpl xrpnfdc 3jaoian sylanb ) AUACZADCZAEFZAGFZUBBUACZABHIZJZAJZBJZHIZFZAU
    CWQWTXFWRWSWTWQBDCZBEFZBGFZUBXFBUCWQXGXFXHXIWQXGQZABUDIZJZAUEZBUEZHIZXBXEXJ
    XKUEZXMXNUDIZXLXOWQAUFCBUFCXPXQFXGAUGBUGABUKUHXJXKDCXLXPFABUIXKUJKWQXMDCXND
    CXOXQFXGAULZBULXMXNUMUHUNXJXAXKFXBXLFABUMXAXKLKWQXGXCXMXDXNHAUJZBUJUOUNWQXH
    QZEJZGXBXEMXTXAEFXBYAFXHWQXAAEHIZEBEAHUPWQWPAGNYBEFAUQZAURAVAUSUTXAELKXHWQX
    EXCGHIZGXHXDGXCHXHXDYAGBELMOZRWQXCDCZYDGFZWQXCXMDXSXRVBZYFXCUACZXCENYGXCUQZ
    XCVCXCVDUSKUTPWQXIQZGJZEXBXESYKXAGFXBYLFXIWQXAAGHIZGBGAHUPWQWPAENYMGFYCAVCA
    VDUSUTXAGLKXIWQXEXCEHIZEXIXDEXCHXIXDYLEBGLSOZRWQYFYNEFZYHYFYIXCGNYPYJXCURXC
    VAUSKUTPVEVFWRWTQZEBHIZJZGXDHIZXBXEWTYSYTFZWRWTXIUUABGNZWTXIQZTJZTYSYTVGUUC
    YRTFYSUUDFUUCYREGHIZTUUCBGEHWTXIVJRVHOYRTLKUUCYTGEHIZTUUCXDEGHXIXDEFZWTYOVK
    RVIOPWTUUBQZYAGYSYTMUUHYREFYSYAFBVLYRELKWTXDUACZUUBXDENZYTGFBVMZWTUUBUUJWTX
    DEBGUUGXDJZGFWTXIUUGUULYAGXDELMOWTUULBGBVNZVTVOVPVQXDVRVSPWTXIXIWAZWBZXIUUB
    WBWTXIWCUUOBWDXIWEKUUBUUNXIBGWFWGWHWIVKYQXAYRFXBYSFYQAEBHWRWTWLWJXAYRLKYQXC
    GXDHYQXCYAGWRXCYAFWTAELWKMOWJUNWSWTQZGBHIZJZEXDHIZXBXEWTUURUUSFZWSWTXHUUTBE
    NZWTXHQZUUDTUURUUSVGUVBUUQTFUURUUDFUVBUUQUUFTUVBBEGHWTXHVJRVIOUUQTLKUVBUUSU
    UETUVBXDGEHXHXDGFZWTYEVKRVHOPWTUVAQZYLEUURUUSSUVDUUQGFUURYLFBVRUUQGLKWTUUIU
    VAXDGNZUUSEFUUKWTUVAUVEWTXDGBEUVCUULEFWTXHUVCUULYLEXDGLSOWTUULBEUUMVTVOVPVQ
    XDVLVSPWTXHXHWAZWBZXHUVAWBWTXHWCUVGBWMXHWEKUVAUVFXHBEWFWGWHWIVKUUPXAUUQFXBU
    URFUUPAGBHWSWTWLWJXAUUQLKUUPXCEXDHUUPXCYLEWSXCYLFWTAGLWKSOWJUNWNWO $.

  $( Associativity of extended real addition.  The correct condition here is
     "it is not the case that both ` +oo ` and ` -oo ` appear as one of
     ` A , B , C ` , i.e. ` -. { +oo , -oo } C_ { A , B , C } ` ", but this
     condition is difficult to work with, so we break the theorem into two
     parts: this one, where ` -oo ` is not present in ` A , B , C ` , and
     ~ xaddass2 , where ` +oo ` is not present.  (Contributed by Mario
     Carneiro, 20-Aug-2015.) $)
  xaddass $p |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\
  ( C e. RR* /\ C =/= -oo ) ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) ) $=
    ( cxr wcel cmnf wne wa cr cxad wceq cpnf caddc rexadd 3eqtr4d adantr oveq1d
    co syl2anc sylan9eqr w3a cc recn addass syl3an 3expa readdcl sylan2 anassrs
    sylan adantll oveq2d oveq2 simp1l simp2l xaddcl xaddnemnf xaddpnf1 3ad2ant1
    3adant3 eqtr4d 3ad2ant2 adantlr wo simp3 xrnemnf mpjaodan xaddpnf2 3ad2ant3
    sylib oveq1 simpl2 simpl3 syl simpl2l simpl3l simpr eqtrd simp1 ) ADEZAFGZH
    ZBDEZBFGZHZCDEZCFGZHZUAZAIEZABJRZCJRZABCJRZJRZKZALKZWIWJHZBIEZWOBLKZWIWJWRW
    OWIWJWRHZHCIEZWOCLKZWTXAWOWIWTXAHZABMRZCJRZABCMRZJRZWLWNXCXDCMRZAXFMRZXEXGW
    JWRXAXHXIKZWJAUBEWRBUBEXACUBEXJAUCBUCCUCABCUDUEUFWTXDIEXAXEXHKABUGXDCNUJWJW
    RXAXGXIKZWRXAHWJXFIEXKBCUGAXFNUHUIOXCWKXDCJWTWKXDKXAABNPQXCWMXFAJWRXAWMXFKW
    JBCNUKULOUKWIXBWOWTWIXBHZWLALJRZWNXLWLLXMXBWIWLWKLJRZLCLWKJUMWIWKDEZWKFGZXN
    LKWIVTWCXOVTWAWEWHUNWBWCWDWHUOABUPSWBWEXPWHABUQUTWKURSTWIXMLKZXBWBWEXQWHAUR
    USZPVAXLWMLAJXBWIWMBLJRZLCLBJUMWEWBXSLKWHBURVBTULVAVCWIXAXBVDZWTWIWHXTWBWEW
    HVECVFVJPVGUIWIWSWOWJWIWSHZLCJRZXMWLWNWIYBXMKWSWIYBLXMWHWBYBLKZWECVHZVIZXRV
    APYAWKLCJWSWIWKXMLBLAJUMXRTQYAWMLAJWSWIWMYBLBLCJVKYETULOVCWQWEWRWSVDWBWEWHW
    JVLBVFVJVGWIWPHZYBLWMJRZWLWNYFYBLYGYFWHYCWBWEWHWPVMZYDVNYFWMDEZWMFGZYGLKYFW
    CWFYIWCWDWBWHWPVOWFWGWBWEWPVPBCUPSYFWEWHYJWBWEWHWPVLZYHBCUQSWMVHSVAYFWKLCJY
    FWKLBJRZLYFALBJWIWPVQZQYFWEYLLKYKBVHVNVRQYFALWMJYMQOWIWBWJWPVDWBWEWHVSAVFVJ
    VG $.

  $( Associativity of extended real addition.  See ~ xaddass for notes on the
     hypotheses.  (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xaddass2 $p |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) /\
  ( C e. RR* /\ C =/= +oo ) ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) ) $=
    ( cxr wcel cpnf wne wa cxad cxne wceq cmnf xnegcl syl xneg11 xnegdi syl2anc
    co wb xaddcl w3a simp1l simp1r sylancl necon3bid mpbird xnegpnf a1i neeqtrd
    pnfxr simp2l simp2r simp3l simp3r xaddass syl222anc oveq1d oveq2d 3eqtr4d
    mpbid ) ADEZAFGZHZBDEZBFGZHZCDEZCFGZHZUAZABIRZCIRZJZABCIRZIRZJZKZVLVOKZVJVK
    JZCJZIRZAJZVNJZIRZVMVPVJWBBJZIRZVTIRZWBWEVTIRZIRZWAWDVJWBDEZWBLGWEDEZWELGVT
    DEZVTLGWGWIKVJVAWJVAVBVFVIUBZAMNVJWBFJZLVJWBWNGVBVAVBVFVIUCVJWBWNAFVJVAFDEZ
    WBWNKAFKSWMUJAFOUDUEUFWNLKVJUGUHZUIVJVDWKVCVDVEVIUKZBMNVJWEWNLVJWEWNGVEVCVD
    VEVIULVJWEWNBFVJVDWOWEWNKBFKSWQUJBFOUDUEUFWPUIVJVGWLVCVFVGVHUMZCMNVJVTWNLVJ
    VTWNGVHVCVFVGVHUNVJVTWNCFVJVGWOVTWNKCFKSWRUJCFOUDUEUFWPUIWBWEVTUOUPVJVSWFVT
    IVJVAVDVSWFKWMWQABPQUQVJWCWHWBIVJVDVGWCWHKWQWRBCPQURUSVJVKDEZVGVMWAKVJVAVDW
    SWMWQABTQZWRVKCPQVJVAVNDEZVPWDKWMVJVDVGXAWQWRBCTQZAVNPQUSVJVLDEZVODEZVQVRSV
    JWSVGXCWTWRVKCTQVJVAXAXDWMXBAVNTQVLVOOQUT $.

  $( Extended real version of ~ pncan .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xpncan $p |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A ) $=
    ( cxr wcel cr wa cxad co wceq adantl oveq2d cmnf ad2antlr cpnf rexr renepnf
    wne syl cc0 eqtrd cxne cneg rexneg renegcl xaddmnf2 syl2anc oveq1 sylan9eqr
    oveq1d simpr 3eqtr4d simpll renemnf xaddass syl222anc simplr rexaddd negidd
    caddc recnd xaddid1 ad2antrr wo wn wdc xrmnfdc exmiddc orbi2i sylibr adantr
    df-ne mpjaodan ) ACDZBEDZFZABGHZBUAZGHVPBUBZGHZAVOVQVRVPGVNVQVRIVMBUCJKVOAL
    IZVSAIALQZVOVTFZLVRGHZLVSAWBVREDZWCLIZVNWDVMVTBUDZMWDVRCDZVRNQWEVROZVRPVRUE
    UFRWBVPLVRGVTVOVPLBGHZLALBGUGVNWILIZVMVNBCDZBNQWJBOZBPBUEUFJUHUIVOVTUJUKVOW
    AFZVSABVRGHZGHZAWMVMWAWKBLQZWGVRLQZVSWOIVMVNWAULVOWAUJVNWKVMWAWLMVNWPVMWABU
    MMWMWDWGVNWDVMWAWFMZWHRWMWDWQWRVRUMRABVRUNUOWMWOASGHZAWMWNSAGWMWNBVRUSHSWMB
    VRVMVNWAUPZWRUQWMBWMBWTUTURTKVMWSAIVNWAAVAVBTTVMVTWAVCZVNVMVTVTVDZVCZXAVMVT
    VEXCAVFVTVGRWAXBVTALVKVHVIVJVLT $.

  $( Extended real version of ~ npcan .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xnpcan $p |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A ) $=
    ( cxr wcel cr wa cxne cxad wceq rexr xnegneg syl adantl oveq2d cneg renegcl
    co rexneg eqeltrd xpncan sylan2 eqtr3d ) ACDZBEDZFZABGZHQZUFGZHQZUGBHQAUEUH
    BUGHUDUHBIZUCUDBCDUJBJBKLMNUDUCUFEDUIAIUDUFBOEBRBPSAUFTUAUB $.

  $( Extended real version of ~ leadd1 ; note that the converse implication is
     not true, unlike the real version (for example ` 0 < 1 ` but
     ` ( 1 +e +oo ) <_ ( 0 +e +oo ) ` ).  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xleadd1a $p |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) ->
    ( A +e C ) <_ ( B +e C ) ) $=
    ( cxr wcel cle wbr wa cxad co cpnf wceq cmnf syl2anc ad2antrr adantr simplr
    syl wne adantlr w3a cr caddc simplrr simpr simplrl simpllr leadd1dd rexaddd
    3brtr4d simpl1 simpl3 xaddcl pnfge rexr renemnf xaddpnf2 ad2antrl sylan9eqr
    oveq1 breqtrrd xrleidd mnfle eqbrtrd wb simpl2 mpbir2and oveq1d breqtrd w3o
    xrletri3 elxr sylib mpjao3dan anassrs xaddmnf2 adantl oveq2d xaddpnf1 sylan
    renepnf eqtrd wo wdc xrmnfdc exmiddc df-ne orbi2i mpjaodan xaddmnf1 xrpnfdc
    wn sylibr ) ADEZBDEZCDEZUAZABFGZHZCUBEZACIJZBCIJZFGZCKLZCMLZWSWTHZAUBEZXCAK
    LZAMLZWSWTXGXCWSWTXGHZHZBUBEZXCBKLZBMLZXKXLHZACUCJBCUCJXAXBFXOABCWSWTXGXLUD
    ZXKXLUEZWSWTXGXLUFZWQWRXJXLUGUHXOACXPXRUIXOBCXQXRUIUJXKXMHZXAKXBFXSXADEZXAK
    FGZWSXTXJXMWSWNWPXTWNWOWPWRUKZWNWOWPWRULZACUMNZOXAUNZRXMXKXBKCIJZKBKCIUTWTY
    FKLZWSXGWTWPCMSYGCUOZCUPCUQNURUSVAWSXNXCXJWSXNHZXAXAXBFYIXAWSXTXNYDPVBYIABC
    IYIABLZWRBAFGZWQWRXNQYIBMAFWSXNUEYIWNMAFGWSWNXNYBPAVCRVDWSYJWRYKHVEZXNWSWNW
    OYLYBWNWOWPWRVFZABVKNZPVGVHVIZTWSXLXMXNVJZXJWSWOYPYMBVLVMPVNVOWSXHXCWTWSXHH
    ZXAXAXBFYQXAWSXTXHYDPVBYQABCIYQYJWRYKWQWRXHQYQBKAFWSBKFGZXHWSWOYRYMBUNRPWSX
    HUEVAWSYLXHYNPVGVHVIZTXFXIHZXAMXBFXIXFXAMCIJZMAMCIUTWTUUAMLZWSWTWPCKSUUBYHC
    WACVPNVQUSYTXBDEZMXBFGZWSUUCWTXIWSWOWPUUCYMYCBCUMNZOXBVCZRVDWSXGXHXIVJZWTWS
    WNUUGYBAVLVMPVNWSXDHZXNXCBMSZWSXNXCXDYOTUUHUUIHZXAKXBFUUJXTYAWSXTXDUUIYDOYE
    RUUJXBBKIJZKUUJCKBIWSXDUUIQVRUUHWOUUIUUKKLWSWOXDYMPZBVSVTWBVAUUHWOXNUUIWCZU
    ULWOXNXNWLZWCZUUMWOXNWDUUOBWEXNWFRUUIUUNXNBMWGWHWMRWIWSXEHZXHXCAKSZWSXHXCXE
    YSTUUPUUQHZXAMXBFUURXAAMIJZMUURCMAIWSXEUUQQVRUUPWNUUQUUSMLWSWNXEYBPZAWJVTWB
    UURUUCUUDWSUUCXEUUQUUEOUUFRVDUUPWNXHUUQWCZUUTWNXHXHWLZWCZUVAWNXHWDUVCAWKXHW
    FRUUQUVBXHAKWGWHWMRWIWSWPWTXDXEVJYCCVLVMVN $.

  $( Commuted form of ~ xleadd1a .  (Contributed by Mario Carneiro,
     20-Aug-2015.) $)
  xleadd2a $p |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) ->
    ( C +e A ) <_ ( C +e B ) ) $=
    ( cxr wcel w3a cle wbr wa cxad xleadd1a wceq xaddcom 3adant2 adantr 3adant1
    co 3brtr3d ) ADEZBDEZCDEZFZABGHZIACJQZBCJQZCAJQZCBJQZGABCKUBUDUFLZUCSUAUHTA
    CMNOUBUEUGLZUCTUAUISBCMPOR $.

  $( Weakened version of ~ xleadd1a under which the reverse implication is
     true.  (Contributed by Mario Carneiro, 20-Aug-2015.) $)
  xleadd1 $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) ->
                  ( A <_ B <-> ( A +e C ) <_ ( B +e C ) ) ) $=
    ( cxr wcel cr w3a cle wbr cxad co rexr xleadd1a syl3an3 xaddcl syl2anc wceq
    wi ex xpncan cxne simp1 3ad2ant3 xnegcl syl syl3anc 3adant2 3adant1 breq12d
    simp2 sylibd impbid ) ADEZBDEZCFEZGZABHIZACJKZBCJKZHIZUOUMUNCDEZUQUTRCLZUMU
    NVAGUQUTABCMSNUPUTURCUAZJKZUSVCJKZHIZUQUPURDEZUSDEZVCDEZUTVFRUPUMVAVGUMUNUO
    UBUOUMVAUNVBUCZACOPUPUNVAVHUMUNUOUJVJBCOPUPVAVIVJCUDUEVGVHVIGUTVFURUSVCMSUF
    UPVDAVEBHUMUOVDAQUNACTUGUNUOVEBQUMBCTUHUIUKUL $.

  $( Extended real version of ~ ltadd1 .  (Contributed by Mario Carneiro,
     23-Aug-2015.)  (Revised by Jim Kingdon, 16-Apr-2023.) $)
  xltadd1 $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) ->
                  ( A < B <-> ( A +e C ) < ( B +e C ) ) ) $=
    ( cxr wcel cr clt wbr cxad co wb cpnf wceq cmnf wa simplr simpr syl2anc syl
    cle w3a simpll3 caddc ltadd1 simp1 simp3 rexaddd simp2 breq12d bitr4d ltpnf
    syl3anc ad2antlr breq2 adantl mpbird readdcl eqeltrd oveq1 wne rexr renemnf
    rexadd xaddpnf2 eqtrd breqtrrd 2thd mnfle 3ad2ant1 ad2antrr simpll1 xrlenlt
    wn mnfxr sylancr mpbid mtbird 3ad2ant3 xaddcl oveq1d renepnf breq2d 2falsed
    xaddmnf2 w3o biimpi 3ad2ant2 adantr mpjao3dan simpl2 pnfge pnfxr a1i breq1d
    elxr mnflt eqbrtrd sylan9eqr eqeq1d ) ADEZBDEZCFEZUAZAFEZABGHZACIJZBCIJZGHZ
    KZALMZANMZXCXDOZBFEZXIBLMZBNMZXLXMOXDXMXBXIXCXDXMPXLXMQWTXAXBXDXMUBXDXMXBUA
    ZXEACUCJZBCUCJZGHXHABCUDXPXFXQXGXRGXPACXDXMXBUEXDXMXBUFZUGXPBCXDXMXBUHXSUGU
    IUJULXLXNOZXEXHXTXEALGHZXDYAXCXNAUKUMXNXEYAKXLBLAGUNUOUPXTXFLXGGXTXFFEZXFLG
    HXTXDXBYBXCXDXNPWTXAXBXDXNUBZXDXBOXFXQFACVCACUQURRXFUKSXTXGLCIJZLXNXGYDMZXL
    BLCIUSZUOXTXBYDLMZYCXBCDEZCNUTZYGCVAZCVBZCVDZRSVEVFVGXLXOOZXEXHYMXEANGHZYMN
    ATHZYNVMZXCYOXDXOWTXAYOXBAVHVIVJYMNDEZWTYOYPKVNWTXAXBXDXOVKZNAVLVOVPXOXEYNK
    XLBNAGUNUOVQYMXHXFNGHZYMNXFTHZYSVMZYMXFDEZYTYMWTYHUUBYRXCYHXDXOXBWTYHXAYJVR
    ZVJZACVSRZXFVHSYMYQUUBYTUUAKVNUUENXFVLVOVPYMXGNXFGYMXGNCIJZNYMBNCIXLXOQVTYM
    YHCLUTZUUFNMZUUDXCUUGXDXOXBWTUUGXACWAZVRZVJCWDZRVEWBVQWCXCXMXNXOWEZXDXAWTUU
    LXBXAUULBWOWFWGZWHWIXCXJOZXEXHUUNXELBGHZUUNBLTHZUUOVMZUUNXAUUPWTXAXBXJWJZBW
    KSUUNXALDEZUUPUUQKUURUUSUUNWLWMBLVLRVPUUNALBGXCXJQZWNVQUUNXHYDXGGHZUUNXGYDT
    HZUVAVMZUUNXGLYDTUUNXGDEZXGLTHUUNXAYHUVDUURXCYHXJUUCWHZBCVSRZXGWKSUUNYHYIYG
    UVEXCYIXJXBWTYIXAYKVRZWHYLRVFUUNUVDYDDEZUVBUVCKUVFUUNUUSYHUVHWLUVELCVSVOXGY
    DVLRVPUUNXFYDXGGUUNALCIUUTVTWNVQWCXCXKOZXMXIXNXOUVIXMOZXEXHUVJANBGXCXKXMPZX
    MNBGHUVIBWPUOWQUVJXFNXGGUVJXFUUFNUVJANCIUVKVTUVJYHUUGUUHUVJXBYHWTXAXBXKXMUB
    ZYJSUVJXBUUGUVLUUISUUKRVEUVJXGFEZNXGGHUVJXMXBUVMUVIXMQUVLXMXBOXGXRFBCVCBCUQ
    URRXGWPSWQVGUVIXNOZXENLGHXHUVNANBLGXCXKXNPUVIXNQUIUVNXFNXGLGUVIXFNMZXNXKXCX
    FUUFNANCIUSZXCYHUUGUUHUUCUUJUUKRZWRWHUVNXGYDLXNYEUVIYFUOXCYGXKXNXCYHYIYGUUC
    UVGYLRVJVEUIUJUVIXOOZXENNGHXHUVRANBNGXCXKXOPUVIXOQZUIUVRXFNXGNGUVRUVOUUHXCU
    UHXKXOUVQVJZXKUVOUUHKXCXOXKXFUUFNUVPWSUMUPUVRXGUUFNUVRBNCIUVSVTUVTVEUIUJXCU
    ULXKUUMWHWIWTXAXDXJXKWEZXBWTUWAAWOWFVIWI $.

  $( Extended real version of ~ ltadd2 .  (Contributed by Mario Carneiro,
     23-Aug-2015.) $)
  xltadd2 $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) ->
                  ( A < B <-> ( C +e A ) < ( C +e B ) ) ) $=
    ( cxr wcel cr w3a clt wbr cxad co xltadd1 rexr wceq xaddcom 3adant2 3adant1
    wb breq12d syl3an3 bitrd ) ADEZBDEZCFEZGABHIACJKZBCJKZHIZCAJKZCBJKZHIZABCLU
    DUBUCCDEZUGUJRCMUBUCUKGUEUHUFUIHUBUKUEUHNUCACOPUCUKUFUINUBBCOQSTUA $.

  $( The sum of nonnegative extended reals is nonnegative.  (Contributed by
     Mario Carneiro, 21-Aug-2015.) $)
  xaddge0 $p |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( 0 <_ A /\ 0 <_ B ) ) ->
                  0 <_ ( A +e B ) ) $=
    ( cxr wcel wa cc0 cle wbr cxad co 0xr a1i simplr xaddcl adantr wceq xaddid2
    simprr syl simpll simprl xleadd1a syl31anc eqbrtrrd xrletrd ) ACDZBCDZEZFAG
    HZFBGHZEZEZFBABIJZFCDZULKLZUFUGUKMZUHUMCDUKABNOUHUIUJRULFBIJZBUMGULUGUQBPUP
    BQSULUNUFUGUIUQUMGHUOUFUGUKTUPUHUIUJUAFABUBUCUDUE $.

  $( Extended real version of ~ le2add .  (Contributed by Mario Carneiro,
     23-Aug-2015.) $)
  xle2add $p |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) ->
     ( ( A <_ C /\ B <_ D ) -> ( A +e B ) <_ ( C +e D ) ) ) $=
    ( cxr wcel wa cle wbr cxad simpll simprl simplr w3a xleadd1a syl3anc xaddcl
    co wi ex simprr xleadd2a adantr syl2anc adantl xrletr syl2and ) AEFZBEFZGZC
    EFZDEFZGZGZACHIZABJRZCBJRZHIZBDHIZUQCDJRZHIZUPUTHIZUNUHUKUIUOURSUHUIUMKUJUK
    ULLZUHUIUMMZUHUKUINUOURACBOTPUNUIULUKUSVASVDUJUKULUAVCUIULUKNUSVABDCUBTPUNU
    PEFZUQEFZUTEFZURVAGVBSUJVEUMABQUCUNUKUIVFVCVDCBQUDUMVGUJCDQUEUPUQUTUFPUG $.

  $( Extended real version of ~ lt2add .  Note that ~ ltleadd , which has
     weaker assumptions, is not true for the extended reals (since
     ` 0 + +oo < 1 + +oo ` fails).  (Contributed by Mario Carneiro,
     23-Aug-2015.) $)
  xlt2add $p |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) ->
     ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) ) $=
    ( cxr wcel wa clt wbr cxad co cpnf wceq cmnf adantr wb mpbid wne cle syl cr
    xaddcl 3ad2ant1 simp1l simp2r syl2anc 3ad2ant2 simp3r simp1r simprl xltadd2
    w3a syl3anc simp3l simp2l xltadd1 xrlttrd anassrs pnfxr a1i pnfge xrltletrd
    simprr npnflt xaddnepnf mpbird oveq2 mnfxr mnfle xrlelttrd nmnfgt sylan9eqr
    syl22anc xaddpnf1 breqtrrd adantlr simpr pm2.21ddne w3o elxr sylib xaddmnf2
    mpjao3dan oveq1 xaddnemnf eqbrtrd 3expia ) AEFZBEFZGZCEFZDEFZGZACHIZBDHIZGZ
    ABJKZCDJKZHIZWJWMWPULZAUAFZWSALMZANMZWTXAGDUAFZWSDLMZDNMZWTXAXDWSWTXAXDGZGZ
    WQADJKZWRWTWQEFZXGWJWMXJWPABUBUCZOWTXIEFZXGWTWHWLXLWHWIWMWPUDZWJWKWLWPUEZAD
    UBUFOWTWREFZXGWMWJXOWPCDUBUGZOXHWOWQXIHIZWTWOXGWJWMWNWOUHZOXHWIWLXAWOXQPWTW
    IXGWHWIWMWPUIZOWTWLXGXNOWTXAXDUJBDAUKUMQXHWNXIWRHIZWTWNXGWJWMWNWOUNZOXHWHWK
    XDWNXTPWTWHXGXMOWTWKXGWJWKWLWPUOZOWTXAXDVCACDUPUMQUQURWTXEWSXAWTXEGWQLWRHWT
    WQLHIZXEWTYCWQLRZWTWHALRZWIBLRZYDXMWTALHIZYEWTACLXMYBLEFWTUSUTZYAWTWKCLSIYB
    CVATVBWTWHYGYEPXMAVDTQZXSWTBLHIZYFWTBDLXSXNYHXRWTWLDLSIXNDVATVBWTWIYJYFPXSB
    VDTQZABVEVMWTXJYCYDPXKWQVDTVFOXEWTWRCLJKZLDLCJVGWTWKCNRZYLLMYBWTNCHIZYMWTNA
    CNEFWTVHUTZXMYBWTWHNASIXMAVITYAVJWTWKYNYMPYBCVKTQZCVNUFVLVOVPWTXFWSXAWTXFGW
    SDNWTXFVQWTDNRZXFWTNDHIZYQWTNBDYOXSXNWTWINBSIXSBVITXRVJWTWLYRYQPXNDVKTQZOVR
    VPWTXDXEXFVSZXAWTWLYTXNDVTWAOWCWTXBGWSALWTXBVQWTYEXBYIOVRWTXCGWQNWRHXCWTWQN
    BJKZNANBJWDWTWIYFUUANMXSYKBWBUFVLWTNWRHIZXCWTUUBWRNRZWTWKYMWLYQUUCYBYPXNYSC
    DWEVMWTXOUUBUUCPXPWRVKTVFOWFWTWHXAXBXCVSXMAVTWAWCWG $.

  $( Extended real version of ~ subge0 .  (Contributed by Mario Carneiro,
     24-Aug-2015.) $)
  xsubge0 $p |- ( ( A e. RR* /\ B e. RR* ) ->
      ( 0 <_ ( A +e -e B ) <-> B <_ A ) ) $=
    ( cxr wcel cpnf wceq cmnf cc0 cxne cxad co cle wbr wb adantl syl wne eqtrdi
    wa breqtrrid w3o elxr 0xr rexr xnegcl xaddcl sylan2 xleadd1 mp3an2i xaddid2
    cr simpr xnpcan breq12d bitrd pnfxr xrletri3 mpan2 renepnf xaddmnf1 syl2anc
    wi clt mnflt0 mnfxr xrlenlt mp2an biimpi mt2 breq2 mtbiri pm2.21d a1d eleq1
    wn mpbiri mnfnepnf neeq1 mpjao3dan 0le0 pnfaddmnf impbid1 biantrurd 3bitr4d
    oveq1 pnfge adantr xnegeq xnegpnf oveq2d breq2d breq1 df-ne 0lepnf xaddpnf1
    mnfaddpnf sylan2br wo xrmnfdc exmiddc mpjaodan 2thd xnegmnf 3jaodan sylan2b
    wdc mnfle ) BCDZACDZBUKDZBEFZBGFZUAHABIZJKZLMZBALMZNZBUBXIXJXQXKXLXIXJSZXOH
    BJKZXNBJKZLMZXPHCDZXRXNCDZXJXOYANUCXJXIXHYCBUDZXHXIXMCDYCBUEAXMUFUGUGXIXJUL
    HXNBUHUIXRXSBXTALXRXHXSBFXJXHXIYDOBUJPABUMUNUOXIXKSZHAGJKZLMZEALMZXOXPXIYGY
    HNXKXIAEFZAELMZYHSZYGYHXIECDYIYKNUPAEUQURXIYGYIXIAUKDZYGYIVBZYIAGFZYLYMXIYL
    YFGFZYMYLXIAEQZYOAUDAUSAUTZVAYOYGYIYOYGHGLMZYRGHVCMZVDYRYSVOZYBGCDZYRYTNUCV
    EHGVFVGVHVIYFGHLVJVKVLZPOXIYISYIYGXIYIULVMYNYMXIYNYOYMYNXIYPYOYNXIUUAVEAGCV
    NVPYNYPGEQVQAGEVRVPYQVAUUBPOXIYLYIYNUAAUBVHVSYIHHYFLVTYIYFEGJKHAEGJWEWARTWB
    XIYJYHAWFWCWDWGYEXNYFHLYEXMGAJXKXMGFXIXKXMEIGBEWHWIROWJWKXKXPYHNXIBEALWLOWD
    XIXLSZHAEJKZLMZGALMZXOXPXIUUEUUFNXLXIUUEUUFXIYNUUEYNVOZXIYNSHHUUDLVTYNUUDHF
    XIYNUUDGEJKHAGEJWEWPROTUUGXIAGQZUUEAGWMXIUUHSHEUUDLWNAWOTWQXIYNXFYNUUGWRAWS
    YNWTPXAAXGXBWGUUCXNUUDHLUUCXMEAJXLXMEFXIXLXMGIEBGWHXCROWJWKXLXPUUFNXIBGALWL
    OWDXDXE $.

  $( Extended real version of ~ posdif .  (Contributed by Mario Carneiro,
     24-Aug-2015.)  (Revised by Jim Kingdon, 17-Apr-2023.) $)
  xposdif $p |- ( ( A e. RR* /\ B e. RR* ) ->
      ( A < B <-> 0 < ( B +e -e A ) ) ) $=
    ( cxr wcel clt wbr cc0 cxad co cpnf wceq cmnf wa breq2d ex wn adantr eqtrdi
    wne mtbiri cxne wb cr w3o elxr cmin posdif rexsub ancoms bitr4d rexr pnfnlt
    wi adantl sylan2 simpl breq1d mtbird 0xr nltmnf ax-mp xnegeq xnegpnf oveq2d
    renepnf xaddmnf1 syl2an2 eqtrd 2falsed mnflt eqbrtrd 0ltpnf xnegmnf renemnf
    xaddpnf1 breqtrrid 2thd 3jaoi sylbi ltpnf simpr breqtrrd oveq1d cneg rexneg
    renegcl eqeltrd rexrd renemnfd xaddpnf2 syl2anc pnfxr xrltnr breq12 oveq12d
    0re ltnri pnfaddmnf mnfltpnf mpbiri oveq1 sylan9eq pnfnemnf mp2an syl eleq1
    xaddmnf2 mnfxr mnfnepnf neeq1 mnfaddpnf 3jaod biimtrid imp ) ACDZBCDZABEFZG
    BAUAZHIZEFZUBZXPBUCDZBJKZBLKZUDXOYABUEXOYBYAYCYDXOAUCDZAJKZALKZUDZYBYAUMZAU
    EZYEYIYFYGYEYBYAYEYBMZXQGBAUFIZEFXTABUGYKXSYLGEYBYEXSYLKBAUHUINUJOYFYBYAYFY
    BMZXQXTYMXQJBEFZYBYFXPYNPZBUKZXPYOYFBULUNUOYMAJBEYFYBUPUQURYMXTGLEFZGCDYQPU
    SGUTVAZYMXSLGEYMXSBLHIZLYMXRLBHYMXRJUAZLYFXRYTKYBAJVBZQVCRVDYBXPYFBJSZYSLKZ
    YPYBUUBYFBVEUNBVFZVGVHNTVIOYGYBYAYGYBMZXQXTUUEALBEYGYBUPYBLBEFYGBVJUNVKUUEG
    JXSEVLUUEXSBJHIZJYGXSUUFKYBYGXRJBHYGXRLUAJALVBVMRVDZQYBXPYGBLSZUUFJKYPYBUUH
    YGBVNUNBVOVGVHVPVQOVRVSXOYHYCYAUMZYJYEUUIYFYGYEYCYAYEYCMZXQXTUUJAJBEYEAJEFY
    CAVTQYEYCWAZWBUUJGJXSEVLUUJXSJXRHIZJUUJBJXRHUUKWCUUJXRCDZXRLSZUULJKYEUUMYCY
    EXRYEXRAWDUCAWEAWFWGZWHQYEUUNYCYEXRUUOWIQXRWJWKVHVPVQOYFYCYAYFYCMZXQXTUUPXQ
    JJEFZJCDZUUQPWLJWMVAAJBJEWNTUUPXTGGEFZGWPWQZUUPXSGGEUUPXSJLHIGUUPBJXRLHYFYC
    WAYFXRLKYCYFXRYTLUUAVCRZQWOWRRNTVIOYGYCYAYGYCMZXQXTUVBXQLJEFWSALBJEWNWTUVBG
    JXSEVLUVBXSJJHIZJYGYCXSUUFUVCUUGBJJHXAXBUURJLSUVCJKWLXCJVOXDRVPVQOVRVSXOYHY
    DYAUMZYJYEUVDYFYGYEYDYAYEYDMZXQXTUVEXQALEFZUVEXOUVFPZYEXOYDAUKQAUTZXEUVEBLA
    EYEYDWAZNURUVEXTYQYRUVEXSLGEUVEXSLXRHIZLUVEBLXRHUVIWCYEUVJLKZYDYEXRUCDZUVKU
    UOUVLUUMXRJSUVKXRUKXRVEXRXGWKXEQVHNTVIOYFYDYAYFYDMZXQXTUVMXQUVFUVMXOUVGYFXO
    YDYFXOUURWLAJCXFWTQUVHXEUVMBLAEYFYDWANURUVMXTYQYRUVMXSLGEUVMXSYSLYFXSYSKYDY
    FXRLBHUVAVDQYDXPYFUUBUUCYDXPLCDZXHBLCXFWTYDUUBYFYDUUBLJSXIBLJXJWTUNUUDVGVHN
    TVIOYGYDYAYGYDMZXQXTUVOXQLLEFZUVNUVPPXHLWMVAALBLEWNTUVOXTUUSUUTUVOXSGGEUVOX
    SLJHIZGYGYDXSUUFUVQUUGBLJHXAXBXKRNTVIOVRVSXLXMXN $.

  $( Under certain conditions, the conclusion of ~ lesubadd is true even in the
     extended reals.  (Contributed by Mario Carneiro, 4-Sep-2015.) $)
  xlesubadd $p |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\
                      ( 0 <_ A /\ B =/= -oo /\ 0 <_ C ) ) ->
                    ( ( A +e -e B ) <_ C <-> A <_ ( C +e B ) ) ) $=
    ( cxr wcel cc0 cle wbr cmnf wne wa cxad cpnf wceq syl2anc breq1d syl5ibrcom
    co wb syl cr cxne simpl1 simpl2 xnegcld xaddcl adantr simpll3 simpr xleadd1
    w3a syl3anc xnpcan sylan bitrd simpr3 oveq1 pnfaddmnf eqtrdi wi xaddmnf1 ex
    simpl3 mnfle breq1 syld wo wdc xrpnfdc sylib mpjaod pnfge ge0nemnf xaddpnf1
    dcne breqtrrd xnegeq xnegpnf oveq2d oveq2 breq2d bibi12d imp simpr2 xrnemnf
    2thd jca mpjaodan ) ADEZBDEZCDEZUKZFAGHZBIJZFCGHZUKZKZBUAEZABUBZLRZCGHZACBL
    RZGHZSZBMNZWQWRKZXAWTBLRZXBGHZXCXFWTDEZWKWRXAXHSWQXIWRWQWIWSDEXIWIWJWKWPUCZ
    WQBWIWJWKWPUDZUEAWSUFOUGWIWJWKWPWRUHWQWRUIWTCBUJULXFXGAXBGWQWIWRXGANXJABUMU
    NPUOWQXEXDWQXDXEAILRZCGHZACMLRZGHZSWQXMXOWQAMNZXMAMJZWQXMXPWOWLWMWNWOUPZXPX
    LFCGXPXLMILRFAMILUQURUSPQWQXQXLINZXMWQWIXQXSUTXJWIXQXSAVAVBTWQXMXSICGHZWQWK
    XTWIWJWKWPVCZCVDTXLICGVEQVFWQWIXPXQVGZXJWIXPVHYBAVIAMVOVJTVKWQAMXNGWQWIAMGH
    XJAVLTWQWKCIJZXNMNYAWQWKWOYCYAXRCVMOCVNOVPWFXEXAXMXCXOXEWTXLCGXEWSIALXEWSMU
    BIBMVQVRUSVSPXEXBXNAGBMCLVTWAWBQWCWQWJWNKWRXEVGWQWJWNXKWLWMWNWOWDWGBWEVJWH
    $.

  ${
    xaddcld.1 $e |- ( ph -> A e. RR* ) $.
    xaddcld.2 $e |- ( ph -> B e. RR* ) $.
    $( The extended real addition operation is closed in extended reals.
       (Contributed by Mario Carneiro, 28-May-2016.) $)
    xaddcld $p |- ( ph -> ( A +e B ) e. RR* ) $=
      ( cxr wcel cxad co xaddcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
  $}

  ${
    xadd4d.1 $e |- ( ph -> ( A e. RR* /\ A =/= -oo ) ) $.
    xadd4d.2 $e |- ( ph -> ( B e. RR* /\ B =/= -oo ) ) $.
    xadd4d.3 $e |- ( ph -> ( C e. RR* /\ C =/= -oo ) ) $.
    xadd4d.4 $e |- ( ph -> ( D e. RR* /\ D =/= -oo ) ) $.
    $( Rearrangement of 4 terms in a sum for extended addition, analogous to
       ~ add4d .  (Contributed by Alexander van der Vekens, 21-Dec-2017.) $)
    xadd4d $p |- ( ph ->
      ( ( A +e B ) +e ( C +e D ) ) = ( ( A +e C ) +e ( B +e D ) ) ) $=
      ( cxad co cxr wcel cmnf wne wa wceq xaddass simpld syl2anc syl3anc oveq2d
      xaddcld xaddnemnf syl112anc xaddcom oveq1d eqtr2d eqtrd 3eqtr4d ) ABDCJKZ
      EJKZJKZBDCEJKZJKZJKZBCJKDEJKZJKZBDJKUNJKZAULUOBJADLMZDNOZPZCLMZCNOZPZELMZ
      ENOZPZULUOQHGIDCERUAUBAURBCUQJKZJKZUMABLMBNOPZVEUQLMUQNOZURVJQFGADEAUTVAH
      SZAVFVGISZUCAVBVHVLHIDEUDTBCUQRUEAVIULBJAULCDJKZEJKZVIAUKVOEJAUTVCUKVOQVM
      AVCVDGSZDCUFTUGAVEVBVHVPVIQGHICDERUAUHUBUIAVKVBUNLMUNNOZUSUPQFHACEVQVNUCA
      VEVHVRGICEUDTBDUNRUEUJ $.
  $}

  ${
    xnn0add4d.1 $e |- ( ph -> A e. NN0* ) $.
    xnn0add4d.2 $e |- ( ph -> B e. NN0* ) $.
    xnn0add4d.3 $e |- ( ph -> C e. NN0* ) $.
    xnn0add4d.4 $e |- ( ph -> D e. NN0* ) $.
    $( Rearrangement of 4 terms in a sum for extended addition of extended
       nonnegative integers, analogous to ~ xadd4d .  (Contributed by AV,
       12-Dec-2020.) $)
    xnn0add4d $p |- ( ph -> ( ( A +e B ) +e ( C +e D ) )
                            = ( ( A +e C ) +e ( B +e D ) ) ) $=
      ( cxnn0 wcel cxr cmnf wne wa xnn0xrnemnf syl xadd4d ) ABCDEABJKBLKBMNOFBP
      QACJKCLKCMNOGCPQADJKDLKDMNOHDPQAEJKELKEMNOIEPQR $.
  $}

  ${
    $( Cancelling a factor of two in ` <_ ` (expressed as addition rather than
       as a factor to avoid extended real multiplication).  (Contributed by Jim
       Kingdon, 18-Apr-2023.) $)
    xleaddadd $p |- ( ( A e. RR* /\ B e. RR* )
        -> ( A <_ B <-> ( A +e A ) <_ ( B +e B ) ) ) $=
      ( cxr wcel wa cr cle wbr cxad co wb cpnf wceq cmnf c2 adantl simpr neneqd
      syl wne cmul cc recn 2timesd ad2antlr breq12d cc0 clt simplr 2re a1i 2pos
      caddc lemul2 syl112anc rexaddd 3bitr4d wn xgepnf ad3antlr mtbird readdcld
      renepnf breq1 eqeltrd simpllr xaddcld oveq12d pnfxr pnfnemnf mp2an eqtrdi
      xaddpnf2 breq1d 2falsed mnfle mpbird oveq1 mnfnepnf neeq1 mpbiri xaddmnf2
      simplll syl2anc eqtrd ad2antrr 2thd w3o elxr biimpi mpjao3dan pnfge breq2
      eqbrtrd simpll eleq1 breqtrrd renemnfd ngtmnft mnfxr xrlenlt mpan2 bitr4d
      adantr mtbid oveq2 necon2i xaddmnf1 breq2d 3bitr4rd neii mtbiri xaddnemnf
      eqeq1 neqned syl22anc biimpa xrleidd eqtr4d breqtrd ) ACDZBCDZEZBFDZABGHZ
      AAIJZBBIJZGHZKZBLMZBNMZYCYDEZAFDZYIALMZANMZYLYMEZOAUAJZOBUAJZGHZAAUMJZBBU
      MJZGHYEYHYPYQYTYRUUAGYPAYMAUBDYLAUCPUDYPBYDBUBDYCYMBUCUEUDUFYPYMYDOFDZUGO
      UHHZYEYSKYLYMQZYCYDYMUIZUUBYPUJUKUUCYPULUKABOUNUOYPYFYTYGUUAGYPAAUUDUUDUP
      YPBBUUEUUEUPUFUQYLYNEZYEYHUUFYELBGHZUUFUUGYJYDYJURYCYNYDBLBVCRUEYBUUGYJKY
      AYDYNBUSUTVAYNYEUUGKYLALBGVDPVAUUFYHLYGGHZUUFUUHYGLMZUUFYGFDZUUIURUUFYGUU
      AFUUFBBYCYDYNUIZUUKUPUUFBBUUKUUKVBVEUUJYGLYGVCRSUUFYGCDZUUHUUIKUUFBBYAYBY
      DYNVFZUUMVGYGUSSVAUUFYFLYGGUUFYFLLIJZLUUFALALIYLYNQZUUOVHLCDZLNTZUUNLMVIV
      JLVMVKVLVNVAVOYLYOEZYEYHUURYENBGHZYBUUSYAYDYOBVPUTYOYEUUSKYLANBGVDPVQUURY
      FNYGGUURYFNAIJZNYOYFUUTMYLANAIVRPUURYAALTZUUTNMYAYBYDYOWCYOUVAYLYOUVANLTV
      SANLVTWAPAWBWDWEUURUULNYGGHYCUULYDYOYCBBYAYBQZUVBVGWFYGVPSWNWGYAYMYNYOWHZ
      YBYDYAUVCAWIWJZWFWKYCYJEZYEYHUVEYEALGHZYAUVFYBYJAWLWFYJYEUVFKYCBLAGWMPVQU
      VEYFLYGGUVEYFCDZYFLGHUVEAAYAYBYJWOZUVHVGYFWLSYJUUIYCYJYGLBIJZLBLBIVRYJYBB
      NTZUVILMYJYBUUPVIBLCWPWAYJUVJUUQVJBLNVTWAZBVMWDWEPWQWGYCYKEZYMYIYNYOUVLYM
      EZYEYHUVMYOYEUVMANUVMAUVLYMQZWRRUVLYOYEKZYMUVLYOANGHZYEYAYOUVPKYBYKYAYONA
      UHHURZUVPAWSYANCDZUVPUVQKWTANXAXBXCWFYKYEUVPKYCBNAGWMPXCZXDXEUVMYFNMZYHUV
      MYFNUVMYFUVMYFYTFUVMAAUVNUVNUPUVMAAUVNUVNVBVEWRRUVLUVTYHKZYMUVLYFNGHZNYFU
      HHURZYHUVTUVLUVGUWBUWCKZUVLAAYAYBYKWOZUWEVGZUVGUVRUWDWTYFNXAXBSUVLYGNYFGY
      KYGNMYCYKYGBNIJZNBNBIXFYKYBBLTUWGNMYKYBUVRWTBNCWPWABLBNUVKXGBXHWDWEPXIUVL
      UVGUVTUWCKUWFYFWSSXJZXDXEVOUVLYNEZYEYHUWIYOYEUWIYOLNMZLNVJXKYNYOUWJKUVLAL
      NXNPXLZUVLUVOYNUVSXDXEUWIUVTYHUWIYFNUWIYAANTZYAUWLYFNTYAYBYKYNWCZUWIANUWK
      XOZUWMUWNAAXMXPRUVLUWAYNUWHXDXEVOUVLYOEZYEYHUVLYOYEUVSXQUWOYFYFYGGUWOYFUW
      OAAYAYBYKYOWCZUWPVGXRUWOABABIUWOANBUVLYOQYCYKYOUIXSZUWQVHXTWGYAUVCYBYKUVD
      WFWKYBYDYJYKWHZYAYBUWRBWIWJPWK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Real number intervals
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c (,) $.
  $c (,] $.
  $c [,) $.
  $c [,] $.

  $( Extend class notation with the set of open intervals of extended reals. $)
  cioo $a class (,) $.

  $( Extend class notation with the set of open-below, closed-above intervals
     of extended reals. $)
  cioc $a class (,] $.

  $( Extend class notation with the set of closed-below, open-above intervals
     of extended reals. $)
  cico $a class [,) $.

  $( Extend class notation with the set of closed intervals of extended
     reals. $)
  cicc $a class [,] $.

  ${
    $d x y z $.
    $( Define the set of open intervals of extended reals.  (Contributed by NM,
       24-Dec-2006.) $)
    df-ioo $a |- (,) = ( x e. RR* , y e. RR* |->
                   { z e. RR* | ( x < z /\ z < y ) } ) $.

    $( Define the set of open-below, closed-above intervals of extended reals.
       (Contributed by NM, 24-Dec-2006.) $)
    df-ioc $a |- (,] = ( x e. RR* , y e. RR* |->
                   { z e. RR* | ( x < z /\ z <_ y ) } ) $.

    $( Define the set of closed-below, open-above intervals of extended reals.
       (Contributed by NM, 24-Dec-2006.) $)
    df-ico $a |- [,) = ( x e. RR* , y e. RR* |->
                   { z e. RR* | ( x <_ z /\ z < y ) } ) $.

    $( Define the set of closed intervals of extended reals.  (Contributed by
       NM, 24-Dec-2006.) $)
    df-icc $a |- [,] = ( x e. RR* , y e. RR* |->
                   { z e. RR* | ( x <_ z /\ z <_ y ) } ) $.
  $}

  ${
    $d x y z A $.  $d x y z C $.  $d x y z B $.  $d x y z R $.  $d x y z S $.
    ixx.1 $e |- O = ( x e. RR* , y e. RR* |->
                   { z e. RR* | ( x R z /\ z S y ) } ) $.
    $( Value of the interval function.  (Contributed by Mario Carneiro,
       3-Nov-2013.) $)
    ixxval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A O B ) =
                 { z e. RR* | ( A R z /\ z S B ) } ) $=
      ( cxr cv wbr wa crab wceq breq1 anbi1d rabbidv breq2 anbi2d rabex ovmpo
      xrex ) ABDEJJAKZCKZFLZUEBKZGLZMZCJNDUEFLZUEEGLZMZCJNHUJUHMZCJNUDDOZUIUMCJ
      UNUFUJUHUDDUEFPQRUGEOZUMULCJUOUHUKUJUGEUEGSTRIULCJUCUAUB $.

    $( Membership in an interval of extended reals.  (Contributed by Mario
       Carneiro, 3-Nov-2013.) $)
    elixx1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <->
                 ( C e. RR* /\ A R C /\ C S B ) ) ) $=
      ( cxr wcel wa co cv wbr crab w3a ixxval eleq2d breq2 breq1 anbi12d 3anass
      wceq elrab bitr4i bitrdi ) DKLEKLMZFDEINZLFDCOZGPZUKEHPZMZCKQZLZFKLZDFGPZ
      FEHPZRZUIUJUOFABCDEGHIJSTUPUQURUSMZMUTUNVACFKUKFUEULURUMUSUKFDGUAUKFEHUBU
      CUFUQURUSUDUGUH $.

    $( The set of intervals of extended reals maps to subsets of extended
       reals.  (Contributed by FL, 14-Jun-2007.)  (Revised by Mario Carneiro,
       16-Nov-2013.) $)
    ixxf $p |- O : ( RR* X. RR* ) --> ~P RR* $=
      ( cv wbr wa cxr crab cpw wcel wral cxp wf wss ssrab2 xrex elpw2 fmpo mpbi
      mpbir rgen2w ) AHCHZDIUFBHEIJZCKLZKMZNZBKOAKOKKPUIFQUJABKKUJUHKRUGCKSUHKT
      UAUDUEABKKUHUIFGUBUC $.

    $( The set of intervals of extended reals exists.  (Contributed by Mario
       Carneiro, 3-Nov-2013.)  (Revised by Mario Carneiro, 17-Nov-2014.) $)
    ixxex $p |- O e. _V $=
      ( cxr cxp cpw xrex xpex pwex wf wss ixxf fssxp ax-mp ssexi ) FHHIZHJZIZTU
      AHHKKLHKMLTUAFNFUBOABCDEFGPTUAFQRS $.
  $}

  ${
    $d x y z R $.  $d x y z S $.  $d x y z A $.  $d x y z B $.  $d x y z C $.
    ixxssxr.1 $e |- O = ( x e. RR* , y e. RR* |->
                   { z e. RR* | ( x R z /\ z S y ) } ) $.
    ${
      $d x y z O $.
      $( The set of intervals of extended reals maps to subsets of extended
         reals.  (Contributed by Mario Carneiro, 4-Jul-2014.) $)
      ixxssxr $p |- ( A O B ) C_ RR* $=
        ( co cxr wcel wa cv wbr crab elmpocl cpw ixxf fovcl elpwid sseld mpcom
        ssriv ) ADEHJZKDKLEKLMZANZUELUGKLABKKUGCNZFOUHBNGOMCKPDEHUGIQUFUEKUGUFU
        EKDEKRKKHABCFGHISTUAUBUCUD $.
    $}

    $( Membership in a set of open intervals of extended reals.  We use the
       fact that an operation's value is empty outside of its domain to show
       ` A e. RR* ` and ` B e. RR* ` .  (Contributed by Mario Carneiro,
       3-Nov-2013.) $)
    elixx3g $p |- ( C e. ( A O B ) <->
          ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A R C /\ C S B ) ) ) $=
      ( cxr wcel wa wbr w3a co anass df-3an anbi1i cv crab elixx1 3anass bitrdi
      elmpocl biadan2 3bitr4ri ) DKLZEKLZMZFKLZMZDFGNZFEHNZMZMUJUKUOMZMUHUIUKOZ
      UOMFDEIPLZUJUKUOQUQULUOUHUIUKRSURUJUPABKKATCTZGNUSBTHNMCKUADEIFJUEUJURUKU
      MUNOUPABCDEFGHIJUBUKUMUNUCUDUFUG $.
  $}

  ${
    $d w x y z A $.  $d w x y z C $.  $d w x y z D $.  $d w x O $.
    $d w x y z B $.  $d w P $.  $d x y z R $.  $d x y z S $.  $d x y z T $.
    $d x y z U $.  $d w W $.  $d w X $.
    ixxssixx.1 $e |- O = ( x e. RR* , y e. RR* |->
                   { z e. RR* | ( x R z /\ z S y ) } ) $.
    ${
      ixx.2 $e |- P = ( x e. RR* , y e. RR* |->
                       { z e. RR* | ( x T z /\ z U y ) } ) $.
      ixx.3 $e |- ( ( A e. RR* /\ w e. RR* ) -> ( A R w -> A T w ) ) $.
      ixx.4 $e |- ( ( w e. RR* /\ B e. RR* ) -> ( w S B -> w U B ) ) $.
      $( An interval is a subset of its closure.  (Contributed by Paul Chapman,
         18-Oct-2007.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
      ixxssixx $p |- ( A O B ) C_ ( A P B ) $=
        ( cxr wcel wa wbr co cv crab elmpocl w3a simp1 a1i simpl 3simpa expimpd
        wi syl2im simpr 3simpb ancoms 3jcad elixx1 3imtr4d mpcom ssriv ) DEFLUA
        ZEFGUAZEQRZFQRZSZDUBZVARZVFVBRZABQQAUBCUBZHTVIBUBITSCQUCEFLVFMUDVEVFQRZ
        EVFHTZVFFITZUEZVJEVFJTZVFFKTZUEVGVHVEVMVJVNVOVMVJUKVEVJVKVLUFUGVEVCVMVJ
        VKSVNVCVDUHVJVKVLUIVCVJVKVNOUJULVEVDVMVJVLSVOVCVDUMVJVKVLUNVDVJVLVOVJVD
        VLVOUKPUOUJULUPABCEFVFHILMUQABCEFVFJKGNUQURUSUT $.
    $}

    ${
      ixxun.2 $e |- P = ( x e. RR* , y e. RR* |->
                         { z e. RR* | ( x T z /\ z U y ) } ) $.
      ixxun.3 $e |- ( ( B e. RR* /\ w e. RR* ) -> ( B T w <-> -. w S B ) ) $.
      $( Split an interval into disjoint pieces.  (Contributed by Mario
         Carneiro, 16-Jun-2014.) $)
      ixxdisj $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
                      ( ( A O B ) i^i ( B P C ) ) = (/) ) $=
        ( cxr wcel c0 wbr w3a co cin wss wceq elin elixx1 3adant3 biimpa simp3d
        cv wa wb adantrr wn 3adant1 simp2d simpl2 simp1d syl2anc mpbid pm2.65da
        adantrl pm2.21d biimtrid ssrdv ss0 syl ) EQRZFQRZGQRZUAZEFMUBZFGHUBZUCZ
        SUDVOSUEVLDVOSDUKZVORVPVMRZVPVNRZULZVLVPSRZVPVMVNUFVLVSVTVLVSVPFJTZVLVQ
        WAVRVLVQULVPQRZEVPITZWAVLVQWBWCWAUAZVIVJVQWDUMVKABCEFVPIJMNUGUHUIUJUNVL
        VRWAUOZVQVLVRULZFVPKTZWEWFWBWGVPGLTZVLVRWBWGWHUAZVJVKVRWIUMVIABCFGVPKLH
        OUGUPUIZUQWFVJWBWGWEUMVIVJVKVRURWFWBWGWHWJUSPUTVAVCVBVDVEVFVOVGVH $.
    $}

    ${
      ixxss1.2 $e |- P = ( x e. RR* , y e. RR* |->
                         { z e. RR* | ( x T z /\ z S y ) } ) $.
      ixxss1.3 $e |- ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) ->
                       ( ( A W B /\ B T w ) -> A R w ) ) $.
      $( Subset relationship for intervals of extended reals.  (Contributed by
         Mario Carneiro, 3-Nov-2013.)  (Revised by Mario Carneiro,
         28-Apr-2015.) $)
      ixxss1 $p |- ( ( A e. RR* /\ A W B ) ->
                   ( B P C ) C_ ( A O C ) ) $=
        ( cxr wcel wbr wa co cv w3a elixx3g simplbi adantl simp3d simplr simpld
        simprbi wi simpll simp1d syl3anc mp2and simprd wb simp2d elixx1 syl2anc
        mpbir3and ex ssrdv ) EQRZEFMSZTZDFGHUAZEGLUAZVFDUBZVGRZVIVHRZVFVJTZVKVI
        QRZEVIISZVIGJSZVLFQRZGQRZVMVJVPVQVMUCZVFVJVRFVIKSZVOTZABCFGVIKJHOUDZUEU
        FZUGZVLVEVSVNVDVEVJUHVLVSVOVJVTVFVJVRVTWAUJUFZUIVLVDVPVMVEVSTVNUKVDVEVJ
        ULZVLVPVQVMWBUMWCPUNUOVLVSVOWDUPVLVDVQVKVMVNVOUCUQWEVLVPVQVMWBURABCEGVI
        IJLNUSUTVAVBVC $.
    $}

    ${
      ixxss2.2 $e |- P = ( x e. RR* , y e. RR* |->
                         { z e. RR* | ( x R z /\ z T y ) } ) $.
      ixxss2.3 $e |- ( ( w e. RR* /\ B e. RR* /\ C e. RR* ) ->
                       ( ( w T B /\ B W C ) -> w S C ) ) $.
      $( Subset relationship for intervals of extended reals.  (Contributed by
         Mario Carneiro, 3-Nov-2013.)  (Revised by Mario Carneiro,
         28-Apr-2015.) $)
      ixxss2 $p |- ( ( C e. RR* /\ B W C ) ->
                   ( A P B ) C_ ( A O C ) ) $=
        ( cxr wcel wbr wa co cv w3a elixx3g simplbi adantl simp3d simpld simprd
        simprbi simplr wi simp2d simpll syl3anc mp2and wb simp1d elixx1 syl2anc
        mpbir3and ex ssrdv ) GQRZFGMSZTZDEFHUAZEGLUAZVFDUBZVGRZVIVHRZVFVJTZVKVI
        QRZEVIISZVIGJSZVLEQRZFQRZVMVJVPVQVMUCZVFVJVRVNVIFKSZTZABCEFVIIKHOUDZUEU
        FZUGZVLVNVSVJVTVFVJVRVTWAUJUFZUHVLVSVEVOVLVNVSWDUIVDVEVJUKVLVMVQVDVSVET
        VOULWCVLVPVQVMWBUMVDVEVJUNZPUOUPVLVPVDVKVMVNVOUCUQVLVPVQVMWBURWEABCEGVI
        IJLNUSUTVAVBVC $.
    $}

    ${
      ixxss12.2 $e |- P = ( x e. RR* , y e. RR* |->
                         { z e. RR* | ( x T z /\ z U y ) } ) $.
      ixxss12.3 $e |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) ->
                        ( ( A W C /\ C T w ) -> A R w ) ) $.
      ixxss12.4 $e |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) ->
                        ( ( w U D /\ D X B ) -> w S B ) ) $.
      $( Subset relationship for intervals of extended reals.  (Contributed by
         Mario Carneiro, 20-Feb-2015.)  (Revised by Mario Carneiro,
         28-Apr-2015.) $)
      ixxss12 $p |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) ->
        ( C P D ) C_ ( A O B ) ) $=
        ( cxr wa wbr co cv elixx3g simplbi adantl simp3d simplrl simprbi simpld
        wcel w3a wi simplll simp1d syl3anc mp2and simprd simplrr simp2d simpllr
        wb elixx1 ad2antrr mpbir3and ex ssrdv ) EUAUMZFUAUMZUBZEGOUCZHFPUCZUBZU
        BZDGHIUDZEFNUDZVPDUEZVQUMZVSVRUMZVPVTUBZWAVSUAUMZEVSJUCZVSFKUCZWBGUAUMZ
        HUAUMZWCVTWFWGWCUNZVPVTWHGVSLUCZVSHMUCZUBZABCGHVSLMIRUFZUGUHZUIZWBVMWIW
        DVLVMVNVTUJWBWIWJVTWKVPVTWHWKWLUKUHZULWBVJWFWCVMWIUBWDUOVJVKVOVTUPWBWFW
        GWCWMUQWNSURUSWBWJVNWEWBWIWJWOUTVLVMVNVTVAWBWCWGVKWJVNUBWEUOWNWBWFWGWCW
        MVBVJVKVOVTVCTURUSVLWAWCWDWEUNVDVOVTABCEFVSJKNQVEVFVGVHVI $.
    $}
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z C $.
    $( The set of open intervals of extended reals exists.  (Contributed by NM,
       6-Feb-2007.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    iooex $p |- (,) e. _V $=
      ( vx vy vz clt cioo df-ioo ixxex ) ABCDDEABCFG $.

    $( Value of the open interval function.  (Contributed by NM, 24-Dec-2006.)
       (Revised by Mario Carneiro, 3-Nov-2013.) $)
    iooval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) =
                 { x e. RR* | ( A < x /\ x < B ) } ) $=
      ( vy vz clt cioo df-ioo ixxval ) DEABCFFGDEAHI $.

    $( An open interval with identical lower and upper bounds is empty.
       (Contributed by Jim Kingdon, 29-Mar-2020.) $)
    iooidg $p |- ( A e. RR* -> ( A (,) A ) = (/) ) $=
      ( vx cxr wcel cioo co cv clt wa crab c0 wceq iooval anidms wral xrltnsym2
      wbr wn ralrimiva rabeq0 sylibr eqtrd ) ACDZAAEFZABGZHQUEAHQIZBCJZKUCUDUGL
      BAAMNUCUFRZBCOUGKLUCUHBCAUEPSUFBCTUAUB $.

    $( Membership in a set of open intervals of extended reals.  We use the
       fact that an operation's value is empty outside of its domain to show
       ` A e. RR* ` and ` B e. RR* ` .  (Contributed by NM, 24-Dec-2006.)
       (Revised by Mario Carneiro, 3-Nov-2013.) $)
    elioo3g $p |- ( C e. ( A (,) B ) <->
          ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ C < B ) ) ) $=
      ( vx vy vz clt cioo df-ioo elixx3g ) DEFABCGGHDEFIJ $.

    $( Membership in an open interval of extended reals.  (Contributed by NM,
       24-Dec-2006.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    elioo1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <->
                 ( C e. RR* /\ A < C /\ C < B ) ) ) $=
      ( vx vy vz clt cioo df-ioo elixx1 ) DEFABCGGHDEFIJ $.

    $( A member of an open interval of reals is a real.  (Contributed by NM,
       17-Aug-2008.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    elioore $p |- ( A e. ( B (,) C ) -> A e. RR ) $=
      ( cioo co wcel cxr w3a clt wbr wa cr elioo3g 3ancomb xrre2 sylanb sylbi )
      ABCDEFBGFZCGFZAGFZHZBAIJACIJKZKALFZBCAMUARTSHUBUCRSTNBACOPQ $.

    $( An open interval does not contain its left endpoint.  (Contributed by
       Jim Kingdon, 30-Mar-2020.) $)
    lbioog $p |- ( ( A e. RR* /\ B e. RR* ) -> -. A e. ( A (,) B ) ) $=
      ( cxr wcel wa cioo co clt wbr w3a xrltnr simp2 nsyl adantr elioo1 mtbird
      wn ) ACDZBCDZEAABFGDRAAHIZABHIZJZRUBQSRTUBAKRTUALMNABAOP $.

    $( An open interval does not contain its right endpoint.  (Contributed by
       Jim Kingdon, 30-Mar-2020.) $)
    ubioog $p |- ( ( A e. RR* /\ B e. RR* ) -> -. B e. ( A (,) B ) ) $=
      ( cxr wcel wa cioo co clt wbr w3a xrltnr simp3 nsyl adantl elioo1 mtbird
      wn ) ACDZBCDZEBABFGDSABHIZBBHIZJZSUBQRSUAUBBKSTUALMNABBOP $.

    $( Value of the open interval function.  (Contributed by NM, 6-Feb-2007.)
       (Revised by Mario Carneiro, 3-Nov-2013.) $)
    iooval2 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) =
                 { x e. RR | ( A < x /\ x < B ) } ) $=
      ( cxr wcel wa cioo co cv clt wbr crab cr iooval cin wss elioore eqsstrrdi
      wceq ssriv df-ss sylib inrab2 ressxr sseqin2 mpbi rabeq ax-mp eqtri eqtrd
      eqtr3di ) BDECDEFZBCGHZBAIZJKUNCJKFZADLZUOAMLZABCNZULUPMOZUPUQULUPMPUSUPS
      ULUPUMMURAUMMUNBCQTRUPMUAUBUSUOADMOZLZUQUOADMUCUTMSZVAUQSMDPVBUDMDUEUFUOA
      UTMUGUHUIUKUJ $.

    $( Subset relationship for open intervals of extended reals.  (Contributed
       by NM, 7-Feb-2007.)  (Revised by Mario Carneiro, 20-Feb-2015.) $)
    iooss1 $p |- ( ( A e. RR* /\ A <_ B ) ->
                 ( B (,) C ) C_ ( A (,) C ) ) $=
      ( vx vy vz vw cioo clt cle df-ioo cv xrlelttr ixxss1 ) DEFGABCHIIIHJDEFKZ
      OABGLMN $.

    $( Subset relationship for open intervals of extended reals.  (Contributed
       by NM, 7-Feb-2007.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    iooss2 $p |- ( ( C e. RR* /\ B <_ C ) ->
                 ( A (,) B ) C_ ( A (,) C ) ) $=
      ( vx vy vz vw cioo clt cle df-ioo cv xrltletr ixxss2 ) DEFGABCHIIIHJDEFKZ
      OGLBCMN $.

    $( Value of the open-below, closed-above interval function.  (Contributed
       by NM, 24-Dec-2006.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    iocval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,] B ) =
                 { x e. RR* | ( A < x /\ x <_ B ) } ) $=
      ( vy vz clt cle cioc df-ioc ixxval ) DEABCFGHDEAIJ $.

    $( Value of the closed-below, open-above interval function.  (Contributed
       by NM, 24-Dec-2006.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    icoval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) =
                 { x e. RR* | ( A <_ x /\ x < B ) } ) $=
      ( vy vz cle clt cico df-ico ixxval ) DEABCFGHDEAIJ $.

    $( Value of the closed interval function.  (Contributed by NM,
       24-Dec-2006.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    iccval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,] B ) =
                 { x e. RR* | ( A <_ x /\ x <_ B ) } ) $=
      ( vy vz cle cicc df-icc ixxval ) DEABCFFGDEAHI $.

    $( Membership in an open interval of extended reals.  (Contributed by NM,
       6-Feb-2007.) $)
    elioo2 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <->
                 ( C e. RR /\ A < C /\ C < B ) ) ) $=
      ( vx cxr wcel wa cioo co cv clt wbr cr crab w3a iooval2 eleq2d wceq breq2
      breq1 anbi12d elrab 3anass bitr4i bitrdi ) AEFBEFGZCABHIZFCADJZKLZUHBKLZG
      ZDMNZFZCMFZACKLZCBKLZOZUFUGULCDABPQUMUNUOUPGZGUQUKURDCMUHCRUIUOUJUPUHCAKS
      UHCBKTUAUBUNUOUPUCUDUE $.

    $( Membership in an open-below, closed-above interval of extended reals.
       (Contributed by NM, 24-Dec-2006.)  (Revised by Mario Carneiro,
       3-Nov-2013.) $)
    elioc1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <->
                 ( C e. RR* /\ A < C /\ C <_ B ) ) ) $=
      ( vx vy vz clt cle cioc df-ioc elixx1 ) DEFABCGHIDEFJK $.

    $( Membership in a closed-below, open-above interval of extended reals.
       (Contributed by NM, 24-Dec-2006.)  (Revised by Mario Carneiro,
       3-Nov-2013.) $)
    elico1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,) B ) <->
                 ( C e. RR* /\ A <_ C /\ C < B ) ) ) $=
      ( vx vy vz cle clt cico df-ico elixx1 ) DEFABCGHIDEFJK $.

    $( Membership in a closed interval of extended reals.  (Contributed by NM,
       24-Dec-2006.)  (Revised by Mario Carneiro, 3-Nov-2013.) $)
    elicc1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,] B ) <->
                 ( C e. RR* /\ A <_ C /\ C <_ B ) ) ) $=
      ( vx vy vz cle cicc df-icc elixx1 ) DEFABCGGHDEFIJ $.

    $( A closed interval with identical lower and upper bounds is a singleton.
       (Contributed by Jeff Hankins, 13-Jul-2009.) $)
    iccid $p |- ( A e. RR* -> ( A [,] A ) = { A } ) $=
      ( vx cxr wcel co csn cv cle wbr w3a wb wi wa clt wn xrlenlt ancoms sylbid
      cicc syl5ibrcom elicc1 anidms xrlttri3 biimprd expcomd com23 3impd eleq1a
      wceq ex xrleid breq2 breq1 3jcad impbid velsn bitr4di bitrd eqrdv ) ACDZB
      AASEZAFZUTBGZVADZVCCDZAVCHIZVCAHIZJZVCVBDZUTVDVHKAAVCUAUBUTVHVCAUIZVIUTVH
      VJUTVEVFVGVJUTVEVFVGVJLZLUTVEMZVFVCANIOZVKAVCPVLVGVMVJVLVGAVCNIOZVMVJLVEU
      TVGVNKVCAPQVLVMVNVJVEUTVMVNMZVJLVEUTMVJVOVCAUCUDQUERUFRUJUGUTVJVEVFVGACVC
      UHUTVFVJAAHIZAUKZVCAAHULTUTVGVJVPVQVCAAHUMTUNUOBAUPUQURUS $.

    $( An empty closed interval of extended reals.  (Contributed by Jim
       Kingdon, 30-Mar-2020.) $)
    icc0r $p |- ( ( A e. RR* /\ B e. RR* ) ->
        ( B < A -> ( A [,] B ) = (/) ) ) $=
      ( vx cxr wcel wa clt wbr cv cle wrex wn cicc co c0 wi xrletr 3com23 3expa
      wceq rexlimdva xrlenlt sylibd con2d crab iccval eqeq1d wral rabeq0 ralnex
      bitri bitrdi sylibrd ) ADEZBDEZFZBAGHZACIZJHURBJHFZCDKZLZABMNZOTZUPUTUQUP
      UTABJHZUQLUPUSVDCDUNUOURDEZUSVDPZUNVEUOVFAURBQRSUAABUBUCUDUPVCUSCDUEZOTZV
      AUPVBVGOCABUFUGVHUSLCDUHVAUSCDUIUSCDUJUKULUM $.
  $}

  ${
    $d x y z $.
    $( An inhabited open interval spans an interval of extended reals.
       (Contributed by NM, 17-Aug-2008.) $)
    eliooxr $p |- ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) ) $=
      ( vx vy vz cxr cv clt wbr wa crab cioo df-ioo elmpocl ) DEGGDHFHZIJPEHIJK
      FGLBCMADEFNO $.
  $}

  $( Ordering implied by a member of an open interval of reals.  (Contributed
     by NM, 17-Aug-2008.)  (Revised by Mario Carneiro, 9-May-2014.) $)
  eliooord $p |- ( A e. ( B (,) C ) -> ( B < A /\ A < C ) ) $=
    ( cioo co wcel cr clt wbr w3a wa cxr wb eliooxr elioo2 syl ibi 3simpc ) ABC
    DEFZAGFZBAHIZACHIZJZUAUBKSUCSBLFCLFKSUCMABCNBCAOPQTUAUBRP $.

  $( The upper bound belongs to an open-below, closed-above interval.  See
     ~ ubicc2 .  (Contributed by FL, 29-May-2014.) $)
  ubioc1 $p |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) ) $=
    ( cxr wcel clt wbr w3a co cle simp2 simp3 xrleid 3ad2ant2 wb elioc1 3adant3
    cioc mpbir3and ) ACDZBCDZABEFZGBABQHDZTUABBIFZSTUAJSTUAKTSUCUABLMSTUBTUAUCG
    NUAABBOPR $.

  $( The lower bound belongs to a closed-below, open-above interval.  See
     ~ lbicc2 .  (Contributed by FL, 29-May-2014.) $)
  lbico1 $p |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) ) $=
    ( cxr wcel clt wbr w3a co cle simp1 xrleid 3ad2ant1 simp3 wb elico1 3adant3
    cico mpbir3and ) ACDZBCDZABEFZGAABQHDZSAAIFZUASTUAJSTUCUAAKLSTUAMSTUBSUCUAG
    NUAABAOPR $.

  $( An element of a closed interval is less than or equal to its upper bound.
     (Contributed by Jeff Hankins, 14-Jul-2009.) $)
  iccleub $p |- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> C <_ B ) $=
    ( cxr wcel cicc co cle wbr wa w3a elicc1 simp3 biimtrdi 3impia ) ADEZBDEZCA
    BFGEZCBHIZPQJRCDEZACHIZSKSABCLTUASMNO $.

  $( An element of a closed interval is more than or equal to its lower bound
     (Contributed by Thierry Arnoux, 23-Dec-2016.) $)
  iccgelb $p |- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> A <_ C ) $=
    ( cxr wcel cicc co cle wbr wa w3a elicc1 biimpa simp2d 3impa ) ADEZBDEZCABF
    GEZACHIZPQJZRJCDEZSCBHIZTRUASUBKABCLMNO $.

  $( TODO - see if this shortens the uses of any other
     elioo*'s $)
  $( Membership in an open interval of extended reals.  (Contributed by NM,
     17-Aug-2008.) $)
  elioo5 $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* )
           -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) ) $=
    ( cxr wcel w3a co clt wbr wa wb elioo1 3adant3 3anass baibr 3ad2ant3 bitr4d
    cioo ) ADEZBDEZCDEZFCABRGEZUAACHIZCBHIZFZUCUDJZSTUBUEKUAABCLMUASUFUEKTUEUAU
    FUAUCUDNOPQ $.

  $( Membership in an open interval of extended reals.  (Contributed by NM,
     8-Jun-2007.)  (Revised by Mario Carneiro, 28-Apr-2015.) $)
  elioo4g $p |- ( C e. ( A (,) B ) <->
        ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ ( A < C /\ C < B ) ) ) $=
    ( cioo co wcel cxr cr w3a clt wbr wa eliooxr elioore df-3an sylibr eliooord
    jca rexr 3anim3i anim1i elioo3g impbii ) CABDEFZAGFZBGFZCHFZIZACJKCBJKLZLZU
    DUHUIUDUEUFLZUGLUHUDUKUGCABMCABNRUEUFUGOPCABQRUJUEUFCGFZIZUILUDUHUMUIUGULUE
    UFCSTUAABCUBPUC $.

  ${
    $d x A $.  $d x B $.
    $( An open interval is a set of reals.  (Contributed by NM,
       31-May-2007.) $)
    ioossre $p |- ( A (,) B ) C_ RR $=
      ( vx cioo co cr cv elioore ssriv ) CABDEFCGABHI $.
  $}

  $( Membership in an open-below, closed-above real interval.  (Contributed by
     Paul Chapman, 30-Dec-2007.)  (Revised by Mario Carneiro, 14-Jun-2014.) $)
  elioc2 $p |- ( ( A e. RR* /\ B e. RR ) ->
      ( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) ) $=
    ( cxr wcel cr wa cioc clt wbr cle w3a rexr cmnf cpnf a1i xrlelttrd ad2antlr
    co wb elioc1 sylan2 mnfxr simpll simpr1 mnfle ad2antrr simpr2 pnfxr xrrebnd
    simpr3 ltpnf syl mpbir2and 3jca ex 3anim1i impbid1 bitrd ) ADEZBFEZGZCABHSE
    ZCDEZACIJZCBKJZLZCFEZVEVFLZVAUTBDEZVCVGTBMZABCUAUBVBVGVIVBVGVIVBVGGZVHVEVFV
    LVHNCIJZCOIJZVLNACNDEVLUCPUTVAVGUDVBVDVEVFUEZUTNAKJVAVGAUFUGVBVDVEVFUHZQVLC
    BOVOVAVJUTVGVKRODEVLUIPVBVDVEVFUKZVABOIJUTVGBULRQVLVDVHVMVNGTVOCUJUMUNVPVQU
    OUPVHVDVEVFCMUQURUS $.

  $( Membership in a closed-below, open-above real interval.  (Contributed by
     Paul Chapman, 21-Jan-2008.)  (Revised by Mario Carneiro, 14-Jun-2014.) $)
  elico2 $p |- ( ( A e. RR /\ B e. RR* ) ->
      ( C e. ( A [,) B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) ) $=
    ( cr wcel cxr wa cico cle wbr clt w3a rexr cmnf cpnf a1i ad2antrr xrltletrd
    co wb elico1 sylan mnfxr simpr1 mnflt simpr2 simplr simpr3 ad2antlr xrrebnd
    pnfxr pnfge syl mpbir2and 3jca ex 3anim1i impbid1 bitrd ) ADEZBFEZGZCABHSEZ
    CFEZACIJZCBKJZLZCDEZVEVFLZUTAFEZVAVCVGTAMZABCUAUBVBVGVIVBVGVIVBVGGZVHVEVFVL
    VHNCKJZCOKJZVLNACNFEVLUCPUTVJVAVGVKQVBVDVEVFUDZUTNAKJVAVGAUEQVBVDVEVFUFZRVL
    CBOVOUTVAVGUGOFEVLUKPVBVDVEVFUHZVABOIJUTVGBULUIRVLVDVHVMVNGTVOCUJUMUNVPVQUO
    UPVHVDVEVFCMUQURUS $.

  $( Membership in a closed real interval.  (Contributed by Paul Chapman,
     21-Sep-2007.)  (Revised by Mario Carneiro, 14-Jun-2014.) $)
  elicc2 $p |- ( ( A e. RR /\ B e. RR ) -> ( C e. ( A [,] B ) <->
                    ( C e. RR /\ A <_ C /\ C <_ B ) ) ) $=
    ( cr wcel wa cicc co cxr cle wbr w3a wb rexr cmnf clt a1i ad2antrr ad2antlr
    cpnf elicc1 syl2an mnfxr simpr1 mnflt simpr2 xrltletrd pnfxr simpr3 xrrebnd
    ltpnf xrlelttrd syl mpbir2and 3jca ex 3anim1i impbid1 bitrd ) ADEZBDEZFZCAB
    GHEZCIEZACJKZCBJKZLZCDEZVEVFLZUTAIEZBIEZVCVGMVAANZBNZABCUAUBVBVGVIVBVGVIVBV
    GFZVHVEVFVNVHOCPKZCTPKZVNOACOIEVNUCQUTVJVAVGVLRVBVDVEVFUDZUTOAPKVAVGAUERVBV
    DVEVFUFZUGVNCBTVQVAVKUTVGVMSTIEVNUHQVBVDVEVFUIZVABTPKUTVGBUKSULVNVDVHVOVPFM
    VQCUJUMUNVRVSUOUPVHVDVEVFCNUQURUS $.

  ${
    elicc2i.1 $e |- A e. RR $.
    elicc2i.2 $e |- B e. RR $.
    $( Inference for membership in a closed interval.  (Contributed by Scott
       Fenton, 3-Jun-2013.) $)
    elicc2i $p |- ( C e. ( A [,] B ) <-> ( C e. RR /\ A <_ C /\ C <_ B ) ) $=
      ( cr wcel cicc co cle wbr w3a wb elicc2 mp2an ) AFGBFGCABHIGCFGACJKCBJKLM
      DEABCNO $.
  $}

  $( Membership in a closed real interval.  (Contributed by Stefan O'Rear,
     16-Nov-2014.)  (Proof shortened by Mario Carneiro, 1-Jan-2017.) $)
  elicc4 $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
      ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) ) $=
    ( cxr wcel cicc co cle wbr wa wb w3a elicc1 3anass bitrdi baibd 3impa ) ADE
    ZBDEZCDEZCABFGEZACHIZCBHIZJZKRSJZUATUDUEUATUBUCLTUDJABCMTUBUCNOPQ $.

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z C $.  $d w x y z D $.
    $( Condition for a closed interval to be a subset of another closed
       interval.  (Contributed by Jeff Madsen, 2-Sep-2009.)  (Revised by Mario
       Carneiro, 20-Feb-2015.) $)
    iccss $p |- ( ( ( A e. RR /\ B e. RR ) /\
                    ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) ) $=
      ( vx vy vz vw cr wcel wa cxr cle wbr cicc co wss rexr anim12i xrletr cv
      df-icc ixxss12 sylan ) AIJZBIJZKALJZBLJZKACMNDBMNKCDOPABOPQUEUGUFUHARBRSE
      FGHABCDOMMMMOMMEFGUBZUIACHUAZTUJDBTUCUD $.

    $( Condition for a closed interval to be a subset of an open interval.
       (Contributed by Mario Carneiro, 20-Feb-2015.) $)
    iccssioo $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
      ( A < C /\ D < B ) ) -> ( C [,] D ) C_ ( A (,) B ) ) $=
      ( vx vy vz vw cicc clt cle cioo df-ioo df-icc xrltletr xrlelttr ixxss12
      cv ) EFGHABCDIJJKKLJJEFGMEFGNACHRZOSDBPQ $.

    $( Condition for a closed-below, open-above interval to be a subset of a
       closed-below, open-above interval.  (Contributed by Thierry Arnoux,
       21-Sep-2017.) $)
    icossico $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
      ( A <_ C /\ D <_ B ) ) -> ( C [,) D ) C_ ( A [,) B ) ) $=
      ( vx vy vz vw cico cle clt df-ico cv xrletr xrltletr ixxss12 ) EFGHABCDIJ
      KJKIJJEFGLZQACHMZNRDBOP $.

    $( Condition for a closed interval to be a subset of another closed
       interval.  (Contributed by Jeff Madsen, 2-Sep-2009.)  (Revised by Mario
       Carneiro, 28-Apr-2015.) $)
    iccss2 $p |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ->
      ( C [,] D ) C_ ( A [,] B ) ) $=
      ( vx vy vz vw cicc co wcel cxr cle wbr w3a elixx3g adantr simprbi xrletr
      wa wss df-icc simplbi simp1d simp2d simpld simprd adantl ixxss12 syl22anc
      cv ) CABIJZKZDULKZTZALKZBLKZACMNZDBMNZCDIJULUAUOUPUQCLKZUMUPUQUTOZUNUMVAU
      RCBMNZTZEFGABCMMIEFGUBZPZUCQZUDUOUPUQUTVFUEUOURVBUMVCUNUMVAVCVERQUFUNUSUM
      UNADMNZUSUNUPUQDLKOVGUSTEFGABDMMIVDPRUGUHEFGHABCDIMMMMIMMVDVDACHUKZSVHDBS
      UIUJ $.

    $( Condition for a closed interval to be a subset of a half-open interval.
       (Contributed by Mario Carneiro, 9-Sep-2015.) $)
    iccssico $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
      ( A <_ C /\ D < B ) ) -> ( C [,] D ) C_ ( A [,) B ) ) $=
      ( vx vy vz vw cicc cle clt cico df-ico df-icc cv xrletr xrlelttr ixxss12
      ) EFGHABCDIJKJJLJKEFGMEFGNACHOZPSDBQR $.

    $( Condition for a closed interval to be a subset of an open interval.
       (Contributed by Mario Carneiro, 20-Feb-2015.) $)
    iccssioo2 $p |- ( ( C e. ( A (,) B ) /\ D e. ( A (,) B ) ) ->
      ( C [,] D ) C_ ( A (,) B ) ) $=
      ( cioo co wcel cxr clt wbr cicc wss eliooxr adantr eliooord simpld adantl
      wa simprd iccssioo syl12anc ) CABEFZGZDUBGZRZAHGBHGRZACIJZDBIJZCDKFUBLUCU
      FUDCABMNUEUGCBIJZUCUGUIRUDCABONPUEADIJZUHUDUJUHRUCDABOQSABCDTUA $.

    $( Condition for a closed interval to be a subset of a closed-below,
       open-above interval.  (Contributed by Mario Carneiro, 20-Feb-2015.) $)
    iccssico2 $p |- ( ( C e. ( A [,) B ) /\ D e. ( A [,) B ) ) ->
      ( C [,] D ) C_ ( A [,) B ) ) $=
      ( vx vy vz cico co wcel wa cxr cle wbr clt cv adantr w3a elixx3g simprbi
      cicc crab df-ico elmpocl1 elmpocl2 simpld simprd adantl iccssico syl22anc
      wss ) CABHIZJZDULJZKALJZBLJZACMNZDBONZCDUAIULUKUMUOUNEFLLEPGPZMNUSFPONKGL
      UBZABHCEFGUCZUDQUMUPUNEFLLUTABHCVAUEQUMUQUNUMUQCBONZUMUOUPCLJRUQVBKEFGABC
      MOHVASTUFQUNURUMUNADMNZURUNUOUPDLJRVCURKEFGABDMOHVASTUGUHABCDUIUJ $.
  $}

  $( The open interval from minus to plus infinity.  (Contributed by NM,
     6-Feb-2007.) $)
  ioomax $p |- ( -oo (,) +oo ) = RR $=
    ( vx cmnf cpnf cioo co cv clt wbr wa crab cxr wcel wceq mnfxr pnfxr iooval2
    cr mp2an rabid2 mnflt ltpnf jca mprgbir eqtr4i ) BCDEZBAFZGHZUFCGHZIZAQJZQB
    KLCKLUEUJMNOABCPRQUJMUIAQUIAQSUFQLUGUHUFTUFUAUBUCUD $.

  $( The closed interval from minus to plus infinity.  (Contributed by Mario
     Carneiro, 4-Jul-2014.) $)
  iccmax $p |- ( -oo [,] +oo ) = RR* $=
    ( vx cmnf cpnf cicc co cv cle wbr wa cxr crab wcel mnfxr pnfxr iccval mp2an
    wceq rabid2 mnfle pnfge jca mprgbir eqtr4i ) BCDEZBAFZGHZUECGHZIZAJKZJBJLCJ
    LUDUIQMNABCOPJUIQUHAJUHAJRUEJLUFUGUESUETUAUBUC $.

  $( The set of positive reals expressed as an open interval.  (Contributed by
     NM, 7-May-2007.) $)
  ioopos $p |- ( 0 (,) +oo ) = { x e. RR | 0 < x } $=
    ( cc0 cpnf cioo co cv clt wbr wa crab cxr wcel wceq 0xr pnfxr iooval2 mp2an
    cr ltpnf biantrud rabbiia eqtr4i ) BCDEZBAFZGHZUDCGHZIZARJZUEARJBKLCKLUCUHM
    NOABCPQUEUGARUDRLUFUEUDSTUAUB $.

  $( The set of positive reals expressed as an open interval.  (Contributed by
     Steve Rodriguez, 25-Nov-2007.) $)
  ioorp $p |- ( 0 (,) +oo ) = RR+ $=
    ( vx cc0 cpnf cioo co cv clt wbr cr crab crp ioopos df-rp eqtr4i ) BCDEBAFG
    HAIJKALAMN $.

  $( Shift the arguments of the open interval function.  (Contributed by NM,
     17-Aug-2008.) $)
  iooshf $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) )
     -> ( ( A - B ) e. ( C (,) D ) <-> A e. ( ( C + B ) (,) ( D + B ) ) ) ) $=
    ( cr wcel wa caddc co clt wbr cioo wb 3expa cxr readdcl rexrd rexr ad2antrl
    elioo5 ltaddsub 3com13 adantrr w3a ltsubadd bicomd adantrl anbi12d ad2ant2l
    cmin ad2ant2rl syl3anc ancoms ad2antll resubcl adantr 3bitr4rd ) AEFZBEFZGZ
    CEFZDEFZGZGZCBHIZAJKZADBHIZJKZGZCABUJIZJKZVJDJKZGZAVEVGLIFZVJCDLIFZVDVFVKVH
    VLUTVAVFVKMZVBURUSVAVPVAUSURVPCBAUAUBNUCUTVBVHVLMZVAURUSVBVQURUSVBUDVLVHABD
    UEUFNUGUHVCUTVNVIMZVCUTGVEOFZVGOFZAOFZVRVAUSVSVBURVAUSGVECBPQUKVBUSVTVAURVB
    USGVGDBPQUIURWAVCUSARSVEVGATULUMVDCOFZDOFZVJOFZVOVMMVAWBUTVBCRSVBWCUTVADRUN
    UTWDVCUTVJABUOQUPCDVJTULUQ $.

  ${
    $d w x y z A $.  $d w x y z B $.
    $( A closed-above interval with real upper bound is a set of reals.
       (Contributed by FL, 29-May-2014.) $)
    iocssre $p |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR ) $=
      ( vx cxr wcel cr wa cioc co cv w3a clt wbr cle elioc2 simp1d 3expia ssrdv
      biimp3a ) ADEZBFEZGCABHIZFTUACJZUBEZUCFEZTUAUDKUEAUCLMZUCBNMZTUAUDUEUFUGK
      ABUCOSPQR $.

    $( A closed-below interval with real lower bound is a set of reals.
       (Contributed by Mario Carneiro, 14-Jun-2014.) $)
    icossre $p |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR ) $=
      ( vx cr wcel cxr wa cico co cv w3a cle wbr clt elico2 simp1d 3expia ssrdv
      biimp3a ) ADEZBFEZGCABHIZDTUACJZUBEZUCDEZTUAUDKUEAUCLMZUCBNMZTUAUDUEUFUGK
      ABUCOSPQR $.

    $( A closed real interval is a set of reals.  (Contributed by FL,
       6-Jun-2007.)  (Proof shortened by Paul Chapman, 21-Jan-2008.) $)
    iccssre $p |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR ) $=
      ( vx cr wcel wa cicc co cv w3a cle wbr elicc2 biimp3a simp1d 3expia ssrdv
      ) ADEZBDEZFCABGHZDRSCIZTEZUADEZRSUBJUCAUAKLZUABKLZRSUBUCUDUEJABUAMNOPQ $.

    $( A closed interval is a set of extended reals.  (Contributed by FL,
       28-Jul-2008.)  (Revised by Mario Carneiro, 4-Jul-2014.) $)
    iccssxr $p |- ( A [,] B ) C_ RR* $=
      ( vx vy vz cle cicc df-icc ixxssxr ) CDEABFFGCDEHI $.

    $( An open-below, closed-above interval is a subset of the extended reals.
       (Contributed by FL, 29-May-2014.)  (Revised by Mario Carneiro,
       4-Jul-2014.) $)
    iocssxr $p |- ( A (,] B ) C_ RR* $=
      ( vx vy vz clt cle cioc df-ioc ixxssxr ) CDEABFGHCDEIJ $.

    $( A closed-below, open-above interval is a subset of the extended reals.
       (Contributed by FL, 29-May-2014.)  (Revised by Mario Carneiro,
       4-Jul-2014.) $)
    icossxr $p |- ( A [,) B ) C_ RR* $=
      ( vx vy vz cle clt cico df-ico ixxssxr ) CDEABFGHCDEIJ $.

    $( An open interval is a subset of its closure.  (Contributed by Paul
       Chapman, 18-Oct-2007.) $)
    ioossicc $p |- ( A (,) B ) C_ ( A [,] B ) $=
      ( vx vy vz vw cicc clt cle cioo df-ioo df-icc cv xrltle ixxssixx ) CDEFAB
      GHHIIJCDEKCDELAFMZNPBNO $.
  $}

  ${
    $d a b w x A $.  $d a b w x B $.  $d a b w x C $.  $d a b w x D $.
    $( A closed-below, open-above interval is a subset of its closure.
       (Contributed by Thierry Arnoux, 25-Oct-2016.) $)
    icossicc $p |- ( A [,) B ) C_ ( A [,] B ) $=
      ( va vb vx vw cicc cle clt cico df-ico df-icc cxr wcel cv wa wbr ixxssixx
      idd xrltle ) CDEFABGHIHHJCDEKCDELAMNFOZMNPAUAHQSUABTR $.

    $( A closed-above, open-below interval is a subset of its closure.
       (Contributed by Thierry Arnoux, 1-Apr-2017.) $)
    iocssicc $p |- ( A (,] B ) C_ ( A [,] B ) $=
      ( va vb vx vw cicc clt cle cioc df-ioc df-icc cv xrltle cxr wcel ixxssixx
      wa wbr idd ) CDEFABGHIIIJCDEKCDELAFMZNUAOPBOPRUABISTQ $.

    $( An open interval is a subset of its closure-below.  (Contributed by
       Thierry Arnoux, 3-Mar-2017.) $)
    ioossico $p |- ( A (,) B ) C_ ( A [,) B ) $=
      ( va vb vx vw cico clt cle cioo df-ioo df-ico cv xrltle cxr wcel ixxssixx
      wa wbr idd ) CDEFABGHHIHJCDEKCDELAFMZNUAOPBOPRUABHSTQ $.

    $( Condition for a closed interval to be a subset of an open interval.
       (Contributed by Thierry Arnoux, 29-Mar-2017.) $)
    iocssioo $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
      ( A <_ C /\ D < B ) ) -> ( C (,] D ) C_ ( A (,) B ) ) $=
      ( va vb vx vw cioc clt cle cioo df-ioo df-ioc cv xrlelttr ixxss12 ) EFGHA
      BCDIJJJKLKJEFGMEFGNACHOZPRDBPQ $.

    $( Condition for a closed interval to be a subset of an open interval.
       (Contributed by Thierry Arnoux, 29-Mar-2017.) $)
    icossioo $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
      ( A < C /\ D <_ B ) ) -> ( C [,) D ) C_ ( A (,) B ) ) $=
      ( va vb vx vw cico clt cle cioo df-ioo df-ico cv xrltletr ixxss12 ) EFGHA
      BCDIJJKJLJKEFGMEFGNACHOZPRDBPQ $.

    $( Condition for an open interval to be a subset of an open interval.
       (Contributed by Thierry Arnoux, 26-Sep-2017.) $)
    ioossioo $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
      ( A <_ C /\ D <_ B ) ) -> ( C (,) D ) C_ ( A (,) B ) ) $=
      ( va vb vx vw cioo clt cle df-ioo cv xrlelttr xrltletr ixxss12 ) EFGHABCD
      IJJJJIKKEFGLZQACHMZNRDBOP $.
  $}

  ${
    $d A y $.  $d B x y $.  $d S x y $.
    $( A nonempty subset of a closed real interval satisfies the conditions for
       the existence of its supremum.  To be useful without excluded middle,
       we'll probably need to change not equal to apart, and perhaps make other
       changes, but the theorem does hold as stated here.  (Contributed by Paul
       Chapman, 21-Jan-2008.) $)
    iccsupr $p |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) ->
                   ( S C_ RR /\ S =/= (/) /\ E. x e. RR A. y e. S y <_ x ) ) $=
      ( cr wcel wa cicc co wss w3a c0 wne cv cle wbr wral 3adant3 iccssre sylan
      wrex sstr ancoms ne0i 3ad2ant3 simplr elicc2 biimpd sylan9r imp ralrimiva
      ssel simp3d wceq breq2 ralbidv rspcev syl2anc 3jca ) CGHZDGHZIZFCDJKZLZEF
      HZMFGLZFNOZBPZAPZQRZBFSZAGUCZVDVFVHVGVDVEGLZVFVHCDUAVFVOVHFVEGUDUEUBTVGVD
      VIVFFEUFUGVDVFVNVGVDVFIZVCVJDQRZBFSZVNVBVCVFUHVPVQBFVPVJFHZIVJGHZCVJQRZVQ
      VPVSVTWAVQMZVFVSVJVEHZVDWBFVEVJUNVDWCWBCDVJUIUJUKULUOUMVMVRADGVKDUPVLVQBF
      VKDVJQUQURUSUTTVA $.
  $}

  ${
    $d w x y z A $.  $d x B $.
    $( Membership in an unbounded interval of extended reals.  (Contributed by
       Mario Carneiro, 18-Jun-2014.) $)
    elioopnf $p |- ( A e. RR* ->
                     ( B e. ( A (,) +oo ) <-> ( B e. RR /\ A < B ) ) ) $=
      ( cxr wcel cpnf cioo co cr clt wbr w3a wa pnfxr elioo2 mpan2 df-3an ltpnf
      wb adantr pm4.71i bitr4i bitrdi ) ACDZBAEFGDZBHDZABIJZBEIJZKZUEUFLZUCECDU
      DUHRMAEBNOUHUIUGLUIUEUFUGPUIUGUEUGUFBQSTUAUB $.

    $( Membership in an unbounded interval of extended reals.  (Contributed by
       Mario Carneiro, 18-Jun-2014.) $)
    elioomnf $p |- ( A e. RR* ->
                     ( B e. ( -oo (,) A ) <-> ( B e. RR /\ B < A ) ) ) $=
      ( cxr wcel cmnf cioo co cr clt wbr w3a wa wb mnfxr elioo2 mpan an32 mnflt
      df-3an adantr pm4.71i 3bitr4i bitrdi ) ACDZBEAFGDZBHDZEBIJZBAIJZKZUFUHLZE
      CDUDUEUIMNEABOPUFUGLUHLUJUGLUIUJUFUGUHQUFUGUHSUJUGUFUGUHBRTUAUBUC $.

    $( Membership in a closed unbounded interval of reals.  (Contributed by
       Mario Carneiro, 16-Sep-2014.) $)
    elicopnf $p |- ( A e. RR ->
                     ( B e. ( A [,) +oo ) <-> ( B e. RR /\ A <_ B ) ) ) $=
      ( cr wcel cpnf cico co cle wbr clt w3a wa cxr wb pnfxr elico2 mpan2 ltpnf
      adantr pm4.71i df-3an bitr4i bitr4di ) ACDZBAEFGDZBCDZABHIZBEJIZKZUFUGLZU
      DEMDUEUINOAEBPQUJUJUHLUIUJUHUFUHUGBRSTUFUGUHUAUBUC $.

    $( Two ways of saying that a real number is positive.  (Contributed by NM,
       7-May-2007.) $)
    repos $p |- ( A e. ( 0 (,) +oo ) <-> ( A e. RR /\ 0 < A ) ) $=
      ( vx cc0 cv clt wbr cr cpnf cioo co breq2 ioopos elrab2 ) CBDZEFCAEFBAGCH
      IJNACEKBLM $.

    $( The set of open intervals of extended reals maps to subsets of reals.
       (Contributed by NM, 7-Feb-2007.)  (Revised by Mario Carneiro,
       16-Nov-2013.) $)
    ioof $p |- (,) : ( RR* X. RR* ) --> ~P RR $=
      ( vx vz vy cv clt wbr wa cxr crab cr cpw wcel wral cxp cioo wf iooval cvv
      co vex wss ioossre cop df-ov iooex opex fvex eqeltri elpw mpbir eqeltrrdi
      cfv rgen2a df-ioo fmpo mpbi ) ADZBDZEFURCDZEFGBHIZJKZLZCHMAHMHHNVAOPVBACH
      UQHLUSHLGUTUQUSOSZVABUQUSQVCVALVCJUAUQUSUBVCJVCUQUSUCZOULRUQUSOUDVDORRUEU
      QUSATCTUFUGUHUIUJUKUMACHHUTVAOACBUNUOUP $.

    $( The set of closed intervals of extended reals maps to subsets of
       extended reals.  (Contributed by FL, 14-Jun-2007.)  (Revised by Mario
       Carneiro, 3-Nov-2013.) $)
    iccf $p |- [,] : ( RR* X. RR* ) --> ~P RR* $=
      ( vx vy vz cle cicc df-icc ixxf ) ABCDDEABCFG $.

    $( The union of the range of the open interval function.  (Contributed by
       NM, 7-May-2007.)  (Revised by Mario Carneiro, 30-Jan-2014.) $)
    unirnioo $p |- RR = U. ran (,) $=
      ( cr cioo crn cuni wcel wss cmnf cpnf co ioomax cxr cxp wfn wf ioof ax-mp
      cpw ffn mnfxr pnfxr fnovrn mp3an eqeltrri elssuni frn sspwuni mpbi eqssi
      ) ABCZDZAUIEAUJFGHBIZAUIJBKKLZMZGKEHKEUKUIEULAQZBNZUMOULUNBRPSTKKGHBUAUBU
      CAUIUDPUIUNFZUJAFUOUPOULUNBUEPUIAUFUGUH $.

    $( Alternate definition of the set of open intervals of extended reals.
       (Contributed by NM, 1-Mar-2007.)  (Revised by Mario Carneiro,
       1-Sep-2015.) $)
    dfioo2 $p |- (,) = ( x e. RR* , y e. RR* |->
                         { w e. RR | ( x < w /\ w < y ) } ) $=
      ( cioo cxr cv co cmpo clt wbr wa cr crab cxp cpw wf wfn wceq ioof ffn
      fnovim mp2b iooval2 mpoeq3ia eqtri ) DABEEAFZBFZDGZHZABEEUFCFZIJUJUGIJKCL
      MZHEENZLOZDPDULQDUIRSULUMDTABEEDUAUBABEEUHUKCUFUGUCUDUE $.

    $( Open intervals are elements of the set of all open intervals.
       (Contributed by Jim Kingdon, 4-Apr-2020.) $)
    ioorebasg $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) e. ran (,) ) $=
      ( cioo cxr cxp wfn wcel co crn cr cpw wf ioof ffn ax-mp fnovrn mp3an1 ) C
      DDEZFZADGBDGABCHCIGRJKZCLSMRTCNODDABCPQ $.
  $}

  $( The predicate "is a nonnegative real".  (Contributed by Jeff Madsen,
     2-Sep-2009.)  (Proof shortened by Mario Carneiro, 18-Jun-2014.) $)
  elrege0 $p |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) ) $=
    ( cc0 cr wcel cpnf cico co cle wbr wa wb 0re elicopnf ax-mp ) BCDABEFGDACDB
    AHIJKLBAMN $.

  $( Nonnegative real numbers are real numbers.  (Contributed by Thierry
     Arnoux, 9-Sep-2018.)  (Proof shortened by AV, 8-Sep-2019.) $)
  rge0ssre $p |- ( 0 [,) +oo ) C_ RR $=
    ( vx cc0 cpnf cico co cr cv wcel cle wbr elrege0 simplbi ssriv ) ABCDEZFAGZ
    NHOFHBOIJOKLM $.

  $( Elementhood in the set of nonnegative extended reals.  (Contributed by
     Mario Carneiro, 28-Jun-2014.) $)
  elxrge0 $p |- ( A e. ( 0 [,] +oo ) <-> ( A e. RR* /\ 0 <_ A ) ) $=
    ( cxr wcel cc0 cle wbr cpnf w3a wa cicc co df-3an wb 0xr pnfxr elicc1 mp2an
    pnfge adantr pm4.71i 3bitr4i ) ABCZDAEFZAGEFZHZUBUCIZUDIADGJKCZUFUBUCUDLDBC
    GBCUGUEMNODGAPQUFUDUBUDUCARSTUA $.

  $( 0 is a member of ` ( 0 [,) +oo ) ` (common case).  (Contributed by David
     A. Wheeler, 8-Dec-2018.) $)
  0e0icopnf $p |- 0 e. ( 0 [,) +oo ) $=
    ( cc0 cpnf cico co wcel cr cle wbr 0re 0le0 elrege0 mpbir2an ) AABCDEAFEAAG
    HIJAKL $.

  $( 0 is a member of ` ( 0 [,] +oo ) ` (common case).  (Contributed by David
     A. Wheeler, 8-Dec-2018.) $)
  0e0iccpnf $p |- 0 e. ( 0 [,] +oo ) $=
    ( cc0 cpnf cicc co wcel cxr cle wbr 0xr 0le0 elxrge0 mpbir2an ) AABCDEAFEAA
    GHIJAKL $.

  $( The nonnegative reals are closed under addition.  (Contributed by Mario
     Carneiro, 19-Jun-2014.) $)
  ge0addcl $p |- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) ->
                   ( A + B ) e. ( 0 [,) +oo ) ) $=
    ( cc0 cpnf cico co wcel cr cle wbr wa caddc elrege0 readdcl ad2ant2r addge0
    an4s sylanbrc syl2anb ) ACDEFZGAHGZCAIJZKZBHGZCBIJZKZABLFZTGZBTGAMBMUCUFKUG
    HGZCUGIJZUHUAUDUIUBUEABNOUAUDUBUEUJABPQUGMRS $.

  $( The nonnegative reals are closed under multiplication.  (Contributed by
     Mario Carneiro, 19-Jun-2014.) $)
  ge0mulcl $p |- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) ->
                   ( A x. B ) e. ( 0 [,) +oo ) ) $=
    ( cc0 cpnf cico co wcel cr cle wbr wa cmul elrege0 ad2ant2r mulge0 sylanbrc
    remulcl syl2anb ) ACDEFZGAHGZCAIJZKZBHGZCBIJZKZABLFZSGZBSGAMBMUBUEKUFHGZCUF
    IJUGTUCUHUAUDABQNABOUFMPR $.

  $( The nonnegative reals are closed under addition.  (Contributed by Mario
     Carneiro, 26-Aug-2015.) $)
  ge0xaddcl $p |- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) ) ->
                   ( A +e B ) e. ( 0 [,] +oo ) ) $=
    ( cc0 cpnf cicc co wcel cxr cle wbr wa cxad elxrge0 xaddcl ad2ant2r xaddge0
    an4s sylanbrc syl2anb ) ACDEFZGAHGZCAIJZKZBHGZCBIJZKZABLFZTGZBTGAMBMUCUFKUG
    HGZCUGIJZUHUAUDUIUBUEABNOUAUDUBUEUJABPQUGMRS $.

  $( The lower bound of a closed interval is a member of it.  (Contributed by
     Paul Chapman, 26-Nov-2007.)  (Revised by FL, 29-May-2014.)  (Revised by
     Mario Carneiro, 9-Sep-2015.) $)
  lbicc2 $p |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) ) $=
    ( cxr wcel cle wbr w3a cicc co simp1 xrleid 3ad2ant1 simp3 elicc1 mpbir3and
    wb 3adant3 ) ACDZBCDZABEFZGAABHIDZRAAEFZTRSTJRSUBTAKLRSTMRSUARUBTGPTABANQO
    $.

  $( The upper bound of a closed interval is a member of it.  (Contributed by
     Paul Chapman, 26-Nov-2007.)  (Revised by FL, 29-May-2014.) $)
  ubicc2 $p |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) ) $=
    ( cxr wcel cle wbr w3a cicc co simp2 simp3 xrleid 3ad2ant2 elicc1 mpbir3and
    wb 3adant3 ) ACDZBCDZABEFZGBABHIDZSTBBEFZRSTJRSTKSRUBTBLMRSUASTUBGPTABBNQO
    $.

  $( Zero is an element of the closed unit.  (Contributed by Scott Fenton,
     11-Jun-2013.) $)
  0elunit $p |- 0 e. ( 0 [,] 1 ) $=
    ( cc0 c1 cicc co wcel cr cle wbr 0re 0le0 0le1 1re elicc2i mpbir3an ) AABCD
    EAFEAAGHABGHIJKABAILMN $.

  $( One is an element of the closed unit.  (Contributed by Scott Fenton,
     11-Jun-2013.) $)
  1elunit $p |- 1 e. ( 0 [,] 1 ) $=
    ( c1 cc0 cicc co wcel cr cle wbr 1re 0le1 1le1 0re elicc2i mpbir3an ) ABACD
    EAFEBAGHAAGHIJKBAALIMN $.

  $( Membership in a negated open real interval.  (Contributed by Paul Chapman,
     26-Nov-2007.) $)
  iooneg $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                  ( C e. ( A (,) B ) <-> -u C e. ( -u B (,) -u A ) ) ) $=
    ( cr wcel w3a clt wbr wa cneg cioo co ltneg 3adant2 cxr rexr elioo5 renegcl
    wb syl3an ancoms 3adant1 anbi12d ancom bitrdi 3com12 3bitr4d ) ADEZBDEZCDEZ
    FZACGHZCBGHZIZBJZCJZGHZUPAJZGHZIZCABKLEZUPUOURKLEZUKUNUSUQIUTUKULUSUMUQUHUJ
    ULUSSUIACMNUIUJUMUQSZUHUJUIVCCBMUAUBUCUSUQUDUEUHAOEUIBOEUJCOEVAUNSAPBPCPABC
    QTUIUHUJVBUTSZUIUODEZUHURDEZUJUPDEZVDBRARCRVEUOOEVFUROEVGUPOEVDUOPURPUPPUOU
    RUPQTTUFUG $.

  $( Membership in a negated closed real interval.  (Contributed by Paul
     Chapman, 26-Nov-2007.) $)
  iccneg $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
                  ( C e. ( A [,] B ) <-> -u C e. ( -u B [,] -u A ) ) ) $=
    ( cr wcel w3a cle wbr wa cneg cicc co renegcl anbi12d elicc2 3adant3 3anass
    wb leneg bitrdi ax-1 impbid2 3ad2ant3 ancom 3adant1 3adant2 bitr3id syl2anr
    ancoms 3bitr4d ) ADEZBDEZCDEZFZUMACGHZCBGHZIZIZCJZDEZBJZUSGHZUSAJZGHZIZIZCA
    BKLEZUSVAVCKLEZUNUMUTUQVEUMUKUMUTRULUMUMUTCMUMUTUAUBUCUQUPUOIUNVEUPUOUDUNUP
    VBUOVDULUMUPVBRZUKUMULVICBSUIUEUKUMUOVDRULACSUFNUGNUNVGUMUOUPFZURUKULVGVJRU
    MABCOPUMUOUPQTUNVHUTVBVDFZVFUKULVHVKRZUMULVADEVCDEVLUKBMAMVAVCUSOUHPUTVBVDQ
    TUJ $.

  $( A shifted real is a member of a shifted, closed-below, open-above real
     interval.  (Contributed by Paul Chapman, 25-Mar-2008.) $)
  icoshft $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) ->
                  ( X + C ) e. ( ( A + C ) [,) ( B + C ) ) ) ) $=
    ( cr wcel w3a cico co cle wbr clt wa caddc wi cxr wb rexr elico2 readdcl
    sylan2 biimpd 3adant3 3anass imbitrdi leadd1 3com12 3expib com12 imp ltadd1
    3adant2 3adant1 anbi12d pm5.32da expcom anim1d 3ad2ant3 biimprd syld sylbid
    imbitrrdi syl2anc ) AEFZBEFZCEFZGZDABHIFZDEFZADJKZDBLKZMZMZDCNIZACNIZBCNIZH
    IFZVGVHVIVJVKGZVMVDVEVHVROVFVDVEMVHVRVEVDBPFVHVRQBRABDSUAUBUCVIVJVKUDUEVGVM
    VIVOVNJKZVNVPLKZMZMZVQVGVIVLWAVGVIMVJVSVKVTVGVIVJVSQZVDVFVIWCOVEVIVDVFMWCVI
    VDVFWCVDVIVFWCADCUFUGUHUIULUJVGVIVKVTQZVEVFVIWDOVDVIVEVFMWDVIVEVFWDDBCUKUHU
    IUMUJUNUOVGWBVNEFZVSVTGZVQVFVDWBWFOVEVFWBWEWAMWFVFVIWEWAVIVFWEDCTUPUQWEVSVT
    UDVBURVGVOEFZVPEFZWFVQOVDVFWGVEACTULVEVFWHVDBCTUMWGWHMVQWFWHWGVPPFVQWFQVPRV
    OVPVNSUAUSVCUTVAUT $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d y F $.
    icoshftf1o.1 $e |- F = ( x e. ( A [,) B ) |-> ( x + C ) ) $.
    $( Shifting a closed-below, open-above interval is one-to-one onto.
       (Contributed by Paul Chapman, 25-Mar-2008.)  (Proof shortened by Mario
       Carneiro, 1-Sep-2015.) $)
    icoshftf1o $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
        F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) ) $=
      ( vy cr wcel cv caddc co cico wral wceq wreu icoshft cmin recnd negsubd
      wf1o ralrimiv wa cneg wi readdcl 3adant2 3adant1 renegcl 3ad2ant3 syl3anc
      w3a imp cxr rexrd icossre syl2anc sselda simpl3 simp3 simp1 eqtrd oveq12d
      wss pncand simp2 adantr 3eltr3d reueq sylib simpll3 simpl1 subadd2d eqcom
      simpl2 3bitr4g reubidva mpbid ralrimiva f1ompt sylanbrc ) BHIZCHIZDHIZULZ
      AJZDKLZBDKLZCDKLZMLZIZABCMLZNGJZWGOZAWLPZGWJNWLWJEUAWEWKAWLBCDWFQUBWEWOGW
      JWEWMWJIZUCZWFWMDRLZOZAWLPZWOWQWRWLIWTWQWMDUDZKLZWHXAKLZWIXAKLZMLZWRWLWEW
      PXBXEIZWEWHHIZWIHIZXAHIZWPXFUEWBWDXGWCBDUFUGZWCWDXHWBCDUFUHZWDWBXIWCDUIUJ
      WHWIXAWMQUKUMWQWMDWQWMWEWJHWMWEXGWIUNIWJHVDXJWEWIXKUOWHWIUPUQURZSWQDWBWCW
      DWPUSSTWEXEWLOWPWEXCBXDCMWEXCWHDRLBWEWHDWEWHXJSWEDWBWCWDUTSZTWEBDWEBWBWCW
      DVASXMVEVBWEXDWIDRLCWEWIDWEWIXKSXMTWECDWECWBWCWDVFSXMVEVBVCVGVHAWLWRVIVJW
      QWSWNAWLWQWFWLIZUCZWRWFOWGWMOWSWNXOWMDWFXOWMWQWMHIXNXLVGSXODWBWCWDWPXNVKS
      XOWFWQWLHWFWQWBCUNIWLHVDWBWCWDWPVLWQCWBWCWDWPVOUOBCUPUQURSVMWFWRVNWMWGVNV
      PVQVRVSAGWLWJWGEFVTWA $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z C $.  $d w x y z D $.
    $( End-to-end closed-below, open-above real intervals are disjoint.
       (Contributed by Mario Carneiro, 16-Jun-2014.) $)
    icodisj $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
                    ( ( A [,) B ) i^i ( B [,) C ) ) = (/) ) $=
      ( vx cxr wcel w3a cico co cin c0 wss wceq wa clt wbr cle wb elico1 biimpa
      cv 3adant3 simp3d adantrr wn 3adant1 simp2d simpl2 simp1d xrlenlt syl2anc
      elin mpbid adantrl pm2.65da pm2.21d biimtrid ssrdv ss0 syl ) AEFZBEFZCEFZ
      GZABHIZBCHIZJZKLVGKMVDDVGKDUAZVGFVHVEFZVHVFFZNZVDVHKFZVHVEVFULVDVKVLVDVKV
      HBOPZVDVIVMVJVDVINVHEFZAVHQPZVMVDVIVNVOVMGZVAVBVIVPRVCABVHSUBTUCUDVDVJVMU
      EZVIVDVJNZBVHQPZVQVRVNVSVHCOPZVDVJVNVSVTGZVBVCVJWARVABCVHSUFTZUGVRVBVNVSV
      QRVAVBVCVJUHVRVNVSVTWBUIBVHUJUKUMUNUOUPUQURVGUSUT $.

    $( If the upper bound of one open interval is less than or equal to the
       lower bound of the other, the intervals are disjoint.  (Contributed by
       Jeff Hankins, 13-Jul-2009.) $)
    ioodisj $p |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) )
     /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = (/) ) $=
      ( vx vy vz vw cxr wcel wa cle cioo co cin c0 wss wceq cicc syl wbr iooss1
      simpllr sylancom ioossicc sstrdi sslin simplll simplrr clt df-ioo xrlenlt
      df-icc cv ixxdisj syl3anc sseqtrd ss0 ) AIJZBIJZKZCIJZDIJZKZKZBCLUAZKZABM
      NZCDMNZOZPQVJPRVGVJVHBDSNZOZPVGVIVKQVJVLQVGVIBDMNZVKVEVFUTVIVMQUSUTVDVFUC
      ZBCDUBUDBDUEUFVIVKVHUGTVGUSUTVCVLPRUSUTVDVFUHVNVAVBVCVFUIEFGHABDSUJUJLLME
      FGUKEFGUMBHUNULUOUPUQVJURT $.
  $}

  ${
    iccshftr.1 $e |- ( A + R ) = C $.
    iccshftr.2 $e |- ( B + R ) = D $.
    $( Membership in a shifted interval.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    iccshftr $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) )
                  -> ( X e. ( A [,] B ) <-> ( X + R ) e. ( C [,] D ) ) ) $=
      ( cr wcel wa cle wbr w3a caddc co cicc wb readdcl leadd1 simpl 2thd 3expb
      adantl adantlr breq1i bitrdi an12s adantll breq2i 3anbi123d elicc2 adantr
      eqeltrrid syl2an anandirs adantrl 3bitr4d ) AIJZBIJZKZFIJZEIJZKZKZVBAFLMZ
      FBLMZNZFEOPZIJZCVILMZVIDLMZNZFABQPJZVICDQPJZVEVBVJVFVKVGVLVDVBVJRVAVDVBVJ
      VBVCUAFESUBUDVEVFAEOPZVILMZVKUSVDVFVQRZUTUSVBVCVRAFETUCUEVPCVILGUFUGVEVGV
      IBEOPZLMZVLUTVDVGVTRZUSVBUTVCWAVBUTVCWAFBETUCUHUIVSDVILHUJUGUKVAVNVHRVDAB
      FULUMVAVCVOVMRZVBUSUTVCWBUSVCKZCIJDIJWBUTVCKZWCCVPIGAESUNWDDVSIHBESUNCDVI
      ULUOUPUQUR $.
  $}

  ${
    iccshftri.1 $e |- A e. RR $.
    iccshftri.2 $e |- B e. RR $.
    iccshftri.3 $e |- R e. RR $.
    iccshftri.4 $e |- ( A + R ) = C $.
    iccshftri.5 $e |- ( B + R ) = D $.
    $( Membership in a shifted interval.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    iccshftri $p |- ( X e. ( A [,] B ) -> ( X + R ) e. ( C [,] D ) ) $=
      ( cr wcel cicc co caddc wss iccssre mp2an sseli wb iccshftr mpanl12 mpan2
      wa biimpd mpcom ) FLMZFABNOZMZFEPOCDNOMZUILFALMZBLMZUILQGHABRSTUHUJUKUHEL
      MZUJUKUAZIULUMUHUNUEUOGHABCDEFJKUBUCUDUFUG $.
  $}

  ${
    iccshftl.1 $e |- ( A - R ) = C $.
    iccshftl.2 $e |- ( B - R ) = D $.
    $( Membership in a shifted interval.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    iccshftl $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) )
                  -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) ) $=
      ( cr wcel wa cle wbr w3a cmin co cicc wb resubcl lesub1 simpl 2thd adantl
      3expb adantlr breq1i bitrdi adantll breq2i 3anbi123d elicc2 adantr syl2an
      an12s eqeltrrid anandirs adantrl 3bitr4d ) AIJZBIJZKZFIJZEIJZKZKZVBAFLMZF
      BLMZNZFEOPZIJZCVILMZVIDLMZNZFABQPJZVICDQPJZVEVBVJVFVKVGVLVDVBVJRVAVDVBVJV
      BVCUAFESUBUCVEVFAEOPZVILMZVKUSVDVFVQRZUTUSVBVCVRAFETUDUEVPCVILGUFUGVEVGVI
      BEOPZLMZVLUTVDVGVTRZUSVBUTVCWAVBUTVCWAFBETUDUNUHVSDVILHUIUGUJVAVNVHRVDABF
      UKULVAVCVOVMRZVBUSUTVCWBUSVCKZCIJDIJWBUTVCKZWCCVPIGAESUOWDDVSIHBESUOCDVIU
      KUMUPUQUR $.
  $}

  ${
    iccshftli.1 $e |- A e. RR $.
    iccshftli.2 $e |- B e. RR $.
    iccshftli.3 $e |- R e. RR $.
    iccshftli.4 $e |- ( A - R ) = C $.
    iccshftli.5 $e |- ( B - R ) = D $.
    $( Membership in a shifted interval.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    iccshftli $p |- ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) ) $=
      ( cr wcel cicc co cmin wss iccssre mp2an sseli wb iccshftl mpanl12 biimpd
      wa mpan2 mpcom ) FLMZFABNOZMZFEPOCDNOMZUILFALMZBLMZUILQGHABRSTUHUJUKUHELM
      ZUJUKUAZIULUMUHUNUEUOGHABCDEFJKUBUCUFUDUG $.
  $}

  ${
    iccdil.1 $e |- ( A x. R ) = C $.
    iccdil.2 $e |- ( B x. R ) = D $.
    $( Membership in a dilated interval.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    iccdil $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) )
                  -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) ) $=
      ( cr wcel wa cle wbr w3a cmul co cicc wb remulcl sylan2 simpl rpre adantl
      crp 2thd cc0 clt elrp lemul1 syl3an3b 3expb adantlr breq1i bitrdi adantll
      breq2i 3anbi123d elicc2 adantr eqeltrrid syl2an anandirs adantrl 3bitr4d
      an12s ) AIJZBIJZKZFIJZEUDJZKZKZVIAFLMZFBLMZNZFEOPZIJZCVPLMZVPDLMZNZFABQPJ
      ZVPCDQPJZVLVIVQVMVRVNVSVKVIVQRVHVKVIVQVIVJUAVJVIEIJZVQEUBZFESTUEUCVLVMAEO
      PZVPLMZVRVFVKVMWFRZVGVFVIVJWGVJVFVIWCUFEUGMKZWGEUHZAFEUIUJUKULWECVPLGUMUN
      VLVNVPBEOPZLMZVSVGVKVNWKRZVFVIVGVJWLVIVGVJWLVJVIVGWHWLWIFBEUIUJUKVEUOWJDV
      PLHUPUNUQVHWAVORVKABFURUSVHVJWBVTRZVIVJVHWCWMWDVFVGWCWMVFWCKZCIJDIJWMVGWC
      KZWNCWEIGAESUTWODWJIHBESUTCDVPURVAVBTVCVD $.
  $}

  ${
    iccdili.1 $e |- A e. RR $.
    iccdili.2 $e |- B e. RR $.
    iccdili.3 $e |- R e. RR+ $.
    iccdili.4 $e |- ( A x. R ) = C $.
    iccdili.5 $e |- ( B x. R ) = D $.
    $( Membership in a dilated interval.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    iccdili $p |- ( X e. ( A [,] B ) -> ( X x. R ) e. ( C [,] D ) ) $=
      ( cr wcel cicc co cmul wss iccssre mp2an sseli wb wa iccdil mpanl12 mpan2
      crp biimpd mpcom ) FLMZFABNOZMZFEPOCDNOMZUJLFALMZBLMZUJLQGHABRSTUIUKULUIE
      UFMZUKULUAZIUMUNUIUOUBUPGHABCDEFJKUCUDUEUGUH $.
  $}

  ${
    icccntr.1 $e |- ( A / R ) = C $.
    icccntr.2 $e |- ( B / R ) = D $.
    $( Membership in a contracted interval.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    icccntr $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) )
                  -> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) ) $=
      ( cr wcel wa cle wbr w3a cdiv co cicc wb rerpdivcl lediv1 crp 2thd adantl
      simpl cc0 clt elrp syl3an3b 3expb adantlr breq1i bitrdi adantll 3anbi123d
      an12s breq2i elicc2 adantr eqeltrrid syl2an anandirs adantrl 3bitr4d ) AI
      JZBIJZKZFIJZEUAJZKZKZVGAFLMZFBLMZNZFEOPZIJZCVNLMZVNDLMZNZFABQPJZVNCDQPJZV
      JVGVOVKVPVLVQVIVGVORVFVIVGVOVGVHUDFESUBUCVJVKAEOPZVNLMZVPVDVIVKWBRZVEVDVG
      VHWCVHVDVGEIJUEEUFMKZWCEUGZAFETUHUIUJWACVNLGUKULVJVLVNBEOPZLMZVQVEVIVLWGR
      ZVDVGVEVHWHVGVEVHWHVHVGVEWDWHWEFBETUHUIUOUMWFDVNLHUPULUNVFVSVMRVIABFUQURV
      FVHVTVRRZVGVDVEVHWIVDVHKZCIJDIJWIVEVHKZWJCWAIGAESUSWKDWFIHBESUSCDVNUQUTVA
      VBVC $.
  $}

  ${
    icccntri.1 $e |- A e. RR $.
    icccntri.2 $e |- B e. RR $.
    icccntri.3 $e |- R e. RR+ $.
    icccntri.4 $e |- ( A / R ) = C $.
    icccntri.5 $e |- ( B / R ) = D $.
    $( Membership in a contracted interval.  (Contributed by Jeff Madsen,
       2-Sep-2009.) $)
    icccntri $p |- ( X e. ( A [,] B ) -> ( X / R ) e. ( C [,] D ) ) $=
      ( cr wcel cicc co cdiv wss iccssre mp2an sseli crp icccntr mpanl12 biimpd
      wb wa mpan2 mpcom ) FLMZFABNOZMZFEPOCDNOMZUJLFALMZBLMZUJLQGHABRSTUIUKULUI
      EUAMZUKULUEZIUMUNUIUOUFUPGHABCDEFJKUBUCUGUDUH $.
  $}

  $( A condition for a ratio to be a member of the closed unit.  (Contributed
     by Scott Fenton, 11-Jun-2013.) $)
  divelunit $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) ->
      ( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) ) $=
    ( cdiv co cc0 c1 cicc wcel cr cle wbr wa clt w3a 0re 1re elicc2i bitri cmul
    df-3an wb ledivmul mp3an2 simpll simprl cap gt0ap0 adantl redivclapd divge0
    adantlr jca biantrurd cc recn ad2antrl mulridd breq2d 3bitr3d bitrid ) ABCD
    ZEFGDHZVAIHZEVAJKZLZVAFJKZLZAIHZEAJKZLZBIHZEBMKZLZLZABJKZVBVCVDVFNVGEFVAOPQ
    VCVDVFTRVNVFABFSDZJKZVGVOVHVMVFVQUAZVIVHFIHVMVRPAFBUBUCUKVNVEVFVNVCVDVNABVH
    VIVMUDVJVKVLUEVMBEUFKVJBUGUHUIABUJULUMVNVPBAJVNBVKBUNHVJVLBUOUPUQURUSUT $.

  $( A linear combination of two reals which lies in the interval between them.
     (Contributed by Jeff Madsen, 2-Sep-2009.)  (Proof shortened by Mario
     Carneiro, 8-Sep-2015.) $)
  lincmb01cmp $p |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) )
                    -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) ) $=
    ( cr wcel wbr cc0 c1 cicc co cmin cmul caddc wb 0re a1i 1re cle eqid recnd
    clt w3a wa simpr elicc2i simp1bi adantl difrp biimp3a adantr syl22anc mpbid
    crp iccdil simpl2 simpl1 resubcld mullidd oveq12d eleqtrd remulcld iccshftr
    mul02d mulcld subadd23d subdid oveq1d resubcl sylancr addcomd subdird eqtrd
    1cnd oveq2d 3eqtr4d addlidd npcand 3eltr3d ) ADEZBDEZABUAFZUBZCGHIJEZUCZCBA
    KJZLJZAMJZGAMJZWEAMJZIJZHCKJZALJZCBLJZMJZABIJWDWFGWEIJZEZWGWJEZWDWFGWELJZHW
    ELJZIJZWOWDWCWFWTEZWBWCUDWDGDEZHDEZCDEZWEUMEZWCXANXBWDOPZXCWDQPWCXDWBWCXDGC
    RFCHRFGHCOQUEUFUGZWBXEWCVSVTWAXEABUHUIUJGHWRWSWECWRSWSSUNUKULWDWRGWSWEIWDWE
    WDWEWDBAVSVTWAWCUOZVSVTWAWCUPZUQZTZVCWDWEXKURUSUTWDXBWEDEWFDEVSWPWQNXFXJWDC
    WEXGXJVAXIGWEWHWIAWFWHSWISVBUKULWDWMCALJZKJZAMJWMAXLKJZMJZWGWNWDWMXLAWDCBWD
    CXGTZWDBXHTZVDZWDCAXPWDAXITZVDXSVEWDWFXMAMWDCBAXPXQXSVFVGWDWNWMWLMJXOWDWLWM
    WDWLWDWKAWDXCXDWKDEQXGHCVHVIXIVATXRVJWDWLXNWMMWDWLHALJZXLKJXNWDHCAWDVMXPXSV
    KWDXTAXLKWDAXSURVGVLVNVLVOWDWHAWIBIWDAXSVPWDBAXQXSVQUSVR $.

  $( A linear combination of two reals which lies in the interval between them.
     Like ~ lincmb01cmp but generalized to require merely ` A <_ B ` not
     ` A < B ` .  (Contributed by Jim Kingdon, 13-May-2026.) $)
  lincmble $p |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) )
                    -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) ) $=
    ( cr wcel cle wbr w3a cc0 c1 cicc co cmul caddc 1red adantl remulcld oveq1d
    recnd adddird wa cmin 0re 1re elicc2i simp1bi resubcld simpl1 readdcld 1cnd
    mullidd 3eqtr3rd simp2bi lemul2ad leadd2dd eqbrtrd simp3bi subge0d leadd1dd
    simpl2 npcand simpl3 mpbird 3eqtr3d breqtrd elicc2 3adant3 adantr mpbir3and
    wb ) ADEZBDEZABFGZHZCIJKLEZUAZJCUBLZAMLZCBMLZNLZABKLEZVTDEZAVTFGZVTBFGZVPVR
    VSVPVQAVPJCVPOVOCDEZVNVOWEICFGZCJFGZIJCUCUDUEZUFZPZUGZVKVLVMVOUHZQZVPCBWJVK
    VLVMVOUTZQZUIVPAVRCAMLZNLZVTFVPVQCNLZAMLJAMLWQAVPWRJAMVPJCVPUJVPCWJSZVAZRVP
    VQCAVPVQWKSZWSVPAWLSZTVPAXBUKULVPWPVSVRVPCAWJWLQWOWMVPABCWLWNWJVOWFVNVOWEWF
    WGWHUMPVKVLVMVOVBZUNUOUPVPVTVQBMLZVSNLZBFVPVRXDVSWMVPVQBWKWNQWOVPABVQWLWNWK
    VOIVQFGZVNVOXFWGVOWEWFWGWHUQVOJCVOOWIURVCPXCUNUSVPWRBMLJBMLXEBVPWRJBMWTRVPV
    QCBXAWSVPBWNSZTVPBXGUKVDVEVNWAWBWCWDHVJZVOVKVLXHVMABVTVFVGVHVI $.

  ${
    $d x y A $.  $d x y B $.
    iccf1o.1 $e |- F = ( x e. ( 0 [,] 1 ) |->
      ( ( x x. B ) + ( ( 1 - x ) x. A ) ) ) $.
    $( Describe a bijection from ` [ 0 , 1 ] ` to an arbitrary nontrivial
       closed interval ` [